Mathematics › Applied Mathematics

Algebraic and Geometric Analysis

Works Count: 32372

Wikipedia

Description

This cluster of papers explores advances in quaternionic analysis, geometric algebra, slice regular functions, dark matter, spinor fields, functional calculus, Clifford analysis, octonions, Dirac operators, and hypercomplex numbers.

Keywords

Quaternionic Analysis; Geometric Algebra; Slice Regular Functions; Dark Matter; Spinor Fields; Functional Calculus; Clifford Analysis; Octonions; Dirac Operators; Hypercomplex Numbers

Analysis on Symmetric Cones

1994-12-29

Jacques Faraut , Adam Korányi

Abstract The present book is the first to treat analysis on symmetric cones in a systematic way. It starts by describing, with the simplest available proofs, the Jordan algebra approach … Abstract The present book is the first to treat analysis on symmetric cones in a systematic way. It starts by describing, with the simplest available proofs, the Jordan algebra approach to the geometric and algebraic foundations of the theory due to M. Koecher and his school. In subsequent parts it discusses harmonic analysis and special functions associated to symmetric cones; it also tries these results together with the study of holomorphic functions on bounded symmetric domains of tube type. It contains a number of new results and new proofs of old results.

Clifford Algebra to Geometric Calculus

1984-01-01

David Hestenes , Garret Sobczyk

Geometric Theory of Functions of a Complex Variable

1969-12-31

G. Goluzin

An introduction to theoretical kinematics

1990-10-01

J. Michael McCarthy

Introduction to Theoretical Kinematics provides a uniform presentation of the mathematical foundations required for studying the movement of a kinematic chain that makes up robot arms, mechanical hands, walking machines, … Introduction to Theoretical Kinematics provides a uniform presentation of the mathematical foundations required for studying the movement of a kinematic chain that makes up robot arms, mechanical hands, walking machines, and similar mechanisms. It is a concise and readable introduction that takes a more modern approach than other kinematics texts and introduces several useful derivations that are new to the literature. The author employees a unique format, highlighting the similarity of the mathematical results for planar, spherical, and spatial cases by studying them all in each chapter rather than as separate topics. For the first time, he applies to kinematic theory two tools of modern mathematics--the theory of multivectors and the theory of Clifford algebras--that serve to clarify the seemingly arbitrary nature of the construction of screws and dual quaternions. The first two chapters formulate the matrices that represent planar, spherical, and spatial displacements and examine a continuous set of displacements which define a continuous movement of a body, introducing the operator. Chapter 3 focuses on the tangent operators of spatial motion as they are reassembled into six-dimensional vectors or screws, placing these in the modern setting of multivector algebra. Clifford algebras are used in chapter 4 to unify the construction of various hypercomplex numbers. Chapter 5 presents the elementary formulas that compute the degrees of freedom, or mobility, of kinematic chains, and chapter 6 defines the structure equations of these chains in terms of matrix transformations. The last chapter computes the quaternion form ofthe structure equations for ten specific mechanisms. These equations define parameterized manifolds in the Clifford algebras, or image spaces, associated with planar, spherical, and spatial displacements. McCarthy reveals a particularly interesting result by showing that these parameters can be mathematically manipulated to yield hyperboloids or intersections of hyperboloids.

Harmonic Analysis on Semi-Simple Lie Groups II

1972-01-01

Garth Warner

Groups and Geometric Analysis

2000-10-03

Sigurđur Helgason

Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces … Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces Appendix Some details Bibliography Symbols frequently used Index Errata.

