On Diophantine definability and decidability in some infinite totally real extensions of

Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any 0$">, there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

     

The cohomology of certain Hopf algebras associated with -groups

We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .

     

On the divergence of the Rogers-Ramanujan continued fraction on the unit circle

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction.

Let the continued fraction expansion of any irrational number be denoted by and let the -th convergent of this continued fraction expansion be denoted by . Let

where . Let . It is shown that if , then the Rogers-Ramanujan continued fraction diverges at . is an uncountable set of measure zero. It is also shown that there is an uncountable set of points such that if , then does not converge generally.

It is further shown that does not converge generally for 1$">. However we show that does converge generally if is a primitive -th root of unity, for some . Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.

     

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.

     

Existence of local sufficiently smooth solutions to the complex Monge-Ampère equation

We prove the local solvability of the -dimensional complex Monge-Ampère equation , , in a neighborhood of any point where .

     

Infinitely many solutions to fourth order superlinear periodic problems

We present a new min-max approach to the search of multiple -periodic solutions to a class of fourth order equations

where is continuous, -periodic in and satisfies a superlinearity assumption when . For every , we prove the existence of a -periodic solution having exactly zeroes in .

     

Semi-linear homology -spheres and their equivariant inertia groups

This paper introduces an abelian group for all semi-linear homology -spheres, which corresponds to a known abelian group for all semi-linear homotopy -spheres, where is a compact Lie group and is a -representation with 0$">. Then using equivariant surgery techniques, we study the relation between both and when is finite. The main result is that under the conditions that -action is semi-free and with 0$">, the homomorphism defined by is an isomorphism if , and a monomorphism if . This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology -spheres.

     

Complete hyperelliptic integrals of the first kind and their non-oscillation

Let be a real polynomial of degree , and be an oval contained in the level set . We study complete Abelian integrals of the form

where are real and is a maximal open interval on which a continuous family of ovals exists. We show that the -dimensional real vector space of these integrals is not Chebyshev in general: for any 1$">, there are hyperelliptic Hamiltonians and continuous families of ovals , , such that the Abelian integral can have at least zeros in . Our main result is Theorem 1 in which we show that when , exceptional families of ovals exist, such that the corresponding vector space is still Chebyshev.

     

On the Clifford algebra of a binary form

The Clifford algebra of a binary form of degree is the -algebra , where is the ideal generated by . has a natural homomorphic image that is a rank Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit -divisor in , where is the curve and is the genus of .

     

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

The periodic isoperimetric problem

Given a discrete group of isometries of , we study the -isoperimetric problem, which consists of minimizing area (modulo ) among surfaces in which enclose a -invariant region with a prescribed volume fraction. If is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where (the group of symmetries of the integer rank three lattice ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than , and we give an isoperimetric inequality for -invariant regions that, for instance, implies that the area (modulo ) of a surface dividing the three space in two -invariant regions with equal volume fractions, is at least (the conjectured solution is the classical Schwarz triply periodic minimal surface whose area is ). Another consequence of this isoperimetric inequality is that -symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group .

     

Symmetrization, symmetric stable processes, and Riesz capacities

Let be a symmetric -stable process killed on exiting an open subset of . We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set in the complement of in the first exit moment from increases when and are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets in with given volume, the balls have the least -capacity ( ).

     

Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields

Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

     

Positive laws in fixed points

Let be an elementary abelian group of order at least acting on a finite -group in such a manner that satisfies a positive law of degree for any . It is proved that the entire group satisfies a positive law of degree bounded by a function of and only.

     

Uncorrelatedness and orthogonality for vector-valued processes

For a square integrable vector-valued process on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for are presented. The process is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for involving the biorthogonal representation for the conditional expectation of with respect to the usual product -algebra is presented.

     

Backward stability for polynomial maps with locally connected Julia sets

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure this easily implies that one of the following holds: 1. for -a.e. , ; 2. for -a.e. , for a critical point depending on .

     

Lattice invariants and the center of the generic division ring

Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

     

Integrals, partitions, and cellular automata

We prove that

where is the decreasing function that satisfies , for . When is an integer and we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

     

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.

     

On restrictions of modular spin representations of symmetric and alternating groups

Let be an algebraically closed field of characteristic and be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups of and -modules such that the restriction is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where is the Schur's double cover or .