Fundamental Tensor Research Articles

Determination of elastic resonance frequencies and eigenmodes using the method of fundamental solutions

In this paper, we present the method of fundamental solutions applied to the determination of elastic resonance frequencies and associated eigenmodes. The method uses the fundamental solution tensor of the Navier equations of elastodynamics in an isotropic material. The applicability of the the method is justified in terms of density results. The accuracy of the method is illustrated through 2D numerical examples for the disk and non trivial shapes.

A Bonnet–Myers type theorem for quaternionic contact structures

We prove a Bonnet–Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.

On Holomorphic Curvature of Complex Finsler Square Metric

The notion of the holomorphic curvature for a Complex Finsler space (M,F) is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. This paper is about the fundamental metric tensor, inverse tensor and as a special approach of the pull-back bundle is devoted to obtaining the holomorphic curvature of Complex Finsler Square metrics. Further, it proved that it is not a weakly Kahler.

Symmetries in 4-dimensional manifolds

In this paper some algebraic and geometrical properties of symmetries (taken here as Lie algebras of smooth Killing vector fields) on a 4??dimensional manifold of arbitrary signature will be described. The discussion will include the theory of the distributions arising from such vector fields, their resulting orbit and isotropy structure and certain stability properties which these orbits may, or may not, possess. A link between the isotropies and the restrictions on the fundamental tensors of Ricci and Weyl (in terms of the subalgebras of the Lie algebras o(4), o(1,3) and o(2,2)) will be briefly discussed.

Markov fundamental tensor and its applications to network analysis

We first present a comprehensive review of various Markov metrics used in the literature and express them in a consistent framework. We then introduce the fundamental tensor – a generalization of the well-known fundamental matrix – and show that classical Markov metrics can be derived from it in a unified manner. We provide a collection of useful relations for Markov metrics that are useful and insightful for network studies. To demonstrate the usefulness and efficacy of the proposed fundamental tensor in network analysis, we present four important applications: 1) unification of network centrality measures, 2) characterization of (generalized) network articulation points, 3) identification of network's most influential nodes, and 4) fast computation of network reachability after failures.

General form of the full electromagnetic Green function in materials physics

In this article, we present the general form of the full electromagnetic Green function which is suitable for the application in bulk materials physics. In particular, we show how the seven adjustable parameter functions of the free Green function translate into seven corresponding parameter functions of the full Green function. Furthermore, for both the fundamental response tensor and the electromagnetic Green function, we discuss the reduction of the Dyson equation on the four-dimensional Minkowski space to an equivalent, three-dimensional Cartesian Dyson equation.

On a Complex Randers Space

In the present paper, a complex Randers space with the metric $$ F = \alpha + \varepsilon \left| \beta \right| + k\frac{{\left| \beta \right|^{2} }}{\alpha },\varepsilon ,k \ne 0 $$ is introduced and expressions for fundamental metric tensor, angular metric tensor, Chern–Finsler connection coefficients and curvature are obtained.

A Counterexample to Matsumoto's Conjecture Regarding Absolute Length vs. Relative Length in Finsler Manifolds

Matsumoto conjectured that for any Finsler manifold $(M, F)$ for which the restriction of the fundamental tensor to the indicatrix of $F$ is positive definite, the absolute length $F(X)$ of any tangent vector $X \in T_xM$ is the global minimum for the relative length $|X|_y$ as $y$ varies along the indicatrix $I_x \subset T_xM$ of $F$. In this note, we disprove this conjecture by presenting a counterexample.

Design, concepts, and applications of electromagnetic metasurfaces

AbstractThe paper overviews our recent work on the synthesis of metasurfaces and related concepts and applications. The synthesis is based on generalized sheet transition conditions (GSTCs) with a bianisotropic surface susceptibility tensor model of the metasurface structure. We first place metasurfaces in a proper historical context and describe the GSTC technique with some fundamental susceptibility tensor considerations. On this basis, we next provide an in-depth development of our susceptibility-GSTC synthesis technique. Finally, we present five recent metasurface concepts and applications, which cover the topics of birefringent transformations, bianisotropic refraction, light emission enhancement, remote spatial processing, and nonlinear second-harmonic generation.