Classical Fourier Analysis

2014-01-01

Loukas Grafakos

View PDF Chat

Composition Operators on Spaces of Analytic Functions

2019-03-04

Carl C. Cowen , Barbara D. MacCluer

Introduction Analysis Background A Menagerie of Spaces Some Theorems on Integration Geometric Function Theory in the Disk Iteration of Functions in the Disk The Automorphisms of the Ball Julia-Caratheodory Theory … Introduction Analysis Background A Menagerie of Spaces Some Theorems on Integration Geometric Function Theory in the Disk Iteration of Functions in the Disk The Automorphisms of the Ball Julia-Caratheodory Theory in the Ball Norms Boundedness in Classical Spaces on the Disk Compactness and Essential Norms in Classical Spaces on the Disk Hilbert-Schmidt Operators Composition Operators with Closed Range Boundedness on Hp (BN) Small Spaces Compactness on Small Spaces Boundedness on Small Spaces Large Spaces Boundedness on Large Spaces Compactness on Large Spaces Hilbert-Schmidt Operators Special Results for Several Variables Compactness Revisited Wogen's Theorem Spectral Properties Introduction Invertible Operators on the Classical Spaces on the Disk Invertible Operators on the Classical Spaces on the Ball Spectra of Compact Composition Operators Spectra: Boundary Fixed Point, j'(a)

Formulas and Theorems for the Special Functions of Mathematical Physics

1966-01-01

Wilhelm Magnus , Fritz Oberhettinger , Raj Pal Soni

Space-Time Algebra

2015-01-01

David Hestenes

Linear Operator Theory in Engineering and Science

1982-01-01

Arch W. Naylor , George R. Sell

On Quaternions and Octonions

2003-01-23

John H. Conway , Derek A. Smith

This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading … This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less f

The Index of Elliptic Operators: III

1968-05-01

Michael Atiyah , I. M. Singer

Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

1977-09-01

Robert S. Strichartz

Differential Equations--Geometric Theory

1964-04-01

Harry Pollard , Solomon Lefschetz

Rapid solution of integral equations of classical potential theory

1985-09-01

Vladimir Rokhlin

Fourier Analysis on Groups

2006-04-26

Walter Rudin , L. Bers , R. Courant +2 more

Irreducible Unitary Representations of the Lorentz Group

1947-07-01

V. Bargmann

Spin-<i>s</i> Spherical Harmonics and ð

1967-11-01

Joshua N. Goldberg , Alan Macfarlane , Ezra T. Newman +2 more

Recent work on the Bondi-Metzner-Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related … Recent work on the Bondi-Metzner-Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular-momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

�ber Modifikationen und exzeptionelle analytische Mengen

1962-08-01

Hans Grauert

Quaternions and matrices of quaternions

1997-01-01

Fuzhen Zhang

Bounded Analytic Functions.

1982-08-01

Peter W. Jones , John B. Garnett

Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals. Preliminaries.- Hp Spaces.- Conjugate Functions.- Some Extremal Problems.- Some Uniform Algebra.- Bounded Mean Oscillation.- Interpolating Sequences.- The Corona Construction.- Douglas Algebras.- Interpolating Sequences and Maximal Ideals.

View PDF Chat

Symplectic Geometry and Analytical Mechanics

1987-01-01

Paulette Libermann , Charles-Michel Marle

Analysis, Manifolds and Physics

2000-01-01

Yvonne Choquet‐Bruhat , Cécile DeWitt-Morette , A. S. Wightman

Hamiltonian reduction of unconstrained and constrained systems

1988-04-25

Людвиг Дмитриевич Фаддеев , R. Jackiw

Recent Letters and Comments discuss the quantization of self-dual two-dimensional Lagrangeans which describe systems that are not explicitly canonical. We make some remarks on the most efficient method for exhibiting … Recent Letters and Comments discuss the quantization of self-dual two-dimensional Lagrangeans which describe systems that are not explicitly canonical. We make some remarks on the most efficient method for exhibiting the canonical structure.

Clifford Algebras and Spinors

1986-01-01

Pertti Lounesto

The Feynman integral for singular Lagrangians

1969-10-01

Lyudvig Dmitrievich Faddeev

Zero rest-mass fields including gravitation: asymptotic behaviour

1965-02-23

Roger Penrose

A zero rest-mass field of arbitrary spin s determines, at each event in space-time, a set of 2 s principal null directions which are related to the radiative behaviour of … A zero rest-mass field of arbitrary spin s determines, at each event in space-time, a set of 2 s principal null directions which are related to the radiative behaviour of the field. These directions exhibit the characteristic ‘peeling-off' behaviour of Sachs, namely that to order r - k -1 ( k = 0, . . . , 2 s ), 2 s - k of them coincide radially, r being a linear parameter in any advanced or retarded radial direction. This result is obtained in part I for fields of any spin in special relativity, by means of an inductive spinor argument which depends ultimately on the appropriate asymptotic behaviour of a very simple Hertz-type complex scalar potential. Spin ( s - ½) fields are used as potentials for spin s fields, etc. Several examples are given to illustrate this, In particular, the method is used to obtain physically sensible singularity-free waves for each spin which can be of any desired algebraic type. In part II, a general technique is described, for discussing asymptotic properties of fields in curved space-times which is applicable to all asymptotically flat or asymptotically de Sitter space-times. This involves the introduction of ‘points at infinity’ in a consistent way. These points constitute a hypersurface boundary I to a manifold whose interior is conformally identical with the original space-time. Zero rest-mass fields exhibit an essential conformal invariance, so their behaviour at ‘infinity’ can be studied at this hypersurface. Continuity at I for the transformed field implies that the ‘peeling-off’ property holds. Furthermore, if the Einstein empty-space equations hold near I then continuity at I for the transformed gravitational field is a consequence. This leads to generalizations of results due to Bondi and Sachs. The case when the Einstein-Maxwell equations hold near I is also similarly treated here. The hypersurface I is space-like, time-like or null according as the cosmological constant is positive, negative or absent. The technique affords a covariant approach to the definition of radiation fields in general relativity. If I is not null, however, the radiation field concept emerges as necessarily origin dependent. Further applications of the technique are also indicated.

Fine Structure of the Hydrogen Atom by a Microwave Method

1947-08-01

Willis E. Lamb , Robert C. Retherford

View PDF Chat

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949-06-01

S. Minakshisundaram , Åke Pleijel

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S . In terms of local coordinates ( x i ), … Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S . In terms of local coordinates ( x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x 1 , x 2 , … x N ) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

View PDF Chat

Representations of the Rotation and Lorentz Groups and Their Applications.

1966-03-01

Róbert Hermann , I. M. Gel'fand , R. A. Minlos +1 more

Clifford Algebra and Spinor-Valued Functions

1992-01-01

Richard Delanghe , F. Sommen , Vladimı́r Souček

Singular Integral Operators

1986-01-01

Solomon G. Mikhlin , Siegfried Prößdorf

The Octonions

2001-05-17

John C. Baez

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here … The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

Formulas and Theorems for the Special Functions of Mathematical Physics

1967-06-01

Wilhelm Magnus , Fritz Oberhettinger , Raj Pal Soni +1 more

First Page First Page

View PDF Chat

Functions of One Complex Variable

1973-01-01

John B. Conway

The Electromagnetic Shift of Energy Levels

1947-08-15

Hans A. Bethe

B very beautiful experiments, Lamb and Retherford1 have shown that the fine structure of the second quantum state of hydrogen does not agree with the prediction of the Dirac theory. … B very beautiful experiments, Lamb and Retherford1 have shown that the fine structure of the second quantum state of hydrogen does not agree with the prediction of the Dirac theory. The 2s level, which according to Dirac’s theory should coincide with the 2p 1 2 level, is actually higher than the latter by an amount of about 0.033 cm−1 or 1000 megacycles. This discrepancy had long been suspected from spectroscopic measurements.23 However, so far no satisfactory theoretical explanation has been given. Kemble and Present, and Pasternack4 have shown that the shift of the Is level cannot be explained by a nuclear interaction of reasonable magnitude, and Uehling5 has investigated the effect of the “polarization of the vacuum” in the Dirac hole theory, and has found that this effect also is much too small and has, in addition, the wrong sign. Schwinger and Weisskopf, and Oppenheimer have suggested that a possible explanation might be the shift of energy levels by the interaction of the

