Automorphisms of Extensions of ℚ by a Direct Sum of Finitely Many Copies of ℚ

Let G be an extension of ℚ by a direct sum of r copies of ℚ. (1) If G is abelian, then G is a direct sum of r + 1 copies of ℚ and AutG ≅ GL(r + 1, Q); (2) If G is non-abelian, then G is a direct product of an extraspecial ℚ-group E and m copies of ℚ, where E/ζE is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let AutG′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and AutG/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → AutG′G → AutG → AutG′ → 1 is split; (ii) AutG′G/AutG/ζG,ζGG ≅ Sp(2n,Q) × (GL(m, Q) ⋉ ℚ(m)); (iii) AutG/ζG,ζGG/InnG ≅ ℚ(2nm).     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Resolving Extensions of Finitely Presented Systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map π ₊:X→Y. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.     

有限个有理函数的随机复动力系统的Julia集

本文讨论有限个有理函数生成的随机复动力系统，得到Julia集有内点的充分条件和必要条件．证明了对任意的正数，可以构造有限个多项式，彼此的Julia集之间的距离大干L，但J(f1，…，fm)含有内点但不是全平面。     

无限循环群被有限生成Abel群的中心扩张

设G是无限循环群被有限生成Abel群的中心扩张,T是G的中心ζG的挠子群.如果T的阶与ζG/(G'⊕T)的挠子群的阶互素,那么群G可分解为G=S×F×T,其中S= 这里d_i都是正整数,满足d_1|d_2|…|d_r,F是秩为s的自由Abel群,T是有限Abel群,T=Z_(e_1)⊕Z_(e_2)⊕…⊕Z_e_t,e_11,满足e_1|e_2|…|e_t,并且(d_1,e_t)=1.进一步,(d_1,d_2,…,d_T;s;e_1,e_2,…,e_t)是群G的同构不变量,即若群H也是无限循环群被有限生成Abel群的中心扩张,T_H是ζH的挠子群.如果T_H的阶与ζH/(H'⊕T_H)的挠子群的阶互索,那么G同构于H的充要条件是它们有相同的不变量.显然,这个结果涵盖了有限生成Abel群的结构定理.     

Extensions of Rings with Many Units

Abstract

In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e ₁,…, e _n } of idempotents such that all e _i Re _i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range.     

Finitely Additive Measures and Extensions of Abstract Control Problems

This work is concerned with certain problems of the theory of well-posed expansions in the class of finitely additive measures and related topics of measure theory itself. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 17, Optimal Control, 2004.     

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

Let G be an extension of a finite characteristically simple group by an abelian group or a finite simple group.It is shown that every Coleman automorphism of G is an inner automorphism.Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.     

All Finitely Axiomatizable Normal Extensions of K4.3 are Decidable

We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of (possibly infinite) frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then apply a Harropstyle argument to establish the decidability of L, provided of course that it has finitely many axioms.     

Ascending Hnn Extensions of Finitely Generated Free Groups Are Hopfian

It is shown that every ascending HNN extension of a finitelygenerated free group is Hopfian. An important ingredient inthe proof is that under certain hypotheses on the group H, ifG is an ascending HNN extension of H, then cd(G) = cd(H) + 1.     

Representations of Positive Integers by a Direct Sum of Quadratic forms

The number of representation of positive integers by quadratic forms $ F_{1}=x_{1}^{2}+3x_{1}x_{2}+8x_{2}^{2} $ and $ G_{1}=2x_{1}^{2}+3x_{1}x_{2}+4x_{2}^{2} $ of discriminant —23 are given. Moreover, a basis for the cusp form space S ₄(Γ₀(23), 1) are constructed. Furthermore, formulas for the representation of positive integers by direct sum of copies of F ₁ and G ₁, i.e. formulas for $ r(n; F_{4}), r(n; G_{4}), r(n; F_{3} \oplus G_{1}), r(n; F_{2} \oplus G_{2}), {\rm and}\ r(n; F_{1} \oplus G_{3}) $ , are derived using the elements of the space S ₄(Γ(23), 1).     

Some Extensions to a Direct Optimization Method

A direct method for the global extremization of a class of integrals, introduced in Refs. 1–3, is generalized to allow for constraints in the form of differential conditions and by considering the so-called infinite-horizon case.     

迭代修复含有限个同类型间断函数的连续性

我们已证明具有一个间断点的函数有连续的二次迭代.它实际上表明在迭代之下它的间断点能被自己函数对修复为连续点.如果一个函数含至少两个间断点,那么,在迭代之下,它的间断点或者被它自己函数对修复为连续点或者被其它间断点的函数对修复为连续点.本文研究具有多于一个但是只含有限个同类型间断点的不连续函数,给出了这些函数二次迭代连续...     

线性变换的特征多项式诱导的直和分解

从线性变换的特征多项式的因式分解入手,研究由特征多项式诱导的全空间的若干种直和分解,得到Jordan标准形理论的一个几何证明,全空间关于线性变换不可分解的刻画以及只存在平凡的不变子空间的刻画.     

On the Representation of Band Limited Functions Using Finitely Many Bits

In this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted L^p norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted L^p spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties.     

A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities

We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

具有有限组不相交压缩圆片的Heegaard分解

设$V\cup_SW$是一个闭的三维流形亏格为$g$的, 弱可约的Heegaard分解, 并且在合痕意义下只有有限组位于曲面不同侧的不相交的压缩圆片, 则它存在一个广义的Heegaard分解: $V\cup_SW=(V_1\cup_{S_1}W_1)\cup_F(W_2\cup_{S_2}V_2)$, 并且满足对于每个$i=1,2$, 压缩体$W_i$都只有一个分离的压缩圆片且$d(S_i)\geq 2$. 进一步的, 如果有有限且多于1组不相交的压缩圆片, 则至少一个$d(S_i)$等于2, 并且Heegaard曲面满足临界性质.     

Geometrical Decomposition of the Free Loop Space on a Manifold with Finitely Many Closed Geodesics

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.