Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}\left(t\right)\equiv -(3/p){\sum }_{x=0}^{p-1}((x³-3x+2t)/p)\left(modp\right)$ and $\left({\sum }_{x=0}^{p-1}((x³+mx+n)/p)\right)²\equiv ((-3m)/p){\sum }_{k=0}^{[p/6]}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right){((4m³+27n²)/\left(12³·4m³\right))}^k\left(modp\right)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right)/m^k\left(modp²\right)$, where m is an integer not divisible by p.

The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral ${\int }_a^{b}f\left(e^{it}\right)du\left(t\right)$ of continuous complex valued integrands $f:𝒞\left(0,1\right)\to ℂ$ defined on the complex unit circle $𝒞\left(0,1\right)$ and various subclasses of integrators $u:\left[a,b\right]\subseteq \left[0,2\pi \right]\to ℂ$ of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.

The contents of the paper is concerned with the two-sample problem where $F_m\left(x\right)$ and $G_n\left(x\right)$ are two empirical distribution functions. The difference $F_m\left(x\right)-G_n\left(x\right)$ changes only at an $x_i,i=1,2,...,m+n$, corresponding to one of the observations. Let $R_{mn}^{+}\left(j\right)$ denote the subscript $i$ for which $F_m\left(x_i\right)-G_n\left(x_i\right)$ achieves its maximum value $D_{mn}^{+}$ for the $j$th time $(j=1,2,...)$. The paper deals with the probabilities for $R_{mn}^{+}\left(j\right)$ and for the vector $(D_{mn}^{+},R_{mn}^{+}\left(j\right))$ under $H_0:F=G$, thus generalizing the results of Steck-Simmons (1973). These results have been derived by applying the random walk model. ...

For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for every positive integer n, where ac’(Pₙ) denotes the nearly antipodal chromatic number of Pₙ. In this paper we show that $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)-n/2-⎣10/n⎦+7$ if n is even positive integer and n ≥ 10, and $a{c}^{\text{'}}\left(Pₙ\right)\le \left(\genfrac{}{}{0pt}{}{n-2}{2}\right)-(n-1)/2-⎣13/n⎦+8$ if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pₙ.

Let: $𝐘=\left({𝐘}_i\right)$, where ${𝐘}_i=\left(Y_{i,1},...,Y_{i,d}\right)$, $i=1,2,\cdots$, be a $d$-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf $F$, and $F_n\left(𝐱\right):=\frac{1}{n}{\sum }_{i=1}^n𝕀\left(Y_{i,1}\le x_1,\cdots ,Y_{i,d}\le x_d\right)$ denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process $B_n=\sqrt{n}\left(F_n-F\right)$ under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.

Let $\left(A,B\right)$ be a linear time dependent control process, defined on an open interval $J=]a,\omega [$ with $a\ge -\mathrm{\infty }$ and $\omega \le \mathrm{\infty }$; in this paper we give a description of the function $\tau :I\to J$, $\tau \left(t\right)=inf\mathit{\{}t{}^{\prime }>t:(A,B)$ is $\left[t,t^{\prime }\right]$-globally controllable from $0\mathit{\}}$ where $I=\mathit{\{}t\in J:\mathrm{\exists }t{}^{\prime }\in J$ with $\left(A,B\right)\left[t,t^{\prime }\right]$-globally controllable from $0\mathit{\}}$.