Quaternions are a mathematical structure used to represent rotations and orientations in 3D space. The document discusses the history, theory, and applications of quaternions. It was invented in 1843 by Sir William Rowan Hamilton and has found modern applications in computer graphics, where it is used for 3D animation and rotations due to advantages over other representations like Euler angles. The theory section covers properties like multiplication and identities. Applications discussed include physics, group theory, and using quaternions in linear interpolation algorithms for smooth 3D animation.
This document presents a five dimensional cosmological model with a perfect fluid coupled to a massless scalar field in general relativity. The field equations are solved assuming an equation of state of p=ρ and a relation between the metric potentials of R=kAn, where k and n are constants. The solutions show the scale factors and scalar field as functions of time. The model expands anisotropically with no initial singularity and decelerates similarly to standard cosmology. Physical quantities like density and pressure diverge initially but vanish at later times.
Impacts of a New Spatial Variable on a Black Hole Metric SolutionIJSRED
This document discusses the impacts of introducing a new spatial variable in black hole metrics. It begins by summarizing Einstein and Rosen's 1935 paper which introduced a variable ρ = r - 2M in the Schwarzschild metric to remove the singularity. The document then introduces a similar new variable p = r - 2√M and analyzes how this impacts the Schwarzschild metric. Specifically, it notes that this new variable allows for negative radii values and multiple asymptotic regions beyond just two, introducing concepts of probability and imaginary spatial coordinates. Overall, the document explores how different mathematical variables can impact theoretical physics concepts like wormholes.
1. The document presents two theorems regarding conditions under which a topological space is metrizable.
2. Theorem 1 states that if a topological space satisfies four conditions, including being a T1 space and having a neighborhood basis with certain properties, then it is metrizable.
3. Theorem 2 also provides conditions for a space to be metrizable, including being a T1 space and having a neighborhood basis at each point with one additional property. The proof shows that Theorem 1 implies the conditions of Theorem 2.
- The author derives the Schwarzschild metric in D spatial dimensions and one time dimension to investigate the effects of a fractal deviation from three dimensions.
- Using the metric, the author considers phenomena like perihelion precession, bending of light, and gravitational redshift to set bounds on the parameter δ that quantifies the deviation from three dimensions.
- The best upper bound found is |δ| < 5.0 × 10-9, from measurements of perihelion precession in binary pulsar systems. This is consistent with zero deviation from three dimensions.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
The document discusses groups and graphs in probability theory. It provides background on the probability that a group element fixes a set and related work applying this concept to graph theory, specifically generalized conjugacy class graphs and orbit graphs. The main results are:
1) For a dihedral group of order 2n acting regularly on the set of commuting pairs of elements, the probability that a group element fixes a set is 1/n.
2) Under this action, the generalized conjugacy class graph of the dihedral group has n isolated vertices and is empty.
3) The orbit graph vertices correspond to the n orbits of sizes 1 or 2n, with an edge between vertices if their corresponding orbits are conjugate.
The document presents some fixed point results for maps satisfying certain contractive conditions in ordered G-metric spaces. It begins with an introduction discussing the background and history of fixed point theory. It then provides definitions related to G-metric spaces. The main result is a fixed point theorem for maps satisfying a rational-type contractive condition in a complete ordered G-metric space. It proves that such maps have at least one fixed point and sequences converge to a fixed point. If there are two distinct fixed points, their G-metric is bounded below by 1/2.
This document presents a five dimensional cosmological model with a perfect fluid coupled to a massless scalar field in general relativity. The field equations are solved assuming an equation of state of p=ρ and a relation between the metric potentials of R=kAn, where k and n are constants. The solutions show the scale factors and scalar field as functions of time. The model expands anisotropically with no initial singularity and decelerates similarly to standard cosmology. Physical quantities like density and pressure diverge initially but vanish at later times.
Impacts of a New Spatial Variable on a Black Hole Metric SolutionIJSRED
This document discusses the impacts of introducing a new spatial variable in black hole metrics. It begins by summarizing Einstein and Rosen's 1935 paper which introduced a variable ρ = r - 2M in the Schwarzschild metric to remove the singularity. The document then introduces a similar new variable p = r - 2√M and analyzes how this impacts the Schwarzschild metric. Specifically, it notes that this new variable allows for negative radii values and multiple asymptotic regions beyond just two, introducing concepts of probability and imaginary spatial coordinates. Overall, the document explores how different mathematical variables can impact theoretical physics concepts like wormholes.
1. The document presents two theorems regarding conditions under which a topological space is metrizable.
2. Theorem 1 states that if a topological space satisfies four conditions, including being a T1 space and having a neighborhood basis with certain properties, then it is metrizable.
