Filtration of the group of principal units of a local function field as a Galois module

Let k be the field of formal power series in one variable over a finite field of constants of characteristic p and K/k be a completely ramified extension of degree p. For the group E₁ of principal units of the field K, considered as a Galois module, generators are constructed effectively. The operator structure (as a Galois module) of the subgroups E_m generating a natural filtration of the group E₁ is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 96–99, 1980.     

A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ _j ?E[ξ _j ], 0<α?Var[ξ _j ] andE[|ξ _j ?μ _j |^2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ₀ are two random variables such thatE[ξ ₀ ² ]<∞ andE[|U|ξ ₀ ² ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ _j :j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereS _n ?ξ₁+ξ₂+?+ξ _n ,μ _j ?E[ξ _j ],s _n ² ?Var[S _n ], andc _n =O(s _n ).     

On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?

     

Summation of weakly dependent sequences by Cesàro methods

Пусть (X, A, u) — пространст во с конечной мерой, (ξ_k) ₁ ^∞ — последовательност ь функций, \(\xi _k \varepsilon L_{2r} (X), r > 1, \int\limits_X {\xi _k d\mu = 0} \) . Изучаются условия, п ри которых справедли вgа - у. з. б.ч., т. e. (ξ _k) суммируется к ну лю почти всюду методо м (С, а),а > 0. Приведем два резу льтата. 1) Если (ξ _k) — слабо мульт ипликативная систем а (в частности, мартингал-разности или независимая сист ема), то условие $$\mathop \sum \limits_1^\infty \mathop {\smallint }\limits_X \left| {\xi _k } \right|^{2r} d\mu \cdot c_r (k,\alpha )< \infty $$ влечетβ - у.з.б.ч. Здесьc _r(k,α)=k ^-2rα при 0<α<(r+1)/2r, c_r=k^?(r+1) In3r-1 k приа=(r+1)/2r, с_r=k^?(r+1) при а >(r+1)/2r. 2) Если (ξ _k) независимы,Mξ _k=0, (r+1)/2r<α=1, то условия $$\mathop \sum \limits_{k = 1}^\infty \frac{{(M\xi _k^2 )^r }}{{k^{r + 1} }}< \infty ,\mathop \sum \limits_{k = 1}^\infty \frac{{M|\xi _k |^{2r} }}{{k^{2r\alpha } }}< \infty $$ влекут за собой а - у. з. б. ч.     

On embedding problems with restricted ramifications

Archiv der Mathematik - Let L / k be a Galois extension with Galois group G, and $(\varepsilon ): 1\to A\to E\to G\to 1$ a central extension. We study the existence of the Galois extension M / L /...     

整数距离图G(D_(m,k,2))的点荫度

图G的点荫度va(G)是顶点集合V(G)能划分成的这样一些子集的最少数目,其中任一子集的点导出子图都是森林.整数距离图G(D)以全体整数作为顶点集,顶点u,v相邻当且仅当|u-v|∈D,其中D是一个正整数集.对于m2k≥2,令D_(m,k,2)=[1,m]\{k,2k}.该文得出了整数距离图G(D_(m,k,2))的点荫度的几个上、下界;进而,对于m≥4,有va(G(D_(m,1,2)))=[(m+4)/5];对于m=10q+j,j=0,1,2,3,5,6,有va(G(D_(m,2,2)))=[(m+1)/5]+1.     

正則的嘉當空間極小超曲面的柯什密德不變量和附屬微分方程

<正> 本文是繼作者前篇論文之後的;目的在於詳細研究該文末節所論的關於拓廣的微小變形問題.和芬斯拉空間相類似地有E.Cartan所建立的以面積概念為基礎的     

On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), G_o=Aut(k K) and fix in K a basis ω₁,...,ω_n over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎G_o is a maximal torus in G′ are studied. The calculation of N_G′(G′⊎G_o) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎G_o is a maximal torus in G′. Bibligraphy: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995. pp. 15–22.     

Ideals and free Pairs in the semigroup β ℤ

We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ? for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ?.     

