The arithmetic complexity counts the number of algebraically independent entries in periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_{\theta}$ is a noncommutative torus corresponding to rational elliptic curve $\mathscr{E}(K)$, then rank $\mathscr{E}(K)$ given by simple formula $r(\mathscr{E}(K))= c(\mathscr{A}_{\theta})-1$, where $c(\mathscr{A}_{...

Abstract In this section, we study representation theory of DAHA, strictly speaking, the spherical subalgebra DAHA type $$A_1$$ A 1 , in terms brane quantization 2d A -model [85] on moduli space flat $${{\,\textrm{SL}\,}}(2,\mathbb {C})$$ <mml:...

Monotonicity properties of the ratio $$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$where f is an entire function are investigated. Earlier results for Euler’s gamma and other functions genus 1 generalised to p with negative zeros. Derivatives order comparable expression above related Stieltjes \(p+1\). Our applied Barnes multiple functions. We al...

Let f $f$ be a noncommutative polynomial of degree m ⩾ 1 $m\geqslant 1$ over an algebraically closed field F $F$ characteristic 0. If n − $n\geqslant m-1$ and α , 2 3 $\alpha _1,\alpha _2,\alpha _3$ are nonzero elements from such that + = 0 _1+\alpha _2+\alpha _3=0$ then every trace zero × $n\times n$ matrix can written as A _1 A_1+\alpha _2A_2+\alpha _3A_3$ for some i $A_i$ in the image M ( ) ...

We study the number of values taken by sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation $1,2,\dots,n$ and $1 \leq u < v n+1$. In particular, we show that for random choice permutation, with high probability there are $(\frac{1+e^{-2}}{4} +o(1)) n^2$ such sums. This answers an old question Erdős Harzheim. also obtain non-trivial bounds on maximum possible distinct sums, ra...

Abstract We study the temperature dependence of $$a_1$$ a 1 meson-nucleon coupling constant in framework soft-wall AdS/QCD model with thermal dilaton field. Profile functions for axial-vector and fermion fields AdS-Schwarzschild metric are presented. It is constructed a...

Abstract In this paper we introduce the concept of translation invariant functions: considering an arbitrary set $$\emptyset \ne S \subset {\mathbb {R}}^n$$ ∅ ≠ S ⊂ R n , function...