Le nombre des diviseurs d'un entier dans les progressions arithmétiques

Let d_k,?(n) be the function number of divisors of the integer n?1, in arithmetic progressions {?+mk}, with 1???k and ?,k coprime, and let F(n;k,?) defined as follows:

$\text{F(n;k,?)=}\text{ln}\text{d}{}_{\mathrm{k,?}}\text{(n)}\text{ln}\text{(?(k)}\text{ln}\text{n)}\text{ln}\text{2}\text{ln}\text{n}\text{.}$

In this Note, we study and give the structure of d_k,?-superior, highly composite numbers, which generalize those defined by S. Ramanujan. We prove that F(n;k,?) reaches its maximum among these numbers. We give it explicitly for k=2,…,13. This generalizes the study of Nicolas and Robin, in which the case k=1 is treated. To cite this article: A. Derbal, A. Smati, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the control of stochastic systems

A general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the “iterative” method of the authors or its more recent adaptations for stochastic operator equations. It is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations; the method, therefore, is applicable to the control problem. Application is made to the state space equation $\text{x}\text{?}\text{= Ax + Bu + C,}\text{where}\text{A, B, C}$ are stochastic matrices corresponding to stochastic operators, i.e., involving randomly time varying elements, e.g., $\text{a}{}_{\mathrm{ij}}\text{(t, ω) ? A, t ? T, ω ? (Ω, F,μ)}$, a p.s. It is emphasized that the processes are arbitrary stochatic processes with known statistics. No assumption is made of Wiener or Markov behavior or of smallness of fluctuations and no closure approximations are necessary. The method differs in interesting aspects from Volterra series expansions used by Wiener and others and has advantages over the other methods. Because of recent progress in the solution of the nonlinear case, it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages.     

An ∞-dimensional inhomogeneous Langevin's equation

On a modified space Φ′ from the space $\text{J}$′ of tempered distributions, it is proven that a stochastic equation, $\text{X(t) = γ + W(t) + ∝}{}_0{}^{\mathrm{t}}\text{L}{}^1\text{(s) X(s) ds}$, has a unique solution, where W(t) is a Φ′-valued Brownian motion independent of a Φ′-valued Gaussian random variable γ and $\text{L}{}^1\text{(s)}$ is an integro-differential operator. As an application, a fluctuaton result (or central limit theorem) is shown for interacting diffusions.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

A note on Ramsey numbers

Upper bounds are found for the Ramsey function. We prove $\text{R(3, x) <}\text{cx}{}^2\text{ln}\text{x}$ and, for each k ? 3, $\text{R(k, x) <}\text{c}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k\; ?\; 1}}\text{(}\text{ln}\text{x)}{}^{\mathrm{k\; ?\; 2}}$ asymptotically in x.     

Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

Existence and uniqueness of statistical measures for solution processes for linear-stochastic differential equations

Structure is developed on the set of real-valued stochastic processes in terms of the authors recently defined statistical measures making explicit an $\text{L}{}_{\mathrm{p}}{}^{\mathrm{n}}\text{(Ω, T)}$-calculus over the structure. This proves that the stochastic-differential equation $\text{L}$y=x, where x is a stochastic process and $\text{L}$ is an nth order linear-stochastic differential operator with up to n ? 1 stochastic-process coefficients, is solved by Adomian's series, and finally, establishes the existence and uniqueness of the statistical measures of the solution process.     

Operator-theoretic solution of stochastic systems

A (stochastic) operator-theoretic approach leads to expresssions for inverses of linear and nonlinear stochastic operators—useful for the solution of linear or nonlinear stochastic differential equations. Operator equations are developed for inverses of linear or nonlinear stochastic operators. Series expressions are obtained which allow writing the solution y=$\text{F}$^?1x of the operator equation $\text{F}$y=x. Special cases are studied in which $\text{F}$ may be linear or nonlinear, deterministic or stochastic in various combinations.     

Representation of functions of Markov processes as solutions of stochastic equations

Let X_t be a homogeneous Markov process generated by the weak infinitesimal operator A. Let $\text{H}$ be the class of functions f such that f, f²?D_A, the domain of A. The main result of this paper states that for ? ∈ $\text{H}$ can be represented by a stochastic integral and other terms. If the process is generated by a second order differential operator (with ‘poor’ coefficients possibly) on C₀²(R^d) then the process itself can be represented as the solution of an Itô stochastic differential equation.     

Refined analysis and improvements on some factoring algorithms

By combining the principles of known factoring algorithms we obtain some improved algorithms which by heuristic arguments all have a time bound $\text{O(}\text{exp}\text{c}\text{ln}\text{n}\text{ln ln}\text{n}$) for various constants c ?3. In particular, Miller's method of solving index equations and Shanks' method of computing ambiguous quadratic forms with discriminant ?n can be modified in this way. We show how to speed up the factorization of n by using preprocessed lists of those numbers in [?u, u] and [n - u, n + u], 0 ? u ? n which only have small prime factors. These lists can be uniformly used for the factorization of all numbers in [n - u, n + u]. Given these lists, factorization takes $\text{O(}\text{exp[2(ln}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{ln ln}\text{n)}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{]}$ steps. We slightly improve Dixon's rigorous analysis of his Monte Carlo factoring algorithm. We prove that this algorithm with probability $\text{sol1}\text{2}$ detects a proper factor of every composite n within $\text{o(}\text{exp}\text{6}\text{ln}\text{n}\text{ln ln}\text{n}$) steps.     

