On the optimal approximation rates for criss-cross finite element spaces

Let Δ denote the triangulation of the plane obtained by multi-integer translates of the four lines x=0, y=0, x=y and x=?y. By $\text{l}{}_{\mathrm{k,\; h}}{}^{\mathrm{\mu }}$ we mean the space of all piecewise polynomials of degree ?k with respect to the scaled triangulation hΔ having continuous partial derivatives of order $\text{?μ}\text{on}\text{R}{}^2$. We show that the approximation properties of $\text{l}{}_{\mathrm{k,\; h}}{}^{\mathrm{\mu }}$ are completely governed by those of the space spanned by the translates of all so called box splines contained in $\text{l}{}_{\mathrm{k,h}}{}^{\mathrm{\mu }}$. Combining this fact with Fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where μ is the largest degree of smoothness for which these spaces are dense as h tends to zero. Furthermore, we study the question of local linear dependence of the translates of the box splines for the above criss-cross triangulations.     

On a family of connectives for fuzzy sets

In this paper we find the general solution of the functional equation $\text{S}{}^1\text{(T(x, y), T(x, N(y))) = x}$, where $\text{S}{}^1$ is a t-conorm, T is a t-norm and N is a strong negation on the unit interval. In particular the result yields a family of connectives for fuzzy sets.     

Improved error bounds for the Liouville-Green (or WKB) approximation

A variant of the Gronwall-Bellman inequality is used to develop new bounds on solutions to the fundamental singular integral equations that arise in the error analysis of the Liouville-Green (or WKB) approximation. Consequently, new improved error bounds are obtained from the Liouville-Green approximation to the solution of the differential equation $\text{d}{}^2\text{W}\text{dT}{}^2\text{? q(T) W = 0}$, where the function q is real and twice continuously differentiable and does not vanish.     

Generalization of Fenchel's duality theorem for convex vector optimization

For a vector-valued function f, Sup f and Inf f are defined from the Yu's domination theory and the Pareto's efficiency. A notion of conjugate is proposed for convex vector-valued function, this construction gives once more the usual conjugate function: $\text{f}{}^1\text{(x}{}^1\text{) =}\text{sup}\text{[〈x}{}^1\text{, x〉 ? f(x)]}$ when the function f is scalar. Then, this concept is used to write the Fenchel's problem in convex multiple objective optimization and to prove the associated duality theorem.     

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.     

Duality theorems for linear systems and convex systems

We show that, if (F →^uX) is a linear system, $\text{Ω ? X}$ a convex target set and $\text{h: X →}\text{R}\text{?}$ a convex functional, then, under suitable assumptions, the computation of inf $\text{h({y ? F ¦ u(y) ? Ω}}$) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of $\text{u}{}^1\text{(X}{}^1\text{)}$, or of the infima on F of Lagrangians, involving elements of $\text{u}{}^1\text{(X}{}^1\text{)}$. Also, we prove similar results for a convex system (F →^uX) and the convex cone Ω of all non-positive elements in X.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

La valeur optimale des programmes entiersThe optimal value of integer programs

We present a formula for the optimal value f_c(y) of the integer program $\text{max}\text{{c′x∣x∈}\text{Ω}\text{(y)∩}\text{N}{}^{\mathrm{n}}\text{}}$ where $\text{Ω}\text{(y)}$ is the convex polyhedron $\text{{x∈}\text{R}{}^{\mathrm{n}}\text{∣Ax=y,}\text{x?0}}$. It is a consequence of Brion and Vergne's formula which evaluates the sum . As in linear programming, f_c(y) can be obtained by inspection of the reduced-costs at the vertices of the polyhedron. We also provide an explicit result that relates f_c(ty) and the optimal value of the associated continous linear program, for large values of $\text{t∈}\text{N}$. To cite this article: J.B. Lasserre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 863–866.     

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 $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

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

An algorithm for the electromagnetic scattering due to an axially symmetric body with an impedance boundary condition

Let B be a body in R³ and let S denote the boundary of B. The surface S is described by $\text{S = {(x, y, z): (x}{}^2\text{+ y}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= f(z), ?1 ? z ? 1}}$, where f is an analytic function that is real and positive on (?1, 1) and f(±1) = 0. An algorithm is described for computing the scattered field due to a plane wave incident field, under Leontovich boundary conditions. The Galerkin method of solution used here leads to a block diagonal matrix involving 2M + 1 blocks, each block being of order 2(2N + 1). If, e.g., N = O(M²), the computed scattered field is accurate to within an error bounded by $\text{Ce}{}^{\mathrm{<mtext>?cN</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where C and c are positive constants depending only on f.     

