Abstract For any triple of positive integers $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ and $c \in{{\mathbb{C}}}^*$, cusp polynomial ${ f_{A^\prime }} x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$ is known to be mirror Geigle–Lenzing orbifold projective line ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. More precisely, with a suitable choi...

Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\!:=\!\sum^{n+k}_{j=1} c_j e^{a_j\cdot y}$. We prove that, for generic $c_j$, the number of connected components of the real ze...

Let $A_1$ be any spectrum in a class of finite spectra whose mod $2$ cohomology is isomorphic to free module rank one over the subalgebra $\mathcal{A}(1)$ Steenrod algebra. $E_{C}$ second Morava-$E$ theory associated universal deformation formal completion supersingular elliptic curve $(C) : y^{2}+y = x^{3}$ defined $\mathbb{F}_{4}$ and $G_{24}$ maximal subgroup automorphism group $\mathbb{S}_{...

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let R be an domain. We say that a bounded factorization it and for \(x \in R\), there positive integer N such any = a_1 \cdots a_n\) of x irreducibles \(a_1, \dots , in R, the inequality \(n \le N\) holds. In addition, we finite only finitely many ways (up to order associates). The notions domains were introduced b...

For a distance-regular graph \(\Gamma\) of diameter 3, the \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found parameters \(\Gamma_2\) given intersection array (independently, were by A.A. Makhnev D.V.Paduchikh). In this case, has an eigenvalue \(a_2-c_3\). paper, we study graphs with \(\theta=1\). particular, prove that, \(Q\)-polynomial from series arrays \(\{2...

‎Let $lambda_1,dots,lambda_n$  be positive real numbers such that $sum_{k=1}^n lambda_k=1$. In this paper, we prove that for any positive operators $a_1,a_2,ldots, a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $t(1)