Stability conditions for linear retarded functional differential equations

The purpose of this note is to study the exponential stability for the linear retarded functional differential equation $\text{x}\text{?}\text{(t) = ∫}{}_{\mathrm{?1}}{}^0\text{[dη(θ)] x(t ? r(θ))}$, where the delay function r(θ) ? 0 is continuous and η(θ) is of bounded variation on the interval [?1, 0]. It is shown that the spectral limit function for the equation above has a continuous dependence on the pair (η, r). The set of all functions of bounded variation η for which the equation above is exponentially stable for every delay function r, the so-called region of stability globally in the delays, is a cone. Therefore for a fixed r, the set of all η which make our equation exponentially stable, that is, the region of stability for the delay function r, contains a cone. A discussion of the characterization of these regions of stability, as well as of the largest cone contained in each region of stability for a fixed delay function r, is given. Some remarks are made with respect to a similar question for the equation $\text{x}\text{?}\text{(t) = Ax(t) + ∫}{}_{\mathrm{?\; 1}}{}^0\text{[dμ(θ)] x(t?r(θ))}$, where A is a real n by n matrix, μ(θ) is bounded variation on [?1, 0] and r(θ) as before. Several examples illustrate the results obtained.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

Does Huygens' principle hold for small perturbations of the wave equation?

The perturbed wave equation □u + q(x)u = 0 in $\text{R}$³ × $\text{R}$ with C∞ ($\text{R}$³) compactly supported initial data at t = 0 is considered. It is proven that the Huygens' principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.     

Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems

We consider a spatially homogeneous system of reaction-diffusion equation defined on the interval (?∞, ∞) of the one-dimensional spatial variable x. It is known that this equation has a one-parameter family of periodic travelling wave solutions Ψ(x + ct; c) if this equation has a spatially homogeneous periodic solution φ(t). The spatial period L(c) of the travelling wave solution satisfies $\text{L(c)}\text{c}\text{→ T}\text{if}\text{c → +∞}$, where c is the propagation speed and T is the period of φ(t). We prove that, in the case c > 0 is sufficiently large, Ψ(x + ct; c) is unstable if φ(t) is “strongly unstable” and Ψ(x + ct; c) is “marginally stable” if φ(t) is “strongly stable.” If the equation is defined on a finite interval [0, l] of the variable x with the periodic boundary conditions, we can obtain a more precise result regarding the stability of $\text{Ψ(x +}\text{c}\text{?}\text{t;}\text{c}\text{?}\text{)}$, where $\text{c}\text{?}\text{> 0}$ is a speed which satisfies $\text{l = mL(}\text{c}\text{?}\text{)}$ for some positive integer m. We prove that this solution is asymptotically stable in the sense of waveform stability if $\text{c}\text{?}\text{> 0}$ is sufficiently large and if φ(t) is “strongly stable.”     

Invariant imbedding applied to a class of freholm integral equations

The objective of this paper is the rigorous derivation of an invariant imbedding algorithm for the solution of the integral equation

$\text{ø(z)g(z)+γ?}{}^{\mathrm{\infty }}{}_0\text{K}\text{(∣z-z′∣)ø(z′)dz′}$

for

$\text{z?0}$

, under suitable restrictions on g, K, and γ. First a set of conditions is determined under which Eq. (1) has a unique solution. The function ø(z) is shown to be approximated almost uniformly for Y=0 and as x→∞ by the solution of     

Geometry of the numerical range of matrices

Some techniques for the study of the algebraic curve C(A) which generates the numerical range W(A) of an n×n matrix A as its convex hull are developed. These enable one to give an explicit point equation of C(A) and a formula for the curvature of C(A) at a boundary point of W(A). Applied to the case of a nonnegative matrix A, a simple relation is found between the curvature of the function Φ(A)=p((1?α)A+ αA^T) (pbeingthePerronroot) at $\text{α=}\text{1}\text{2}$ and the curvature of W(A) at the Perron root of $\text{1}\text{2}\text{(A+A}{}^{\mathrm{T}}\text{)}$. A connection with 2-dimensional pencils of Hermitian matrices is mentioned and a conjecture formulated.     

The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy

Let M be the n-dimensional Minkowski space, n ? 3. One consequence of [1] is that the null space of the equation $\text{{(n ? 2k + 2)d}{}^1\text{d + (n ? 2k ? 2)dd}{}^1\text{} Φ = 0}$ on differential k-forms Φ in M is conformally covariant. The same is true of a nonlinear equation obtained by adding to the above a term homogeneous of degree $\text{(n + 2)}\text{(n ? 2)}$. This generalizes the well-known conformal covariance properties of the wave equation and the equations $\text{φ ± φ}{}^{\mathrm{<mtext>(n\; +\; 2)</mtext><mtext>(n\; ?\; 2)</mtext>}}\text{= 0}$ when k = 0, and of Maxwell's equations on a vector potential when $\text{k =}\text{(n ± 2)}\text{2}$ (and n is even). We define a natural (conformally invariant) symplectic structure for the new equations, and use it to calculate the $\text{(n + 1)(n + 2)}\text{2}$ conserved quantities corresponding to the standard conformal group generators.     

Uniqueness for algebraically regular solutions to pathological differential equations

A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices $\text{M}$ with standard norm. It is not assumed that $\text{A}$ ∈ $\text{M}$, but it is required that A have an inverse in $\text{M}$, a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary.The theorem may be easily generalized to any Banach algebra with identity.     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.