70 results on '"Hodge dual"'
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2. Quasi-Quanta Language Package
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Emmerson, Parker
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additive ,procedure ,homomorphism ,complex number ,domain ,Transform ,Numeric Energy ,group functor ,sharp-logics ,Quasi-Quanta ,Infinity meaning ,charge distribution ,orientability ,transcendental numbers ,logic vector ,Entanglement ,Energy of Number ,quantum field ,gauge ,Vector-Wave ,coordinates ,boundaries ,Language ,Energy Numbers ,manifold ,algebraic law ,element ,coboundary ,field ,multiplicative ,curvature ,range ,metric tensor ,real-valued function ,Quasi-Quanta Extended Operational-Integrable Function ,iteratives ,Fractal ,energy vector ,smooth ,imaginary gauge artefact ,differential ,topological counting ,Morphism ,Geometry ,projection ,hodge dual ,pattern ,connectedness ,embedding ,FOS: Mathematics ,intersection ,algorithm ,Pre-numeric Quasi-Quanta ,algebras ,Cross-fractal ,quantum gravity ,quasi-quanta logic ,cohomology ,Integral Field ,Mathematics ,omega sub lambda, the highest energy level - Abstract
I investigate combinations of quasi-quanta expressions and how they yield alternatesolutions for expressions inMorphic Topology of Numeric Energy: A Fractal Morphism of Topological Counting Shows Real Differentiation of Numeric Energy. For Praising Jehovah, I do publish these mathematical gesturing forms from the infinity meaning of His word. Thanks mom! This quasi-quanta language package outlines methods for combining by topo- logical functor entanglement, symbolic, numeric-energy components. Methods, guidelines and algebraic rules for combining the quasi-quanta into the energy number equivalencies are also notated herein. The Quasi-Quanta Language Package is intended to show the symbolic pat- terns for configuring the quasi quanta symbology into the numeric energy ex- pressions. This should put to rest any doubt that Energy Numbers are indeed a real, logically configured phenomenon a priori to real or complex numbers, but optionally mappable to the real or complex plane. Pre-numeric energy symbol configurations offer a broad language of pat- tern detection and logical symbol operation delineated with particular solving methods herein. This hopefully provides a new way to looking at the branches of mathematics and their inter-operable analog functions. So, inevitably, we decompose the current perspective on numbers and prove a novel method for ordering and combining symbolic orientations.
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- 2023
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3. Geometrical modeling and numerical analysis of edge dislocation
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Shunsuke Kobayashi and Ryuichi Tarumi
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Stress field ,Physics ,Finite strain theory ,Mathematical analysis ,Torsion (mechanics) ,Tensor ,Dislocation ,Affine connection ,Galerkin method ,Hodge dual - Abstract
In this study, we conduct modeling and numerical analysis of a straight edge dislocation in a three-dimensional geometrically nonlinear elastic medium. Based on the fundamental framework of geometrical elasto-plasticity, kinematics of the continuum is represented by the reference R, intermediate B and current S states and local deformations are described by the multiplicative decomposition of the deformation gradient; F = Fe·Fp. Here, the reference and current states are Euclidean submanifolds while the intermediate state B is represented by Weitzenbock manifold with non-zero torsion in the affine connection. Following to the equivalence of torsion and dislocation density tensor through Hodge star operation, a plastic deformation gradient Fp is obtained from the Cartan first structure equation using the homotopy operator. The current state S is determined so that it minimizes the strain energy functional. We solve the variational problem by using isogeometric analysis; Galerkin method with non-uniform B-spline basis function. The present analysis shows that all stress components agree quantitatively well with those of the Volterra dislocation model. Around the dislocation core, stress fields are non-singular on the contrary to the Volterra dislocation. In addition, we found that all stress components distribute asymmetrically due to the geometrical nonlinearity of the model.
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- 2021
4. Hodge dual operators and model algebras for rational representations of the general linear group
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Sangjib Kim and Soo Teck Lee
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Pure mathematics ,Algebra and Number Theory ,Operator (computer programming) ,010102 general mathematics ,0103 physical sciences ,Multiplicity (mathematics) ,General linear group ,010307 mathematical physics ,0101 mathematics ,Hodge dual ,01 natural sciences ,Mathematics - Abstract
In this paper, we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group GL n ( C ) . We then use the properties of a generalized version of the Hodge dual operator to determine an explicit basis for each of these algebras, and by restriction, we obtain an explicit basis for each of the irreducible rational representations of GL n ( C ) . Our results cleanly extends classical algebro-combinatorial results on polynomial representations of GL n ( C ) to rational representations.
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- 2020
5. Generalized Hodge dual for torsion in teleparallel gravity.
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Huang, Peng and Yuan, Fang-Fang
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- *
HODGE theory , *TORSION theory (Algebra) , *OPERATOR theory , *DIMENSION theory (Algebra) , *GRAVITY - Abstract
For teleparallel gravity in four dimensions, Lucas and Pereira have shown that its action can be constructed via a generalized Hodge dual for torsion tensor. In this paper, we demonstrate that a direct generalization of this approach to other dimensions fails due to the fact that no generalized Hodge dual operator could be given in general dimensions. Furthermore, if one enforces the definition of a generalized Hodge dual to be consistent with the action of teleparallel gravity in general dimensions, the basic identity for any sensible Hodge dual would require an ad hoc definition for the second Hodge dual operation which is totally unexpected. Therefore, we conclude that at least for the torsion tensor, the observation of Lucas and Pereira only applies to four dimensions. [ABSTRACT FROM AUTHOR]
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- 2016
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6. Adaptive estimation of Hodge star operator on simplicial surfaces
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A. F. El Ouafdi, Djemel Ziou, and H. El Houari
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Surface (mathematics) ,Pure mathematics ,Diagonal ,020207 software engineering ,02 engineering and technology ,Geometry processing ,Positive-definite matrix ,Differential operator ,Computer Graphics and Computer-Aided Design ,symbols.namesake ,Mathematics::Algebraic Geometry ,Operator (computer programming) ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,Computer Vision and Pattern Recognition ,Hodge dual ,Software ,Mathematics - Abstract
The Hodge star operator is a fundamental component of the second-order differential operators that bridges constitutive physical laws, as a matter of fact, it plays a central role in geometry processing and physical simulation. To be admissible, the discrete Hodge operators should be regular, symmetric, sparse and positive definite. Unfortunately, the last criteria is rarely met in the literature, which leads to inconsistent physical simulation behavior. In this paper, we exploit the intrinsic relationship between Hodge operator and the physical Fourier’s constitutive laws, to construct an adaptive discrete Hodge operator by expressing the Fourier’s laws on a surface mesh. As by-product, the new discrete Hodge operator is diagonal, regular and positive definite. Various comparative examples are presented to demonstrate the performance of our approach. The results show that the proposed operator performs better compared with the standard discrete Hodge.
