We study the $L^p$ mapping properties of a class of singular integral operators $T_{p,\mathrm{\Omega },h}$ related to polynomial mappings. We prove that this class of singular operators and some of its related maximal operators are bounded on $L^p$ when the kernel function $\text{Ω}$ in $L{(\text{log }L)}^{\alpha }({\mathbf{S}}^{n-1})$ for same and the radial function $h(|x|)$ satisfia mild integrability condition.

The main purpose of this paper is to use a variant of Grüss inequality to obtain a sharp perturbed generalized three point inequality for functions whose $n$th derivative is bounded both above and below. Thus we provide improvement of a previous result.

By using Lyapunov’s function approach [13], some new results were established, which guarantee that the zero solution of non-linear vector differential equations of the form

$$X^{(5)}+a(t)\text{Ψ}(\dot{X},\ddot{X})\stackrel{⃛}{X}+b(t)\text{Φ}(X,\dot{X},\ddot{X},\stackrel{⃛}{X},X^{(4)})+c(t)\text{Θ}(\dot{X})+F(X)=0$$

is unstable.

The Johnson scheme is one of the most famous association schemes. The structure is based on binomial coefficients. On the other hand, the generalized Johnson scheme ($q$-analog of the Johnson scheme) is based on $q$-binomial coefficients. In this paper, we consider the association scheme which is based on multinomial coefficients and compute intersection numbers (structure constants) of it.

The object of the present paper is to study weakly quasi-conformally symmetric Riemannian manifolds. Among others we obtain various sufficient conditions for such a manifold to be of weakly symmetric. The decomposable weakly quasi-conformally symmetric manifolds are studied and classified regorously. The existence of a weakly quasi-conformally symmetric and decomposable weakly quasi-conformally symmetric manifolds have been ensured by several non-trivial examples.

The cycle-complete graph Ramsey number $r(C_m,K_n)$ is the smallest integer $N$ such that every graph $G$ of order $N$ contains a cycle $C_m$ on $m$ vertices or has independent number $\alpha (G)\ge n$. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that $r(C_m,K_n)=(m-1)(n-1)+1$ for all $m\ge n\ge 3$ (except $r(C_3,K_3)=6$). In this paper we will present a proof for the conjecture in the case $n=m=8$.

A subset $D$ of vertices in a graph $G$ is $k$-dependent if the maximum degree of a vertex in the subgraph $\langle D\rangle$ induced by $D$ is at most $k$. The $k$-dependent domination number ${\gamma }^k(G)$ of a graph $G$ is the minimum cardinality of a $k$-dependent dominating set of $G$. Any $k$-dependent dominating set $D$ of a graph $G$ with $\left|D\right|={\gamma }^k(G)$ is called a ${\gamma }^k$-set of $G$. A vertex $x$ of a graph $G$ is called: (i) ${\gamma }^k$-good if $x$ belongs to some ${\gamma }^k$-set, (ii) ${\gamma }^k$-fixed if $x$ belongs to every ${\gamma }^k$-set, (iii) ${\gamma }^k$-free if $x$ belongs to some ${\gamma }^k$-set but not to all ${\gamma }^k$-sets, (iv) ${\gamma }^k$-bad if $x$ belongs to no ${\gamma }^k$-set. In this paper we deal with ${\gamma }^k$-good/bad/fixed/free vertices and present results on changing and unchanging of the $k$-dependent domination number when a graph is modified by adding an edge or deleting a vertex.

We obtain the curvature form $F^{\nabla }=P\circ d^{\nabla }\circ \nabla -d^{\nabla }\circ P\circ \nabla +d^{\nabla }\circ \nabla \circ P$ for a vector bundle pseudoconnection $\nabla$, where $d^{\nabla }$ is the exterior derivative associated to $\nabla$. We use $F^{\nabla }$ to obtain the curvature of $\nabla$. We also prove that $F^{\nabla }=0$ is a necessary (but not sufficient) condition for $d^{\nabla }$ to be a chain complex. Instead we prove that $F^{\nabla }=0$ and $d^{\nabla }\circ d^{\nabla }\circ \nabla =0$ are necessary and sufficient conditions for $d^{\nabla }$ to be a chain 2-complex, i.e., $d^{\nabla }\circ d^{\nabla }\circ d^{\nabla }=0$.

The paper deals with extended pseudo projective curvature tensor $P^e$ of contact metric manifolds. We prove that ($k,\mu$)-manifold with vanishing extended pseudo projective curvature tensor $P^e$ is a Sasakian manifold. Several interesting corollaries of this result are drawn. Non-Sasakian $(k,\mu )$-manifold with pseudo projective curvature tensor $P$ satisfying $P(\xi ,X)·S=0$, where $S$ is the Ricci tensor, are classified.