3. Theorem 2 also provides conditions for a space to be metrizable, including being a T1 space and having a neighborhood basis at each point with one additional property. The proof shows that Theorem 1 implies the conditions of Theorem 2.
- The author derives the Schwarzschild metric in D spatial dimensions and one time dimension to investigate the effects of a fractal deviation from three dimensions.
- Using the metric, the author considers phenomena like perihelion precession, bending of light, and gravitational redshift to set bounds on the parameter δ that quantifies the deviation from three dimensions.
- The best upper bound found is |δ| < 5.0 × 10-9, from measurements of perihelion precession in binary pulsar systems. This is consistent with zero deviation from three dimensions.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
The document discusses groups and graphs in probability theory. It provides background on the probability that a group element fixes a set and related work applying this concept to graph theory, specifically generalized conjugacy class graphs and orbit graphs. The main results are:
1) For a dihedral group of order 2n acting regularly on the set of commuting pairs of elements, the probability that a group element fixes a set is 1/n.
2) Under this action, the generalized conjugacy class graph of the dihedral group has n isolated vertices and is empty.
3) The orbit graph vertices correspond to the n orbits of sizes 1 or 2n, with an edge between vertices if their corresponding orbits are conjugate.
The document presents some fixed point results for maps satisfying certain contractive conditions in ordered G-metric spaces. It begins with an introduction discussing the background and history of fixed point theory. It then provides definitions related to G-metric spaces. The main result is a fixed point theorem for maps satisfying a rational-type contractive condition in a complete ordered G-metric space. It proves that such maps have at least one fixed point and sequences converge to a fixed point. If there are two distinct fixed points, their G-metric is bounded below by 1/2.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) th...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document discusses group theory and its applications in physics. It begins by introducing symmetry groups that are important in physics, including translations, rotations, and Lorentz transformations. It then discusses the use of group theory in formulating fundamental forces and the Standard Model of particle physics. The document provides definitions of group theory concepts like groups, operations, identity, and inverse. It explains how group theory provides a mathematical framework for describing physical symmetries.
1) The document presents a new argument for proving that closed timelike curves cannot exist in physical spacetime.
2) It introduces spacetime as a four-dimensional manifold with a Lorentzian metric and performs an "ADM 3+1 split" to divide spacetime into spatial and temporal components.
3) The argument then defines what would constitute a "closed timelike curve" and proves that the coordinate shift introduced in the 3+1 split would eliminate any such curves, making spacetime globally hyperbolic without closed timelike curves.
A Fixed Point Theorem Using Common Property (E. A.) In PM Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses the probability that an element of a metacyclic 3-group of negative type fixes a set and its orbit graph. It begins by providing background on commutativity degree, metacyclic p-groups, and basic graph theory concepts. Previous related work calculating the probability that an element fixes a set is summarized. The document then presents the main results, which are computing the probability that an element of a metacyclic 3-group of negative type fixes a set, and applying this to construct the orbit graph.
D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and...SEENET-MTP
This document summarizes a presentation given at the 2013 Balkan Workshop in Vrnjacka Banja, Serbia on using the "reverse engineering method" (REM) to model cosmology. The presentation reviewed REM and how it can be used to determine scalar field potentials from a given scale factor evolution. Computer programs for numerically and graphically processing REM with different cosmologies were discussed. Examples presented included regular and tachyonic potentials, and cosmology with non-minimally coupled scalar fields and f(R) gravity. Specific examples plotted potentials and scale factors for exponential and linear expansion universes. The presentation concluded with references for further reading on REM and its applications in cosmology.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
A common fixed point theorem for six mappings in g banach space with weak-com...Alexander Decker
The document presents a theorem proving the existence of a common fixed point for six mappings (P, Q, A, B, S, T) in a G-Banach space under certain conditions. Some key points:
- Defines concepts of G-Banach space, which generalizes the ordinary Banach space.
- States a theorem that proves four mappings have a unique common fixed point in a G-Banach space if they satisfy certain contraction conditions.
- The main result extends this to prove that six mappings (P, Q, A, B, S, T) have a unique common fixed point in a G-Banach space if they satisfy new generalized contraction conditions and are weakly compatible.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
This document presents a new generalized Lindley distribution (NGLD). The NGLD contains the gamma, exponential, and Lindley distributions as special cases. Statistical properties of the NGLD like the hazard function, moments, and moment generating function are derived. Maximum likelihood estimation is discussed to estimate the parameters of the NGLD. Two real data sets are analyzed to illustrate the usefulness of the new distribution.
Fixed point theorem in fuzzy metric space with e.a propertyAlexander Decker
This document presents a theorem proving the existence of a common fixed point for four self-mappings (A, B, S, T) on a fuzzy metric space under certain conditions. Specifically:
1) The mappings satisfy containment and weakly compatible conditions, as well as property (E.A).
