On Saint‐Venant's principle in a poroelastic arch‐like region

In this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch‐like region described by R: a<r<b, 0<θ<ω, where r and θ are polar coordinates and a, b, and ω (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self‐equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ω is lower than the critical value $\pi\sqrt{2}$. The intended applications of these results concern various branches of medicine as for example the bone implants. Copyright © 2010 John Wiley & Sons, Ltd.     

On a new method for the study of the spatial behavior in a homogeneous elastic arch-like region

We consider a two-dimensional homogeneous elastic state in the arch-like region a?≤?r?≤?b, 0?≤?θ?≤?α, where (r,θ) denotes plane polar coordinates. We assume that three of the edges are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load. The Airy stress function ‘?’ satisfies a fourth-order differential equation in the plane polar coordinates with appropriate boundary conditions. We develop a method which allows us to treat in a unitary way the two problems corresponding to the self-equilibrated loads distributed on the straight and curved edges of the region. In fact, we introduce an appropriate change for the variable r and for the Airy stress functions to reduce the corresponding boundary value problem to a simpler one which allows us to indicate an appropriate measure of the solution valuable for both the types of boundary value problems. In terms of such measures we are able to establish some spatial estimates describing the spatial behavior of the Airy stress function. In particular, our spatial decay estimates prove a clear relationship with the Saint-Venant's principle on such regions.     

On Saint-Venant's principle for a linear poroelastic material in plane strain

In this paper we consider the state of plane strain in an elastic material with voids occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, while the other three edges are traction-free and subjected to zero volumetric fraction or zero equilibrated force. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The both cases of homogeneous and inhomogeneous poroelastic materials are considered. The results of the present paper prove how the spatial-decay rate varies with the constitutive profile.     

Saint-Venant decay rates for an inhomogeneous isotropic linear thermoelastic strip

In this paper we consider the state of plane strain in an isotropic and inhomogeneous thermoelastic material occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction applied on one of the heated edges, while the other three edges are thermally insulated and traction-free. Our aim is to derive some explicit spatial estimates describing how certain appropriate measures of the Airy stress function and temperature evolve with respect to the distance from the loaded and heated edge, provided specific assumptions are made upon the derivatives of the thermoelastic coefficients. The results of the present paper prove how the spatial decay rate varies with the inhomogeneous constitutive profile.     

Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the L_q(T^d)-sense, where T^d is the d-dimensional torus. The functions to be approximated are in the unit ball SB^r_p, θ of the mixed smoothness Besov space or in the unit ball SW^r_p of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)₊ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SB^r_p, θ are both n^−r(log n)^{(d−1)(r+1/2−1/θ)}.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

The median function on graphs with bounded profiles

The median of a profile π=(u₁,…,u_k) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of π. It is shown that for profiles π with diameter θ the median set can be computed within an isometric subgraph of G that contains a vertex x of π and the r-ball around x, where r>2θ−1−2θ/|π|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

On some properties of base-matroids

Let M=(E,F) be a rank-n matroid on a set E and B one of its bases. A closed set θ⊆E is saturated with respect to B, or B-saturated, when |θ∩B|=r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ|?r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by M_B. In this paper we prove some properties of M_B, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M_B for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice B_n.     

Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ? ^d which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr ⁿ for all x∈? ^d , r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L ²(µ) is also bounded from L ^∞(µ) into RBMO(µ) and from H _atb ^1,∞ (µ) into L ¹(µ). According to the interpolation theorem introduced by Tolsa, the L ^p (µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on L ^p (µ) (1 < p < ∞).     

Symmetric duality with (p, r) − ρ − (η, θ)-invexity

In this paper we introduce a new type of generalized invex function, called (p, r) − ρ − (η, θ)-invex function and study symmetric duality results under these assumptions. In our study the nonnegative orthants for the constraints are replaced by closed convex cones and their polars. We establish weak and strong duality theorems under (p, r) − ρ − (η, θ)-invexity assumptions for the symmetric dual problems. We also give many examples to justify our results.     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

A problem ? multi-colorings of graphs

In this paper we consider colorings of the edges of the complete graph K_m with n colors such that the edges of any color form a non-trivial complete subgraph of K_m. We allow an edge of K_m to have more than one color. Such a coloring will be called r-admissible if no cycle of length r has a different color for each edge. Let E (m, n, r) be the maximum number of incidences of colors and edges, taken over all r-admissible colorings of K_m with n colors. Then for r = 3,4, and 5 we give an upper bound for E (m, n, r); as well as a lower bound for E (m, n, r) for all r. An analogue to a problem of Zarankiewicz concerning 0, 1-matrices is mentioned.     

Complex interpolation between two weighted Bergman spaces on tubes over symmetric cones

We prove that the complex interpolation space [A_ν^p₀,A_ν^p₁]_θ, 0<θ<1, between two weighted Bergman spaces A_ν^p₀ and A_ν^p₁ on the tube in $\text{C}{}^{\mathrm{n}}$, n?3, over an irreducible symmetric cone of $\text{R}{}^{\mathrm{n}}$ is the weighted Bergman space A_ν^p with 1/p=(1?θ)/p₀+θ/p₁. Here, ν>n/r?1 and 1?p₀<p₁<2+ν/(n/r?1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D. Békollé et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables one to construct embeddings of ? _p ⁿ , p > 0, into various types of Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed ? _p ⁿ into $\ell _r^{(1 + \eta )n}$ , with sharp polynomial bounds in η > 0.     

Bold graph drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r<w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r?w, a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time.     

On the order of a class of meromorphic functions

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ _k )(k=1,2, ...,q;0≤θ ₁<θ₂<...<θ _q<θ _q+1=θ ₁+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ _k+1 ?θ _k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities.     

A stochastic Ramsey theorem

We establish a stochastic extension of Ramsey's theorem. Any Markov chain generates a filtration relative to which one may define a notion of stopping times. A stochastic colouring is any k-valued (k<∞) colour function defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time θ and any infinite history ω of the Markov chain, let ω|θ denote the finite partial history up to and including the time θ(ω). Given k=2, for every ?>0, we prove that there is an increasing sequence θ₁<θ₂<? of bounded stopping times having the property that, with probability greater than 1−?, the history ω is such that the values assigned to all pairs (ω|θi,θj), with i<j, are the same. Just as with the classical Ramsey theorem, we also obtain an analogous finitary stochastic Ramsey theorem. Furthermore, with appropriate finiteness assumptions, the time one must wait for the last stopping time (in the finitary case) is uniformly bounded, independently of the probability transitions. We generalise the results to any finite number k of colours.     

On linear codes whose weights and length have a common divisor

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1<r<q=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r<q and whose dual minimum distance is at least 3, has length at least (r−1)q+(p−1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.