An infinite class of generalized room squares

A generalized Room square $\text{G}$ of order n and degree k is an ${}^{\mathrm{n?1}}{}^{\mathrm{k?1}}\text{×}{}^{\mathrm{n?1}}{}^{\mathrm{k?1}}$ array, each cell of which is either empty or contains an unordered k-tuple of a set S, |S| = n, such that each row and each column of the array contains each element of S exactly once and $\text{G}$ contains each unordered k-tuple of S exactly once. Using a class of Steiner systems and a generalized Room square of order 18 and degree 3 constructed by ad hoc methods, an infinite class of degree 3 squares is constructed.     

A continuation method for solving separable nonlinear least squares problems

Given the data (x_i, y_i), i=1, 2, …, n, the problem is to find the values of the linear and nonlinear parameters â and b? which minimize the nonlinear functional |F(b)a?y|₂² over $\text{a ?}\text{R}{}^{\mathrm{p}}\text{, b ?}\text{R}{}^{\mathrm{q}}\text{,}\text{where}\text{F ?}\text{R}{}^{\mathrm{n\times p}}$ is a variable matrix and assumed to be of full rank, and $\text{y ?}\text{R}{}^{\mathrm{n}}$ is a constant vector.In this paper, we present a method for solving this problem by imbedding it into a one-parameter family of problems and by following its solution path using a predictor-corrector algorithm. In the course of iterations, the original problem containing p+q+1 variables is transformed into a problem with q+1 nonlinear variables by taking the separable structure of the problem into account. By doing so, the method reduces to solving a series of equations of smaller size and a considerable saving in the storage is obtained.Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method.     

Sobolev inequalities with remainder terms

The usual Sobolev inequality in $\text{R}$ⁿ, n ? 3, asserts that , with S_n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? $\text{R}$ⁿ. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with $\text{p =}\text{2}{}^1\text{2}$ and with $\text{q =}\text{n}\text{(n ? 1)}$. (ii) If ? ≠ 0 on ?Ω, then with $\text{q =}\text{2(n ? 1)}\text{(n ? 2)}$. Some further results and open problems in this area are also presented.     

All directed BIBDs with k = 3 exist

A directed BIBD with parameters ($\text{υ, b, r, k, λ}{}^1$) is a BIBD with parameters ($\text{υ, b, r, k, 2λ}{}^1$) in which each ordered pair of varieties occurs together in exactly $\text{λ}{}^1$ blocks. It is shown that $\text{λ}{}^1\text{υ(υ ? 1) ≡ 0}$ (mod 3) is a necessary and sufficient condition for the existence of a directed ($\text{υ, b, r, k, λ}{}^1$) BIBD with k = 3.     

A sharp inequality for Sobolev functionsUne inégalité dans un espace de Sobolev

Let N?5, a>0, $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$, $\text{2}{}^1\text{=}\text{2N}\text{N?2}$, $\text{2}{}^{\mathrm{\#}}\text{=}\text{2(N?1)}\text{N?2}$ and 6u6²=|?u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $\text{u∈}\text{H}{}^1\text{(Ω)?{0}}$,

This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

The torsion group of a field defined by radicals

Let $\text{L}\text{F}$ be a finite separable extension, $\text{L}{}^1\text{= L{0}}$, and $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$ the torsion subgroup of $\text{L}{}^1\text{F}{}^1$. When $\text{L}\text{F}$ is an abelian extension $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$ is explicitly determined. This information is used to study the structure of $\text{T(}\text{L}{}^1\text{F}{}^1\text{)}$. In particular, $\text{T(}\text{F(α)}{}^1\text{F}{}^1\text{)}$ when a^m = a ∈ F is explicitly determined.     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

On the number of zeros of diagonal cubic forms

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

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.     

A family of difference sets

A construction is given for difference sets with parameters $\text{v =}\text{1}\text{2}\text{3}{}^{\mathrm{s+1}}\text{(3}{}^{\mathrm{s+1}}\text{? 1)}$, $\text{k =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s+1}}\text{+ 1), λ =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s}}\text{+ 1), n = 3}{}^{\mathrm{2s}}$ in certain noncyclic groups of order v. For s = 1 it is shown that the construction yields all possible difference sets with parameters (36, 15, 6, 9) in an abelian group of order 36.     