Physics of the non-Abelian Coulomb phase: Insights from Padé approximants

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Pad\'e approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$, and the derivative of the beta function, $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $\gamma_{\bar\psi\psi,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.

Gravitational Wave as Symmetry Breaking within a New Model: An Overview

I outline a new hypothetical approach issuing a second gravitational equation in the scope of a promising model tackling the gravitational wave problem. This wave equation for graviton is framed in the endeavour to bridge the puzzling missing link to allow for quantum scale physics in a unifying gravity theory, through a new coupling constant S: thus wave is regarded as a symmetry breaking of general covariance of field equations through contraction of Riemann tensor by a constant tensor. That also allows an inertial mass to be assigned to the graviton (OE-25 eV/c2). This extension of General Relativity stems from self-evident considerations on the differential conditions of compatibility involving the two fundamental tensors on the curvature of the Space-Time continuum. Some considerations about last detected events are broached on the gauging of S constant, bringing forth a value that differs of two orders of magnitude with respect to the fitting of known binary star systems, unless source parameters are revised.

On a Riemannian manifold with a circulant structure whose third power is the identity

It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.

Infrared fixed point physics in SO(Nc) and Sp(Nc) gauge theories

We study properties of asymptotically free vectorial gauge theories with gauge groups $G={\rm SO}(N_c)$ and $G={\rm Sp}(N_c)$ and $N_f$ fermions in a representation $R$ of $G$, at an infrared (IR) zero of the beta function, $\alpha_{IR}$, in the non-Abelian Coulomb phase. The fundamental, adjoint, and rank-2 symmetric and antisymmetric tensor fermion representations are considered. We present scheme-independent calculations of the anomalous dimensions of (gauge-invariant) fermion bilinear operators $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and of the derivative of the beta function at $\alpha_{IR}$, denoted $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. It is shown that all coefficients in the expansion of $\gamma_{\bar\psi\psi,IR}$ that we calculate are positive for all representations considered, so that to $O(\Delta_f^4)$, $\gamma_{\bar\psi\psi,IR}$ increases monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase. Using this property, we give a new estimate of the lower end of this phase for some specific realizations of these theories.

Notes on the Nonextensive-Entropy Derived Evolution and Fundamental Metric Tensors in the Continuum Mechanics of Media and of Special and General Relativity

Many successes of special and general relativity in describing the universe have occurred as a result of the derivation of flat and curved spacetime metrics compatible with the Einstein equation of general relativity. These metrics were written on, or rather multiplied linear and nonlinear scalar coefficient functions to the background of a flat extensive spacetime (‘special relativity’) metric. In this letter we present a new perspective complementary to the one of fractional derivatives, this where fractional deformations of spacetime are supposed, which in this letter is obtained from a q-deformed non-extensive entropy derived type of metric that reduces to the flat Minkowski or Euclidean metric depending on application dimension and signature. This q-deformed metric then, is a parametrically deviated-from flat spacetime metric that encodes parametrically a scalar 'curvature' which tends to zero (flat) as q->1 for several instances and under minimal assumptions. Its derivation & some of its qualities will be presented. Also presented are calculations of the Ricci tensors, and scalar curvatures and therefore the stress-energy tensors for some of these cases, with at least one case closely corresponding to a q-deformed ADS metric.

Curvature coordinates to describe the explosion of Chernobyl’s reactor core in April 1986, using the tensor equations

This research analyzes the process of the explosion of the reactor core of Chernobyl nuclear plant in April 1986, using the tensor equations. These tensor equations show a movement of a vector in the three dimensional curvature coordinates of time, water flow, and void. The equations shows that this vector moves along the geodesic in the curvature coordinates, which is described by fundamental tensor ( g µν ), Christoffel symbol (Γ α µνσ ) and Ricci tensor ( R µν ), where µ, ν, σ, α are suffixes that indicate the coordinates. The solution of the tensor equations shows that the geodesic of the vector has a singular point, which describes a turning point of the reactor core from the normal operation to the explosion, which we reported in our previous articles [1, 2].

