Abstract: We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a \(\mathcal {C}\)-selfadjoint operator generates a contraction \(C_0\)-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of complex symmetric, unbounded semigroups and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all \(\mathcal {C}\)-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a \(\mathcal {C}\)-selfadjoint, unbounded semigroup is not necessarily \(\mathcal {C}\)-selfadjoint. PubDate: 2020-02-21

Abstract: Abstract We study closed n-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifolds. Moreover, under some curvature conditions, the result that a critical metric must be Einstein is proved. PubDate: 2020-02-05

Abstract: Abstract In the first part of this work we study the set of bicomplex numbers from the point of view of a hyperbolic module. We make use of the partial order defined on the set of hyperbolic numbers. We recall some properties of the hyperbolic geometrical objects that were defined in previous papers. With the help of these notions some fractal-type sets in the four dimensional space are constructed. PubDate: 2020-02-05

Abstract: Abstract In this paper we obtain subordination, superordination, and sandwich-type results related to certain family of integral operators defined on the space of meromorphic functions in the open unit disk. Also, an application of the subordination and superordination theorems to the Gauss hypergeometric function are considered, and the main new results generalize some previously well-known sandwich-type theorems. PubDate: 2020-01-21

Abstract: Abstract The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary. PubDate: 2020-01-05

Abstract: Abstract In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe–Ritter to the sphere \({{\mathbb {S}}}^n\). We determine the resulting Osterwalder–Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group \(G^c = \mathop {{\mathrm{O}}{}}\nolimits _{1,n}({{\mathbb {R}}})^{\uparrow }\) and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain \(\Xi \), known as the crown of the hyperboloid, containing a half-sphere \({{\mathbb {S}}}^n_+\) and the hyperboloid \({{\mathbb {H}}}^n\) as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of \(G^c\) in spaces of holomorphic functions on \(\Xi \). We connect this analysis with the boundary components which are the de Sitter space and a bundle over the space of future pointing lightlike vectors. PubDate: 2020-01-03

Abstract: Abstract Suppose \(\left( \phi _t \right) _{t \ge 0}\) is a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) with Denjoy–Wolff point \(\tau =1\). Suppose K is a compact subset of \(\mathbb {D}\). We prove that the capacity of the condenser \((\mathbb {D}, \phi _t(K) )\) is a decreasing function of t. Moreover, we study its asymptotic behavior as \(t \rightarrow + \infty \) in relation with the type of the semigroup. PubDate: 2020-01-03

Abstract: Abstract We study the Dirichlet problem for the quasilinear partial differential equation \(\triangle u(z) = h(z)\cdot f(u(z))\) in the unit disk \({\mathbb {D}}\subset {\mathbb {C}}\) with arbitrary continuous boundary data \(\varphi :\partial {\mathbb {D}}\rightarrow {\mathbb {R}}\). The multiplier \(h:{\mathbb {D}}\rightarrow {\mathbb {R}}\) is assumed to be in the class \(L^p({\mathbb {D}}),\)\(p>1,\) and the continuous function \(f:\mathbb {R}\rightarrow {\mathbb {R}}\) is such that \(f(t)/t\rightarrow 0\) as \(t\rightarrow \infty .\) Applying the potential theory and the Leray–Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class \(W^{2,p}_{\mathrm{loc}}({\mathbb {D}})\). Furthermore, we show that \(u\in C^{1,\alpha }_{\mathrm{loc}}({\mathbb {D}})\) with \(\alpha = (p-2)/p\) if \(p>2\) and, in particular, with arbitrary \(\alpha \in (0,1)\) if the multiplier h is essentially bounded. In the latter case, if in addition \(\varphi \) is Hölder continuous of some order \(\beta \in (0,1)\), then u is Hölder continuous of the same order in \(\overline{{\mathbb {D}}}\). We extend these results to arbitrary smooth (\(C^1\)) domains. PubDate: 2020-01-01

