Existence of an Almost Periodic Solution in a Difference Equation with Infinite Delay

In order to obtain the existence of an almost periodic functional difference equation x(n + 1) = ?(n,x_n ),n ∈ Z ⁺ and where x_n is defined by x_n (s) = x(n + s) for s ∈ Z ^?, on an axiomatic phase space B, we consider a certain stability property, which is referred to as BS-stable under disturbances from Ω(?) with respect to K, this stability implies ρ-stable under disturbances from Ω(?) with respect to compact set K.     

Extension of holomorphic and pluriharmonic functions with thin singularities on parallel sections

The paper is of survey character. We present and discuss recent results concerning the extension of functions that admit holomorphic or plurisubharmonic extension in a fixed direction. These results are closely related to Hartogs’ fundamental theorem, which states that if a function f(z), z = (z ₁, z ₂, ..., z _z ), is holomorphic in a domain D ? ?ⁿ in each variable z _j , then it is holomorphic in D in the n-variable sense.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

Differential Inequalities and Starlikeness

We study some differential inequalities in the unit disc which imply starlikeness: for example if ? (z) = z + Σ^∞ _n=2 a_n (?)zⁿ is analytic in D = {z | |z| < 1} and |z(?′′(z) ? 1)| ≤ 1 ? α, z?D for some α ] [0, 1), then ? is one-to-one on D and ? (D) is a starlike domain with respect to the origin.     

Classification(s) of Danielewski hypersurfaces

The Danielewski hypersurfaces are the hypersurfaces X _Q,n in \mathbbC³ {\mathbb{C}^3} defined by an equation of the form x ⁿ y = Q(x, z) where n ⩾ 1 and Q(x, z) is a polynomial such that Q(0, z) is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface X _Q,n with n ⩾ 2 admits at least two nonequivalent embeddings into \mathbbC³ {\mathbb{C}^3} .     

Functions of Two Variables with Large Tangent Plane Sets

We show that there exist a C¹function,f, of two variables and a setE ⊆ R²of zero Lebesgue measure such that using the natural three-dimensional parametrization of planesz = ax + by + ctangent to the surfacez = f(x, y), the (three-dimensional) interior of the set of parameter values, (a, b, c), of tangent planes corresponding to points (x, y) inEis nonempty. From the Morse–Sard theorem it follows that there are no such C²functions. We also study briefly the relationship of our example with the Denjoy–Young–Saks theorem.     

Some solutions (linear in the spatial variables) and generalized Chapman–Enskog functions basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

A Legendre expansion of the (matrix) scattering kernel relevant to the (vector- valued) linearized Boltzmann equation for a binary mixture of rigid spheres is used to define twelve solutions that are linear in the spatial variables {x, y, z}. The twelve (asymptotic) solutions are expressed in terms of three vector-valued functions A ⁽¹⁾(c), A⁽²⁾(c), and B(c). These functions are generalizations of the Chapman–Enskog functions used to define asymptotic solutions and viscosity and heat conduction coefficients for the case of a single-species gas. To provide evidence that the three Chapman–Enskog vectors exist as solutions of the defining linear integral equations, numerical results developed in terms of expansions based on Hermite cubic splines and a collocation scheme are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations.     

Some solutions (linear in the spatial variables) and generalized Chapman–Enskog functions basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

A Legendre expansion of the (matrix) scattering kernel relevant to the (vector- valued) linearized Boltzmann equation for a binary mixture of rigid spheres is used to define twelve solutions that are linear in the spatial variables {x, y, z}. The twelve (asymptotic) solutions are expressed in terms of three vector-valued functions A ⁽¹⁾(c), A⁽²⁾(c), and B(c). These functions are generalizations of the Chapman–Enskog functions used to define asymptotic solutions and viscosity and heat conduction coefficients for the case of a single-species gas. To provide evidence that the three Chapman–Enskog vectors exist as solutions of the defining linear integral equations, numerical results developed in terms of expansions based on Hermite cubic splines and a collocation scheme are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

The cyclic minimum correlation problem

For variables (x,y,z) in [0, 1]³, three functionsA(y,z),B(z,x),C(x,y), with values in [0, 1], are to be chosen to minimize the integral, over (x,y,z) in the unit cube, ofAB+BC+CA, subject to prescribed values for the integral of each function. It is shown that a minimum can be achieved by dividing each of thex,y,z intervals into three or fewer subintervals and taking each ofA,B,C as indicator function of the union of some of the nine (or fewer) rectangles into which this divides its domain. Several specializations and generalizations of this problem are given consideration. It can be considered as a decision problem with distributed information.     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.     

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

Analytic properties of conditional curvatures of convex hypersurfaces and the dirichlet problem for the Monge-Ampère equation

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z _ij || = ϕ(x, z, p) in Λ ⁿ , wherez = z(x ₁,...,z _n ) is a convex function,p = (p ₁,...,P _n) = (∂z/∂x ₁,...,ϖz/ϖx _n), andz _ij =ϖ ² z/ϖx _i ϖx _j. We consider the Cayley-Klein model of the space Λ ⁿ and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Irreducible Divisor Graphs II

Let R be an integral domain, and let x ∈ R be a nonzero nonunit that can be written as a product of irreducibles. Coykendall and Maney (to appear), defined the irreducible divisor graph of x, denoted G(x), as follows. The vertices of G(x) are the nonassociate irreducible divisors of x (each from a pre-chosen coset of the form π U(R) for π ∈ R irreducible). Given distinct vertices y and z, we put an edge between y and z if and only if yz|x. Finally, if y ⁿ |x but y ⁿ⁺¹ ? x, then we put n ? 1 loops on the vertex y.

In this article, inspired by the approach of the authors from Akhtar and Lee (to appear Akhtar , R. , Lee , L. Homology of zero divisors . To appear in Rocky Mountain J. Math.  [Google Scholar]), we study G(x) using homology. A connection is found between H ₁ and the cycle space of G(x). We also characterize UFDs via these homology groups.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Multipliers for logarithmic Cauchy integrals in the ball

Let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥ 1. Let K \mathcal{K} ₀(n) denote the class of functions defined for z ∈ B _n as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈ \mathbbC \mathbb{C} ⁿ ,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K} ₀(n) for all f ∈ K \mathcal{K} ₀(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K} ₀(n), n ≥ 1. Bibliography: 15 titles.     

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

Abstract

If P(h) is a h-pseudodifferential operator in R ⁿ associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e ^?tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.