Finite-dimensional calculus
Identifieur interne : 003956 ( Main/Curation ); précédent : 003955; suivant : 003957Finite-dimensional calculus
Auteurs : Philip Feinsilver [États-Unis, France] ; Ren Schott [France]Source :
- Journal of Physics A Mathematical and Theoretical [ 1751-8113 ] ; 2009.
Descripteurs français
- Pascal (Inist)
English descriptors
- KwdEn :
- Teeft :
- Algebra, Analytic representations, Appell system, Binomial distribution, Calculus, Canonical, Canonical appell system, Canonical commutation relations, Canonical polynomials, Canonical variables, Column vector, Commutation, Commutation relations, Commutation rule, Corresponding polynomial, Cosh, Current work, Dual vector, Feinsilver, Gegenbauer polynomials, Hilbert space, Image analysis, Krawtchouk, Krawtchouk expansion, Krawtchouk expansions, Krawtchouk ghosts, Krawtchouk polynomials, Math, Matrix, Matrix approach, Negative jumps, Nite operator calculus, Operator calculus, Orthofermion algebra, Other words, Phys, Polynomials orthogonal, Probability theory, Quantum information, Quantum mechanics, Recurrence relation, Schott, Sech, Special functions, Tanh, Theor, Vacuum state, Vector space.
- mix :
Abstract
We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated HeisenbergWeyl algebra is called a TAA algebra after Tekin, Aydin and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the HeisenbergWeyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the HeisenbergWeyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement infinite terms Rota's finite operator calculus.
Url:
- https://api.istex.fr/ark:/67375/0T8-7MQGN0RB-9/fulltext.pdf
- https://hal.archives-ouvertes.fr/hal-00186130
DOI: 10.1088/1751-8113/42/37/375214
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wicri:auto="newGroup">Université de Lorraine</orgName></affiliation></author></analytic><series><title level="j">Journal of Physics A: Mathematical and Theoretical</title><idno type="ISSN">1751-8113</idno><imprint><date type="datePub">2009</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>Heisenberg-Weyl algebra</term><term>inversion formulas</term><term>orthofermions</term><term>polynomial systems</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We develop finite-dimensional calculus using matrices. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin, and Arik who formulated it in terms of orthofermions. the matrix approach is used to implement our method of polynomial inversion in one-variable and multivariable settings. Here we establish notations, present some algebraic developments, and discuss the univariate case in detail.</div></front></TEI></HAL><INIST><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en" level="a">Finite-dimensional calculus</title><author><name sortKey="Feinsilver, Philip" sort="Feinsilver, Philip" uniqKey="Feinsilver P" first="Philip" last="Feinsilver">Philip Feinsilver</name><affiliation wicri:level="1"><inist:fA14 i1="01"><s1>Department of Mathematics, Southern Illinois University</s1><s2>Carbondale, IL 62901</s2><s3>USA</s3><sZ>1 aut.</sZ></inist:fA14><country>États-Unis</country><wicri:noRegion>Carbondale, IL 62901</wicri:noRegion></affiliation></author><author><name sortKey="Schott, Rene" sort="Schott, Rene" uniqKey="Schott R" first="René" last="Schott">René Schott</name><affiliation wicri:level="3"><inist:fA14 i1="02"><s1>IECN and LORIA, Université Henri Poincaré-Nancy 1, BP 239</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>2 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">INIST</idno><idno type="inist">09-0393015</idno><date when="2009">2009</date><idno type="stanalyst">PASCAL 09-0393015 INIST</idno><idno type="RBID">Pascal:09-0393015</idno><idno type="wicri:Area/PascalFrancis/Corpus">000260</idno><idno type="wicri:Area/PascalFrancis/Curation">000767</idno><idno type="wicri:Area/PascalFrancis/Checkpoint">000242</idno><idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">000242</idno><idno type="wicri:doubleKey">1751-8113:2009:Feinsilver P:finite:dimensional:calculus</idno><idno type="wicri:Area/Main/Merge">003C76</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Finite-dimensional calculus</title><author><name sortKey="Feinsilver, Philip" sort="Feinsilver, Philip" uniqKey="Feinsilver P" first="Philip" last="Feinsilver">Philip Feinsilver</name><affiliation wicri:level="1"><inist:fA14 i1="01"><s1>Department of Mathematics, Southern Illinois University</s1><s2>Carbondale, IL 62901</s2><s3>USA</s3><sZ>1 aut.</sZ></inist:fA14><country>États-Unis</country><wicri:noRegion>Carbondale, IL 62901</wicri:noRegion></affiliation></author><author><name sortKey="Schott, Rene" sort="Schott, Rene" uniqKey="Schott R" first="René" last="Schott">René Schott</name><affiliation wicri:level="3"><inist:fA14 i1="02"><s1>IECN and LORIA, Université Henri Poincaré-Nancy 1, BP 239</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>2 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></analytic><series><title level="j" type="main">Journal of physics. A, Mathematical and theoretical : (Print)</title><title level="j" type="abbreviated">J. phys., A, math. theor. : (Print)</title><idno type="ISSN">1751-8113</idno><imprint><date when="2009">2009</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Journal of physics. A, Mathematical and theoretical : (Print)</title><title level="j" type="abbreviated">J. phys., A, math. theor. : (Print)</title><idno type="ISSN">1751-8113</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Analytic representation</term><term>Operator</term><term>Quantum mechanics</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Mécanique quantique</term><term>Représentation analytique</term><term>Opérateur</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated Heisenberg-Weyl algebra is called a TAA algebra after Tekin, Aydin and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the Heisenberg-Weyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the Heisenberg-Weyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement infinite terms Rota's 'finite operator calculus'.