On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .     

Bahadur asymptotic efficiency of ω2 -type tests in the many-sample case

Consider independent samples of sizen ₁,n ₂,... ,n ₂ and the empirical distribtion constructed using these samples. We test the hypothesis that all the samples are drawn from a population with the same continuous distribution function F (x). The test statistic is the weight function) is defined as generalizes Kiefer's well-known statistic originally proposed for the case K=2 and q,= 1. A rough asymptotics is obtained for the probability of large deviations of the statistic , which allows to give explicit expressions for the Bahadur local exact slopes and to compare the statistics for various K and q in the sense of Bahadur efficiency. In conclusion, the question posed by Renyi is considered: to what extent is it advisable to use statistics of the type instead of pooling all the samples and using a one-sample test?     

Metrische Räume mit Dreiseitverbindbaren

This paper presents a system of axioms for n-dimensional metric geometry. For every group satisfying the axioms there exist a group-space and an embedding of into a projective-metric space Ω. We construct an isomorphism of onto a subgroup of a special orthogonal group O _n+1 ^* (K,f). This group belongs to a metric vector space (V,f) over a field K of characteristic ≠ 2 where dim rad V≦1. The (full) groups o _n+1 ^* (K,f) are models of the system of axioms.     

For which p and is the equation true?

It is explained when the classes of p-absolutely summing and p-integral operators given on the space coincide. For a Banach space there is considered the following subset of the real line: . In the case when is an infinite-dimensional subspace of the space , it is proved that J_X=(1,2] if , and J_X?{2} if and χ is not isomorphic with a Hilbert space.     

Extension of functions from Sobolev spaces

By definition, the domain Ω ??ⁿ belongs to the class EW _p ^l if there exists a continuous linear extension operator . An example is given of a domain Ω ??² with compact closure and Jordan boundary, having the following properties: (1) The curve ?Ω is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) Ω ε EW _p ¹ for p<2. and Ω ? EW _p ¹ for p?2. (3) for p>2 and for p?2.     

Completely- matrices

A matrix is a real square matrixM such that for everyq the linear complementarity problem: Findw andz satisfyingw = q + Mz, w ≥ 0, z ≥ 0, w ^T z = 0, has a solution. We characterize the class of completely- matrices, defined here as the class of -matrices all of whose nonempty principal submatrices are also -matrices.     

Ein zerlegungssatz für P(ϰ)

We shall prove the following partition theorems:

1. For every setS and for each cardinal ? ≥ ω, |S| ≥ ? there exists a partitionT: [S]^? → 2^? such that for every pairwise disjoint familie and everyα < 2^? there exists a set
2. Suppose ? ≥ ω, 2 ^{andS an arbitrary set, 2|S| ≤ (2?)+ω Then there exists a partitionT: P(S) → 2? such that for every pairwise disjoint family and everyα < 2? there exists a set Both theorems will give partial answers to an Erd?s problem.}

     

VMO spaces not having Schauder basis

ПустьA _n (n=0, 1, 2, ...) — последо вательность δ-алгебр, порождаемых двоичны ми отрезками [k2^?n , (k+1)2^?n ) (0≦k<2 ⁿ ). Замыкание множества двоичных с тупенчатых функций по норме      

Analytic continuation from a continuum to its neighborhood

Let be a positive number. A function f analytic in an open set ? ?? is called -analytic on the set E, E ??, if . THEOREM. Let K be a compact connected subset of the plane. For any >0 there exists an open neighborhood V of the set K such that any function -analytic on K coincides in some neighborhood of the set K with a function analytic in V. This theorem answers a question posed in the collection (RZhMat., 1979, 3B536, pp. 33–35 of the book).     

Singular strictly increasing functions and a problem on partitions of closed intervals

We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets and of a closed interval [a; b] ? ? such that (1) ∩ = ø; (2) ∪ = [a; b]; (3) the Lebesgue measures of the intersections of and with an arbitrary interval J ? [a; b] are positive.     

Local convergence in measure on semifinite von Neumann algebras

Suppose that ? is a von Neumann algebra of operators on a Hilbert space $\mathcal{H}$ and τ is a faithful normal semifinite trace on ?. The set of all τ-measurable operators with the topology t _τ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t _τ and are denoted by t _τ1 and t _wτ1, respectively. The set with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in with respect to the topologies t _τ1 and t _wτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra $\mathcal{B}(\mathcal{H})$ to semifinite von Neumann algebras. The following theorem is proved: For a von Neumann algebra ? with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ? is finite; (ii) t _wτ1 = t _τ1; (iii) the multiplication is jointly t _τ1-continuous from to ; (iv) the multiplication is jointly t _τ1-continuous from to ; (v) the involution is t _τ1-continuous from to .     

Algebraic extensions of finite corank of hilbertian fields

We consider here a hilbertian fieldk and its Galois group (k _s/k). For a natural numbere we prove that almost all (σ) ∈ (k_s/k)^e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ₁, …, σ_e is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk _s (σ) of 〈σ〉, …, σ_e ink _s, which containsk. Then the group (k _s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [k _s (σ):k _s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk?K?k _s (σ) such that 1<[k _s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (k_s/k)^e.     

Un teorema sulla rappresentazione degli integrali

Let be a De Possel differentiation basis in a complete measure space , with μ≥0, μ(X)<+∞; letB be a Banach space and . When the F′ exists μ-a.e. onX, we prove that for eachp>0, the μ-integral of‖F′‖ ^p is the lower-bound of a class ofp-variations.     

The space BMO and strong means of Fourier series

пУстьS _k (f,x) — ЧАстНАь сУ ММА РьДА ФУРьЕ ФУНкцИ Иf пО тРИгОНОМЕтРИЧЕскОИ сИстЕМЕ,s _k (f,x) — ЧАстНАь сУММА сО пРьжЕННОгО РьДА. Дль \(\Delta _k^n = \left[ {\frac{n}{n},\frac{{k + 1}}{n}} \right)\) , гДЕk=0, 1, ...,n?1, пОлОжИМ , ЕслИt?δ _k ⁿ И , ЕслИt?[0, 1)δ _k ⁿ . пОкАжАНО, ЧтО ОпЕРАтО Ры ИМЕУт слАБыИ тИп (1,1). РАссМОтР ЕН РьД слЕДстВИИ О пОВ ЕДЕНИИ сИльНых сРЕДНИх РьДА ФУРьЕ сУММИРУЕМОИ ФУНкцИИ.     

A class of limit distributions for maximum cumulative sum

Let X₁, X₂, ... be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variances. S_o=0 and S_n=∑ _i=1 ⁿ X_i. It is proved that is the limit distribution function of the normalized random variable a(k, n)} for some sequence of centering constants a (k,n).