9,450 research outputs found
On the theta operator for modular forms modulo prime powers
We consider the classical theta operator on modular forms modulo
and level prime to where is a prime greater than 3. Our main
result is that mod will map forms of weight to forms of
weight and that this weight is optimal in certain cases
when is at least 2. Thus, the natural expectation that mod
should map to weight is shown to be false.
The primary motivation for this study is that application of the
operator on eigenforms mod corresponds to twisting the attached Galois
representations with the cyclotomic character. Our construction of the
-operator mod gives an explicit weight bound on the twist of a
modular mod Galois representation by the cyclotomic character
On modular Galois representations modulo prime powers
We study modular Galois representations mod . We show that there are
three progressively weaker notions of modularity for a Galois representation
mod : we have named these `strongly', `weakly', and `dc-weakly' modular.
Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These
notions of modularity are relative to a fixed level .
Using results of Hida we display a `stripping-of-powers of away from the
level' type of result: A mod strongly modular representation of some
level is always dc-weakly modular of level (here, is a natural
number not divisible by ).
We also study eigenforms mod corresponding to the above three notions.
Assuming residual irreducibility, we utilize a theorem of Carayol to show that
one can attach a Galois representation mod to any `dc-weak' eigenform,
and hence to any eigenform mod in any of the three senses.
We show that the three notions of modularity coincide when (as well as
in other, particular cases), but not in general
Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras
We present a new method for constructing -dimensional minimally
superintegrable systems based on block coordinate separation of variables. We
give two new families of superintegrable systems with () singular
terms of the partitioned coordinates and involving arbitrary functions. These
Hamiltonians generalize the singular oscillator and Kepler systems. We derive
their exact energy spectra via separation of variables. We also obtain the
quadratic algebras satisfied by the integrals of motion of these models. We
show how the quadratic symmetry algebras can be constructed by novel
application of the gauge transformations from those of the non-partitioned
cases. We demonstrate that these quadratic algebraic structures display an
universal nature to the extent that their forms are independent of the
functions in the singular potentials.Comment: 13 pages, no figure; Version to appear in Annals of Physic
Sequential importance sampling for multiway tables
We describe an algorithm for the sequential sampling of entries in multiway
contingency tables with given constraints. The algorithm can be used for
computations in exact conditional inference. To justify the algorithm, a theory
relates sampling values at each step to properties of the associated toric
ideal using computational commutative algebra. In particular, the property of
interval cell counts at each step is related to exponents on lead
indeterminates of a lexicographic Gr\"{o}bner basis. Also, the approximation of
integer programming by linear programming for sampling is related to initial
terms of a toric ideal. We apply the algorithm to examples of contingency
tables which appear in the social and medical sciences. The numerical results
demonstrate that the theory is applicable and that the algorithm performs well.Comment: Published at http://dx.doi.org/10.1214/009053605000000822 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras
We present a useful proposition for discovering extended Laplace-Runge-Lentz
vectors of certain quantum mechanical systems. We propose a new family of
superintegrable systems and construct their integrals of motion. We solve these
systems via separation of variables in spherical coordinates and obtain their
exact energy eigenvalues and the corresponding eigenfunctions. We give the
quadratic algebra relations satisfied by the integrals of motion. Remarkably
these algebra relations involve the Casimir operators of certain higher rank
Lie algebras in the structure constants.Comment: Latex 12 pages, no figure
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