Solutions of some functional equation bounded on nonzero Christensen measurable sets

Let n be a positive integer. We characterize solutions f: X → ? of the equation f (x + f(x) ⁿ y = f(x)f(y) mapping a real separable F-space X into ?, which are bounded on nonzero Christensen measurable sets.     

Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = (X,E) be a digraph without loops or multiple edges, |X| ?3, and h be an integer ?1, if G contains a spanning arborescence and if d⁺_G(x)+d^?_G(x)+d^?_G(y)+d^?_G(y)? 2|X |?2h?1 for all x, y?X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ?h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.     

An alternative-conditional functional equation and the orthogonally additive functionals

Let X be a real inner product space of dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the recent studies made on the alternative-conditional functional equation

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Invariance principle for integral type functionals of square-integrable martingales

Let X_n = {X_n(t): 0 ⩽ t ⩽1}, n ⩾ 0, be a sequence of square-integrable martingales. The main aim of this paper is to give sufficient conditions under which ∫^·₀f_n (A_n(t), X_n(t)) dX_n(t) converges weakly in D[0, 1] to ∫^·₀f₀(A₀(t), X₀(t)) dX₀ (t) as n → ∞, where {A_n, n ⩾ 0} is some sequence of increasing processes corresponding to the sequence {X_n, n ⩾ 0}.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

On an approximation property for nondiscrete inequivalent representations of finitely generated groups

Let R(Γ, G) be the variety of representations of a finitely generated group Γ in a simple complex algebraic group G. We establish some sufficient conditions for the image of the diagonal representation ϱ = (ϱ₁, …, ϱ_t), ϱ_i ε R(Γ, G), to be dense in G^f in the complex topology (“weak approximation”).     

Stochastic gradient algorithm with random truncations

Let f : R^d × R^d → R be a Borel-measurable function which satisfies ∫_R^d′|f(θ, x) < ∞, ∨θ ϵ R^d, where q₀(·) is a probability measure on (R^d′, B^d′). The problem of minimization of the function f₀(θ) = ∫_R^d′(θ, x)q₀(d), θ ϵ R^d, is considered for the case when the probability measure q₀(·) is unknown, but a realization of a non-stationary random process {X_n}_n⩾1 whose single probability measures in a certain sense tend to q₀(·), is available. The random process {X_n}_n⩾1 is defined on a common probability space, R^′-valued, correlated and satisfies certain uniform mix conditions. The function f(·, ·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f₀(·), and its almost sure convergence is proved.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.     

Classes of matrices associated with the optimal assignment problem

Let E=[e_ij] be a matrix with integral elements, and let x be an indeterminate defined over the rational field Q. We investigate matrices of the form X=[x^e_ij] (i = 1,…, m; j = 1,…, n; m ⩽ n). We may multiply the lines (rows or columns) of the matrix X by suitable integral powers of x in various ways and thereby transform X into a matrix Y=[x^f_ij] such that the f_ij are nonnegative integers and each line of Y contains at least one element x⁰ = 1. We call Y a normalized form of X, and we denote by S(X) the class of all normalized forms associated with a given matrix X. The classes S(X) have a fascinating combinatorial structure, and the present paper is a natural outgrowth and extension of an earlier study. We introduce new concepts such as an elementary transformation called an interchange. We prove, for example, that two matrices in the same class are transformable into one another by interchanges. Our analysis of the class S(X) also yields new insights into the structure of the optimal assignments of the matrix E by way of the diagonal products of the matrix X.     

Representing measures for R(X) and their Green's functions

Let X be a compact subset of the complex plane with a nonempty interior, R(X) the uniform closure in C(X) of the rational functions with poles off X, and m a representing measure on ∂X for the functional on R(X) of evaluation at a point a in int X. Let N² be the space of functions f in L²(m) satisfying ∝ fdm = ∝ fKdm = 0 for all h in R(X), and let T be the operator on N² of multiplication by z followed by projection onto N². The spectral properties of T are investigated and shown to depend in part on the behavior of the so-called Green's function of m. In case m is the harmonic measure on ∂X for a the latter function is the classical Green's function for int X with singularity at a. Special attention is paid to the case where X is the closure of a finitely connected Jordan domain whose boundary curves are analytic. In that context, new proofs are given of Beurling's invariant subspace theorem and of Forelli's theorem on extreme points in the unit ball of the Hardy space H¹(m).     

Quantitative form of the Luzin <Emphasis Type="Italic">C</Emphasis>-property

We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t ^−a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which

Central limit theorem and increment conditions

Let φ be a convex function defined on R⁺, with φ(0) = 0 and lim_x→0φ(x)/x=0. We show that there exists a uniformly bounded process (X_t) on [0,1] with continuous sample paths that satisfies the increment condition for every u < t, E(φ(| X_t− X_u|)) ⩽ t − u. but that fails the CLT.     

Continuous on rays solutions of an equation of the Go?a?b-Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14] and J. Brzd?k (2003) [11].