Recent zbMATH articles in MSC 46https://zbmath.org/atom/cc/462024-02-15T19:53:11.284213ZWerkzeugSobolev's worldline and memeshttps://zbmath.org/1526.010192024-02-15T19:53:11.284213Z"Kutateladze, S. S."https://zbmath.org/authors/?q=ai:kutateladze.semen-sSummary: This is a brief overview of the worldline and memes of Sergei Sobolev (1908--1989), a cofounder of distribution theory.The structure of algebraic Baer \(\ast \)-algebrashttps://zbmath.org/1526.160332024-02-15T19:53:11.284213Z"Szűcs, Zsolt"https://zbmath.org/authors/?q=ai:szucs.zsolt"Takács, Balázs"https://zbmath.org/authors/?q=ai:takacs.balazsThe authors study properties of complex \(\ast\)-algebras in which every element is algebraic, that is, the root of a polynomial with complex coefficients. In particular, this class includes all finite-dimensional complex \(\ast\)-algebras.
The first main result is a characterization of when such a \(\ast\)-algebra \(A\) admits a \(C^\ast\)-seminorm. Among the equivalent conditions are the following:
\begin{itemize}
\item If \(a\in A\) with \(a^\ast a=0\), then \(a=0\).
\item \(A\) is semi-simple and every self-adjoint element has spectrum contained in \(\mathbb R\).
\item Every self-adjoint element is a linear combination of orthogonal projections.
\end{itemize}
The second main result is a structure theorem for algebraic Baer \(\ast\)-algebras. Following \textit{S. K. Berberian} [Baer \(^*\)-rings. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0242.16008)], a \(\ast\)-algebra is called a Baer \(\ast\)-algebra if every right annihilator is the principal right ideal generated by a projection. The finite-dimensional Baer \(\ast\)-algebras are exactly the (underlying \(\ast\)-algebras of) finite-dimensional \(C^\ast\)-algebras.
The authors show that a \(\ast\)-algebra is a Baer \(\ast\)-algebra in which every element is algebraic if and only if it is the direct sum of a finite-dimensional Baer \(\ast\)-algebra and a commutative Baer \(\ast\)-algebra in which every element is algebraic.
As a corollary it is shown that the group algebra \(\mathbb C[G]\) is a Baer \(\ast\)-algebra in which every element is algebraic if and only if \(G\) is finite.
Reviewer: Melchior Wirth (Klosterneuburg)Hilbert evolution algebras and its connection with discrete-time Markov chainshttps://zbmath.org/1526.170512024-02-15T19:53:11.284213Z"Vidal, Sebastian J."https://zbmath.org/authors/?q=ai:vidal.sebastian-j"Cadavid, Paula"https://zbmath.org/authors/?q=ai:cadavid.paula"Rodriguez, Pablo M."https://zbmath.org/authors/?q=ai:rodriguez.pablo-mIn this paper, the authors introduce Hilbert evolution algebras as an extension of evolution algebras in a separable Hilbert space, so that an infinite number of non-zero structure constants is feasible. Then, they define the evolution operator of these new algebras, and study under which conditions it is continuous. The iterative application of this operator provides certain boundedness conditions on transition probabilities so that the whole dynamics of a discrete-time Markov chain with infinite countable state space can be described by a particular type of Hilbert evolution algebra. Some properties concerning the boundedness of the evolution operator of this algebra are studied.
Reviewer: Raúl M. Falcón (Sevilla)Duality for \(K\)-analytic group cohomology of \(p\)-adic Lie groupshttps://zbmath.org/1526.220112024-02-15T19:53:11.284213Z"Thomas, Oliver"https://zbmath.org/authors/?q=ai:thomas.oliverSummary: We prove a duality result for the analytic cohomology of Lie groups over non-archimedean fields acting on locally convex vector spaces by combining Tamme's non-archimedean van Est comparison morphism with Hazewinkel's duality result for Lie algebra cohomology.Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groupshttps://zbmath.org/1526.220132024-02-15T19:53:11.284213Z"Gloeckner, Helge"https://zbmath.org/authors/?q=ai:glockner.helge"Tárrega, Luis"https://zbmath.org/authors/?q=ai:tarrega.luisSummary: Let \(M\) be a compact smooth manifold of dimension \(m\) (without boundary) and \(G\) be a finite-dimensional Lie group, with Lie algebra \(\mathfrak{g}\). Let \(H^{>\frac{m}{2}} (M, G)\) be the group of all
mappings \(\gamma : M \to G\) which are \(H^s\) for some \(s >\frac{m}{2}\). We show that \(H^{>\frac{m}{2}} (M, G)\) can be made a regular Lie group in Milnor's sense, modelled on the Silva space \(H^{>\frac{m}{2}} (M, \mathfrak{g}) := \varinjlim_{s>\frac{m}{2}} H^s(M, \mathfrak{g})\), such that
\[
H^{>\frac{m}{2}} (M, G) =\varinjlim{}_{s>\frac{m}{2}} H^s(M, G)
\]
as a Lie group (where \(H^s(M, G)\) is the Hilbert-Lie group of all \(G\)-valued \(H^s\)-mappings on \(M\)). We also explain how the (known) Lie group structure on \(H^s(M, G)\) can be obtained as a special case of a general construction of Lie groups \(\mathcal{F}(M, G)\) whenever function spaces \(\mathcal{F}(U, \mathbb{R})\) on open subsets \(U \subseteq \mathbb{R}^m\) are given, subject to simple axioms.Paatero's classes \(V(k)\) as subsets of the Hornich spacehttps://zbmath.org/1526.300212024-02-15T19:53:11.284213Z"Andreev, Valentin V."https://zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we consider Paatero's classes \(V(k)\) of functions of bounded boundary rotation as subsets of the Hornich space \(\mathcal{H}\). We show that for a fixed \(k\ge 2\) the set \(V(k)\) is a closed and convex subset of \(\mathcal{H}\) and is not compact. We identify the extreme points of \(V(k)\) in \(\mathcal{H}\).Sectorial extensions for ultraholomorphic classes defined by weight functionshttps://zbmath.org/1526.300492024-02-15T19:53:11.284213Z"Jiménez-Garrido, J."https://zbmath.org/authors/?q=ai:jimenez-garrido.javier"Sanz, J."https://zbmath.org/authors/?q=ai:sanz.javier"Schindl, Gerhard"https://zbmath.org/authors/?q=ai:schindl.gerhardSummary: We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions \(\omega\) and transfer the proofs from the single weight sequence case from [\textit{V. Thilliez}, Result. Math. 44, No. 1--2, 169--188 (2003; Zbl 1056.30054)] to the weight function setting. We are following a different approach than the results obtained in a recent paper by the authors [Result. Math. 74, No. 1, Paper No. 27, 44 p. (2019; Zbl 1419.46020)], more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.Relative controllability of quaternion differential equations with delayhttps://zbmath.org/1526.340542024-02-15T19:53:11.284213Z"Fu, Teng"https://zbmath.org/authors/?q=ai:fu.teng"Kou, Kit Ian"https://zbmath.org/authors/?q=ai:kou.kitian|kou.kit-ian"Wang, Jinrong"https://zbmath.org/authors/?q=ai:wang.jinrongSummary: The main focus of this study is on the relative controllability of linear and semilinear quaternion differential equations with delay. The Gram criterion and the rank criterion for the relative controllability of linear quaternion differential equations (LQDEs) with pure delay are presented by the delayed quaternion Gram matrix and the maximal right linearly independent group, respectively. Furthermore, a pure delay-based minimal norm control of LQDEs is achieved. For LQDEs with delay, we provide a necessary and sufficient condition for relative controllability and eliminate the requirement for permutation matrices. The relative controllability requirement for semi-LQDEs (SLQDEs) is then established using the Krasnoselskii's fixed point theorem. Some examples are given to illustrate our theoretical results.Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argumenthttps://zbmath.org/1526.340582024-02-15T19:53:11.284213Z"Wang, Jingjing"https://zbmath.org/authors/?q=ai:wang.jingjing"Zhu, Song"https://zbmath.org/authors/?q=ai:zhu.song"Liu, Xiaoyang"https://zbmath.org/authors/?q=ai:liu.xiaoyang"Wen, Shiping"https://zbmath.org/authors/?q=ai:wen.shipingSummary: This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.Trudinger-type inequalities in \(\mathbb{R}^N\) with radial increasing mass-weighthttps://zbmath.org/1526.350162024-02-15T19:53:11.284213Z"Tarsi, Cristina"https://zbmath.org/authors/?q=ai:tarsi.cristinaSummary: We prove Trudinger-type inequalities with radial increasing mass-weights in the whole \(\mathbb{R}^N\), in the setting of mass-weighted Sobolev spaces \(W^{1,N}_w (\mathbb{R}^N)\). Due to the presence of increasing weights, we will not apply symmetrization tools: the proofs of our inequalities mainly rely on a proper transformation of variables, which allows us to reduce the weighted case to the unweighted classical one.
For the entire collection see [Zbl 1515.31001].Large-time behaviour for anisotropic stable nonlocal diffusion problems with convectionhttps://zbmath.org/1526.350582024-02-15T19:53:11.284213Z"Endal, Jørgen"https://zbmath.org/authors/?q=ai:endal.jorgen"Ignat, Liviu I."https://zbmath.org/authors/?q=ai:ignat.liviu-i"Quirós, Fernando"https://zbmath.org/authors/?q=ai:quiros-gracian.fernandoSummary: We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a ``projection'' of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.
Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.Hardy-Leray inequalities in variable Lebesgue spaceshttps://zbmath.org/1526.351932024-02-15T19:53:11.284213Z"Cruz-Uribe, David"https://zbmath.org/authors/?q=ai:cruz-uribe.david-v"Suragan, Durvudkhan"https://zbmath.org/authors/?q=ai:suragan.durvudkhanSummary: In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate \(p(\cdot)\)-Laplace operator.The multiplication of distributions in the study of delta shock waves for zero-pressure gasdynamics system with energy conservation lawshttps://zbmath.org/1526.352342024-02-15T19:53:11.284213Z"Sen, Anupam"https://zbmath.org/authors/?q=ai:sen.anupam"Raja Sekhar, T."https://zbmath.org/authors/?q=ai:sekhar.t-rajaSummary: In this article, we study the delta shock wave for zero-pressure gasdynamics system with energy conservation laws in the frame of \(\alpha\)-solutions defined in the setting of distributional products. By reformulating the system, we construct within a convenient space of distributions, all solutions which include discontinuous solutions and Dirac delta measures. We also establish the generalized Rankine-Hugoniot jump conditions for delta shock waves. The \(\alpha\)-solutions which we constructed coincide with the solution obtained through different methods.Some notes on functions of least \(W^{s,1}\)-fractional seminormhttps://zbmath.org/1526.352872024-02-15T19:53:11.284213Z"Bucur, Claudia"https://zbmath.org/authors/?q=ai:bucur.claudiaSummary: In this survey we discuss some existence and asymptotic results, originally obtained in
[\textit{C. Bucur} et al., Interfaces Free Bound. 22, No. 4, 465--504 (2020; Zbl 1458.35447); Int. Math. Res. Not. 2023, No. 2, 1173--1235 (2023; Zbl 1507.35315)],
for functions of least \(W^{s,1}\)-fractional seminorm. We present the connection between these functions and nonlocal minimal surfaces, leveraging this relation to build a function of least fractional seminorm. We further prove that a function of least fractional seminorm is the limit for \(p\rightarrow 1\) of the sequence of minimizers of the \(W^{s,p}\)-energy. Additionally, we consider the fractional 1-Laplace operator and study the equivalence between weak solutions and functions of least fractional seminorm.
For the entire collection see [Zbl 1522.35003].Small perturbations of critical nonlocal equations with variable exponentshttps://zbmath.org/1526.352992024-02-15T19:53:11.284213Z"Tao, Lulu"https://zbmath.org/authors/?q=ai:tao.lulu"He, Rui"https://zbmath.org/authors/?q=ai:he.rui"Liang, Sihua"https://zbmath.org/authors/?q=ai:liang.sihuaSummary: In this article, we are concerned with the following critical nonlocal equation with variable exponents:
\[
\begin{cases}
(-\Delta )_{p(x, y)}^su = \lambda f(x, u) + |u|^{q(x) - 2}u &\text{in }\Omega,\\
u=0& \text{in }\mathbb{R}^N\setminus\Omega,
\end{cases}
\]
where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with Lipschitz boundary, \(N \geq 2\), \(p\in C(\Omega\times \Omega)\) is symmetric, \(f:C(\Omega\times\mathbb{R})\to\mathbb{R}\) is a continuous function, and \(\lambda\) is a real positive parameter. We also assume that \(\{x\in\mathbb{R}^N: q(x) = p_s^\ast(x)\}\neq\varnothing\), and \(p_s^\ast(x) = N\tilde{p}(x)/(N - s\tilde{p}(x))\) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations (\(\lambda\) small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function \(f\) does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of \(s = 1\) and subcritical case.Fractional Calderón problems and Poincaré inequalities on unbounded domainshttps://zbmath.org/1526.353232024-02-15T19:53:11.284213Z"Railo, Jesse"https://zbmath.org/authors/?q=ai:railo.jesse"Zimmermann, Philipp"https://zbmath.org/authors/?q=ai:zimmermann.philippSummary: We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.Boundary and rigidity of nonsingular Bernoulli actionshttps://zbmath.org/1526.370052024-02-15T19:53:11.284213Z"Hasegawa, Kei"https://zbmath.org/authors/?q=ai:hasegawa.kei"Isono, Yusuke"https://zbmath.org/authors/?q=ai:isono.yusuke"Kanda, Tomohiro"https://zbmath.org/authors/?q=ai:kanda.tomohiroThe paper presents a first rigidity result for Bernoulli shift actions that are not measure-preserving, indeed a significant contribution to the field. The authors achieve this by establishing solidity for specific non-singular Bernoulli actions while introducing a novel boundary concept associated with such actions.
The introduction provides an overview of the research area, emphasizing the importance of Bernoulli shift actions in understanding rigidity phenomena. Solidity, a well-studied concept in the context of probability measure-preserving (pmp) Bernoulli actions, is defined and discussed.
