Recent zbMATH articles in MSC 37https://zbmath.org/atom/cc/372024-03-13T18:33:02.981707ZUnknown authorWerkzeugInvariant measures concentrated on countable structureshttps://zbmath.org/1528.031532024-03-13T18:33:02.981707Z"Ackerman, Nathanael"https://zbmath.org/authors/?q=ai:ackerman.nathanael-leedom"Freer, Cameron"https://zbmath.org/authors/?q=ai:freer.cameron-e"Patel, Rehana"https://zbmath.org/authors/?q=ai:patel.rehanaSummary: Let \(L\) be a countable language. We say that a countable infinite \(L\)-structure \(\mathcal{M}\) admits an invariant measure when there is a probability measure on the space of \(L\)-structures with the same underlying set as \(\mathcal{M}\) that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of \(\mathcal{M}\). We show that \(\mathcal{M}\) admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in \(\Aut(\mathcal{M})\) of an arbitrary finite tuple of \(\mathcal{M}\) fixes no additional points. When \(\mathcal{M}\) is a Fraïssé limit in a relational language, this amounts to requiring that the age of \(\mathcal{M}\) have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.Brooks' theorem for measurable coloringshttps://zbmath.org/1528.031962024-03-13T18:33:02.981707Z"Conley, Clinton T."https://zbmath.org/authors/?q=ai:conley.clinton-taylor"Marks, Andrew S."https://zbmath.org/authors/?q=ai:marks.andrew-s"Tucker-Drob, Robin D."https://zbmath.org/authors/?q=ai:tucker-drob.robin-dSummary: We generalize Brooks' theorem to show that if \(G\) is a Borel graph on a standard Borel space \(X\) of degree bounded by \(d\geqslant 3\) which contains no \((d+1)\)-cliques, then \(G\) admits a \(\mu\)-measurable \(d\)-coloring with respect to any Borel probability measure \(\mu\) on \(X\), and a Baire measurable \(d\)-coloring with respect to any compatible Polish topology on \(X\). The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID \(d\)-colorings of Cayley graphs of degree \(d\), except in two exceptional cases.Group actions and harmonic analysis in number theory. Abstracts from the workshop held May 7--12, 2023https://zbmath.org/1528.110012024-03-13T18:33:02.981707ZSummary: This workshop focuses on new problems and new methods at the interface of harmonic analysis (taken in a very broad sense) and ergodic theory, with applications focused on number theory. Special emphasis is put on equidistribution problems on arithmetic symmetric spaces, effective methods in homogeneous dynamics, periods of automorphic forms, families of \(L\)-functions over number fields and function fields, and applications of Fourier uniqueness.Singular vectors on manifolds over totally real number fieldshttps://zbmath.org/1528.110562024-03-13T18:33:02.981707Z"Datta, Shreyasi"https://zbmath.org/authors/?q=ai:datta.shreyasi"Radhika, M. M."https://zbmath.org/authors/?q=ai:radhika.m-mDirichlet's approximation theorem for a single linear form states that for any \(x \in\mathbb{R}^m\) and any large integer \(T\), there is a \(p\in\mathbb{Z}\) and a \(q \in \mathbb{Z}^m\setminus \{0\}\) with \(\vert q \cdot x - p \vert < 1/T^m\) and \(\Vert q \Vert \le T\). Here, \(\Vert \cdot \Vert\) denotes the supremum norm. A vector is said to be singular if for any \(c > 0\), for any sufficiently large \(T\), one can improve these inequalities to \(\vert q \cdot x - p \vert < c/T^m\) and \(\Vert q \Vert \le T\).
In the present paper, these notions are extended to approximation by numbers from a fixed totally number field and several results results from the classical theory are extended to this setting. Let \(K\) be a totally real number field of degree \(d\), let \(\mathcal{O}_K\) be its ring of integers and let \(S\) be the normalised inequivalent valuations of \(K\). For each \(\sigma \in S\), let \(K_\sigma \cong \mathbb{R}\) denote the completion of \(K\) at \(\sigma\) and let \(K_S = \prod_{\sigma \in S} K_\sigma\). Identify \(K\) with its diagonal embedding into \(K_S\), and for \(m >1\), identify \(K^m\) with the coordinate wise diagonal embedding of \(K\) into \(K_S^m\). In this setting, a vector \(x \in K_S^m\) is singular if for any \(c>0\), for \(Q>0\) sufficiently large, there is a \(q_0 \in \mathcal{O}_K\) and \(q \in \mathcal{O}_K^m \setminus \{0\}\), such that \(\Vert q \cdot x - q_0 \Vert < c/Q^m\) and \(\Vert q \Vert \le Q\).
The authors prove that this notion is indeed the correct extension of the notion of singular vectors by proving a version of Dirichlet's theorem. They subsequently prove that the set of singular measures is a null set with respect to a large class of measures -- the so-called friendly measures -- which include the natural measures on a large family of non-degenerate manifolds as well as many fractal measures. They also prove, that if \(m=1\) the set of singular elements in \(K_S\) coincides with the field \(K\), while for \(m > 1\), singular vectors exist in abundance. The results are extensions of known results for the case \(K= \mathbb{Q}\). They are proved using homogeneous dynamics via establishing an analogue of Mahler's compactness criterion and Dani's correspondence in the present setup.
Reviewer: Simon Kristensen (Aarhus)A dynamical analogue of Sen's theoremhttps://zbmath.org/1528.111242024-03-13T18:33:02.981707Z"Sing, Mark O.-S."https://zbmath.org/authors/?q=ai:sing.mark-o-sLet \(K\) be a local field of characteristic 0 with perfect residue field of characteristic \(p\), and let \(\bar{K}\) be an algebraic closure of \(K\). Let \(\mathcal{O}_K\) be the ring of integers of \(K\), and let \(P(x)\) be a monic polynomial of degree \(q=p^r\) with coefficients in \(\mathcal{O}_K\). Let \(\alpha_0\in K\smallsetminus{\mathcal{O}_K^{\times}}\) and let \((\alpha_n)_{n\ge0}\) be a sequence of elements of \(\bar{K}\), not all 0, such that \(P(\alpha_n)=\alpha_{n-1}\) for all \(n\ge1\). For \(n\ge0\) set \(K_n=K(\alpha_n)\). Then \(K_{\infty}=\bigcup_{n\ge0}K_n\) is a ``branch extension'' associated to \(P(x)\) and \(\alpha_0\).
This paper investigates properties of the ramification filtration of \(K_{\infty}/K\). The main theorem states that if the polynomial \(P(x)\) and the sequence \((\alpha_n)_{n\ge0}\) satisfy certain technical conditions then \(K_n/K\) is a ramification subextension of \(K_{\infty}/K\) for all sufficiently large \(n\). This result can be viewed as an analogue of Sen's theorem [\textit{S. Sen}, Invent. Math. 17, 44--50 (1972; Zbl 0242.12012)]. The hypotheses of the main theorem are shown to hold in two important cases: First, if \(P(x)\) has degree \(p\), and second, if \(P(x)\) is post-critically bounded. (One says that \(P(x)\) is post-critically bounded if for every \(\beta\in\bar{K}\) such that \(P'(\beta)=0\) the sequence \((P^{\circ n}(\beta))_{n\ge1}\) is bounded with respect to the \(p\)-adic metric on \(\bar{K}\).)
Reviewer: Kevin Keating (Gainesville)On characteristic polynomials of automorphisms of Enriques surfaceshttps://zbmath.org/1528.140472024-03-13T18:33:02.981707Z"Brandhorst, Simon"https://zbmath.org/authors/?q=ai:brandhorst.simon"Rams, Sławomir"https://zbmath.org/authors/?q=ai:rams.slawomir"Shimada, Ichiro"https://zbmath.org/authors/?q=ai:shimada.ichiroLet \(f^*\) be an isometry of the numerical Néron-Severi lattice of a complex Enriques surface \(Y\), induced by an automoprhism \(f\in\mathrm{Aut}(Y)\) of \(Y\) and let \(p_f(x)\) be its characteristic polynomial. The authors in the present paper study the possible factors of the modulo-2 reduction of \(p_f(x)\) and show in their main theorem that \(p_f(x)\) is the product of (some of) the modulo-2 reductions of the cyclotomic polynomials \(\Phi_1 \Phi_3,\Phi_5,\Phi_7\) and \(\Phi_9\). In particular, each of the previous polynomial appears as factor of the modulo-2 reduction of \(p_f(x)\) for some automorphism \(f\) of an Enriques surface and examples are given. The authors also provide the algorithms than can be used to prove the results, based on McMullen's and Borcherds'methods.
The main result is also used to prove results on the K3 surface \(X\) which is the 2-cover of \(Y\): in fact, given \(f\in\mathrm{Aut}(Y)\), let \(\tilde f\) be its non-unique lift to \(X\) and let \(N\) be the orthogonal complement of the sublattice of \(H^2(X,\mathbb Z)\) invariant for the cover involution. It is known by [\textit{Y. Matsumoto} et al., Math. Nachr. 291, No. 13, 2084--2098 (2018; Zbl 1401.14171)] that the restriction \(f_N\) of \(\tilde f^*\) to \(N\) has finite order and the order is a divisor of one among 120, 90, 84, 72, 56, 48. This result is sharpened in the present paper and the authors show that, on the one hand, some orders are not possible and, on the other hand, some other orders do appear as orders of some \(f_N\).
Reviewer: Paola Comparin (Temuco)Uniform simplicity for subgroups of piecewise continuous bijections of the unit intervalhttps://zbmath.org/1528.200392024-03-13T18:33:02.981707Z"Guelman, Nancy"https://zbmath.org/authors/?q=ai:guelman.nancy"Liousse, Isabelle"https://zbmath.org/authors/?q=ai:liousse.isabelleLet \(I=[0, 1)\) and \(\mathcal{PC}(I)\) (resp. \(\mathcal{PC}^{+}(I)\)) be the quotient group of the group of all piecewise continuous (resp. piecewise continuous and orientation preserving) bijections of \(I\) by its normal subgroup consisting in elements with finite support.
A perfect group \(G\) is \(N\)-uniformly perfect if any product of commutators in \(G\) can be written as a product of at most \(N\) commutators in \(G\).
\textit{P. Arnoux} has proven in his unpublished PhD. Thesis [Un invariant pour les échanges d'intervalles et les flots sur les surfaces. Reims: Université de Reims (PhD Thesis) (1981)] that \(\mathcal{PC}^{+}(I)\) and certain other groups of interval exchanges are simple.
In the paper under review, the author proves the simplicity of the group \(\mathcal{A}^{+}(I)\) of locally orientation preserving, piecewise continuous, piecewise affine maps of \(I\). Furthermore, the author provide conditions, which guarantee that a subgroup \(G\) of \(\mathcal{PC}(I)\) is uniformly simple. As corollaries, he obtains that \(\mathcal{PC}(I)\), \(\mathcal{PC}^{+}(I)\), \(\mathcal{A}^{+}(I)\) and other related groups as well as the Thompson group \(T\) are uniformly simple.
Reviewer: Enrico Jabara (Venezia)Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic spacehttps://zbmath.org/1528.200722024-03-13T18:33:02.981707Z"Matsuzaki, Katsuhiko"https://zbmath.org/authors/?q=ai:matsuzaki.katsuhiko"Yabuki, Yasuhiro"https://zbmath.org/authors/?q=ai:yabuki.yasuhiro"Jaerisch, Johannes"https://zbmath.org/authors/?q=ai:jaerisch.johannesSummary: For a non-elementary discrete isometry group \(G\) of divergence type acting on a proper geodesic \(delta\)-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of \(G\). As applications of this result, we have: (1) under a minor assumption, such a discrete group \(G\) admits no proper conjugation, that is, if the conjugate of \(G\) is contained in \(G\), then it coincides with \(G\); (2) the critical exponent of any non-elementary normal subgroup of \(G\) is strictly greater than the half of that for \(G\).Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real mapshttps://zbmath.org/1528.280112024-03-13T18:33:02.981707Z"Moreira, Carlos Gustavo"https://zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-aSummary: We prove dimension formulas for arithmetic sums of regular Cantor sets, and, more generally, for images of cartesian products of regular Cantor sets by differentiable real maps.Intersections of middle-\(\alpha\) Cantor sets with a fixed translationhttps://zbmath.org/1528.280122024-03-13T18:33:02.981707Z"Huang, Yan"https://zbmath.org/authors/?q=ai:huang.yan"Kong, Derong"https://zbmath.org/authors/?q=ai:kong.derongSummary: For \(\lambda\in(0,1/3\) let \(C_\lambda\) be the middle-\(1-2\lambda\) Cantor set in \(\mathbb{R}\). Given \(t\in[-1,1]\), excluding the trivial case we show that
\[
\Lambda(t):=\{\lambda\in(0,1/3]:C_\lambda\cap(C_\lambda+1)\neq\emptyset\}
\]
is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of \(\Lambda(t)\), which reveals a dimensional variation principle. Furthermore, for any \(\beta\in[0,1]\) we show that the level set
\[
\Lambda_\beta(t)=\left\{\lambda\in\Lambda(t):\dim_H(C_\lambda\cap(C_\lambda+t))=\dim_P(C_\lambda\cap(C_\lambda+t))=\beta\frac{\log 2}{-\log\lambda}\right\}
\]
has equal Hausdorff and packing dimension \((-\beta\log\beta-(1-\beta)\log\frac{1-\beta}{2})/\log 3\). We also show that the set of \(\lambda\in\Lambda(t)\) for which \(\dim_H(C_\lambda\cap(C_\lambda+t))\neq\dim_P(C_\lambda\cap(C_\lambda+t))\) has full Hausdorff dimension.On the connectedness of attractors of orbital contractive IFSshttps://zbmath.org/1528.280252024-03-13T18:33:02.981707Z"Mihail, Alexandru"https://zbmath.org/authors/?q=ai:mihail.alexandru"Savu, Irina"https://zbmath.org/authors/?q=ai:savu.irinaSummary: We study certain topological properties of attractors of orbital contractive iterated function systems. We give a necessary and sufficient condition for an attractor to be connected and we present two results that establish when an attractor is arcwise connected. We provide some examples.Equidistribution, potential theory and arithmetic applicationshttps://zbmath.org/1528.300072024-03-13T18:33:02.981707Z"Burgos, José Ignacio"https://zbmath.org/authors/?q=ai:burgos.jose-ignacio"Menares, Ricardo"https://zbmath.org/authors/?q=ai:menares.ricardo(no abstract)The automorphism group of Rauzy diagramshttps://zbmath.org/1528.320182024-03-13T18:33:02.981707Z"Boissy, Corentin"https://zbmath.org/authors/?q=ai:boissy.corentinSummary: We give a description of the automorphism group of a Rauzy diagram as a subgroup of the symmetric group. This is based on an example that appears in some personal notes of Yoccoz that are to be published in the project ``Yoccoz archives''.Flows on the \(\mathrm{PGL}(V)\)-Hitchin componenthttps://zbmath.org/1528.320202024-03-13T18:33:02.981707Z"Sun, Zhe"https://zbmath.org/authors/?q=ai:sun.zhe"Wienhard, Anna"https://zbmath.org/authors/?q=ai:wienhard.anna-katharina"Zhang, Tengren"https://zbmath.org/authors/?q=ai:zhang.tengrenSummary: In this article we define new flows on the Hitchin components for \(\mathrm{PGL}(V)\). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in \(S\) and capture new phenomena which are not present in the case when \(n = 2\). We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article [\textit{Z. Sun} and \textit{T. Zhang}, ``The Goldman symplectic form on the \(\mathrm{PGL}(V)\)-Hitchin component'', Preprint, \url{arXiv:1709.03589}], this gives a global Darboux coordinate system on the Hitchin component.A fixed-point curve theorem for finite-orbits local diffeomorphismshttps://zbmath.org/1528.320222024-03-13T18:33:02.981707Z"Lisboa, Lucivanio"https://zbmath.org/authors/?q=ai:lisboa.lucivanio"Ribón, Javier"https://zbmath.org/authors/?q=ai:ribon.javierSummary: We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism \(F\), in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of \(F\). As an application we obtain that at least one eigenvalue of the linear part of \(F\) at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.The birational invariants of Lins Neto's foliationshttps://zbmath.org/1528.320482024-03-13T18:33:02.981707Z"Ling, Hao"https://zbmath.org/authors/?q=ai:ling.hao"Lu, Jun"https://zbmath.org/authors/?q=ai:lu.jun.1"Tan, Sheng-Li"https://zbmath.org/authors/?q=ai:tan.sheng-li\textit{A. Lins Neto} constructed in [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 2, 231--266 (2002; Zbl 1130.34301)] families of singular holomorphic foliations in \(\mathbb{CP}^2\) which are counterexamples to two famous questions.
Poincaré's Problem: is it possible to decide if an algebraic differential equation is algebraically integrable?
Painlevé's Problem: is it possible to decide if an algebraic differential equation has a rational first integral of a given genus g?
In the present work, the authors determine the minimal models of these families of singular holomorphic foliations in \(\mathbb{CP}^2\), calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. The work proves that the slopes of Lins Neto's foliations are at least 6, and their limits are bigger than 7.
Reviewer: Jesus Muciño Raymundo (Morelia)A toolbox of averaging theorems. Ordinary and partial differential equationshttps://zbmath.org/1528.340012024-03-13T18:33:02.981707Z"Verhulst, Ferdinand"https://zbmath.org/authors/?q=ai:verhulst.ferdinandThe book is intended to present a concise survey of averaging theorems as a toolbox for applied mathematicians, physicists and engineers, with emphasis on the practical use. For this reason, proofs are mostly omitted, although an extensive source of bibliographic references is provided. Some discussion on the results is added and examples, from the most elementary to the most complex, are included to illustrate the theoretical results.
The book is organized as follows. The first chapter discusses, at a very elementary level, possible approaches to perturbation problems. Chapters 2--4 introduce the first and second order periodic averaging, with several examples and some qualitative statements. Chapter 5 is devoted to the general first and second order averaging theory, with several applications. New theory and new applications can be found in Chapters 6--9, which include averaging on large timescales, averaging over spatial variables, applications to Hamiltonian systems and the study of more general dynamics, e.g. invariant manifolds and quasi-periodic solutions. Finally, in Chapter 10 an account of the averaging methods for PDEs is given.
In summary, this is an enjoyable and very clear textbook for scientists, engineers, students and practitioners, which reflects the author's vast expertise on the subject.
Reviewer: Pablo Amster (Buenos Aires)Darboux and analytic integrability of five-dimensional semiclassical Jaynes-Cummings systemhttps://zbmath.org/1528.340032024-03-13T18:33:02.981707Z"Taha, Rebaz Yaseen"https://zbmath.org/authors/?q=ai:taha.rebaz-yaseen"Amen, Azad Ibrahim"https://zbmath.org/authors/?q=ai:amen.azad-ibrahim"Hussein, Niazy Hady"https://zbmath.org/authors/?q=ai:hussein.niazy-hadySummary: We study the dynamics of a version of the Jaynes-Cummings system without the rotating wave approximation, which essentially consists of the interaction of two systems, \(n\) two-level atoms and a single system of electromagnetic field. In particular, we investigate some types of first integrals such as Darboux, analytic, and time-dependent first integrals of this system. We prove that only two Darboux first integrals and two analytic first integrals exist for this system. Moreover, we determine all exponential factors and invariant algebraic hypersurfaces of this system.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Existence of anti-periodic solutions for \(\Psi\)-Caputo-type fractional \(p\)-Laplacian problems via Leray-Schauder degree theoryhttps://zbmath.org/1528.340072024-03-13T18:33:02.981707Z"El Mfadel, Ali"https://zbmath.org/authors/?q=ai:el-mfadel.ali"Melliani, Said"https://zbmath.org/authors/?q=ai:melliani.said"Elomari, M'hamed"https://zbmath.org/authors/?q=ai:elomari.mhamedIn this work, authors consider a nonliner fractional differential equation with a very general kind of Caputo derivative. The main aim is to study the exitenece of anti periodic solution of the considered problem. In order to obtain the result, the Leray-Schauder degree theory is applied. One example is also provided to illustrate the analytical findings.
Reviewer: Syed Abbas (Mandi)On existence and classification of singular points for higher order differential algebraic equationshttps://zbmath.org/1528.340152024-03-13T18:33:02.981707Z"Chistyakov, V. F."https://zbmath.org/authors/?q=ai:chistyakov.viktor-filimonovich"Chistyakova, E. V."https://zbmath.org/authors/?q=ai:chistyakova.elena-viktorovnaThis paper is devoted to the study of singular points for linear differential-algebraic equations (DAEs) of higher order
\[
\Lambda_k x:=\sum_{i=0}^k A_i(t)x^{(i)}(t)=f,\quad t\in T=[\alpha,\,\beta],\tag{1}
\]
where \(A_i(t)\) are \((n\times n)\)-matrices defined on \(T\), \(f=f(t)\) is a given function, \(x\) is the unknown function, \(x^{(i)}(t)\) denotes the \(i\)-th derivative \((d/dt)^i x(t)\). We assume that the coefficients of (1) are sufficiently smooth and \(A_k(t)\) is singular for all \(t\in T\). A point in \(T\) is said to be a singular point of (1) if at this point the existence or the uniqueness of solutions is violated, or the dimension of the solution subspace changes. First, the authors present some solvability results for DAEs without singular points in the domain. A notion of index is given by assuming the existence of a left regularizing operator that brings (1) into a system of ordinary differential equations. Next, a formal definition of (isolated) singular points of (1) is given. Then, singular points are characterized and classified by two types: differential singular points and algebraic singular points. Several specially constructed examples are given for illustrating how to find singular points and that if (1) has a singular point, then it may be unsolvable.
Reviewer: Vu Hoang Linh (Hà Nội)Convolution equations with variable time nonlocal coefficientshttps://zbmath.org/1528.340222024-03-13T18:33:02.981707Z"Goodrich, Christopher S."https://zbmath.org/authors/?q=ai:goodrich.christopher-sThe author studies the existence of positive solutions for one-dimensional nonlocal equation of the form
\[
-M\big( (a\star(g\circ u)(t)\big)u''(t)=\lambda f(t,u(t)
\]
subject to \(u(0)=u'(1)=0\). The proof of the main results is based upon a classical topological fixed point theorem and a nonstandard order cone.
Reviewer: Ruyun Ma (Lanzhou)Oscillators at resonancehttps://zbmath.org/1528.340352024-03-13T18:33:02.981707Z"Rojas, David"https://zbmath.org/authors/?q=ai:rojas.davidSummary: An oscillator is isochronous if all motions are periodic with a common period. When the system is forced by a time-dependent periodic perturbation with the same period, the dynamics may change drastically and the phenomenon of resonance can appear. In this article we will study which properties the perturbations must fulfil in order to obtain unbounded solutions. We will consider different oscillators, from harmonic to nonlinear generalizations, and we will set out a number of remarks about the concept of auto-parametric resonance.New results on periodic solutions for second order damped vibration systemshttps://zbmath.org/1528.340382024-03-13T18:33:02.981707Z"Khaled, Khachnaoui"https://zbmath.org/authors/?q=ai:khaled.khachnaouiIn this paper, the author considers the following second-order damped vibration system
\[
\begin{cases} \ddot{u}(t)+A\dot{u}(t)+\nabla_{u} V(s,u(t))ds,\ \forall t\in\mathbb{R},\\
u(0)=u(T)=\dot{u}(0)-\dot{u}(T)=0,\ T>0 \end{cases} \tag{1}
\]
where \(A\) is a skew-symmetric matrix, \(V(t,x)=-K(t,x)+W(t,x)\) and \(K,W\in C^{2}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})\) are \(T\)-periodic in the first variable. Using variational methods and the Fountain Theorem, he proves under suitable conditions on \(A,K,W\) that problem (1) possesses a sequence \((u_{k})\) of odd \(T\)-periodic solutions satisfying \(\left\|u_{k}\right\|\longrightarrow\infty\) as \(k\longrightarrow\infty\).
