Related papers: A discrete Faa di Bruno's formula
The essential variables in a finite function $f$ are defined as variables which occur in $f$ and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When…
Determinants and symmetric functions of the eigenvalues of matrices characterizing stochastic processes with indepedent increments. Relationships with Fibonacci numbers are derived.
The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar…
For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine…
We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula, and study their limit in the continuum.
We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra,…
We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a…
In this paper, we obtained the sufficient conditions for the existence of solutions to the discrete boundary value problems of fractional difference equation depending on parameters. We use Krasnoselskii fixed point theorem to establish the…
In this paper, by introducing the degenerate Fubini-type polynomials, we give several relations with the help of the Fa\`a di Bruno formula and some properties of Bell polynomials, and generating function methods. Also, we derive some new…
For a permutation $\pi$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $\pi$. We characterize the possible derivatives of…
Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $\tau$ functions for the Painlev\'e equations is also discussed.
We define discrete generating series for arbitrary functions \( f \colon \mathbb{Z}^n \rightarrow \mathbb{C} \) and derive functional relations that these series satisfy. For linear difference equations with constant coefficients, we…
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…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
We derived the formulae of central differentiation for the finding of the first and second derivatives of functions given in discrete points, with the number of points being arbitrary. The obtained formulae for the derivative calculation do…
A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order $\delta \in (1,2)$ is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on…
In this paper we prove a Thomae derivative formula for trigonal curves admitting a non-singular affine model. This formula relates the derivatives of theta functions with rational characteristics on the curve to explicit expressions in the…
The calculus of finite differences is a solid foundation for the development of operations such as the derivative and the integral for infinite sequences. Here we showed a way to extend it for finite sequences. We could then define…