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- 2020
7. A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus
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Odysseas Kosmas, Andrey P. Jivkov, Lee Margetts, and Pieter D. Boom
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Constitutive equation ,Structure (category theory) ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,Displacement (vector) ,General Materials Science ,0101 mathematics ,Hodge dual ,Mathematical Physics ,Mathematics ,Elastic materials ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Linear elasticity ,Mathematical Physics (math-ph) ,Condensed Matter Physics ,Cohomology ,0104 chemical sciences ,010404 medicinal & biomolecular chemistry ,Discrete exterior calculus ,Mechanics of Materials ,Modeling and Simulation ,Dissipative system - Abstract
A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknowns are displacements, represented by a primal vector-valued 0-cochain. Displacement differences and internal forces are represented by a primal vector-valued 1-cochain and a dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Poisson's equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Excellent agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.
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- 2021
8. The computational algorithm and numerical analysis of the signed diagonal Hodge star operator
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Satoshi Noguchi and Kenji Oguni
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Physics ,Numerical analysis ,Diagonal ,Applied mathematics ,Computational algorithm ,Hodge dual - Published
- 2019
9. Stability of magnetic black holes in general nonlinear electrodynamics
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Daisuke Yoshida, Jiro Soda, and Kimihiro Nomura
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Electromagnetic field ,Physics ,High Energy Physics - Theory ,010308 nuclear & particles physics ,General function ,Astrophysics::High Energy Astrophysical Phenomena ,Magnetic monopole ,FOS: Physical sciences ,Field strength ,General Relativity and Quantum Cosmology (gr-qc) ,01 natural sciences ,Stability (probability) ,General Relativity and Quantum Cosmology ,Nonlinear system ,Magnetic Black ,High Energy Physics - Theory (hep-th) ,Quantum electrodynamics ,0103 physical sciences ,010306 general physics ,Hodge dual - Abstract
We study the perturbative stability of magnetic black holes in a general class of nonlinear electrodynamics, where the Lagrangian is given by a general function of the field strength of electromagnetic field $F_{\mu\nu}$ and its Hodge dual $\widetilde{F}_{\mu\nu}$. We derive sufficient conditions for the stability of the black holes. We apply the stability conditions to Bardeen's regular black holes, black holes in Euler-Heisenberg theory, and black holes in Born-Infeld theory. As a result, we obtain a sufficient condition for the stability of Bardeen's black holes, which restricts $F_{\mu\nu}\widetilde{F}^{\mu\nu}$ dependence of the Lagrangian. We also show that black holes in Euler-Heisenberg theory are stable for a sufficiently small magnetic charge. Moreover, we prove the stability of black holes in the Born-Infeld electrodynamics even when including $F_{\mu\nu}\widetilde{F}^{\mu\nu}$ dependence., Comment: 25 pages, 2 figures
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- 2020
10. A new Hodge operator in Discrete Exterior Calculus. Application to fluid mechanics
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Dina Razafindralandy, Rama Ayoub, Aziz Hamdouni, Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356 (LaSIE), and Université de La Rochelle (ULR)-Centre National de la Recherche Scientifique (CNRS)
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FOS: Computer and information sciences ,Diagonal ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Computational Engineering, Finance, and Science (cs.CE) ,Operator (computer programming) ,Mathematics::Algebraic Geometry ,0202 electrical engineering, electronic engineering, information engineering ,[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph] ,0101 mathematics ,Computer Science - Computational Engineering, Finance, and Science ,Hodge dual ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Applied Mathematics ,Triangulation (social science) ,020207 software engineering ,General Medicine ,Types of mesh ,Algebra ,Discrete exterior calculus ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,[NLIN.NLIN-CD]Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD] ,Discrete differential geometry ,Analysis ,Interior point method ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.
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- 2020
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11. The $${\mathcal {L}}_B$$ L B -cohomology on compact torsion-free $$\mathrm {G}_2$$ G 2 manifolds and an application to ‘almost’ formality
- Author
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Spiro Karigiannis, Chi Cheuk Tsang, and Ki Fung Chan
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Hodge theory ,010102 general mathematics ,01 natural sciences ,Cohomology ,Combinatorics ,symbols.namesake ,Differential geometry ,0103 physical sciences ,Torsion (algebra) ,Exterior derivative ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Hodge dual ,Analysis ,Poincaré duality ,Mathematics - Abstract
We study a cohomology theory $$H^{\bullet }_{\varphi }$$ , which we call the $${\mathcal {L}}_B$$ -cohomology, on compact torsion-free $$\mathrm {G}_2$$ manifolds. We show that $$H^k_{\varphi } \cong H^k_{\mathrm {dR}}$$ for $$k \ne 3, 4$$ , but that $$H^k_{\varphi }$$ is infinite-dimensional for $$k = 3,4$$ . Nevertheless, there is a canonical injection $$H^k_{\mathrm {dR}} \rightarrow H^k_{\varphi }$$ . The $${\mathcal {L}}_B$$ -cohomology also satisfies a Poincare duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $$\mathrm {d}$$ and the derivation $${\mathcal {L}}_B$$ and uses both Hodge theory and the special properties of $$\mathrm {G}_2$$ -structures in an essential way. As an application of our results, we prove that compact torsion-free $$\mathrm {G}_2$$ manifolds are ‘almost formal’ in the sense that most of the Massey triple products necessarily must vanish.
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- 2018
12. Two-form gauge theory dual to scalar-tensor theory
- Author
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Daisuke Yoshida
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Physics ,Field (physics) ,010308 nuclear & particles physics ,Duality (optimization) ,Kinetic term ,01 natural sciences ,Einstein tensor ,symbols.namesake ,Scalar–tensor theory ,General Relativity and Quantum Cosmology ,0103 physical sciences ,symbols ,Gauge theory ,010306 general physics ,Hodge dual ,Scalar field ,Mathematical physics - Abstract
We generalize the electromagnetic duality between a massless free scalar field and a two-form gauge field in a four-dimensional spacetime to scalar-tensor theories. We derive the action of a two-form gauge field that is dual to two kinds of scalar-tensor theories: the shift symmetric K-essence theory and the shift symmetric Horndeski theory up to quadratic in a scalar field. The former case, the dual two-form, has a nonlinear kinetic term. The latter case, the dual two-form, has nontrivial interactions with gravity through the Einstein tensor. In both cases, the duality relation is modified from the usual case, that is, the dual two-form field is not simply given by the Hodge dual of the gradient of the scalar field.
