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Symmetry and the Zeros of Riemann's Zeta Function: Two finite mirror image vector series restrict the nontrivial zeros of Riemann's zeta function to t

Symmetry and the Zeros of Riemann's Zeta Function: Two finite mirror image vector series restrict the nontrivial zeros of Riemann's zeta function to t

Anthony D. Lander

Createspace Independent Publishing Platform
2018
nidottu
The famous "nontrivial zeros" are a set of complex numbers that produce zero when given to Riemann's zeta function. This set of numbers influences the distribution of the prime numbers. The nontrivial zeros therefore lie at the very heart of mathematics, since every integer greater than 1 is a unique product of primes. Riemann's hypothesis is that the real part of each nontrivial zero is a half. The author, Anthony Lander, is a paediatric surgeon and not a mathematician. However, Anthony has had a longstanding interest in symmetry and symmetry breaking in biological systems. He came across Riemann's hypothesis in 2012 and believes that a symmetry evident in Euler's zeta underlies the truth of Riemann's hypothesis and why the zeros repel.
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Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization

Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization

Springer Nature Switzerland AG
2019
nidottu
This book gathers papers presented at the 13th International Workshop on Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization (WSOM+), which was held in Barcelona, Spain, from the 26th to the 28th of June 2019. Since being founded in 1997, the conference has showcased the state of the art in unsupervised machine learning methods related to the successful and widely used self-organizing map (SOM) method, and extending its scope to clustering and data visualization. In this installment of the AISC series, the reader will find theoretical research on SOM, LVQ and related methods, as well as numerous applications to problems in fields ranging from business and engineering to the life sciences. Given the scope of its coverage, the book will be of interest to machine learning researchers and practitioners in general and, more specifically, to those looking for the latest developments in unsupervised learning and data visualization.
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Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization

Advances in Self-Organizing Maps, Learning Vector Quantization, Clustering and Data Visualization

Springer International Publishing AG
2022
nidottu
In this collection, the reader can ?nd recent advancements in self-organizing maps (SOMs) and learning vector quantization (LVQ), including progressive ideas on exploiting features of parallel computing. The collection is balanced in presenting novel theoretical contributions with applied results in traditional ?elds of SOMs, such as visualization problems and data analysis. Besides, the collection further includes less traditional deployments in trajectory clustering and recent results on exploiting quantum computation. The presented book is worth interest to data analysis and machine learning researchers and practitioners, speci?cally those interested in being updated with current developments in unsupervised learning, data visualization, and self-organization.
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Advances in Self-Organizing Maps, Learning Vector Quantization, Interpretable Machine Learning, and Beyond

Advances in Self-Organizing Maps, Learning Vector Quantization, Interpretable Machine Learning, and Beyond

Springer International Publishing AG
2024
nidottu
The book presents the peer-reviewed contributions of the 15th International Workshop on Self-Organizing Maps, Learning Vector Quantization and Beyond (WSOM$+$ 2024), held at the University of Applied Sciences Mittweida (UAS Mitt\-weida), Germany, on July 10–12, 2024. The book highlights new developments in the field of interpretable and explainable machine learning for classification tasks, data compression and visualization. Thereby, the main focus is on prototype-based methods with inherent interpretability, computational sparseness and robustness making them as favorite methods for advanced machine learning tasks in a wide variety of applications ranging from biomedicine, space science, engineering to economics and social sciences, for example. The flexibility and simplicity of those approaches also allow the integration of modern aspects such as deep architectures, probabilistic methods and reasoning as well as relevance learning. The book reflects both new theoretical aspects in this research area and interesting application cases. Thus, this book is recommended for researchers and practitioners in data analytics and machine learning, especially those who are interested in the latest developments in interpretable and robust unsupervised learning, data visualization, classification and self-organization.
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Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

Maresuke Shiraishi

Springer Verlag, Japan
2013
sidottu
The non-Gaussianity in the primordial density fluctuations is a key feature to clarify the early Universe and it has been probed with the Cosmic Microwave Background (CMB) bispectrum. In recent years, we have treated the novel-type CMB bispectra, which originate from the vector- and tensor-mode perturbations and include the violation of the rotational or parity invariance. On the basis of our current works, this thesis provides the general formalism for the CMB bispectrum sourced by the non-Gaussianity in the scalar, vector and tensor-mode perturbations. Applying this formalism, we calculate the CMB bispectra from the two scalars and a graviton correlation and primordial magnetic fields, and then outline new constraints on these magnitudes. Furthermore, this formalism can be easily extended to the cases where the rotational or parity invariance is broken. We also compute the CMB bispectra from the scalar-mode non-Gaussianities with a preferred direction and the tensor-mode non-Gaussianities induced by the parity-violating Weyl cubic terms. Here, we show that these bispectra include unique signals, which any symmetry-invariant models can never produce.
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Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum

Maresuke Shiraishi

Springer Verlag, Japan
2015
nidottu
The non-Gaussianity in the primordial density fluctuations is a key feature to clarify the early Universe and it has been probed with the Cosmic Microwave Background (CMB) bispectrum. In recent years, we have treated the novel-type CMB bispectra, which originate from the vector- and tensor-mode perturbations and include the violation of the rotational or parity invariance. On the basis of our current works, this thesis provides the general formalism for the CMB bispectrum sourced by the non-Gaussianity in the scalar, vector and tensor-mode perturbations. Applying this formalism, we calculate the CMB bispectra from the two scalars and a graviton correlation and primordial magnetic fields, and then outline new constraints on these magnitudes. Furthermore, this formalism can be easily extended to the cases where the rotational or parity invariance is broken. We also compute the CMB bispectra from the scalar-mode non-Gaussianities with a preferred direction and the tensor-mode non-Gaussianities induced by the parity-violating Weyl cubic terms. Here, we show that these bispectra include unique signals, which any symmetry-invariant models can never produce.
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Integration Between The Lebesgue Integral And The Henstock-kurzweil Integral: Its Relation To Local Convex Vector Spaces

Integration Between The Lebesgue Integral And The Henstock-kurzweil Integral: Its Relation To Local Convex Vector Spaces

Jaroslav Kurzweil

World Scientific Publishing Co Pte Ltd
2002
sidottu
The main topics of this book are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.The book is self-contained. It is of interest to specialists in the field of real functions, and it can also be read by students, since only the basics of mathematical analysis and vector spaces are required.
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Groups, Matrices, and Vector Spaces

Groups, Matrices, and Vector Spaces

James B. Carrell

Springer-Verlag New York Inc.
2017
sidottu
This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symmetry groups, determinants, linear coding theory and cryptography are interwoven throughout. Each section ends with ample practice problems assisting the reader to better understand the material. Some of the applications are illustrated in the chapter appendices. The author's unique melding of topics evolved from a two semester course that he taught at the University of British Columbia consisting of an undergraduate honors course on abstract linear algebra and a similar course on the theory of groups. The combined content from both makes this rare text ideal for a year-long course, covering more material than most linear algebra texts. It is also optimal for independent study and as a supplementary text for various professional applications. Advanced undergraduate or graduate students in mathematics, physics, computer science and engineering will find this book both useful and enjoyable.
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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

John Guckenheimer; Philip Holmes

Springer-Verlag New York Inc.
1983
sidottu
From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #Book Review - Engineering Societies Library, New York#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #American Mathematical Monthly#2
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Rings, Fields, and Vector Spaces

Rings, Fields, and Vector Spaces

B.A. Sethuraman

Springer-Verlag New York Inc.
1996
sidottu
This book is an attempt to communicate to undergraduate math­ ematics majors my enjoyment of abstract algebra. It grew out of a course offered at California State University, Northridge, in our teacher preparation program, titled Foundations of Algebra, that was intended to provide an advanced perspective on high-school mathe­ matics. When I first prepared to teach this course, I needed to select a set of topics to cover. The material that I selected would clearly have to have some bearing on school-level mathematics, but at the same time would have to be substantial enough for a university-level course. It would have to be something that would give the students a perspective into abstract mathematics, a feel for the conceptual elegance and grand simplifications brought about by the study of structure. It would have to be of a kind that would enable the stu­ dents to develop their creative powers and their reasoning abilities. And of course, it would all have to fit into a sixteen-week semester. The choice to me was clear: we should study constructibility. The mathematics that leads to the proof of the nontrisectibility of an arbitrary angle is beautiful, it is accessible, and it is worthwhile. Every teacher of mathematics would profit from knowing it. Now that I had decided on the topic, I had to decide on how to develop it. All the students in my course had taken an earlier course . .
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Geometry, Algebra, and Trigonometry by Vector Methods

Geometry, Algebra, and Trigonometry by Vector Methods

Arthur Herbert Copeland; Carl B. Allendoerfer

Literary Licensing, LLC
2013
sidottu
""Geometry, Algebra, and Trigonometry by Vector Methods"" is a comprehensive textbook written by Arthur Herbert Copeland. The book is designed to provide students with a thorough understanding of the fundamental concepts of geometry, algebra, and trigonometry using vector methods. The book is divided into three parts, each focusing on one of these subjects.Part one of the book covers the basics of geometry, including points, lines, angles, and polygons. The author uses vector methods to explain these concepts, making it easier for students to visualize and understand them. The section also includes a discussion on vectors and their properties.Part two of the book focuses on algebra, covering topics such as equations, functions, and graphs. The author uses vector methods to explain these concepts as well, making it easier for students to understand the relationship between algebra and geometry.Part three of the book covers trigonometry, including the properties of triangles, trigonometric functions, and identities. The author again uses vector methods to explain these concepts, making it easier for students to visualize and understand them.Throughout the book, the author provides numerous examples and exercises to help students practice and reinforce their understanding of the concepts presented. The book is suitable for students studying geometry, algebra, and trigonometry at the high school or college level.Allendoerfer Mathematics Series.This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.
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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

John Guckenheimer; Philip Holmes

Springer-Verlag New York Inc.
2013
nidottu
From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #Book Review - Engineering Societies Library, New York#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #American Mathematical Monthly#2
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