On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Congruence properties of Apéry numbers

Apéry introduced a recurrence relation for a proof of the irrationality of ζ(3). Let a_n (n ≥ 0) satisfy the relation n³a_n ? (34n³ ? 51n² + 27n ? 5)a_{n ? 1} + (n ? 1)³a_{n ? 2} = 0. Which values of a₀ and a₁ cause each a_n to be an integer? This question is answered and some congruence properties of the a_n are given.     

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

Let (M_r)_{r∈? 0} be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function ? on ?ⁿ belonging to the non quasi-analytic (M_r)-class of Roumieu type, there is an element g of the same class which is analytic on ?ⁿ F and such that D^α ?(x) = D^αg(x) for every σ ∈ ?₀ ⁿ SBAP and x ∈ F.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ?: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² ? r ⁿ ? 3r² ? 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.     

Behrend's theorem for sequences containing no k-element arithmetic progression of a certain type

Let k and n be positive integers, and let d(n, k) be the maximum density in {0, 1, 2,…, kⁿ ? 1} of a set containing no arithmetic progression of k terms with first term a = Σa_ikⁱ and common difference d = Σ?_ikⁱ, where 0 ? a_i ? k ? 1, ?_i = 0 or 1, and ?_i = 1 ? a_i = 0. Setting β_k = lim_n→∞d(n, k), we show that lim_k→∞β_k is either 0 or 1.     

On the simplex method and a class of linear complementary problems

We consider the linear complementarity problem of finding vectors w?Rⁿ, z?Rⁿ satisfying w ? Mz = q, w ? 0, z ? 0, w^Tz = 0. We show that if the off diagonal elements of M are nonpositive, then the above problem is solved by applying the simplex method to the problem Minimize z₀ subject to w ? Mz ? e_nz₀ = q, (z₀, w, z) ? 0, where e_n is a column vector of 1's. In fact the sequence of basic feasible solutions obtained by the simplex method and by Lemke's algorithm are the same. We also obtain necessary and sufficient conditions for the problem to have solutions for all q.     

An ?-dependent formulation of the Kadomtsev-Petviashvili hierarchy

We briefly review a recursive construction of ?-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ?-expansion of the operator X = X ₀ + ?X ₁ + ? ² X ₂ + ... for which the dressing operator W is expressed in the exponential form W = e^X/?. The wave function ?? associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form ?? = e^S/kh, and the coefficients S_n of the ?-expansion S = S ₀ + ?S ₁ + ? ² S ₂ + ... are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an ?-expansion of the form log ?? = ?^?2 F ₀ + ?^?1 F ₁ + F ₂ + ....     

There are no de bruijn sequences of span n with complexity 2n − 1 + n + 1

If s = (s₀, s₁,…, s_2ⁿ?1) is a binary de Bruijn sequence of span n, then it has been shown that the least length of a linear recursion that generates s, called the complexity of s and denoted by c(s), is bounded for n ? 3 by 2^{n ? 1} + n ? c(s) ? 2ⁿ ?1. A numerical study of the allowable values of c(s) for 3 ? n ? 6 found that all values in this range occurred except for 2^n?1 + n + 1. It is proven in this note that there are no de Bruijn sequences of complexity 2^n?1 + n + 1 for all n ? 3.     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Cauchy-Riemann Automorphism Groups that cannot be projectively realized

LetM ? ?ⁿ be a real-analytic, nonspherical hypersurface passing through the origin and having nondegenerate Levi form. Let Aut₀ M be the stability group of 0. Whenn = 12 an example is constructed for which Aut₀ M cannot be linearized.     

