Integration Of Differential Forms Research Articles

Estimates of Weighted Hardy-Littlewood Inequality for Differential Forms

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields. As an important tool the Hardy-Littlewood inequality have been playing critical roles in many mathematics, including potential analysis, partial differential equations and the theory of elasticity.in this paper, the local and the global parametric weighted Hardy-Littlewood inequality had been proved on Riemannian manifolds by using the generalized weak reverse Holder inequality for A-harmonic tensors. The required versions can be obtained by selecting the suitable values for the parameters contained in each theorem. These results can be considered as generalizations of the classical Hardy-Littlewood inequality. As applications , the global versions of the Hardy-Littlewood inequality had been extended in L s -averaging domain and L s (w,l) averaging domain. Finally, the global parametric Hardy-Littlewood inequality for the projection operator had been obtained using the global weighted inequality for A -harmonic tensors. These results can also be used to study the integrability of differential forms and estimate the integrals for differential forms.

Book review: Marcelo Epstein, The Geometrical Language of Continuum Mechanics

Intended mainly for continuum mechanicists, Epstein's book introduces modern geometry and some of its applications to theoretical continuum mechanics. Thus, examples for the mathematical objects introduced are chosen from the realm of mechanics. In particular, differentiable manifolds, tangent and cotangent bundles, Riemannian manifolds, Lie derivatives, Lie groups, Lie algebras, differential forms and integration theory are presented in the main part of the book. Once the reader's familiarity with continuum mechanics is used for the introduction of basic geometry, geometry is used in order to generalize notions of continuum mechanics. Integration of differential forms is used to formulate flux theory on manifolds devoid of a Riemannian structure. More specialized topics, namely, Whitney's geometric integration theory and Sikorski's differential spaces are used to relax smoothness assumptions for bodies and fields defined on them. Finally, an overview is given of the work that Epstein and co-workers carried out in recent years where the theory of inhomogeneity of constitutive relations is developed using the geometry of principal fiber bundles, G-structures and connections.

Weighted Decomposition Estimates for Differential Forms

After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.

On the ternary complex analysis and its applications

In previous works, a possible extension of the complex numbers together with its connected trigonometry was introduced. In the present paper, we focus on the simplest case of ternary complex numbers. Then, some types of holomorphy adapted to the ternary complex numbers and the corresponding results upon integration of differential forms are given. Several physical applications are discussed and, in particular, one type of holomorphic function gives rise to a new form of stationary magnetic field. The movement of a monopole-type object in this field is then studied and shown to be integrable. The monopole scattering in the ternary field is finally studied.

Integration of differential forms on manifolds with locally-finite variations. II

In part I of the paper, we have defined n-dimensional C0-manifolds in ℝn(m ≥ n) with locally-finite n-dimensional variations (a generalization of locally-rectifiable curves to dimensionn > 1) and integration of measurable differential n-forms over such manifolds. The main result of part II states that an n-dimensional manifold that is C1-embedded into ℝm has locally-finite variations and the integral of a measurable differential n-form defined in part I can be calculated by the well-known formula. Bibliography: 5 titles.

Integration of differential forms on manifolds with locally-finite variations

It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C1-manifolds in ℝm (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C0-manifolds in ℝm with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main result states that any such manifold generates an n-dimensional current in ℝm (i.e., any compactly supported C∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles.

The Geometry of Physics: An Introduction

Preface Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields 2. Tensors and exterior forms 3. Integration of differential forms 4. The Lie derivative 5. The Poincare Lemma and potentials 6. Holonomic and non-holonomic constraints Part II. Geometry and Topology: 7. R3 and Minkowski space 8. The geometry of surfaces in R3 9. Covariant differentiation and curvature 10. Geodesics 11. Relativity, tensors, and curvature 12. Curvature and topology: Synge's theorem 13. Betti numbers and De Rham's theorem 14. Harmonic forms Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups 16. Vector bundles in geometry and physics 17. Fiber bundles, Gauss-Bonnet, and topological quantization 18. Connections and associated bundles 19. The Dirac equation 20. Yang-Mills fields 21. Betti numbers and covering spaces 22. Chern forms and homotopy groups Appendix: forms in continuum mechanics.

