Showing papers in "Crelle's Journal in 1994"

Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties.

Abstract: The basic object for the sequel is a fixed regul r Dirichlet form S with domain Q) ( } on a real Hubert space H = L(X, m). The underlying topological space X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[w] = X. The form S is always assumed to be strongly local (i.e. S (u, v) = 0 whenever u e Q) ((?) is constant on a neighborhood of the support of v €&(£)) and to be irreducible (i.e. we ^loc(0} without restriction this version is always chosen quasi-continuous.

The geometry and topology of 3-Sasakian manifolds.

Abstract: In recent years quaternionic Kahler and hyperkähler manifolds have received a great deal of attention. They appear in many different areas of mathematics and mathematical physics. It has been argued that these recent advances in quaternionic geometry vindicate Hamilton's conviction that the algebra of quaternions should play an important role in mathematical physics [At], [Hil]. The purpose of this paper is to describe the geometry and topology of a class of Riemannian Einstein manifolds that is closely related to both hyperkähler and quaternionic Kahler manifolds. These manifolds, known äs manifolds with a Sasakian 3-structure, first appeared in a paper by Kuo in 1970 [Ku] which was published a few years before Ishihara and Calabi introduced the now commonly accepted terms \"quaternionic Kahler\" and \"hyperkähler\", respectively. We shall refer to manifolds with a Sasakian 3-structure äs 3-Sasakian manifolds.

Stochastic two-scale convergence in the mean and applications

Abstract: It is well known that the modelling of physical processes in strongly inhomogeneous media leads to the study of differential equations with rapidly varying coefficients. More precisely, if the scale of inhomogeneity of the medium is of order ε, then the coefficients of the differential operators are of the form a(s~x), where the function a depends on the microstructure. For small ε, a direct numerical solution of such problems is for all practical purposes impossible, and this necessitates the construction of averaged models. The main goal of homogenization theory is the analysis of these averaged models s ε -» 0, with particular emphasis on the establishment of convergence to a limit in an appropriate sense and the derivation of averaged equations for the limit. One expects such results if there is a certain regularity in the microstructure, present for example when the coefficients are periodic, almost periodic, or are homogeneous random fields.

A characterization of Hermitian function fields over finite fields.

Abstract: Let F be a function field of one variable of genus g over a finite field Fr with r elements. Then the number 7V of prime divisors of degree l of F is bounded by the HasseWeil bound N ^ r + l + 2g]/r. A function field is called maximal if N is equal to the upper bound r + l + 2gyr. For applications to coding theory, maximal function fields are of high interest [Stil], [Tsf-Vla]. A necessary condition for maximality is that r is a square. Moreover, it is known that function fields over fq2 of genus g > q(q — l)/ 2 are not maximal [Stil], Ch.V.3.

Fourier integral operators with fold singularities.

Abstract: are locally diffeomorphisms. In particular dx = dY-=d. Then &εΙ(Χ,Υ,<€') maps ^a.compOO into L> tloc(X) if β ^ a — μ. This was shown by H rmander s a consequence of the calculus in [7]. By composing 3F with a fractional integral operator it is easy to see that ^G/*(jr,r,

Commutants of unitaries in UHF algebras and functorial properties of exactness.

Abstract: Since Grothendieck showed in [18] that the theory of tensor products of locally convex spaces gives important informations on functional analytic properties of locally convex vector spaces, it was expected that tensor products in other categories (äs e.g. of C*algebras) lead to new insights. An early belief in the simplicity of the theory of tensor products of C*-algebras was destroyed in 1964 when Takesaki [32] discovered non-nuclear C*-algebras (non-nuclear with respect to the category of C*-algebras). A C*-algebra B is nuclear if, for every C*-algebra C, on the algebraic tensor product BQC there exists only one C*-norm. The study of nuclear C*-algebras by Takesaki [32], Lance [27], Choi and Effros [8] (inspired by ideas of Connes [11]) led to the characterization of unital nuclear C*-algebras by the nuclearity of its identity map (cf. [21] for an independent simple proof).

Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties.

Abstract: Classes of varieties satisfying this principle are known other than quadrics (e.g. SeveriBrauer varieties and more generally complete varieties which are homogeneous spaces under a connected linear algebraic group). However counterexamples to the Hasse principle are known already in the class of rational varieties (varieties which become birational to projective space over a finite extension of the ground field). In 1970, Manin ([Ma71], [Ma86]) showed how most known counterexamples could be accounted for in terms of an obstruction based on the Brauer group of varieties. Further work (see [CT92] for a survey) has shown that for some classes of rational varieties this obstruction, henceforth referred to äs the Brauer-Manin obstruction, is the only obstruction to the Hasse principle. A reasonable class to investigate in this respect is that of varieties which are geometrically unirational, and also that of varieties which are geometrically birational to Fano varieties (a Fano variety is a smooth projective variety whose anticanonical class is ample; it is an open question whether such a variety is geometrically unirational).

Quasi-projections in Teichmüller space.

Abstract: where d(x, C) = inf d(x, C). (Provided Xis proper, i.e. /?-balls are compact, nc(x) is never yeC empty.) Suppose now that X is a simply-connected Riemannian manifold with non-positive sectional curvatures, and that C is a geodesic segment, ray or line. In this case nc(x) always consists of a single point. If the sectional curvatures are bounded above by a negative constant, then \\\\dnc\\\\ -> 0 äs d(x, C) goes to oo (where dnc is the differential of nc). In other words, is contracting at large distances.

The Betti numbers of the moduli space of stable sheaves of rank 2 on P^2

Abstract: Several authors have studied the moduli spaces M(cl5 c2) of stable sheaves of rank 2 on P. In particular in the case that c\\ — 4c2 φ 0 mod 8, M(cl9 c2) is a smooth projective rational variety defined over Z([M1], [S]) and the Picard group Pic(M(cl9c2)) is isomorphic to Ζ. Since W(M(c^c2\\&) = 0, i ^ l , H(M(c1?c2),Z) is isomorphic to Ζ. In bis master thesis the author determined Weil's zeta function for the cases where cl= —i and c2's are small by using elementary transformation and induction on c2. Then the Weil conjecture teils us the Betti numbers of M(c1?c2) with ^ = -l and c2 small. Klyachko [K] computed independently the Betti numbers of M( — i,ri) for small n by considering a torus action on M( — l, n). In this paper, by adding a new idea to the above, we shall give a complete form of Weil's zeta function. Then, since M(c1? c2) is defined over Z, the Betti numbers ^(Μ(—\\,η)) can be, in principle, read out from the zeta function. Our main result is stated s follows.

Lagrange's Four Squares Theorem with almost prime variables.

Abstract: Representations of integers by sums of integral squares belong to the small stock of ancient problems in number theory. Lagrange succeeded in proving that all natural numbers can be written äs the sum of at most four squares of positive integers. Nowadays various proofs of this result are known, of which we single out the explicit formula for the number of representations of an integer by sums of four integral squares due to Jacobi. Later Kloosterman [K] considered the positive definite quadratic form

On the analogue of the division polynomials for hyperelliptic curves.

Abstract: The group laws of (the Jacobians of) elliptic and, to some extent, hyperelliptic curves are being used or proposed for a variety of applications, including primality testing, e.g., Adleman and Huang [2] and Goldwasser and Kilian [20], factoring, e.g., Lenstra [32], and cryptography, e.g. Koblitz [29]. In addition, there is a good deal of current interest in calculating in Jacobians in general. See, e.g., Cassels [9], [10], Chudnovsky and Chudnovsky [14], Flynn [15], Gordan and Grant [21], Huang and lerardi [24], Mazur [33], and Zimmer [46].

Counting characters in blocks, II.

