Related papers: A Topological View on Integration and Exterior Cal…
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of…
An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that…
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations.…
In this work we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on…
We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.
In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a…
Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…
The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
Could elementary complex analysis, which covers the topics such as algebra of complex numbers, elementary complex functions, complex differentiation and integration, series expansions of complex functions, residues and singularities, and…
We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…
Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This…
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…
We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…
Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection…