论文标题
空间组合制剂中的Stirling操作员
Stirling operators in spatial combinatorics
论文作者
论文摘要
我们定义和研究斯特林数字的空间（无限维）对应物。在古典组合学中，可以从\ mathbb n $ in \ mathbb n $中扩展到pochhammer符号$（m）_n $，延长到跌落的fortorials $（z）_n = z（z-1）\ dotsm（z-n+1）$ $ z $的参数$ z $ from $第二种是$（z）_n $通过$ z^k $，$ k \ leq n $的扩展系数，反之亦然。当考虑到本地紧凑的波兰$ x $中元素的空间位置时，我们用配置的空间代替了$ \ mathbb n $ - $ x $上的离散radon测量$γ= \sum_iδ_{x_i} $ $（γ）_n：= \ sum_ {i_1} \ sum_ {i_2 \ ne i_1} \ dotsm \ sum_ {i_n \ ne i_1，\ dots，\ dots，i_n \ ne \ ne \ ne \ ne i_ {n-1}}δ_ {（x_ {i_1}，x_ {i_2}，\ dots，x_ {i_n}）} $可以自然地扩展到mappings $ m^{（1）}（x）\niΩ $ m^{（n）}（x）$表示$ \ mathbb f $ - 价值的空间，对称（适用于$ n \ ge2 $）radon radon radon radon Mesuals在$ x^n $上。 $ m^{（n）}（x）$与空间$ \ mathcal {cf}^{（n）}（x）$之间的$ m^{（n）}（x）$之间存在自然双重性。第一和第二类的stirling操作员，$ \ mathbf {s}（n，k）$和$ \ mathbf {s}（s}（n，k）$是线性操作员，作用于空间之间从$ m^{（k）}（x）$中表现为$ m^{（n）}（x）$，满足$（ω）_n = \ sum_ {k = 1}^n \ mathbf {s}（s}（n，n，k） n} = \ sum_ {k = 1}^n \ mathbf {s}（n，k）^*（ω）_k $。我们得出了斯特林操作员的组合特性，以泊松点过程的概括以及在规范的换向关系下的灯芯顺序呈现他们的联系。
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.
