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1.
Wilhelm P. A. Klingenberg 《Transactions of the American Mathematical Society》2004,356(6):2545-2556
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 .
2.
Jacques Boulanger Jean-Luc Chabert 《Transactions of the American Mathematical Society》2004,356(12):5071-5088
Let be a cyclotomic field with ring of integers and let be a polynomial whose values on belong to . If the ideal of generated by the values of on is itself, then every algebraic integer of may be written in the following form:
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
3.
Jó zef H. Przytycki Akira Yasuhara 《Transactions of the American Mathematical Society》2004,356(9):3669-3685
We study the linking numbers in a rational homology -sphere and in the infinite cyclic cover of the complement of a knot. They take values in and in , respectively, where denotes the quotient field of . It is known that the modulo- linking number in the rational homology -sphere is determined by the linking matrix of the framed link and that the modulo- linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo ' and `modulo '. When the finite cyclic cover of the -sphere branched over a knot is a rational homology -sphere, the linking number of a pair in the preimage of a link in the -sphere is determined by the Goeritz/Seifert matrix of the knot.
4.
5.
Alexander Gorodnik 《Transactions of the American Mathematical Society》2004,356(11):4447-4463
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.
6.
Jerry R. Muir Jr. Ted J. Suffridge 《Transactions of the American Mathematical Society》2007,359(4):1485-1498
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 .
7.
Gautam Bharali 《Transactions of the American Mathematical Society》2004,356(12):4811-4827
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 .
8.
First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.
9.
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 .
10.
Alberto Elduque 《Transactions of the American Mathematical Society》2004,356(10):3963-3983
The centralizer algebra of the action of on the real tensor powers of its natural module, , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for and with the decomposition of into irreducible submodules is considered.
11.
In this note we discuss the effect of the -nullification and the -cellularization over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe by means of a covering fibration, and we classify all finite groups for which is -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of and
12.
Filippo Gazzola Hans-Christoph Grunau Enzo Mitidieri 《Transactions of the American Mathematical Society》2004,356(6):2149-2168
We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces and in higher-order Sobolev spaces on a bounded domain can be refined by adding remainder terms which involve norms. In the higher-order case further norms with lower-order singular weights arise. The case being more involved requires a different technique and is developed only in the space .
13.
Boris Rubin 《Transactions of the American Mathematical Society》2004,356(12):5045-5070
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.
14.
Shulim Kaliman Sté phane Vé né reau Mikhail Zaidenberg 《Transactions of the American Mathematical Society》2004,356(2):509-555
The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .
15.
J. Cichon J. D. Mitchell M. Morayne 《Transactions of the American Mathematical Society》2007,359(5):2059-2074
If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.
16.
Jerzy Bialkowski Karin Erdmann Andrzej Skowronski 《Transactions of the American Mathematical Society》2007,359(6):2625-2650
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 .
17.
Werner Ehm Tilmann Gneiting Donald Richards 《Transactions of the American Mathematical Society》2004,356(11):4655-4685
A classical theorem of Boas, Kac, and Krein states that a characteristic function with for admits a representation of the form
where the convolution root is complex-valued with for . The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If is real-valued and even, can the convolution root be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of is obtained. Furthermore, the analogous problem for radially symmetric functions defined on is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if is a probability density on whose characteristic function vanishes outside the unit ball, then
where denotes the first positive zero of the Bessel function , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in does not exist.
where the convolution root is complex-valued with for . The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If is real-valued and even, can the convolution root be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of is obtained. Furthermore, the analogous problem for radially symmetric functions defined on is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if is a probability density on whose characteristic function vanishes outside the unit ball, then
where denotes the first positive zero of the Bessel function , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in does not exist.
18.
Maria Isabel Cortez Jean-Marc Gambaudo Alejandro Maass 《Transactions of the American Mathematical Society》2007,359(5):2305-2315
In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.
19.
Laurent Hauswirth Joaquí n Pé rez Pascal Romon Antonio Ros 《Transactions of the American Mathematical Society》2004,356(5):2025-2047
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 .
20.
Joana Ventura 《Transactions of the American Mathematical Society》2005,357(6):2253-2289
In this paper we compute some derived functors of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.
When the group is a cyclic -group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor .
When the group is , we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired functors.