Download or read book Construction of Quaternions and Cayley Numbers as Examples of Linear Algebras written by Marlene Tellinghuisen Tate and published by . This book was released on 1964 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On Quaternions and Octonions written by John H. Conway and published by CRC Press. This book was released on 2003-01-23 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less f
Download or read book Rethinking Quaternions written by Ron Goldman and published by Morgan & Claypool Publishers. This book was released on 2010 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolate between two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text."--P. 4 of cover.
Download or read book Introduction to Quaternions written by Philip Kelland and published by . This book was released on 1882 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Introduction to Modern Algebra and Its Applications written by Nadiya Gubareni and published by CRC Press. This book was released on 2021-06-23 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchberger’s algorithm which is one of the important algorithms for constructing Gröbner basis. Key Features: Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications. Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty. Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics.
Download or read book The American Mathematical Monthly written by and published by . This book was released on 1978 with total page 1064 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes articles, as well as notes and other features, about mathematics and the profession.
Download or read book Elements of Quaternions written by A. S. Hardy and published by READ BOOKS. This book was released on 2008-11 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: ELEMENTS OF QUATERNIONS by A. S. HARDY. Originally published in 1891. PREFACE: object of the following treatise is to exhibit the elementary principles and notation of the Quaternion Calculus, so as to meet the wants of beginners in the class-room. The Elements and Lectures of Sir William Rowan Hamilton, while they may be said to contain the suggestion of all that will be done in the way of Quaternion research and application, are not, for this reason, as also on account of their diffuseness of style, suitable for the purposes of elementary instruction. Taits work on Quaternions is also, in its originality and conciseness, beyond the time and needs of the beginner. In addition to the above, the following works have been consulted Calcolo dei Quaternione. Bellavitis Modena, 1858. Exposition de la MetTiode des fiquipollences Traduit de 1 Italien de Giusto Bellavitis, par C.-A. Laisant Paris, 1874. Original memoir in the Memoirs of the Italian Society. 1854. Thorie lementaire des Quantites Complexes. J. Hoiiel Paris, 1874. Essai sur une Manure de Representer les Quantites Imaginaires dans les Construction G-eometriques. Par R. Argand Paris, 1806. Second edition, with preface. by J. Hoiiel Paris, 1874. Translated, with notes, from the French, by A. S. Hardy. Van Nostrands Science Series, No. 52 1881. Kurze Anleitunff zum Rechnen mit den Hamilton sclieri Quaternionen. J. Odstrcil Halle, 18T9. Applications Mecaniques du Qalcul des Quaternions. Laisant Paris, 1877. Introduction to Quaternions. Kelland and Tait Lon don, 1873. A free use has been made of the examples and exercises of the last work and, in Article 87, is given, by permis sion, the substance of a paper from Volume L, page 379, American Journal of Mathematics, illustrating admirably the simplicity and brevity of the Quaternion method. If this presentation of the principles shall afford the undergraduate student a glimpse of this elegant and pow erful instrument of analytical research, or lead him to follow their more extended application in the works above cited, the aim of this treatise will have been accomplished. The author expresses his obligation to Mr. T. W. D. Worthen for valuable assistance in the preparation of this work, and to Mr, J. S. Gushing for whatever of typographical excellence it possesses. A. S. HABDY. HANOVER, N. H., June 21, 1881. Contents include: CHAPTER I. Addition and Subtraction of Vectors or, Geometric Addition and Subtraction. Article Paga 1. Definition of a vector. Effect of the minus sign before a vector 1 2. Equal vectors 2 3. Unequal vectors. Vector addition 2 4. Vector addition, commutative 3 5. Vector addition, associative 3 6. Transposition of terms in a vector equation 4 7. Definition of a tensor 4 8. Definition of a scalar 5 9. Distributive law in the multiplication of vector by scalar quantities 6 10. If Sa S 3 0, then 2a and 2 5 7 11. Examples 8 12. Complanar vectors Condition of complanarity 15 13. Co-initial vectors Condition of collinearity 16 14. Examples 17 15. Expression for a medial vector 24 16. Expression for an angle-bisector 25 17. Examples 26 18. Mean point 28 19. Examples 28 20. Exercises 30 CHAPTER II. Multiplication and Division of Vectors or, Geometric Multiplication and Division. 21. Elements of a quaternion 32 22. Equal quaternions 34 28. Positive rotation 35 VI. Article Page 24 Analytical expression for a quaternion. Product and quo tient of rectangular unit-vectors. Tensor and versor of a quaternion ............... ....... 36 25 Symbolic notation q TqVq ................ 39 26. Reciprocal of a quaternion ................ 39 27. Quadrantal versors, i j, ft ................ 40 28. Whole powers of unit vectors. Square of a unit vector is 1, 41 29...
Download or read book Utility of Quaternions in Physics written by Alexander McAulay and published by Createspace Independent Publishing Platform. This book was released on 2017-06-18 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: In math, the quaternions are a number method that extends the complex numbers. They were originally described by the mathematician William Rowan Hamilton and applied to mechanics in space (3D). Quaternions characteristics are that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two lines in 3D (the quotient of two vectors). Quaternions find uses in theoretical and applied mathematics, in particular for calculations involving 3D rotations such as in computer graphics, computer vision, and crystallographic texture analysis. In useful applications, they find use alongside other methods, like Euler angles and rotation matrices, depending on the application. In contemporary mathematical language, quaternions form a 4D associative normed division algebra over the real numbers, and consequently also a domain. In fact, the quaternions were the elementary noncommutative division algebra to be discovered. According to the Frobenius theorem, it is one of only two finite-dimensional dividing rings containing the real numbers as a proper subring, and the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of whichever quaternions are the largest associative algebra.
Download or read book On Quaternions and Octonions written by John Horton Conway and published by . This book was released on 2005 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numbers written by Heinz-Dieter Ebbinghaus and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about all kinds of numbers, from rationals to octonians, reals to infinitesimals. It is a story about a major thread of mathematics over thousands of years, and it answers everything from why Hamilton was obsessed with quaternions to what the prospect was for quaternionic analysis in the 19th century. It glimpses the mystery surrounding imaginary numbers in the 17th century and views some major developments of the 20th century.
Download or read book Linear Algebra written by Michael E. Taylor and published by American Mathematical Soc.. This book was released on 2020-07-06 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text develops linear algebra with the view that it is an important gateway connecting elementary mathematics to more advanced subjects, such as advanced calculus, systems of differential equations, differential geometry, and group representations. The purpose of this book is to provide a treatment of this subject in sufficient depth to prepare the reader to tackle such further material. The text starts with vector spaces, over the sets of real and complex numbers, and linear transformations between such vector spaces. Later on, this setting is extended to general fields. The reader will be in a position to appreciate the early material on this more general level with minimal effort. Notable features of the text include a treatment of determinants, which is cleaner than one often sees, and a high degree of contact with geometry and analysis, particularly in the chapter on linear algebra on inner product spaces. In addition to studying linear algebra over general fields, the text has a chapter on linear algebra over rings. There is also a chapter on special structures, such as quaternions, Clifford algebras, and octonions.
Download or read book Introduction to Modern Algebra and Its Applications written by Nadiya Gubareni and published by CRC Press. This book was released on 2021-06-23 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchberger’s algorithm which is one of the important algorithms for constructing Gröbner basis. Key Features: Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications. Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty. Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics.
Download or read book A Taste of Jordan Algebras written by Kevin McCrimmon and published by Springer Science & Business Media. This book was released on 2006-05-29 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes the history of Jordan algebras and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.
Download or read book Scientific Legacy Of Professor Zbigniew Oziewicz Selected Papers From The International Conference Applied Category Theory Graph operad logic written by Hilda Maria Colin Garcia and published by World Scientific. This book was released on 2023-09-27 with total page 771 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dedicated to the memory of the late Professor Zbigniew Oziewicz from Universidad Nacional Autónoma de México, the book consists of papers on a wide variety of topics related to the work of Professor Oziewicz, which were presented at the special conference on Graph-Operads-Logic (GOL 2021), selected through peer review to promote his scientific legacy.Professor Oziewicz was a great enthusiast and supporter of category theory and its applications in physics, as well as in various areas of mathematics (topology, noncommutative geometry, etc.). In particular, he made significant contributions to the theory of Frobenius algebras, which now are becoming more important due to their connection with topological quantum field theories that are used in mathematical physics and in quantum topology. Professor Oziewicz was a great and very generous teacher, who immersed his students in the beautiful ideas of category theory as well as mathematical physics and computation. It was his idea to start a series of conferences under the title Graphs-Operads-Logic, most of them held in Mexico, with some of them in the USA, which were a great platform to discuss various ideas connected with category theory and its various applications, and to make friends with other scientists. Despite his passing, the GOL 2021 conference is included in this series to pay tribute to his many contributions to diverse areas of science.The book is laid out in twelve main topics where we can find relevant works from distinguished experts.
Download or read book Mathematics in Victorian Britain written by photographer and broadcaster Foreword by Dr Adam Hart-Davis and published by Oxford University Press. This book was released on 2011-09-29 with total page 477 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the Victorian era, industrial and economic growth led to a phenomenal rise in productivity and invention. That spirit of creativity and ingenuity was reflected in the massive expansion in scope and complexity of many scientific disciplines during this time, with subjects evolving rapidly and the creation of many new disciplines. The subject of mathematics was no exception and many of the advances made by mathematicians during the Victorian period are still familiar today; matrices, vectors, Boolean algebra, histograms, and standard deviation were just some of the innovations pioneered by these mathematicians. This book constitutes perhaps the first general survey of the mathematics of the Victorian period. It assembles in a single source research on the history of Victorian mathematics that would otherwise be out of the reach of the general reader. It charts the growth and institutional development of mathematics as a profession through the course of the 19th century in England, Scotland, Ireland, and across the British Empire. It then focuses on developments in specific mathematical areas, with chapters ranging from developments in pure mathematical topics (such as geometry, algebra, and logic) to Victorian work in the applied side of the subject (including statistics, calculating machines, and astronomy). Along the way, we encounter a host of mathematical scholars, some very well known (such as Charles Babbage, James Clerk Maxwell, Florence Nightingale, and Lewis Carroll), others largely forgotten, but who all contributed to the development of Victorian mathematics.
Download or read book Finite Dimensional Algebras written by Yurj A. Drozd and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This English edition has an additional chapter "Elements of Homological Al gebra". Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras.
Download or read book The Theory of Spinors written by Elie Cartan and published by Courier Corporation. This book was released on 1981-02-01 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: The French mathematician Élie Cartan (1869–1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities. The book is divided into two parts. The first is devoted to generalities on the group of rotations in n-dimensional space and on the linear representations of groups, and to the theory of spinors in three-dimensional space. Finally, the linear representations of the group of rotations in that space (of particular importance to quantum mechanics) are also examined. The second part is devoted to the theory of spinors in spaces of any number of dimensions, and particularly in the space of special relativity (Minkowski space). While the basic orientation of the book as a whole is mathematical, physicists will be especially interested in the final chapters treating the applications of spinors in the rotation and Lorentz groups. In this connection, Cartan shows how to derive the "Dirac" equation for any group, and extends the equation to general relativity. One of the greatest mathematicians of the 20th century, Cartan made notable contributions in mathematical physics, differential geometry, and group theory. Although a profound theorist, he was able to explain difficult concepts with clarity and simplicity. In this detailed, explicit treatise, mathematicians specializing in quantum mechanics will find his lucid approach a great value.
