Showing papers in "Mathematische Zeitschrift in 2000"

On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

Abstract: We consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution remains bounded but its slope becomes unbounded in finite time. A quite detailed description of the wave-breaking phenomenon is given: there is at least a point (in general depending on time) where the slope becomes infinite exactly at breaking time. The precise blow-up rate is established and for a large class of initial data we also determine the blow-up set.

Bilinear estimates in BMO and the Navier-Stokes equations

Abstract: We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

The number of simple modules of the Hecke algebras of type G ( r ,1, n )

Abstract: This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r,1,n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of “Kleshchev multipartitions”.

A Sobolev mapping property of the Bergman kernel

Abstract: We prove that if D is a pseudoconvex domain with Lipschitz boundary having an exhaustion function \(\rho\) such that \(-(-\rho)^{\eta}\) is plurisubharmonic, then the Bergman projection maps the Sobolev space \(W_s\) boundedly to itself for any \(s<\eta/2\).

Gromov-Witten invariants of blow-ups along points and curves

Abstract: In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve

A local Riemann hypothesis, I

Abstract: Daniel Bump1, Kwok-Kwong Choi2, Par Kurlberg3, Jeffrey Vaaler4 1 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA (e-mail: bump@math.stanford.edu) 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong (e-mail: choi@maths.hku.hk) 3 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail: kurlberg@math.uga.edu) 4 University of Texas at Austin, Department of Mathematics RLM 8.100, Austin, TX 78712-1082, USA (e-mail: vaaler@math.utexas.edu)

A solution to the problem of L p −maximal regularity

Abstract: We give a negative solution to the problem of the $L^p$ -maximal regularity on various classes of Banach spaces including $L^q$ -spaces with $1 < q eq 2 < +\infty$ .

The similarity degree of an operator algebra, II

Abstract: For every integer \(d\ge 1\), there is a unital closed subalgebra \(A_d\subset B(H)\) with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism \(u\colon A_d\to B(H)\) we have \(\|u\|_{cb} \le K\|u\|^d\) with \(K>0\) independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than \(d-1\) crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbjornsen. We also include several complements to our previous publications on the same subject.

Vector-valued holomorphic functions revisited

Abstract: . Let $\Omega \subset \C$ be open,X a Banach space and $W\subset X^\prime$ . We show that every $\sigma (X,W)\mbox{-holomorphic function } f: \Omega \to X$ is holomorphic if and only if every $\sigma(X,W)\mbox{-bounded}$ set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that $\varphi \circ f$ is holomorphic for all $\varphi \in W$ , where W is a separating subspace of $X^\prime$ to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of $\Omega$ , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers).

Linear equivalence of ideal topologies

Abstract: It is proved that whenever P is a prime ideal in a commutative Noetherian ring such that the P-adic and theP-symbolic topologies are equivalent, then the two topologies are equivalent linearly Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves

Algebraic analysis of offsets to hypersurfaces

Abstract: In this paper, we present a complete algebraic analysis of degeneration and existence of simple and special components of generalized offsets to irreducible hypersurfaces over algebraically closed fields of characteristic zero. More precisely, we analyze the degeneration situations when offseting, and we state that there exist, at most, a finite set of distances for which the offset of a hypersurface may degenerate. As a consequence of this analysis, an algorithmic method to determine such distances is derived. Furthermore, as an application of these results, a complete degeneration analysis of the generalized offset to the sphere is developed. In addition, we study the existence of simple and special components of the offset. In this context we prove that, in the case of classical offsets, there always exists at least one simple component and, in the case of generalized offsets, we prove that for almost every distance and for almost every isometry, all components of the offset are simple.

Monotonicity of the cd-index for polytopes

Abstract: We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional polytope with n vertices is at most that of C(n,d), the d-dimensional cyclic polytope with n vertices.

Separate weak*-continuity of the triple product in dual real JB*-triples

Abstract: We prove that, if E is a real JB*-triple having a predual \(E_{*_{}},\) then \(E_{*_{}}\) is the unique predual of E and the triple product on E is separately $\sigma (E,E_{*_{}})-$continuous

Anderson model with decaying randomness: mobility edge

Abstract: In this paper we consider the Anderson model with decaying randomness \(a_nq_{\omega}(n)\), \(a_n > 0, n \in {\mathbb Z}^{ u}\) and \(q_{\omega}(n)\), i.i.d. random variables with an absolutely continuous distribution \(\mu\). For a class of \(\mu\) we show the following results on a set \(\omega\) of full measure.

Multiple periodic points of the Poincare map of Lagrangian systems on tori

Abstract: In this paper, suppose \(L(t,x,p)={1\over 2}A(x)p\cdot p+V(t,x)\), A is positive definite and symmetric, and both A and V are \(C^3\) and 1-periodic in all of their variables. We prove that the Poincare map (i.e. the time-1-solution map) of the Lagrangian system

Decompositions of simplicial balls and spheres with knots consisting of few edges

Abstract: Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every $d \geq 3$ . This answers a question of Danaraj and Klee [10]; it also strengthens a result of Lickorish [16] about non-shellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with “few edges” in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of “few edges.”

Some results on a class of nonlinear Schrodinger equations

Abstract: Abstract. By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schrödinger equation $-h^2\\Delta u+V(x)u=f(x,u)$ in $R^N$ where $f(x,u)$ satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions.

Hypersurfaces of prescribed mean curvature in Lorentzian manifolds

Abstract: We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.

Pointwise multipliers and decomposition theorems in analytic Besov spaces

Abstract: For the holomorphic Besov spaces $\mathcal B^p_s$ in the unit ball of $\mathbb C^n,$ we study necessary and sufficient conditions on holomorphic functions $g_1,\,g_2$ so that the operator $(f_1,f_2) \mapsto g_1f_1+g_2f_2$ maps $\mathcal B^p_s\times \mathcal B^p_s $ onto $ \mathcal B^p_s.$ Related to this problem, we also obtain some characterizations of the pointwise multipliers for these spaces.

Some new examples in the theory of quadratic forms

Abstract: We construct a 6-dimensional anisotropic quadratic form \(\phi\) and a 4-dimensional quadratic form \(\psi\) over some fieldF such that \(\phi\) becomes isotropic over the function field \(F(\psi)\) but every proper subform of \(\phi\) is still anisotropic over \(F(\psi)\). It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form \(\phi\) with trivial determinant such that the index of the Clifford invariant of \(\phi\) is 4 but \(\phi\) can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest.

Nombre maximal d'hyperplans instables pour un fibré de Steiner

Abstract: Soit ${\cal S}_{n,k}$ la famille des fibres de Steiner S sur ${\bf P}_n$ definis par une suite exacte ( $k>0$ ) \[ 0\rightarrow kO_{{\bf P}_n}(-1) \longrightarrow (n+k)O_{{\bf P}_n}\longrightarrow S \rightarrow 0 \] Nous montrons le resultat suivant : Soient $S\in{\cal S}_{n,k}$ et $H_1,\cdots,H_{n+k+2}$ des hyperplans distincts tels que $h^0(S^{\vee}_{H_i}) eq 0$ . Alors il existe une courbe rationnelle normale $C_n\subset{\bf P}_{n}^{\vee}$ telle que $H_{i}\in C_n$ pour $i=1, ..., n+k+2$ et $S\simeq E_{n+k-1}(C_n)$ , ou $E_{n+k-1}(C_n)$ est le fibre de Schwarzenberger sur ${\bf P}_n$ appartenant a ${\cal S}_{n,k}$ associea la courbe $C_n\subset{\bf P}_{n}^{\vee}$ . On en deduit qu'un fibre de Steiner $S\in{\cal S}_{n,k}$ , s'il n'est pas un fibre de Schwarzenberger, possede au plus (n+k+1) hyperplans instables; ceci prouve dans tous les cas un resultat de Dolgachev et Kapranov ([DK], thm. 7.2) concernant les fibres logarithmiques.

Note on mod p Siegel modular forms II

Abstract: The structure of the ring of mod p Siegel modular forms of degree two is determined in the cases where the prime p is 2 or 3.

Rational obstruction theory and rational homotopy sets

Abstract: We develop an obstruction theory for homotopy of homomorphisms \(f,g : {\mathcal M }\to{\mathcal N }\) between minimal differential graded algebras. We assume that \({\mathcal M }=\Lambda V\) has an obstruction decomposition given by \(V=V_0\oplus V_1\) and that f and g are homotopic on \(\Lambda V_0\). An obstruction is then obtained as a vector space homomorphism \(V_1\to H^*({\mathcal N})\). We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes \([{\mathcal M},{\mathcal N }]\). This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps.

Saturation for valuations on two-dimensional regular local rings

Abstract: We study valuations centered in a regualar local ring of dimension two. We define the notion of saturation with respect to such a valuation, extending the classical definitions. The invariants associated in a natural way to the valuation are related with the saturated ring and som geometric properties are deduced.

Some results on a class of nonliner Schrödinger equations

Abstract: By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schrodinger equation \(-h^2\Delta u+V(x)u=f(x,u)\) in \(R^N\) where \(f(x,u)\) satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions.

The spectrum is continuous on the set of p-hyponormal operators

Abstract: In this paper it is shown that the spectrum \(\sigma\), a set valued function, is continuous when the function is restricted to the set of all p-hyponormal operators on a Hilbert space