The authors highlight the challenge of extending results involving solidity to non-pmp Bernoulli actions, as existing proofs heavily rely on the measure-preserving condition. The main result of the states the following. Let \(G\) be a countable discrete group and consider a product measure space with two base points
\[
(\Omega , \mu):=\prod_{g\in G} (\{0,1\}, p_g\delta_0 + q_g \delta_1),
\]
where \(p_g\in (0,1)\) and \(p_g+q_g=1\) for all \(g\in G\).
Assume that \((\Omega,\mu)\) has no atoms and satisfies Kakutani's condition, so that the nonsingular Bernoulli action \(G \curvearrowright (\Omega,\mu)\) is defined. Assume further that:
(i) \(G\) is exact;
(ii) For any \(g\in G\), \(p_h = p_{gh}\) for all but finitely many \(h\in G\).
Then the Bernoulli action \(G \curvearrowright (\Omega,\mu)\) is solid.
Concrete examples are provided, showing that the result applies to groups with more than one end and groups acting on trees. Additionally, the paper introduces the concept of ``condition AO'' and discusses its role in the context of the proof, emphasizing the technical challenges associated with actions that do not preserve the measure.
Reviewer: Alcides Buss (Florianópolis)Strong orbit equivalence in Cantor dynamics and simple locally finite groupshttps://zbmath.org/1526.370062024-02-15T19:53:11.284213Z"Robert, Simon"https://zbmath.org/authors/?q=ai:robert.simonMany notions in dynamics, notably those related to orbit structure, find their roots in operator algebraic considerations. Strong orbit equivalence is one of these notions, and is defined as follows.
First, if two minimal homeomorphisms \(\varphi,\psi\) of the Cantor space \(X\) have the same orbits, one can uniquely define a cocycle \(c:X\to \mathbb Z\) from \(\psi\) to \(\varphi\) by the equation
\[
\psi(x)=\varphi^{c(x)}(x).
\]
Now, given two minimal homeomorphisms of the Cantor space, we say that they are \emph{strongly orbit equivalent} if they can be conjugated so that they share the same orbits, and both cocycles from the second to the first and from the first to the second have only one discontinuity point.
On the face of it, it is not clear why strong orbit equivalence should be transitive, or why one should consider it at all. A fundamental result of \textit{T. Giordano} et al. [J. Reine Angew. Math. 469, 51--111 (1995; Zbl 0834.46053)] answers both questions. It states that \(\varphi\) and \(\psi\) are strongly orbit equivalent iff the associated crossed products \(C^*\) algebras \(C^*(X,\varphi)\) and \(C^*(X,\psi)\) are isomorphic.
The present paper is motivated by a third characterization of strong orbit equivalence discovered in [\textit{T. Giordano} et al., Isr. J. Math. 111, 285--320 (1999; Zbl 0942.46040)], in terms of a locally finite group \(\Gamma_x^\varphi\). This group is defined as follows: first, the \textit{full group} of a minimal homeomorphism \(\varphi\) is the group of all homeomorphisms whose orbits are contained in those of \(\varphi\). To any \(\psi\) of the full group of \(\varphi\) we can associate a cocycle just as before, and the \textit{topological full group} of \(\varphi\) is the group of all \(\psi\) in the full group of \(\varphi\) whose cocyle is \textit{continuous}. Next, for a given \(x\in X\), the \(\varphi\)-orbit of \(x\) carries a natural linear order given by \(y\leq_{\varphi}z\) iff \(z=\varphi^n(y)\) for some \(n\geq 0\). We can finally define \(\Gamma_x^\varphi\) as the setwise stabilizer of the forward orbit of \(x\): it is the group of all \(\psi\) in the topological full group of \(\varphi\) such that for all \(y\geq_\varphi x\) we have \(\psi(y)\geq_\varphi x\).
Two minimal homeomorphisms \(\varphi\) and \(\psi\) of the Cantor space \(X\) are strongly orbit equivalent if and only if for all \(x,y\in X\), \(\Gamma^\varphi_x\) is isomorphic to \(\Gamma^\psi_y\), as proved in [T. Giordano et al., loc. cit.]. Their proof however relies crucially on deep operator algebraic techniques.
The main result of the present paper is a purely dynamical and very elegant proof of the aforementioned characterization of strong orbit equivalence. It relies on results on ample groups of homeomorphisms found in [\textit{W. Krieger}, Math. Ann. 252, 87--95 (1980; Zbl 0472.54028)].
As a byproduct, using the fact that the derived subgroup of \(\Gamma^\varphi_x\) still resembles strong orbit equivalence and is a simple locally finite group, and the \(S_\infty\)-universality of strong orbit equivalence (see [\textit{J. Melleray}, Isr. J. Math. 236, No. 1, 317--344 (2020; Zbl 1479.03020)]), the author shows that isomorphism of locally finite simple countable groups is a \(S_\infty\)-universal equivalence relation for Borel reducibility.
Reviewer: François Le Maître (Paris)Elements of hyperbolic theory on an infinite-dimensional torushttps://zbmath.org/1526.370332024-02-15T19:53:11.284213Z"Glyzin, Sergey D."https://zbmath.org/authors/?q=ai:glyzin.sergei-dmitrievich"Kolesov, Andrei Yu."https://zbmath.org/authors/?q=ai:kolesov.andrei-yuSummary: On the infinite-dimensional torus \(\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty} \), where \(E\) is an infinite-dimensional Banach torus and \(\mathbb{Z}^{\infty}\) is an abstract integer lattice, a special class of diffeomorphisms \(\operatorname{Diff}(\mathbb{T}^{\infty})\) is considered. It consists of the maps \(G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}\) whose differentials \(DG\) and \(D(G^{-1})\) are uniformly bounded and uniformly continuous on \(\mathbb{T}^{\infty} \). For diffeomorphisms in \(\operatorname{Diff}(\mathbb{T}^{\infty})\) elements of hyperbolic theory are presented systematically, starting with definitions and some auxiliary facts and ending by more advanced results. The latter include a criterion for hyperbolicity, a theorem on the \(C^1\)-roughness of hyperbolicity for diffeomorphisms in \(\operatorname{Diff}(\mathbb{T}^{\infty})\), the Hadamard-Perron theorem, as well as a fundamental result of hyperbolic theory, the fact that each Anosov diffeomorphism \(G\in\operatorname{Diff}(\mathbb{T}^{\infty})\) has a stable and an unstable invariant foliation.Korovkin-type theorems and local approximation problemshttps://zbmath.org/1526.410072024-02-15T19:53:11.284213Z"Altomare, Francesco"https://zbmath.org/authors/?q=ai:altomare.francescoSummary: Of concern are local approximation problems for sequences of positive linear operators acting on linear subspaces of functions defined on a metric space. A Korovkin-type theorem is established in such a framework together with several consequences related to one dimensional, multidimensional and infinite dimensional settings (Hilbert spaces). Furthermore, some applications are discussed which concern classical sequences of positive linear operators including (one dimensional and multidimensional) Bernstein operators, Kantorovich operators, Szász-Mirakyan operators, Gauss-Weierstrass operators and Bernstein-Schnabl operators on convex subsets of Hilbert spaces. Finally the paper ends with a reassessment of a result of Korovkin concerning subspaces of bounded \(2 \pi\)-periodic functions on \(\mathbb{R}\) and with an application related to sequences of convolution operators generated by positive approximate identities.On some characterizations of greedy-type baseshttps://zbmath.org/1526.410092024-02-15T19:53:11.284213Z"Berná, Pablo M."https://zbmath.org/authors/?q=ai:berna.pablo-manuel"Chu, Hùng Việt"https://zbmath.org/authors/?q=ai:chu.hung-viet.1Summary: In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding Greedy Algorithm. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak Thresholding Greedy Algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.Geometric harmonic analysis IV. Boundary layer potentials in uniformly rectifiable domains, and applications to complex analysishttps://zbmath.org/1526.420012024-02-15T19:53:11.284213Z"Mitrea, Dorina"https://zbmath.org/authors/?q=ai:mitrea.dorina"Mitrea, Irina"https://zbmath.org/authors/?q=ai:mitrea.irina"Mitrea, Marius"https://zbmath.org/authors/?q=ai:mitrea.mariusThe present book is the fourth in a series of five volumes, at the confluence of Harmonic Analysis,
Geometric Measure Theory, Function Space Theory, and Partial Differential Equations. The series is generically
branded as Geometric Harmonic Analysis, with the individual volumes carrying the following subtitles:
Volume~I: A Sharp Divergence Theorem with Nontangential Pointwise Traces;
Volume~II: Function Spaces Measuring Size and Smoothness on Rough Sets;
Volume~III: Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering;
Volume~IV: Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis;
Volume~V: Fredholm Theory and Finer Estimates for Integral Operators, with Applications to Boundary Problems.
The main objective of the series is to produce tools that can treat efficiently boundary value problems
for elliptic systems in inclusive geometric settings, beyond the category of Lipschitz domains.
In this fourth volume, the bulk of the results amounts to a versatile Calderón-Zygmund
theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries.
The picture that emerges is that Calderón-Zygmund theory is a multi-faceted body of results
aimed at describing how singular integral operators behave in many geometric and analytic settings.
Applications to Complex Analysis in several variables are subsequently presented, starting from the realization that many natural integral operators in this setting, such as the Bochner-Martinelli operator, are actual particular cases of double-layer potential operators associated with the complex Laplacian. What follows is a concise description of the contents of each chapter.
Chapter 1 focuses on singular integral operators (SIOs) of boundary layer type on Lebesgue and Sobolev spaces.
Generic Calderón-Zygmund convolution-type SIOs [Volume~III, Chapter~2] are not expected to induce well-defined
mappings on Sobolev spaces on uniformly rectifiable (UR) sets, as this requires a special algebraic structure of
their integral kernels. Topics treated in this chapter include the history and physical interpretations of the
classical harmonic layer potentials, ``tangential'' singular integral operators, volume and integral operators of
boundary layer type associated with a given open set of locally finite perimeter and a given weakly elliptic system,
a multitude of relevant examples and alternative points of view, a rich function theory of Calderón-Zygmund type for
boundary layer potentials associated with a given weakly elliptic system and an open set with a uniformly rectifiable boundary,
the interpretation of the Cauchy and Cauchy-Clifford operators as double-layer potential operators,
and how to modify boundary layer potential operators to increase the class of functions to which they may be applied.
Chapter 2 concentrates on layer potential operators acting on Hardy, BMO, VMO, and Hölder spaces defined on boundaries of UR domains.
A fundamental aspect of this analysis is that a special algebraic structure is required of the integral kernel for a singular integral operator to map either of these spaces into itself and the brand of Divergence Theorem produced in Volume I plays a significant part.
In fact, the same type of philosophy prevails in relation to the action of double-layer potential operators on Calderón,
Morrey-Campanato, and Morrey spaces discussed in Chapter 3, and also for the action of double layer potential operators on Besov and
Triebel-Lizorkin spaces, treated in Chapter 4.
Chapter 5 addresses the following basic question: describe the most general classes of singular integral operators
on the boundary of an arbitrary given UR domain $\Omega\subset{\mathbb{R}}^n$ which map Hardy, BMO, VMO, Hölder, Besov, and
Triebel-Lizorkin spaces defined on $\partial\Omega$ boundedly into themselves. The authors provide an answer through the
introduction of what they call generalized double-layer operators. They also take a look at Riesz transforms
from the point of view of generalized double layers.
In Chapter 6 the authors develop a theory of boundary layer potentials associated with the Stokes system of linear hydrostatics,
and related topics. Among other things, they establish Green-type formulas, derive mapping properties for the aforementioned
boundary layer potential operators, and prove Fatou-type results, in settings that are sharp from a geometric/analytic
point of view. Once again, the brand of Divergence Theorem discussed in Volume~I plays a prominent role in carrying out this program.
Chapter 7 contains a multitude of applications of the body of results developed so far in the area of Geometric Harmonic Analysis
to the field of Complex Analysis of Several Variables. As is well known, Complex Analysis, Geometric
Measure Theory and Harmonic Analysis interface tightly in the complex plane. However, this rich interplay
between these branches of mathematics has been considerably less explored in the higher-dimensional setting,
involving several complex variables. The main goal of the current chapter is to further the present understanding
of this aspect. Themes covered include CR-functions and differential forms on boundaries of sets of locally finite perimeter,
integration by parts formulas involving the \(d\)-bar operator on sets of locally finite perimeter,
the Bochner-Martinelli integral operator, a sharp version of the Bochner-Martinelli-Koppelman formula,
the Extension Problem for CR-functions in a variety of spaces on boundaries of Ahlfors regular (and UR) domains.
Chapter 8 focuses on the study of Hardy spaces for certain second-order weakly elliptic operators in the complex plane, such as
the Bitsadze operator in the unit disk. The purpose of the chapter is to characterize the space of null solutions and to
identify precisely the corresponding spaces of boundary traces.