Distribution of an analog of Sherman's statistics under rank-censored observations

Let U_n(1),..., U_n(n) be a variational series constructed from a sequence of n aggregate-independent random variables distributed uniformly on (0, 1). Let 0 = k₀, k₁,..., k_m, k_m+1= n+1 be an increasing sequence of nonnegative integers, λ= k_r+1?k_r, r=0,..., m, and $$\xi _n = \frac{1}{2}\sum\nolimits_{r = 0}^m {\left| {U_n (k_{r + 1} ) - U_n^\prime (k_r ) - \frac{{k_{r + 1} - k_r }}{{n + 1}}} \right|.}$$ Under certain restrictions on the numbers λ_r= k{r+1}?k_r, in this paper we have shown the asymptotic normality (with an appropriate norming) of the quantity ξ_n as n, m →∞ such that lim sup (m/√n) ar ∞.     

Genus number and l-rank of genus group of cyclic extensions of degree l

Let l be an odd prime, and k an algebraic number field of a finite degree. Let S be a finite product of distinct prime ideals g of k such that Ng1 (mod l). Let I(s) (resp. P(s)) denote the group of ideals (resp. principal ideals) of k prime to S, and let P_S denote the ray modulo S. In this paper we prove that the order (resp. the l-rank) of I(S)/P(S)^lP_S is expressed by the decomposition groups of prime factors of S in a Galois extension K_o (resp. K_r) over k. As an application of this, some results about genus theory are obtained.     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

Cyclic algebras,complete fields,and crossed products

Theorem 1

Let q=char(k). Let M be a subfield of D which is Galois over K of degree m with Galois group H.

1. If q/m then H has a normal q-Sylow subgroup.
2. Iq q ? m then H is an abelian group with one or two generators, an extension of a cyclic group by a cyclic group of order e where k contains a primitive e-th root of unity.

Letk(X) be the generic division ring overk of indexn as defined by Amitsur.

Theorem 2

If n is divisible by the square of a prime p≠char(k) and k does not contain a primitive p-th root of unity, then k(X) is not a crossed product.     

An example of a function which is Denjoy integrable but not khinchin summable

The following result is proven: if ξ is an irrational number “anomalously badly“ approximable by rationals, then there are functions which are not Khinchin ξ-summable but which are Denjoy integrable. Let I be the interval 0 ≤ x ≤ 1, and let ξ be an irrational, 0 < ξ< 1. Let T _ξ denote the transformation of I into itself defined as follows: $$T_\xi x = \left\{ {\begin{array}{*{20}c} {x + \xi ,ifx + \xi \in I;} \\ {x + \xi - 1} \\ \end{array} } \right.$$ otherwise.     

The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk

We study the asymptotic tail behavior of the maximum M = max{0,S _n ,n ≥ = 1} of partial sums S _n = ξ₁ + ? + ξ _n of independent identically distributed random variables ξ₁,ξ₂,... with negative mean. We consider the so-called Cramer case when there exists a β > 0 such that E e ^βξ1 = 1. The celebrated Cramer-Lundberg approximation states the exponential decay of the large deviation probabilities of M provided that Eξ₁ e ^βξ1 is finite. In the present article we basically study the critical case Eξ₁ e ^βξ1 = ∞.     

On Embeddings of AG(tn,k) into AG(t,K) with t≥2, n≥1

Let k and K be commutative fields with dim_kK=2n, n1 and char(k)2, 3. If k satisfies one of the following conditions: (1) k is a finite field and k contains a 3-th root of unity, or (2) K is a cyclic extension of k where k is not 3-closed, and k contains a 2^a3^b-th root of unity, where 6n=2^a3^bc and c is coprime to 2, 3, then there exists an embedding of the tn-dimensional affine space AG(tn,k) into the t-dimensional affine space AG(t,K), for all t2.     

图是超限制性边连通的充分条件

设G=（V,E）是连通图.边集S E是一个限制性边割,如果G-S是不连通的且G—S的每个分支至少有两个点.G的限制性连通度λ＇（G）是G的一个最小限制性边割的基数.G是λ＇-连通的,如果G存在限制性边割.G是λ＇-最优的,如果λ＇（G）=ζ（G）,其中ζ（G）是min{d（x）＋d（y）-2：xy是G的一条边}.进一步,如果每个最小的限制性边割都孤立一条边,则称G是超限制性边连通的或是超-λ＇.G的逆度R（G）=∑_（v∈V） 1/d（v）,其中d（v）是点v的度数.我们证明了G是λ＇-连通的且不含三角形,如果R（G）≤2＋1/ζ-ζ/（（2δ-2）（2δ-3））＋（n-2δ-ζ＋2）/（（n-2δ＋1）（n-2δ＋2））,则G是超-λ＇.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.