Abstract: The original version of the article unfortunately contained few errors under Preliminaries section. The corrected text is given below. PubDate: 2019-12-01

Abstract: Abstract It is known that if \(f: D_1 \rightarrow D_2\) is a polynomial biholomorphism with polynomial inverse and constant Jacobian then \(D_1\) is a 1-point poly-quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever \(D_2\) is a complete circular domain. Bell conjectured that all 1-point poly-quadrature domains arise in this manner. In this note, we construct a 1-point poly-quadrature domain of order 1 that is not biholomorphic to any complete circular domain. PubDate: 2019-12-01

Abstract: In this paper, the existence and uniqueness of solution of the Cauchy problem for abstract Boussinesq equation is obtained. The leading part of the equation include general elliptic operator and abstract positive operator in a Banach space E. Since the Banach space E and linear operators are sufficiently large classes, by choosing their we obtain the existence and uniqueness of solution of numerous classes of generalized Boussinesq type equations which occur in a wide variety of physical systems. By applying this result, the Wentzell–Robin type mixed problem for Boussinesq equations and the Cauchy problem for finite or infinite systems of Boussinesq equations are studied. PubDate: 2019-12-01

Abstract: Abstract In this paper, we study the weak asymptotic in the \(\mathbb {C}\)-plane of some wave functions resulting from the WKB-techniques applied to a Schrödinger equation with quartic oscillator and having some boundary condition. As a first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions. Especially, we investigate the properties of the Cauchy transform of the root counting measure of re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form \({\mathcal {C}}^{2}(z) +r(z){\mathcal {C}}(z)+s(z)=0\), where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential \(-p(z)dz^{2}\) where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential. PubDate: 2019-12-01

Abstract: Abstract We prove local solvability and stability for the inverse problem of recovering a complex-valued square integrable potential in the Sturm–Liouville equation on a finite interval from spectra of two boundary value problems with one common boundary condition. For this purpose we generalize classical Borg’s method to the case of multiple spectra. PubDate: 2019-12-01

Abstract: Abstract In this article, we first establish a global Carleman estimate for an ultrahyperbolic Schrödinger equation. Next, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the equation by some lateral boundary data. PubDate: 2019-12-01

Abstract: Abstract We introduce and analyze the concept of infinitesimal relative position vector field between “infinitesimally nearby” observers, showing the equivalence between different definitions. Through the Fermi–Walker derivative of infinitesimal relative position vector fields along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Sufficient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson–Walker spacetimes is characterized. PubDate: 2019-12-01

Abstract: Abstract Let \(L=-\Delta + x ^2\) be the Hermite operator, where \(\Delta \) is the Laplacian on \({\mathbb {R}}^{d}\). In this paper, we define fractional Carleson measure associated with Hermite operator, which is adapted to the operator L. Then, we will use it to characterize the dual spaces and predual spaces of the Hardy spaces \(H_L^p({\mathbb {R}}^d)\) associated with L. PubDate: 2019-12-01

Abstract: Abstract Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus \(g\ge 2\) with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra. PubDate: 2019-12-01

Abstract: Abstract We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter \(\lambda \) varies. PubDate: 2019-12-01

Abstract: Abstract We develop the Titchmarsh–Weyl theory for vector-valued discrete Schrödinger operators. We show that the Weyl m functions associated with these operators are matrix valued Herglotz functions that map complex upper half plane to the Siegel upper half space. We discuss about the Weyl disk and Weyl circle corresponding to these operators by defining these functions on a bounded interval. We also discuss the geometric properties of Weyl disk and find the center and radius of the Weyl disk explicitly in terms of matrices. PubDate: 2019-12-01

Abstract: Abstract Given a bounded function \(\varphi \) on the unit disk in the complex plane, we consider the operator \(T_{\varphi }\), defined on the Bergman space of the disk and given by \(T_{\varphi }(f)=P(\varphi f)\), where P denotes the orthogonal projection to the Bergman space in \(L^2({\mathbb {D}},dA)\). For algebraic symbols \(\varphi \), we provide new necessary conditions on \(\varphi \) for \(T_{\varphi }\) to be hyponormal, extending recent results of Fleeman and Liaw. Our approach is perturbative and aims to understand how small changes to a symbol preserve or destroy hyponormality of the corresponding operator. We consider both additive and multiplicative perturbations of a variety of algebraic symbols. One of our main results provides a necessary condition on the complex constant C for the operator \(T_{z^n+C z ^s}\) to be hyponormal. This condition is also sufficient if \(s\ge 2n\). PubDate: 2019-12-01

Abstract: Abstract Any ramified holomorphic covering of a closed unit disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give two explicit constructions for a holomorphic ramified covering of a disc by other bordered Riemann surface. The machinery used here strongly resembles the description of magnetic configurations in submicron planar magnets. PubDate: 2019-12-01

Abstract: Abstract In this paper sufficient conditions for continuous and compact embeddings of generalized higher order Lipschitz classes on a compact subset of n-dimensional real Euclidean spaces are obtained. PubDate: 2019-12-01

Abstract: Abstract We show the maximum principle for exponential energy minimizing maps. We then estimate the distance of two image points of an exponentially harmonic map between surfaces. We also study the existence of an exponentially harmonic map between surfaces if the image is contained in a convex disc. We finally investigate the existence of an exponentially harmonic map \(f:M_1\rightarrow M_2\) between surfaces in case \(\pi _2 (M_2) = \emptyset \). PubDate: 2019-12-01

Abstract: Abstract We investigate an inverse problem referring to roulettes in normed planes, thus generalizing analogous results of Bloom and Whitt on the Euclidean subcase. More precisely, we prove that a given curve can be traced by rolling another curve along a line if two natural conditions are satisfied. Our access involves details from a metric theory of trigonometric functions, which was recently developed for normed planes. Based on this, our approach differs from other ones in the literature. PubDate: 2019-09-26

Abstract: Abstract In this paper we show several sufficient conditions for close-to-convex functions to be strongly starlike of some order. The results continue the line of study from the first author’s paper on the order of strong starlikeness of strongly convex functions, (Nunokawa in Proc Japan Acad Ser A 69(7):234–237, 1993). Also it appears an small improvement of a certain classical results of Ch. Pommerenke. As an application, we also derive estimates for the radii of star-likeness for close-to-convex functions. PubDate: 2019-09-13

Abstract: Abstract In this paper, we study abundant exact solutions including the lump and interaction solutions to the (2 + 1)-dimensional Yu–Toda–Sasa–Fukuyama equation. With symbolic computation, lump solutions and the interaction solutions are generated directly based on the Hirota bilinear formulation. Analyticity and well-definedness is guaranteed through some conditions posed on the parameters. With special choices of the involved parameters, the interaction phenomena are simulated and discussed. We find the lump moves from one hump to the other hump of the two-soliton, while the lump separates from the hump of the one-soliton. PubDate: 2019-09-03