An iterative method for a system of linear complementarity problems with perturbations and interval data

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^∗],[y^∗]) which includes the solution set for the system of linear complementarity problems defined by any fixed A∈[A],B∈[B],b∈[b] and c∈[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.     

On direct product cancellation of graphs

The direct product of graphs obeys a limited cancellation property. Lovász proved that if C has an odd cycle then A×C≅B×C if and only if A≅B, but cancellation can fail if C is bipartite. This note investigates the ways cancellation can fail. Given a graph A and a bipartite graph C, we classify the graphs B for which A×C≅B×C. Further, we give exact conditions on A that guarantee A×C≅B×C implies A≅B. Combined with Lovász’s result, this completely characterizes the situations in which cancellation holds or fails.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

Families intersecting on an interval

We shall be interested in the following Erd?s-Ko-Rado-type question. Fix some set B⊂[n]={1,2,…,n}. How large a subfamily A of the power set P[n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B=[t] is a block of length t, we can do no better than taking A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more ‘direct’.     

A quasicancellation property for the direct product of graphs

We are motivated by the following question concerning the direct product of graphs. If A×C≅B×C, what can be said about the relationship between A and B? If cancellation fails, what properties must A and B share? We define a structural equivalence relation ∼ (called similarity) on graphs, weaker than isomorphism, for which A×C≅B×C implies A∼B. Thus cancellation holds, up to similarity. Moreover, if C is bipartite, then A×C≅B×C if and only if A∼B. We conjecture that the prime factorization of connected bipartite graphs is unique up to similarity of factors, and we offer some results supporting this conjecture.     

Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

Existence and uniqueness of best proximity points in geodesic metric spaces

A mapping T:A∪B→A∪B such that T(A)⊆B and T(B)⊆A is called a cyclic mapping. A best proximity point x for such a mapping T is a point such that d(x,Tx)= dist(A,B). In this work we provide different existence and uniqueness results of best proximity points in both Banach and geodesic metric spaces. We improve and extend some results on this recent theory and give an affirmative partial answer to a recently posed question by Eldred and Veeramani in [A.A. Eldred, P. Veeramani Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2) (2006) 1001-1006].     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

Group rings over Dedekind rings

Seghal posed the following question: IfA andB are rings, doesA[X,X ⁻¹] ℞B[X,X ⁻¹] implyA ℞B. In general the answer to this question is no. In this note we give an affirmative answer in the case thatA andB are Dedekind rings. The author is research assistant at the NFWO.     

Generalized base representations

Let B∈Z[x] be a polynomial with b=B(0). Let S be a complete residue class modulo b containing 0. We attempt to classify the polynomials B and residue classes S so that for every polynomial P∈Z[x] there exists a polynomial Q with coefficients in S such that .     

Bounds on global secure sets in cactus trees

Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.     

On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.     

Formally smooth bimodules

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M] of M-subgenerated modules provide natural examples of formally smooth bimodules.     

The recognition of the class of indecomposable digraphs under low hemimorphy

Given a digraph G=(V,A), the subdigraph of G induced by a subset X of V is denoted by G[X]. With each digraph G=(V,A) is associated its dual G^?=(V,A^?) defined as follows: for any x,y∈V, (x,y)∈A^? if (y,x)∈A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H^?. Given k>0, the digraphs G=(V,A) and H=(V,B) are k-hemimorphic if for every X⊆V, with |X|≤k, G[X] and H[X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G=(V,A), a subset X of V is an interval of G provided that for a,b∈X and x∈V−X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For example, 0?, {x}, where x∈V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.     

Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|?n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x₁,…,xn)⊆K[x₁,…,xn] (K a field) is ⌈n/2⌉.     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

Convergence to the equilibrium for the Pauli equation without detailed balance condition

Let $A?B$ be an extension of commutative rings with identity, X an analytic indeterminate over B, and $R\text{:}=A+XB[[X]]$, the subring of the formal power series ring $B[[X]]$, consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module. To cite this article: S. Hizem, A. Benhissi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On free quadratic extensions of rings

The quaternion algebraB[j] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetB[j] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.