{"id":{"repo_id":"windsor","oai_identifier":"oai:uwindsor.scholaris.ca:20.500.14776/2582"},"canonical_url":"https://search.dev.ndltd.org/etd/windsor/oai:uwindsor.scholaris.ca:20.500.14776/2582","repository":{"repo_id":"windsor","name":"University of Windsor","base_url":"https://uwindsor.scholaris.ca/server/oai/request"},"display":{"title":"OPTIMALITY OF CHEMICAL BALANCE WEIGHING DESIGNS.","abstract":"Net N and n be positive integers with N (GREATERTHEQ) n and let D(N,n) denote the set of all N x n matrices X = (x(,ij)) with x(,ij) = -1,0 or 1. Let D'(N,n) be the set of all matrices in D(N,n) with entries -1,1. Each such matrix in D(N,n) or D'(N,n) will be called a weighing design matrix. If X(,0) minimizes (PHI)(X('T)X) over D(N,n) for some real valued function (PHI), then X(,0) is said to be (PHI)-optimum over D(N,n). The characterization of such X(,0) arises from the statistical problems of weighing designs, certain block designs and 2('n) fractional factorial designs. The well-known D-, A- and E-optimality criteria are obtained by taking (PHI)(X('T)X) = det(X('T)X)('-1), tr(X('T)X)('-1) and the maximum of the eigenvalue of (X('T)X)('-1) respectively. All these criteria are functions of the spectrum of X('T)X. Let (lamda)(,1),(lamda)(,2),...,(lamda)(,n) be the eigenvalues of X('T)X. Then a more general family of criteria are the following (PHI)(,p)-criteria: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The A-criterion is the (PHI)(,1)-criterion, the E-criterion is the limit of the (PHI)(,p)-criterion as p (--->) (INFIN) and the D-criterion is the limit of the (PHI)(,p)-criterion as p (--->) 0('+). In this thesis techniques are developed for proving the optimality of designs with respect to the A- and (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. In particular, A-optimal designs in D(N,n) are classified for n = 6 and N arbitrary. In addition, for the cases N (TBOND) 2(mod 4) and N (TBOND) 3(mod 4), certain designs are shown to be (PHI)(,p)-optimal in D(N,n) if N is sufficiently larger than n. In the latter case, it is also shown that A-optimality of certain designs in D(N,n) implies optimality with respect to the (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. Analogous results are shown to hold for designs in D'(N,n). Further, in the case N (TBOND) 2(mod 4), designs in D'(N,n) are shown to be optimal with respect to a large class of criteria which includes the (PHI)(,p)-criteria, 0 (LESSTHEQ) p (LESSTHEQ) (INFIN), for all n and N.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1983 .M273. Source: Dissertation Abstracts International, Volume: 44-09, Section: B, page: 2778. Thesis (Ph.D.)--University of Windsor (Canada), 1983.","abstract_html":"Net N and n be positive integers with N (GREATERTHEQ) n and let D(N,n) denote the set of all N x n matrices X = (x(,ij)) with x(,ij) = -1,0 or 1. Let D'(N,n) be the set of all matrices in D(N,n) with entries -1,1. Each such matrix in D(N,n) or D'(N,n) will be called a weighing design matrix. If X(,0) minimizes (PHI)(X('T)X) over D(N,n) for some real valued function (PHI), then X(,0) is said to be (PHI)-optimum over D(N,n). The characterization of such X(,0) arises from the statistical problems of weighing designs, certain block designs and 2('n) fractional factorial designs. The well-known D-, A- and E-optimality criteria are obtained by taking (PHI)(X('T)X) = det(X('T)X)('-1), tr(X('T)X)('-1) and the maximum of the eigenvalue of (X('T)X)('-1) respectively. All these criteria are functions of the spectrum of X('T)X. Let (lamda)(,1),(lamda)(,2),...,(lamda)(,n) be the eigenvalues of X('T)X. Then a more general family of criteria are the following (PHI)(,p)-criteria: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The A-criterion is the (PHI)(,1)-criterion, the E-criterion is the limit of the (PHI)(,p)-criterion as p (--->) (INFIN) and the D-criterion is the limit of the (PHI)(,p)-criterion as p (--->) 0('+). In this thesis techniques are developed for proving the optimality of designs with respect to the A- and (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. In particular, A-optimal designs in D(N,n) are classified for n = 6 and N arbitrary. In addition, for the cases N (TBOND) 2(mod 4) and N (TBOND) 3(mod 4), certain designs are shown to be (PHI)(,p)-optimal in D(N,n) if N is sufficiently larger than n. In the latter case, it is also shown that A-optimality of certain designs in D(N,n) implies optimality with respect to the (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. Analogous results are shown to hold for designs in D'(N,n). Further, in the case N (TBOND) 2(mod 4), designs in D'(N,n) are shown to be optimal with respect to a large class of criteria which includes the (PHI)(,p)-criteria, 0 (LESSTHEQ) p (LESSTHEQ) (INFIN), for all n and N.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1983 .M273. Source: Dissertation Abstracts International, Volume: 44-09, Section: B, page: 2778. Thesis (Ph.D.)--University of Windsor (Canada), 1983.","abstract_has_math":false,"creators":["MASARO, JOSEPH COSTANTINO."],"institution":"University of Windsor","degree_name":"Ph.D.","degree_level":"Doctoral","degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1983,"date_issued":"1983-01-01","date_published":"1983-01-01","updated_at":"2026-07-27T22:04:36Z","subjects":[],"languages":["en_CA"],"rights":[],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14776/2582","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["MASARO, JOSEPH COSTANTINO."