Absolute regularity and functions of Markov chains

We first give an extension of a theorem of Volkonskii and Rozanov characterizing the strictly stationary random sequences satisfying ‘absolute regularity’. Then a strictly stationary sequence {X_k, k = …, ?1, 0, 1,…} is constructed which is a 0?1 instantaneous function of an aperiodic Markov chain with countable irreducible state space, such that n^?2 var (X₁ + ? + X_n) approaches 0 arbitrarily slowly as n → ∞ and (X₁ + ? + X_n) is partially attracted to every infinitely divisible law.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

The product of linear nonhomogeneous forms

We show that for an arbitrary unimodular lattice Λ of dimension n and an arbitrary point C =(c₁, c₂...c_n) ? Rⁿ a point Y = (y₁, y₂,..., y_n) ε Λ can be found and also a number h, satisfying the condition 1 ?h ? 2^?n/2 θ^?1 + 1 (0 < θ ? 2^?n/2), such that the inequality $$\prod\nolimits_{i = 1}^n {\left| {Y_i + hc_i } \right|}< \theta $$ will be satisfied.     

Order of starlikeness for multipliers of univalent functions

A function f(z) = z ? ∑_{n = 2}^∞a_nzⁿ, a_n ? 0, is said to be in the family F({b_n}) if there exists a sequence {b_n} of positive real numbers such that ∑_{n = 2}^∞b_na_n ? 1. All functions in F({b_n}) are univalent (and starlike) if and only if b_n ? n for every n. The extreme points, distortion properties, order of starlikeness, and radius of convexity for such families are determined. By specializing {b_n}, the results reduce to those of some well-known families. Information is also obtained when the arguments of the coefficients are unrestricted. All results are sharp.     

Padé-type approximants with orthogonal generating polynomials

The interpolation of the function x → 1/(1 ? xt) generating the series f(t) = ∑^∞_{i = 0}c_itⁱ at the zeros of an orthogonal polynomial with respect to a distribution d α satisfying some conditions will give us a process for accelerating the convergence of f_n(t) = ∑ⁿ_{i = 0}c_itⁱ. Then, we shall see that the polynomial of best approximation of x → 1/(1 ? xt) over some interval or its development in Chebyshev polynomials T_n or U_n are only particular cases of the main theorem.At last, we shall show that all these processes accelerate linear combinations with positive coefficients of totally monotonic and oscillating sequences.     

Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

Suppose that ? ⁿ is the p-dimensional space with Euclidean norm ∥ ? ∥, K (? ^p ) is the set of nonempty compact sets in ? ^p , ?₊ = [0, +∞), D = ?₊ × ? ^m × ? ⁿ × [0, a], D ₀ = ?₊ × ? ^m , F ₀: D ₀ → K (? ^m ), and co F ₀ is the convex cover of the mapping F ₀. We consider the Cauchy problem for the system of differential inclusions $$\dot x \in \mu F(t,x,y,\mu ),\quad \dot y \in G(t,x,y,\mu ),\quad x(0) = x_0 ,\quad y(0) = y_0$$ with slow x and fast y variables; here F: D → K (? ^m ), G: D → K (? ⁿ ), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F ₀, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ₀ > 0 such that for any μ ∈ (0, μ₀] and any solution (x _μ(t), y _μ(t)) of the problem under consideration, there exists a solution u _μ(t) of the problem ${\dot u}$ ∈ μ co F ₀ (t, u), u(0) = x ₀ for which the inequality ∥x _μ(t) ? u _μ(t)∥ < ε holds for each t ∈ [0, 1/μ].     

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π and let Nn be the number of real roots of Tn(θ) in(0,2π).In this paper it is proved that limn →∞ Var(Nn)/n=c0,where 0相似文献   

On hypergraphs without two edges intersecting in a given number of vertices

Let X be a finite set of n-melements and suppose t ? 0 is an integer. In 1975, P. Erdös asked for the determination of the maximum number of sets in a family $\text{F}$ = {F₁,…, F_m}, F_i ? X, such that ∥F_i ∩ F_j∥ ≠ t for 1 ? i ≠ j ? m. This problem is solved for n ? n₀(t). Let us mention that the case t = 0 is trivial, the answer being 2^{n ? 1}. For t = 1 the problem was solved in [3]. For the proof a result of independent interest (Theorem 1.5) is used, which exhibits connections between linear algebra and extremal set theory.