Abstract: In the first paper [7] of this series we announced a conjecture expressing the number of ordinary irreducible characters of fixed defect in a block of a finite group G in terms of similar numbers for blocks of proper subgroups of G. Now we extend that conjecture from ordinary characters to projective characters of G, i.e. to characters of twisted group algebras ?l of G over suitable fields 5 of characteristic zero. We also use Clifford theory to cancel a large number of terms in the alternating sums used in that conjecture. The result is the equivalent Conjecture 17.10 expressing the number of ordinary irreducible characters of fixed defect d in a block B of 2l in terms of the numbers of rather special irreducible characters with the same defect in related blocks of twisted subgroup algebras for proper subgroups of G. We call these special characters \"weights\" by analogy with Alperin's well-known weights [2], which are a special case of ours. For characters of height zero our weight conjecture is equivalent to the projective form of the Alperin-McKay conjecture (see §18 below). We saw in Theorem 8.3 of [7] that our conjecture for groups implied the Alperin Weight Conjecture [2]. These two conjectures of Alperin together imply half of the Brauer Height Conjecture by Proposition 5.6 in [14]. So that, too, would be a consequence of our conjectures, if they turned out to be true.

A strong minimax property of nondegenerate minimal submanifolds.

Abstract: Clearly any nondegenerate critical point of a smooth function/: U\" -> R that is stable is also a strict local minimum. For functionals on infinite dimensional spaces, such äs the area functional on spaces of surfaces, the Situation is more complicated because \"local\" can be interpreted according to various topologies (strong, weak, etc.). Let F be the area functional (or more generally a smooth parametric elliptic functional) in a riemannian manifold N. In this paper we show (theorem 2) that if M is a smooth compact embedded submanifold (with or without boundary) that is stationary and strictly stable for F, then M is locally F-minimizing in a very strong sense: there is an open subset U of N containing M such that M has less area than any other surface (integral current) homologous in U to M. (In case every connected component of M has nonempty boundary, M has less area than any other surface in U having the same boundary.)

Geometry and topology of the boundary in the critical Neumann problem.

Abstract: Solvability of (1.1) in case N = 2 has been proven in [3], under the assumption limsupfA(/) = -f oo. In contrast, problem (1.1) has in general no solution (see [2]), if i-* oo N ^ 3, unless some curvature assumption is satisfied by Γχ . More precisely, denoted by H(x), χβδΩ, the mean curvature of du at x9 (1.1) turns out to be solvable, provided H(x) > 0 for some χ in the interior of /\\ (see [2], and also [21]).

Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces.

Abstract: The notions of ergodicity (absence of non-trivial invariant sets) and conservativity (absence of non-trivial wandering sets) are basic for the theory of measure preserving transformations. Ergodicity implies conservativity, but the converse is not true in general. Nonetheless, transformations from some classes always happen to be either ergodic (hence, conservative), or completely dissipative (i.e., their ergodic components are just orbits in the state space). The first Statement of this type was proved by Hopf [Ho] for the geodesic flow on Riemannian surfaces with constant negative curvature (whence the name \"the Hopf dichotomy\"). It was generalized by Sullivan to the geodesic flow on higher dimensional Riemannian manifolds with constant negative curvature [Su2] (both Hopfand Sullivan considered the Liouville invariant measure of the geodesic flow). It turned out that ergodicity of the geodesic flow is equivalent to recurrence of the Brownian motion on the manifold (in this form it is called the Hopf-Tsuji-Sullivan theorem). Later Sullivan generalized this result to invariant measures arising from a conformal density on the sphere at infinity [Su3] (see also a survey in [N] and a recent paper [Y]).

On the Fourier transforms of weighted orbital integrals.

Abstract: Suppose for a moment that G is a finite group. There are two canonical bases for the vector space of class functions on G. One is parametrized by the set Γ (G) of conjugacy classes in G, the other by the set Π (G) of (equivalence classes of) irreducible representations. Consider the elements of these bases s G-invariant linear functionals on C(G). In other words, set /G(y) = IGr 1 I/(*~V), y xeG and

On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a cone.

