Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n?m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.     

On minima of rational indefinite quadratic forms

Let f(x) be an indefinite quadratic form with real coefficients in n variables with nonzero determinant d. The collection of all values of $\text{v(f) = |d|}{}^{\mathrm{<mtext>?1</mtext><mtext>n</mtext>}}\text{inf}\text{|f(x)|}$, where infimum is taken over x ∈ Zⁿ such that f(x) ≠ 0 (x ≠ 0) is called the spectrum of nonzero minima (spectrum of minima) of such forms. The spectrum is said to be discrete if for every δ > 0, there are only finitely many values of v(f) > δ. It is proved that for rational quadratic forms in n ≥ 3 variables and real quadratic forms in n ≥ 21 variables the spectra of nonzero minima are discrete. Also the spectra of minima of indefinite ternary and quaternary rational quadratic forms are discrete.     

On the oscillations of multiplicative functions taking values ±1

For multiplicative functions f(n), which take on the values ±1, we show that under certain conditions on f(n), for all x sufficiently large, there are at least values of n?x for which f(n(n+1))=−1.     

Near automorphisms of cycles

Let f be a permutation of V(G). Define δ_f(x,y)=|d_G(x,y)-d_G(f(x),f(y))| and δ_f(G)=∑δ_f(x,y) over all the unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among all the permutations f of V(G). The permutation f with δ_f(G)=π(G) is called a near automorphism of G. In this paper, we study the near automorphisms of cycles C_n and we prove that π(C_n)=4⌊n/2⌋-4, moreover, we obtain the set of near automorphisms of C_n.     

Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Roll cutting in the curtain industry,or: A well-solvable allocation problem

We study the problem of cutting a number of pieces of the same length from n rolls of different lengths so that the remaining part of each utilized roll is either sufficiently short or sufficiently long. A piece is ‘sufficiently short’, if it is shorter than a pre-specified threshold value δ_min, so that it can be thrown away as it cannot be used again for cutting future orders. And a piece is ‘sufficiently long’, if it is longer than a pre-specified threshold value δ_max (with δ_max > δ_min), so that it can reasonably be expected to be usable for cutting future orders of almost any length. We show that this problem, faced by a curtaining wholesaler, is solvable in O(nlogn) time by analyzing a non-trivial class of allocation problems.     

Existence of positive solution for some class of nonlinear fractional differential equations

In this paper, we investigate the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t)=t^νf(u), 0<t<1, where D=t^−βδD_β^γ−δ,δ, β>0, γ?0, 0<δ<1, ν>−β(γ+1). Our main work is to deal with limit case of f(s)/s as s→0 or s→∞ and Φ(s)/s, Ψ(s)/s as s→0 or s→∞, where Φ(s), Ψ(s) are functions connected with function f. In J. Math. Appl. 252 (2000) 804-812, we consider the existence of a positive solution for the particular case of Eq. (1.1), i.e., the Riemann-Liouville type (β=1, γ=0) nonlinear fractional differential equation, using the super-lower solutions method. Here, we devote to the existence of positive solution and multi-positive solutions for Eq. (1.1) by means of the fixed point theorems for the cone.     

Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type

In this paper, we study the global L^∞ solutions for the Cauchy problem of nonsymmetric system (1.1) of Keyfitz-Kranzer type. When n=1, (1.1) is the Aw-Rascle traffic flow model. First, we introduce a new flux approximation to obtain a lower bound ρε,δ?δ>0 for the parabolic system generated by adding “artificial viscosity” to the Aw-Rascle system. Then using the compensated compactness method with the help of L¹ estimate of wε,δx(⋅,t) we prove the pointwise convergence of the viscosity solutions under the general conditions on the function P(ρ), which includes prototype function , where γ∈(−1,0)∪(0,∞), A is a constant. Second, by means of BV estimates on the Riemann invariants and the compensated compactness method, we prove the global existence of bounded entropy weak solutions for the Cauchy problem of general nonsymmetric systems (1.1).     

Mappings of Finite Distortion of Polynomial Type

Suppose f:? ⁿ →? ⁿ is a mapping of K-bounded p-mean distortion for some p>n?1. We prove the equivalence of the following properties of f: the doubling condition for J(x,f) over big balls centered at the origin, the boundedness of the multiplicity function N(f,? ⁿ ), the polynomial type of f, and the polynomial growth condition for f.     

Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class

Let us consider the problem whether there does exist a finite-time self-similar solution of the backward type to the semilinear Keller-Segel system. In the case of parabolic-elliptic type for n?3 we show that there is no such a solution with a finite mass in the scaling invariant class. On the other hand, in the case of parabolic-parabolic type for n?2, non-existence of finite-time self-similar solutions is proved in a larger class of a finite mass with some local bounds.     

Gauges of Baire class one functions

Let K be a compact metric space and be a bounded Baire class one function. We proved that for any ε>0 there exists an upper semicontinuous positive function δ of f with finite oscillation index and |f(x)−f(y)|<ε whenever d(x,y)<min{δ(x),δ(y)}.     

Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let V? be a closed subspace of L₂(R) generated from the integer shifts of a continuous function ? with a certain decay at infinity and a non-vanishing property for the function Φ^†(γ)=_∑n∈Z?(n)e−inγ on [−π,π]. In this paper we determine a positive number δ? so that the set {n+δn}_n∈Z is a set of stable sampling for the space V? for any selection of the elements δn within the ranges ±δ?. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈V? and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

On Value Distribution of Δ~nf

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.     

Arithmetic properties of certain functions in several variables

If T is an n × n matrix with nonnegative integral entries, we define a transformation T: Cⁿ → Cⁿ by w = Tz where

$\text{W}{}_1\text{=}\text{∏}\text{j=1}\text{n}\text{z}{}_{\mathrm{j}}{}^{\mathrm{t}}\text{ij}\text{(1?i?n).}$

We consider functions f(z) of n complex variables which satisfy a functional equation giving f(Tz) as a rational function of ¹f(z) and we obtain conditions under which such a function f(z) takes transcendental values at algebraic points.     

Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition

In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε²Δu+f(u) in a smooth bounded domain Ω⊂R³ under the boundary condition εν_∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior.     

Löwner expansions

A fundamental problem is to estimate the logarithmic coefficients of a power series with constant coefficient zero which represents a function which has distinct values at distinct points of the unit disk. A source of estimates is an expansion theorem for the Löwner equations which is obtained from a study of contractive substitutions in Hilbert spaces of analytic functions. The methods are an outgrowth of the theory of square summable power series [1]. Assume that σ_n is a given function of nonnegative integers n, with nonnegative values, such that σ₀ = 0 and such that σ_{n ? 1} ? σ_n when n is positive. Infinite values are allowed. The underlying Hilbert space is the set $\text{G}$_σ(0) of equivalence classes of power series f(z) = ∑ a_nzⁿ with constant coefficient zero such that f(z)²_{$\text{G}$σ⁽⁰⁾} = ∑(n/σ_n)|a_n|² is finite. Equivalence of power series f(z) and g(z) means that the coefficient of zⁿ in f(z) is equal to the coefficient of zⁿ in g(z) when σ_n is finite.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f(u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f(u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.     

On locally repeated values of certain arithmetic functions,I

Let ν(n) denote the number of distinct prime factors of n. We show that the equation n + ν(n) = m + ν(m) has many solutions with n ≠ m. We also show that if ν is replaced by an arbitrary, integer-valued function f with certain properties assumed about its average order, then the equation n + f(n) = m + f(m) has infinitely many solutions with n ≠ m.