Reviewer: Mohsen Timoumi (Monastir)Traveling waves of predator-prey system with a sedentary predatorhttps://zbmath.org/1528.340392024-03-13T18:33:02.981707Z"Li, Hongliang"https://zbmath.org/authors/?q=ai:li.hongliang.1|li.hongliang"Wang, Yang"https://zbmath.org/authors/?q=ai:wang.yang.18"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rong"Ma, Zhaohai"https://zbmath.org/authors/?q=ai:ma.zhaohaiPredator-prey systems with spatial diffusion are in many cases known to possess traveling wave solutions. In the present article a predator-prey system of the form
\[
\begin{aligned} u_t & = d_1u_{xx} + f(u) (p(u)-g(w)), \\
w_t & = g(w) (\beta f(u) -\mu(w)) \end{aligned}
\]
is considered in which only the prey species is moving via the diffusion term while the predator is sedentary. Here, \(f(u)p(u)\) describes the growth rate of the prey in the absence of the predator, while \(g(w)\mu(w)\) models the density-dependent death rate of the predator in the absence of the prey.
Under certain assumptions on the nonlinearities there exist precisely three stationary solutions for \(d_1=0\): \(E_0=(0,0)\), \(E_K=(K,0)\) where only prey is present and a positive equilibrium \(E_\ast\). The main result states that there are no traveling waves connecting \(E_0\) to \(E_\ast\), there are no traveling waves connecting \(E_K\) to \(E_\ast\) with wave speed \(c\leq 0\), while for \(c > 0\) there exist traveling waves connecting \(E_K\) to \(E_\ast\).
The proof uses the construction of a Ważewski set combined with a shooting argument where a part of the unstable manifold of \(E_K\) is traced forward. The article concludes with a short section on a specific system with Holling type-III functional response, including numerical experiments for this system.
Reviewer: Jörg Härterich (Bochum)Hysteresis and stabilityhttps://zbmath.org/1528.340412024-03-13T18:33:02.981707Z"Chow, Amenda N."https://zbmath.org/authors/?q=ai:chow.amenda-n-f"Morris, Kirsten A."https://zbmath.org/authors/?q=ai:morris.kirsten-a"Rabbah, Gina F."https://zbmath.org/authors/?q=ai:rabbah.gina-farajSummary: A common definition of hysteresis is that the graph of the state of the system displays looping behavior as the input of the system varies. Alternatively, a dynamical systems perspective can be used to define hysteresis as a phenomenon arising from multiple equilibrium points. Consequently, hysteresis is a topic that can be used to illustrate and extend concepts in a dynamical systems course. The concept is illustrated in this paper through examples of ordinary differential equations, most motivated by applications. Simulations are presented to complement the analysis. The examples can be used to construct student exercises, and specific additional questions are listed in an appendix. The paper concludes with a discussion of possible extensions, including hysteresis in partial differential equations.Bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equationshttps://zbmath.org/1528.350082024-03-13T18:33:02.981707Z"Yagasaki, Kazuyuki"https://zbmath.org/authors/?q=ai:yagasaki.kazuyuki"Yamazoe, Shotaro"https://zbmath.org/authors/?q=ai:yamazoe.shotaroIn this paper, the authors study pitchfork bifurcations of fundamental solitary waves (one component is identically zero) detected by the Melnikov analysis and determine the spectral and/or orbital stability of the fundamental and bifurcated solitary waves under some nondegenerate conditions. The main tool to determine their spectral stability is the Evans function technique and the Hamiltonian-Krein index theory. Moreover, approximate expressions of the bifurcated solitary waves, which are needed for application of these techniques, are provided.
In summary, the authors study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations. They establish criteria under which the fundamental solitary wave undergoes a pitchfork bifurcation, and utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral and/or orbital stability of the bifurcated solitary waves as well as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. Finally, they apply the theory to a cubic nonlinearity case and give numerical evidences for the theoretical results.
Reviewer: Liming Ling (Guangzhou)Spectral characteristics of Schrödinger operators generated by product systemshttps://zbmath.org/1528.350372024-03-13T18:33:02.981707Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jake"Gohlke, Philipp"https://zbmath.org/authors/?q=ai:gohlke.philippThe paper under review deals with discrete one-dimensional Schrödinger operators, i.e., operators in \(\ell^2(\mathbb{Z})\) of the form
\[
H := \Delta + V,
\]
where \(\Delta\) is the discrete Laplacian and \(V\) is the multiplication operator by the potential \(V\colon \mathbb{Z}\to \mathbb{R}\), which is dynamically defined in the following sense: given a dynamical system \((\mathbb{X},S)\) (i.e.,\ \(\mathbb{X}\) is a compact metric space and \(S\) is a homeomorphism on \(\mathbb{X}\)) and \(f\colon \mathbb{X}\to \mathbb{R}\) continuous, then
\[
V(n):=f(S^nx) ,\quad n\in\mathbb{Z}
\]
for some \(x\in\mathbb{X}\).
The key point of interest are now such operators which come from product dynamical systems; that is \(\mathbb{X} = \mathbb{X}_1 \times \mathbb{X}_2\) and \(S = S_1\times S_2\). Examples contained in this class may be sums of two potentials (e.g., periodic plus random, or periodic plus periodic with incommenuate frequencies) which are of the form
\[
V(n) = V_1(n) + V_2(n) , \quad n\in\mathbb{Z},
\]
where \(f(x^1,x^2):=f_1(x^1) + f_2(x^2)\) and \(f_j\) generates \(V_j\) as above, but also \(V\) generated by
\[
f(x^1,x^2):= f_1(x^1)\cdot f_2(x^2),
\]
i.e., multiplicative modulation.
The paper investigates spectral results for such operator (or operator families) \(H\), in particular Cantor spectra of zero Lebesgue measure (sum of a potential generated by a locally constant function sampled over a minimal subshift satisfying Boshernitzans condition and a periodic potential; as well as their product instead of sum), absence of eigenvalues, representation of the spectrum by spectra of periodic modification of random potentials, and (sub-/super-)criticality in case of periodic perturbations of quasiperiodic potentials with applications of the almost Mathieu operator.
Reviewer: Christian Seifert (Hamburg)Quantifying the dissipation enhancement of cellular flowshttps://zbmath.org/1528.351222024-03-13T18:33:02.981707Z"Iyer, Gautam"https://zbmath.org/authors/?q=ai:iyer.gautam"Zhou, Hongyi"https://zbmath.org/authors/?q=ai:zhou.hongyiSummary: We study the dissipation enhancement by cellular flows. Previous work by \textit{G. Iyer} et al. [Trans. Am. Math. Soc. 374, No. 9, 6039--6058 (2021; Zbl 1477.35044)] produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size, and diffusivity. Explicitly we show that the \textit{mixing time} is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the \textit{dissipation time} of incompressible flows to the \textit{mixing time.} The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling.Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamicshttps://zbmath.org/1528.351442024-03-13T18:33:02.981707Z"Ablowitz, Mark J."https://zbmath.org/authors/?q=ai:ablowitz.mark-j"Cole, Justin T."https://zbmath.org/authors/?q=ai:cole.justin-t"El, Gennady A."https://zbmath.org/authors/?q=ai:el.gennady-a"Hoefer, Mark A."https://zbmath.org/authors/?q=ai:hoefer.mark-a"Luo, Xu-Dan"https://zbmath.org/authors/?q=ai:luo.xudanThe authors consider the Korteweg-de Vries (KdV) equation \( u_{t}+6uu_{x}+\varepsilon ^{2}u_{xxx}=0\), where \(t>0\) is the time variable, \( x\in \mathbb{R}\) the space variable, \(u\) is proportional to the wave amplitude, and \(\varepsilon >0\) is the dispersion parameter measuring the relative strength of dispersion and nonlinearity. The soliton solution to the KdV equation is: \(u_{s}(x,t)=B+A_{0}sech^{2}(\sqrt{\frac{A_{0}}{ 2\varepsilon ^{2}}}[x-(2A_{0}+6B)t-x_{0}])\), where the parameter \(B\in \mathbb{R}\) corresponds to the background, constant mean field, \(A_{0}>0\) is the soliton amplitude and \(x_{0}\in \mathbb{R}\) is the soliton's initial position. The purpose of the paper is to analyze through different methods the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave. The authors write the solution of the KdV equation as the sum \( u(x,t)=w(x,t)+v(x,t;w(x,t))\), where \(w(x,t)\) is an approximation of either through a rarefaction wave or disperse shock wave and \(v(x,t)\) is a solitary mode ansatz, with boundary conditions that decay to zero as \(\left\vert x\right\vert \rightarrow \infty \). This leads to the equation for \(v\): \( v_{t}+6(wv)_{x}+6vv_{x}+\varepsilon ^{2}v_{xxx}=-F[w(x,t)]\), where \( F[w(x,t)]=u_{t}+6(w^{2})_{x}+\varepsilon ^{2}w_{xxx}\). The authors look for soliton modes with the secant hyperbolic form \(v(x,t)=2\kappa (t)sech^{2}( \frac{\kappa (t)}{\xi }[x-z(t)])\). In the case of an interaction with a rarefaction wave, with initial condition \(u(x,0)=c^{2}H(x)+v(x,0;x_{0})\), for a soliton located to the left or right of the origin at time \(t=0\), \(H\) being the Heaviside step function. The rarefaction wave is approximated by: \( w(x,t)=0\), \(x\leq 0\), \(w(x,t)=\frac{x}{6t}\), \(0<x\leq 6c^{2}t\), \(w(x,t)=c^{2} \), \(6c^{2}t<x\), which is a continuous, global, and weak solution to the inviscid Burgers equation \(u_{t}+6uu_{x}=0\). The authors compute the expressions of the parameters \(\kappa \) and \(z\) in the above expression of \(v \), whence that of \(u\). They compare this expression with results of numerical simulations. They then analyze the situation of a soliton interaction with a dispersive shock wave, considering the same initial condition.\ The evolution of the step down initial condition can be split into three regions: \(w(x,t)=0\), \(x<-12c^{2}t\), \(w(x,t)=w_{D}(x,t)\), \( -12c^{2}t\leq x<-2c^{2}t\), \(w(x,t)=-c^{2}\), \(-2c^{2}t\leq x\), where the interval \((-12c^{2}t,-2c^{2}t)\) is the disperse shock wave region with the mean field \(w_{D}(x,t)\), which is chosen as \(-\frac{x+12c^{2}t}{10t}\), \(t>0\) . The authors again compute the expressions of the parameters \(\kappa \) and \( z\) in the above expression of \(v\), whence that of \(u\), which is compared to that obtained through numerical computations. The next stage consists to study the problem of interaction between soliton and rarefaction or dispersive shock waves using the multiphase Whitham modulation theory, observing that the KdV equation admits a family of quasi-periodic or multiphase solutions in the form \(u(x,t)=F_{N}(\theta _{1}/\varepsilon ,\theta _{2}/\varepsilon ,\ldots ,\theta _{N}/\varepsilon )\), where the integer \(N\in \mathbb{N}\) corresponds to the number of nontrivial, independent variables \(\theta _{j}=k_{j}x-\omega _{j}t+\theta _{0j}\), \( j=1,\ldots ,N\), required to describe the solution. This allows building an expression of \(u\) which involves parameters \(\lambda _{j}\) that are the band edges of the Schrödinger operator \(\mathcal{L}=\varepsilon ^{2}\partial _{xx}+u(x,t)\). The authors draw computations first in the case of zero-phase modulations, then for one-phase modulations and for soliton-mean field interaction, for two-phase modulations and for soliton-dispersive mean field interaction, or linear wavepacket-dispersive shock wave interaction. In the last part of their paper, the authors use the inverse scattering transform to solve the KdV\ equation. They introduce the Lax pair: \(v_{xx}+(\frac{ u(x,t)}{\varepsilon ^{2}}+\frac{k^{2}}{\varepsilon ^{2}})v=0\), \( v_{t}=(u_{x}+\gamma )v+(4k^{2}-2u)v_{x}\), where \(k\) is a time-independent spectral parameter and \(\gamma \) a constant. They build the right and left scattering problems and they derive expressions of the solution in the cases of an interaction with a rarefaction or a dispersive shock wave.
Reviewer: Alain Brillard (Riedisheim)Parametrix problem for the Korteweg-de Vries equation with steplike initial datahttps://zbmath.org/1528.351522024-03-13T18:33:02.981707Z"Piorkowski, Mateusz"https://zbmath.org/authors/?q=ai:piorkowski.mateuszThe KdV equation is considered with sufficiently smooth steplike initial data. In order to obtain precise asymptotics of solutions to the Cauchy problem, an approach based on the direct comparison of resolvents of the associated Riemann-Hilbert problem is applied. These solutions develop shock waves between asymptotically constant and soliton regions.
Reviewer: Piotr Biler (Wrocław)Nonlocal symmetries and solutions of the \((2+1)\) dimension integrable Burgers equationhttps://zbmath.org/1528.351532024-03-13T18:33:02.981707Z"Xin, Xiangpeng"https://zbmath.org/authors/?q=ai:xin.xiangpeng"Jin, Meng"https://zbmath.org/authors/?q=ai:jin.meng"Yang, Jiajia"https://zbmath.org/authors/?q=ai:yang.jiajia"Xia, Yarong"https://zbmath.org/authors/?q=ai:xia.yarongSummary: The conservation laws of low-dimensional partial differential equations have been employed by \textit{S. Y. Lou} et al. [J. Phys. A, Math. Theor. 45, No. 15, Article ID 155209, 14 p. (2012; Zbl 1248.37069)] to construct various high-dimensional equations, particularly focusing on high-dimensional integrable equations. However, solving these high-dimensional equations poses significant challenges. In this paper, the \((2+1)\)-dimensional Burgers equation is studied by means of nonlocal symmetry method for the first time. For this high-dimensional equation, we establish a link with the exact solution of the \((1+1)\)-dimensional Burgers equation through nonlocal symmetry. Furthermore, we successfully construct multiple exact solutions for the high-dimensional Burgers equation by leveraging the exact solution of the low-dimensional counterpart. We also give the corresponding images of several solutions to study the dynamic behavior of the equations.Riemann-Hilbert approach to the focusing and defocusing nonlocal derivative nonlinear Schrödinger equation with step-like initial datahttps://zbmath.org/1528.351592024-03-13T18:33:02.981707Z"Hu, Beibei"https://zbmath.org/authors/?q=ai:hu.beibei"Shen, Zuyi"https://zbmath.org/authors/?q=ai:shen.zuyi"Zhang, Ling"https://zbmath.org/authors/?q=ai:zhang.ling.2"Fang, Fang"https://zbmath.org/authors/?q=ai:fang.fangSummary: In this paper, we consider the Cauchy problem for the integrable nonlocal derivative nonlinear Schrödinger (DNLS) equation with step-like initial data. The main aim is to obtain a solution for the prescribed initial conditions. To do this, we adapt the Riemann-Hilbert (RH) approach to discuss of the step-like initial value problem associated with the nonlocal DNLS equation. The main idea is the representation of the solution of the Cauchy problem in terms of the solution of an associated \(2 \times 2\) matrix RH problem and obtain the solution of the Cauchy problem of the nonlocal DNLS equation.Semirational rogue waves of the three coupled higher-order nonlinear Schrödinger equationshttps://zbmath.org/1528.351602024-03-13T18:33:02.981707Z"Lan, Zhong-Zhou"https://zbmath.org/authors/?q=ai:lan.zhongzhouSummary: Under investigation in this paper is the three coupled Hirota equations in the long distance communication or ultrafast signal routing system. Semirational solutions are derived by virtue of the Darboux dressing transformation. Based on the discussion of the parameters in solutions, we see that the semirational rogue waves are formed by the interaction between the rogue wave and the dark-dark-bright/breather soliton.Multi-elliptic-dark soliton solutions of the defocusing nonlinear Schrödinger equationhttps://zbmath.org/1528.351612024-03-13T18:33:02.981707Z"Ling, Liming"https://zbmath.org/authors/?q=ai:ling.liming"Sun, Xuan"https://zbmath.org/authors/?q=ai:sun.xuanSummary: We derive multi-elliptic-dark soliton solutions for the defocusing nonlinear Schrödinger (NLS) equation by using the Darboux-Bäcklund transformation. Additionally, we investigate the asymptotic dynamic behaviors of these solutions along the trajectories of dark solitons, which reveals that the \(N\)-elliptic-dark soliton solution can be decomposed into \(N\) single-elliptic-dark soliton solutions, as \(t \to \pm \infty \).Recovery of a spatially-dependent coefficient from the NLS scattering maphttps://zbmath.org/1528.351662024-03-13T18:33:02.981707Z"Murphy, Jason"https://zbmath.org/authors/?q=ai:murphy.jasonSummary: We follow up on work of
\textit{W. A. Strauss} [in: Scattering theory in mathematical physics. Proceedings of the NATO Advanced Study Institute held at Denver, CO, USA, June 11--29, 1973. Dordrecht: Springer Netherlands. 53--78 (1974; Zbl 0297.35062)],
\textit{R. Weder} [Math. Methods Appl. Sci. 24, No. 4, 245--254 (2001; Zbl 0988.35128); Proc. Am. Math. Soc. 129, No. 12, 3637--3645 (2001; Zbl 0986.35125)], and
\textit{M. Watanabe} [J. Math. Anal. Appl. 459, No. 2, 932--944 (2018; Zbl 1382.35280)]
concerning scattering and inverse scattering for nonlinear Schrödinger equations with nonlinearities of the form \(\alpha(x)|u|^p u\).Modulational instability and rogue wave solutions for the mixed focusing-defocusing semi-discrete coherently coupled nonlinear Schrödinger system with \(4 \times 4\) Lax pairhttps://zbmath.org/1528.351702024-03-13T18:33:02.981707Z"Wen, Xiao-Yong"https://zbmath.org/authors/?q=ai:wen.xiaoyong"Liu, Xue-Ke"https://zbmath.org/authors/?q=ai:liu.xue-keSummary: Under consideration is the semi-discrete mixed focusing-defocusing coherently coupled nonlinear Schrödinger system with the fourth-order auxiliary linear problem, which may describe some dynamic behaviors of light pulses. Firstly, the modulational instability is studied to analyze the possible generation reason of diverse localized waves from plane wave solution. Secondly, based on the known \(4 \times 4\) Lax pair, the discrete generalized \(( m , N-m)\)-fold Darboux transformation is constructed and extended to solve this discrete system. Finally, as an application of the resulting Darboux transformation, some exact solutions with position control parameters are derived and shown graphically, in particular, we obtain some novel rogue wave and periodic wave solutions with only one peak and no valleys on vanishing background, which are different from the usual fundamental rogue wave with one peak and two valleys. These results might be useful for understanding propagation of light pulses.Dynamic analysis on optical pulses via modified PINNs: soliton solutions, rogue waves and parameter discovery of the CQ-NLSEhttps://zbmath.org/1528.351722024-03-13T18:33:02.981707Z"Yin, Yu-Hang"https://zbmath.org/authors/?q=ai:yin.yuhang"Lü, Xing"https://zbmath.org/authors/?q=ai:lu.xingSummary: Under investigation in this paper is the cubic-quintic nonlinear Schrödinger equation, which describes the propagation of optical on resonant-frequency fields in the inhomogeneous fiber. According to abundant previous researches on the model, exact soliton solutions and rogue wave solutions have been derived through Darboux transformation. The modulation instability phenomenon has been analyzed to evaluate the ability of an initially perturbed plane wave to split into localized energy packets when propagating in a dispersive and nonlinear medium.
Numerical solutions with high accuracy are needed in fields of production and engineering. Nonetheless, the data acquisition costs of the optical pulse transmission system is high, which will limit the accuracy and the efficiency of typical numerical and data-driven methods. With the physical knowledge embedded into neural networks in the form of loss function, the problem of big data dependence has been solved. For dynamic analysis on optical pulses with small amount of known information, we strive to obtain high accuracy numerical solutions. Considering the case that the cubic-quintic nonlinear Schrödinger equation is converted to the Kundu-Eckhaus equation with simplified coefficient constraints through variable transformation, we construct modified physics-informed neural networks, where conversions on the input and output are attached to deep neural networks. Training networks with the given initial and boundary data, we effectively derive the expected soliton and rogue wave solutions, where the approximated one-soliton, two-soliton, first-order and second-order rogue waves are included. In general, the modified network reaches low prediction errors with small data available. With the coefficients of equations, the weights and the bias of networks combined as parameters to be trained, we deduce the corresponding value of condition settings for different systems. Moreover, we simulate diverse localized waves in the context of nonlinear electrical transmissions with different environment settings and compare the evolution process to reach conclusions on the parameter discovery.Stability and local bifurcations of single-mode equilibrium states of the Ginzburg-Landau variational equationhttps://zbmath.org/1528.351752024-03-13T18:33:02.981707Z"Kulikov, Dmitriĭ Anatol'evich"https://zbmath.org/authors/?q=ai:kulikov.dmitrii-anatolevichThis work addresses the Ginzburg-Landau equation written for the complex function \(u\), having real-valued coefficients and a generalized type of nonlinearity, i.e. in the form \(u|u|^{2p}\) with \(p\in \mathbb{N}\). The problem is considered under periodic boundary conditions and the main question explored is the equilibrium and bifurcation of the periodic solutions. The main discussion is devoted to the stability of invariant manifolds and the respective bifurcations in an auxiliary linear as well as the nonlinear problem based on the consideration of a set of single-mode spatially periodic components.
Reviewer: Eugene Postnikov (Kursk)Bifurcation from infinity and multiplicity results for an elliptic system from biologyhttps://zbmath.org/1528.352122024-03-13T18:33:02.981707Z"Li, Chunqiu"https://zbmath.org/authors/?q=ai:li.chunqiu"Peng, Zhen"https://zbmath.org/authors/?q=ai:peng.zhen.1|peng.zhenSummary: This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology
\[
\begin{aligned}
&- \kappa \Delta u = \lambda u + f (x, u) - v,\\
&- \Delta v = u - v,
\end{aligned}
\]
in a bounded domain \(\Omega \subset \mathbb{R}^N\). We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.Regularity of transition densities and ergodicity for affine jump-diffusionshttps://zbmath.org/1528.370012024-03-13T18:33:02.981707Z"Friesen, Martin"https://zbmath.org/authors/?q=ai:friesen.martin"Jin, Peng"https://zbmath.org/authors/?q=ai:jin.peng"Kremer, Jonas"https://zbmath.org/authors/?q=ai:kremer.jonas"Rüdiger, Barbara"https://zbmath.org/authors/?q=ai:rudiger.barbaraSummary: This paper studies the transition density and exponential ergodicity for affine processes on the canonical state space \(\mathbb{R}_{\geq 0}^m \times \mathbb{R}^n\). Under a Hörmander-type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance.
{{\copyright} 2022 Wiley-VCH GmbH.}Multiple recurrence and convergence without commutativityhttps://zbmath.org/1528.370022024-03-13T18:33:02.981707Z"Frantzikinakis, Nikos"https://zbmath.org/authors/?q=ai:frantzikinakis.nikos"Host, Bernard"https://zbmath.org/authors/?q=ai:host.bernard\textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 34, 275--291 (1978; Zbl 0426.28014)] showed that if \(S\) and \(T\) are commuting measure-preserving transformations on a probability space \((X,\mu)\) then they exhibit `multiple recurrence': For any set \(A\) of positive measure there is some \(n\in\mathbb{N}\) with \(\mu(A\cap T^{-n}A\cap S^{-n}A)>0\). This has led to enormous generalizations in several directions, including relaxing the commutativity assumption, proving convergence of the related non-conventional ergodic averages, multiple recurrence involving more transformations, IP formulations, and so on.
