On non-Kählerianity of nonuniform lattices¶in SO(n, 1)(n≥ 4)

In this note, we will show that no nonuniform lattice of SO(n, 1)(n≥ 4) is the fundamental group of a quasi-compact K?hler manifold Received: 18 July 2000 / Revised version: 4 September 2000     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 2000     

Normality of orbit closures for Dynkin quivers¶of type ?n

It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? _n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? _n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001     

Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001     

The representation of quadratic forms by almost universal forms of higher rank

 In this article, we prove that there are only finitely many positive definite integral quadratic forms of rank n+3(n≥2) that represent all positive definite integral quadratic forms of rank n but finitely many exceptions. Furthermore we determine all diagonal quadratic forms having such property and its exceptions remaining four as candidates. Received: 29 November 2000 ; in final form: 8 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 11E12, 11E20.     

The braid group B4 is linear

We study a linear representation ρ:B _n ? GL _m (Z[q ^±1,t ^±1]) with m=n(n-1)/2. We will show that for n=4, this representation is faithful. We prove a relation with the new Charney length function. We formulate a conjecture implying that ρ is faithful for all n. Oblatum 15-VI-1999 & 24-II-2000?Published online: 18 September 2000     

Images of eigenvalue distributions under power maps

 In [9], it was shown that if U is a random n×n unitary matrix, then for any p≥n, the eigenvalues of U ^p are i.i.d. uniform; similar results were also shown for general compact Lie groups. We study what happens when p<n instead. For the classical groups, we find that we can describe the eigenvalue distribution of U ^p in terms of the eigenvalue distributions of smaller classical groups; the earlier result is then a special case. The proofs rely on the fact that a certain subgroup of the Weyl group is itself a Weyl group. We generalize this fact, and use it to study the power-map problem for general compact Lie groups. Received: 19 December 2000 / Revised version: 7 August 2002 / Published online: 21 February 2003 Permanent address: Center for Communications Research, Princeton, NJ, USA. e-mail: rains@idaccr.org     

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

We show the following two results on a set of n points in the plane, thus answering questions posed by Erdos and Purdy [11]: 1. The maximum number of triangles of maximum area (or of maximum perimeter) in a set of n points in the plane is exactly n . 2. The maximum possible number of triangles of minimum positive area in a set of n points in the plane is Θ(n ² ) . Received April 19, 1999, and in revised form February 29, 2000, and September 12, 2000. Online publication April 6, 2001.     

Unique solvability of the Dirichlet problem¶for weakly harmonic maps

We prove existence and uniqueness of weakly harmonic maps from the unit ball in ℝ ⁿ (with n≥ 3) to a smooth compact target manifold within the class of maps with small scaled energy for suitable boundary data. Received: 9 June 2000 / Revised version: 17 April 2001     

Cycles in Newton´s means

In the dynamical system defined by Newton′s means for n complex variables, n≥2 there are invariant, planar curves with (chaotic) dynamics conjugated to the dynamics of z↦z ⁿ on the unit circle in the complex plane. Oblatum 8-III-2000 & 8-XI-2000?Published online: 5 March 2001     

A second-order bundle method to minimize the maximum eigenvalue function

In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n×n symmetric matrices. We show how a simple bundle method, the approximate eigenvalue method can be used to globalize the second-order method developed by M.L. Overton in the eighties and recently revisited in the framework of the ?-Lagrangian theory. With no additional assumption, the resulting algorithm generates a minimizing sequence. A geometrical and constructive proof is given. To prove that quadratic convergence is achieved asymptotically, some strict complementarity and non-degeneracy assumptions are needed. We also introduce new variants of bundle methods for semidefinite programming. Received: February 9, 1998 / Accepted: May 2, 2000?Published online September 20, 2000     

Density results in the Δ20 e-degrees

We show that the Δ⁰ ₂ enumeration degrees are dense. We also show that for every nonzero n-c. e. e-degree a, with n≥ 3, one can always find a nonzero 3-c. e. e-degree b such that b < a on the other hand there is a nonzero ωc. e. e-degree which bounds no nonzero n-c. e. e-degree. Received: 13 June 2000 / Published online: 3 October 2001     

Exponentially Many Hadamard Designs

If there is a Hadamard design of order n, then there are at least 2^{8n−16−9log n} non-isomorphic Hadamard designs of order 2n. Mathematics Subject Classificaion 2000: 05B05     

Each complexity-two word represents every sufficiently long spike

When is an integer for which then r _l denotes the partial function , and denotes the composition monoid of all partial functions . For a pair of positive integers we study the set S(m,n) of all integers such that there exists no which satisfies . Main Theorem: S(m,n) is finite. Received June 13, 2000; accepted in final form September 25, 2000.     

Eigenvalue distributions of random unitary matrices

 Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices and random unitary matrices, and show that (X -E [X])/(\Var (X))^1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I ₁ , ..., I _m , the corresponding random variables are jointly normal in the large n limit. Received: 15 November 2000 / Revised version: 27 September 2001 / Published online: 17 May 2002     

Cancelling periodic points

We prove that a self map of a PL-manifold of dimension is homotopic to a map with no periodic points of period n iff the Nielsen numbers (k divides n) disappear. Received: 15 December 1999 / Accepted: 25 September 2000 / Published online: 18 June 2001     

Intersecting Curves in the Plane

 We prove that for every family of n pairwise intersecting simple closed planar curves in general position, at least (4/5)n ²−O(n) points lie on more than one curve. This improves the previous lower bound of (3/4)n ²−O(n) due to Richter and Thomassen. Received: March 29, 2000 Final version received: August 30, 2001 RID="*" ID="*" Research supported in part by NSF grant DMS-9970325 Acknowledgments. I thank Bruce Richter for informing me about this problem, Gelasio Salazar for reading a preliminary version of the paper, and a Referee for useful comments. Current Address: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA. e-mail: mubayi@microsoft.com 1991 Mathematics Subject Classification. 05C35, 52C10