Rank and Conjugation for a Second Frobenius Representation of an Overpartition

The notions of rank and conjugation are developed in the context of a second Frobenius representation of an overpartition. Some identities for overpartitions that are invariant under this conjugation are presented. Jeremy Lovejoy: The author was partially supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”.     

Dyson’s New Symmetry and Generalized Rogers-Ramanujan Identities

We present a generalization, which we call (k, m)-rank, of Dyson’s notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Rogers-Ramanujan identities. We also describe the relationship between (k, m)-rank and Garvan’s k-rank.     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

Partitions,Durfee symbols,and the Atkin–Garvan moments of ranks

Atkin and Garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks. Here we consider a combinatorial interpretation of these moments. This requires the introduction of a new representation for partitions, the Durfee symbol, and subsequent refinements. This in turn leads us to a variety of new congruences for our ‘marked’ Durfee symbols much in the spirit of Dyson’s original conjectures on the ranks of partitions. Mathematics Subject Classification (2000) 05A17, 05A19, 11P83     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Analytic representations in the three-dimensional Frobenius problem

We consider the Diophantine problem of Frobenius for the semigroup , where d ³ denotes the triple (d ₁,d ₂,d ₃), gcd (d ₁,d ₂,d ₃)=1. Based on the Hadamard product of analytic functions, we find the analytic representation of the diagonal elements a _kk (d ³) of Johnson’s matrix of minimal relations in terms of d ₁, d ₂, and d ₃. With our recent results, this gives the analytic representation of the Frobenius number F(d ³), genus G(d ³), and Hilbert series H(d ³;z) for the semigroups . This representation complements Curtis’s theorem on the nonalgebraic representation of the Frobenius number F(d ³). We also give a procedure for calculating the diagonal and off-diagonal elements of Johnson’s matrix.      

Regular weights and the Johnson scheme

This work addresses existence of weighted coherent configurations (cc’s) in the special case of the Johnson schemes,J(n, k). Many examples of weighted cc’s occur in the context of a permutation representation of a group which affords the cc. We determine the ranks of the weighted Johnson schemes which arise in this “group case” in connection with the alternating groupA _n . The association schemeJ(n, 2) has rank 3, with the triangular graphT(n) as one of its graphs. Conditions on feasible parameters for regular, full rank weights onT(n) are established. We describe two infinite families of such feasible parameters. Research supported by the Department of Education and the Rackham Graduate School of the University of Michigan as part of Ph.D. thesis written under the direction of D. G. Higman.     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

Frobenius criteria of freeness and Gorensteinness

Let F ⁿ (−) be the Frobenius functor of Peskine and Szpiro. In this note, we show that the maximal Cohen–Macaulayness of F ⁿ (M) forces M to be free, provided M has a rank. We apply this result to obtain several Frobenius-related criteria for the Gorensteinness of a local ring R, one of which improves a previous characterization due to Hanes and Huneke. We also establish a special class of finite length modules over Cohen–Macaulay rings, which are rigid against Frobenius.     

Limit correlation functions at zero for fixed trace random matrix ensembles

The limit as N → ∞ of the eigenvalue correlation function is studied in a neighborhood of zero for N × N Hermitian matrices chosen at random from the Hilbert-Schmidt sphere of an appropriate radius. Dyson’s famous sin π(t₁ − t₂)/π(t₁ − t₂)-kernel asymptotic is extended to this class of matrix ensembles. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 68–80.     

Exponential Renormalization

Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization. Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counter-factors and of order n bare coupling constants).     

Multi-fuzzy sets: An extension of fuzzy sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.     

A generalization of Kendall’s tau and the asymptotic efficiency of the corresponding independence test

We consider a generalization of Kendall’s rank correlation coefficient proposed by Kochar and Gupta in 1987. This generalization is a nondegenerate U-statistic with a special kernel. We calculate Bahadur’s local efficiency for the independence test based on this statistic. It is shown that this test is locally efficient for alternatives described by the Woodworth dependence function. Bibligraphy: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 227–237. Translated by S. Yu. Pilyugin.     

Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.     

Averaging sequences and abelian rank in amenable groups

We investigate the connection between the abelian rank of a countable amenable group and the existence of good averaging sequences (e.g., for the ergodic theorem). We show that if G is a group with finite abelian rank r(G), then 2^r(G) is a lower bound on the constant associated to a Tempel’man sequence, and if G is abelain there is a Tempel’man sequence in G with this constant. On the other hand, infinite rank precludes the existence of Tempel’man sequences and forces all tempered sequences to grow super-exponentially.     

Some properties of Frobenius numbers and a fraction of symmetric semigroups in the weak limit for <Emphasis Type="Italic">n</Emphasis>=3

We generalize and prove a conjecture by V.I. Arnold on the parity of Frobenius numbers. For the case of symmetric semigroups with three generators we give an exact formula for Frobenius numbers, which is, in a sense, a sum of two Sylvester’s formulae. We prove that a fraction of symmetric semigroups vanishes in the weak limit.     

Inequalities between rank moments of overpartitions

F.G. Garvan proved an inequality between crank moments and rank moments of partitions which utilizes an inequality for symmetrized rank and crank moments. In this paper, we study two symmetrized rank moments of overpartitions and prove an inequality between two rank moments of overpartitions.     

Irreducible subrepresentations of the conjugation representation of finitep-groups

For finitep-groupsG we study the conjugation representation γG which is defined by lettingG act on itself by conjugation. Roth conjectured that every irreducible representation ofG which is trivial on the center ofG, occurs in γG. However, this is not true in general. We construct minimal counterexamples and verify Roth's conjecture for various classes of finitep-groups, for instance those of maximal class. Dedicated to Professor George Maltese on the occasion of his 60th birthday