具有最小弧数且基指数为3的本原不可幂对称无环带号有向图

Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 1215-1227], authors conjectured that D is the underlying digraph of S with exp(D) = 3 if and only if D is isomorphic to ED n,3,3 , where ED n,3,3 = (V, A) is a digraph with V = {1, 2, . . . , n}, A = {(1, i), (i, 1) | 3≤i≤n} ∪ {(2i-1, 2i), (2i, 2i-1) | 2≤i≤ n/2 } ∪ {(2, 3), (3, 2), (2, 4), (4, 2)}). In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.     

Comparative analysis of automotive paints by laser induced breakdown spectroscopy and nonparametric permutation tests

Laser-induced breakdown spectroscopy (LIBS) has been investigated for the discrimination of automobile paint samples. Paint samples from automobiles of different makes, models, and years were collected and separated into sets based on the color, presence or absence of effect pigments and the number of paint layers. Twelve LIBS spectra were obtained for each paint sample, each an average of a five single shot “drill down” spectra from consecutive laser ablations in the same spot on the sample. Analyses by a nonparametric permutation test and a parametric Wald test were performed to determine the extent of discrimination within each set of paint samples. The discrimination power and Type I error were assessed for each data analysis method. Conversion of the spectral intensity to a log-scale (base 10) resulted in a higher overall discrimination power while observing the same significance level. Working on the log-scale, the nonparametric permutation tests gave an overall 89.83% discrimination power with a size of Type I error being 4.44% at the nominal significance level of 5%. White paint samples, as a group, were the most difficult to differentiate with the power being only 86.56% followed by 95.83% for black paint samples. Parametric analysis of the data set produced lower discrimination (85.17%) with 3.33% Type I errors, which is not recommended for both theoretical and practical considerations. The nonparametric testing method is applicable across many analytical comparisons, with the specific application described here being the pairwise comparison of automotive paint samples.     

Non‐parametric permutation test for the discrimination of float glass samples based on LIBS spectra

Laser‐induced breakdown spectroscopy (LIBS) coupled with non‐parametric permutation based hypothesis testing is demonstrated to have good performance in discriminating float glass samples. This type of pairwise sample comparison is important in manufacturing process quality control, forensic science and other applications where determination of a match probability between two samples is required. Analysis of the pairwise comparisons between multiple LIBS spectra from a single glass sample shows that some assumptions required by parametric methods may not hold in practice, motivating the adoption of a non‐parametric permutation test. Without rigid distributional assumptions, the permutation test exhibits excellent discriminating power while holding the actual size of Type I error at the nominal level. Copyright © 2010 John Wiley & Sons, Ltd.     

Representations of quantum permutation algebras

We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π:A_s(n)→B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n=6.     

On bijections for pattern-avoiding permutations

By considering bijections from the set of Dyck paths of length 2n onto each of Sn(321) and Sn(132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207-219] gave a bijection that preserves the number of fixed points and the number of excedances in each σ∈Sn(321). We show that a direct bijection Γ:Sn(321)→Sn(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ. We also show that a bijection ?^∗:Sn(213)→Sn(321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133-148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383-409] preserves these same statistics, and we show that an analogous bijection from Sn(132) onto Sn(213) does the same.     

Signed graph factors and degree sequences

For a signed graph G and function , a signed f‐factor of G is a spanning subgraph F such that sdeg_F(υ) = f(υ) for every vertex υ of G, where sdeg(υ) is the number of positive edges incident with v less the number of negative edges incident with υ, with loops counting twice in either case. For a given vertex‐function f, we provide necessary and sufficient conditions for a signed graph G to have a signed f‐factor. As a consequence of this result, an Erd?s‐Gallai‐type result is given for a sequence of integers to be the degree sequence of a signed r‐graph, the graph with at most r positive and r negative edges between a given pair of distinct vertices. We discuss how the theory can be altered when mixed edges (i.e., edges with one positive and one negative end) are allowed, and how it applies to bidirected graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 27–36, 2006     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Regular symmetric groups of boolean functions

We show that with the exception of four known cases: C₃, C₄, C₅, and , all regular permutation groups can be represented as symmetric groups of boolean functions. This solves the problem posed by A. Kisielewicz in the paper [A. Kisielewicz, Symmetry groups of boolean functions and constructions of permutation groups, J. Algebra 199 (1998) 379-403]. A slight extension of our proof yields the same result for semiregular groups.