Recent zbMATH articles in MSC 47https://zbmath.org/atom/cc/472024-02-28T19:32:02.718555ZWerkzeugReverse Cholesky factorization and tensor products of nest algebrashttps://zbmath.org/1527.150092024-02-28T19:32:02.718555Z"Paulsen, Vern I."https://zbmath.org/authors/?q=ai:paulsen.vern-ival"Woerdeman, Hugo J."https://zbmath.org/authors/?q=ai:woerdeman.hugo-jSummary: We prove that every positive semidefinite matrix over the natural numbers that is eventually 0 in each row and column can be factored as the product of an upper triangular matrix times a lower triangular matrix. We also extend some known results about factorization with respect to tensor products of nest algebras. Our proofs use the theory of reproducing kernel Hilbert spaces.Study on Birkhoff orthogonality and symmetry of matrix operatorshttps://zbmath.org/1527.150192024-02-28T19:32:02.718555Z"Wei, Yueyue"https://zbmath.org/authors/?q=ai:wei.yueyue"Ji, Donghai"https://zbmath.org/authors/?q=ai:ji.donghai"Tang, Li"https://zbmath.org/authors/?q=ai:tang.liSummary: We focus on the problem of generalized orthogonality of matrix operators in operator spaces. Especially, on \(\mathcal{B}(l_1^n,l_p^n)\) (\(1\le p\le \infty\)), we characterize Birkhoff orthogonal elements of a certain class of matrix operators and point out the conditions for matrix operators which satisfy the Bhatia-Šemrl property. Furthermore, we give some conclusions which are related to the Bhatia-Šemrl property. In a certain class of matrix operator space, such as \(\mathcal{B}(l_\infty^n)\), the properties of the left and right symmetry are discussed. Moreover, the equivalence condition for the left symmetry of Birkhoff orthogonality of matrix operators on \(\mathcal{B}(l_p^n)\) (\(1< p< \infty\)) is obtained.A note on numerical ranges of tensorshttps://zbmath.org/1527.150242024-02-28T19:32:02.718555Z"Chandra Rout, Nirmal"https://zbmath.org/authors/?q=ai:rout.nirmal-chandra"Panigrahy, Krushnachandra"https://zbmath.org/authors/?q=ai:panigrahy.krushnachandra"Mishra, Debasisha"https://zbmath.org/authors/?q=ai:mishra.debasisha\textit{R. Ke} et al. [Linear Algebra Appl. 508, 100--132 (2016; Zbl 1346.15025)] introduced tensor numerical ranges using tensor inner products and tensor norms via \(k\)-mode product. The authors study the numerical range and the numerical radius for even-order square tensors via the Einstein product, and establish the convexity of the numerical range. They also develop algorithms to compute the numerical ranges of tensors, thus designing very efficient algorithms for the calculation of their eigenvalues. Finally, they investigate some properties of the numerical range of the Moore-Penrose inverse of a tensor.
Reviewer: Mohammad Sal Moslehian (Mashhad)On linear preservers of permanental rankhttps://zbmath.org/1527.150322024-02-28T19:32:02.718555Z"Guterman, A. E."https://zbmath.org/authors/?q=ai:guterman.alexander-e"Spiridonov, I. A."https://zbmath.org/authors/?q=ai:spiridonov.igor-aSummary: Let \(\operatorname{Mat}_n (\mathbb{F})\) denote the set of square \(n \times n\) matrices over a field \(\mathbb{F}\) of characteristic different from two. The permanental rank \(\operatorname{prk}(A)\) of a matrix \(A \in \operatorname{Mat}_n (\mathbb{F})\) is the size of the maximal square submatrix in \(A\) with nonzero permanent. By \(\Lambda^k\) and \(\Lambda^{\leq k}\) we denote the subsets of matrices \(A \in \operatorname{Mat}_n (\mathbb{F})\) with \(\operatorname{prk}(A) = k\) and \(\operatorname{prk}(A) \leq k\), respectively. In this paper for each \(1 \leq k \leq n - 1\) we obtain a complete characterization of linear maps \(T : \operatorname{Mat}_n (\mathbb{F}) \to \operatorname{Mat}_n (\mathbb{F})\) satisfying \(T(\Lambda^{\leq k}) = \Lambda^{\leq k}\) or bijective linear maps satisfying \(T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq k}\). Moreover, we show that if \(\mathbb{F}\) is an infinite field, then \(\Lambda^k\) is Zariski dense in \(\Lambda^{\leq k}\) and apply this to describe such bijective linear maps satisfying \(T(\Lambda^k) \subseteq \Lambda^k\).Correction to: ``Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum''https://zbmath.org/1527.150342024-02-28T19:32:02.718555Z"Shapiro, Boris"https://zbmath.org/authors/?q=ai:shapiro.boris-zalmanovich"Štampach, František"https://zbmath.org/authors/?q=ai:stampach.frantisekThe authors provide a corrected proof of Theorem 8, which was previously published in [the authors, Constr. Approx. 49, No. 2, 191--226 (2019; Zbl 1416.15022)].
Reviewer: D. L. Suthar (Dessie)\(n\)-Jordan homomorphisms into fields and integral domainshttps://zbmath.org/1527.160362024-02-28T19:32:02.718555Z"Honary, Taher Ghasemi"https://zbmath.org/authors/?q=ai:ghasemi-honary.taher"Mahyar, Hakimeh"https://zbmath.org/authors/?q=ai:mahyar.hakimeh"Sadri, Mehrad Mostofi"https://zbmath.org/authors/?q=ai:sadri.mehrad-mostofiSummary: We investigate under what conditions an \(n\)-Jordan homomorphism between rings or algebras is an \(n\)-homomorphism. More specifically, if \(R\) and \(S\) are rings and \(S\) is of characteristic greater than \(n\), along with some other results for an \(n\)-Jordan homomorphism \(\psi :R \rightarrow S\), we prove the following results: If \(S\) is a field and either \(char S = 0\) or \(char S > 2 card \{\alpha \in S: {\alpha}^{n - 1} = 1\}\), then \(\psi\) is an \(n\)-homomorphism, where \(char S\) and \(card\) denote the characteristic of \(S\) and the cardinal number (of a set), respectively. If \(S\) is an integral domain, then \(\psi\) is an \(n\)-homomorphism if \(S\) is an algebra, or either one of the conditions \(char S = 0, char S > 2(n - 1), n = 3\), holds.Boundedness of operators on Campanato spaces related with Schrödinger operators on Heisenberg groupshttps://zbmath.org/1527.220152024-02-28T19:32:02.718555Z"Dai, Tiantian"https://zbmath.org/authors/?q=ai:dai.tiantianSummary: Let \(L=-\Delta_{\mathbb{H}^n}+V\) be a Schrödinger operator on Heisenberg groups \(\mathbb{H}^n\), where \(\Delta_{\mathbb{H}^n}\) is the sub-Laplacian and the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_q, q\geq\mathcal{Q}/2\) and \(\mathcal{Q}=2n+2\) is the homogeneous dimension of \(\mathbb{H}^n\). We establish a \(T1\) criterion for the boundedness of \(\gamma\)-Schrödinger-Calderón-Zygmund operators on Campanato type spaces \(\mathrm{BMO}^{\alpha}_L (\mathbb{H}^n)\). As an application, by the aid of regularity estimate for fractional heat semigroup \(\{e^{-tL^{\beta}}\}_{t>0}\), we prove the \(\mathrm{BMO}^{\alpha}_L\)-boundedness of operators generated by fractional heat semigroups including the maximal operators, the square functions, the Laplace transform type multipliers and the fractional integral associated with Schrödinger operator \(L\) via \(T1\) theorem, respectively.Unitary group orbits versus groupoid orbits of normal operatorshttps://zbmath.org/1527.220272024-02-28T19:32:02.718555Z"Beltiţă, Daniel"https://zbmath.org/authors/?q=ai:beltita.daniel"Larotonda, Gabriel"https://zbmath.org/authors/?q=ai:larotonda.gabrielSummary: We study the unitary orbit of a normal operator \(a\in\mathcal{B}(\mathcal{H})\), regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the various ambient spaces if and only if the spectrum of \(a\) is finite, and in that case, it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of \(a\) is not finite, we describe the closure of the orbits of \(a\) for the different norm topologies involved. We relate these results to the action of the groupoid of partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator \(a\), we also describe the norm closure of its groupoid orbit \(\mathcal{O}_a\), which leads to necessary and sufficient spectral conditions on \(a\) ensuring that \(\mathcal{O}_a\) is norm closed and that \(\mathcal{O}_a\) is a closed embedded submanifold of \(\mathcal{B}(\mathcal{H})\).Minimization of functionals with adjustability quasiconvexity and its applications to eigenvector problem and fixed point theoryhttps://zbmath.org/1527.260062024-02-28T19:32:02.718555Z"Du, Wei-Shih"https://zbmath.org/authors/?q=ai:du.wei-shihIn this paper, the concepts of quasiconvexity and strictly quasiconvexity are generalized to the adjustability quasiconvexity and strictly adjustability quasiconvexity, and some minimization theorems of functions with respect to the adjustability quasiconvexity are presented. Simultaneously, some fixed point theorems and existence theorems for eigenvector problem are established.
Reviewer: Shengkun Zhu (Chengdu)Avatars of Stein's theorem in the complex settinghttps://zbmath.org/1527.300072024-02-28T19:32:02.718555Z"Bonami, Aline"https://zbmath.org/authors/?q=ai:bonami.aline"Grellier, Sandrine"https://zbmath.org/authors/?q=ai:grellier.sandrine"Sehba, Benoît"https://zbmath.org/authors/?q=ai:sehba.benoit-florentSummary: In this paper, we establish some variants of Stein's theorem, which states that a non-negative function belongs to the Hardy space \(H^1(\mathbb{T})\) if and only if it belongs to \(L\log L(\mathbb{T})\). We consider Bergman spaces of holomorphic functions in the upper half plane and develop avatars of Stein's theorem and relative results in this context. We are led to consider weighted Bergman spaces and Bergman spaces of Musielak-Orlicz type. Eventually, we characterize bounded Hankel operators on \(A^1(\mathbb{C}_+)\).Boundary values in spaces spanned by rational functions and the index of invariant subspaceshttps://zbmath.org/1527.300222024-02-28T19:32:02.718555Z"Brennan, J. E."https://zbmath.org/authors/?q=ai:brennan.james-eLet \(\mu\) be a finite positive measure with compact support in the complex plane \(\mathbb{C}\). The points of an open set \(\Omega\subset\mathbb{C}\) are called analytic bounded point evaluations if there exists \(C>0\) such that
\[
|P(z)|\leq C\|P\|_{L^{p}(\mu)}
\]
for every \(z\in\Omega\) and for all polynomials \(P\). For \(1\leq p<\infty\), let \(H^{p}(\mu)\) denote the closed subspace of \(L^{p}(\mu)\) that is spanned by the complex polynomials and let \(R^{p}(\mu)\) denote the closed subspace of \(L^{p}(\mu)\) spanned by the rational functions having no poles on the support of \(\mu\).
In the paper under review the author extends to \(R^{p}(\mu)\) the results of \textit{A. Aleman} et al. [Ann. Math. (2) 169, No. 2, 449--490 (2009; Zbl 1179.46020)] for the nontangential limits of functions in \(H^{p}(\mu)\) and the index of the invariant subspaces of \(H^{p}(\mu)\) under the operator \(M_{z}\) of multiplication by \(z\). In the main result it is shown that, under certain hypothesis for the measure \(\mu\), \(i)\) every function in \(R^{p}(\mu)\) admits nontangential limits almost everywhere with respect to harmonic measure on large portions of the boundary of the set of points that are analytic bounded point evaluations for \(R^{p}(\mu)\) and \(ii)\) every closed nontrivial subspace of \(R^{p}(\mu)\) that is invariant under multiplication by \(z\) has index 1. The role played by the semiadditivity property of analytic capacity and F. and M. Riesz' theorem in the proof of the main results is also described thoroughly in separate sections of the paper.
Reviewer: Stamatis Pouliasis (Thessaloniki)Complex symmetric weighted composition-differentiation operators on \(H^2\)https://zbmath.org/1527.300362024-02-28T19:32:02.718555Z"Hu, Lian"https://zbmath.org/authors/?q=ai:hu.lian"Li, Songxiao"https://zbmath.org/authors/?q=ai:li.songxiao"Yang, Rong"https://zbmath.org/authors/?q=ai:yang.rongSummary: In this paper, we study the complex symmetric weighted composition-differentiation operator \(D_{\psi,\phi}\) with respect to the conjugation \(JW_{\xi, \tau}\) on the Hardy space \(H^2\). As an application, we characterize the necessary and sufficient conditions for such an operator to be normal under some mild conditions. Finally, the spectrum of \(D_{\psi,\phi}\) is also investigated.One weight inequality for Bergman projection and Calderón operator induced by radial weighthttps://zbmath.org/1527.300372024-02-28T19:32:02.718555Z"Reyes, Francisco J. Martín"https://zbmath.org/authors/?q=ai:reyes.francisco-j-martin"Ortega, Pedro"https://zbmath.org/authors/?q=ai:ortega.pedro-a"Peláez, José Ángel"https://zbmath.org/authors/?q=ai:pelaez.jose-angel"Rättyä, Jouni"https://zbmath.org/authors/?q=ai:rattya.jouniFor a nonnegative function \(\omega\in L^1([0,1))\), its extension to the unit disc \(\mathbb D\), defined by \(\omega(z):=\omega(|z|)\) for all \(z\in \mathbb D\), is called a radial weight. For such a radial weight \(\omega\) and any \(0<p<\infty\), the weighted Lebesgue space \(L^p_{\omega}\) is defined as the space of all complex-valued measurable functions \(f\) on \(\mathbb D\) such that
\[
\|f\|_{L^p_{\omega}}:=\left(\int_{\mathbb D} |f(z)|^p\omega(z)dA(z)\right)^{1/p}<\infty.
\]
The weighted Bergman space \(L^p_{\omega}\cap\mathcal{H}(\mathbb D)\), where \(\mathcal{H}(\mathbb D)\) is the space of all analytic functions on \(\mathbb D\), is denoted by \(A^p_{\omega}\).
When \(p=2\), the orthogonal Bergman projection from \(L^2_{\omega}\) onto \(A^2_{\omega}\) is the following integral operator
\[
P_{\omega}(f)(z):=\int_{\mathbb D} f(\xi) \overline{B_z^{\omega}(\xi)}\omega(\xi)dA(\xi),
\]
where \(\{B_z^{\omega}(\xi)\}_{z\in\mathbb D}\) are the reproducing kernels of \(A^2_{\omega}\)\,.
The paper under review shows that when \(\omega\) admits the doubling property
\[
\sup_{0\leq r<1} \frac{\int_{r}^1\omega(s)ds}{\int_{\frac{1+r}{2}}^1 \omega(s)ds}<\infty
\]
and \(\nu\) is any radial weight, then the boundedness of the Bergman projection \(P_{\omega}\) as an operator from \(L^p_{\omega}\) to \(A^p_{\omega}\), for some \(p\in(1,\infty)\), implies a Muckenhoupt-type condition. Moreover, if \(\nu\) is of the form
\[
\nu(s)=\omega(s)\left(\int_{r}^1 s\omega(s)ds\right)^\alpha
\]
for some \(\alpha\in(-1,\infty)\), then such Muckenhoupt-type condition is also sufficient, which extends the classical theorem due to \textit{F. Forelli} and \textit{W. Rudin} [Indiana Univ. Math. J. 24, 593--602 (1974; Zbl 0297.47041)] to a much larger class of weights.
At last, the boundedness of the Calderón operator is also investigated.
The paper is well written and interesting.
Reviewer: Qingze Lin (Guangzhou)Products of composition, differentiation and multiplication from the Cauchy spaces to the Zygmund spacehttps://zbmath.org/1527.300392024-02-28T19:32:02.718555Z"Hibschweiler, Rita"https://zbmath.org/authors/?q=ai:hibschweiler.rita-aSummary: In this paper, we study products of composition, multiplication and differentiation acting on the fractional Cauchy spaces and mapping into the Zygmund space. Characterizations are provided for boundedness and compactness of these operators.Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomialshttps://zbmath.org/1527.330072024-02-28T19:32:02.718555Z"Yafaev, D. R."https://zbmath.org/authors/?q=ai:yafaev.dimitri-rSummary: We find and discuss asymptotic formulas for orthonormal polynomials \(P_n(z)\) with recurrence coefficients \(a_n, b_n\). Our main goal is to consider the case where off-diagonal elements \(a_n\to\infty\) as \(n\to\infty\). Formulas obtained are essentially different for relatively small and large diagonal elements \(b_n\). Our analysis is intimately linked with spectral theory of Jacobi operators \(J\) with coefficients \(a_n, b_n\) and a study of the corresponding second order difference equations. We introduce the Jost solutions \(f_n(z), n\geq - 1\), of such equations by a condition for \(n\to\infty\) and suggest an Ansatz for them playing the role of the semiclassical Liouville-Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions \(P_n(z)\) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for \(P_n(z)\) as \(n\to\infty\) in terms of the Wronskian of the solutions \(P_n(z)\) and \(f_n(z)\). The formulas obtained for \(P_n(z)\) generalize the asymptotic formulas for the classical Hermite polynomials where \(a_n=\sqrt{(n+1)/2}\) and \(b_n=0\). The spectral structure of Jacobi operators \(J\) depends crucially on a rate of growth of the off-diagonal elements \(a_n\) as \(n\to\infty\). If the Carleman condition is satisfied, which, roughly speaking, means that \(a_n=O(n)\), and the diagonal elements \(b_n\) are small compared to \(a_n\), then \(J\) has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values \(| f_{- 1}(\lambda\pm i0)|\) of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of \(J\) is discrete. We also review the case of stabilizing recurrence coefficients when \(a_n\) tend to a positive constant and \(b_n\to 0\) as \(n\to\infty\). It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.Uniqueness of solutions for a \(\psi\)-Hilfer fractional integral boundary value problem with the \(p\)-Laplacian operatorhttps://zbmath.org/1527.340102024-02-28T19:32:02.718555Z"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmed"Alghanmi, Madeaha"https://zbmath.org/authors/?q=ai:alghanmi.madeaha"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Alharbi, Boshra"https://zbmath.org/authors/?q=ai:alharbi.boshraThis work investigates the existence of a unique solution to a \(\Psi\)-Hilfer fractional boundary value problem involving a \(p\)-Laplacian operator and subject to a \(\Psi\)-Hilfer Riemann-Liouville fractional integral boundary conditions. The Banach contraction mapping theorem is used to prove uniqueness of solution in a bounded interval.
The authors express the solution to a \(\Psi\)-Hilfer fractional boundary value problem in an integral form while bounds for the \(p\)-Laplacian operator was given in Lemma 3.
