On the monodromy group for the contact problem for plane curves

Summary Fixr≥2,N=r(r+3)/2 andN smooth plane curvesA ₁…,A _N with degA _i>-2 fori=l,…,N. Then the monodromy group for the plane curves of degreer tangent tog _iA_i, g_i∈PGL(3), is the full symmetric group.

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Riassunto Sianor≥2,N=r(r+3)/2 eA ₁…,A _N curve piane lisce di grado almeno 2. Si dimostra che la monodromia per le curve piane di grador tangenti ag ₁ A _i,g _i∈PGL(3), è il gruppo simmetrico.

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Supported in part by NATO junior fellowship at M.I.T.     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

Bounds for the crossing number of the N-cube

Let Q_n denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number v(Q_n), i.e., the minimum number of edge-crossings in any planar drawing of Q_n. The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let N = 2ⁿ be the number of vertices of Q_n. We show that v(Q_n) < 1/6N². For the lower bound we prove that v(Q_n) = Ω(N(lg N)^{c lg lg N}), where c > 0 is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is v(Q_n) = Ω(N(lg N)²). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.     

Asymptotics for the standard and the Capelli identities

Let {c _n (St _k )} and {c _n (C _k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold:

On the Grassmann modules for the unitary groups

Let V be 2n-dimensional vector space over a field 𝕂 equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let 𝕂₀ ? 𝕂 be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, …, 2n}, there exist natural representations of the groups G ? U(2n, 𝕂/𝕂₀) and H = G ∩ SL(V) ? SU(2n, 𝕂/𝕂₀) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.     

Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ?^d ? Sp (d, ?), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d² + 2d, whereas if H ? Sp (d, R), then dim H ≤ d² + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth for the Central Polynomials

We study the growth of the central polynomials for the infinite dimensional Grassmann algebra G, and for the algebra M_k(F) of the k × k matrices, both over a field F of characteristic zero.     

Gauges for the cookie-cutter sets

Let E be a cookie-cutter set with dimH E =s. It is known that the Hausdorff s-measure and the packing s-measure of the set E are positive and finite. In this paper, we prove that for a gauge function g the set E has positive and finite Hausdorff g-measure if and only if 0 〈 liminft→0 g（t）/ts 〈 ∞. Also, we prove that for a doubling gauge function g the set E has positive and finite packing g-measure if and only if 0 〈 lim supt→0 g（t）/ts 〈 ∞.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

A Technique for the Search for Vertices of the Components of a k G-Permutation Module

In this paper we present a necessary condition for a p-group V ≤ G to be a vertex of some indecomposable direct summand of the permutation module k _H  ↑ ^G , where H ≤ G, and G is a finite group. We call this condition H-suitability and present a method how to check for it. In an example, we determine all H-suitable groups. In fact, in this example every H-suitable group is the vertex of some indecomposable direct summand of k _H  ↑ ^G .     

Brown representability for the dual

Let T be the homotopy category of all spectra. Brown proved that a homological functor H: T ^{o p} → Ab is representable if it takes coproducts to products. That is, the functors [−,h] may be characterised as the homological functors taking coproducts to products. In this article, we will prove the dual. A covariant functor H:T → Ab which takes products to products is representable; it is of the form [h,−]. Oblatum 10-VII-1997 & 23-VII-1997     

A method for computing the Perron root for primitive matrices

Following the Perron theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix A, there is a positive rank one matrix X such that B = A ° X , where ° denotes the Hadamard product of matrices, and such that the row (column) sums of matrix B are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix B without an explicit knowledge of X. The convergence rate of this algorithm is similar to that of the power method but it uses less computational load. A byproduct of the proposed algorithm is a new method for calculating the first eigenvector.     

Asymptotic Formulas for the Zeros of the Meixner Polynomials

The zeros of the Meixner polynomialm_n(x; β, c) are real, distinct, and lie in (0, ∞). Letα_{n, s}denote thesth zero ofm_n(nα; β, c), counted from the right; and letα_{n, s}denote thesth zero ofm_n(nα; β, c), counted from the left. For each fixeds, asymptotic formulas are obtained for bothα_{n, s}andα_{n, s}, asn→∞.     

Anti-integrability for the Logistic Maps

The embedding of the Bernoulli shift into the logistic map x→μx(1- x) forμ> 4 is reinterpreted by the theory of anti-integrability: it is inherited from the anti-integrable limitμ→∞.     

Approximation Algorithms for the Achromatic Number

The achromatic number for a graph G = V, E is the largest integer m such that there is a partition of V into disjoint independent sets {V₁, …, V_m} such that for each pair of distinct sets V_i, V_j, V_i V_j is not an independent set in G. Yannakakis and Gavril (1980, SIAM J. Appl. Math.38, 364–372) proved that determining this value for general graphs is NP-complete. For n-vertex graphs we present the first o(n) approximation algorithm for this problem. We also present an O(n^5/12) approximation algorithm for graphs with girth at least 5 and a constant approximation algorithm for trees.     

Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f _n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f _n ’s is involved and we assume f _n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f _n ) to f is assumed.     

Bounds for the vertex linear arboricity

The vertex linear arboricity vla(G) of a nonempty graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces a subgraph whose connected components are paths. This paper provides an upper bound for vla(G) of a connected nonempty graph G, namely vla(G) ≦ 1 + ?δ(G)/2? where δ(G) denotes the maximum degree of G. Moreover, if δ(G) is even, then vla(G) = 1 + ?δ(G)/2? if and only if G is either a cycle or a complete graph.     

Criteria for Convergence of the Number of Near Maxima for Long Tails

Let M_n denote the maximum of a random sample of size n and K_n(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (M_n – a, M_n]. There is a known integral criterion for almost sure convergence (to unity) of K_n(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(K_n(a > 1)) tends to zero.AMS 2000 Subject Classification. 62G30, 60F15     

Fast graphs for the random walker

 Consider the time T _oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T _oz in terms of the volume of z and the graph distance between o and z. The bounds are for expected value and large deviations, and are asymptotically sharp. We deduce rate of escape results for random walks on infinite graphs of exponential or polynomial growth, and resolve a conjecture of Benjamini and Peres. Received: 31 October 2000 / Revised version: 5 January 2002 / Published online: 22 August 2002     

Blow-up examples for the Yamabe problem

It has been conjectured that if solutions to the Yamabe PDE on a smooth Riemannian manifold (M ⁿ , g) blow-up at a point p ? M{p \in M} , then all derivatives of the Weyl tensor W _g of g, of order less than or equal to [\fracn-62]{[\frac{n-6}{2}]} , vanish at p ? M{p \in M} . In this paper, we will construct smooth counterexamples to the Weyl Vanishing Conjecture for any n ≥ 25.