2) There exists a contractive inequality relating the mappings.
3) The range of one mapping (T) is a closed subspace.
Under these assumptions, the theorem proves the mappings have a unique common fixed point. The proof constructs sequences to show the mappings share a single fixed point. References at the end provide background on fuzzy metric spaces and related fixed point results.
A weaker version of continuity and a common fixed point theoremAlexander Decker
This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.
Second Order Parallel Tensors and Ricci Solitons in S-space forminventionjournals
In this paper, we prove that a symmetric parallel second order covariant tensor in (2m+s)- dimensional S-space form is a constant multiple of the associated metric tensor. Then we apply this result to study Ricci solitons for S-space form and Sasakian space form of dimension 3
Causal set theory is an approach to quantum gravity that represents spacetime as a locally finite partially ordered set of points with causal relations. It is a minimalist approach that does not assume an underlying spacetime continuum. There are two main methods to reconstruct a manifold from a causal set: 1) extracting manifold properties like dimension from causal sets that can be embedded in a manifold, and 2) sprinkling points randomly into an existing manifold to produce an embedded causal set. To study dynamics, an action must be defined on causal sets that reproduces the Einstein-Hilbert action in the continuum limit. Several proposals have been made to define nonlocal operators on causal sets that approach the d'Alembertian operator in the limit. Overall causal set
This document provides an introduction to matrix algebra concepts needed for a systems biology course, including matrices, determinants, inverses, eigenvalues and eigenvectors. It discusses how matrices first arose from solving systems of linear equations and how the modern approach is to transform linear systems into matrix equations. Key concepts introduced include matrix operations like addition and multiplication, properties of the matrix multiplication like it being non-commutative, and how determinants are important for solving linear systems. The document also notes how complex numbers allow solving equations that have no real number solutions.
The document discusses the Reconstruction Conjecture in graph theory, which states that any graph of order 3 or more can be uniquely reconstructed (up to isomorphism) from its vertex-deleted subgraphs. The conjecture was originally posed by Paul Kelly and Stanislaw Ulam in the 1950s. Since then, significant progress has been made in determining classes of graphs that are reconstructible, such as trees and regular graphs. While a full proof remains elusive, mathematicians have identified certain graph properties that can be determined from a graph's vertex-deleted subgraphs alone. Approaches involving edge reconstruction have also been explored as an alternative way to approach the conjecture.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) th...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
The document discusses group theory and its applications in physics. It begins by introducing symmetry groups that are important in physics, including translations, rotations, and Lorentz transformations. It then discusses the use of group theory in formulating fundamental forces and the Standard Model of particle physics. The document provides definitions of group theory concepts like groups, operations, identity, and inverse. It explains how group theory provides a mathematical framework for describing physical symmetries.
1) The document presents a new argument for proving that closed timelike curves cannot exist in physical spacetime.
2) It introduces spacetime as a four-dimensional manifold with a Lorentzian metric and performs an "ADM 3+1 split" to divide spacetime into spatial and temporal components.
3) The argument then defines what would constitute a "closed timelike curve" and proves that the coordinate shift introduced in the 3+1 split would eliminate any such curves, making spacetime globally hyperbolic without closed timelike curves.
A Fixed Point Theorem Using Common Property (E. A.) In PM Spacesinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses the probability that an element of a metacyclic 3-group of negative type fixes a set and its orbit graph. It begins by providing background on commutativity degree, metacyclic p-groups, and basic graph theory concepts. Previous related work calculating the probability that an element fixes a set is summarized. The document then presents the main results, which are computing the probability that an element of a metacyclic 3-group of negative type fixes a set, and applying this to construct the orbit graph.
D. Vulcanov, REM — the Shape of Potentials for f(R) Theories in Cosmology and...SEENET-MTP
This document summarizes a presentation given at the 2013 Balkan Workshop in Vrnjacka Banja, Serbia on using the "reverse engineering method" (REM) to model cosmology. The presentation reviewed REM and how it can be used to determine scalar field potentials from a given scale factor evolution. Computer programs for numerically and graphically processing REM with different cosmologies were discussed. Examples presented included regular and tachyonic potentials, and cosmology with non-minimally coupled scalar fields and f(R) gravity. Specific examples plotted potentials and scale factors for exponential and linear expansion universes. The presentation concluded with references for further reading on REM and its applications in cosmology.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
A common fixed point theorem for six mappings in g banach space with weak-com...Alexander Decker
The document presents a theorem proving the existence of a common fixed point for six mappings (P, Q, A, B, S, T) in a G-Banach space under certain conditions. Some key points:
- Defines concepts of G-Banach space, which generalizes the ordinary Banach space.
- States a theorem that proves four mappings have a unique common fixed point in a G-Banach space if they satisfy certain contraction conditions.
- The main result extends this to prove that six mappings (P, Q, A, B, S, T) have a unique common fixed point in a G-Banach space if they satisfy new generalized contraction conditions and are weakly compatible.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
This document presents a new generalized Lindley distribution (NGLD). The NGLD contains the gamma, exponential, and Lindley distributions as special cases. Statistical properties of the NGLD like the hazard function, moments, and moment generating function are derived. Maximum likelihood estimation is discussed to estimate the parameters of the NGLD. Two real data sets are analyzed to illustrate the usefulness of the new distribution.
Fixed point theorem in fuzzy metric space with e.a propertyAlexander Decker
This document presents a theorem proving the existence of a common fixed point for four self-mappings (A, B, S, T) on a fuzzy metric space under certain conditions. Specifically:
1) The mappings satisfy containment and weakly compatible conditions, as well as property (E.A).
2) There exists a contractive inequality relating the mappings.
3) The range of one mapping (T) is a closed subspace.
Under these assumptions, the theorem proves the mappings have a unique common fixed point. The proof constructs sequences to show the mappings share a single fixed point. References at the end provide background on fuzzy metric spaces and related fixed point results.
A weaker version of continuity and a common fixed point theoremAlexander Decker
This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.
Second Order Parallel Tensors and Ricci Solitons in S-space forminventionjournals
In this paper, we prove that a symmetric parallel second order covariant tensor in (2m+s)- dimensional S-space form is a constant multiple of the associated metric tensor. Then we apply this result to study Ricci solitons for S-space form and Sasakian space form of dimension 3
Causal set theory is an approach to quantum gravity that represents spacetime as a locally finite partially ordered set of points with causal relations. It is a minimalist approach that does not assume an underlying spacetime continuum. There are two main methods to reconstruct a manifold from a causal set: 1) extracting manifold properties like dimension from causal sets that can be embedded in a manifold, and 2) sprinkling points randomly into an existing manifold to produce an embedded causal set. To study dynamics, an action must be defined on causal sets that reproduces the Einstein-Hilbert action in the continuum limit. Several proposals have been made to define nonlocal operators on causal sets that approach the d'Alembertian operator in the limit. Overall causal set
This document provides an introduction to matrix algebra concepts needed for a systems biology course, including matrices, determinants, inverses, eigenvalues and eigenvectors. It discusses how matrices first arose from solving systems of linear equations and how the modern approach is to transform linear systems into matrix equations. Key concepts introduced include matrix operations like addition and multiplication, properties of the matrix multiplication like it being non-commutative, and how determinants are important for solving linear systems. The document also notes how complex numbers allow solving equations that have no real number solutions.
The document discusses the Reconstruction Conjecture in graph theory, which states that any graph of order 3 or more can be uniquely reconstructed (up to isomorphism) from its vertex-deleted subgraphs. The conjecture was originally posed by Paul Kelly and Stanislaw Ulam in the 1950s. Since then, significant progress has been made in determining classes of graphs that are reconstructible, such as trees and regular graphs. While a full proof remains elusive, mathematicians have identified certain graph properties that can be determined from a graph's vertex-deleted subgraphs alone. Approaches involving edge reconstruction have also been explored as an alternative way to approach the conjecture.
The mathematical and philosophical concept of vectorGeorge Mpantes
What is behind the physical phenomenon of the velocity; of the force; there is the mathematical concept of the vector. This is a new concept, since force has direction, sense, and magnitude, and we accept the physical principle that the forces exerted on a body can be added to the rule of the parallelogram. This is the first axiom of Newton. Newton essentially requires that the power is a " vectorial " size , without writing clearly , and Galileo that applies the principle of the independence of forces .
Cyclic groups are common in everyday life and appear in patterns found in nature, geometry, and music. The document discusses several applications of cyclic groups, including:
1) Number theory, where cyclic groups are used in the division algorithm and Chinese Remainder Theorem.
2) Bell ringing methods, which form cyclic groups by permuting the order of bells rung.
3) Music, where octaves form a cyclic group through rotational symmetry.
This document presents a study of five dimensional string cosmological models with bulk viscosity in general relativity. It investigates three cases of the field equations and determines solutions for each. For case I, the solution reduces the model to Minkowski spacetime. For case II, the solution yields a model that expands from a big bang and stops expanding at a finite time, with the volume increasing over time. The energy density is only due to bulk viscosity and Lyra geometry. For case III, three specific solutions are presented: one that reduces to a flat 5D spacetime; one that contracts/expands depending on constants; and one that follows power law inflation. Physical properties like expansion, shear, and density are also analyzed for each case.
