Some asymptotic formulas on generalized divisor functions,II

Let A be an infinite sequence of positive integers a₁ < a₂ <… and put $\text{f}{}_{\mathrm{A}}\text{(x) =}\text{Σ}{}_{\mathrm{a\in A,\; a\le x}}\text{(}\text{1}\text{a}\text{)}$, $\text{D}{}_{\mathrm{A}}\text{(x) =}\text{max}{}_{\mathrm{1\le n\le x}}\text{Σ}{}_{\mathrm{<mtext>a\in A,</mtext><mtext>a</mtext><mtext>n</mtext>}}\text{1}$. In Part I, it was proved that $\text{lim}{}_{\mathrm{x\to +\infty }}\text{sup}\text{D}{}_{\mathrm{A}}\text{(x)}\text{f}{}_{\mathrm{A}}\text{(x)}\text{= +∞}$. In this paper, this theorem is sharpened by estimating D_A(x) in terms of f_A(x). It is shown that lim_x→+∞sup D_A(x) exp(?c₁(logf_A(x))²) = +∞ and that this assertion is not true if c₁ is replaced by a large constant c₂.     

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: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

“Integer-making” theorems

Let X = {x₁, x₂,…} be a finite set and associate to every x_i a real number α_i. Let f(n) [g (n)] be the least value such that given any family $\text{F}$ of subsets of X having maximum degree n [cardinality n], one can find integers α_i, i=1,2,… so that α_i ? α_i|<1 and

$\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤?(n)}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤}\text{g(n)}$

for all $\text{E ?}\text{F}$. We prove

$\text{f(n)≤n ? 1 and g(n)≤c(n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

An asymptotic evaluation of the cycle index of a symmetric group

Let $\text{Z(S}{}_{\mathrm{n}}\text{;?(x))}$ denote the polynomial obtained from the cycle index of the symmetric group Z(S_n) by replacing each variable s_i by f(x¹). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=x^k + a_k+1x^k+1 + a_k+2x^k+2+? with every a₁?0. Finally, let 0<x<1 and let x??. We prove that

This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of $\text{1}\text{n!e}$.In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented.     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

On differences and sums of integers,I

A set {b₁,b₂,…,b_i} ? {1,2,…,N} is said to be a difference intersector set if {a₁,a₂,…,a_s} ? {1,2,…,N}, j > ?N imply the solvability of the equation a_x ? a_y = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b₁,b₂,…,b_i} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations ($\text{a}{}_{\mathrm{x}}\text{?}\text{a}{}_{\mathrm{y}}\text{p}\text{= + 1}$, $\text{(a}{}_{\mathrm{u}}\text{?}\text{a}{}_{\mathrm{v}}\text{p}\text{) = ? 1}$, $\text{(a}{}_{\mathrm{r}}\text{+}\text{a}{}_{\mathrm{s}}\text{p}\text{) = + 1}$, $\text{(a}{}_{\mathrm{t}}\text{+}\text{a}{}_{\mathrm{z}}\text{p}\text{) = ? 1}$ (where ($\text{a}\text{p}$) denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets.     

Comparaison de deux indicateurs de la régularité en variation des queues de distributionsComparison between two indicators for the variation regularity of tails of distributions

We compare assumptions used in [4] in order to study the rate of convergence to 0, as u→s₊(F), of , where $\text{F}{}_{\mathrm{u}}$ is the survival function of the excesses over u, s₊(F)=sup{x,F(x)<1} is the upper end point of the distribution function (d.f.) F and $\text{G}{}_{\mathrm{\gamma }}$ is the survival function of the Generalized Pareto Distribution, with assumptions used in [2] in order to study the rate of convergence to 0, as n→+∞, of $\text{d}\text{?}{}_{\mathrm{n}}\text{=}\text{sup}{}_{\mathrm{<mtext>x\in </mtext><mtext>R</mtext>}}\text{|F}{}^{\mathrm{n}}\text{(x)?H}{}_{\mathrm{\gamma }}\text{(}\text{x?α}{}_{\mathrm{n}}\text{σ}{}_{\mathrm{n}}\text{)|}$, where H_γ is the d.f. of an extreme value distribution. In each case, an indicator linked to regular variation assumptions had been introduced. We characterize situations where these two indicators coincide, and others where they are different. To cite this article: R. Worms, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 709–712.     

