The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these \"even index-sequences\" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the ordered even d-simplex numbers for any finite dimension.
","abstract_html":"<p>The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the ordered even d-simplex numbers for any finite dimension.</p>","abstract_has_math":false,"creators":["Hannula, Hunter DM"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis","degree_discipline":"Math and Computer Science","degree_department":null,"school":null,"contributors":["Daniel Rowe"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2026,"date_issued":"2026-05-01T07:00:00Z","date_published":"2026-05-01T07:00:00Z","updated_at":"2026-07-24T03:24:38Z","subjects":["simplex numbers","triangular numbers","parity","harmonic analysis","quasiperiodic","affine space","Fourier analysis","representation theory","Algebra","Discrete Mathematics and Combinatorics","Harmonic Analysis and Representation","Number Theory"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.nmu.edu/theses/910","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Daniel Rowe"]},{"key":"dc:creator","label":"Author","values":["Hannula, Hunter DM"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2026-04-03T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Math and Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["simplex numbers","triangular numbers","parity","harmonic analysis","quasiperiodic","affine space","Fourier analysis","representation theory","Algebra","Discrete Mathematics and Combinatorics","Harmonic Analysis and Representation","Number Theory"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.nmu.edu/theses/910"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these \"even index-sequences\" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the ordered even d-simplex numbers for any finite dimension.
"]},{"key":"dc:title","label":"Title","values":["Numerical and Harmonic Analysis of Simplex Number Parity"]}]}],"canonical_facts":{"dc:contributor":["Daniel Rowe"],"dc:creator":["Hannula, Hunter DM"],"dc:date.available":["2026-04-03T07:00:00Z"],"dc:description.abstract":["The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these \"even index-sequences\" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the ordered even d-simplex numbers for any finite dimension.
"],"dc:identifier":["https://commons.nmu.edu/theses/910"],"dc:subject":["simplex numbers","triangular numbers","parity","harmonic analysis","quasiperiodic","affine space","Fourier analysis","representation theory","Algebra","Discrete Mathematics and Combinatorics","Harmonic Analysis and Representation","Number Theory"],"dc:title":["Numerical and Harmonic Analysis of Simplex Number Parity"],"thesis:degree_discipline":["Math and Computer Science"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T03:24:38Z"}