284 research outputs found

    Harmonic analysis on a finite homogeneous space

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    In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the introduction of three types of spherical functions, we develop a theory of Gelfand Tsetlin bases for permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme; we also consider statistical and probabilistic applications. After that, we consider the composition of two permutation representations, giving a non commutative generalization of the Gelfand pair associated to the ultrametric space; actually, we study the more general notion of crested product. Finally, we consider the exponentiation action, generalizing the decomposition of the Gelfand pair of the Hamming scheme; actually, we study a more general construction that we call wreath product of permutation representations, suggested by the study of finite lamplighter random walks. We give several examples of concrete decompositions of permutation representations and several explicit 'rules' of decomposition.Comment: 69 page

    Commutative association schemes

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    Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

    Partial geometric designs and difference families

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    We examine the designs produced by different types of difference families. Difference families have long been known to produce designs with well behaved automorphism groups. These designs provide the elegant solutions desired for applications. In this work, we explore the following question: Does every (named) design have a difference family analogue? We answer this question in the affirmative for partial geometric designs

    European Journal of Combinatorics Index, Volume 27

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    BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p

    On symmetric association schemes and associated quotient-polynomial graphs

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    Let denote an undirected, connected, regular graph with vertex set , adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of generated by . We refer to as the adjacency algebra of . In this paper we investigate algebraic and combinatorial structure of for which the adjacency algebra is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) has a standard basis ; (ii) for every vertex there exists identical distance-faithful intersection diagram of with cells; (iii) the graph is quotient-polynomial; and (iv) if we pick then has distinct eigenvalues if and only if . We describe the combinatorial structure of quotient-polynomial graphs with diameter and distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether is distance-regular or not and, more generally, which distance- matrices are polynomial in , giving also these polynomials.This research has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The second author acknowledges the financial support from the Slovenian Research Agency (research program P1-0285 and research project J1-1695).Peer ReviewedPostprint (published version

    On symmetric association schemes and associated quotient-polynomial graphs

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    Let Γ\Gamma denote an undirected, connected, regular graph with vertex set XX, adjacency matrix AA, and d+1{d+1} distinct eigenvalues. Let A=A(Γ){\mathcal A}={\mathcal A}(\Gamma) denote the subalgebra of MatX(C)_X({\mathbb C}) generated by AA. We refer to A{\mathcal A} as the {\it adjacency algebra} of Γ\Gamma. In this paper we investigate algebraic and combinatorial structure of Γ\Gamma for which the adjacency algebra A{\mathcal A} is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) A{\mathcal A} has a standard basis {I,F1,…,Fd}\{I,F_1,\ldots,F_d\}; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ\Gamma with d+1d+1 cells; (iii) the graph Γ\Gamma is quotient-polynomial; and (iv) if we pick F∈{I,F1,…,Fd}F\in \{I,F_1,\ldots,F_d\} then FF has d+1d+1 distinct eigenvalues if and only if span{I,F1,…,Fd}=\{I,F_1,\ldots,F_d\}=span{I,F,…,Fd}\{I,F,\ldots,F^d\}. We describe the combinatorial structure of quotient-polynomial graphs with diameter 22 and 44 distinct eigenvalues. As a consequence of the technique from the paper we give an algorithm which computes the number of distinct eigenvalues of any Hermitian matrix using only elementary operations. When such a matrix is the adjacency matrix of a graph Γ\Gamma, a simple variation of the algorithm allow us to decide wheter Γ\Gamma is distance-regular or not. In this context, we also propose an algorithm to find which distance-ii matrices are polynomial in AA, giving also these polynomials.Comment: 22 pages plus 4 pages of reference
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