Related papers: When all Permutations are Combinatorial Similariti…

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y…

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y…

The group of combinatorial self-similarities of a pseudometric space $(X, d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym} (X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{IP}$…

Let (X,d) be a metric space and m\in X. Suppose that \phi:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{\phi,m} on X which is equivalent to d. If d^{\phi,m} is totally bounded, its completion is a…

Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings \phi of the polynomial algebra K[x,y] such that for the quotient fields, one has K(\phi(x),…

We characterize finitary coarse spaces $X$ such that every permutation of $X$ is an asymorphism.

Let H be a complex infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms $\phi$ of B(H) which are uniformly continuous on every commutative C*-subalgebra. In particular, we obtain that if $\phi$ satisfies…

We establish a complete algebraic characterization of self-similar iterated function systems $\Phi$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure…

Let $S(H)$ be the set of all self-adjoint bonded linear operators on $H$ and $\mathcal{V} \subset S(H)$ a subset that is pertinent in mathematical foundations of quantum mechanics. A symmetry is a bijective map $\phi :\mathcal{V} \to…

We characterize all permutations on even-by-even grids that maximally separate neighboring vertices. More precisely, let $n_1$, $n_2$ be positive even integers, let $I(n_1,n_2)=\{1,\dots,n_1\}\times\{1,\dots,n_2\}$ be the $n_1\times n_2$…

We call an objective function or algorithm symmetric with respect to an input if after swapping two parts of the input in any algorithm, the solution of the algorithm and the output remain the same. More formally, for a permutation $\pi$ of…

For $0< \rho < 1/3$ and $\rho \le \lambda \le 1-2\rho$, let $E$ be the self-similar set generated by the iterated function system $$\Phi = \big\{ \varphi_1(x) = \rho x ,\; \varphi_2(x) = \rho x + \lambda, \; \varphi_3(x) = \rho x + 1- \rho…

An algebraic structure is said to be congruence permutable if its arbitrary congruences $\alpha$ and $\beta$ satisfy the equation $\alpha \circ \beta =\beta \circ \alpha$, where $\circ$ denotes the usual composition of binary relations. For…

A permutation on an alphabet $ \Sigma $, is a sequence where every element in $ \Sigma $ occurs precisely once. Given a permutation $ \pi $= ($\pi_{1} $, $ \pi_{2} $, $ \pi_{3} $,....., $ \pi_{n} $) over the alphabet $ \Sigma $ =$\{ $0, 1,…

Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY…

If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e., \[ f(x^2+xy+y^2) = f(x)^2 + f(x)\,f(y) + f(y)^2, \] then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form…

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal…

Let $G$ be a countable group, $\operatorname{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and $\operatorname{Is}(G) \subset \operatorname{Sub}(G)$ the collection of isolated points. We denote by…

A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using…

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f)…