Introduction to graph theory (All chapter)sobia1122
1) Graph theory can be used to model and solve problems in many fields like physics, chemistry, computer science, and more. Certain problems can be formulated as problems in graph theory.
2) Graph theory has developed from puzzles and practical problems, like the Königsberg bridge problem inspiring Eulerian graph theory and the "Around the World" game inspiring Hamiltonian graph theory.
3) Connectivity in graphs measures how connected a graph is, and how the removal of vertices or edges affects connectivity. Connectivity is important for applications like communication networks.
This document summarizes a presentation about the history and applications of quaternions. It begins with quotes from Jean Piaget about the importance of mathematics in understanding knowledge development. It then provides background on quaternions, including their creation in 1843, rise and fall in popularity, and recent resurgence. Applications discussed include aerospace guidance, computer graphics, signal processing, bio-logging, and modeling molecular docking. The document argues quaternions are well-suited for tasks involving rotation, orientation, viewpoint shifting, and control processes.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
Trigonometry developed from studying right triangles in ancient Egypt and Babylon, with early work done by Hipparchus and Ptolemy. It was further advanced by Indian, Islamic, and Chinese mathematicians. Key developments include Madhava's sine table, al-Khwarizmi's sine and cosine tables, and Shen Kuo and Guo Shoujing's work in spherical trigonometry. European mathematicians like Regiomontanus, Rheticus, and Euler established trigonometry as a distinct field and defined functions analytically. Trigonometry is now used in many areas beyond triangle calculations.
This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.
Quantum Geometry: A reunion of math and physicsRafa Spoladore
Caltech's professor Anton Kapustin "describes the relationship between mathematics and physics, mathematicians and physicists, and so on. He focuses on the noncommutative character of algebras of observables in quantum mechanics." via http://motls.blogspot.com.br/2014/11/anton-kapustin-quantum-geometry-reunion.html
Talk at 2013 WSC, ISI Conference in Hong Kong, August 26, 2013Christian Robert
Those are the slides for my conference talk at 2013 WSC, in the "Jacob Bernoulli's "Ars Conjectandi" and the emergence of probability" session organised by Adam Jakubowski
A Survey On The Weierstrass Approximation TheoremMichele Thomas
The document provides a survey of the Weierstrass approximation theorem and related results in approximation theory over the past century. It begins with an introduction to the theorem proved by Weierstrass in 1885, which showed that continuous functions can be uniformly approximated by polynomials on compact intervals. The document then discusses several improvements, generalizations, and ramifications of the theorem developed in subsequent decades, including results on approximating functions by trigonometric polynomials, Bernstein polynomials, and rational functions. It concludes by mentioning several influential theorems in approximation theory from the 20th century, such as Stone's theorem on uniform approximation by collections of functions.
Euclidean Equivalent of Minkowski’s Space-Time Theory and the Corresponding M...Premier Publishers
This document communicates some of the main results obtained from a theoretical work which performs a type of Wick’s rotation, where Lorentz’s group is connected in the resulting Euclidean metric, and as a consequence models the particles with rest mass as photons in a compacted additional dimension (for a photon of the ordinary 3-dimensional space, they do not go through the 4-dimension due to null angle in this dimension). Among its reported results are new explanations, much more elegant than the current ones, of the material waves of De Broglie, the uncertainty principle, the dilation of the proper time, the Higgs field, the existence of the antiparticles and specifically of the electron-positron annihilation, among others. It also leaves open the possibility of unifying at least three of the fundamental forces and the different types of particles under a single model of photon and compact dimension. Additionally, two experimental results are proposed that can only currently be explained by this theory.
This document provides an outline and introduction to a course on mathematics for artificial intelligence, with a focus on vector spaces and linear algebra. It discusses:
1. A brief history of linear algebra, from ancient Babylonians solving systems of equations to modern definitions of matrices.
2. The definition of a vector space as a set that can be added and multiplied by elements of a field, with properties like closure under addition and scalar multiplication.
3. Examples of using matrices and vectors to model systems of linear equations and probabilities of transitions between web pages.
4. The importance of linear algebra concepts like bases, dimensions, and eigenvectors/eigenvalues for machine learning applications involving feature vectors and least squares error.
This report summarizes research on the motion of particles on curves. It was found that:
1) The center of mass of 3 points on an ellipse that divide its perimeter evenly traces out a smaller ellipse of the same shape.
2) The maximum product of distances between 4 particles on a rectangle occurs when particles are at the corners for small rectangles, but 2 particles move off the corners for larger rectangles.
3) The center of mass of n points on a square that divide its perimeter evenly traces out a smaller square n times for odd n, and remains fixed at the center for even n.
