The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Commuting Traces of Biadditive Maps on Invertible Elements

Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R² → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx^?1y^?1) = θ(x)θ(y)θ(x)^?1θ(y)^?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Non-existence of nodal solution form-Laplace equation involving critical Sobolev exponents

In this paper we study the non-existence of nodal solutions for critical Sobolev exponent problem-div(|∇u| ^m−2∇u)=|u| ^p-1 u+|u| ^q-1 u inB(R)u = 0 on ∂B(R) whereB(R) is a ball of radiusR in ℝⁿ.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

On stable range conditions

In this paper, we show that a ring R satisfies unit 1-stable range if and only if a₁R + ? + a_mR = dR with m ≥ 2,a ₁, ?a_m ?R implies that there exist u₁ , ?u_m ? U(R) such that a₁u₁ +?+a_mu_m = d and an exchange ring R has stable range one if and only if a₁R+?+a_mR = dR with m ≥ 2,a ₁,?,a_m ? R implies that there exist unit-regular w ₁,?,w_m ? R such that a₁w₁ +?+ a_mw_m = d. Also we show that an exchange ring R satisfies the n-stable range condition if and only if a( ⁿR)+bR = dR with a ? Rⁿ,b,d ? R implies that there exist unimodular regular w ? ⁿ R and: y ? R such that aw+by = d.     

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ?₂ and m ² = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

Cycles in bipartite graphs and an application in number theory

Let G = G(A, B) be a bipartite graph with |A| = u, |B| = v, and let / be a positive integer. In this paper we prove the following result: If v ≤ u, uv ≤ n, m = |E(G)|, and m ≥ max{180/u, 20/n^{1/2(1+(1/l))},} then G contains a C_2/. © 1995 John Wiley & Sons, Inc.     

On the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V(G)) = ?_{u ? V(G)} f (u){f (V(G)) = \sum_{u\in V(G)} f (u)}. The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b _R (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E^￠ í E(G){E^{\prime} \subseteq E(G)} for which γ _R (G − E′) > γ _R (G). In this paper we present different bounds on the Roman bondage number of planar graphs.     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

On the Elementary Retarded,Ultrahyperbolic Solution of the Klein-Gordon Operator,Iterated k Times

Let t = (t₁,…,t_n) be a point of ?ⁿ. We shall write . We put, by the definition, W_α(u, m) = (m^?2u)^{(α ? n)/4}[π^{(n ? 2)/2}2^{(α + n ? 2)/2}Г(α/2)]J_{(α ? n)/2}(m²u)^1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W_α(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m²}W_{α + 2}(u, m) = W_α(u, m), where {□ + m²} is the ultrahyperbolic operator. Then we express W_α(u, m) as a linear combination of R_α(u) of differntial orders; R_α(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W_?2k(u, m) = {□ + m²}^kδ, k = 0, 1,…; W₀(u, m) = δ; and {□ + m²}^kW_2k(u, m) = δ. Finally we prove that W_α(u, m = 0) = R_α(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.     

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R^N × R₊ u_t = Δu^m + u^p with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation u_t = Δu^m − u^p, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.     

The Bass-Quillen problem on a class of local rings with weak global dimension two

Let (R,m) be a local GCD domain. R is called a U2 ring if there is an element u ∈ m-m2 such that R/(u) is a valuation domain and Ru is a B′ezout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U2 ring, then R and R[x] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over R[X] is free.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.     

Definable group extensions in semi‐bounded o‐minimal structures

In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈R^m, +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Power cocentralizing generalized derivations on prime rings

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u)) ⁿ = 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s ₄ and char (R) = 2; (3) R satisfies s ₄ and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b) ⁿ = 0.     

On the structure of linear graphs

Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n ²/4]+1) contains a circuit ofl edges for every 3 ≦l<c ₂ n, also that everyG(n; [n ²/4]+1) contains ak _e(u _n, u_n) withu _n=[c ₁ logn] (for the definition ofk _e(u _n, u_n) see the introduction). Finally fort>t ₀ everyG(n; [tn ^3/2]) contains a circuit of 2l edges for 2≦l<c ₃ t ². This work was done while the author received support from the National Science Foundation, N.S.F. G.88.     

Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^\fracm+2m-2=0{c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 \tfrac2mm-2{\Gamma\geq\tfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.