Sufficient Conditions and Duality for a Class of Multiobjective Fractional Programming Problems with Higher-Order (F,α,ρ,d)-Convexity

In this paper, a class of multiobjective fractional programming problems (denoted by (MFP)) is considered. First, the concept of higher-order (F,α,ρ,d)-convexity of a function f:C→R with respect to the differentiable function φ:R ⁿ ×R ⁿ →R is introduced, where C is an open convex set in R ⁿ and α:C×C→R ₊?{0} is a positive value function. And an important property, which the ratio of higher-order (F,α,ρ,d)-convex functions is also higher-order (F,α ^′,ρ ^′,d ^′)-convex, is proved. Under the higher-order (F,α,ρ,d)-convexity assumptions, an alternative theorem is also given. Then, some sufficient conditions characterizing properly (or weakly) efficient solutions of (MFP) are obtained from the above property and alternative theorem. Finally, a class of dual problems is formulated and appropriate duality theorems are proved.     

Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=r^γ for γ>0, γ2 or φ(r)=r^γ ln r for γ2 ₊. For each positive integer N, let h=N⁻¹ and let {x_i: i =1, 2, …, (N+1)^d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]^d. Given f: [0, 1]^d→ , let s_h be its unique RBF interpolant at the grid vertices: s_h(x_i)=f(x_i), i=1, 2, …, (N+1)^d. For h→0, we show that the uniform norm of the error f−s_h on a compact subset K of the interior of [0, 1]^d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid h ^d, provided that f is a data function whose partial derivatives in the interior of [0, 1]^d up to a certain order can be extended to Lipschitz functions on [0, 1]^d.     

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (V_n, g) be a C^∞ compact Riemannian manifold. For a suitable function on V_n, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦⁻¹. Using the method of continuity, we first solve in C^∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C^∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C^∞(V_n × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Positive periodic solutions of higher-dimensional functional difference equations with a parameter

By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A(n)=diag[a₁(n),a₂(n),…,a_m(n)], h(n)=diag[h₁(n),h₂(n),…,h_m(n)], a_j,h_j :Z→R⁺, τ :Z→Z are T -periodic, j=1,2,…,m, T1, λ>0, x :Z→R^m, f :R₊^m→R₊^m, where R₊^m={(x₁,…,x_m)^TR^m, x_j0, j=1,2,…,m}, R⁺={xR, x>0}.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

The Faber Operator and its Boundedness

Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A( ) to A(G) mapping wⁿ onto the nth Faber polynomial F_n(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

Stationary distributions of the simplest dynamical systems in R perturbed by generalized Poisson process

Let z(t) Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Null spaces and ranges for the classical and spherical Radon transforms

Let R be the classical Radon transform that integrates a function over hyperplanes in Rⁿ and let SM be the transform that integrates a function over spheres containing the origin in Rⁿ. We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in Rⁿ as well as the null space of SM for square integrable functions on a ball. We show SM: L²(Rⁿ) → L²(Rⁿ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g(x) is a Schwartz function on Rⁿ that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM(C^∞(Rⁿ)). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

An Entire Holomorphic Function Associated to an Entire Harmonic Function

Let h be a harmonic function on ^N. Then there exists a holomorphic function f on such that f(t)=h(t, 0, …, 0) for all real t. Precise inequalities relating the growth rate of f to that of h are proved. These results are applied to deduce uniqueness theorems for harmonic functions of sufficiently slow growth that vanish at certain lattice points. Another application concerns the rate at which a harmonic function of finite order can decay along a ray.     

On <Emphasis Type="Italic">n</Emphasis>-commuting and <Emphasis Type="Italic">n</Emphasis>-skew-commuting maps with generalized derivations in prime and semiprime rings

Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h: R → R is called n-centralizing (n-commuting) on a subset S of R if [h(x),x ⁿ ] ∈ Z(R) ([h(x),x ⁿ ] = 0 respectively) for all x ∈ S. The following are proved:

Asymptotic Properties of Zeros of Hypergeometric Polynomials

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |z^k(1−z)|=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.     

Variations of the norms of differential operators in spaces of entire functions

The paper is devoted to the investigation of the norms of the operators F(D), where , in the Banach spaces B_k, consisting of the restrictions toR ⁿ of the entire functions inC ⁿ , whose Fourier transforms, understood in the sense of distribution theory, are concentrated on a compactum K R ⁿ . The class from whichone recruits the functions F is the class of the Fourier transforms of regular Borel measures with finite variation onR ⁿ .Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 86–93, 1988.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

On optimal polynomial interpolation of analytic functions

This paper discusses the problem of choosing the Lagrange interpolation points T = (t₀, t₁,…, t_n) in the interval −1 t 1 to minimize the norm of the error, considered as an operator from the Hardy space H²(R) of analytic functions to the space C[−1, 1]. It is shown that such optimal choices converge for fixed n, as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R. These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also.