Hodge Star Research Articles

A Davenport dual angles approach for minimal parameterization of the rigid body displacement and motion

In this paper, the problem of the decomposition of a rigid body displacement by a succession of three rigid body displacements with specified screw-axis is completely solved. Using the isomorphism between the Lie group of the rigid body displacements and the Lie group of the orthogonal dual tensors, the necessary and sufficient conditions for which the decomposition is possible are given. It will be determined either the closed form the dual angles of the screw displacements (translation and rotation). The results are coordinate-free, and they are obtained using only algebraic elements of tensor calculus. In the specific case of a screw axis is perpendicular to the other two, the decomposition is possible for any rigid body displacement. So, a result that generalizes the Davenport decomposition in case of rotation is obtained. A Davenport dual angles decomposition is a new minimal parameterization of the rigid body displacement.For rigid body motion, the kinematic equations that link the instantaneous dual angular velocity to the time variation of Davenport dual angles are deduced. The cases of singularity of the decomposition are identified and a physical interpretation of these cases is given. The results interest the inverse kinematic of robotics, control theory and astrodynamics (full body relative orbital motion problem).

Four-dimensional traversable wormholes and bouncing cosmologies in vacuum

In this letter we point out the existence of solutions to General Relativity with a negative cosmological constant in four dimensions, which contain solitons as well as traversable wormholes. The latter connect two asymptotically locally AdS4 spacetimes. At every constant value of the radial coordinate the spacetime is a spacelike warped AdS3. We compute the dual energy momentum tensor at each boundary showing that it yields different results. We also show that these vacuum wormholes can have more than one throat and that they are indeed traversable by computing the time it takes for a light signal to go from one boundary to the other, as seen by a geodesic observer. We generalize the wormholes to include rotation and charge. When the cosmological constant is positive we find a cosmology that is everywhere regular, has either one or two bounces and that for late and early times matches the Friedmann-Lemaître-Robertson-Walker metric with spherical topology and an exponential scale factor.

Tensor product dual frames

To construct dual frames with good structure for a given frame is a fundamental problem in the theory of frames. The tensor product duals of tensor product frames can provide a rank-one decomposition of bounded antilinear operators between two Hilbert spaces. This paper addresses tensor product dual frames. We derive a necessary and sufficient condition for two tensor product Bessel sequences to be a pair of dual frames; obtain explicit expressions of all dual frames and tensor product dual frames of tensor product frames; and demonstrate the existence of non-tensor product dual frames of tensor product frames.

Arity Shape of Polyadic Algebraic Structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.

Inflation driven by massive vector fields with derivative self-interactions

Inspired by the Generalized Proca Theory, we study a vector–tensor model of inflation with massive vector fields and derivative self-interactions. The action under consideration contains a usual Maxwell-like kinetic term, a general potential term and a term with nonminimal derivative coupling between the vector field and gravity, via the dual Riemann tensor. In this theory, the last term contains a free parameter, [Formula: see text], which quantifies the nonminimal derivative coupling. In this scenario, taking into account a spatially flat Friedmann–Robertson–Walker (FRW) universe and a general vector field, we obtain the general expressions for the equation of motion and the total energy–momentum tensor. Choosing a Proca-type potential, a suitable inflationary regimen driven by massive vector fields is studied. In this model, the isotropy of expansion is guaranteed by considering a triplet of orthogonal vector fields. In order to obtain an inflationary solution with this model, the quasi de Sitter expansion was considered. In this case, the vector field behaves as a constant. Finally, slow-roll analysis is performed and slow-roll conditions are defined for this model, which, for suitable constraints of the model parameters, can give the required number of e-folds for sufficient inflation.

Geometry of exceptional super Yang-Mills theories

Some time ago, Sezgin, Bars and Nishino have proposed super Yang-Mills theories (SYM's) in $D=11+3$ and beyond. Using the "Magic Star" projection of $\mathfrak{e}_{8(-24)}$, we show that the geometric structure of SYM's in $11+3$ and $12+4$ space-time dimensions is recovered from the affine symmetry of the space $AdS_{4}\otimes S^{8}$, with the $8$-sphere being a line in the Cayley plane. By reducing to transverse transformations, along maximal embeddings, the near horizon geometries of the M2-brane ($AdS_{4}\otimes S^{7}$) and M5-brane ($AdS_{7}\otimes S^{4}$) are recovered. Generalizing the construction to higher, generic levels of the recently introduced "Exceptional Periodicity" (EP) and exploiting the embedding of semi-simple rank-3 Jordan algebras into rank-3 T-algebras of special type, yields the spaces $AdS_{4}\otimes S^{8n}$ and $AdS_{7}\otimes S^{8n-3}$, with reduced subspaces $AdS_{4}\otimes S^{8n-1}$ and $AdS_{7}\otimes S^{8n-4}$, respectively. Within EP, this suggests generalizations of the near horizon geometry of the M2-brane and its Hodge (magnetic) duals, related to $(1,0)$ SYM's in $(8n+3)+3$ dimensions, constituting a particular class of novel higher-dimensional SYM's, which we name exceptional SYM's. Remarkably, the $n=3$ level gives $AdS_{4}\otimes S^{23}$, hinting at M2 and M21 branes as solutions of bosonic M-theory, and reduction to $AdS_{3}\otimes S^{23}$ gives support for Witten's monstrous $AdS$/CFT construction.

Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model

We investigate whether the Standard Model, within the accuracy of current experimental measurements, satisfies the regularity in the form of Hodge duality condition introduced and studied in [9]. We show that the neutrino and quark mass-mixing and the difference of fermion masses are necessary for this property. We demonstrate that the current data supports this new geometric feature of the Standard Model, Hodge duality, provided that all neutrinos are massive.

ON THE COVARIANT REPRESENTATION OF INTEGRAL EQUATIONS OF THE ELECTROMAGNETIC FIELD

Gauss integral theorems for electric and magnetic fields, Faradays law of electromagnetic induction, magnetic field circulation theorem, theorems on the flux and circulation of vector potential, which are valid in curved spacetime, are presented in a covariant form. Covariant formulas for magnetic and electric fluxes, for electromotive force and circulation of the vector potential are provided. In particular, the electromotive force is expressed by a line integral over a closed curve, while in the integral, in addition to the vortex electric field strength, a determinant of the metric tensor also appears. Similarly, the magnetic flux is expressed by a surface integral from the product of magnetic field induction by the determinant of the metric tensor. A new physical quantity is introduced - the integral scalar potential, the rate of change of which over time determines the flux of vector potential through a closed surface. It is shown that the commonly used four-dimensional Kelvin-Stokes theorem does not allow one to deduce fully the integral laws of the electromagnetic field and in the covariant notation requires the addition of determinant of the metric tensor, besides the validity of the Kelvin-Stokes theorem is limited to the cases when determinant of metric tensor and the contour area are independent from time. This disadvantage is not present in the approach that uses the divergence theorem and equation for the dual electromagnetic field tensor. A new effect is predicted, according to which the circulation of magnetic field can appear even in the absence of electric current and with a constant electric field through the contour, if the area of this contour would change. By analogy with electromagnetic induction, for the magnetic field circulation to appear it is important that electric field flux that passes through the area of the contour would change over time.

The computational algorithm and numerical analysis of the signed diagonal Hodge star operator

The computational algorithm and numerical analysis of the signed diagonal Hodge star operator

Holographic thermodynamics of accelerating black holes

We present a careful study of accelerating black holes in anti-de Sitter spacetime, formulating the thermodynamics and resolving discrepancies that have appeared in previous investigations of the topic. We compute the dual stress-energy tensor for the spacetime and identify the energy density associated with a static observer at infinity. The dual energy-momentum tensor can be written as a three-dimensional perfect fluid plus a non-hydrodynamic contribution with a universal coefficient which is given in gauge theory variables. We demonstrate that both the holographic computation and the method of conformal completion yield the same result for the mass. We compare to previous work on black funnels and droplets, showing that the boundary region can be endowed with non-compact geometry, and comment on this novel holographic dual geometry.

On the equivalence of the norms of the discrete diffrential forms in discrete exterior calculus

For analysis of electromagnetic fields, differential forms are useful mathematical tools. Discrete exterior calculus (DEC) is a finite-difference-type framework of discretization of the discrete differential forms. Although several numerical estimations of errors of DEC have been reported, a tangible theoretical error bound is not established. One reason of this is that there are two norms that have been commonly used for the discrete differential forms. In this paper, we show that on any quasi-uniform mesh, the two norms of the discrete differential forms in DEC are equivalent, and moreover, the order of accuracy of the discrete differential forms are independent of the choice of the norm. As an application, it is shown that the accuracy of the discrete Hodge operator of DEC is of the second-order if any quasi-uniform Delaunay mesh is adopted.

Paravectors and the Geometry of 3D Euclidean Space

We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors and triparavectors to describe line segments and plane fragments in this space. A key point in this product of paravectors is the notion of the orientation of a point, in such a way that biparavectors representing line segments are the result of the product of points with opposite orientations. Incidence relations can also be formulated in terms of the product of paravectors. To study the transformations of points, lines, and planes, we introduce an algebra of transformations that is analogous to the algebra of creation and annihilation operators in quantum theory. The paravectors, biparavectors and triparavectors are mapped into this algebra and their transformations are studied; we show that this formalism describes in an unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale transformation. Using the concept of Hodge duality, we define a new operation called cotranslation, and show that the operation of perspective projection can be written as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation.