Clifford modules

1964-07-01

Michael Atiyah , R. Bott , Alexander Shapiro

The octonions

2001-12-21

John C. Baez

View PDF Chat

Handbook of Elliptic Integrals for Engineers and Scientists

1971-01-01

Paul F. Byrd , Morris D. Friedman

Classical Fourier Analysis

2008-01-01

Loukas Grafakos

View PDF Chat

Geometric Invariant Theory

1965-01-01

David Mumford

Additional Properties of Trigonometric Functions

2025-06-25

Nicolas A. Pereyra | CRC Press eBooks

Derivatives of Trigonometric Functions

2025-06-25

Nicolas A. Pereyra | CRC Press eBooks

Scaled-Hyperbolic Clifford Algebras

2025-06-24

Ilwoo Cho | Advances in Applied Clifford Algebras

Representation Structure of the $\mathrm {SL}(2, \mathbb {C})$ Acting in the Hilbert Space of the Quantum Coulomb Field

2025-06-24

Jarosław Wawrzycki , T. Wawrzycki | Acta Physica Polonica B

We give a complete description of the representation of $\mathrm {SL}(2,\mathbb {C})$ acting in the Hilbert space of the quantum Coulomb field and a constructive consistency proof of the axioms … We give a complete description of the representation of $\mathrm {SL}(2,\mathbb {C})$ acting in the Hilbert space of the quantum Coulomb field and a constructive consistency proof of the axioms of the quantum theory of the Coulomb field. Abstract Published by the Jagiellonian University 2025 authors

New Operators and Their Applications in Geometric Function Theory

2025-06-24

Alina Alb Lupaş | Series on concrete and applicable mathematics

Dirihlet problem for the generalized Beltrami equation

2025-06-24

Pelin Ayşe Gökgöz | Istanbul Journal of Mathematics

Real algorithms for special least squares solutions of the complex split quaternion matrix equations $$AX=B$$ and $$AXC=B$$

2025-06-24

Zhihan Zhou , Zixiang Meng , Yuqing Zhang +2 more | Numerical Algorithms

RECOVERING A SURFACE IN ISOTROPIC SPACE USING DUAL MAPPING ACCORDING TO CURVATURE INVARIANTS

2025-06-22

A. Artykbaev , Шерзодбек Исмоилов , Gulnoza Kholmurodova | Journal of Mathematics Mechanics and Computer Science

The problem of recovering a surface according to its curvature is one of the fundamental problems of differential geometry. Problems of recovering surfaces in various spaces by their total or … The problem of recovering a surface according to its curvature is one of the fundamental problems of differential geometry. Problems of recovering surfaces in various spaces by their total or mean curvature have been widely studied in many works.Recovering of a surface by its total curvature is equivalent to solving the Monge-Ampere equation of elliptic type; such problems are solved in special cases. When the right part is given concretely.The Monge-Ampere equation is solved using a dual mapping of isotropic space, in which the dual surface is a transfer surface. Also, some special cases are used to find the surface equation.The connection between dual mean curvature and amalgamatic curvature is studied.The equivalence of the problem of recovering by dual mean and amalgamatic curvature is shown. In particular, the problem of recovering surfaces with total negative constant curvature, the mean curvature of which is a function of one variable, is solved. Furthermore, the problems of the recovering surfaces are solved according to their dual mean curvature, amalgamatic and Casorati curvatures.

Serret-Frenet Formula of Quaternionic Framed Curves

2025-06-20

Zülal Derin , Mehmet Ali̇ Güngör | Mathematical Sciences and Applications E-Notes

In this study, we present an approach by introducing the quaternionic structure of framed curves. Furthermore, we derive Serret-Frenet formulas and give specific results for quaternionic framed curves. Initially, we … In this study, we present an approach by introducing the quaternionic structure of framed curves. Furthermore, we derive Serret-Frenet formulas and give specific results for quaternionic framed curves. Initially, we focus on the moving frame and its curvatures corresponding to the frame $ T,N,B $ along the quaternionic framed base curve in three-dimensional Euclidean space ${\mathbb{R}^3}$. Then, we establish the Serret-Frenet type formulas of quaternionic framed curves. We then generalize these formulas and the definition of quaternionic framed curves to four-dimensional Euclidean space, highlighting the relationship between the curvatures in both 3-dimensional and four-dimensional Euclidean spaces. In addition, the theorems are supported by examples, demonstrating the applicability of the proposed results.