This document discusses dimensions in mathematics and physics. It begins by explaining one-dimensional, two-dimensional, and three-dimensional objects like lines, squares, cubes, and tesseracts. It then discusses higher dimensions posited by theories like string theory and M-theory. Key definitions of dimension discussed include topological dimension, Hausdorff dimension, and covering dimension. In physics, it discusses the three spatial dimensions and time as the fourth dimension, as well as theories proposing additional curled up dimensions to explain phenomena.
This document summarizes research on simplifying calculations of scattering amplitudes, especially for tree-level amplitudes. It introduces the spinor-helicity formalism for writing compact expressions for amplitudes. It then discusses color decomposition in SU(N) gauge theory and the Yang-Mills Lagrangian. Specific techniques explored include BCFW recursion relations, an inductive proof of the Parke-Taylor formula, the 4-graviton amplitude and KLT relations, multi-leg shifts, and the MHV vertex expansion. The goal is to develop recursion techniques that vastly simplify calculations compared to traditional Feynman diagrams.
This document provides an overview of the history of mathematics, outlining some of the key developments in different time periods and civilizations. It discusses the origins and early developments of mathematics in ancient Babylon, Egypt, India, Greece, and beyond. Some of the important concepts covered include early algebra and geometry developed by civilizations like the Babylonians, as well as later advances in areas like calculus, trigonometry, and abstract algebra made from the 16th century onward by mathematicians such as Newton, Leibniz, Descartes, and others. It also profiles several influential mathematicians and their contributions to fields like algebra, geometry, and number theory.
Similar to Quaternions, Alexander Armstrong, Harold Baker, Owen Williams (20)
3. Chapter 1
Introduction
Quaternions are a Victorian mathematical invention that is used in modern com-
puter gaming applications today. This report provides an overview of quater-
nions, giving a brief history, the theory and describing their uses. The history
section explains how they were discovered and how they were used over the next
170 years. The theory section takes a look at the basic properties and identi-
ties of quaternions. The final section is an exploration of their application in
mathematics, physics and computer graphics.
2
4. Chapter 2
History
When it finally came time for quaternions to enter the world they did so via a
blunted pen knife. In 1843, while walking along the Royal Canal in Dublin, Sir
William Rowan Hamilton suddenly arrived at the concept. He rushed over to
the Broom Bridge and inscribed on it the famous equation:
i2
= k2
= j2
= ijk = 1
Having worked towards this discovery for the past fifteen years, Hamilton had
made a significant step forwards for mathematics at the time.
He was trying to extend the complex numbers in two dimensions to three dimen-
sions. This was known as his theory of triplets. By abandoning commutativity
and adding a fourth dimension Hamilton had managed to find a way round
the theory of triplets and discovered quaternions. Hamilton was already a very
sucessful mathematician. Quaternions were the topic to which he would dedi-
cate the next and last twenty-two years of his life. [5]
While Hamilton is known as the creator and champion of quaternions he was
not the only one at the time to have made steps in this field. Gauss, in papers
unpublished until 1900, showed he had discovered quaternions as early as 1819.
Rodruigues’ work on rotation, published in 1840, was a significant precursor. [1]
Hamilton went on to publish 109 of the 151 papers published on quaternions by
his death in 1865. He was working on his book, Elements of Quaternions, until
a few days before he died. [5]
Quaternions found their way into a scientific context via James Clerk Maxwell
when he used the vector part of quaternions in his Treatise on Electricity and
Magnetism. He was influenced by Peter Guthrie Tait who expressed the theo-
rems of Gauss, Stokes and Green in quaternion notation. [10]
3
5. Following the invention of vector algebra by Josiah Willard Gibbs, quaternions
fell into disuse. Gibbs recognised that it was not necessary to introduce the
imaginary part of quaternions when applying them to the world of physics.
In 1891, in the journal Nature, there was a vigorous and prolonged argument
between Gibbs and Tait with Gibbs preferring the minimalist approach in not
including the imaginary part of quaternions in his vector algebra. Tait, however,
was a staunch supporter of the elegance and notational superiority of quater-
nions and so disagreed with stripping them down for the parts needed in physics.
He commented ’Prof Gibbs must be ranked as one of the retarders of quaternion
progress, in view of his pamphlet on Vector Analysis, a sort of hermaphrodite
monster, compounded of the notations of Hamilton and Grassman.’ [4]
After that, quaternions took a back seat only finding application in specific ar-
eas of physics such as quantum mechanics and quantum field theory. Writing
in 1966, however, Bork predicted ’Perhaps we shall again see a great resurgence
in the quaternion.’ [4]
He was right. Today quaternions have found themselves a new home in comput-
ing. Any video game that requires a player to look around at objects requires a
graphics card to rotate continuously objects in three dimensions for a realistic
experience. The SLERP algorithm, discussed later in this report, uses quater-
nions to speed up graphical output. One hundred and seventy years later we
can still feel the e↵ects of Hamilton’s discovery in 1843.
