Regularity of Poisson equation in some logarithmic space

In this note, the regularity of Poisson equation -△u = f with f lying in logarithmic function space Lp(LogL)a(Ω)(1＜p ＜∞, a ∈ R) is studied. The result of the note generalizes the W2,p estimate of Poisson equation in Lp(Ω).     

Supports of fourier transforms of refinable frame functions and their applications to FMRA

Let M be a d × d expansive matrix, and FL ²(??) be a reducing subspace of L ²(? ^d ). This paper characterizes bounded measurable sets in ? ^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL ²(??), respectively. Some examples are also provided.     

The crooked property

Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over ${\mathbb{F}}_{2^n}$ with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F, from ${\mathbb{F}}_{p^n}$ to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem.     

N元Stolarsky平均的凸性和几何凸性

讨论了n个正数的Stolarsky平均的S-凸性和S-几何凸性,证明了:n元Stolarsky平均在r>1时是S-凸的和S-几何凸的;在r<1时是S-凹的.作为推论,此文也比较了n个正数的Stolarsky平均和算术平均的大小.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

Regularity of weakly subquadratic functions

Related to the theory of convex and subadditive functions, we investigate weakly subquadratic mappings, that is, solutions of the inequality

     

The Three-Dimensional ℤ2 Electrodynamics and the Two-Dimensional Ising Model

We calculate the correlation functions for the three-dimensional ₂ electrodynamics for particular values of the interaction energies and with free boundary conditions.     

A complete monotonicity property of the gamma function

A logarithmically completely monotonic function is completely monotonic. The function is strictly completely monotonic on (0,∞). The function is strictly logarithmically completely monotonic on (0,∞).     

Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

与例外函数和分担函数相关的亚纯函数的正规族

设F为区域D内的一族亚纯函数,对于每个f∈F,f的所有极点重数至少是2,a(z)和b(z)为两个在D内满足a(z)■b(z)的全纯函数.若对于每个f∈F,f(z)≠a(z)和f(z)≠b(z),则F在D内正规.这个结果改进了经典的Montel定则.此外,我们也讨论了亚纯函数族中每个函数与其导函数分担两个全纯函数的正规性.     

Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

Integrals and derivatives connected to the r-Lambert function

In this paper, we derive some formulas for higher order derivatives of r-Lambert functions. Moreover, an integration formula involving powers of r-Lambert function is obtained.     

The limiting distribution of the divisor function

Let σα(n) be the sum of the αth power of the positive divisors of n. We establish an asymptotic formula for the natural density of the set of integers n that satisfy σα(n)/nα?t, as t→∞. Two other limiting distributions considered are based on Jordan's totient function and Dedekind's psi function.     

关于Hayman问题与分担值

该文主要讨论Hayman的一个问题与分担值的联系，并运用新颖的方法证明了：设f 为非常数的整函数，n,k均为正整数，n≥k+1,a为非零的有穷复数，若a为f^n与(f^n)^(k)的分担值，则nf′=ωf，ω为方程t^k=1的根.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.