The goal here to is to introduce irreducible root systems, and through them try to classify certain lattices. As we’ve seen, lattices are just about the best thing there is in number theory (maybe), since it’s easy to pop out a modular form from a lattice under the right circumstances. Luckily, we’ll be working in those very circumstances! To get there, we’re going to have to define what a code...

From Last few year, internet became the first priority of everyone. It is a very easy and fast way to transfer and access data and information throughout the world. This information basically in the form of digital data (text, images, audio, video). Everyone using internet for their personal or professional use. Due to this it is important to protect user data from unauthorized access. When we ...

This is an introduction to Shannon’s Information Theory. It is more like a long note so that it is by no means a complete survey or completely mathematically rigorous. It covers two main topics: entropy and channel capacity. All related concepts will be developed in a totally combinatorial flavor. Some issues usually not addressed in the literature will be discussed here as well. Hopefully, it ...

The eld of statistics can be divided into two branches, descriptive statistics and inferential statistics. Descriptive statistics, as its name suggests, is concerned with describing a population, or a sample from that population, in terms of numerical measures that are computed from data. Such a numerical measure is called a statistic. If the data used in the computation comes from the whole po...

We give a brief survey of higher algebraic K-theory and its connection to motivic cohomology. We start with limits and colimits, and then pass to the combinatorial construction of topological spaces by means of “systems of simplices”, usefully mediated by simplicial sets. Definitions of K-theory are offered, and the main theorems are stated. A definition of motivic cohomology is offered and its...

The following is a introduction to a method, called signed difference analysis (or SDA), that I have been developing, off and on, over the last seven years. SDA is a general technique for evaluating psychological models using only the “non-metric” or ordinal properties of dependent variables. Although still a work in progress, the following is an outline of its main elements which, hopefully, w...

This article is a description of the techniques and tools used in scientific writing, as encountered in three submitted articles ([Leroy and Weis 1991], [Tofte 1998], and [Tofte and Birkedal 1998]), in order to better define the guidelines one should abide by when writing such literature. This report will be largely biased by the scientific formation of the author, who is particularly subjectiv...

Programming languages and direct manipulation user interfaces are distinct approaches for creating digital objects that combine visual and textual elements, such as documents, graphics, web applications, games, and data visualizations. At one end of the spectrum, programming languages provide experts an array of abstraction mechanisms to generate complex output, but the reliance on text-based r...

This brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result ...

Matrix algebra is one of the most useful tools in many disciplines, in particularly those related to data analysis (statistics, signal processing, machine learning, etc.). This short tutorial intends to provide a brief summary of the most important facts relating to matrix algebra for a person that has taken a linear algebra course a long time ago and has forgotten most of the details. Although...