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...

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$ $[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma }_i\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

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.

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 $f$.

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 $0\mathit{\}}$.

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 $t$.

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 $M$ be a free module over a noetherian ring. For ${\omega }_1,...,{\omega }_k\in M$, let $𝒜$ be the ideal generated by coefficients of ${\omega }_1\wedge ...\wedge {\omega }_k$. For an element $\omega \in {\bigwedge }^{p}M$ with $p\<\mathrm{prof}.𝒜$, if $\omega \wedge {\omega }_1\wedge ...\wedge {\omega }_k=0$, there exists ${\eta }_1,...,{\eta }_k\in {\bigwedge }^{p-1}M$ such that $\omega ={\sum }_{i=1}^k{\eta }_i\wedge {\omega }_i$. This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.