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).

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.

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.

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.

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.

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.

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.

The scope of this note is a self-contained presentation of a mathematical method that enables us to give an absolute upper bound for the difference of the Gini coefficients $$\left|G({\sigma }_1,\cdots ,{\sigma }_n)-G({\gamma }_1,\cdots ,{\gamma }_n)\right|,$$ where $({\gamma }_1,\cdots ,{\gamma }_n)$ represents the vector of the gross wages and $({\sigma }_1,\cdots ,{\sigma }_n)$ represents the vector of the corresponding super-gross wages that is used in the Czech Republic for calculating the net wage. Since (as of June 2019) ${\sigma }_i=100·⌈1.34{\gamma }_i/100⌉$, the study of the above difference seems to be somewhat inaccessible for many economists. However, our estimate...

In this paper we prove that for each $1<p,\tilde{p}<\infty$, the Banach space $(l^{\tilde{p}},{∥·∥}_{\tilde{p}})$ can be equivalently renormed in such a way that  the Banach space $(l^{\tilde{p}},{∥·∥}_{L,\alpha ,\beta ,p,\tilde{p}})$ is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in $l^2$ with the Day norm. We also show that the Banach space $(l^{\tilde{p}},{∥·∥}_{L,\alpha ,\beta ,p,\tilde{p}})$ has the weak fixed point property for nonexpansive mappings.

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 $V^{\mathbb{P}}$.