空间组合学中的Stirling算子
Stirling operators in spatial combinatorics
摘要：
我们定义和研究的空间（无限维）对应的斯特林数。在经典组合学中，Pochhammer符号$(m)_n$可以从自然数$m\in\mathbb N$扩展到来自$\mathbb F=\mathbb R\text{ or }\mathbb C$的自变量$z$的下降因子$(z)_n=z(z-1)\dotsm (z-n+1)$，并且第一和第二类斯特林数是$(z)_n$通过$z^k$，$k\leq n$的扩展的系数，反之亦然。当考虑局部紧波兰空间$X$中元素的空间位置时，我们用构形空间--$X$上的离散Radon测度$γ=\sum_iδ_{x_i}$来代替$\mathbb N$，其中$δ_{x_i}$是质量在$x_i$的Dirac测度。空间下降因子$(γ)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}δ_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$可以自然地推广到映射$M^{(1)}(X)\niω\mapsto (ω)_n\in M^{(n)}(X)$，其中$M^{(n)}(X)$表示$\mathbb F$值的空间，$X^n$上的对称（对于$n\ge2$）氡测量。$M^{(n)}(X)$与$X^n$上的$\mathbb F$值对称连续函数空间$\mathcal {CF}^{(n)}(X)$之间具有自然的对偶性。第一类和第二类斯特林算子$\mathbf{s}(n,k)$和$\mathbf{S}(n,k)$是作用在空间$\mathcal {CF}^{(n)}(X)$和$\mathcal {CF}^{(k)}(X)$之间的线性算子，使得它们的对偶算子从$M^{(k)}(X)$作用到$M^{(n)}(X)$，分别满足$(ω)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*ω^{\otimes k}$和$ω^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(ω)_k$。我们得到了Stirling算子的组合性质，给出了它们与Poisson点过程的推广以及在正则对易关系下与Wick序的联系。
英文摘要：
We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations -- discrete Radon measures $γ=\sum_iδ_{x_i}$ on $X$, where $δ_{x_i}$ is the Dirac measure with mass at $x_i$.The spatial falling factorials $(γ)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}δ_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\niω\mapsto (ω)_n\in M^{(n)}(X)$, where $M^{(n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space $\mathcal {CF}^{(n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$, are linear operators, acting between spaces $\mathcal {CF}^{(n)}(X)$ and $\mathcal {CF}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(ω)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*ω^{\otimes k}$ and $ω^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(ω)_k$, respectively. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