A composition operator \(G = G_1 \circ G_2\) is used in the work to prove the uniqueness of solution, where both \(G_1\) and \(G_2\) are defined in the work.
Conditions for the existence of a unique solution to the \(p\)-Laplacian \(\Psi\)-Hilfer fractional boundary value problem for \(1<p\leq 2\) is stated in Theorem 1. In order to prove this theorem, the authors defined a set \(B_r=\{y \in C[0,1],\mathbb{R}): \Vert y \Vert \leq r\}\) where \(r \geq M^{q-1}\Omega\), \(M\) and \(\Omega\) are defined in the work. The operator \(G\) is shown to map the set \(B_r\) into itself. It is also shown to be a contraction map. Theorem 2 states and proves existence of solution when \(p>2\) using the operator \(G\). A space is defined as \(B_{\bar{r}}=\{y \in C[0,1],\mathbb{R}): \Vert y \Vert \leq \bar{r}\}\) where \(\bar{r}\geq \max\left\{\frac{1-f_0}{k},\frac{f_0\Omega}{1-k\Omega} \right\}\), and \(f_0=\max _{t\in [0,1]}|f(t,0)|\). Theorem 3 states and proves uniqueness of solution for the case \(p>2\).
The authors use an example to illustrate the results.
Reviewer: Ogbu F. Imaga (Ota)Incommensurate multi-term fractional differential equations with variable coefficients with respect to functionshttps://zbmath.org/1527.340142024-02-28T19:32:02.718555Z"Emin, Sedef"https://zbmath.org/authors/?q=ai:emin.sedef"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran(no abstract)Existence of solutions for a three-point Hadamard fractional resonant boundary value problemhttps://zbmath.org/1527.340172024-02-28T19:32:02.718555Z"Gholami, Yousef"https://zbmath.org/authors/?q=ai:gholami.yousefIn this article, the authors study a class of three point boundary value problems of Hadamard fractional nonlinear differential equation as
\[
\begin{cases} (^H D^\alpha_{a^+}u)(t)=f(t,u,^{H} D^{\alpha-1}_{a^+}u)),~~ 1<\alpha\leq 2,~0<a<t<b, \\
(^{H} D^{\alpha-2}_{a^+}u)(a)=0,~~\eta u(\xi)=u(b),~~ a<\xi<b,~\eta>0, \end{cases}
\]
where the parameters satisfy
\[
(\log\frac{b}{a})^{\alpha-1}=\eta (\log\frac{\xi}{a})^{\alpha-1}.
\]
The existence of solutions for the boundary value problems is established, and is proven by using the theory of coincidence degree. Finally, a numerical example is provided to illustrate the theoretical results obtained.
Reviewer: Xiping Liu (Shanghai)Analysis of multipoint impulsive problem of fractional-order differential equationshttps://zbmath.org/1527.340232024-02-28T19:32:02.718555Z"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Abdalla, Bahaaeldin"https://zbmath.org/authors/?q=ai:abdalla.bahaaeldin"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Gul, Rozi"https://zbmath.org/authors/?q=ai:gul.roziIn this article, there is considered the following four point impulsive nonlocal boundary value problem
\[
\begin{cases} & ^C \mathbf{D}^\beta_{x_k} w(x)= f(x,w(x), ^C \mathbf{D}^\beta_{x_k} w(x) ), 1 < \beta \leq 2, x \in \mathcal{J}_1= \mathcal{J} \setminus \{ x_1, x_2, \dots, x_p \}, \\
& \Delta w(x_k)=I_k(w(x_k^-)), \ \Delta w'(x_k)=\bar{I}_k(w(x_k^-)), \ x_k \in (0,1), k=1,2,\dots,p, \\
& w'(0)+c w(\eta)=0, \ \ d w'(1) + w(\xi)=0, \ \ \eta, \xi \in (0,1), \end{cases} \tag{1}
\]
where \(^C \mathbf{D}^\beta_{x_k}\) denotes the fractional Caputo derivative of order \(\beta\) at the point \(x_k\), \(\mathcal{J}=[0,1]\), \(f : \mathcal{J} \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) and \(I_k, \bar{I}_k : \mathbb{R} \to \mathbb{R}\) are continuous functions and \( \Delta w(x_k)=w(x_k^+)-w(x_k^-)\) with \(w(x_k^+) =\lim_{h \to 0^+} w(x_k +h), w(x_k^-) =\lim_{h \to 0^-} w(x_k +h), k=1,2,\dots,p\) for \(0=x_0 < x_1 < x_2 < \dots < x_{p+1}=1\).
As first main result the authors present sufficient conditions for the existence of a unique solution of the problem (1) on \(\mathcal{J}\). The proof is based on the Banach's fixed point theorem.
Using Schaefer's fixed point theorem the authors prove their second main result in form of sufficient conditions for the existence of at least one solution of the same problem (1).
Two examples are provided to illustrate the obtained results.
Reviewer: Hristo S. Kiskinov (Plovdiv)Reconstruction of the Strum-Liouville operator with periodic boundary conditions and discontinuityhttps://zbmath.org/1527.340402024-02-28T19:32:02.718555Z"Zhang, Ran"https://zbmath.org/authors/?q=ai:zhang.ran.4"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfu(no abstract)Periodic boundary value problems for fractional dynamic equations on time scaleshttps://zbmath.org/1527.340472024-02-28T19:32:02.718555Z"Gogoi, Bikash"https://zbmath.org/authors/?q=ai:gogoi.bikash"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Saha, Utpal Kumar"https://zbmath.org/authors/?q=ai:saha.utpal-kumar"Tikare, Sanket"https://zbmath.org/authors/?q=ai:tikare.sanket-aThe paper deals with the existence, uniqueness, and Ulam stability of solutions of a nonlinear fractional dynamic equation involving the Caputo fractional nabla derivative with periodic boundary conditions
\[
\begin{aligned}
&^C{\mathcal D}^\gamma h(x)=G(x,h(x),^C{\mathcal D}^\gamma h(x)),\\
&h(0)=h(T)=0,\qquad T\in{\mathbb R},
\end{aligned}
\]
on time scales \({\mathbb T}\), where \(x\in[0,T]\cap{\mathbb T}\), \(T>0\) and \(F:{\mathcal J}\times{\mathbb R}\times{\mathbb R}\to{\mathbb R}\) is an ld-continuous function in its first variable. In the paper, the existence of a solution is investigated based on the fixed point theory with Green's function, and then the uniqueness result is obtained by employing a dynamic inequality. Some Ulam stability results are presented. As an application of the theoretical results, an appropriate example is given for demonstration.
Reviewer: Abdullah Özbekler (Ankara)Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary conditionhttps://zbmath.org/1527.340502024-02-28T19:32:02.718555Z"Aliyev, Ziyatkhan S."https://zbmath.org/authors/?q=ai:aliyev.ziyatkhan-seyfaddin|aliyev.ziyatkhan-s"Aliyeva, Yagut N."https://zbmath.org/authors/?q=ai:aliyeva.yagut-n(no abstract)Spectral properties of the nonsectorial Sturm-Liouville operator on the semiaxishttps://zbmath.org/1527.340512024-02-28T19:32:02.718555Z"Ishkin, Kh. K."https://zbmath.org/authors/?q=ai:ishkin.khabir-kabirovichAuthor's abstract: The paper deals with some spectral properties of the Sturm-Liouville operator on the semiaxis \(\mathbb{R}_+\) with a complex potential growing at infinity. Instead of the well-known V. B. Lidskii conditions concerning the boundedness from below of the real part or the semiboundedness of the imaginary part of the potential, it is assumed that the range of the potential is disjoint from some small sector containing the negative real semiaxis. Under some additional conditions on the potential, of the type of smoothness and regularity of the growth at infinity, it is shown that the numerical range of the operator fills the entire complex plane, the spectrum is discrete, there exists a sector free from the spectrum, and any ray in this sector is a ray of the best decay of the resolvent. These facts are used to establish the Abel-Lidskii basis property of the root vector system.
Reviewer: Hüseyin Tuna (Burdur)The Christoffel problem in the hyperbolic planehttps://zbmath.org/1527.340692024-02-28T19:32:02.718555Z"Li, Haizhong"https://zbmath.org/authors/?q=ai:li.haizhong"Wan, Yao"https://zbmath.org/authors/?q=ai:wan.yaoIn this paper, the authors investigate the following nonlinear ODE
\[
\varphi \left(\varphi_{\theta\theta}-\frac{\varphi^2_\theta}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)= f(\theta ), \quad \theta \in [0, 2\pi ), \tag{1.1}
\]
where \(f\) is a given \(2\pi\)-periodic \(C^2\)-function and is not necessarily positive.
The athors obtain some existence result of equation (1.1) that is based on a variational method. They consider the functional:
\[
I(\varphi )=\int\limits^{2\pi}_0 \frac{1 + 2f(\theta )}{ \varphi}\, d\theta \cdot \int\limits^{2\pi}_0 \left(\varphi- \frac{\varphi^2_\theta}{\varphi} \right) \,d\theta. \tag{1.8}
\]
whose critical points correspond to solutions of equation (1.1).
In order to estimate the upper bound of \(I(\varphi )\), the authors make use of a generalized Blaschke-Santaló inequality without the convexity assumption.
As a result, the authors prove the existence of solutions under some periodic assumption and an integral condition. One of their main results is the following.
Theorem 1.1. Assume that \(f\) is a \(\frac{2\pi}{k}\)-periodic \(C^2\)-function for some integer \(k > 1\). If
\[
\int\limits^{2\pi}_0 f(\theta )\,d\theta > -\pi. \tag{1.9}
\]
Then equation (1.1) has a positive \(\frac{2\pi}{k}\)-periodic solution.
In particular, when \(k = 2\) they obtain the following corollary.
Corollary 1.2. Assume that \(f\) is a positive \(\pi\)-periodic \(C^2\)-function. Then equation (1.1) has a positive \(\pi\)-periodic solution.
Furthermore, the authors apply the transformation \(\varphi (\theta ) = u^2(\tau )\) with \(\theta = 2\tau\), which connects equation (1.1) and the equation of the centro-affine Minkowski problem in \(\mathbb{R}^2\) as follows
\[
u_{\tau\tau} + u = \frac{g(\tau)}{u^3}, \quad g(\tau)= 2f(2\tau)+1. \tag{1.10}
\]
Then some existence results of the centro-affine Minkowski problem can be used for equation (1.1), the authors obtain
Theorem 1.4. Assume that \(f\) is a positive \(2\pi\)-periodic \(C^2\)-function. Suppose that at least one of the following conditions holds:
\begin{itemize}
\item[(i)] \(\overline{B}(\theta ) \not= 0\) at any critical point \(\theta\) of \(f\), and deg\((\overline{G}, \mathbb{S}^1) \not= -1\).
\item[(ii)] \(\overline{B}(\theta ) > 0\) at any maximum point \(\theta\) of \(f\).
\end{itemize}
Here \(\overline{B}(\theta )\) and \(\overline{G}(\theta )\) are given by
\[
\overline{B}(\theta ) =\int\limits^{2\pi}_0 \frac{f(\theta+s)-f(\theta)-f_\theta(\theta ) \sin s}{ \sin^2\frac{s}{2}}\,ds, \overline{G}(\theta ) =\left(-\overline{B}(\theta ), f_\theta (\theta )\right). \tag{1.11}
\]
Then equation (1.1) has a positive \(2\pi\)-periodic solution.
Reviewer: Petr Tomiczek (Plzeň)Ambrosetti-Prodi type result of first-order differential equations with locally coercive nonlinearitieshttps://zbmath.org/1527.340702024-02-28T19:32:02.718555Z"Lu, Yanqiong"https://zbmath.org/authors/?q=ai:lu.yanqiong"Wang, Rui"https://zbmath.org/authors/?q=ai:wang.rui.31The authors obtain an Ambrosetti-Prodi-type result for the \(T\)-periodic problem associated with the first-order differential equation
\[
u'(t)=a(t)u(t)-f(t,u(t))+s.
\]
Here, \(a :{\mathbb R} \to [0, +\infty)\) is a \(T\)-periodic function with \(\int_0^T a(t)\,dt = 0\), the function \(f\) is of Carathéodory type, \(T\)-periodic with respect to its first variable, and satisfies some local coercivity at infinity, while \(s\) is a real parameter. The proof makes use of topological degree theory.
Reviewer: Alessandro Fonda (Trieste)On one approach to the study of the periodic problem for random differential equationshttps://zbmath.org/1527.341002024-02-28T19:32:02.718555Z"Kornev, S. V."https://zbmath.org/authors/?q=ai:kornev.sergei-viktorovich"Korneva, P. S."https://zbmath.org/authors/?q=ai:korneva.p-s"Iakusheva, N. E."https://zbmath.org/authors/?q=ai:iakusheva.n-eSummary: Geometric and topological methods of analysis applied to problems of nonlinear oscillations of dynamical systems go back to the names of Poincaré, Brauer, Alexandrov, Hopf, Leray, and Schauder. Later, these methods were developed and demonstrated their effectiveness in the works of many mathematicians. Note, in particular, an extremely fruitful direction associated with the notion of a guiding function, whose base was laid by Krasnosel'skii and Perov. In this paper, to study the periodic problem of random differential equations, we use a modification of the classical notion of a guiding function, which is a random nonsmooth multivalent guiding function. A significant advantage over the classical approach is the ability to ``localize'' the verification of the main condition of ``directionality'' on a domain that depends on the guiding function itself and on the domain not of the whole space, but of its subspace of lower dimension. In classical works on the method of guiding functions, as a rule, it is assumed that these functions are smooth over the whole phase space. This condition may seem restrictive, for example, in situations where the guiding potentials are different in different domains of the space. To remove this restriction, the paper considers nonsmooth direction potentials and their generalized gradients.Abstract degenerate Volterra inclusions in locally convex spaceshttps://zbmath.org/1527.341042024-02-28T19:32:02.718555Z"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this article, we analyze the abstract degenerate Volterra integro-differential equations in sequentially complete locally convex spaces by using multivalued linear operators and vector-valued Laplace transform. We follow the method which is based on the use of \((a, k)\)-regularized \(C\)-resolvent families generated by multivalued linear operators and which suggests a very general way of approaching abstract Volterra equations. Among many other themes, we consider the Hille-Yosida type theorems for \((a, k)\)-regularized \(C\)-resolvent families, differential and analytical properties of \((a, k)\)-regularized \(C\)-resolvent families, the generalized variation of parameters formula, and subordination principles. We also introduce and analyze the class of \((a, k)\)-regularized \((C_1, C_2)\)-existence and uniqueness families. The main purpose of third section, which can be viewed of some independent interest, is to introduce a relatively simple and new theoretical concept useful in the analysis of operational properties of Laplace transform of non-continuous functions with values in sequentially complete locally convex spaces. This concept coincides with the classical concept of vector-valued Laplace transform in the case that \(X\) is a Banach space.Resolvent family for the evolution process with memoryhttps://zbmath.org/1527.341062024-02-28T19:32:02.718555Z"Xu, Gen Qi"https://zbmath.org/authors/?q=ai:xu.gen-qiAuthor's abstract: In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, \(\{(G(t), F(t)), ~t\ge 0\},\) that is, called the resolvent family for the linear evolution process with memory, the \(F(t)\) is called the memory effect family. In this paper, we prove that the families \(G(t)\) and \(F(t)\) are exponentially bounded, and the family \((G(t), F(t))\) associate with an operator pair \((A,L)\) that is called generator of the resolvent family. Using \((A,L),\) we derive associated differential equation with memory and representation of \(F(t)\) via \(L.\) These results give necessary conditions of the well-posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on \((A,L).\) The obtained results can be directly applied to linear delay differential equation, integro-differential equation and functional differential equations.
Reviewer: Sotiris K. Ntouyas (Ioannina)Trace formula of the differential operator with delays on a quantum graphhttps://zbmath.org/1527.341082024-02-28T19:32:02.718555Z"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfu"Wei, Li-Xiao"https://zbmath.org/authors/?q=ai:wei.li-xiao"Xu, Xin-Jian"https://zbmath.org/authors/?q=ai:xu.xinjianThe paper deals with the system of the Sturm-Liouville-type equations with constant delays on the lasso graph
\[
-y_j''(x) + q_j(x) y_j(x-a_j) = \lambda^2 y_j(x_j), \quad j = 1,2,
\]
with the standard matching conditions
\[
y_1(1) = y_2(0) = y_2(1), \quad y_1'(1) - y_2'(0) + y_2'(1) = 0
\]
and the Dirichlet boundary condition \(y_1(0) = 0\). Here \(a_j \in(0,1)\), \(q_j(x) = 0\) a.e. on \((0,a_j)\), and \(q_j \in L^1(a_j,1)\), \(j = 1, 2\). A feature of this boundary value problem is the non-locality.
The authors derive the eigenvalue asymptotics and the trace formula for the considered operator. The methods are standard, they are based on obtaining the characteristic function and on the contour integration in the \(\lambda\)-plane.
Reviewer: Natalia Bondarenko (Saratov)On qualitative analysis of boundary value problem of variable order fractional delay differential equationshttps://zbmath.org/1527.341092024-02-28T19:32:02.718555Z"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Ali, Gauhar"https://zbmath.org/authors/?q=ai:ali.gauhar"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamal"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Meganathan, M."https://zbmath.org/authors/?q=ai:meganathan.murugesan"Abdalla, Bahaaeldin"https://zbmath.org/authors/?q=ai:abdalla.bahaaeldinIn this article, a class of integral boundary value problems for variable order fractional differential equations with time delays are investigated, where the boundary condition is nonlinear. The existence and uniqueness theorem of solutions for boundary value problems are established by using Schauder fixed point theorem and Banach theorem. In addition, stability analysis was conducted on the boundary value problem, and relevant concepts of Ulam Hyers and Ulam Hyers Rassias stability are given. Sufficient conditions for the stability of Ulam Hyers and Ulam Hyers Rassias are established. Finally, some examples are provided to validate the main conclusions obtained in this paper.
Reviewer: Xiping Liu (Shanghai)Periodic solutions of second-order degenerate differential equations with infinite delay in Banach spaceshttps://zbmath.org/1527.341122024-02-28T19:32:02.718555Z"Bu, Shangquan"https://zbmath.org/authors/?q=ai:bu.shangquan"Cai, Gang"https://zbmath.org/authors/?q=ai:cai.gangFrom the authors' summary: ``We consider the well-posedness of the second-order degenerate differential equations
\[
(Mu^\prime)^\prime(t) + \Lambda u^\prime (t) + \int_{-\infty}^t a(t-s)u^\prime (s)\,ds
= Au(t) + \int_{-\infty}^t b(t-s)Bu(s)\,ds + f(t)
\]
with infinite delay on $[0, 2\pi]$ in Lebesgue-Bochner spaces $L^p([0, 2\pi];X)$ and periodic Besov spaces $B^s_{p,q}([0, 2\pi];X)$, where $A,B,\Lambda,$ and $M$ are closed linear operators in a Banach space $X$ satisfying $D(A) \cap D(B) \subset D(M) \cap D(\Lambda)$ and the kernels $a,b \in L^1(\mathbb{R}_+)$.''
