171 research outputs found
The Terwilliger Algebra of an Almost-Bipartite P- and Q-Polynomial Association Scheme
Get PDFLet Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. We compute this multiplicity for those irreducible T-modules which have diameter at least D−3
Taut distance-regular graphs and the subconstituent algebra
No full textAbstractWe consider a bipartite distance-regular graph Γ with diameter D⩾4 and valency k⩾3. Let X denote the vertex set of Γ and fix x∈X. Let Γ22 denote the graph with vertex set X˘={y∈X|∂(x,y)=2}, and edge set R˘={yz|y,z∈X˘,∂(y,z)=2}, where ∂ is the path-length distance function for Γ. The graph Γ22 has exactly k2 vertices, where k2 is the second valency of Γ. Let η1,η2,…,ηk2 denote the eigenvalues of the adjacency matrix of Γ22; we call these the local eigenvalues of Γ. Let A denote the adjacency matrix of Γ. We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of Γ and the eigenvalues of A. Let T=T(x) denote the subalgebra of MatX(C) generated by A,E0*,E1*,…,ED*, where for 0⩽i⩽D, Ei* represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimEi*W⩽1 for 0⩽i⩽D. By the endpoint of W we mean min{i|Ei*W≠0}. We give a detailed description of the thin irreducible T-modules that have endpoint 2 and dimension D-3. MacLean [An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216] defined what it means for Γ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T-modules
OIMB Term Photo: Fall 1985
No full textFall 1985
Back Row: Laura Mann, Honore Brandis, Tod Perkins, Sue Ross, Jon Temte, Jo Williams, Jeff Goddard, â Stine Brown, Wendy Manley, Lynn Rudy, Richard Castenholz.
4th Row: Mark Devlin, Sylvia Pauly, Lisa Haggblom, Renee Salant.
3rd Row: Betty Rogers, David Robinson, Debbie Beutler, Steve Allison, Ginger Claflin, Holly Colon, Phil Nyren, Martin Posey, Glen Chen.
2nd Row: Don Rogers, Kathy Cottrell, Glyc' Pimentel, Jan Hodder, Marge Price, Nora Terwilliger, Sonia Javier, Bob Terwilliger.
Front Row: Rich Gorham, John Gross, Andy Bruckner, Jerry Rudy, Rega Lapar
The subconstituent algebra of a distance-regular graph; thin modules with endpoint one
No full textAbstractWe consider a distance-regular graph Γ with diameter D⩾3, intersection numbers ai,bi,ci and eigenvalues θ0>θ1>⋯>θD. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of MatX(C) generated by A,E*0,E*1,…,E*D, where A denotes the adjacency matrix of Γ and E*i denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE*iW⩽1 for 0⩽i⩽D. By the endpoint of W we mean min{i|E*iW≠0}. We describe the thin irreducible T-modules with endpoint 1. Let W denote a thin irreducible T-module with endpoint 1. Observe E*1W is a one-dimensional eigenspace for E*1AE*1; let η denote the corresponding eigenvalue. It is known θ̃1⩽η⩽θ̃D where θ̃1=−1−b1(1+θ1)−1 and θ̃D=−1−b1(1+θD)−1. For η=θ̃1 and η=θ̃D the structure of W was worked out by Go and the present author [Tight distance-regular graphs and the subconstituent algebra, preprint]. For θ̃1<η<θ̃D we obtain the following results. We show the dimension of W is D. Let v denote a nonzero vector in E*1W. We show W has a basis Eiv (1⩽i⩽D), where Ei denotes the primitive idempotent of A associated with θi. We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square norm of each basis vector. We show W has a basis E*i+1Aiv (0⩽i⩽D−1), where Ai denotes the ith distance matrix for Γ. We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W
The Generalized Terwilliger Algebra and its Finite-dimensional Modules when d=2
No full textAbstractTerwilliger [J. Algebraic Combin.1 (1992), 363–388] considered the C-algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M*. This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations are often called the triple product relations. They determine much of the structure of T, though not all of it in general. To illuminate the role these relations play, the current author introduced [J. Algebra233 (2000), 213–252] a generalization T of T. To go from T to T, we replace M and M* with a pair of dual character algebras C and C*. The dimensions of C and C* are equal; let d+1 denote this common dimension. Intuitively, T is the associative C-algebra with identity generated by C and C* subject to the analogues of Terwilliger's triple product relations. T is infinite-dimensional and noncommutative in general. In this paper we study T and its finite-dimensional modules when d=2 and T has no “extra” vanishing intersection numbers or dual intersection numbers. In this case we show T is C-algebra isomorphic to M3(C)⊕A, where M3(C) denotes the C-algebra consisting of all 3-by-3 matrices with entries in C and A denotes the associative C-algebra with identity generated by the symbols e and f subject to the relations e2=e and f2=f. We find a basis for A and we determine the center of A. We classify the finite-dimensional indecomposable A-modules up to isomorphism. There are four such A-modules in every odd dimension, and in every even dimension these modules are parameterized by a single complex number. We also classify the finite-dimensional irreducible A-modules up to isomorphism. Using our results concerning A, we find a basis for T, we describe the center of T, and we classify both the finite-dimensional indecomposable and the finite-dimensional irreducible T-modules up to isomorphism
