Normal families of holomorphic functions and shared values

LetF be a family of holomorphic functions in a domainD, and leta, b be two distinct finite complex numbers. If, for anyf ∈F,f andf′ sharea IM, andf′(z)=b wheneverf(z)=b, thenF is normal inD. This improves results due to Pang, Pang and Zalcman, Xu, etc. Supported by the NNSF of China (Grant No. 10071038) and “Qing Lan Project” of the Educational Department of Jiangsu Province.     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Normal families and uniqueness of entire functions and their derivatives

In this paper we use the theory of normal families to prove: Let a and b be two complex numbers such that b ≠ a, 0, and let f be a non-constant entire function. If f and f′ share the value a CM and if f = b whenever f′ = b, then f ≡ f′. This result improves a result due to Rubel and Yang. Received: 13 June 2005     

Shared values and normality

This paper investigates the relationship between the normality and the shared values for a meromorphic function on the unit disc Δ. Based on Marty’s normality criterion and through a detailed analysis of the meromorphic functions, it is shown that if for every f ∈ , f and f ^(k) share a and b on Δ and the zeros of f(z) − a are of multiplicity k ⩾ 3, then is normal on Δ, where is a family of meromorphic functions on the unit disc Δ, and a and b are distinct values. Selected from Journal of East China Normal University (Natural Science), 2003, 4: 12–18. This work was supported by the National Natural Science Foundation of China under grant number 10271122 and by Shanghai City Foundation for selected academic research     

Groups of typeF a,b,−c

For integersa, b andc, the groupF ^a,b,−c is defined to be the group 〈R, S : R ²=RS ^aRS^bRS^−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp ⁿ (for certainp andn) as groups of typeF ^a,b,−c for certain (not unique) choices ofa, b andc.     

The lattice of flats and its underlying flag matroid polytope

LetM be a matroid andF the collection of all linear orderings of bases ofM, orflags ofM. We define the flag matroid polytope Δ(F). We determine when two vertices of Δ(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats ofM and certain maximal faces of Δ(F). Supported in part by NSA grant MDA904-95-1-1056.     

On homomorphisms from the Hamming cube to Z

WriteF for the set of homomorphisms from {0, 1} ^d toZ which send0 to 0 (think of members ofF as labellings of {0, 1} ^d in which adjacent strings get labels differing by exactly 1), andF ₁ for those which take on exactlyi values. We give asymptotic formulae for |F| and |F|. In particular, we show that the probability that a uniformly chosen memberf ofF takes more than five values tends to 0 asd→∞. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constantb such thatf a.s. takes at mostb values. This in turn verified a conjecture of I. Benjaminiet al., that for eacht>0,f a.s. takes at mosttd values. Determining |F| is equivalent both to counting the number of rank functions on the Boolean lattice 2^[d] (functionsf: 2^[d]→N satisfyingf( ) andf(A)≤f(A∪x)≤f(A)+1 for allA∈2^[d] andx∈[d]) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from {0, 1} ^d toK ₃, the complete graph on 3 vertices). Our proof uses the main lemma from Kahn’s proof of constant range, together with some combinatorial approximation techniques introduced by A. Sapozhenko. Research supported by a Graduate School Fellowship from Rutgers University.     

The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem

A Cohen-Macaulay complex is said to be balanced of typea=(a ₁,a ₂, ...,a _s ) if its vertices can be colored usings colors so that every maximal face gets exactlya _i vertices of thei:th color. Forb=(b ₁,b ₂, ...,b _s ), 0≦b≦a, letf _b denote the number of faces havingb _i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f _b )_0≦b≦a′ or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result. First and third authors partially supported by the National Science Foundation.     

The distance between certainn-dimensional Banach spaces

IfE andF aren-dimensional Banach spaces, ifE has cotype 2, and if the ball ofF* has a small number of extreme points, then the Banach-Mazur distanced(E, F)≦C √nlogn. The techniques lead to the formally stronger result: IfE andF* have type 2 constantsa andb, respectively, thend(E, F)≦√n(a+b). IfE isn-dimensional, the identity map onE, when restricted to a large subspace ofE, factors through with normC √n. The authors’ work was supported in part, respectively, by NSF grant numbers MCS 78-02194, MCS 79-02489, and MCS 77-04174.     

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

On a difference equation arising in a learning-theory model

An analysis is presented of the equationf(x+a)−f(x)=e ^−x {f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use of bilateral Laplace transform. Research supported by the National Science Foundation, Grant GP-2558.     

A note on Weil index

Let F be a non-archimedean local field of characteristic 0 and(?)a nontrivial additive character.Weil first defined the Weil indexγ(a,(?))(a∈F~*)in his famous paper,from which we know thatγ(a,(?))γ(b,(?))=γ(ab,(?))γ(1,(?))(a,b)andγ(a,(?))~4 =(-1,-1),where(a,b)is the Hilbert symbol for F.The Weil index plays an important role in the theory of theta series and in the general representation theory.In this paper,we establish an identity relating the Weil indexγ(a,(?))and the Gauss sum.     

On the mean square of the error term for the asymmetric two-dimensional divisor problem (I)

Suppose a and b are two fixed positive integers such that (a, b) = 1. In this paper we shall establish an asymptotic formula for the mean square of the error term Δ _a,b (x) of the general two-dimensional divisor problem.     

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.     

Maximal Betti numbers of Cohen–Macaulay complexes with a given <Emphasis Type="Italic">f</Emphasis>-vector

Given the f-vector f = (f₀, f₁, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δ_f with f(Δ_f) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has for all i and j, where f(Δ) is the f-vector of Δ and where β _ij (I _Δ) are graded Betti numbers of the Stanley–Reisner ideal I _Δ of Δ. The first author is supported by JSPS Research Fellowships for Young Scientists. Received: 23 January 2006     

Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can be chosen so that the surface bundles M(F′,f′) and M(F,f) have the same first Betti numbers. Y. Ni is partially supported by a Centennial fellowship of the Graduate School at Princeton University. S.C. Wang is partially supported by MSTC     

On the dynamics of composite entire functions

Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains. The second author was supported by Max-Planck-Gessellschaft ZFDW, and by Tian Yuan Foundation, NSFC.