On Characters and Superdimensions of Some Infinite-Dimensional Irreducible Representations of osp(m|n)

Chiral spinors and self dual tensors of the Lie superalgebra $\mathfrak{osp}(m|n)$ are infinite dimensional representations belonging to the class of representations with Dynkin labels $[0,\ldots,0,p]$. We have shown that the superdimension of $[0,\ldots,0,p]$ coincides with the dimension of a $\mathfrak{so}(m-n)$ representation. When the superdimension is finite, these representations could play a role in supergravity models. Our technique is based on expansions of characters in terms of supersymmetric Schur functions. In the process of studying these representations, we obtain new character expansions.

The $${\mathcal {L}}_B$$ L B -cohomology on compact torsion-free $$\mathrm {G}_2$$ G 2 manifolds and an application to ‘almost’ formality

We study a cohomology theory $H^{\bullet}_{\varphi}$, called 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 $k \neq 3, 4$, but that $H^k_{\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_{\varphi}$. The $\mathcal L_B$-cohomology also satisfies a Poincar\'e 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.

An action for and hydrodynamics from the improved Large D membrane

It has recently been demonstrated that black hole dynamics at large D is dual to the motion of a probe membrane propagating in the background of a spacetime that solves Einstein’s equations. The equation of motion of this membrane is determined by the membrane stress tensor. In this paper we ‘improve’ the membrane stress tensor derived in earlier work to ensure that it defines consistent probe membrane dynamics even at finite D while reducing to previous results at large D. Our improved stress tensor is the sum of a Brown York term and a fluid energy momentum tensor. The fluid has an unusual equation of state; its pressure is nontrivial but its energy density vanishes. We demonstrate that all stationary solutions of our membrane equations are produced by the extremization of an action functional of the membrane shape. Our action is an offshell generalization of the membrane’s thermodynamical partition function. We demonstrate that the thermodynamics of static spherical membranes in flat space and global AdS space exactly reproduces the thermodynamics of the dual Schwarzschild black holes even at finite D. We study the long wavelength dynamics of membranes in AdS space that are everywhere approximately ‘parallel’ to the boundary, and demonstrate that the boundary ‘shadow’ of this membrane dynamics is boundary hydrodynamics with a definite constitutive relation. We determine the explicit form of shadow dual boundary stress tensor upto second order in derivatives of the boundary temperature and velocity, and verify that this stress tensor agrees exactly with the fluid gravity stress tensor to first order in derivatives, but deviates from the later at second order and finite D.

The Graf Product: A Clifford Structure Framework on the Exterior Bundle

The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent bundle, besides the volume element centrality, the Hodge operator and the so called truncated subalgebra as well.

Numerical convergence of discrete exterior calculus on arbitrary surface meshes

ABSTRACTDiscrete 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 for curved surfaces. This paper presents numerical evidence 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.

Exceptional M-brane sigma models and η-symbols

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.

Homogeneous and Inhomogeneous Maxwell’s Equations in Terms of Hodge Star Operator

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

Impact of time-of-day on diffusivity measures of brain tissue derived from diffusion tensor imaging

Diurnal fluctuations in MRI measures of structural and functional properties of the brain have been reported recently. These fluctuations may have a physiological origin, since they have been detected using different MRI modalities, and cannot be explained by factors that are typically known to confound MRI measures. While preliminary evidence suggests that measures of structural properties of the brain based on diffusion tensor imaging (DTI) fluctuate as a function of time-of-day (TOD), the underlying mechanism has not been investigated. Here, we used a longitudinal within-subjects design to investigate the impact of time-of-day on DTI measures. In addition to using the conventional monoexponential tensor model to assess TOD-related fluctuations, we used a dual compartment tensor model that allowed us to directly assess if any change in DTI measures is due to an increase in CSF/free-water volume fraction or due to an increase in water diffusivity within the parenchyma. Our results show that Trace or mean diffusivity, as measured using the conventional monoexponential tensor model tends to increase systematically from morning to afternoon scans at the interface of grey matter/CSF, most prominently in the major fissures and the sulci of the brain. Interestingly, in a recent study of the glymphatic system, these same regions were found to show late enhancement after intrathecal injection of a CSF contrast agent. The increase in Trace also impacts DTI measures of diffusivity such as radial and axial diffusivity, but does not affect fractional anisotropy. The dual compartment analysis revealed that the increase in diffusivity measures from PM to AM was driven by an increase in the volume fraction of CSF-like free-water. Taken together, our findings provide important insight into the likely physiological origins of diurnal fluctuations in MRI measurements of structural properties of the brain.