Related papers: $C^r$-right equivalence of analytic functions

Let $C$ be an affine plane curve. We consider additive functions $f: K\rightarrow K$ for which $f(x)f(y)=0$, whenever $(x,y)\in C$. We show that if $K=\mathbb{R}$ and $C$ is the hyperbola with defining equation $xy=1$, then there exist…

Let $f :\R\to\R$ be a continuous function. We prove that under some additional assumptions on $f$ and $A:\R\to\R_{+}$, weak $\Cuno$ solutions of the differential inequality $-\diver(A(\abs{\nabla u})\nabla u)\ge f(u)$ on $\RN$ are…

Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $\phi$ on the edges of $E$, we define a C*-algebra denoted ${\cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain…

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the…

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left|…

Let $(f, g)$ be a pair of complex analytic functions on a singular analytic space $X$. We give ``the correct'' definition of the relative polar curve of $(f, g)$, and we give a very formal generalization of L\^e's attaching result, which…

Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors…

It is proved that for every $r,s \in \mathbb{N}\backslash \{0,1\}$ the action of $\mathbb{F}_{r+s}$ on $\partial_\beta (\mathbb{F}_{r+s}/\mathbb{F}_r)$ is topologically amenable. In particular the $C^*$-algebra associated to the…

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary…

We show that under mild conditions, a Gaussian analytic function $\boldsymbol F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock space has the property that a.s. no non-zero function in that space vanishes where…

A realization is a triple, $(A,b,c)$, consisting of a $d-$tuple, $A= (A =_1, \cdots, A_d )$, $d\in \mathbb{N}$, of bounded linear operators on a separable, complex Hilbert space, $\mathcal{H}$, and vectors $b,c \in \mathcal{H}$. Any such…

We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the…

Let $x \in \mathbb{R}^d$, $d \geq 3,$ and $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a twice differentiable function with all second partial derivatives being continuous. For $1\leq i,j \leq d$, let $a_{ij} : \mathbb{R}^d \rightarrow…

The notion of right semi-equivalence in a right $(n+2)$-angulated category is defined in this article. Let $\mathscr C$ be an $n$-exangulated category and $\mathscr X$ is a strongly covariantly finite subcategory of $\mathscr C$. We prove…

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(x\sigma(y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed…

Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with stationary increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a symmetric function with Ef(\eta)<\ff, where \eta=N(0,1). When \si^2(s) is concave or when \si^2(s)=s^r$,…

We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\frac{\pi}{2})$, the complete elliptical integral of the first kind, and its…

In this paper, we consider the characterization of norm--parallelism problem in some classical Banach spaces. In particular, for two continuous functions $f, g$ on a compact Hausdorff space $K$, we show that $f$ is norm--parallel to $g$ if…

In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| < 1$, in which we replace the coefficients $a_n$ $(n= 0,1,\ldots ,N)$ of the power…