Topological quantum field theory in quaternionic geometry

A topological quantum field theory on a 4 k-dimensional manifold M admitting an almost quaternionic structure is proposed. Expectation values of certain operators on M are proved to be independent of the choice of an almost quaternionic structure used in calculations and thus carry only smooth information about M. These invariants are explicitly expressed as integrals of differential forms over the instanton moduli space associated with a chosen almost quaternionic structure. When M admits a hyperKähler structure the topological field theory has three additional supersymmetries which induce three complex structures on the associated instanton moduli space proving thus that the latter is a hypercomplex manifold. In this case an analogue of the four-dimensional Donaldson map is constructed which provides a number of candidates for new invariants of 4 k-dimensional hypercomplex structures. In the case k = 1 the proposed topological theory on an almost quaternionic manifold reproduces the Witten interpretation of the four-dimensional Donaldson invariants.

ℒ2-cohomologie des espaces stratifiésdes espaces stratifiés

The first results relating intersection homology with ℒ2-cohomology were found by Cheeger, Goresky and MacPhersson (cf.[4] and [5]). The first spaces considered were the compact stratified pseudomanifolds with isolated singularities. Later, Nagase extended this result to any compact stratified spaceA possessing a Cheeger type riemannian metric μ (cf. [12]). The proof of the isomorphism\(H_{(2)}^* \left( {A - \Sigma ,\mu } \right) \cong IH_*^{\bar p} \left( A \right)\) uses the axiomatic caracterisation of the intersection homology of [2]. In this work we show how to realize this isomorphism by the usual integration of differential forms on simplices. The main tool used is the blow up of A into a smooth manifold, introduced in [2]. We also prove that any stratified space possesses a Cheeger type riemannian metric.

Abelian and nonabelian mathematics

Many theories that are part of the most beautiful chapters in the mathematics of the nineteenth and twentieth centuries have a common algebraic nucleus: the duality theory of abelian groups and their groups of characters. It is natural to assume that a series of basic difficulties in mathematics can be explained by the lack of a generalization of this duality to nonabelian groups. Such a point of view has been presented in A. Weil's paper [6]. The "nonabelian mathematics of the future" philosophy also inspired me when I was just starting to work on mathematics. Now, half a century later, I think it would be interesting to sum up the achievements in this direction, the more so since, just as before, the unsolved problems ou tnumber the solved ones, so that the old program may apparently inspire yet another generation. The goal of the following survey is to give a general idea of the intertwining of ideas that arise in this area. The most important example of a pairing is provided by the integration of differential forms. If, for example, 00 is a 1-form, and r is a curve on a manifold X, then f~00 can be considered as a pairing (r More precisely, the pairing is defined between the space H~(X;R) of one-dimensional homologies and the quotient space {o~ : d00 = 0}/{00 = df}. The nondegeneracy of this pairing and its generalization to m-dimensional chains is guaranteed by the de Rham theorem. (One could also substitute C for R everywhere.) A nondegenerate symmetric form on V determines an isomorphism V* -~ V. If the real vector space V is oriented, then the operation W ~ W • carries over to

Integration of differential forms on schemes.

Integration of differential forms on schemes.

Integration of differential forms of the classesW*p,q

Integration of differential forms of the classesW*p,q

A problem of integral geometry in RPn, connected with the integration of differential forms

A problem of integral geometry in RPn, connected with the integration of differential forms

Integration of differential forms on supermanifolds

Integration of differential forms on supermanifolds

Plateau's Problem; an Invitation to Varifold Geometry.

The phenomena of least area problems Integration of differential forms over rectifiable sets Varifolds Variational problems involving varifolds References Additional references Index.

Differential forms and nuclear structure

The solution of a set of Schrodinger’s equation can be considered from the point of view of integration of differential forms over simplices in higher dimensional differentiable manifolds. In this way, the problems of nuclear structure are transferred from conjecturing potential wells and nuclear models to one of integration using the appropriate simplices. Using this approach, it is shown that without any reference to physical models it is possible to obtain the neutron-proton ratio of stable spinless nuclei their coulomb and binding energies, which is in good agreement with that obtained by experiment. It is also shown from the nature of the solution that a nucleus consists of only two types of particles, one having an energy of the coulomb type and the other neutral. Possible extension to the theory for nuclear systems having spin is also discussed.