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- 2019
13. Numerical convergence of discrete exterior calculus on arbitrary surface meshes
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Mamdouh S. Mohamed, Anil N. Hirani, and Ravi Samtaney
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Computational Mechanics ,010103 numerical & computational mathematics ,02 engineering and technology ,Computer Science::Computational Geometry ,01 natural sciences ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Convergence (routing) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,medicine ,Applied mathematics ,Polygon mesh ,Mathematics - Numerical Analysis ,0101 mathematics ,Hodge dual ,Calculus (medicine) ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics ,Partial differential equation ,020207 software engineering ,Numerical Analysis (math.NA) ,Surface (topology) ,medicine.disease ,Computational Mathematics ,Discrete exterior calculus ,Poisson's equation ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.
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- 2018
14. Homogeneous and Inhomogeneous Maxwell’s Equations in Terms of Hodge Star Operator
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Zakir Hossine and Showkat Ali
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Coordinate-free ,symbols.namesake ,Work (thermodynamics) ,Maxwell's equations ,Homogeneous ,Differential form ,Operator (physics) ,symbols ,Hodge dual ,Exterior algebra ,Mathematical physics ,Mathematics - Abstract
The main purpose of this work is to provide application of differential forms in physics. For this purpose, we describe differential forms, exterior algebra in details and then we express Maxwell’s equations by using differential forms. In the theory of pseudo-Riemannian manifolds there will be an important operator, called Hodge Star Operator. Hodge Star Operator arises in the coordinate free formulation of Maxwell’s equation in flat space-time. This operator is an important ingredient in the formulation of Stoke’stheorem.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 15-27
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- 2018
15. Electromagnetic Wave Equation on Differential Form Representation
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I Gusti Ngurah Yudi Handayana
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Physics ,Electromagnetic wave equation ,Differential form ,Operator (physics) ,Minkowski space ,Exterior derivative ,Hodge dual ,Wave equation ,Differential (mathematics) ,Mathematical physics - Abstract
One of the indispensable part of the theoretical physics interest is geometry differential. This one interest of physical area has been developed such as in electromagnetism. Maxwell's equations have been generalized in two covariant forms in differential form representation. A beautiful calculus vector in this representation, such as exterior derivative and Hodge star operator, lead this study. Electromagnetic wave equation has been expressed in differential form representation using Laplace-de Rham operator. Explicitly, wave equation shows the same form in Minkowski space-time like vector representation. This study is able to introduce us to learn application of differential form in physics.
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- 2018
16. A simple and complete discrete exterior calculus on general polygonal meshes
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Luiz Velho and Lenka Ptackova
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Discretization ,Aerospace Engineering ,020207 software engineering ,02 engineering and technology ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,0104 chemical sciences ,Algebra ,010404 medicinal & biomolecular chemistry ,Discrete exterior calculus ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Modeling and Simulation ,Automotive Engineering ,0202 electrical engineering, electronic engineering, information engineering ,Exterior derivative ,Polygon mesh ,Lie derivative ,Hodge dual ,Laplace operator ,Exterior algebra ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.
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- 2021
17. Hodge Decomposition Methods for a Quad-Curl Problem on Planar Domains
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Jiguang Sun, Li Yeng Sung, and Susanne C. Brenner
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Discrete mathematics ,Curl (mathematics) ,Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Hodge decomposition ,General Engineering ,010103 numerical & computational mathematics ,Mixed finite element method ,01 natural sciences ,Finite element method ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Planar ,Computational Theory and Mathematics ,Vector field ,0101 mathematics ,Hodge dual ,Software ,Mathematics - Abstract
We develop and analyze $$P_k$$ Lagrange finite element methods for a quad-curl problem on planar domains that is based on the Hodge decomposition of divergence-free vector fields. Numerical results that illustrate the performance of the finite element methods are also presented.
- Published
- 2017
18. Pseudo-Spectral Methods for the Laplace-Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One
- Author
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Lise-Marie Imbert-Gérard and Leslie Greengard
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Numerical Analysis ,Laplace transform ,Applied Mathematics ,Hodge theory ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Beltrami equation ,010101 applied mathematics ,Computational Mathematics ,Laplace–Beltrami operator ,Partial derivative ,Vector field ,0101 mathematics ,Hodge dual ,Spectral method ,Analysis ,Mathematics - Abstract
The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three-dimensional space, with a view toward applications in plasma physics and fluid dynamics. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017
- Published
- 2017
19. Collocated electrodynamic FDTD schemes using overlapping Yee grids and higher-order Hodge duals
- Author
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Michal Okoniewski, M. E. Potter, and Chris Deimert
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Physics and Astronomy (miscellaneous) ,02 engineering and technology ,FDTD methods ,01 natural sciences ,Lebedev quadrature ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Hodge dual ,Computer Science::Distributed, Parallel, and Cluster Computing ,Mathematics ,010302 applied physics ,Coupling ,Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Isotropy ,Finite-difference time-domain method ,020206 networking & telecommunications ,Grid ,Computer Science Applications ,Computational Mathematics ,Discrete exterior calculus ,Modeling and Simulation ,Dual polyhedron - Abstract
The collocated Lebedev grid has previously been proposed as an alternative to the Yee grid for electromagnetic finite-difference time-domain (FDTD) simulations. While it performs better in anisotropic media, it performs poorly in isotropic media because it is equivalent to four overlapping, uncoupled Yee grids. We propose to couple the four Yee grids and fix the Lebedev method using discrete exterior calculus (DEC) with higher-order Hodge duals. We find that higher-order Hodge duals do improve the performance of the Lebedev grid, but they also improve the Yee grid by a similar amount. The effectiveness of coupling overlapping Yee grids with a higher-order Hodge dual is thus questionable. However, the theoretical foundations developed to derive these methods may be of interest in other problems.
- Published
- 2016
20. Integral representations on supermanifolds: super Hodge duals, PCOs and Liouville forms
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R. Catenacci, Leonardo Castellani, and Pietro Antonio Grassi
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010308 nuclear & particles physics ,Differential form ,010102 general mathematics ,Duality (mathematics) ,Statistical and Nonlinear Physics ,01 natural sciences ,String (physics) ,Algebra ,High Energy Physics::Theory ,Differential geometry ,0103 physical sciences ,Supermanifold ,Supergeometry ,0101 mathematics ,Hodge dual ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Symplectic geometry - Abstract
We present a few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.
