Localization for a porous medium type equation in high dimensions

We prove the strict localization for a porous medium type equation with a source term, , , 1$">, \sigma +1$">, 0,$"> in the case of arbitrary compactly supported initial functions . We also otain an estimate of the size of the localization in terms of the support of the initial data and the blow-up time . Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.

     

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra and Dyer-Lashof algebra have both striking similarities and differences arising from their common origins in ``lower-indexed' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra , whose module actions are equivalent to, but quite different from, those of and . The exact relationships emerge as ``sheared algebra bijections', which also illuminate the role of the cohomology of . As a bialgebra, has a particularly attractive and potentially useful structure, providing a bridge between those of and , and suggesting possible applications to the Miller spectral sequence and the structure of Dickson algebras.

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

The peak algebra and the descent algebras of types B and D

We show the existence of a unital subalgebra of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra contains a two-sided ideal which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form . We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions and by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces and over and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.

     

Poincaré's closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface . This result is due originally to Poincaré. The proof uses the -dimensional Riemannian manifold of piecewise geodesic closed curves on with a fixed number of corners, chosen sufficiently large. In we consider a submanifold formed by those elements of which are simple regular and divide into two parts of equal total curvature . The main burden of the proof is to show that the energy integral , restricted to , assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on .

     

On peak-interpolation manifolds for

Let be a bounded, weakly convex domain in , , having real-analytic boundary. is the algebra of all functions holomorphic in and continuous up to the boundary. A submanifold is said to be complex-tangential if lies in the maximal complex subspace of for each . We show that for real-analytic submanifolds , if is complex-tangential, then every compact subset of is a peak-interpolation set for .

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

On orbital partitions and exceptionality of primitive permutation groups

Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

     

How to make a triangulation of polytopal

We introduce a numerical isomorphism invariant for any triangulation of . Although its definition is purely topological (inspired by the bridge number of knots), reflects the geometric properties of . Specifically, if is polytopal or shellable, then is ``small' in the sense that we obtain a linear upper bound for in the number of tetrahedra of . Conversely, if is ``small', then is ``almost' polytopal, since we show how to transform into a polytopal triangulation by local subdivisions. The minimal number of local subdivisions needed to transform into a polytopal triangulation is at least . Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for exponential in . We prove here by explicit constructions that there is no general subexponential upper bound for in . Thus, we obtain triangulations that are ``very far' from being polytopal. Our results yield a recognition algorithm for that is conceptually simpler, although somewhat slower, than the famous Rubinstein-Thompson algorithm.

     

A join theorem for the computably enumerable degrees

It is shown that for any computably enumerable (c.e.) degree , if , then there is a c.e. degree such that (so is lowand is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low c.e. degrees are not elementarily equivalent as partial orderings.

     

The flat model structure on

Given a cotorsion pair in an abelian category with enough objects and enough objects, we define two cotorsion pairs in the category of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when is hereditary. We then show that both of these induced cotorsion pairs are complete when is the ``flat' cotorsion pair of -modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat' model category structure on . In the last section we use the theory of model categories to show that we can define using a flat resolution of and a cotorsion coresolution of .

     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.

     

Luzin gaps

We isolate a class of ideals on that includes all analytic P-ideals and all ideals, and introduce `Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients . We also prove that the ideals and have the Radon-Nikodým property, and (using OCA) a uniformization result for -coherent families of continuous partial functions.

     

Sharp dimension estimates of holomorphic functions and rigidity

Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that

for all , with equality for some positive integer if and only if is holomorphically isometric to . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

     

Uniform Sobolev inequalities and absolute continuity of periodic operators

We establish certain uniform inequalities for a family of second order elliptic operators of the form on the -torus, where and is a symmetric, positive definite matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on with singular potential. The absolute continuity of the elliptic operator div on with a positive periodic scalar function is also studied.

     

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Let be a nondegenerate quadratic form and a nonzero linear form of dimension 3$">. As a generalization of the Oppenheim conjecture, we prove that the set is dense in provided that and satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.

     

A separable Brown-Douglas-Fillmore theorem and weak stability

We give a separable Brown-Douglas-Fillmore theorem. Let be a separable amenable -algebra which satisfies the approximate UCT, be a unital separable amenable purely infinite simple -algebra and be two monomorphisms. We show that and are approximately unitarily equivalent if and only if We prove that, for any 0$"> and any finite subset , there exist 0$"> and a finite subset satisfying the following: for any amenable purely infinite simple -algebra and for any contractive positive linear map such that

for all there exists a homomorphism such that

provided, in addition, that are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number and any 0$"> there is 0$"> such that in any unital amenable purely infinite simple -algebra if

for a pair of unitaries, then there exists a pair of unitaries and in such that

     

Radon transforms on affine Grassmannians

We develop an analytic approach to the Radon transform , where is a function on the affine Grassmann manifold of -dimensional planes in , and is a -dimensional plane in the similar manifold k$">. For , we prove that this transform is finite almost everywhere on if and only if , and obtain explicit inversion formulas. We establish correspondence between Radon transforms on affine Grassmann manifolds and similar transforms on standard Grassmann manifolds of linear subspaces of . It is proved that the dual Radon transform can be explicitly inverted for , and interpreted as a direct, ``quasi-orthogonal" Radon transform for another pair of affine Grassmannians. As a consequence we obtain that the Radon transform and the dual Radon transform are injective simultaneously if and only if . The investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity.

     

Cohomology operations for Lie algebras

If is a Lie algebra over and its centre, the natural inclusion extends to a representation of the exterior algebra of in the cohomology of . We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of , and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to .