Cellular algorithms for binary matrix operations

A Characterization of Perfect Matrices

Attribute reduction plays an important role in rough set theory. It is an important application in data mining. In this paper, we focus on discussing the relation between set covering and attribute reduction in rough set theory. Based on the equivalence between minimal set covering and minimal attribute reduction sets, attribute reduction graph (ARG) is constructed. A novel algorithm to find the minimal attribute reduction sets, which is based on combination of binary discernibility matrix and graph theory is proposed in this paper. This algorithm demonstrates its efficiency and feasibility by an example.

NAS FD Lung: A novel lung assist diagnostic system based on neural architecture search

In the paper we investigate an algorithmic associative binary operation * on the set ℒℛ1 of Littlewood-Richardson tableaux with entries equal to one. We extend * to an algorithmic nonassociative binary operation on the set ℒℛ1 × ℕ and show that it is equivalent to the operation of taking the generic extensions of objects in the category of homomorphisms from semisimple nilpotent linear operators to nilpotent linear operators. Thus we get a combinatorial algorithm computing generic extensions in this category.

In this work, we bring together object tracking and digital watermarking to solve the spatio-temporal object adjacency problem in image sequences. Spatio-temporal relationships are established by embedding objects with unique digital watermarks and then by propagating the watermark frame by frame. Watermark propagation is accomplished by an existing object tracking module so that a tracked object acquires its watermark from the correspondences established by the object tracker. The spatio-temporally marked image sequences can then be searched to establish spatial and temporal adjacency among objects without using traditional spatio-temporal graphs. Borrowing from graph theory, we construct binary adjacency matrices among tracked objects and develop interpretation rules to establish a track history for each object. Track history can be used to determine the arrival of new objects in frames or the changing of spatial and temporal positions of objects with respect to each other as they move through time and space.

Binary relations, matrices and inference developments

X-codes are special linear maps for a compression technique, called X-compact. In the context of integrated circuit testing, an (m, n, d, x) X-code is an $m\times n$ binary matrix which compresses n-bit output from the circuit under test into $m$ bits while allowing for detecting the existence of up to $d$ erroneous output bits even if up to $x$ bits of the correct behavior are unknowable. In particular, X-codes of constant column weight x+1 are of special interest for practical reasons. While it is known that there exist infinite series of (m, n, 1, 2) X-codes of constant weight 3 with $n=\Theta(m^{2})$ , here we show that for $d\geq 4$ the largest possible $n$ is $o(m^{2})$ . This is an application of extremal graph theory and the first asymptotic improvement on this problem. We also investigate a special class of X-codes of constant weight 3 with a design theoretic property that boosts test quality when there are fewer unknowable bits than anticipated. We improve the tightest known lower bound on the largest number $n$ of columns of an X-code of this kind through probabilistic combinatorics. A deterministic algorithm is also given that produces X-codes of this type attaining the improved bound and runs in time polynomial in $m$ .

Contributors

Channel coding linear programming decoding (CC- LPD) and compressed sensing linear programming decoding (CS- LPD) are two setups that are formally tightly related. Recently, a connection between CC-LPD and CS-LPD was exhibited that goes beyond this formal relationship. The main ingredient was a lemma that allowed one to map vectors in the nullspace of some zero-one measurement matrix into vectors of the fundamental cone defined by that matrix. The aim of the present paper is to extend this connection along several directions. In particular, the above-mentioned lemma is extended from real measurement matrices where every entry is equal to either zero or one to complex measurement matrices where the absolute value of every entry is a non-negative integer. Moreover, this lemma and its generalizations are used to translate performance guarantees from CC-LPD to CS-LPD. In addition, the present paper extends the formal relationship between CC-LPD and CS-LPD with the help of graph covers. First, this graph-cover viewpoint is used to obtain new connec- tions between, on the one hand, CC-LPD for binary parity- check matrices, and, on the other hand, CS-LPD for complex measurement matrices. Secondly, this graph-cover viewpoint is used to see CS-LPD not only as a well-known relaxation of some zero-norm minimization problem but (at least in the case of real measurement matrices with only zeros, ones, and minus ones) also as a relaxation of a problem we call the zero-infinity operator minimization problem.

Sociometrics of Macaca Mulatta III: n-path centrality in grooming networks

Morphological operations are commonly used tools for the extraction of regional features and shape information in pattern recognition and image understanding tasks. In the present paper, a general approach for low-cost hardware-realization of multidimensional binary morphological operations is proposed. The underlying definitions of Minkowski addition and Minkowski subtraction are reduced to linear n-D convolutions, which lead to several alternative implementations. The flexible architecture of modern FPGAs allows the construction of operations with fixed and variable sized structuring elements that handle regions of up to 1000 object elements (e.g. pixels) in parallel. This leads to low-cost hardware realizations of real-time computer vision tasks. Possible clock frequencies of more than 100 MHz provide morphological image processing for camera resolution of 320/spl times/240 pixel at frame rates of over 1000 Hz.

This paper deals with studying some algebraic structures of the graphs as an attempt to visualize abstract mathematics. We have used some binary graph operations to investigated the algebraic structures of graphs with examples. This work emphasizes specifically the construction of semigroup or monoid and semiring, and their properties. This manuscript also aims to give a focused introduction of a class of homomorphism on the semiring of graphs. Some instances of real-life decision problems are consequently discussed. This article is also in a nascent stage of relating number theory and graph theory through mappings.

Computation of dominant energy transmission paths for ship structure using a graph theory algorithm

This chapter presents the terminology and concepts of graph theory , and describes basic matrix operations that are used in social network analysis. Both graph theory and matrix operations have served as the foundations of many concepts in the analysis of social networks (Hage and Harary 1983; Harary, Norman, and Cartwright 1965). In this chapter, the notation presented in Chapter 3 is used, and more concepts and ideas from graph theory are described and illustrated with examples. The topics covered in this chapter are important for the methods discussed in the remaining chapters of the book, but they are especially important in Chapter 5 (Centrality, Prestige, and Related Actor and Group Measures), Chapter 6 (Structural Balance, Clusterability, and Transitivity), Chapter 7 (Cohesive Subgroups), and Chapter 8 (Affiliations, Co-memberships, and Overlapping Subgroups). We start this chapter with a discussion of some reasons why graph theory and graph theoretic concepts are important for social network analysis. We then define a graph for representing a nondirectional relation. We begin with simple concepts, and progressively build on these to achieve more complicated, and more interesting, graph theoretic concepts. We then define and discuss directed graphs, for representing directional relations. Again, we begin with simple directed graph concepts and build to more complicated ideas. Following this, we discuss signed and valued graphs. We then define and discuss hypergraphs, which are used to represent affiliation networks.

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.