Retarded equations on the sphere induced by linear equations

Using a Poincaré compactification, the linear homogeneous system of delay equations {x = Ax(t ? 1) (A is an n × n real matrix) induces a delay system π(A) on the sphere Sⁿ. The points at infinity belong to an invariant submanifold S^{n ? 1} of Sⁿ. For an open and dense set of 2 × 2 matrices A with distinct eigenvalues, the system π(A) has only hyperbolic critical points (including the critical points at infinity). For an open and dense set of 2 × 2matrices $\text{A}$ with complex eigenvalues, the nonwandering set at infinity is the union of an odd number of hyperbolic periodic orbits; if $\text{(}\text{det}\text{A}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{<}\text{3π}\text{2}$, the restriction of $\text{π(}\text{A}\text{)}$ to S¹ is Morse-Smale. For n = 1 there exist periodic orbits of period 4 provided that $\text{?A >}\text{π}\text{2}$ and Hopf bifurcation of a center occurs for ?A near $\text{(}\text{π}\text{2}\text{) + 2kπ, k ? Z}$.     

On the special values of automorphic L-functions

Let π be a cuspidal representation of $\text{GL}{}_{\mathrm{n}}\text{(}\text{A}{}_{\mathrm{<mtext>Q</mtext>}}\text{)}$ with non-vanishing cohomology and denote by L(π,s) its L-function. Under a certain local non-vanishing assumption, we prove the rationality of the values of L(π?χ,0) for characters χ, which are critical for π. Note that conjecturally any motivic L-function should coincide with an automorphic L-function on GL_n; hence, our result corresponds to a conjecture of Deligne for motivic L-functions. To cite this article: J. Mahnkopf, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A rank characterization of the number of final classes of a nonnegative matrix

Let A be a nonnegative square matrix, and let D be a diagonal matrix whose iith element is $\text{(Ax)}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}$, where x is a (fixed) positive vector. It is shown that the number of final classes of A equals n?rank(A?D). We also show that null(A?D) = null(A?D)², and that this subspace is spanned by a set of nonnegative elements. Our proof uses a characterization of nonnegative matrices having a positive eigenvector corresponding to their spectral radius.     

Anti-Hadamard matrices

An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λ_i, singular values σ_i, and inverse B = (b_ij). We are interested in the four closely related problems of finding λ(n) = min_{A, i}|λ_i|, σ(n) = min_{A, i}σ_i, χ(n) = max_{A, i, j} |b_ij|, and μ(n) = max_AΣ_ijb²_ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between $\text{(2n)}{}^{\mathrm{?1}}\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and c√n (2.274)^?n, where c is a constant, $\text{c}\text{(2.274)}{}^{\mathrm{n}}\text{?χ(n)?2(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, and $\text{c}\text{(5.172)}{}^{\mathrm{n}}\text{?μ(n)?4n}{}^2\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{n}}$. We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry.     

Extensions of the Ostrowski-Reich theorem for SOR iterations

The Ostrowski-Reich theorem states that for a system Ax =b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0,2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A=M?N with A,M nonsingular, if A is hermitian and $\text{M}{}^1\text{+N}$ is positive definite, then M^?1N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when $\text{A+A}{}^1$ is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations.     

On ranges of Lyapunov transformations

A Lyapunov transformation is a linear transformation on the set $\text{H}$_n of hermitian matrices H ? $\text{C}$^n,n of the form $\text{L}$_A(H) = A¹H + HA, where A ?$\text{C}$^n,n. Given a positive stable A ?$\text{C}$^n,n, the Stein-Pfeffer Theorem characterizes those K ? $\text{H}$_n for which K = $\text{L}$_B(H), where B is similar to A and H is positive definite. We give a new proof of this result, and extend it in several directions. The proofs involve the idea of a controllability subspace, employed previously in this context by Snyders and Zakai.     

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.     

The voltage and current transfer ratios of RLC operator networks

The subject of this work is the voltage and current transfer ratios of three-terminal networks having no mutual coupling and whose impedances are analytic functions taking their values in an abelian self-adjoint algebra of bounded linear operators on a Hilbert space. Each such value is also assumed to be invertible. It is shown that these ratios have the form [I + A(ζ)]^?1, where, for each ζ in a sufficiently small open cone in the right-half complex plane with apex at the origin and the real axis as its bisector, the numerical range of A(ζ) is contained in a compact subset of the open right-half plane. This implies that the ratios are strictly contractive for each ζ in the cone. The angle of the cone is $\text{π}\text{(2k + 2)}$, where k is the number of internal nodes of a certain “surrogate” network; this result is best possible. For two-terminal-pair networks the ratios are shown to be strictly contractive for each ζ in a similar cone with angle $\text{π}\text{(2k + 4)}$.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…     

Permanental Polytopes of Doubly Stochastic Matrices

A subpolytope Γ of the polytope Ω_n of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω₂ is contained in any permanental polytope of Ω₃ with positive dimension. There is no 3-dimensional permanental polytope of Ω₃, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω₃ (a square of side $\text{1}\text{3}$). Permanental polytopes of dimension $\text{(n}{}^2\text{?3n+4)}\text{2}$ are exhibited for each n?4.     

