Methods for generating random variates with Polya characteristic functions

Polya has shown that real even continuous functions that are convex on (0,∞), for 1 t = 0, and decreasing to 0 as t → ∞ are characteristic functions. Dugué and Girault (1955) have shown that the corresponding random variables are distributed as $\text{Y}\text{Z}$ where Y is a random variable with density (2π)^?1$\text{(}\text{sin}\text{(}\text{x}\text{2}\text{)/(}\text{x}\text{2}\text{))}{}^2$, and Z is independent of Y and has distribution function 1 ? φ + tφ′, t > 0. This property allows us to develop fast algorithms for this class of distributions. This is illustrated for the symmetric stable distribution, Linnik's distribution and a few other distributions. We pay special attention to the generation of Y.     

Version continue de l'algorithme d'Uzawa

In Carlier et al. (ESAIM Proceedings, CEMRACS 1999), an algorithm was proposed to approximate the projection of a function $\text{f∈H}{}^1{}_0\text{(Ω)}$ (where $\text{Ω}$ is a convex domain) onto the cone of convex functions. This algorithm is based on a dual expression of the constraint, which leads to a saddle-point problem which has no solution in general. We show here that the Uzawa algorithm for this saddle-point problem can be seen as the semi-discretization of an evolution equation

$\text{d}\text{λ}\text{d}\text{t}\text{+?Ψ(λ)?0,}$

where Ψ is a convex, l.s.c., proper function. In case the saddle-point problem has no solution, one has $\text{0∈}\text{R(?Ψ)}$ but ?Ψ^?1(0)=?. We establish that λ(t) is then divergent, and that a subsequence of the associated trajectory in the primal space converges weakly to the solution of the initial projection problem. To cite this article: B. Maury, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Optimal approximation of convex curves by functions which are piecewise linear

In this paper an efficient method is presented for solving the problem of approximation of convex curves by functions that are piecewise linear, in such a manner that the maximum absolute value of the approximation error is minimized. The method requires the curves to be convex on the approximation interval only. The boundary values of the approximation function can be either free or specified. The method is based on the property of the optimal solution to be such that each linear segment approximates the curve on its interval optimally while the optimal error is uniformly distributed among the linear segments of the approximation function. Using this method the optimal solution can be determined analytically to the full extent in certain cases, as it was done for functions x² and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In general, the optimal solution has to be computed numerically following the procedure suggested in the paper. Using this procedure, optimal solutions were computed for functions sin x, tg x, and arc tg x. Optimal solutions to these functions were used in practical applications.     

A class of Orlicz-Sobolev spaces with applications to variational problems involving nonlinear Hill's equations

Necessary and sufficient conditions are proved for a $\text{b}$⁽²⁾-Young function G (with independent variable t) to be convex (resp. concave) in t² in terms of inequalities between the second derivative of G and the first derivative of its Legendre transform G? (with independent variable s). It is then proven that a Young function G is convex (resp. concave) in t² if and only if G? is concave (resp. convex) in s². These results, along with another set of inequalities for functions G convex (resp. concave) in t², allow the proof of the uniform convexity and thereby of the reflexivity with respect to Luxemburg's norm $\text{∥f∥}{}_{\mathrm{G}}\text{=}\text{inf}\text{{k > 0: ∝}{}_{\mathrm{\Omega }}\text{dξ G(f}\text{(ξ)}\text{k}\text{) ? 1}}$ of the Orlicz space $\text{L}{}_{\mathrm{G}}\text{(Ω)}$ over an open domain Ω ?$\text{R}$_N with Lebesgue measure dξ. When applied to $\text{G(t) =}\text{¦t¦}{}^{\mathrm{p}}\text{p}$ and $\text{G}\text{?}\text{(s) =}\text{¦s¦}{}^{\mathrm{p\prime }}\text{p′}$ with p^?1 + (p′)^?1 = 1, the preceding results lead to the shortest proof to date of two Clarkson's inequalities and of the reflexivity of L^p-spaces for 1 < p < +∞. Finally, some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear Hill's equations; these variational problems are formulated on suitable Orlicz-Sobolev spaces $\text{W}{}^{\mathrm{m}}\text{L}{}_{\mathrm{G}}\text{(Ω)}$ and thereby allow for nonlinear terms which may grow faster than any power of the variable.     

On the ∂-Neumann problem for generalized functions

Using the new theory of generalized functions developed by one of the authors the $\text{?}$ equation in $\text{C}$ⁿ is studied. In particular it is proven that if G is any generalized function on $\text{C}$ (in the above sense) then there is a generalized function S on $\text{C}$ such that $\text{?S}\text{?}\text{z}\text{?}\text{= G}$. Several other results are proven valid in polydiscs of $\text{C}$ⁿ, for which differential forms whose coefficients are generalized functions are introduced.     

Projections of polynomial hulls

The following theorem is discussed. Let X be a compact subset of the unit sphere in $\text{C}$ⁿ whose polynomially convex hull, $\text{X}\text{?}$, contains the origin, then the sum of the areas of the n coordinate projections of $\text{X}\text{?}$ is bounded below by π. This applies, in particular, when $\text{X}\text{?}$ is a one-dimensional analytic subvariety V containing the origin, and in this case generalizes the fact that the “area” of V is at least π; in fact, the area of V is the sum of the areas of the n coordinate projections when these areas are counted with multiplicity. A convex analog of the theorem is obtained. Hartog's theorem that separate analyticity implies analyticity, usually proved with the use of subharmonic functions (Hartog's lemma), will be derived as a consequence of the theorem, the proof of which is based upon the elements of uniform algebras.     