Reviewer: Mohammed El Aïdi (Bogotá)Approximation of the classes of periodic functions of one and many variables from the Nikol'skii-Besov and Sobolev spaceshttps://zbmath.org/1526.420042024-02-15T19:53:11.284213Z"Romanyuk, A. S."https://zbmath.org/authors/?q=ai:romanyuk.anatolii-sergiiovych"Yanchenko, S. Ya."https://zbmath.org/authors/?q=ai:yanchenko.serhii-yaSummary: We establish the exact-order estimates for the best orthogonal trigonometric approximations of the Nikol'skii-Besov classes \({B}_{1,\theta}^r ( \mathbb{T}^d)\), \(1 \leq \theta \leq \infty ,\) of periodic functions of one and many variables with predominant mixed derivative in the space \(B_{ \theta , 1} ( \mathbb{T}^d)\). In the multidimensional case, \( d \geq 2,\) we establish the exact-order estimates for the approximations of the indicated classes of functions by their step-hyperbolic Fourier sums and determine the orders of orthoprojection widths in the same space. The behaviors of the corresponding approximation characteristics of the Sobolev classes \({W}_{1,\alpha}^r ( \mathbb{T}^d )\) with \(d \in\{1, 2\}\) are also investigated.On (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth modulihttps://zbmath.org/1526.420092024-02-15T19:53:11.284213Z"Chen, Shaolin"https://zbmath.org/authors/?q=ai:chen.shaolin"Hamada, Hidetaka"https://zbmath.org/authors/?q=ai:hamada.hidetakaSummary: The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in [\textit{D. Kalaj}, Trans. Am. Math. Soc. 372, No. 6, 4031--4051 (2019; Zbl 1422.30002)]. Next, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established, which give analogies and extensions of a result in [\textit{G. H. Hardy} and \textit{J. E. Littlewood}, J. Reine Angew. Math. 167, 405--423 (1932; JFM 58.0333.03)]. Furthermore, we establish a Fejér-Riesz type inequality on pluriharmonic functions in the Euclidean unit ball in \(\mathbb{C}^n\), which extends the main result in [\textit{P. Melentijević} and \textit{V. Božin}, Potential Anal. 54, No. 4, 575--580 (2021; Zbl 1460.31005)]. Additionally, we also discuss the Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and extend the corresponding results in \textit{K. M. Dyakonov} [Acta Math. 178, No. 2, 143--167 (1997; Zbl 0898.30040)], \textit{G. H. Hardy} and \textit{J. E. Littlewood}, [Math. Z. 34, 403--439 (1931; JFM 57.0476.01)], \textit{K. M. Dyakonov} [Adv. Math. 187, No. 1, 146--172 (2004; Zbl 1056.30018)] and \textit{M. Pavlović} [Rev. Mat. Iberoam. 23, No. 3, 831--845 (2007; Zbl 1148.31003)].On basicity of a certain trigonometric system in a weighted Lebesgue spacehttps://zbmath.org/1526.420112024-02-15T19:53:11.284213Z"Guliyeva, A. E."https://zbmath.org/authors/?q=ai:guliyeva.aysel-eSummary: In this paper one perturbed system of exponents \(1\cup\left\{e^{\pm i\lambda_nt}\right\}_{n\in\mathbb{N}}\) is considered, where \(\lambda_n=\sqrt{n^2+\alpha n+\beta}\), \(\forall\,n\in\mathbb{N} \). A weighted Lebesgue space \(L_{p,w}\left(-\pi,\pi\right)\), \(1<p<+\infty\) is considered, where \(w:\left[-\pi,\pi\right]\to\left[0,+\infty\right]\) is a weight function. A sufficient condition for the basicity of this system depending on the parameters \(\alpha;\beta\in\mathbb{R}\) in \(L_{p,w}\left(-\pi,\pi\right)\) is founded for the case when weight \(w\) satisfies the Muckenhoupt condition.Time-frequency analysis associated with \(k\)-Hankel-Wigner transformshttps://zbmath.org/1526.420152024-02-15T19:53:11.284213Z"Boubatra, Mohamed Amine"https://zbmath.org/authors/?q=ai:boubatra.mohamed-amineSummary: In this paper, we introduce the \(k\)-Hankel-Wigner transform on \(\mathbb{R}\) in some problems of time-frequency analysis. As a first point, we present some harmonic analysis results such as Plancherel's, Parseval's and an inversion formulas for this transform. Next, we prove a Heisenberg's uncertainty principle and a Calderón's reproducing formula for this transform. We conclude this paper by studying an extremal function for this transform.Singular integral operators with rough kernel on function spaces over local fieldshttps://zbmath.org/1526.420202024-02-15T19:53:11.284213Z"Ashraf, Salman"https://zbmath.org/authors/?q=ai:ashraf.salman"Jahan, Qaiser"https://zbmath.org/authors/?q=ai:jahan.qaiserSummary: In this article we study the boundedness of classical singular integral operator over local fields with rough kernel. We are relaxing the smoothness condition on kernel by block spaces and found the boundedness of truncated singular integral operator on different function spaces such as Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Unlike Euclidean space, our bound of truncated singular integral operator depends on the constant which leads the unboundedness of singular integral operator over local fields.Commutators of multilinear Calderón-Zygmund operators with kernels of Dini's type on generalized weighted Morrey spaces and applicationshttps://zbmath.org/1526.420222024-02-15T19:53:11.284213Z"Guliyev, V. S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabirThe paper establishes weighted Morrey strong and weak \(L(\log L)\)-type endpoint estimates for multilinear Calderón-Zygmund operators with kernels of Dini type, and for iterated commutators of such operators with functions in BMO\(^m\). The results are applied to give weighted Morrey estimates for iterated commutators of bilinear pseudo-differential operators and paraproducts with mild regularity.
Reviewer: Guorong Hu (Nanchang)Some remarks on convex body dominationhttps://zbmath.org/1526.420232024-02-15T19:53:11.284213Z"Hytönen, Tuomas P."https://zbmath.org/authors/?q=ai:hytonen.tuomas-pSummary: Convex body domination is an important elaboration of the technique of sparse domination that has seen significant development and applications over the past ten years. In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued functions, and yields matrix-weighted norm inequalities in this setting. We explore applications to ``generalised commutators'', obtaining new examples of bounded operators among linear combinations of compositions of the form \(a_i T b_i\), where \(a_i\), \(b_i\) are pointwise multipliers and \(T\) is a singular integral operator.Parabolic non-singular integral operator and its commutators on parabolic vanishing generalized Orlicz-Morrey spaceshttps://zbmath.org/1526.420242024-02-15T19:53:11.284213Z"Omarova, M. N."https://zbmath.org/authors/?q=ai:omarova.mehriban-n|omarova.meriban-nSummary: We obtain the sufficient conditions for the boundedness of the parabolic nonsingular integral operator and its commutators on the parabolic vanishing generalized Orlicz Morrey spaces \(M^{\Phi,\varphi}(\mathbb{D}^{n+1}_+)\) including their weak versions.A \(T(P)\)-theorem for Zygmund spaces on domainshttps://zbmath.org/1526.420262024-02-15T19:53:11.284213Z"Vasin, A. V."https://zbmath.org/authors/?q=ai:vasin.andrei-v"Dubtsov, E. S."https://zbmath.org/authors/?q=ai:doubtsov.evgueniSummary: Given a bounded Lipschitz domain \(D\subset \mathbb{R}^d\), a higher-order modulus of continuity \(\omega \), and a convolution Calderón-Zygmund operator \(T\), the restricted operators \(T_D\) that are bounded on the Zygmund space \(\mathcal{C}_{\omega}(D)\) are described. The description is based on properties of the functions \(T_D P\) for appropriate polynomials \(P\) restricted to \(D\).Remarks on vector-valued Gagliardo and Poincaré-Sobolev-type inequalities with weightshttps://zbmath.org/1526.420322024-02-15T19:53:11.284213Z"Perales, Javier Martínez"https://zbmath.org/authors/?q=ai:perales.javier-martinez"Pérez, Carlos"https://zbmath.org/authors/?q=ai:perez.carlos-javier|perez.carlos-e|perez-moreno.carlos|perez.carlos-aSummary: In this paper, we review certain extensions of the Gagliardo and Poincaré- Sobolev-type inequalities to later explore the possibility of extending them to the vectorvalued setting. We restrict ourselves to the most classical case of \(\ell_q\)-valued functions, where already some difficulties arise, due to the lack of a vector-valued variant of the truncation method, both on the classical and the fractional case. We think that these difficulties may be overcome in the future, and we pose some conjectures in this direction.
For the entire collection see [Zbl 1515.31001].A remark on the atomic decomposition in Hardy spaces based on the convexification of ball Banach spaceshttps://zbmath.org/1526.420352024-02-15T19:53:11.284213Z"Sawano, Yoshihiro"https://zbmath.org/authors/?q=ai:sawano.yoshihiro"Kobayashi, Kazuki"https://zbmath.org/authors/?q=ai:kobayashi.kazukiSummary: The purpose of the present note is to slightly shorten the proof of the atomic decomposition based on the paper by \textit{S. Dekel} et al. [in: Constructive theory of functions. Proceedings of the 12th international conference, Sozopol, Bulgaria, June 11--17, 2016. Sofia: Prof. Marin Drinov Academic Publishing House. 59--73 (2018; Zbl 1447.42020)]. The atomic decomposition in the present paper is applicable to Hardy spaces based on the convexification of ball Banach spaces. The decomposition is rather canonical although it does not depend linearly on functions. Also, this decomposition is applicable under a rather weak condition as we will see.
For the entire collection see [Zbl 1515.31001].Extrapolation in new weighted grand Morrey spaces beyond the Muckenhoupt classeshttps://zbmath.org/1526.420382024-02-15T19:53:11.284213Z"Meskhi, Alexander"https://zbmath.org/authors/?q=ai:meskhi.alexanderSummary: Rubio de Francia's extrapolation theorem for new weighted grand Morrey spaces \(\mathcal{M}_w^{p), \lambda, \theta}(X)\) with weights \(w\) beyond the Muckenhoupt \(\mathcal{A}_p\) classes is established. This result, in particular, implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. Necessary conditions for the boundedness of the Hardy-Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are also investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.Approximation via gradients on the ball. The Zernike casehttps://zbmath.org/1526.420402024-02-15T19:53:11.284213Z"Marriaga, Misael E."https://zbmath.org/authors/?q=ai:marriaga.misael-e"Pérez, Teresa E."https://zbmath.org/authors/?q=ai:perez.teresa-e"Piñar, Miguel A."https://zbmath.org/authors/?q=ai:pinar.miguel-a"Recarte, Marlon J."https://zbmath.org/authors/?q=ai:recarte.marlon-jThe authors consider the inner product
\[
\langle f,g\rangle_{\nabla,\mu}=f(0)g(0)+\lambda\int_{\mathcal{B}^d}\nabla f(x)\cdot\nabla g(x) W_{\mu}(x)dx
\]
on the unit ball \( \mathcal{B}^d \) in \( \mathbb{R}^d \) with normalizing factor \( \lambda>0 \) and \( W_{\mu}(x)=(1-\Vert x\Vert^2)^{\mu}\), \(\mu\geq 0 \). They determine an explicit orthogonal polynomial basis and deduce relations between the partial Fourier sums in terms of those polynomials and the partial Fourier sums in terms of the classical ball polynomials. Approximation properties of the Fourier series in the corresponding Sobolev space are studied, and numerical examples are given.
Reviewer: Alexei Lukashov (Moskva)Maltsev equal-norm tight frameshttps://zbmath.org/1526.420442024-02-15T19:53:11.284213Z"Novikov, Sergey Ya."https://zbmath.org/authors/?q=ai:novikov.sergey-ya"Sevost'yanova, Victoria V."https://zbmath.org/authors/?q=ai:sevostyanova.victoria-vSummary: A frame in \(\mathbb{R}^d\) is a set of \(n\geqslant d\) vectors whose linear span coincides with \(\mathbb{R}^d\). A frame is said to be equal-norm if the norms of all its vectors are equal. Tight frames enable one to represent vectors in \(\mathbb{R}^d\) in the form closest to the representation in an orthonormal basis. Every equal-norm tight frame is a useful tool for constructing efficient computational algorithms. The construction of such frames in \(\mathbb{C}^d\) uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in \(\mathbb{R}^d\) appeared only at the beginning of the 21st century. The present paper shows that Mal'tsev's note of 1947 [\textit{A. I. Mal'tsev}, Izv. Akad. Nauk SSSR, Ser. Mat. 11, 567--568 (1947; Zbl 0029.40502)] was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in \(\mathbb{R}^d\). Our main purpose is to show the historical significance of Maltsev's discovery. We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces. Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsev construction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence of any subset of \(d\) vectors in the frame.Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditionshttps://zbmath.org/1526.430022024-02-15T19:53:11.284213Z"Feichtinger, Hans G."https://zbmath.org/authors/?q=ai:feichtinger.hans-georg"Gumber, Anupam"https://zbmath.org/authors/?q=ai:gumber.anupamSummary: This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a ``Tauberian generator'', which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces \(( \boldsymbol{B} , \| \cdot \|_{\boldsymbol{B}} )\). The key assumption is availability of a double module structure on \(( \boldsymbol{B} , \| \cdot \|_{\boldsymbol{B}} )\), meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz-Kolmogorov Theorem characterizing compact subsets in such Banach spaces. The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space \(( \boldsymbol{B} , \| \cdot \|_{\boldsymbol{B}} )\). Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.Noetherian solvability and explicit solution of a singular integral equation with weighted Carleman shift in Besov spaceshttps://zbmath.org/1526.450012024-02-15T19:53:11.284213Z"Bliev, N. K."https://zbmath.org/authors/?q=ai:bliev.nazarbai-kadyrovich"Tulenov, K. S."https://zbmath.org/authors/?q=ai:tulenov.kanat-serikovich"Yerkinbayev, N. M."https://zbmath.org/authors/?q=ai:yerkinbayev.n-mCauchy singular integral equations with shifts are a classical topic in the field of integral equations. Conditions for the Fredholm property and computations of the index have been established in different contexts, see, e.g., [\textit{V. G. Kravchenko} and \textit{G. S. Litvinchuk}, Introduction to the theory of singular integral operators with shift. Dordrecht: Kluwer Academic Publishers (1994; Zbl 0811.47049)]. Here the authors consider the problem in the framework of Besov spaces. A detailed analysis is presented, improving the recent results of the first two authors [Complex Var. Elliptic Equ. 66, No. 2, 336--346 (2021; Zbl 1462.30067)]. In particular explicit solutions are given in Besov spaces.
Reviewer: Luigi Rodino (Torino)On fragments on lattice normed vector latticeshttps://zbmath.org/1526.460012024-02-15T19:53:11.284213Z"Bolat, Sezer"https://zbmath.org/authors/?q=ai:bolat.sezer"Erkurşun-Özcan, Nazife"https://zbmath.org/authors/?q=ai:erkursun-ozcan.nazife"Gezer, Niyazi Anıl"https://zbmath.org/authors/?q=ai:gezer.niyazi-anilConsidering a lattice normed vector space \((V,\|\cdot\|, E)\) with \(E\) a vector lattice, the codomain of \(\|\cdot\|\), the authors call \(z\in V\) a \((bo)\)-fragment of \(x\in V\) provided that \(\|z\|+\|x - z\|=0\). The elementary properties of \((bo)\)-fragments are demonstrated and specified for \(C[0,1]\).
