Matrix scaling, explicit Sinkhorn limits, and arithmetic

Matrix scaling and explicit doubly stochastic limits

The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, often called the \emph{Sinkhorn limit} of $A$. The main result in this paper is the computation of exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices.

Matrix scaling limits in finitely many iterations

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter family of row but not column stochastic $n\times n$ positive matrices is constructed that become doubly stochastic after exactly one column scaling.

连分数的算术-几何标度谱

The arithmetic-geometric scaling spectrum for continued fractions

为了比较连分数数与相应逼近式的逼近子，我们引入了算术-几何标度。我们将利用数论自由能函数完全确定它的多重分形谱，并证明由具有相同标度指数的无理数组成的集合的豪斯多夫维数与该自由能函数的Legendre变换一致.此外，我们确定的局部行为的频谱在右边边界点点的渐近性，并讨论了连接到一组无理数的连分数数字超过一个给定的数字趋于无穷大。

求和乘法算术函数：标度与重整化

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions F{f;N,p^m}= \sum_{k\leq N} f(p^m k), m \in \mathbb Z_+ \cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k \in \mathbb Z_+, where p is a prime number . Assuming an asymptotic behavior of summatory function, F{f;N,1} \stackrel{N\to \infty}{=} G_1(N) [1+ { \cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2 \geq 0, - \infty < b_1, b_2< \infty , we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\inf…

The Scaling Site

We investigate the semi-ringed topos obtained by extension of scalars from the arithmetic site of our previous work, by replacing the smallest Boolean semifield by the tropical semifield of real numbers with the max-plus operations. The obtained site is the semi-direct product of the Euclidean half-line by the action of the monoid of positive integers by multiplication. Its points are the same as the points of the arithmetic site over the tropical semifield of real numbers, and coincide with the quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele c…