Abstract: While a general theory of regularity for elliptic boundary value problems has been developed for some time, the investigation of the location of the spectrum for corresponding operator pencils was started rather recently in [16]. Other papers addressing this problem are [3], [6]-[14], [18]. The operator pencil corresponding to linearized three dimensional elasticity was studied in detail in [l 5]. The present article is devoted to a detailed analysis of the operator pencil corresponding to the first boundary value problem of linearized three dimensional hydrodynamics.

Modèles minimaux des courbes de genre deux.

Abstract: Notre but est de determiner explicitement la fibre speciale 3CS de X en fonction des coefficients a{, de fa9on analogue au cas des courbes elliptiques ([Ta], §6-8). Plus precisement, Namikawa et Ueno [N, U] ont classifie tous les types possibles de X (completant ainsi la liste d'Ogg [Og]). II s'agit ici de reconnaitre le type de X dans la liste de Namikawa et Ueno ä partir de l'equation (1). Notons que plus generalement, si R est un anneau de Dedekind ä corps residuels parfaits, l'etude des fibres geometriques de X aux places p ne divisant pas 2 se ramene de fa9on Standard ä la Situation ci-dessus. Plus precisement, soit Rf l'henselise strict de Äp, alors SCS fc(p)fc(p) n'est autre chose que la fibre speciale du modele minimal de C sur Ä!. p

Quillen metrics and higher analytic torsion forms.

Abstract: Let || \\λ(ξ) and || \\λ(Κ·^ξ} be the Quillen metrics [Q2], [BGS 3] on λ (ξ) and which are respectively associated to gTM, Η and g , A ****. Quillen metrics are the product of the obvious L2 metric on the considered line (which one obtains by identifying cohomology classes with harmonic forms) by the Ray-Singer torsion of the associated Dolbeault complex [RS]. Let | | |Ιλ-ΐ(κ%ί)®λ(ο ^ ̂ e corresponding metric on

...k-rectifiable sets.

Abstract: Now, although a study of rectifiable sets of \"higher order of differentiability\" appears to be a natural one, it seems to be missing in the literature. Such a work should lead to some characterization, similar to (1.1), of ^*-rectifiable sets in terms of the existence of suitable \"approximate tangent paraboloids\" of order k. Moreover, one should expect a good behaviour of such sets under #*\"\" maps, and one would like to have that the Solutions of variational problems involving fc-order derivatives of the surfaces are ^-rectifiable.

Periodic solutions of non-autonomous Hamiltonian systems with symmetries.

Abstract: has no 1-periodic solution. In [10] Li and Willem obtained the same conclusion under a weaker assumption on H near u = 0. In the case of autonomous Hamiltonian Systems where H = H (u) Rabinowitz [14] showed that for each T> 0 there exist Tperiodic Solutions with arbitrarily large L°°-norm without needing an assumption near 0. This result depends in an essential way on the invariance of the autonomous equation under time shifts: If u (t) solves u = JVH(u) then clearly ue(t) -·= u(t + ) is also a solution. Working on a space of periodic functions Rabinowitz was able to use an index theory for the group F = i?/ TZ.

On characters of coprime operator groups and the Glaubermann character correspondence.

Abstract: An air injection sub for use with a dual conduit drill pipe string is provided having inner and outer concentric tubular members which are connected to the corresponding inner and outer conduits of a dual conduit string to provide isolated annular and central passageways and is particularly characterized by improved injection means in the inner tubular member which allows fluid to pass from the annular passageway into the central passageway through a plurality of apertures which open into the central passageway at a plurality of levels and angular positions, is field adjustable and provides improved erosion resistance.

Gamma cohomology and the Selberg zeta function

Abstract: We propose a new method for studying $n$- and $\Gamma$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $\Gamma$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $\Gamma$-cohomology of principal series representations if $\Gamma$ is cocompact and torsion free.