In this very short paper, we show that the average overlap density of a union-closed family $\mathcal{F}$ subsets $\{1,2,\ldots,n\}$ may be as small \[\Theta((\log_2 \log_2 |\mathcal{F}|)/(\log_2 |\mathcal{F}|)),\] for infinitely many positive integers $n$.

A space $Y$ is called an {em extension} of a space $X$, if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {em equivalent}, if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Yleq Y'$, if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ ...

in this paper, we define and study the notion of the real-valued functions on a frame $l$. we show that $f(l) $, consisting of all frame homomorphisms from the power set of $mathbb{r}$ to a frame $ l$, is an $f$-ring, as a generalization of all functions from a set $x$ into $mathbb r$. also, we show that $f(l) $ is isomorphic to a sub-$f$-ring of $mathcal{r}(l)$, the ring of real-valued continu...

For a constant $alphain left(-frac{pi}{2},frac{pi}{2}right)$,  we definea  subclass of the spirallike functions, $SP_{p}(alpha)$, the setof all functions $fin mathcal{A}$[releft{e^{-ialpha}frac{zf'(z)}{f(z)}right}geqleft|frac{zf'(z)}{f(z)}-1right|.]In  the present paper, we shall give the estimate of the norm of the pre-Schwarzian derivative  $mathrm{T}...

Given a partially order set (poset) $P$, and pair of families ideals $\mathcal{I}$ filters $\mathcal{F}$ in $P$ such that each $(I,F)\in \mathcal{I}\times\mathcal{F}$ has non-empty intersection, the dualization problem over is to check whether there an ideal $X$ which intersects every member does not contain any $\mathcal{I}$. Equivalently, for distributive lattice $L=L(P)$, given by poset its ...

We consider $\mathcal{N}=1$ superconformal field theories in three dimensions possessing a conserved current multiplet ${\mathcal{F}}_{({\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}{\ensuremath{\alpha}}_{3}{\ensuremath{\alpha}}_{4})}$, which we refer to as the superspin-2 multiplet. At component level it contains spin-2 different from energy-momentum tensor and fermionic higher-spin of spi...

Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of rational elliptic curve $E$ and $p$-adic Lie extension $\mathcal{F}_\infty$ $\mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ over $\mathcal{F}_\infty$. In this paper, we study arithmetic statistics for algebraic structure group. The results provide insights into asymptotics ...

We establish, for $1<p<\\infty$, higher order $\\mathcal{S}^{p}$-differentiability results of the function $\\varphi \\colon t\\in \\mathbb{R} \\mapsto f(A+tK) - f(A)$ selfadjoint operators $A$ and $K$ on a separable Hilbert space $\\mathcal{H}$ with element Schatten class $\\mathcal{S}^{p}(\\mathcal{H})$ $f$ $n$-times differentiable $\\mathbb{R}$. prove that if either $f^{(n)}$ are bound...

Let \(\psi\) be univalent, we introduce a newly defined class of analytic functions, given by $$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ (0)=0 \right\} . \end{aligned}$$Note that functions in this may not univalent. Further, establish the growth theorem with some geometrical conditions on and obtain Koebe domain related shar...