Reviewer: Valerii V. Obukhovskij (Voronezh)Pseudo almost periodic synchronization of OVCNNs with time-varying delays and distributed delays on time scaleshttps://zbmath.org/1527.341172024-02-28T19:32:02.718555Z"Shen, Shiping"https://zbmath.org/authors/?q=ai:shen.shiping"Meng, Xiaofang"https://zbmath.org/authors/?q=ai:meng.xiaofang"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.2|yang.li.7|yang.li.5|yang.li.1|yang.liSummary: This paper investigates the problem of the pseudo almost periodic synchronization in octonion-valued cellular neural networks with time-varying and distributed delays on time scales, employing a non-decomposition method. Secondly, by using the differential inequality technique on the time scale, Banach fixed point theorem, and calculus theory on the time scale are utilized to derive a sufficient condition for the existence of pseudo almost periodic solutions in the neural network on the time scale. Thirdly, the forensic method is employed to achieve the pseudo almost periodic synchronization in the network error system. Finally, a numerical example is given to illustrate the effectiveness of the results. The results obtained in this paper are new even for differential equations and difference equations.Existence and uniqueness of global solution for abstract second order differential equations with state-dependent delayhttps://zbmath.org/1527.341202024-02-28T19:32:02.718555Z"Hernández, Eduardo"https://zbmath.org/authors/?q=ai:hernandez.eduardo-mIn the paper, the problem of the existence and uniqueness of a MILD solution to the initial value problem for a certain second-order nonlinear abstract differential equation with a state-dependent delay is studied. Well-posedness of the given problem is also discussed. The results obtained are applied and further discussed in Section 3, where two examples concerning the partial differential equations are presented.
Reviewer: Jiří Šremr (Brno)On periodic solutions for some nonlinear fractional pantograph problems with \(\Psi \)-Hilfer derivativehttps://zbmath.org/1527.341252024-02-28T19:32:02.718555Z"Benzenati, Djilali"https://zbmath.org/authors/?q=ai:benzenati.djilali"Bouriah, Soufyane"https://zbmath.org/authors/?q=ai:bouriah.soufyane"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffakIn the paper, a certain periodic type problem is studied for a non-linear fractional differential equation with the \(\Psi\)-Hilfer fractional derivative and a proportional delay. The authors provide sufficient conditions for the existence as well as uniqueness of a solution to the given problem and show a possible use of the stated results.
Reviewer: Jiří Šremr (Brno)A note on approximate controllability of second-order neutral stochastic delay integro-differential evolution inclusions with impulseshttps://zbmath.org/1527.341262024-02-28T19:32:02.718555Z"Sivasankar, Sivajiganesan"https://zbmath.org/authors/?q=ai:sivasankar.sivajiganesan"Udhayakumar, Ramalingam"https://zbmath.org/authors/?q=ai:udhayakumar.r(no abstract)Existence of periodic solutions for a class of dynamic equations with multiple time varying delays on time scaleshttps://zbmath.org/1527.341272024-02-28T19:32:02.718555Z"Agrawal, Divya"https://zbmath.org/authors/?q=ai:agrawal.divya"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed-alam|abbas.syed-saiden|abbas.syed|abbas.syed-afsar|abbas.syed-zaheer|abbas.syed-muzahir|abbas.syed-wasim|abbas.syed-f|abbas.syed-mohsin|abbas.syed-hussnainSummary: This manuscript considers general coupled dynamic equations on time scales with multiple and time-varying delays. The considered equations describe several continuous and discrete models as special cases. Using the coincidence degree theory approach, we investigate the existence of periodic solutions. The particular case describes the mathematical model of interacting phytoplankton when both species produce a chemical that is toxin to each other. The presented results extend and complement the existing results. Finally, some examples are presented to illustrate the analytical results.Spectral asymptotics and a trace formula for a fourth-order differential operator corresponding to thin film equationhttps://zbmath.org/1527.341312024-02-28T19:32:02.718555Z"Polyakov, Dmitry M."https://zbmath.org/authors/?q=ai:polyakov.dmitrii-mikhailovichIn the paper, the high-energy asymptotics of the eigenvalues and a trace formula are found for a self-adjoint fourth-order operator on the interval with Neumann-Dirichlet boundary conditions and periodic coefficients.
The operator considered here acts as
\[
H y = y^{(4)}+(py')'+qy
\]
on \(L^2(0,1)\) with the boundary conditions
\[
y'(0) = y'''(0)+p(0) y'(0) = y(1) = y''(1) = 0
\]
and a certain domain (for brevity, we do not state it here explicitly). The coefficients \(p\) and \(q\) are 1-periodic. This operator is connected with the one-dimensional thin film equation.
Reviewer: Jiři Lipovský (Hradec Králové)Conformable fractional Sturm-Liouville problems on time scaleshttps://zbmath.org/1527.341352024-02-28T19:32:02.718555Z"Allahverdiev, Bilender P."https://zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyin(no abstract)Geometric harmonic analysis V. Fredholm theory and finer estimates for integral operators, with applications to boundary problemshttps://zbmath.org/1527.350052024-02-28T19:32:02.718555Z"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 last installment 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 [Zbl 1517.42001];
Volume~II: Function spaces measuring size and smoothness on rough sets [Zbl 1521.42003];
Volume~III: Integral representations, Calderón-Zygmund theory, Fatou theorems, and applications to scattering [1523.35001];
Volume~IV: Boundary layer potentials in uniformly rectifiable domains, and applications to complex analysis [Zbl 1526.42001];
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 having the ability to treat efficiently boundary value problems for elliptic systems in inclusive geometric settings, beyond the category of Lipschitz domains.
The present volume focuses on proving well-posedness and Fredholm solvability results regarding boundary value problems for elliptic second-order homogeneous constant complex coefficient systems, and domains of a rather general geometric character. The formulation of the boundary value problems treated here is optimal from a multitude of aspects, pertaining to Functional Analysis, Geometry, Partial Differential Equations, and Topology. The work in the present volume aligns with the program stemming from A.P. Calderón's 1978 ICM address which advocates the use of layer potentials for much more general elliptic systems than the Laplacian.
In Chapter~1, the authors introduce the notion of distinguished coefficient tensor, which is central to all subsequent developments in this volume. Relevant examples of weakly elliptic homogeneous constant (complex) coefficient second-order systems possessing distinguished coefficient tensors are given, including certain Lamé-like systems, and the entire class of scalar weakly elliptic homogeneous constant (complex) coefficient second-order operators in dimensions \(\geq 3\). Also, the authors study whether the quality of being a distinguished coefficient tensor is stable under transposition, and discuss the issue of the existence and uniqueness of a distinguished coefficient tensor.
Chapter~2 focuses on providing examples of weakly elliptic homogeneous constant coefficient second-order systems with the property that their associated \(L^p\) Dirichlet Problems in the upper half-space fail to be Fredholm solvable. The manner in which this connects with earlier work in Chapter~1 is that one looks for such pathological weakly elliptic systems in the class of those which lack a distinguished coefficient tensor.
Chapter~3 is devoted to the task of quantifying global and infinitesimal flatness in classes of Euclidean sets of locally finite perimeter which may otherwise lack structural qualities which have traditionally been used to describe regularity.
Chapter~4 deals with the class of singular integral operators (SIO's) of chord-dot-normal type (as defined in Volume~IV, \S 5.2) associated with Ahlfors regular domains \(\Omega\subset\mathbb{R}^n\) with unbounded boundaries. These operators are sensitive to flatness, in the sense that they become identically zero when the underlying domain is a half-space in \(\mathbb{R}^n\).
In Chapter~5, the authors clarify how the flatness of a surface is related to the functional analytic properties of singular integral operators defined on it by setting up a two-way street between geometry and analysis. Working now in domains with compact boundaries, the authors establish estimates for the essential norm of SIO's of chord-dot-normal type in terms of the proximity of the unit normal to the surface in question to Sarason's space VMO.
Chapter~6 deals with the Radon-Carleman Problem, which the authors define as the task of computing and/or estimating the essential norm and/or Fredholm radius of singular integral operators of double layer type, associated with elliptic PDE, on function spaces naturally intervening in the formulation of boundary problems for said PDE.
In Chapter~7, the authors prove Fredholm and invertibility properties of boundary layer potentials on compact surfaces, which are subsequently used to treat boundary value problems in domains with compact boundaries. This body of work points to the fact that the category of Ahlfors regular domains with a compact and sufficiently flat boundary at infinitesimal level is a most natural geometric environment where Fredholm theory becomes applicable to boundary layer potential operators.
In Chapter~8, as a culmination of the work in this volume, the authors formulate and solve boundary value problems for elliptic second-order systems, involving a large variety of boundary conditions and function spaces, in a geometric setting displaying new levels of generality and inclusiveness. Also, they take the first steps in the direction of combining geometric measure theory with scattering theory, by solving the basic boundary value problems in acoustic theory in novel geometric settings. Significantly, the authors successfully implement the method of boundary layer potentials for the Dirichlet and Neumann Problems for elliptic systems with data in Muckenhoupt weighted Lebesgue spaces, as well as Hardy, Sobolev, BMO, VMO, Hölder, Morrey, Besov, Triebel-Lizorkin spaces, and Generalized Banach Function Spaces.
Reviewer: Mohammed El Aïdi (Bogotá)Harnack inequalities and Hölder estimates for fully nonlinear integro-differential equations with weak scaling conditionshttps://zbmath.org/1527.350152024-02-28T19:32:02.718555Z"Kitano, Shuhei"https://zbmath.org/authors/?q=ai:kitano.shuheiHölder estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some reasonable and commonly used assumptions using the viscosity solution approach. The aim of this article is to study what can be demonstrated when the kernels are of variable order and critically close to 2. Many examples of possible applications of these results are given.
Reviewer: Vincenzo Vespri (Firenze)On the regularized Moore-Gibson-Thompson equationhttps://zbmath.org/1527.350452024-02-28T19:32:02.718555Z"Dell'Oro, Filippo"https://zbmath.org/authors/?q=ai:delloro.filippo"Liverani, Lorenzo"https://zbmath.org/authors/?q=ai:liverani.lorenzo"Pata, Vittorino"https://zbmath.org/authors/?q=ai:pata.vittorinoSummary: We study the regularized MGT equation \(u_{ttt} + \alpha u_{tt} +\beta Au_t +\gamma A u +\delta A u_{tt} = 0\) where \(A\) is a strictly positive unbounded operator and \(\alpha, \beta, \gamma, \delta>0 \). The effect of the regularizing term \(\delta A u_{tt}\) translates into having an analytic semigroup \(S(t) = e^{t{{\mathbb A}}}\) of solutions. Moreover, the asymptotic properties of the semigroup are ruled by the stability number
\[\varkappa = \beta - \frac{\gamma}{\alpha +\delta \lambda_0}\]
which, contrary to the case of the standard MGT equation, depends also on the minimum \(\lambda_0>0\) of the spectrum of \(A \).Existence and stability results for a class of nonlocal semilinear equations involving delayshttps://zbmath.org/1527.350492024-02-28T19:32:02.718555Z"Nguyen Nhu Quan"https://zbmath.org/authors/?q=ai:nguyen-nhu-quan.Summary: We investigate the existence and weak stability of mild solutions for a class of nonlocal semilinear equations involving finite delays. Based on local estimates and fixed point arguments, we prove the global existence of mild solutions to problems in which the nonlinearity is superlinear or sublinear without the smallness condition on initial data or on the coefficients. Then by using a special measure of noncompactness, we show some sufficient conditions ensuring the weak stability of mild solutions.Interactions of \((m,n)\) and \((m+1,n)\) modes with real eigenvalues: a dynamic transition approachhttps://zbmath.org/1527.350532024-02-28T19:32:02.718555Z"Şengül, Taylan"https://zbmath.org/authors/?q=ai:sengul.taylan"Tiryakioglu, Burhan"https://zbmath.org/authors/?q=ai:tiryakioglu.burhanSummary: In this work, we consider the multiplicity two dynamic transitions of a broad class of problems. The first main assumption is the existence of two critical eigenmodes of the linear operator which depend on at least two wave indices one of which are consecutive \(m,m+1\) and the other identical \(n\). The second main assumption is an orthogonality condition on the nonlinear interactions of the basis vectors of the phase space which is typical in many applications. Under this assumption we obtain a reduced system of ODE's which describe the first dynamic transitions. We make a careful analysis of this reduced system to classify all possible transition behavior. We then apply our main theoretical findings to the 2D Rayleigh-Bénard convection with free-slip boundary conditions and show that this problem displays an \(S^1\) attractor bifurcation.Ergodicity of the Fisher infinitesimal model with quadratic selectionhttps://zbmath.org/1527.350582024-02-28T19:32:02.718555Z"Calvez, Vincent"https://zbmath.org/authors/?q=ai:calvez.vincent"Lepoutre, Thomas"https://zbmath.org/authors/?q=ai:lepoutre.thomas"Poyato, David"https://zbmath.org/authors/?q=ai:poyato.davidSummary: We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher's infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogs, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein-Rutman theorem in a genuinely non-linear, and non-monotone setting.Frequency theorem and inertial manifolds for neutral delay equationshttps://zbmath.org/1527.350942024-02-28T19:32:02.718555Z"Anikushin, Mikhail"https://zbmath.org/authors/?q=ai:anikushin.mikhail-mikhailovichSummary: We study the infinite-horizon quadratic regulator problem for linear control systems in Hilbert spaces, where the cost functional is in some sense unbounded. Our motivation comes from delay equations with the feedback part containing discrete delays or, in other words, measurements given by \(\delta \)-functionals, which are unbounded in \(L_2\). Working in an abstract context in which such (and many others, including parabolic boundary control problems) equations can be treated, we obtain a version of the Frequency Theorem. It guarantees the existence of a unique optimal process and shows that the optimal cost is given by a quadratic Lyapunov-like functional. In our adjacent works it is shown that such functionals can be used to construct inertial manifolds and allow to treat and extend many works in the field in a unified manner. Here we concentrate on applications to delay equations and especially mention the works of R.A. Smith on developments of convergence theorems and the Poincaré-Bendixson theory; and also the works of Yu.A. Ryabov, R.D. Driver and C. Chicone on inertial manifolds for equations with small delays and their recent generalization for equations of neutral type given by S. Chen and J. Shen.Higher Hölder regularity for nonlocal parabolic equations with irregular kernelshttps://zbmath.org/1527.351082024-02-28T19:32:02.718555Z"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Kim, Hyojin"https://zbmath.org/authors/?q=ai:kim.hyojin"Kim, Kyeongbae"https://zbmath.org/authors/?q=ai:kim.kyeongbaeIn this paper, the authors prove Hölder regularity for the parabolic equation
\begin{align*}
\partial_t u - \mathrm{pv}\int\limits_{\mathbb{R}^n}\Phi(u(x,t)-u(y,t))\frac{A(x,y,t)}{|x-y|^{n+2s}}\,dy=f,
\end{align*}
where \(\Phi:\mathbb{R}\to\mathbb{R}\) satisfies \(\Phi(0)=0\) and standard growth conditions
\begin{align*}
(\Phi(\xi)-\Phi(\xi'))(\xi-\xi')\gtrsim |\xi-\xi'|^2,\\
|\Phi(\xi)-\Phi(\xi')|\lesssim|\xi-\xi'|,
\end{align*}
\(A:\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) is measurable function bounded from above and below, which is ``locally close enough to being translation invariant'', and \(f\) satisfies a suitable integrability condition. This condition allows for discontinuous kernels.
For \(\Phi(t)=t\) and \(A=1\), the equation is the fractional heat equation.
The condition on \(A\) is a parabolic version of the condition presented for nonlocal elliptic equations in [\textit{S. Nowak}, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)].
The proofs use a perturbation argument similar to [\textit{S. Nowak}, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)] and the method of iterated discrete differentiation from [\textit{L. Brasco} et al., J. Evol. Equ. 21, No. 4, 4319--4381 (2021; Zbl 1486.35084)].
Additionally, the authors prove existence and local boundedness for this equation without the extra constraint on \(A\).
Reviewer: Vivek Tewary (Sri City)The Cauchy problem for a quasilinear system of equations with coupling in the linearizationhttps://zbmath.org/1527.351442024-02-28T19:32:02.718555Z"Angeles, Felipe"https://zbmath.org/authors/?q=ai:angeles.felipeSummary: The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed in the linearization and we do not assume the Friedrich's symmetrizability of the system. This coupling results in linear energy estimates that prevent the application of Banach's contraction principle. A metric fixed point theorem is developed in order to conclude the local existence and uniqueness of solutions. We show that the boundedness in the high norm and contraction in the low norm can be incorporated into the formulation of the fixed point by introducing the notion of a closed extension of the solution map. We apply our results to the Cattaneo-Christov system for viscous compressible fluid flow, a system of equations whose inviscid part is not hyperbolic.Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensionshttps://zbmath.org/1527.351472024-02-28T19:32:02.718555Z"Goldberg, Michael"https://zbmath.org/authors/?q=ai:goldberg.michael-joseph"Green, William R."https://zbmath.org/authors/?q=ai:green.william-rThe authors consider the following fourth-order Schrödinger equation
\[
i\psi_t=(\Delta^2+V)\psi\,,\quad \psi(0,x)=f(x),\quad x\in\mathbb R^3.
\]
Here \(V\) is a real-valued potential which decays as \(|x|\to\infty\). In the case \(V=0\) it is well-known that the solution operator satisfies the following decay estimate:
\[
\|e^{-it\Delta^2}f\|_{L^{\infty}(\mathbb R^3)}\leq Ct^{-3/4}\|f\|_{L^1(\mathbb R^3)},
\]
for an absolute constant \(C>0\). The authors improve this decay estimate under some additional conditions. Namely, if \(0\) is a regular point of the spectrum of \(\Delta^2+V\), there are no embedded eigenvalues in the spectrum of \(\Delta^2+V\), and \(|V(x)|\leq C(1+|x|^2)^{-\beta/2}\) for some \(\beta>7\), then
\[
\|e^{-it(\Delta^2+V)}f\|_{L^{\infty}(\mathbb R^3)}\leq Ct^{-5/4}\|f\|_{L^{1,1}(\mathbb R^3)},
\]
where \(L^{1,1}(\mathbb R^3)\) is the space \(L^1(\mathbb R^3)\) weitghed by \((1+|x|^2)^{1/2}\). Here \(C\) is denoting an absolute positive constant which is possibly re-defined line by line.
