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Reasoning --- Euclidean algorithm

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This is a comprehensive two-volumes text on plane and space geometry, transformations and conics, using a synthetic approach. The first volume focuses on Euclidean Geometry of the plane, and the second volume on Circle measurement, Transformations, Space geometry, Conics. The book is based on lecture notes from more than 30 courses which have been taught over the last 25 years. Using a synthetic approach, it discusses topics in Euclidean geometry ranging from the elementary (axioms and their first consequences), to the complex (the famous theorems of Pappus, Ptolemy, Euler, Steiner, Fermat, Morley, etc.). Through its coverage of a wealth of general and specialized subjects, it provides a comprehensive account of the theory, with chapters devoted to basic properties of simple planar and spatial shapes, transformations of the plane and space, and conic sections. As a result of repeated exposure of the material to students, it answers many frequently asked questions. Particularattention has been given to the didactic method; the text is accompanied by a plethora of figures (more than 2000) and exercises (more than 1400), most of them with solutions or expanded hints. Each chapter also includes numerous references to alternative approaches and specialized literature. The book is mainly addressed to students in mathematics, physics, engineering, school teachers in these areas, as well as, amateurs and lovers of geometry. Offering a sound and self-sufficient basis for the study of any possible problem in Euclidean geometry, the book can be used to support lectures to the most advanced level, or for self-study.

Geometry. --- Euclidean algorithm. --- Geometria euclidiana

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This book mainly deals with the Bochner-Riesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the Bochner-Riesz means and important achievements attained in the last 50 years. For the Bochner-Riesz means of multiple Fourier integral, it includes the Fefferman theorem which negates the Disc multiplier conjecture, the famous Carleson-Sjölin theorem, and Carbery-Rubio de Francia-Vega's work on almost everywhere convergence of the Bochner-Riesz means below the critical index. For the Bochner-Riesz means o

Fourier series. --- Euclidean algorithm. --- Algorithm of Euclid --- Continued division --- Division, Continued --- Euclid algorithm --- Euclidian algorithm --- Euclid's algorithm --- Algorithms --- Number theory --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Fourier analysis --- Harmonic analysis --- Harmonic functions --- Fourier series --- Euclidean algorithm --- Mathematical models.

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Cryptography lies at the heart of most technologies deployed today for secure communications. At the same time, mathematics lies at the heart of cryptography, as cryptographic constructions are based on algebraic scenarios ruled by group or number theoretical laws. Understanding the involved algebraic structures is, thus, essential to design robust cryptographic schemes. This Special Issue is concerned with the interplay between group theory, symmetry and cryptography. The book highlights four exciting areas of research in which these fields intertwine: post-quantum cryptography, coding theory, computational group theory and symmetric cryptography. The articles presented demonstrate the relevance of rigorously analyzing the computational hardness of the mathematical problems used as a base for cryptographic constructions. For instance, decoding problems related to algebraic codes and rewriting problems in non-abelian groups are explored with cryptographic applications in mind. New results on the algebraic properties or symmetric cryptographic tools are also presented, moving ahead in the understanding of their security properties. In addition, post-quantum constructions for digital signatures and key exchange are explored in this Special Issue, exemplifying how (and how not) group theory may be used for developing robust cryptographic tools to withstand quantum attacks.

NP-Completeness --- protocol compiler --- post-quantum cryptography --- Reed–Solomon codes --- key equation --- euclidean algorithm --- permutation group --- t-modified self-shrinking generator --- ideal cipher model --- algorithms in groups --- lightweight cryptography --- generalized self-shrinking generator --- numerical semigroup --- pseudo-random number generator --- symmetry --- pseudorandom permutation --- Berlekamp–Massey algorithm --- semigroup ideal --- algebraic-geometry code --- non-commutative cryptography --- provable security --- Engel words --- block cipher --- cryptography --- beyond birthday bound --- Weierstrass semigroup --- group theory --- braid groups --- statistical randomness tests --- group-based cryptography --- alternating group --- WalnutDSA --- Sugiyama et al. algorithm --- cryptanalysis --- digital signatures --- one-way functions --- key agreement protocol --- error-correcting code --- group key establishment

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The first book to offer a comprehensive view of the LLL algorithm, this text surveys computational aspects of Euclidean lattices and their main applications. It includes many detailed motivations, explanations and examples.

Cryptography -- Mathematics. --- Euclidean algorithm. --- Integer programming. --- Lattice theory. --- Cryptography --- Lattice theory --- Integer programming --- Euclidean algorithm --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Computer Science --- Mathematical Theory --- Algorithms. --- Mathematics. --- Algorithm of Euclid --- Continued division --- Division, Continued --- Euclid algorithm --- Euclidian algorithm --- Euclid's algorithm --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Cryptanalysis --- Cryptology --- Secret writing --- Steganography --- Algorism --- Computer science. --- Data structures (Computer science). --- Data encryption (Computer science). --- Computer science --- Computer Science. --- Data Structures. --- Data Encryption. --- Mathematics of Computing. --- Data Structures, Cryptology and Information Theory. --- Algorithm Analysis and Problem Complexity. --- Algebra --- Arithmetic --- Foundations --- Algorithms --- Number theory --- Programming (Mathematics) --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Signs and symbols --- Symbolism --- Writing --- Ciphers --- Data encryption (Computer science) --- Data structures (Computer scienc. --- Computer software. --- Cryptology. --- Data Structures and Information Theory. --- Informatics --- Science --- Data encoding (Computer science) --- Encryption of data (Computer science) --- Computer security --- Software, Computer --- Computer systems --- Computer science—Mathematics. --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Artificial intelligence --- Cryptography. --- Information theory. --- Data Science. --- Data processing. --- Communication theory --- Communication --- Cybernetics --- Computer mathematics