Exterior Differentiation Research Articles

Geometry of concircular curvature tensor of Nearly Kahler manifold

In this paper, we study the necessary condition where a nearly Kahler manifold of flat concircular tensor has been found. And the relationship between these invariants and additional properties of symmetry concircular tensor, as well as geometrical sense of the reference (manipulation) in zero of these invariants are studied.
 Let M - smooth manifold of dimension 2n; - algebra of smooth functions on M; - the module of smooth vector fields on manifold of M; g = <·, > - Riemannian metrics; Ñ – Riemannian connection of the metrics g on M; d - the operator of exterior differentiation. In the further all manifold, Tensor fields, etc. objects are assumed smooth a class .

Managing Reference‐Group Effects in Sequential Product Upgrades

In this study, we examine how firms can leverage reference‐group effects (RGEs) in sequential introduction of upgraded products. RGEs arise when consumers engage in social comparisons: leaders desire to distinguish themselves from followers and followers desire to mimic leaders. We find that RGEs have a non‐monotonic impact on the profitability of product upgrades: a stronger RGE first increases and then decreases profits. This inverted‐U effect is fundamentally driven by two opposite effects of RGEs: a positive push effect on leaders’ upgrading to a new generation when followers adopt an existing product that leaders possess, and a negative pull effect on followers’ adoption of the existing product when leaders upgrade to the new product. We also offer insights on inter‐generational differentiation in the presence of RGEs. We find that by coordinating product differentiation in two dimensions, functional improvement and exterior form, the firm is able to eliminate the negative pull effect, which occurs when the strength of RGEs is intermediate. In such cases, both functional improvement and exterior differentiation increase with the strength of RGEs. When the strength of RGEs is sufficiently high, however, the firm will focus on exterior differentiation rather than functional improvement. Managerial implications are discussed.

Uniqueness of exterior differentiation on locally finite graphs

In this short note, we prove the uniqueness of exterior differentiation on locally finite graphs.

A Gromov’s dimension comparison estimate for rectifiable sets

We extend the validity of a Gromov’s dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.

Riesz transforms and spectral multipliers of the Hodge–Laguerre operator

On $\mathbb{R}^d_+$, endowed with the Laguerre probability measure $\mu_\alpha$, we define a Hodge-Laguerre operator $\mathbb{L}_\alpha=\delta\delta^*+\delta^* \delta$ acting on differential forms. Here $\delta$ is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives $\partial_{x_i}$ are replaced by the "Laguerre derivatives" $\sqrt{x_i}\partial_{x_i}$, and $\delta^*$ is the adjoint of $\delta$ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure $\mu_\alpha$. We prove dimension-free bounds on $L^p$, $1<p<\infty$, for the Riesz transforms $\delta \mathbb{L}_\alpha^{-1/2}$ and $\delta^* \mathbb{L}_ \alpha^{-1/2}$. As applications we prove the strong Hodge-de Rahm-Kodaira decomposition for forms in $L^p$ and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in $L^p$. We also prove that for suitable functions $m$ the operator $m(\mathbb{L}^\alpha)$ is bounded on $L^p$, $1<p<\infty$.

Differential geometry for physicists and mathematicians: moving frames and differential forms: from Euclid past Riemann

Introduction: Orientations Tools: Differential Forms Vector Spaces and Tensor Products Exterior Differentiation Two Klein Geometries: Affine Klein Geometry Euclidean Klein Geometry Cartan Connections: Generalized Geometry Made Simple Affine Connections Euclidean Connections Riemannian Spaces and Pseudo-Spaces The Future?: Extensions of Cartan Understand the Past to Imagine the Future A Book of Farewells.

A Low Rank Property and Nonexistence of Higher-Dimensional Horizontal Sobolev Sets

We establish a “low rank property” for Sobolev mappings that pointwise solve a first-order nonlinear system of PDEs, whose smooth solutions have the so-called “contact property”. As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group \(\mathbb {H}^1\), and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of \(\mathbb {H}^1\). This provides a complete solution to a question raised in a paper by Balogh et al. (Ergodic Theory Dynam Syst 26(3):621–651, 2006). Our approach differs from the previous ones. Its technical aspect consists in performing an “exterior differentiation by blow-up”, where the standard distributional exterior differentiation is not possible. This method extends to higher-dimensional Sobolev mappings, taking values in higher-dimensional Heisenberg groups.