Reviewer: Luigi Provenzano (Padova)Doubly nonlinear equations for the 1-Laplacianhttps://zbmath.org/1527.351642024-02-28T19:32:02.718555Z"Mazón, J. M."https://zbmath.org/authors/?q=ai:mazon-ruiz.jose-m"Molino, A."https://zbmath.org/authors/?q=ai:molino.alexis"Toledo, J."https://zbmath.org/authors/?q=ai:toledo.j-julian|toledo.julianSummary: This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian,
\[
\frac{\partial v}{\partial t} - \Delta_1 u \ni 0 \text{ in } (0, \infty) \times \Omega, \quad v\in \gamma (u),
\]
and initial data in \(L^1 (\Omega)\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) and \(\gamma\) is a maximal monotone graph in \(\mathbb{R}\times\mathbb{R}\). We prove that, under certain assumptions on the graph \(\gamma\), there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.Persistence of regularity of the solution to a hyperbolic boundary value problem in domain with cornerhttps://zbmath.org/1527.351692024-02-28T19:32:02.718555Z"Benoit, Antoine"https://zbmath.org/authors/?q=ai:benoit.antoineSummary: This article is about the question of the persistence of regularity for the solution to hyperbolic boundary value problem in the quarter-space. More precisely we will both consider the pure boundary value problem and the initial boundary value problem and we propose a functional space, based upon the high order Sobolev space in which a control of the data of the problem leads to a control of the solution (in the same space). The space proposed here contains the tangential Sobolev space. The analysis borrows some ideas of the study of characteristic boundary value problems in the half-space for which the good derivative to consider is known to be the tangential derivatives \(x_1 \partial_1\) instead of the normal derivative \(\partial_1\). For quarter-space problems the good quantity to consider will be the radial derivative \(x_1 \partial_1 + x_2 \partial_2\) and then we recover the control of tangential derivatives \(x_1 \partial_1\) and \(x_2 \partial_2\) using explicit formulas in polar coordinates. The regularity of the solution is then established intrinsically by adapting the method introduced by the author to deal with half-space problems without using regularization methods.Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flowshttps://zbmath.org/1527.352702024-02-28T19:32:02.718555Z"Coti Zelati, Michele"https://zbmath.org/authors/?q=ai:coti-zelati.michele"Gallay, Thierry"https://zbmath.org/authors/?q=ai:gallay.thierrySummary: We consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity \(\nu\), which is assumed to be small, and the wave number \(k\) in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity \(v\), we obtain optimal decay estimates for large times, both in the enhanced dissipation regime \(\nu \ll |k|\) and in the Taylor dispersion regime \(|k| \ll \nu\). Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart-Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.
{\copyright} 2023 The Authors. \textit{Journal of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.Mild ill-posedness in \(L^\infty\) for 2D magnetohydrodynamic system with dampinghttps://zbmath.org/1527.352752024-02-28T19:32:02.718555Z"Fang, Hui"https://zbmath.org/authors/?q=ai:fang.hui"Bie, Qunyi"https://zbmath.org/authors/?q=ai:bie.qunyiSummary: The present paper is devoted to the initial-value problem for the incompressible magnetohydrodynamic system in two-dimensional space where the velocity equation involves no dissipation and the vertical component equation has a damping. Here we concentrate on the stability problem for the magnetohydrodynamic system. By constructing a sequence of special initial data, we show that the solution map of vorticity is discontinuous. Our result indicates that the \(L^\infty\)-norm of the vorticity is mildly ill-posed.Semigroup wellposedness and asymptotic stability of a compressible Oseen-structure interaction via a pointwise resolvent criterionhttps://zbmath.org/1527.352782024-02-28T19:32:02.718555Z"Geredeli, Pelin G."https://zbmath.org/authors/?q=ai:geredeli.pelin-guvenSummary: In this study, we consider the Oseen structure of the linearization of a compressible fluid-structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra ``convective derivative'' and ``material derivative'' terms, which render the coupled system highly nondissipative. We show first a new well-posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one-dimensional less) of the entire state space, where the continuous semigroup is \textit{uniformly bounded}. For this, we appeal to the pointwise resolvent condition introduced in \textit{R. Chill} and \textit{Y. Tomilov} [Banach Cent. Publ. 75, 71--109 (2007; Zbl 1136.47026)], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy-to-follow proof.
{{\copyright} 2023 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH.}The tension determination problem for an inextensible interface in 2D Stokes flowhttps://zbmath.org/1527.352942024-02-28T19:32:02.718555Z"Kuo, Po-Chun"https://zbmath.org/authors/?q=ai:kuo.po-chun"Lai, Ming-Chih"https://zbmath.org/authors/?q=ai:lai.mingchih"Mori, Yoichiro"https://zbmath.org/authors/?q=ai:mori.yoichiro"Rodenberg, Analise"https://zbmath.org/authors/?q=ai:rodenberg.analiseSummary: Consider an inextensible closed filament immersed in a 2D Stokes fluid. Given a force density \(\boldsymbol{F}\) defined on this filament, we consider the problem of determining the tension \(\sigma\) on this filament that ensures the filament is inextensible. This is a subproblem of dynamic inextensible vesicle and membrane problems, which appear in engineering and biological applications. We study the well-posedness and regularity properties of this problem in Hölder spaces. We find that the tension determination problem admits a unique solution if and only if the closed filament is \textit{not} a circle. Furthermore, we show that the tension \(\sigma\) gains one derivative with respect to the imposed line force density \(\boldsymbol{F}\) and show that the tangential and normal components of \(\boldsymbol{F}\) affect the regularity of \(\sigma\) in different ways. We also study the near singularity of the tension determination problem as the interface approaches a circle and verify our analytical results against numerical experiment.A nonlinear Klein-Gordon equation on a star graphhttps://zbmath.org/1527.353182024-02-28T19:32:02.718555Z"Goloshchapova, Nataliia"https://zbmath.org/authors/?q=ai:goloshchapova.nataliiaSummary: We study local well-posedness and orbital stability/instability of standing waves for a first order system associated with a nonlinear Klein-Gordon equation on a star graph. The proof of the well-posedness uses a classical fixed point argument and the Hille-Yosida theorem. Stability study relies on the linearization approach and recent results for the NLS equation with the \(\delta\)-interaction on a star graph.
{{\copyright} 2021 Wiley-VCH GmbH}On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolutionhttps://zbmath.org/1527.353222024-02-28T19:32:02.718555Z"Lee, Jinyeop"https://zbmath.org/authors/?q=ai:lee.jinyeop"Michelangeli, Alessandro"https://zbmath.org/authors/?q=ai:michelangeli.alessandroSummary: Fragmented Bose-Einstein condensates are large systems of identical bosons displaying \textit{multiple} macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}Normalized solutions for the Klein-Gordon-Dirac systemhttps://zbmath.org/1527.353242024-02-28T19:32:02.718555Z"Zelati, Vittorio Coti"https://zbmath.org/authors/?q=ai:coti-zelati.vittorio"Nolasco, Margherita"https://zbmath.org/authors/?q=ai:nolasco.margheritaSummary: We prove the existence of a stationary solution for the system describing the interaction between an electron coupled with a massless scalar field (a photon). We find a solution, with fixed \(L^2\)-norm, by variational methods, as a critical point of an energy functional.Restriction estimates in a conical singular space: Schrödinger equationhttps://zbmath.org/1527.353252024-02-28T19:32:02.718555Z"Chen, Jingdan"https://zbmath.org/authors/?q=ai:chen.jingdan"Gao, Xiaofen"https://zbmath.org/authors/?q=ai:gao.xiaofen"Xu, Chengbin"https://zbmath.org/authors/?q=ai:xu.chengbinSummary: This paper continues our previous program to study the restriction estimates in a class of conical singular spaces \(X = C(Y) = (0, \infty)_r \times Y\) equipped with the metric \(g=\mathrm{d}r^2+r^2h\), where the cross section \(Y\) is a compact \((n-1)\)-dimensional closed Riemannian manifold \((Y, h)\). Assuming the initial data possesses additional regularity in the angular variable \(\theta \in Y\), we prove some linear restriction estimates for the solutions of Schrödinger equations on the cone \(X\). The smallest positive eigenvalue of the operator \(\Delta_h + V_0 + (n-2)^2/4\) plays an important role in the result. As applications, we prove local energy estimates and Keel-Smith-Sogge estimates for the Schrödinger equation in this setting.Restriction theorem fails for the Fourier-Hermite transform associated with the normalized Hermite polynomials with respect to a discrete surfacehttps://zbmath.org/1527.353282024-02-28T19:32:02.718555Z"Ghosh, Sunit"https://zbmath.org/authors/?q=ai:ghosh.sunit"Swain, Jitendriya"https://zbmath.org/authors/?q=ai:swain.jitendriyaSummary: The aim of the article is to show the invalidity of the Strichartz estimate for the free Schrödinger equation associated with the Ornstein-Uhlenbeck operator \(L=-\frac{1}{2} \Delta + \langle x, \nabla \rangle\) in \(\mathbb{R}^n\). As a consequence we obtain a negative answer to the Fourier-Hermite restriction problem associated with the normalized Hermite polynomials for a discrete surface in \(\mathbb{Z} \times \mathbb{N}_0^n\).Properties of the support of solutions of a class of nonlinear evolution equationshttps://zbmath.org/1527.353432024-02-28T19:32:02.718555Z"Bustamante, Eddye"https://zbmath.org/authors/?q=ai:bustamante.eddye"Jiménez Urrea, José"https://zbmath.org/authors/?q=ai:jimenez-urrea.joseSummary: In this work we consider equations of the form
\[
\partial_t u+P\big(\partial_x \big) u+G\big( u,\partial_x u,\ldots,\partial_x^l u\big)=0,
\]
where \(P\) is any polynomial without constant term, and \(G\) is any polynomial without constant or linear terms. We prove that if \(u\) is a sufficiently smooth solution of the equation, such that \(\mathrm{supp}\, u(0),\mathrm{supp}\, u(T)\subset (-\infty,B]\) for some \(B>0\), then there exists \(R_0 >0\) such that \(\mathrm{supp}\, u(t)\subset (-\infty, R_0]\) for every \(t\in [0,T]\). Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation,
\[
\partial_t u+\partial_x^5 u+\partial_x^3 u+u\partial_x u=0,
\]
and for the generalized KdV hierarchy
\[
\partial_t u+ (-1)^{k+1}\partial_x^{2k+1} u+G\big( u,\partial_x u,\ldots, \partial_x^{2k}u\big) =0.
\]
{{\copyright} 2022 Wiley-VCH GmbH.}Integration of the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functionshttps://zbmath.org/1527.353552024-02-28T19:32:02.718555Z"Hoitmetov, Umid Azadovich"https://zbmath.org/authors/?q=ai:khoitmetov.umid-azadovich"Hasanov, Temur Gafurjonovich"https://zbmath.org/authors/?q=ai:hasanov.temur-gafurjonovichSummary: In this paper, we solve the Cauchy problem for the Korteweg-de Vries equation with loaded terms and a self-consistent source in the class of rapidly decreasing functions. To solve this problem, the method of the inverse scattering problem is used. The evolution of the scattering data of the self-adjoint Sturm-Liouville operator, whose coefficient is a solution of the Korteweg-de Vries equation with loaded terms and a self-consistent source, is obtained. Examples are given to illustrate the application of the obtained results.Focusing nonlinear Hartree equation with inverse-square potentialhttps://zbmath.org/1527.353682024-02-28T19:32:02.718555Z"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.10|chen.yu.16|chen.yu.4|chen.yu.17|chen.yu.6|chen.yu.2|chen.yuqun|chen.yu.3|chen.yu.1|chen.yu.8"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jing"Meng, Fanfei"https://zbmath.org/authors/?q=ai:meng.fanfeiSummary: In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space \(H^1(\mathbb{R}^d )\) using the method from [\textit{B. Dodson} and \textit{J. Murphy}, Proc. Am. Math. Soc. 145, No. 11, 4859--4867 (2017; Zbl 1373.35287)]. The main difficulties are due to the fact that the equation is not space-translation invariant and that the nonlinearity is non-local. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when \(a < 0\), using the dispersive estimate established by [\textit{J. Zheng}, J. Math. Phys. 59, No. 11, 111502, 14 p. (2018; Zbl 1408.35178)].Mixed type boundary-transmission problems with Interior cracks of the thermo-piezo-electricity theory without energy dissipationhttps://zbmath.org/1527.354082024-02-28T19:32:02.718555Z"Chkadua, Otar"https://zbmath.org/authors/?q=ai:chkadua.otar"Toloraia, Anika"https://zbmath.org/authors/?q=ai:toloraia.anikaSummary: In the paper, we study mixed type interaction problem of pseudo-oscillations between thermo-elastic and thermo-piezo-elastic bodies with interior cracks. The model under consideration is based on the Green-Haghdi theory of thermo-piezo-electricity without energy dissipation. This theory permits propagation of thermal waves only with a finite speed. Using the potential theory and boundary pseudodifferential equations method, we prove the existence and uniqueness of solutions and analyze their smoothness.Radially symmetric spiral flows of the compressible Euler-Poisson system for semiconductorshttps://zbmath.org/1527.354192024-02-28T19:32:02.718555Z"Chen, Liang"https://zbmath.org/authors/?q=ai:chen.liang.11"Mei, Ming"https://zbmath.org/authors/?q=ai:mei.ming"Zhang, Guojing"https://zbmath.org/authors/?q=ai:zhang.guojingSummary: In this paper, we study the steady flows to the compressible Euler-Poisson system for semiconductors with the nonzero angular velocity in a radially symmetric way in an annulus. The main purpose here is to elucidate the effect of the angular velocity in the structure of the steady flows. We show the well-posedness of all kinds of types of radially symmetric spiral flows including radial subsonic/supersonic/transonic flows, and further give a specific classification of the flow patterns under the assumption of various boundary conditions at the inner and the outer circle. Additionally, different from the purely radial case, the uniqueness of radial subsonic flow can not be obtained due to the nonlocal effect caused by the angular velocity, consequently we prove the uniqueness of the radial subsonic solution in the case without the semiconductor effect or with a small current assumption. Moreover, some new patterns of spiral flows with or without shock are observed, such as a smooth transonic flow and a supersonic-supersonic shock flow for a large relaxation time parameter.Semiconductor full quantum hydrodynamic model with non-flat doping profile. I: Stability of steady statehttps://zbmath.org/1527.354202024-02-28T19:32:02.718555Z"Hu, Haifeng"https://zbmath.org/authors/?q=ai:hu.haifeng"Zhang, Kaijun"https://zbmath.org/authors/?q=ai:zhang.kaijunSummary: This is the first part of our series of studies concerning the full quantum hydrodynamic model for semiconductors with non-flat doping profile. In this paper, we are concerned with the existence, uniqueness and asymptotic stability of subsonic steady states to the model in a bounded interval, which is subject to physical boundary conditions. The main results are proved by Stampacchia's truncation method, the Leray-Schauder Fixed Point Theorem, Schauder's Fixed Point Theorem and intricate energy estimates.On numerical approximations of fractional and nonlocal mean field gameshttps://zbmath.org/1527.354282024-02-28T19:32:02.718555Z"Chowdhury, Indranil"https://zbmath.org/authors/?q=ai:chowdhury.indranil"Ersland, Olav"https://zbmath.org/authors/?q=ai:ersland.olav"Jakobsen, Espen R."https://zbmath.org/authors/?q=ai:jakobsen.espen-robstadSummary: We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.Well-posedness and stability analysis of an epidemic model with infection age and spatial diffusionhttps://zbmath.org/1527.354462024-02-28T19:32:02.718555Z"Walker, Christoph"https://zbmath.org/authors/?q=ai:walker.christophSummary: A compartment epidemic model for infectious disease spreading is investigated, where movement of individuals is governed by spatial diffusion. The model includes infection age of the infected individuals and assumes a logistic growth of the susceptibles. Global well-posedness of the equations within the class of nonnegative smooth solutions is shown. Moreover, spectral properties of the linearization around a steady state are derived. This yields the notion of linear stability which is used to determine stability properties of the disease-free and the endemic steady state.Exponential behaviour of nonlinear fractional Schrödinger evolution equation with complex potential and Poisson jumpshttps://zbmath.org/1527.354672024-02-28T19:32:02.718555Z"Durga, N."https://zbmath.org/authors/?q=ai:durga.nagarajan"Muthukumar, P."https://zbmath.org/authors/?q=ai:muthukumar.palanisamySummary: This paper aims to investigate stochastic fractional Schrödinger evolution equations with potential and Poisson jumps in Hilbert space. The solvability of the proposed system is established by using fractional calculus, semigroup theory, Krasnoselskii's fixed point theorems and stochastic analysis. Furthermore, sufficient conditions are formulated and proved to assure that the mild solution decays exponentially to zero in the square mean. Lastly, an application is given to demonstrate the developed theory.Random differential hyperbolic equations of fractional order in Fréchet spaceshttps://zbmath.org/1527.354702024-02-28T19:32:02.718555Z"Helal, Mohamed"https://zbmath.org/authors/?q=ai:helal.mohamed-atefSummary: In the present paper, we provide some existence results for the Darboux problem of partial fractional random differential equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem and the concept of measure of noncompactness.On nonlinear fractional Choquard equation with indefinite potential and general nonlinearityhttps://zbmath.org/1527.354722024-02-28T19:32:02.718555Z"Liao, Fangfang"https://zbmath.org/authors/?q=ai:liao.fangfang"Chen, Fulai"https://zbmath.org/authors/?q=ai:chen.fulai"Geng, Shifeng"https://zbmath.org/authors/?q=ai:geng.shifeng"Liu, Dong"https://zbmath.org/authors/?q=ai:liu.dong.3|liu.dong.2|liu.dong|liu.dong.1Summary: In this paper, we consider a class of fractional Choquard equations with indefinite potential
\[
(-\Delta)^{\alpha} u + V(x)u = \bigg[\int_{{\mathbb{R}}^N} \frac{M(\epsilon y) G(u)}{|x-y|^{\mu}}\,\mathrm{d}y \bigg] M(\epsilon x) g(u), \quad x \in {\mathbb{R}}^N,
\]
where \(\alpha \in (0, 1)\), \(N > 2 \alpha\), \(0 < \mu < 2\alpha\), \(\epsilon\) is a positive parameter. Here \((-\Delta)^{\alpha}\) stands for the fractional Laplacian, \(V\) is a linear potential with periodicity condition, and \(M\) is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.On BMO and Hardy regularity estimates for a class of non-local elliptic equationshttps://zbmath.org/1527.354742024-02-28T19:32:02.718555Z"Ma, Wenxian"https://zbmath.org/authors/?q=ai:ma.wenxian"Yang, Sibei"https://zbmath.org/authors/?q=ai:yang.sibeiSummary: Let \(\sigma \in (0,\,2), \chi^{(\sigma)}(y):=\mathbf{1}_{\sigma \in (1,2)}+\mathbf{1}_{\sigma =1}\mathbf{1}_{y\in B(\mathbf{0},\,1)}\), where \(\mathbf{0}\) denotes the origin of \(\mathbb{R}^n\), and \(a\) be a non-negative and bounded measurable function on \(\mathbb{R}^n\). In this paper, we obtain the boundedness of the non-local elliptic operator
\[
Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{\mathrm{d}y}{|y|^{n+\sigma}}
\]
from the Sobolev space based on \(\mathrm{BMO}(\mathbb{R}^n) \cap (\bigcup_{p\in (1,\infty)}L^p (\mathbb{R}^n))\) to the space \(\mathrm{BMO} (\mathbb{R}^n)\), and from the Sobolev space based on the Hardy space \(H^1 (\mathbb{R}^n)\) to \(H^1 (\mathbb{R}^n)\). Moreover, for any \(\lambda \in (0,\,\infty)\), we also obtain the unique solvability of the non-local elliptic equation \(Lu-\lambda u=f\) in \(\mathbb{R}^n\), with \(f\in \mathrm{BMO}(\mathbb{R}^n) \cap (\bigcup_{p\in (1,\infty)}L^p (\mathbb{R}^n))\) or \(H^1 (\mathbb{R}^n)\), in the Sobolev space based on \(\mathrm{BMO}(\mathbb{R}^n)\) or \(H^1 (\mathbb{R}^n)\). The boundedness and unique solvability results given in this paper are further devolvement for the corresponding results in the scale of the Lebesgue space \(L^p (\mathbb{R}^n)\) with \(p\in (1,\,\infty)\), established by \textit{H. Dong} and \textit{D. Kim} [J. Funct. Anal. 262, No. 3, 1166--1199 (2012; Zbl 1232.35182)], in the endpoint cases of \(p=1\) and \(p=\infty\).Generalized telegraph equation with fractional \(p(x)\)-Laplacianhttps://zbmath.org/1527.354812024-02-28T19:32:02.718555Z"Vanterler da C. Sousa, José"https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Lamine, Mbarki"https://zbmath.org/authors/?q=ai:lamine.mbarki"Tavares, Leandro S."https://zbmath.org/authors/?q=ai:tavares.leandro-sSummary: The purpose of this paper is devoted to discussing the existence of solutions for a generalized fractional telegraph equation involving a class of \(\psi\)-Hilfer fractional with \(p(x)\)-Laplacian differential equation.Existence of classical solutions for a class of nonlinear impulsive evolution partial differential equationshttps://zbmath.org/1527.354842024-02-28T19:32:02.718555Z"Cherfaoui, Saïda"https://zbmath.org/authors/?q=ai:cherfaoui.saida"Georgiev, Svetlin Georgiev"https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Kheloufi, Arezki"https://zbmath.org/authors/?q=ai:kheloufi.arezki"Mebarki, Karima"https://zbmath.org/authors/?q=ai:mebarki.karimaSummary: This paper is devoted to the study of a class of impulsive nonlinear evolution partial differential equations. We give new results about existence and multiplicity of global classical solutions. The method used is based on the use of fixed points for the sum of two operators. Our main results will be illustrated by an application to an impulsive Burgers equation.Inverse problems for some fractional equations with general nonlinearityhttps://zbmath.org/1527.354982024-02-28T19:32:02.718555Z"Kow, Pu-Zhao"https://zbmath.org/authors/?q=ai:kow.pu-zhao"Wang, Jenn-Nan"https://zbmath.org/authors/?q=ai:wang.jenn-nanSummary: Inspired by some interesting equations modeling anomalous diffusion and nonlinear phenomena, we will study the inverse problems of uniquely identifying coefficients in nonlinear terms from over-determined data. Precisely, we consider a semilinear fractional Schrödinger operator \((-\Delta)^su+Q(x,u)=0\) in \(\Omega\) with \(0<s<1\). The fractional Laplacian arises due to the anomalous diffusion, e.g., the motion of particles described by Lévy flights. Here, we consider the semilinear term \(Q(x,u)=q(x,|u|)u\), which appears naturally in the study of nonlinear optics with cubic Kerr-type nonlinearity, the complex Ginzburg-Landau equation with cubic-quintic nonlinearity, or even the Hartree equation with convolution-type nonlinearity, etc. In this article, we consider the time-independent and the time-evolution semilinear fractional Schrödinger equations with ``Dirichlet'' condition given on the complement of \(\Omega\). We prove both the well-posedness of the forward problems and the unique determination of the inverse problems with measurements taken on the complement of \(\Omega\).Young equations with singularitieshttps://zbmath.org/1527.355102024-02-28T19:32:02.718555Z"Addona, Davide"https://zbmath.org/authors/?q=ai:addona.davide"Lorenzi, Luca"https://zbmath.org/authors/?q=ai:lorenzi.luca"Tessitore, Gianmario"https://zbmath.org/authors/?q=ai:tessitore.gianmarioSummary: In this paper we prove existence and uniqueness of a mild solution to the Young equation \(dy(t) = Ay(t) dt + \sigma (y(t)) dx(t)\), \(t \in [0, T]\), \(y(0) = \psi\). Here, \(A\) is an unbounded operator which generates a semigroup of bounded linear operators \((S(t))_{t \geq 0}\) on a Banach space \(X\), \(x\) is a real-valued \(\eta\)-Hölder continuous. Our aim is to reduce, in comparison to \textit{M. Gubinelli} et al. [Potential Anal. 25, No. 4, 307--326 (2006; Zbl 1103.60062)] and \textit{D. Addona} et al. [J. Evol. Equ. 22, No. 1, Paper No. 3, 34 p. (2022; Zbl 1485.35439)] (see also [\textit{A. Deya} et al., Probab. Theory Relat. Fields 153, No. 1--2, 97--147 (2012; Zbl 1255.60106); \textit{M. Gubinelli} and \textit{S. Tindel}, Ann. Probab. 38, No. 1, 1--75 (2010; Zbl 1193.60070)]), the regularity requirement on the initial datum \(\psi\) eventually dropping it.
The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval \([a, b] \subset \mathbb{R}\) when the \(X_\alpha\)-norm of the function under the integral sign blows up approaching \(a\) and \(X_\alpha\) is an intermediate space between \(X\) and \(D(A)\).On a non-local Sobolev-Galpern-type equation associated with random noisehttps://zbmath.org/1527.355152024-02-28T19:32:02.718555Z"Long Le Dinh"https://zbmath.org/authors/?q=ai:long-le-dinh."Duc Phuong Nguyen"https://zbmath.org/authors/?q=ai:duc-phuong-nguyen."Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: This paper aims to retrieve the initial value for a non-local fractional Sobolev-Galpern problem. The given data are subject to noise by the discrete random model. We show that the solution to the problem is ill-posed in the sense of Hadamard. In this paper, we applied the Fourier truncation method to construct the regularized solution. We estimate the convergence between the solution and the regularized solution. In addition, the numerical example is also proposed to assess the efficiency of the theory.The existence of a unique fixed point of mappings generated by a multidimensional system with relay hysteresishttps://zbmath.org/1527.370232024-02-28T19:32:02.718555Z"Kamachkin, A. M."https://zbmath.org/authors/?q=ai:kamachkin.aleksandr-michailovich"Yevstafyeva, V. V."https://zbmath.org/authors/?q=ai:yevstafyeva.victoria-v"Potapov, D. K."https://zbmath.org/authors/?q=ai:potapov.dmitrii-konstantinovichSummary: A multidimensional system of ordinary differential equations with relay hysteresis is considered. The system parameters are assumed to be such that there exists a family of continuous operators each of which maps some connected compact set into itself. In this case, the operator corresponds to a periodic orbit with an even number of switching points in the phase space of the system. For the operator family, a necessary and sufficient condition for the existence of a single fixed point is obtained.The response solution for a class of nonlinear periodic systems under small perturbationshttps://zbmath.org/1527.370302024-02-28T19:32:02.718555Z"Li, Qi"https://zbmath.org/authors/?q=ai:li.qi.6|li.qi.2|li.qi.1|li.qi"Xu, Junxiang"https://zbmath.org/authors/?q=ai:xu.junxiang|xu.junxiang.1Summary: This paper considers periodic perturbations of a class of nonlinear degenerate systems. By the technique of introducing external parameters, the implicit function theorem and the theory of Brouwer topological degree, it is proved that the perturbed systems have response solutions if the perturbations are sufficiently small.Singular dynamic mode decompositionhttps://zbmath.org/1527.370962024-02-28T19:32:02.718555Z"Rosenfeld, Joel A."https://zbmath.org/authors/?q=ai:rosenfeld.joel-a"Kamalapurkar, Rushikesh"https://zbmath.org/authors/?q=ai:kamalapurkar.rushikeshSummary: This manuscript is aimed at addressing several long-standing limitations of dynamic mode decomposition in the application of Koopman analysis. Principal among these limitations are the convergence of associated dynamic mode decomposition (DMD) algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in the general analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of two DMD algorithms that converge when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.On the Hyers-Ulam solution and stability problem for general set-valued Euler-Lagrange quadratic functional equationshttps://zbmath.org/1527.390142024-02-28T19:32:02.718555Z"Zhang, Dongwen"https://zbmath.org/authors/?q=ai:zhang.dongwen"Rassias, John Michael"https://zbmath.org/authors/?q=ai:rassias.john-michael"Li, Yongjin"https://zbmath.org/authors/?q=ai:li.yongjinSummary: By established a Banach space with the Hausdorff distance, we introduce the alternative fixed-point theorem to explore the existence and uniqueness of a fixed subset of Y and investigate the stability of set-valued Euler-Lagrange functional equations in this space. Some properties of the Hausdorff distance are furthermore explored by a short and simple way.Littlewood-Paley-Stein theory and Banach spaces in the inverse Gaussian settinghttps://zbmath.org/1527.420122024-02-28T19:32:02.718555Z"Almeida, Víctor"https://zbmath.org/authors/?q=ai:almeida.victor-m"Betancor, Jorge J."https://zbmath.org/authors/?q=ai:betancor.jorge-j"Fariña, Juan C."https://zbmath.org/authors/?q=ai:farina.juan-carlos"Rodríguez-Mesa, Lourdes"https://zbmath.org/authors/?q=ai:rodriguez-mesa.lourdesSummary: In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator \(\mathcal{A}=-\frac{1}{2}{\Delta}-x\nabla\) in the inverse Gaussian setting for Banach valued functions. We characterize the uniformly convex and smooth Banach spaces by using \(L^p(\mathbb{R}^n,\gamma_{-1})\)- properties of the \(\mathcal{A} \)-Littlewood-Paley functions. We also use Littlewood-Paley functions associated with \(\mathcal{A}\) to characterize the Köthe function spaces with the UMD property.Extrapolation of compactness for certain pseudodifferential operatorshttps://zbmath.org/1527.420152024-02-28T19:32:02.718555Z"Carro, María J."https://zbmath.org/authors/?q=ai:carro.maria-jesus"Soria, Javier"https://zbmath.org/authors/?q=ai:soria.javier"Torres, Rodolfo H."https://zbmath.org/authors/?q=ai:torres.rodolfo-hSummary: A recently developed extrapolation of compactness on weighted Lebesgue spaces is revisited and a new application to a class of compact pseudodifferential operators is presented.Invertibility of positive Toeplitz operators and associated uncertainty principlehttps://zbmath.org/1527.420482024-02-28T19:32:02.718555Z"Green, A. Walton"https://zbmath.org/authors/?q=ai:green.a-walton"Mitkovski, Mishko"https://zbmath.org/authors/?q=ai:mitkovski.mishkoSummary: We study invertibility and compactness of positive Toeplitz operators associated to a continuous Parseval frame on a Hilbert space. As applications, we characterize compactness of affine and Weyl-Heisenberg localization operators as well as give uncertainty principles for the associated transforms.On wavelet type Bernstein operatorshttps://zbmath.org/1527.420522024-02-28T19:32:02.718555Z"Karsli, H."https://zbmath.org/authors/?q=ai:karsli.harunIt is well-known that the Bernstein operators are of pivotal importance in approximation theory. This paper specifically proposes the wavelet-type Bernstein operators by using the compactly supported Daubechies wavelets of a given function \(f\). The basis used in the process of construction is the wavelet expansion of the function \(f\) in lieu of its rational sampling values \(f(k/n), n\ge 1\) with \(k=0,1,2,\dots, n\). These new operators are more flexible than the usual ones and serve as a natural extension of the classical Bernstein operators, and their Kantorovich and Durrmeyer type modifications. The article also envisages some important properties and convergence results pertaining to the novel operators.
Reviewer: Azhar Y. Tantary (Srinagar)Wavelets on the spectrumhttps://zbmath.org/1527.420532024-02-28T19:32:02.718555Z"Shukla, Niraj K."https://zbmath.org/authors/?q=ai:shukla.niraj-k"Mittal, Shiva"https://zbmath.org/authors/?q=ai:mittal.shivaSummary: \textit{J.-P. Gabardo} and \textit{M. Z. Nashed} [J. Funct. Anal. 158, No. 1, 209--241 (1998; Zbl 0910.42018); Contemp. Math. 216, 41--61 (1998; Zbl 0893.42018)] have introduced a generalized notion of multiresolution analysis, called \textit{nonuniform multiresolution analysis} (NUMRA), based on the theory of spectral pairs (\(\Omega, \Lambda\)) in which the translation set is a spectrum \(\Lambda\) which is not necessarily a group nor a uniform discrete set, given by \(\Lambda=\{0, \frac{r}{N}\}+2\mathbb{Z}\), where \(N\geq 1\) (an integer) and \(r\) is an odd integer with \(1 \leq r \leq 2 N-1\) such that \(r\) and \(N\) are relatively prime and \(\mathbb{Z}\) is the set of integers. In this article, the theory of wavelets on the spectrum is developed. To describe wavelets in the nonuniform discrete setting, first we provide a characterization of an orthonormal basis for \(l^2(\Lambda)\) and then show that the Hilbert space \(l^2(\Lambda)\) can be expressed as an orthogonal decomposition in terms of countable number of its closed subspaces. In addition, we show that the wavelets associated with NUMRA are connected with the wavelets on the spectrum.The Laplace transform on the rearrangement of invariant spaceshttps://zbmath.org/1527.440022024-02-28T19:32:02.718555Z"Castillo, René Erlin"https://zbmath.org/authors/?q=ai:castillo.rene-erlin"Miranda B., A. Ricardo"https://zbmath.org/authors/?q=ai:miranda-b.a-ricardo"Ramos-Fernández, Julio C."https://zbmath.org/authors/?q=ai:ramos-fernandez.julio-cSummary: In a self contained presentation we study the boundedness of the Laplace transform on the rearrangement invariant Lebesgue \(L_p\)-space, we use real analysis techniques provided by convex analysis and the \(K\)-method.A Banach space with an infinite dimensional reflexive quotient algebra \(\mathcal{L}(X) / \mathcal{SS}(X)\)https://zbmath.org/1527.460102024-02-28T19:32:02.718555Z"Pelczar-Barwacz, Anna"https://zbmath.org/authors/?q=ai:pelczar-barwacz.annaThe author constructs a Banach space \(X\) of Gowers-Maurey type such that the algebra \(\mathcal L(X)\) of bounded operators on \(X\) admits a direct sum decomposition \(\mathcal L(X) = \mathcal S(X) \oplus V\), where \(S(X)\) is the closed ideal of the strictly singular operators and \(V\) is an infinite-dimensional reflexive space with an unconditional basis \((I_s)_{s \in \mathbb N}\).
The complicated example is based on a twofold modification of constructions of Gowers-Maurey type [\textit{W. T. Gowers} and \textit{B. Maurey}, Math. Ann. 307, No. 4, 543--568 (1997; Zbl 0876.46006)]. Firstly, \(I_s\) is a projection \(X \to X_s\), where the \(X_s \subset X\) form a sequence of \(1\)-complemented totally incomparable closed subspaces of Gowers-Maurey type for which the linear span of \(\bigcup_{s\in\mathbb N} X_s\) is dense in \(X\). Secondly, by a careful augmentation of the norming set of \(X\), the author ensures that the sequence \((I_s)_{s \in \mathbb N}\) is equivalent to the canonical basis of \(Y^*\) for a certain mixed Tsirelson space \(Y\). The second ingredient prevents \((X_s)\) from being a Schauder decomposition of \(X\), and the verification of the decomposition of \(\mathcal L(X)\) requires very precise use of techniques and estimates for mixed Tsirelson spaces [\textit{S. A. Argyros} and \textit{S. Todorcevic}, Ramsey methods in analysis. Basel: Birkhäuser (2005; Zbl 1092.46002)].
