on with and . In 1964 Kanel proved that if is an ignition non-linearity, then as when , and when L_1$">. We answer the open question of the relation of and by showing that . We also determine the large time limit of in the critical case , thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.
There is a positive constant such that for any diagram representing the unknot, there is a sequence of at most Reidemeister moves that will convert it to a trivial knot diagram, where is the number of crossings in . A similar result holds for elementary moves on a polygonal knot embedded in the 1-skeleton of the interior of a compact, orientable, triangulated 3-manifold . There is a positive constant such that for each , if consists of tetrahedra and is unknotted, then there is a sequence of at most elementary moves in which transforms to a triangle contained inside one tetrahedron of . We obtain explicit values for and .
Let be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers and . Using a new degree Blaschke product model for the dynamics of and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers and .
Let be a Polish group. We characterize when there is a Polish space with a continuous -action and an analytic set (that is, the Borel image of some Borel set in some Polish space) having uncountably many orbits but no perfect set of orbit inequivalent points.
Such a Polish -space and analytic exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of onto the infinite symmetric group, , consisting of all permutations of equipped with the topology of pointwise convergence.
The set of fixed points of is denoted by and we suppose that they are -rational and that for Let be the minimal semi-stable model of the -adic open disc over in which specializes to distinct smooth points. We study the differential data that can be associated to each irreducible component of the special fibre of Using this data we show that if , then the fixed points are equidistant, and that there are only a finite number of conjugacy classes of order automorphisms in which are not the identity
where depends only on . However one cannot choose depending linearly on .
Our proof is constructive, but nonlinear--which is quite surprising for such an elementary linear PDE. When there is a simpler proof by duality--hence nonconstructive.
We give an interpretation of the double affine Hecke algebra of Cherednik as a (suitably regularized) algebra of double cosets of a group by a subgroup , extending the well-known interpretations of the finite and affine Hecke algebras. In this interpretation, consists of -points of a simple algebraic group, where is a 2-dimensional local field such as or , and is a certain analog of the Iwahori subgroup.
is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories , using the theory of complex representations of finite groups as a model for what one would like to know.
An analogue of Artin's Theorem is proved for all complex oriented : the abelian subgroups of serve as a detecting family for , modulo torsion dividing the order of .
When is a complete local ring, with residue field of characteristic and associated formal group of height , we construct a character ring of class functions that computes . The domain of the characters is , the set of -tuples of elements in each of which has order a power of . A formula for induction is also found. The ideas we use are related to the Lubin-Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, -theory, etc.
The th Morava K-theory Euler characteristic for is computed to be the number of -orbits in . For various groups , including all symmetric groups, we prove that is concentrated in even degrees.
Our results about extend to theorems about , where is a finite -CW complex.
in of a function on the unit circle which is either continuous and quasianalytic in the sense of Bernstein or and quasianalytic in the sense of Denjoy is pluripolar.
Let be a finite group of automorphisms of a nonsingular three-dimensional complex variety , whose canonical bundle is locally trivial as a -sheaf. We prove that the Hilbert scheme parametrising -clusters in is a crepant resolution of and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on and coherent -sheaves on . This identifies the K theory of with the equivariant K theory of , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.
Let be a sequence of polynomials of degree in variables over a field . The zero-pattern of at is the set of those ( ) for which . Let denote the number of zero-patterns of as ranges over . We prove that for and
for . For , these bounds are optimal within a factor of . The bound () improves the bound proved by J. Heintz (1983) using the dimension theory of affine varieties. Over the field of real numbers, bounds stronger than Heintz's but slightly weaker than () follow from results of J. Milnor (1964), H.E. Warren (1968), and others; their proofs use techniques from real algebraic geometry. In contrast, our half-page proof is a simple application of the elementary ``linear algebra bound'.
Heintz applied his bound to estimate the complexity of his quantifier elimination algorithm for algebraically closed fields. We give several additional applications. The first two establish the existence of certain combinatorial objects. Our first application, motivated by the ``branching program' model in the theory of computing, asserts that over any field , most graphs with vertices have projective dimension (the implied constant is absolute). This result was previously known over the reals (Pudlák-Rödl). The second application concerns a lower bound in the span program model for computing Boolean functions. The third application, motivated by a paper by N. Alon, gives nearly tight Ramsey bounds for matrices whose entries are defined by zero-patterns of a sequence of polynomials. We conclude the paper with a number of open problems.