An identity for the divisor generating function arising from sorting theory

In the present paper we prove

$\text{∑}\text{n=1}\text{∞}\text{nx}{}^{\mathrm{n}}\text{∏}\text{j=n+I}\text{∞}\text{(1?x}{}^{\mathrm{j}}\text{)}$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{(?1)}{}^{\mathrm{n?1}}\text{x}{}^{\mathrm{<mtext>n(n+1)</mtext><mtext>2</mtext>}}\text{(1?x)(1?x}{}^2\text{)?(1?x}{}^{\mathrm{n?1}}\text{)(1?x}{}^{\mathrm{n}}\text{)}{}^2$

$\text{=}\text{∑}\text{n=1}\text{∞}\text{d(n)x}{}^{\mathrm{n}}$

     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

Invariant subspaces of certain subnormal operators

Let T be a subnormal, nonnormal operator on a Hilbert space and suppose that the point spectrum of $\text{T}{}^1$ is empty. Then there exist vectors x ≠ 0 for which $\text{(T}{}^1\text{? zI)}{}^{\mathrm{?1}}\text{x}$ exists and is weakly continuous for all z. It is shown that under certain conditions, the Cauchy integral of this vector function taken around an appropriate contour, not necessarily lying in the resolvent set of $\text{T}{}^1$, leads to a proper (nontrivial) invariant subspace of $\text{T}{}^1$.     

Loeb-measurable solutions to 1finite games

Let ¹M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal ¹finite subset of ¹N such that $\text{F = {1,…,γ}, γ?}{}^1\text{N?N}$, we define a class of ¹finite cooperative games having the form $\text{Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν) = 〈F,A(F),}{}^1\text{ν〉}$, where A(F) is the internal algebra of the internal subsets of F, and ${}^1\text{ν}$ is a set-function with $\text{Dom}{}^1\text{ν=A(F)}$, $\text{Rng}{}^1\text{ν =}{}^1\text{R}{}_{\mathrm{+}}$, and ${}^1\text{ν(Ø) = 0}$. If $\text{SI(}{}^1\text{ν)}$ is the space of S-imputations of a game Γ_F(¹ν) such that ${}^1\text{ν(F)<η}$, for some $\text{η?}{}^1\text{N}$, then we prove that $\text{SI(}{}^1\text{ν)}$ contains two nonempty subsets: $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$, termed the quasi-kernel and S-bargaining set, respectively. Both $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ are external solution concepts for games of the form Γ_F (¹ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L(Θ) is the Loeb space generated by the ¹finitely additive measure space 〈F, A(F), U_F〉, and if a game Γ_F(¹ν) has a nonatomic representation $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ on L(Θ) with respect to S-bounded transformations, then the standard part of any element in $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ is Loeb-measurable and belongs to the quasi-kernel of $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ defined in standard terms.     

Minimax estimators for a multinormal precision matrix

Let S_p×p ～ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ^?1 are given for L₁, an Empirical Bayes loss function; and L₂, a standard loss function (R_i ≡ E(L_i ∣ Σ), i = 1, 2). The estimators are $\text{Σ}\text{?}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+}\text{br}\text{(S)I}{}_{\mathrm{p\times p}}$, a, b ≥ 0, r(·) a functional on $\text{R}{}^{\mathrm{<mtext>p(p+2)</mtext><mtext>2</mtext>}}$. Stein, Efron, and Morris studied the special cases $\text{Σ}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{(E}\text{Σ}\text{?}{}_{\mathrm{<mtext>k?p?1</mtext>}}{}^{\mathrm{?1}}\text{= Σ}{}^{\mathrm{?1}}\text{)}$ and $\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+ (b/tr S)I}$, for certain, a, b. From their work $\text{R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤ R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, a = k ? p ? 1, b = p² + p ? 2; whereas, we prove $\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) (?Σ)}$. The reversal is surprising because $\text{L}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) →}\text{L}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S)}$ a.e. (for a particular L₂). Assume $\text{R}$ (compact) ? $\text{S}$, $\text{S}$ the set of p × p p.s.d. matrices. A “divergence theorem” on functions F_p×p : $\text{R}$ → $\text{S}$ implies identities for R_i, i = 1, 2. Then, conditions are given for $\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣^1/p/tr(S).     

A Hamiltonian decomposition of K2m1, 2m ≥ 8

It is shown that $\text{K}{}_{\mathrm{2m}}{}^1$, 2m ≥ 8, can be decomposed into Hamiltonian circuits. A direct construction utilizing difference methods is given for 2m ≡ 0 (mod 4). The case 2m ≡ 2 (mod 4) is handled inductively by means of a construction which shows that $\text{K}{}_{\mathrm{4m\; ?\; 2}}{}^1$ admits such a decomposition if $\text{K}{}_{\mathrm{2m}}{}^1$ does.