On the Equi-nuclearity of Roe Algebras of Metric Spaces

The authors define the equi-nuclearity of uniform Roe algebras of a family of metric spaces. For a discrete metric space X with bounded geometry which is covered by a family of subspaces {Xi}∞i=1, if {Cu*(Xi)}∞i=1 are equi-nuclear and under some proper gluing conditions, it is proved that Cu*(X) is nuclear. Furthermore, it is claimed that in general, the coarse Roe algebra C*(X) is not nuclear.     

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.     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

Effect of the initial estimator on the asymptotic behavior of one-step M-estimator

For a general (real) parameter, let M _nbe the M-estimator and M _n ⁽¹⁾ be its one-step version (based on a suitable initial estimator M _n ⁽⁰⁾). It is known that, under certain regularity conditions, n(M _n ⁽¹⁾-M _n)=O _p(1). The asymptotic distribution of n(M _n ⁽¹⁾-M _n) is studied; it is typically non-normal and it reveals the role of the initial estimator M _n ⁽⁰⁾.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0387     

On the classification of Hilbert modular threefolds

Letk be a totally real number field with ring of integersO _k . The Hilbert modular variety overk is a desingularization of the (natural) compactification of PSL₂(O _k )∖H ^k . The purpose of this paper is to present specific numerical bounds on the size of the discriminantd _k of a cubic fieldk with Hilbert modular variety of particular classifications. specifically, it is shown that ifd _k>2.12×10⁷, then the Hilbert modular variety overk is not rational and further, ifd _k>2.77×10⁸, then Hilbert modular variety overk is of general type. This material is based on work supported by the National Science Foundation under Grant No. DMS-9008689     

On the diameter and radius of randon subgraphs of the cube

The n-dimensional cube Qⁿ is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qⁿ has diameter and radious n. Write M = n2^n-1 = e(Qⁿ) for the size of Qⁿ. Let Q = (Q_t)_o^M be a random Qⁿ-process. Thus Q^t is a spanning subgraph of Qⁿ of size t, and Q_t is obtained from Q_t–1 by the random addition of an edge of Qⁿ not in Q_t–1, Let t^(k) = τ(Q;δ?k) be the hitting time of the property of having minimal degree at least k. We show that the diameter d_t = diam (Q_t) of Q_t in almost every Q? behaves as follows: d_t starts infinite and is first finite at time t⁽¹⁾, it equals n + 1 for t⁽¹⁾ ? t⁽²⁾ and d_t, = n for t ? t⁽²⁾. We also show that the radius of Q_t, is first finite for t = t⁽¹⁾, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q_t, for t = Ω(M). © 1994 John Wiley & Sons, Inc.     

Approximation of the vth Root of N

We present a class of functions g_K(w), K ≥ 2, for which the recursive sequences w_{n + 1} = g_K(w_n) converge to N^1/v with relative error . Newton's method results when K = 2. The coefficients of the g_K(w) form a triangle, which is Pascal's for v = 2. In this case, if w₁ = x₁/y₁, where x₁, y₁ is the first positive solution of Pell's equation x² ? Ny² = 1, then w_{n + 1} = x_{n + 1}/y_{n + 1} is the Kⁿpth or 2Kⁿpth convergent of the continued fraction for , its period p being even or odd.     

On the comparison of PBIB designs with two associate classes

A method to compare two-associate-class PBIB designs is discussed. As an application, it is shown that ifd ^* is a group-divisible design withλ ₂=λ₁+1, a group divisible design with group size two andλ ₂=λ₁+1>1, a design based on a triangular scheme andv=10 andλ ₁=λ₂+1, a design with anL ₂ scheme andλ ₂=λ₁+1, a design with anL _s scheme,v=(s+1) ², andλ ₂=λ₁+1, wheres is a positive integer, or a design with a cyclic schemev=5, andλ ₁=λ₂±1, thend ^* is optimum with respect to a very general class of criteria over all the two-associate-class PBIB designs with the same values ofv, b andk asd ^*. The best two-associate-class PBIB design, however, is not necessarily optimal over all designs. This paper was prepared with the support of Office of Naval Research Contract No. N00014-75-C-0444/NR 042-036 and National Science Foundation Grant No. MCS-79-09502.     

