{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98364"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98364","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Asymptotically optimal shapes for counting lattice points and eigenvalues","abstract":"\"In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the \"\"radius\"\" approaches infinity. In particular, the result implies when 1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity. The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 < p < 1 by building on our results here. The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity? In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates. Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.\"","abstract_html":"&quot;In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the &quot;&quot;radius&quot;&quot; approaches infinity. In particular, the result implies when 1 &lt; p &lt; ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity. The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 &lt; p &lt; 1 by building on our results here. The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity? In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates. Our results handle the p-circle when p &gt; 1 (concave) and also when 0 &lt; p &lt; 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.&quot;","abstract_has_math":false,"creators":["Liu, Shiya"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Laugesen, Richard S.","DeVille, Lee","Kirr, Eduard","Li, Xiaochun"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:56:41Z","date_published":"2017-09-29T17:56:41Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Lattice points","Planar convex domain","P-ellipse","Spectral optimization","Laplacian","Dirichlet eigenvalues","Neumann eigenvalues","Translated lattice","Schrödinger eigenvalues","Harmonic oscillator"],"languages":["en"],"rights":["Copyright 2017 Shiya Liu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98364","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Laugesen, Richard S.","DeVille, Lee","Kirr, Eduard","Li, Xiaochun"]},{"key":"dc:creator","label":"Author","values":["Liu, Shiya"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T17:56:41Z","2017-07-11","2017-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Lattice points","Planar convex domain","P-ellipse","Spectral optimization","Laplacian","Dirichlet eigenvalues","Neumann eigenvalues","Translated lattice","Schrödinger eigenvalues","Harmonic oscillator"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Shiya Liu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98364"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the \"\"radius\"\" approaches infinity. In particular, the result implies when 1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity. The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 < p < 1 by building on our results here. The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity? In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates. Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Shiya Liu, accepted the attached license on 2017-07-10 at 19:16.","The student, Shiya Liu, submitted this Dissertation for approval on 2017-07-10 at 19:27.","This Dissertation was approved for publication on 2017-07-11 at 11:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11384 on 2017-09-29 at 11:29:38","Made available in DSpace on 2017-09-29T17:56:41Z (GMT). No. of bitstreams: 2 LIU-DISSERTATION-2017.pdf: 852588 bytes, checksum: bd0bd7cfeae3f1aa29eeff5f4275d32e (MD5) LICENSE.txt: 4206 bytes, checksum: 49315b75a24cb3adbb75a64a89af0682 (MD5) Previous issue date: 2017-07-11"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Asymptotically optimal shapes for counting lattice points and eigenvalues"]}]}],"canonical_facts":{"dc:contributor":["Laugesen, Richard S.","DeVille, Lee","Kirr, Eduard","Li, Xiaochun"],"dc:creator":["Liu, Shiya"],"dc:date":["2017-09-29T17:56:41Z","2017-07-11","2017-08"],"dc:description":["\"In Part I, we aim to maximize the number of first-quadrant lattice points under a concave (or convex) curve with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the optimal stretch factor approaches 1 as the \"\"radius\"\" approaches infinity. In particular, the result implies when 1 < p < ∞ that among all p-ellipses (or Lamé curves), the p-circle x^p+y^p=r^p is asymptotically optimal for enclosing the most first-quadrant lattice points as the radius approaches infinity. The case p = 2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0 < p < 1 by building on our results here. The case p = 1 remains open: which right triangles in the first quadrant (with two sides along the axes) will enclose the most lattice points for given area, and what are the limiting shapes of those triangles as the area tends to infinity? In Part II, we translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. We seek to identify the limiting shape of the curve that encloses the greatest number of shifted lattice points in the same family of reciprocal stretching curves as in Part I. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead it degenerates. Our results handle the p-circle when p > 1 (concave) and also when 0 < p < 1 (convex). The straight line case (p = 1) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials.\"","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Shiya Liu, accepted the attached license on 2017-07-10 at 19:16.","The student, Shiya Liu, submitted this Dissertation for approval on 2017-07-10 at 19:27.","This Dissertation was approved for publication on 2017-07-11 at 11:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #11384 on 2017-09-29 at 11:29:38","Made available in DSpace on 2017-09-29T17:56:41Z (GMT). No. of bitstreams: 2 LIU-DISSERTATION-2017.pdf: 852588 bytes, checksum: bd0bd7cfeae3f1aa29eeff5f4275d32e (MD5) LICENSE.txt: 4206 bytes, checksum: 49315b75a24cb3adbb75a64a89af0682 (MD5) Previous issue date: 2017-07-11"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/98364"],"dc:language":["en"],"dc:rights":["Copyright 2017 Shiya Liu"],"dc:subject":["Lattice points","Planar convex domain","P-ellipse","Spectral optimization","Laplacian","Dirichlet eigenvalues","Neumann eigenvalues","Translated lattice","Schrödinger eigenvalues","Harmonic oscillator"],"dc:title":["Asymptotically optimal shapes for counting lattice points and eigenvalues"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:35Z"}