Margulis numbers for Haken manifolds

For every closed orientable hyperbolic Haken 3-manifold and, more generally, for any orientable hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If H ₁(M;ℚ) ≠ 0, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.     

Special points on products of two Shimura curves

Let S ₁ and S ₂ be two Shimura curves over ℚ attached to rational indefinite quaternion algebras B _{1 ℚ} and B _{1 ℚ} with maximal orders B ₁ and B ₂ respectively. We consider an irreducible closed algebraic curve C in the product (S ₁×S ₂)_ℂ such that C(ℂ) ∩ (S ₁×S ₂)(ℂ) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields, that C is of Hodge type. Received: 3 January 2000 / Revised version: 2 October 2000     

A new characterization of Anderson’s inequality in <Emphasis Type="Italic">C</Emphasis><Subscript>1</Subscript>-classes

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.     

On the first pontrjagin class of homotopy complex projective spaces

Let M ²ⁿ be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p ₁(M) of M ²ⁿ has the form (n+1+24α(M))u ² for some integer α(M) where u is a generator of H ²(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Sur le 2-groupe de classes d'idéaux de ℚ(√d,i)

Let ℕ,i=√−1,k=ℚ(√d,i) andC ₂ the 2-part of the class group ofk. Our goal is to determine alld such thatC ₂⋍ℤ/2ℤ×ℤ/2ℤ. Soientd∈ℕ,i=√−1,k=ℚ(√d,i), etC ₂ la 2-partie du groupe de classes dek. On s'intéresse à déterminer tous lesd tel queC ₂⋍ℤ/2ℤ×ℤ/2ℤ.      

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

Some Elements of Finite Order in K2Q

Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.     

Polynomial hulls and an optimization problem

We say that a subset of Cⁿ is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B_n be the open unit ball in Cⁿ.Suppose K is a C^∞ compact manifold in ∂B₁ × Cⁿ, n > 1, diffeomorphic to ∂B₁ × ∂B_n, each of whose fibers K_z over ∂B₁ bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B₁. This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H^∞ optimization problem. If pis a C^∞ real-valued function on ∂B₁× Cⁿ, we show that the infimum γ_ρ = inf_ƒ∈H ^∞ (B₁)ⁿ ‖ρ(z, ƒ (z))‖_∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B₁ × C ⁿ|ρ(z, w) ≤ γ_ρ has bounded connected strictly hypoconvex fibers over the circle.     

Nonlinear and linear instability of the Rossby-Haurwitz wave

The dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves, being of great meteorological importance, are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H ₁ ⊕ H _n , where 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 M ₀ ⁿ − 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 the 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 H ₁ ⊕ H _n (n ≥ 2) is Lyapunov 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 M ₀ ⁿ − H _n . A necessary condition for this instability is given. The condition states that the spectral number η(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 are shown in two Hilbert spaces. The instability in the invariant set M ₊ ⁿ of small-scale perturbations (χ(t) > n(n + 1)) is still an open problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

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)).     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

The Lindel?f number of Cp(X)×Cp(X) for strongly zero-dimensional X

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C _p (X, M) is a continuous image of a closed subspace of C _p (X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindel?f number of C _p (X)×C _p (X) coincides with the Lindel?f number of C _p (X). We also prove that l(C _p (X ⁿ )^κ) ≤ l(C _p (X)^κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.     

Geometric realization of Adams maps

Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l _k : H*(G/P, ℚ) → H*(G/P, ℚ), l _k (u) = k ⁱ u, u ∈ H ²ⁱ (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l _k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.     

Composition Operators on the Bloch Space of Several Complex Variables

Abstract In this paper, we study the boundedness and compactness of composition operator C _φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B _n, we give the necessary and sufficient conditions for a composition operator C _φ to be compact on β(B _n) or β ₀(B _n). Supported by the National Natural Science Foundation and the National Education Committee Doctoral Foundation     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds

For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N ^{n +1},g), with non-trivial n dimensional integer homology, such that for some Γ∈H _n (N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities. Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998