Theory Of Differential Forms Research Articles

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.

Conserved momenta of ferromagnetic solitons through the prism of differential geometry

The relation between symmetries and conservation laws for solitons in a ferromagnet is complicated by the presence of gyroscopic (precessional) forces, whose description in the Lagrangian framework involves a background gauge field. This makes canonical momenta gauge-dependent and requires a careful application of Noether’s theorem. We show that Cartan’s theory of differential forms is a natural language for this task. We use it to derive conserved momenta of the Belavin–Polyakov skyrmion, whose symmetries include translation, global spin rotation, and dilation.

Моделювання скінченних неоднорідностей дискретними особливостями

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

Normal forms and singularities of non-holonomic robotic systems: A study of free-floating space robots

This paper promotes an idea of describing singularities of non-holonomic robotic systems by normal forms under feedback of associated control systems. In order to increase the explanatory power of the presentation this idea has been applied to the dynamics of a planar free-floating space robot composed of a base (a spacecraft) and a k-DOF (Degrees Of Freedom) on board manipulator. A general Lagrangian dynamics model of the robot is provided. Relying on the preservation of the linear and angular momenta, the affine Pfaffian constraints of motion are obtained. These constraints are expressed in specific coordinates resulting in a pre-normal form of the dynamics model of the robot, represented as a control-affine system, and an associated differential form. The pre-normal form is further transformed by feedback to a normal form, using some tools of the theory of differential forms. Normal forms of the space robot dynamics for k=2 up to k=8 DOF on board are computed that describe the ultimate structure of the dynamics model as well as characterize configuration singularities of the space robot.

On Weak Solutions to Dirac-Harmonic Equations for Differential Forms

In this paper, we consider the Dirac-harmonic equations for differential forms with Dirichlet boundary data in weak sense. Using the Hodge-decomposition theory of differential forms and Browder–Minty theory of monotone operators, the existence is obtained. According to the Poincare-type inequality of Dirac–Sobolev space, we get the higher integrability.

Modeling continuous distributions of dislocations in multi-walled nanotube composites

A modeling paradigm for continuous distributions of defects in multi-walled nanotube composites is proposed based on the theory of differential forms. A necessary and sufficient condition for a defect-free composite is derived and, concomitantly, a measure of defect density is proposed. Mathematically, this condition ensures the consistency of an overdetermined system of partial differential equations. Physically, the condition expresses the fact that each multilayered nanotube can find mutually coherent nanotubes in its neighborhood. Several examples of defect-free arrangements are presented and discussed.

Besov and Triebel–Lizorkin regularity for the Hodge decomposition and applications to magnetic potentials

The Hodge decomposition is a fundamental result in the theory of differential forms. It has been extensively studied from different perspectives and it is the source of several applications in physics and mathematics. For example, it is important in the analysis of Navier–Stokes' and Maxwell's equations, it can be used to prove results on inverse scattering problems in quantum mechanics and it is useful for the study of magnetic potentials associated to a magnetic field. The regularity properties of the Hodge decomposition have been studied in several works in the last decades, most prominently it has been proved that Sobolev regularity holds true. In this paper we prove regularity for the Hodge decomposition in a much larger scale of function spaces: Besov and Triebel–Lizorkin spaces on compact Riemannian manifolds with boundary. We prove regularity in such spaces for the Hodge–Morrey decomposition and the Hodge–Morrey–Friedrichs decomposition. We, furthermore, state and prove a refinement for the Hodge–Morrey–Friedrichs decomposition (with regularity) that allows expressing every differential form as the sum of the differential of a form plus the co-differential of another form plus a harmonic field belonging to a finite dimensional space. This holds true even if the original form is barely differentiable (with sth order of differentiability, for every real number s>0). We finally provide an application to the study of magnetic potentials associated to magnetic fields. More precisely, we prove the existence of a magnetic potential associated to a magnetic field. The former being one time more differentiable than the latter, in terms of Besov and Triebel–Lizorkin regularity. This is an important motivation for us, since we have the intention to apply our results to further investigations in relativistic inverse (time dependent) scattering problems in quantum mechanics, where Besov and Triebel–Lizorkin regularity for magnetic potentials is relevant.

Spin-invariant valuations on the octonionic plane

The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of Val2Spin (9).

Material description of fluxes in terms of differential forms

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.

