Estimates of Wiman–Valiron type for evolution equations

We establish estimates of Wiman–Valiron type for solutions of evolution equations with a pseudodifferential operator of the Hörmander class in a Hilbert space. Estimates of this type characterize the behavior of the solution of the problem as t→∞ or t → 0 depending on the decay or growth rate of the Fourier coefficients of the initial data.     

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x₀ + ?_i=1ⁿ e_i\frac?f?x_i=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.     

Classes of functions subharmonic in ℝm and bounded on certain sets

Let Z_j be the Euclidean space of vectors \((z_{j,1,...,} z_{j_{j \cdot n_j + 1} } ), Z = \mathop \oplus \limits_{j = 1}^P Z_j\) . The function u: Z → ?₊, u ?0, is said to be logarithmically p-subharmonic if log u(z) is upper semicontinuous with respect to the totality of the variables and subharmonic or identically equal to ?∞ with respect to each z_j when the remaining ones are fixed. For such functions, with the growth estimate $$log u(z) \leqslant \delta \mathop \Pi \limits_{j = 1}^P (1 + |z_{j,n_j + 1} |) + N(\mathop {\sum\limits_{\mathop {1 \leqslant j \leqslant p}\limits_{} } {z_{j,k}^2 } }\limits_{1 \leqslant k \leqslant n_j } )^{1/2} + C; \delta ,N \geqslant 0, C \in \mathbb{R}$$ one proves theorems on equivalence of^∞) (L_q)-norms of their restrictions to \(X = \mathop \oplus \limits_{j = 1}^P (Z_{j,1} ,...,z_{j,n_j } )\) and to a relatively dense subset of it, generalizing the known Cartwright and Plancherel-Pólya results.     

Wiman–Valiron Type Inequalities for Entire and Random Entire Functions of Finite Logarithmic Order

Siberian Mathematical Journal -     

The genus distributions for a certain type of permutation graphs in orientable surfaces

A circuit is a connected nontrivial 2-regular graph.A graph G is a permutation graph over a circuit C,if G can be obtained from two copies of C by joining these two copies with a perfect matching.In this paper,based on the joint tree method introduced by Liu,the genus polynomials for a certain type of permutation graphs in orientable surfaces are given.     

Outer limits of subdifferentials for min–max type functions

Abstract

We generalize the outer subdifferential construction suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.     

Several identities for certain products of theta functions

By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44, 1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products of theta functions φ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.     

Entire Dirichlet Series of Rapid Growth and New Estimates for the Measure of Exceptional Sets in Theorems of the Wiman–Valiron Type

For entire Dirichlet series of the form , we establish conditions under which the relation

Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szeg? weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ? 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006).     

The error norm of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szegö weight functions consisting of any one of the four Chebyshev weights divided by the polynomial On certain spaces of analytic functions, the error term of these formulae is a continuous linear functional. We compute explicitly the norm of the error functional.     

The universality for linear combinations of Lerch zeta functions and the Tornheim�CHurwitz type of double zeta functions

In this paper, we consider the universality for linear combinations of Lerch zeta functions. J. Kaczorowski, A. Laurin?ikas and J. Steuding treated universal Dirichlet series with the case that the compact sets ${\mathcal{K}_l}$ are disjoint. But we consider the both cases that the compact subset ${\mathcal{K}_l}$ is disjoint and not disjoint. Next, we will show the non-trivial zeros of the Tornheim?CHurwitz type of double zeta functions in the region of absolute convergence. Moreover we show the universality for the Tornheim?CHurwitz type of double zeta function.     

The Paley–Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤^* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotic form of the Hermite-Padé approximations for a system of mittag-leffler functions

We use the Laplace method for investigation of asymptotic properties of the Hermite integrals. In particular, we find asymptotic form for diagonal Hermite-Padé approximations for a system of exponents. Analogous results are obtained for a system of degenerate hypergeometric functions. These theorems supplement the well-known results of F. Wielinnsky, A. I. Aptekarev and others.