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- 2016
21. The Haydys monopole equation
- Author
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Goncalo Oliveira and Ákos Nagy
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Tangent bundle ,Mathematics - Differential Geometry ,Instanton ,General Mathematics ,High Energy Physics::Lattice ,Magnetic monopole ,Holomorphic function ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,symbols.namesake ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,0101 mathematics ,Hodge dual ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Mathematical physics ,Riemann surface ,010102 general mathematics ,53C07, 58D27, 58E15, 70S15 ,Mathematical Physics (math-ph) ,Submanifold ,Moduli space ,Differential Geometry (math.DG) ,symbols - Abstract
We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12]., Comment: 28 pages, no figures. Published version (with a few extra typo/reference fixes)
- Published
- 2019
22. Comparison of discrete Hodge star operators for surfaces
- Author
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Anil N. Hirani, Ravi Samtaney, and Mamdouh S. Mohamed
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Hodge theory ,Diagonal ,Mathematical analysis ,020207 software engineering ,02 engineering and technology ,Barycentric coordinate system ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Industrial and Manufacturing Engineering ,Mathematics::Numerical Analysis ,Computer Science Applications ,Physics::Fluid Dynamics ,010101 applied mathematics ,Computer Science::Graphics ,Mathematics::Algebraic Geometry ,Discrete exterior calculus ,Rate of convergence ,Incompressible flow ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Galerkin method ,Hodge dual ,Mathematics - Abstract
We investigate the performance of various discrete Hodge star operators for discrete exterior calculus (DEC) using circumcentric and barycentric dual meshes. The performance is evaluated through the DEC solution of Darcy and incompressible Navier-Stokes flows over surfaces. While the circumcentric Hodge operators may be favorable due to their diagonal structure, the barycentric (geometric) and the Galerkin Hodge operators have the advantage of admitting arbitrary simplicial meshes. Numerical experiments reveal that the barycentric and the Galerkin Hodge operators retain the numerical convergence order attained through the circumcentric (diagonal) Hodge operators. Furthermore, when the barycentric or the Galerkin Hodge operators are employed, a super-convergence behavior is observed for the incompressible flow solution over unstructured simplicial surface meshes generated by successive subdivision of coarser meshes. Insofar as the computational cost is concerned, the Darcy flow solutions exhibit a moderate increase in the solution time when using the barycentric or the Galerkin Hodge operators due to a modest decrease in the linear system sparsity. On the other hand, for the incompressible flow simulations, both the solution time and the linear system sparsity do not change for either the circumcentric or the barycentric and the Galerkin Hodge operators. Various definitions for discrete Hodge star operators are investigated.Darcy and incompressible Navier-Stokes flows are used for numerical experiments.Barycentric Hodge star reproduces the convergence rate of the circumcentric Hodge.The difference in the computational cost is generally minor.
- Published
- 2016
23. Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes
- Author
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Ravi Samtaney, Mamdouh S. Mohamed, and Anil N. Hirani
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Mathematics - Differential Geometry ,Physics and Astronomy (miscellaneous) ,Discretization ,Differential form ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Hodge dual ,Vector calculus ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics ,Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Fluid Dynamics (physics.flu-dyn) ,Numerical Analysis (math.NA) ,Physics - Fluid Dynamics ,Time-scale calculus ,Computational Physics (physics.comp-ph) ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Discrete exterior calculus ,Differential Geometry (math.DG) ,Modeling and Simulation ,Exterior derivative ,Interior product ,Physics - Computational Physics - Abstract
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step., 35 pages, 8 figures
- Published
- 2016
24. Some results for the Hodge decomposition theorem in Euclidean three-space
- Author
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Paul Bracken
- Subjects
Discrete mathematics ,Hodge conjecture ,Pure mathematics ,p-adic Hodge theory ,Hodge theory ,Euclidean geometry ,Euclidean domain ,Hodge dual ,Space (mathematics) ,Atiyah–Singer index theorem ,Mathematics - Published
- 2016
25. Conformal Killing forms on nearly Kähler manifolds
- Author
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Uwe Semmelmann and Antonio Martínez Naveira
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Pure mathematics ,Degree (graph theory) ,010102 general mathematics ,Structure (category theory) ,Conformal map ,01 natural sciences ,Computational Theory and Mathematics ,0103 physical sciences ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Hodge dual ,Linear combination ,Analysis ,Mathematics - Abstract
We study conformal Killing forms on compact 6-dimensional nearly Kahler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of dω and its Hodge dual ⁎ d ω , where ω is the fundamental 2-form of the nearly Kahler structure. The proof is based on a fundamental integrability condition for conformal Killing forms. We have partial results in the case of conformal Killing 2-forms. In particular we show the non-existence of J-anti-invariant Killing 2-forms.
- Published
- 2020
26. Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative
- Author
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Jun-Jin Peng
- Subjects
High Energy Physics - Theory ,Pure mathematics ,Physics and Astronomy (miscellaneous) ,010308 nuclear & particles physics ,Scalar (mathematics) ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,01 natural sciences ,General Relativity and Quantum Cosmology ,symbols.namesake ,Operator (computer programming) ,High Energy Physics - Theory (hep-th) ,Laplace–Beltrami operator ,0103 physical sciences ,symbols ,Exterior derivative ,010306 general physics ,Hodge dual ,Lagrangian ,Mathematics - Abstract
In this paper, we aim to explore the properties and applications on the operators consisting of the Hodge star operator together with the exterior derivative, whose action on an arbitrary $p$-form field in $n$-dimensional spacetimes makes its form degree remain invariant. Such operations are able to generate a variety of $p$-forms with the even-order derivatives of the $p$-form. To do this, we first investigate the properties of the operators, such as the Laplace-de Rham operator, the codifferential and their combinations, as well as the applications of the operators in the construction of conserved currents. On basis of two general p-forms, then we construct a general n-form with higher-order derivatives. Finally, we propose that such an n-form could be applied to define a generalized Lagrangian with respect to a p-form field according to the fact that it incudes the ordinary Lagrangians for the $p$-form and scalar fields as special cases., 23 pages, two tables, accepted by Communications in Theoretical Physics
- Published
- 2020
27. Gauge Field Mimetic Cosmology
- Author
-
Hassan Firouzjahi, Shinji Mukohyama, Seyed Ali Hosseini Mansoori, and Mohammad Ali Gorji
- Subjects
Physics ,High Energy Physics - Theory ,Cosmology and Nongalactic Astrophysics (astro-ph.CO) ,Spacetime ,010308 nuclear & particles physics ,Dark matter ,FOS: Physical sciences ,Astronomy and Astrophysics ,General Relativity and Quantum Cosmology (gr-qc) ,Gauge (firearms) ,Curvature ,01 natural sciences ,Cosmology ,General Relativity and Quantum Cosmology ,Theoretical physics ,High Energy Physics - Theory (hep-th) ,0103 physical sciences ,Gauge theory ,Invariant (mathematics) ,010306 general physics ,Hodge dual ,Astrophysics - Cosmology and Nongalactic Astrophysics - Abstract
We extend the mimetic cosmology to models containing gauge invariant $p$-forms. The $0$-form case reproduces the well-known results of the mimetic dark matter, the $1$-form corresponds to the gauge field mimetic model while the $2$-form model is the Hodge dual of the $0$-form model in $4$ spacetime dimensions. We study the cosmological applications of the new gauge field mimetic model and show that it generates an energy density component which mimics the roles of spatial curvature. In the presence of the Maxwell term, the model also supports the flat, open and closed de Sitter-like cosmological backgrounds while the spatial geometry is flat for all three cases. We perform the cosmological perturbations analysis and show that the model is stable in the case of open de Sitter-like solution while it suffers from ghost instabilities in the case of the closed de Sitter-like solution., 21 pages, references added, matchs published version
- Published
- 2018
28. Development of Geometric Formulation of Elasticity
- Author
-
Andrey P. Jivkov and Odysseas Kosmas
- Subjects
Discrete exterior calculus ,Geometric mechanics ,Discretization ,Mathematical analysis ,Boundary value problem ,Elasticity (economics) ,Hodge dual ,Ambient space ,Mathematics ,Moduli - Abstract
Recently, new techniques have been presented that discretize continuous elasticity variables as cochains over a primal mesh, representing the solid, and an appropriately defined dual one. Discrete strain and stress can be then thought of as a vector-valued 1-form (or vector-valued 1-cochain on the primal mesh) and covector-valued 2-form (or vector-valued 2-cochain on the dual mesh) respectively. The governing equations can be formulated by requiring energy balance and invariance under time-dependent rigid translations and rotations of the ambient space. To obtain those, we project the discrete stress into normal and tangential components and formulate the boundary value problem with a system of two matrix equations. This allow for treating both classical and coupled-stress (micro-polar) elasticity. The link between discrete strains and stresses is provided by a material discrete Hodge star operator, which we define to include geometric and physical factors, such as lengths, areas, and moduli of elasticity and rigidity. The performance of the proposed formulation is demonstrated by a simple example.
- Published
- 2018
29. Exterior calculus and fermionic quantum computation
- Author
-
Apostolos Vourdas
- Subjects
0301 basic medicine ,Statistics and Probability ,Computer science ,High Energy Physics::Lattice ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,03 medical and health sciences ,Computer Science::Hardware Architecture ,Computer Science::Emerging Technologies ,0103 physical sciences ,Calculus ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,010306 general physics ,Hodge dual ,Representation (mathematics) ,Mathematical Physics ,Complement (set theory) ,Quantum computer ,Condensed Matter::Quantum Gases ,Quantum Physics ,Statistical and Nonlinear Physics ,030104 developmental biology ,Modeling and Simulation ,Scheme (mathematics) ,Logic gate ,Join (sigma algebra) ,Quantum Physics (quant-ph) ,Hardware_LOGICDESIGN - Abstract
Exterior calculus with its three operations meet, join and hodge star complement, is used for the representation of fermion-hole systems and for fermionic analogues of logical gates. Two different schemes that implement fermionic quantum computation, are proposed. The first scheme compares fermionic gates with Boolean gates, and leads to novel electronic devices that simulate fermionic gates. The second scheme usesa well known map between fermionic and multi-qubit systems, to simulate fermionic gates within multi-qubit systems.
- Published
- 2018
- Full Text
- View/download PDF
30. The Hodge conjecture: the complications of understanding the shape of geometric spaces
- Author
-
Vicente Muñoz Velázquez
- Subjects
Pure mathematics ,Topología ,Multidisciplinary ,Hodge theory ,Complex differential form ,Geometría diferencial ,Positive form ,Hodge conjecture ,Geometria algebraica ,History and Philosophy of Science ,Algebraic surface ,Hodge dual ,Geometry and topology ,Algebraic geometry and analytic geometry ,Mathematics - Abstract
The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.
- Published
- 2018
31. Visualizing One-, Two-, and Three-Forms
- Author
-
Jon Pierre Fortney
- Subjects
Gravitation ,Section (fiber bundle) ,Pure mathematics ,Alpha (programming language) ,Perspective (geometry) ,Differential form ,Hodge dual ,Vector space - Abstract
In this chapter we will introduce and discuss at length one of the ways that physicists sometimes visualize “nice” differential forms. In essence, we will be considering ways of visualizing one-forms, two-forms, and three-forms on a vector space. That is, we will find a “cartoon picture” of \(\alpha _p \in T^*_p\mathbb {R}^2\) and \(\alpha _p \in T^*_p\mathbb {R}^3\). Our picture of αp will be superimposed on the vector space \(T_p\mathbb {R}^3\). This perspective is developed extensively in Misner, Thorne, and Wheeler’s giant book Gravitation. In fact, they make some efforts to develop the four-dimensional picture (for space-time) as well, however we will primarily stick to the two and three-dimensional cases here. Section one focuses on the two-dimensional case and sections two through four focuses on the three-dimensional case. Then in section five we expand our cartoon picture to general two and three-dimensional manifolds. Again, this cartoon picture really only applies to “nice” differential forms, of the kind physicists are more likely to encounter, but it is still useful for forms and manifolds that are not overly complicated. Finally, in section six we introduce the Hodge star operator. Our visualizations of forms in three dimensions provide a nice way to visualize what the Hodge star operator does, which makes this a nice place to introduce it. Despite the power and usefulness of the way of visualizing differential forms in physics developed in this chapter, it is rarely encountered in mathematics. One of the reasons for this is that in reality it is not a completely general way of considering forms; when dealing with dimensions greater than four or with more abstract manifolds or with forms that are not “nice” in some sense it breaks down.
