Let $f$ be a continuous function on $I$ and $A$, $B\in {\mathrm{𝒮𝒜}}_I\left(H\right)$, the convex set of selfadjoint operators with spectra in $I$. If $A\ne B$ and $f$, as an operator function, is Gateaux differentiable on $$[A,B]:=\left\{(1-t)A+tB\mid t\in \left[0,1\right]\right\},$$ while $p:\left[0,1\right]\to ℝ$ is Lebesgue integrable, then we have the inequalities $$\begin{array}{cc}\hfill \parallel {\int }_0^{1}p\left(\tau \right)& f\left(\left(1-\tau \right)A+\tau B\right)d\tau -{\int }_0^{1}p\left(\tau \right)d\tau {\int }_0^{1}f\left(\left(1-\tau \right)A+\tau B\right)d\tau \parallel \hfill \\ & \le {\int }_0^1\tau (1-\tau )|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau \hfill \\ & \le \frac{1}{4}{\int }_0^1|\frac{{\int }_{\tau }^{1}p\left(s\right)ds}{1-\tau }-\frac{{\int }_0^{\tau }p\left(s\right)ds}{\tau }|∥\nabla f_{\left(1-\tau \right)A+\tau B}\left(B-A\right)∥d\tau ,\hfill \end{array}$$ where $\nabla f$ is the Gateaux derivative of $f$.

For positive integers $n$, Euler’s phi function and Dedekind’s psi function are given by $$\phi \left(n\right)=n\prod _{\begin{array}{c}p\mid n\\ p\mathrm{prime}\end{array}}\left(1-\frac{1}{p}\right)\text{and}\psi \left(n\right)=n\prod _{\begin{array}{c}p\mid n\\ p\mathrm{prime}\end{array}}\left(1+\frac{1}{p}\right),$$ respectively. We prove that for all $n\ge 2$ we have $${\left(1-\frac{1}{n}\right)}^{n-1}{\left(1+\frac{1}{n}\right)}^{n+1}\le {\left(\frac{\phi \left(n\right)}{n}\right)}^{\phi \left(n\right)}{\left(\frac{\psi \left(n\right)}{n}\right)}^{\psi \left(n\right)}$$ and $${\left(\frac{\phi \left(n\right)}{n}\right)}^{\psi \left(n\right)}{\left(\frac{\psi \left(n\right)}{n}\right)}^{\phi \left(n\right)}\le {\left(1-\frac{1}{n}\right)}^{n+1}{\left(1+\frac{1}{n}\right)}^{n-1}.$$ The sign of equality holds if and only if $n$ is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

This work deals with the analysis pertaining some dynamic behavior of vector solutions of first order two-dimensional neutral delay differential systems of the form $$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}u\left(t\right)+pu(t-\tau )\\ v\left(t\right)+pv(t-\tau )\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}u(t-\alpha )\\ v(t-\beta )\end{array}\right].$$ The effort has been made to study $$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}x\left(t\right)-p\left(t\right)h_1\left(x(t-\tau )\right)\\ y\left(t\right)-p\left(t\right)h_2\left(y(t-\tau )\right)\end{array}\right]+\left[\begin{array}{cc}a\left(t\right)& b\left(t\right)\\ c\left(t\right)& d\left(t\right)\end{array}\right]\left[\begin{array}{c}{f}_1\left(x(t-\alpha )\right)\\ f_2\left(y(t-\beta )\right)\end{array}\right]=0,$$ where $p,a,b,c,d,h_1,h_2,f_1,f_2\in C(ℝ,ℝ)$; $\alpha ,\beta ,\tau \in {ℝ}^{+}$. We verify our results with the examples.

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 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.

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 [...] f(z)=∑n=0∞αnzn $f\left(z\right)={\sum }_{n=0}^{\infty }{\alpha }_n{z}^n$ be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that [...] ‖f(y)−f(x)‖≤‖y−x‖∫01fa′(‖(1−t)x+ty‖)dt $$∥f\left(y\right)-f\left(x\right)∥\le ∥y-x∥{\int }_0^1{f}_a^{\text{'}}\left(∥(1-t)x+ty∥\right)dt$$ where [...] fa(z)=∑n=0∞|αn| zn $f_a\left(z\right)={\sum }_{n=0}^{\infty }|{\alpha }_n|z^n$ . Inequalities for the commutator such as [...] ‖f(x)f(y)−f(y)f(x)‖≤2fa(M)fa′(M)‖y−x‖, $$∥f\left(x\right)f\left(y\right)-f\left(y\right)f\left(x\right)∥\le 2f_a\left(M\right)f_a^{\text{'}}\left(M\right)∥y-x∥,$$ if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided. ...

We give several different $q$-analogues of the following two congruences of Z.-W. Sun: $$\sum _{k=0}^{(p^r-1)/2}\frac{1}{8^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{2}{p^r}\right)(mod{p}^2)\text{and}\sum _{k=0}^{(p^r-1)/2}\frac{1}{{16}^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{3}{p^r}\right)(mod{p}^2),$$ where $p$ is an odd prime, $r$ is a positive integer, and $\left(\frac{m}{n}\right)$ is the Jacobi symbol. The proofs of them require the use of some curious $q$-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

We list some open problems concerning the polarized partition relation. We solve a couple of them, by showing that for every limit non-inaccessible ordinal α there exists a forcing notion ℙ such that the strong polarized relation $\left(\genfrac{}{}{0pt}{}{{\aleph }_{\alpha +1}}{{\aleph }_{\alpha }}\right)\to {\left(\genfrac{}{}{0pt}{}{{\aleph }_{\alpha +1}}{{\aleph }_{\alpha }}\right)}_2^{1,1}$ holds in $V^{\mathbb{P}}$.

In this note, we estimate the distance between two $q$-nomial coefficients $\left|{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_q-{\left(\genfrac{}{}{0pt}{}{n^{\text{'}}}{k^{\text{'}}}\right)}_q\right|$, where $(n,k)\ne (n^{\text{'}},k^{\text{'}})$ and $q\ge 2$ is an integer.

Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(a+ba\right)$ voting sequences are equally probable. Denote by ${\alpha }_r$ and by ${\beta }_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1,\cdots ,a+b$. The purpose of this paper is to derive, for $a\ge b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1,\cdots ,a+b$ for which (i) ${\alpha }_r={\beta }_r-c$, (ii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$, (iii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c\pm 1$, where $c=0,\pm 1,\pm 2,\cdots$.

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.