Related papers: On the spherical derivative of a rational function

In the first part of this paper we establish, in terms of so called k-tangential sets, a kind of optimal estimate for the size and structure of the set of non-differentiability of Lipshitz functions with one-sided directional derivatives.…

We treat shared value problems for rational functions $R(z)$ and their derivative $R'(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and $\partial_{z} R = \lambda w \cdot R'(w)$ on ${\mathbb C}…

In this note, we propose a formula for the subdifferential of the maximum functional $m(f) = \max_K f$ on the space of real-valued continuous functions $f$ defined on an arbitrary metric compact $K$. We show that, given $f$, the…

A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…

Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f)…

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply…

The functional derivative of the effective action with respect to an external field is part of the equation of motion of this field if one-loop effects induced by quantum fluctuations or thermal fluctuations are included when minimizing the…

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…

Consider the surface measure $\mu$ on a sphere in a nonvertical hyperplane on the Heisenberg group $\mathbb{H}^n$, $n\ge 2$, and the convolution $f*\mu$. Form the associated maximal function $Mf=\sup_{t>0}|f*\mu_t|$ generated by the…

Fixed a continuous kernel K on the $d$-dimensional torus, we consider a generalization of the univariate $sk$-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given…

This paper extends the classical Ostrogradsky-Hermite reduction for rational functions to more general functions in primitive extensions of certain types. For an element $f$ in such an extension $K$, the extended reduction decomposes $f$ as…

Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function…

Let $\phi$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The…

In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate…

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let $n\in\mathbb{Z}$, $f, g\colon\mathbb{R}\to\mathbb{R}$ be…

Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…

In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to…

We introduce a new model of linear regression for random functional inputs taking into account the first order derivative of the data. We propose an estimation method which comes down to solving a special linear inverse problem. Our…

Let $\phi$ be a an endomorphism of degree $d\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $\phi$ has good reduction outside of $S$.…

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…