The Asymptotic Solution of the Korteweg-deVries Equation in the Absence of Solitons

The asymptotic solution of the Korteweg-de Vries equation u_τ + ?u_xxx + 2uu_x = 0 for initial conditions from which no solitons evolve is obtained as a slowly varying similarity solution of the form τ^?2/3(V_z?V₂, where V = V(z/τ) and z = τ^?1/3x. The results are consistent with, but go somewhat beyond, those recently obtained by Ablowitz and Segur [2] through a rather different approach.     

Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on ??₂(?^2d), the set of probability measures with finite quadratic moments on the phase space ?^2d = ?^d × ?^d, which is a metric space when endowed with the Wasserstein distance W₂. We study the initial value problem dμ_t/dt + ? · (??_d v _tμ_t) = 0, where ??_d is the canonical symplectic matrix, μ₀ is prescribed, and v _t is a tangent vector to ??₂(?^2d) at μ_t, belonging to ?H(μ_t), the subdifferential of H at μ_t. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ₀ is absolutely continuous. It ensures that μ_t remains absolutely continuous and v _t = ?H(μ_t) is the element of minimal norm in ?H(μ_t). The second method handles any initial measure μ₀. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on ??₂(?^2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μ_t) = H(μ₀). © 2007 Wiley Periodicals, Inc.     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

Eigenvalue Bounds for the Orr–Sommerfeld Equation and Their Relevance to the Existence of Backward Wave Motion

Theoretical estimates of the phase velocity c_r of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr–Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow or nearly parallel viscous shear flows in straight channels with velocity U(z) (=1?z², z∈[?1, +1] for plane Poiseuille flow), leave open the possibility that these phase velocities lie outside the range U_min<c_r<U_max but not a single experimental or numerical investigation, concerned with unstable waves in the context of flows with (d²U/dz²)_max≤0, has supported such a possibility as yet. U_min, U_max and (d²U/dz²)_max are, respectively, the minimum value of U(z), the maximum value of U(z), and the maximum value of (d²U/dz²) for z∈[?1, +1]. This gap between the theory on one hand and experiment and computation on the other has remained unexplained ever since Joseph [3] derived these estimates, first in 1968, and has even led to the speculation of a negative phase velocity in plane Poiseuille flow (i.e., c_r<U_min=0) and hence the possibility of a “backward” wave as in Jeffrey-Hamel flow in a diverging channel with backflow [1]. A simple mathematical proof of the nonexistence of such a possibility is given herein by showing that if (d²U/dz²)_max≤0 and (d⁴U/dz⁴)_min≥0 for z∈[?1, +1], then the phase velocity c_r of an arbitrary unstable wave must satisfy the inequality U_min<c_r<U_max, (d⁴U/dz⁴)_min is the minimum value of (d⁴U/dz⁴) for z∈[?1, +1], and therefore c_r cannot be negative when U_min=0. Another result that provides valuable insight into the general modal structure of the problem of instability of the above class of flows with U_min≥0 (e.g., plane Poiseuille flow) is that all standing waves, that is, modes for which c_r=0, are stable.     

Holomorphic Dependence of the Heins Map

Let D be a bounded convex domain and Hol _c (D,D) the set of holomorphic maps from D to C ⁿ with image relatively compact in D. Consider Hol _c (D,D) as a open set in the complex Banach space H ^∞ _n (D) of bounded holomorphic maps from D to C ⁿ . We show that the map τ: Hol _c (D,D) → D (called the Heins map for D equals to the unit disc of C) which associates to ? ∈ Hol _c (D,D) its unique fixed point τ? ∈ D is holomorphic and its differential is given by dτ_?(v) = (Id-df_τ(?))^?1 v(τ(?)) for v ∈ H ^∞ _n (D).     

The Iwasava decomposition and intermediate subgroups of the Steinberg groups over the field of fractions of a principal ideal ring

Abstract The Iwasava decomposition is proved for the Steinberg groups of types ² A _2l−1, ² D _l , ² E ₆, ³ D ₄ over the field of fractions of a principal ideal ring. By using this decomposition, it is described that subgroups exist between the Steinberg groups over the rings D and K under some restrictions on the ring D. This work was partially supported by RFFI (Grant No. 08-01-00824)     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.