3,013 research outputs found
Diversity in Parametric Families of Number Fields
Let X be a projective curve defined over Q and t a non-constant Q-rational
function on X of degree at least 2. For every integer n pick a point P_n on X
such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large
N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/\log N
distinct, where c>0. We prove that there are at least N/(\log N)^{1-c} distinct
fields, where c>0.Comment: Minor inaccuracies detected by the referees are correcte
Some genus 3 curves with many points
Using an explicit family of plane quartic curves, we prove the existence of a
genus 3 curve over any finite field of characteristic 3 whose number of
rational points stays within a fixed distance from the Hasse-Weil-Serre upper
bound. We also provide an intrinsic characterization of so-called Legendre
elliptic curves
Generalized wordlength patterns and strength
Xu and Wu (2001) defined the \emph{generalized wordlength pattern} of an arbitrary fractional factorial design (or orthogonal array) on
factors. They gave a coding-theoretic proof of the property that the design
has strength if and only if . The quantities are
defined in terms of characters of cyclic groups, and so one might seek a direct
character-theoretic proof of this result. We give such a proof, in which the
specific group structure (such as cyclicity) plays essentially no role.
Nonabelian groups can be used if the counting function of the design satisfies
one assumption, as illustrated by a couple of examples
Serre's "formule de masse" in prime degree
For a local field F with finite residue field of characteristic p, we
describe completely the structure of the filtered F_p[G]-module K^*/K^*p in
characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of
F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's
mass formula in degree p. We also determine the compositum C of all degree p
separable extensions with solvable galoisian closure over an arbitrary base
field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in
the case of the local field F. Our method allows us to compute the contribution
of each character G\to\F_p^* to the degree p mass formula, and, for any given
group \Gamma, the contribution of those degree p separable extensions of F
whose galoisian closure has group \Gamma.Comment: 36 pages; most of the new material has been moved to the new Section
Alternating groups and moduli space lifting Invariants
Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of
degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves
Fried-Serre on deciding when sphere covers with odd-order branching lift to
unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on
Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces
of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp.
odd) theta functions when r is even (resp. odd). For inner spaces the result is
independent of r. Another use appears in,
http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of
families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for
the prime p=2 lying over Hurwitz spaces first studied by,
http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and
Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*)
High tower levels are general-type varieties and have no rational points.For
infinitely many of those MTs, the tree of cusps contains a subtree -- a spire
-- isomorphic to the tree of cusps on a modular curve tower. This makes
plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs.
Establishing these modular curve-like properties opens, to MTs, modular
curve-like thinking where modular curves have never gone before. A fuller html
description of this paper is at
http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from
proof sheets, but does include some proof simplification in \S
On p-adic lattices and Grassmannians
It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive
group G over a field k, carry the geometric structure of an inductive limit of
projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for
G. From the point of view of number theory it would be interesting to obtain an
analogous geometric interpretation of quotients of the form
G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of
p-typical Witt vectors, k is a perfect field of characteristic p and G is a
reductive group scheme over W(k). The present paper is an attempt to describe
which constructions carry over from the function field case to the p-adic case,
more precisely to the situation of the p-adic affine Grassmannian for the
special linear group G=SL_n. We start with a description of the R-valued points
of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R),
where R is a perfect k-algebra. In order to obtain a link with geometry we
further construct projective k-subvarieties of the multigraded Hilbert scheme
which map equivariantly to the p-adic affine Grassmannian. The images of these
morphisms play the role of Schubert varieties in the p-adic setting. Further,
for any reduced k-algebra R these morphisms induce bijective maps between the
sets of R-valued points of the respective open orbits in the multigraded
Hilbert scheme and the corresponding Schubert cells of the p-adic affine
Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math.
Zeitschrift, of the previously published preprint "On p-adic loop groups and
Grassmannians
HMDB: A Large Video Database for Human Motion Recognition
With nearly one billion online videos viewed everyday, an emerging new frontier in computer vision research is recognition and search in video. While much effort has been devoted to the collection and annotation of large scalable static image datasets containing thousands of image categories, human action datasets lag far behind. Current action recognition databases contain on the order of ten different action categories collected under fairly controlled conditions. State-of-the-art performance on these datasets is now near ceiling and thus there is a need for the design and creation of new benchmarks. To address this issue we collected the largest action video database to-date with 51 action categories, which in total contain around 7,000 manually annotated clips extracted from a variety of sources ranging from digitized movies to YouTube. We use this database to evaluate the performance of two representative computer vision systems for action recognition and explore the robustness of these methods under various conditions such as camera motion, viewpoint, video quality and occlusion.United States. Defense Advanced Research Projects Agency. Information Processing Techniques OfficeUnited States. Defense Advanced Research Projects Agency. System Science Division. Defense Sciences OfficeNational Science Foundation (U.S.) (NSF-0640097)National Science Foundation (U.S.) (NSF-0827427)United States. Air Force Office of Scientific Research (FA8650-05- C-7262)Adobe SystemsKing Abdullah University of Science and TechnologyNEC ElectronicsSony CorporationEugene McDermott FoundationBrown University. Center for Computing and VisualizationRobert J. and Nancy D. Carney Fund for Scientific InnovationUnited States. Defense Advanced Research Projects Agency (DARPA-BAA-09-31)United States. Office of Naval Research (ONR-BAA-11-001)Ministry of Science, Research and the Arts of Baden Württemberg, German
Finite-dimensional representations of twisted hyper loop algebras
We investigate the category of finite-dimensional representations of twisted
hyper loop algebras, i.e., the hyperalgebras associated to twisted loop
algebras over finite-dimensional simple Lie algebras. The main results are the
classification of the irreducible modules, the definition of the universal
highest-weight modules, called the Weyl modules, and, under a certain mild
restriction on the characteristic of the ground field, a proof that the simple
modules and the Weyl modules for the twisted hyper loop algebras are isomorphic
to appropriate simple and Weyl modules for the non-twisted hyper loop algebras,
respectively, via restriction of the action
Eta invariants for flat manifolds
Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we
calculate the eta invariants of the signature operator for almost all
7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta
invariants are an integer numbers. The article was motivated by D. D. Long and
A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom.
Topology 4, 2000, 171-178Comment: 18 pages, a new version with referees comment
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