Monotonicity of permanents of certain doubly stochastic circulant matrices

Let p_k(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions p_k((1?θ)J_n+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here J_n is the n×n matrix all whose entries are equal $\text{1}\text{n}$. In the present paper it is proved that the conjecture holds true for the circulant matrices A=αI_n+ βP_n, α, β?0, α+;β=1, and $\text{A=}\text{(nJ}{}_{\mathrm{n}}\text{?I}{}_{\mathrm{n}}\text{?P}{}_{\mathrm{n}}\text{)}\text{(n?2)}$, where I_n and P_n are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1).     

Simple product colorings

If A is a set colored with m colors, and B is colored with n colors, the coloring of A × B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m × n simple product coloring (SPC) of A × B. SPC's of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 × 2, and 2 × 2 × 2 SPC's of $\text{Q}$² and $\text{Q}$³ which forbid the distance one; that there is no 2^k SPC of $\text{Q}$^k forbidding the distance one, for k > 3; and that there is no 2 × 2 SPC of $\text{Q}$ × $\text{Q}$(√15), and thus none of $\text{R}$², forbidding the distance 1.     

Third order efficiency of conditional tests for exponential families

Let P_η, η = (θ, γ) ∈ Θ × Γ ? $\text{R}$ × $\text{R}$^k, be a (k + 1)-dimensional exponential family. Let $\text{?}{}_{\mathrm{n}}{}^1$, n ∈ $\text{N}$, be an optimal similar test for the hypothesis {P_(θ,γ)ⁿ: γ ∈ Γ} (θ ∈ Θ fixed) against alternatives P_(θ1,γ1)ⁿ, θ₁ > θ, γ₁ ∈ Γ. It is shown that (?_n¹)_{n∈$\text{N}$} is third order efficient in the class of all test-sequences that are asymptotically similar of level α + o(n^?1) (locally uniformly in the nuisance parameter γ).     

A generalization of the fast LUP matrix decomposition algorithm and applications

We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(m^α?1n) time, where the complexity of matrix multiplication is O(m^α). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(m^α?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations $\text{A}\text{x}\text{=}\text{b}$ (where $\text{b}$ is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, $\text{A}{}^1$, of A (i.e., $\text{AA}{}^1\text{A = A}$). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.     

Non existence de solutions globales de certaines équations d'ondes non linéairesNonexistence of global solutions to some nonlinear wave equations

We study the semilinear wave equation u_tt?Δu=p^?k|u|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$, where p is a conformal factor approaching 0 at infinity. We prove that the solutions blow-up in finite time for small powers m, while having an arbitrarily long life-span for large m. Furthermore, we study the finite time blow-up of solutions for the class of quasilinear wave equations u_tt?Δu=p^?k|Lu|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$. To cite this article: M. Aassila, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961–966.     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.     

Minimally k-connected graphs of low order and maximal size

Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) $\text{e(G)?kn?(}\text{k+1}\text{2}\text{)}$; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph K_k,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove $\text{e(G)??}\text{(n+k)}{}^2\text{8}\text{?}$ and characterize all minimally k-connected graphs of order n and size $\text{?(}\text{(n+k)}{}^2\text{8}\text{?}$.     

Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The permanent function is used to determine geometrical properties of the set $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then $\text{F}$ corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of $\text{F}$. If A is fully indecomposable, then the dimension of $\text{F}$ equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of $\text{Ω}{}_{\mathrm{n}}$ are triangles and rectangles. For n ? 6, $\text{Ω}{}_{\mathrm{n}}$ has four types of three-dimensional faces. The facets of the faces of $\text{Ω}{}_{\mathrm{n}}$ are characterized. Faces of $\text{Ω}{}_{\mathrm{n}}$ which are simplices are determined. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is two-neighborly but not a simplex, then $\text{F}$ has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2^k. All k-dimensional faces with at least 2^k?1 + 1 vertices are determined.     

On a problem of C. C. Chen and D. E. Daykin

Let α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that $\text{α(k, p, h) ≤ h + 1 + (k ? 1){(p ? h ? 1) × (h}{}^{\mathrm{p}}\text{+ 1)}}{}^{\mathrm{<mtext>1</mtext><mtext>h</mtext>}}$ and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if

$\text{(}\text{n}\text{3}\text{)>(}\text{p}\text{3}\text{)}\text{Σ}\text{i=1}\text{t}\text{(}\text{e}{}_{\mathrm{i}}\text{2}\text{)}$

where e_i is the number of edges of color i, 1 ≤ i ≤ t.     