Orthogonal polynomials on the negative multinomial distribution

Orthogonal polynomials on the multivariate negative binomial distribution,

$\text{(1 + Θ)}{}^{\mathrm{?\alpha ?x}}\text{(}\text{π}\text{j=0}\text{p}\text{Θ}{}_{\mathrm{j}}{}^{\mathrm{x}}\text{j}\text{x}{}_{\mathrm{j}}\text{!}\text{)}\text{Γ(α + x)}\text{Γ(α)}$

where α > 0, Θ₁ > 0, x = ΣΘ_i, x₀, x₁, …, x_p = 0,1, … are constructed and their properties studied.     

Positive dependence properties of elliptically symmetric distributions

Let X₁, …, X_p have p.d.f. g(x₁² + … + x_p²). It is shown that (a) X₁, …, X_p are positively lower orthant dependent or positively upper orthant dependent if, and only if, X₁,…, X_p are i.i.d. N(0, σ²); and (b) the p.d.f. of |X₁|,…, |X_p| is TP₂ in pairs if, and only if, In g(u) is convex. Let X₁, X₂ have p.d.f. $\text{f(x}{}_1\text{, x}{}_2\text{) =}\text{|Σ|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{g((x}{}_1\text{, x}{}_2\text{)}\text{Σ}{}^{\mathrm{?1}}\text{(x}{}_1\text{, x}{}_2\text{)′)}$. Necessary and sufficient conditions are given for f(x₁, x₂) to be TP₂ for fixed correlation ?. It is shown that if f is TP₂ for all ? >0. then (X₁, X₂)′ ～ N(0, Σ). Related positive dependence results and applications are also considered.     

The existence,uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system $\text{?x}\text{?t}\text{= Δx + F(x,y),}\text{?y}\text{?t}\text{= G(x,y)}$ is investigated, where x and y are scalar functions of time (t ? 0), and n space variables $\text{(ξ}{}_1\text{,…, ξ}{}_{\mathrm{n}}\text{), Δx ≡ ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{x}\text{?ξ}{}_{\mathrm{i}}{}^2$, and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution $\text{(x(r),y(r)),}\text{where}\text{r = (ξ}{}_1{}^2\text{+ … + ξ}{}_{\mathrm{n}}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which is bounded for r ? 0 and satisfies x(0) >x₀, y(0) > y₀, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.     

Note on lower bounds for the rank of a matrix

The following results are proved: Let A = (a_ij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f₁,…,f_n) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have

$\text{Π}\text{i∈S}\text{f}{}_{\mathrm{i}}\text{(A)<}\text{Π}\text{i∈S}\text{|a}{}_{\mathrm{ii}}\text{|,}$

then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1.     

Hilbert space representations of general discrete time stochastic processes

We show that under mild conditions the joint densities Px₁,…,x_n) of the general discrete time stochastic process X_n on $\text{pH}$ can be computed via

$\text{Px}{}_1\text{,…,x}{}_{\mathrm{n}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{) = 6?T(x}{}_1\text{)…T(x}{}_{\mathrm{n}}\text{)6}{}^2$

where ? is in a Hilbert space $\text{pH}$, and T (x), x ? $\text{pH}$ are linear operators on $\text{pH}$. We then show how the Central Limit Theorem can easily be derived from such representations.     

Compositions of singular integral operators

The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L¹(Rⁿ) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on R^{n + 1} homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x₀ = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε₁) Sing(ψ, ε₂)f = ?2π²(∝ φ)(∝ ψ)f + Sing(A, ε₁, + ε₂)f, where

(with notation $\text{¦t¦}{}_0{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{and}\text{¦t¦}{}_1{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{sgn}\text{t}$). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol

$\text{ω}{}_0\text{(x,?)=?πiθ(?)∫θ(x?y)}\text{sgn}\text{y·?dy}\text{if}\text{?}{}_1\text{=1}$

,

$\text{=?2θ(?)∫θ(x?y)}\text{log}\text{|y·?|dy}\text{if}\text{?}{}_1\text{=0}$

where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity.     

A survey of published programs for best approximation

We list and discuss published programs for best approximation of functions by linear and nonlinear families in all standard forms.In this note we list and discuss the published programs for obtaining best approximations. Let X be a set on which we wish to approximate. Most sets will be finite (an equivalent terms is discrete). Let 6 6 be a norm on the continuous functions on X. Let G be a familiy of continuous functions on X. For a given basis {φ₁…,φ_n}, the linear family G is the set of all functions of the form

$\text{L(A,x)=}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{φ}{}_{\mathrm{k}}\text{(x)}$

The problem of best approximation is given a continuous function f, to find g¹ to minimize e(g) = ∥f-g∥ over g?G. Such g¹ is called a best approximation to f.Discrete linear approximation problems are sometimes formulated as solution of an overdetermined system of linear equations Ax = b with respect to a norm where a_ij = φ_j(x_i) and b_i = f(x)_i).     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].