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.     

Un analogue d'un théorème de Hardy pour la transformation de DunklAn analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform $\text{F}{}_{\mathrm{D}}$ on $\text{R}{}^{\mathrm{d}}$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that and , where $\text{L}{}_{\mathrm{k}}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ are the L^p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.     

On a function due to S. Chowla

Let θ(k, pⁿ) be the least s such that the congruence $\text{x}{}_1{}^{\mathrm{k}}\text{+ ? + x}{}_{\mathrm{s}}{}^{\mathrm{k}}\text{≡ 0}$ (mod pⁿ) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then $\text{θ(k, p}{}^{\mathrm{n}}\text{) ≤ k}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n.     

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.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

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).     

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.     

The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

New bounds on exponential sums related to the Diffie–Hellman distributions

Given $\text{θ∈}\text{F}{}_{\mathrm{p}}{}^1$ (p prime) of multiplicative order t>p^δ, we obtain nontrivial bounds on exponential sums

$\text{∑}\text{s′=1}\text{t}\text{∑}\text{s=1}\text{t}\text{e}{}_{\mathrm{p}}\text{aθ}{}^{\mathrm{s}}\text{+cθ}{}^{\mathrm{ss\prime }}$

as well as the corresponding incomplete sums. These estimates are of relevance to several issues, such as the Diffie–Hellman distributions in cryptography, prime divisors of ‘sparse integers’, the distribution $\text{mod}\text{p}$ of Mersenne numbers M_q=2^q?1 (q prime). The method is closely related to that of Bourgain and Konyagin (C. R. Acad. Sci. Paris, Ser. I 337 (2) (2003) 75–80). To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

A divided difference inequality for n-convex functions

For a₍₁₎ ? a₍₂₎ ? ··· ? a_(n) ? 0, b₍₁₎ ? b₍₂₎ ? ··· ? b_(n) ? 0, the ordered values of a_i, b_i, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that

$\text{∑}\text{1}\text{n}\text{a}{}_{\mathrm{i}}\text{b}{}_{\mathrm{i}}\text{?}\text{∑}\text{1}\text{m}\text{a}{}^{\mathrm{p}}{}_{\mathrm{(i)}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{∑}\text{1}\text{m?k?1}\text{b}{}^{\mathrm{q}}{}_{\mathrm{(i)}}\text{+}\text{b}{}^{\mathrm{q}}{}_{\mathrm{[m?k]}}\text{(k+1)}{}^{\mathrm{<mtext>q</mtext><mtext>p</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>q</mtext>}}$

where $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1, b}{}_{\mathrm{[j]}}\text{= b}{}_{\mathrm{(j)}}\text{+ b}{}_{\mathrm{(j\; +\; 1)}}\text{+ ··· + b}{}_{\mathrm{(n)}}\text{,}\text{and}\text{k}$ is the integer such that $\text{b}{}_{\mathrm{(m\; ?\; k\; ?\; 1)}}\text{?}\text{b[m ? k]}\text{(k + 1)}$ and $\text{b}{}_{\mathrm{(m\; ?\; k)}}\text{<}\text{b}{}_{\mathrm{[m\; ?\; k\; +\; 1]}}\text{k}$. The inequality is shown to be sharp. When p < 1 and a_(i)'s are in increasing order then the inequality is reversed.     

Nouvelle preuve d'un théorème de Yoccoz

If θ is an irrational real number such that ∑(lnq_n+1)/q_n=+∞ where p_n/q_n are the convergents of θ, then the quadratic polynomial $\text{P}{}_{\mathrm{\theta }}\text{(z)=}\text{e}{}^{\mathrm{<mtext>i</mtext><mtext>2\pi \theta </mtext>}}\text{z+z}{}^2$ is not linearizable at 0. This theorem has been proved in 1988 by J.C. Yoccoz, who first constructs a nonlinearizable germ by inverting a renormalisation procedure, and then proves universality of the quadratic family for that question. We give an alternative proof, based on the study of the explosion of parabolic cycles. To cite this article: A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Estimates on the first eigenvalue of an integral equation

We show how inequalities of the type $\text{∥F∥}{}_{\mathrm{p}}\text{? C(p, q) a}{}^{\mathrm{<mtext>1\; +\; (</mtext><mtext>1</mtext><mtext>p</mtext><mtext>)?\; (</mtext><mtext>1</mtext><mtext>q</mtext><mtext>)</mtext>}}\text{∥ F ′ ∥}{}_{\mathrm{q\prime }}$ when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝₀^ak(s, z)F(s) ds.     

A note on character sums

Let $\text{S(k) = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{p?1}}\text{(}\text{n}\text{p}\text{)n}{}^{\mathrm{k}}$ where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign.     

Binomial self-inverse sequences and tangent coefficients

This paper treats the class of sequences {a_n} that satisfy the recurrence relation

$\text{a}{}_{\mathrm{2n+1}}\text{=∑}{}_{\mathrm{<mtext>k=0</mtext>}}{}^{\mathrm{<mtext>n</mtext>}}\text{(?1)}{}^{\mathrm{<mtext>k</mtext>}}\text{(}\text{n}\text{k}\text{a}{}_{\mathrm{k}}\text{d}{}^{\mathrm{n?k}}$

between the odd and even terms of {a_n} that involves the coefficients of tan(t), namely

$\text{a}{}_{\mathrm{2n+1}}\text{=∑}{}_{\mathrm{<mtext>k=0</mtext>}}{}^{\mathrm{<mtext>n</mtext>}}\text{(?1)}{}^{\mathrm{<mtext>k</mtext>}}\text{(}\text{2}\text{n}\text{+1}\text{2}\text{k}\text{+1}\text{)}\text{T}{}_{\mathrm{k}}\text{(d/2)}{}^{\mathrm{2k+1}}\text{a}{}_{\mathrm{2n?2k}}$

A combinatorial setting is then provided to elucidate the appearance of the tangent coefficients in this equation.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

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).     

On Lp-spectrum of elliptic operators

For elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{a}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$ on Rⁿ and certain of their singular perturbations $\text{B = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x)D}{}^{\mathrm{\alpha }}$ relative compactness of B with respect to A is established. This result applies to the study of L^p-spectra of elliptic operators for different p.     

Infinite 0–1-sequences without long adjacent identical blocks

This paper deals with sequences a₁a₂a₃ ··· of symbols 0 and 1 with the property that they contain no arbitrary long blocks of the form a_i+1 ? a_i+k = ww. The behaviour of this class of sequences with respect to some operations is examined. Especially the following is shown: Let be $\text{a}{}^{\mathrm{(0)}}{}_{\mathrm{i}}\text{= a}{}_{\mathrm{i}}\text{, a}{}^{\mathrm{(n+1)}}{}_{\mathrm{i}}\text{= (}\text{1}\text{i}\text{) ∑}{}^{\mathrm{i}}{}_{\mathrm{k\; =\; 1}}\text{a}{}^{\mathrm{(n)}}{}_{\mathrm{k}}$, then there exists a sequence without arbitrary long adjacent identical blocks such that no lim_k→∞a⁽ⁿ⁾_k exists. Let be α? (0, 1), then there exists such a sequence with lim_k→∞a⁽¹⁾_k = α. Furthermore a class of sequences appearing in computer graphics is considered.