Binary matrix factorisations under Boolean arithmetic

For a binary matrix X, the Boolean rank br(X) is the smallest integer for which X can be factorised into the Boolean matrix product of two binary matrices A and B with inner dimension br(X). The isolation number i(X) of X is the maximum number of 1s no two of which are in a same row, column or a 2 x 2 submatrix of all 1s. In Part I. of this thesis, we continue Anna Lubiw's study of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds for all its submatrices. We show that the stronger concept of superfirmness of X is equivalent to having no odd holes in the rectangle cover graph of X, the graph in which br(X) and i(X) translate to the clique cover number and the independence number, respectively. A binary matrix is minimally non-firm if it is not firm but all of its proper submatrices are. We introduce a matrix operation that leads to generalised binary matrices and, under some conditions, preserves firmness and superfirmness. Then we use this matrix operation to derive several infinite families of minimally non-firm matrices. To the best of our knowledge, minimally non-firm matrices have not been studied before and our constructions provide the first infinite families of them. In Part II. of this thesis, we explore rank-k binary matrix factorisation (k-BMF). In k-BMF, we are given an m x n binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension m x k and k x n respectively, which minimise the distance between X and the Boolean matrix product of A and B in the squared Frobenius norm. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times a zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

Loop Optimizations in C and C++ Compilers: An Overview

The evolution of computer hardware in the past decades has truly been remarkable. From scalar instruction execution through superscalar and vector to parallel, processors are able to reach astonishing speeds – if programmed accordingly. Now, writing programs that take all the hardware details into consideration for the sake of efficiency is extremely difficult and error-prone. Therefore we increasingly rely on compilers to do the heavy-lifting for us. A significant part of optimizations done by compilers are loop optimiza- tions. Loops are inherently expensive parts of a program in terms of run time, and it is important that they exploit superscalar and vector instructions. In this paper, we give an overview of the scientific literature on loop optimization technology, and summarize the status of current implementations in the most widely used C and C++ compilers in the industry

Economics students’ migrations in the Hungarian higher education system

The Hungarian Higher Education has faced several structural challenges since the regime change of 1990. The ‘golden age’ seems to be over and the government tends to impose severe limits on institutions regarding the number of state-financed students, the minimal application points and the institutions’ missions. These new aspects influence the application procedures, as students are eager to achieve the highest price-value combination on the education market, which leads to internal and external migrations. In this study, we focus on the former, by using the agglomeration analysis of higher education institutions. We apply a modified Universal Law of Gravity to gather information about social and/or economic phenomena. On the level of single individuals, these types of decisions are random, but on the collective level, they can be characterized by certain principles and rules. This study explores the changes in the agglomeration areas and the limiting factors related to colleges and universities, offering economic education between 2004 and 2014. This period is adequate, as to identify the peculiarities and different influences of the market, the government, and the global trends and to identify the new spatial roles of the institutions