Hopf Algebras of Type <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>, Twistings and the FRT-construction

We study pointed Hopf algebras of the form U(R _Q ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where R _Q is the Yang–Baxter operator associated with the multiparameter deformation of GL _n supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(R _Q ) is of type A _n in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(R _Q ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(R _Q ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(R _Q ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(R _Q ), U(R _Q′) can be obtained from each other by twisting the comultiplication if and only if In the last part we show that U _Q is always a quotient of a double crossproduct. I wish to thank UIC, where some of the work was done, for hospitality.     

Bispectrality of Multivariable Racah–Wilson Polynomials

We construct a commutative algebra A_x{\mathcal{A}}_{x} of difference operators in ℝ ^p , depending on p+3 parameters, which is diagonalized by the multivariable Racah polynomials R _p (n;x) considered by Tratnik (J. Math. Phys. 32(9):2337–2342, 1991). It is shown that for specific values of the variables x=(x ₁,x ₂,…,x _p ) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n{\mathcal{A}}_{n} in the variables n=(n ₁,n ₂,…,n _p ) which is also diagonalized by R _p (n;x). Thus, R _p (n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grünbaum (Commun. Math. Phys. 103(2):177–240, 1986). Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials (Tratnik in J. Math. Phys. 32(8):2065–2073, 1991), this change of variables and parameters in A_x{\mathcal{A}}_{x} and A_n{\mathcal{A}}_{n} leads to bispectral commutative algebras for the multivariable Wilson polynomials.     

Variants of Kazhdan’s property for subgroups of semisimple groups

Some variants of Kazhdan’s property (T) for discrete groups are presented. It is shown that some groups (e.g. SL _n (Q),n≧3) which do not have property (T) still have some of these weaker properties. Applications to cohomology and infinitesimal rigidity for certain actions on manifolds are derived. Research partially supported by a grant from the Israel-United States Binational Science Foundation. Research partially supported by NSF Grant.     

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.     

On Ramsey Numbers for Trees Versus Wheels of Five or Six Vertices

 For given two graphs G dan H, the Ramsey number R(G,H) is the smallest positive integer n such that every graph F of order n must contain G or the complement of F must contain H. In [12], the Ramsey numbers for the combination between a star S _n and a wheel W _m for m=4,5 were shown, namely, R(S _n ,W ₄)=2n−1 for odd n and n≥3, otherwise R(S _n ,W ₄)=2n+1, and R(S _n ,W ₅)=3n−2 for n≥3. In this paper, we shall study the Ramsey number R(G,W _m ) for G any tree T _n . We show that if T _n is not a star then the Ramsey number R(T _n ,W ₄)=2n−1 for n≥4 and R(T _n ,W ₅)=3n−2 for n≥3. We also list some open problems. Received: October, 2001 Final version received: July 11, 2002 RID="*" ID="*" This work was supported by the QUE Project, Department of Mathematics ITB Indonesia Acknowledgments. We would like to thank the referees for several helpful comments.     

Thompson’s conjecture for simple groups with connected prime graph

We deal with finite simple groups G with the property π(G) ⊆ {2, 3, 5, 7, 11, 13, 17}, where π(G) is the set of all prime divisors of the order of a group G. The set of all such groups is denoted by ζ ₁₇. Thompson’s conjecture in [1, Question 12.38] is proved valid for all groups in ζ ₁₇ whose prime graph is connected.     

Extremal dependence analysis of network sessions

We refine a stimulating study by Sarvotham et al. (Comput Networks 48:335–350, 2005) which highlighted the influence of peak transmission rate on network burstiness. From TCP packet headers, we amalgamate packets into sessions where each session is characterized by a 5-tuple (S,D,R,R ^∨,Γ)=(total payload, duration, average transmission rate, peak transmission rate, initiation time). After careful consideration, a new definition of peak rate is required. Unlike Sarvotham et al. (Comput Networks 48:335–350, 2005) who segmented sessions into two groups labelled alpha and beta, we segment into 10 sessions according to the empirical quantiles of the peak rate variable as a demonstration that the beta group is far from homogeneous. Our more refined segmentation reveals additional structure that is missed by segmentation into two groups. In each segment, we study the dependence structure of (S,D,R) and find that it varies across the groups. Furthermore, within each segment, session initiation times are well approximated by a Poisson process whereas this property does not hold for the data set taken as a whole. Therefore, we conclude that the peak rate level is important for understanding structure and for constructing accurate simulations of data in the wild. We outline a simple method of simulating network traffic based on our findings.     

