The Geometry Of Higher Dimensional Polytopes

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The Geometry Of Higher Dimensional Polytopes

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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2018-08-03

The Geometry Of Higher Dimensional Polytopes written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-03 with Technology & Engineering categories.

The majority of the chemical elements form chemical compounds with molecules of higher dimension (i.e., substantially exceeding three). This fact is very important for the analysis of molecular interactions in various areas: nanomedicine, nanotoxicology, and quantum biology. The Geometry of Higher-Dimensional Polytopes contains innovative research on the methods and applications of the structures of binary compounds. It explores the study of geometry polytopes from a higher-dimensional perspective, taking into account the features of polytopes that are models of chemical compounds. While highlighting topics including chemical compounds, symmetry transformation, and DNA structures, this book is ideally designed for researchers, academicians, and students seeking current research on dimensions present in binary compounds.

The Geometry Of Higher Dimensional Polytopes

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Author : Gennadiy Vladimirovich Zhizhin
language : en
Publisher: IGI Global, Engineering Science Reference
Release Date : 2018-08-03

The Geometry Of Higher Dimensional Polytopes written by Gennadiy Vladimirovich Zhizhin and has been published by IGI Global, Engineering Science Reference this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-03 with Mathematics categories.

The majority of the chemical elements form chemical compounds with molecules of higher dimension (i.e., substantially exceeding three). This fact is very important for the analysis of molecular interactions in various areas: nanomedicine, nanotoxicology, and quantum biology. The Geometry of Higher-Dimensional Polytopes contains innovative research on the methods and applications of the structures of binary compounds. It explores the study of geometry polytopes from a higher-dimensional perspective, taking into account the features of polytopes that are models of chemical compounds. While highlighting topics including chemical compounds, symmetry transformation, and DNA structures, this book is ideally designed for researchers, academicians, and students seeking current research on dimensions present in binary compounds.

The Classes Of Higher Dimensional Polytopes In Chemical Physical And Biological Systems

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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2022-04-08

The Classes Of Higher Dimensional Polytopes In Chemical Physical And Biological Systems written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-08 with Mathematics categories.

The study of the geometry of structures that arise in a variety of specific natural systems, such as chemical, physical, biological, and geological, revealed the existence of a wide range of types of polytopes of the highest dimension that were unknown in classical geometry. At the same time, new properties of polytopes were discovered as well as the geometric patterns to which they obey. There is a need to classify these types of polytopes of the highest dimension by listing their properties and formulating the laws to which they obey. The Classes of Higher Dimensional Polytopes in Chemical, Physical, and Biological Systems explains the meaning of higher dimensions and systematically generalizes the results of geometric research in various fields of knowledge. This book is useful both for the fundamental development of geometry and for the development of branches of science related to human activities. It builds upon previous books published by the author on this topic. Covering areas such as heredity, geometry, and dimensions, this reference work is ideal for researchers, scholars, academicians, practitioners, industry professionals, instructors, and students.

Lectures On Polytopes

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Author : Günter M. Ziegler
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-05-03

Lectures On Polytopes written by Günter M. Ziegler and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-03 with Mathematics categories.

Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.

Normal Partitions And Hierarchical Fillings Of N Dimensional Spaces

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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2020-12-25

Normal Partitions And Hierarchical Fillings Of N Dimensional Spaces written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-25 with Mathematics categories.

In the study of the structure of substances in recent decades, phenomena in the higher dimension was discovered that was previously unknown. These include spontaneous zooming (scaling processes), discovery of crystals with the absence of translational symmetry in three-dimensional space, detection of the fractal nature of matter, hierarchical filling of space with polytopes of higher dimension, and the highest dimension of most molecules of chemical compounds. This forces research to expand the formulation of the question of constructing n-dimensional spaces, posed by David Hilbert in 1900, and to abandon the methods of considering the construction of spaces by geometric figures that do not take into account the accumulated discoveries in the physics of the structure of substances. There is a need for research that accounts for the new paradigm of the discrete world and provides a solution to Hilbert's 18th problem of constructing spaces of higher dimension using congruent figures. Normal Partitions and Hierarchical Fillings of N-Dimensional Spaces aims to consider the construction of spaces of various dimensions from two to any finite dimension n, taking into account the indicated conditions, including zooming in on shapes, properties of geometric figures of higher dimensions, which have no analogue in three-dimensional space. This book considers the conditions of existence of polytopes of higher dimension, clusters of chemical compounds as polytopes of the highest dimension, higher dimensions in the theory of heredity, the geometric structure of the product of polytopes, the products of polytopes on clusters and molecules, parallelohedron and stereohedron of Delaunay, parallelohedron of higher dimension and partition of n-dimensional spaces, hierarchical filling of n-dimensional spaces, joint normal partitions, and hierarchical fillings of n-dimensional spaces. In addition, it pays considerable attention to biological problems. This book is a valuable reference tool for practitioners, stakeholders, researchers, academicians, and students who are interested in learning more about the latest research on normal partitions and hierarchical fillings of n-dimensional spaces.

