Cosheaf representations of relations and Dowker complexes

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

An O(n3) time algorithm for the maximum weight b-matching problem on bipartite graphs

Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results: We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A∪B,E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the first O(n3) time algorithm for finding the maximum weight b-matching of G, where |A|+|B| = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the first algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.

Parameterized Algorithms for Finding a Collective Set of Items

We extend the work of Skowron et al. (AIJ, 2016) by considering the parameterized complexity of the following problem. We are given a set of items and a set of agents, where each agent assigns an integer utility value to each item. The goal is to find a set of k items that these agents would collectively use. For each such collective set of items, each agent provides a score that can be described using an OWA (ordered weighted average) operator and we seek a set with the highest total score. We focus on the parameterization by the number of agents and we find numerous fixed-parameter tractability results (however, we also find some W[1]-hardness results). It turns out that most of our algorithms even apply to the setting where each agent has an integer weight.

An O(n^3) time algorithm for the maximum weight b-matching problem on bipartite graphs

Background: A matching between two sets A and B assigns some elements of A to some elements of B. Finding the similarity between two sets of elements by advantage of the matching is widely used in computational biology for example in the contexts of genome-wide and sequencing association studies. Frequently, the capacities of the elements are limited. That is, the number of the elements that can be matched to each element should not exceed a given number. Results: We use bipartite graphs to model relationships between pairs of objects. Given an undirected bipartite graph G = (A [ B;E), the b-matching of G matches each vertex v in A (resp. B) to at least 1 and at most b(v) vertices in B (resp. A), where b(v) denotes the capacity of v. We propose the rst O(n3) time algorithm for nding the maximum weight b-matching of G, where jAj + jBj = O(n). Conclusions: The b-matching has been studied widely for the bipartite graphs with integer weight edges. But our algorithm is the rst algorithm for the maximum (respectively minimum) b-matching problem with non positive real (respectively non negative real) edge weights.

The Computational Power of Neural Networks and Representations of Numbers in Non-Integer Bases

We briefly survey the basic concepts and results concerning the computational power of neural net-orks which basically depends on the information content of eight parameters. In particular, recurrent neural networks with integer, rational, and arbitrary real weights are classi ed within the Chomsky and finer complexity hierarchies. Then we re ne the analysis between integer and rational weights by investigating an intermediate model of integer-weight neural networks with an extra analog rational-weight neuron (1ANN). We show a representation theorem which characterizes the classification problems solvable by 1ANNs, by using so-called cut languages. Our analysis reveals an interesting link to an active research field on non-standard positional numeral systems with non-integer bases. Within this framework, we introduce a new concept of quasi-periodic numbers which is used to classify the computational power of 1ANNs within the Chomsky hierarchy.

A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.