相关论文: Combinatorics of Partial Derivatives
In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
Time-dependent fractional-derivative problems $D_t^\delta u + Au = f$ are considered, where $D_t^\delta$ is a Caputo fractional derivative of order $\delta\in (0,1)\cup (1,2)$ and~$A$ is a classical elliptic operator, and appropriate…
We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which…
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann-Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their…
In this paper a Ward-Fonten\'e differential universal algebra is constructed. In this algebra it is possible to obtain a product $\psi$-rule and a general $\psi$-rule of Leibniz for any calculus on sequences. In particular, the simplicial…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted…
In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a…
Motivated by applications to stochastic programming, we introduce and study the expected-integral functionals, which are mappings given in an integral form depending on two variables, the first a finite dimensional decision vector and the…
A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function that can be expanded by Taylor series, we show that D^Elafa*D^Beta f(t)=D^(Elafa+Beta)f(t). GFD is applied for some functions in…
The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…
Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a…
The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of…
Differential categories were introduced by Blute, Cockett, and Seely as categorical models of differential linear logic and have since lead to abstract formulations of many notions involving differentiation such as the directional…
In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then…
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…
Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a condition for the inclusion of a regular fraction as follows. Let $F = \sum_\alpha b_\alpha X^\alpha$ be the…