Here special structural properties of recurrence questions along iterates of the form \(T^{n^k}\) for \(k\ge2\) are exploited to show multiple recurrence and non-conventional ergodic convergence results for pairs of measure-preserving transformations of zero entropy without commutativity assumptions under the hypothesis that the iterates of the two transformations are \(T^n\) and \(S^{n^k}\) with \(k\ge2\). Well-known examples show that results of this sort cannot hold in general for the case \(T^n\) and \(S^n\).
Reviewer: Thomas B. Ward (Durham)Multiply minimal points for the product of iterateshttps://zbmath.org/1528.370032024-03-13T18:33:02.981707Z"Huang, Wen"https://zbmath.org/authors/?q=ai:huang.wen.1"Shao, Song"https://zbmath.org/authors/?q=ai:shao.song"Ye, Xiangdong"https://zbmath.org/authors/?q=ai:ye.xiangdongSummary: The multiple Birkhoff recurrence theorem states that for any \(d \in \mathbb{N}\), every system \((X, T)\) has a multiply recurrent point \(x\), i.e., \((x, x, \ldots, x)\) is recurrent under \(\tau_d =: T \times T^2 \times \cdots \times T^d\). It is natural to ask if there always is a multiply minimal point, i.e., a point \(x\) such that \((x, x, \ldots, x)\) is \(\tau_d\)-minimal. A negative answer is presented in this paper via studying the horocycle flows.
However, it is shown that for any minimal system \((X, T)\) and any nonempty open set \(U\), there is \(x \in U\) such that \(\{ n \in \mathbb{Z} :T^n x \in U, \ldots, T^{dn}x \in U\}\) is piecewise syndetic; and that for a PI minimal system, any \(M\)-subsystem of \((X^d, \tau_d)\) is minimal.A note on entropy of Delone setshttps://zbmath.org/1528.370042024-03-13T18:33:02.981707Z"Hauser, Till"https://zbmath.org/authors/?q=ai:hauser.tillSummary: In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non-compact) locally compact abelian groups that contain Meyer sets.
{{\copyright} 2022 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.}Quantitative disjointness of nilflows from horospherical flowshttps://zbmath.org/1528.370052024-03-13T18:33:02.981707Z"Katz, Asaf"https://zbmath.org/authors/?q=ai:katz.asafSummary: We prove a quantitative variant of a disjointness theorem of nilflows from horospherical flows following a technique of \textit{A. Venkatesh} [Ann. Math. (2) 172, No. 2, 989--1094 (2010; Zbl 1214.11051)], combined with the structural theorems for nilflows by \textit{B. Green} et al. [Ann. Math. (2) 176, No. 2, 1231--1372 (2012; Zbl 1282.11007)].Polynomial effective equidistributionhttps://zbmath.org/1528.370062024-03-13T18:33:02.981707Z"Lindenstrauss, Elon"https://zbmath.org/authors/?q=ai:lindenstrauss.elon"Mohammadi, Amir"https://zbmath.org/authors/?q=ai:mohammadi.amir"Wang, Zhiren"https://zbmath.org/authors/?q=ai:wang.zhirenSummary: We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of \(\mathrm{SL}_2(\mathfrak{l})\) in arithmetic quotients of \(\mathrm{SL}_2(\mathbb{C})\) and \(\mathrm{SL}_2(\mathfrak{l})\times\mathrm{SL}_2(\mathfrak{l})\).
The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.Product type potential on the one-dimensional lattice systems: selection of maximizing probability and a large deviation principlehttps://zbmath.org/1528.370072024-03-13T18:33:02.981707Z"Mohr, J."https://zbmath.org/authors/?q=ai:mohr.jurgen|mohr.john-w|mohr.jonathan|mohr.james|mohr.joanaOne of the main results that the author proves is the existence of an equilibrium probability measure for a one-dimensional lattice system. More precisely, consider the set of functions \(\Omega=[0,1]^{\mathbb{N}}\) from the natural numbers to the closed interval \([0,1]\). Here \(\mathbb{N}=\{1,2,3,\dots\}\). A real-valued function \(f:\Omega\to\mathbb{R}\) is assumed to take the form \(f(x_1,x_2,\dots)=\sum^\infty_{j=1}f_j(x_j)\), where each real-valued function \(f_j:[0,1]\to\mathbb{R}\) is Lipschitz continuous with Lipschitz constant being less than \(\frac{1}{2^j}\) and periodic, \(f_j(0)=f_j(1)\). Consider the real-valued function defined as \(F(a)=f(a,a,a,\dots)\) with some differentiability conditions (please see the conditions in Theorem 7). This is assumed to achieve maximum value at one point \(a_1\). Define the probability measure \[\tilde{\mu}_{0,\beta}=\frac{e^{\beta F(a)}}{\int_0^1e^{\beta F(a)}da}da\] on the closed interval \([0,1]\) and the probability measure \(\tilde{\mu}_\beta=\bigotimes^\infty_{i=1}\tilde{\mu}_{0,\beta}\) on \(\Omega\). Then the author proves the following two results:
(1) There holds:
\[\lim\limits_{\beta\to+\infty}\tilde{\mu}_\beta\bigoplus^\infty_{i=1}\delta_{a_1};\]
(2) There holds:
\[\sum^\infty_{j=1}f_j(a_1)=\max\limits_{\mu\in\mathcal{M}_\sigma}\int_{\Omega}fd\mu,\]
where \(\mathcal{M}_\sigma\) is the set of invariant probability measures with respect to \(\sigma:\Omega\to\Omega\), \(\sigma(x_1,x_2,\dots)=(x_2,x_3,\dots)\).
The ideas of the proofs are as follows. Regarding statement (1), since the function \(F(a)\) achieves its maximum at one point \(a_1\) as \(\beta\to+\infty\), the support for the probability measure \(\tilde{\mu}_{0,\beta}\) gets concentrated near \(a_1\), and in the limit \(\beta\to+\infty\), one gets the Dirac delta function \(\delta_{a_1}\). Regarding statement (2), the assumptions on the function \(f(x_1,x_2,\dots)\) and statement (1) imply the conditions of Theorem 5 in [\textit{A. O. Lopes} et al., Ergodic Theory Dyn. Syst. 35, No. 6, 1925--1961 (2015; Zbl 1352.37090)], that in turn implies statement (2).
Reviewer: Haru Pinson (Tucson)Random walks on convergence groupshttps://zbmath.org/1528.370082024-03-13T18:33:02.981707Z"Azemar, Aitor"https://zbmath.org/authors/?q=ai:azemar.aitorLet \(G\) be a countable group equipped with a probability measure \(\mu\). One can then consider the associated random walk and ask whether almost every trajectory of the random walks converges inside some natural ``compactification'' of \(G\). If this is the case one can wonder whether the compactification, endowed with the hitting measure, can be identified with the Poisson boundary of \((G,\mu)\).
When \(G\) is a hyperbolic group, results of this kind have been proved by \textit{V. A. Kaimanovich} [Ann. Math. (2) 152, No. 3, 659--692 (2000; Zbl 0984.60088)], the natural compactification being the one obtained by adding to \(G\) its Gromov boundary. Similar results have been later obtained by many authors under some weaker form of hyperbolicity of \(G\) (or considering other boundaries, as for example Martin boundaries, Floyd boundaries, etc.). See, e.g., the work of \textit{J. Maher} and \textit{G. Tiozzo} [Proc. Lond. Math. Soc. (3) 123, No. 2, 153--202 (2021; Zbl 1495.60034)], and that of \textit{I. Gekhtman} et al. [Invent. Math. 223, No. 2, 759--809 (2021; Zbl 1498.20102)], among many others.
In this work, the author considers a group \(G\) acting on a compact metric space \(M\) as a convergence group. This property is in particular satisfied for the action of a hyperbolic group on its Gromov boundary. He proves that if the action is non-elementary and minimal, then there exists a topology on \(G\cup M\) such that for any measure \(\mu\) on \(G\), whose support generates \(G\), almost every trajectory of the corresponding random walk converges to a point of \(M\). Under additional assumptions on \(G\) and \(\mu\), the author also proves that \(M\), with a stationary measure, can be identified with the Poisson boundary of \((G,\mu)\). The last section of the article contains further applications, as for instance to the Dirichlet problem, i.e., finding a harmonic extension to \(G\cup M \) for some continuous function on \(M\).
Reviewer: Pierre Py (Grenoble)Semiflows: prolongation sets, point-transitive, topologically transitivehttps://zbmath.org/1528.370092024-03-13T18:33:02.981707Z"Barzanouni, Ali"https://zbmath.org/authors/?q=ai:barzanouni.aliSummary: Let \(\varphi : T \times X \to X\), or simply \((T, X)\), be any topological semiflow on a space \(X\) with a topological semigroup \(T\). We give some new results about prolongation sets for the semiflow \((T, X)\). Also, we recall notions of point-transitivity and topological transitivity for \((T, X)\) and give some examples to study the relations between them. Let \(X\) be a locally compact Hausdorff space and let \((T, X)\) be point-transitive. Then \(M_\gamma = \{x \in X : \gamma(x) = \min(\gamma)\}\) is dense in \(X\) for every \(T\)-invariant upper semicontinuous function \(\gamma : X \to [0, \infty)\). The converse does hold if \((T, X)\) is a flow on the compact metric space \((X, d)\). We show that the semiflow \((T, X)\) on a Hausdorff space \(X\) with \(\overline{TX} = X\) is topologically transitive if and only if every \(T\)-invariant function \(\gamma : X \to [0, \infty)\), that is continuous on a comeagre set, is constant on a comeagre set. We introduce a \(TP\)-point in a topological space \(X\), which is weaker than an isolated point, and show that if \(X\) is a regular topological space with a \(TP\)-point \(p \in X\) and \((T, X)\) is syndetically transitive, then \(p\) is a periodic point and \(X = Kp\) for some compact set \(K \subseteq T\). Finally, we show that every expansive semiflow \((T, [0, 1])\) is point-transitive if \(T\) is a right \(C\)-semigroup.Glasner property for linear group actions and their productshttps://zbmath.org/1528.370102024-03-13T18:33:02.981707Z"Bulinski, Kamil"https://zbmath.org/authors/?q=ai:bulinski.kamil"Fish, Alexander"https://zbmath.org/authors/?q=ai:fish.alexanderSummary: A theorem of \textit{S. Glasner} from 1979 [Isr. J. Math. 32, 161--172 (1979; Zbl 0406.54023)]
shows that if \(Y \subset \mathbb{T}= \mathbb{R}/\mathbb{Z}\) is infinite then for each \(\epsilon > 0\) there exists an integer \(n\) such that \(\mathit{nY}\) is \(\epsilon\)-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on \(\mathbb{T}^d\) also satisfy such a \textit{Glasner property} where each infinite set (in fact, sufficiently large finite set) will have an \(\epsilon\)-dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski connected Zariski closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the Cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.On \(N\)-distal homeomorphismshttps://zbmath.org/1528.370112024-03-13T18:33:02.981707Z"Rego, E."https://zbmath.org/authors/?q=ai:rego.eduardo-francisco|rego.elias|rego.e-r"Salcedo, J. C."https://zbmath.org/authors/?q=ai:salcedo.j-cThe authors introduce \(N\)-distal, \(N\)-equicontinuous and \(N\)-expansive properties and establish many interrelations with the corresponding well-known notions. Interesting examples are provided to show the difference between \(N\)- and \(M\)-distal. Here \(N\) and \(M\) are positive integers.
For clarification: item (ii) of Lemma 3.2 could have been a definition as usually sup of an empty set is taken to be \(-\infty\).
Reviewer: C. R. E. Raja (Bangalore)Multidimensional multiplicative large sets in totally minimal systemshttps://zbmath.org/1528.370122024-03-13T18:33:02.981707Z"Zhao, Jian Jie"https://zbmath.org/authors/?q=ai:zhao.jianjieSummary: In this paper, it is shown that for a residual set of points in a totally minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains a subset with positive multidimensional multiplicative upper Banach density, extending a previous result by \textit{D. Glasscock} et al. [Discrete Contin. Dyn. Syst. 39, No. 10, 5891--5921 (2019; Zbl 1436.37013)]. Meanwhile, we give some combinatorial properties of the sets with positive multidimensional multiplicative upper Banach density.Symbolic systems with prescribed partial digit densityhttps://zbmath.org/1528.370132024-03-13T18:33:02.981707Z"Becker, Alex J."https://zbmath.org/authors/?q=ai:becker.alex-jenaro"Baraviera, Alexandre T."https://zbmath.org/authors/?q=ai:baraviera.alexandre-tavaresSummary: In this paper, we present a class of subshifts on \(\{0,1\}^{\mathbb{N}}\) defined with partial digit density. Fixed a real number \(p_c\in [0,1]\), initially we consider the sets \(P(p_c)\) that satisfy the following rules of transition: \(0\rightarrow 0\) and \(0\rightarrow 1\) are always possible; \(1\rightarrow 0\) is allowed if density of ones (ratio between amount of ones and length of the word) is bigger than \(p_c\); and \(1\rightarrow 1\) is allowed if density of ones is less or equal than \(p_c\). Those subsets are not shift invariant; in order to get a set with this property, define \(\Sigma (p_c)\) as the closure of all images of the \(P(p_c)\) by the shift; we show that, for every \(p_c \in (0, 1)\), \(\Sigma (p_c)\) is not a subshift of finite type. Moreover, it is possible to show that \(\Sigma (p_c)\) have positive topological entropy for all \(p_c\in [0,1)\).Characteristic measures for language stable subshiftshttps://zbmath.org/1528.370142024-03-13T18:33:02.981707Z"Cyr, Van"https://zbmath.org/authors/?q=ai:cyr.van"Kra, Bryna"https://zbmath.org/authors/?q=ai:kra.brynaGiven a homeomorphism \(T\) on a compact metrizable space \(X\), the automorphism group \(\operatorname{Aut}(X,T)\) is given by the set of all homeomorphisms of \(X\) which commute with \(T\). A Borel measure \(\mu\) on \(X\) is called characteristic if it \(\operatorname{Aut}(X,T)\)-invariant.
A few years ago,
\textit{J. Frisch} and \textit{O. Tamuz} [Ergodic Theory Dyn. Syst. 42, No. 5, 1655--1661 (2022; Zbl 1501.37014)]
asked whether it is true that every \(\mathbb{Z}\)-subshift admits a characteristic measure. We know of four classes of subshifts which admit a characteristic measure for relatively simple reasons, but these classes are far from covering the space of all \(\mathbb{Z}\)-subshifts. We remark that in general the automorphism group of a subshift is non-amenable, so the existence of characteristic measures is not clear at all.
The contribution of this article is to define a new class of subshifts which admit characteristic measures. These are called the ``language stable'' subshifts. Essentially, these are the subshifts which can be approximated by subshifts of finite type in such a way that a particular approximation is the best possible for arbitrarily long gaps, infinitely often (for readers familiar with ``first offender words'', a subshift is language stable if the set of lengths for which there are no first offenders has Banach upper density 1).
The proof that language stable subshifts admit characteristic measures is short and clear, and most of the article is devoted to constructing examples that are language stable but do not fall into the previously known classes that admit characteristic measures.
I think this article provides a very interesting class of examples and I will be looking forward to hearing more about language stable subshifts and characteristic measures in the future.
Reviewer: Sebastián Barbieri (Santiago)Generic \(\delta\)-chaos for erasing interval mapshttps://zbmath.org/1528.370152024-03-13T18:33:02.981707Z"Della Corte, Alessandro"https://zbmath.org/authors/?q=ai:della-corte.alessandroSummary: We prove that completely erasing interval maps exhibit generic \(\delta\)-chaos, in the sense that there exists a dense \(G_\delta\) subset \(M\subset [0,1]^2\) such that every element of \(M\) is a Li-Yorke pair of modulus \(\delta\).Effective intrinsic ergodicity for countable state Markov shiftshttps://zbmath.org/1528.370162024-03-13T18:33:02.981707Z"Rühr, René"https://zbmath.org/authors/?q=ai:ruhr.rene"Sarig, Omri"https://zbmath.org/authors/?q=ai:sarig.omri-mSummary: For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to \textit{S. Kadyrov} [Colloq. Math. 149, No. 1, 93--101 (2017; Zbl 1380.37014)]. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.Interval rearrangement ensembleshttps://zbmath.org/1528.370172024-03-13T18:33:02.981707Z"Teplinsky, A."https://zbmath.org/authors/?q=ai:teplinskij.a-yu|teplinsky.alexeySummary: We introduce a new concept of interval rearrangement ensembles (IRE), which is a generalization of interval exchange transformations (IET). This construction extends the space of IETs in accordance with the pinpointed natural duality. Induction of Rauzy-Veech kind is applicable to IREs. It is conjugate to the reverse operation by the duality mentioned above. A natural extension of an IRE is associated with two transverse flows on a flat translation surface with branching points.Variational principle for polynomial entropy on subsets of free semigroup actionshttps://zbmath.org/1528.370182024-03-13T18:33:02.981707Z"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.12"Peng, Dongmei"https://zbmath.org/authors/?q=ai:peng.dongmeiLet \(X\) be a compact metric space and denote by \(\mathcal M(X)\) the family of all Borel probability measures on \(X\); moreover, let \(G\) be a monoid of continuous self-maps of \(X\), finitely generated by \(\mathcal G\subseteq G\). By extending previous similar notions, for any subset \(Z\) of \(X\), the authors introduce the notion \(h^B_{\mathrm{pol}}(Z,\mathcal G)\) of Bowen polynomial entropy of \(\mathcal G\) on \(Z\). In case \(G\) is in addition a free semigroup and \(\mu\in\mathcal M(X)\), they introduce the lower and the upper local polynomial entropy \(\underline h_\mu^{\mathrm{pol}}(\mathcal G)\) and \(\overline h^{\mathrm{pol}}_\mu(\mathcal G)\). Some properties of these new notions of entropy are established. The main result of the paper is the following variational principle: in the above setting, for any non-empty compact subset \(Z\) of \(X\), there holds
\[
h^B_{\mathrm{pol}}(Z,\mathcal G)=\sup\{\underline h_\mu^{\mathrm{pol}}(\mathcal G):\mu\in\mathcal M(X),\mu(Z)=1\}.
\]
Reviewer: Anna Giordano Bruno (Udine)Entropy for semigroup actions on general topological spaceshttps://zbmath.org/1528.370192024-03-13T18:33:02.981707Z"Souza, Josiney A."https://zbmath.org/authors/?q=ai:souza.josiney-a"Santana, Alexandre J."https://zbmath.org/authors/?q=ai:santana.alexandre-jSummary: This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler-Konheim-McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.Branched coverings of the sphere having a completely invariant continuum with infinitely many Wada lakeshttps://zbmath.org/1528.370202024-03-13T18:33:02.981707Z"Iglesias, J."https://zbmath.org/authors/?q=ai:iglesias.jorge"Portela, A."https://zbmath.org/authors/?q=ai:portela.aldo"Rovella, A."https://zbmath.org/authors/?q=ai:rovella.alvaro"Xavier, J."https://zbmath.org/authors/?q=ai:xavier.juliana-cSummary: We construct a family of smooth branched coverings of degree 2 of the sphere \(S^2\) having a completely invariant indecomposable continuum \(K\) and infinitely many Wada Lakes.Some properties of dynamical systems having shadowinghttps://zbmath.org/1528.370212024-03-13T18:33:02.981707Z"Askri, Ghassen"https://zbmath.org/authors/?q=ai:askri.ghassenSummary: Let \(X\) be a compact metric space and \(f : X \to X\) be a continuous map having the shadowing property. In this paper, first, we prove that if \(X\) is connected, then \(\omega_f\) is continuous if, and only if, \(f\) is minimal. Second, we prove that the continuity of \(\omega_f\) yield \(CR(f) = R(f) = AP(f)\). Third, it is shown that if any point is recurrent then the collection of minimal sets is closed in \((2^X, d_H)\). Finally, we show that the existence of a Li-Yorke pair is sufficient to have positive topological sequence entropy and if furthermore, \(f\) has \(s\)-shadowing property then \(f\) is Li-Yorke chaotic.On almost shadowable measureshttps://zbmath.org/1528.370222024-03-13T18:33:02.981707Z"Lee, Keonhee"https://zbmath.org/authors/?q=ai:lee.keonhee"Rojas, Arnoldo"https://zbmath.org/authors/?q=ai:morales.carlos-arnoldoSummary: In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are \textit{weakly shadowable}, namely, that there are Borelians with a measure close to 1 such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an \(F_{\sigma \delta}\) subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces.On a classification of chaotic laminations which are nontrivial basic sets of axiom a flowshttps://zbmath.org/1528.370232024-03-13T18:33:02.981707Z"Medvedev, Vladislav S."https://zbmath.org/authors/?q=ai:medvedev.vladislav-s"Zhuzhoma, E. V."https://zbmath.org/authors/?q=ai:zhuzhoma.evgenii-vSummary: We prove that, given any \(n\geqslant 3\) and \(2\leqslant q\leqslant n-1\), there is a closed \(n\)-manifold \(M^n\) admitting a chaotic lamination of codimension \(q\) whose support has the topological dimension \({n-q+1}\). For \(n=3\) and \(q=2\), such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by \textit{V. Z. Grines} [J. Dyn. Control Syst. 6, No. 1, 97--126 (2000; Zbl 0984.37020)], \textit{R. V. Plykin} [Russ. Math. Surv. 39, No. 6, 85--131 (1984; Zbl 0584.58038); translation from Usp. Mat. Nauk 39, No. 6(240), 75--113 (1984)], \textit{A. Yu. Zhirov} [J. Dyn. Control Syst. 6, No. 3, 397--430 (2000; Zbl 1063.37035)].Oscillations in dynamic systems with an entropy operatorhttps://zbmath.org/1528.370242024-03-13T18:33:02.981707Z"Popkov, Yuri S."https://zbmath.org/authors/?q=ai:popkov.yuri-sSummary: This paper considers dynamic systems with an entropy operator described by a perturbed constrained optimization problem. Oscillatory processes are studied for periodic systems with the following property: the entire system has the same period as the process generated by its linear part. Existence and uniqueness conditions are established for such oscillatory processes, and a method is developed to determine their form and parameters. Also, the general case of noncoincident periods is analyzed, and a method is proposed to determine the form, parameters, and the period of such oscillations. Almost periodic processes are investigated, and existence and uniqueness conditions are proved for them as well.Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flowshttps://zbmath.org/1528.370252024-03-13T18:33:02.981707Z"Call, Benjamin"https://zbmath.org/authors/?q=ai:call.benjamin"Thompson, Daniel J."https://zbmath.org/authors/?q=ai:thompson.daniel-jThis paper considers equilibrium states for geodesic flows on closed rank-one manifolds. It extends results found in [\textit{K.Burns} et al., Geom. Funct. Anal. 28, No. 5, 1209--1259 (2018; Zbl 1401.37038)]. The authors work on an \(n\)-dimensional closed connected \(C^\infty\) Riemannian manifold \(M\) with metric \(g\) and non-positive sectional curvature. They consider the geodesic flow \(g_t\) for real values of \(t\) on the unit tangent bundle \(T^1M\). Equilibrium states are analyzed for Hölder continuous potentials or scalar multiples of the geometric potential \(\varphi^\mu\).
