Upper and lower bounds for a Čebyšev type functional in terms of Riemann-Stieltjes integral are given. Applications for functions of selfadjoint operators in Hilbert spaces are also provided.

An observation is to be classified into one of two multivariate normal populations with equal covariance matrix. In this paper, we consider the confidence intervals for expected probability of misclassification (EPMC) for improved linear discriminant rule in two types of data: namely, large sample data and high dimensional data. Our approximate confidence interval is based on the asymptotic normality of consistent estimator of EPMC. Using results of stochastic expression for two bilinear forms and two quadratic forms, we prove asymptotic normality under two different frameworks. Through simulation study, it is observed that our approximate confidence interval has a good performance not only in high dimensional and large sample settings, but also in large sample settings.

A graph $G\left(p,q\right)$ is said to be odd harmonious if there exists an injection $f:V\left(G\right)\to \left\{0,1,2,\cdots ,2q-1\right\}$ such that the induced function $f^{*}:E\left(G\right)\to \left\{1,3,\cdots ,2q-1\right\}$ defined by $f^{*}\left(uv\right)=f\left(u\right)+f\left(v\right)$ is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper, we prove that shadow and splitting of graph $K_{2,n,}{C}_n$ for $n\equiv 0$ (mod 4), the graph $H_{n,n},$ double quadrilateral snakes $DQ\left(n\right),n\ge 2,$ the graph $P_{r,m}$ if $m$ is odd, banana tree and the path union of cycles $C_n$ for $n\equiv 0$ (mod 4) are odd harmonious.

As a generalization of quasi-Einstein manifold, De and Ghosh introduced the notion of generalized quasi-Einstein manifold. The object of the present paper is to study Ricci pseudosymmetric generalized quasi-Einstein manifolds (briefly, $G{\left(QE\right)}_n$) in the framework of pseudo-Riemannian geometry. Specifically, we study the concircular Ricci pseudosymmetric $G{\left(QE\right)}_n$, projective Ricci pseudosymmetric $G{\left(QE\right)}_n$, $W_3$-Ricci pseudosymmetric $G{\left(QE\right)}_n$, conharmonic Ricci pseudosymmetric $G{\left(QE\right)}_n$, conformal Ricci pseudosymmetric $G{\left(QE\right)}_n$ and quasi-conformal Ricci pseudosymmetric $G{\left(QE\right)}_n$.

In this note, we consider when a plane curve given by a polynomial of the form

$$x^3+a_1\left(t\right)x^2+a_2\left(t\right)x+a_3\left(t\right)=0,$$

where ${\text{deg}}_t{a}_i\left(t\right)\le id\left(d\mathrm{:\hspace{0.17em}}\text{even}\right),$ has degenerated $(2,3)$ torus decompositions by using arithmetic properties of elliptic surfaces and show that a 3-cuspidal quartic has infinitely many degenerated $(2,3)$ torus decompositions.

We prove that every $C1$ generic three-dimensional flow has either infinitely many sinks or finitely many sectional-hyperbolic attractors whose basins form a full Lebesgue measure set. In particular, all such flows exhibit sectional-hyperbolic attractors.