Abstract: Abstract In this paper, we establish five new sharp versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing Schwarz function in place of the initial coefficients in the power series representations of the functions involved and thereby, we generalize several related results of earlier authors. PubDate: 2020-09-15

Abstract: Abstract The present paper deals with modifications of Bernstein, Kantorovich, Durrmeyer and genuine Bernstein–Durrmeyer operators. Some previous results are improved in this study. Direct estimates for these operators by means of the first and second modulus of continuity are given. Also the asymptotic formulas for the new operators are proved. PubDate: 2020-09-15

Abstract: Abstract In this paper we investigate the boundary behavior of finitely bi-Lipschitz homeomorphisms between Finsler manifolds. Our study involves the module technique and classes of mappings whose moduli of the curve/surface families are integrally controlled from above and below. The Lusin (N)-property with respect to the k-dimensional Hausdorff measure for the finitely bi-Lipschitz mappings is also established. PubDate: 2020-09-14

Abstract: Abstract We study the h-transform of Doob for nonlocal branching processes, we show that the branching property is preserved provided that h is a coherent state, and we emphasize the probabilistic representation of the solution to the associated nonlinear evolution equation. The tools are from the analytic and probabilistic potential theory. We also investigate the h-transform of a subordinate \(C_0\) -semigroup of sub-Markovian operators on an \(L^p\) space. PubDate: 2020-09-12

Abstract: Abstract In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier–Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier–Stokes equations have no Type I blowups. PubDate: 2020-09-10

Abstract: Abstract We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon–John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center. PubDate: 2020-09-09

Abstract: Abstract In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda u ^{q-2}u\ln u ^2 + u ^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where \(N >sp\) with \(s \in (0, 1)\) , \(p>1\) , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{ u(x)-u(y) ^p}{ x-y ^{N+ps}}dxdy, \end{aligned}$$ \(p_s^*=Np/(N-ps)\) is the fractional critical Sobolev exponent, \(\Omega \subset {\mathbb {R}}^N\) \((N\ge 3)\) is a bounded domain with Lipschitz boundary and \(\lambda \) is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution \(u_b\) . Moreover, for any \(\lambda > 0\) , we show that the energy of \(u_b\) is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as \(b \rightarrow 0\) . PubDate: 2020-09-09

Abstract: Abstract We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Fuglede (Potential Anal 51:197–217, 2019). Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates. PubDate: 2020-09-08

Abstract: Abstract Based on the extended homoclinic test technique, we introduce an ansätz functions to construct double periodic-soliton solutions of new (2 + 1)-Dimensional KdV Equation. Some entirely new double periodic-soliton solutions are obtained. The obtained solutions show that there exist multiple-periodic solitary waves in the different directions for the new (2 + 1)-Dimensional KdV Equation. With the help of symbolic computation, the properties for these new solutions are presented with some figures. PubDate: 2020-09-04

Abstract: Abstract We analyze generalized \((3+1)\) -dimensional Yu–Toda–Sasa–Fukuyama(YTSF) equation, a nonlinear evolution equation to understand pulse behavior when variations are strong. Using the Lie symmetry reduction, the generalized form of (3+1)-dimensional YTSF equation is reduced to ordinary differential equations. We introduce the main result for the analysis of soliton solutions that accounts for perturbation and dispersion of the waveform including linear and nonlinear effects. We discuss soliton interactions as a key feature of soliton based telecommunication transmission systems. Solitons propagate at distinct speed and interact quite strongly with each other having beaming correspondence. Though the interaction is transient, the coherence is diagonally placed. The solitons after perfectly elastic collisions recover their shape, amplitude and velocity except phase shift. PubDate: 2020-09-04

Abstract: Abstract We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner–Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in \(C^l\) , \(l\geqslant 1\) , up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in \(C^1\) and \(H^1\) . PubDate: 2020-08-26

Abstract: Abstract The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class of convex functions are computed. PubDate: 2020-08-12

Abstract: Abstract The Funk–Radon transform, also known as the spherical Radon transform, assigns to a function on the sphere its mean values along all great circles. Since its invention by Paul Funk in 1911, the Funk–Radon transform has been generalized to other families of circles as well as to higher dimensions. We are particularly interested in the following generalization: we consider the intersections of the sphere with hyperplanes containing a common point inside the sphere. If this point is the origin, this is the same as the aforementioned Funk–Radon transform. We give an injectivity result and a range characterization of this generalized Radon transform by finding a relation with the classical Funk–Radon transform. PubDate: 2020-08-11

Abstract: Abstract The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class \(C_{0}.\) Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Carathéodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established. PubDate: 2020-08-10

Abstract: Abstract We study infinitesimal generators of one-parameter semigroups in the unit disk \(\mathbb {D}\) having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein–Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning. We also give a new proof of the classical Cowen–Pommerenke inequalities for univalent self-maps of \(\mathbb {D}\) . PubDate: 2020-08-08

Abstract: Abstract Let \(\varphi _{1},\varphi _{2},\varphi _{3}\) be three functions meromorphic in the unit disc \(\Delta \) and continuous on the closure of \(\Delta \) such that \(\varphi _{i} (z)\ne \varphi _{j}(z)\) on the unit circle \( z =1.\) Let \(\mathcal {F}\) be a family of meromorphic functions such that \(f \ne \varphi _{j}\) on \(\Delta \) for \(j=1,2,3 ~\text{ and }~f\in \mathcal {F}.\) Then there exists a constant M such that \((1- z ^{2})f^{\#}(z)\le M \) for each \(z\in \Delta ~\text{ and }~f\in \mathcal {F}.\) In addition, this constant M depends only on the three functions \(\varphi _{1},\varphi _{2}~\text{ and }~\varphi _{3}.\) This generalizes the related theorems due to Lappan (In: Progress in analysis: proceedings of the 3rd international ISAAC congress in progress in analysis, vol 1, pp 221–228. World Scientific, 2003), and Xu and Qiu (C R Math 349:1159–1160, 2011), respectively. PubDate: 2020-08-04

Abstract: Abstract In this paper, we study Schrödinger equations on elliptic curves called generalized Lamé equations. We suggest a method of finding integrable potentials for Schrödinger type equations. We apply this method to the Lamé equations and provide a sequence of integrable potentials for which the eigenvalue problem is solved explicitly. PubDate: 2020-07-15

Abstract: Abstract In this paper we refine the re-expansion problems for the one-dimensional torus and extend them to the multidimensional tori and to compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group \(\mathrm{SU(2)}\) a simple sufficient condition is given. PubDate: 2020-07-13

Abstract: Abstract There are many articles in the literature dealing with a first, second and third-order differential inequalities, inclusions or subordinations in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential inequalities, inclusions and subordinations in two complex functions p and q in the complex plane. A typical example of a system of differential inequalities is $$ \left\{ {\begin{array}{*{20}l} {\text{Re} [2p(z) + zp^{{\prime }} (z) + z^{2} p^{{\prime \prime }} (z) - q(z)] > 0,} \hfill \\ {\text{Re} [p(z) + 7zq^{{\prime }} (z)] > 4.} \hfill \\ \end{array} } \right. $$ The authors determine properties of the functions p and q satisfying some special systems of differential inequalities, and extend their results to differential inclusions and subordinations. PubDate: 2020-07-03

Abstract: Abstract In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with \(N=1\) , from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena. PubDate: 2020-06-27