Abstract: Abstract Let Y be a complex Banach space and let \(r\ge 1\). In this paper, we are concerned with an extension operator \(\varPhi _{\alpha , \beta }\) that provides a way of extending a locally univalent function f on the unit disc \(\mathbb {U}\) to a locally biholomorphic mapping \(F\in H(\varOmega _r)\), where \(\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: z_1 ^2+\Vert w\Vert _Y^r<1\}\). We prove that if f can be embedded as the first element of a g-Loewner chain on \(\mathbb {U}\), where g is a convex (univalent) function on \(\mathbb {U}\) such that \(g(0)=1\) and \(\mathfrak {R}g(\zeta )>0\), \(\zeta \in \mathbb {U}\), then \(F =\varPhi _{\alpha , \beta }(f)\) can be embedded as the first element of a g-Loewner chain on \(\varOmega _r\), for \(\alpha \in [0, 1]\), \(\beta \in [0, 1/r]\), \(\alpha +\beta \le 1\). We also show that normalized univalent Bloch functions on \(\mathbb {U}\) (resp. normalized uniformly locally univalent Bloch functions on \(\mathbb {U}\)) are extended to Bloch mappings on \(\varOmega _r\) by \(\varPhi _{\alpha ,\beta }\), for \(\alpha >0\) and \(\beta \in [0,1/r)\) (resp. for \(\alpha =0\) and \(\beta \in [0,1/r]\)). In the case of the Muir type extension operator \(\varPhi _{P_k}\), where \(k\ge 2\) is an integer and \(P_k:Y\rightarrow \mathbb {C}\) is a homogeneous polynomial mapping of degree k with \(\Vert P_k\Vert \le d(1,\partial g(\mathbb {U}))/4\), we prove a similar extension result for the first elements of g-Loewner chains on \(\varOmega _k\). Next, we consider a modification of the Muir type extension operator \(\varPhi _{G,k}\), where \(k\ge 2\) is an integer and \(G:Y\rightarrow \mathbb {C}\) is a holomorphic function such that \(G(0)=0\) and PubDate: 2020-01-01

Abstract: Abstract The class of holomorphic self-mappings of the open unit disk (which are contractions with respect to the Poincaré metric) admits a natural extension to the class of holomorphic pseudo-contractions. In this paper, we study various inequalities involving the values of derivatives of holomorphic pseudo-contractions at fixed points (particularly, at the Denjoy–Wolff fixed point). PubDate: 2020-01-01

Abstract: Abstract Let the vector fields \(X_1, \ldots , X_{6}\) form an orthonormal basis of \(\mathcal {H}\), the orthogonal complement of a Cartan subalgebra (of dimension 2) in \({{\,\mathrm{SU}\,}}(3)\). We prove that weak solutions u to the degenerate subelliptic p-Laplacian $$\begin{aligned} \Delta _{\mathcal {H},{p}} u(x)=\sum _{i=1}^{6} X_i^{*}\left( \nabla _{\!{\mathcal {H}}}u ^{p-2}X_{i}u \right) =0, \end{aligned}$$have Hölder continuous horizontal derivatives \(\nabla _{\!{\mathcal {H}}}u=(X_1u, \ldots , X_{6}u)\) for \(p\ge 2\). We also prove that a similar result holds for all compact connected semisimple Lie groups. PubDate: 2019-12-24

Abstract: Abstract We establish a criterion for a set of eigenfunctions of the one-dimensional Schrödinger operator with distributional potentials and boundary conditions containing the eigenvalue parameter to be a Riesz basis for \({\mathscr {L}}_2(0,\pi )\). PubDate: 2019-12-23

Abstract: Abstract Stationary adiabatic flows of real gases issued from a source of given intensity are studied. Thermodynamic states of gases are described by Legendrian or Lagrangian manifolds. Solutions of Euler equations are given implicitly for any equation of state and the behavior of solutions of the Navier–Stokes equations with the viscosity considered as a small parameter is discussed. For different intensities of the source we introduce a small parameter into the Navier–Stokes equation and construct corresponding asymptotic expansions. We consider the most popular model of real gases—the van der Waals model, and ideal gases as well. PubDate: 2019-12-23

Abstract: Abstract An operator I is said to be an averaging (or mean-value) operator on a set \({\mathcal {K}}\) of analytic functions in \(\Delta =\{z: z <1\}\), if \(I[f](0)=f(0)\) and \(I[f](\Delta )\) is contained in the convex hull of \(f(\Delta )\) for all \(f\in {\mathcal {K}}\). In this work we consider the class \(\mathcal {SP}(\alpha )\) of functions defined by us (Folia Sci Univ Technol Resov 28:35–42, 1993), which is connected with the class of uniformly convex functions introduced by Goodman (Ann Polon Math 56:87–92, 1991). We describe an interesting new construction of averaging operators which might attract a considerable attention of mathematicians working in the field. PubDate: 2019-12-23

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 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