We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\ cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ ...

We study the positivity of first Chern class a rank r Ulrich vector bundle E on smooth n-dimensional variety $X \subseteq \mathbb P^N$. prove that $c_1(E)$ is very positive every subvariety not contained in union lines X. In particular if X covered by lines, then big and $c_1(E)^n \ge r^n$. Moreover we classify bundles with $c_1(E)^2=0$ surfaces or $c_1(E)^3=0$ threefolds (with some exceptions).

Assume that $${\mathbb {F}}$$ is an algebraically closed field and let q denote a nonzero scalar in not root of unity. The universal Askey–Wilson algebra $$\triangle _q$$ unital associative -algebra defined by generators relations. are A, B, C the relations state each $$\begin{aligned} A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, C+\frac{q AB-q^{-1} BA}{q^2-q...

We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers–Ramanujan identities modulo $8$ quadruple sums, related work of Andrews, Schilling, Warnaar [J. Amer. Math. Soc. 12 (1999), pp. 677–702].

The Riemannian product $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ , where $${\mathbb{M}}_i(c_i)$$ denotes the 2-dimensional space form of constant sectional curvature $$c_i \in {\mathbb{R}}$$ has two different $${\mathrm{Spin}^{\mathrm{c}}}$$ structures carrying each a parallel spinor. restriction these spinor fields to 3-dimensional hypersurface M characterizes isometric immersion into...

Let A $A$ be an n × $n\times n$ real matrix, and let M $M$ random matrix whose entries are independent identically distributed sub-Gaussian variables with mean 0 variance 1. We make two contributions to the study of s ( + ) $s_n(A+M)$ , smallest singular value $A+M$ . (1) show that for all ε ⩾ $\epsilon \geqslant 0$ P [ ⩽ ] = O 2 e − Ω \begin{equation*} \mathbb {P}[s_n(A M) \leqslant \epsilon O...