A central limit theorem for D(A)-valued processes

Let D(A) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D(A) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.     

Properties of Euclidean and non-Euclidean distance matrices

A distance matrix D of order n is symmetric with elements $\text{?}\text{1}\text{2}\text{d}{}_{\mathrm{ij}}{}^2$, where d_ii=0. D is Euclidean when the $\text{1}\text{2}\text{n(n?1)}$ quantities d_ij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether e^TD^?e is not or is zero, the generating points lie in either p=ρ?1 dimensions, in which case they lie on a hypersphere, or in p=ρ?2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X.     

Analysis of Basis Pursuit via Capacity Sets

Finding the sparsest solution α for an under-determined linear system of equations D α=s is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of α that allow its recovery using ? ₁-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of D called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given D, called the capacity sets. We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of D α=s.     

Parallel X-ray tomography of convex domains as a search problem in two dimensions

In 1961, at A.M.S. Symposium on Convexity, P.C. Hammer proposed the following problem: how many X-ray pictures of a convex planar domain D must be taken to permit its exact reconstruction? Richard Gardner writes in his fundamental 2006 book [4] that X-rays in four different directions would do the job. The present paper points at the possibility that in certain asymptotical sense X-rays in only three different directions can be enough for approximate reconstruction of centrally symmetric convex domains. The accuracy of reconstruction would tend to become perfect in the limit, as the directions of the three X-rays change, all three converging to some given direction. The analysis leading to that conclusion is based on two lemmas of Section 1 and Pleijel type identity for parallel X-rays derived in Sections 2 and 3. These tools together supply a systemof two differential equations with respect to two unknown functions that describe the two branches of the domain boundary D. The system is easily resolved. The solution intended to provide a complete tomography reconstruction of D, happens however to depend on a two dimensional parameter, whose “real value” remains unknown. So tomography reconstruction of D becomes possible if a satisfactory approximation to that unknown “real value” can be found. In the last section a test procedure for the individual candidates for “approximate real value” of the parameter is described. A uniqueness theorem concerning tomography of circular discs is proved.     

On the numerical solutions for the fractional diffusion equation

Fractional differential equations have recently been applied in various area of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FDE to a system of ordinary differential equations, which solved by the finite difference method. Numerical simulation of FDE is presented and the results are compared with the exact solution and other methods.     

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on Rⁿ let ((B_t, P^μ) be Brownian motion in Rⁿ with initial distribution μ. Let D be an open subset of Rⁿ with exit time ζ ≡ inf {t > 0: B_t ? D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G_μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν(dx) = P^μ(B_T ? dx, T <ζ), where T is some natural stopping time for (B_t), and we have applied this characterization to show that a measure ν in D satisfies G_ν ? G_μ iff ν is of the form ν(dx) = P^α(B_T ? dx, T <ζ) + β(dx), where T is some natural stopping time for (B_t) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R² and μ is a finite measure on R² such that ∫ log⁺ ∥x∥ du(x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R² such that (log⁺ ∥B_T∧t∥)_0<t<∞ is P^μ-uniformly integrable.     

Congruence by Dissection of Topological Discs—An Elementary Approach to Tarski's Circle Squaring Problem

Let G be a group of affine transformations of the plane R ² and let the family F consist of all topological discs in R ² whose boundary is subject to some smoothness condition (general, rectifiable, piecewise C ¹ , piecewise C ² ). Are any two members D,E ∈ F congruent by dissection with respect to G such that all the pieces in the corresponding dissections of D and E belong to F as well? We give an affirmative answer if G contains all affine transformations and F consists of the discs whose boundary is piecewise C ¹ . An example shows that C ¹ cannot be replaced by C ² . Moreover, if G is either the group of equiaffine transformations or the group of similarities, then congruence by dissection of two convex discs D and E turns out to be essentially equivalent to congruence by dissection of the boundaries bd(D ) and bd(E ).     

On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R ⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ? R ⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.     

Norm convergence of normalized iterates and the growth of Kœnigs maps

Let ? be an analytic function defined on the unit diskD, with ?(D)?D, ?(0)=0, and ?′(0)=λ≠0. Then by a classical result of G. K?nigs, the sequence of normalized iterates Φ _n /λ ⁿ converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH _p , the sequence Φ _n /λ ⁿ converges to σ in the norm ofH _p . We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ? is univalent. When ? is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyH _p . It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceL _a ^p .     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

A generalization of a result of Sylvester's

Let v₁,…,v_n be vectors in Zⁿ with D = det(v₁,…,v_n) > 0. Let v_{n + 1} be in the cone generated by v₁,…,v_n and such that v₁,…,v_v, v_{n + 1} generate Zⁿ as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v₁,…,v_n, v_{n + 1} and χ = Dv_{n + 1} ? (v₁ + … v_n + v_{n + 1}).     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

Twisted unknots

Let K be a knot in the 3-sphere S³, and D a disk in S³ meeting K transversely in the interior. For non-triviality we assume that |D∩K|?2 over all isotopies of K in S³??D. Let K_D,n(?S³) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S³, we call K_D,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot K_D,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. A??t Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Espace des modules des variétés hyperboliquement plongéesThe moduli space of hyperbolically imbedded manifolds

In this Note, we construct the moduli space of hyperbolically imbedded manifolds. We recall that the moduli space of compact hyperbolic manifolds has been constructed by Brody and Wright. To construct our moduli space, we use a general criterion to represent analytic functors by coarse moduli spaces due to Schumacher. The objects to deform are couples (X,D) where X is a compact manifold and D is a normal crossing divisor in X such that X?D is hyperbolically imbedded in X. This criterion is based on two ingredients: in our case, the first is the existence of semi-universal logarithmic deformation due to Kawamata. The second is a consequence of a theorem of stability of hyperbolically imbedded spaces through logarithmic deformations. We use the relative-distance of Kobayashi to simplify the proof. To cite this article: A. Khalfallah, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 237–242.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

An Extended Faber Operator

Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

Some discontinuous translation-invariant linear forms

The author proved in [3] that every translation-invariant linear form on $\text{D}$(Rⁿ), as well as on other spaces of test functions and distributions, is necessarily continuous. The same result has also been proved for the Hilbert space L²(G) where G is a compact connected Abelian group. In contrast to this it is proved here that there do exist discontinuous translation-invariant linear forms on the Banach spaces l₁(Z) and L¹(R), and on the Hibert spaces L²(D) and L²(R). Here Z denotes the additive group of the integers, D denotes the totally disconnected compact Abelian Cantor discontinuum group, and R denotes the additive group of the real numbers. The proofs divide into two parts: A general criterion (Theorem 1) and proofs that the spaces l₁(Z), L²(D), L²(R), and L¹(R) satisfy this criterion (Theorems 2, 3, 4, and 5, respectively).     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.