Related papers: Geometric Calculus of the Gauss Map
We outline an alternative approach to the geometric notion of a saddle point for real-valued functions of two variables. It is argued that this is more natural compared to the usual treatment of this topic in standard texts on Calculus.
In an earlier paper we constructed a Cartier divisor on the theta divisor of a principally polarised abelian variety whose support is precisely the ramification locus of the Gauss map. In this note we discuss a Green's function associated…
It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…
We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
Malliavin Calculus can be seen as a differential calculus on Wiener spaces. We present the notion of stochastic manifold for which the Malliavin Calculus plays the same role as the classical differential calculus for the differential…
Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving $p$-Laplacian. In this paper,…
Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products…
The main purpose of this paper is to introduce the geometric difference sequence space $l_\infty^{G} (\Delta_G)$ and prove that $l_\infty^{G} ({\Delta}_{G})$ is a Banach space with respect to the norm $\left\|.\right\|^G_{{\Delta}_G}.$ Also…
We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.
Employing a class of generalized connections, we describe certain differential complices $\left(\tilde \Omega^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic…
For differential calculi over certain right coideal subalgebras of quantum groups the notion of quantum tangent space is introduced. In generalization of a result by Woronowicz a one to one correspondence between quantum tangent spaces and…
This manuscript introduces novel approaches to three phenomena. First, we extend the algebraic formulation of kinetic theory within the contact framework by making explicit the gauge freedom, thereby obtaining a formulation in which the…
Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic…
The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description…
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our…
We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
For a class of non compact Riemannian manifolds with ends, we give pseudo-differential expansions of bounded functions of the semi-classical Laplacian and study related Lp boundedness properties.
The inverse problem of calculus of variations and $s$-equivalence are re-examined by using results obtained from non-commutative geometry ideas. The role played by the structure of the modified Poisson brackets is discussed in a general…
A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in…