- Published
- 2018
32. Linking numbers in local quantum field theory
- Author
-
Giuseppe Ruzzi, Detlev Buchholz, Fabio Ciolli, and Ezio Vasselli
- Subjects
Electromagnetic field ,High Energy Physics - Theory ,FOS: Physical sciences ,01 natural sciences ,law.invention ,Tensor field ,law ,Settore MAT/05 - Analisi Matematica ,0103 physical sciences ,Minkowski space ,0101 mathematics ,Local quantum field theory ,Hodge dual ,Settore MAT/07 - Fisica Matematica ,Mathematical Physics ,Mathematical physics ,Physics ,Quantum Physics ,010102 general mathematics ,Zero (complex analysis) ,Commutator (electric) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Intrinsic vector potential ,Linking numbers ,Massless particles ,Massless particle ,High Energy Physics - Theory (hep-th) ,010307 mathematical physics ,Quantum Physics (quant-ph) - Abstract
Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article., Comment: 15 pages, no figures, v2 as to appear in Lett. Math. Phys
- Published
- 2018
- Full Text
- View/download PDF
33. Exceptional M-brane sigma models and $\eta$-symbols
- Author
-
Shozo Uehara and Yuho Sakatani
- Subjects
Physics ,High Energy Physics - Theory ,Sigma model ,010308 nuclear & particles physics ,Formalism (philosophy) ,Magnetic monopole ,General Physics and Astronomy ,Sigma ,01 natural sciences ,Action (physics) ,High Energy Physics::Theory ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Brane cosmology ,Brane ,010306 general physics ,Hodge dual ,Mathematics::Symplectic Geometry ,Mathematical physics - Abstract
We develop the M-brane actions proposed in arXiv:1607.04265 by using $\eta$-symbols determined in arXiv:1708.06342. Introducing $\eta$-forms that are defined with the $\eta$-symbols, we present U-duality-covariant M-brane actions which describe the known brane worldvolume theories for M$p$-branes with $p=0,2,5$. We show that the self-duality relation known in the double sigma model is naturally generalized to M-branes. In particular, for an M5-brane, the self-duality relation is nontrivially realized, where the Hodge star operator is defined with the familiar M5-brane metric while the $\eta$-form contains the self-dual 3-form field strength. The action for a Kaluza-Klein monopole is also partially reproduced. Moreover, we explain how to treat type IIB branes in our general formalism. As a demonstration, we reproduce the known action for a $(p,q)$-string., Comment: 55 pages; v2: references added, minor improvements; v3: clarifications added, to appear in PTEP; v4: typos corrected
- Published
- 2017
34. Axial torsion waves in metric-affine gravity
- Author
-
Vedad Pasic and Elvis Barakovic
- Subjects
Physics ,Gravity (chemistry) ,FOS: Physical sciences ,Torsion (mechanics) ,General Relativity and Quantum Cosmology (gr-qc) ,Affine connection ,General Relativity and Quantum Cosmology ,Gravitation ,Spinor field ,Affine transformation ,83C15 83C35 83D05 53Z05 ,Hodge dual ,Vector potential ,Mathematical physics - Abstract
We construct new explicit vacuum solutions of quadratic metric-affine gravity. The approach of metric-affine gravity in using an independent affine connection produces a theory with 10+64 unknowns, which implies admitting torsion and possible nonmetricity. Our spacetimes are generalisations of classical pp-waves, four-dimensional Lorentzian spacetimes which admit a nonvanishing parallel spinor field. We generalize this definition to metric compatible spacetimes with pp-metric and purely axial torsion. It has been suggested that one can interpret that the axial component of torsion as the Hodge dual of the electromagnetic vector potential. We compare these solutions with our previous results and other solutions of classical models describing the interaction of gravitational and neutrino fields., Comment: 6 pages. Proceedings of the MG14 Meeting on General Relativity, University of Rome "La Sapienza", Italy, 12 - 18 July 2015. Edited by: Massimo Bianchi (Universit\`a degli Studi di Roma "Tor Vergata", Italy), Robert T Jantzen (Villanova University, USA), Remo Ruffini (International Center for Relativistic Astrophysics Network (ICRANet), Italy and University of Rome "La Sapienza", Italy)
- Published
- 2017
35. A Hodge theory for Alexandrov spaces with curvature bounded from above
- Author
-
Nat Smale
- Subjects
Pure mathematics ,Metric space ,Specialization (pre)order ,Applied Mathematics ,Hodge theory ,Bounded function ,Mathematical analysis ,Hodge dual ,Curvature ,Space (mathematics) ,Analysis ,Cohomology ,Mathematics - Abstract
It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains.
- Published
- 2015
36. On algebraic cohomology classes on a smooth model of a fiber product of families of K3 surfaces
- Author
-
O. V. Nikol’skaya
- Subjects
Hodge conjecture ,Algebra ,Algebraic cycle ,Pure mathematics ,General Mathematics ,Product (mathematics) ,Hodge theory ,Algebraic surface ,Hodge dual ,Cohomology ,K3 surface ,Mathematics - Abstract
Hodge’s conjecture on algebraic cycles is proved for a smooth projective model X of the fiber product X1 ×CX2 of nonisotrivial one-parameter families of K3 surfaces (possibly with degeneracies) under certain constraints on the ranks of the transcendental cycle lattices of the general geometric fibers Xks and representations of the Hodge groups Hg(Xks).
