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Stirling operators in spatial combinatorics
Dmitri Finkelshtein 
,
Yuri Kondratiev,
Eugene Lytvynov ,
Maria João Oliveira
,
Yuri Kondratiev,
Eugene Lytvynov ,
Maria João Oliveira
Journal of Functional Analysis, Volume: 282, Issue: 2, Start page: 109285
Swansea University Authors:
Dmitri Finkelshtein , Eugene Lytvynov
-
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DOI (Published version): 10.1016/j.jfa.2021.109285
Abstract
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\te...
| Published in: | Journal of Functional Analysis |
|---|---|
| ISSN: | 0022-1236 |
| Published: |
Elsevier BV
2022
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa58358 |
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2021-10-15T14:49:33Z |
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2023-01-11T14:38:54Z |
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<?xml version="1.0"?><rfc1807><datestamp>2022-10-27T11:33:56.3961982</datestamp><bib-version>v2</bib-version><id>58358</id><entry>2021-10-15</entry><title>Stirling operators in spatial combinatorics</title><swanseaauthors><author><sid>4dc251ebcd7a89a15b71c846cd0ddaaf</sid><ORCID>0000-0001-7136-9399</ORCID><firstname>Dmitri</firstname><surname>Finkelshtein</surname><name>Dmitri Finkelshtein</name><active>true</active><ethesisStudent>false</ethesisStudent></author><author><sid>e5b4fef159d90a480b1961cef89a17b7</sid><ORCID>0000-0001-9685-7727</ORCID><firstname>Eugene</firstname><surname>Lytvynov</surname><name>Eugene Lytvynov</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2021-10-15</date><deptcode>MACS</deptcode><abstract>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 $\gamma=\sum_i\delta_{x_i}$ on $X$, where $\delta_{x_i}$ is the Dirac measure with mass at $x_i$. The spatial falling factorials $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, 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 $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ and $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. In the case where $X$ has only a single point, the Stirling operators can be identified with Stirling numbers. 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.</abstract><type>Journal Article</type><journal>Journal of Functional Analysis</journal><volume>282</volume><journalNumber>2</journalNumber><paginationStart>109285</paginationStart><paginationEnd/><publisher>Elsevier BV</publisher><placeOfPublication/><isbnPrint/><isbnElectronic/><issnPrint>0022-1236</issnPrint><issnElectronic/><keywords>Spatial falling factorials; Stirling operators; Poisson functional; Wick ordering for canonical commutation relations</keywords><publishedDay>15</publishedDay><publishedMonth>1</publishedMonth><publishedYear>2022</publishedYear><publishedDate>2022-01-15</publishedDate><doi>10.1016/j.jfa.2021.109285</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics and Computer Science School</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>MACS</DepartmentCode><institution>Swansea University</institution><apcterm/><funders/><projectreference/><lastEdited>2022-10-27T11:33:56.3961982</lastEdited><Created>2021-10-15T15:42:58.8682689</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Dmitri</firstname><surname>Finkelshtein</surname><orcid>0000-0001-7136-9399</orcid><order>1</order></author><author><firstname>Yuri</firstname><surname>Kondratiev</surname><order>2</order></author><author><firstname>Eugene</firstname><surname>Lytvynov</surname><orcid>0000-0001-9685-7727</orcid><order>3</order></author><author><firstname>Maria João</firstname><surname>Oliveira</surname><order>4</order></author></authors><documents><document><filename>58358__21177__028b8315bfa94591a245ab1865d5cbbd.pdf</filename><originalFilename>Second revision.pdf</originalFilename><uploaded>2021-10-15T15:47:16.9161153</uploaded><type>Output</type><contentLength>406045</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2022-10-19T00:00:00.0000000</embargoDate><documentNotes>©2021 All rights reserved. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND)</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language><licence>https://creativecommons.org/licenses/by-nc-nd/4.0/</licence></document></documents><OutputDurs/></rfc1807> |
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2022-10-27T11:33:56.3961982 v2 58358 2021-10-15 Stirling operators in spatial combinatorics 4dc251ebcd7a89a15b71c846cd0ddaaf 0000-0001-7136-9399 Dmitri Finkelshtein Dmitri Finkelshtein true false e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2021-10-15 MACS 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 $\gamma=\sum_i\delta_{x_i}$ on $X$, where $\delta_{x_i}$ is the Dirac measure with mass at $x_i$. The spatial falling factorials $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, 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 $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ and $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. In the case where $X$ has only a single point, the Stirling operators can be identified with Stirling numbers. 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. Journal Article Journal of Functional Analysis 282 2 109285 Elsevier BV 0022-1236 Spatial falling factorials; Stirling operators; Poisson functional; Wick ordering for canonical commutation relations 15 1 2022 2022-01-15 10.1016/j.jfa.2021.109285 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2022-10-27T11:33:56.3961982 2021-10-15T15:42:58.8682689 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Dmitri Finkelshtein 0000-0001-7136-9399 1 Yuri Kondratiev 2 Eugene Lytvynov 0000-0001-9685-7727 3 Maria João Oliveira 4 58358__21177__028b8315bfa94591a245ab1865d5cbbd.pdf Second revision.pdf 2021-10-15T15:47:16.9161153 Output 406045 application/pdf Accepted Manuscript true 2022-10-19T00:00:00.0000000 ©2021 All rights reserved. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND) true eng https://creativecommons.org/licenses/by-nc-nd/4.0/ |
| title |
Stirling operators in spatial combinatorics |
| spellingShingle |
Stirling operators in spatial combinatorics Dmitri Finkelshtein Eugene Lytvynov |
| title_short |
Stirling operators in spatial combinatorics |
| title_full |
Stirling operators in spatial combinatorics |
| title_fullStr |
Stirling operators in spatial combinatorics |
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Stirling operators in spatial combinatorics |
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Stirling operators in spatial combinatorics |
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4dc251ebcd7a89a15b71c846cd0ddaaf e5b4fef159d90a480b1961cef89a17b7 |
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4dc251ebcd7a89a15b71c846cd0ddaaf_***_Dmitri Finkelshtein e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
| author |
Dmitri Finkelshtein Eugene Lytvynov |
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Dmitri Finkelshtein Yuri Kondratiev Eugene Lytvynov Maria João Oliveira |
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Journal of Functional Analysis |
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282 |
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109285 |
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0022-1236 |
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10.1016/j.jfa.2021.109285 |
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Elsevier BV |
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Faculty of Science and Engineering |
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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 $\gamma=\sum_i\delta_{x_i}$ on $X$, where $\delta_{x_i}$ is the Dirac measure with mass at $x_i$. The spatial falling factorials $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, 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 $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ and $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. In the case where $X$ has only a single point, the Stirling operators can be identified with Stirling numbers. 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. |
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2022-01-15T05:15:03Z |
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11.294684 |