OIMB Term Photo: Spring 1973
No full textSpring 1973
Back Row: Bob Terwilliger, Ron Faisch, Judy Glover, George Williamson, Jon Christenson, Terry Kay, Pete Sorenson, Greg Wolf, Greg Walther, Jud Guitteau, Randy Eisendorf, Judy Dixon, Betty Rogers, Shirley Leegard, Jean Hanna, Don Albiston, Becky Swanson, Mark Clemons, Marc Firestone, Char Skibinski, Scott English.
Middle Row: Betsy Neeley, Barb Liner, Virginia Bonnerlin, Deena Mault, Barbara Boczar, Sue Adams, Marcia Peterson, Pete Leslie, Laurie Dull, John Deckard, Henry Kunowski.
Front Row: Beth Waldow, Bill Loy, Pat Russell, Jerry Rudy, Chris Rains, Liz Gassell, Debbie Gentling, Mary Leitschuh, Debra Walker, Gerry Diethelm, Ann Brooke, Pat Piccione, Nora Terwilliger, Janine Carlson, Fred Tepfer
A study of the new agricultural education curriculum in the secondary schools of Kenya
No full textThe purposes of this study were to investigate how the new Kenyan agricultural education curriculum was being implemented, factors influencing the implementation process and how the implementation affected students' achievement in agricultural education.A sample of 53 secondary schools was randomly drawn from a total of 969 schools in two provinces. The sample represented rural and urban schools as well as government maintained, government aided and private schools. An achievement test was developed based on the first two years of the secondary school agriculture curriculum. The test was administered to 3,264 Form III students who were supposed to have covered the Forms I and II parts of the curriculum. One survey instrument was administered to the agriculture teacher and a second instrument was administered to the headmaster/headmistress of each participating school.Results indicated that school location and category had no significant influence on methods used in teaching agriculture or content coverage. More topics were either partially or not covered as instruction progressed. Lack of resources and time limitation were reported as the main reasons for partial or no content coverage. Teacher qualifications, school location and school category were observed to have a significant effect on students' achievement. Content coverage and extent of using objectives in instructional planning had no significant influence on students' achievement. An analysis of 13 selected variables related to school educational environment indicated significant influence of the variables on students' achievement but not on content coverage or extent of using objectives.In conclusion, school location, school category, teacher qualification and availability of books were significant factors related to students' achievement. Second, there was no relationship between instructional planning and curriculum implementation. Third, teachers used more theoretical than practical-oriented teaching methods. Consequently, curriculum implementation did not match the syllabus in both content coverage and development of practical skills in agriculture. Fourth, resources available for teaching agriculture varied in secondary schools and most schools lacked these resources.These conclusions call for increased resources for agriculture teacher education, especially in practical skills. The agriculture curriculum should be reviewed to ensure that it is consistent with available resources and time. These actions need to be accompanied by a strong program monitoring system.Made available in DSpace on 2011-05-07T13:59:02Z (GMT). No. of bitstreams: 2
license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5)
9021704.pdf: 6585282 bytes, checksum: 7784896e539b6dd1bade5eae13a23eab (MD5)
Previous issue date: 1990Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T15:01:35Z
Item is restricted indefinitely.Restriction data tranferred 2014-07-01T11:29:13-05:00
Original Data
Group with Access UIUC Users [automated]
Release Date: none
Reason: ETDs are only available to UIUC Users without author permissionETDs are only available to UIUC Users without author permissionU of I Onl
The influence of organizational level and ambiguity on evaluative criteria utilized during organizational consolidation