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-06-20 10:39"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2013-03-21 18:25","2025-06-20T14:39:24Z"]},{"key":"dc:date.issued","label":"Date","values":["1983-01-01"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Windsor"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_CA"]},{"key":"dc:rights","label":"Dc Rights","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14776/2582"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Net N and n be positive integers with N (GREATERTHEQ) n and let D(N,n) denote the set of all N x n matrices X = (x(,ij)) with x(,ij) = -1,0 or 1. Let D'(N,n) be the set of all matrices in D(N,n) with entries -1,1. Each such matrix in D(N,n) or D'(N,n) will be called a weighing design matrix. If X(,0) minimizes (PHI)(X('T)X) over D(N,n) for some real valued function (PHI), then X(,0) is said to be (PHI)-optimum over D(N,n). The characterization of such X(,0) arises from the statistical problems of weighing designs, certain block designs and 2('n) fractional factorial designs. The well-known D-, A- and E-optimality criteria are obtained by taking (PHI)(X('T)X) = det(X('T)X)('-1), tr(X('T)X)('-1) and the maximum of the eigenvalue of (X('T)X)('-1) respectively. All these criteria are functions of the spectrum of X('T)X. Let (lamda)(,1),(lamda)(,2),...,(lamda)(,n) be the eigenvalues of X('T)X. Then a more general family of criteria are the following (PHI)(,p)-criteria: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The A-criterion is the (PHI)(,1)-criterion, the E-criterion is the limit of the (PHI)(,p)-criterion as p (--->) (INFIN) and the D-criterion is the limit of the (PHI)(,p)-criterion as p (--->) 0('+). In this thesis techniques are developed for proving the optimality of designs with respect to the A- and (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. In particular, A-optimal designs in D(N,n) are classified for n = 6 and N arbitrary. In addition, for the cases N (TBOND) 2(mod 4) and N (TBOND) 3(mod 4), certain designs are shown to be (PHI)(,p)-optimal in D(N,n) if N is sufficiently larger than n. In the latter case, it is also shown that A-optimality of certain designs in D(N,n) implies optimality with respect to the (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. Analogous results are shown to hold for designs in D'(N,n). Further, in the case N (TBOND) 2(mod 4), designs in D'(N,n) are shown to be optimal with respect to a large class of criteria which includes the (PHI)(,p)-criteria, 0 (LESSTHEQ) p (LESSTHEQ) (INFIN), for all n and N.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1983 .M273. Source: Dissertation Abstracts International, Volume: 44-09, Section: B, page: 2778. Thesis (Ph.D.)--University of Windsor (Canada), 1983."]},{"key":"dc:title","label":"Title","values":["OPTIMALITY OF CHEMICAL BALANCE WEIGHING DESIGNS."]}]}],"canonical_facts":{"dc:creator":["MASARO, JOSEPH COSTANTINO."],"dc:date.accessioned":["2025-06-20 10:39"],"dc:date.available":["2013-03-21 18:25","2025-06-20T14:39:24Z"],"dc:date.issued":["1983-01-01"],"dc:description.abstract":["Net N and n be positive integers with N (GREATERTHEQ) n and let D(N,n) denote the set of all N x n matrices X = (x(,ij)) with x(,ij) = -1,0 or 1. Let D'(N,n) be the set of all matrices in D(N,n) with entries -1,1. Each such matrix in D(N,n) or D'(N,n) will be called a weighing design matrix. If X(,0) minimizes (PHI)(X('T)X) over D(N,n) for some real valued function (PHI), then X(,0) is said to be (PHI)-optimum over D(N,n). The characterization of such X(,0) arises from the statistical problems of weighing designs, certain block designs and 2('n) fractional factorial designs. The well-known D-, A- and E-optimality criteria are obtained by taking (PHI)(X('T)X) = det(X('T)X)('-1), tr(X('T)X)('-1) and the maximum of the eigenvalue of (X('T)X)('-1) respectively. All these criteria are functions of the spectrum of X('T)X. Let (lamda)(,1),(lamda)(,2),...,(lamda)(,n) be the eigenvalues of X('T)X. Then a more general family of criteria are the following (PHI)(,p)-criteria: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The A-criterion is the (PHI)(,1)-criterion, the E-criterion is the limit of the (PHI)(,p)-criterion as p (--->) (INFIN) and the D-criterion is the limit of the (PHI)(,p)-criterion as p (--->) 0('+). In this thesis techniques are developed for proving the optimality of designs with respect to the A- and (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. In particular, A-optimal designs in D(N,n) are classified for n = 6 and N arbitrary. In addition, for the cases N (TBOND) 2(mod 4) and N (TBOND) 3(mod 4), certain designs are shown to be (PHI)(,p)-optimal in D(N,n) if N is sufficiently larger than n. In the latter case, it is also shown that A-optimality of certain designs in D(N,n) implies optimality with respect to the (PHI)(,p)-criteria 0 (LESSTHEQ) p (LESSTHEQ) 1. Analogous results are shown to hold for designs in D'(N,n). Further, in the case N (TBOND) 2(mod 4), designs in D'(N,n) are shown to be optimal with respect to a large class of criteria which includes the (PHI)(,p)-criteria, 0 (LESSTHEQ) p (LESSTHEQ) (INFIN), for all n and N.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1983 .M273. Source: Dissertation Abstracts International, Volume: 44-09, Section: B, page: 2778. Thesis (Ph.D.)--University of Windsor (Canada), 1983."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14776/2582"],"dc:language.iso":["en_CA"],"dc:rights":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:title":["OPTIMALITY OF CHEMICAL BALANCE WEIGHING DESIGNS."],"dc:type":["info:eu-repo/semantics/doctoralThesis"],"thesis:degree_discipline":["Mathematics and Statistics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Windsor"]},"updated_at":"2026-07-27T22:04:36Z"}