On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

-theorems for fusion systems

For an odd prime, we generalise the Glauberman-Thompson -nilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We define a notion of -free fusion systems and show that if is a -free fusion system on some finite -group , then is controlled by for any Glauberman functor , generalising Glauberman's -theorem (Glauberman, 1968) to arbitrary fusion systems.

     

Prime specialization in genus 0

For a prime polynomial , a classical conjecture predicts how often has prime values. For a finite field and a prime polynomial , the natural analogue of this conjecture (a prediction for how often takes prime values on ) is not generally true when is a polynomial in ( the characteristic of ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values as varies. We prove the surprising fact that this ``Möbius average,' which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve .

The periodic Möbius average behavior implies in specific examples that a polynomial in does not take prime values as often as analogies with suggest, and it leads to a modified conjecture for how often prime values occur.

     

A tensor norm preserving unconditionality for -spaces

We show that, for each , there is an -tensor norm (in the sense of Grothendieck) with the surprising property that the -tensor product has local unconditional structure for each choice of arbitrary -spaces . In fact, is the tensor norm associated to the ideal of multiple -summing -linear forms on Banach spaces.

     

On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.

     

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with being a critical frequency in the sense that We show that if the zero set of has isolated connected components such that the interior of is not empty and is smooth, has isolated zero points, , , and has critical points such that , then for small, there exists a standing wave solution which is trapped in a neighborhood of Moreover the amplitudes of the standing wave around , and are of a different order of . This type of multi-scale solution has never before been obtained.

     

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Let be the solution operator for in , Tr on , where is a bounded domain in . B. E. J. Dahlberg proved that for a bounded Lipschitz domain maps boundedly into weak- and that there exists such that is bounded for . In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.

     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .

     

Torsion in coinvariants of certain Cantor minimal -systems

Let be a finite abelian group. We will consider a skew product extension of a product of two Cantor minimal -systems associated with a -valued cocycle. When is non-cyclic and the cocycle is non-degenerate, it will be shown that the skew product system has torsion in its coinvariants.

     

Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function , the number of bound states of the operator in below . Here is a bounded potential behaving asymptotically like where is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator on the sphere has negative eigenvalues less than , we prove that may be estimated as

Thus, in particular, if there are no such negative eigenvalues, then has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that and that there is exactly one eigenvalue less than , with all others , we show that the negative spectrum is asymptotic to a geometric progression with ratio .

     

The center of the category of -modules

Let be a semi-simple connected Lie group. Let be a maximal compact subgroup of and the complexified Lie algebra of . In this paper we describe the center of the category of -modules.

     

Rigid complexes via DG algebras

Let be a commutative ring, a commutative -algebra and a complex of -modules. We begin by constructing the square , which is also a complex of -modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism , then the pair is called a rigid complex over relative to (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes.

We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks (resp. ) along a finite (resp. essentially smooth) ring homomorphism .

In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in our paper Rigid Dualizing Complexes and Perverse Sheaves on Schemes, where we give a comprehensive version of Grothendieck duality for schemes.

The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh (1997).

     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth. For any positive number there exists a radially symmetric stationary solution with free boundary . The system depends on a positive parameter , and for a sequence of values there also exist branches of symmetric-breaking stationary solutions, parameterized by , small, which bifurcate from these values. In particular, for near the free boundary has the form where is the spherical harmonic of mode . It was recently proved by the authors that the stationary solution is asymptotically stable for any , but linearly unstable if , where if and if ; . In this paper we prove that for each of the stationary solutions which bifurcates from is linearly stable if and linearly unstable if . We also prove, for , that the point is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in .

     

Large orbits in coprime actions of solvable groups

Let be a solvable group of automorphisms of a finite group . If and are coprime, then there exists an orbit of on of size at least . It is also proved that in a -solvable group, the largest normal -subgroup is the intersection of at most three Hall -subgroups.

     

Bimodules and -rationality of vertex operator algebras

This paper studies the twisted representations of vertex operator algebras. Let be a vertex operator algebra and an automorphism of of finite order For any , an - -bimodule is constructed. The collection of these bimodules determines any admissible -twisted -module completely. A Verma type admissible -twisted -module is constructed naturally from any -module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra is -rational if and only if its twisted associative algebra is semisimple and each irreducible admissible -twisted -module is ordinary.

     

The zero set of semi-invariants for extended Dynkin quivers

We show that the set of common zeros of all semi-invariants vanishing at 0 on the variety of all representations with dimension vector of an extended Dynkin quiver under the group is a complete intersection if is ``big enough'. In case does not contain an open -orbit, which is the case not considered so far, the number of irreducible components of grows with , except if is an oriented cycle.

     

Homology of generalized Steinberg varieties and Weyl group invariants

Let be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and anti-invariants of the cohomology of the Springer fibres of the cone of nilpotent elements, , of for the Steinberg variety of triples.

Using a general specialization argument we show that for a parabolic subgroup of the space of -invariants and the space of -anti-invariants of are isomorphic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced by Douglass and Röhrle (2004).

The rational group algebra of the Weyl group of is isomorphic to the opposite of the top Borel-Moore homology of , where . Suppose is a parabolic subgroup of . We show that the space of -invariants of is , where is the idempotent in the group algebra of affording the trivial representation of and is defined similarly. We also show that the space of -anti-invariants of is , where is the idempotent in the group algebra of affording the sign representation of and is defined similarly.

     

Simple derivations of differentiably simple Noetherian commutative rings in prime characteristic

Let be a differentiably simple Noetherian commutative ring of characteristic (then is local with ). A short proof is given of the Theorem of Harper (1961) on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there exists a nilpotent simple derivation of the ring such that if , then for some . The derivation is given explicitly, and it is unique up to the action of the group of ring automorphisms of . Let be the set of all such derivations. Then . The proof is based on existence and uniqueness of an iterative -descent (for each ), i.e., a sequence in such that , and for all . For each , and .

     

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type and with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type with coefficients over the module Here the first standard generators of the group act by -multiplication, while the last one acts by -multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type as well as the cohomology of the classical braid group with coefficients in the -dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.