Related papers: Product Calculus and Stokes Theorem
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Let $V$ be a quasi-conformal grading-restricted vertex algebra, $W$ be its module, and $\W_{z_1, \ldots, z_n}$ be the space of rational differential forms with complex parameters $(z_1, \ldots, z_n)$ for $n \ge 0$. Using geometric…
We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional)…
Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral join complexes. Then we compute the universal algebra of polyhedral join complexes under certain split conditions and…
We study the product formula $(fg)(A) = f(A)g(A)$ in the framework of (unbounded) functional calculus of sectorial operators $A$. We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional…
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…
Weighted logic programming, a generalization of bottom-up logic programming, is a well-suited framework for specifying dynamic programming algorithms. In this setting, proofs correspond to the algorithm's output space, such as a path…
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular…
We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is…
The motivation of this paper is to construct the theory of vector calculus of multivariate arithmetical functions. We prove analogues of integral theorems and Poincare's lemma.
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…