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Dimension reduction techniques for the minimization of theta functions on lattices
We consider the minimization of theta functions Λ ( α ) = ∑ p ∈ Λ e − π α | p | 2 amongst periodic configurations Λ ⊂ R d , by reducing the dimension of the problem, following as a motivation the case d = 3, where minimizers are supposed to be either the body-centered cubic or the face-centered cubi...
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| Published in: | Journal of mathematical physics 2017-07, Vol.58 (7), p.1 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c320t-bcd9cefaaf4f4e2fcc95dcc819428108b1435c1d589c1900742447c892ed9fce3 |
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| container_issue | 7 |
| container_start_page | 1 |
| container_title | Journal of mathematical physics |
| container_volume | 58 |
| creator | Bétermin, Laurent Petrache, Mircea |
| description | We consider the minimization of theta functions
Λ
(
α
)
=
∑
p
∈
Λ
e
−
π
α
|
p
|
2
amongst periodic configurations
Λ
⊂
R
d
, by reducing the dimension of the problem, following as a
motivation the case d = 3, where minimizers are supposed to be either the
body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension
is by considering layered lattices, and minimize either among competitors presenting
different sequences of repetitions of the layers, or among competitors presenting
different shifts of the layers with respect to each other. The second case presents the
problem of minimizing theta functions also on translated lattices, namely, minimizing
(
Λ
,
u
)
↦
Λ
+
u
(
α
)
, relevant to the study of two-component Bose-Einstein
condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is
by considering lattices with a product structure or by successively minimizing over
concentric layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we
study in detail in two dimensions. |
| doi_str_mv | 10.1063/1.4995401 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1063_1_4995401</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2116079714</sourcerecordid><originalsourceid>FETCH-LOGICAL-c320t-bcd9cefaaf4f4e2fcc95dcc819428108b1435c1d589c1900742447c892ed9fce3</originalsourceid><addsrcrecordid>eNp9kE9LAzEQxYMoWKsHv8GCJ4WtmWyySY5S6x8oCKLnJZ1NaEp3tybZg356d9uee5rhzW9mHo-QW6AzoGXxCDOuteAUzsgEqNK5LIU6JxNKGcsZV-qSXMW4oRRAcT4hn8--sW30XZsFW_eYxi5ZXLf-p7cxc13I0tpmjW994__Mft65UUsmc32734jZoG5NSh5tvCYXzmyjvTnWKfl-WXzN3_Llx-v7_GmZY8FoyldYa7TOGMcdt8whalEjKtCcqcH6CnghEGqhNIKmVHLGuUSlma21Q1tMyd3h7i50o9dUbbo-tMPLigGUVGo5nDhBgWaSl1IINlD3BwpDF2OwrtoF35jwWwGtxmArqI7BDuzDgY3o0z6QE_A_3wZ3_A</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1927467552</pqid></control><display><type>article</type><title>Dimension reduction techniques for the minimization of theta functions on lattices</title><source>American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list)</source><source>AIP Journals (American Institute of Physics)</source><creator>Bétermin, Laurent ; Petrache, Mircea</creator><creatorcontrib>Bétermin, Laurent ; Petrache, Mircea</creatorcontrib><description>We consider the minimization of theta functions
Λ
(
α
)
=
∑
p
∈
Λ
e
−
π
α
|
p
|
2
amongst periodic configurations
Λ
⊂
R
d
, by reducing the dimension of the problem, following as a
motivation the case d = 3, where minimizers are supposed to be either the
body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension
is by considering layered lattices, and minimize either among competitors presenting
different sequences of repetitions of the layers, or among competitors presenting
different shifts of the layers with respect to each other. The second case presents the
problem of minimizing theta functions also on translated lattices, namely, minimizing
(
Λ
,
u
)
↦
Λ
+
u
(
α
)
, relevant to the study of two-component Bose-Einstein
condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is
by considering lattices with a product structure or by successively minimizing over
concentric layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we
study in detail in two dimensions.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.4995401</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>New York: American Institute of Physics</publisher><subject>Body centered cubic lattice ; Bose-Einstein condensates ; Condensates ; Crystal lattices ; Crystals ; Face centered cubic lattice ; Lattice theory ; Mathematics ; Optimization ; Physics</subject><ispartof>Journal of mathematical physics, 2017-07, Vol.58 (7), p.1</ispartof><rights>Author(s)</rights><rights>Copyright American Institute of Physics Jul 2017</rights><rights>2017 Author(s). Published by AIP Publishing.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c320t-bcd9cefaaf4f4e2fcc95dcc819428108b1435c1d589c1900742447c892ed9fce3</citedby><cites>FETCH-LOGICAL-c320t-bcd9cefaaf4f4e2fcc95dcc819428108b1435c1d589c1900742447c892ed9fce3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.4995401$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>314,780,782,784,795,27923,27924,76254</link.rule.ids></links><search><creatorcontrib>Bétermin, Laurent</creatorcontrib><creatorcontrib>Petrache, Mircea</creatorcontrib><title>Dimension reduction techniques for the minimization of theta functions on lattices</title><title>Journal of mathematical physics</title><description>We consider the minimization of theta functions
