Schedule-at-a-Glance

51 PRIMA 2013 AbstractsJongyook ParkUniversity of Science and Technology of China, Chinajongyook@hanmail.netIn 2006, Bannai and Bannai classified primitive symmetricassociation schemes with m 1 = 3. In their paper, thehardest case was k 1 = 3. Even though, (primitive) symmetricassociation schemes with k 1 = 3 were classified byYamazaki in 1998, they avoided the use of the difficultand deep result of Yamazaki. In this talk, we generalizethe result of Bannai and Bannai.Triangulated categories from categories ofspecail exact squencesKeyan Song, Yuehui ZhangShanghai Jiao Tong University, Chinazyh@sjtu.edu.cnLet A be an artinian algebra over an algebraically closedfield k. Let F be the category consisting of all four-termexact sequences in modA. Let C be the full subcategory ofF consisting of all exact sequences with two middle termsprojective modules in modA. It is proved that C is contravariantlyfinite. As applications, it is proved that C is aFrobenius category and has Auslander-Reiten sequencesprovided A is an selfinjective algebra. Furthermore, it isproved that the stable category C equals to C/H as factorcategories, where H is the set of all homotopic relations,and they are both equivalent to the stable category modA.An interesting result is that these three categories C, C/Hand modA are equivalent as triangulated categories.Contributed Talks Group 3Discrete MathematicsNumerical verification method for well conditionedspherical t-designsCongpei AnJinan University, Chinaandbach@163.comA set X N of N points on the unit sphere is a sphericalt-design if the average value of any polynomial of degreeat most t over X N is equal to the average value of thepolynomial over the sphere. In this talk, we focus on thecomputational construction of spherical t-designs on theunit sphere S 2 ⊂ R 3 when N ≥ (t + 1) 2 , the dimension ofthe space P t of spherical polynomials of degree at mostt. We show how to construct well conditioned sphericaldesigns with N ≥ (t+1) 2 points by maximizing the determinantof a matrix while satisfying a system of nonlinearconstraints for 1 ≤ t ≤ 100. Interval methods are thenused to prove the existence of a true spherical t-designvery close to the calculated points and to provide a guaranteedinterval containing the determinant. The resultingspherical designs have good geometrical properties (separationand mesh norm). We discuss some open problemson the distribution of point set over the sphere.This is a joint work with X.Chen, I. H. Sloan and R. S.Womersley.Edge colouring graphs with bounded colourclassesDavid CariolaroXi’an Jiaotong-Liverpool University, Chinadavid.cariolaro@xjtlu.edu.cnAn edge colouring of a graph G is a function which assignsa colour to each edge of G in such a way that adjacentedges receive distinct colours. The minimum number ofcolours used by an edge colouring of G is a well knownand studied graph parameter called chromatic index anddenoted by χ ′ (G). It was proved by Holyer in 1981 thatcomputing χ ′ (G) is in general NP-hard. We consider avariation of the edge colouring problem, which may beof interest for applications. Specifically, we consider edgecolourings of G such that no colour is assigned to morethan B edges, where B is a positive integer fixed in advance(clearly such edge colourings always exist). We callany such colouring a B-bounded edge colouring of G. Theminimum number of colours in a B-bounded edge colouringof G is a graph parameter which we denote by χ ′ B (G)and call B-bounded chromatic index. A B-bounded edgecolouring which uses exactly χ ′ B (G) colours is called anoptimal B-bounded edge colouring of G.It is not difficult to see that, for every graph G,χ ′ B (G) = max{χ′ (G), ⌈ |E(G)|B ⌉.This prompts the question of whether the parameter χ ′ Band an optimal B-bounded edge colouring can be computedin polynomial time for a fixed value of B. Ourgoal in this talk will be to answer this question in theaffirmative. This is joint work with Romeo Rizzi [1].[1] R. Rizzi and D. Cariolaro, Polynomial time complexityof Edge Colouring Graphs with Bounded Colour Classes,Algorithmica (2013), in press, doi: 10.1007/s00453-013-9746-7.Realizing joint degree matricesAaron DutleUniversity of South Carolina, USAdutle@math.sc.eduThe Joint Degree Matrix of a graph is a matrix whose(i, j) entry is the number of edges connecting vertices ofdegree i to vertices of degree j. A given matrix is calledrealizable if it is the joint degree matrix of a simple graph.As in the case of degree sequences, there is a simple set ofconditions for when a matrix is realizable, as well as anoperation that can be used to transform any realizationinto any other realization. We discuss these problems,as well as the extremal question of when a joint degreematrix is uniquely realizable.Generalization of Legendre polynomials theoryLemin GuTongji University, Chinagulemin@tongji.edu.cnIn all of n-order polynomials that highest coefficientis 1čňLegendre polynomials are in the interval [−1, 1]with the least square error of zero polynomials, that forP∫n(x) = x n + a(n − 1)x n−1 + ... + a1x + a0, making[P n(x)] 2 dx = min. Promotes the Legendre multinomialbasicprinciple: In all of n-order polynomials that highestcoefficient is not 1, namely an ≠ 1,there must be a newpolynomials Gn(x) = x n + a(n − 1)x n−1 + ... + a1x + a0,making ∫ [Gn(x)] 2 dx = min. Through discussion of thesquare error minimization, leads to the new polynomialsestablishment necessity. For convenient on elaborationin this abstract,called it "Gn(x) polynomial" (Generalizationof Legendre polynomial). First the existenceof Gn(x) polynomials issues are studied: By establishingGn(x) equation the ordinary solution expressions in the