Little Grothendieck`s theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H, then the set of those bounded linear operators T from E to H, such that there exists a norm one functional and corresponding pre-Hilbertian semi-norm on E such that for all , is norm dense in the set of all bounded linear operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and is a bounded linear operator then there exists a state such that for all . Received June 28, 1999; in final form January 28, 2000 / Published online March 12, 2001     

A reduction method to find spanning Eulerian subgraphs

We ask, When does a graph G have a subgraph Γ such that the vertices of odd degree in Γ form a specified set S ? V(G), such that G - E(Γ) is connected? If such a subgraph can be found for a suitable choice of S, then this can be applied to problems such as finding a spanning eulerian subgraph of G. We provide a general method, with applications.     

Rooted Spanning Trees in Tournaments

A tournament of order n is an orientation of a complete graph with n vertices, and is specified by its vertex set V(T) and edge set E(T). A rooted tree is a directed tree such that every vertex except the root has in-degree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature. This result was strengthened to the following one: Every tournament contains a rooted spanning 2-tree of depth at most 2. We prove that every tournament of order n≥800 contains a spanning rooted special 2-tree in this paper, where a rooted special 2-tree is a rooted 2-tree of depth 2 such that all except possibly one, non-root, non-leaf vertices, have out-degree 2 in the tree. Revised: November 9, 1998     

Decomposition of convex figures into similar pieces

If a convex plane figureP can be decomposed into finitely many nonoverlapping convex figures such that one of these pieces is similar toP, thenP is a polygon. Also, ifP can be decomposed into infinitely many nonoverlapping sets such that each of the pieces is similar toP, thenP is a polygon.     

On finite groups whose Sylow subgroups have a bounded number of generators

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most δ elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and of index bounded by a function of δ and p. This result will be used to prove that G has a nilpotent normal subgroup of index bounded in terms of δ and p.     

Uniform partial regularity of quasi minimizers for the perimeter

Quasi minimizers for the perimeter are measurable subsets G of such that for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties. Received: July 12, 1999 / Accepted: October 1, 1999     

The Clique Complex and Hypergraph Matching

The width of a hypergraph is the minimal for which there exist such that for any , for some . The matching width of is the minimal such that for any matching there exist such that for any , for some . The following extension of the Aharoni-Haxell matching Theorem [3] is proved: Let be a family of hypergraphs such that for each either or , then there exists a matching such that for all . This is a consequence of a more general result on colored cliques in graphs. The proofs are topological and use the Nerve Theorem. Received June 14, 1999     

On arithmetic macaulayfication of certain local rings

Let A be a Noetherian ring.We consider the existence of Cohen-Macaulay Rees algebras of A. If the non-Cohen-Macaulay locus of A is of dimension 0, then we already know that such a Rees algebra exists. In the present paper, we show that such a Rees algebra also exists when the non-Cohen-Macaulay locus of A is of dimension 1.     

Symmetric Graphs and Flag Graphs

 With any G-symmetric graph Γ admitting a nontrivial G-invariant partition , we may associate a natural “cross-sectional” geometry, namely the 1-design in which for and if and only if α is adjacent to at least one vertex in C, where and is the neighbourhood of B in the quotient graph of Γ with respect to . In a vast number of cases, the dual 1-design of contains no repeated blocks, that is, distinct vertices of B are incident in with distinct subsets of blocks of . The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them. In particular, we show that such graphs can be reconstructed from and the induced action of G on . The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of . This leads to, in a subsequent paper, a construction of G-symmetric graphs such that and each is incident in with vertices of B. The work was supported by a discovery-project grant from the Australian Research Council. Received April 24, 2001; in revised form October 9, 2002 Published online May 9, 2003     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

Alternating Hamiltonian cycles

Colour the edges of a complete graph withn vertices in such a way that no vertex is on more thank edges of the same colour. We prove that for everyk there is a constantc _ksuch that ifn>c _kthen there is a Hamiltonian cycle with adjacent edges having different colours. We prove a number of other results in the same vein and mention some unsolved problems.     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Blockings-dimensional subspaces by lines inPG(2s,q)

We investigate sets of lines inPG(2s,q) such that everys-dimensional subspace contains a line of this set. We determine the minimum number of lines in such a set and show that there is only one type of such a set with this minimum number of lines.     

VC-Dimension of Sets of Permutations

VC-dimension of a set of permutations to be the maximal k such that there exist distinct that appear in A in all possible linear orders, that is, every linear order of is equivalent to the standard order of for at least one permutation . In other words, the VC-dimension of A is the maximal k such that for some the restriction of A to contains all possible linear orders. This is analogous to the VC-dimension of a set of strings. Our main result is that there exists a universal constant C such that any set of permutations with VC-dimension 2 is of size . This is analogous to Sauer's lemma for the case of VC-dimension 2. One corollary of our main result is that any acyclic set of linear orders of is of size , (a set A of linear orders on is called acyclic if no 3 elements appear in A in all 3 orders (i,j,k), (k,i,j) and (j,k,i)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of n alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive. Received August 6, 1998     

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Planar graph colorings without short monochromatic cycles

It is well known that every planar graph G is 2‐colorable in such a way that no 3‐cycle of G is monochromatic. In this paper, we prove that G has a 2‐coloring such that no cycle of length 3 or 4 is monochromatic. The complete graph K₅ does not admit such a coloring. On the other hand, we extend the result to K₅‐minor‐free graphs. There are planar graphs with the property that each of their 2‐colorings has a monochromatic cycle of length 3, 4, or 5. In this sense, our result is best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 25–38, 2004     

Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency

We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is (m, k)-colorable. We include an elementary proof (not assuming the truth of the four-color theorem) that every planar graph is (4, 1)-colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, k)-colored; we conjecture that 4 can be replaced by 3 in this statement.     

On the structure of some infinite dimensional linear groups

If G is a group and if the upper hypercenter, Z, of G is such that G∕Z is finite then a recent theorem shows that G contains a finite normal subgroup L such that G∕L is hypercentral. The purpose of the current paper is to obtain a version of this result for subgroups G of GL(F,A), when A is an infinite dimensionalF-vector space.     

On b-perfect Chordal Graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph. This research was supported by Algerian-French program CMEP/Tassili 05 MDU 639.