Remarques sur les systemes dynamiques donnes avec plusieurs facteurs

Two Bernoulli shifts are given, (X, T) and (X′, T′), with independent generatorsR=P ∨Q andR′=P′ ∨Q′ respectively. (R andR′ are finite). One can chooseR such that if (X′, T′) can be made a factor of (X, T) in such a way that (P′) _T′ and (Q′) _T′ are full entropy factors of (P) _T and (Q) _T respectively thend (P ∨Q)=d(P′ ∨Q′). In addition it is proved that if (X, T) is a Bernoulli shift and ifS is a measure preserving transformation ofX that has the same factor algebras asT thenS=T orS=T ⁻¹. A tool for this proof, which may be of independent interest is a relative version for very weak Bernoullicity.

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Equipe de Recherche n^o 1 “Processus stochastique et applications” dépendant de la Section n^o 1 “Mathématiques, Informatique” associée au C.N.R.S.     

A Volumetric Penrose Inequality for Conformally Flat Manifolds

We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to \mathbbRⁿ\ W, n 3 3{\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}, and so that their boundary is a minimal hypersurface. (Here, W ì \mathbbRⁿ{\Omega\subset \mathbb{R}^{n}} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by \frac12(V/b_n)^(n-2)/n{\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}, where V is the Euclidean volume of Ω and β _n is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost.     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

To any finite symmetric subsetR ⊂ O₃ corresponds a Hecke operatorT _R on L²(S ²) which leaves the eigenspaces ℋ_n (n ≥ 0) of the Laplacian invariant. We compute the trace ofT _R | ℋ_n and prove that the sum of the positive eigenvalues ofT _R on ⊕_k=0 ^n-1ℋ_k prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator , and renormalizing the traces ofT _{R m} | ℋ _n slightly, we obtain sequences of Fourier coefficients of modular forms on Γ₀(4). Dedicated to R. Remmert on the occasion of his seventieth birthday     

Towards a Characterization of Markov Processes Enjoying the Time-Inversion Property

We give a necessary and sufficient condition for a homogeneous Markov process taking values in ℝ ⁿ to enjoy the time-inversion property of degree α. The condition sets the shape for the semigroup densities of the process and allows to further extend the class of known processes satisfying the time-inversion property. As an application we recover the result of Watanabe (Z. Wahrscheinlichkeitstheor. Verwandte Geb. 31:115–124, 1975) for continuous and conservative Markov processes on ℝ₊. As new examples we generalize Dunkl processes and construct a matrix-valued process with jumps related to the Wishart process by a skew-product representation.      

Universal lattices and property tau

We prove that the universal lattices – the groups G=SL_d(R) where R=ℤ[x₁,...,x_k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K₂ of some finite commutative rings and a relative property T of . Mathematics Subject Classification (2000) 20F69, 13M05, 19C20, 20G05, 20H05     

Orders on Multisets and Discrete Cones

We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.      

Morphisms between the moduli spaces of curves with generalized Teichmüller structure

 Let C _g,n be a compact surface of genus g having n punctures, T _g,n the associated Teichmüller space, and Γ _g,n the corresponding Teichmüller modular group. We will show that for any subgroup Γ≤Γ _g,n , _Γ T _g,n  ≔ T _g,n /Γ is a moduli space of curves having Γ-structure. Furthermore, we will investigate for which Γ≤Γ _g,n and for which finite coverings P : C _g′,n′ →C _g,n there exists a holomorphic map Φ _P  : _Γ T _g,n  → _Γ′T _g′,n′ such that for any R ? _Γ T _g,n , Φ _P (R) → R is topologically equivalent to P. Received: 14 May 2001 / Revised version: 2 November 2001     

Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n ^−1/2 β _n log ^3/2 n in which β _n can be particularly taken as (log log n)^1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n ^−1/2(log log n)^1/2log ² n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n ^−1/2 β _n log ^1/2 n for the above β _n under the given covariance function, which improves the relevant one n ^−1/2(log log n)^1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.     

A note on the existence of singular integrals

Abstract. In this note the existence of a singular integral operator T acting on Lipo(R“) spacesis studied. Suppose     

Hitting Simplices with Points in ℝ3

The so-called first selection lemma states the following: given any set P of n points in ℝ ^d , there exists a point in ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) simplices spanned by P, where the constant c _d depends on d. We present improved bounds on the first selection lemma in ℝ³. In particular, we prove that c ₃≥0.00227, improving the previous best result of c ₃≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c ₃≤1/4⁴≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled).