Related papers: L'Hopital rules for complex-valued functions in hi…

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients…

We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an…

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…

In this note we give a quantitative version of the following simple observation: a discrete harmonic function on the lattice may vanish at each point of a large cube without being zero identically, at the same time there is a version of…

This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space…

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are…

A fundamental open question asking whether all real-valued strongly quasiconvex functions defined on $\mathbb R^n$ are necessarily continuous, akin to their convex counterparts, is answered in detail in this paper. Among other things, we…

As was the case in a previous paper, the differential form x+ydxdy plays the role that the variable z plays in the standard calculus of complex variable. The role of holomorphic functions will now be played by strict harmonic differential…

We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…

Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show…

We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…

Quantum work fluctuation theorems are known to hold when the work is defined as the difference between the outcomes of projective measurements carried out on the Hamiltonian of the system at the initial and the final time instants of the…

We present both the Lagrangian and Hamiltonian procedures for treating higher-order equations of motion for mechanical models by adopting the Riemann-Liouville Fractional integral to describe their action. We point out and discuss its…

We present examples of holomorphic functions that vanish to in- finite order at points at the boundary of their domain of definition. They give rise to examples of Dirichlet minimizing Q-valued functions indicating that "higher"-regularity…

In this paper, we study the Lambert-Tsallis function, which is a generalization of the Lambert function with two real parameters. We give a condition on the parameters such that there exists a complex domain touching zero on boundary which…

Leibniz's rule for the $n$-th derivative of a product is a very well known and extremely useful formula. In this article, we introduce an analogous explicit formula for the $n$-th derivative of a quotient of two functions. Later, we use…

We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to…

The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order…

We show that every finite Boolean combination of polynomial equalities and inequalities in C^n admits two uniform normal forms: an $\exists\forall$ form and a $\forall\exists$ form, each using a single polynomial equation. Both forms use…

The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…