Solutions of the Dual Quaternion Matrix Equation $$AXB = C$$

2025-06-20

Yinping Li , Ying Li , Xiaochen Liu +1 more | Communications on Applied Mathematics and Computation

EXTENSIONS OF THE SCHWARZ PROBLEM FOR THE BELTRAMI OPERATOR IN THE HALF UNIT DISC

2025-06-18

Bahriye Karaca | Journal of Mathematical Sciences

Theorem on vacuum stability

2025-06-17

NULL AUTHOR_ID | Physical review. D/Physical review. D.

On Exterior-Algebraic Quaternions with Application to Maxwell Equations

2025-06-16

Ivano Colombaro | Acta Applicandae Mathematicae

Probabilities with Values in Scaled Hyperbolic Numbers

2025-06-15

Daniel Alpay , Ilwoo Cho | Advances in Applied Clifford Algebras

Abstract In this paper, we introduce a notion of a probabilistic measure which takes values in t -scaled hyperbolic numbers for $$t\in \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> … Abstract In this paper, we introduce a notion of a probabilistic measure which takes values in t -scaled hyperbolic numbers for $$t\in \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> , with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set $$\mathbb {D}_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> of all t -scaled hyperbolic numbers for arbitrarily fixed $$t\in \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> .

On hyperbolicity of spiral-like circular-like domain

2025-06-14

Sanjoy Chatterjee , Golam Mostafa Mondal | Proceedings - Mathematical Sciences

Geometric analysis for the Pontryagin action and boundary terms

2025-06-12

Jasel Berra–Montiel , Iná S. Santos , Alberto Molgado | Classical and Quantum Gravity

Abstract In this article, we analyze the Pontryagin model adopting different geometric-covariant approaches. In particular, we focus on the manner in which boundary conditions must be imposed on the background … Abstract In this article, we analyze the Pontryagin model adopting different geometric-covariant approaches. In particular, we focus on the manner in which boundary conditions must be imposed on the background manifold in order to reproduce an unambiguous theory on the boundary. At a Lagrangian level, we describe the symmetries of the theory and construct the Lagrangian covariant momentum map which allows for an extension of Noether's theorems. Through the multisymplectic analysis we obtain the covariant momentum map associated with the action of the gauge group on the covariant multimomenta phase-space. By performing a space plus time decomposition by means of a foliation of the appropriate bundles, we are able to recover not only the $t$-instantaneous Lagrangian and Hamiltonian of the theory, but also the generator of the gauge transformations. In the polysymplectic framework we perform a Poisson-Hamilton analysis with the help of the De Donder-Weyl Hamiltonian and the Poisson-Gerstenhaber bracket. Remarkably, as long as we consider a background manifold with boundary, in all of these geometric formulations, we are able to recover the so-called differentiability conditions as a straightforward consequence of Noether's theorem.

Paley–Wiener theorems for slice regular functions

2025-06-12

Yanshuai Hao , Pei Dang , Weixiong Mai | Annali di Matematica Pura ed Applicata (1923 -)

View PDF Chat

Planar six-point Feynman integrals for four-dimensional gauge theories

2025-06-12

Samuel Abreu , Pier Francesco Monni , Ben Page +1 more | Journal of High Energy Physics

A bstract We compute all planar two-loop six-point Feynman integrals entering scattering observables in massless gauge theories such as QCD. A central result of this paper is the formulation of … A bstract We compute all planar two-loop six-point Feynman integrals entering scattering observables in massless gauge theories such as QCD. A central result of this paper is the formulation of the differential-equations method under the algebraic constraints stemming from four-dimensional kinematics, which in this case leaves only 8 independent scales. We show that these constraints imply that one must compute topologies with only up to 8 propagators, instead of the expected 9. This leads to the decoupling of entire classes of integrals that do not contribute to scattering amplitudes in four dimensional gauge theories. We construct a pure basis and derive their canonical differential equations, of which we discuss the numerical solution. This work marks an important step towards the calculation of massless 2 → 4 scattering processes at two loops.

