Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Total colorings of graphs of order 2n having maximum degree 2n−2

Let χ _t (G) and †(G) denote respectively the total chromatic number and maximum degree of graphG. Yap, Wang and Zhang proved in 1989 that ifG is a graph of orderp having †(G)≥p−4, then χ _t (G≤Δ(G)+2. Hilton has characterized the class of graphG of order 2n having †(G)=2n−1 such that χ _t (G=Δ(G)+2. In this paper, we characterize the class of graphsG of order 2n having †(G)=2n−2 such that χ _t (G=Δ(G)+2 Research supported by National Science Council of the Republic of China (NSC 79-0208-M009-15)     

On the distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.     

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f _s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K _s }}, where the minimum is taken over all K _t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n ^1/2+o(1)) ≤ f _s,s+1(n) ≤ O(n ^1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f _s,s+1(n) ≤ O(n ^2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n ^1/2−ɛ ) ≤ f _s,s+k (n) ≤ O(n ^1/2+ɛ . In addition, we also discuss some connections between the function f _s,t and vertex Folkman numbers.     

Stability Theory of General Contractions for Delay Equations

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)v _t with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))v _t , thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)v _t + f(t, v _t ). In addition, we consider general contractions e ^−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.     

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Semiclassical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L ² normalized family of functions such that P(h)u(h) is O(h) in L ²(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L ^p norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch ^−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

On Kneser solutions of higher order nonlinear ordinary differential equations

The equationx ⁽ⁿ⁾(t)=(−1) ⁿ │x(t)│ ^k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t₀)^−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t ₀)^−n/(k−1). It is shown that they do not necessarily coincide withC(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.