On strictly positively invariant cones

For a matrix A ∈ R^{n × n}, it is shown that strict positive invariance of a proper cone $\text{C}$ ? Rⁿ (that is, e^tA$\text{C}\text{/{0}}$ ? int $\text{C}$ ?t 0) implies the existence of a certain direct sum decomposition of Rⁿ into A-invariant subspaces. Our results lead to a characterization of the set of initial points which give rise to solution curves that reach $\text{S}$, under the differential equation ? = Ax. Also given is an application in stability theory.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

On some properties of the Cesàro limit of a stochastic matrix

Let$\text{?(x}{}_1\text{,…,x}{}_{\mathrm{p}}\text{)}$ be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let $\text{?}\text{?}\text{(A}{}_1\text{,…,A}{}_{\mathrm{p}}\text{)}$ be the stochastic matrix which is obtained from ? by substituting the Kronecker product of An11,…,Anppfor each term Xn11·?·Xnpp. In this paper, we present necessary and sufficient conditions for the Cesàro limit of the sequence of the powers of $\text{?}\text{?}\text{(A}{}_1\text{,…,A}{}_{\mathrm{p}}\text{)}$ to be equal to the Kronecker product of the Cesàro limits associated with each of A1,…,Ap. These conditions show that the equality of these two matrices depends only on the number of ergodic sets under$\text{?}\text{?}\text{(A}{}_1\text{,…,A}{}_{\mathrm{p}}\text{)}$ and?or the cyclic structure of the ergodic sets under A1,…,Ap, respectively. As a special case of these results, we obtain necessary and sufficient conditions for the interchangeability of the Kronecker product and the Cesàro limit operator.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

On the analytic continuation of a certain Dirichlet series

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by

$\text{D}{}_{\mathrm{F}}\text{(s,p,α)}\text{=}\text{∑}\text{α∈Z}{}^{\mathrm{n}}\text{?{0}}\text{F(α)}{}^{\mathrm{?s}}\text{e(ρF(α)+〈α, α〉)}$

where ? ∈ $\text{Q}$, α ∈ $\text{Q}$ⁿ, 〈x, y〉 = x₁y₁ + ? + x_ny_n, e(a) = exp (2πia) for a ∈ $\text{R}$, and s = σ + ti is a complex number. The author proves that: (1) D_F(s, ?, α) has analytic continuation into the whole s-plane, (2) D_F(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at $\text{s =}\text{n}\text{δ}$. The residue at $\text{s =}\text{n}\text{δ}$ is given explicitly. (3) ? = 0, α ? $\text{Z}$ⁿ, D_F(s, 0, α) is analytic for $\text{α>,}\text{n}\text{(δ ? 1)}$.     

Some permutation problems

Put Z_n = {1, 2,…, n} and let π denote an arbitrary permutation of Z_n. Problem I. Let π = (π(1), π(2), …, π(n)). π has an up, down, or fixed point at a according as a < π(a), a > π(a), or a = π(a). Let $\text{A}\text{(r, s, t)}$ be the number of π ∈ Z_n with r ups, s downs, and t fixed points. Problem II. Consider the triple π^?1(a), a, π(a). Let R denote an up and F a down of π and let B(n, r, s) denote the number of π ∈ Z_n with r occurrences of π^?1(a)RaRπ(a) and s occurrences of π^?1(a)FaFπ(a). Generating functions are obtained for each enumerant as well as for a refinement of the second. In each case use is made of the cycle structure of permutations.     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

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.     

Solution of a system of functional-differential equations arising from an optimization problem

In connection with an optimization problem, all functions ?: Iⁿ → $\text{R}$ with continuous nonzero partial derivatives and satisfying

$\text{??}\text{?x,}{}_{\mathrm{i}}\text{??}\text{?x}{}_{\mathrm{j}}$

for all x_i, x_j ≠ I, i, j = 1,2,…, n (n > 2) are determined (I is an interval of positive real numbers).