Reviewer: Hans-Olav Tylli (Helsinki)Weakly unbounded norm topology and \textit{wun}-Dunford-Pettis operatorshttps://zbmath.org/1527.460112024-02-28T19:32:02.718555Z"Haghnejad Azar, Kazem"https://zbmath.org/authors/?q=ai:azar.kazem-haghnejad"Matin, Mina"https://zbmath.org/authors/?q=ai:matin.mina"Alavizadeh, Razi"https://zbmath.org/authors/?q=ai:alavizadeh.raziSummary: A functional \(f\) on a Banach lattice \(E\) is \textit{un}-continuous, if \(x_\alpha \xrightarrow{un}0\) implies \(f(x_\alpha)\rightarrow 0\) for each norm bounded net \((x_\alpha)\subseteq E\). We denote the vector space of all \textit{un}-continuous functionals on \(E\) by \({{E}^{\diamond}}\) and we call it \textit{un}-dual of Banach lattice \(E\). In this paper, we study the \textit{un}-dual of Banach lattice \(E\) and compare it with the topological dual; i.e. \(E^*\). For example, we show that if \(E^*\) has order continuous norm, then \(E^* = {{E}^{\diamond}}\). We introduce and study weakly unbounded norm topology (\textit{wun}-topology) on a Banach lattice and compare it with the weak topology and the \textit{uaw}-topology. Finally, we introduce and study the class of \textit{wun}-Dunford-Pettis operators defined from a Banach lattice \(E\) into a Banach space \(X\) and investigate some of its properties and its relationships with some well-known classes of operators.Complex interpolation between two mixed norm Bergman spaces in tube domains over homogeneous coneshttps://zbmath.org/1527.460122024-02-28T19:32:02.718555Z"Gonessa, Jocelyn"https://zbmath.org/authors/?q=ai:gonessa.jocelyn"Mbapte, Rostand Franck"https://zbmath.org/authors/?q=ai:mbapte.rostand-franck"Nana, Cyrille"https://zbmath.org/authors/?q=ai:nana.cyrilleSummary: We use a family of atomic decomposition operators to determine an explicit function realising the complex interpolation between two mixed norm weighted Bergman spaces in tube domains over open convex homogeneous cones. Our results extend and improve earlier work from
[\textit{D.~Békollé} et al., C. R., Math., Acad. Sci. Paris 337, No.~1, 13--18 (2003; Zbl 1033.32004)],
where the problem was considered for scalar powers \(\boldsymbol{\alpha} = (\alpha, \dots, \alpha)\) and symmetric cones \(\Omega\).Characterizations of seminorms satisfying some inequalities with respect to a finite-dimensional subspacehttps://zbmath.org/1527.460142024-02-28T19:32:02.718555Z"Imekraz, Rafik"https://zbmath.org/authors/?q=ai:imekraz.rafikSummary: We prove several characterizations of seminorms which complete (in a specific sense) the distance to a fixed finite-dimensional subspace of a normed space. We also give a few extensions for closed subspaces of a Banach space.Noncommutative Mulholland inequalities associated with factors and their applicationshttps://zbmath.org/1527.460372024-02-28T19:32:02.718555Z"Yang, Yongqiang"https://zbmath.org/authors/?q=ai:yang.yongqiang"Yan, Cheng"https://zbmath.org/authors/?q=ai:yan.cheng"Han, Yazhou"https://zbmath.org/authors/?q=ai:han.yazhou"Liu, Shuting"https://zbmath.org/authors/?q=ai:liu.shutingSummary: In this paper, we obtain noncommutative Mulholland inequalities associated with type I and type \(\mathrm{II}_1\) factors, which are generalizations of noncommutative Minkowski inequalities, and conclude that the noncommutative Mulholland inequalities with non-power functions fail to exist for type \(\mathrm{II}_\infty\) factors. As applications of the noncommutative Mulholland inequalities, we prove that functionals \(\phi^{-1}\circ\tau\circ\phi(|\cdot|)\) are noncommutative \(F\)-norms under same conditions of establishing the noncommutative Mulholland inequalities, and the conclusion also fails to hold for type \(\mathrm{II}_\infty\) factors if \(\phi\) are non-power functions. Furthermore, we prove that the functionals \(\phi^{-1}\circ\tau\circ\phi(|\cdot|)\) associated with type I or type II factors, are norms if and only if \(\phi(t) = \phi (1)t^p\), (\(t \geq 0\)), for some \(p \geq 1\). In addition, we define noncommutative \(F\)-normed spaces by the above noncommutative \(F\)-norms and give a positive answer about the uniform convexity of the noncommutative \(F\)-normed spaces.Ring derivations of Murray-von Neumann algebrashttps://zbmath.org/1527.460442024-02-28T19:32:02.718555Z"Huang, Jinghao"https://zbmath.org/authors/?q=ai:huang.jinghao"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aSummary: Let \(\mathcal{M}\) be a type \(\mathrm{II}_1\) von Neumann algebra, \(S(\mathcal{M})\) be the Murray-von Neumann algebra associated with \(\mathcal{M}\) and let \(\mathcal{A}\) be a \(\ast \)-subalgebra in \(S(\mathcal{M})\) with \(\mathcal{M} \subseteq \mathcal{A} \). We prove that any ring derivation \(D\) from \(\mathcal{A}\) into \(S(\mathcal{M})\) is necessarily inner. Further, we prove that if \(\mathcal{A}\) is an \(E W^\ast \)-algebra such that its bounded part \(\mathcal{A}_b\) is a \(W^\ast \)-algebra without finite type I direct summands, then any ring derivation \(D\) from \(\mathcal{A}\) into \(L S( \mathcal{A}_b)\) -- the algebra of all locally measurable operators affiliated with \(\mathcal{A}_b\), is an inner derivation. We also give an example showing that the condition \(\mathcal{M} \subseteq \mathcal{A}\) is essential. At the end of this paper, we provide several criteria for an abelian extended \(W^\ast \)-algebra such that all ring derivations on it are linear.Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaceshttps://zbmath.org/1527.460492024-02-28T19:32:02.718555Z"Cobzaş, S."https://zbmath.org/authors/?q=ai:cobzas.stefanSummary: We prove versions of Ekeland, Takahashi and Caristi principles in pre-ordered quasi-metric spaces, the equivalence between these principles, as well as their equivalence to some completeness results for the underlying quasi-metric space. These extend the results proved in [\textit{S.~Cobzaş}, Topology Appl. 265, Article ID 106831, 22~p. (2019; Zbl 1423.58009)]
for quasi-metric spaces.
The key tools are Picard sequences for some special set-valued mappings on a pre-ordered quasi-metric space \(X\), defined in terms of the preorder and of a function \(\varphi\) on \(X\).Poisson's equation for \(A\)-mean ergodic operatorshttps://zbmath.org/1527.470012024-02-28T19:32:02.718555Z"Oğuz, Gencay"https://zbmath.org/authors/?q=ai:oguz.gencay"Orhan, Cihan"https://zbmath.org/authors/?q=ai:orhan.cihanSummary: Let \(X\) be a Banach space and \(T\in B(X)\). For (weakly) mean ergodic operator \(T\), Poisson's equation \(y=(I-T)x\) can be solved for a given \(y\in X\) if and only if \(x_n:=\dfrac{1}{n}\sum_{k=1}^n \sum_{j=0}^{k-1}T^jy\) (weakly) converges. In the present paper, replacing Cesàro means by a general nonnegative regular matrix \(A=(a_{nk})\), we study weak convergence of \(x_n:=\sum \nolimits_{j=1}^\infty a_{nj} \sum \nolimits_{k=1}^jT^{k-1}y\) in order to solve the equation \(y=(I-T)x\). Then we show that \(A\)-mean ergodicity and weak \(A\)-mean ergodicity are equivalent for a power-bounded operator \(T\).Automorphisms of effect algebras with respect to convex sequential producthttps://zbmath.org/1527.470022024-02-28T19:32:02.718555Z"Zhang, Jinhua"https://zbmath.org/authors/?q=ai:zhang.jinhua"Ji, Guoxing"https://zbmath.org/authors/?q=ai:ji.guoxingSummary: Let \(\mathcal{H}\) be a complex Hilbert space with dim \(\mathcal{H}\geq 3\) and \(\mathcal{B}(\mathcal{H})\) the algebra of all bounded linear operators on \(\mathcal{H}\). The effect algebra \(E(\mathcal{H})\) on \(\mathcal{H}\) is the set of all positive contractions in \(\mathcal{B}(\mathcal{H})\). We consider the automorphism of \(E(\mathcal{H})\) with respect to convex sequential product ox on \(E(\mathcal{H})\) for some \(\lambda\in[0,1]\) defined by \(A\circ_\lambda B=\lambda A^{\frac{1}{2}}BA^{\frac{1}{2}}+(1-\lambda)B^{\frac{1}{2}}AB^{\frac{1}{2}}\), \(\forall A,B\in E(\mathcal{H})\). We show that an automorphism of \(E(\mathcal{H})\) with respect to convex sequential product is just reduced by a unitary or an anti-unitary operator \(U\) on \(\mathcal{H}\).A note on the fixed point theorem of \(F\)-contraction mappings in rectangular \(M\)-metric spacehttps://zbmath.org/1527.470032024-02-28T19:32:02.718555Z"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Anjum, Rizwan"https://zbmath.org/authors/?q=ai:anjum.rizwan"Anwar, Rabia"https://zbmath.org/authors/?q=ai:anwar.rabiaSummary: In this note, we show that the main result (Theorem 3.2) due to \textit{M. Asim} et al. [Appl. Gen. Topol. 23, No. 2, 363--376 (2022; Zbl 1498.54056)] is still valid if we remove the assumption of continuity of the mapping.Fixed point theorems of enriched multivalued mappings via sequentially equivalent Hausdorff metrichttps://zbmath.org/1527.470042024-02-28T19:32:02.718555Z"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Anjum, Rizwan"https://zbmath.org/authors/?q=ai:anjum.rizwan"Tahir, Muhammad Haris"https://zbmath.org/authors/?q=ai:tahir.muhammad-haris(no abstract)Solvability of some fractional differential equations in the Hölder space \(\mathcal{H}_{\gamma}(\mathbb{R_+})\) and their numerical treatment via measures of noncompactnesshttps://zbmath.org/1527.470052024-02-28T19:32:02.718555Z"Amiri Kayvanloo, Hojjatollah"https://zbmath.org/authors/?q=ai:kayvanloo.hojjatollah-amiri"Mursaleen, Mohammad"https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Mehrabinezhad, Mohammad"https://zbmath.org/authors/?q=ai:mehrabinezhad.mohammad"Pouladi Najafabadi, Farzaneh"https://zbmath.org/authors/?q=ai:najafabadi.farzaneh-pouladiSummary: We study the following fractional boundary value problem:
\[
\begin{cases}
D^{\alpha}\upsilon (t)+f(t,\upsilon (t))=0,\quad \alpha \in (1,2],\quad 0<t<+\infty, \\
\upsilon (0)=0,\quad D^{\alpha -1}\upsilon (\infty)=\lambda \int_0^{\tau} \upsilon (t)\mathrm{d}t.
\end{cases}
\]
The goal of this paper is to bring forward a new family of measures of noncompactness and prove a fixed point theorem of Darbo type in the Hölder space \(\mathcal{H}_{\gamma}(\mathbb{R_+})\). Moreover, we provide an example which supports our main result and in carrying out an proximate solution for the mentioned example with high precision we apply several numerical methods.A fixed point theorem for \(p\)-contraction mappings in partially ordered metric spaces and application to ordinary differential equationshttps://zbmath.org/1527.470062024-02-28T19:32:02.718555Z"Aouine, Ahmed Chaouki"https://zbmath.org/authors/?q=ai:aouine.ahmed-chaoukiSummary: In this paper, we prove a fixed point theorem for \(p\)-contraction mappings in partially ordered metric spaces. As an application, we investigate the possibility of optimally controlling the solution of the ordinary differential equations.Banach, Kannan, Chatterjea, and Reich-type contractive inequalities for multivalued mappings and their common fixed pointshttps://zbmath.org/1527.470072024-02-28T19:32:02.718555Z"Debnath, Pradip"https://zbmath.org/authors/?q=ai:debnath.pradip(no abstract)On existence and uniqueness of a solution of an integral equation using contractive mappinghttps://zbmath.org/1527.470082024-02-28T19:32:02.718555Z"Dhariwal, Rishi"https://zbmath.org/authors/?q=ai:dhariwal.rishi"Kumar, Deepak"https://zbmath.org/authors/?q=ai:kumar.deepakSummary: In the present manuscript, we introduced \(\alpha-\psi\) type contractive mapping in \(C^\ast\)-algebra valued partial metric space for unital \(C^\ast\)-algebra \(\mathbb{A}\), and proved some fixed point theorems. To discuss the usability of the proved results, we established the existence and uniqueness of a solution of an integral equations.A note on the split common fixed point equality problems in Hilbert spaceshttps://zbmath.org/1527.470092024-02-28T19:32:02.718555Z"Kılıçman, A."https://zbmath.org/authors/?q=ai:kilicman.adem"Mohammed, L. B."https://zbmath.org/authors/?q=ai:mohammed.lawan-bulamaSummary: We study the split common fixed point equality problems (SCFPEP). Furthermore, we formulate and analyse the algorithms for solving this SCFPEP for the finite family of quasi-nonexpansive operators in Hilbert spaces and shows how it unifies and generalizes previously discussed problems. In the end, we give numerical example that illustrates our theoretical results.Ćirić-generalized contraction via \(wt\)-distancehttps://zbmath.org/1527.470102024-02-28T19:32:02.718555Z"Lakzian, Hosein"https://zbmath.org/authors/?q=ai:lakzian.hosein"Kocev, Darko"https://zbmath.org/authors/?q=ai:kocev.darko"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In this present paper, besides other things, we introduce the concept of Ćirić-generalized contractions via \(wt\)-distance and then we will prove some new fixed point results for these mappings, which generalize and improve fixed point theorems by \textit{L. B. Ćirić} in [Proc. Am. Math. Soc. 45, 267--273 (1974; Zbl 0291.54056); Publ. Inst. Math., Nouv. Sér. 12(26), 19--26 (1971; Zbl 0234.54029); Publ. Inst. Math., Nouv. Sér. 26(40), 79--82 (1979; Zbl 0448.54047)] and also, \textit{B. E. Rhoades} in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 47, No. 4, 2683--2693 (2001; Zbl 1042.47521)]. Some examples illustrate usefulness of the new results. At the end, we will give some applications to nonlinear fractional differential equations.Mann-Dotson's algorithm for a countable family of non-self Lipschitz mappings in hyperbolic metric spacehttps://zbmath.org/1527.470112024-02-28T19:32:02.718555Z"Patel, Prashant"https://zbmath.org/authors/?q=ai:patel.prashant"Shukla, Rahul"https://zbmath.org/authors/?q=ai:shukla.rahul(no abstract)Fixed points of set-valued mappings in Menger probabilistic metric spaces endowed with an amorphous binary relationhttps://zbmath.org/1527.470122024-02-28T19:32:02.718555Z"Prasad, Gopi"https://zbmath.org/authors/?q=ai:prasad.gopi"Deshwal, Sheetal"https://zbmath.org/authors/?q=ai:deshwal.sheetal"Srivastav, Rupesh K."https://zbmath.org/authors/?q=ai:srivastav.rupesh-kSummary: In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadžić-type \(t\)-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by \textit{N. Deo} et al. [Appl. Math. Comput. 201, No. 1--2, 604--612 (2008; Zbl 1173.41011)] on the space \(C([0, \frac{n}{n+1}])\). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature.Some results for cyclic weak contractions in modular metric spacehttps://zbmath.org/1527.470132024-02-28T19:32:02.718555Z"Rahimpoor, H."https://zbmath.org/authors/?q=ai:rahimpoor.hossein"Nikoufar, I."https://zbmath.org/authors/?q=ai:nikoufar.ismailSummary: In this paper, we present some fixed point results for cyclic weak \(\phi\)-contractions in \(\omega\)-complete modular metric spaces and \(\omega\)-compact modular metric spaces, respectively. Some results for contractions that have zero cyclic properties are also provided.Solvability of Infinite systems of differential equations of general order in the sequence space \(bv_\infty\)https://zbmath.org/1527.470142024-02-28T19:32:02.718555Z"Saboori, M. H."https://zbmath.org/authors/?q=ai:saboori.mohammad-hassan"Hassani, M."https://zbmath.org/authors/?q=ai:hassani.mahmoud"Allahyari, R."https://zbmath.org/authors/?q=ai:allahyari.rezaSummary: We introduce the Hausdorff measure of noncompactness in the sequence space \(bv_\infty\) and investigate the existence of solution of infinite systems of differential equations with respect to Hausdorff measure of noncompactness. Finally, we present an example to defend of theorem of existential.On \(\varphi\)-contractions and fixed point results in fuzzy metric spaceshttps://zbmath.org/1527.470152024-02-28T19:32:02.718555Z"Saheli, Morteza"https://zbmath.org/authors/?q=ai:saheli.morteza"Mohsenialhosseini, Seyed Ali Mohammad"https://zbmath.org/authors/?q=ai:mohsenialhosseini.seyed-ali-mohammad"Goraghani, Hadi Saeidi"https://zbmath.org/authors/?q=ai:goraghani.hadi-saeidiSummary: In this paper, \(\varphi\)-contractions are defined and then, some new fixed point theorems are established for certain nonlinear mappings associated with one-dimensional \((c)\)-comparison functions in fuzzy metric spaces. Next, generalized \(\varphi\)-contractions are defined by using five-dimensional \((c)\)-comparison functions, and the existence of fixed points for nonlinear maps on fuzzy metric spaces is studied. Moreover, some examples are given to illustrate our results.The analysis of some special solutions of Mackey-Glass equation with variable delayshttps://zbmath.org/1527.470162024-02-28T19:32:02.718555Z"Yazgan, Ramazan"https://zbmath.org/authors/?q=ai:yazgan.ramazan(no abstract)A fixed point approach for tuning circuit problem in dislocated \(b\)-metric spaceshttps://zbmath.org/1527.470172024-02-28T19:32:02.718555Z"Younis, Mudasir"https://zbmath.org/authors/?q=ai:younis.mudasir"Singh, Deepak"https://zbmath.org/authors/?q=ai:singh.deepak-kumar"Abdou, Afrah A. N."https://zbmath.org/authors/?q=ai:abdou.afrah-ahmad-noan(no abstract)Adaptive hybrid steepest descent algorithms involving an inertial extrapolation term for split monotone variational inclusion problemshttps://zbmath.org/1527.470182024-02-28T19:32:02.718555Z"Zhou, Zheng"https://zbmath.org/authors/?q=ai:zhou.zheng"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Li, Songxiao"https://zbmath.org/authors/?q=ai:li.songxiao(no abstract)Corrigendum to: ``On Lau's conjecture''https://zbmath.org/1527.470192024-02-28T19:32:02.718555Z"Salame, Khadime"https://zbmath.org/authors/?q=ai:salame.khadimeCorrigendum to the author's paper [ibid. 148, No. 1, 343--350 (2020; Zbl 1506.47090)].Majorization of quadratic stochastic operatorshttps://zbmath.org/1527.470202024-02-28T19:32:02.718555Z"Masharipov, Sirojiddin I."https://zbmath.org/authors/?q=ai:masharipov.sirojiddin-iSummary: The main goal of this article is to show for which values of \(P_{ij,k}V(x)\) is a majorize with \(x\). In this article the necessary concepts are given, and the application to majorization is presented.A relaxed projection method using a new linesearch for the split feasibility problemhttps://zbmath.org/1527.470212024-02-28T19:32:02.718555Z"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthep"Kesornprom, Suparat"https://zbmath.org/authors/?q=ai:kesornprom.suparat"Pholasa, Nattawut"https://zbmath.org/authors/?q=ai:pholasa.nattawut"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-je"Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasit(no abstract)An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order \(1 < r < 2\) using sectorial operatorshttps://zbmath.org/1527.490082024-02-28T19:32:02.718555Z"Raja, Marimuthu Mohan"https://zbmath.org/authors/?q=ai:raja.marimuthu-mohan"Vijayakumar, Velusamy"https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Panda, Sumati Kumari"https://zbmath.org/authors/?q=ai:panda.sumati-kumari"Shukla, Anurag"https://zbmath.org/authors/?q=ai:shukla.anurag"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type \((P, \eta, r, \gamma)\), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired.Exit from singularity. New optimization methods and the \(p\)-regularity theory applicationshttps://zbmath.org/1527.490142024-02-28T19:32:02.718555Z"Evtushenko, Yuri"https://zbmath.org/authors/?q=ai:evtushenko.yuri-g"Malkova, Vlasta"https://zbmath.org/authors/?q=ai:malkova.vlasta"Tret'yakov, Alexey"https://zbmath.org/authors/?q=ai:tretyakov.alexey-aSummary: In the paper, we introduce a new nonsingular operator instead of a degenerate operator of the first derivative in a singular case for solving and describing nonregular optimization problems and some problems in calculus. Such operator is called \(p\)-factor-operator and its construction is based on the derivatives up to order \(p\) as well as on some element \(h\), which we call the ``exit from singularity''. The special variant of the method of the Modified Lagrangian Functions for optimization problems with inequality constraints is justified on the basis of the 2-factor transformation and constructions of \(p\)-regularity theory. These results are used in some classical branches of calculus: implicit function theorem is given for the singular case and is shown the existence of solutions to a boundary-valued problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the \(p\)-factor method for solving ODEs with a small parameter and new formula is obtained for the solutions of such type equations.
