-manifolds with planar presentations and the width of satellite knots

We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

Corollary.

The width of a satellite knot is no less than the width of its pattern knot and so

.

     

Constructive recognition of

Existing black box and other algorithms for explicitly recognising groups of Lie type over have asymptotic running times which are polynomial in , whereas the input size involves only . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises explicitly.

The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising in its natural representation in polynomial time, given a discrete logarithm oracle for . The algorithm presented here takes as input a generating set for a subgroup of that is isomorphic modulo scalars to , where is a finite field of the same characteristic as ; it returns the natural representation of modulo scalars. Since a faithful projective representation of in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in rather than in , elementary algorithms will recognise explicitly in polynomial time in these cases. Given a discrete logarithm oracle for , our algorithm thus provides the required polynomial time oracle for recognising explicitly in the remaining case, namely for representations in the natural characteristic.

This leads to a partial solution of a question posed by Babai and Shalev: if is a matrix group in characteristic , determine in polynomial time whether or not is trivial.

     

Quivers with relations arising from clusters case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let be a cluster algebra of type . We associate to each cluster of an abelian category such that the indecomposable objects of are in natural correspondence with the cluster variables of which are not in . We give an algebraic realization and a geometric realization of . Then, we generalize the ``denominator theorem' of Fomin and Zelevinsky to any cluster.

     

Quantifier elimination for algebraic -groups

We prove that if is an algebraic -group (in the sense of Buium over a differentially closed field of characteristic , then the first order structure consisting of together with the algebraic -subvarieties of , has quantifier-elimination. In other words, the projection on of a -constructible subset of is -constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

     

On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod :

where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are -cores and discuss an intriguing relation between -cores and the Andrews-Garvan crank.

     

The invariant factors of the incidence matrices of points and subspaces in and

We determine the Smith normal forms of the incidence matrices of points and projective -dimensional subspaces of and of the incidence matrices of points and -dimensional affine subspaces of for all , , and arbitrary prime power .

     

Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on

We investigate the relationship between the decay at infinity of the right-hand side and solutions of an equation when is a second order elliptic operator on It is shown that when is Fredholm, inherits the type of decay of (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.

     

Automorphisms of fiber surfaces of genus , inducing the identity in cohomology

Let be a complex non-singular projective surface of general type with a genus fibration and . Let be a non-trivial subgroup of automorphisms of , inducing trivial actions on for all . Then , and . Examples of such surfaces are given.

     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.