Linear Continuous Functionals and Representation of Functions by Series in the Spaces E ϕ

We give a criterion of the existence of linear continuous nontrivial functionals in a space E with a continuous measure. We also give a criterion that x _n} _n=1 be a representation system in a space E with lim _t/(t)/t=0.     

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.     

Pre-dual of the Function Algebra A ?��(D) and Representation of Functions in Dirichlet Series

In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A ^−∞(D) (D being either a bounded C ²-smooth convex domain in ${\mathbb{C}^N (N > 1)}${\mathbb{C}^N (N > 1)} , or a bounded convex domain in \mathbbC{\mathbb{C}}) as a space of entire functions with certain growth. A possibility of representation of functions from the pre-dual space in a form of Dirichlet series with frequencies from D, is also studied.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

A New Proof for the Existence of an Equivariant Entire Solution Connecting the Minima of the Potential for the System Δu − W u (u) = 0

Recently, Giorgio Fusco and the author in [2 Alikakos , N.D. , Fusco , G. ( 2011 ). Entire solutions to equivariant elliptic systems with variational structure . Arch. Rat. Mech. Anal. 202 : 567 – 597 .[Crossref], [Web of Science ®] , [Google Scholar]] studied the system Δu ? W _u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of equivariant solutions connecting the minima in certain directions at infinity, together with an estimate. In this paper a new proof is given which, in particular, avoids both the introduction of a pointwise constraint in the minimization process and the equivariant extensions of the various test functions.     

The New Extensions of the Henstock–Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake–Henstock–Kurzweil and the Hake–McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space \(\mathbb {R}^{m}\). These are ”natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of \(\mathbb {R}^{m}\). Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock–Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock–Kurzweil and the McShane integrals in terms of our integrals.     

Representation of Functions in the Weighted Space L 1 μ by Trigonometric and Walsh Series

We prove that there exists a series of the form (*)

Stirling’s Formula for Gamma Functions,Bounds for Ratios of Gamma Functions,Beta Functions and Percentiles of a Studentized Sample Mean: A Synthesis with New Results

In the context of Stirling’s formula for gamma functions and bounds for ratios of gamma functions, this work has a threefold purpose: (1) Outline recently published literature; (2) Synthesize techniques and results from Bhattacharjee and Mukhopadhyay (Commun Stat, Theory & Methods 39:1046–1053, 2010) and Mukhopadhyay (Commun Stat, Theory & Methods 40:1283–1297, 2011) which have gone perhaps unnoticed by some recent researchers; and (3) Incorporate new results for beta functions and useful bounds for the percentiles of a Studentized sample mean obtained from a normal distribution. This synthesized review may help in gaining a wider perspective about this area.     

On the Generalization of Some Hermite–Hadamard Inequalities for Functions with Convex Absolute Values of the Second Derivatives Via Fractional Integrals

Ukrainian Mathematical Journal - We provide a unified approach to getting Hermite–Hadamard inequalities for functions with convex absolute values of the second derivatives via the...     

Nonequilibrium Green’s Functions in the Atomic Representation and the Problem of Quantum Transport of Electrons Through Systems With Internal Degrees of Freedom

We develop the theory of quantum transport of electrons through systems with strong correlations between fermionic and internal spin degrees of freedom. The atomic representation for the Hamiltonian of a device and nonequilibrium Green’s functions constructed using the Hubbard operators allow overcoming difficulties in the perturbation theory encountered in the traditional approach because of a larger number of bare scattering amplitudes. Representing the matrix elements of effective interactions as a superposition of terms each of which is split in matrix indices, we obtain a simple method for solving systems of very many equations for nonequilibrium Green’s functions in the atomic representation. As a result, we obtain an expression describing the electron currents in a device one of whose sites is in tunnel coupling with the left contact and the other, with the right contact. We derive closed kinetic equations for the occupation numbers under conditions where the electron flow leads to significant renormalization of them.     

p-adic Transcendence and p-adic Transcendence Measures for the Values of Mahler Type Functions

We prove the p-adic transcendence and p-adic transcendence measures for the values of some Mahler type functions.     

H~1-Estimates of the Littlewood-Paley and Lusin Functions for Jacobi Analysis Ⅱ

Let $({\Bbb R}_+,*,\Delta)$ be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley $g$ function and the Lusin area function for the Jacobi hypergroup and consider their $(H^1, L^1)$ boundedness. Although the $g$ operator for $({\Bbb R}_+,*,\Delta)$ possesses better property than the classical $g$ operator, the Lusin area operator has an obstacle arisen from a second convolution. Hence, in order to obtain the $(H^1, L^1)$ estimate for the Lusin area operator, a slight modification in its form is required.     

On the strong law of large numbers for ��-mixing and ��-mixing random variables

We consider the classical Kolmogorov condition for strong law of large numbers for sequences of dependent random variables; the so-called ??-mixing and Rademacher?CMenchoff condition for ??-mixing sequences.     

Representation of the β-function and anomalous dimensions by nonsingular integrals: Proof of the main relation

A method for calculating the β-function and anomalous dimensions, convenient for numerical calculations in the ?-expansion framework, was previously proposed, and the relation underlying the method was verified up to the four-loop approximation. We prove this relation in all orders of the perturbation theory.     

A modification of Newton's method to solve the Dirichlet problem for the equation Δu=f(x,u)

The Dirichlet problem for a weakly nonlinear equation u=f(x, u) is investigated. We use successive approximations constructed by modified Newton's scheme and apply the extremal properties of the solutions of the elliptic equation of the form u – c(x)u=F(x), where c(x) 0. Numerical solution of the resulting sequence of linear boundary-value problems is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 27–30, 1987.     

σ-Moderate solutions of Lu = uα and fine trace on the boundary

We investigate the class of all positive solutions of the equation Lu = u^α, 1 < α ≤ 2, in a smooth domain E ⊂ℝ^d. We define the fine trace tr(u) of a solution u as a pair (Γ, v), where Γ is a set of singular boundary points of u, and v is a certain σ-finite measure on the complement of Γ. We describe all possible traces and we construct the minimal solution with the given trace.     

Comparison theorems for nonlinear evolution equations and their application to the investigation of the dependence of the solutions of the equation ut=δϕ(u) ON ϕ and on the boundary conditions

One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u₁=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions.