Value sharing and differential equations

Suppose that f is a nonconstant entire function and L[f] a linear differential polynomial in f with constant coefficients. In this paper, by considering the existence of the solutions of some differential equations, we find all the forms of entire functions f in most cases when f and L[f] share two values counting multiplicities jointly. This result generalize some known results due to Rubel–Yang and Li–Yang.     

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).     

β级星像函数类和凸像函数类的扩展问题

本文利用文[2,3]的引理和算子L(a,c)f(z)的一些性质.结合Hadamard乘积,研究了算子L(a,c)f(z),获得了L(a,c)f(z)∈S*(β)和L(a,c)f(z)∈K(β)的充分条件,推广了文[2,3]的相关结论.     

Linear Equations for the Number of Intervals Which are Isomorphic with Boolean Lattices and the Dehn–Sommerville Equations

Let P be a finite poset. Let L: = J(P) denote the lattice of order ideals of P. Let b _i (L) denote the number of Boolean intervals of L of rank i.

We construct a simple graph G(P) from our poset P. Denote by f _i (P) the number of the cliques K _i+1, contained in the graph G(P).

Our main results are some linear equations connecting the numbers f _i (P) and b _i (L).

We reprove the Dehn–Sommerville equations for simplicial polytopes.

In our proof, we use free resolutions and the theory of Stanley–Reisner rings.     

Finitely arithmetically fixed elements of a field

An element r of a field L will be called finitely arithmetically fixed (f.a.f.) if there exists some finite subset A of L containing r such that every map f from A to L which behaves “like a homomorphism” on A, leaves r fixed. This notion will be generalized to a relative one for any field extension L/K, and several results describing the set of f.a.f. elements are obtained. Received: 21 December 2005     

On submanifolds of Lxf F satisfying Chen's basic equality

It is well known that the warped product Lx_f F of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine submanifolds of Lx_f F which are tangent to the structure vector field and satisfy Chen's basic equality. Also, we investigate tubular hypersurfaces of Lx_f CE ^m which satisfy Chen's basic equality where CE ^m is a complex Euclidean m-space. This revised version was published online in June 2006 with corrections to the Cover Date.     

An algebra of absolutely continuous functions and its multipliers

The aim of this paper is to study the algebraAC _p of absolutely continuous functionsf on [0,1] satisfying f(0) = 0,f ’ ∈ L^p[0, 1] and the multipliers ofAC _p .     

A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn

Let L_n(f, x)(f ε C)[0, ∞)) be a Bernstein-type approximation operator as defined and studied by Bleimann, Butzer, and Hahn. Probabilistic arguments are used to simplify and sharpen some of their results. The rates of convergence are given in terms of the first and second moduli of continuity. Moreover, an appropriate limit of L_n is shown to be the well-known Szasz operator.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.     

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.     

Properties of Certain Integral Operators

For analytic functions f() belonging to the class A_p, integral operators I_n^a (f()) are introduced. The object of the present paper is to derive some interesting properties of integral operators I_n^a(f(z)). Our results contain some previous results by M. Obradovié [4], S. Owa, M. Obradovi, and M. Nunokawa [6], and by D.K. Thomas [7].AMS Subject Classification (2000) primary 30C45 secondary 32A20Supported, in part, by the Japanese Ministry of Education, Science and Culture under a grant-in-aid for general scientific research.     

On the structure of the solution set of a third kind boundary value problem

We show that given any closed subset C of a real Banach space E, there is a continuous function f(t, x) which is Lipschitz continuous in its second variable such that the solution set of the corresponding third kind boundary value problem is homeomorphic to C (Theorem 1.1). In the special problem we give the infimum of Lipschitz constants L_f of such functions f(t, x) (Theorem 1.3).     

The Cluster Function of Single-Valued Functions

Given a single-valued function f between topological spaces X and Y, we interpret the cluster set C(f;x) as a multivalued function F=C(f;⋅) associated to f – the cluster function of f. For appropriate metrizable spaces X and Y, we characterize cluster functions C(f;⋅) among arbitrary set-valued functions F and show that every cluster function F=C(f;⋅) admits a selection h of Baire class 2 such that F=C(h;⋅). Mathematics Subject Classifications (2000) Primary: 54C50, 54C60; secondary: 26A21, 54C65.This research was partially supported by DFG Grant RI 1087/2.     

A remark on Weierstrass points and linear series on multiple covering curves

Here we study multiple coverings of rational and irational curves. We give a theorem about the non-gap sequence on m-gonal curves. We then study general irrational covering f ： X→ C, and say when h^0（X, f^＊（L）） = h^0（C,L） for L line bundle on C.     

On the topology of a generic fibre of a polynomial function

In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

On Dirichlet Series with Periodic Coefficients

We investigate Dirichlet series L(s, f) = _n=1 with q-periodic coefficients f(n), i.e. f(n+q) = f(n) for all integers n and some fixed integer q, and we prove an asymptotic formula for the number of nontrivial zeros of L(s, f). Further, we give a necessary condition for L(s, f) to have a distribution of the nontrivial zeros symmetrical with respect to the critical line.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.