Recent zbMATH articles in MSC 30Ghttps://www.zbmath.org/atom/cc/30G2022-05-16T20:40:13.078697ZWerkzeugOpen and surjective mapping theorems for differentiable maps with critical pointshttps://www.zbmath.org/1483.260122022-05-16T20:40:13.078697Z"Li, Liangpan"https://www.zbmath.org/authors/?q=ai:li.liangpanSummary: Let \(\Omega\) be an open subset of \(\mathbb R^n\) \((n \ge 2)\), and let \(F : \Omega \rightarrow \mathbb R^n\) be a continuously differentiable map with countably many critical points. We show that \(F\) is an open map. Let \(G :\mathbb R^n \rightarrow \mathbb R^n\) \((n \ge 1)\) be a continuously differentiable map such that \(G(x) \rightarrow \infty\) as \(x \rightarrow \infty \). Then it is proved that \(G\) is surjective if and only if each connected component of the complement of the set of critical values of \(G\) contains at least one image of \(G\). Several applications of both theorems especially to complex analysis are presented.Polyanalytic boundary value problems for planar domains with harmonic Green functionhttps://www.zbmath.org/1483.300722022-05-16T20:40:13.078697Z"Begehr, Heinrich"https://www.zbmath.org/authors/?q=ai:begehr.heinrich"Shupeyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:shupeyeva.bibinurThe authors characterize the solvability of three boundary value problems for the inhomogeneous polyanalytic equation in planar domains (having a harmonic Green function), namely the well-posed iterated Schwarz problem, and two over-determined iterated problems of Dirichlet and Neumann type. Solutions formulas are also obtained and, in particular, it is concluded that the polyanalytic Cauchy-Pompeiu representation formula provides the solution to the Dirichlet problem (for any degree \(n\), and in the cases for which the solution exists).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the extended class of SUPM and their generating URSM over non-Archimedean fieldhttps://www.zbmath.org/1483.300882022-05-16T20:40:13.078697Z"Banerjee, Abhijit"https://www.zbmath.org/authors/?q=ai:banerjee.abhijit"Maity, Sayantan"https://www.zbmath.org/authors/?q=ai:maity.sayantanSummary: In this article, we investigate an extended class of strong uniqueness polynomial over non-Archimedean field than that was recently studied by \textit{H.H. Khoai} and \textit{V. H. An} [\(p\)-Adic Numbers Ultrametric Anal. Appl. 12, No. 4, 276--284 (2020; Zbl 1456.30082)]. We also find the unique range set of weight 2 corresponding to the SUPM which improve and generalize significantly the results of the paper [loc. cit.] and an earlier one due to \textit{P.-C. Hu} and \textit{C.-C. Yang} [Acta Math. Vietnam. 24, No. 1, 95--108 (1999; Zbl 0986.30025)].Bounded extremal problems in Bergman and Bergman-Vekua spaceshttps://www.zbmath.org/1483.300892022-05-16T20:40:13.078697Z"Delgado, Briceyda B."https://www.zbmath.org/authors/?q=ai:delgado.briceyda-b"Leblond, Juliette"https://www.zbmath.org/authors/?q=ai:leblond.julietteSummary: We analyze Bergman spaces \(A_f^p(\mathbb{D})\) of generalized analytic functions of solutions to the Vekua equation \(\bar{\partial}w = (\bar{\partial}f/f)\bar{w}\) in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions \(f\) and \(1<p<\infty\). We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space \(A^p(\mathbb{D})\) and in its generalized version \(A^p_f(\mathbb{D})\), that consists in approximating a function in subsets of \(\mathbb{D}\) by the restriction of a function belonging to \(A^p(\mathbb{D})\) or \(A^p_f(\mathbb{D})\) subject to a norm constraint. Preliminary constructive results are provided for \(p = 2\).On the investigation of isotropic thick-walled shellshttps://www.zbmath.org/1483.300902022-05-16T20:40:13.078697Z"Khvoles, A."https://www.zbmath.org/authors/?q=ai:khvoles.a-r|khvoles.alexander|khvoles.a-a"Zgenti, V."https://www.zbmath.org/authors/?q=ai:zgenti.v"Vashakmadze, T."https://www.zbmath.org/authors/?q=ai:vashakmadze.tamaz-s|vashakmadze.tamara-sSummary: We consider the problems of creating 2-dim models for thin-walled structures and satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models.Generalized growth of special monogenic functions having finite convergence radiushttps://www.zbmath.org/1483.300912022-05-16T20:40:13.078697Z"Kumar, Susheel"https://www.zbmath.org/authors/?q=ai:kumar.susheelSummary: In the present paper, we study the growth of special monogenic functions having finite convergence radius. The characterizations of generalized order and generalized type of special monogenic functions having finite convergence radius have been obtained in terms of their Taylor's series coefficients.\(k\)-CF functions and \(\Box_b\) on the quaternionic Heisenberg grouphttps://www.zbmath.org/1483.300922022-05-16T20:40:13.078697Z"Shi, Yun"https://www.zbmath.org/authors/?q=ai:shi.yun"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.18Summary: The tangential \(k\)-Cauchy-Fueter operator and \(k\)-CF functions on the quaternionic Heisenberg group are quaternionic counterparts of the tangential CR operator \(\overline{\partial}_b\) and CR functions on the Heisenberg group in the theory of several complex variables. We analyze the operator \(\Box_b\) associated the tangential 2-Cauchy-Fueter operator and give its fundamental solution when the coefficients of the group satisfy the condition \(\sum_{l=0}^{n-1}a_l\ne\pm\sum_{l=0}^{n-1}|a_l|\). As an application, we prove that the \(L^p\)-integrable 2-CF function space is trivial in this case. We also discuss the results for general \(k\).Short-time special affine Fourier transform for quaternion-valued functionshttps://www.zbmath.org/1483.300932022-05-16T20:40:13.078697Z"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohan"Shah, Firdous A."https://www.zbmath.org/authors/?q=ai:shah.firdous-ahmad"Teali, Aajaz A."https://www.zbmath.org/authors/?q=ai:teali.aajaz-aSummary: The special affine Fourier transform is a promising tool for analyzing transient signals with more degrees of freedom via a chirp-like basis. In this article, our goal is to introduce a novel quaternion-valued short-time special affine Fourier transform in the context of two-dimensional quaternion-valued signals. In addition to studying all fundamental properties of the proposed transform, we also formulate some notable uncertainty inequalities including the Heisenberg-Weyl inequality, logarithmic inequality and local-type inequalities by employing the machinery of quaternionic Fourier transforms. Nevertheless, an illustrative example is presented to endorse the obtained results.\(q\)-analyticity in the sense of Ahern \& Brunahttps://www.zbmath.org/1483.320032022-05-16T20:40:13.078697Z"Daghighi, Abtin"https://www.zbmath.org/authors/?q=ai:daghighi.abtinSummary: We consider an alternative notion of polyanalyticity in several complex variables based upon a previous work of \textit{P. Ahern} and \textit{J. Bruna} [Rev. Mat. Iberoam. 4, No. 1, 123--153 (1988; Zbl 0685.42008)]. We also generalize this notion to the case of generic embedded
submanifolds of \(\mathbb{C}^n\) and give characterizations of the notions involved.Formalizing basic quaternionic analysishttps://www.zbmath.org/1483.684892022-05-16T20:40:13.078697Z"Gabrielli, Andrea"https://www.zbmath.org/authors/?q=ai:gabrielli.andrea"Maggesi, Marco"https://www.zbmath.org/authors/?q=ai:maggesi.marcoSummary: We present a computer formalization of quaternions in the HOL Light theorem prover. We give an introduction to our library for potential users and we discuss some implementation choices.
As an application, we formalize some basic parts of two recently developed mathematical theories, namely, slice regular functions and Pythagorean-hodograph curves.
For the entire collection see [Zbl 1369.68009].