We define the $k$-dimensional generalized Euler function $\varphi_k(n)$ as number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both product $a_1\cdots a_k$ sum $a_1+\cdots +a_k$ are prime to $n$. investigate some properties $\varphi_k(n)$, obtain a corresponding Menon-type identity.

(3) All calculations are mod m¡ so that the digits of the mixed radix representation of [x]u can be obtained using only calculations mod m¡. (4) The value x x(a) can be obtained from x x(a) = x x(a_1) 4x(°_1) x " so that the modular number x need not be remembered during the whole process. (5) The matrices (do) and (x.j) are computed preliminary to the iteration procedure and are not part of it.

let g_1 and g_2 be simple connected graphs with disjoint vertex sets v(g_1) and v(g_2), respectively. for given vertices a_1in v(g_1) and a_2in v(g_2), a splice of g_1 and g_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the :union: of g_1 and g_2. in this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

this paper deals with the boundary value problem involving the differential equationbegin{equation*}    ell y:=-y''+qy=lambda y, end{equation*} subject to the standard boundary conditions along with the following discontinuity  conditions at a point $ain (0,pi)$ begin{equation*}    y(a+0)=a_1 y(a-0),quad y'(a+0)=a_1^{-1}y'(a-0)+a_2 y(a-0),end{equation*}where $q(x),  a_1 , a_2$ are  real, $qin l...

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d imes d$ such that ${ m Tr}(A_kA_l)= $ for all $k,l=1,..., n$? For such matrices to exist, one must have $ geq 0$ for all $k,l=1,..., n$. We prove that if $n FRENKEL, Peter Erno, WEINER, Mihály. On vector configurations that can ...

We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka-Volterra system for an exponential random time after which it switches to a different deterministic Lotka-Volterra system. This switching procedure is then repeated. The resulting process switches between two deterministic Lotka-Volterra systems at exponential times a...

In this paper, we study combinatorial aspects of permutations \(\{1,\ldots ,n\}\) and related topics. particular, prove that there is a unique permutation \(\pi \) such all the numbers \(k+\pi (k)\) (\(k=1,\ldots ,n\)) are powers two. We also show \(n\mid {\mathrm{per}}[i^{j-1}]_{1\leqslant i,j\leqslant n}\) for any integer \(n>2\). conjecture if group G contains no element order among \(2,\ldo...