Mesonic Differential Forms

When M is a differentiable manifold, the exterior differential k -forms on M are the alternate k -linear forms on the tangent bundle T(M) . The mesonic differential k -forms are the k -linear forms on T(M) that are alternate with respect to the variables of odd rank, and also alternate with respect to the variables of even rank. After a reminder about meson algebras, and after the presentation of elementary properties of mesonic forms, this article introduces the mesonic differentiation of mesonic forms, which can be partially compared to the exterior differentiation of exterior forms. Some applications to riemannian manifolds and flat manifolds follow.

The vanishing of [formula omitted] harmonic one-forms on based path spaces

We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L2 two-forms, respectively; these are considered as extended Clark–Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap.

Exterior Algebra with Differential Forms on Manifolds

The concept of an exterior algebra was originally introduced by H. Grassman for the purpose of studying linear spaces. Subsequently Elie Cartan developed the theory of exterior differentiation and successfully applied it to the study of differential geometry [8], [9] or differential equations. More recently, exterior algebra has become powerful and irreplaceable tools in the study of differential manifolds with differential forms and we develop theorems on exterior algebra with examples.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11528 Dhaka Univ. J. Sci. 60(2): 247-252, 2012 (July)

Hodge-Frobenius equations and the Hodge–Bäcklund transformation

Linear and nonlinear Hodge-like systems for 1-forms are studied with an assumption equivalent to complete integrability substituted for the requirement of closure under exterior differentiation. The systems are placed in a variational context and properties of critical points are investigated. Certain standard choices of energy density are related by Bäcklund transformations which employ basic properties of the Hodge involution. These Hodge-Bäcklund transformations yield invariant forms of classical Bäcklund transformations that arise in diverse contexts. Some extensions to higher-degree forms are indicated.

Uniqueness of rotating charged black holes in five-dimensional minimal gauged supergravity

We study a five-dimensional spacetime admitting, in the presence of torsion, a non-degenerate conformal Killing–Yano 2-form which is closed with respect to both the usual exterior differentiation and the exterior differentiation with torsion. Furthermore, assuming that the torsion is closed and co-closed with respect to the exterior differentiation with torsion, we prove that such a spacetime is the only spacetime given by the Chong–Cvetič–Lü–Pope solution for stationary, rotating charged black holes with two independent angular momenta in five-dimensional minimal gauged supergravity.

Exact stationary solutions independent of energy for strongly nonlinear stochastic systems of multiple degrees of freedom

A new procedure is proposed to construct strongly nonlinear systems of multiple degrees of freedom subjected to parametric and/or external Gaussian white noises, whose exact stationary solutions are independent of energy. Firstly, the equivalent Fokker-Planck-Kolmogorov (FPK) equations are derived by using exterior differentiation. The main difference between the equivalent FPK equation and the original FPK equation lies in the additional arbitrary antisymmetric diffusion matrix. Then the exact stationary solutions and the structures of the original systems can be obtained by using the coefficients of antisymmetric diffusion matrix. The obtained exact stationary solutions, which are generally independent of energy, are for the most general class of strongly nonlinear stochastic systems multiple degrees of freedom (MDOF) so far, and some classes of the known ones dependent on energy belong to the special cases of them.

From Tracial Anomalies to Anomalies in Quantum Field Theory

ζ-regularized traces, resp. super-traces, are defined on a classical pseudo-differential operator A by: \({{{{tr}}^Q(A):= {{f.p.}} \, {{tr}}(A Q^{{-z}})_{{\vert_{{z=0}}}}, \quad{{resp.}} \quad {{str}}^Q(A):= {{f.p.}} \, {{str}}(A Q^{{-z}})_{{\vert_{{z=0}}}},}}\) where f.p. refers to the finite part and Q is an (invertible and admissible) elliptic reference operator with positive order. They are commonly used in quantum field theory in spite of the fact that, unlike ordinary traces on matrices, they are neither cyclic nor do they commute with exterior differentiation, thus giving rise to tracial anomalies. The purpose of this article is to show, on two examples, how tracial anomalies can lead to anomalous phenomena in quantum field theory.

Elie Cartan and pan-geometry of multispatial hyperspace

Elie Cartan has proved that highest dimensionality of any simple geometric space is three and that an exterior differentiation of a 3D+ geometric object gives bivector, which may correspond to some two 2D surfaces as if the 3D+ geometric object comprised two 3D objects. Since one cannot increase the dimensionality of a 3D space even though more than four independently varying physical magnitudes do exist, then an expansion of dimensionality requires a multispatial hyperspace that contains many simple geometric 3D spaces. Presence of such a hyperspace prompts for an entirely new concept of vectors with an isometric operation of vector multiplication of traditional vectors (3-tuples). This new operation on 3-vectors implies presence of a 3D mass-based linear vector space and consequently thus a 9D geometric hyperspace for classical mechanics alone. Also an outline of entirely new, synthetic approach to physics and mathematics is introduced. This synthetic approach can be used to design a computer-aided knowledge extracting system, which could generate entirely new scientific knowledge.