For the entire collection see [Zbl 1508.90001].Boundedness and unboundedness in total variation regularizationhttps://zbmath.org/1527.490362024-02-28T19:32:02.718555Z"Bredies, Kristian"https://zbmath.org/authors/?q=ai:bredies.kristian"Iglesias, José A."https://zbmath.org/authors/?q=ai:iglesias.jose-antonio"Mercier, Gwenael"https://zbmath.org/authors/?q=ai:mercier.gwenaelIn mathematics, the total variation identifies several slightly different concepts, related to the local or global structure of the codomain of a function or a measure. The total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the image gradient magnitude is high. The total variation regularization have had an important role in several classical problems of the geometric measure theory, signal processing and physics.
Several authors studied the total variation regularization [\textit{V. Caselles} et al., Rev. Mat. Iberoam. 27, No. 1, 233--252 (2011; Zbl 1228.94005); \textit{K. Bredies} et al., SIAM J. Imaging Sci. 3, No. 3, 492--526 (2010; Zbl 1195.49025); \textit{A. Chambolle} and \textit{P-L. Lions}, Numer. Math. 76, No. 2, 167--188 (1997; Zbl 0874.68299); \textit{E. Gonzalez} and \textit{U. Massari} Rend. Sem. Mat. Univ. Politec. Torino 52, No. 1, 1--28 (1994; Zbl 0819.49025); \textit{M. Grasmair} and \textit{A. Obereder} Numer. Funct. Anal. Optim. 29, No. 3--4, 346--361 (2008; Zbl 1142.65013); \textit{M. Grasmair}, J. Math. Imaging Vision 27, No. 1, 59--66 (2007; Zbl 1478.94041); \textit{V. Gutev}, J. Math. Anal. Appl. 491, No. 1, Article ID 124242, 12 p. (2020; Zbl 1519.54006); \textit{C. Kirisits} et al., SIAM J. Imaging Sci. 12, No. 4, 1643--1668 (2019; Zbl 1439.49066); \textit{T. Valkonen}, Inverse Probl. 37, No. 4, Article ID 045010, 30 p. (2021; Zbl 1515.65147)].
The principal objective in this paper is to study boundedness and unboundedness in total variation regularization. The authors present a simple proof of boundedness of the minimizer for fixed regularization parameter, and obtain a boundedness result for the case of infimal convolution of first and second order total variation regularizers, for which the optimality conditions are closely related to subgradients of total variation.
Reviewer: Lakehal Belarbi (Mostaganem)The Weitzenböck formula for the divgrad operatorhttps://zbmath.org/1527.530122024-02-28T19:32:02.718555Z"Kimaczyńska, Anna"https://zbmath.org/authors/?q=ai:kimaczynska.annaSummary: The paper concerns symmetric tensors and consists of interesting formulas having also consequences in geometry. First, the differential operators of gradient and of divergence in the bundles of symmetric tensors and symmetric tensors with values in the tangent bundle, respectively, are investigated, leading to a new, simple proof of the Weitzenböck formula. In the second part of this paper the differential operators and symmetric forms on \(\mathbb{R}^n\) are investigated as an application of previous considerations.On a new generalized Tsallis relative operator entropyhttps://zbmath.org/1527.540052024-02-28T19:32:02.718555Z"Tarik, Lahcen"https://zbmath.org/authors/?q=ai:tarik.lahcen"Chergui, Mohamed"https://zbmath.org/authors/?q=ai:chergui.mohamed-el-amine"El Wahbi, Bouazza"https://zbmath.org/authors/?q=ai:el-wahbi.bouazzaSummary: In this paper, we present a generalization of Tsallis relative operator entropy defined for positive operators and we investigate some related properties. Some inequalities involving the generalized Tsallis relative operator entropy are pointed out as well.Some fixed point theorems in the complex valued metric-like spaceshttps://zbmath.org/1527.540452024-02-28T19:32:02.718555Z"Hosseini, Amin"https://zbmath.org/authors/?q=ai:hosseini.aminSummary: The main purpose of this article is to study fixed point theorem in the complex valued metric-like spaces. In this article, we prove that if \((\mathfrak{X},\Delta)\) is a complete complex valued metric-like space and the mappings \(U,V:\mathfrak{X}\longrightarrow\mathfrak{X}\) satisfy \[\Delta(Ux,Vy)\precsim\alpha\Delta(x,y)+\frac{\beta\Delta\left(x,Ux\right)\Delta(y,Vy)}{1+\Delta(x,y)}\]
for all \(x,y\in\mathfrak{X} \), where \(\alpha, \beta\) are non-negative real numbers and \(\alpha+\beta<1\), then \(U\) and \(V\) have a unique common fixed point. Another main result of this article reads as follows. Let \((\mathfrak{X},\Delta)\) be a complete complex valued metric-like space and let \(V:\mathfrak{X}\longrightarrow\mathfrak{X}\) be a map such that \(\Delta(Vx,Vy)\precsim\Delta(x,y)-\varphi(\Delta(x,y))\) for all \(x,y\in\mathfrak{X} \), where \(\varphi:\mathbb{C}\succsim 0\longrightarrow\mathbb{C}\succsim 0\) is a non-decreasing continuous function such that \(\varphi(z)=0\) if and only if \(z=0\). Then, \(V\) has a unique fixed point. Some other related results are also discussed.A new generalization of metric spaces satisfying the \(T_2\)-separation axiom and some related fixed point resultshttps://zbmath.org/1527.540712024-02-28T19:32:02.718555Z"Touail, Y."https://zbmath.org/authors/?q=ai:touail.youssefSummary: In this paper, without using neither the compactness nor the uniform convexity, some fixed point theorems are proved by using a binary relation in the setting of a new class of spaces called \(T\)-partial metric spaces. This class of spaces can be considered the first generalization of metric spaces such that the generated topology is a Hausdorff topology. Our theorems generalize and improve very recent fixed point results in the literature. Finally, we show the existence of a solution for a class of differential equations under new weak conditions.Solution of fractional integral equations via fixed point resultshttps://zbmath.org/1527.540772024-02-28T19:32:02.718555Z"Zhou, Mi"https://zbmath.org/authors/?q=ai:zhou.mi"Saleem, Naeem"https://zbmath.org/authors/?q=ai:saleem.naeem"Bashir, Shahid"https://zbmath.org/authors/?q=ai:bashir.shahid(no abstract)Monotone ODEs with discontinuous vector fields in sequence spaceshttps://zbmath.org/1527.580052024-02-28T19:32:02.718555Z"Zubelevich, Oleg"https://zbmath.org/authors/?q=ai:zubelevich.oleg-eduardovichAn ODE system in the Fréchet space is considered with unconditional Schauder basis. The right hand of the concerned ODE is involved in a discontinuous function. An existence theorem for IVP is shown under some monotonic conditions. The main technique consists in the idea of the partial order.
Reviewer: Yong-Kui Chang (Xi'an)The Mellin-edge quantisation for corner operatorshttps://zbmath.org/1527.580082024-02-28T19:32:02.718555Z"Schulze, B.-W."https://zbmath.org/authors/?q=ai:schulze.bert-wolfgang"Wei, Yawei"https://zbmath.org/authors/?q=ai:wei.yaweiSummary: We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold \(M\) with second order singularities. The typical ingredients come from the ``most singular'' stratum of \(M\) which is a second order edge where the infinite transversal cone has a base \(B\) that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over \(B\). In this respect, our result is formally analogous to a quantisation rule of \textit{J. B. Gil} et al. [Osaka J. Math. 37, No. 1, 221--260 (2000; Zbl 1005.58010)] for the simpler case of edge-degenerate symbols that corresponds to the singularity order~1. However, from the singularity order~2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over \(B\) that tend to infinity.Feynman-Kac formula for perturbations of order \(\leq 1\), and noncommutative geometryhttps://zbmath.org/1527.580092024-02-28T19:32:02.718555Z"Boldt, Sebastian"https://zbmath.org/authors/?q=ai:boldt.sebastian"Güneysu, Batu"https://zbmath.org/authors/?q=ai:guneysu.batuSummary: Let \(Q\) be a differential operator of order \(\leq 1\) on a complex metric vector bundle \(\mathscr{E}\rightarrow\mathscr{M}\) with metric connection \(\nabla\) over a possibly noncompact Riemannian manifold \(\mathscr{M}\). Under very mild regularity assumptions on \(Q\) that guarantee that \(\nabla^\dagger\nabla/2+Q\) canonically induces a holomorphic semigroup \(\mathrm{e}^{-zH^\nabla_Q}\) in \(\Gamma_{L^2}(\mathscr{M}, \mathscr{E})\) (where \(z\) runs through a complex sector which contains \([0,\infty)\)), we prove an explicit Feynman-Kac type formula for \(\mathrm{e}^{-tH^\nabla_Q}\), \(t > 0\), generalizing the standard self-adjoint theory where \(Q\) is a self-adjoint zeroth order operator. For compact \(\mathscr{M}\)'s we combine this formula with Berezin integration to derive a Feynman-Kac type formula for an operator trace of the form
\[
\operatorname{Tr}\left(\widetilde{V}\int^t_0\mathrm{e}^{-sH^\nabla_V}P\mathrm{e}^{-(t-s)H^\nabla_V}\mathrm{d}s\right),
\]
where \(V\), \(\widetilde{V}\) are of zeroth order and \(P\) is of order \(\leq 1\). These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat-Heckmann localization formula on the loop space of such a manifold.The cutoff phenomenon for the stochastic heat and wave equation subject to small Lévy noisehttps://zbmath.org/1527.600432024-02-28T19:32:02.718555Z"Barrera, Gerardo"https://zbmath.org/authors/?q=ai:barrera.gerardo"Högele, Michael A."https://zbmath.org/authors/?q=ai:hoegele.michael"Pardo, Juan Carlos"https://zbmath.org/authors/?q=ai:pardo.juan-carlosThis paper establishes the small noise cutoff profile to mild solutions of the stochastic heat and stochastic wave equations driven by additive and multiplicative Wiener and Lévy noises, respectively. The methods rely on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the multiplicative stochastic solution flows in terms of stochastic exponentials. This interesting work generalizes the small noise cutoff phenomenon to a class of stochastic partial differential equations.
Reviewer: Guanggan Chen (Chengdu)Extended convergence for a fifth-order novel scheme free from derivativeshttps://zbmath.org/1527.650382024-02-28T19:32:02.718555Z"Behl, Ramandeep"https://zbmath.org/authors/?q=ai:behl.ramandeep"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"Martínez, Eulalia"https://zbmath.org/authors/?q=ai:martinez.eulalia"Joshi, Janak"https://zbmath.org/authors/?q=ai:joshi.janak(no abstract)New improved convergence analysis for the secant methodhttps://zbmath.org/1527.650392024-02-28T19:32:02.718555Z"Magreñán, Á. Alberto"https://zbmath.org/authors/?q=ai:magrenan.angel-alberto"Argyros, Ioannis K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinosSummary: We present a new convergence analysis, for the secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies. Numerical examples validating the theoretical results are also provided in this study.Total least squares problems on infinite dimensional spaceshttps://zbmath.org/1527.650402024-02-28T19:32:02.718555Z"Contino, Maximiliano"https://zbmath.org/authors/?q=ai:contino.maximiliano"Fongi, Guillermina"https://zbmath.org/authors/?q=ai:fongi.guillermina"Maestripieri, Alejandra"https://zbmath.org/authors/?q=ai:maestripieri.alejandra-l"Muro, Santiago"https://zbmath.org/authors/?q=ai:muro.santiagoSummary: We study weighted total least squares problems on infinite dimensional spaces. We present some necessary and sufficient conditions for the regularized problem to have a solution. The existence of solution can also be assured for the regularized minimization problem with a constraint to special subsets. Furthermore, we show that a regularization in infinite dimensional total least squares problems is necessary, since in most cases the problem without regularization does not admit a solution.On the convergence rate of Fletcher-Reeves nonlinear conjugate gradient methods satisfying strong Wolfe conditions: application to parameter identification in problems governed by general dynamicshttps://zbmath.org/1527.650482024-02-28T19:32:02.718555Z"Riahi, Mohamed Kamel"https://zbmath.org/authors/?q=ai:riahi.mohamed-kamel"Qattan, Issam A."https://zbmath.org/authors/?q=ai:qattan.issam-a(no abstract)Iterative approximation of a common solution of split equilibrium, split variational inequality, and fixed point problem for a nonexpansive semigrouphttps://zbmath.org/1527.650502024-02-28T19:32:02.718555Z"Rizvi, Shuja H."https://zbmath.org/authors/?q=ai:rizvi.shuja-haider"Sikander, Fahad"https://zbmath.org/authors/?q=ai:sikander.fahad(no abstract)An effective approach for numerical solution of linear and nonlinear singular boundary value problemshttps://zbmath.org/1527.650592024-02-28T19:32:02.718555Z"Saldır, Onur"https://zbmath.org/authors/?q=ai:saldir.onur"Giyas Sakar, Mehmet"https://zbmath.org/authors/?q=ai:sakar.mehmet-giyas(no abstract)Computing eigenvalues of the Laplacian on rough domainshttps://zbmath.org/1527.651232024-02-28T19:32:02.718555Z"Rösler, Frank"https://zbmath.org/authors/?q=ai:rosler.frank"Stepanenko, Alexei"https://zbmath.org/authors/?q=ai:stepanenko.alexeiSummary: We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A~key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.A Banach spaces-based mixed finite element method for the stationary convective Brinkman-Forchheimer problemhttps://zbmath.org/1527.651252024-02-28T19:32:02.718555Z"Caucao, Sergio"https://zbmath.org/authors/?q=ai:caucao.sergio"Gatica, Gabriel N."https://zbmath.org/authors/?q=ai:gatica.gabriel-n"Gatica, Luis F."https://zbmath.org/authors/?q=ai:gatica.luis-fSummary: We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman-Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach-Nečas-Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart-Thomas elements of order \(k \geq 0\) for the pseudostress tensor and discontinuous piecewise polynomial elements of degree \(\leq k\) for the velocity. Stability, convergence, and optimal \textit{a priori} error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.Linear orthosets and orthogeometrieshttps://zbmath.org/1527.810012024-02-28T19:32:02.718555Z"Paseka, Jan"https://zbmath.org/authors/?q=ai:paseka.jan"Vetterlein, Thomas"https://zbmath.org/authors/?q=ai:vetterlein.thomasSummary: Anisotropic Hermitian spaces can be characterised as anisotropic orthogeometries, that is, as projective spaces that are additionally endowed with a suitable orthogonality relation. But linear dependence is uniquely determined by the orthogonality relation and hence it makes sense to investigate solely the latter. It turns out that by means of orthosets, which are structures based on a symmetric, irreflexive binary relation, we can achieve a quite compact description of the inner-product spaces under consideration. In particular, Pasch's axiom, or any of its variants, is no longer needed. Having established the correspondence between anisotropic Hermitian spaces on the one hand and so-called linear orthosets on the other hand, we moreover consider the respective symmetries. We present a version of Wigner's Theorem adapted to the present context.Bell bi-inequalities for Bell local correlation tensorshttps://zbmath.org/1527.810022024-02-28T19:32:02.718555Z"Zhu, Wen-Qian"https://zbmath.org/authors/?q=ai:zhu.wen-qian"Hu, Di"https://zbmath.org/authors/?q=ai:hu.di"Guo, Zhi-Hua"https://zbmath.org/authors/?q=ai:guo.zhihua"Cao, Huai-Xin"https://zbmath.org/authors/?q=ai:cao.huaixinSummary: When an \(n\)-partite physical system is measured by \(n\) observers, the joint probabilities of outcomes conditioned on the observables chosen by the \(n\)-parties form a nonnegative tensor \textbf{P}, called an \(n\)-partite correlation tensor (CT). According to the special relativity, CTs can be classified as signaling and nonsignaling ones; and from the point of view of hidden variable theory, CTs can be divided into Bell local and Bell nonlocal ones. In this paper, we aim to establish some Bell bi-inequalities for \(n\)-partite Bell local correlation tensors. A Bell bi-inequality consists of two Bell inequalities that hold only for Bell local CTs. First, we obtain an inequality for \(n\)-partite nonsignaling CTs, which can be used to check the nonsignaling property of a CT. Second, we recall the mathematical definition of Bell locality of CTs and prove global properties of the set of all Bell local CTs over an index set \(\Delta_n\). Then we establish a series tight Bell bi-inequalities and prove that a CT \textbf{P} is Bell local if and only if it satisfies all tight Bell bi-inequalities. Lastly, we list some examples to illustrate how to establish a Bell bi-inequality.Influence of the adiabatic process on a quantum system demonstrated using the energy conservation principlehttps://zbmath.org/1527.810112024-02-28T19:32:02.718555Z"Zhou, Yun-Song"https://zbmath.org/authors/?q=ai:zhou.yunsong"Zhao, Li-Ming"https://zbmath.org/authors/?q=ai:zhao.liming"Wei, Gong-Min"https://zbmath.org/authors/?q=ai:wei.gong-min(no abstract)Global quantum discord and entanglement in two coupled double quantum dots AlGaAs/GaAshttps://zbmath.org/1527.810212024-02-28T19:32:02.718555Z"Ait Mansour, Hicham"https://zbmath.org/authors/?q=ai:mansour.hicham-ait"Faqir, Mustapha"https://zbmath.org/authors/?q=ai:faqir.mustapha"El Baz, Morad"https://zbmath.org/authors/?q=ai:baz.morad-elSummary: This work presents a theoretical study on the behavior of global quantum discord and entanglement in two coupled double quantum dots made of AlGaAs/GaAs as a function of temperature. We use each double quantum dot as a qubit, where the electron can occupy either the right or left dot. The goal of our investigation is to understand the impact of the energy offset of each qubit and the tunneling coupling energy on quantum correlations. Our findings show that the energy offset and tunneling coupling energy significantly affect the variations of entanglement of formation, standard discord, and global quantum discord. Our results provide insights into the interplay between quantum correlations and environmental parameters and have important implications for the development of quantum technologies.Quantum-dynamical semigroups and the church of the larger Hilbert spacehttps://zbmath.org/1527.810252024-02-28T19:32:02.718555Z"vom Ende, Frederik"https://zbmath.org/authors/?q=ai:vom-ende.frederikSummary: In this work we investigate Stinespring dilations of quantum-dynamical semigroups, which are known to exist by means of a constructive proof given by Davies in the early 70s. We show that if the semigroup describes an open system, that is, if it does not consist of only unitary channels, then the evolution of the dilated closed system has to be generated by an unbounded Hamiltonian; subsequently the environment has to correspond to an infinite-dimensional Hilbert space, regardless of the original system. Moreover, we prove that the second derivative of Stinespring dilations with a bounded total Hamiltonian yields the dissipative part of some quantum-dynamical semigroup -- and vice versa. In particular this characterizes the generators of quantum-dynamical semigroups via Stinespring dilations.Quaternionic quantum automatahttps://zbmath.org/1527.810312024-02-28T19:32:02.718555Z"Dai, Songsong"https://zbmath.org/authors/?q=ai:dai.songsongSummary: Quaternionic quantum theory is a generalization of the standard complex quantum theory. Inspired by this, we study the quaternionic quantum computation using quaternions. We first develop a theory of quaternionic quantum automata as a model of quaternionic quantum computation. Quaternionic quantum automata also can be seen as an extension of complex quantum automata. Then we introduce some operations of quaternionic quantum automata and establish some of their basic properties.Quantum oscillator as a minimization problemhttps://zbmath.org/1527.810472024-02-28T19:32:02.718555Z"D'Eliseo, Maurizio M."https://zbmath.org/authors/?q=ai:deliseo.maurizio-m(no abstract)Upper bounds on quantum dynamics in arbitrary dimensionhttps://zbmath.org/1527.810522024-02-28T19:32:02.718555Z"Shamis, Mira"https://zbmath.org/authors/?q=ai:shamis.mira"Sodin, Sasha"https://zbmath.org/authors/?q=ai:sodin.sashaSummary: Motivated by the research on upper bounds on the rate of quantum transport for one-dimensional operators, particularly, the recent works of \textit{S. Jitomirskaya} and \textit{W. Liu} [J. Math. Phys. 62, No. 7, Article ID 073506, 9 p. (2021; Zbl 1469.81022)] and \textit{S. Jitomirskaya} and \textit{M. Powell} [in: Analysis at large. Dedicated to the life and work of Jean Bourgain. Cham: Springer. 173--201 (2022; Zbl 1521.81068)] and the earlier ones of \textit{D. Damanik} and \textit{S. Tcheremchantsev} [J. Am. Math. Soc. 20, No. 3, 799--827 (2007; Zbl 1114.81036); J. Funct. Anal. 255, No. 10, 2872--2887 (2008; Zbl 1153.81011)], we propose a method to prove similar bounds in arbitrary dimension. The method applies both to Schrödinger and to long-range operators.