Higher-order scheme-independent series expansions of γψ¯ψ,IR and βIR′ in conformal field theories

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling $\alpha=\alpha_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$ and (ii) the derivative $\beta' = d\beta/d\alpha$, to $O(\Delta_f^5)$, both evaluated at $\alpha_{IR}$, where $\Delta_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G={\rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, $\gamma_{\bar\psi\psi,IR}$, calculated to the order $\Delta_f^p$, with $1 \le p \le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed. We also present expansions for $\alpha_{IR}$ calculated to $O(\Delta_f^4)$.

The Ether Theory Unifying the Relativistic Gravito-Electromagnetism Including also the Gravitons and the Gravitational Waves

In previous publications, we showed that Maxwell’s equations are an approximation to those of General Relativity when V<<c, where V is the velocity of the particle submitted to the electromagnetic field. This was demonstrated by showing that the Lienard-Wiechert potential four-vector A_u created by an electric charge is the equivalent of the gravitational four-vector G_u created by a massive neutral point when V<<c. In the present paper, we generalize these results for V non-restricted to be small. To this purpose, we show first that the exact Lagrange-Einstein function of an electric charge q submitted to the field due an immobile charge q_0 is of the same form as that of a particle of mass m submitted to the field created by an immobile particle of mass m_0. Maxwell’s electrostatics is then generalized as a case of the Einstein’s general relativity. In particular, it appears that an immobile q_0 creates also an electromagnetic horizon that behaves like a Schwarzschild horizon. Then, there exist ether gravitational waves constituted by gravitons in the same way as the electromagnetic waves are constituted by photons. Now, since A_u and G_u, are equivalent, and as we show, G_u produces the approximation, for V<<c, of g_u4 created by m_0 mobile, where the g_uv are the components of Einstein’s fundamental tensor, it follows that A_u+u_u produces the approximation, for V<<c, of Bet_u4 , where the Bet_uv created by m_0 and by q_0, generalize the g_uv.

Effects of Mesoscopic Microstructural Features on the Effective Ionic Diffusivity of Solid Electrolytes for Li Batteries

The kinetics of ion transport in solid electrolytes is known to be highly sensitive to the topological characteristics of ion conduction pathways. The conduction mechanisms usually involve concurrent ionic diffusion along a variety of mesoscopic features of internal microstructures such as bulk grain, structural domains, and associated boundaries as well as their network. Due to the complexity of these structural features of the solid electrolytes during operation, it is significantly challenging to characterize the microstructure-ionic diffusion property relationship. For better mechanistic understanding of relevant conduction mechanisms, it is necessary to thoroughly explore the impacts of individual microstructural factors on the overall kinetic properties and performances. In this presentation, we will present our development of an efficient mesoscopic computational method for extracting the effective diffusivity of the solid electrolyte containing arbitrarily distributed microstructural inhomogeneities such as differently oriented multiple grains, several structural variants of phase domains, and grain/domain boundaries. Using two- and three-dimensional digital microstructures generated by phase-field simulations as well as the fundamental diffusivity tensors of reference phases and boundaries derived from atomistic calculations as inputs, the method enable us to efficiently compute the effective diffusivity tensor of the entire system. We will then discuss the applications of this framework to investigating the relationship between relevant individual microstructural features and effective diffusivity of highly conductive solid electrolytes (e.g., LLTO and LLZO) focusing on the impacts of topological features of grains, internal mesoscopic structures of high- and low-temperature phase domains, and grain/phase boundary networks. *This work of was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Higher order scheme-independent calculations of physical quantities in the conformal phase of a gauge theory

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to $O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $\Delta_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function.

Microscopic theory of the refractive index

We examine the refractive index from the viewpoint of modern first-principles materials physics. We first argue that the standard formula, n2=ɛrμr, is generally in conflict with fundamental principles on the microscopic level. Instead, it turns out that an allegedly approximate relation, n2=ɛr, which is already being used for most practical purposes, can be justified theoretically at optical wavelengths. More generally, starting from the fundamental, Lorentz-covariant electromagnetic wave equation in materials as used in plasma physics, we rederive a well-known, three-dimensional form of the wave equation in materials and thereby clarify the connection between the covariant fundamental response tensor and the various Cartesian tensors used to describe optical properties. Finally, we prove a general theorem by which the fundamental, covariant wave equation can be reformulated concisely in terms of the microscopic dielectric tensor.