On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ ℝ ⁿ if and only if the formal (m − 1)-Taylor remainder R ^m−1 f(x,y) of f satisfies the pointwise inequality |R ^m−1 f(x,y)| ≤ |x − y| ^m [a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.     

Non-isogenous superelliptic Jacobians

Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set of roots of f and that Gal(h) acts doubly transitively on as well. Let J(C_f,ℓ) and J(C_h,ℓ) be the Jacobians of the superelliptic curves C_f,ℓ:y^ℓ=f(x) and C_h,ℓ:y^ℓ=h(x) respectively. We prove that J(C_f,ℓ) and J(C_h,ℓ) are not isogenous over K_a if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity.     

On the Cohen–Lusk theorem

Let G be a finite group and X be a G-space. For a map f: X → ℝ ^m , the partial coincidence set A(f, k), k ≤ |G|, is the set of points x ∈ X such that there exist k elements g ₁,…, g _k of the group G, for which f(g ₁ x) = ⋅⋅⋅ = f(g _k x) holds. We prove that the partial coincidence set is nonempty for G = ℤ _p ⁿ under some additional assumptions. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 61–67, 2007.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

On mappings preserving pentagons

Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿ(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝⁿ→ℝⁿup to translation such that f(x) = (b/a) I(x).     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, the ring of integers in the pth cyclotomic field, C _{f, p} : y ^p  =  f(x) the corresponding superelliptic curve and J(C _{f, p} ) its jacobian. Assuming that either n  =  p + 1 or p does not divide n(n  −  1), we prove that the ring of all endomorphisms of J(C _{f, p} ) coincides with . The same is true if n  =  4, the Galois group of f(x) is the full symmetric group S ₄ and K contains a primitive pth root of unity. An erratum to this article can be found at     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.     

On the total curvatures of a tame function

Given a definable function f: ℝ ⁿ ↦ ℝ, enough differentiable, we study the continuity of the total curvature function t → K(t), total curvature of the level f ⁻¹(t), and the total absolute curvature function t → |K|(t), total absolute curvature of the level f ⁻¹(t). We show they admits at most finitely many discontinuities. Partially supported by the European research network IHP-RAAG contract number HPRN-CT-2001-00271 and partially supported by Deutsche Forschungs-Gemeinschaft in the Priority Program Global Differential Geometry.     

Existence, uniqueness and stability of C m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C ^m (J ⁿ⁺¹, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.     

Existence of solution of the pullback equation involving volume forms

Let Ω ⊂ ℝ ⁿ be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of Ω satisfying the volume form equation

Extension of Floquet's theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

Second order necessary and sufficient conditions for convex composite NDO

In convex composite NDO one studies the problem of minimizing functions of the formF:=h ○f whereh:ℝ ^m → ℝ is a finite valued convex function andf:ℝ ⁿ → ℝ ^m is continuously differentiable. This problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework, e.g. convex inclusions, minimax problems, and penalty methods for constrained optimization. In the present work we extend the second order theory developed by A.D. Ioffe in [11, 12, 13] for the case in whichh is sublinear, to arbitrary finite valued convex functionsh. Moreover, a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature.     

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.