Quelques Résultats sur les Algèbres Flexibles Préhilbertiennes sans Diviseurs de Zéro Vérifiant ∥a 2∥ = ∥a∥2

Résumé. Soit A une algèbre réelle. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ = ∥a∥² pour tout . Si A est flexible, sans diviseurs de zéro et de dimension ≤ 4, alors A est isomorphe à ou , ce qui généralise un théorème d’El-Mallah [1]. Si A est flexible, sans diviseurs de zéro, contenant un idempotent central et vérifiant la propriété d’Osborn, alors A est de dimension finie et isomorphe à , ou . Enfin nous montrons qu’une algèbre normée préhilbertienne unitaire d’unité e telle que ∥e∥ = 1 est flexible et vérifie ∥a ²∥ = ∥ a∥².

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Let A be a real algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a ²∥ = ∥a∥² for all . If A is flexible, without divisor of zero and of a dimension ≤ 4, then A is isomorphic to or , which generalize El-Mallah’s theorem [1]. If A is flexible, without divisor of zero, containing a central idempotent and satisfying Osborn’s properties, then A is finite dimensional and isomorphic to , or . Finally we prove that a normed pre-Hilbert algebra with unit e such that ∥e∥ = 1 is flexible and satisfies ∥a ²∥ = ∥a∥².

     

Sur les anneaux tels que tout produit de copies d'un module quasi-injectif soit un module quasi injectif. — II

Résumé On étudie dans cet article les anneaux noethériens commutatifs tels que tout produit de copies d'un module quasi injectif soit un module quasi-injectif, un tel anneau est produit fini d'anneaux locaux vérifiant certaines propriétés. On étudie également des anneaux noethériens commutatifs un peu plus généraux: les C1-anneaux, qui sont caractérisés comme étant des produits finis d'anneaux locaux A_i,1≤i≤n tels que tout idéal ′ du complété R(A_i)-adique ?_i de A_i vérifie ′ = ( ′∩A_i)?_i. On donne des exemples de tels anneaux. Entrata in Redazione il 12 febbraio 1975.     

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.     

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Invertibility of linear f-order preservers

In this paper, we prove that monotonic linear transformations with respect to partial orders and are invertible. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 3–11, 2003.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

An asymptotic analysis method for the linearly shell theory

In this paper, we consider a linearly elastic shell, i.e. a three-dimensional linearly elastic body with a small thickness denoted by 2ε, which is clamped along its part of the lateral boundary and subjected to the regular loads. In the linear case, one can use the two-dimensional models of Ciarlet or Koiter to calculate the displacement for the shell. Some error estimates between the approximate solution of these models and the three-dimensional displacement vector field of a flexural or membrane shell have been obtained. Here we give a new model for a linear and nonlinear shell, prove that there exists a unique solution U of the two-dimensional variational problem and construct a three-dimensional approximate solutions UKT(x,ξ) in terms of U: We also provide the error estimates between our model and the three-dimensional displacement vector field :‖u-UKT‖1,Ω≤C∈r,r=3/2, an elliptic membrane, r = 1/2, a general membrane, where C is a constant dependent only upon the data‖u‖3,Ω,‖UKT‖3,Ω,θ.     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

Strong law of large numbers for functionals of products of random matrices

For the sequence A₁, A₂, ..., A_n, ... of m×m independent random matrices such that for each k there exists a joint density function P_k(X) of the elements _ij ^k, we prove the following theorem: if and for some positive constants ₁ and ₂, then with probability 1,

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.     

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}.     

Local gradient estimates for solutions of a simplest regularization of a class of nonuniformly elliptic equations

An estimate of , Ω′⊃⊃Ω, for solutions u^ε of the family of equations

Some characterizations of weighted Herz-type Hardy spaces and their applications

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-weight, 1<q>∞,n(1−1/q)≤α<∞ and 0<p<∞. Then, using these new characterizations, they investigate the convergence of a bounded set in these spaces, and study the boundedness of some potential operators on these spaces. Supported by the NNSF of China     

A Weighted Erdős-Ginzburg-Ziv Theorem

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct so that they can be considered as sets. If S is a sequence of m+n−1 elements from a finite abelian group G of order m and exponent k, and if is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence of S such that . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and w_i = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that if S has an n-set partition A=A₁, . . .,A_n such that |w_iA_i| = |A_i| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A′₁, . . .,A′_n of S such that and for all i, where w_iA_i={w_ia_i |a_i∈A_i}.     

