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We introduce a new generalization of Euler's $\varphi$-function associated with a system of polynomials of several variables. We reprove by a short direct approach certain known related identities, and study some other special cases that do…
We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict…
We study the operad of associative algebras equipped with a derivation. We show that it is determined by polynomials in several variables and substitution. Replacing polynomials by rational functions gives an operad which is isomorphic to…
One discovers why the solution of generalized umbral calculus difference nonhomogeneous equation in the form recently proposed by the author extends here now to generalized appellian delta operator and corresponding polynomials case almost…
The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations…
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of…
In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we treat non-doubling functions of the distance to the boundary of bounded domain.
An abstract theory of ultradifferentiable sheafs is developed. Moreover, various applications to the theory of linear partial differential equations, differential geometry and, in particular, CR geometry are discussed.
We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…
We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincar\'{e} and log-Sobolev inequalities for the…
Here we define a Caputo like discrete fractional difference and we compare it to the earlier defined Riemann-Liouville fractional discrete analog. Then we produce discrete fractional Taylor formulae for the first time, and we estimate their…
Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several…
In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $\nu_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable…
We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…
In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results…
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and…
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential…
In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…