On the recovery of a manifold with boundary in Rn

If the Riemann curvature tensor associated with a smooth field $\text{C}$ of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset $\text{Ω?}\text{R}{}^{\mathrm{n}}$, then $\text{C}$ is the metric tensor field of a manifold isometrically immersed in $\text{R}{}^{\mathrm{n}}$.In this Note, we first show how, under a mild smoothness assumption on the boundary of $\text{Ω}$, this classical result can be extended “up to the boundary”. When $\text{Ω}$ is bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spaces $\text{C}{}^{\mathrm{?}}\text{(}\text{Ω}\text{)}$. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Monotone Splitting of matrices

Given the iterative scheme xⁱ⁺¹ = BTxⁱ + r where B, T are fixed n×:n real matrices, r a fixed real n-vector and xⁱ a real n-vector we investigate the convergence and monotonicity of schemes of the type

$\text{v}{}^{\mathrm{i+1}}\text{w}{}^{\mathrm{i+1}}\text{=}\text{B}\text{O}\text{O}\text{B}\text{S}{}_{11}\text{?S}{}_{12}\text{?S}{}_{21}\text{S}{}_{22}\text{v}{}^{\mathrm{i}}\text{w}{}^{\mathrm{i}}\text{+}\text{r}\text{r}$

, where S_ij are n×:n real matrices related to T. The n-vector iterates vⁱ,wⁱ bracket in a certain sense solutions x of x = BTx + r. We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself xⁱ⁺¹ = BTxⁱ + r. This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.     

General scheme for solving linear algebraic problems by direct methods

Assume that A is an m × n real matrix with m?n and rank(A)=n. Let b be a real vector with m components and consider the problem of finding $\text{x=A}{}^2\text{b}$, where $\text{A}{}^2\text{=(A}{}^{\mathrm{T}}\text{A)}{}^{\mathrm{?1}}\text{A}{}^{\mathrm{T}}$. A general scheme for solving this problem is described. Many well-known and commonly used in practice direct methods can be found as special cases within the general scheme. This is illustrated by four examples. The general scheme can be used to study the common properties of the direct methods. The usefulness of this approach in the efforts to improve the efficiency of the direct methods for sparse matrices is demonstrated by formulating an algorithm that can be applied to any particular method belonging to the general scheme. This algorithm is implemented in several subroutines solving linear algebraic problems by different direct methods. Numerical results, obtained in a wide range of runs with these subroutines, are given.     

Up to isometries,a deformation is a continuous function of its metric tensorAux isométries près,une déformation est une fonction continue de son tenseur métrique

If the Riemann–Christoffel tensor associated with a field of class $\text{C}{}^2$ of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset $\text{Ω?}\text{R}{}^3$, then this field is the metric tensor field associated with a deformation of class $\text{C}{}^3$ of the set $\text{Ω}$, uniquely determined up to isometries of $\text{R}{}^3$. We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies. To cite this article: P.G. Ciarlet, F. Laurent, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 489–493.     

An inequality involving permanents of certain direct products

Let A denote a decomposable symmetric complex valued n-linear function on C^m. We prove

$\text{6A·A6}{}^2\text{?2}{}^{\mathrm{n}}\text{2n}\text{n}{}^{\mathrm{?1}}\text{6A?A6}{}^2$

, where · denotes the symmetric product and ? the tensor product. As a consequence we have per

$\text{M}\text{M}\text{M}\text{M}\text{?2}{}^{\mathrm{n}}\text{[per(M)]}{}^2$

, where M is a positive semidefinite Hermitian matrix and per denotes the permanent function. A sufficient condition for equality in the matrix inequality is that M is a nonnegative diagonal matrix.     

A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ(Λ,Λ′) of all such solutions.     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

On the semi-definiteness of the real pencil of two Hermitian matrices

Let A and B be two n×n real symmetric matrices. A theorem of Calabi and Greub-Milnor states that if n?3 and A and B satisfy the condition

$\text{(uAu′)}{}^2\text{+ (uBu′)}{}^2\text{≠ 0}$

for all nonzero vectors u, then there is a linear combination of A and B that is definite. In this note, the author proves two theorems of the semi-definiteness of a nontrivial linear combination of A and B by replacing the condition (¹) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi.     

