Let $$A_1, A_2\in {{\mathbb {C}}}(z)$$ be rational functions of degree at least two that are neither Lattès maps nor conjugate to $$z^{\pm n}$$ or $$\pm T_n.$$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms $$({{\mathbb {P}}}^1({{\mathbb {C}}}))^2$$ the form $$(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$$ In particular, we show if $$A\in is not a “generalized ...

We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For case $N=1$ such are introduced into index computations by an action $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. (re)derive this fact using three different field theoretic descriptions four dimensional models. The naturally associa...

The symmetrization map $$\pi :{\mathbb{C}}^2\rightarrow {\mathbb{C}}^2$$ is defined by (z_1,z_2)=(z_1+z_2,z_1z_2).$$ closed symmetrized bidisc $$\Gamma$$ the of unit $$\overline{{\mathbb{D}}^2}$$ , that is, $$\begin{aligned} \Gamma = \pi (\overline{{\mathbb{D}}^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\le 1, i=1,2 \}. \end{aligned}$$ A pair commuting Hilbert space operators (S, P) for which a spectral...

We establish a generalization of Sturm–Picone comparison theorem for pair fractional nonlocal equations: $$\begin{aligned} (-div. (A_1(x)\nabla ))^{s} u= & {} C_{1}(x) u \,\,\,\text { in }\,\,\Omega ,\\ 0 \,\,\,\,\text on }\,\,\,\,\,\partial \Omega , \end{aligned}$$ and (A_2(x)\nabla v= C_{2}(x) v in}\,\,\Omega on}\,\,\,\,\,\partial where $$\Omega \subset \mathbb {R}^n$$ is an open bounded subs...

Let $$[a_1(x),a_2(x),a_3(x),\cdots ]$$ be the continued fraction expansion of $$x\in (0,1)$$ . This paper is concerned with certain sets fractions non-decreasing partial quotients. As a main result, we obtain Hausdorff dimension set $$\begin{aligned} \left\{ x\in (0,1): a_1(x)\le a_2(x)\le \cdots ,\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} \end{aligned}$$ fo...