The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

New integral representations in the linear theory of viscoelastic materials with voids

We investigate the two basic internal BVPs related to the linear theory of viscoelasticity for Kelvin-Voigt materials with voids by means of the potential theory. By using an indirect boundary integral method, we represent the solution of the first (second) BVP of steady vibrations in terms of a simple (double) layer elastopotential. The representations achieved are different from the previously known ones. Our approach hinges on the theory of reducible operators and on the theory of differential forms.

Mimetic discretization of the Abelian Chern-Simons theory and link invariants

A mimetic discretization of the Abelian Chern-Simons theory is presented. The study relies on the formulation of a theory of differential forms in the lattice, including a consistent definition of the Hodge duality operation. Explicit expressions for the Gauss Linking Number in the lattice, which correspond to their continuum counterparts are given. A discussion of the discretization of metric structures in the space of transverse vector densities is presented. The study of these metrics could serve to obtain explicit formulae for knot an link invariants in the lattice.

ON AN INTEGRAL EQUATION OF THE FIRST KIND ARISING IN THE THEORY OF COSSERAT

In this paper we study an integral equation of the first kind concerning an indirect boundary integral method for the Dirichlet problem in the theory of Cosserat continuum. Our method hinges on the theory of reducible operators and on the theory of differential forms.

A Recursive Sparsification of the Inverse Hodge Matrix

When applying the theory of differential forms to solve wave propagation problems in time domain, we must solve at each time step a sparse linear system defined by the insertion of constitutive laws via the mass matrices. In this paper, we describe a recursive technique to efficiently calculate the approximated inverse of Hodge matrix. The fundamental idea is to recursively decompose the mass matrix in to a decreasing size sequence of matrices using block matrix inversion. During the recomposition process, the matrix is sparsified. Numerical results are presented to validate our approach.

Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

The present paper is concerned with an indirect method to solve the Dirichlet and the traction problems for Lamé system in a multiply connected bounded domain of ℝn, n ≥ 2. It hinges on the theory of reducible operators and on the theory of differential forms. Differently from the more usual approach, the solutions are sought in the form of a simple layer potential for the Dirichlet problem and a double layer potential for the traction problem.2000 Mathematics Subject Classification. 74B05; 35C15; 31A10; 31B10; 35J57.

Differential forms on log canonical spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.

Multivariate tensor-based morphometry on surfaces: Application to mapping ventricular abnormalities in HIV/AIDS

Here we developed a new method, called multivariate tensor-based surface morphometry (TBM), and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using concepts from differential geometry and the theory of differential forms, we created mathematical structures known as holomorphic one-forms, to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also provides a natural way to register anatomical surfaces across subjects, and improves on prior methods as it handles surfaces that branch and join at complex 3D junctions. To analyze anatomical differences, we computed new statistics from the Riemannian surface metrics—these retain multivariate information on local surface geometry. We applied this framework to analyze lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8 healthy controls. Our method detected a 3D profile of surface abnormalities even in this small sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group differences, relative to other TBM-based methods including analysis of the Jacobian determinant, the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based morphometry studies of the brain.

Freud’s Identity of Differential Geometry, the Einstein-Hilbert Equations and the Vexatious Problem of the Energy-Momentum Conservation in GR

We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud’s identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud’s identity (which is a decomposition of the Einstein indexed 3-forms $$*{\mathcal{G}^{a}}$$ in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud’s identity is usually applied in the formulation of energy-momentum “conservation laws” in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the “tricks of the trade” of the subject).

Conservation laws in curvilinear coordinates: A short proof of Vinokur’s Theorem using differential forms

In computational fluid dynamics it is important to maintain strong conservation form in the equations of motion regardless of the coordinate system used. Vinokur’s Theorem says that conservation form can always be maintained. However, the proof is long and has non-obvious steps. In this note a short proof of Vinokur’s Theorem is given which is both simple and illuminating. It uses the theory of differential forms which may also be useful in other algorithmic constructions.

An [formula omitted] theory for differential forms on path spaces I

An L 2 theory of differential forms is proposed for the Banach manifold of continuous paths on a Riemannian manifold M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M. A Hodge decomposition is given for L 2 H-one-forms, and the structure of H-two-forms is described. The dual operator d ∗ is analysed in terms of a natural connection on the H-tangent spaces. Malliavin calculus is a basic tool.