Properties of Quantum Languages

The class of quantum languages Q() over an alphabet is the class of languages accepted by quantum automata. We study properties of Q() and compare Q() with the class of regular languages R(). It is shown that Q() is closed under union, intersection, and reversal but is not closed under complementation, concatenation, or Kleene star. It is also shown that Q() and R() are incomparable. Finally, we prove that L Q() if and only if L admits a transition amplitude function satisfying a certain property and a similar characterization is given for R().     

Conservative Means of Orthogonal Series and the Spaces $$L^p [0;1] $$ , $$p \in (1;\infty ) $$

Necessary and sufficient conditions for an orthogonal series to be the Fourier series of a function in the space , , are obtained. In the special case of regular summation methods we recover the classical results of Orlicz and Lomnicki.     

A uniform refinement property for congruence lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice with compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice cannot be obtained using the Pudlák, Tischendorf, Tuma approach.

The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

     

A Basis for the Top Homology of a Generalized Partition Lattice

For a fixed positive integer k, consider the collection of all affine hyperplanes in n-space given by xi – xj = m, where i, j [n], i j, and m {0, 1,..., k}. Let Ln,k be the set of all nonempty affine subspaces (including the empty space) which can be obtained by intersecting some subset of these affine hyperplanes. Now give Ln,k a lattice structure by ordering its elements by reverse inclusion. The symmetric group Gn acts naturally on Ln,k by permuting the coordinates of the space, and this action extends to an action on the top homology of Ln,k. It is easy to show by computing the character of this action that the top homology is isomorphic as an Gn-module to a direct sum of copies of the regular representation, CGn. In this paper, we construct an explicit basis for the top homology of Ln,k, where the basis elements are indexed by all labelled, rooted, (k + 1)-ary trees on n-vertices in which the root has no 0-child. This construction gives an explicit Gn-equivariant isomorphism between the top homology of Ln,k and a direct sum of copies of CGn.     

Lipschitzian optimization without the Lipschitz constant

We present a new algorithm for finding the global minimum of a multivariate function subject to simple bounds. The algorithm is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. This is done by carrying out simultaneous searches using all possible constants from zero to infinity. On nine standard test functions, the new algorithm converges in fewer function evaluations than most competing methods.The motivation for the new algorithm stems from a different way of looking at the Lipschitz constant. In particular, the Lipschitz constant is viewed as a weighting parameter that indicates how much emphasis to place on global versus local search. In standard Lipschitzian methods, this constant is usually large because it must equal or exceed the maximum rate of change of the objective function. As a result, these methods place a high emphasis on global search and exhibit slow convergence. In contrast, the new algorithm carries out simultaneous searches using all possible constants, and therefore operates at both the global and local level. Once the global part of the algorithm finds the basin of convergence of the optimum, the local part of the algorithm quickly and automatically exploits it. This accounts for the fast convergence of the new algorithm on the test functions.     

A Classification of Regular Embeddings of Graphs of Order a Product of Two Primes

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primitive permutation groups of degree p or degree pq for p q and it is also independent of the classification of the arc-transitive graphs of order pq for p q.     

On Domination Number of 4-Regular Graphs

Let G be a simple graph. A subset S V is a dominating set of G, if for any vertex v V – S there exists a vertex u S such that uv E(G). The domination number, denoted by (G), is the minimum cardinality of a dominating set. In this paper we prove that if G is a 4-regular graph with order n, then (G) ⁴/₁₁ n     

Decomposition Rank of Subhomogeneous C*-Algebras

We analyze the decomposition rank (a notion of covering dimensionfor nuclear C*-algebras introduced by E. Kirchberg and the author)of subhomogeneous C*-algebras. In particular, we show that asubhomogeneous C*-algebra has decomposition rank n if and onlyif it is recursive subhomogeneous of topological dimension n,and that n is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C.Phillips to show the following. Let A be the crossed productC*-algebra coming from a compact smooth manifold and a minimaldiffeomorphism. Then the decomposition rank of A is dominatedby the covering dimension of the underlying manifold. 2000 MathematicsSubject Classification 46L85, 46L35.     

Parametric decomposition of powers of ideals versus regularity of sequences

Let be an ideal in a Noetherian local ring . Then the sequence is -regular if every is a non-zerodivisor in and if for all integers , where runs over the elements of the set .

     

-adic -basis and regular local ring

We introduce the concept of -adic -basis as an extension of the concept of -basis. Let be a regular local ring of prime characteristic and a ring such that . Then we prove that is a regular local ring if and only if there exists an -adic -basis of and is Noetherian.