The authors' main theorem is as follows: If \( g_t\) is the geodesic flow on a closed rank-1 manifold \(M\), \(\varphi\) is defined by \(\varphi : T^1M \rightarrow \mathbb{R}\), where \(\varphi = q \varphi^u\) (where \(q\) is a real number) or is Hölder continuous, and if \(P(\mathrm{Sing}, \varphi) < P(\varphi)\), then the unique equilibrium state \(\mu_\varphi\) has the Kolmogorov property. Here \(P(\phi)\) represents the topological pressure with respect to the geodesic flow and \(\mathrm{Sing}\) is the set of vectors whose rank is greater than 1. The rank of a vector \(v\) in \(T^1M\) is the dimension of the space of parallel Jacobi vector fields for the geodesic through \(v\).
The Kolmogorov property, a mixing property in ergodic theory that is somewhat weaker than Bernoulli mixing, is described in [\textit{I. P. Kornfel'd} et al., Ergodic theory. (Ehrgodicheskaya teoriya) (Russian). Moskva: ``Nauka'' (1980; Zbl 0508.28008)].
Reviewer: William J. Satzer Jr. (St. Paul)A new proof of the dimension gap for the Gauss maphttps://zbmath.org/1528.370262024-03-13T18:33:02.981707Z"Jurga, Natalia"https://zbmath.org/authors/?q=ai:jurga.nataliaSummary: In [Isr. J. Math. 124, 61--76 (2001; Zbl 1015.11040)], \textit{Y. Kifer} et al. showed that the Bernoulli measures for the Gauss map \(T(x)=1/x\bmod 1\) satisfy a `dimension gap' meaning that for some \(c > 0\), \(\sup_{\mathbf{p}}\dim\mu_{\mathbf{p}}<1-c\), where \(\mu_p\) denotes the (pushforward) Bernoulli measure for the countable probability vector \(\mathbf{p}\). In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.Erratum: A corrected proof of the scale recurrence lemma from the paper ``Stable intersections of regular Cantor sets with large Hausdorff dimensions''https://zbmath.org/1528.370272024-03-13T18:33:02.981707Z"Moreira, Carlos"https://zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-a"Zamudio, Alex"https://zbmath.org/authors/?q=ai:zamudio.alex-mauricioSummary: This is an erratum for the paper by the first author and \textit{J.-C. Yoccoz} [ibid. 154, No. 1, 45--96 (2001; Zbl 1195.37015)]. We show how to fix a flaw -- a bad choice of parameters -- in the proof of the \textit{scale recurrence lemma}. This lemma is an important step towards establishing the main theorem.Chaotic topological foliationshttps://zbmath.org/1528.370282024-03-13T18:33:02.981707Z"Zhukova, N. I."https://zbmath.org/authors/?q=ai:zhukova.nina-ivanovna"Levin, G. S."https://zbmath.org/authors/?q=ai:levin.g-s"Tonysheva, N. S."https://zbmath.org/authors/?q=ai:tonysheva.n-sSummary: We call a foliation \((M, F)\) on a manifold \(M\) chaotic if it is topologically transitive and the union of closed leaves is dense in \(M\). The chaotic topological foliations of arbitrary codimension on \(n\)-dimensional manifolds can be considered as a multidimensional generalization of chaotic dynamical systems in the Devaney sense. For topological foliations \((M, F)\) covered by bundles, we prove that a foliation \((M, F)\) is chaotic if and only if its global holonomy group is chaotic. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic topological foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed.Upper bound on the regularity of the Lyapunov exponent for random products of matriceshttps://zbmath.org/1528.370292024-03-13T18:33:02.981707Z"Bezerra, Jamerson"https://zbmath.org/authors/?q=ai:bezerra.jamerson"Duarte, Pedro"https://zbmath.org/authors/?q=ai:duarte.pedroSummary: We prove that if \({\boldsymbol{\mu}}\) is a finitely supported measure on \({\boldsymbol{SL}}_{\mathbf{2}}({\mathbb{R}})\) with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not \({\boldsymbol{\alpha}}\)-Hölder around \({\boldsymbol{\mu}}\) for any \({\boldsymbol{\alpha}}\) exceeding the Shannon entropy of \({\boldsymbol{\mu}}\) over the Lyapunov exponent of \({\boldsymbol{\mu}}\).Bursting solutions of the Rössler equationshttps://zbmath.org/1528.370302024-03-13T18:33:02.981707Z"Fowler, A. C."https://zbmath.org/authors/?q=ai:fowler.andrew-c"McGuinness, M. J."https://zbmath.org/authors/?q=ai:mcguinness.mark-jSummary: We provide an analytic solution of the Rössler equations based on the asymptotic limit \(c\to \infty\) and we show in this limit that the solution takes the form of multiple pulses, similar to ``burst'' firing of neurons. We are able to derive an approximate Poincaré map for the solutions, which compares reasonably with a numerically derived map.Distortion in the group of circle homeomorphismshttps://zbmath.org/1528.370312024-03-13T18:33:02.981707Z"Banecki, Juliusz"https://zbmath.org/authors/?q=ai:banecki.juliusz"Szarek, Tomasz"https://zbmath.org/authors/?q=ai:szarek.tomasz-jakub|szarek.tomasz-zacharyLet \(H\) be an arbitrary group. An element \(f\in H\) is called distorted in \(H\) if there exists a finitely generated subgroup \(G\subset H\) containing \(f\) such that \[\lim_{n\rightarrow\infty}\frac{d_{\mathcal{G}}(f^n,\mathrm{id})}{n}=0\] for some (and hence every) generating set \( \mathcal{G} \). The problem of the existence of distorted elements in some groups of homeomorphisms has been intensively studied for many years and substantial progress has been achieved for groups of diffeomorphisms of manifolds. In [``Distortion elements in Diff\(^{\infty}(\mathbb{R}/\mathbb{Z})\)'', Preprint (2008)], \textit{A. Avila} shows that rotations with irrational rotation number are distorted in the group of smooth diffeomorphisms of the circle.
In the present paper, the authors give a constructive proof that all irrational rotations are distorted both in the group of piecewise affine circle homeomorphisms \(\mathrm{PAff}_+(\mathbb{R}/\mathbb{Z})\), and in the group of smooth circle diffeomorphisms, \(\mathrm{Diff}^{\infty}(\mathbb{R}/\mathbb{Z})\). The proof is obtained by constructing a subsequence where above limit is zero. That answers the question: ``Does the group of piecewise-affine circle homeomorphisms contain distorted elements?'' which was addressed in [\textit{A. Navas}, in: Proceedings of the international congress of mathematicians 2018, ICM 2018, Rio de Janeiro, Brazil, August 1--9, 2018. Volume III. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 2035--2062 (2018; Zbl 1451.37039)].
Reviewer: Semra Pamuk (Ankara)Hitting functions for mixed partitionshttps://zbmath.org/1528.370322024-03-13T18:33:02.981707Z"Dzhalilov, Akhtam Abdurakhmanovich"https://zbmath.org/authors/?q=ai:dzhalilov.akhtam-abdurakhmanovich"Khomidov, Mukhriddin Karimjon ugli"https://zbmath.org/authors/?q=ai:khomidov.mukhriddin-karimjon-ugliSummary: Let \(T_{\rho}\) be an irrational rotation on a unit circle \(S^1\simeq [0,1)\). Consider the sequence \(\{\mathcal{P}_n\}\) of increasing partitions on \(S^1\). Define the hitting times \(N_n(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^j_{\rho}(y)\in P_n(x)\} \), where \(P_n(x)\) is an element of \(\mathcal{P}_n\) containing \(x\). \textit{D. H. Kim} and \textit{B. K. Seo} in [Nonlinearity 16, No. 5, 1861--1868 (2003; Zbl 1046.37023)] proved that the rescaled hitting times \(K_n(\mathcal{Q}_n;x,y):= \frac{\log N_n(\mathcal{Q}_n;x,y)}{n}\) a.e. (with respect to the Lebesgue measure) converge to \(\log2\), where the sequence of partitions \(\{\mathcal{Q}_n\}\) is associated with chaotic map \(f_2(x):=2x \bmod 1\). The map \(f_2(x)\) has positive entropy \(\log2\). A natural question is what if the sequence of partitions \(\{\mathcal{P}_n\}\) is associated with a map with zero entropy. In present work we study the behavior of \(K_n(\tau_n;x,y)\) with the sequence of mixed partitions \(\{\tau_n\}\) such that \(\mathcal{P}_n\cap [0,\frac{1}{2}]\) is associated with map \(f_2\) and \(\mathcal{D}_n\cap [\frac{1}{2},1]\) is associated with irrational rotation \(T_{\rho} \). It is proved that \(K_n(\tau_n;x,y)\) a.e. converges to a piecewise constant function with two values. Also, it is shown that there are some irrational rotations that exhibit different behavior.The thermodynamic formalism and the central limit theorem for stochastic perturbations of circle maps with a breakhttps://zbmath.org/1528.370332024-03-13T18:33:02.981707Z"Dzhalilov, Akhtam"https://zbmath.org/authors/?q=ai:dzhalilov.akhtam-abdurakhmanovich"Mayer, Dieter"https://zbmath.org/authors/?q=ai:mayer.dieter-h"Aliyev, Abdurahmon"https://zbmath.org/authors/?q=ai:aliyev.abdurahmonSummary: Let \(T\in C^{2+\varepsilon}(S^1\setminus\{x_b\})\), \(\varepsilon>0\), be an orientation preserving circle homeomorphism with rotation number \(\rho_T=[k_1,\,k_2,\,\ldots,\,k_m,\,1,\,1,\,\ldots]\), \(m\ge1\), and a single break point \(x_b\). Stochastic perturbations \(\overline{z}_{n+1} = T(\overline{z}_n) + \sigma \xi_{n+1}\), \(\overline{z}_0:=z\in S^1\) of critical circle maps have been studied some time ago by \textit{O. Díaz-Espinosa} and \textit{R. de la Llave} [J. Mod. Dyn. 1, No. 3, 477--543 (2007; Zbl 1130.37022)], who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by \textit{A. Dzhalilov} et al. [Discrete Contin. Dyn. Syst. 24, No. 2, 381--403 (2009; Zbl 1168.37009)], to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of \textit{E. B. Vul} et al. [Russ. Math. Surv. 39, No. 3, 1--40 (1984; Zbl 0561.58033); translation from Usp. Mat. Nauk 39, No. 3(237), 3--37 (1984)], to establish a symbolic dynamics for any point \({z\in S^1}\) and to define a transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables \(\xi_i\) into the linear term \(L_n(z_0)= \xi_n+\sum\limits_{k=1}^{n-1}\xi_k\prod\limits_{j=k}^{n-1} T'(z_j)\), \(z_0\in S^1\) and a higher order term, which is possible in a neighbourhood \(A_k^n\) of the points \(z_k\), \(k\le n-1\), not containing the break points of \(T^n\). For this we construct for a certain sequence \(\{n_m\}\) a series of neighbourhoods \(A_k^{n_m}\) of the points \(z_k\) which do not contain any break point of the map \(T^{q_{n_m}}\), \(q_{n_m}\) the first return times of \(T\). The proof of our results follows from the proof of the central limit theorem for the linearized process.Quasi-graphs, zero entropy and measures with discrete spectrumhttps://zbmath.org/1528.370342024-03-13T18:33:02.981707Z"Li, Jian"https://zbmath.org/authors/?q=ai:li.jian"Oprocha, Piotr"https://zbmath.org/authors/?q=ai:oprocha.piotr"Zhang, Guohua"https://zbmath.org/authors/?q=ai:zhang.guohua.1Summary: In this paper, we study dynamics of maps on quasi-graphs and characterise their invariant measures. In particular, we prove that every invariant measure of a quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre-Misiurewicz's result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum.On a classification of periodic maps on the 2-torushttps://zbmath.org/1528.370352024-03-13T18:33:02.981707Z"Baranov, Denis A."https://zbmath.org/authors/?q=ai:baranov.denis-alekseevich"Grines, Vyacheslav Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Pochinka, Olga V."https://zbmath.org/authors/?q=ai:pochinka.olga-v"Chilina, Ekaterina E."https://zbmath.org/authors/?q=ai:chilina.ekaterina-evgenevnaSummary: In this paper, following \textit{J. Nielsen} [Mat.-Fys. Medd., Danske Vid. Selsk. 15, No.1, 1--77 (1937; Zbl 0017.13302)], we introduce a complete characteristic of orientation- preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to \textit{V. Z. Grines} and \textit{A. Bezdenezhnykh} [Diffeomorphisms with Orientable Heteroclinic Sets on Two-Dimensional Manifolds, Methods of the Qualitative Theory of Differential Equations, ed. E. A. Leontovich-Andronova, GGU, Gorki, 139--152 (1985)], any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.Anosov endomorphisms on the two-torus: regularity of foliations and rigidityhttps://zbmath.org/1528.370362024-03-13T18:33:02.981707Z"Cantarino, Marisa"https://zbmath.org/authors/?q=ai:cantarino.marisa"Varão, Régis"https://zbmath.org/authors/?q=ai:varao.regisSummary: We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the two-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterisation of smooth conjugacy between special Anosov endomorphisms (defined as those having only one unstable direction for each point) and their linearisations.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}Dependency of the positive and negative long-time behaviors of flows on surfaceshttps://zbmath.org/1528.370372024-03-13T18:33:02.981707Z"Yokoyama, Tomoo"https://zbmath.org/authors/?q=ai:yokoyama.tomooIn this paper, the author studies the structure of \(\omega\)-limit sets of flows defined on compact surfaces. In particular, it is shown that the \(\omega\)-limit of an orbit whose closure has non-empty interior (\textit{locally dense} orbit), is either a nowhere dense subset of singular points or the closure of a non-closed locally dense recurrent trajectory (Theorem~A). Further, the author observes that this is also the structure depicted by the \(\omega\)-limit sets of non-closed trajectories of non-wandering flows (Theorem~B). For Hamiltonian flows, Theorem~C establishes that \(\omega\)-limits of non-closed trajectories on possibly non-compact surfaces consist of singular points.
The author also introduces the notion of \textit{pre-Hamiltonian flow} which is a flow whose non-singular trajectories are the connected components of the level sets of the restriction of some real-valued function to the surface obtained by removing the singular points, and comprise a codimension-one foliaton of this new surface. Theorem~E states that if a flow has a finite number of singular points being pre-Hamiltonian is equivalent to being Hamiltonian.
Reviewer: Héctor Barge (Madrid)On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinderhttps://zbmath.org/1528.370382024-03-13T18:33:02.981707Z"Chen, Qinbo"https://zbmath.org/authors/?q=ai:chen.qinbo"Damjanović, Danijela"https://zbmath.org/authors/?q=ai:damjanovic.danijela"Petković, Boris"https://zbmath.org/authors/?q=ai:petkovic.borisSummary: Let \(\mathcal{F}\) and \(\mathcal{K}\) be commuting \(C^\infty\) diffeomorphisms of the cylinder \(\mathbb{T}\times \mathbb{R}\) that are, respectively, close to \(\mathcal{F}_0 (x, y)=(x+\omega (y), y)\) and \(T_\alpha (x, y)=(x+\alpha , y)\), where \(\omega (y)\) is non-degenerate and \(\alpha\) is Diophantine. Using the KAM iterative scheme for the group action we show that \(\mathcal{F}\) and \(\mathcal{K}\) are simultaneously \(C^\infty \)-linearizable if \(\mathcal{F}\) has the intersection property (including the exact symplectic maps) and \(\mathcal{K}\) satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of \(\mathbb{Z}^2\)-actions on the cylinder, generated by commuting twist maps.Interior dynamics of Fatou setshttps://zbmath.org/1528.370392024-03-13T18:33:02.981707Z"Hu, Mi"https://zbmath.org/authors/?q=ai:hu.mi.1Summary: In this paper, we investigate the precise behavior of orbits inside attracting basins. Let \(f\) be a holomorphic polynomial of degree \(m \geq 2\) in \(\mathbb{C}\), \(\mathcal{A}(p)\) be the basin of attraction of an attracting fixed point \(p\) of \(f\), and \(\Omega_i\)\((i=1, 2, \cdots)\) be the connected components of \(\mathcal{A}(p)\). Assume \(\Omega_1\) contains \(p\) and \(\{f^{-1}(p)\} \cap \Omega_1 \neq \{p\}\). Then there is a constant \(C\) so that for every point \(z_0\) inside any \(\Omega_i\), there exists a point \(q \in \cup_k f^{-k} (p)\) inside \(\Omega_i\) such that \(d_{\Omega_i} (z_0, q) \leq C\), where \(d_{\Omega_i}\) is the Kobayashi distance on \(\Omega_i\). In [the author, ``Dynamics inside parabolic basins'', Preprint, \url{arXiv:2208.03756}] we proved that this result is not valid for parabolic basins.Perturbations of graphs for Newton maps. I: bounded hyperbolic componentshttps://zbmath.org/1528.370402024-03-13T18:33:02.981707Z"Gao, Yan"https://zbmath.org/authors/?q=ai:gao.yan.2|gao.yan"Nie, Hongming"https://zbmath.org/authors/?q=ai:nie.hongmingAuthors' abstract: We consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree 2 on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.
Reviewer: Walter Bergweiler (Kiel)Attractors of a weakly dissipative system allowing transition to the stochastic web in the conservative limithttps://zbmath.org/1528.370412024-03-13T18:33:02.981707Z"Golokolenov, Alexander V."https://zbmath.org/authors/?q=ai:golokolenov.alexander-v"Savin, Dmitry V."https://zbmath.org/authors/?q=ai:savin.dmitrii-vladimirovichSummary: This article deals with the dynamics of a pulse-driven self-oscillating system -- the Van der Pol oscillator -- with the pulse amplitude depending on the oscillator coordinate. In the conservative limit the ``stochastic web'' can be obtained in the phase space when the function defining this dependence is a harmonic one. The paper focuses on the case where the frequency of external pulses is four times greater than the frequency of the autonomous system. The results of a numerical study of the structure of both parameter and phase planes are presented for systems with different forms of external pulses: the harmonic amplitude function and its power series expansions. Complication of the pulse amplitude function results in the complication of the parameter plane structure, while typical scenarios of transition to chaos visible in the parameter plane remain the same in different cases. In all cases the structure of bifurcation lines near the border of chaos is typical of the existence of the Hamiltonian type critical point. Changes in the number and the relative position of coexisting attractors are investigated while the system approaches the conservative limit. A typical scenario of destruction of attractors with a decrease in nonlinear dissipation is revealed, and it is shown to be in good agreement with the theory of 1:4 resonance. The number of attractors of period 4 seems to grow infinitely with the decrease of dissipation when the pulse amplitude function is harmonic, while in other cases all attractors undergo destruction at certain values of dissipation parameters after the birth of high-period periodic attractors.On some aspects of the response to stochastic and deterministic forcingshttps://zbmath.org/1528.370422024-03-13T18:33:02.981707Z"Santos Gutiérrez, Manuel"https://zbmath.org/authors/?q=ai:santos-gutierrez.manuel"Lucarini, Valerio"https://zbmath.org/authors/?q=ai:lucarini.valerioSummary: The perturbation theory of operator semigroups is used to derive response formulas for a variety of combinations of acting forcings and reference background dynamics. In the case of background stochastic dynamics, we decompose the response formulas using the Koopman operator generator eigenfunctions and the corresponding eigenvalues, thus providing a functional basis towards identifying relaxation timescales and modes and towards relating forced and natural fluctuations in physically relevant systems. To leading order, linear response gives the correction to expectation values due to extra deterministic forcings acting on either stochastic or chaotic dynamical systems. When considering the impact of weak noise, the response is linear in the intensity of the (extra) noise for background stochastic dynamics, while the second order response given the leading order correction when the reference dynamics is chaotic. In this latter case we clarify that previously published diverging results can be brought to common ground when a suitable interpretation -- Stratonovich vs Itô -- of the noise is given. Finally, the response of two-point correlations to perturbations is studied through the resolvent formalism via a perturbative approach. Our results allow, among other things, to estimate how the correlations of a chaotic dynamical system changes as a results of adding stochastic forcing.Piecewise monotonic maps with a common piecewise constant stationary densityhttps://zbmath.org/1528.370432024-03-13T18:33:02.981707Z"Wang, Zi"https://zbmath.org/authors/?q=ai:wang.zi"Ding, Jiu"https://zbmath.org/authors/?q=ai:ding.jiu"Rhee, Noah"https://zbmath.org/authors/?q=ai:rhee.noah-hSummary: For a prescribed piecewise constant density function defined on the unit interval, we construct piecewise strictly monotonic maps, consisting of piecewise stretching linear functions, from the interval to itself whose stationary density is the given function. We also show the statistical stability of such maps under some natural condition.Existence and stability results of stochastic differential equations with non-instantaneous impulse and Poisson jumpshttps://zbmath.org/1528.370442024-03-13T18:33:02.981707Z"Varshini, S."https://zbmath.org/authors/?q=ai:varshini.s"Banupriya, K."https://zbmath.org/authors/?q=ai:banupriya.k"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Ravikumar, K."https://zbmath.org/authors/?q=ai:ravikumar.kasinathanSummary: This paper focuses on a new class of non-instantaneous impulsive stochastic differential equations generated by mixed fractional Brownian motion with poisson jump in real separable Hilbert space. A set of sufficient conditions are generated based on the stochastic analysis technique, analytic semigroup theory of linear operators, fractional power of operators, and fixed point theory to obtain existence and uniqueness results of mild solutions for the considered system. Furthermore, the asymptotic behaviour of the system is investigated. Finally, an example is proposed to validate the obtained results.Non-limit-circle and limit-point criteria for symplectic and linear Hamiltonian systemshttps://zbmath.org/1528.370452024-03-13T18:33:02.981707Z"Zemánek, Petr"https://zbmath.org/authors/?q=ai:zemanek.petrSummary: Several necessary and/or sufficient conditions for the existence of a non-square-integrable solution of symplectic dynamic systems with general linear dependence on the spectral parameter on time scales are established and a sufficient condition for the limit-point case is derived. Almost all presented results are new even in the continuous and discrete cases, that is, for the linear Hamiltonian differential systems and for the discrete symplectic systems, respectively.