No full textThe study was focused on evaluative criteria utilized at four organizational levels within two consolidating organizations. Sources of ambiguity present in the decision making process were delineated, the influence of ambiguity on the evaluative criteria was analyzed, and current assessments of organizational participants were examined. The research method was a case study that utilized written documents and interviews with organizational participants. A content analysis of these documents was conducted to discern evaluative criteria and sources of ambiguity. The data were then analyzed to determine if ambiguity influenced the criteria adopted during the consolidation process.Findings indicated that evaluative criteria employed at all levels included need, organizational mission, merit, and resources. Sources of ambiguity that emerged during the process included differing value orientations, lack of time and attention, the nature of the problem, problematical information, and symbols and metaphors. Evaluative criteria and sources of ambiguity spanned both organizations and all four levels and varied in interpretation by level. At all four levels, ambiguity influenced evaluative criteria in the sense that, as ambiguity increased, criteria tended to become less specific and more general. The broadening of the criteria permitted the utilization of the criteria by more organizational participants.Conclusions included that during periods of increasing uncertainty, informal networks and core groups are significant in decision making processes, participants who are intolerant of ambiguity may become immobilized, and participants who tolerate ambiguity can influence decision making processes. Implications for research, organizations, and leaders were included. Recommendations included further research on the influence of ambiguity on evaluative criteria, the influence of time on organizational decision making, and the influence of key relationships among decision makers on decision making processes and outcomes.Made available in DSpace on 2011-05-07T13:55:16Z (GMT). No. of bitstreams: 2
license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5)
9136592.pdf: 12941370 bytes, checksum: 85027a98802f54421ef8dd6a887e3c00 (MD5)
Previous issue date: 1991Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T15:00:48Z
Item is restricted indefinitely.Restriction data tranferred 2014-07-01T11:28:47-05:00
Original Data
Group with Access UIUC Users [automated]
Release Date: none
Reason: ETDs are only available to UIUC Users without author permissionETDs are only available to UIUC Users without author permissionU of I Onl
An individual's correspondence preference as related to work-related complaints, neuroticism, and job satisfaction
No full textThe purposes of the study were (a) to establish some discriminate validity for the Internal-External Correspondence Scale by comparing participants' responses on a measure of extraversion and a measure of locus of control to participants' responses on the Internal-External Correspondence Scale, and (b) to see if an individual's relative correspondence preference was related to an individual's endorsement of external work-related complaints. Individual were asked to complete six paper and pencil measures at one time.Participants in this study were 59 individuals enrolled in graduate classes at a private teachers' college in South Carolina. Questionnaires were distributed to 140 individuals and 59 of these individuals returned questionnaires for a response rate of 42%. The participants were 90% female and 54% were teachers. Data analysis included Pearson correlations and descriptive statistics.Results suggested that correspondence preference is a personality construct which is independent of extraversion and locus of control and that correspondence preference may function as a moderator variable. Individuals with a preference for external correspondence had a characteristic pattern of significant correlations which was theoretically consistent as did individuals with a preference for internal correspondence. Significant differences between the correlation coefficients for the two groups were not demonstrated perhaps because of the small sample size.Discussion pointed out that the Internal-External Correspondence Scale needs further study. Behavioral correlates for several of the measures need to be established.Made available in DSpace on 2011-05-07T14:15:17Z (GMT). No. of bitstreams: 2
license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5)
9124419.pdf: 3678432 bytes, checksum: e20180ad83c09e395ef0535d2591e61c (MD5)
Previous issue date: 1991Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding ([email protected]) on 2011-05-07T15:04:41Z
Item is restricted indefinitely.Restriction data tranferred 2014-07-01T11:30:55-05:00
Original Data
Group with Access UIUC Users [automated]
Release Date: none
Reason: ETDs are only available to UIUC Users without author permissionETDs are only available to UIUC Users without author permissionU of I Onl
OIMB Term Photo: Spring 1995
No full textSpring 1995
Back Row: Alan Shanks, Kevin Johnson, Nora Terwilliger, Lynda Shapiro, Sharon Clark, Troy Mutchler, Alan Osborn, Mark Amend.
Middle Row: Jody Falkenstern, Julia Gervasi, Christina Ross, Steve Sadro, Margie Ryan, Peggy Hughes, Karen Dumler, John Jansen.
Front Row: Jeff Goddard, Kristin Bigler, Tara Hopman, Laura Verhegge, Barbara Butler, Jan Hodder, Richard Emlet
- …
