we investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers.  in particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Infinitely many elliptic curves over $$\mathbf{Q }$$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect their 2. Let $$N_j(X)$$ denote the number with at least j pairs 4, and height most X. In this article we show that $$N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$$ . We also show, as $$X\rightarrow \infty $$ , $$N_2(X)=c_{2,1}X^{1/6}+o(X^{1/1...

Abstract Let $$g_1H_1,\ldots ,g_nH_n$$ g 1 H , … n be cosets of subgroups $$H_1,\ldots ,H_n$$ a finite group G such that $$g_1H_1\cup \ldots \cup g_nH_n\ne G$$ ∪ ...

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.

Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. edge $\{e_1,\ldots,e_n\}$ such $e_i\in E(G_i)$ for $1\leq i \leq n$. This can viewed as version well-known Dirac theorem. In this paper, we prove asymptotically by showing every $\varepsilon...

‎Let $lambda_1,dots,lambda_n$  be positive real numbers such that $sum_{k=1}^n lambda_k=1$. In this paper, we prove that for any positive operators $a_1,a_2,ldots, a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $t(1)

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. Th...

Given $$n,m\in \mathbb {N}$$ and a simply connected uniform domain $$U\subset {C}$$ with sufficiently smooth boundary $$\partial U$$ , we study the $$C^{\ast}$$ -algebra $$\begin{aligned} {\mathfrak B}_{U,n,m}({\mathfrak L}):=\mathrm{alg}\{aI,B_{U,1},\ldots ,B_{U,n},\widetilde{B}_{U,1},\ldots ,\widetilde{B}_{U,m}: a\in X}({\mathfrak L})\}, \end{aligned}$$ generated by operators of multiplicatio...

Abstract Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in {Z}$ primitive root modulo p . $0<a_1<\cdots <a_n<p$ all the cubic residues in interval $(0,p)$ Then clearly sequence $a_1 \bmod p,\, a_2 p,\ldots , a_n p$ is permutation of $g^3 p,\,g^6 g^{3n} We determine sign this permutation.