{{\copyright} 2022 Wiley-VCH GmbH.}On the adjoint action of the group of symplectic diffeomorphismshttps://zbmath.org/1528.370462024-03-13T18:33:02.981707Z"Lempert, László"https://zbmath.org/authors/?q=ai:lempert.laszloSummary: We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold \((X,\omega)\) on \(C^\infty(X)\) and on functions \(C^\infty(X)\to\mathbb{R}\). We describe various properties of invariant convex functions on \(C^\infty(X)\). Among other things we show that continuous convex functions \(C^\infty (X)\to\mathbb{R}\) that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of \(C^\infty(X)\); but this is not generally true if the convexity condition is dropped.The \(0:1\) resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcationhttps://zbmath.org/1528.370472024-03-13T18:33:02.981707Z"Zhou, Xing"https://zbmath.org/authors/?q=ai:zhou.xingSummary: We consider the non-semisimple \(0:1\) resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues \(\pm i\alpha\; (\alpha \in\mathbb{R}\) and \(\alpha >0)\) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by \textit{H. W. Broer} et al. [Z. Angew. Math. Phys. 44, No. 3, 389--432 (1993; Zbl 0805.58047)], this bifurcation essentially triggered by the reversible universal unfolding
\[
M=\frac{1}{2} p^2 +\frac{1}{4}q^4 +(\lambda +I_1) q^2
\]
with respect to BCKV-restricted morphisms of the planar non-semisimple singularity \(\frac{1}{2}p^2 +\frac{1}{4}q^4\) (the \(I_1\) is regarded as distinguished parameter with respect to the external parameter \(\lambda)\). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the \(S_1\)-symmetry generated by the additional pair of imaginary eigenvalues \(\pm i\alpha\) to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by H. W. Broer et al. [loc. cit.]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.On some properties of semi-Hamiltonian systems arising in the problem of integrable geodesic flows on the two-dimensional torushttps://zbmath.org/1528.370482024-03-13T18:33:02.981707Z"Agapov, S. V."https://zbmath.org/authors/?q=ai:agapov.sergei-vadimovich"Fakhriddinov, Zh. Sh."https://zbmath.org/authors/?q=ai:fakhriddinov.zh-shSummary: \textit{M. Bialy} and \textit{A. E. Mironov} [Discrete Contin. Dyn. Syst. 29, No. 1, 81--90 (2011; Zbl 1232.37035); Nonlinearity 24, No. 12, 3541--3554 (2011; Zbl 1232.35092); J. Geom. Phys. 87, 39--47 (2015; Zbl 1304.53084)]
demonstrated in a recent series of works that the search for polynomial first integrals of a geodesic flow on the 2-torus reduces to the search for solutions to a system of quasilinear equations which is semi-Hamiltonian. We study the various properties of this system.Topological analysis of pseudo-Euclidean Euler top for special values of the parametershttps://zbmath.org/1528.370492024-03-13T18:33:02.981707Z"Altuev, Murat K."https://zbmath.org/authors/?q=ai:altuev.murat-k"Kibkalo, Vladislav A."https://zbmath.org/authors/?q=ai:kibkalo.vladislav-aleksandrovichSummary: An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations.Periodic solutions, KAM tori, and bifurcations in the planar anisotropic Schwarzschild-type problemhttps://zbmath.org/1528.370502024-03-13T18:33:02.981707Z"Alberti, Angelo"https://zbmath.org/authors/?q=ai:alberti.angelo"Vidal, Claudio"https://zbmath.org/authors/?q=ai:vidal.claudio"Vidarte, Jhon"https://zbmath.org/authors/?q=ai:vidarte.jhonThe anisotropic Schwarzschild-type problem in the plane considered here is determined by the Hamiltonian
\[
H=\frac 12 (p_x^2+p_y^2)-\frac 1{\sqrt{\mu x^2+y^2}}-\frac b{(\mu x^2+y^2)^{\frac 32}}, \quad \mu>0.
\]
This becomes the standard Schwarzschild Hamiltonian for \(\mu=0\). Here the anisotropy is assumed to be small, namely \(\mu\) close to 1 and \(b\) small. In this case, a small parameter \(\epsilon\) can be introduced such that \(b=\epsilon B\) and \(\mu=1-\mu_0\epsilon\) for suitable \(B\) and \(\mu_0\). The Hamiltonian \(H\) is then written as a perturbation of the Kepler problem by developing it in Taylor series in \(\epsilon\). By using polar and Delauney coordinates, \(H\) is normalized by averaging perturbations over the periodic solutions of the Kepler system. The normalized Hamiltonian is then reduced, by truncating terms of degree \(\geq 2\) in \(\epsilon\), to the reduced space of the planar Kepler problem, which is diffeomorphic to \(\mathbb S^2\). The critical points of the equations of motion are then determined. Symplectic coordinates are introduced in the neighborhoods of the critical points. In these coordinates periodic solutions are found via Reeb's theorem and their stability is discussed. Moreover, periodic bifurcations and symmetric periodic solutions are studied. It is remarked that certain symmetric solutions of the system found in literature depend on \(\epsilon\), while the ones determined here do not. The existence of KAM tori enclosing some periodic solutions is proved for certain values of the energy, by using a theorem by \textit{Y. Han} et al. [Ann. Henri Poincaré 10, No. 8, 1419--1436 (2010; Zbl 1238.37018)]. At last, for a Hill-type Hamiltonian obtained from \(H\), the existence of periodic solutions and their stability is discussed.
Reviewer: Giovanni Rastelli (Vercelli)Complex Arnol'd-Liouville mapshttps://zbmath.org/1528.370512024-03-13T18:33:02.981707Z"Biasco, Luca"https://zbmath.org/authors/?q=ai:biasco.luca"Chierchia, Luigi"https://zbmath.org/authors/?q=ai:chierchia.luigiThis paper considers Arnol'd-Liouville action-angle variables that depend on adiabatic invariants for real analytic Hamiltonian systems with one degree-of-freedom. The specific goal is studying the holomorphic properties of the complex continuation of these action-angle variables, and especially their behavior near separatrices.
Two primary results are as follows. The first shows that, in the vicinity of the separatrices, the actions -- considered as functions of the energy -- have a special representation in terms of the affine functions of the logarithm with coefficients that are analytic functions. This result allows estimation of derivatives of the action function.
The second result allows the computation of the analyticity radii of the action-angle variables in arbitrary neighborhoods of the separatrices and their behavior in a suitably scaled distance from separatrices.
Later in the paper the authors consider the convexity of the inverse of the action functions near separatrices. It is also shown that there are inflection points within separatrices.
Reviewer: William J. Satzer Jr. (St. Paul)Generic KAM Hamiltonians are not quantum ergodichttps://zbmath.org/1528.370522024-03-13T18:33:02.981707Z"Gomes, Seán"https://zbmath.org/authors/?q=ai:gomes.sean-pThe author investigates a converse to quantum ergodicity, namely generic failure of quantum ergodicity for the quantization of a KAM perturbation of a completely integrable classical Hamiltonian. More precisely, for a KAM perturbation of a completely integrable Kolmogorov nondegenerate Gevrey smooth classical Hamiltonian (e.g., a completely integrable Schrödinger operator \(-\Delta +V\)), the main result establishes failure of quantum ergodicity for almost every perturbation size parameter.
The paper builds on results by \textit{G. Popov} [Mat. Contemp. 26, 87--107 (2004; Zbl 1074.37031); Ergodic Theory Dyn. Syst. 24, No. 5, 1753--1786 (2004; Zbl 1088.37030); Ann. Henri Poincaré 1, No. 2, 249--279 (2000; Zbl 1002.37028); Ann. Henri Poincaré 1, No. 2, 223--248 (2000; Zbl 0970.37050)], who constructed a normal form for Gevrey smooth Hamiltonians (which is refined here) and who proved that perturbed elliptic operators in a closely related setting have quasimodes which are microlocalized in phase space near suitable KAM tori. The author extends the results from microlocalization of quasimodes to eigenfunctions by controlling the spectral concentration.
Reviewer: Marius Lemm (Cambridge)Two different sequences of infinitely many homoclinic solutions for a class of fractional Hamiltonian systemshttps://zbmath.org/1528.370532024-03-13T18:33:02.981707Z"Benhassine, A."https://zbmath.org/authors/?q=ai:benhassine.abderrazekSummary: We consider the problem of existence of infinitely many homoclinic solutions for the following fractional Hamiltonian systems (FHS):
\[ \begin{array}{c}{-}_t{D}_{\infty }^{\alpha }\left.{(}_{-\infty }{D}_t^{\alpha }x\left(t\right)\right)-L\left(t\right)x\left(t\right)+\nabla W\left(t,x\left(t\right)\right)=0,\\
x \in{H}^{\alpha }\left({\mathbb{R}},{\mathbb{R}}^N\right),\end{array}\]
where \(\alpha \in \left(\left.\frac{1}{2},1\right]\right.,t\in{\mathbb{R}},x\in{\mathbb{R}}^N,\) and \({-}_t{D}_t^{\alpha }\) and \({}_t{D}_{\infty }^{\alpha }\) are the left and right Liouville-Weyl fractional derivatives of order \(\alpha\) on the entire axis \(\mathbb{R} ,\) respectively. The novelty of our results is that, under the assumption that the nonlinearity \(W\in{C}^1\left({\mathbb{R}}\times{\mathbb{R}}^N,{\mathbb{R}}\right)\) involves a combination of superquadratic and subquadratic terms, for the first time, we show that the FHS possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of the FHS goes to infinity and zero, respectively. Some recent results available in the literature are generalized and significantly improved.Monotonicity and asymptotic analysis of the period function of nearly parallel vortex filaments modelhttps://zbmath.org/1528.370542024-03-13T18:33:02.981707Z"Lu, Lin"https://zbmath.org/authors/?q=ai:lu.lin"He, Xiaokai"https://zbmath.org/authors/?q=ai:he.xiaokai"Chen, Aiyong"https://zbmath.org/authors/?q=ai:chen.aiyongSummary: The periods of the periodic wave solutions for nearly parallel vortex filaments model are discussed. By the transformation of variables, the vortex filaments model is reduced to the planar Hamiltonian system whose Hamiltonian function includes a logarithm term. We successfully handle the logarithm term in the study of the monotonicity of the period function of periodic solutions. Moreover, the asymptotic behavior of the period function is revealed.A nonholonomic model and complete controllability of a three-link wheeled snake robothttps://zbmath.org/1528.370552024-03-13T18:33:02.981707Z"Artemova, Elizaveta M."https://zbmath.org/authors/?q=ai:artemova.elizaveta-markovna"Kilin, Alexander A."https://zbmath.org/authors/?q=ai:kilin.aleksandr-aleksandrovichSummary: This paper is concerned with the controlled motion of a three-link wheeled snake robot propelled by changing the angles between the central and lateral links. The limits on the applicability of the nonholonomic model for the problem of interest are revealed. It is shown that the system under consideration is completely controllable according to the Rashevsky-Chow theorem. Possible types of motion of the system under periodic snake-like controls are presented using Fourier expansions. The relation of the form of the trajectory in the space of controls to the type of motion involved is found. It is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction.Anti-integrability for three-dimensional quadratic mapshttps://zbmath.org/1528.370562024-03-13T18:33:02.981707Z"Hampton, Amanda E."https://zbmath.org/authors/?q=ai:hampton.amanda-e"Meiss, James D."https://zbmath.org/authors/?q=ai:meiss.james-dSummary: We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced 30 years ago for the Frenkel-Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a \textit{quadratic correspondence} whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit becoming an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations to obtain orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node and period-doubling bifurcations have symbol sequences that differ in exactly one position.Euler-Lagrange-Herglotz equations on Lie algebroidshttps://zbmath.org/1528.370572024-03-13T18:33:02.981707Z"Anahory Simoes, Alexandre"https://zbmath.org/authors/?q=ai:anahory-simoes.alexandre"Colombo, Leonardo"https://zbmath.org/authors/?q=ai:colombo.leonardo-jesus"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuel"Salgado, Modesto"https://zbmath.org/authors/?q=ai:salgado.modesto-r"Souto, Silvia"https://zbmath.org/authors/?q=ai:souto.silviaSummary: We introduce Euler-Lagrange-Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from \(TQ\times \mathbb{R}\) and \(T^\ast Q \times \mathbb{R}\) to \(A\times \mathbb{R}\) and \(A^\ast\times\mathbb{R}\), respectively, where \(A\) is a Lie algebroid and \(A^\ast\) carries the associated Poisson structure. We see that \(A^\ast\times\mathbb{R}\) possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on \(TQ\times\mathbb{R}\) and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.Integrable equations associated with the finite-temperature deformation of the discrete Bessel point processhttps://zbmath.org/1528.370582024-03-13T18:33:02.981707Z"Cafasso, Mattia"https://zbmath.org/authors/?q=ai:cafasso.mattia"Ruzza, Giulio"https://zbmath.org/authors/?q=ai:ruzza.giulioSummary: We study the finite-temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro-differential Painlevé II equation of \textit{G. Amir} et al. [Commun. Pure Appl. Math. 64, No. 4, 466--537 (2011; Zbl 1222.82070)], and we compute initial conditions for the Poissonization parameter equal to 0. As proved by \textit{D. Betea} and \textit{J. Bouttier} [Math. Phys. Anal. Geom. 22, No. 1, Paper No. 3, 47 p. (2019; Zbl 1409.82010)], in a suitable continuum limit the last particle distribution converges to that of the finite-temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg-de Vries equation, as well as the discrete integro-differential Painlevé II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its-Izergin-Korepin-Slavnov theory of integrable operators developed by \textit{A. Borodin} [Int. Math. Res. Not. 2000, No. 9, 467--494 (2000; Zbl 0964.39015)]
and \textit{J. Baik} et al. [J. Am. Math. Soc. 12, No. 4, 1119--1178 (1999; Zbl 0932.05001)].The 3-wave resonant interaction model: spectra and instabilities of plane waveshttps://zbmath.org/1528.370592024-03-13T18:33:02.981707Z"Romano, Marzia"https://zbmath.org/authors/?q=ai:romano.marzia"Lombardo, Sara"https://zbmath.org/authors/?q=ai:lombardo.sara"Sommacal, Matteo"https://zbmath.org/authors/?q=ai:sommacal.matteoSummary: The three wave resonant interaction model (3WRI) is a non-dispersive system with quadratic coupling between the components that finds application in many areas, including nonlinear optics, fluids and plasma physics. Using its integrability, and in particular its Lax Pair representation, we carry out the linear stability analysis of the plane wave solutions interacting under resonant conditions when they are perturbed via localised perturbations. A topological classification of the so-called \textit{stability spectra} is provided with respect to the physical parameters appearing both in the system itself and in its plane wave solution. Alongside the stability spectra, we compute the corresponding gain function, from which we deduce that this system is linearly unstable for any generic choice of the physical parameters. In addition to stability spectra of the same kind observed in the system of two coupled nonlinear Schrödinger equations, whose non-vanishing gain functions detect the occurrence of the modulational instability, the stability spectra of the 3WRI system possess new topological components, whose associated gain functions are different from those characterising the modulational instability. By drawing on a recent link between modulational instability and the occurrence of rogue waves, we speculate that linear instability of baseband-type can be a necessary condition for the onset of rogue wave types in the 3WRI system, thus providing a tool to predict the subsequent nonlinear evolution of the perturbation.Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetryhttps://zbmath.org/1528.370602024-03-13T18:33:02.981707Z"Parker, Ross"https://zbmath.org/authors/?q=ai:parker.ross"Sandstede, Björn"https://zbmath.org/authors/?q=ai:sandstede.bjornSummary: We consider the existence and spectral stability of periodic multi-pulse solutions in Hamiltonian systems which are translation invariant and reversible, for which the fifth-order Korteweg-de Vries equation is a prototypical example. We use Lin's method to construct multi-pulses on a periodic domain, and in particular demonstrate a pitchfork bifurcation structure for periodic double pulses. We also use Lin's method to reduce the spectral problem for periodic multi-pulses to computing the determinant of a block matrix, which encodes both eigenvalues resulting from interactions between neighboring pulses and eigenvalues associated with the essential spectrum. We then use this matrix to compute the spectrum associated with periodic single and double pulses. Most notably, we prove that brief instability bubbles form when eigenvalues collide on the imaginary axis as the periodic domain size is altered. These analytical results are all in good agreement with numerical computations, and numerical timestepping experiments demonstrate that these instability bubbles correspond to oscillatory instabilities.On existence and concentration of solutions for Hamiltonian systems involving fractional operator with critical exponential growthhttps://zbmath.org/1528.370612024-03-13T18:33:02.981707Z"Costa, Augusto C. R."https://zbmath.org/authors/?q=ai:costa.augusto-c-r"Maia, Bráulio B. V."https://zbmath.org/authors/?q=ai:maia.braulio-b-v"Miyagaki, Olímpio H."https://zbmath.org/authors/?q=ai:miyagaki.olimpio-hiroshiSummary: This paper is concerned with the existence and concentration of ground state solutions for the following class of fractional Schrödinger system
\[
\begin{aligned}
(-\Delta )^{1/2}u + (\lambda a(x) +1)u= H_v (u,v) \text{ in } \mathbb{R}, u,v \in H^{1/2}(\mathbb{R}), \\
(-\Delta )^{1/2}v + (\lambda a(x) + 1)v= H_u (u,v) \text{ in } \mathbb{R}, u,v \in H^{1/2}(\mathbb{R}),
\end{aligned}
\]
where \(H\) has exponential critical growth, \(\lambda\) is a positive parameter and \(a(x)\) has a potential well with \(\mathrm{int}\big( a^{-1}(0)\big)\) consisting of \(k\) disjoint components \(\Omega_1, \ldots, \Omega_k\). The proof relies on variational methods and combines truncation arguments and the Moser iteration technique.
{{\copyright} 2022 Wiley-VCH GmbH.}On the norm equivalence of Lyapunov exponents for regularizing linear evolution equationshttps://zbmath.org/1528.370622024-03-13T18:33:02.981707Z"Blumenthal, Alex"https://zbmath.org/authors/?q=ai:blumenthal.alex"Punshon-Smith, Sam"https://zbmath.org/authors/?q=ai:punshon-smith.samSummary: We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability \((L^p)\) or differentiability \((W^{s, p})\). In contrast to finite dimensions, the Lyapunov exponent could apriori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier-Stokes equations with stochastic or periodic forcing.Energy decay for evolution equations with delay feedbackshttps://zbmath.org/1528.370632024-03-13T18:33:02.981707Z"Komornik, Vilmos"https://zbmath.org/authors/?q=ai:komornik.vilmos"Pignotti, Cristina"https://zbmath.org/authors/?q=ai:pignotti.cristinaSummary: We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time. A step by step procedure combined with Gronwall's inequality allows us to prove the existence and uniqueness of solutions. Furthermore, under appropriate conditions we obtain exponential decay estimates.
{{\copyright} 2022 Wiley-VCH GmbH}Attractor bifurcation for positive solutions of evolution equationshttps://zbmath.org/1528.370642024-03-13T18:33:02.981707Z"Jia, Mo"https://zbmath.org/authors/?q=ai:jia.mo"Li, Desheng"https://zbmath.org/authors/?q=ai:li.desheng"Bai, Jinlong"https://zbmath.org/authors/?q=ai:bai.jinlong"Zhang, Yuxiang"https://zbmath.org/authors/?q=ai:zhang.yuxiangSummary: We establish an attractor bifurcation theorem and prove a global dynamic bifurcation theorem for positive solutions of evolution equations in ordered Banach spaces. These results may give a new insight into some fundamental problems in mathematical biology such as the persistence and the reproducing numbers of species.Dynamics of stochastic FitzHugh-Nagumo system on unbounded domains with memoryhttps://zbmath.org/1528.370652024-03-13T18:33:02.981707Z"My, Bui Kim"https://zbmath.org/authors/?q=ai:my.bui-kim"Toan, Nguyen Duong"https://zbmath.org/authors/?q=ai:toan.nguyen-duongAuthors' abstract: In this paper, we consider the non-autonomous stochastic FitzHugh-Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all \(N\)-dimensional space.
Reviewer: Anhui Gu (Chongqing)Limiting dynamics of stochastic heat equations with memory on thin domainshttps://zbmath.org/1528.370662024-03-13T18:33:02.981707Z"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Li, Hui"https://zbmath.org/authors/?q=ai:li.hui.9"Huang, Xin"https://zbmath.org/authors/?q=ai:huang.xin.1"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian(no abstract)Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: theoretical results and applicationshttps://zbmath.org/1528.370672024-03-13T18:33:02.981707Z"Wang, Renhai"https://zbmath.org/authors/?q=ai:wang.renhai"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to \textit{G. da Prato} and \textit{M. Röckner} [Prog. Probab. 59, 115--122 (2008; Zbl 1154.60050)] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future.Steady states of two-dimensional granular systems are unique, stable, and sometimes satisfy detailed balancehttps://zbmath.org/1528.370682024-03-13T18:33:02.981707Z"Myhill, Alex D. C."https://zbmath.org/authors/?q=ai:myhill.alex-d-c"Blumenfeld, Raphael"https://zbmath.org/authors/?q=ai:blumenfeld.raphaelSummary: Understanding the structural evolution of granular systems is a long-standing problem. A recently proposed theory for such dynamics in two dimensions predicts that steady states of very dense systems satisfy detailed-balance. We analyse analytically and numerically the steady states of this theory in systems of arbitrary density and report the following. (1) We discover that all such dynamics almost certainly possess only one physical steady state, which may or may not satisfy detailed balance. (2) We show rigorously that, if a detailed balance solution is possible then it is unique. The above two results correct an erroneous conjecture in the literature. (3) We show rigorously that the detailed-balance solutions in very dense systems are globally stable, extending the local stability found for these solutions in the literature. (4) In view of recent experimental observations of robust detailed balance steady states in very dilute cyclically sheared systems, our results point to a self-organisation of process rates in dynamic granular systems.Data-driven discovery of governing equations for coarse-grained heterogeneous network dynamicshttps://zbmath.org/1528.370692024-03-13T18:33:02.981707Z"Owens, Katherine"https://zbmath.org/authors/?q=ai:owens.katherine"Kutz, J. Nathan"https://zbmath.org/authors/?q=ai:kutz.j-nathanSummary: We leverage data-driven model discovery methods to determine governing equations for the emergent behavior of heterogeneous networked dynamical systems. Specifically, we consider networks of coupled nonlinear oscillators whose collective behavior approaches a limit cycle. Stable limit cycles are of interest in many biological applications, as they model self-sustained oscillations (e.g. heartbeats, chemical oscillations, neurons firing, circadian rhythm). For systems that display relaxation oscillations, our method automatically detects boundary (time) layer structures in the dynamics, fitting inner and outer solutions and matching them in a data-driven manner. We demonstrate the method on well-studied systems: the Rayleigh oscillator and the van der Pol oscillator. We then apply the mathematical framework to networks of semisynchronized Kuramoto, Rayleigh, Rössler, and FitzHugh-Nagumo oscillators, as well as heterogeneous combinations thereof, showing that the discovery of coarse-grained variables and dynamics can be accomplished with the proposed architecture.Retraction maps: a seed of geometric integratorshttps://zbmath.org/1528.370702024-03-13T18:33:02.981707Z"Barbero-Liñán, María"https://zbmath.org/authors/?q=ai:barbero-linan.maria"de Diego, David Martín"https://zbmath.org/authors/?q=ai:martin-de-diego.davidThe authors propose a generalization of the classical concept of retraction map. Retraction maps provide a means to select a smooth curve on a differentiable manifold given an initial position and velocity; a curve like this is an approximation of the Riemannian exponential map. The classical version of retraction map can be used in many ways, including approximating geodesics on a Riemannian manifold. More background and additional references for retraction maps are provided in [\textit{P.-A. Absil} et al., in Optimization algorithms on matrix manifolds. Princeton Unversity Press, Princeton NJ (2008; Zbl 1147.65043)].
In this paper the original idea is extended to construct geometric integrators that can take the form of a discretization map. Lifts of such discretization maps to the tangent and cotangent bundles turn out to inherit the properties of the original map, and therefore continue to be discretization maps.
It turns out that the cotangent lift of a discretization map is also a natural symplectomorphism. This is particularly useful for creating geometric integrators for systems defined by either Hamiltonian or Lagrangian functions. The authors show how well-known geometric methods can be derived in their general framework. Furthermore, they believe that their work can enable other geometric integrators, even higher order ones, that might be applied to more complex mechanical systems.
Reviewer: William J. Satzer Jr. (St. Paul)Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systemshttps://zbmath.org/1528.370712024-03-13T18:33:02.981707Z"Bartel, Andreas"https://zbmath.org/authors/?q=ai:bartel.andreas"Günther, Michael"https://zbmath.org/authors/?q=ai:gunther.michael"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgit"Reis, Timo"https://zbmath.org/authors/?q=ai:reis.timoSummary: A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions-Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.Dynamical systems theory and algorithms for NP-hard problemshttps://zbmath.org/1528.370722024-03-13T18:33:02.981707Z"Sahai, Tuhin"https://zbmath.org/authors/?q=ai:sahai.tuhinSummary: This article surveys the burgeoning area at the intersection of dynamical systems theory and algorithms for NP-hard problems. Traditionally, computational complexity and the analysis of non-deterministic polynomial-time (NP)-hard problems have fallen under the purview of computer science and discrete optimization. However, over the past few years, dynamical systems theory has increasingly been used to construct new algorithms and shed light on the hardness of problem instances. We survey a range of examples that illustrate the use of dynamical systems theory in the context of computational complexity analysis and novel algorithm construction. In particular, we summarize a) a novel approach for clustering graphs using the wave equation partial differential equation, b) invariant manifold computations for the traveling salesman problem, c) novel approaches for building quantum networks of Duffing oscillators to solve the MAX-CUT problem, d) applications of the Koopman operator for analyzing optimization algorithms, and e) the use of dynamical systems theory to analyze computational complexity.