- Published
- 2014
37. Numerical Electromagnetic Frequency Domain Analysis with Discrete Exterior Calculus
- Author
-
Shu C. Chen and Weng Cho Chew
- Subjects
Physics and Astronomy (miscellaneous) ,FOS: Physical sciences ,Boundary (topology) ,02 engineering and technology ,Topology ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,Boundary value problem ,Hodge dual ,Mathematics ,010302 applied physics ,Numerical Analysis ,Delaunay triangulation ,Applied Mathematics ,020206 networking & telecommunications ,Computational Physics (physics.comp-ph) ,Computer Science Applications ,Computational Mathematics ,Discrete exterior calculus ,Maxwell's equations ,Modeling and Simulation ,Frequency domain ,symbols ,Perfect conductor ,Physics - Computational Physics - Abstract
In this paper, we perform a numerical analysis in frequency domain for various electromagnetic problems based on discrete exterior calculus (DEC) with an arbitrary 2-D triangular or 3-D tetrahedral mesh. We formulate the governing equations in terms of DEC for 3D and 2D inhomogeneous structures, and also show that the charge continuity relation is naturally satisfied. Then we introduce effective signed dual volume to incorporate material information into Hodge star operators and take into account the case when circumcenters fall outside triangles or tetrahedrons, which may lead to negative dual volume without Delaunay triangulation. Then we demonstrate the implementation of various boundary conditions, including perfect magnetic conductor (PMC), perfect electric conductor (PEC), Dirichlet, periodic, and absorbing boundary conditions (ABC) within this method. An excellent agreement is achieved through the numerical calculation of several problems, including homogeneous waveguides, microstructured fibers, photonic crystals, scattering by a 2-D PEC, and resonant cavities., 41 pages with 19 figures. Briefly presented in IEEE APS 2016 meeting
- Published
- 2017
38. Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
- Author
-
Paul J. Atzberger and Ben Gross
- Subjects
Differential equation ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,Operator (computer programming) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Calculus ,Mathematics - Numerical Analysis ,0101 mathematics ,Hodge dual ,Physics ,Numerical Analysis ,Partial differential equation ,Applied Mathematics ,Numerical analysis ,General Engineering ,Spherical harmonics ,020207 software engineering ,Numerical Analysis (math.NA) ,16. Peace & justice ,Quadrature (mathematics) ,Computational Mathematics ,Computational Theory and Mathematics ,Exterior derivative ,Software - Abstract
We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$ showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace-Beltrami equations demonstrating our approach., 22 pages, 13 figures
- Published
- 2017
39. Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces
- Author
-
Nicolas Besse and Uriel Frisch
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Differential form ,FOS: Physical sciences ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Invariant (mathematics) ,Hodge dual ,Cauchy's integral formula ,Mathematics ,Mechanical Engineering ,010102 general mathematics ,Fluid Dynamics (physics.flu-dyn) ,Cauchy distribution ,Physics - Fluid Dynamics ,16. Peace & justice ,Condensed Matter Physics ,Euler equations ,Differential geometry ,Flow (mathematics) ,Differential Geometry (math.DG) ,Mechanics of Materials ,symbols ,Analysis of PDEs (math.AP) - Abstract
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools of differential geometry and of geodesic flow on the space SDiff of volume-preserving transformations (Arnold, Ann. Inst. Fourier, vol. 16, 1966, pp. 319-361), all manners of generalisations are here derived. The Cauchy invariants equation and the Cauchy formula, relating the vorticity and the Jacobian of the Lagrangian map, are shown to be two expressions of this Lie-advection invariance, which are duals of each other (specifically, Hodge dual). Actually, this is shown to be an instance of a general result, which holds for flow both in flat (Euclidean) space and in a curved Riemannian space: any Lie-advection invariant p-form which is exact (i.e. is a differential of a (p-1)-form) has an associated Cauchy invariants equation and a Cauchy formula. This constitutes a new fundamental result in linear transport theory, providing a Lagrangian formulation of Lie advection for some classes of differential forms. The result has a broad applicability: examples include the magnetohydrodynamics (MHD) equations and various extensions thereof, discussed by Lingam et al. (Phys. Lett. A, vol. 380, 2016, pp. 2400-2406), and include also the equations of Tao (2016, arXiv:1606.08481 [math.AP]), Euler equations with modified Biot-Savart law, displaying finite-time blow-up. Our main result is also used for new derivations, and several new results, concerning local helicity-type invariants for fluids and MHD flow in flat or curved spaces of arbitrary dimension., 50 pages, 1 figure
- Published
- 2017
40. Exterior Product Symbol
- Author
-
Vincent Pavan
- Subjects
Lie coalgebra ,Multivector ,Symbol ,media_common.quotation_subject ,Independence (mathematical logic) ,Cross product ,Topology ,Relation (history of concept) ,Hodge dual ,Exterior algebra ,Mathematical economics ,Mathematics ,media_common - Abstract
One of the most important questions regarding vector families concerns the independence or the relation among them.
- Published
- 2017
41. Hodge Type Theorems for Arithmetic Hyperbolic Manifolds
- Author
-
Nicolas Bergeron, Colette Moeglin, and John J. Millson
- Subjects
Hodge conjecture ,Discrete mathematics ,Pure mathematics ,Automorphic form ,Congruence (manifolds) ,Hyperbolic manifold ,Type (model theory) ,Hodge dual ,Relatively hyperbolic group ,Cohomology ,Mathematics - Abstract
This note is a variation on the lecture given by the first named author at the conference celebrating the work (and sixty-fifth anniversary) of Jean-Michel Bismut. In this lecture a proof of some new cases of the Hodge conjecture for Shimura varieties uniformized by complex balls was sketched following [3]. In this note we exemplify the main ideas of the proof on real hyperbolic manifolds. The Hodge conjecture does not make sense anymore but, somewhat analogously, we prove that classes of totally geodesic submanifolds generate the cohomology groups of degree k of compact congruence n-dimensional hyperbolic manifolds “of simple type” as long as k is strictly smaller than n/3. This is a particular case of the main result of [2].
- Published
- 2017
42. Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces
- Author
-
Atsushi Ikeda
- Subjects
Pure mathematics ,Divisor ,General Mathematics ,Hodge theory ,010102 general mathematics ,01 natural sciences ,Cohomology ,Torelli theorem ,Hodge conjecture ,Algebra ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,0103 physical sciences ,FOS: Mathematics ,14C34, 14J27, 14H40 ,010307 mathematical physics ,0101 mathematics ,Hodge dual ,Algebraic Geometry (math.AG) ,Hodge structure ,Mathematics - Abstract
We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors., Comment: 30 pages
- Published
- 2017
- Full Text
- View/download PDF
43. Noether-Lefschetz Locus and a Special Case of the Variational Hodge Conjecture: Using Elementary Techniques
- Author
-
Ananyo Dan
- Subjects
Pure mathematics ,Hodge theory ,010102 general mathematics ,Complete intersection ,Codimension ,01 natural sciences ,010101 applied mathematics ,Hodge conjecture ,Mathematics::Algebraic Geometry ,0101 mathematics ,Locus (mathematics) ,Commutative algebra ,Hodge dual ,Irreducible component ,Mathematics - Abstract
Fix integers n ≥ 1 and d such that nd > 2n + 2. The Noether-Lefschetz locus NL d,n parametrizes smooth projective hypersurfaces of degree d in ℙ2n+1 satisfying the condition: H n,n (X,ℂ) ∩ H2n(X,ℚ) ≠ ℚ. An irreducible component of the Noether-Lefschetz locus is locally a Hodge locus. One question is to ask under what choice of a Hodge class γ∈ H n,n (X,ℂ) ∩ H2n(X, ℚ) does the variational Hodge conjecture hold true? In this article we use methods coming from commutative algebra and Hodge theory to give an affirmative answer in the case γ is the class of a complete intersection subscheme in X of codimension n. Another problem studied in this article is: In the case n = 1 when is an irreducible component of the Noether-Lefschetz locus nonreduced? Using the theory of infinitesimal variation of Hodge structures of hypersurfaces in ℙ3, we determine all non-reduced components with codimension less than or equal to 3d for d ≫ 0. Here again our primary tool is commutative algebra.
