A sequence ${T_n}_{n=1}^{infty}$ of bounded linear  operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of t...

We give a construction of quantum LDPC codes dimension $\Theta(\log N)$ and distance $\Theta(N/\log as the code length $N\to\infty$. Using product chain complexes this also provides family $\Omega(N^{1-\alpha/2}/\log $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. introduce study new operation called lifted product, which naturally generalizes operations for complexes. Moreover, simple byp...

We consider the jellium model of $N$ particles on a line confined in an external harmonic potential and with pairwise one-dimensional Coulomb repulsion strength $\alpha > 0$. Using gas method, we study statistics $s = (1/N) \sum_{i=1}^N f(x_i)$ where $f(x)$, principle, is arbitrary smooth function. While mean $s$ easy to compute, variance nontrivial due long-range interactions. In this paper de...

We investigate the nonnegative solutions to nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote Riesz potentials of order $\alpha$ $\beta$ respectively. Our approach relies on a nonlocal positivity principle which allows us derive optimal ranges for parameters $\alph...

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe sets of left zero-divisors, right zero-divisors and two sided C_{n}$; their numbers. For $n \geq 4$, define an undirected graph $\Gamma(\mathcal C_{n})$ associated with whose vertices are excluding zero element $\theta$ distinct $\alpha$ $\beta$ joined by edge in case $\alph...

In this work we consider the fractional Kirchhoff equations with singular nonlinearity, $$\displaylines{ M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx dy\Big) (-\Delta)^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\cr \int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}dx v \mu b(x)|v|^{q-2}v +\fra...

Let $ A be a closed linear operator defined on complex Banach space X $. We show novel representation, using strongly continuous families of bounded operators \mathbb{N}_0 $, for the unique solution following time-stepping scheme \begin{document}$ \begin{eqnarray*} (*)\left\{ \begin{array}{rcl} \,_C \nabla^{\alpha} u^n&amp; = &amp;Au^{n}+f^{n}, \quad n\geq 2;\\ u^0&amp; &amp;u_0; \\ u^1&amp; &a...

The Corners Theorem states that for any $\alpha > 0$ there exists an $N_0$ such abelian group $G$ with $|G| = N \geq N_0$ and subset $A \subset G \times G$ $|A| \ge \alpha N^2$ we can find a corner in $A$ , i.e. exist $x, y, d \in $d \neq $(x, y), (x+d, (x, y+d) A$. Here, consider stronger version: given $A$, each define $S_d \{(x, y) : A \}$ . So $|S_d|$ is the number of corners size $d$. Is i...

For $\alpha > 0$ we consider the operator $K_\alpha \colon \ell^2 \to \ell^2$ corresponding to matrix \[\left(\frac{(nm)^{-\frac{1}{2}+\alpha}}{[\max(n,m)]^{2\alpha}}\right)_{n,m=1}^\infty.\] By interpreting $K_\alpha$ as inverse of an unbounded Jacobi matrix, show that absolutely continuous spectrum coincides with $[0, 2/\alpha]$ (multiplicity one), and there is no singular spectrum. There a f...