Exterior Differential Calculus Research Articles

Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge–Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency. In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

New Developments in Aggregation Economics

The goal of this article is to provide a general characterization of the aggregate behavior of a group in a market environment. We allow for public and private consumption, intragroup production, and consumption externalities within a group; we only assume that the group always reaches Pareto-efficient decisions. We show that aggregation problems involve a simple mathematical structure: The aggregate demand of the group, considered as a vector field, can be decomposed into a sum of gradients. We briefly introduce exterior differential calculus as a tool to study this structure. We analyze two main issues. One is testability: What restrictions (if any) on the aggregate demand function characterize the efficient behavior of the group? The second issue relates to identifiability; we investigate the conditions under which it is possible to recover the underlying structure—namely, individual preferences, the decision process, and the resulting intragroup transfers—from the group's aggregate behavior.

On the genesis of the Cartan–Kähler theory

The theory of exterior differential systems plays a crucial role in Cartan’s whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research thus far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan’s early papers, no specific analysis of Cartan’s subsequent works laying the foundations of what nowadays is known as the Cartan–Kahler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan’s work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan’s relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan’s theory of exterior differential systems.

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.

Exterior Difference System on Hypercubic Lattice

Based on the exterior differential calculus, we define the exterior difference system on the hypercubic lattice. The pairing formula, bracket, contract operator and Lie derivative operator for this system are also constructed. The discrete H. Cartan formulas are obtained too. After those preparations, we prove the discrete Frobenius Theorem, discrete retraction theorem and discrete Cartan–Kahler theorem.

Applying exterior differential calculus to economics: a presentation and some new results

The paper provides an introduction to exterior differential calculus, and an application to the standard problem of the characterization of aggregate demand in an economy in which the number of agents is smaller than the number of goods.

Applying exterior differential calculus to economics: a presentation and some new results: A comment

Applying exterior differential calculus to economics: a presentation and some new results: A comment

Aggregation and Market Demand: An Exterior Differential Calculus Viewpoint

We analyze under which conditions a given vector field can be disaggregated as a linear combination of gradients. This problem is typical of aggregation theory, as illustrated by the literature on the characterization of aggregate market demand and excess demand. We argue that exterior differential calculus provides very useful tools to address these problems. In particular, we show, using these techniques, that any analytic mapping in Rn satisfying Walras Law can be locally decomposed as the sum of n individual, utility-maximizing market demand functions. In addition, we show that the result holds for arbitrary (price-dependent) income distributions, and that the decomposition can be chosen such that it varies continuously with the mapping. Finally, when income distribution can be freely chosen, then decomposition requires only n/2 agents.

Axial anomaly in lattice abelian gauge theory in arbitrary dimensions

The axial anomaly of lattice abelian gauge theory on a hyper-cubic regular lattice in arbitrary even dimensions is investigated by applying the method of exterior differential calculus. The topological invariance, gauge invariance and locality of the axial anomaly determine the explicit form of the topological part. The anomaly is obtained up to a multiplicative constant for finite lattice spacing and can be interpreted as the Chern character of the abelian lattice gauge theory.

A calculus based on a q-deformed Heisenberg algebra

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.

On certain realizations of the q-deformed exterior differential calculus

We investigate two particular realizations of a q-deformed differential calculus at q being a primitive root of unity, qN = 1. Particular attention is paid to the Z3-graded case N = 3. First we construct an analogue of the exterior differential calculus on a manifold, then we introduce a discrete realization of such a calculus on generalized Clifford algebras. Finally, combining both constructions, we discuss a ZN-graded generalization of gauge theory.

A calculus based on a

We show how one can construct a differential calculus over an algebra where position variables x and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by x and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo 3. We show how the notion of covariant derivation can be generalized with a 1-form A, and we give the expression in local coordinates of the curvature 3-form. Finally, the introduction of notions of a scalar product and integration of the Z_3-graded exterior forms enables us to define variational principle and to derive the differential equations satisfied by the curvature 3-form. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F_{ik} and its covariant derivatives D_i F_{km}.

Nonstandard quantum groups and superization

We obtain the universal R-matrix of the nonstandard quantum group associated to the Alexander–Conway knot polynomial. We show further that this nonstandard quantum group is related to the super-quantum group Uq gl(1‖1) by a general process of superization, which we describe. We also study a twisted variant of this nonstandard quantum group and obtain, as a result, a twisted version of Uq gl(1‖1) as a q supersymmetry of the exterior differential calculus of any quantum plane of Hecke type, acting by mixing the bosonic xi coordinates and the forms dxi.

QPFT operator algebras and commutative exterior differential calculus

The reduction of the structure theory of the operator algebras of quantum projective (sl(2, ℂ)-invariant) field theory (QPFT operator algebras) to a commutative exterior differential calculus by means of the operation of renormalization of a pointwise product of operator fields is described.

Algebraic structures of the quantum projective [sl(2,C)-invariant] field theory: The commutative version of Elie Cartan’s exterior differential calculus

The commutative exterior differential calculus, appearing in the quantum projective field theory, is described.

The application of Regge calculus to quantum gravity and quantum field theory in a curved background

The application of Regge calculus to quantum gravity and quantum field theory in a curved background is discussed. A discrete form of exterior differential calculus is developed, and this is used to obtain Laplacians for P -forms on the Regge manifold. To assess the accuracy of these approximations, the eigenvalues of the discrete Laplacians were calculated for the regular tesselations of S 2 and S 3 . The results indicate that the methods obtained in this paper may be used in curved space–times with an accuracy comparing with that obtained in lattice gauge theories on a flat background. It also becomes evident that Regge calculus provides particularly suitable lattices for Monte-Carlo techniques.

Non-static electromagnetic field in general relativity VI

The equations of Rainich's “already unified field theory” are investigated for the typeG3 II acting on the spatially homogeneous hypersurfacesx0=constant. Two solutions of a non-static electromagnetic field for the above case are presented here by using the exterior differential calculus. The space-time admits a three-parameter continuous group of motions, the minimum invariant varieties being the geodesically parallel parametric surfacesx1=constant,x2=constant andx3=constant orthogonal tox0=constant at points of the geodesics.

On the covariance of the Fokker-Planck equation

Abstract We study the covariance of the Fokker-Planck equation under general gross variables transformations by means of Riemann differential geometry using an affine connection Γ ϱ μν unsymmetric in their lower indices and without assuming that the covariant derivative of the diffusion tensor D μν ;ϱ be zero. We come to the conclusion that to achieve our aim we only need the value of the contraction Γ ϱ ϱν , all other components of the affine connection remaining completely arbitrary. We argue, therefore, that the most economic way of presenting the covariance of the Fokker-Planck equation is by means of exterior differential calculus. As an application we study physical systems under detailed balance showing that for them the irreversible part of the contravariant drift vector, that is then uniquely determined, is related only to a symmetric tensor while its reversible component is exclusively related to an antisymmetric tensor. A criticism of a compact Fokker-Planck equation is also included.