Bounded factorization property for Fréchet spaces

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by L^G(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ???) if L^G(E, F) ? LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ??? is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K?othe spaces the property ??? is characterized in terms of their K?othe matrices. As an application we prove that in certain cases the relations L(E, G₁) = LB(E, G₁) and L(G₂, F) = LB(G₂, F) imply (E, G, F) ∈ ??? where G is a tensor product of G₁ and G₂.     

Characterization of a class of triangle-free graphs with a certain adjacency property

Let m and n be nonnegative integers. Denote by P(m,n) the set of all triangle-free graphs G such that for any independent m-subset M and any n-subset N of V(G) with M ∩ N = Ø, there exists a unique vertex of G that is adjacent to each vertex in M and nonadjacent to any vertex in N. We prove that if m ? 2 and n ? 1, then P(m,n) = Ø whenever m ? n, and P(m,n) = {K_m,n+1} whenever m > n. We also have P(1,1) = {C₅} and P(1,n) = Ø for n ? 2. In the degenerate cases, the class P(0,n) is completely determined, whereas the class P(m,0), which is most interesting, being rich in graphs, is partially determined.     

One-dimensional Tilings Using Tiles with Two Gap Lengths

Let n,p,k,q,l be positive integers with n=k+l+1. Let x₁,x₂, . . . ,x_n be a sequence of positive integers with x₁<x₂<···<x_n. A set {x₁,x₂, . . . ,x_n} is called a set of type (p,k;q,l) if the set of differences {x₂–x₁,x₃–x₂, . . . ,x_n–x_n–1} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).     

Intersectional properties concerning planarm-convex sets

LetC be a collection of closed sets in the plane, and letS=∩C. (1) If IncC ⊆ kerS for allC inC and if dim kerS≧1, thenS is a union of three (or fever) convex sets. In particular, the results holds when the members ofC are 3-convex sets, all having the same kernelK, provided dimK≧1. (2) IfC is a finite collection ofm-convex sets such that ∩{kerC:C inC inC} ≠ ⊘,S~ IncS is connected, and for someZ inC, lncC⊆ lncZ for allC inC, thenS ism-convex.     

Condorcet proportions and Kelly's conjectures

Let C(m,n) be the proportion of all n-tuples of linear orders on a set of m alternatives such that some alternative x is ranked ahead of y in at least n of the orders, for each y≠x. Kelly proved that C(m,n)<C(m,r+1) for m3 and odd n 3, and that C(m,n)>C(m,n+1) for m3 and even n2. He also conjectured that C(m,n)>C(m+1,n) for m3 and n=3 or n5, and that C(m,n)>C(m,n+2) for m3 and n=1 or n3. The first of these conjectures is shown to be true for n=3. and for m=3 and odd n. The second conjecture is established for mε{3,4} and odd n, and for m=3 and all large even n.     

List circular coloring of trees and cycles

Suppose G=(V, E) is a graph and p ≥ 2q are positive integers. A (p, q)‐coloring of G is a mapping ?: V → {0, 1, …, p‐1} such that for any edge xy of G, q ≤ |?(x)‐?(y)| ≤ p‐q. A color‐list is a mapping L: V → ({0, 1, …, p‐1}) which assigns to each vertex v a set L(v) of permissible colors. An L‐(p, q)‐coloring of G is a (p, q)‐coloring ? of G such that for each vertex v, ?(v) ∈ L(v). We say G is L‐(p, q)‐colorable if there exists an L‐(p, q)‐coloring of G. A color‐size‐list is a mapping ? which assigns to each vertex v a non‐negative integer ?(v). We say G is ?‐(p, q)‐colorable if for every color‐list L with |L(v)| = ?(v), G is L‐(p, q)‐colorable. In this article, we consider list circular coloring of trees and cycles. For any tree T and for any p ≥ 2q, we present a necessary and sufficient condition for T to be ?‐(p, q)‐colorable. For each cycle C and for each positive integer k, we present a condition on ? which is sufficient for C to be ?‐(2k+1, k)‐colorable, and the condition is sharp. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 249–265, 2007     

Bounding the Derived Length for a Given Set of Character Degrees

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, q ^p , pq ^p }, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.     

Common Best Proximity Points: Global Optimal Solutions

Let S:A→B and T:A→B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx=x and Tx=x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common solution, it is attempted to find an element x that is closest to both Sx and Tx in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx=x and Tx=x when there is no common solution. It is remarked that the functions x→d(x,Sx) and x→d(x,Tx) gauge the error involved for an approximate solution of the equations Sx=x and Tx=x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x→d(x,Sx) and x→d(x,Tx) by stipulating a common approximate solution of the equations Sx=x and Tx=x to fulfill the condition that d(x,Sx)=d(x,Tx)=d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yielding common optimal approximate solutions of certain fixed point equations. Besides establishing the existence of common best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Contact Elements on Fibered Manifolds

For every product preserving bundle functor T ^μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle K_k,l^r,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of K_k,l^r,s,q YK_{k,l}^{r,s,q} Y into itself and of T( K_k,l^r,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to K_k,l^r,s,q YK_{k,l}^{r,s,q} Y .     

