Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function f\colon \RR^n \to \RR to be \emph{vertically rigid} if \graph(cf) is isometric to \graph (f) for every $c \neq 0$. It is \emph{horizontally rigid} if \graph(f(c \vec{x})) is isometric to \graph (f) for every $c \neq 0$ (see \cite{CCR}). In an earlier paper the authors of the present paper settled Jankovi\'c's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ (a,b,k \in \RR). Later they proved that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ (a,b,d,k \in \RR, $k \neq 0$, s : \RR \to \RR continuous). The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. C. Richter proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $a+bx$ (a,b\in \RR). The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $a + bx + dy$ (a,b,d \in \RR). This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations

Finite symmetric functions with non-trivial arity gap

Given an $n$-ary $k-$valued function $f$, $gap(f)$ denotes the essential arity gap of $f$ which is the minimal number of essential variables in $f$ which become fictive when identifying any two distinct essential variables in $f$. In the present paper we study the properties of the symmetric function with non-trivial arity gap ($2\leq gap(f)$). We prove several results concerning decomposition of the symmetric functions with non-trivial arity gap with its minors or subfunctions. We show that all non-empty sets of essential variables in symmetric functions with non-trivial arity gap are separable.Comment: 12 page