On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

Properties of matroids characterizable in terms of excluded matroids

Some properties π of matroids are characterizable in terms of a set S(π) of exluded matroids, that is, a matroid M satisfies property π if and only if M has no minor (series-minor, parallel-minor) isomorphic to a matroid in S(π). This note presents a necessary and sufficient condition for a property to be characterizable in terms of excluded 3-connected matroids.     

When is a Coxeter System Determined by its Coxeter Group?

A Coxeter system is a pair (W, S) where W is a group and whereS is a set of involutions in W such that W has a presentationof the form W=     

Continuity andkth order differentiability in Orlicz-Sobolev spaces:W kLA

The paper gives a necessary and sufficient condition for the embedding of the Orlicz-Sobolev spaceW ^kL_A (Ω) inC(Ω). The same condition is also found to be necessary and sufficient so that a continuous function inW ^kL_A (Ω) be differentiable of orderk almost everywhere in Ω.     

Projective Equivalence of Δ-Matroids with Coefficients and Symplectic Geometries

The projective equivalence of matroid representations over fields and of oriented matroids is well studied. This paper is devoted to the study of projective equivalence of Δ-matroids with coefficients, which covers the concept of projective equivalence of matroids with coefficients and thus in particular the projective equivalence of represented and oriented matroids. A necessary and sufficient condition for the projective equivalence of Δ-matroids with coefficients is established in terms of the inner Tutte group T _M ⁽⁰⁾ of the underlying combinatorial geometry M. The structure of T_M ⁽⁰⁾ of symplectic projective spaces M of odd dimensions d≥3 over fields is computed.     

On matroids of the greatestW-connectivity

The Whitney connectivity (W-connectivity) of a matroidM is defined by T. Inukai and L. Weinberg as the least integerk for which there exists a subsetS of the ground setE ofM such that ?(S) ≥ k, ?(E ? S) ≥ k, andp(S)+p(E-S)-pM+1=k where ? is the rank function ofM.Mis called a Whitney matroid if there exists no such integer. In this case, theW-connectivity ofM to be the rank ofM is defined. In this paper, several properties of Whitney matroids are demonstrated. In addition, the Whitney matroids whose duals are also Whitney matroids are characterized, and an interpretation of binaryW-matroids is given.     

The Intersection of Two Infinite Matroids

Conjecture: Let M and N be two matroids (possibly of infiniteranks) on the same set S. Then there exists a set I independentin both M and N, which can be partitioned as I=HK, where sp_M(H)sp_N(K)=S.This conjecture is an extension of Edmonds' matroid intersectiontheorem to the infinite case. We prove the conjecture when oneof the matroids (say M) is the sum of countably many matroidsof finite rank (the other matroid being general). For the proofwe have also to answer the following question: when does thereexist a subset of S which is spanning for M and independentin N?     

Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

The Elimination of All but Two from a Tournament

In a tournament with N participants, the smallest number Q(N)of decisive matches which may be necessary for the identificationof the runner-up has been observed by Schreier (1932) and Steinhaus(1950) to be the number M(N) defined in Section 1. Here we determinethe smallest number P(N) of such matches which may be necessaryfor the identification of the top pair, champion and runner-up,without necessarily distinguishing the champion. In particularwe show that P(N) = M(N)–1 if and only if N is of theform 2ⁿ+1, and that the optimal strategy has some unexpectedfeatures.     

Infinite matroids in graphs

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals.In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

Matroids with given restrictions and contractions

If M is a matroid on a set S and if X is a subset of S, then there are two matroids on X induced by M: namely, the restriction and the contraction of M onto X. Necessary and sufficient conditions are obtained for two matroids on the same set to be of this form and an analogous result is obtained when (X₁,…, X_t) is a partition of S. The corresponding results when all the matroids are binary are also obtained.     

A note on interconnecting matchings in graphs

We prove a sufficient condition for a graph G to have a matching that interconnects all the components of a disconnected spanning subgraph of G. The condition is derived from a recent extension of the Matroid intersection theorem due to Aharoni and Berger. We apply the result to the problem of the existence of a (spanning) 2-walk in sufficiently tough graphs.     

Packing and covering with matroid circuits

We prove a conjecture of Welsh, that for every matroid M without coloops, ν(M) + θ(M) ≤ ?(M) + κ(M) where ν(M) is the maximum number of pairwise disjoint circuits, θ(M) is the minimum number of circuits whose union is E(M), ?(M) is the corank of M, and κ(M) is the number of connected components of M. For binary matroids the result was previously proved by Oxley.     

Chords and disjoint paths in matroids

A chord of a circuit C of a matroid M on E is a cell e ? S\C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths and to characterize a class of matroid sums.     

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

Regular submodules of torsion modules over a discrete valuation domain

A submodule W of a p-primary module M of bounded order is known to be regular if W and M have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.