Related papers: $*$-Logarithm for Slice Regular Functions

Let $(\pi, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $\pi^\#(g,t) = \pi(g)…

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the…

Under suitable asymptotic and convexity conditions on a function $g\colon\mathbb{R}_+\to\mathbb{R}$, the solution to $\Delta f=g$, where $\Delta$ is the forward difference operator, is unique up to an additive constant and is called the…

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…

Given an arithmetic function $g(n)$ write $M_g(x) := \sum_{n \leq x} g(n)$. We extend and strengthen the results of a fundamental paper of Hal\'{a}sz in several ways by proving upper bounds for the ratio of $\frac{|M_g(x)|}{M_{|g|}(x)}$,…

A fractional matching of a graph $G$ is a function $h: E(G) \to [0,1]$ such that $\sum_{e \in E_G(v)} h(e) \leq 1$ for every vertex $v \in V(G)$, where $E_G(v)$ is the set of edges incident to $v$. If $\sum_{e \in E_G(v)} h(e) = 1$ for all…

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $\Delta$. It is well-known that every complex-valued harmonic function in the unit disc $\Delta$ can be uniquely…

In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We…

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we…

In this paper, we obtain subdifferential representation of a proper $w^*$-lower semicontinous convex function on $X^*$ as follows: Let $g$ be a proper convex $w^*$-lower semicontinuous function on $X^*$. Assume that int dom $g$…

Let $f$ be a continuous real function defined in a subset of the real line. The standard definition of continuity at a point $x$ allow us to correlate any given epsilon with a (possibly depending of $x$) delta value. This pairing is known…

In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n…

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…

We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial…

We prove a H\"{o}lder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function $v$ on $\mathbb{R}^d$ from its Fourier transform $\mathcal{F} v$ given on $[-r,r]^d$. This estimate relies…

If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading…

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f g = f$. In case $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we…

Motivated by recent work of Renegar, we present new computational methods and associated computational guarantees for solving convex optimization problems using first-order methods. Our problem of interest is the general convex optimization…

We consider the problem of characterizing all functions $f$ defined on the set of integers modulo $n$ with the property that an average of some $n$th roots of unity determined by $f$ is always an algebraic integer. Examples of such…

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, $A\mapsto A^*$ and $A\mapsto A_*$, of an aggregation function $A$. We prove that if $A^*$ has a slightly stronger property of…