Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1$, $\gcd (a,b)=1, a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\hspace{4.44443pt}(\@mod \; 4)$ and either $b$ or $c$ is an odd prime power, then the equation $x^{2}+b^{y}=c^{z}$ has only the positive integer solution $(x,y,z)=(a,2,r)$ with $\min (y,z)>1$.
If G is a simple graph of size n without isolated vertices and G is its complement, we show that the domination numbers of G and G satisfy γ(G) + γ(G) ≤ { n − δ + 2 if γ(G) > 3, δ + 3 if γ(G) > 3, where δ is the minimum degree of vertices in G.
The paper is devoted to the question whether some kind of additional information makes it possible to determine the fundamental matrix of variational equations in ℝ3 . An application concerning computation of a derivative of a scalar Poincaré mapping is given.
The independent domination number $i(G)$ (independent number $\beta (G)$) is the minimum (maximum) cardinality among all maximal independent sets of $G$. Haviland (1995) conjectured that any connected regular graph $G$ of order $n$ and degree $\delta \le \frac{1}{2}{n}$ satisfies $i(G)\le \lceil \frac{2n}{3\delta }\rceil \frac{1}{2}{\delta }$. For $1\le k\le l\le m$, the subset graph $S_{m}(k,l)$ is the bipartite graph whose vertices are the $k$- and $l$-subsets of an $m$ element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for $i(S_{m}(k,l))$ and prove that if $k+l=m$ then Haviland’s conjecture holds for the subset graph $S_{m}(k,l)$. Furthermore, we give the exact value of $\beta (S_{m}(k,l))$.
Let (G) and i(G) be the domination number and the independent domination number of G, respectively. Rad and Volkmann posted a conjecture that i(G)/ (G) 6 (G)/2 for any graph G, where (G) is its maximum degree (see N. J.Rad, L.Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than (G)/2 are provided as well., Shaohui Wang, Bing Wei., and Seznam literatury
From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251– 257] we know that if S, T are commuting B-Fredholm operators acting on a Banach space X, then ST is a B-Fredholm operator. In this note we show that in general we do not have ind(ST) = ind(S) + ind(T), contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist U, V ∈ L(X) such that S, T, U, V are commuting and US + V T = I, then ind(ST) = ind(S) + ind(T), where ind stands for the index of a B-Fredholm operator.
Let $k\subseteq k'$ be a field extension. We give relations between the kernels of higher derivations on $k[X]$ and $k'[X]$, where $k[X]:=k[x_1,\dots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D=\{D_n\}_{n=0}^\infty $ a higher $k$-derivation on $k[X]$ and $D'=\{D_n'\}_{n=0}^\infty $ a higher $k'$-derivation on $k'[X]$ such that $D'_m(x_i)=D_m(x_i)$ for all $m\geq 0$ and $i=1,2,\dots ,n$. Then (1) $k[X]^D=k$ if and only if $k'[X]^{D'}=k'$; (2) $k[X]^D$ is a finitely generated $k$-algebra if and only if $k'[X]^{D'}$ is a finitely generated $k'$-algebra. Furthermore, we also show that the kernel $k[X]^D$ of a higher derivation $D$ of $k[X]$ can be generated by a set of closed polynomials.
The main purpose of this paper is to use the analytic method to study the calculating problem of the general Kloosterman sums, and give an exact calculating formula for it.