In the case of ergodic operators, one can use large deviation estimates for the Green function in finite volumes to verify the assumptions of our general theorem. Such estimates have been proved for numerous classes of quasiperiodic operators in one and higher dimension, starting from the works of \textit{J. Bourgain} et al. [Commun. Math. Phys. 220, No. 3, 583--621 (2001; Zbl 0994.82044)].
One of the applications is a power-logarithmic bound on the quantum transport defined by a multidimensional discrete Schrödinger (or even long-range) operator associated with an irrational shift, valid for all Diophantine frequencies and uniformly for all phases. To the best of our knowledge, these are the first results on the quantum dynamics for quasiperiodic operators in dimension greater than one that do not require exclusion of a positive measure of phases. Moreover, and in contrast to localisation, the estimates are uniform in the phase.
The arguments are also applicable to ergodic operators corresponding to other kinds of base dynamics, such as the skew-shift.On the number and locations of eigenvalues of the discrete Schrödinger operator on a latticehttps://zbmath.org/1527.810632024-02-28T19:32:02.718555Z"Akhmadova, M. O."https://zbmath.org/authors/?q=ai:akhmadova.mukhayyo-o"Alladustova, I. U."https://zbmath.org/authors/?q=ai:alladustova.i-u"Lakaev, S. N."https://zbmath.org/authors/?q=ai:lakaev.saidakhmat-norjigitovich|lakaev.saidakhmat-nSummary: We consider the family of Schrödinger operators \(\hat{H}_{\gamma\lambda\mu}=\hat{H}_0+\hat{V}_{\gamma\lambda\mu}\) on the one-dimensional lattice \(\mathbb{Z} \), where \(\hat{H}_0\) is a convolution operator with a given Hopping matrix \(\hat{\varepsilon} \), and \(\hat{V}_{\gamma\lambda\mu}\) is a multiplication operator by the function \(\hat{v}\) such that \(\hat{v}(0)=\gamma, \hat{v}(x)=\frac{\lambda}{2}\) for \(|x|=1, \hat{v}(x)=\frac{\mu}{2}\) for \(|x|=2\) and \(\hat{v}(x)=0\) for \(|x|>2, \gamma,\lambda,\mu\in\mathbb{R} \). Under certain conditions on the potential, we prove that the discrete Schrödinger operator \(\hat{H}_{\gamma\lambda\mu}\) can have zero, one, two or three eigenvalues outside the essential spectrum. Moreover, we obtain conditions for the existence of three eigenvalues, two of them situated below the bottom of the essential spectrum and the other one above its top.\(G\)-circulant quantum Markov semigroupshttps://zbmath.org/1527.810722024-02-28T19:32:02.718555Z"Bolaños-Servín, Jorge R."https://zbmath.org/authors/?q=ai:bolanos-servin.jorge-r"Quezada, Roberto"https://zbmath.org/authors/?q=ai:quezada.roberto-b"Vázquez-Becerra, Josué"https://zbmath.org/authors/?q=ai:vazquez-becerra.josueSummary: We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of \(G\)-circulant GKSL generator and \(G\)-circulant QMS from the circulant case, corresponding to \(\mathbb{Z}_n\), to an arbitrary finite group \(G\). Second, we show that each \(G\)-circulant GKSL generator has a block-diagonal representation \(Q\otimes\mathbf{1}_G\), where \(Q\) is a \(G\)-circulant matrix determined by some \(\alpha\in\ell_2(G)\). Denoting by \(H\) the subgroup of \(G\) generated by the support of \(\alpha\), we prove that \(Q\) has its own block-diagonal matrix representation \(\widetilde{Q}\otimes\mathbf{1}_r\) where \(\widetilde{Q}\) is an irreducible \(H\)-circulant matrix and \(r\) is the index of \(H\) in \(G\). Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of \(G\)-circulant QMSs.A construction of quarter BPS coherent states and Brauer algebrashttps://zbmath.org/1527.810792024-02-28T19:32:02.718555Z"Lin, Hai"https://zbmath.org/authors/?q=ai:lin.hai.2|lin.hai|lin.hai.3|lin.hai.1"Zeng, Keyou"https://zbmath.org/authors/?q=ai:zeng.keyouSummary: BPS coherent states have gravity dual descriptions in terms of semiclassical geometries. The half BPS coherent states have been well studied, however less is known about quarter BPS coherent states. Here we provide a construction of quarter BPS coherent states. They are coherent states built with two matrix fields, generalizing the half BPS case. These states are both the eigenstates of the annihilation operators and in the kernel of the anomalous dimension dilatation operator. Another useful labeling of quarter BPS states is by representations of Brauer algebras and their projection onto a subalgebra \(\mathbb{C}[S_n \times S_m]\). Here, the Schur-Weyl duality for the Brauer algebra plays an important role in organizing the operators. One interesting subclass of these Brauer states are labeled by representations involving two Young tableaux. We obtain the overlap between quarter BPS Brauer states and quarter BPS coherent states, where the Schur polynomials are used. We also derive superposition formulas transforming quarter BPS coherent states to quarter BPS Brauer states. The entanglement entropy of Brauer states as well as the overlap between Brauer states and squeezed states are also computed.Shifted quantum groups and matter multiplets in supersymmetric gauge theorieshttps://zbmath.org/1527.810802024-02-28T19:32:02.718555Z"Bourgine, Jean-Emile"https://zbmath.org/authors/?q=ai:bourgine.jean-emileSummary: The notion of \textit{shifted} quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal \(\mathfrak{gl}(1)\) and quantum affine \(\mathfrak{sl}(2)\) algebras (resp. denoted \({\ddot{U}}_{q_1,q_2}^{\boldsymbol{\mu }}(\mathfrak{gl}(1))\) and \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2)))\). It defines several new representations, including finite dimensional highest \(\ell \)-weight representations for the toroidal algebra, and a vertex representation of \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2))\) acting on Hall-Littlewood polynomials. It also explores the relations between representations of \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2))\) and \({\ddot{U}}_{q_1,q_2}^{\boldsymbol{\mu }}(\mathfrak{gl}(1))\) in the limit \(q_1\rightarrow \infty\) (\(q_2\) fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d \({{\mathcal{N}}}=1\) and 3d \({{\mathcal{N}}}=2\) gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.Cherenkov radiation with massive bosons and quantum frictionhttps://zbmath.org/1527.810922024-02-28T19:32:02.718555Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Shirley, Christopher"https://zbmath.org/authors/?q=ai:shirley.christopherSummary: This work is devoted to several translation-invariant models in nonrelativistic quantum field theory (QFT), describing a nonrelativistic quantum particle interacting with a quantized relativistic field of bosons. In this setting, we aim at the rigorous study of Cherenkov radiation or friction effects at small disorder, which amounts to the metastability of the embedded mass shell of the bare nonrelativistic particle when the coupling to the quantized field is turned on. Although this problem is naturally approached by means of Mourre's celebrated commutator method, important regularity issues are known to be inherent to QFT models and restrict the application of the method. In this perspective, we introduce a novel non-standard procedure to construct Mourre conjugate operators, which differs from second quantization and allows to circumvent many regularity issues. To show its versatility, we apply this construction to the Nelson model with massive bosons, to Fröhlich's polaron model, and to a quantum friction model with massless bosons introduced by Bruneau and De Bièvre: for each of those examples, we improve on previous results.A pedagogical introduction to the Schwinger variational principle: an application to low energy positron-atom scatteringhttps://zbmath.org/1527.811082024-02-28T19:32:02.718555Z"Seidel, Eliton Popovicz"https://zbmath.org/authors/?q=ai:seidel.eliton-popovicz"Arretche, Felipe"https://zbmath.org/authors/?q=ai:arretche.felipe(no abstract)Order topology in Minkowski space and applicationshttps://zbmath.org/1527.830022024-02-28T19:32:02.718555Z"Koumantos, Panagiotis N."https://zbmath.org/authors/?q=ai:koumantos.panagiotis-nSummary: In this article, we consider and study the order topology in Minkowski spacetime of the special theory of relativity, \textit{i.e.} the finest locally convex topology \(\tau\) on spacetime for which every order bounded subset of spacetime is \(\tau\)-bounded. This order topology that is introduced into spacetime as an ordered vector space proves to be Hausdorff and differs from Zeeman's order topology. Applying the order topology we obtain new results by applying and extending previous results on the mean ergodic theorem and functional differential evolution equations in the Minkowski space.Twisted self-similarity and the Einstein vacuum equationshttps://zbmath.org/1527.830112024-02-28T19:32:02.718555Z"Shlapentokh-Rothman, Yakov"https://zbmath.org/authors/?q=ai:shlapentokh-rothman.yakovSummary: In the previous works \textit{I. Rodnianski} and \textit{Y. Shlapentokh-Rothman} [``Naked singularities for the Einstein vacuum equations: the exterior solution'', Preprint, \url{arXiv:1912.08478}] and \textit{Y. Shlapentokh-Rothman} [``Naked singularities for the Einstein vacuum equations: the interior solution'', Preprint \url{arXiv:2204.09891}] we have introduced a new type of self-similarity for the Einstein vacuum equations characterized by the fact that the homothetic vector field may be spacelike on the past light cone of the singularity. In this work we give a systematic treatment of this new self-similarity. In particular, we provide geometric characterizations of spacetimes admitting the new symmetry and show the existence and uniqueness of formal expansions around the past null cone of the singularity which may be considered analogues of the well-known Fefferman-Graham expansions. In combination with results from Rodnianski and Shlapentokh-Rothman [loc. cit.] our analysis will show that the twisted self-similar solutions are sufficiently general to describe all possible asymptotic behaviors for spacetimes in the small data regime which are self-similar and whose homothetic vector field is everywhere spacelike on an initial spacelike hypersurface. We present an application of this later fact to the understanding of the global structure of Fefferman-Graham spacetimes and the naked singularities of Rodnianski and Shlapentokh-Rothman [loc. cit.] and Shlapentokh-Rothman [loc. cit.]. Lastly, we observe that by an amalgamation of the techniques from \textit{I. Rodnianski} and \textit{Y. Shlapentokh-Rothman} [Geom. Funct. Anal. 28, No. 3, 755--878 (2018; Zbl 1394.35501)], one may associate true solutions to the Einstein vacuum equations to each of our formal expansions in a suitable region of spacetime.Traversable wormholes in four dimensionshttps://zbmath.org/1527.830182024-02-28T19:32:02.718555Z"Maldacena, Juan"https://zbmath.org/authors/?q=ai:maldacena.juan-m.1"Milekhin, Alexey"https://zbmath.org/authors/?q=ai:milekhin.alexey"Popov, Fedor"https://zbmath.org/authors/?q=ai:popov.fedor-kSummary: We present a wormhole solution in four dimensions. It is a solution of an Einstein Maxwell theory plus charged massless fermions. The fermions give rise to a negative Casimir-like energy, which makes the wormhole possible. It is a long wormhole that does not lead to causality violations in the ambient space. It can be viewed as a pair of entangled near extremal black holes with an interaction term generated by the exchange of fermion fields. The solution can be embedded in the Standard Model by making its overall size small compared to the electroweak scale.Logamediate inflation in DGP cosmology driven by a non-canonical scalar fieldhttps://zbmath.org/1527.831632024-02-28T19:32:02.718555Z"Ravanpak, A."https://zbmath.org/authors/?q=ai:ravanpak.arvin"Fadakar, G. F."https://zbmath.org/authors/?q=ai:fadakar.g-fSummary: The main properties of the logamediate inflation driven by a non-canonical scalar field in the framework of DGP braneworld gravity are investigated. Considering high energy conditions, we analytically calculate the slow-roll parameters. Then, we deal with perturbation theory and calculate the most important respective parameters, such as the scalar spectral index and the tensor-to-scalar ratio. We find that the spectrum of scalar fluctuations is always red-tilted. Also, we understand that the running in the scalar spectral index is nearly zero. Finally, we compare this inflationary scenario with the latest observational results from Planck 2018.Connecting optimization with spectral analysis of tri-diagonal matriceshttps://zbmath.org/1527.901482024-02-28T19:32:02.718555Z"Lasserre, Jean B."https://zbmath.org/authors/?q=ai:lasserre.jean-bernardSummary: We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of \((r\times r)\) tri-diagonal matrices of increasing size. Equivalently it reduces to computing the smallest (resp. largest) root of a certain univariate degree-\(r\) orthonormal polynomial. This provides a strong connection between the fields of optimization, orthogonal polynomials, numerical analysis and linear algebra, via asymptotic spectral analysis of tri-diagonal symmetric matrices.On a new simple algorithm to compute the resolventshttps://zbmath.org/1527.901532024-02-28T19:32:02.718555Z"Ba Khiet Le"https://zbmath.org/authors/?q=ai:ba-khiet-le."Théra, Michel"https://zbmath.org/authors/?q=ai:thera.michel-aSummary: Resolvents of operators are the core of many fundamental algorithms used in optimization. However their computation is in general difficult except for very particular operators. In the paper we provide a new simple algorithm with linear convergence rate to compute the resolvents for the class of operators which can be decomposed as a sum of a maximally monotone operator with a computable resolvent and a single-valued locally Lipschitz continuous mapping.The generalized Bregman distancehttps://zbmath.org/1527.901562024-02-28T19:32:02.718555Z"Burachik, Regina S."https://zbmath.org/authors/?q=ai:burachik.regina-sandra"Dao, Minh N."https://zbmath.org/authors/?q=ai:dao.minh-ngoc"Lindstrom, Scott B."https://zbmath.org/authors/?q=ai:lindstrom.scott-bSummary: Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the \textit{generalized Bregman distance}, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback-Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert \(\mathcal{W}\) function, whose importance in optimization is of growing interest.A viscosity iterative method with alternated inertial terms for solving the split feasibility problemhttps://zbmath.org/1527.901612024-02-28T19:32:02.718555Z"Liu, Lulu"https://zbmath.org/authors/?q=ai:liu.lulu"Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoli"Wang, Shen"https://zbmath.org/authors/?q=ai:wang.shen.1|wang.shen"Rassias, Michael Th."https://zbmath.org/authors/?q=ai:rassias.michael-thSummary: In this paper, we propose a viscosity iterative algorithm with alternated inertial extrapolation step to solve the split feasibility problem, where the self-adaptive stepsize is used. Under appropriate conditions, the proposed algorithm is proved to converge to a solution of the split feasibility problem, which is also the unique solution of a variational inequality problem. Finally, we demonstrate the effectiveness of the algorithm by a numerical example.
For the entire collection see [Zbl 1495.90002].A note on the linear convergence of generalized primal-dual hybrid gradient methodshttps://zbmath.org/1527.902532024-02-28T19:32:02.718555Z"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.3|wang.kai.29|wang.kai.24|wang.kai.20|wang.kai.6|wang.kai.5|wang.kai.8|wang.kai.9|wang.kai.10|wang.kai.12|wang.kai.2|wang.kai.11|wang.kai.4|wang.kai.1"Wang, Qun"https://zbmath.org/authors/?q=ai:wang.qunSummary: We revisit a class of primal-dual algorithms proposed in [\textit{B. He} et al., J. Math. Imaging Vis. 58, No. 2, 279--293 (2017; Zbl 1387.90186)], and focus on investigating the global linear convergence rate of these approaches under two scenarios. One scenario is assuming that one of the objectives is strongly convex and its gradient is Lipschitz continuous, and the other one is the hypothesis of some error bound conditions. Fruthermore, some theoretical results are verified by numerical simulation.