Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials

Nakata, Yukihiko; Urade, Yoshihiro; Nakanishi, Toshihiro

doi:10.3390/sym11111336

Cited by 9 publications

(5 citation statements)

References 91 publications

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“…i.e., the field strengths are (anti-)self-dual (for recent work on self-dual fields see, e.g., [32,33]),. Therefore, the second pair of Maxwell free equations d ⋆ C = 0 follows as an immediate consequence of the first (41).…”

Section: Gauge Symmetry and Fields Associated With Solutions Of B-dif...mentioning

confidence: 99%

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

Kassandrov,

Rizcallah,

Matveev

2023

Axioms

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We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (B) in particular. We demonstrate that any B-differentiable function gives rise to a null shear-free congruence (NSFC) on the B-vector space CM and on its Minkowski subspace M. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of B-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in CM. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”.

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Section: Gauge Symmetry and Fields Associated With Solutions Of B-dif...mentioning

confidence: 99%

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

Kassandrov,

Rizcallah,

Matveev

2023

Axioms

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“…Within these discussions, a discrete analogue of (1.1) is generally constructed, though, as remarked before, the reconstruction of the closed range theorem and the P-A-L type dualities, which are the main focuses of the present paper, has not drawn enough attention. We note that kinds of dualities used to be studied in, e.g., [4,10,15,22,29,35,36,40,43]. These works mainly focus on the dual representations for finite element spaces.…”

Section: Andmentioning

confidence: 99%

Partially adjoint discretizations of adjoint operators

Zhang¹

2022

Preprint

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This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. The validities of the theoretical framework and the formal construction of discretizations are illustrated by a systematic family of in-pair discretizations of the adjoint exterior differential operators.The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincaré-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. Particularly by partially adjoint discretizations of exterior differential operators, the Poincaré-Lefschetz duality is preserved as an identity, which was not yet obtained before. The ABCD, a new kind of discretization method, is motivated by and plays a crucial role in the construction of partially adjoint discretizations. Besides, it can be used for the discretization of single operators; for example, in this paper, de Rham complexes that start with the Crouzeix-Raviart finite element spaces are constructed by series of ABCDs, including both the discretized operators and the domain spaces; commutative diagrams with appropriate regularities are constructed thereon. Equivalences are established between primal and dual discretizations for the elliptic source and eigenvalue problems and the primal and dual mixed discretizations of the Hodge Laplacian problems of exterior differential operators based on the partially adjoint discretizations of adjoint operators. This shows how structure-preserving schemes for equations with linear operators can be lead to by partially adjoint discretizations under the new theoretical framework. SHUO ZHANG 1.3. Organization of the paper 7 1.4. Notations and conventions 8 2. Theory of partially adjoint operators 8 2.1. Basics of adjoint operators revisited 8 2.2. Theory of partially adjoint operators 10 2.3. Some technical proofs 19 3. Partially adjoint discretizations of adjoint operators 23 3.1. Partially adjoint discretizations by CD and ABCD 24 3.2. Characteristics of lo...

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“…The expectation value of O 0 ∈ Λ 0 (G) is denoted by E p [ O 0 ] as in (15). This expectation value is written in terms of the inner product with (22) as…”

Section: Expectation Valuesmentioning

confidence: 99%

“…Several topological approaches to master equations exist in the literature [9,10,11]. Not only master equations, but also random walks on lattices [12], electric circuits [13,14,15], and so on, have been studied from the viewpoint of algebraic topology. By introducing inner products for functions on graphs, one can define adjoint operators and Laplacians as self-adjoint operators [12].…”

Section: Introductionmentioning

confidence: 99%

Diffusion equations from master equations -- A discrete geometric approach --

Goto,

Hino

2019

Preprint

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In this paper, master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology play roles, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.

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Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials

Cited by 9 publications

References 91 publications

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

Partially adjoint discretizations of adjoint operators

Diffusion equations from master equations -- A discrete geometric approach --

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