Асимптотические рав енства для интеграло в от модуля суммы кратн ого ряда из синусов

Assume that the coefficients of the series $$\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i = 1}^m \sin k_i x_i $$ satisfy the following conditions: a) a_k → 0 for k₁ + k₂ + ...+k_m →∞, b) \(\delta _{B,G}^M (a) = \mathop {\mathop \sum \limits_{k_i = 1}^\infty }\limits_{i \in B} \mathop {\mathop \sum \limits_{k_j = 2}^\infty }\limits_{j \in G} \mathop {\mathop \sum \limits_{k_v = 0}^\infty }\limits_{v \in M\backslash (B \cup G)} \mathop \Pi \limits_{i \in B} \frac{1}{{k_i }}|\mathop \sum \limits_{I_j = 1}^{[k_j /2]} (\nabla _{l_G }^G (\Delta _1^{M\backslash B} a_k ))\mathop \Pi \limits_{j \in G} l_j^{ - 1} |< \infty ,\) for ∨B?M, ∨G?M,B∩G, where M={1,2, ...,m}, $$\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\Delta _1^j a_k = a_k - a_{k_{M\backslash \{ j\} } ,k_{j + 1} } ,\Delta _1^B a_k = \Delta _1^{B\backslash \{ j\} } (\Delta _1^j a_k ), \hfill \\ \Delta _{l_j }^j a_k = a_{k_{M\backslash \{ j\} } ,k_j - l_j } - a_{k_{M\backslash \{ j\} } ,k_j + l_j } ,\nabla _{l_G }^G a_k = \nabla _{l_{G\backslash \{ j\} } }^{G\backslash \{ j\} } (\nabla _{l_j }^j a_k ). \hfill \\ \end{gathered} $$ Then for all n∈N^m the following asymptotic equation is valid: $$\mathop \smallint \limits_{{\rm T}_{\pi /(2n + 1)}^m } |\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i \in M} \sin k_i x_i |dx = \mathop \sum \limits_{k = 1}^n \left| {a_k } \right|\mathop \Pi \limits_{i \in M} k^{ - 1} + O(\mathop {\mathop \sum \limits_{B,{\mathbf{ }}G \subset M} }\limits_{B \ne M} \delta _{B,G}^M (a)).$$ Here \(T_{\pi /(2n + 1)}^m = \left\{ {x = (x1,x2,...,xm):\pi /(2n + 1) \leqq xi \leqq \pi ;i = \overline {1,m} } \right\}\) . In the one-dimensional case such an equation was proved by S. A. Teljakovskii.     

A proximal point algorithm for minimax problems

This paper presents a proximal point algorithm for solving discretel _∞ approximation problems of the form minimize ∥Ax−b∥_∞. Let ε_∞ be a preassigned positive constant and let ε _l ,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε _l < ε_∞. Then, starting from an arbitrary pointz ₀, the proposed method generates a sequence of points z _l ,l= 0,1,2,..., via the rule . One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within a finite number of iterations. The smaller are the numbers ε _l the smaller is the number of iterations. In fact, if ε ₀ is sufficiently small then z₁ solves the original minimax problem. The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax problem of the form minimize where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize , and ify solves the dual thenx=A ^T y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate the feasibility of our ideas.     

Intégration analytique d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier,II

Résumé Nous avons formé dans le premier mémoire une solution formelle de(28.1) ordonnée suivant les puissances entière des solutions de(E). En annulant les Z_j qui correspondent aux polynomes de degrés différents de σ_v*(<σ_i*), nous obtenons une solution formelle (F _v). Dans le chapitre III, nous cherchons la condition suffisante pour la convergence uniforme de la série formelle (F _v). Dans le chapi tre IV, ce résultat est étendu au cas où la solution formelle dépend essentiellement des polynomes λ_j(x) de divers degrés. Le dernier chapitre est consacré à l'extension du théorème d'existence deM. M. Hukuhara qui joue un r?le important dans notre Mémoire.     

On the spectral representation of the elementary operator and spectral mapping theorems

LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X) ⁿ ,B∈B(Y) ⁿ , the elementary operator acting onB(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS _p (L _A ,R _B )=σ(A)×σ(B) and .