We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$

Let $\mathbb{P}=\{p_1,p_2,\cdots ,p_i,\cdots \}$ be the set of prime numbers (or more generally a set of pairwise co-prime elements). Let us denote $A_p^{a,b}=\{p^{an+b}m\mid n\in \mathbb{N}\cup \left\{0\right\};m\in \mathbb{N},p\mathrm{does}\mathrm{not}\mathrm{divide}m\}$, where $a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}$. Then for arbitrary finite set $B$, $B\subset \mathbb{P}$ holds $$d\left(\bigcap _{p_i\in B}{A}_{p_i}^{a_i,b_i}\right)=\prod _{p_i\in B}d\left(A_{p_i}^{a_i,b_i}\right),$$ and $$d\left(A_{p_i}^{a_i,b_i}\right)=\frac{\frac{1}{p_i^{b_i}}\left(1-\frac{1}{p_i}\right)}{1-\frac{1}{p_i^{a_i}}}.$$ If we denote $$A=\left\{\frac{\frac{1}{p^b}\left(1-\frac{1}{p}\right)}{1-\frac{1}{p^a}}\mid p\in \mathbb{P},a\in \mathbb{N},b\in \mathbb{N}\cup \left\{0\right\}\right\},$$ where $\mathbb{P}$ is the set of all prime numbers, then for closure of set $A$ holds $$\mathrm{cl}A=A\cup B\cup \{0,1\},$$ where $B=\left\{\frac{1}{p^b}\left(1-\frac{1}{p}\right)\mid p\in \mathbb{P},b\in \mathbb{N}\cup \left\{0\right\}\right\}$.

We establish certain properties for the class $𝒰(\Omega ,{\zeta }_0)$ of universal functions in $\Omega$ with respect to the center ${\zeta }_0\in \Omega$, for certain types of connected non-simply connected domains $\Omega$. In the case where $ℂ/\Omega$ is discrete we prove that this class is $G_{\delta }$-dense in $H\left(\Omega \right)$, depends on the center ${\zeta }_0$ and that the analog of Kahane’s conjecture does not hold.

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).

Let $p$ be a prime number, ${\mathbb{Q}}_p$ the field of $p$-adic numbers and ${ℂ}_p$ the completion of the algebraic closure of ${\mathbb{Q}}_p$. In this paper we obtain a representation theorem for rigid analytic functions on ${𝐏}^1\left({ℂ}_p\right)\setminus C(t,ϵ)$ which are equivariant with respect to the Galois group $G=Ga{l}_{cont}({ℂ}_p/{\mathbb{Q}}_p)$, where $t$ is a lipschitzian element of ${ℂ}_p$ and $C(t,ϵ)$ denotes the $ϵ$-neighborhood of the $G$-orbit of $t$.

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 $f$.

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.

Various techniques are presented for constructing $\Lambda$ (p) sets which are not $\Lambda (p+ϵ)$ for all $ϵ\>0$. The main result is that there is a $\Lambda$ (4) set in the dual of any compact abelian group which is not $\Lambda (4+ϵ)$ for all $ϵ\>0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda$ (p) not $\Lambda (p+ϵ)$ sets, for certain values of $p$. The main new constructions in specific dual groups are: – there is a $\Lambda$ (2k) set which is not $\Lambda (2k+ϵ)$ in $\mathbf{Z}\left(2\right)\oplus \mathbf{Z}\left(2\right)\oplus \cdots$ for all $2\le k$, $k\in \mathbf{N}$ and...