Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

The parity of the Cochran-Harvey invariants of 3-manifolds

Given a finitely presented group and an epimorphism Cochran and Harvey defined a sequence of invariants , which can be viewed as the degrees of higher-order Alexander polynomials. Cochran and Harvey showed that (up to a minor modification) this is a never decreasing sequence of numbers if is the fundamental group of a 3-manifold with empty or toroidal boundary. Furthermore they showed that these invariants give lower bounds on the Thurston norm.

Using a certain Cohn localization and the duality of Reidemeister torsion we show that for a fundamental group of a 3-manifold any jump in the sequence is necessarily even. This answers in particular a question of Cochran. Furthermore using results of Turaev we show that under a mild extra hypothesis the parity of the Cochran-Harvey invariant agrees with the parity of the Thurston norm.

     

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

扭Heisenberg-Virasoro代数上的Poisson结构

Poisson代数是指同时具有代数结构和李代数结构的一类代数,其乘法与李代数乘法满足Leibniz法则.扭Heisenberg-Virasoro代数是一类重要的无限维李代数,是次数不超过1的微分算子李代数W(0)的普遍中心扩张,与曲线的模空间有密切联系.本文主要研究扭Heisenberg-Virasoro代数上的Poisson结构,首先确定了李代数W(0)上的Poisson结构,进而给出了扭Heisenberg-Virasoro代数上的Poisson结构.     

Perturbation bounds for weighted polar decomposition in the weighted unitarily invariant norm

In this paper, by generalizing the ideas of the (generalized) polar decomposition to the weighted polar decomposition and the unitarily invariant norm to the weighted unitarily invariant norm, we present some perturbation bounds for the generalized positive polar factor, generalized nonnegative polar factor, and weighted unitary polar factor of the weighted polar decomposition in the weighted unitarily invariant norm. These bounds extend the corresponding recent results for the (generalized) polar decomposition. In addition, we also give the comparison between the two perturbation bounds for the generalized positive polar factor obtained from two different methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Research problem: on the norms of infinite Cauchy-Toeplitz-plus-Cauchy-Hankel matrices

We give a conjecture and research problem related to ℓ_p operator and spectral norms of the matrix T_n + H_n.     

Boundedness for iterated commutators on the mixed norm spaces

This paper studies the iterated commutators on mixed norm spaces L²(?) characterizing the conjugate holomorphic symbols for which the corresponding iterated commutators are bounded by using the Bergman geometry, properties of holomorphic functions and related analysis.     

Approximating composition operator norms on the Dirichlet space

Let φ be any univalent self-map of the unit disk D whose image Ω≡φ(D) is compactly contained in D. We provide a method for approximating the norm of the composition operator Cφ on the Dirichlet space to any desired degree of accuracy. The approximation uses a special basis which is orthogonal in both the Bergman space on the disk and the Bergman space on Ω.     

A class of integral operators on mixed norm spaces in the unit ball

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball. The author is supported in part by the NNSF China (No. 10671115), grants from Specialized Research Fund for the doctoral program of Higher Education (No. 20060560002) and NSF of Guangdong Province (No. 06105468).     

Boundedness of the Cesàro operator on mixed norm spaces

In this note, the boundedness of the Cesàro operator on mixed norm space , , is proved.

     

Remarks on the lower bounds for the average genus

Let G be a graph of maximum degree at most four. By using the overlap matrix method which is introduced by B. Mohar, we show that the average genus of G is not less than 1/3 of its maximum genus, and the bound is best possible. Also, a new lower bound of average genus in terms of girth is derived.     

Lower bounds for the logarithmic Sobolev constant avoiding uniform lower bounds on the Ricci curvature

In this paper we obtain a lower bound for the logarithmic Sobolev constant of the operator on C^∞(M) given by L^U f = Δ f - (?U|?f), where U ? C^∞(M), M being a finite dimensional compact Riemannian manifold without boundary, in terms of the spectral gap of L^U and the lowest eigenvalue of the operator -L^U + V, where V is a function related to U and the Ricci curvature of M. Under suitable conditions and being U ≡ 0, this result improves a previous one by J.-D. DEUSCHEL and D.W. STROOCK (J. Funct. Anal. 92 (1990), 30–48).     

Improved lower bounds on the length of Davenport-Schinzel sequences

We derive lower bounds on the maximal length _s(n) of (n, s) Davenport Schinzel sequences. These bounds have the form _2s=1(n)=(n^s(n)), where(n) is the extremely slowly growing functional inverse of the Ackermann function. These bounds extend the nonlinear lower bound ₃ (n)=(n(n)) due to Hart and Sharir [5], and are obtained by an inductive construction based upon the construction given in [5].Work on this paper has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.     

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.     

Further results on the lower bounds of mean color numbers

Let G be a graph with n vertices. The mean color number of G, denoted by μ(G), is the average number of colors used in all n‐colorings of G. This paper proves that μ(G) ≥ μ(Q), where Q is any 2‐tree with n vertices and G is any graph whose vertex set has an ordering x₁,x₂,…,x_n such that x_i is contained in a K₃ of G[V_i] for i = 3,4,…,n, where V_i = {x₁,x₂,…,x_i}. This result improves two known results that μ(G) ≥ μ(O_n) where O_n is the empty graph with n vertices, and μ(G) ≥ μ(T) where T is a spanning tree of G. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 51–73, 2005     

Some lower bounds on Shelah rank in the free group

We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.