For the entire collection see [Zbl 1445.37003].Extinction of multiple populations and a team of die-out Lyapunov functionshttps://zbmath.org/1528.370732024-03-13T18:33:02.981707Z"Akhavan, Naghmeh"https://zbmath.org/authors/?q=ai:akhavan.naghmeh"Yorke, James A."https://zbmath.org/authors/?q=ai:yorke.james-aSummary: The extinction of species is a major problem of concern with a large literature. We investigate a differential equations model for population interactions with the goal of determining when several species (i.e., coordinates of a bounded solution) must die out or ``go extinct'' and must do so exponentially fast. Typically each coordinate represents the population density of a different species. For our main tool, we create what we call ``die-out'' Lyapunov functions. A given system may have several or many such functions, each of which is a function of a different set of coordinates. That die-out function implies that one of the species in its subset must die out exponentially fast -- for almost every choice of coefficients of the system. We create a ``team'' of die-out functions that work together to show that \(k\) species must die, where \(k\) is determined separately. Second, we present a ``trophic'' condition for generalized Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting. That implies that all solutions are bounded.Block-pulse integrodifference equationshttps://zbmath.org/1528.370742024-03-13T18:33:02.981707Z"Gilbertson, Nora M."https://zbmath.org/authors/?q=ai:gilbertson.nora-m"Kot, Mark"https://zbmath.org/authors/?q=ai:kot.markSummary: We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing growth, for populations that are confined to a finite habitat-patch. This method is based on approximating the growth function of the IDE with a piecewise-constant function, and we call the resulting model a block-pulse IDE. We explicitly write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. We characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including formal stability analyses, and we explore the bifurcation structure of these models. These simple models display rich dynamics, with numerous fold bifurcations. We then use three-, five-, and ten-step block-pulse IDEs, with a numerical root finder, to approximate models with compensatory Beverton-Holt growth and depensatory, or Allee-effect, growth. Our method provides a good approximation for the equilibrium distributions for compensatory and depensatory growth and offers numerical and analytical advantages over the original growth models.Global extinctions arising from barnacle-algae-mussel interaction modelhttps://zbmath.org/1528.370752024-03-13T18:33:02.981707Z"Zhou, Hui"https://zbmath.org/authors/?q=ai:zhou.hui.2Summary: In this article, based on the result of the uniform persistence for \(4\)-dimensional autonomous system with interactions in a rocky intertidal community by \textit{S.-B. Hsu} et al. [SIAM J. Appl. Math. 79, No. 5, 2032--2053 (2019; Zbl 1427.34062)], the extinction of the model is further explored. According to the existence of the positive equilibrium, we rigorously classify three categories corresponding to extinct states of the three species Mussel, Algae and Barnacle, respectively. The extinction results of the \(4\)-dimensional model in this paper exactly verify further the uniform persistence obtained in [loc. cit.], and the classifications are total for the uniform persistence. What is rather more significant is that the distinguished criteria characterize the three extinct steady state. The global extinctions of the model are helpful to understand the mechanism of the three species coexistence and the cyclic succession fluctuation observed in [\textit{E. Benincà} et al., Proc. Natl. Acad. Sci. USA 112, No. 20, 6389--6394 (2015; doi:10.1073/pnas.1421968112)].Probing 3D chaotic Thomas' cyclically attractor with multimedia encryption and electronic circuitryhttps://zbmath.org/1528.370762024-03-13T18:33:02.981707Z"Khan, Najeebalam"https://zbmath.org/authors/?q=ai:khan.najeebalam"Qureshi, Muhammad Ali"https://zbmath.org/authors/?q=ai:qureshi.muhammad-ali"Akbar, Saeed"https://zbmath.org/authors/?q=ai:akbar.saeed"Ara, Asmat"https://zbmath.org/authors/?q=ai:ara.asmatSummary: This study investigates Thomas' cyclically symmetric attractor dynamics with mathematical and electronic simulations using a proportional fractional derivative to comprehend the dynamics of a given chaotic system. The three-dimensional chaotic flow was examined in detail with Riemann-Liouville derivative for different values of the fractional index to highlight the sensitivity of chaotic systems with initial conditions. Thus, the dynamics of the fractional index system were investigated with Eigenvalues, Kaplan-Yorke dimension, Lyapunov exponent, and NIST testing, and their corresponding trajectories were visualized with phase portraits, 2D density plot, and Poincaré maps. After obtaining the results, we found that the integer index dynamics are more complex than the fractional index dynamics. Furthermore, the chaotic system circuit is simulated with operational amplifiers for different fractional indices to generate analog signals of the symmetric attractor, making it an important aspect of engineering. The qualitative application of our nonlinear chaotic system is then applied to encrypt different data types such as voice, image, and video, to ensure that the developed nonlinear chaotic system can widely applied in the field of cyber security.Dynamic analysis of new two-dimensional fractional-order discrete chaotic map and its application in cryptosystemhttps://zbmath.org/1528.370772024-03-13T18:33:02.981707Z"Liu, Ze-Yu"https://zbmath.org/authors/?q=ai:liu.zeyu"Xia, Tie-Cheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng"Hu, Ye"https://zbmath.org/authors/?q=ai:hu.yeSummary: A new fractional difference equation two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent, and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Dynamical properties of a modified chaotic Colpitts oscillator with triangular wave non-linearityhttps://zbmath.org/1528.370782024-03-13T18:33:02.981707Z"Suresh, Rasappan"https://zbmath.org/authors/?q=ai:suresh.rasappan"Kumar, Kumaravel Sathish"https://zbmath.org/authors/?q=ai:kumar.kumaravel-sathish"Regan, Murugesan"https://zbmath.org/authors/?q=ai:regan.murugesan"Kumar, K. A. Niranjan"https://zbmath.org/authors/?q=ai:kumar.k-a-niranjan"Devi, R. Narmada"https://zbmath.org/authors/?q=ai:narmada-devi.r"Obaid, Ahmed J."https://zbmath.org/authors/?q=ai:obaid.ahmed-jSummary: The purpose of this paper is to introduce a new chaotic oscillator. Although different chaotic systems have been formulated by earlier researchers, only a few chaotic systems exhibit chaotic behaviour. In this work, a new chaotic system with chaotic attractor is introduced for triangular wave non-linearity. It is worth noting that this striking phenomenon rarely occurs in respect of chaotic systems. The system proposed in this paper has been realized with numerical simulation. The results emanating from the numerical simulation indicate the feasibility of the proposed chaotic system. More over, chaos control, stability, diffusion and synchronization of such a system have been dealt with.Optimal aquaculture planning while accounting for the size spectrumhttps://zbmath.org/1528.370792024-03-13T18:33:02.981707Z"Yoshioka, Hidekazu"https://zbmath.org/authors/?q=ai:yoshioka.hidekazuSummary: Aquaculture planning of a fish species requires a balance between the feeding costs and commercial benefits while considering that the resources should not be disposed of at the terminal time. Furthermore, the planning process should account for the size spectrum, namely that different individual fishes have different body sizes. Herein, we present a novel dynamic programming method to anticipate and tackle these issues consistently. The opening time of harvesting and the subsequent harvesting policy after that time are the decision variables. The novelty of the presented model is its ability to efficiently describe the distributed body weights based on a logistic growth model having an unknown maximum body weight distributed according to a probability measure. We start from a discrete-time and discrete-state model on a daily basis and then obtain its continuous-state counterpart. The innovative two models are presented for the first time, and the former serves as the fully implementable numerical discretization of the latter. We present application examples of the proposed models to real data of an ongoing aquaculture system for the ayu sweetfish \textit{Plecoglossus altivelis altivelis} in Japan.Settled elements in profinite groupshttps://zbmath.org/1528.370802024-03-13T18:33:02.981707Z"Cortez, María Isabel"https://zbmath.org/authors/?q=ai:cortez.maria-isabel"Lukina, Olga"https://zbmath.org/authors/?q=ai:lukina.olgaAn infinite rooted tree \(T\) is called a \(d\)-ary tree if it contains a single vertex called the root at level \(0\), and for all \(n\geq 1\), each vertex at level \(n-1\) is joined to \(d\geq 2\) vertices at level \(n\). Denote by \(\mathrm{Aut}(T)\) the set of automorphisms on \(T\). Let \(\sigma\in \mathrm{Aut}(T)\). Then \(\sigma\) restricted on the set of vertices at level \(n\) is a permutation of a set of \(d^n\) elements. A vertex \(v\) at level \(n\) is said to be in a cycle of length \(k \geq 1\) of \(\sigma\) if \(\sigma^k(v) = v\) and \(\sigma^j(v)\neq v\) for every \(1\leq j <k\). It is in a stable cycle if all the descendants of \(\{v, \sigma(v), \dots, \sigma^{k-1}(v)\}\) at level \(m>n\) form a cycle of length \(d^{m-n}k\). We say that \(\sigma\) is settled if the number of vertices in stable cycles at level \(n\) divided by \(d^n\) tends to \(1\). A profinite subgroup \(\mathcal{G}\subset\mathrm{Aut}(T)\) is called densely settled if the set of settled elements in \(\mathcal{G}\) is dense in \(\mathcal{G}\).
Let \(K\) be an algebraic number field (containing \(\mathbb{Q}\)), and \(\overline{K}\) a separable closure of \(K\). For \(\alpha\in K\) and a polynomial \(f\) of degree \(d\) with coefficients in \(K\), we can associate a \(d\)-ary tree \(T\) by choosing \(V_n=\{f^{-n}(\alpha)\}\) as vertices at level \(n\) and taking an edge joining a vertex \(\beta\in V_{n+1}\) and \(\gamma\in V_n\) if \(f(\beta)=\gamma\). An arboreal representation of \(K\) is a homomorphism \(\rho_{f,\alpha}: \mathrm{Gal}(\overline{K}/K) \rightarrow\mathrm{Aut}(T)\), from the Galois group \(\mathrm{Gal}(\overline{K}/K)\) to the space \(\mathrm{Aut}(T)\) of automorphisms on the \(d\)-ary tree \(T\). Let \(Y_n\) be the Galois group of the extension \(K(f^{-n}(\alpha))\). Then the image of the representation is the profinite groupe \(Y_{\infty}=\varprojlim \{Y_{n+1} \rightarrow Y_n\} \).
In a series of papers [Geom. Dedicata 124, 27--35 (2007; Zbl 1206.11069); Pure Appl. Math. Q. 5, No. 1, 213--225 (2009; Zbl 1167.11011); Proc. Am. Math. Soc. 140, No. 6, 1849--1863 (2012; Zbl 1243.11115)], \textit{N. Boston} and \textit{R. Jones} studied the density of settled elements in the image of an arboreal representation and conjectured that for a quadratic polynomial \(f\) and \(\alpha\in \mathbb{Q}\), the set of settled elements is dense in the image of \(\rho_{f,\alpha}\).
The present paper studies the conjecture of N. Boston and R. Jones for quadratic polynomials having a strictly pre-periodic post-critical orbit of length \(2\). Among others, the authors prove that the above defined profinite groupe \(Y_{\infty}\) is densely settled. The authors also provide new evidence that the conjecture of Boston and Jones holds for quadratic polynomials with strictly pre-periodic post-critical orbits of length at least \(3\).
The main tools of the paper are maximal abelian subgroups and their Weyl groups. The novelty is an application of the Weyl group framework to the study of arboreal representations and groups acting on trees.
Reviewer: Lingmin Liao (Créteil)On the relationship between Lozi maps and max-type difference equationshttps://zbmath.org/1528.390012024-03-13T18:33:02.981707Z"Linero Bas, A."https://zbmath.org/authors/?q=ai:linero-bas.antonio"Nieves Roldán, D."https://zbmath.org/authors/?q=ai:nieves-roldan.dIn this paper, the authors formulate two different sets of sufficient conditions on \(\alpha,\beta,\gamma,\delta\in\mathbb{R}\) for which the so-called generalized Lozi difference equation
\[
x_{n+1}=\alpha\left|x_n\right|+\beta x_n+ \gamma x_{n-1}+\delta
\]
is conjugate via a logarithmic change of variable to a family of second-order max-type difference equations. The appearance of the max-function originates from the observation that for all \(x\in\mathbb{R}\) we have \(|x|=\max\left\{x,-x\right\}\). Applying these results, the authors then study the dynamical properties of the family of max-type difference equations
\[
x_{n+1}=\frac{\max\left\{x_n^3,\alpha\right\}}{x_nx_{n-1}},\qquad \alpha>0,
\]
and subsequently the family of Lozi-type difference equations
\[
x_{n+1}=1-\alpha\left|x_n\right|+\alpha x_{n-1},\qquad \alpha\in\mathbb{R}.
\]
Finally, the authors present the results of some numerical simulations, which motivate the formulation of some open problems.
Reviewer: Jonathan Hoseana (Bandung)Non-existence of S-integrable three-point partial difference equations in the lattice planehttps://zbmath.org/1528.390062024-03-13T18:33:02.981707Z"Levi, Decio"https://zbmath.org/authors/?q=ai:levi.decio"Rodríguez, Miguel A."https://zbmath.org/authors/?q=ai:rodriguez.miguel-angelIn this short note the authors use \textit{R. Yamilov}'s theorem [J. Phys. A, Math. Gen. 39, No. 45, R541--R623 (2006; Zbl 1105.35136), Theorem 13] to show that lattice equations defined on three points cannot be S-integrable. An equation (continuous or discrete) is said to be S-integrable if it is integrable by means of the inverse scattering transform method, while it is said to be C-integrable if it is linearisable through a change of variables, see [\textit{F. Calogero}, in: What is integrability, Springer Ser. Nonlinear Dyn. 1--62 (1991; Zbl 0808.35001)]. Yamilov's theorem states that for any evolutionary differential-difference equation of the form
\[
\frac{\mathrm{d}u_{n}}{\mathrm{d}t} = f(u_{n+N},u_{n+N-1},\dots,u_{n+M}), \qquad \frac{\partial f}{\partial u_{n+N}} \frac{\partial f}{\partial u_{n+M}}\neq 0, \tag{1}
\]
where \(f\) is a locally analytic function of its arguments and \(N\geq M\), a necessary condition for S-integrability is \(M=-N\). See also [the authors, Ufim. Mat. Zh. 13, No. 2, 158--165 (2021; Zbl 1488.39050)] and the monograph [\textit{D. Levi} et al., Continuous symmetries and integrability of discrete equations. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1525.39001)] for further details about this theorem.
Let us now be more specific. The paper focuses on partial difference equations defined on a three-point stencil, i.e., functional relations of the following form:
\[
\mathcal{E}_{n,m}(u_{n,m},u_{n+1,m},u_{n,m+1})=0, \tag{2}
\]
for an unknown field \(u_{n,m}\colon\mathbb{Z}^{2}\to\mathbb{C}\). The function \(\mathcal{E}_{n,m}(x,y,z,t)\) is multilinear in its arguments. In the literature some results on the C-integrability of partial difference equations of the form (2) have been obtained in [\textit{C. Scimiterna} and \textit{D. Levi}, J. Phys. A, Math. Theor. 45, No. 45, Article ID 025205, 13 p. (2012; Zbl 1266.39013)].
The question that this paper answers is ``can a partial difference equation of the form (2) be S-integrable?''. The answer is found to be negative. This result is proved by considering the partial continuous limit of Equation (2):
\[
u_{n,m+1}=u_{n}(t+\varepsilon) = u_{n}(t) + \varepsilon\frac{\mathrm{d}u_{n}}{\mathrm{d}t} + \varepsilon\frac{\mathrm{d}^{2}u_{n}}{\mathrm{d}t^{2}} +O(\varepsilon^{3}). \tag{3}
\]
Indeed, plugging (3) into (2), after some algebraic simplification one obtains that the partial continuous limit is a differential-difference equation (1) with \(N=1\) and \(M=0\). This readily implies the statement.
The paper is written concisely, but without sacrificing its clearness, and with a good amount of background material and references.
Reviewer: Giorgio Gubbiotti (Milano)On target-oriented control of Hénon and Lozi mapshttps://zbmath.org/1528.390072024-03-13T18:33:02.981707Z"Braverman, E."https://zbmath.org/authors/?q=ai:braverman.elena"Rodkina, A."https://zbmath.org/authors/?q=ai:rodkina.alexandraSummary: We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the Hénon and the Lozi maps.Moduli of smoothness and generalized canonical Fourier-Bessel differential operator on the half-linehttps://zbmath.org/1528.410652024-03-13T18:33:02.981707Z"Mfadel, Ali El"https://zbmath.org/authors/?q=ai:el-mfadel.ali"Elomari, M'hamed"https://zbmath.org/authors/?q=ai:elomari.mhamedSummary: The main crux of this work is to study the equivalence of modulus of smoothness and K-functionals in the Sobolev space \(W^p_{2,\nu ,\mathfrak{M}}\) constructed by the canonical Fourier-Bessel differential operator \(\Delta_{\nu}^{\mathfrak{M}}\) on the half-line where \(\nu >-1/2\) and \(\mathfrak{M}\in SL(2,\mathbb{R})\). The proofs are based on harmonic analysis related to the generalized canonical Fourier-Bessel differential operator and its associated transform.Deep learning via dynamical systems: an approximation perspectivehttps://zbmath.org/1528.411062024-03-13T18:33:02.981707Z"Li, Qianxiao"https://zbmath.org/authors/?q=ai:li.qianxiao"Lin, Ting"https://zbmath.org/authors/?q=ai:lin.ting"Shen, Zuowei"https://zbmath.org/authors/?q=ai:shen.zuoweiAn important challenge in recent trends in deep learning is to develop a framework for its effectiveness by capturing the effect of sequential function composition in deep neural networks and one important tool used in this regard is a dynamical systems approach whereby one thinks of deep neural networks as a discretization of certain ordinary differential equations. In this interesting paper, the authors study the dynamical systems approach whereby deep residual networks are idealized as continuous-time dynamical systems, from an approximation perspective.
The authors prove several theorems which give sufficient conditions for universal approximation using continuous time deep residual networks which can also be understood as approximation theories in \(L_p\) using certain flow maps of dynamical systems. The authors also establish rates of approximation in terms of some time horizons.
This is a new and interesting idea in approximation theory, namely that composition function approximation through flow maps contributes to building useful mathematical frameworks to investigate deep learning.
The paper is well written with ample references covering both classical and newer work on deep neural networks, their connections to approximation theory, control and differential equations.
Reviewer: Steven B. Damelin (Ann Arbor)Incidence problems in harmonic analysis, geometric measure theory, and ergodic theory. Abstracts from the workshop held June 4--9, 2023https://zbmath.org/1528.420012024-03-13T18:33:02.981707ZSummary: The workshop \textit{Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory} covered interactions between geometric problems involving fractals, dimensions, patterns, projections and incidences, and on the other hand recent developments in Fourier analysis and Ergodic theory which have been inspired by fractal geometric problems, or have been instrumental in solving them.Wasserstein distance between noncommutative dynamical systemshttps://zbmath.org/1528.460532024-03-13T18:33:02.981707Z"Duvenhage, Rocco"https://zbmath.org/authors/?q=ai:duvenhage.roccoSummary: We introduce and study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein distance arises, but also study the asymmetric case. This setup is illustrated in the context of reduced dynamics, and a number of simple examples are also presented.Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigrouphttps://zbmath.org/1528.490262024-03-13T18:33:02.981707Z"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan|gao.yuan.1"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guoSummary: Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton-Jacobi equations (HJE). The discrete Hamiltonian is an \(m\)-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation and the backward equation with ``no reaction'' boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear formhttps://zbmath.org/1528.530362024-03-13T18:33:02.981707Z"Bandyopadhyay, S."https://zbmath.org/authors/?q=ai:bandyopadhyay.saugata"Dacorogna, B."https://zbmath.org/authors/?q=ai:dacorogna.bernard"Matveev, V. S."https://zbmath.org/authors/?q=ai:matveev.vladimir-s"Troyanov, M."https://zbmath.org/authors/?q=ai:troyanov.marcThis important paper completely solves a classical problem of Riemannian geometry, going back to Riemann himself, who solved it in a special case, after having enunciated it in his \textit{Habilitationsvortrag} of 1854, in his 1861 submission for a prize question of the Paris Academy, \textit{Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae}. There Riemann considered what is now called a Riemannian metric, that is, a symmetric positive definite 2-form \(g = g_{ij}(x)\), asked and answered the question under what conditions there exists a coordinate system such that \(g\) is given by a constant matrix. The answer is that such coordinates exist locally if and only if what came to be known as the Riemann curvature tensor is identically zero. One notices that the assumption of positive definiteness is not essential for Riemann's proof, that it is enough for the symmetric form to be nondegenerate.
The case in which the bilinear form is skew-symmetric was considered and solved, thereby laying the foundation for symplectic geometry, by \textit{G. Darboux} [Bull. Sci. Math. 6, 14--36, 49--68 (1882; JFM 14.0294.01)], where he proved that there exists a local coordinate system such that a nondegenerate differential 2-form \(\omega = \omega_{ij}(x)\) is given by a constant matrix, if and only if \(\omega\) is closed.
The aim of the paper under review is to provide necessary and sufficient conditions that offer a complete answer to Riemann's and Darboux's question for an arbitrary bilinear form, i.e., a tensor field of type \((0, 2)\), which may have nontrivial symmetric and skew-symmetric parts that can be degenerate. Since the case in which the symmetric part is nondegenerate can be approached with the methods of Riemann, the results in the paper under review are new only in the case in which \(g\) is degenerate and \(\omega\) is arbitrary.
The results are formulated such that ``the hypothesis on \(g\) and \(\omega\) can effectively be checked using only differentiation and algebraic manipulations, as was the case in the results of Riemann and Darboux''. Moreover, particular care has been taken to use a notational language and methods that ``were available to, and used by, Riemann, Darboux and other fathers of differential geometry. These methods include basic real analysis, basic linear algebra and the standard results on the existence and uniqueness of solutions of systems of ordinary differential equations.''
The proofs are first presented in a form that ``would be understood by Bernhard Riemann and mathematicians coming shortly after him, such as Sophus Lie, Gregorio Ricci-Curbastro, Gaston Darboux, Tullio Levi-Civita and Ferdinand Georg Frobenius'', i.e., under the assumption ``that all objects are sufficiently smooth'', to be later, whenever possible, given in minimal regularity.
There are, however, three concepts used in the paper under review, which were not available in Riemann's or Darboux's time: Levi-Civita's parallel transport, the idea of the holonomy group, together with the Ambrose-Singer Theorem, and ideas from the theory of integrable Hamiltonian systems.
Reviewer: Victor V. Pambuccian (Glendale)Reductions: precontact versus presymplectichttps://zbmath.org/1528.530752024-03-13T18:33:02.981707Z"Grabowska, Katarzyna"https://zbmath.org/authors/?q=ai:grabowska.katarzyna"Grabowski, Janusz"https://zbmath.org/authors/?q=ai:grabowski.januszThe main goal of the paper is to prove a version of Marsden-Weinstein-Meyer symplectic reduction in the contact setting (Theorem 1.1). To achieve this aim, the authors work in greater generalities and allow the constant rank of the structure not to be maximal, i.e., they consider precontact structures. Note also that the underlying distribution is not necessarily defined by a global \(1\)-form. Their major tool consists in working on a presymplectic cover of the precontact manifold, which is an \(\mathbb{R}^\times\)-bundle over the manifold equipped with a presymplectic structure that defines the contact structure through the bundle projection map.
