Differential Forms in , I

Weintraub, Steven H.

doi:10.1016/b978-0-12-394403-0.00001-3

Cited by 7 publications

(4 citation statements)

References 5 publications

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“…where the symbols ´on the right side of this equation are both the usual cross product in R 3 . Defining the reduction cross product in C 3 in terms of conventional cross products in R 3 avoids the complexities confronted in extending the curl to spaces other than R 3 [24][25][26][27]. Again, if C and C¢ are purely real-valued vectors, then C C ´¢ in C 3 corresponds to the standard cross product in R 3 .…”

Section: Generalizing Maxwell's Equations To Complex-valued Fieldsmentioning

confidence: 99%

Generalizing Maxwell’s equations to complex-valued electromagnetic fields

Reggia

2023

Phys. Scr.

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There is a well-known asymmetry in classical electromagnetism, apparent in Maxwell’s equations, that arises from the existence of electric but not magnetic charge. This has motivated numerous experimental searches for magnetic monopoles which have, to date, not been found. To address this asymmetry, the research reported here generalizes these equations to accommodate complex-valued electromagnetic fields, thereby making Maxwell’s equations more symmetric. The resulting generalized equations remain consistent with the experimental predictions of the original Maxwell equations, and they are shown to continue to exhibit conservation of charge. The increased symmetry of the complex-valued equations is demonstrated via a duality transformation that is derived and verified here. Importantly, the generalized theory implies that a novel type of magnetic monopoles exists while simultaneously explaining why their detection has eluded previous experimental searches. Further study of the possibility that electromagnetic fields include imaginary-valued components is clearly merited because of the implications it could have for the foundations of classical electrodynamics.

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Section: Generalizing Maxwell's Equations To Complex-valued Fieldsmentioning

confidence: 99%

Generalizing Maxwell’s equations to complex-valued electromagnetic fields

Reggia

2023

Phys. Scr.

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“…Remark 6. For a vector bundle E over a n-dimensional differentiable manifold B equipped with a connection, the total Pontryagin class is expressed as [48]…”

Section: Application Of the Ssm-connection In Number Theorymentioning

confidence: 99%

Solitons Equipped with a Semi-Symmetric Metric Connection with Some Applications on Number Theory

Hakami,

Siddiqi,

Siddiqui

et al. 2023

Mathematics

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A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of η-Ricci solitons (η-RS) for an interesting manifold called the (ε)-Kenmotsu manifold ((ε)-KM), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and η-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient η-Ricci solitons (a special case of η-Ricci soliton), with a Poisson equation on the same ambient manifold for a SSM-connection. In addition, we derive an inequality for the lower bound of gradient η-Ricci solitons for (ε)-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the (ε)-Kenmotsu manifold equipped with a semi-symmetric metric connection.

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“…Because our domain is contractible, though, Poincaré's lemma guarantees that V exists if its Hessian is symmetric [13]. We obtain the Hessian of V by differentiating (6):…”

Section: Variable Transformationmentioning

confidence: 99%

“…The control depends on ∇V , but not on V itself, and we can compute it from Z and K using (13). However, we did compute Z (for plotting purposes) by selecting a reference point and computing line integrals from that point to every other point.…”

Section: Numeric Examplementioning

confidence: 99%

Solving the Hamilton-Jacobi-Bellman equation for a stochastic system with state constraints

Rutquist

Wik

Breitholtz

2014

53rd IEEE Conference on Decision and Control

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We present a method for solving the Hamilton-Jacobi-Bellman (HJB) equation for a stochastic system with state constraints. A variable transformation is introduced which turns the HJB equation into a combination of a linear eigenvalue problem, a set of partial differential equations (PDE:s), and a point-wise equation. For a fixed solution to the eigenvalue problem, the PDE:s are linear and the point-wise equation is quadratic, indicating that the problem can be solved efficiently using an iterative scheme.As an example, we numerically solve for the optimal control of a Linear Quadratic Gaussian (LQG) system with state constraints. A reasonably accurate solution is obtained even with a very small number of collocation points (three in each dimension), which suggests that the method could be used on high order systems, mitigating the curse of dimensionality.

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Differential Forms in , I

Cited by 7 publications

References 5 publications

Generalizing Maxwell’s equations to complex-valued electromagnetic fields

Generalizing Maxwell’s equations to complex-valued electromagnetic fields

Solitons Equipped with a Semi-Symmetric Metric Connection with Some Applications on Number Theory

Solving the Hamilton-Jacobi-Bellman equation for a stochastic system with state constraints

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