‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0neq ain R$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))in ...

Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,be a simple connected graph, with sequence of vertex degrees$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1}frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning tree...

We obtain asymptotic formulas for the sums $\sum_{n_1,\ldots,n_k\le x} f((n_1,\ldots,n_k))$ and $ \sum_{n_1,\ldots,n_k\le f([n_1,\ldots,n_k])$ involving gcd lcm of integers $n_1,\ldots,n_k$, where $f$ belongs to certain classes additive arithmetic functions. In particular, we consider generalized omega function $\Omega_{\ell}(n)= \sum_{p^\nu \mid\mid n} \nu^{\ell}$ investigated by Duncan (1962)...

Let $P_n(y_1,\ldots,y_n):= \prod_{1\leq i<j\leq n}\left( 1 -\frac{y_i}{y_j}\right) $ and $P_n:= \sup_{(y_1,\ldots,y_n)}P_n(y_1,\ldots,y_n) where the supremum is taken over $n$-ples $(y_1,\ldots,y_n)$ of real numbers satisfying $0 <|y_1| < |y_2|< \cdots |y_n|$. We prove that $P_n \leq 2^{\lfloor n/2\rfloor}$ for every $n$, i.e., we extend to all $n$ bound Pohst proved $n\leq 11$. As a consequenc...

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair vertices $u, v$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$. The span $\phi$ the difference between largest smallest labels assigned to by $\phi$, $\lambda_{h_1, h_l}(G)$ defined as minimum over all h_l)$-labellings $G$. In this pap...

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in way that each vertex $V_i$ adjacent to all the other $V_j$ for $1\leq i&lt;j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote with $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring colors $1,2,\ldots,t$ an \emph{interval...

Let $$A_n=(a_1,a_2,\ldots ,a_n)$$ and $$B_n=(b_1,b_2,\ldots ,b_n)$$ be nonnegative integer sequences with $$A_n\le B_n$$ . The purpose of this note is to give a good characterization such that every sequence $$\pi =(d_1,d_2,\ldots d_n)$$ even sum \pi \le graphic. This solves forcible version problem posed by Niessen generalizes the Erdős–Gallai theorem.

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field rational functions finite $\mathbb{F}_q$ characteristic $p$, then proportion $n$-tuples $(t_1,\ldots, t_n)$ monic polynomials degree $d$ which $f(t_1,\ldots, t_n,X)$ reducible out all $O(dq^{-d/2})$.

Let $p$ be a prime. $R$ regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ Cohen the same residue field $k$ as and $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo does not belong ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ $k[[x_1,\ldots,x_d]]$. We extend result Vasiu-Zink (for $d=2$) show each Barsotti-Tate...

in this paper, an iterative method is proposed for solving matrix equation $sum_{j=1}^s a_jx_jb_j = e$. this method is based on the global least squares (gl-lsqr) method for solving the linear system of equations with the multiple right hand sides. for applying the gl-lsqr algorithm to solve the above matrix equation, a new linear operator, its adjoint and a new inner product are dened. it is ...