Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.     

The order of hypotraceable oriented graphs

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented graphs of order n for every n≥8 except possibly for n=9,11 and that is the only one of order less than 8.Furthermore, we determine all the hypotraceable oriented graphs of order 8 and explain the relevance of these results to the problem of determining, for given k≥2, the maximum order of nontraceable oriented digraphs each of whose induced subdigraphs of order k is traceable.     

On the complexity of crossings in permutations

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π^∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [C. Dwork, R. Kumar, M. Noar, D. Sivakumar, Rank aggregation methods for the Web, Proc. WWW10 (2001) 613-622] and prove the NP-hardness of the common crossing minimization problem for k=4 permutations. Then we establish a 2−2/k-approximation, improving the previous factor of 2. The max version is shown NP-hard for every k≥4, and there is a 2-approximation. Both approximations are optimal, if the common permutation is selected from the given ones. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.     

Matching polynomials for chains of cycles

Došli? and Måløy (2010) [2] obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the h-cactus chains with the most matchings.     

Relations on generalized degree sequences

We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.     

Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H. In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρ and σ on H we investigate when the hypergroups associated with the relations ρ∩σ, ρ∪σ and ρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.     

Parametric duality on minimax programming involving generalized convexity in complex space

In this paper we employ generalized convexity of complex functions to establish several sufficient optimality conditions for minimax programming in complex spaces. Using such criteria, we constitute a parametrical dual, and establish the weak, strong, and strict converse duality theorems in the framework.     

Graphs with the maximum or minimum number of 1-factors

Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs which are unions of complete regular bipartite graphs, and complete graphs with a matching removed. The first family is determined using the Alon and Friedland bound. For the second family, we show that a graph transformation which is known to increase network reliability also increases the number of 1-factors. In fact, more is true: this graph transformation increases the number of k-factors for all k≥1, and “in reverse” also shows that in general, threshold graphs have the fewest k-factors. We are then able to determine precisely which threshold graphs have the fewest 1-factors. We conjecture that the same graphs have the fewest k-factors for all k≥2 as well.     

A note on the generalized min-sum set cover problem

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin (STOC 2009). Bansal, Gupta, and Krishnaswamy (SODA 2010) give a 485-approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α-point scheduling to obtain our improvements.     

On the spectral characterization of some unicyclic graphs

Let H(n;q,n₁,n₂) be a graph with n vertices containing a cycle C_q and two hanging paths P_n1 and P_n2 attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12;6,1,5) and H(12;8,2,2), no two non-isomorphic graphs of the form H(n;q,n₁,n₂) are A-cospectral. It is proved that all graphs H(n;q,n₁,n₂) are determined by their L-spectra. And all graphs H(n;q,n₁,n₂) are proved to be determined by their Q-spectra, except for graphs with a being a positive even number and with b≥4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.     

Extremal infinite graph theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.     

On the demand pooling anomaly in inventory theory

The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

Higher order duality in multiobjective fractional programming with support functions

In this paper a new class of higher order (F,ρ,σ)-type I functions for a multiobjective programming problem is introduced, which subsumes several known studied classes. Higher order Mond-Weir and Schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in Rn. Weak and strong duality results are studied in both the cases assuming the involved functions to be higher order (F,ρ,σ)-type I. A number of previously studied problems appear as special cases.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Bases in Banach spaces of smooth functions on Cantor-type sets

We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used.