Almost commuting matrices

It is shown that if A and B are n × n complex matrices with $\text{A = A}{}^1\text{and}\text{∥AB ? BA∥</}\text{2?}{}^2\text{(n ? 1)}$, then there exist n × n matrices A′ and B′ with $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

Some properties of Q-matrices

This paper deals with the class of Q-matrices, that is, the real n × n matrices M such that for every q ∈ $\text{R}$^n×1, the linear complementarity problem

$\text{Iw ? Mz = q}$

,

$\text{w ? 0, z ? 0, and w}{}^{\mathrm{T}}\text{z = 0}$

, has a solution. In general, the results are of two types. First, sufficient conditions are given on a matrix M so that M ∈ Q. Second, conditions are given so that M ? Q.     

On the convergence of the Neumann series in interval analysis

Let $\text{A}$ be a real or complex n × n interval matrix. Then it is shown that the Neumann series $\text{Σ}{}^{\mathrm{\infty }}{}_{\mathrm{k=0}}\text{A}{}^{\mathrm{k}}$ is convergent iff the sequence {$\text{A}$^k} converges to the null matrix $\text{O}$, i.e., iff the spectral radius of the real comparison matrix $\text{B}$ constructed in [2] is less than one.     

Class numbers of cyclotomic fields

In the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number h_p⁺ of the pth real cyclotomic field satisfies h_p⁺ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that $\text{h}{}_{\mathrm{n}}{}^{\mathrm{+}}\text{> n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; \epsilon </mtext>}}$ for infinitely many n. In the second part of the paper we show that if l ?= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then $\text{l}\text{2}$ (l, if l is odd) divides the class number h_n⁺ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides h_n⁺.     

Trace formulas for a class of Toeplitz-like operators II

Let P_T denote the orthogonal projection of L²(R¹, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure $\text{dΔ(γ) = ¦ h(γ)¦}{}^2\text{dγ}$, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if $\text{(γ}{}^2\text{+ 1)}{}^{\mathrm{?1}}\text{log}\text{¦ h(γ)¦}$ is summable, if $\text{¦ h ¦}{}^{\mathrm{?2}}$ is locally summable, and if $\text{h}\text{h}{}^{\mathrm{\#}}$ belongs to the span in L^∞ of e^?iyTH^∞:T ? 0, in which h is chosen to be an outer function and h^#(γ) agrees with the complex conjugate of h(γ) on the line, then

$\text{lim trace}\text{T↑∞}\text{{(P}{}_{\mathrm{T}}\text{GP}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? P}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{P}{}_{\mathrm{T}}\text{}}$

exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.     

Continuity of positive functionals on topological 1-algebras

It is shown that the set $\text{C}$_{m × n} of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by $\text{A}{}^1\text{A = A}{}^1\text{B,}{}^1\text{AA}{}^1\text{= BA}{}^1\text{,}\text{where}\text{A}{}^1$ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F_{2 × 2}, the corresponding intersections A ∩ B all exist.     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

A note on the proportion of singular (0, 1)-matrices in a class associated with fractional 2n factorial designs

Let the set T = {(x₁, x₂,…, x_n): x_i = 0, 1}. Since the elements of T can be seen as binary representations of integers, we order them with their corresponding integer values. Let $\text{Γ}{}^1$ be the set of (n + 1) × (n + 1) matrices of the form [1 … D], where the n + 1 rows of D are distinct ordered elements of T. We show that the proportion of singular matrices in $\text{Γ}{}^1$ approaches 0 as n → ∞.     

Rôle de l'espace de Besov B∞−1,∞ dans le contrôle de l'explosion éventuelle en temps fini des solutions régulières des équations de Navier–Stokes

Let $\text{u∈C([0,T}{}^1\text{[;L}{}^{\mathrm{n}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{)}$ be a maximal solution of the Navier–Stokes equations. We prove that u is C^∞ on $\text{]0,T}{}^1\text{[×}\text{R}{}^{\mathrm{n}}$ and there exists a constant $\text{ε}{}_1\text{>0}$, which depends only on n, such that if $\text{T}{}^1$ is finite then, for all $\text{ω∈S(}\text{R}{}^{\mathrm{n}}\text{)}{}^{\mathrm{n}}\text{,}$ we have To cite this article: R. May, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the stabilizer subgroup of a fair of matrices

Let (F,G) be a pair of matrices defined over an arbitrary field, Fn × n, Gn × m. Consider the natural action of GL_n x GL_m on this pair given by (F,G) ? (gFg^-1,gGh^-1), where (g,h) ∈ GL_n × GL_m. This action is of interest in system theory as well as the representation theory of quivers. In this paper we study the stabilizer subgroup of this action stab(F,G), i.e.

$\text{{(g,h) ∈ GL}{}_{\mathrm{n}}\text{x GL}{}_{\mathrm{m}}\text{|gFg}{}^{-1}\text{= F,gGh}{}^{-1}\text{= G}}$

.