Superdecomposable E(R)-Algebras

ABSTRACT

An E-ring is a ring that is naturally isomorphic to the endomorphism ring of its additive group. E-rings with various properties have been constructed in the literature; we now consider superdecomposable E-rings. More generally, we construct superdecomposable algebras A over integral domains R, which are at the same time E(R)-algebra in the sense that the ring of R-endomorphisms of the underlying R-module structure is canonically isomorphic to A. We also establish the existence of arbitrarily large superdecomposable modules over such algebras.     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

On the Ramsey numbers R(3, 8) and R(3, 9)

Using methods developed by Graver and Yackel, and various computer algorithms, we show that 28 ≤ R(3, 8) ≤ 29, and R(3, 9) = 36, where R(k, l) is the classical Ramsey number for 2-coloring the edges of a complete graph.     

The Reed-Muller Code R(1,7) Is Normal

We prove that the Reed-Muller code R(1,7) is normal. The normality of R(1,7) was a long-standing open question, and has an important consequence in a conjecture about the function t[n,k], the smallest covering radius of any [n,k] binary linear code.     

R(4, 5) = 25

The Ramsey number R(4, 5) is defined to be the least positive integer n such that every n-vertex graph contains either a clique of order 4 or an independent set of order 5. With the help of a long computation using novel techniques, we prove that R(4, 5) = 25. © 1995 John Wiley & Sons, Inc.     

On the Overgroups of EO(2l,R)

Let R be a commutative ring with 1, let 2 R ^*, and let l 3. We describe the subgroups of the general linear group GL(n,R) that contain the split elementary orthogonal group EO(2l,R). For every intermediate subgroup H, there exists a unique maximal ideal A R such that E(2l,R,A) H and, moreover, H normalizes EO(2l,R)E(2l,R,A). In the case where R = K is a field, similar results were obtained earlier by Dye, King, Li Shangzhi, and Bashkirov. Bibliography: 31 titles.     

Singular semi-linear equations inL 1(R)

Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L ¹(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫ _−∞ ^∞ . Some extensions of this result are also given. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS 75-05501.     

The space Ω m p (R d ) and some properties

Let m be a v-moderate function defined on R ^d and let g ∈ L ²(R ^d ). In this work, we defineΩ _m ^p (R ^d ) to be the vector space of f ∈ L _m ² (R ^d ) such that the Gabor transform V ^f belongs to L ^p (R ^2d ), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω _m ^p (R ^d ). We also discuss inclusion properties and obtain the dual space of Ω _m ^p (R ^d ). At the end of this work, we study multipliers from L _w ¹ (R ^d ) into Ω _w ^p (R ^d ) and from Ω _w ^p (R ^d ) into L _w−1 ^∞ (R ^d ), where w is the Beurling weight function. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004     

Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

Let R be a commutative ring with identity. The set 𝕀(R) of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of 𝕀(R) is called the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). We write 𝒢 for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in 𝒢 that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that 𝔸𝔾(R) ∈ 𝒢 is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with 𝔸𝔾(R) ∈ 𝒢. Also, a connection between finite rings and their corresponding graphs is realized.     

New Lower Bounds on the Multicolor Ramsey Numbers R r (C 2 m )

The multicolor Ramsey number R_r(H) is defined to be the smallest integer n=n(r) with the property that any r-coloring of the edges of the complete graph K_n must result in a monochromatic subgraph of K_n isomorphic to H. It is well known that 2^rm<R_r(C_2m+1)<2(r+2)!m and R_r(C_2m)≥(r−1)(m−1)+1. In this paper, we prove that R_r(C_2m)≥2(r−1)(m−1)+2. This research is supported by NSFC(60373096, 60573022) and SRFDP(20030141003)     

Stochastic flows on the boundaries of SL(n,R)

Summary We study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in SL(n, R) or GL(n, R). These compact spaces are called boundaries of SL(n, R), which include SO(n), the flag manifold, the sphereS ⁿ–1 and the Grassmannians. The one point motions of these flows are Brownian motions. For almost every, , we determine the set of stable points. This is a random open set whose complement has zero Lebesgue measure. The distance between any two points in the same component of this set tends to zero exponentially fast under the flow. The Lyapunov exponents at stable points are computed explicitly. We apply our results to a stochastic flow onS ⁿ–2 generated by a stochastic differential equation which exhibits some nice symmetry.Research supported in part by Hou Yin Dong Education Foundation of China On leave from Nankai University, Tianjin, China     

EXTENSIONS R[α] ∩ R[α−1] AND R[α − a] ∩ R[(α − a)−1] OF A NOETHERIAN DOMAIN R

Let R be a Noetherian domain and α an anti-integral element of degree d over R. Let a and b be elements of R. In this paper, we give a condition for R[(α ? a)^?1] to be coincide with R[(α ? b)^?1]. Moreover, we give a sufficient condition for R[α] ∩ R[α^?1] to be R[α ? a] ∩ R[(α ? a)^?1].     

GENERALIZED ERMAKOV SYSTEMS IN TERMS OF sl(2,R) INVARIANTS

Abstract

Conventional generalized Ermakov systems are shown to be a subset of the class of second order ordinary differential equations invariant under sl(2,R) symmetry. When the system is two-dimensional, it can be reduced to a one-dimensional time-dependent simple harmonic oscillator by a suitable choice of new time and distance variables.     

Property (R) for Bounded Linear Operators

We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid, or left polaroid, operators.     

Der Satz von Tits für PGL2(R), R ein kommutativer Ring vom stabilen Rang 2

Certain permutation groups on sets with distance relation are characterized as groups of projectivities PGL₂(R) on the projective line over a commutative ring R of stable rank 2, thus generalizing a classical result of Tits where R is a field.

     

Isomorphic classification of the spaces $H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R)$

We give a representation of the spaces H^k_loc(\Bbb R)?L²(\Bbb R)H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R) as Köthe sequence spaces and use some recent topological invariants to investigate their isomorphic classification.     

Orthonormal Sequences in L 2(R d ) and Time Frequency Localization

We prove that there does not exist an orthonormal basis {b _n } for L ²(R) such that the sequences {μ(b _n )}, {m([^(b_n)])}\{\mu(\widehat{b_{n}})\} , and {D(b_n)D([^(b_n)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\} are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L ²(R ^d ). On the other hand, for d>1 we construct a basis {b _n } for L ²(R ^d ) such that the sequences {μ(b _n )}, {m([^(b_n)])}\{\mu(\widehat{b_{n}})\} , and {D(b_n)D([^(b_n)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\} are bounded.