Contractive Korovkin approximations

Our results are related to $\text{L}$¹-shadows in L_p-spaces. For p = 1 we will complete the characterization of $\text{L}$¹-shadows and $\text{L}$^1,1-shadows. For 1 < p < ∞ S. J. Bernau has shown that the $\text{L}$¹-shadow of a set in L_p is the range of a contractive projection. We will show that the corresponding theorem is not true for all reflexive spaces, but is true for locally uniformly convex reflexive spaces.     

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.     

On the theory of semigroups of operators on locally convex spaces

The behavior of strongly continuous one-parameter semigroups of operators on locally convex spaces is considered. The emphasis is placed on semigroups that grow too rapidly to be treated by classical Laplace transform methods.A space

of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent $\text{R}$ of an operator A on E is defined as an operator on

. It is noted that $\text{R}$ may exist when the classical resolvent (λ ? A)^?1 fails to exist. Conditions on $\text{R}$ are given that are necessary and sufficient to guarantee that A is the generator of a semigroup T(t). The action of $\text{R}$ is characterized by convolution against the semigroup, and the semigroup is computed as the limit of $\text{R}$ acting on an approximate identity.Conditions on an operator B are introduced that are sufficient to guarantee that A + B is the generator of a semigroup whenever A is. A formula is given for the perturbed semigroup.Two characterizations of semigroups that can be extended holomorphically into some sector of the complex plane are given. One is in terms of the growth of the derivative $\text{(}\text{d}\text{dt}\text{) T(t)}$ as t approaches 0, the other is in terms of the behavior of $\text{R}$ⁿ, the powers of the generalized resolvent.Throughout, the generalized resolvent plays a role analogous to the role of the classical resolvent in the work of Hille, Phillips, Yosida, Miyadera, and others.     

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.     

The Euclidean distance construction of order homomorphisms

The Euclidean distance technique of Arrow and Hahn is used to construct an upper semicontinuous order homomorphism (partial utility function) from (X, ≻) to ($\text{R}$, >), where X is a closed, convex subset of $\text{R}$^N and ≻ is a continuous strict partial order on X. It is also shown that the order homomorphism is upper semicontinuous as a function on $\text{X×}\text{P}\text{(X)}$, where$\text{P}\text{(X)}$ is the set of continuous strict partial orders on X, taken with the topology of closed convergence.     

Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups

It is known that a function on $\text{R}$ⁿ which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense. In this paper an analogous theorem is proved in which $\text{R}$ⁿ is replaced by a set X, the averages over balls are replaced by a family of sublinear operators satisfying certain axioms, and the polynomials are replaced by a class of functions having certain regularity properties with respect to the averaging operators. Applications are given to function theory on domains in $\text{C}$ⁿ, to nilpotent Lie groups, and to the classical Euclidean case. The first application provides a characterization of the duals of Hardy spaces on the ball in $\text{C}$ⁿ.     

A generalization of the implicit function theorem for mappings from Rn + 1 into Rn and its applications

Some regularity results about the local structure of the zero set of a mapping of n + 1 real variables with values in $\text{R}$ⁿ are presented. They all provide improved and generalized versions of the Morse lemma for plane curves as well as generalizations of the implicit function theorem. We describe their applications to one-parameter nonlinear problems, including an analysis of bifurcation phenomena at simple or multiple eigenvalues.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

Bivariate box splines and smooth pp functions on a three direction mesh

Let S denote the space of bivariate piecewise polynomial functions of degree ? k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For $\text{ρ = ?}\text{(2k ? 2)}\text{3}\text{?}$, k ? 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for $\text{S}{}_{\mathrm{<mtext>loc</mtext>}}\text{= {f ∈ S:}\text{supp}\text{f ? Ω}}$, with Ω a convex subset of $\text{R}$².     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

Calculation of the eigenvalues of Schrödinger equations by an extension of Hill's method

The eigenfunctions of the one dimensional Schrödinger equation Ψ″ + [E ? V(x)]Ψ=0, where V(x) is a polynomial, are represented by expansions of the form $\text{∑}\text{k}\text{=0}\text{∞}{}^{\mathrm{<mtext>c</mtext>}}\text{k}{}^{\mathrm{?}}\text{k}\text{(ω,}\text{x}\text{)}$. The functions ?_k (ω, x) are chosen in such a way that recurrence relations hold for the coefficients c_k: examples treated are D_k(ωx) (Weber-Hermite functions), exp (?ωx²)x^k, exp (?cx^q)D_k(ωx). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter ω may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of ω. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x) = x^2m, m = 2(1)7, and the 6 first eigenvalues corresponding to V(x) = x² + λx¹⁰ and x² + λx¹², λ = .01(.01).1(.1)1(1)10(10)100.     

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.     

Hamiltonicity in (0–1)-polyhedra

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? $\text{R}$ⁿ is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? $\text{R}$ⁿ has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? $\text{R}$^d having (0–1)-valued vertices such that $\text{G(P) ? G(P′)}$. Some combinatorial consequences of these results are also discussed.