Hilberdink showed that a continuous prime system for which there exists constant $A$ such the function $N(x)-Ax$ is periodic satisfies $N(x)=c(x-1)+1$. He further $c_0>2$, of this form if and only $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot 10^{14}$. To do so compute representation twisted exponential as sum over roots Riemann zeta function. We t...

The third real de Rham cohomology of compact homogeneous spaces is studied. Given $$M=G/K$$ with G semisimple, we first show that each bi-invariant symmetric bilinear form Q on $${\mathfrak {g}} $$ such $$Q|_{{\mathfrak {k}} \times {\mathfrak }=0$$ naturally defines a G-invariant closed 3-form $$H_Q$$ M, which plays the role so called Cartan $$Q([\cdot ,\cdot ],\cdot )$$ Lie group G. Indeed, ev...

Abstract A Darwinian perspective of the standard model physics (SMP) quantum fields (QFs) is proposed, called physics-cell (PC) approach. Because evolution not deterministic, PC approach allows for violation charge-parity-time symmetry. In approach, SMP laws are contained in PCs which receive and emit QFs through PCs’ outer surface necessarily constrained by Bekenstein’s surface-information lim...

This paper is concerned with the logarithmically improved regularity criterion in terms of middle eigenvalue strain tensor to 3D Boussinesq equations Besov spaces negative indices. It shown that a weak solution regular on $ (0, T] provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (...

Let ${\mathcal A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ)$, where $P$ Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in {\mathbb C}$ $|s|=1$ and all $a,...

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families congruences modulo arbitrary powers $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $k$, have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}...

A number of possible pulsar radio emission mechanisms are based on streaming instabilities in relativistically hot electron-positron pair plasmas. At saturation the unstable waves can form, principle, stable solitary which could emit observed intense signals. We searched for proper plasma parameters would lead to formation solitons, investigated their properties and dynamics as well resulting o...