A multivector data structure for differential forms and equations

We use tools from algebraic topology to show that a class of structural differential equations may be represented combinatorially, and thus, by a computer data structure. In particular, every differential k-form may be represented by a formal k-cochain over a cellular structure that we call a starplex, and exterior differentiation is equivalent to the coboundary operation on the corresponding k-cochain. Furthermore, there is a one-to-one correspondence between this model and the classical finite cellular model supported by the Generalized Stokes’ Theorem and translation between the two models can be completely automated. Our results point the way to a common combinatorial and data structure well-suited for a physical modeling computer algebra that unifies finite and infinitesimal, symbolic and numeric, geometric and physical descriptions of distributed phenomena. We illustrate the advantages of our approach by a prototype interactive physics editor that uses the computer algebra to automatically translate intuitive geometrical/physical descriptions of balance conditions created by the user into the corresponding symbolic differential and integral equations.

Canonical structure of classical field theory in the polymomentum phase space

Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder-Weyl (DW). The bi-vertical ( n + 1)-form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset, it gives rise to the invariantly defined map between horizontal forms playing the role of dynamical variables and the so-called vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Z-graded Lie algebra on the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a “higher order” and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian n-form H vol ( vol ) is the space-time volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed.

EXACT STATIONARY SOLUTIONS OF AVERAGED EQUATIONS OF STOCHASTICALLY AND HARMONICALLY EXCITED MDOF QUASI-LINEAR SYSTEMS WITH INTERNAL AND/OR EXTERNAL RESONANCES

The exact stationary solutions of the averaged equations of stochastically and harmonically excitedndegree-of-freedom quasi-linear systems withminternal and/or external resonances are obtained as functions of bothnindependent amplitudes andmcombinations of phase angles. To make the solutions more general, the equivalent stochastic systems of the averaged equations are obtained by using the differential forms and exterior differentiation. By considering the periodic boundary conditions with respect tomcombinations of phase angles, the probability potentials of the exact stationary solutions of the equivalent stochastic systems are expanded into anm-fold harmonic series ofmcombinations of phase angles, and the exact stationary solutions are obtained for the case where the averaged equations belong to the class of stationary potential. To examples are given to illustrate the application of the proposed procedure.

Descent equations of Yang-Mills anomalies in noncommutative geometry

Consistent Yang-Mills anomalies ∫ ω 2 n− k k−1 ( n ϵ N, k = 1,2,…, 2 n) as described collectively by Zumino's descent equations δω 2 n− k k−1 + d ω 2 n− k−1 k = 0 starting with the Chern character Ch 2 n = d ω 2 n−1 0 of a principal SU ( N) bundle over a 2 n-dimensional manifold are considered (i.e. ∫ ω 2 n− k k−1 are the Chern-Simons terms ( k = 1), axial anomalies ( k = 2), Schwinger terms ( k = 3), etc. in (2 n − k) dimensions). A generalization in the spirit of Conne's non-commutative geometry using a minimum of data is found. For an arbitrary graded differential algebraa Ω = ⊗ k = 0 ∞Ω (k) with exterior differentiation d, form-valued functions ▪ are constructed which are connected by generalized descent equations δω 2 n− k k−1 + d ω 2 n− k−1 k = (…). Here Ch 2 n = ( F A ) n , where F A = d( A) + A 2 for A ϵ Ω (1) , and (…) is not zero but a sum of graded commutators which vanish under integrations (traces). The problem of constructing Yang-Mills anomalies on a given graded differential algebra is thereby reduced to finding an interesting integration ∫ on it. Examples for graded differential algebras with such integrations are given, and thereby noncommutative generalizations of Yang-Mills anomalies are found.

Linearization by Generalized Input-Output Injection and Synthesis of Observers

The problem under interest is the linearization of nonlinear systems by generalized input-output injection in order to built observers with linear error dynamics. The method is based on the study of the structure of the input-output differential equations; thus, the problem is solved as a realization one. A necessary and sufficient condition and an algorithm give the generalized state coordinates transformation, if it exists, and the generalized input-output injections which linearize multi-input multi-output nonlinear systems. This condition uses exterior differentiation and is fully constructive.