Related papers: Product Calculus and Stokes Theorem
We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…
We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four…
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…
Given an algebra $A$ over a differential field $K$, we study derivations on $A$ that are compatible with the derivation on $K$. There is a universal object, which is a twisted version of the usual module of differentials, and we establish…
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…
It is known that a category of many-sorted algebras on pure sets of similarity type is "concretely equivalent" to a category of single-sorted algebras. In this paper, we characterize a single-sorted variety that corresponds to a many-sorted…
We introduce two new binary operations with combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series…
This paper summarizes the core definitions and results regarding the chain differential for functions in locally convex topological vector spaces. In addition, it provides a few elementary calculus rules of practical interest, notably for…
In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial…
We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion…
This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is…
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative…
Boolean calculus has been studied extensively in the past in the context of switching circuits, error-correcting codes etc. This work generalizes several approaches to defining a differential calculus for Boolean functions. A unified theory…
We present here a product between vectors and scalars that mixes them within their own space, using imaginaries to describe geometric products between vectors as complex vectors, rather than introducing higher order/dimensional vector…
A very small amount of K\"ahler algebra (i.e. Clifford algebra of differential forms) in the real plane makes x + ydxdy emerge as a factor between the differentials of the Cartesian and polar coordinates, largely replacing the concept of…
Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…
We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate…
In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time.…
A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related…
We characterize the situations in which certain accumulation properties of topological spaces are preserved under taking products.