4
6. Chapter 3
Theory
Hamilton had already showed that complex numbers form an algebra of couples.
He then began exploring the idea of a complex number with two imaginary parts.
This was his Theory of Triplets that he was unable to make work. However when
he discovered quaternions, from the latin quaternio, meaning four, he realised
a third imaginary component was necessary.
This is a quaternion. [8]
w = a + bi + cj + dk
The addition of quaternions is straightforward, simply adding the coe cients of
i,j and k:
w + z = (aw + az) + (bw + bz)i + (cw + cz)j + (dw + dz)k
The method for multiplication is as follows:
w ⇤ z = (awaz bwbz cwcz dwdz)
+(awbz + awbz + cwdz cwdz)i
+(awcz + awcz bwdz + bwdz)j
+(awdz + awdz + bwcz bwcz)
The conjugate of a quaternion is :
¯w = a bi cj dk
5
7. This is the norm.
||w|| =
p
a2 + b2 + c2 + d2
[11]
Identities are
k = ( 1)k = ijk(k) = ij(k2
) = ij( 1) = ij
j = ( 1)j = (ijk)j = ik(j2
) = ik( 1) = ik
i = (1)i = ( ijk)i = (i2
)( jk) = ( 1)( jk) = jk
The Cayley table below, along with these identities shows that multiplication
of quaternions is non-commutative.
Q ⇥ Q 1 -1 i -i j -j k -k
1 1 -1 i -i j -j k -k
-1 -1 1 -i i -j j -k k
i i -i -1 1 k -k -j j
-i -i i 1 1 -k k j -j
j j -j -k k 1 1 i -i
-j -j j k -k 1 1 -i i
k k -k j -j -i i 1 1
-k -k k -j j i -i 1 1
Table 3.1: Cayley Table
6
8. Chapter 4
Applications
4.1 Group Theory and Number Theory
In group theory, a group is a set combined with a binary operation for which
four ’group axioms’ apply. The set of quaternions is a subgroup of order 8,
defined as {±1, ±i, ±j, ±k}. A ring is a set combined with 2 binary operations,
for which 8 ’ring axioms’ apply. Hurwitz introduced the ring of integral quater-
nions [6], which is a subring of H. H is the set of all quaternions, labelled H
after Hamilton, where for a+bi+cj +dk, either each of a, b, c, d, 2 Z, or each of
a, b, c, d is congruent to modulo Z. This was used to prove Lagrange’s theorem,
that every positive integer is the sum of at most 4 squares. [6]
4.2 Applications in Physics
In physics, the Dirac equation is a wave equation that was derived by British
physicist Paul Dirac in 1928. Objects related to quaternions arise from the
solution of this equation in particle physics. The non-commutativity of quater-
nions is essential here. The Lorentz Transforms are coordinate transformations
between two coordinate frames that move at constant velocity relative to each
other. Quaternions can be used to express this, making them useful for work
on special and general relativity. Quaternion properties in rotations make them
useful in scattering experiments in crystallography, used to determine the ar-
rangement of atoms in crystalline solids. [6] [13]
4.3 Heisenberg Uncertainty Principle
Quaternions can be used in the Heisenberg Uncertainty Principle. This is refers
to any variety of mathematical inequalities, in which certain pairs of physical
7
9. properties of a particle can be known. [7] These pairs are known as complimen-
tary variables. The principle was introduced by the German physicist Werner
Heisenberg in 1927. It states that the more precisely measured one aspect of
the complimentary variables is, the less precisely the other is determined, and
vice versa. For example, if we know a particle’s position and momentum, then
the more precisely the position, the less precisely the momentum is known. In
the case of quaternions, the Uncertainty Principle states that PQ QP = h,
where P and Q are linear operators that represent momentum and position
respectively, and h is Planck’s constant, which is non-zero. [13] This is only
possible because quaternions are non-commutative. If they were commutative,
then PQ QP would be equal to zero.
4.4 3-D Animation And Linear Interpolation
A significant application of quaternions is their use in computer programming.
In particular, they have made an impact in three-dimensional (3D) animation
due to their usefulness in storing and manipulating rotations and orientations
in 3D space. Euler angles may appear suited to this purpose, being three-
dimensional vectors whose entries are either x, y and z coordinates for orien-
tations or the rotation about the x, y and z axis for rotations. However, this
approach poses problems, specifically their proneness to gimbal locking [2] and
their lack of suitability to linear interpolation between orientations.
If Euler angles are used to express certain rotations, when a single axis is rotated
by 90 degrees, say the y-axis, the x- and z-axes are forced into a parallel config-
uration, resulting in a loss of one degree of freedom. This is what is meant by
gimbal locking (see figure 4.1). In particular, it is an issue most problematic in
three-dimensional computer game design where it can restrict the player’s move-
ment to a two-dimensional plane. It can also be inconvenient in animation. [3]
One solution to this issue is the introduction of an additional degree of freedom;
or equivalently, raising the system from three to four dimensions by using a
vector representation of quaternions. Once this is done, if two axes are forced
into a parallel configuration, the system still maintains three degrees of freedom.
With regards to linear interpolation between orientations, quaternions are again
a good solution. This kind of interpolation in 3D graphics refers to an algorithm
which takes two three-dimensional positions as inputs and results in a smoothly
animated rotation between them. Finding and animating a smooth path from
one point to another with Euler angles is considerably less e cient than it is
with quaternions. Cheap Euler angle-based interpolation algorithms result in
non-optimal paths to the final destination and even if more e cient algorithms
are used, the system can still become gimbal locked, resulting in a path which
is also sub-optimal as the object must move in various directions to free itself of
gimbal lock, resulting in seemingly erratic movement. In comparison, interpolat-
8
10. Figure 4.1: A 3 dimensional gimbal in its default state (left) and a gimbal locked
state (right). [12]
ing between quaternion orientations is much more e cient with a comparatively
cheap formula known as Spherical Linear Interpolation (SLERP). SLERP refers
to the the linear interpolation between two points along the surface of a sphere,
rather than as a direct line through three-dimensional space. The reason we
are interested in this is that, much like the set of two-dimensional unit vectors
makes up the edge of a two-dimensional unit circle, the set of four-dimensional
unit quaternions makes up the surface of a four-dimensional unit hypersphere.
Visualising this is di cult, but that does not matter too much as spherical in-
terpolation in four dimensions works just as it does in three.
Let A, B be points on a sphere and let 0 u 1. Then, where ⌦ is the angle
subtended by the arc between A and B,
SLERP(A, B, u) = A
sin(1 u)⌦
sin(⌦)
+ B
sin(⌦u)
sinh(⌦)
[9]
Using u as a parameter to determine how far along the arc we want to be, u = 0
being our start point, A, and u = 1 being our end point, B, we can generate
frames of the animation to match the frame rate of the video or game being
played by incrementally increasing u. This results in perfectly smooth motion.
So, when considering 3D animation, we can take a pure quaternion A (a quater-
nion of the form 0+xi+yj +zk, this being equivalent to the three-dimensional
vector (x,y,z)) and a pure quaternion B and, using SLERP, linearly interpolate
between the two points producing pure quaternions at each step of the way. In
this way, the result will always be equivalent to a three-dimensional position.
9
11. While it may seem counterintuitive to apply the four-dimensional quaternions
to a problem occurring strictly in three dimensions, these methods have simpli-
fied animation considerably and advanced the field of computer graphics.
10
12. Bibliography
[1] S Altmann. Mathematics Magazine, 62:p291–308, 1989.
[2] P Banerjee and D Zetu. Virtual Machine. John Wiley and Sons, Inc., 2001.
[3] N Bobic. Rotating objects using quaternions.
http://www.gamasutra.com/view/feature/131686/rotating objects using quaternions.php,
1998. Last accessed 2nd November 2016.
[4] A Bork. Vectors versus quaternions - the letters in nature. American
Journal of Physics, 34:p202, 1966.
[5] M Crowe. A History of Vector Analysis. Dover Publications, Inc., 1987.
[6] R Eimerl. The Quaternions and their Applications. University of Puget
Sound, 2015.
[7] A Jha. What is heisenberg’s uncertainty principle?
https://www.theguardian.com/science/2013/nov/10/what-is-heisenbergs-
uncertainty-principle, 2013. Last accessed 1st November 2016.
[8] P Kelland and P. G. Tait. Introduction to Quaternions with Numerous
Examples. Macmillan and co., 1873.
[9] K Shoemake. Development of vector analysis from quaternions. SIG-
GRAPH ’85 Proceedings of the 12th annual conference on Computer graph-
ics and interactive techniques, pages p245–254, 1985.
[10] R Stephenson. Development of vector analysis from quaternions. American
Journal of Physics, 34:p194, 1966.
[11] E.W. Weisstein. Quaternion. http://mathworld.wolfram.com/Quaternion.html,
2004. Last accessed 1st November 2016.
[12] Wikipedia. Gimbal lock. https://en.wikipedia.org/wiki/Gimbal lock, 2016.
Last accessed 2nd November 2016.
[13] Zipcon. Applications of quaternions.
http://www.zipcon.net/ swhite/docs/math/quaternions/applications.html,
2010. Last accessed 1st November 2016.
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