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

Über die Wurzelschranke für das Minimalgewicht von codes

Let d be the minimum distance of an (n, k) code $\text{C}$, invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that $\text{C}$ contains the repetition code, that dim($\text{C}$ ∏ $\text{C}$^⊥) = k ? 1 and that the supports of the minimal weight vectors of $\text{C}$ form a 2-design. Then d² ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d² ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d² ? d + 1 > n and F = GF(2) then (d ? 1)² ? n. Examples are the generalized quadratic residue codes.     

On a problem of Chvátal and Erdös on hypergraphs containing no generalized simplex

Let X be an n-element set and $\text{T}$ a family of k-subsets of X. Let r be an integer, k > r ? 2. Suppose that $\text{T}$ does not contain r + 1 members having empty intersection such that any r of them intersect non-trivially. Chvátal and Erdös conjectured that for (r + 1) k ? rn we have |$\text{F}$|$\text{?}\text{n?1}\text{k?1}$. In this paper we first prove that This conjecture holds asymptotically (Theory 1). In Theorems 4 and 5 we prove it for r = 2, K ? 5, n > n_o(k); k ? 3r, n > n_o(k, r), respectively.     

On the C1 normal forms for hyperbolic vector fields

Given two germs of hyperbolic vector fields associated to autonomous ordinary differential equations $\text{x}\text{?}\text{=Ax+?}$ and $\text{y}\text{?}\text{=By+?}$, where $\text{x,y∈}\text{R}{}^{\mathrm{n}}$, and A and B are n×n matrices, we prove that under some algebraic conditions on the eigenvalues of the matrices and genericity condition on the nonlinear terms, they are at least C¹ conjugate if and only if A and B are similar. To cite this article: Z. Ren, J. Yang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On a conjecture about the exponent set of primitive matrices

M. Lewin and Y. Vitek conjecture [7] that every integer $\text{?[}\text{(n>}{}^2\text{?2n+2)}\text{2}\text{]+1}$ is an exponent of some n×n primitive matrix. In this paper, we prove three results related to Lewin and Vitek's conjecture: (1) Every integer $\text{?[}\text{(n}{}^2\text{?2n+2)}\text{4}\text{]+1}$ is an exponent of some n×n primitive matrix. (2) The conjecture is true when n is sufficiently large. (3) We give a counterexample to show that the conjecture is not true in the case when n=11.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

An asymptotic evaluation of the cycle index of a symmetric group

Let $\text{Z(S}{}_{\mathrm{n}}\text{;?(x))}$ denote the polynomial obtained from the cycle index of the symmetric group Z(S_n) by replacing each variable s_i by f(x¹). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=x^k + a_k+1x^k+1 + a_k+2x^k+2+? with every a₁?0. Finally, let 0<x<1 and let x??. We prove that

This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of $\text{1}\text{n!e}$.In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented.     

On a property of the set of radiation patterns

Consider the exterior boundary value problem (▽² + K²) u = 0, in Ω, k >0. $\text{u¦}{}_{\mathrm{\Gamma }}\text{= h}$, where Γ is a smooth closed connected surface in $\text{R}$³, $\text{u ~}\text{exp}\text{(ik ¦x¦)¦x¦}{}^{\mathrm{?1}}\text{∝(k, n)}\text{as}\text{¦X¦→ ∞, n = x¦x¦}{}^{\mathrm{?1}}$, ∝ is called the radiation pattern. We prove that when h runs through any dense set in L²(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L²(S²) for any k >0, where S² is the unit sphere in $\text{R}$³.     

Recurrence relations based on minimization

This paper investigates solutions of the general recurrence M(0) = g(0), M(n + 1) = g(n + 1) + min_0?k?n(αM(k) + βM(n ? k)) for various choices of α, β, and g(n). In a large number of cases it is possible to prove that M(n) is a convex function whose values can be computed much more efficiently than would be suggested by the defining recurrence. The asymptotic behavior of M(n) can be deduced using combinatorial methods in conjunction with analytic techniques. In some cases there are strong connections between M(n) and the function H(x) defined by $\text{H(x) = 1}\text{for}\text{x < 1, H(x) = H(}\text{(x ? 1)}\text{α}\text{) + H(}\text{(x ? 1)}\text{β}\text{)}\text{for}\text{x ? 1}$. Special cases of these recurrences lead to a surprising number of interesting problems involving both discrete and continuous mathematics.