Biochemistry In The Space Of The Highest Dimension

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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2024-04-29

Biochemistry In The Space Of The Highest Dimension written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-29 with Medical categories.

In biochemistry, a critical oversight persists in neglecting molecules' higher dimensionality and its profound implications for living organisms. Despite groundbreaking research revealing the transformative properties imbued by higher dimensions, mainstream biochemistry remains entrenched in a two-dimensional paradigm, failing to recognize the intricate relationship between molecular structure and biological function. This oversight poses a significant obstacle in understanding and addressing complex diseases plaguing humanity, from cancer to neurological disorders. Biochemistry in the Space of the Highest Dimension is a groundbreaking book that pioneers a paradigm shift in biochemistry, systematically unraveling the mysteries of higher-dimensional biomolecules and their impact on living organisms. Through meticulously researched chapters addressing topics from the structure of nucleic acids to the emergence of life itself, this book provides a comprehensive framework for understanding and leveraging the power of higher dimensions in biochemistry. Bridging the gap between theory and application offers scholars and practitioners a transformative roadmap toward innovative therapies and a deeper understanding of life's complexities.

Nanotechnologies And Clusters In The Spaces Of Higher Dimension Emerging Research And Opportunities

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Author : Zhizhin, Gennadiy Vladimirovich
language : en
Publisher: IGI Global
Release Date : 2020-10-09

Nanotechnologies And Clusters In The Spaces Of Higher Dimension Emerging Research And Opportunities written by Zhizhin, Gennadiy Vladimirovich and has been published by IGI Global this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-09 with Technology & Engineering categories.

Research on nanomaterials and their applications has become a trending area in various fields of study and practice. Its properties and abilities open a variety of scientific advancements that weren’t possible in past years. One specific area of research that is benefiting from the implementation of nanotechnology is the study of higher-dimensional compounds that include metallic atoms and other polytypes. There is vast potential in the study of how nanomaterials are currently being used for producing clusters in higher dimensions of space. Nanotechnologies and Clusters in the Spaces of Higher Dimension: Emerging Research and Opportunities provides emerging research exploring the theoretical and practical aspects of the production of intermetallic clusters in high dimensional spaces using nanotechnology. Featuring coverage on a broad range of topics such as intermetallic compounds, incident conservation law, and applied mathematics, this book is ideally designed for practitioners, scientists, engineers, researchers, educators, physicists, mathematicians, students, and academicians seeking current research on the use of nanomaterials in interdimensional science.

Lectures On Discrete Geometry

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Author : Jiri Matousek
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01

Lectures On Discrete Geometry written by Jiri Matousek and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.

Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Stochastic And Integral Geometry

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Author : Rolf Schneider
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-09-08

Stochastic And Integral Geometry written by Rolf Schneider and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-08 with Mathematics categories.

Stochastic geometry deals with models for random geometric structures. Its early beginnings are found in playful geometric probability questions, and it has vigorously developed during recent decades, when an increasing number of real-world applications in various sciences required solid mathematical foundations. Integral geometry studies geometric mean values with respect to invariant measures and is, therefore, the appropriate tool for the investigation of random geometric structures that exhibit invariance under translations or motions. Stochastic and Integral Geometry provides the mathematically oriented reader with a rigorous and detailed introduction to the basic stationary models used in stochastic geometry – random sets, point processes, random mosaics – and to the integral geometry that is needed for their investigation. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

Thinking Geometrically

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Author : Thomas Q. Sibley
language : en
Publisher: The Mathematical Association of America
Release Date : 2015-08-14

Thinking Geometrically written by Thomas Q. Sibley and has been published by The Mathematical Association of America this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-14 with Mathematics categories.

Thinking Geometrically: A Survey of Geometries is a well written and comprehensive survey of college geometry that would serve a wide variety of courses for both mathematics majors and mathematics education majors. Great care and attention is spent on developing visual insights and geometric intuition while stressing the logical structure, historical development, and deep interconnectedness of the ideas. Students with less mathematical preparation than upper-division mathematics majors can successfully study the topics needed for the preparation of high school teachers. There is a multitude of exercises and projects in those chapters developing all aspects of geometric thinking for these students as well as for more advanced students. These chapters include Euclidean Geometry, Axiomatic Systems and Models, Analytic Geometry, Transformational Geometry, and Symmetry. Topics in the other chapters, including Non-Euclidean Geometry, Projective Geometry, Finite Geometry, Differential Geometry, and Discrete Geometry, provide a broader view of geometry. The different chapters are as independent as possible, while the text still manages to highlight the many connections between topics. The text is self-contained, including appendices with the material in Euclid’s first book and a high school axiomatic system as well as Hilbert’s axioms. Appendices give brief summaries of the parts of linear algebra and multivariable calculus needed for certain chapters. While some chapters use the language of groups, no prior experience with abstract algebra is presumed. The text will support an approach emphasizing dynamical geometry software without being tied to any particular software.