- Published
- 2017
44. Computational electromagnetics with discrete exterior calculus
- Author
-
Weng Cho Chew and Shu Chen
- Subjects
Curl (mathematics) ,Pure mathematics ,Discrete exterior calculus ,Differential form ,Exterior derivative ,Computational electromagnetics ,Hodge dual ,Cohomology ,Finite element method - Abstract
A novel computational electromagnetic method developed with discrete exterior calculus (DEC) (M. Desbrun et al., arXiv:math/0508341) on simplicial mesh is presented. As is well known, differential forms can been used to recast Maxwell's equations in a more succinct fashion, which completely separate metric-free and metric-dependent components. Instead of dealing with vectorial field as finite difference method and finite element method (FEM), DEC studies electric and magnetic fields in discrete differential forms, i.e. their integrals over simplicial structures. In DEC, electric and magnetic fields are represented as cochains, which are vectors with finite length. Differential operators, such as curl and divergence, are replaced by discrete exterior derivative, which are highly sparse matrices only containing geometrical connection information with elements 0, 1 or −1. From Yee grid, the necessity and importance of a dual mesh or dual grid has been realized. In DEC, the dual mesh is constructed by connecting circumcenters of simplicial structures (triangles in 2D and tetrahedrons in 3D). With this circumcenter dual or Voronoi dual, the discrete Hodge star operators, which map from primal cochains to dual cochains, can be constructed as diagonal matrices. Effective dual volume is also introduced to incorporate material information into Hodge star operators. Then constitutive relations in an inhomogeneous medium can be described with these Hodge star operators. The case when circumcenters fall outside triangles or tetrahedrons is also considered and treated appropriately. The implementation of various boundary conditions is also illustrated.
- Published
- 2017
45. Hodge Numbers from Picard-Fuchs Equations
- Author
-
Alan Thompson, Andrew Harder, Charles F. Doran, Thompson, Alan [0000-0003-1400-0098], and Apollo - University of Cambridge Repository
- Subjects
Pure mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,p-adic Hodge theory ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Hodge dual ,QA ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics ,14D07, 14D05, 14J32 ,010308 nuclear & particles physics ,Hodge theory ,010102 general mathematics ,Complex differential form ,Cohomology ,Positive form ,Hodge conjecture ,Algebra ,Calabi-Yau manifolds ,Geometry and Topology ,Analysis ,Hodge structure ,variation of Hodge structures - Abstract
Given a variation of Hodge structure over $\mathbb{P}^1$ with Hodge numbers $(1,1,\dots,1)$, we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-M\"oller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.
- Published
- 2017
46. A Mimetic Spectral Element Method for Non-Isotropic Diffusion Problems
- Author
-
Marc Gerritsma, K. Olesen, and Bo Gervang
- Subjects
Physics ,Constraint (information theory) ,Mathematical analysis ,Isotropy ,Spectral element method ,Diffusion (business) ,Hodge dual ,Divergence (statistics) ,Mathematics::Numerical Analysis - Abstract
We present a mimetic spectral element method for the solution of the stationary Darcy’s problem. We show that the divergence constraint is satisfied exactly for both heterogeneous, non-isotropic, and deformed mesh problems.
- Published
- 2017
47. Corrigendum to 'Delaunay Hodge star' [Comput. Aided Des. 45 (2013) 540–544]
- Author
-
Evan B. Vanderzee, Anil N. Hirani, and Kaushik Kalyanaraman
- Subjects
Class (set theory) ,Delaunay triangulation ,Diagonal ,Search engine indexing ,020207 software engineering ,02 engineering and technology ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Industrial and Manufacturing Engineering ,Computer Science Applications ,010101 applied mathematics ,Combinatorics ,Discrete exterior calculus ,0202 electrical engineering, electronic engineering, information engineering ,Polygon mesh ,Pairwise comparison ,0101 mathematics ,Hodge dual ,Mathematics - Abstract
A discrete diagonal Hodge star operator for pairwise Delaunay triangulations was described in Hirani et al. (2013). This note fixes errors in two figures and some indexing mistakes in the original paper. One consequence of this fix is that discrete exterior calculus may be applicable to a wider class of meshes.
- Published
- 2018
48. Discrete electromagnetic theory with exterior calculus
- Author
-
Shu Chen and Weng Cho Chew
- Subjects
Pure mathematics ,Fundamental theorem ,Differential form ,020208 electrical & electronic engineering ,Divergence theorem ,020206 networking & telecommunications ,02 engineering and technology ,Discrete exterior calculus ,Quantum mechanics ,0202 electrical engineering, electronic engineering, information engineering ,Exterior derivative ,Hodge dual ,Exterior algebra ,Vector calculus ,Mathematics - Abstract
A self-contained electromagnetic theory is developed on a simplicial lattice. Instead of dealing with vectorial field, discrete exterior calculus (DEC) studies the discrete differential forms of electric and magnetic fields. Circumcenter dual is adopted to achieve diagonality and simplicity of Hodge star operators. In this paper, Gauss' theorem and Stokes' theorem are shown to be satisfied inherently. Many other electromagnetic theorems, like reciprocity theorem, can be derived on this simplicial lattice consistently with an appropriate definition of wedge product between forms. The preservation of these theorems guarantees that this treatment of Maxwell's equations will not lead to spurious solutions.
- Published
- 2016
49. Generalized modal analysis of waveguides and resonators with discrete exterior calculus
- Author
-
Shu Chen and Weng C. Chew
- Subjects
0209 industrial biotechnology ,Electromagnetics ,Discretization ,Modal analysis ,Diagonal ,Mathematical analysis ,020207 software engineering ,02 engineering and technology ,020901 industrial engineering & automation ,Discrete exterior calculus ,Simple (abstract algebra) ,0202 electrical engineering, electronic engineering, information engineering ,Tetrahedron ,Hodge dual ,Mathematics - Abstract
A generalized modal analysis with discrete exterior calculus for 2-D or 3-D electromagnetic system is presented. For an arbitrarily shaped inhomogeneous system, discrete exterior calculus provides a simple way to formulate the problem based on a triangular or tetrahedral discretization. In this paper, circumcenter dual is adopted to achieve diagonal discrete Hodge star operators. We also consider signed dual volumes for all dimensions to keep the correctness and accuracy of discrete Hodge star operators. Traveling modes of inhomogeneous waveguides and resonant cavity modes are numerically calculated to validate this method.
- Published
- 2016
50. The approach to the Hodge conjecture via normal functions
- Author
-
James Lewis and B. Brent Gordon
- Subjects
Hodge conjecture ,Pure mathematics ,Lefschetz theorem on (1,1)-classes ,Hodge theory ,Mathematical analysis ,Hodge bundle ,Hodge dual ,Mathematics - Published
- 2016
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