On Duality in Filtered Cohomologies

For a symplectic manifold(M~(2n), ω) without boundary(not necessarily compact), we prove Poincaré type duality in filtered cohomology rings of differential forms on M, and we use this result to obtain duality between(d + d~Λ)-and dd~Λ-cohomologies.     

<Emphasis Type="Italic">n</Emphasis>-transitivity of bisection groups of a Lie groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ N by an arbitrary group of C^r-bisections of a Lie groupoid Γ of class C^r, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.     

A Hadamard product involving N-matrices

We show that for any pair M,N of n by n M-matrices, the Hadamard (entry-wise) product M°N_-1 is again an M-matrix. For a single M-matrix M, the matrix M°M^-1 is also considered.     

A note on linear transformations over integral domains

It is an easy fact from linear algebra that if M is a finite-dimensional vector space over a field R, ϕM→M a diagonalizable linear transformation, and N a ϕ-invariant subspace of M, then ϕ∣N is diagonalizable. We show that an appropriate generalization of this holds for M a torsion-free module over an integral domain R.     

On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)−5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)−B, then |E(M)|3r(M)−4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)−c(M).     

Discriminant preserving linear maps

Consider the n-square matrices over an infiniie field Kas an n²-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK)     

Algorithms for finding a Kth best valued assignment

We consider a new problem, the Kth best valued assignment problem. Given a bipartite graph G and a cost vector w on its edge set, this is the problem of finding a perfect matching M_k in G such that there exist perfect matchings M₁,…,M_K−1 satisfying w(M₁) < < w(M_K−1) < w(M_K), and w(M_K) < w(M) for all perfect matchings M with w(M) ≠ w(M₁),…,w(M_K). Here w(M) denotes the sum of costs of edges in M. In this paper, we propose two algorithms for solving this problem and verify the efficiency of our algorithms by our preliminary computational experiments.     

On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σ[M] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σ[M] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σ[M]. Then, for every n ε , there are only finitely many indecomposable modules of length n in σ[M].

With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

The generalized column incidence graph and a matroid base-listing algorithm

The generalized column incidence graph of a matroid base is defined, and it is shown that all elements on a minimal path in this graph lie in a common circuit. Also, an algorithm is provided which lists all bases of a matroid and calculates the Whitney and Tutte polynomials. The complexity of this algorithm is shown to be O(mN(n- m)(c(M) + m)), where Mis a matroid of rank mon a set of cardinality nNis the number of bases of M, and c(M) is the complexity of checking independence in M.     

π-Armendariz rings relative to a monoid

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a₁g₁+ a₂g₂+ · · · + a_ng_n, β = b₁h₁+ b₂h₂+ · · · + b_mh_m ∈ R[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.     

Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

In this paper, we prove that for-1/2 ≤β≤0.suppose M is an invariant subspaces of the Hardy Sobolev spaces H_β~2(D) for T_z~β, then M() zM is a generating wandering subspace of M, that is,M=[MzM]_T_z~β Moreover, any non-trivial invariant subspace M of H_β~2(D) is also generated by the quasi-wandering subspace P_MT_z~βM~⊥ that is,M=[P_MT_z~βM~⊥]_(T_z~β).     

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group Diff_G^∞(M)₀ of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of Diff_G^∞(M)₀ when M is a G-manifold with codimension one orbit.     

Incidence matrices, geometrical bases, combinatorial prebases and matroids

For a relation A (C × D), where C,D are two finite sets, and an ordering σ of C we construct a matroid M(σ) on the set D. For the relation A with the incidence matrix  we also define a geometrical basis with respect to F, where F is a subset of the set of all circuits of the column matroid on Â. Geometrical bases are certain bases of this column matroid. We establish connections between the bases of matroids M(σ) and the geometrical bases of A with respect to F. These connections give a combinatorial way of constructing bases of the column matroid on  using a subset F of its circuits.

We also consider a matroid M and the incidence relation between what we call the extended circuits of M and the bases of M. Applying the technique above we obtain the matroids M(σ) on the set of bases of the matroid M. In case of the incidence relation between vertices and edges of a graph this technique yields a unique matroid, the usual matroid of the graph.

Some particular relations are considered: a class of relations with a certain property (the T-property) and the relation of inclusion of chambers in simplices in an affine point configuration.     

The reconstruction of a matroid from its connectivity function

In this paper, we shall present an algorithm to decide when a connected matroid M is reconstructible from its connectivity function. When M is not reconstructible, this algorithm gives all the matroids with the same connectivity function as M.     

Matrices which Commute with their Transposes over a Field of Characteristic Two

Let M be a square matrix with entries in a field F of characteristic two. I show that a necessary and sufficient condition for M to be similar to a matrix over F which commutes with its transpose is that one of the following statements holds.

i) The minimal polynomial of M is not square-free.

ii) Some elementary divisor of M occurs with multiplicity greater than or equal to its degree of inseparability.     

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group U_hg and is commutative with respect to an involutive operator . Such a multiplication is unique. Let M be a kählerian symmetric space with the canonical Poisson structure. Then we construct a U_hg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.     

How a strongly irreducible Heegaard splitting intersects a handlebody

In [Topology Appl. 90 (1998) 135] Scharleman showed that a strongly irreducible Heegaard splitting surface Q of a 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M.