Reviewer: S. S. Kutateladze (Novosibirsk)A study of Fibonacci difference $\mathcal I$-convergent sequence spaceshttps://zbmath.org/1526.460022024-02-15T19:53:11.284213Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Alshlool, Kamal M. A. S."https://zbmath.org/authors/?q=ai:alshlool.kamal-m-a-s"Abdullah, Sameera A. A."https://zbmath.org/authors/?q=ai:abdullah.sameera-a-aFrom the preface: This chapter discusses the construction of new sequence spaces by means of the matrix domain of a particular limitation method, more precisely, using the Fibonacci difference matrix $\widehat{F}$ defined by Fibonacci sequence $(f_n)$ for $n\in\{0,1,\dots\}$ and the notion of ideal convergence, introduces some new sequence spaces related to the matrix domain of $\widehat{F}$. Some topological and algebraic properties, and inclusion relations of these spaces are studied.
For the entire collection see [Zbl 1495.40002].Vector valued ideal convergent generalized difference sequence spaces associated with multiplier sequenceshttps://zbmath.org/1526.460032024-02-15T19:53:11.284213Z"Tripathy, Binod Chandra"https://zbmath.org/authors/?q=ai:tripathy.binod-chandraFrom the preface: This chapter introduces the vector valued generalized difference ideal convergent sequence spaces associated with the multiplier sequence and investigates some algebraic and topological properties such as solid, symmetry, convergence free, monotone, completeness and compactness. The multiplier problem is characterized, and suitable examples are discussed to justify some failure cases and for the definitions.
For the entire collection see [Zbl 1495.40002].Can one identify two unital \(\mathrm{JB}^*\)-algebras by the metric spaces determined by their sets of unitaries?https://zbmath.org/1526.460042024-02-15T19:53:11.284213Z"Cueto-Avellaneda, María"https://zbmath.org/authors/?q=ai:cueto-avellaneda.maria"Peralta, Antonio M."https://zbmath.org/authors/?q=ai:peralta.antonio-mSummary: Let \(M\) and \(N\) be two unital \(\mathrm{JB}^*\)-algebras and let \(\mathcal{U}(M)\) and \(\mathcal{U}(N)\) denote the sets of all unitaries in \(M\) and \(N\), respectively. We prove that the following statements are equivalent: \par a. \(M\) and \(N\) are isometrically isomorphic as (complex) Banach spaces; \par b. \(M\) and \(N\) are isometrically isomorphic as real Banach spaces; \par c. there exists a surjective isometry \(\Delta:\mathcal{U}(M)\to\mathcal{U}(N)\). \par We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry \(\Delta:\mathcal{U}(M)\to\mathcal{U}(N)\) we can find a surjective real linear isometry \(\Psi:M\to N\) which coincides with \(\Delta\) on the subset \(e^{iM_{sa}}\). If we assume that \(M\) and \(N\) are \(\mathrm{JBW}^*\)-algebras, then every surjective isometry \(\Delta:\mathcal{U}(M)\to\mathcal{U}(N)\) admits a (unique) extension to a surjective real linear isometry from \(M\) onto \(N\). This is an extension of the Hatori-Molnár theorem to the setting of \(\mathrm{JB}^*\)--algebras.Banach limits and their applicationshttps://zbmath.org/1526.460052024-02-15T19:53:11.284213Z"Semenov, Evgenii"https://zbmath.org/authors/?q=ai:semenov.evgenii-m"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-a"Usachev, Alexandr"https://zbmath.org/authors/?q=ai:usachev.aleksandrThis paper initially recalls the origin of the concept of Banach limits. Then, some properties related to the linear space of all almost convergent sequences, the invariance of Banach limits, the geometry and topology of the set of all Banach limits are presented. Finally, some applications of Banach limits are indicated, for example, in the theories of operators, orthogonal series and \(\mathcal{L}_{1, \infty}\) singular traces, where \(\mathcal{L}_{1, \infty}\) is the Macaev ideal of all compact operators in the algebra of all bounded linear operators on a Hilbert space with singular values of size $O(1/n)$.
Reviewer: Elói M. Galego (São Paulo)Extremal structure of projective tensor productshttps://zbmath.org/1526.460062024-02-15T19:53:11.284213Z"García-Lirola, Luis C."https://zbmath.org/authors/?q=ai:garcia-lirola.luis-carlos"Grelier, Guillaume"https://zbmath.org/authors/?q=ai:grelier.guillaume"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Rueda Zoca, Abraham"https://zbmath.org/authors/?q=ai:rueda-zoca.abrahamConsider a Banach space \(X\) as canonically embedded in its bidual. An extreme point of the unit ball of \(X\) that remains an extreme point of the unit ball of the bidual is called a weak\(^\ast\)-extreme point; see the seminal work of \textit{H.~P. Rosenthal} [Adv. Math. 70, No.~1, 1--58 (1988; Zbl 0654.46024)]. The article explores this for projective tensor products. The classification of extremal structure of projective tensor product spaces has a long history. The main result of the article is that, if the space of compact operators \({\mathcal K}(X,Y^\ast)\) separates points of \(X \otimes_{\pi}Y\), then for closed and bounded convex sets \(C,D\) in the component spaces, any non-zero weak\(^\ast\)-extreme point of the closed convex hull of \(C \otimes D\) is of the form \(x \otimes y\), where \(x \in C\), \(y\in D\) are weak\(^\ast\)-extreme points.
For a description of weak\(^\ast\)-extreme points in the case of injective tensor products, see the article by \textit{K.~Jarosz} and the reviewer [Contemp. Math. 328, 231--237 (2003; Zbl 1069.46009)].
Reviewer: T.S.S.R.K. Rao (Bangalore)Covering by planks and avoiding zeros of polynomialshttps://zbmath.org/1526.460072024-02-15T19:53:11.284213Z"Glazyrin, Alexey"https://zbmath.org/authors/?q=ai:glazyrin.alexey"Karasev, Roman"https://zbmath.org/authors/?q=ai:karasev.roman-n"Polyanskii, Alexandr"https://zbmath.org/authors/?q=ai:polyanskii.aleksandr-aAnalysing the proofs in the papers [\textit{Y.-F. Zhao}, Am. Math. Mon. 129, No. 7, 678--680 (2022; Zbl 1494.52018); \textit{O. Ortega-Moreno}, ``The complex plank problem, revisited'', Disc. Comput. Geom. (2022; \url{doi:10.1007/s00454-022-00423-7})], the authors discover the following very general results on zero sets of multivariate polynomials.
Theorem 1. If a polynomial \(P \in \mathbb R [x_1, \dots , x_d]\) of degree \(n\) has a nonzero restriction to the unit sphere \(S^{d-1} \subset \mathbb R^d\) and attains its maximal absolute value on \(S^{d-1}\) at a point \(p\), then \(p\) is at angular distance at least \(\frac{\pi}{2n}\) from the intersection of the zero set of \(P\) with \(S^{d-1}\).
Theorem 2. If a homogeneous polynomial \(P \in \mathbb C[x_1, \dots , x_d]\) of degree \(n\) is not identically zero and attains its maximal absolute value on the unit sphere \(S^{2d-1}\subset \mathbb C^d\) at a point \(p\), then \(p\) is at angular distance at least \(\arcsin\frac{1}{\sqrt{n}}\) from the intersection of the zero set of \(P\) with \(S^{2d-1}\).
They also prove the following result about the zero set inside the unit ball \(B^d \subset \mathbb R^d\): for every nonzero polynomial \(P \in \mathbb R [x_1, \dots , x_d]\) of degree \(n\), there exists a point of \(B^d\) at distance at least \(\frac{1}{n}\) from the zero set of the polynomial~\(P\).
Substituting in the last result a polynomial of the form
\[
P=\prod_{k=1}^n\left(b_k -\sum_{j=1}^d a_{k,j}x_j \right),
\]
one obtains the famous Bang plank covering theorem [\textit{Th. Bang}, Proc. Am. Math. Soc. 2, 990--993 (1951; Zbl 0044.37802)] for the case of planks of equal widths. The authors state the following challenging conjecture which, if true, could be considered as a polynomial generalization of the Bang theorem for arbitrary widths of planks.
Conjecture. Assume that \(P_1, \ldots, P_N \in \mathbb R [x_1, \dots , x_d]\) are nonzero polynomials and \(\delta_1, \ldots, \delta_N > 0\) are such that
\[
\sum_{k=1}^N \delta_k \deg P_k \le 1.
\]
Then there exists a point \(p \in B^d\) such that, for every \(k = 1, \dots ,N\), the point \(p\) is at distance at least \(\delta_k\) from the zero set of \(P_k\).
Reviewer: Vladimir Kadets (Holon)\(\mathcal{K}_1\) and \(\mathcal{K}\)-groups of absolute matrix order unit spaceshttps://zbmath.org/1526.460082024-02-15T19:53:11.284213Z"Kumar, Amit"https://zbmath.org/authors/?q=ai:kumar.amit.3|kumar.amit-nSummary: In this paper, we describe the Grothendieck groups \(\mathcal{K}_1(\mathbb{X})\) and \(\mathcal{K}(\mathbb{X})\) of an absolute matrix order unit space \(\mathbb{X}\) for unitary and partial unitary elements, respectively. For this purpose, we study some basic properties of unitary and partial unitary elements, and define their path homotopy equivalence. We prove that \(\mathcal{K}_1(\mathbb{X})\) and \(\mathcal{K}(\mathbb{X})\) are ordered abelian groups. We also prove that \(\mathcal{K}_1(\mathbb{X})\) and \(\mathcal{K}(\mathbb{X})\) are functors from the category of absolute matrix order unit spaces with morphisms as unital completely absolute value preserving maps to the category of ordered abelian groups. Later, we show that under certain conditions, quotient of \(\mathcal{K}(\mathbb{X})\) is isomorphic to the direct sum of \(\mathcal{K}_0(\mathbb{X})\) and \(\mathcal{K}_1(\mathbb{X}),\) where \(\mathcal{K}_0(\mathbb{X})\) is the Grothendieck group for order projections.Positively limited sets in Banach latticeshttps://zbmath.org/1526.460092024-02-15T19:53:11.284213Z"Ardakani, Halimeh"https://zbmath.org/authors/?q=ai:ardakani.halimeh"Chen, Jin Xi"https://zbmath.org/authors/?q=ai:chen.jinxiSummary: We introduce and study the class of positively limited sets in Banach lattices, that is, sets on which every \(\mathrm{weak}^{\ast}\) null sequence of positive functionals converges uniformly to zero. Moreover the relationships between the class of positively limited sets and other known classes of sets such as almost limited sets and (weakly) compact sets are discussed. Some properties of Banach lattices can be characterized in terms of positively limited sets.Extreme positive operators on topologically truncated Banach latticeshttps://zbmath.org/1526.460102024-02-15T19:53:11.284213Z"Boulabiar, Karim"https://zbmath.org/authors/?q=ai:boulabiar.karim"Hajji, Rawaa"https://zbmath.org/authors/?q=ai:hajji.rawaaSummary: We characterize extreme positive contractions on \(C_0(X)\)'s in terms of lattices and we extend the results to the more general setting of topologically truncated Banach lattices via a new Kakutani type representation theorem.The class of uaw-weak\(^\star\) Dunford-Pettis operatorshttps://zbmath.org/1526.460112024-02-15T19:53:11.284213Z"El Kaddouri, A."https://zbmath.org/authors/?q=ai:el-kaddouri.abdelmonaim"Boumnidel, Sanaa"https://zbmath.org/authors/?q=ai:boumnidel.sanaa"Aboutafail, O."https://zbmath.org/authors/?q=ai:aboutafail.otman"Bouras, K."https://zbmath.org/authors/?q=ai:bouras.khalidSummary: The purpose of this paper is to introduce and characterize a new class of operators that we call unbounded absolutely weak weak\(^\star\) Dunford-Pettis and study some of its properties. We have also included some comments, remarks, and examples.A note on the Banach lattice \(c_0( \ell_2^n)\), its dual and its bidualhttps://zbmath.org/1526.460122024-02-15T19:53:11.284213Z"Lourenço, M. L."https://zbmath.org/authors/?q=ai:lourenco.mary-lilian"Miranda, V. C. C."https://zbmath.org/authors/?q=ai:miranda.vinicius-c-cThe Banach space \(E=c_0(\ell_2^n)\) is known because \textit{C. Stegall} [``Duals of certain spaces with the Dunford-Pettis property'', Notices Am. Math. Soc. 19, No. 11, Preliminary Report, A--799 (1972), \url{https://www.ams.org/journals/notices/197211/197211FullIssue.pdf}] showed that \(E\) and \(E^*\) have the Dunford-Pettis (DP, for short) property, while \(E^{**}\) fails it. Moreover, \(E\) has a natural order that makes it a Banach lattice.
In this paper, the authors study the Banach lattices \(E\), \(E^*\) and \(E^{**}\). Among other results, they show that
\begin{itemize}
\item \(E\) has the strong Gelfand-Phillips property, but fails the weak DP\(^*\), the weak Grothendieck and the positive Grothendieck properties;
\item \(E^*\) has the weak Grothendieck property, but fails the positive Grothendieck property;
\item \(E^{**}\) has the weak and the positive Grothendieck properties, but fails the weak DP-property.
\end{itemize}
We refer to the paper for the definitions of these properties.
