On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

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 .

     

A note on -estimates for stable integrals with drift

Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .

     

Structural interactions of conjugacy closed loops

We study conjugacy closed loops by means of their multiplication groups. Let be a conjugacy closed loop, its nucleus, the associator subloop, and and the left and right multiplication groups, respectively. Put . We prove that the cosets of agree with orbits of , that and that one can define an abelian group on . We also explain why the study of finite conjugacy closed loops can be restricted to the case of nilpotent. Group is shown to be a subgroup of a power of (which is abelian), and we prove that can be embedded into . Finally, we describe all conjugacy closed loops of order .

     

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.

     

Quantum symmetric derivatives

For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative.

     

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

where is a bounded open subset of , , is the so-called Laplace operator, , is a Radon measure with bounded variation on , , , and and belong to the Lorentz spaces , , and , respectively. In particular we prove the existence under the assumptions that , belongs to the Lorentz space , , and is small enough.

     

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, should be biholomorphic to a rational homogeneous manifold , where is a simple Lie group, and is a maximal parabolic subgroup.

In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents , and (b) recovering the structure of a rational homogeneous manifold from . The author proves that, when and the generic variety of minimal rational tangents is 1-dimensional, is biholomorphic to the projective plane , the 3-dimensional hyperquadric , or the 5-dimensional Fano homogeneous contact manifold of type , to be denoted by .

The principal difficulty is part (a) of the scheme. We prove that is a rational curve of degrees , and show that resp. 2 resp. 3 corresponds precisely to the cases of resp. resp. . Let be the normalization of a choice of a Chow component of minimal rational curves on . Nefness of the tangent bundle implies that is smooth. Furthermore, it implies that at any point , the normalization of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that is a rational curve, our principal object of study is the universal family of , giving a double fibration , which gives -bundles. There is a rank-2 holomorphic vector bundle on whose projectivization is isomorphic to . We prove that is stable, and deduce the inequality from the inequality resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on in the special case where .

     

Profinite and pro- completions of Poincaré duality groups of dimension 3

We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic).

We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .

     

On symplectic fillings of lens spaces

Let be the contact structure naturally induced on the lens space by the standard contact structure on the three-sphere . We obtain a complete classification of the symplectic fillings of up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

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.

     

Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Complex linear differential equations of the form

with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient of belongs to the weighted Bergman space , where , for all , then all solutions are of order of growth at most , measured according to the Nevanlinna characteristic. In the case when all solutions are shown to be not only of order of growth zero, but of bounded characteristic. Conversely, if all solutions are of order of growth at most , then the coefficient is shown to belong to for all and .

Analogous results, when the coefficients belong to certain weighted Hardy spaces, are obtained. The non-homogeneous equation associated to is also briefly discussed.

     

Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras

Let be a field, a non-zero element of and the Iwahori-Hecke algebra of the symmetric group . If is a block of of -weight and the characteristic of is at least , we prove that the decomposition numbers for are all at most . In particular, the decomposition numbers for a -block of of defect are all at most .

     

Pure subrings of regular rings are pseudo-rational

We prove a generalization conjectured by Aschenbrenner and Schoutens (2003) of the Hochster-Roberts-Boutot-Kawamata Theorem: let be a pure homomorphism of equicharacteristic zero Noetherian local rings. If is regular, then is pseudo-rational, and if is moreover -Gorenstein, then it is pseudo-log-terminal.

     

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs (), (), , , and () and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift of every non-projective simple module is isomorphic to its third syzygy .

     

On the essential commutant of QC

Let (QC) (resp. ) be the -algebra generated by the Toeplitz operators QC (resp. ) on the Hardy space of the unit circle. A well-known theorem of Davidson asserts that (QC) is the essential commutant of . We show that the essential commutant of (QC) is strictly larger than . Thus the image of in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of (QC).

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.