The optimal solution to the problem of complex extrapolation of a first-order scheme

Given the linear stationary first-order iterative scheme $\text{x(}{}^{\mathrm{m+1}}\text{= Tx(}{}^{\mathrm{m}}\text{+ c}$ for the solution of the linear complex system (I ? T)x = c, its extrapolated complex scheme $\text{x(}{}^{\mathrm{m+1}}\text{) = T}{}_{\mathrm{\omega }}\text{x(}{}^{\mathrm{m}}\text{) + ωc [T}{}_{\mathrm{\omega }}\text{≡ (1 ? ω)I + ωT]}$ is considered. The problem which is studied and solved is that of determining an optimum value for ω, over the set of complex numbers, such that the extrapolated scheme considered converges asymptotically as fast as possible.     

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.     

Some aspects of the finite moment problem

The approximate solution of the finite moment problem $\text{μ}{}_{\mathrm{k}}\text{= ∫}{}_0{}^1\text{x}{}^{\mathrm{k?1}}\text{?(x)}\text{d}\text{x}$, k = 1, 2, 3, …, is considered. This problem is related to the problem of finding a best polynomial least squares approximation to a given function $\text{?(x)}$ in [0,1]. The connection with Laplace transform inversion is emphasized, and a set of special square matrices with integral elements is introduced, which has an intimate relation to the above two problems. These matrices are the well-known inverses of finite segments of the infinite Hilbert matrix.     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.     

Algebraic braided model of the affine line and difference calculus on a topological spaceModèle algébrique tressé de la droite affine et calcul aux différences sur un espace topologique

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions $\text{f=f(x):}\text{Z}\text{→k}$ (with an arbitrary commutative ring k) which are constant when x?+∞ or x??∞ and their “differentials” considered as functions x?f(x+1)?f(x) (difference calculus). Thanks to this new “differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra $\text{D}{}^1\text{(X)}$ which is similar in spirit to the algebra $\text{W}{}^1\text{(X)}$ introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of $\text{D}{}^1\text{(X)}$. In particular, if $\text{k=}\text{Z}$, we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.     

Bounce law at the corners of convex billiards

Let C be a convex subset of $\text{R}{}^{\mathrm{n}}$. Given any elastic shock solution x(·) of the differential inclusion$\text{x}\text{̈}\text{(t)+N}{}_{\mathrm{C}}\text{(x(t))∋0,}\text{t>0,}$the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the corners of the boundary. For that purpose, we define a sequence (C_ε) of regular sets tending to C as ε→0, then we consider the approximate differential inclusion , and finally we pass to the limit when ε→0. For approximate sets defined by $\text{C}{}_{\mathrm{\epsilon }}\text{=C+ε}\text{B}$ (where $\text{B}$ is the unit euclidean ball of $\text{R}{}^{\mathrm{n}}$), we recover the bounce law associated with the Moreau–Yosida regularization.     

Nonclassical eigenvalue asymptotics

The leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {$\text{(x,y)∥x¦}{}^{\mathrm{\mu }}\text{¦y¦? 1}$} and for $\text{?Δ + ¦x¦}{}^{\mathrm{\alpha }}\text{¦ y¦}{}^{\mathrm{\beta }}$ all of which are non-Weyl because of infinite phase space volumes are computed. Along the way, a general inequality on quantum partition functions computed in a kind of Born-Oppenheimer approximation is proved.     

Nonstandard points on algebraic curves

Let Γ be an algebraic curve which is given by an equation f(x, y) = 0, f(x, y) ∈ k[x, y] where k is an algebraic number field and f(x, y) is irreducible. Suppose that there exists an ${}^1\text{Γ}$ a nonstandard point $\text{(ξ, η) ∈}{}^1\text{k ×}{}^1\text{k}$. Then k(ξ, η) is (isomorphic to) the algebraic function field of Γ and, at the same time, is a subfield of ${}^1\text{k}$. Correlating the divisors of the function field k(ξ, η) and of the number field ${}^1\text{k}$, we develop an analogue of the Artin-Whaples theory of the product formula. This leads to one of Siegel's basic inequalities for rational points on algebraic curves.