Cohen-macaulay bipartite graphs

Let G be a graph on the vertex set V={x ₁, ..., x _n}. Let k be a field and let R be the polynomial ring k[x ₁, ..., x _n]. The graph ideal I(G), associated to G, is the ideal of R generated by the set of square-free monomials x _ix_j so that x _i, is adjacent to x _j. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring. Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs. Partially supported by COFAA-IPN, CONACyT and SNI, México.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

Filtrations,rees rings,and ideal transforms

Let I be an ideal, and let f = {K_n|n ≥ 0 } be a filtration of the Noetherian ring R, such that Iⁿ ⊆ K_n for all n ≥ 0. We study when the Rees ring $\text{R}$(f) is either finite or integral over the Rees ring $\text{R}$(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (Iⁿ : J^{m(n) + 1}), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.     

Arithmetic-progression-weighted subsequence sums

Let G be an abelian group, let s be a sequence of terms s ₁, s ₂, …, s _n ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let $$W \odot S = \left\{ {w_1 s_1 + \cdots + w_n s_n :w_i a term of W,w_i \ne w_j for i \ne j} \right\},$$ which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| ? 1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S ≠ G. This result then allows us to characterize when a linear equation $$a_1 x_1 + \cdots + a_r x_r \equiv \alpha mod n,$$ where α, a ₁, …, a _r ∈ ? are given, has a solution (x ₁, …, x _r ) ∈ ? ^r modulo n with all x _i distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G \cong C_{n_1 } \oplus C_{n_2 }$ (where n ₁ |n ₂ and n ₂ ≥ 3) having k distinct terms, for any k ε [3, min{n ₁ + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

The analytic spread of codimension two monomial varieties

Let ${\rm} A=k[{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k[{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? k[x₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.     

On Modules with Finite Exchange Property

The purpose of this paper is to answer an open problem proposed by Matlis on modules with the finite exchange property. The problem is: Let {M _i | i ∈ I} be an indexed family of indecomposable injective modules and M a direct sum of M _i , i.e. M = ⊕{M _i | i ∈ I} and N a summand of M. Is N a direct sum of indecomposable injective modules, i.e. N = ⊕{M _j |j ∈ J, J ? I}? The answer to this problem is affirmative for module M with finite exchange property.     

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 ).     

Nil-automorphisms of groups with residual properties

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g ∈ G, there exists n = n(g) such that [g, _n x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G〈x〉. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e.g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.     

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J (R) (or CGJ (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

Gotzmann Edge Ideals

Let P = 𝕜[x ₁,…, x _n ] be the polynomial ring in n variables. A homogeneous ideal I ? P generated in degree d is called Gotzmann if it has the smallest possible Hilbert function out of all homogeneous ideals with the same dimension in degree d. The edge ideal of a simple graph G on vertices x ₁,…, x _n is the quadratic square-free monomial ideal generated by all x _i x _j where {x _i , x _j } is an edge of G. The only edge ideals that are Gotzmann are those edge ideals corresponding to star graphs.     

Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
       
    

     

Cohen–Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals

Let 𝕂 be a field, and let R = 𝕂[x ₁,…, x _n ] be the polynomial ring over 𝕂 in n indeterminates x ₁,…, x _n . Let G be a graph with vertex-set {x ₁,…, x _n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J ^(k) and J ^[k], respectively. In this paper, we give necessary and sufficient conditions for R/J ^k , R/J ^(k), and R/J ^[k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.     

Fractal functions and interpolation

Let a data set {(x _i,y _i) ∈I×R;i=0,1,?,N} be given, whereI=[x ₀,x _N]?R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x _i)=y _i fori ε {0,1,?,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.     

Simultaneous weighted sums of elements of finitely generated multiplicative groups

Let {G_j}_jεJ be a finite set of finitely generated subgroups of the multiplicative group of complex numbers C^x. Write H=∩ _jεJ G_j. Let n be a positive integer and a_ij a complex number for i = 1, ..., n and j ε J. Then there exists a set W with the following properties. The cardinality of W depends only on {G_j}_jεJ and n. If, for each jεJ, α has a representation α = Σ _iⁿ = ₁a _ijg_ij in elements g_ij of G_j, then α has a representation a= Σ_k=1ⁿ w_kh_k with w_kεW, h_k εH for k = 1,..., n. The theorem in this note gives information on such representations.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

Some remarks on the existence of optimal controls for quasilinear systems

Let a quasilinear control system having the state space \(\bar X \subseteq R^n \) be governed by the vector differential equation $$\dot x = G(u(t))x,$$ wherex(0) =x ₀ andU is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR ^m.LetG:U ?R be a bounded measurable nonlinear function, such thatG(U) is compact and convex.G ^?1 can be convex onG(U) or concave. The main results of the paper establish the existence of a controlu ∈U which minimizes the cost functional $$I(u) = \int_0^T {L(u(t))x(t)dt,} $$ whereL(·) is convex. A practical example of application for chemical reactions is worked out in detail.     

Spherical Means Associated with Non-isotropic Dilation Groups

Let {δ_t}_t>0 be a non-isotropic dilation group on R ⁿ . Let τ: R ⁿ → [0,∞) be a continuous function that vanishes only at the origin and satisfies τ(δ _t x) = tτ(x), t > 0, x ∈ R ⁿ . In this paper we obtain two-sided inequalities for spherical means of the form $\int_{S^{n-1}}\tau(r_1\omega_1,\cdots,r_n\omega_n)^{-\alpha}d\sigma (\omega),$ where α is a positive constant, and r₁,…, r_n are positive parameters.