Diffusions with holding and jumping boundary

Consider a family of probability measures {νξ} on a bounded open region D R d with a smooth boundary and a positive parameter set {βξ},all indexed by ξ∈D.For any starting point inside D,we run a diffusion until it first exits D,at which time it stays at the exit point ξ for an independent exponential holding time with rate βξ and then leaves ξ by a jump into D according to the distribution νξ.Once the process jumps inside,it starts the diffusion afresh.The same evolution is repeated independently each time the process jumped into the domain.The resulting Markov process is called diffusion with holding and jumping boundary(DHJ),which is not reversible due to the jumping.In this paper we provide a study of DHJ on its generator,stationary distribution and the speed of convergence.     

A method for solving stochastic eigenvalue problems

We present a novel approach for calculating stochastic eigenvalues of differential and integral equations as well as for random matrices. Five examples based on very different types of problem have been analysed and detailed numerical results obtained. It would seem that the method has considerable promise. The essence of the method is to replace the stochastic eigenvalue problem λ(ξ)?(ξ)=A(ξ)?(ξ), where ξ is a set of random variables, by the introduction of an auxiliary equation in which . This changes the problem from an eigenvalue one to an initial value problem in the new pseudo-time variable t. The new linear time-dependent equation may then be solved by a polynomial chaos expansion (PCE) and the stochastic eigenvalue and its moments recovered by a limiting process. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation by PCE, but it does introduce an additional, ‘pseudo-time’, independent variable t. The paper illustrates the viability of this approach by application to several examples based on realistic problems.     

On invertible substitutions with two fixed points

Let ? be a primitive substitution on a two-letter alphabet {a,b} having two fixed points ξ_a and ξ_b. We show that the substitution ? is invertible if and only if one has ξ_a=abξ and ξ_b=baξ. To cite this article: Z.-X. Wen et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 727–731.     

Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion

We consider a diffusion (ξ _t ) _t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ? of interest. We consider sequences of local models at ? corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ?. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ?, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n ^?3/2 at ?, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n ^?2 at ?, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.     

Diffusion equation for multivalued stochastic differential equations

Deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type ξ′ + kξ?b(ξ) + g, where k is maximal monotonic and b is Lipschitzian. An existence and uniqueness result is proven for corresponding stochastic equation. The diffusion equation satisfied by the laws of ξ(t) is established. In the particular case k = 0, this equation is equivalent to the Fokker-Planck equation.     

On Beurling’s boundary differential relation

We prove that for an arbitrary positive continuous function Φ on the complex plane ? there exists an injective disc algebra function φ and n ∈ ? such that ξ ? φ(ξ ⁿ ) solves Beurling’s boundary differential relation |f′(ξ)| = Φ(f(ξ)) on ?Δ. Moreover, if the growth of Φ is sublinear, the existence of univalent solutions of Beurling’s boundary differential relation is shown.     

On a question of A. Koldobsky

We construct an example of a non-convex star-shaped origin-symmetric body D⊂R³ such that its section function AD,ξ(t):=area(D∩{ξ^⊥+tξ}) is decreasing in t?0 for every fixed direction ξ∈S².     

A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ _j ?E[ξ _j ], 0<α?Var[ξ _j ] andE[|ξ _j ?μ _j |^2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ₀ are two random variables such thatE[ξ ₀ ² ]<∞ andE[|U|ξ ₀ ² ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ _j :j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereS _n ?ξ₁+ξ₂+?+ξ _n ,μ _j ?E[ξ _j ],s _n ² ?Var[S _n ], andc _n =O(s _n ).     

A remark on the stabilization of the 1-d wave equation

We consider the wave equation on an interval of length 1 with an interior damping at ξ and with Dirichlet boundary condition at the two ends. It is well known that, if ξ is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify.     

Patterson-Wiedemann construction revisited

In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2^n-1-2^(n-1)/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2^ab) which acts on PG(2,2^a).     

Solutions of a transcendental equation basic to the study of composite media heat transfer

The theory of complex variables is used to develop exact, closed-form solutions of the transcendental equation: a tan(ξ)+tan(cξ?b) = 0.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

Invertible substitutions and local isomorphisms

Let ?₁ and ?₂ be two primitive invertible substitutions over a two-letter alphabet. Let ξ_?1 and ξ_?2 be fixed points of ?₁ and ?₂, respectively. We show that ξ_?1 and ξ_?2 are locally isomorphic if and only if there exists a primitive invertible substitution ?₀ and two positive integers m and n such that M_?1=M_?0^m and M_?2=M_?0ⁿ, where M_? is the substitutive matrix of the substitution ?. To cite this article: Z.-X. Wen et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 629–634.     

Effective Fast Algorithms for Polynomial Spectral Factorization

Let p(z) be a polynomial of degree n having zeros |ξ₁|≤???≤|ξ _m |<1<|ξ _m+1|≤???≤|ξ _n |. This paper is concerned with the problem of efficiently computing the coefficients of the factors u(z)=∏ _i=1 ^m (z?ξ _i ) and l(z)=∏ _i=m+1 ⁿ (z?ξ _i ) of p(z) such that a(z)=z ^?m p(z)=(z ^?m u(z))l(z) is the spectral factorization of a(z). To perform this task the following two-stage approach is considered: first we approximate the central coefficients x _?n+1,. . .x _n?1 of the Laurent series x(z)=∑ _i=?∞ ^+∞ x _i z ⁱ satisfying x(z)a(z)=1; then we determine the entries in the first column and in the first row of the inverse of the Toeplitz matrix T=(x _i?j ) _i,j=?n+1,n?1 which provide the sought coefficients of u(z) and l(z). Two different algorithms are analyzed for the reciprocation of Laurent polynomials. One algorithm makes use of Graeffe's iteration which is quadratically convergent. Differently, the second algorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed.     

A family of orthonormal wavelet bases with dilation factor 4

In this paper, we study a method for the construction of orthonormal wavelet bases with dilation factor 4. More precisely, for any integer M>0, we construct an orthonormal scaling filter mM(ξ) that generates a mother scaling function ?M, associated with the dilation factor 4. The computation of the different coefficients of ²|mM(ξ)| is done by the use of a simple iterative method. Also, this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity. A first estimate of α(M), the asymptotic regularity of ?M is given by α(M)∼0.25M. Examples are provided to illustrate the results of this work.     

Fractional parts of powers and Sturmian words

Let $b?2$ be an integer. In terms of combinatorics on words we describe all irrational numbers $\xi >0$ with the property that the fractional parts $\{\xi b^n\}$, $n?0$, all belong to a semi-open or an open interval of length $1/b$. The length of such an interval cannot be smaller, that is, for irrational ξ, the fractional parts $\{\xi b^n\}$, $n?0$, cannot all belong to an interval of length smaller than $1/b$. To cite this article: Y. Bugeaud, A. Dubickas, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

How to determine the maximum genus of a graph

The maximum genus, γ_M(G), of a connected graph G is the largest genus γ(S) for orientable surfaces S in which G has a 2-cell embedding. In this paper, we define a new combinatorial invariant ξ(G), the Betti deficiency of G, to be ξ(C) = min_C?G{ξ(C) 6 ξ(C) = number of odd components of a cotree C of G (by odd component we mean one with an odd number of edges). We formalize a new embedding technique to obtain the formula:

$\text{γ}{}_{\mathrm{M}}\text{(G)=}\text{1}\text{2}\text{(β(G)?ξ(G))}$

where β(G) denotes the Betti number of G.In a further paper, various consequences will be given.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).