Recent zbMATH articles in MSC 30C40https://zbmath.org/atom/cc/30C402023-03-23T18:28:47.107421ZWerkzeugSkew-orthogonal polynomials in the complex plane and their Bergman-like kernelshttps://zbmath.org/1503.330082023-03-23T18:28:47.107421Z"Akemann, Gernot"https://zbmath.org/authors/?q=ai:akemann.gernot"Ebke, Markus"https://zbmath.org/authors/?q=ai:ebke.markus"Parra, Iván"https://zbmath.org/authors/?q=ai:parra.ivanSummary: Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.Geometric properties of the Bernatsky integral operatorhttps://zbmath.org/1503.450142023-03-23T18:28:47.107421Z"Maĭer, Fedor Fedorovich"https://zbmath.org/authors/?q=ai:maier.fedor-fedorovich"Tastanov, Maĭrambek Gabulievich"https://zbmath.org/authors/?q=ai:tastanov.mairambek-gabulievich"Utemisova, Anar Altaevna"https://zbmath.org/authors/?q=ai:utemisova.anar-altaevnaSummary: In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection \(f(z)\in S^o\Leftrightarrow g(z) = zf'(z) \in S^*\) of the classes \(S^o\) and \(S^*\) of convex and star-shaped functions can be considered as mapping using the differential operator \(G[f](x) = zf'(z)\) of class \(S^o\) to class \(S^*\), that is, \(G: S^o \to S^*\) or \(G(S^o) = S^*\). The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator \(G^{-1}[f](x)\), which translates \(S^* \to S^o\) and thereby ``improves'' the properties of functions, maps the entire class \(S\) of single-leaf functions into itself.
At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class \(S\) or its subclasses to themselves or to other subclasses.
This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition \(a < \operatorname{Re}z f'(z)/f(z) < b\) (\(0 < a < 1 < b\)), in the class \(K(\gamma)\) of functions, almost convex in order \(\gamma \). The results of the article summarize or reinforce previously known effects.