In Section 2, the yoga between precontact manifolds and presymplectic \(\mathbb{R}^\times\)-bundles is spelled out. For the paper, a precontact structure of rank \(2r+1\) on a manifold \(M\) is a distribution \(C \subset TM\) whose fibers are of codimension \(1\) and such that, if \(C=\operatorname{ker}(\eta)\) locally for some (locally defined) \(1\)-form \(\eta\), then the \(2\)-form \(d\eta\) has rank \(2r\) on \(C\). An \(\mathbb{R}^\times\)-bundle \(\tau:P \to M\) is a principal bundle over \(M\) with structure group \(\mathbb{R}^\times\). It is called a presymplectic \(\mathbb{R}^\times\)-bundle of rank \(2r+1\) if \(P\) is endowed with a closed \(2\)-form \(\omega\) of rank \(2(r+1)\) which is \(1\)-homogeneous, i.e., pulling back \(\omega\) by the action map \(h_s:P\to P\) for \(s\in \mathbb{R}^\times\) amounts to rescaling by \(s\): \(h_s^\ast(\omega)=s\, \omega\). (To recover contact structures or symplectic \(\mathbb{R}^\times\)-bundles, one needs \(\operatorname{dim}(M)=\operatorname{dim}(P)-1=2r+1\).) The main result of the section is Theorem 2.17. In an abridged form, it states the one-to-one correspondence between precontact manifolds \((M,C)\) of rank \(2r+1\) and isomorphism classes of presymplectic \(\mathbb{R}^\times\)-bundles \((\tau:P\to M, h_\bullet, \omega)\) of rank \(2(r+1)\). Moreover, it is shown that a contactomorphism can be covered by an isomorphism of the corresponding covering presymplectic \(\mathbb{R}^\times\)-bundles, see Proposition 2.19.
In Section 3, the authors study a precontact-to-contact reduction theory. It consists in reducing a precontact manifold \((M,C)\) with respect to the foliation \(\mathcal{F}_C\) by maximal integral submanifolds of the characteristic distribution of \(C\). This endows \(M/\mathcal{F}_C\) with a canonical contact structure if it is smooth and the projection \(M\to M/\mathcal{F}_C\) is a surjective submersion (Theorem 3.1). This result is then reinterpreted as a presymplectic-to-symplectic reduction using the constructions from Section 2. Furthermore, variants of these reductions with respect to suitable smaller foliations are stated in Section 3.2 and are a key ingredient for proving the main theorem.
A brief review of contact Hamiltonian mechanics is given in Section 4. It is viewed as a simple instance of the precontact/presymplectic setting.
The task of Section 5 is twofold. First, it characterizes (co)isotropic and Legendrian/Lagrangian submanifolds in the presence of a precontact/presymplectic structure. Second, it is to define constant rank submanifolds in a contact manifold and to provide a reduction method with respect to those as a precontact-to-contact reduction.
Eventually, Theorem 1.1 is proved as part of Section 6. It requires the notion of a contact moment map for a precontact manifold \((M,C)\) equipped with an action of a Lie group \(G\) by contactomorphisms. It is important to remark that a contact moment map \(J:P \to {g}^\ast\) (here \({g}^\ast\) is the dual of the Lie algebra \({g}\) of \(G\)) is defined on a covering presymplectic \(\mathbb{R}^\times\)-bundle \(\tau:P\to M\). The contact moment map is both \(\mathbb{R}^\times\)- and \(G\)-equivariant, where the action \(G\times P \to P\) lifts the one on \(M\). Due to the \(\mathbb{R}^\times\)-equivariance we are forced to consider, for any \(\mu \in {g}^\ast\setminus \{0\}\), the \(\mathbb{R}^\times\)-subbundle \(P_{[\mu]}:= J^{-1}(\{c \mu \mid c \in \mathbb{R}^\times\})\) of \(P\) over \(M_{\mu}:=\tau (P_{[\mu]})\subset M\). Write \({g}_\mu^\circ := \{\xi \in {g} \mid \mu(\xi)=0\) and \(\operatorname{ad}_{\xi}^\ast(\mu)=0\}\), which is a Lie algebra, and let \(G_\mu^\circ\) be the connected Lie subgroup of \(G\) integrating \({g}_\mu^\circ\). Then, under some natural assumptions (see Theorem 6.3), the reduction \(P(\mu):=P_{[\mu]}/G_\mu^\circ\) is a (pre)symplectic \(\mathbb{R}^\times\)-bundle over \(M(\mu)=M_\mu/G_\mu^\circ\) which induces a (pre)contact structure on the latter. Theorem 1.1 then follows as the special `maximal rank' case. There are two illuminating simple examples that conclude the section.
Reviewer: Maxime Fairon (Glasgow)Primary singularities of vector fields on surfaceshttps://zbmath.org/1528.580132024-03-13T18:33:02.981707Z"Hirsch, M. W."https://zbmath.org/authors/?q=ai:hirsch.morris-w"Turiel, F. J."https://zbmath.org/authors/?q=ai:turiel.francisco-javierSummary: Unless another thing is stated one works in the \(C^\infty\) category and manifolds have empty boundary. Let \(X\) and \(Y\) be vector fields on a manifold \(M\). We say that \(Y\) tracks \(X\) if \([Y, X]=fX\) for some continuous function \(f:M \rightarrow \mathbb{R}\). A subset \(K\) of the zero set \(\mathsf{Z} (X)\) is an essential block for \(X\) if it is non-empty, compact, open in \(\mathsf{Z}(X)\) and its Poincaré-Hopf index does not vanishes. One says that \(X\) is non-flat at \(p\) if its \(\infty\)-jet at \(p\) is non-trivial. A point \(p\) of \(\mathsf{Z}(X)\) is called a primary singularity of \(X\) if any vector field defined about \(p\) and tracking \(X\) vanishes at \(p\). This is our main result: consider an essential block \(K\) of a vector field \(X\) defined on a surface \(M\). Assume that \(X\) is non-flat at every point of \(K\). Then \(K\) contains a primary singularity of \(X\). As a consequence, if \(M\) is a compact surface with non-zero characteristic and \(X\) is nowhere flat, then there exists a primary singularity of \(X\).Random periodicity for stochastic Liénard equationshttps://zbmath.org/1528.600622024-03-13T18:33:02.981707Z"Uda, Kenneth"https://zbmath.org/authors/?q=ai:uda.kennethSummary: In this paper, we establish some sufficient conditions for the existence of random limit cycle generated by stochastic Liénard equation. Our technique involve Lyapunov functions and truncation arguments. Furthermore, using polar coordinate transformation and rigid rotation, we further established existence (non-existence) of a possible minimal period of random periodic solution of stochastic van der Pol oscillator.Rates of convergence for Gibbs sampling in the analysis of almost exchangeable datahttps://zbmath.org/1528.600752024-03-13T18:33:02.981707Z"Gerencsér, Balázs"https://zbmath.org/authors/?q=ai:gerencser.balazs"Ottolini, Andrea"https://zbmath.org/authors/?q=ai:ottolini.andreaThe paper analyses a Gibbs sampler for sampling \(\mathbf p = (p_1, ..., p_d) \in [0,1]^d\) from a distribution with density proportional to
\[
\exp\Bigl( - A^2 \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 \Bigr),
\]
where \(A\) is large and the \(c_{i,j}\) are non-negative weights. The Gibbs sampler used is \textit{Glauber dynamics}: a single, uniformly chosen coordinate is updated in each step according to the conditional distribution given all the others.
The non-negative weights can be viewed as a weighted graph. Extending these weights to a symmetric matrix \(C = (c_{i,j})_{i,j=1}^d\) with diagonal \(c_{i,i} := \sum_{j \in [d] : j \ne i} c_{i,j}\), the weights generate a reversible, discrete-time Markov chain on \([d]\):
\[
\textstyle p_{i,j} = c_{i,j} / c_i, \quad\text{where}\quad c_i := \sum_{j \in [d] : j \ne i} c_{i,j}.
\]
It has equilibrium distribution \(c = (c_i)_{i=1}^d\).
Letting \(\Delta = I - P\) denote the Laplacian of the Markov chain,
\[
\textstyle \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 = \langle \mathbf p, \Delta \mathbf p \rangle, \quad\text{where}\quad \langle \mathbf x, \mathbf y \rangle := \sum_{i \in [d]} \pi_i x_i y_i.
\]
The main result analyses the mixing time of the Gibbs sampler, in the limit \(A \to \infty\) with other variables fixed. The mixing metric is the \(\ell_\infty\)-Wasserstein distance:
\[
\textstyle d_\infty(\mu, \pi) := \inf_{(X, Y)} \mathbb E( \| X - Y \|_\infty ),
\]
where the infimum is over all couplings \((X, Y)\) of \(\mu\) and \(\pi\). For \(\varepsilon \in (0,1)\), define
\[
\textstyle t_\mathsf{mix}(\varepsilon) := \inf\{ t \ge 0 \mid \sup_{\mathbf p \in [0,1]^d} d_\infty(K_t(\mathbf p), \pi_{A,C}) \le \varepsilon \},
\]
where \(K_t(\mathbf p)\) are the time-\(t\) transition probabilities of the Gibbs sampler started from \(\mathbf p \in [0,1]^d\) and \(\pi_{A,C}\) is the target distribution:
\[
\frac{d\pi_{A,C}}{d\mathbf p} \propto \exp\Bigl( - A^2 \sum_{i,j \in [d] : i < j} c_{i,j} (p_i - p_j)^2 \Bigr).
\]
In short, the mixing time is shown to be of order \(A^2\):
\[
t_\mathsf{mix}(\tfrac14 - \varepsilon) \gtrsim \tfrac \lambda\gamma A^2 \quad\text{and}\quad t_\mathsf{mix}(\varepsilon) \lesssim d A^2,
\]
where \(\lambda\) is the spectral gap of \(\Delta\) and \(\gamma\) the absolute spectral gap. The implicit constants in \(\gtrsim\)/\(\lesssim\) depend only on \(\varepsilon\).
This paper is motivated by work of de Finetti, who was particularly interested in understanding situations where \(\mathbf p\) is approximately concentrated around a main diagonal of the hypercube:
\[
\mathcal D := \{ p \mathbf 1 \mid p \in [0,1] \} \subseteq [0,1]^d, \quad\text{where}\quad \mathbf 1 := (1, \ldots, 1).
\]
This situation is referred to as \textit{almost exchangeability}, because it captures the idea that the \(d\) variables are almost indistinguishable.
Reviewer: Sam Olesker-Taylor (Coventry)Center manifolds for rough partial differential equationshttps://zbmath.org/1528.601122024-03-13T18:33:02.981707Z"Kuehn, Christian"https://zbmath.org/authors/?q=ai:kuhn.christian"Neamţu, Alexandra"https://zbmath.org/authors/?q=ai:neamtu.alexandraIn this paper, the authors deal with the existence of center manifolds for stochastic partial differential equations with nonlinear drift driven by \(\gamma\)-Holder rough paths with \(\gamma \in ({\frac 13}, {\frac 12})\). The main result is proved by the Lyapunov-Perron method along with the rough paths theory and the semigroup theory. As an application, the authors show the existence of center manifolds for a class of reaction-diffusion equations and the Swift-Hohenberg equations.
Reviewer: Bixiang Wang (Socorro)Deformation spaces and static animationshttps://zbmath.org/1528.650142024-03-13T18:33:02.981707Z"Dorfsman-Hopkins, Gabriel"https://zbmath.org/authors/?q=ai:dorfsman-hopkins.gabrielSummary: We study applications of 3D printing to the broad goal of understanding how mathematical objects vary continuously in families. To do so, we model the varying parameter as the vertical axis of a 3D print, introducing the notion of a \textit{static animation}: a 3D printed object each of whose layers is a member of the continuously deforming family. We survey examples and draw connections to algebraic geometry, complex dynamics, chaos theory, and more. We also include a detailed tutorial (with accompanying code and files) so that the reader can create static animations of their own.
For the entire collection see [Zbl 1525.65002].Extracting structured dynamical systems using sparse optimization with very few sampleshttps://zbmath.org/1528.650352024-03-13T18:33:02.981707Z"Schaeffer, Hayden"https://zbmath.org/authors/?q=ai:schaeffer.hayden"Tran, Giang"https://zbmath.org/authors/?q=ai:tran.giang-n"Ward, Rachel"https://zbmath.org/authors/?q=ai:ward.rachel-a"Zhang, Linan"https://zbmath.org/authors/?q=ai:zhang.linanSummary: Learning governing equations allows for deeper understanding of the structure and dynamics of data. We present a random sampling method for learning structured dynamical systems from undersampled and possibly noisy state-space measurements. The learning problem takes the form of a sparse least-squares fitting over a large set of candidate functions. Based on a Bernstein-like inequality for partly dependent random variables, we provide theoretical guarantees on the recovery rate of the sparse coefficients and the identification of the candidate functions for the corresponding problem. Computational results are demonstrated on datasets generated by the Lorenz 96 equation, the viscous Burgers' equation, and the two-component reaction-diffusion equations. Our formulation includes theoretical guarantees of success and is shown to be efficient with respect to the ambient dimension and the number of candidate functions.Dynamically consistent nonstandard numerical schemes for solving some computer virus and malware propagation modelshttps://zbmath.org/1528.650362024-03-13T18:33:02.981707Z"Hoang, Manh Tuan"https://zbmath.org/authors/?q=ai:hoang.manh-tuan"Ngo, Thi Kim Quy"https://zbmath.org/authors/?q=ai:ngo.thi-kim-quy"Tran, Dinh Hung"https://zbmath.org/authors/?q=ai:tran.dinh-hungSummary: This work is devoted to constructing reliable numerical schemes for some computer virus and malware propagation models. We apply the Mickens' methodology to formulate nonstandard finite difference (NSFD) schemes for some epidemiological models describing the spread of computer viruses and malware. Positivity, boundedness and global asymptotic stability (GAS) of the proposed NSFD schemes are studied rigorously. It should be emphasized that the GAS of the NSFD models are established based on an extension of the classical Lyapunov's direct method. As an important consequence, we conclude that the constructed NSFD schemes are dynamically consistent with respect to the positivity, boundedness and GAS of the continuous models. Finally, a set of numerical examples is conducted to support the theoretical findings and to demonstrate advantages of the NSFD schemes over some well-known standard ones. The numerical examples show that the used standard numerical schemes fail to preserve the qualitative dynamical properties of the continuous models for all finite step sizes; consequently, they can generate numerical approximations that are completely different from the exact solutions. Conversely, the NSFD schemes can provide reliable numerical solutions regardless of chosen step sizes.Rigorous FEM for one-dimensional Burgers equationhttps://zbmath.org/1528.650722024-03-13T18:33:02.981707Z"Kalita, Piotr"https://zbmath.org/authors/?q=ai:kalita.piotr"Zgliczyński, Piotr"https://zbmath.org/authors/?q=ai:zgliczynski.piotrSummary: We propose a method to integrate dissipative PDEs rigorously forward in time with the use of the finite element method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The proposed approach is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two particular choices of the forcing we prove the existence of the periodic globally attracting trajectory and give precise bounds on its shape.Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin schemehttps://zbmath.org/1528.650822024-03-13T18:33:02.981707Z"Schlachter, Louisa"https://zbmath.org/authors/?q=ai:schlachter.louisa"Totzeck, Claudia"https://zbmath.org/authors/?q=ai:totzeck.claudiaSummary: We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.A multiple-relaxation-time lattice Boltzmann model for Burgers equationhttps://zbmath.org/1528.650912024-03-13T18:33:02.981707Z"Yu, Xiaomei"https://zbmath.org/authors/?q=ai:yu.xiaomei"Zhang, Ling"https://zbmath.org/authors/?q=ai:zhang.ling.2"Hu, Beibei"https://zbmath.org/authors/?q=ai:hu.beibei"Hu, Ye"https://zbmath.org/authors/?q=ai:hu.yeSummary: A multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed to solve Burgers equation. The general MRT-LB model can be thought as a three-level nonlinear finite difference scheme. Maximum value principle has been proved, and the existence, uniqueness, and stability parameters are discussed for the three-level finite difference scheme of MRT-LB model. Numerical stability analysis shows that the MRT-LB model is not only a three-level nonlinear finite difference one but also has the good numerical stability. Numerical solutions have been compared with the exact solutions and other LB models reported in previous studies. The experiments results show that the scheme is accurate and effective for Burgers equation.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: a computational studyhttps://zbmath.org/1528.650942024-03-13T18:33:02.981707Z"Larios, Adam"https://zbmath.org/authors/?q=ai:larios.adam"Victor, Collin"https://zbmath.org/authors/?q=ai:victor.collinSummary: Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equation, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to the estimation of a physical length scale in the case of a uniform static grid.Filtering methods for coupled inverse problemshttps://zbmath.org/1528.650972024-03-13T18:33:02.981707Z"Herty, Michael"https://zbmath.org/authors/?q=ai:herty.michael-matthias"Iacomini, Elisa"https://zbmath.org/authors/?q=ai:iacomini.elisaThe paper discusses the use of the ensemble Kalman filter (EnKF), an iterative filtering method, for solving inverse problems in a finite dimensional setting. The paper proposes a possible adaptation of the EnKF towards a multiobjective minimization formulation, in which competing models are given data and a choice of parameters must be determined. The paper defines the concept of Pareto optimality and a Pareto set for vector-valued optimization problems. The paper also introduces a weighted objective function and a convex combination of observations to approximate the Pareto set. The paper compares two algorithms, a direct approach and an adaptive approach, for solving coupled inverse problems using the EnKF procedure. The adaptive approach decreases the number of iterations needed to recover a good approximation of the Pareto front and leads to a better approximation of the Pareto set by updating the initial ensemble at each step. Numerical results demonstrate the improvement of the adaptive strategy, even in the nonlinear case.
Reviewer: Hongliang Li (Chengdu)Reconstruction of an interface between the fluid and piezoelectric solid by acoustic measurementshttps://zbmath.org/1528.650982024-03-13T18:33:02.981707Z"Wu, Chengyu"https://zbmath.org/authors/?q=ai:wu.chengyu"Yang, Jiaqing"https://zbmath.org/authors/?q=ai:yang.jiaqingSummary: In this paper, we consider an inverse interaction scattering problem of recovering an interface between the fluid and piezoelectric solid from acoustic measurements. First, the well-posedness of the interaction model is shown in associated function spaces by the variational method. Then new uniqueness results are proved for the inverse problem by taking far-field data at one fixed frequency, based on a uniform a priori estimate of the solutions of the interaction model. With these results, the factorization method is then justified to reconstruct the shape and location of the interface between the fluid and piezoelectric solid. Finally, we investigate an associated interior transmission eigenvalue problem, and show that the set of interior transmission eigenvalues is at most discrete and with no finite accumulation point under a natural assumption on physical coefficients.What's decidable about discrete linear dynamical systems?https://zbmath.org/1528.682282024-03-13T18:33:02.981707Z"Karimov, Toghrul"https://zbmath.org/authors/?q=ai:karimov.toghrul"Kelmendi, Edon"https://zbmath.org/authors/?q=ai:kelmendi.edon"Ouaknine, Joël"https://zbmath.org/authors/?q=ai:ouaknine.joel-o"Worrell, James"https://zbmath.org/authors/?q=ai:worrell.james-bSummary: We survey the state of the art on the algorithmic analysis of discrete linear dynamical systems, focussing in particular on reachability, model-checking, and invariant-generation questions, both unconditionally as well as relative to oracles for the Skolem Problem.
For the entire collection see [Zbl 1516.68022].Finiteness of spatial central configurations with fixed subconfigurationshttps://zbmath.org/1528.700162024-03-13T18:33:02.981707Z"Deng, Yiyang"https://zbmath.org/authors/?q=ai:deng.yiyang"Hampton, Marshall"https://zbmath.org/authors/?q=ai:hampton.marshallFor a generalized \(N\)-body problem with a rational homogeneous central potential, by using some techniques of tropical geometry, the authors show that ``any fixed configuration of \(N-1\) masses in space with no three points collinear and no four points cocircular can be extended to a central configuration of \(N\) masses by adding a specified additional mass only in finitely many ways.''