Reviewer: Manuel González (Santander)Arazy-Cwikel and Calderón-Mityagin type properties of the couples \((\ell^p, \ell^q)\), \(0 \leq p<q \leq \infty\)https://zbmath.org/1526.460132024-02-15T19:53:11.284213Z"Astashkin, Sergey V."https://zbmath.org/authors/?q=ai:astashkin.sergey-v"Cwikel, Michael"https://zbmath.org/authors/?q=ai:cwikel.michael"Nilsson, Per G."https://zbmath.org/authors/?q=ai:nilsson.per-gSummary: We establish Arazy-Cwikel type properties for the family of couples \((\ell^p, \ell^q)\), \(0 \leq p < q \leq \infty\), and show that \((\ell^p,\ell^q)\) is a Calderón-Mityagin couple if and only if \(q \geq 1\). Moreover, we identify interpolation orbits of elements with respect to this couple for all \(p\) and \(q\) such that \(0\leq p<q\leq \infty\) and obtain a simple positive solution of a Levitina-Sukochev-Zanin problem, clarifying its connections with whether \((\ell^p, \ell^q)\) has the Calderón-Mityagin property or not.Continuous nowhere Hölder functions on \(\mathbb{Z}_p\)https://zbmath.org/1526.460142024-02-15T19:53:11.284213Z"Araújo, G."https://zbmath.org/authors/?q=ai:araujo.gustavo"Bernal-González, L."https://zbmath.org/authors/?q=ai:bernal-gonzalez.luis"Fernández-Sánchez, J."https://zbmath.org/authors/?q=ai:fernandez-sanchez.juan"Seoane-Sepúlveda, J. B."https://zbmath.org/authors/?q=ai:seoane-sepulveda.juan-benignoSummary: \textit{K. Weierstraß} (1872) [in: Mathematische Werke. II. Berlin: Mayer \& Müller, 71--74 (1895; JFM 26.0041.01)] was probably the first to present the existence of continuous nowhere differentiable functions (although \textit{B. Bolzano} in 1830 [Schriften. I: Functionenlehre (1930; JFM 56.0901.01)] was the first to come up with such a construction). Almost a century later,
\textit{V.~I. Gurarij} [Sov. Math., Dokl. 7, 500--502 (1966; Zbl 0185.20203); translation from Dokl. Akad. Nauk SSSR 167, 971--973 (1966)]
observed that the family of continuous functions on \([0,1]\) that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction,
\textit{S.~Hencl} [Proc. Am. Math. Soc. 128, No. 12, 3505--3511 (2000; Zbl 0956.26008)]
generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of \(\mathbb{R}\), but in subsets of the \(p\)-adic field \(\mathbb{Q}_p\).Invariant spaces of holomorphic functions on the Siegel upper half-spacehttps://zbmath.org/1526.460152024-02-15T19:53:11.284213Z"Calzi, Mattia"https://zbmath.org/authors/?q=ai:calzi.mattia"Peloso, Marco M."https://zbmath.org/authors/?q=ai:peloso.marco-mariaSummary: In this paper we consider the (ray) representations of the group Aut of biholomorphisms of the Siegel upper half-space \(\mathcal{U}\) defined by \(U_s(\varphi) f = (f \circ \varphi^{- 1}) (J \varphi^{- 1})^{s / 2}\), \(s \in \mathbb{R}\), and characterize the semi-Hilbert spaces \(H\) of holomorphic functions on \(\mathcal{U}\) satisfying the following assumptions:
\begin{itemize}
\item[(a)] \(H\) is strongly decent;
\item[(b)] \(U_s\) induces a bounded ray representation of the group Aff of affine automorphisms of \(\mathcal{U}\) in \(H\).
\end{itemize}
We use this description to improve the known characterization of the semi-Hilbert spaces of holomorphic functions on \(\mathcal{U}\) satisfying (a) and (b) with Aff replaced by Aut.
In addition, we characterize the mean-periodic holomorphic functions on \(\mathcal{U}\) under the representation \(U_0\) of Aff.On Hardy kernels as reproducing kernelshttps://zbmath.org/1526.460162024-02-15T19:53:11.284213Z"Oliva-Maza, Jesús"https://zbmath.org/authors/?q=ai:oliva-maza.jesusSummary: Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like \(L^2(\mathbb{R}^+)\) or \(H^2(\mathbb{C}^+)\). These kernels entail an algebraic \(L^1\)-structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the \(L^2(\mathbb{R}^+)\) case turn out to be Hardy kernels as well. In the \(H^2(\mathbb{C}^+)\) scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley-Wiener type, and a connection with one-sided Hilbert transforms.Further applications of two minimax theoremshttps://zbmath.org/1526.460172024-02-15T19:53:11.284213Z"Giandinoto, D."https://zbmath.org/authors/?q=ai:giandinoto.dSummary: In this paper, we deal with new applications of two minimax theorems of
\textit{B.~Ricceri} [Arch. Math. 60, No.~4, 367--377 (1993; Zbl 0778.49008), J. Nonlinear Var. Anal. 3, No.~1, 45--52 (2019; Zbl 1435.49002)].
Here is a particular case of one of the results that we obtain: Let \((T,F,\mu)\) be a non-atomic measure space, with \(\mu (T) <+\infty\), \((E, \|\cdot\|)\) a real Banach space, \(I \subseteq E\) an unbounded set whose closure does not contain 0. Moreover, let \(p, q, r, s\) be four numbers such that \(0 < s < q \leq p\), \(p \geq 1\), \(r > 1\). Set \(X := \{ f \in L^p (T,E) : f(T) \subseteq I\}\). Then, one has
\[
\inf\limits_{u\in X} \frac{(\int_T \| u(t)\|^s \,d\mu )^r}{\int_T \| u(t)\|^q \,d\mu} = 0.
\]Anisotropic ball Campanato-type function spaces and their applicationshttps://zbmath.org/1526.460182024-02-15T19:53:11.284213Z"Li, Chaoan"https://zbmath.org/authors/?q=ai:li.chaoan"Yan, Xianjie"https://zbmath.org/authors/?q=ai:yan.xianjie"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunThe paper deals with the anisotropic ball Campanato-type function spaces associated with a dilation given by \(A\), a general expansive \(n\times n\) matrix, and \(X\), a ball quasi-Banach function space of \(\mathbb{R}^n\) (following the definition included in [\textit{Y.~Sawano} et al., Diss. Math. 525, 102~p. (2017; Zbl 1392.42021)]).
The authors show that these spaces are duals of anisotropic Hardy spaces \(H^{A}_X(\mathbb{R}^n)\), obtaining in this way anisotropic Littlewood-Paley function characterization of \(H^{A}_X(\mathbb{R}^n)\). Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both \(A\) and \(X\), further lead to their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-Slice spaces, some of the obtained results are new. Applications to the case of anisotropic Hardy-variable spaces are also included.
Reviewer: Santiago Boza (Barcelona)Real interpolation of variable martingale Hardy spaces and BMO spaceshttps://zbmath.org/1526.460192024-02-15T19:53:11.284213Z"Lu, Jianzhong"https://zbmath.org/authors/?q=ai:lu.jianzhong"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Zhou, Dejian"https://zbmath.org/authors/?q=ai:zhou.dejianSummary: In this paper, we mainly consider the real interpolation spaces for variable Lebesgue spaces defined by the decreasing rearrangement function and for the corresponding martingale Hardy spaces. Let \(0<q\leq \infty\) and \(0<\theta <1\). Our three main results are the following:
\[
\begin{aligned} ({\mathcal{L}}_{p(\cdot)}({\mathbb{R}}^n),L_{\infty}({\mathbb{R}}^n))_{\theta, q} &= {\mathcal{L}}_{{p(\cdot)}/(1-\theta),q}({\mathbb{R}}^n),\\
({\mathcal{H}}_{p(\cdot)}^s(\Omega),H_{\infty}^s(\Omega))_{\theta,q} &= {\mathcal{H}}_{{p(\cdot)}/(1-\theta),q}^s(\Omega) \end{aligned}
\]
and
\[
\begin{aligned} ({\mathcal{H}}_{p(\cdot)}^s(\Omega), \mathrm{BMO}_2(\Omega))_{\theta, q} = {\mathcal{H}}_{{p(\cdot)}/(1- \theta),q}^s(\Omega), \end{aligned}
\]
where the variable exponent \(p(\cdot)\) is a measurable function.Marcinkiewicz sampling theorem for Orlicz spaceshttps://zbmath.org/1526.460202024-02-15T19:53:11.284213Z"Pawlewicz, Aleksander"https://zbmath.org/authors/?q=ai:pawlewicz.aleksander"Wojciechowski, Michał"https://zbmath.org/authors/?q=ai:wojciechowski.michalSummary: In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an \(N\)-function \(\varphi\), boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.Sharp stability of log-Sobolev and Moser-Onofri inequalities on the spherehttps://zbmath.org/1526.460212024-02-15T19:53:11.284213Z"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhen"Tang, Hanli"https://zbmath.org/authors/?q=ai:tang.hanliSummary: In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere \(\mathbb{S}^n\). Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere \(\mathbb{S}^n\) must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on the sphere \(\mathbb{S}^n\).On a sharp inequality of Adimurthi-Druet type and extremal functionshttps://zbmath.org/1526.460222024-02-15T19:53:11.284213Z"de Oliveira, José Francisco"https://zbmath.org/authors/?q=ai:de-oliveira.jose-francisco"do Ó, João Marcos"https://zbmath.org/authors/?q=ai:do-o.joao-m-bezerraSummary: Our main purpose is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser type inequality in
[\textit{J.~F. De Oliveira} and \textit{J.~M. Do Ó}, Calc. Var. Partial Differ. Equ. 52, No.~1--2, 125--163 (2015; Zbl 1333.46033)]
for the entire space. In addition, we perform the two-step strategy due to \textit{L. Carleson} and \textit{S.-Y. A. Chang} [Bull. Sci. Math., II. Sér. 110, 113--127 (1986; Zbl 0619.58013)] together with blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under a subcritical regime for some special cases.Haar frame characterizations of Besov-Sobolev spaces and optimal embeddings into their dyadic counterpartshttps://zbmath.org/1526.460232024-02-15T19:53:11.284213Z"Garrigós, Gustavo"https://zbmath.org/authors/?q=ai:garrigos.gustavo"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreas"Ullrich, Tino"https://zbmath.org/authors/?q=ai:ullrich.tinoThe authors investigate the norm characterization for elements in Besov spaces \(B^s_{p,q}(\mathbb R)\) and Triebel-Lizorkin spaces \(F^s_{p,q}(\mathbb R)\) in terms of expressions involving their Haar coefficients or suitable variations thereof. The paper deals with the range of parameters \((s,p,q)\) in which the Haar system is not an unconditional basis. This complements previous work of the authors, e.g., [\textit{G.~Garrigós} et al., J. Geom. Anal. 31, No.~9, 9045--9089 (2021; Zbl 1478.46032)], where a complete description was given for the parameter range where the Haar system forms an unconditional basis or a Schauder basis.
The main aim of the paper under review is to see that the range of those characterizations previously shown in terms of Haar coefficients can be extended to other parameters provided that they doubly oversample with Haar coefficients obtained by a shift, namely, if \(h_{j,\mu}(x)= h(2^jx-\mu)\) for \(j=0,1,2, \dots\) and \(\mu\in \mathbb Z\), where \(h=\chi_{[0,1/2)}- \chi_{[1/2,1)}\) and \(h_{-1,\mu}=\chi_{[\mu,\mu+1)}\) stands for the Haar system in \(\mathbb R\), they consider \(\tilde h_{j,\nu}(x)= h(2^jx-\nu/2)\) for \(j=0,1,2, \dots\) and \(\nu\in \mathbb Z\) and \(\tilde h_{-1,\nu}=\chi_{[\nu,\nu+1)}\) and look at the extended Haar system \(\{\tilde h_{j,\nu}: h\ge -1,\,\nu\in \mathbb Z\}\). They use the notation \(c_{j,\mu}(f)= 2^j |\langle f, \tilde h_{j,2\mu}\rangle|+ 2^j |\langle f, \tilde h_{j,2\mu+1}\rangle|\) for \(j=0,1,2, \dots\) and \(c_{-1,\mu}(f)=\langle f, \chi_{[\mu,\mu+1)}\rangle\) and get characterizations of the norms in Besov and Triebel-Lizorkin spaces using these sequences, extending some previously known results which simply used the coefficients \(2^{j}\langle f, h_{j,\mu}\rangle\). They also show that, in the case \(1/p<s<1\), the classical Besov space \(B^s_{p,q}\) is a closed subset of its dyadic counterpart. They also provide equivalent norms for the Sobolev space \(W^1_p(\mathbb R)\) for \(1<p<\infty\) and some optimal inclusions between \(B^s_{p,q}\) and \(F^s_{p,q}\) and their dyadic counterparts.
Reviewer: Oscar Blasco (València)Gradual improvement of the \(L_p\) moment-entropy inequalityhttps://zbmath.org/1526.460242024-02-15T19:53:11.284213Z"Lv, Songjun"https://zbmath.org/authors/?q=ai:lv.songjunSummary: We introduce a family of general \(L_p\)-moments of a continuous function with compact support on \(\mathbb{R}^n\) and prove their associated \(L_{k, p}\) moment-entropy inequalities. We show that these inequalities not only directly imply but also gradually strengthen the classical \(L_p\) moment-entropy inequality.Variable Besov-type spaceshttps://zbmath.org/1526.460252024-02-15T19:53:11.284213Z"Zeghad, Zouheyr"https://zbmath.org/authors/?q=ai:zeghad.zouheyr"Drihem, Douadi"https://zbmath.org/authors/?q=ai:drihem.douadiSummary: In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the \(\varphi\)-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained.Trudinger-type inequalities in Musielak-Orlicz spaceshttps://zbmath.org/1526.460262024-02-15T19:53:11.284213Z"Ohno, Takao"https://zbmath.org/authors/?q=ai:ohno.takao"Shimomura, Tetsu"https://zbmath.org/authors/?q=ai:shimomura.tetsuSummary: We are concerned with Trudinger-type inequalities for variable Riesz potentials \(J_{\alpha (\cdot),\tau} f\) of functions in Musielak-Orlicz spaces \(L^{\Phi}(X)\) over bounded metric measure spaces equipped with lower Ahlfors \(Q(x)\)-regular measures. As an application and example we obtain Trudinger's inequality for double phase functionals with variable exponents.On sequence space representation and extension of vector-valued functionshttps://zbmath.org/1526.460272024-02-15T19:53:11.284213Z"Kruse, Karsten"https://zbmath.org/authors/?q=ai:kruse.karstenThe following extension problem is considered in this article: Let \(\Gamma\) be a subset of \(\Omega\). Assume that a function \(f:\Gamma \rightarrow E\), with values in a locally convex space \(E\), satisfies that \(u \circ f\) can be extended to \(\Omega\) for each \(u\) in a subspace \(G\) of the dual \(E'\) of \(E\), and that each extension belongs to a certain function space \(\mathcal{F}(\Omega)\) of scalar functions. The question is to give conditions on \(\Gamma\), \(\Omega\), \(G\) and \(E\) to ensure that there is a vector-valued function \(F:\Omega \rightarrow E\), belonging to a related space \(\mathcal{F}(\Omega,E)\) of vector-valued functions, such that its restriction to \(\Gamma\) coincides with~\(f\).
