Liftable <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>-covers

Let k be an algebraically closed field of characteristic p and let G \hookrightarrow Aut_k(k[[ t ]] ){G \hookrightarrow {\rm Aut}_k(k\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G \hookrightarrow Aut_R(R[[ t ]] ){G \hookrightarrow {\rm Aut}_R(R\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} which reduces to the given action, i.e., a lift to characteristic 0. We show that liftable actions exists in the case that G  =  D ₄ and p  =  2. In fact we introduce a family, the supersimple D ₄-actions, which can always be lifted to characteristic 0.     

On the solvability in Hilbert space of certain nonlinear operator equations depending on parameters

The equation (*) Au − λTu + μCu = f is studied in a real separable Hilbert space H. Here, λ, μ > 0 are fixed constants and f ε H is fixed. The operators A: D H → H, C: D H → H are monotone and compact, respectively, where D denotes a closed ball in H. The operator T: H → H is linear, compact, self-adjoint and positive-definite. Degree-theoretic arguments are used for the existence of solutions of (*) and extensions of recent results of Kesavan are established. For example, it is shown, under additional assumptions, that there exists a constant μ₀ > 0 such that (*) is solvable for all μ μ₀ and all λ ε R.     

Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

Full-size image

Then

Full-size image

18(n+m+1)(λ_n+γ_m).In particular ,

Full-size image

Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

Julia Sets of Certain Exponential Functions

We characterize the Julia sets of certain exponential functions. We show that the Julia sets J(F_λn) of F_λn(z) = λ_ne^zn where λ_n > 0 is the whole plane , provided that lim_k → ∞ F^k_λn(0) = ∞. In particular, this is true when λ_n are real numbers such that . On the other hand, if , then J(F_λn) is nowhere dense in and is the complement of the basin of attraction of the unique real attractive fixed point of F_λn. We then prove similar results for the functions[formula] where λ_i    − {0}, 1 ≤ i ≤ n + 1, a_j > 1, 1 ≤ j ≤ n, and m, n ≥ 1.     

Approximation mit einer erweiterten Klasse von Exponentialsummen

Wir behandeln das Problem, eine stetige Funktion f im Intervall [0, 1] mit einer erweiterten Klasse von Exponentialsummen gleichmäβig zu approximieren. Die Klasse V_n^τ(S) besteht dabei aus allen reellwertigen Lösungen von homogenen, linearen Differentialgleichungen n-ter Ordnung mit konstanten Koeffizienten, bei denen das charakteristische Polynom nur Nullstellen in einer Menge S der komplexen Zahlen besitzt. Wir geben einen sehr kurzen Beweis dafür, daβ jede solche Summe n-ter Ordnung höchstens n − 1 Nullstellen in [0, 1] besitzt, wenn die Frequenzen im Streifen T={λC:|Imλ|<π} liegen. Bei Beschränkung auf T={λC:0<|Imλ|≤π} läβt sich eine Minimallösung notwendig und hinreichend charakterisieren durch eine Alternante der Länge n + k + 1 und die Minimallösung ist eindeutig bestimmt, falls die Frequenzen im Innern von T^* liegen.     

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ₁(m): m ≥ 1} and {λ₂(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

Analysis of steady viscous flow in slender tubes

The computer extended perturbation series method is used to analyze the problem of steady viscous flow in slender tubes. The objective is to obtain an expansion in a power series of λ (= ɛ R, ɛ is a small parameter and is a streamwise Reynolds number) and look for its analytic continuation. Such an expansion was usually terminated at the second or third order term and consequently they have a very limited utility. Sufficiently large number of terms in the series, representing physical quantities are, generated for the detail analysis which enables to get converging Pade’ sums for large λ. Domb-Sykes plot enables in finding singularity restricting the convergence of the series. Useful results valid up to λ = 15 are obtained for different derived quantities whereas in earlier findings [6], analysis could be done only up to λ = 10 resulting into a substantial improvement in the present study.Received: September 17, 2003; revised: July 5, 2004     

The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V₊  − V₋ , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N₊ (V; λ, α), N₋ (V; λ, α), and N₀(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators.     

