Schedule-at-a-Glance

52 PRIMA 2013 Abstractsinterval |x| < 1 are given, have proven that when n is positiveinteger and the interval is in |x| ≤ 1, two linear independencespecial solutions of them are Gn(x) polynomials,and provide the general term expression; Second, thegeometric meaning of Gn(x) polynomials are discussed,have proven that Gn(x)polynomials are the polynomialsthat in [−1, 1] with the error squares value are the least;Third, the main properties of Gn(x) polynomials are introducedčňwhichincluding Gn(x)polynomials recurrenceformula, the orthogonality, the odevity, the squares valuearea, the former n=9 polynomials, the corresponding geometry,and so on. Last, some special applications ofGn(x) polynomials under different boundary conditionsare given: Combined with the best Chebyshev approximationtheory they can constitute maximum error minimizingleast square approximation; Combined with theLeast Absolute Deviation approximation theory can constitutezero error type least square approximation; Underthe conditions of an=1 form the Legendre polynomial,andso on.The subsets counting problems and their applicationsJiyou LiShanghai Jiao Tong University, Chinalijiyou@sjtu.edu.cnLet G be an abelian group and D be a finite subset of G.∑Denote N(b) to be the number of subsets S in D such thatx∈S x = b and N k(b) to be the number of k-subsets Sin D such that ∑ x∈S x = b. When G is the set of integers,the decision version of the subset sum problem, i.e.,determining if N(b) > 0, is a well-known NP-completeproblem. And its counting version, the explicit enumerationof N(b) or N k (b), is a #P-complete problem. Inthis talk, we will investigate these problems from a mathematicalpoint of view, and introduce their relations toadditive combinatorics, coding theory and computer sciences.Paths and cycles of interval graphsPeng LiShanghai Jiao Tong University, Chinalidetiansjtu@sjtu.edu.cnWe recently find a linear time algorithm for solving the 1-Fixed-Endpoint Path Cover Problem on interval graphs.The design and the analysis of this algorithm motivateus to develop many properties on the paths and cycles ofinterval graphs and graphs. In this short presentation, wereport a series of equivalent characterizations of Hamiltonianconnectedness and other path/cycle properties ofinterval graphs.This is joint work with Yaokun Wu.The spectrum of 3-way k-homogeneous LatintradesTrent G. MarbachThe University of Queensland, Australiatrent.marbach@uqconnect.edu.auThe theory of Latin trades has a deep and interestingstructure, with connections to permutation groups, geometryand topology. The study of µ-way k-homogeneousLatin trades has received attention recently. The effortsof five publications completed the spectrum of the ordersof 2-way k-homogeneous Latin trades. This was followedby a recent paper (Bagheri Gh., et al. 2012), whichfocused particularly on the 3-way k-homogeneous Latintrades with k ≤ 15. In this talk, I will give a brief reporton the existence of µ-way k-homogeneous Latin trades,including three new constructions that complete the 3-way k-homogeneous Latin trade spectrum for all but ahandful of values. We will also provide the directions forfurther research motivated by this work.On the fractional metric dimension of treegraphsSuhadi Wido SaputroInstitut Teknologi Bandung, Indonesiasuhadi@math.itb.ac.idLet G be a finite, simple, and connected graph. The distancebetween any two vertices u, v ∈ V (G), denoted byd G (u, v), is the length of a shortest (u − v) path in G.A vertex x ∈ V (G) is called resolve a pair {u, v} of verticesof G if d G (u, x) ≠ d G (v, x). For u, v ∈ V (G), theset of all vertices which resolve the pair {u, v} is denotedby R G {u, v}. A real value function g : V (G) → [0, 1]is called a resolving function of G if g(R G {u, v}) ≥ 1for any two distinct vertices u, v ∈ V (G). The fractionalmetric dimension, denoted by dim f (G), is givenby dim f (G) = min{|g| : g is a resolving function of G},where |g| = g(V (G)). In this paper, we determine thefractional metric dimension for tree graph T n of ordern ≥ 3.Unimodality of genus distribution of laddersLiangxia WanBeijing Jiaotong University, Chinalxwan@bjtu.edu.cnGiven a graph,its genus distribution is unimodal? Asthough many results about genus distribution of graphshave obtained, there are only several those about unimodalityof genus distribution of graphs. In this talk,unimodality of genus distri- bution of some graphs areverified.The lit-only σ-gameZiqing XiangShanghai Jiao Tong University, ChinaZiqingXiang@gmail.comLet G be a finite graph with vertex set V which may notnecessarily be loopless. For each v ∈ V , let α v ∈ F V 2 bethe map which takes value 1 on w ∈ V if and only if thereis an edge between v and w in G; let T v be the linear mapon F V 2 that sends x ∈ FV 2 to itself if x(v) = 0 and sendsx ∈ F V 2 to x + αv if x(v) = 1. Note that Tv is a transvectionwhen v is not a loop in G while T v is an idempotentwhen v is a loop in G. We consider the digraph Γ withvertex set F V 2 and arc set {(x, Tv(x) : x ∈ FV 2 , v ∈ V },which is the phase space of the lit-only σ-game on G.We determine the reachability relation for the digraph Γ.A surprising corollary of this work is that, for α, β ∈ F V 2 ,basically, α can reach β in Γ if and only if α − β lies inthe binary linear subspace spanned by {α v : v ∈ V }.An important step of our work is to define the line graphof a multigraph and to provide a forbidden subgraph characterization.If the graph G is loopless, as an applicationof our knowledge of the corresponding digraph Γ, we areable to determine the multiplicative group generated by{T v : v ∈ V }. We also indicate possible approaches onextending the work here to the general case of G being adigraph.This is joint work with Yaokun Wu.