A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

On the probability that given polynomials have a specified highest common factor

The polynomial functions f₁, f₂,…, f_m are found to have highest common factor h for a set of values of the variables x₁, x₂,…,x_m whose asymptotic density is

$\text{1}\text{h}{}^{\mathrm{n}}\text{∑}\text{d∣h}\text{μ(d)Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, dh)}\text{d}{}^{\mathrm{m}}\text{Π}\text{p∣h}\text{1?}\text{Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, p)}\text{p}{}^{\mathrm{m}}$

For the special case f₁(x) = f₂(x) = … = f_m(x) = x and h = 1 the above formula reduces to $\text{Π}{}_{\mathrm{?}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{m}}\text{) =}\text{1}\text{ζ}\text{(m)}$, the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f₁, f₂,…, f_m for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density.     

Estimées Ck pour les couronnes de convexes de type fini de Cn

C^k estimates for convex domains of finite type in $\text{C}{}^{\mathrm{n}}$ are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of $\text{C}{}^{\mathrm{n}}$, of finite type m and m′ such that $\text{D}\text{?D′}$, for all q=1,…,n?2, there exists a linear operator $\text{T}{}^1{}_{\mathrm{q}}$ from $\text{C}{}_{\mathrm{0,q}}\text{(}\text{D′?D}\text{)}$ to $\text{C}{}_{\mathrm{0,q?1}}\text{(}\text{D′?D}\text{)}$ such that for all $\text{k∈}\text{N}$ and all (0,q)-form f, $\text{?}\text{?}$-closed of regularity C^k up to the boundary, $\text{T}{}^1{}_{\mathrm{q}}\text{f}$ is of regularity C^{k+1/max(m,m′)} up to the boundary and $\text{?}\text{?}\text{T}{}_{\mathrm{q}}{}^1\text{f=f}$. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the $\text{?}\text{?}$ problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Certain conjugacy classes of units in integral group rings of metacyclic groups

Let G be the metacyclic group of order pq given by

$\text{G = 〈σ, τ: σ}{}^{\mathrm{p}}\text{= 1 = τ}{}^{\mathrm{q}}\text{, τστ}{}^{\mathrm{?}}\text{= σ}{}^{\mathrm{j}}\text{〉}$

where p is an odd prime, q ≥ 2 a divisor of p ? 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring $\text{Z}$G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is

$\text{(p ? 1)}{}^{\mathrm{q?1}}\text{μ}{}_0\text{H}\text{vq}$

where ν, μ₀, and H are suitably defined numbers.     

Inverse Hölder inequalities in one and several dimensions

We study certain functionals and obtain an inverse Hölder inequality for n functions f₁^a1,…,f_n^an (f_k concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f₁,…,f_n, where $\text{?}{}^2\text{f}{}_{\mathrm{k}}\text{?x}{}_{\mathrm{i}}{}^2\text{? 0, i = 1,…, d, k = 1,…, n}$.Finally we give an inverse Minkowski inequality for concave functions.     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

About some problems of spectral synthesis

For a given pair $\text{(A,b)∈}\text{R}{}^{\mathrm{n\times n}}\text{×}\text{R}{}^{\mathrm{n\times 1}}$ such that A is cyclic and b is a cyclic generator (with respect to A) of $\text{R}{}^{\mathrm{n\times 1}}$, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence $\text{{f}{}_{\mathrm{j\}}}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{,f}{}_{\mathrm{j}}\text{∈}\text{R}{}^{\mathrm{1\times n}}$,so that a the zeros of the rational function det P(z), where $\text{P(z) = zI ? A ? ∑}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{z}{}^{\mathrm{-(m+j)}}\text{b?}$_f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.     

A queueing model and a set of orthogonal polynomials

A single serving queueing model is studied where potential customers are discouraged at the rate λ_n = λqⁿ, 0 < q < 1, n is the queue length. The serving rate is μ_n = μ(1 ? qⁿ), n = 0, 1,…. The spectral function is computed and the corresponding set of orthogonal polynomials is studied in detail. The slightly more general model with $\text{λ}{}_{\mathrm{n}}\text{=}\text{λq}{}^{\mathrm{n}}\text{(1 + bq}{}^{\mathrm{n}}\text{)}\text{, μ}{}_{\mathrm{n}}\text{=}\text{μ(1 ? q}{}^{\mathrm{n}}\text{)}\text{(1 + bq}{}^{\mathrm{n}}\text{)}$ and the analogous orthogonal polynomials are also investigated. In both cases a method developed by Pollaczek is used which has been used very successfully to study new sets of orthogonal polynomials by Askey and Ismail.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

A remark to the preceding paper by Chernoff

It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C₀^∞(R^m) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C^∞(R^m) and if $\text{T ? ?a ? b ¦ x ¦}{}^2\text{on}\text{C}{}_0{}^{\mathrm{\infty }}\text{(R}{}^{\mathrm{m}}\text{)}$.     

Singular perturbations in the interaction representation,II

Perturbations of $\text{(?Δ)}{}^{\mathrm{<mtext>m</mtext><mtext>2</mtext>}}\text{in}\text{L}{}^2\text{(R}{}^{\mathrm{k}}\text{),}\text{for}\text{k ? 1}$ and suitable m, by distributions V for which $\text{V}\text{?}\text{(k) = 0(¦k¦}{}^{\mathrm{\alpha }}\text{),}\text{where}\text{α =}\text{(m + 1 ? ε)}\text{2 ? k}\text{, 0 < ε ? m + 1 ? 2k}$, are shown to correspond to self-adjoint operators H_v, in such a way that H_v depends continuously on V, and agrees with H + V when V is sufficiently regular. These results extend joint work with Irving E. Segal [J. Functional Analysis38 (1980), 71–98], in which perturbations of $\text{(}\text{?id}\text{dx}\text{)}{}^{\mathrm{m}}$ by distributions V with bounded Fourier transforms in L²(R¹) were considered.     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

Measure algebras associated with a second order differential operator

Let $\text{L =}\text{d}{}^2\text{dx}{}^2\text{? q(x)}$ be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that $\text{L(f 1 g) = (Lf) 1 g}$ for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and $\text{x → (1 + ¦ x ¦) q(x)}$ is integrable, or that q is of bounded variation. A function ψ is then found such that $\text{M}$ψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of $\text{M}$_ψ is discussed and conditions for its semisimplicity are obtained.