‎let $lambda_1,dots,lambda_n$ be positive real numbers such that‎‎$sum_{k=1}^n lambda_k=1$‎. ‎we prove that for‎‎any positive operators $a_1,a_2,cdots‎, ‎a_n$ in semifinite von‎‎neumann algebra $m$ with faithful normal trace that $t(1)


 I will describe recent joint work with Arkady Vaintrob on Fan-JarvisRuan theory, which is an analog of Gromov-Witten theory where a target space is replaced by a quasihomogeneous isolated hypersurface singularity. In the case of simple singularities of type A it corresponds to the intersection theory on the moduli space of higher spin curves, and constitutes a framework for the famous Witten ...

For every positive integer n, Sita Ramaiah’s identity states that $$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; _2(n)= \sum 1, \end{aligned}$$ $$(\mathbb {Z})^*$$ is the multiplicative group of units ring $$\mathbb {Z}$$ and $$\sigma _s(n) \displaystyle \nolimits _{d\mid n}d^s$$ . This can als...

A Theorem of V.Ovsienko characterizes sequences positive integers $(a_1,a_2,\ldots,a_n)$ such that the $(2\times2)$-matrix $\begin{pmatrix} a_n & -1 \\ 1 0 \end{pmatrix}\cdots \begin{pmatrix} a_1 \end{pmatrix}$ is equal to $\pm Id$. In this paper, we study equation when replace Id$ by M$. particular, give a combinatorial description solutions in terms dissections convex polygons cases $M=\begin...

Abstract Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in {Z}$ primitive root modulo p . $0<a_1<\cdots <a_n<p$ all the cubic residues in interval $(0,p)$ Then clearly sequence $a_1 \bmod p,\, a_2 p,\ldots , a_n p$ is permutation of $g^3 p,\,g^6 g^{3n} We determine sign this permutation.

A Theorem of V.Ovsienko characterizes sequences positive integers $(a_1,a_2,\ldots,a_n)$ such that the $(2\times2)$-matrix $\begin{pmatrix} a_n & -1 \\ 1 0 \end{pmatrix}\cdots \begin{pmatrix} a_1 \end{pmatrix}$ is equal to $\pm Id$. In this paper, we study equation when replace Id$ by M$. particular, give a combinatorial description solutions in terms dissections convex polygons cases $M=\begin...