Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

There are two algebraic lower bounds of the number of n-periodic points of a self-map f : M → M of a compact smooth manifold of dimension at least 3: NF_n(f) = min{#Fix(g~n); g ～ f; g continuous} and NJD_n(f) = min{#Fix(g~n); g ～ f; g smooth}. In general, NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism,the equality NF_n(f) = NJD_n(f) holds for all n ? all eigenvalues of a quotient cohomology homomorphism induced by f have moduli 1.     

The Fixed Points of Contractions of <Emphasis Type="Italic">f</Emphasis>-Quasimetric Spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.     

Some Existence Theorems on All Fractional (<Emphasis Type="Italic">g</Emphasis>, <Emphasis Type="Italic">f</Emphasis>)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) → Z⁺ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F _h such that E(H) ∩ E(F _h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.     

Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G. For a positive integer n, let E _n (g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if E _n (g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of E _n (g). For soluble G we prove that if, for some n, the Fitting height of E _n (g) is equal to k, then g belongs to the (k+1)th Fitting subgroup F_k+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

On the rate of decay of concentration functions of <Emphasis Type="Italic">n</Emphasis>-fold convolutions of probability distributions

Concentration functions of n-fold convolutions of probability distributions is shown to exhibit the following behavior. Let φ(n) be an arbitrary sequence tending to infinity as n tends to infinity, and ψ(x) be an arbitrary function tending to infinity as x tends to infinity. Then there exists a probability distribution F of a random variable X such that the mathematical expectation E _ψ(|X|) is infinite and, moreover, the upper limit of the sequence \(\sqrt n \phi \left( n \right)Q_n\) is equal to infinity, where Q _n is the maximal atom of the n-fold convolution of distribution F. Thus, no infinity conditions imposed on the moments can force the concentration functions of n-fold convolutions decay essentially faster than o(n ^?1/2).     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ? and α_j, β_j: ? → ? are described.     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV _n ? ?²/? _n ,n ∈ {2,3,4,?} defined by the immersions of ?² into ? ⁿ⁺¹ X ⁿ : (z, w) → (z ⁿ ,z ^n?1 w, ?,zw ^n?1 ,w ⁿ )     

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).     

Normal families of meromorphic functions and shared functions

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f^(k) and g^(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities m_f for f and m_h for h satisfy m_f ≥ m_h + k + 1 for k > 1 and m_f ≥ 2m_h + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n_f ≥ n_h + 1, then the family F is normal on D.     

Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z ₀) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) ? α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β ₀ and α(z) ? α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.     

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: G ^m → G and word maps with constants w _Σ: G ^m → G of a simple algebraic group G, where w is a nontrivial word in the free group F _m of rank m, w _Σ = w ₁ σ ₁ w ₂ ··· w _r σ _r w _{r + 1}, w ₁, …, w _{r + 1} ∈ F _m , w ₂, …, w _r ≠ 1, Σ = {σ ₁, …, σ _r | σ _i ∈ G Z(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γ_w, G) of the group Γ_w = F _m /<w>.     

Some relations among multiplicative and <Emphasis Type="Italic">q</Emphasis>-additive functions

We give all solutions of the equation f(n) = g(n) + h(n) for every n ∈ ?, where f is a completely multiplicative, g is a 2-additive, and h is a 3-additive function. We also determine all completely multiplicative functions f and all q-additive functions g for which f(n) = g ²(n) for every n ∈ ?.     

Functional equations and Weierstrass sigma-functions

It is proved that if an entire function f: ? → ? satisfies an equation of the form α ₁(x)β ₁(y) + α ₂(x)β ₂(y) + α ₃(x)β ₃(y), x,y ∈ C, for some α _j , β _j : ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az ² + Bz + C) ? σ _Γ(z - z ₁) ? σ _Γ(z - z ₂), where Γ is a lattice in ?; σ _Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z ₁, z ₂ ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).     

The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Semi-Heavy Tails

In this paper, we study properties of functions and sequences with a semi-heavy tail, that is, functions and sequences of the form w(x) = e^?βxf(x), β > 0, resp., w_n = cⁿf_n, 0 < c < 1, where the function f(x), resp., the sequence (f_n), is regularly varying. Among others, we give a representation theorem and study convolution properties. The paper includes several examples and applications in probability theory.