‎For any archimedean$f$-ring $A$ with unit in whichbreak$awedge‎ ‎(1-a)leq 0$ for all $ain A$‎, ‎the following are shown to be‎ ‎equivalent‎: ‎ ‎1‎. ‎$A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all‎ ‎integer-valued continuous functions on some frame $L$‎. 2‎. ‎$A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$‎ ‎of all integer-valued continuous functions‎, ‎in the usual se...

The relation between Hamiltonicity and toughness of a graph is long standing research problem. paper studies the Cartesian product $$G_1\square G_2$$ graphs $$G_1$$ $$G_2$$ satisfying that traceable connected with path factor. Let $$P_n$$ be order n H bipartite graph. With certain requirements n, we show following three statements are equivalent: (i) $$P_n\square H$$ Hamiltonian; (ii) 1-tough; ...

We obtain the criterion of best approximant for multivariable functions in $L_{p_1,\ldots p_n}$ space.

We prove the existence of critical points vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1}\atop{i\ne j}} ^N \Gamma_i\Gamma_jG(p_i,p_j)+\Psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to sphere or projective plane. Here $G$ denotes Green function Laplace-Beltrami operator in $\Sigma$, $\Psi\colon \Sigma^N\to\mathbb{R}$ may be any class $...

Favard's theorem characterizes bases of functions $\{p_n\}_{n\in\mathbb{Z}_+}$ for which $x p_n(x)$ is a linear combination $p_{n-1}(x)$, $p_n(x)$, and $p_{n+1}(x)$ all $n \geq 0$ with $p_{0}\equiv1$ (and $p_{-1}\equiv by convention). In this paper we explore the differential analogue theorem, that is, $\{\varphi_n\}_{n\in\mathbb{Z}_+}$ $\varphi_n'(x)$ $\varphi_{n-1}(x)$, $\varphi_n(x)$, $\varp...