Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

Sul comportamento quasi periodico del resto di una certa classe di funzioni fortemente moltiplicative

Riassunto Siaf(x) una funzione periodica di periodo 1 ed a variazione limitata. Nel presente lavoro si dimostrano risultati diB ² quasi periodicità per convoluzioni aritmetiche del tipo dove la successione soddisfa opportuna ipotesi. I risultati ottenuti si possono applicare ai resti di certe funzioni fortemente moltiplicative.

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Summary Letf(x) be a periodic function with period 1 and of bounded variation. In this paper we prove results ofB ² almost periodicity for arithmetical convolutions of the type , where the sequence satisfies certain conditions. The results obtained can be applied to the remainder term of strongly multiplicative functions.

     

The individual ergodic theorem forp-sequences

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L ₁ there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have for everyp-sequence {k_n}.     

Chaos and asymptotical stability in discrete-time recurrent neural networks with generalized input-output function

We theoretically investigate the asymptotical stability, local bifurcations and chaos of discrete-time recurrent neural networks with the form of

Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here md T = ∂/∂t with a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderón-Zygmund-Mikhlin operators. Estimates for these operators in L _k ^p (H_n), H^P(H_n), and S _k ^pv (H_n) spaces can therefore be deduced.     

On the DirichletL-functions

The main purpose of this paper is to give a sharper asymptotic formula for the mean square value , whereL′(s, χ) denotes the derivative ofL (s, χ) with respect tos.     

Precise estimate of total deficiency of meromorphic derivatives

Let f(z) be a transcendental meromorphic function in the finite plane andk be a positive integer. Then we have . Moreover, if the order of f(z) is finite, then we also have , where δ(a, f^(k)) denotes the deficiency of the valuea with respect to f^(k) and θ(∞,f) is the ramification index of ∞ with respect tof.     

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Integration of a zonal function related to the local harmonic analysis on spheres

In studying local harmonic analysis on the sphere Sⁿ, R.S. Strichartz introduced certain zonal functions ϕ₂(d(x, y)) which satisfy the equation , where Δ_z is the Laplace operator and δ_−y the Dirac measure. The explicit expression of the constant a (λ) is given by R.S. Strichartz in the case that n is odd. Appyling the Apéry identity, we show in this paper that for n even, where w_n-1 is the surface area of S^n-1, . The author's research was supported by a grant from NSFC.     

Sufficient and necessary conditions for limit laws on lag sums of I. I. D. R. V. S

Let {X, X _n ;n>-1} be a sequence of i.i.d.r.v.s withEX=0 andEX ²=σ²(0 < σ < ∞). we obtain some sufficient and necessary conditions for

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

On the structure of Noetherian symbolic rees algebras

Let A be a Noetherian local ring and I an ideal of A. In this paper we use a slightly generalized notion of a symbolic power I⁽ⁿ⁾ of I and considerR = . First we characterize the property of R to be Noetherian by an equimultiplicity condition of some symbolic power I^(k). The main purpose of this note is to explore the problem whenR,R′ = and are Cohen-Macaulay or Gorenstein algebras in the case that A is a normal domain and ht I=1. Partially supported by the Max-Planck-Institute of Mathematics Bonn. Supported by a grant of the Heinrich Hertz-Stiftung. Supported by a grant of the Alexander von Humboldt-Stiftung.     

The true order of magnitude of Lamé rounding error sums

Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

Asymptotic behavior of the solutions of an integrodifferential system

Summary We consider the system(L): , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet where is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁[0, ∞), thendet forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp [ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose exp [at]B(t) ∈ ∈ L₁[0, ∞) for some a>0 anddet forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Some sharp estimates for the second continuity modulus for periodic functions and for functions extended from a segment

Let C be the space of 2π-periodic continuous real functions with the uniform norm, let H_n be the set of trigonometric polynomials of order not more than n, let ω₂(f) be the second continuity modulus for a function f∈C, and let T_n(f) be the best approximation polynomial of order n for f∈C. Set ; U:C→C; . In this paper, for h sufficiently large we find the values C(U,h) for some positive operators U. For example, C(A₀,h) and C(T₀,h) are found. For n=1,2,3 we find the values for some linear positive operators U:C→H_n. We establish relations between C(T₀,h) and exact constants in the inequality ω₂(f,h₁)≤C(h₁;h)ω₂(f,h) for some h and h₁ such that 0<h<h₁≤π. For a seminorm P invariant with respect to the shift and majorized by the uniform norm, analogs of C(U,h) are estimated from above. We investigate the problem of extension of a function defined on a segment with preservation of the second continuity modulus. The relation

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

A semigroup characterization of the multiparameter Wiener process

It is well known that the infinitesimal operator characterizes the Wiener process {W(t), t≥0}. In this paper we show that the family of operators plays the same role for the q-parameter Wiener process. Partially supported by CNPq Grant No. 200. 607/82.     

Construction of representations of the Poincaré Lie algebra by extension of direct integrals of representations ofSL(2,C)

Summary Continuing the program initiated in the first paper of this series, representations of the Lie algebra of the Poincaré group are constructed by forming suitable direct integrals of representations of SL(2, C) and extending them to by means of four additional momentum operators whose matrix elements are derived in a basis adapted to the direct integral structure. The representations obtained here include Wigner’s representations with positive mass and spin and Wigner’s representations with zero mass, spin and helicity ± . Lavoro eseguito con contributo del C.N.R. nell’ambito dell’attività di ricerca del Gruppo Nazionale per la Fisica Matematica e per le applicazioni della Matematica alla Fisica ed all’Ingegneria. Entrata in Redazione il 12 marzo 1972.