Angular distribution in complex oscillation theory

Let f_1 and f_2 be two linearly independent solutions of the differential equationf″ Af=0,where A is an entire function.Set E=f_1f_2.In this paper,we shall studythe angular distribution of E and establish a relation between zero accumulation rays andBorel directions of E.Consequently we can obtain some results in the complex differentialequation by using known results in angular distribution theory of meromorphic functions.     

On Extending Peano Derivatives

Let E be a closed set with inf E = a and sup E = b, and k be a positive integer. Let f : E Rbe such that the k-th Peano derivative of f relative to E, f _(k) (x, E), exists. It is proved under certain condition on the function f, that an extension F : [a, b] Rof f exists such that the ordinary derivative of F of order k, F ^<k> (x) exists on [a, b] and is continuous on [a, b], and f _<> (x, E) = F ^<i> (x) on E, for i = 1, 2, &, k.     

On meromorphic solutions of <Emphasis Type="Italic">f</Emphasis><Superscript>2</Superscript> + <Emphasis Type="Italic">g</Emphasis><Superscript>2</Superscript> = 1

We show that meromorphic solutions f, g of f ² + g ² = 1 in C² must be constant, if f _z2 and g _z1 have the same zeros (counting multiplicities). We also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

Another Note on Polynomial vs Rational Approximation

LetEbe a subspace ofC(X) and letR(E)=g/h : g, hE; h>0}. We make a simple, yet intriguing observation: if zero is a best approximation toffromE, then zero is a best approximation toffromR(E). We also prove that if {E_n} is dense inC(X) then for almost allf(in the sense of category)[formula]That extends the results of P. Borwein and S. Zhou who proved it for the case whenE_nis the space of algebraic or trigonometric polynomials of degreen.     

Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x ₁,…, x _n ]?R. For a polynomial f ∈ R[x ₁,…, x _n ]?R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x ₁,…, x _n ] and Q(R)[f] ∩ R[x ₁,…, x _n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x ₁,…, x _n ] whose kernel equals R[f].     

On zeros of polynomials of best approximation to holomorphic andC ∞ functions

LetE be a compact subset of the complex planeC such that Leja's extremal functionL _E forE is continuous. If almost all zeros of the polynomials of best approximation to a functionfC(E) are outside the setE _R ={zC:L _E (z<R)}, for someR>1, thenf is extendible to a holomorphic function inE _R . If the zeros ofn-th, polynomial of best approximation tof are outside and the sequence {R _n ^–n } rapidly decreases to zero thenf can be extended to aC function on 075-4}.     

The cotype constant and an almost euclidean decomposition for finite-dimensional normed spaces

Every 2n-dimensional normed spaceE contains twon-dimensional subspacesE ₁ andE ₂ which are orthogonal with respect to the inner product induced by the John ellipsoid ofE and which satisfyd(E _i, l ₂ ⁿ )≦f(K ₂(E)), wheref(K ₂(E)) is some number that depends only on the cotype constant ofE, denotedK ₂(E). Supported in part by NSF grant DMS 8401906.     

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x ₁,…, x _n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x ₁,…, x _n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x ₁,…, x _n )² is central valued on R;

2. R satisfies s ₄, the standard identity of degree 4.

     

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE ⁰ can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin. Partially supported by a grant from NSERC of Canada.     

Functions and line digraphs

Consider two maps f and g from a set E into a set F such that f(x) ≠ g(x) for every x in E. Suppose that there exists a positive integer n such that for any element z in F either f^?1(z) or g^?1(z) has at most n elements. Then, E can be partitioned into 2n + 1 subsets E₁, E₂,…,E_{2n + 1} such that f(E_i)∩ g(E_i) = ?, 1 ≤ i ≤ 2n + 1. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 296–303, 2003     

m-Power Commuting MAPS on Semiprime Rings

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 ? e)U ? M₂(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Quadratic Central Differential Identities on a Multilinear Polynomial

Let K be a commutative ring with unity, R a prime K-algebra, Z(R) the center of R, d and δ nonzero derivations of R, and f(x ₁,…, x _n ) a multilinear polynomial over K. If [d(f(r ₁,…, r _n )), δ (f(r ₁,…, r _n ))] ? Z(R), for all r ₁,…, r _n  ? R, then either f(x ₁,…, x _n ) is central valued on R or {d, δ} are linearly dependent over C, the extended centroid of R, except when char(R) = 2 and dim _C RC = 4.     

Spherical derivatives and normal families

We show that a family of functions meromorphic in a plane domain D whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each f meromorphic in the unit disk D, inf _z∈D f ^#(z) ≤ 1/2, where f ^# denotes the spherical derivative of f.     

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E^X is the equivalence relation which is defined on the Polish group C(X,R₊*) by where f, g are in C(X,R₊*), then E^X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere Sⁿ with the one-point compactification of Rⁿ via the stereographic projection, while E_n,r is the equivalence relation which is defined on the Polish group C^r(Rⁿ,R₊*) by where f, g are in C^r(Rⁿ,R₊*), then E_n,r is also induced by a turbulent Polish group action. Dedicated to my sister Alexandra and to her daughter Marianthi.     

An extremal problem for Dirichlet-finite holomorphic functions on Riemann surfaces

Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|² has the least harmonic majorantf ₂ onR. We show Σf ₂(a ≦π ⁻¹ D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬℓ₁ fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Growth Behavior and Zero Distribution of Rational Approximants

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m _n for meromorphic functions f on a compact set E of ℂ where m _n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E _ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E _ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.     

The rationality of the Hilbert-Kunz multiplicity in graded dimension two

We show that the Hilbert-Kunz multiplicity is a rational number for an R₊−primary homogeneous ideal I=(f₁, . . . , f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. More specific, we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f₁, . . . , f_n) on the projective curve Y=ProjR.