Rate of Convergence of Positive Series

We investigate the rate of convergence of series of the form

Omega result for the mean square of the Riemann zeta function

A recent method of Soundararajan enables one to obtain improved Ω-result for finite series of the form ∑_nf(n) cos (2πλ_nx+β) where 0≤λ₁≤λ₂≤. . . and β are real numbers and the coefficients f(n) are all non-negative. In this paper, Soundararajan’s method is adapted to obtain improved Ω-result for E(t), the remainder term in the mean-square formula for the Riemann zeta-function on the critical line. The Atkinson series for E(t) is of the above type, but with an oscillating factor (−1)ⁿ attached to each of its terms.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes

For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ_{2m, n}(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ_{2m, n} max{λ_{2m, n}(x): −1 ≤ x ≤ 1}, and show that Λ_{2m, n} = λ_{m, n}(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ_{2m, n)} as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ_{2, n}.     

Orientations Acycliques et le Polynôme Chromatique

On attache à tout graphe G son polynôme chromatiqueχ_G (λ), qui dénombre ses colorations régulières avecλ couleurs. D’après Stanley, on sait que |χ_G ( − 1)| est égal au nombre d’orientations acycliques du graphe, un résultat qui fut raffiné par Greene et Zaslavsky. Nous nous proposons de l’affiner davantage en interprétant, avec l’aide de certaines orientations acycliques, les coefficients deχ_G (λ) développé en puissances de λ et surtout en puissances de (λ −  1). L’utilisation systématique des fonctions génératrices des fonctions d’ensembles permet d’avoir des démonstrations très courtes et explicatives. Elles se veulent une réponse à la suggestion faite par Gebhard et Sagan, qui ont déjà trouvé des démonstrations combinatoires de deux résultats de Greene et Zaslavsky. Les fonctions d’ensembles permettent aussi d’établir une série d’interprétations nouvelles de l’invariant β_Gde Crapo. Cet article donne également un nouvel éclat aux résultats classiques de Cartier, Foata, Viennot, Brenti, Gessel et Stanley. The chromatic polynomialχ_G (λ), which is associated with each graph G, enumerates its regular colorations with λ colors. Stanley showed that |χ_G ( − 1)| is equal to the number of acyclic orientations of the graph, a result that was refined by Greene and Zaslavsky. The purpose of the paper is to show that a further refinement can be obtained by interpreting each coefficient of χ_G(λ), when the polynomial is developed with respect to powers of λ and (λ −  1). A systematic use of the generating functions for set functions enables us to have very short and instructive proofs. Gebhard and Sagan, who had already found combinatorial proofs of two results by Greene and Zaslavsky, suggested that further proofs were to be found. Finally, the set functions algebra allows us to establish a series of new interpretations for Crapo’sβ_G invariant. This paper also brings a new light to the classical results due to Cartier, Foata, Viennot, Brenti, Gessel and Stanley.     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

Planar binary trees and perturbative calculus of observables in classical field theory

We study the Klein–Gordon equation coupled with an interaction term (□+m²)φ+λφ^p=0. In the linear case (λ=0) a kind of generalized Noether's theorem gives us a conserved quantity. The purpose of this paper is to find an analogue of this conserved quantity in the interacting case. We will see that we can do this perturbatively, and we define explicitly a conserved quantity, using a perturbative expansion based on Planar Trees and a kind of Feynman rule. Only the case p=2 is treated but our approach can be generalized to any ^p-theory.     

A new class of symmetric polynomials defined in terms of tableaux

A new class of symmetric polynomials in n variables z = (z₁,…, z_n), denoted t_λ(z), and labelled by partitions λ = [λ₁ … λ_n] is defined in terms of standard tableaux (equivalently, in terms of Gel'fand-Weyl patterns of the general linear group GL(n,C)). The t_λ(z) are shown to be a -basis of the ring of all symmetric polynomials in n variables. In contrast to the usual basis sets such as the Schur functions e_λ(z), which are homogeneous polynomials in the z_i, the t_λ(z) are inhomogeneous. This property is reflected in the fact that the t_λ(z) are a natural basis for the expansion of certain (inhomogeneous) symmetric polynomials constructed from rising factorials. This and several other properties of the t_λ(z) are proved. Two generalizations of the t_λ(z) are also given. The first generalizes the t_λ(z) to a 1-parameter family of symmetric polynomials, T_λ(α; z), where α is an arbitrary parameter. The T_λ(α; z) are shown to possess properties similar to those of the t_λ(z). The second generalizes the t_λ(z) to a class of skew-tableau symmetric polynomials, t_λ/μ(z), for which only a few preliminary results are given.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,