Helly EPT graphs on bounded degree trees: Characterization and recognition

The edge-intersection graph of a family of paths on a host tree is called an $EPT$ graph. When the tree has maximum degree $h$, we say that the graph is $[h,2,2]$. If, in addition, the family of paths satisfies the Helly property, then the graph is Helly $[h,2,2]$. In this paper, we present a family of $EPT$ graphs called gates which are forbidden induced subgraphs for $[h,2,2]$ graphs. Using these we characterize by forbidden induced subgraphs the Helly $[h,2,2]$ graphs. As a byproduct we prove that in getting a Helly $EPT$-representation, it is not necessary to increase the maximum degree of the host tree. In addition, we give an efficient algorithm to recognize Helly $[h,2,2]$ graphs based on their decomposition by maximal clique separators.     

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

Constructing spherical CR manifolds by gluing tetrahedra

We propose a general method of constructing spherical CR manifolds by gluing tetrahedra adapted to CR geometry. We obtain spherical CR structures on the complement of the figure eight knot and the Whitehead link complement with holonomy in $\mathrm{PU}(2,1,\mathbb{Z}[\omega ])$ and $\mathrm{PU}(2,1,\mathbb{Z}[\mathrm{i}])$ respectively (the same integer rings appearing in real hyperbolic geometry). To cite this article: E. Falbel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Asymptotic expansions for Laplace transforms of Markov processes

Let ${\mu }^{?}$ be the probability measures on $D[0,T]$ of suitable Markov processes ${\{{\xi }_t^{?}\}}_{0\le t\le T}$ (possibly with small jumps) depending on a small parameter $?>0$, where $D[0,T]$ denotes the space of all functions on $[0,T]$ which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms ${\int }_{D[0,T]}\exp ?\{{?}^{?1}F(x)\}{\mu }^{?}(dx)$ as $?\to 0$ for smooth functionals F on $D[0,T]$. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylor's expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13].     

Locality of optimal binary codes

The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with $r=1,2\text{ and }3$. Based on these results, for $2\le k\le 7$ and $n\ge k+2$, we give an optimal linear $[n,k,d]$ code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for $n<2(2^k-1)$ and by the Griesmer bound for $n\ge 2(2^k-1)$. Almost all the $[n,k,d]$ codes ($2\le k\le 7$) have locality $r\le 3$ except for the three codes, and most of the $[n,k,d]$ code with $n<2(2^k-1)$ achieves the Cadambe–Mazumdar bound for LRCs.     

Une propriété de composition dans l'espace Hs

Let us assume that $1<p?q?+\infty$, $1/p<s?[s]<1$, and $[s]?1$. If f and g are functions in the Besov space $B_p^{s\text{,}q}(\mathbb{R})$, such that g is real valued and such that $f(0)=0$, then the composed function $f○g$ belongs to $B_p^{s\text{,}q}(\mathbb{R})$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that $V\in L^2([0,\pi ];dx)$, we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators $?d^2/d{x}^2+V$ in $L^2([0,\pi ];dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $?d^2/d{x}^2+V$ in $L^p([0,\pi ];dx)$, $p\in (1,\infty )$.     

A failing eigenfunction expansion associated with an indefinite Sturm–Liouville problem

We consider the indefinite Sturm–Liouville problem $?f^{″}=\lambda rf$, $f^{\prime }(?1)=f^{\prime }(1)=0$ where $r\in L^1[?1,1]$ satisfies $xr(x)>0$. Conditions are presented such that the (normed) eigenfunctions $f_n$ form a Riesz basis of the Hilbert space $L_{|r|}^2[?1,1]$ (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function $f\in L_{|r|}^2[?1,1]$ having no eigenfunction expansion $f=\sum {\beta }_n{f}_n$. Furthermore, a sequence $({\alpha }_n)\in l^2$ is constructed such that the “Fourier series” $\sum {\alpha }_n{f}_n$ does not converge in $L_{|r|}^2[?1,1]$. These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form $\mathfrak{t}[u,v]=\int u^{\prime }{\overline{v}}^{\prime }pdx$ with Dirichlet boundary conditions in $L^2[?1,1]$ where $p=1/r$. For the associated operator $T_{\mathfrak{t}}$ we construct elements in the difference between $\mathrm{dom}\mathfrak{t}$ and the domain of the associated regular closed form, i.e. $\mathrm{dom}{|T_{\mathfrak{t}}|}^{1/2}$.