This type of problems have been considered in the literature by many authors since the work of Dunford and Grothendieck. The author develops a unified approach to the extension problem for a large class of function spaces. The results obtained are based on a representation of vector-valued functions as linear continuous operators, which extend results of Bierstedt, Bonet, Frerick, Gramsch, Jordá and others. In the last section, these results are applied to represent function spaces \(\mathcal{F}(\Omega,E)\) by sequence spaces if one knows such a sequence space representation for \(\mathcal{F}(\Omega)\).
Reviewer: José Bonet (València)Generalized Orlicz spaces of Banach-valued functions: basic theory and dualityhttps://zbmath.org/1526.460282024-02-15T19:53:11.284213Z"Ruf, Thomas"https://zbmath.org/authors/?q=ai:ruf.thomasSummary: For a measure space \(\Omega\), we extend the theory of Orlicz spaces generated by an even convex integrand \(\varphi : \Omega \times X \to [0, \infty]\) to the case when the range Banach space \(X\) is arbitrary. We settle fundamental structural properties such as completeness, characterize separability, reflexivity and represent the dual space. This representation includes the case when \(X^\prime\) has no Radon-Nikodym property or \(\varphi\) is unbounded. We apply our theory to represent convex conjugates and Fenchel-Moreau subdifferentials of integral functionals, leading to the first general such result on function spaces with non-separable range space. For this, we prove a new interchange criterion between infimum and integral for non-separable range spaces, which we consider to be of independent interest.Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functionshttps://zbmath.org/1526.460292024-02-15T19:53:11.284213Z"Maan, Jeetendrasingh"https://zbmath.org/authors/?q=ai:maan.jeetendrasingh"Negrín, E. R."https://zbmath.org/authors/?q=ai:negrin.emilio-ramon|negrin.emilio-rSummary: Our goal in this article is to derive Abelian theorems for the two-sided Laplace transform, Mellin transform, one-sided real Laplace transform and Stieltjes transform over distributions of compact support and over certain function spaces of generalized functions.Approximately invertible elements in non-unital normed algebrashttps://zbmath.org/1526.460302024-02-15T19:53:11.284213Z"Esmeral, Kevin"https://zbmath.org/authors/?q=ai:esmeral.kevin"Feichtinger, Hans G."https://zbmath.org/authors/?q=ai:feichtinger.hans-georg"Hutník, Ondrej"https://zbmath.org/authors/?q=ai:hutnik.ondrej"Maximenko, Egor A."https://zbmath.org/authors/?q=ai:maksimenko.egor-aSummary: We introduce the concept of \textit{approximately invertible elements} in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian \(\mathrm{C}^*\)-algebras.Maximal abelian subalgebras of Banach algebrashttps://zbmath.org/1526.460312024-02-15T19:53:11.284213Z"Dales, H. G."https://zbmath.org/authors/?q=ai:dales.h-garth"Pham, H. L."https://zbmath.org/authors/?q=ai:pham.hung-le"Żelazko, W."https://zbmath.org/authors/?q=ai:zelazko.wieslawThe present paper begins by showing that if \(A\) is a commutative unital Banach algebra whose character space has cardinality greater than one, then there are families of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras that contain \(A\) as a maximal abelian subalgebra. Motivated by this property, the authors address the following questions for an infinite-dimensional, commutative, unital Banach algebra \(A\): (i) How many pairwise non-isomorphic, closed, non-commutative, unital subalgebras \(C\) of \(\mathcal{B}(A)\) (the Banach algebra of all bounded linear operators on \(A\)) are such that \(A\) is a maximal abelian subalgebra of \(C\)? (ii) How many pairwise non-isomorphic, non-commutative, unital Banach algebras \(C\) are there that contain \(\mathcal{B}(A)\) as a closed, unital subalgebra and are such that \(A\) is a maximal abelian subalgebra of \(C\)?
Concerning the first question, the authors show that in the case where \(A\) is an infinite-dimensional function algebra, \(A\) is a maximal abelian subalgebra of infinitely-many pairwise non-isomorphic closed subalgebras of \(\mathcal{B}(A)\).
The authors give a significant answer to the second question by showing that there are compact spaces \(K\) and a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras \(C\) such that each \(C\) contains \(\mathcal{B}(C(K))\) as a closed subalgebra and such that \(C(K)\) is a maximal abelian subalgebra in each \(C\).
Reviewer: Armando R. Villena (Granada)Hypomodules and amenability of pseudo-complete locally convex algebrashttps://zbmath.org/1526.460322024-02-15T19:53:11.284213Z"Ayinde, S. A."https://zbmath.org/authors/?q=ai:ayinde.s-a"Agboola, S. O."https://zbmath.org/authors/?q=ai:agboola.s-o"Adesina, A. K."https://zbmath.org/authors/?q=ai:adesina.a-k"Agunbiade, S. A."https://zbmath.org/authors/?q=ai:agunbiade.s-aSummary: Given a pseudo-complete locally convex algebra $A$, we define for $A$ flat hypomodules and cyclic flat hypomodules in line with hypocontinuous multiplication in $A$. We generalize results available on amenability of Fréchet algebras by using locally bounded approximate identity for pseudo-complete locally convex algebras endowed with the strict inductive limit topology.Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injectivehttps://zbmath.org/1526.460332024-02-15T19:53:11.284213Z"Horváth, Bence"https://zbmath.org/authors/?q=ai:horvath.bence"Kania, Tomasz"https://zbmath.org/authors/?q=ai:kania.tomaszSummary: We study automatic injectivity of surjective algebra homomorphisms from \(\mathscr{B}(X)\), the algebra of (bounded, linear) operators on \(X\), to \(\mathscr{B}(Y)\), where \(X\) is one of the following \textit{long} sequence spaces: \(c_0(\lambda),\ell_{\infty}^c(\lambda)\), and \(\ell_p(\lambda)\) (\(1\leqslant p<\infty\)) and \(Y\) is arbitrary. \textit{En route} to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the `sequential strong operator topology'.Picard-Borel ideals in Fréchet algebras and Michael's problemhttps://zbmath.org/1526.460342024-02-15T19:53:11.284213Z"Esterle, Jean"https://zbmath.org/authors/?q=ai:esterle.jeanIt is well known that every character of a Banach algebra is continuous, and it is a major open problem in automatic continuity theory, first asked by \textit{E. A. Michael} in [Locally multiplicatively-convex topological algebras. Providence, RI: American Mathematical Society (AMS) (1952; Zbl 0047.35502)], whether or not this result extends to commutative Fréchet algebras. In [\textit{J.~Esterle}, Ann. Sci. Éc. Norm. Supér. (4) 29, No.~5, 539--582 (1996; Zbl 0890.46039)], the author proved that for a specific concrete commutative Fréchet algebra \(\mathcal{U}\) and for each of the specific concrete dense Picard-Borel ideals \(\mathcal{I}_{\lambda,\infty}\), the existence of a discontinuous character on any commutative Fréchet algebra is equivalent to the existence of a character on \(\mathcal{U}\) that vanishes on \(\mathcal{I}_{\lambda,\infty}\). Since every Picard-Borel ideal is contained in a maximal Picard-Borel ideal, if a Picard-Borel ideal is a maximal ideal, it is necessarily of codimension \(1\) and hence the kernel of a character of the algebra. In this paper, the author provides a major step towards that end by proving that every Picard-Borel ideal is a prime ideal.
Reviewer: Hung Le Pham (Wellington)Representation theorems for closed ideals of \(C_B(X)\)https://zbmath.org/1526.460352024-02-15T19:53:11.284213Z"Olfati, Alireza"https://zbmath.org/authors/?q=ai:olfati.ali-rezaThe paper studies the properties of the Banach algebra \(C_B(X)\) of all continuous bounded \(\mathbb C\)-valued functions on a completely regular Hausdorff space \(X\).
Let \(\beta X\) denote the Stone-Čech compactification of \(X\). For a function \(h\in C_B(X)\), one considers its unique extension \(h^\beta:\beta X\rightarrow\mathbb C\). The cozero-set of a map \(f\) is denoted by \(\mathrm{coz}(f)\). The collection of all maps of \(C_B(X)\) with compact support is denoted by \(C_{00}(X)\). The set
\[
l_C(X)=\{x\in X: x \ \text{has a compact neigbourhood}\}
\]
is the set of all locally compact points of \(X\).
For an ideal \(H\) of \(C_B(X)\), the following notation is used:
\begin{itemize}
\item
\(\mathfrak {sp}(H)=\bigcup_{h\in H} \mathrm{coz}(h^\beta)\);
\item
\({\mathfrak k}(H)=\{f\in C_B(X): \text{there exists} \ h\in H \ \text{such that} \ f=fh\}\).
\end{itemize}
As the first main result, the author shows that, for a closed ideal \(H\) of \(C_B(X)\), the following claims hold:
\begin{enumerate}
\item
\({\mathfrak k}(H)\) is isometrically isomorphic to \(C_{00}(\mathfrak {sp}(H))=\{f\in C_B({\mathfrak sp}(H)): f \text{ has a compact support}\}\).
\item
\({\mathfrak k}(H)\) is dense in \(H\).
\item
\(H\) is non-vanishing if and only if \({\mathfrak k}(H)\) is non-vanishing.
\end{enumerate}
In the third part of the paper, several properties of this set \(l_C(X)\) are studied.
It is also shown that, if \(X\) is a completely regular Hausdorff space, then the sum of any two closed ideals of \(C_B(X)\) is closed.
In the fourth section, it is shown that, if \(X\) is a completely regular Hausdorff space, then the partially ordered set of all closed ideals of \(C_B(X)\) is a distributive lattice.
Many properties of the set of all locally null points of \(X\), with respect to an ideal, are also studied in this paper.
Reviewer: Mart Abel (Tartu)A Gelfand-type duality for coarse metric spaces with property Ahttps://zbmath.org/1526.460362024-02-15T19:53:11.284213Z"Braga, Bruno M."https://zbmath.org/authors/?q=ai:de-mendonca-braga.bruno"Vignati, Alessandro"https://zbmath.org/authors/?q=ai:vignati.alessandroSummary: We prove the following two results for a given uniformly locally finite metric space with Yu's property A [\textit{G.-L. Yu}, Invent. Math. 139, No. 1, 201--240 (2000; Zbl 0956.19004)]: 1. The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences modulo closeness. 2. The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness. The main difficulty lies in the latter. To prove that, we obtain several uniform approximability results for maps between Roe algebras and use them to obtain a theorem about the ``uniqueness'' of Cartan masas of Roe algebras. We finish the paper with several applications of the results above to concrete metric spaces.The structure of KMS weights on étale groupoid \(C^\ast\)-algebrashttps://zbmath.org/1526.460372024-02-15T19:53:11.284213Z"Christensen, Johannes"https://zbmath.org/authors/?q=ai:christensen.johannesThe author studies KMS weights for \(C^\ast\)-dynamical systems constructed from continuous real-valued homomorphisms on second-countable locally compact Hausdorff étale groupoids. This work is motivated by the growing trend (ignited by the work of \textit{J.~B. Bost} and \textit{A.~Connes} [Sel. Math., New Ser. 1, No.~3, 411--457 (1995; Zbl 0842.46040)]) of giving concrete descriptions of KMS states for different examples of \(C^\ast\)-dynamical systems.
In the context of unital \(C^\ast\)-algebras, KMS weights are just scaled KMS states, and so studying KMS weights is essentially the same as studying KMS states. However, in the context of non-unital \(C^\ast\)-algebras (such as those studied in this paper), the two notions do not coincide. Moreover, as indicated in \textit{K.~Thomsen}'s work [Adv. Math. 309, 334--391 (2017; Zbl 1358.81120)] on KMS weights on graph \(C^\ast\)-algebras, KMS weights are a more appropriate invariant for non-unital \(C^\ast\)-dynamical systems than KMS states. For this reason, the author extends several important results regarding KMS states to KMS weights in the setting of the aforementioned \(C^\ast\)-dynamical systems.
In particular, the author finds an embedding of the set of \(\beta\)-KMS weights in a certain locally convex topological vector space, and proves that many important properties of \(\beta\)-KMS states also hold for \(\beta\)\nobreakdash-KMS weights. The author then proves that there is a bijective correspondence between \(\beta\)-KMS weights for \(C^\ast\)-dynamical systems associated to continuous real-valued Hausdorff étale groupoid homomorphisms and pairs consisting of a regular Borel measure \(\mu\) on the unit space of the groupoid and a \(\mu\)-measurable field of quasi-invariant states on the \(C^\ast\)-algebras of the isotropy groups of the groupoid satisfying certain properties. This correspondence restricts to a bijection between KMS states and the pairs for which \(\mu\) is a probability measure, and thus the result is an extension of Neshveyev's theorem [\textit{S.~Neshveyev}, J. Oper. Theory 70, No.~2, 513--530 (2013; Zbl 1299.46067), Theorem~1.3] for KMS states to the more general setting of KMS weights. The author uses this result to resolve an open question posed by Thomsen at the end of Section 2 of \textit{K.~Thomsen} [J. Funct. Anal. 266, No.~5, 2959--2988 (2014; Zbl 1308.46073)], and also to solve a problem left open in [\textit{J.~Christensen} and \textit{K.~Thomsen}, J. Oper. Theory 76, No.~2, 449--471 (2016; Zbl 1389.46084)].
The author establishes two key technical results in order to prove the main theorems of the paper. The first of these results is that all KMS weights for a \(C^\ast\)-dynamical system are finite on a certain subset of the Pedersen ideal. This result allows the author to extend Neshveyev's theorem to KMS weights. The second of these results is that the extremal measures in the convex set of quasi-invariant regular Borel measures on Hausdorff étale groupoids are exactly the ergodic measures. This observation allows the author to use ideas from ergodic theory to study KMS weights. The author uses this result to extend the refinement of Neshveyev's theorem given in [\textit{J.~Christensen}, Commun. Math. Phys. 364, No.~1, 357--383 (2018; Zbl 1408.46058), Theorem~5.2] from the setting of KMS states to the setting of KMS weights.