View PDF Chat

Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory

2025-06-10

Rainer Kühne

I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechanics, quantum electrodynamics, quantum flavourdynamics, quantum chromodynamics, and general relativity. This examination includes the rotations … I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechanics, quantum electrodynamics, quantum flavourdynamics, quantum chromodynamics, and general relativity. This examination includes the rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge transformations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that Dirac spinors can be described within the framework of gravitation theory.

View PDF Chat

Weights for mod <i>p</i> Quaternionic Forms in the Unramified Case

2025-06-03

Shalini Bhattacharya , Abhik Ganguli

QBCR2O method for solving quaternion matrix equation with its applications

2025-06-01

Xianzhe Zhang , Caiqin Song | Journal of Mathematical Analysis and Applications

The Symmetry of Hilbert Transformation in $$\mathbb {R}^3$$

2025-06-01

Pei Dang , Hua Liu , Tao Qian | Advances in Applied Clifford Algebras

Possible Solution for the Gauge Hierarchy Problem and Vacuum Catastrophe

2025-06-01

On three summation equations for functions that are holomorphic in the plane with a cut along a polygonal line

2025-06-01

N. N. Garif’yanov , E. V. Strezhneva | Problemy analiza

Exterior-algebraic formulation of quaternions with applications

2025-06-01

Ivano Colombaro | Journal of Physics Conference Series

Abstract The purpose of this paper is to describe the formulation of quaternion algebra by means of exterior algebra and calculus, in a three dimensional time-like spacetime. A formal structure … Abstract The purpose of this paper is to describe the formulation of quaternion algebra by means of exterior algebra and calculus, in a three dimensional time-like spacetime. A formal structure is provided, corroborating the equivalence with existing concepts and formulas known in literature. A first application is thus presented by depicting the description of rotations expressed with exterior-algebraic quaternionic notation. Secondly, a formal equivalence between exterior-algebraic quaternions and the classical theory of electromagnetism is recovered, too.

Structure Preserving Quaternion Biconjugate Residual Algorithm with Application

2025-05-31

Xianzhe Zhang , Caiqin Song , В. И. Васильев +1 more | Journal of Scientific Computing

Non-Newtonian caustics and wavefronts in multiplicative Euclidean 2-space

2025-05-31

Zhong-Qi Ma , Yao Xiaohua , Jiaxin Li +1 more | Modern Physics Letters A

The geometric topological structures pertaining to caustics and wavefronts play a significant and crucial role in the realms of optics, astrophysics, as well as general relativity, serving as fundamental elements … The geometric topological structures pertaining to caustics and wavefronts play a significant and crucial role in the realms of optics, astrophysics, as well as general relativity, serving as fundamental elements in the theoretical frameworks and analyses within these scientific domains. This paper aims at analyzing geometric topological structure of multiplicative caustics and multiplicative wavefronts in the non-Newtonian(multiplicative) Euclidean space [Formula: see text]. First, we provide the definitions of multiplicative caustic and multiplicative wavefronts of a multiplicative plane curve [Formula: see text]. By taking the derivatives of all orders of parametric expressions, we obtain that the point where multiplicative caustic has a multiplicative cusp corresponds to the ordinary multiplicative vertex of [Formula: see text], and it also corresponds to the point where multiplicative wavefront has [Formula: see text]-singularity. Then we establish the relationships between singularities and multiplicative geometric invariants of curves by using multiplicative distance squared function and singularity theory. Finally, we use an example to illustrate these results.