Related papers: The classification of generalized Riemann derivati…
We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.
It is known that if $f\colon {\mathbb R}^2 \to {\mathbb R}$ is a polynomial in each variable, then $f$ is a polynomial. We present generalizations of this fact, when ${\mathbb R}^2$ is replaced by $G\times H$, where $G$ and $H$ are…
This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…
The systems of differential equations whose solutions coincide with Bethe ansatz solutions of generalized Gaudin models are constructed. These equations we call the {\it generalized spectral Riccati equations}, because the simplest equation…
We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\dots, x_{n}]$ \ with \ $(n+2)$ \ monomials. We give two general factorizations theorem in the algebra \ $\C< z, (\frac{\partial}{\partial…
We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes…
In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for…
Ambiguity is shown in the context of the differential calculus of several variables and with the help of the language of category theory, a way to solve it in its most general form is offered. It is also shown that this new definition is…
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to…
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
A different proof to a known criterion of derived equivalence implying birationality is given. Derived equivalent smooth projective curves over an algebraically closed field are proved to be isomorphic. A different proof of derived…
The main aim of this note is to provide characterization theorems concerning real derivations. Among others the following implication will be verified: Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and…
Initialization of fractional differential equations remains an ongoing problem. In recent years, the initialization function approach and the infinite state approach provide two effective ways to deal with this problem. The purpose of this…
We show that the $n$th derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for $n$ odd/even, respectively. We show this by giving a full asymptotic expansion of…
We present a generalization of a formula of higher order derivatives and give a short proof.
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…
Using a concept of filter we propose one generalization of Riemann integral, that is integration with respect to filter. We study this problem, demonstrate different properties and phenomena of filter integration.
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation $\mathsf{K}$, which all…