Reviewer: Becky Armstrong (Münster)Induced Stinespring factorization and the Wittstock support theoremhttps://zbmath.org/1526.460382024-02-15T19:53:11.284213Z"Pascoe, J. E."https://zbmath.org/authors/?q=ai:pascoe.james-eldred"Tully-Doyle, Ryan"https://zbmath.org/authors/?q=ai:tully-doyle.ryanSummary: Given a pair of self-adjoint-preserving completely bounded maps on the same \(C^*\)-algebra, say that \(\varphi \leq \psi\) if the kernel of \(\varphi\) is a subset of the kernel of \(\psi\) and \(\psi \circ \varphi^{-1}\) is completely positive. The \textit{Agler class} of a map \(\varphi\) is the class of \(\psi \geq \varphi\). Such maps admit colligation formulae, and, in Lyapunov type situations, transfer function type realizations on the Stinespring coefficients of their Wittstock decompositions. As an application, we prove that the support of an extremal Wittstock decomposition is unique.Some applications of group-theoretic Rips constructions to the classification of von Neumann algebrashttps://zbmath.org/1526.460392024-02-15T19:53:11.284213Z"Chifan, Ionuţ"https://zbmath.org/authors/?q=ai:chifan.ionut"Das, Sayan"https://zbmath.org/authors/?q=ai:das.sayan-kumar"Khan, Krishnendu"https://zbmath.org/authors/?q=ai:khan.krishnenduThe left group von Neumann algebra \(L(G)\) of a discrete group \(G\) is the bicommutant of the range of the left regular representation of \(G\) inside the algebra of all bounded linear operators on \(\ell^2(G)\). This is known to be a II\(_1\)-factor precisely when \(G\) is icc (i.e., has infinite nontrivial conjugacy classes). A central question in the classification theory of von Neumann algebras is when \(G\) can be completely reconstructed from \(L(G)\). Such groups are called \(W^*\)-superrigid. A conjecture of Alain Connes predicts that all icc property \((T)\) (Kazhdan property) groups are \(W^*\)-superrigid; note that currently no examples of such (i.e., $W^*$-superrigid property~$(T)$) groups are known yet, and more significantly, there are not so many algebraic features of property \((T)\) groups recognizable at the von Neumann algebraic level.
This is the main focus of the present paper. Some progress is made for property \((T)\) groups that appear as certain fiber products of Belegradek-Osin Rips-type constructions
[\textit{I.~Belegradek} and \textit{D.~Osin}, Groups Geom. Dyn. 2, No.~1, 1--12 (2008; Zbl 1152.20039)].
Using \textit{S.~Popa}'s deformation theory
[\textit{S. Popa}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume~I: Plenary lectures and ceremonies. Zürich: European Mathematical Society (EMS). 445--477 (2007; Zbl 1132.46038)],
the authors show that certain algebraic features of property \((T)\) groups are recognizable via their left von Neumann algebra. In particular, as an evidence for the Connes rigidity conjecture, they obtain infinite families of pairwise non-isomorphic property \((T)\) group factors. They use the Rips construction to build examples of property \((T)\) II\(_1\)-factors with masa's without property \((T)\), answering a question of
\textit{Y.-L. Jiang} and \textit{A.~Skalski} [Groups Geom. Dyn. 15, No.~3, 849--892 (2021; Zbl 1489.46066)].
Reviewer: Massoud Amini (Tehran)Interpolated family of non-group-like simple integral fusion rings of Lie typehttps://zbmath.org/1526.460402024-02-15T19:53:11.284213Z"Liu, Zhengwei"https://zbmath.org/authors/?q=ai:liu.zhengwei"Palcoux, Sebastien"https://zbmath.org/authors/?q=ai:palcoux.sebastien"Ren, Yunxiang"https://zbmath.org/authors/?q=ai:ren.yunxiangSummary: This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL\((2,q))\), \(q\) prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings \((\mathcal{R}_q)\) interpolates to all integers \(q\geq 2\), providing (when \(q\) is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring \(\mathcal{R}_q\) (with \(q\) non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.Colored planar algebras for commuting squares and applications to Hadamard subfactorshttps://zbmath.org/1526.460412024-02-15T19:53:11.284213Z"Montgomery, Michael"https://zbmath.org/authors/?q=ai:montgomery.michael-t|montgomery.michael-rSummary: We define a colored planar algebra associated to a non-degenerate commuting square and identify the biunitary of the square as an element of the planar algebra. We prove a variation of the graph planar algebra embedding theorem and use the biunitary to construct representations of annular algebras and quantum groups from the commuting square. When the commuting square subfactor is amenable we can compute elements in the spectrum of the principal graph of the subfactor. This leads to two criteria which imply non-flatness of the biunitary and infinite depth of the subfactor. Computations with these criteria are performed with a continuous family of biunitaries on the 3311 principal graph, Petrescu's continuous family of \(7 \times 7\) complex Hadamard matrices, and type {II} Paley Hadamard matrices. We conclude that all of Petrescu's \(7 \times 7\) complex Hadamard matrices and all type {II} Paley Hadamard matrices yield infinite depth subfactors.Regularity for free multiplicative convolution on the unit circlehttps://zbmath.org/1526.460422024-02-15T19:53:11.284213Z"Belinschi, Serban T."https://zbmath.org/authors/?q=ai:belinschi.serban-teodor"Bercovici, Hari"https://zbmath.org/authors/?q=ai:bercovici.hari"Ho, Ching-Wei"https://zbmath.org/authors/?q=ai:ho.ching-weiSummary: Suppose that \(\mu_1\) and \(\mu_2\) are Borel probability measures on the unit circle, both different from unit point masses, and let \(\mu\) denote their free multiplicative convolution. We show that \(\mu\) has no continuous singular part (relative to arclength measure) and that its density can only be locally unbounded at a finite number of points, entirely determined by the point masses of \(\mu_1\) and \(\mu_2\). Analogous results were proved earlier for the free additive convolution on \(\mathbb{R}\) and for the free multiplicative convolution of Borel probability measures on the positive half-line.The free field: realization via unbounded operators and Atiyah propertyhttps://zbmath.org/1526.460432024-02-15T19:53:11.284213Z"Mai, Tobias"https://zbmath.org/authors/?q=ai:mai.tobias"Speicher, Roland"https://zbmath.org/authors/?q=ai:speicher.roland"Yin, Sheng"https://zbmath.org/authors/?q=ai:yin.shengSummary: Let \(X_1, \dots, X_n\) be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the noncommutative distribution of \(X_1, \dots, X_n\). More precisely, \(X_1, \dots, X_n\) generate the free skew field if and only if there exist no non-zero finite rank operators \(T_1, \dots, T_n\) such that \(\sum_i [T_i, X_i] = 0\). Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of \(X_1, \dots, X_n\). Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field [\textit{P. A. Linnell}, Forum Math. 5, No. 6, 561--576 (1993; Zbl 0794.22008)].
We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of
\textit{I.~Charlesworth} and \textit{D.~Shlyakhtenko} [J. Funct. Anal. 271, No.~8, 2274--2292 (2016; Zbl 1376.46051)].Decomposability of multiparameter CAR flowshttps://zbmath.org/1526.460442024-02-15T19:53:11.284213Z"Arjunan, Anbu"https://zbmath.org/authors/?q=ai:arjunan.anbuSummary: Let \(P\) be a closed convex cone in \(\mathbb{R}^d\) which is assumed to be spanning \(\mathbb{R}^d\) and contains no line. In this article, we consider a family of CAR flows over \(P\) and study the decomposability of the associated product systems. We establish a necessary and sufficient condition for CAR flow to be decomposable. As a consequence, we show that there are uncountable many CAR flows which are cocycle conjugate to the corresponding CCR flows.Chaotic tracial dynamicshttps://zbmath.org/1526.460452024-02-15T19:53:11.284213Z"Jacelon, Bhishan"https://zbmath.org/authors/?q=ai:jacelon.bhishanSummary: The action on the trace space induced by a generic automorphism of a suitable finite classifiable \({\mathrm{C}^*}\)-algebra is shown to be chaotic and weakly mixing. Model \({\mathrm{C}^*}\)-algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe \(KK\)-contractible stably projectionless \({\mathrm{C}^*}\)-algebras as crossed products.The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebrashttps://zbmath.org/1526.460462024-02-15T19:53:11.284213Z"Guo, Liang"https://zbmath.org/authors/?q=ai:guo.liang"Luo, Zheng"https://zbmath.org/authors/?q=ai:luo.zheng"Wang, Qin"https://zbmath.org/authors/?q=ai:wang.qin.1"Zhang, Yazhou"https://zbmath.org/authors/?q=ai:zhang.yazhouIt was noticed by \textit{G.-L. Yu} in [Invent. Math. 139, 201--240 (2000; Zbl 0956.19004)] that the index of the Dirac operator is more easily computable when the Dirac operator is twisted by a family of ``almost flat Bott bundles''. Thus one can reduce the coarse Novikov conjecture to arguing the evaluation map between the twisted algebras with ``nice coefficients''. This approach is called the geometric Dirac-dual-Dirac method.
Compared to the Dirac-dual-Dirac method for groups, there are no conceptual descriptions of what a ``nice coefficient'' for twisted algebras is. The authors introduce a notion of coarsely proper algebra for a coarse embedding of one metric space into another. They also introduce the twisted localization/Roe algebra with coefficients in a coarsely proper algebra and a twisted assembly map between the \(K\)-theories of these two algebras. Using these constructions, they prove that, given a group extension \(1 \to N \to G \to Q \to 1\) with \(N\) coarsely embeddable into a Hilbert space and \(Q\) coarsely embeddable into an admissible Hilbert-Hadamard space, the coarse Novikov conjecture holds for \(G\).
Reviewer: Vladimir M. Manuilov (Moskva)Yang-Mills-scalar-matter fields in the quantum Hopf fibrationhttps://zbmath.org/1526.460472024-02-15T19:53:11.284213Z"Moncada, Gustavo Amilcar Saldaña"https://zbmath.org/authors/?q=ai:moncada.gustavo-amilcar-saldanaIn a separate article, [``Quantum principal bundles and Yang-Mills-scalar-matter fields'', Preprint (2021), \url{arXiv:2109.01554}], the author presented a noncommutative geometric theory of interaction between magnetic monopoles and space-time scalar matter fields, in the framework of differential structure for quantum principal bundles developed by \textit{M. Đurđević} [Rep. Math. Phys. 41, No. 1, 91--115 (1998; Zbl 0931.58009)].
In this paper, the author works out details for a significant concrete example to illustrate the nontriviality of this theory. More precisely, focusing on the noncommutative version of the Hopf fibration, namely, the quantum principal bundle \(\mathbb{S}_{q}^{2}\hookrightarrow SU_{q}\left( 2\right) \twoheadrightarrow U_{q}\left( 1\right) \equiv\mathbb{C}\left[ z,z^{-1}\right] \), with a canonical quantum principal connection arising from the Levi-Civita connection, the author presents solutions to the quantum Yang-Mills-scalar-matter field equations and computes the spectrum of the left/right Laplace-de Rham operator for the associated quantum vector bundles.
Reviewer: Albert Sheu (Lawrence)Asymptotics of singular values for quantised derivatives on noncommutative torihttps://zbmath.org/1526.460482024-02-15T19:53:11.284213Z"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-a"Xiong, Xiao"https://zbmath.org/authors/?q=ai:xiong.xiao"Zanin, Dmitriy"https://zbmath.org/authors/?q=ai:zanin.dmitriy-vSummary: We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus \(\mathbb{T}_\theta^d\) of the form \(T I_\theta^{- 1}\). Here \(T\) stands for classical pseudodifferential operators on quantum torus of order 0, and \(I_\theta^{- 1}\) is the Riesz potential of order \(-1\). As an application, we obtain Weyl type asymptotics for the quantised derivative \textit{đ}\(x\) of an operator \(x\) from the homogeneous Sobolev space \(\dot{W}_d^1(\mathbb{T}_\theta^d)\). The asymptotic coefficient is equivalent to the norm of in the principal ideal \(\mathcal{L}_{d, \infty}\), as well as the norm of \(x\) in \(\dot{W}_d^1(\mathbb{T}_\theta^d)\). We precise and rectify earlier results in our previous paper
[\textit{E.~McDonald} et al., Commun. Math. Phys. 371, No.~3, 1231--1260 (2019; Zbl 1446.46051)]
on quantum integration formula for \textit{đ}\(x\).Second order derivative of a functional associated to an optimal transport maphttps://zbmath.org/1526.460492024-02-15T19:53:11.284213Z"Wences, Giovanni"https://zbmath.org/authors/?q=ai:wences.giovanni"Delgado, Joaquín"https://zbmath.org/authors/?q=ai:delgado.joaquin-fSummary: In this article we investigate the second order differentiability of a functional associated to a Monge's optimal transportation problem, namely the case of the quadratic cost, in its dual formulation. The application problem that motivates the present research is an algorithm for warping of images that uses the first derivative of this functional for a gradient-descent method which is proposed in
[\textit{R.~Chartrand} et al., ``A gradient descent solution to the Monge-Kantorovich problem'', Appl. Math. Sci., Ruse 3, No. 21--24, 1071--1080 (2009; MR2524965), \url{https://www.m-hikari.com/ams/ams-password-2009/ams-password21-24-2009/chartrandAMS21-24-2009.pdf}].
We prove the second order Gâteaux differentiability of the functional at the Monge's potential. With this result we also prove that this functional is first order differentiable in the strong sense, that is, Fréchet differentiable. Our results can be used for a further improvement of a Newton-like algorithm in this and other problems.DoNOF: an open-source implementation of natural-orbital-functional-based methods for quantum chemistryhttps://zbmath.org/1526.460502024-02-15T19:53:11.284213Z"Piris, Mario"https://zbmath.org/authors/?q=ai:piris.mario"Mitxelena, Ion"https://zbmath.org/authors/?q=ai:mitxelena.ionSummary: The natural orbital functional theory (NOFT) has emerged as an alternative formalism to both density functional (DF) and wavefunction methods. In NOFT, the electronic structure is described in terms of the natural orbitals (NOs) and their occupation numbers (ONs). The approximate NOFs have proven to be more accurate than those of the density for systems with a significant multiconfigurational character, on one side, and scale better with the number of basis functions than correlated wavefunction methods, on the other side. A challenging task in NOFT is to efficiently perform orbital optimization. In this article we present DoNOF, our open source implementation based on diagonalizations that allows to obtain the resulting orbitals automatically orthogonal. The one-particle reduced-density matrix (1RDM) of the ensemble of pure-spin states provides the proper description of spin multiplets. The capabilities of the code are tested on the water molecule, namely, geometry optimization, natural and canonical representations of molecular orbitals, ionization potential, and electric moments. In DoNOF, the electron-pair-based NOFs developed in our group PNOF5, PNOF7 and PNOF7s are implemented. These \(\mathcal{JKL}\)-only NOFs take into account most non-dynamic effects plus intrapair-dynamic electron correlation, but lack a significant part of interpair-dynamic correlation. Correlation corrections are estimated by the single-reference NOF-MP2 method that simultaneously calculates static and dynamic electron correlations taking as a reference the Slater determinant formed with the NOs of a previous PNOF calculation. The NOF-MP2 method is used to analyze the potential energy surface (PES) and the binding energy for the symmetric dissociation of the water molecule, and compare it with accurate wavefunction-based methods.Operators induced by certain hypercomplex systemshttps://zbmath.org/1526.460512024-02-15T19:53:11.284213Z"Alpay, Daniel"https://zbmath.org/authors/?q=ai:alpay.daniel"Cho, Ilwoo"https://zbmath.org/authors/?q=ai:cho.ilwooSummary: In this paper, we consider a family \(\{\mathbb{H}_t\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^2, \pi_t)\}_{t \in \mathbb{R}}\) of the hypercomplex system \(\{\mathbb{H}_t\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_t(h)\) of hypercomplex numbers \(h \in \mathbb{H}_t\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^2\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.Estimates for the deviations of integral operators in semilinear metric spaces and their applicationshttps://zbmath.org/1526.460522024-02-15T19:53:11.284213Z"Babenko, V. F."https://zbmath.org/authors/?q=ai:babenko.vladislav-f"Babenko, V. V."https://zbmath.org/authors/?q=ai:babenko.viktor-v"Kovalenko, O. V."https://zbmath.org/authors/?q=ai:kovalenko.oleg-v"Parfinovych, N. V."https://zbmath.org/authors/?q=ai:parfinovych.nataliia-viktorivnaSummary: We develop the theory of approximations in functional semilinear metric spaces that allows us to consider the classes of multi- and fuzzy-valued functions, as well as the classes of functions with values in Banach spaces, including the classes of random processes. For integral operators on the classes of functions with values in semilinear metric spaces, we obtain estimates of their deviations and discuss possible applications of these estimates to the investigation of the problems of approximation by generalized trigonometric polynomials, optimization of approximate integration formulas, and reconstruction of functions according to incomplete information.The geometry of discrete \(L\)-algebrashttps://zbmath.org/1526.510012024-02-15T19:53:11.284213Z"Rump, Wolfgang"https://zbmath.org/authors/?q=ai:rump.wolfgangSummary: The relationship of discrete \(L\)-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete \(L\)-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete \(L\)-algebra \(X\) is determined and shown to be a complete invariant. It is proved that \(X \setminus \{1\}\) is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality \(n > 3\), a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated \(L\)-algebra is determined.Lipschitz functions on unions and quotients of metric spaceshttps://zbmath.org/1526.510052024-02-15T19:53:11.284213Z"Freeman, David"https://zbmath.org/authors/?q=ai:freeman.david-mandell"Gartland, Chris"https://zbmath.org/authors/?q=ai:gartland.chrisSummary: Given a finite collection \(\{X_i\}_{i \in I}\) of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to \(L^1\), we prove that their union has Lipschitz free space isomorphic to \(L^1\). The short proof we provide is based on the Pełczyński decomposition method. A corollary is a solution to a question of \textit{P. L. Kaufmann} [Stud. Math. 226, No. 3, 213--227 (2015; Zbl 1344.46008)] about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to \(L^1\). These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to \(L^1\). Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.Metric spaces where geodesics are never uniquehttps://zbmath.org/1526.530402024-02-15T19:53:11.284213Z"Banaji, Amlan"https://zbmath.org/authors/?q=ai:banaji.amlanSummary: This article concerns a class of metric spaces, which we call \textit{multigeodesic spaces}, where between any two distinct points there exist multiple distinct minimizing geodesics. We provide a simple characterization of multigeodesic normed spaces and deduce that \((C([0,1], \Vert \cdot \Vert_1)\) is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.Characterizing compact sets in \(\mathrm{L}^\mathrm{p}\)-spaces and its applicationhttps://zbmath.org/1526.540052024-02-15T19:53:11.284213Z"Koshino, Katsuhisa"https://zbmath.org/authors/?q=ai:koshino.katsuhisaThe main goal of the paper is to obtain an equivalent condition for subsets of \(L^p\)-spaces on metric measure spaces to be compact. The result obtained is a generalization of the Kolmogorov-Riesz theorem which is useful for characterizing compact sets in \(L^p(\mathbb{R}^n)\). Additionally, it is proved that LIP\(_b(X)\) is a \(\sigma\)-compact subset of \(L^p(X)\) containing topological copies of \(\mathbb{Q}\) under certain assumptions. Here note that LIP\(_b(X) = \{f \in \mbox{LIP}(X) : f \mbox{ has a bounded support}\}\), where LIP\((X)\) is the subspace in \(L^p(X)\) consisting of Lipschitz maps.
Reviewer: Manisha Aggarwal (Delhi)Geometric inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniqueshttps://zbmath.org/1526.580112024-02-15T19:53:11.284213Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabriceIn this survey paper, some functional inequalities are introduced for elliptic diffusion operators using curvature-dimension conditions, which include the Soblove/isoperimetric inequalities and Li-Yau type parabolic Harnack inequality. These inequalties are also extended to the sub-Riemannian setting by using generalized curvature-dimension conditions.
For the entire collection see [Zbl 1481.28001].
Reviewer: Feng-Yu Wang (Tianjin)Dual spaces for martingale Musielak-Orlicz Lorentz Hardy spaceshttps://zbmath.org/1526.600332024-02-15T19:53:11.284213Z"Weisz, Ferenc"https://zbmath.org/authors/?q=ai:weisz.ferenc"Xie, Guangheng"https://zbmath.org/authors/?q=ai:xie.guangheng"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachunThis paper deals with various subspaces of martingales on a stochastic probability space with discrete time. A Musielak-Orlicz space on the underlying probability space generalizes the \(L^p\)-space by a more informative locally depending function instead of a global parameter \(p \ge 1\), and a Musielak-Orlicz Lorentz space controls further by a parameter \(q > 0\). Five martingale spaces are considered, which are called Martingale Musielak-Orlicz Lorentz Hardy spaces, depending on whether the maximal function, the quadratic variation and conditional quadratic variation, etc. of a martingale lies in the Musielak-Orlicz Lorentz space.
An atomic representation of such a martingale is one which is given by an infinite sum of special simpler functions. The authors show how, under some technical assumptions, these martingales of the martingale Musielak-Orlicz Lorentz Hardy spaces can be presented by such infinite sums of atoms. For the atomic presentations it is required that the sigma algebras of the filtration of the underlying probability space is generated by countably many atoms.
After some quasinorm estimates among the various martingale spaces, the paper continues with the introduction of martingale Musielak-Orlicz BMO-type spaces, and show how they are the dual spaces of martingale Musielak-Orlicz Lorentz Hardy spaces specified by conditional quadratic variation and maximal function, respectively.
Even if the paper appears somewhat technical the authors emphasize in a last discussion how the requirements can be easily checked in certain cases and how a vast class of similar martingale Hardy spaces known from the literature are covered by their paper.
The proof is by longer direct estimates and computations, and stopping times play an important role.
Reviewer: Bernhard Burgstaller (Florianópolis)Deep neural networks on diffeomorphism groups for optimal shape reparametrizationhttps://zbmath.org/1526.650272024-02-15T19:53:11.284213Z"Celledoni, Elena"https://zbmath.org/authors/?q=ai:celledoni.elena"Glöckner, Helge"https://zbmath.org/authors/?q=ai:glockner.helge"Riseth, Jørgen N."https://zbmath.org/authors/?q=ai:riseth.jorgen-n"Schmeding, Alexander"https://zbmath.org/authors/?q=ai:schmeding.alexanderSummary: One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.Weak limit of homeomorphisms in \(W^{1, n-1}\) and (INV) conditionhttps://zbmath.org/1526.740082024-02-15T19:53:11.284213Z"Doležalová, Anna"https://zbmath.org/authors/?q=ai:dolezalova.anna"Hencl, Stanislav"https://zbmath.org/authors/?q=ai:hencl.stanislav"Malý, Jan"https://zbmath.org/authors/?q=ai:maly.janSummary: Let \(\Omega\), \(\Omega' \subset \mathbb{R}^3\) be Lipschitz domains, let \(f_m : \Omega \to \Omega'\) be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and \(\sup_m \int_{\Omega} (|Df_m|^2+1/J^2_{f_m})<\infty\). Let \(f\) be a weak limit of \(f_m\) in \(W^{1, 2}\). We show that \(f\) is invertible a.e., and more precisely that it satisfies the (INV) condition of Conti and De Lellis, and thus that it has all of the nice properties of mappings in this class. Generalization to higher dimensions and an example showing sharpness of the condition \(1/J^2_f \in L^1\) are also given. Using this example we also show that, unlike the planar case, the class of weak limits and the class of strong limits of \(W^{1, 2}\) Sobolev homeomorphisms in \(\mathbb{R}^3\) are not the same.Complementarity in quantum walkshttps://zbmath.org/1526.810012024-02-15T19:53:11.284213Z"Grudka, Andrzej"https://zbmath.org/authors/?q=ai:grudka.andrzej"Kurzyński, Paweł"https://zbmath.org/authors/?q=ai:kurzynski.pawel"Polak, Tomasz P."https://zbmath.org/authors/?q=ai:polak.tomasz-p"Sajna, Adam S."https://zbmath.org/authors/?q=ai:sajna.adam-s"Wójcik, Jan"https://zbmath.org/authors/?q=ai:wojcik.jan"Wójcik, Antoni"https://zbmath.org/authors/?q=ai:wojcik.antoniSummary: The eigenbases of two quantum observables, \(\{|a_i\rangle\}^D_{i=1}\) and \(\{|b_j\rangle\}^D_{j=1}\), form mutually unbiased bases (MUB) if \(|\langle a_i|b_j\rangle| = 1/\sqrt{D}\) for all \(i\) and \(j\). In realistic situations MUB are hard to obtain and one looks for approximate MUB (AMUB), in which case the corresponding eigenbases obey \(|\langle a_i|b_j\rangle|\leqslant c/\sqrt{D}\), where \(c\) is some positive constant independent of \(D\). In majority of cases observables corresponding to MUB and AMUB do not have clear physical interpretation. Here we study discrete-time quantum walks (QWs) on \(d\)-cycles with a position and coin-dependent phase-shift. Such a model simulates a dynamics of a quantum particle moving on a ring with an artificial gauge field. In our case the amplitude of the phase-shift is governed by a single discrete parameter \(q\). We solve the model analytically and observe that for prime \(d\) the eigenvectors of two QW evolution operators form AMUB. Namely, if \(d\) is prime the corresponding eigenvectors of the evolution operators, that act in the \(D\)-dimensional Hilbert space (\(D = 2d\)), obey \(|\langle v_q|v^\prime_{q^\prime}\rangle| \leqslant \sqrt{2}/\sqrt{D}\) for \(q \neq q^\prime\) and for all \(|v_q\rangle\) and \(|v^\prime_{q^\prime}\rangle\). Finally, we show that the analogous AMUB relation still holds in the continuous version of this model, which corresponds to a one-dimensional Dirac particle.Addendum to: ``Weyl cocycles''https://zbmath.org/1526.810372024-02-15T19:53:11.284213Z"Bonora, L."https://zbmath.org/authors/?q=ai:bonora.losiano|bonora.lorianoSummary: Weyl 0- and 1-cocycles of canonical dimension 6 in six dimensions, which were computed earlier in [\textit{L. Bonora} et al., Classical Quantum Gravity 3, 635--649 (1986; Zbl 0615.58046)], are recalculated from scratch. The analysis yields five Weyl invariants (0-cocycles), instead of four, and the same four non-trivial 1-cocycles (possible trace anomalies), like in that reference (up to the correction of one typo).An inner product for 4D quantum gravity and the Chern-Simons-Kodama statehttps://zbmath.org/1526.830052024-02-15T19:53:11.284213Z"Alexander, Stephon"https://zbmath.org/authors/?q=ai:alexander.stephon-h-s"Herczeg, Gabriel"https://zbmath.org/authors/?q=ai:herczeg.gabriel"Freidel, Laurent"https://zbmath.org/authors/?q=ai:freidel.laurentSummary: We demonstrate that reality conditions for the Ashtekar connection imply a non-trivial measure for the inner product of gravitational states in the polarization where the Ashtekar connection is diagonal, and we express the measure as the determinant of a certain first-order differential operator. This result opens the possibility to perform a non-perturbative analysis of the quantum gravity scalar product. In this polarization, the Chern-Simons-Kodama state, which solves the constraints of quantum gravity for a certain factor ordering, and which has de Sitter space as a semiclassical limit, is perturbatively non-normalizable with respect to the naïve inner product. Our work reopens the question of whether this state might be normalizable when the correct non-perturbative inner product and choice of integration contour are taken into account. As a first step, we perform a semi-classical treatment of the measure by evaluating it on the round three-sphere, viewed as a closed spatial slice of de Sitter. The result is a simple, albeit divergent, infinite product that might serve as a regulator for a more complete treatment of the problem. Additionally, our results suggest deep connections between the problem of computing the norm of the CSK state in quantum gravity and computing the Chern-Simons partition function for a complex group.Recovery of rapidly decaying source terms from dynamical samples in evolution equationshttps://zbmath.org/1526.940152024-02-15T19:53:11.284213Z"Aldroubi, Akram"https://zbmath.org/authors/?q=ai:aldroubi.akram"Gong, Le"https://zbmath.org/authors/?q=ai:gong.le"Krishtal, Ilya"https://zbmath.org/authors/?q=ai:krishtal.ilya-arkadievichSummary: We analyze the problem of recovering a source term of the form \(h(t)=\sum_jh_j\phi (t-t_j)\chi_{[t_j, \infty)}(t)\) from space-time samples of the solution \(u\) of an initial value problem in a Hilbert space of functions. In the expression of \(h\), the terms \(h_j\) belong to the Hilbert space, while \(\phi\) is a generic real-valued function with exponential decay at \(\infty\). The design of the sampling strategy takes into account noise in measurements and the existence of a background source.