Λ
(
α
)
=
∑
p
∈
Λ
e
−
π
α
|
p
|
2
amongst periodic configurations
Λ
⊂
R
d
, by reducing the dimension of the problem, following as a
motivation the case d = 3, where minimizers are supposed to be either the
body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension
is by considering layered lattices, and minimize either among competitors presenting
different sequences of repetitions of the layers, or among competitors presenting
different shifts of the layers with respect to each other. The second case presents the
problem of minimizing theta functions also on translated lattices, namely, minimizing
(
Λ
,
u
)
↦
Λ
+
u
(
α
)
, relevant to the study of two-component Bose-Einstein
condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is
by considering lattices with a product structure or by successively minimizing over
concentric layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we
study in detail in two dimensions.</description><subject>Body centered cubic lattice</subject><subject>Bose-Einstein condensates</subject><subject>Condensates</subject><subject>Crystal lattices</subject><subject>Crystals</subject><subject>Face centered cubic lattice</subject><subject>Lattice theory</subject><subject>Mathematics</subject><subject>Optimization</subject><subject>Physics</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LAzEQxYMoWKsHv8GCJ4WtmWyySY5S6x8oCKLnJZ1NaEp3tybZg356d9uee5rhzW9mHo-QW6AzoGXxCDOuteAUzsgEqNK5LIU6JxNKGcsZV-qSXMW4oRRAcT4hn8--sW30XZsFW_eYxi5ZXLf-p7cxc13I0tpmjW994__Mft65UUsmc32734jZoG5NSh5tvCYXzmyjvTnWKfl-WXzN3_Llx-v7_GmZY8FoyldYa7TOGMcdt8whalEjKtCcqcH6CnghEGqhNIKmVHLGuUSlma21Q1tMyd3h7i50o9dUbbo-tMPLigGUVGo5nDhBgWaSl1IINlD3BwpDF2OwrtoF35jwWwGtxmArqI7BDuzDgY3o0z6QE_A_3wZ3_A</recordid><startdate>201707</startdate><enddate>201707</enddate><creator>Bétermin, Laurent</creator><creator>Petrache, Mircea</creator><general>American Institute of Physics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope></search><sort><creationdate>201707</creationdate><title>Dimension reduction techniques for the minimization of theta functions on lattices</title><author>Bétermin, Laurent ; Petrache, Mircea</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c320t-bcd9cefaaf4f4e2fcc95dcc819428108b1435c1d589c1900742447c892ed9fce3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Body centered cubic lattice</topic><topic>Bose-Einstein condensates</topic><topic>Condensates</topic><topic>Crystal lattices</topic><topic>Crystals</topic><topic>Face centered cubic lattice</topic><topic>Lattice theory</topic><topic>Mathematics</topic><topic>Optimization</topic><topic>Physics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bétermin, Laurent</creatorcontrib><creatorcontrib>Petrache, Mircea</creatorcontrib><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><jtitle>Journal of mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bétermin, Laurent</au><au>Petrache, Mircea</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Dimension reduction techniques for the minimization of theta functions on lattices</atitle><jtitle>Journal of mathematical physics</jtitle><date>2017-07</date><risdate>2017</risdate><volume>58</volume><issue>7</issue><spage>1</spage><pages>1-</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>We consider the minimization of theta functions
Λ
(
α
)
=
∑
p
∈
Λ
e
−
π
α
|
p
|
2
amongst periodic configurations
Λ
⊂
R
d
, by reducing the dimension of the problem, following as a
motivation the case d = 3, where minimizers are supposed to be either the
body-centered cubic or the face-centered cubic lattices. A first way to reduce dimension
is by considering layered lattices, and minimize either among competitors presenting
different sequences of repetitions of the layers, or among competitors presenting
different shifts of the layers with respect to each other. The second case presents the
problem of minimizing theta functions also on translated lattices, namely, minimizing
(
Λ
,
u
)
↦
Λ
+
u
(
α
)
, relevant to the study of two-component Bose-Einstein
condensates, Wigner bilayers and of general crystals. Another way to reduce dimension is
by considering lattices with a product structure or by successively minimizing over
concentric layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we
study in detail in two dimensions.</abstract><cop>New York</cop><pub>American Institute of Physics</pub><doi>10.1063/1.4995401</doi><tpages>40</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-2488 |
| ispartof | Journal of mathematical physics, 2017-07, Vol.58 (7), p.1 |
| issn | 0022-2488 1089-7658 |
| language | eng |
| recordid | cdi_crossref_primary_10_1063_1_4995401 |
| source | American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list); AIP Journals (American Institute of Physics) |
| subjects | Body centered cubic lattice Bose-Einstein condensates Condensates Crystal lattices Crystals Face centered cubic lattice Lattice theory Mathematics Optimization Physics |
| title | Dimension reduction techniques for the minimization of theta functions on lattices |
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