1 Basic Concepts and Notation Linear algebra provides away of compactly representing and operating onsets of linear equations. For example, consider the following system of equations: 4x 1 ?5x 2 =?13 ?2x 1 +3x 2 =9. This is two equations and two variables, so as you know from highschool algebra, you can find a unique solution forx 1 andx 2 (unless the equations are somehow degenerate, for example if the second equation (more…)

My goals in these lectures are: ? To present enough linear algebra to understand the Singular Value Decomposition and related methods such as the Principal Components Analysis . ? To present some applications, especially concerning Latent Semantic Analysis and facial recognition . ? The linear algebra concepts that I’dliketoteachare: linear independence, span, basis, eigenvalue, eigenvector, singular (more…)

The Mathematica function DSolve finds symbolic solutions to differential equations. (The Mathe- matica function NDSolve, on the other hand, is a general numerical differential equation solver.) DSolve can handle the following types of equations: † Ordinary Differential Equations (ODEs), in which there is a single independent variable t and one or more dependent variables x i Ht L. DSolve is equipped with (more…)

In this paper, we present a tutorial on two powerful cryptanalysis techniques applied to symmetric-key block ciphers: linear cryptanalysis [1] and differential cryptanalysis [2]. Linear cryptanalysis was introduced by Matsui at EUROCRYPT ’93 as a theoretical attack on the Data Encryption Standard (DES) [3] and later successfully used in the practical cryptanalysis of DES [4]; differential cryptanalysis was first (more…)

Abstract This paper is a short and painless introduction to the  calculus. Originally developed in order to study some mathematical properties of eectively computable functions, this formalism has provided a strong theoretical foundation for the family of functional programming languages. We show how to perform some arithmetical computations using the  calculus and how to dene recursive functions, even though functions in  calculus are not given (more…)

ABSTRACT The problems of finding a longest common subsequence of two sequences A and B and a shortest edit script for transforming A into B have long been known to be dual problems. In this paper, they are shown to be equivalent to finding a shortest/longest path in an edit graph. Using this perspective, a simple O(ND) time and space algorithm is developed where N is the sum of the lengths of A and B and D is the size of the minimum edit script for A and B. The algorithm (more…)

operation that is a bottleneck for many important algorithms. Faster matrix multiplication would give more efficient algorithms for many standard linear algebra problems, such as inverting matrices, solving systems of linear equations, and finding determinants. Even some basic graph algorithms run only as fast as matrix multiplication. The standard method for multiplying n × n matrices requires O(n3) multiplications. The fastest known algorithm, devised in 1987 by Don (more…)

Abstract In this paper, we propose a new progressionmechanism for a restricted form of incomplete knowledge formulated as a basic action theory in the situation calculus. Specifically, we focus on functional fluents and deal directly with the possible values these fluents may have and how these values are affected by both physical and sensing actions. The method we propose is logically complete and can be calculated efficiently using database techniques under (more…)

Gottfried Wilhelm Leibniz and Isaac Newton were geniuses who lived quite different lives and invented quite different versions of the infinitesimal calculus, each to suit his own interests and purposes. Newton discovered his fundamental ideas in 1664–1666, while a student at Cambridge University. During a good part of these years the University was closed due to the plague, and Newton worked at his family home in Woolsthorpe, Lincolnshire. However, his (more…)

Abstract. The pure pattern calculus generalises the pure lambda-calculus by basing computation on pattern-matching instead of beta-reduction. The simplicity and power of the calculus derive from allowing any term to be a pattern. As well as supporting a uniform approach to functions, it supports a uniform approach to data structures which underpins two new forms of polymorphism. Path polymorphism supports searches or queries along all paths through an (more…)