Reviewer: Xiang Yu (Chengdu)Fast-forward generation of non-equilibrium steady states of a charged particle under the magnetic fieldhttps://zbmath.org/1528.810322024-03-13T18:33:02.981707Z"Setiawan, Iwan"https://zbmath.org/authors/?q=ai:setiawan.iwan"Sugihakim, Ryan"https://zbmath.org/authors/?q=ai:sugihakim.ryan"Gunara, Bobby Eka"https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Masuda, Shumpei"https://zbmath.org/authors/?q=ai:masuda.shumpei"Nakamura, Katsuhiro"https://zbmath.org/authors/?q=ai:nakamura.katsuhiroSummary: The fast-forward (FF) is one of the ideas to speed up the dynamics of given systems, and reproduces series of events on a shortened time scale, just like rapid projection of movie films on a screen. Considering a charged particle under the electromagnetic field, we present a scheme of FF generation of its non-equilibrium steady state, which realizes with complete fidelity the underlying quantum adiabatic dynamics throughout the FF protocol. We then apply the scheme to Landau states of a clean spin-less electron gas in a 2D \(x\)-\(y\) plane under the constant magnetic field \(B\) in the \(z\) direction. We have found how the electric field should be applied in rapid preparation of the quantum-mechanical Hall state as a non-equilibrium steady state. The FF electric field expressed in terms of the time-scaling function is found to be common to both the ground and excited Landau states. The FF driving avoids the decoherence inevitable in slow adiabatic procedures and eliminates the undesired mixing among Landau states with different quantum numbers that usually occurs in fast control.Multi-party entanglement generation through superconducting circuitshttps://zbmath.org/1528.810452024-03-13T18:33:02.981707Z"Shahmir, Syed"https://zbmath.org/authors/?q=ai:shahmir.syed"Khan, Mughees Ahmad"https://zbmath.org/authors/?q=ai:khan.mughees-ahmad"Abbas, Tasawar"https://zbmath.org/authors/?q=ai:abbas.tasawar"Alvi, Sajid Hussain"https://zbmath.org/authors/?q=ai:alvi.sajid-hussain"Islam, Rameez-ul"https://zbmath.org/authors/?q=ai:ul-islam.rameezSummary: In this article multipartite entanglement generation schemes are suggested using Circuit Quantum Electrodynamics (Circuit-QED). These proposed setups are achieved by building an interactive connection among many charge qubits, which act as an artificial atom as well as coupler. By initializing these charge qubits in a circuit, one can obtain the multiparty entanglement among these qubits. Further, the charge qubits used for coupling the circuit are traced out. These maximally entangled states are experimentally feasible and can exhibit good fidelity and can further be utilized for quantum information processing tasks.Physical thinking and the GHZ theoremhttps://zbmath.org/1528.810632024-03-13T18:33:02.981707Z"Nikulov, Alexey"https://zbmath.org/authors/?q=ai:nikulov.alexeySummary: Quantum mechanics is one of the most successful theories of physics. But the creators of quantum mechanics had to reject realism in order to describe some paradoxical quantum phenomena. Einstein considered the rejection of realism unacceptable, since according to his understanding, realism is the presupposition of every kind of physical thinking. The dispute about the permissibility of rejecting realism has largely determined the modern understanding of quantum theory and even led to the emergence new quantum information technologies. Many modern authors are sure that realism can be refute experimentally. Some authors are sure even that real technologies can be created on the basis of this refutation. The well-known GHZ theorem is considered critically in this work the authors of which are sure that realism can be refute experimentally.On the calculation of bound-state energies supported by hyperbolic double well potentialshttps://zbmath.org/1528.811292024-03-13T18:33:02.981707Z"Fernández, Francisco M."https://zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: We obtain eigenvalues and eigenfunctions of the Schrödinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical energies for arbitrary values of this model parameter. We test the numerical method by means of a suitable exact asymptotic expression for the eigenvalues and also calculate critical values of the strength parameter that are related to the number of bound states supported by the potential.Quasimonochromatic dynamical system and optical soliton cooling with triple-power law of self-phase modulationhttps://zbmath.org/1528.811402024-03-13T18:33:02.981707Z"Biswas, Anjan"https://zbmath.org/authors/?q=ai:biswas.anjan"Bagchi, Bijan K."https://zbmath.org/authors/?q=ai:bagchi.bijan-kumar"Yıldırım, Yakup"https://zbmath.org/authors/?q=ai:yildirim.yakup"Khan, Salam"https://zbmath.org/authors/?q=ai:khan.salam"Asiri, Asim"https://zbmath.org/authors/?q=ai:asiri.asim-mSummary: Soliton perturbation theory is used in this paper to uncover the adiabatic conservation laws. The effect of soliton cooling arises from a fixed point in the corresponding dynamical system.Dirac theory in hydrodynamic formhttps://zbmath.org/1528.811622024-03-13T18:33:02.981707Z"Fabbri, Luca"https://zbmath.org/authors/?q=ai:fabbri.lucaSummary: We consider quantum mechanics written in hydrodynamic formulation for the case of relativistic spinor fields to study their velocity: within such a hydrodynamic formulation it is possible to see that the velocity as is usually defined can not actually represent the tangent vector to the trajectories of particles. We propose an alternative definition for this tangent vector and hence for the trajectories of particles, which we believe to be new and the only one possible. We discuss how these results are a necessary step to take in order to face further problems, like the definition of trajectories for multi-particle systems or ensembles, as they happen to be useful in many applications and interpretations of quantum mechanics.Generalized coherent states of light interacting with a nonlinear medium, quantum superpositions and dissipative processeshttps://zbmath.org/1528.811662024-03-13T18:33:02.981707Z"Giraldi, Filippo"https://zbmath.org/authors/?q=ai:giraldi.filippoSummary: The interaction of a single-mode quantized light field with a low-dissipative nonlinear medium is analyzed in case the field mode is initially prepared in a generalized coherent state (GCS). Similarly to the case of canonical coherent states (CSs) of light, the time evolution is periodic and becomes a superposition of a finite number of GCSs at determined time instants. The effects of the dissipative process on these superpositions are studied by evaluating the loss of one photon and the corresponding rate. Special conditions are determined such that the superpositions of GCSs created in the time evolution of a GCS are stronger or weaker against the dissipative process than the superpositions of optical CSs generated in the evolution of a canonical CS. This selection is performed by comparing the corresponding dissipation rates of loosing one photon. In this way, perturbations of coherent light are found such that the quantum superpositions of GCSs created in the time evolution are stronger against the dissipative process than the superpositions of optical CSs generated with coherent light.GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controlshttps://zbmath.org/1528.811772024-03-13T18:33:02.981707Z"Petruhanov, V. N."https://zbmath.org/authors/?q=ai:petruhanov.v-n"Pechen, A. N."https://zbmath.org/authors/?q=ai:pechen.aleksandr-nikolaevichSummary: The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to various coherently controlled closed and open quantum systems. In this work, we adopt the GRAPE method for optimizing objective functionals in open quantum systems driven by both coherent and incoherent controls. In our case, a tailored or engineered environment acts on the controlled system as control via \textit{time-dependent decoherence rates} \(\gamma_i(t)\) or, equivalently, via \textit{spectral density} \(n_\omega(t)\) \textit{of the environment}. To develop the GRAPE approach for this problem, we compute gradient of various objectives for general \(N\)-level open quantum systems for the piecewise constant class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a \(3\times3\) matrix determining the system's dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano's method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated. Robustness of the optimal controls is also studied.From symmetries to commutant algebras in standard Hamiltonianshttps://zbmath.org/1528.812292024-03-13T18:33:02.981707Z"Moudgalya, Sanjay"https://zbmath.org/authors/?q=ai:moudgalya.sanjay"Motrunich, Olexei I."https://zbmath.org/authors/?q=ai:motrunich.olexei-iSummary: In this work, we revisit several families of standard Hamiltonians that appear in the literature and discuss their symmetries and conserved quantities in the language of commutant algebras. In particular, we start with families of Hamiltonians defined by parts that are local, and study the algebra of operators that separately commute with each part. The families of models we discuss include the spin-1/2 Heisenberg model and its deformations, several types of spinless and spinful free-fermion models, and the Hubbard model. This language enables a decomposition of the Hilbert space into dynamically disconnected sectors that reduce to the conventional quantum number sectors for regular symmetries. In addition, we find examples of non-standard conserved quantities even in some simple cases, which demonstrates the need to enlarge the usual definitions of symmetries and conserved quantities. In the case of free-fermion models, this decomposition is related to the decompositions of Hilbert space via irreducible representations of certain Lie groups proposed in earlier works, while the algebra perspective applies more broadly, in particular also to arbitrary interacting models. Further, the von Neumann Double Commutant Theorem (DCT) enables a systematic construction of local operators with a given symmetry or commutant algebra, potentially eliminating the need for ``brute-force'' numerical searches carried out in the literature, and we show examples of such applications of the DCT. This paper paves the way for both systematic construction of families of models with exact scars and characterization of such families in terms of non-standard symmetries, pursued in a parallel paper \textit{S. Moudgalya} and \textit{O. I. Motrunich} [``Exhaustive characterization of quantum many-body scars using commutant algebras'', Preprint, \url{arXiv:2209.03377}].Thermodynamics of the classical periodic Toda lattice.https://zbmath.org/1528.820082024-03-13T18:33:02.981707Z"Matsuyama, Akihiko"https://zbmath.org/authors/?q=ai:matsuyama.akihiko(no abstract)Zero-temperature stochastic Ising model on planar quasi-transitive graphshttps://zbmath.org/1528.820222024-03-13T18:33:02.981707Z"De Santis, Emilio"https://zbmath.org/authors/?q=ai:de-santis.emilio"Lelli, Leonardo"https://zbmath.org/authors/?q=ai:lelli.leonardoThis research paper makes a significant contribution to the understanding of the zero-temperature stochastic Ising model applied to connected planar quasi-transitive graphs with rotational and translational invariance. The abstract adeptly articulates the study's focused inquiry, centering on the utilization of a Bernoulli product measure with a parameter $p$ in the range (0, 1) for the initial spin configuration. Of particular interest is the paper's pivotal result, rigorously establishing that when $p=1/2$ and the graph satisfies the planar shrink property, all vertices exhibit infinite flipping almost surely. This outcome represents a noteworthy advancement in comprehending the behavior of the Ising model under specified conditions. Furthermore, the research's interdisciplinary approach, bridging statistical physics and graph theory through the examination of planar quasi-transitive graphs, enhances its scholarly relevance. The abstract's structure is commendable, presenting a well-organized and succinct overview of the research objectives, methodologies, and key findings. The technical yet accessible language employed is conducive to both specialists and those possessing a general background in statistical physics. In conclusion, the clarity, structure, and substantive findings conveyed in the abstract collectively underscore the scholarly merit of the manuscript.
Reviewer: Abdelkader Boudjemline (Annaba)Transition to fully or partially arrested state in coupled logistic maps on a ladderhttps://zbmath.org/1528.820312024-03-13T18:33:02.981707Z"Shambharkar, Nitesh D."https://zbmath.org/authors/?q=ai:shambharkar.nitesh-d"Deshmukh, Ankosh D."https://zbmath.org/authors/?q=ai:deshmukh.ankosh-d"Gade, Prashant M."https://zbmath.org/authors/?q=ai:gade.prashant-mSummary: Layered structures are an object of interest for theoretical and experimental reasons. In this work, we study coupled map lattice on a ladder. The ladder consists of two one-dimensional chains coupled at every point. We study linearly and nonlinearly coupled logistic maps in this system and study transition to nonzero persistence, in particular. We coarse-grain the variable value by assigning spin \(+1(-1)\) to sites that have value greater (less) than the fixed point and compute the number of sites that have not changed their spin values at all even times till the given time \(t\). The fraction of such sites at a given time \(t\) is known as persistence. In our system, we observe a power-law of persistence at the critical value of coupling. This transition is also accompanied by long-range antiferromagnetic ordering for nonlinear coupling and long-range ferromagnetic ordering for linear coupling. The number of domain walls decay as \(1/\sqrt{t}\) at the critical point in both cases. The persistence exponent is 0.375 for a nonlinear case with two layers which is an exponent for the voter model on the ladder as well as for the Ising model at zero temperature or voter model in 1D. For linear coupling, we obtain a smaller persistence exponent.Formalism for stochastic perturbations and analysis in relativistic starshttps://zbmath.org/1528.830032024-03-13T18:33:02.981707Z"Satin, Seema"https://zbmath.org/authors/?q=ai:satin.seema-eSummary: Perturbed Einstein's equations with a linear response relation and a stochastic source, applicable to a relativistic star model are worked out. These perturbations which are stochastic in nature, are of significance for building a non-equilibrium statistical mechanics theory in connections with relativistic astrophysics. A fluctuation dissipation relation for a spherically symmetric star in its simplest form is obtained. The FD relation shows how the random velocity fluctuations in the background of the unperturbed star can dissipate into Lagrangian displacement of fluid trajectories of the dense matter. Interestingly in a simple way, a constant (in time) coefficient of dissipation is obtained without a delta correlated noise. This formalism is also extended for perturbed TOV equations which have a stochastic contribution, and show up in terms of the effective or root mean square pressure perturbations. Such contributions can shed light on new ways of analysing the equation of state for dense matter. One may obtain contributions of first and second order in the equation of state using this stochastic approach.Dispersive Friedmann universes and synchronizationhttps://zbmath.org/1528.830062024-03-13T18:33:02.981707Z"Cotsakis, Spiros"https://zbmath.org/authors/?q=ai:cotsakis.spirosSummary: We introduce consideration of dispersive aspects of standard perfect fluid Friedmann cosmology and study the new qualitative behaviours of cosmological solutions that emerge as the fluid parameter changes and zero eigenvalues appear in the linear part of the Friedmann equations. We find that due to their insufficient degeneracy, the Milne, flat, Einstein-static, and de Sitter solutions cannot properly bifurcate. However, the dispersive versions of Milne and flat universes contained in the versal unfolding of the standard Friedmann equations possess novel long-term properties not met in their standard counterparts. We apply these results to the horizon problem and show that unlike their hyperbolic versions, the dispersive Milne and flat solutions completely synchronize in the future, hence offering a solution to the homogeneity, isotropy, and causal connectedness puzzles.Revisiting the renormalization of Einstein-Maxwell theory at one-loophttps://zbmath.org/1528.830082024-03-13T18:33:02.981707Z"Park, I. Y."https://zbmath.org/authors/?q=ai:park.il-young|park.inyong-ySummary: In a series of recent works based on foliation-based quantization in which renormalizability has been achieved for the physical sector of the theory, we have shown that the use of the standard graviton propagator interferes, due to the presence of the trace mode, with the four-dimensional covariance. A subtlety in the background field method also requires careful handling. This status of the matter motivated us to revisit an Einstein-scalar system in one of the sequels. Continuing the endeavors, we revisit the one-loop renormalization of an Einstein-Maxwell system in the present work. The systematic renormalization of the cosmological and Newton constants is carried out by applying the refined background field method. The one-loop beta function of the vector coupling constant is explicitly computed and compared with the literature. The longstanding problem of the gauge choice dependence of the effective action is addressed, and the manner in which gauge choice independence is restored in the present framework is discussed. The formalism also sheds light on background independent analysis. The renormalization involves a metric field redefinition originally introduced by 't Hooft; with the field redefinition the theory should be predictive.Wormhole time machines and multiple historieshttps://zbmath.org/1528.830352024-03-13T18:33:02.981707Z"Shoshany, Barak"https://zbmath.org/authors/?q=ai:shoshany.barak"Wogan, Jared"https://zbmath.org/authors/?q=ai:wogan.jaredSummary: In a previous paper, we showed that a class of time travel paradoxes which cannot be resolved using Novikov's self-consistency conjecture can be resolved by assuming the existence of multiple histories or parallel timelines. However, our proof was obtained using a simplistic toy model, which was formulated using contrived laws of physics. In the present paper we define and analyze a new model of time travel paradoxes, which is more compatible with known physics. This model consists of a traversable Morris-Thorne wormhole time machine in 3+1 spacetime dimensions. We define the spacetime topology and geometry of the model, calculate the geodesics of objects passing through the time machine, and prove that this model inevitably leads to paradoxes which cannot be resolved using Novikov's conjecture, but can be resolved using multiple histories. An open-source simulation of our new model using Mathematica is available for download on GitHub. We also provide additional arguments against the Novikov self-consistency conjecture by considering two new paradoxes, the switch paradox and the password paradox, for which assuming self-consistency inevitably leads to counter-intuitive consequences. Our new results provide more substantial support to our claim that if time travel is possible, then multiple histories or parallel timelines must also be possible.Dynamics of self-gravitating systems in non-linearly magnetized chameleonic Brans-Dicke gravityhttps://zbmath.org/1528.830442024-03-13T18:33:02.981707Z"Yousaf, Z."https://zbmath.org/authors/?q=ai:yousaf.z"Bhatti, M. Z."https://zbmath.org/authors/?q=ai:bhatti.muhammad-zaeem-ul-haq"Rehman, S."https://zbmath.org/authors/?q=ai:rehman.salim-u|rehman.semeen|ur-rehman.shafiq.1|rehman.shaista-amat-ur|rehman.sajjad-ur|rehman.shakeel-ul|rehman.sana|rehman.sirajur|rehman.shafiq-ur|rehman.sajid|rehman.sumaira|rehman.shafqat-ur|ur-rehman.shafiq"Bamba, Kazuharu"https://zbmath.org/authors/?q=ai:bamba.kazuharuSummary: We study the effects of magnetic fields of non-linear electrodynamics in chameleonic Brans-Dicke theory under the existence of anisotropic spherical fluid. In particular, we explore dissipative and non-dissipative self-gravitating systems in the quasi-homologous regime with the minimal complexity constraint. As a result, under the aforementioned circumstances, several analytic solutions are found. Furthermore, by analyzing the dynamics of a dissipative fluid, it is demonstrated that a void covering the center can satisfy the Darmois criteria. The temperature of the self gravitating systems is also investigated.Cosmic consequences of Barrow holographic dark energy with Granda-Oliveros cut-off in fractal cosmologyhttps://zbmath.org/1528.830452024-03-13T18:33:02.981707Z"Al Mamon, Abdulla"https://zbmath.org/authors/?q=ai:al-mamon.abdulla"Sharma, Umesh Kumar"https://zbmath.org/authors/?q=ai:sharma.umesh-kumar"Kumar, Mukesh"https://zbmath.org/authors/?q=ai:kumar.mukesh"Mishra, Ambuj Kumar"https://zbmath.org/authors/?q=ai:mishra.ambuj-kumarSummary: In a fractal world with matter (pressureless) and dark energy, we examine the recently proposed Barrow holographic dark energy model with the Granda-Oliveros IR cutoff. We depict our model Hubble parameter evolution by comparing it with the most recent cosmic chronometer data, which consists of \(31H(z)\) data points with \(1\sigma\) error bars. Also, we compare the derived model against the concordance \(\Lambda\) CDM model by using the dimensionless Hubble parameter \(E(z)\). Additionally, we show the evolution of the distance modulus \(\mu (z)\) for the derived model and compare with the 580 data points of Type Ia Supernovae (Union 2.1 compilation) dataset. The consequences of the model are discussed through different cosmological parameters which describe that in the recent past, the transition of the universe's expansion from the decelerated to an accelerated stage happened smoothly. Moreover, we demonstrate that this could be an answer for the thermal history of the universe, including the order of matter and dark energy phases. The equation of state for dark energy is also impacted by the new Barrow exponent \(\Delta\), which, depending on its value, may cause it to lie in the quintessence regime, the phantom regime, or undergo the phantom-divide crossing during evolution.Quantum modified gravity at low energy in the Ricci flow of quantum spacetimehttps://zbmath.org/1528.830512024-03-13T18:33:02.981707Z"Luo, M. J."https://zbmath.org/authors/?q=ai:luo.ming-jian|luo.meijin|luo.minjieSummary: Quantum treatment of physical reference frame leads to the Ricci flow of quantum spacetime, which is a quite rigid framework to quantum and renormalization effect of gravity. The theory has a low characteristic energy scale described by a unique constant: the critical density of the universe. At low energy long distance (cosmic or galactic) scale, the theory modifies Einstein's gravity which naturally gives rise to a cosmological constant as a counter term of the Ricci flow at leading order and an effective scale dependent Einstein-Hilbert action. In the weak and static gravity limit, the framework gives rise to a transition trend away from Newtonian gravity and similar to the MOdified Newtonian Dynamics (MOND) around the characteristic scale. When local curvature is large, Newtonian gravity is recovered. When local curvature is low enough to be comparable with the asymptotic background curvature corresponding to the characteristic energy scale, the transition trend produces the baryonic Tully-Fisher relation. For intermediate general curvature around the background curvature, the interpolating Lagrangian function yields a similar transition trend to the observed radial acceleration relation of galaxies. When the baryonic matter density is much lower than the critical density at the outskirt of a galaxy, there may be a universal ``acceleration floor'' corresponding to the acceleration expansion of the universe, which differs from MOND at its deep-MOND limit. The critical acceleration constant \(a_0\) introduced in MOND is related to the low characteristic energy scale of the theory. The cosmological constant gives a universal leading order contribution to \(a_0\) and the flow effect gives the next order scale dependent contribution, which equivalently induces the ``cold dark matter'' to the theory. \(a_0\) is consistent with galaxian data when the ``dark matter'' is about 5 times the baryonic matter.Holographic space-time, Newton's law, and the dynamics of horizonshttps://zbmath.org/1528.831232024-03-13T18:33:02.981707Z"Banks, Tom"https://zbmath.org/authors/?q=ai:banks.thomas"Fischler, Willy"https://zbmath.org/authors/?q=ai:fischler.willySummary: We revisit the construction of models of quantum gravity in \(d\) dimensional Minkowski space in terms of random tensor models, and correct some mistakes in our previous treatment of the subject. We find a large class of models in which the large impact parameter scattering scales with energy and impact parameter like Newton's law. The scattering amplitudes in these models describe scattering of jets of particles, and also include amplitudes for the production of highly meta-stable states with all the parametric properties of black holes. These models have emergent energy, momentum and angular conservation laws, despite being based on time dependent Hamiltonians. The scattering amplitudes in which no intermediate black holes are produced have a time-ordered Feynman diagram space-time structure: local interaction vertices connected by propagation of free particles (really Sterman-Weinberg jets of particles). However, there are also amplitudes where jets collide to form large meta-stable objects, with all the scaling properties of black holes: energy, entropy and temperature, as well as the characteristic time scale for the decay of perturbations. We generalize the conjecture of Sekino and Susskind, to claim that all of these models are fast scramblers. The rationale for this claim is that the interactions are invariant under fuzzy subgroups of the group of volume preserving diffeomorphisms, so that they are highly non-local on the holographic screen. We review how this formalism resolves the Firewall Paradox.Modified Brans-Dicke cosmology with minimum length uncertaintyhttps://zbmath.org/1528.831552024-03-13T18:33:02.981707Z"Paliathanasis, Andronikos"https://zbmath.org/authors/?q=ai:paliathanasis.andronikos"Leon, Genly"https://zbmath.org/authors/?q=ai:leon.genlySummary: We consider a modification of the Brans-Dicke gravitational Action Integral inspired by the existence of a minimum length uncertainty for the scalar field. In particular, the kinetic part of the Brans-Dicke scalar field is modified such that the equation of motion for the scalar field is modified according to the quadratic Generalized Uncertainty Principle (GUP). For the background geometry, we assume the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker metric. We investigate the dynamics and the cosmological evolution of the dynamical variables of the theory, and we compare the results with the unmodified Brans-Dicke theory. It follows that in consideration because of the additional degrees of freedom in the energy-momentum tensor the dynamical variables describe various aspects of the cosmological history. This is one of the first studies on the effects of GUP in a Machian gravitational theory.Exact parallel waves in general relativityhttps://zbmath.org/1528.831562024-03-13T18:33:02.981707Z"Roche, Cian"https://zbmath.org/authors/?q=ai:roche.cian"Aazami, Amir Babak"https://zbmath.org/authors/?q=ai:aazami.amir-babak"Cederbaum, Carla"https://zbmath.org/authors/?q=ai:cederbaum.carlaSummary: We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries \textit{with parallel rays}. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of ``null coordinates'' and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing an open question in the field -- the Ehlers-Kundt conjecture.Deep-learning-based upscaling method for geologic models via theory-guided convolutional neural networkhttps://zbmath.org/1528.860022024-03-13T18:33:02.981707Z"Wang, Nanzhe"https://zbmath.org/authors/?q=ai:wang.nanzhe"Liao, Qinzhuo"https://zbmath.org/authors/?q=ai:liao.qinzhuo"Chang, Haibin"https://zbmath.org/authors/?q=ai:chang.haibin"Zhang, Dongxiao"https://zbmath.org/authors/?q=ai:zhang.dongxiaoSummary: Large-scale or high-resolution geologic models usually comprise a huge number of grid blocks, which can be computationally demanding and time-consuming to solve with numerical simulators. Therefore, it is advantageous to upscale geologic models (e.g., hydraulic conductivity) from fine-scale (high-resolution grids) to coarse-scale systems. Numerical upscaling methods have been proven to be effective and robust for coarsening geologic models, but their efficiency remains to be improved. In this work, a deep-learning-based method is proposed to upscale the fine-scale geologic models, which can assist to improve upscaling efficiency significantly. In the deep learning method, a deep convolutional neural network (CNN) is trained to approximate the relationship between the coarse block of fine-scale hydraulic conductivity fields and the corresponding hydraulic heads, which can then be utilized to replace the numerical solvers while solving the flow equations for each coarse block. In addition, physical laws (e.g., governing equations and periodic boundary conditions) can also be incorporated into the training process of the deep CNN model, which is termed the theory-guided convolutional neural network (TgCNN). With the physical information considered, dependence on the data volume of training the deep learning models can be reduced greatly. Several cases of subsurface flow, with varying two-dimensional and three-dimensional structures and isotropic and anisotropic conditions, are used to evaluate the performance of the proposed deep-learning-based upscaling method. The results show that the deep learning method can provide equivalent upscaling accuracy to the numerical method, and efficiency can be improved significantly compared to numerical upscaling.The p-AAA algorithm for data-driven modeling of parametric dynamical systemshttps://zbmath.org/1528.930012024-03-13T18:33:02.981707Z"Rodriguez, Andrea Carracedo"https://zbmath.org/authors/?q=ai:carracedo-rodriguez.andrea"Balicki, Linus"https://zbmath.org/authors/?q=ai:balicki.linus"Gugercin, Serkan"https://zbmath.org/authors/?q=ai:gugercin.serkanSummary: The AAA algorithm has become a popular tool for data-driven rational approximation of single-variable functions, such as transfer functions of a linear dynamical system. In the setting of parametric dynamical systems appearing in many prominent applications, the underlying (transfer) function to be modeled is a multivariate function. With this in mind, we develop the AAA framework for approximating multivariate functions where the approximant is constructed in the multivariate barycentric form. The method is data driven, in the sense that it does not require access to the full state-space model and requires only function evaluations. We discuss an extension to the case of matrix-valued functions, i.e., multi-input/multi-output dynamical systems, and provide a connection to the tangential interpolation theory. Several numerical examples illustrate the effectiveness of the proposed approach.