Best proximity point theorems

Let us assume that A and B are non-empty subsets of a metric space. In view of the fact that a non-self mapping T:A?B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element x that is as close to Tx as possible. In other words, when the fixed point equation Tx=x has no solution, then it is attempted to determine an approximate solution x such that the error d(x,Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, known as best proximity points, of the fixed point equation Tx=x when there is no solution. Because d(x,Tx) is at least d(A,B), a best proximity point theorem ascertains an absolute minimum of the error d(x,Tx) by stipulating an approximate solution x of the fixed point equation Tx=x to satisfy the condition that d(x,Tx)=d(A,B). This article establishes best proximity point theorems for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings.     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method

Let G(n,c/n) and G_r(n) be an n‐node sparse random graph and a sparse random r‐regular graph, respectively, and let I(n,r) and I(n,c) be the sizes of the largest independent set in G(n,c/n) and G_r(n). The asymptotic value of I(n,c)/n as n → ∞, can be computed using the Karp‐Sipser algorithm when c ≤ e. For random cubic graphs, r = 3, it is only known that .432 ≤ lim inf_n I(n,3)/n ≤ lim sup_n I(n,3)/n ≤ .4591 with high probability (w.h.p.) as n → ∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit lim_n I(n,c)/n can be computed exactly even when c > e, and lim_n I(n,r)/n can be computed exactly for some r ≥ 1. For example, when the weights are exponentially distributed with parameter 1, lim_n I(n,2e)/n ≈ .5517, and lim_n I(n,3)/n ≈ .6077. Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we consider. We extend our results to maximum weight matchings in G(n,c/n) and G_r(n). For the case of exponential distributions, we compute the corresponding limits for every c > 0 and every r ≥ 2. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

On the number of (r,r+1)‐ factors in an (r,r+1)‐factorization of a simple graph

For integers d≥0, s≥0, a (d, d+s)‐graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)‐factor of a graph G is a spanning (r, r+1)‐subgraph of G. An (r, r+1)‐factorization of a graph G is the expression of G as the edge‐disjoint union of (r, r+1)‐factors. For integers r, s≥0, t≥1, let f(r, s, t) be the smallest integer such that, for each integer d≥f(r, s, t), each simple (d, d+s) ‐graph has an (r, r+1) ‐factorization with x (r, r+1) ‐factors for at least t different values of x. In this note we evaluate f(r, s, t). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 257‐268, 2009     

Minimum Cost Homomorphism Dichotomy for Oriented Cycles

For digraphs D and H, a mapping f : V(D) → V(H) is a homomorphism of D to H if uv ∈ A(D) implies f(u) f(v) ∈ A(H). If, moreover, each vertex u ∈ V(D) is associated with costs c _i (u), i ∈ V(H), then the cost of the homomorphism f is ∑ _{u ∈V(D)} c _f(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem for H (abbreviated MinHOM(H)). The problem is to decide, for an input graph D with costs c _i (u), u ∈ V(D), i ∈ V(H), whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when H is an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.     

Necessary and sufficient condition in Lyapunov robust control: Multi-input case

We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A ^T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB ^T PB>0 and that the determinant ofB ^T PW be Hurwitz, whereW=block diag[w ₁,...,w _m ], with elementw _i =[s ^k _i –1,s ^k _i –2,...,s, 1] ^T ; here, the symbolsk _i ,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems.     

The existence of howell designs of siden+1 and order 2n

AHowell design of side s andorder 2n, or more briefly, anH(s, 2n), is ans×s array in which each cell either is empty or contains an unordered pair of elements from some 2n-set, sayX, such that (a) each row and each column is Latin (that is, every element ofX is in precisely one cell of each row and each column) and (b) every unordered pair of elements fromX is in at most one cell of the array. Atrivial Howell design is anH(s, 0) havingX=? and consisting of ans×s array of empty cells. A necessary condition onn ands for the existence of a nontrivialH(s, 2n) is that 0<n≦s≦2n-1. AnH(n+t, 2n) is said to contain a maximum trivial subdesign if somet×t subarray is theH(t, 0). This paper describes a recursive construction for Howell designs containing maximum trivial subdesigns and applies it to settle the existence question forH(n+1, 2n)’s: forn+1 a positive integer, there is anH(n+1, 2n) if and only ifn+1 ∉ {2, 3, 5}.     

A note on the Picard bundle over a moduli space of vector bundles

Let ??(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ??_ξ ? ??(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by ?? the vector bundle over X ×??(n , d ) defined by the kernel of the evaluation map H ⁰(X , E ) → E_x , where E ∈??(n , d ) and x ∈ X . We prove that ?? and its restriction ??_ξ to X × ??ξ are stable. The space of all infinitesimal deformations of ?? over X ×??(n , d ) is proved to be of dimension 3g and that of ??ξ over X × ??ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

A note on k-th commutators in an associative ring

Il commutatorek-esimo dix, y in un anelloR è definito induttivamente da: [x, y]₁=[x, y]=xy−yx and [x, y] _k =[[x, y] _k−1 ,y]. Sia oraR un anello libero da 2-torsione e senza ideali destri nil non nulli. Si prova che se [[a, b], [c, d]] _k è nilpotente, per ogni scelta dia, b, c, d inR, alloraR è commutativo, se [[a, b], [c, d]] _k ha potenze nel centro diR alloraR soddisfa l'identità standard di grado 4. Inoltre si caratterizzano gli anelli in cui [[a, b], [c, d]] _k è nilpotente o regolare per ogni scelta dia, b, c, d inR.