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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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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)

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.

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

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.

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.

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

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.

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.

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.

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

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.

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

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

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.

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.

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.

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.

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

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

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)

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.

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).

We hope to understand the Hodge theoretic aspect of mirror symmetry in the framework of the fundamental diagram of log mixed Hodge theory. We give a formulation of mirror conjecture for Calabi–Yau threefolds as the coincidence of log period maps with specified sections under the mirror map. Since a variation of Hodge structure with unipotent monodromies on a product of punctured discs uniquely extends over the puncture to a log Hodge structure, we can work on the boundary point over which the log Riemann–Hilbert correspondence exists, and we can observe clearly in high resolution the behavior of Z-structure over the boundary point (cf. notes in Introduction below). This is an advantage of log Hodge theory.

In this work, we present a Hodge-type decomposition for variable exponent spaces of Clifford-valued functions, where one of the components is the kernel of the parabolic-type Dirac operator.

We show that the Suslin forms are themselves a product of Suslin matrices (over the integers) and that their corresponding linear transformation is the Hodge star operator. The Fossum–Foxby–Iversen theory of acyclic complexes, as developed by Suslin, plays a vital role in this analysis.

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

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

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

One of the most important questions regarding vector families concerns the independence or the relation among them.

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].

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

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.

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.

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.

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.

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.