</div></front></TEI></INIST><ISTEX><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Finite-dimensional calculus</title><author><name sortKey="Feinsilver, Philip" sort="Feinsilver, Philip" uniqKey="Feinsilver P" first="Philip" last="Feinsilver">Philip Feinsilver</name></author><author><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:C4ACD704C3D937AD1E9C7959EA389BA072326DCD</idno><date when="2009" year="2009">2009</date><idno type="doi">10.1088/1751-8113/42/37/375214</idno><idno type="url">https://api.istex.fr/ark:/67375/0T8-7MQGN0RB-9/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">002E71</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">002E71</idno><idno type="wicri:Area/Istex/Curation">002E33</idno><idno type="wicri:Area/Istex/Checkpoint">000A63</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">000A63</idno><idno type="wicri:doubleKey">1751-8113:2009:Feinsilver P:finite:dimensional:calculus</idno><idno type="wicri:Area/Main/Merge">003A34</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Finite-dimensional calculus</title><author><name sortKey="Feinsilver, Philip" sort="Feinsilver, Philip" uniqKey="Feinsilver P" first="Philip" last="Feinsilver">Philip Feinsilver</name><affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics, Southern Illinois University, Carbondale, IL 62901</wicri:regionArea><placeName><region type="state">Illinois</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">France</country></affiliation></author><author><name sortKey="Schott, Ren" sort="Schott, Ren" uniqKey="Schott R" first="Ren" last="Schott">Ren Schott</name><affiliation wicri:level="3"><country xml:lang="fr">France</country><wicri:regionArea>IECN and LORIA, Universit Henri Poincar-Nancy 1, BP 239, 54506 Vandoeuvre-ls-Nancy</wicri:regionArea><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandoeuvre-ls-Nancy</settlement></placeName></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Physics A Mathematical and Theoretical</title><title level="j" type="abbrev">J. Phys. A Math. Theor.</title><idno type="ISSN">1751-8113</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2009">2009</date><biblScope unit="volume">42</biblScope><biblScope unit="issue">37</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="16">16</biblScope><biblScope unit="production">Printed in the UK</biblScope></imprint><idno type="ISSN">1751-8113</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1751-8113</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Algebra</term><term>Analytic representations</term><term>Appell system</term><term>Binomial distribution</term><term>Calculus</term><term>Canonical</term><term>Canonical appell system</term><term>Canonical commutation relations</term><term>Canonical polynomials</term><term>Canonical variables</term><term>Column vector</term><term>Commutation</term><term>Commutation relations</term><term>Commutation rule</term><term>Corresponding polynomial</term><term>Cosh</term><term>Current work</term><term>Dual vector</term><term>Feinsilver</term><term>Gegenbauer polynomials</term><term>Hilbert space</term><term>Image analysis</term><term>Krawtchouk</term><term>Krawtchouk expansion</term><term>Krawtchouk expansions</term><term>Krawtchouk ghosts</term><term>Krawtchouk polynomials</term><term>Math</term><term>Matrix</term><term>Matrix approach</term><term>Negative jumps</term><term>Nite operator calculus</term><term>Operator calculus</term><term>Orthofermion algebra</term><term>Other words</term><term>Phys</term><term>Polynomials orthogonal</term><term>Probability theory</term><term>Quantum information</term><term>Quantum mechanics</term><term>Recurrence relation</term><term>Schott</term><term>Sech</term><term>Special functions</term><term>Tanh</term><term>Theor</term><term>Vacuum state</term><term>Vector space</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">We discuss topics related to finite-dimensional calculus in the context of finite-dimensional quantum mechanics. The truncated HeisenbergWeyl algebra is called a TAA algebra after Tekin, Aydin and Arik who formulated it in terms of orthofermions. It is shown how to use a matrix approach to implement analytic representations of the HeisenbergWeyl algebra in univariate and multivariate settings. We provide examples for the univariate case. Krawtchouk polynomials are presented in detail, including a review of Krawtchouk polynomials that illustrates some curious properties of the HeisenbergWeyl algebra, as well as presenting an approach to computing Krawtchouk expansions. From a mathematical perspective, we are providing indications as to how to implement infinite terms Rota's finite operator calculus.</div></front></TEI></ISTEX></double></record>Pour manipuler ce document sous Unix (Dilib)
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