On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and change

DEFINING THE RISK, PREVALENCE, AND PATHOLOGICAL THRESOLD OF LOW RUMINAL pH IN FEELOT CATTLE

The diet transition phase is thought to be the highest risk period for development of low ruminal pH, while pathology associated with low reticulo-ruminal pH (RRpH) induced ruminal acidosis (RA) is often found at slaughter, months after the diet transition. Two experiments were conducted to 1) determine the risk of low RRpH during the transition phase and 2) explore the association of rumen fermentation and acute phase protein response during finishing with pathology identified post mortem. In experiment 1, RRpH was measured in 32 mixed breed steers (n = 16) and heifers (n = 16) housed in commercial feedlot pens with 227 ±13 and 249 ± 6 hd/pen cohort steers and heifers, respectively. Cattle were transitioned from a diet containing 46.5% forage and 53.5% concentrate to a diet containing 9.5% forage and 90.5% concentrate dry matter (DM) basis) over 40 d. In addition, wheat replaced barley as the grain source during the dietary transition. Both mean and minimum RRpH decreased as the proportion of concentrate in the diet increased. The area (duration severity) that RRpH was 180 min), increased with increasing concentrate. Despite having a high risk for low RRpH, most cattle had only 1-3 bouts of low RRpH during the diet transition, and extent was mild. Steers had greater dry matter intake (DMI), lower RRpH, and greater standard deviation of RRpH than heifers, suggesting that susceptibility to RA may differ between steers and heifers. In experiment 2, ruminal pH, short-chain fatty acid concentrations and serum acute phase proteins were measured in 28 cannulated steers during the final 5 wk of finishing when fed a diet containing 5% forage and 95% concentrate (DM basis). Rumen and livers were examined and pathology scores were determined at slaughter. There was no difference in minimum pH, mean pH, or duration that ruminal pH was < 5.5 between steers with or without pathology. However, steers with pathology spent more time with ruminal pH < 5.2 and tended to spend more time with ruminal pH < 5.8. Acetate concentration tended to be greater in steers with pathology than without pathology. Serum amyloid A was greater and haptoglobin tended to be greater in steers with pathology than those without. Overall, liver and rumen pathology was associated with a greater duration that ruminal pH is < 5.2 and a chronic systemic acute phase protein response. In summary, feedlot cattle experience low RRpH during dietary transition and that the risk increases with increasing levels of concentrate. However, during the dietary transition the extent of low RRpH was mild. During the last 5 wk of finishing, the duration that ruminal pH was < 5.2 and the plasma concentration of serum amyloid A, were associated with greater rumen and liver pathology scores, suggesting that low ruminal pH occurring during the latter part of finishing may have an impact on risk for rumenitis and liver abscesses

Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.Comment: 26 page

Statics and dynamics of elastic manifolds in media with long-range correlated disorder

We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects correlated as r^{-a} for large separation r. We derive the functional renormalization-group equations to one-loop order, which allow us to describe the universal properties of the system in equilibrium and at the depinning transition. Using a double epsilon=4-d and delta=4-a expansion, we compute the fixed points characterizing different universality classes and analyze their regions of stability. The long-range disorder-correlator remains analytic but generates short-range disorder whose correlator exhibits the usual cusp. The critical exponents and universal amplitudes are computed to first order in epsilon and delta at the fixed points. At depinning, a velocity-versus-force exponent beta larger than unity can occur. We discuss possible realizations using extended defects.Comment: 16 pages, 11 figures, revtex

Random field spin models beyond one loop: a mechanism for decreasing the lower critical dimension

The functional RG for the random field and random anisotropy O(N) sigma-models is studied to two loop. The ferromagnetic/disordered (F/D) transition fixed point is found to next order in d=4+epsilon for N > N_c (N_c=2.8347408 for random field, N_c=9.44121 for random anisotropy). For N < N_c the lower critical dimension plunges below d=4: we find two fixed points, one describing the quasi-ordered phase, the other is novel and describes the F/D transition. The lower critical dimension can be obtained in an (N_c-N)-expansion. The theory is also analyzed at large N and a glassy regime is found.Comment: 4 pages, 5 figure

Lattice Fluid Dynamics from Perfect Discretizations of Continuum Flows

We use renormalization group methods to derive equations of motion for large scale variables in fluid dynamics. The large scale variables are averages of the underlying continuum variables over cubic volumes, and naturally live on a lattice. The resulting lattice dynamics represents a perfect discretization of continuum physics, i.e. grid artifacts are completely eliminated. Perfect equations of motion are derived for static, slow flows of incompressible, viscous fluids. For Hagen-Poiseuille flow in a channel with square cross section the equations reduce to a perfect discretization of the Poisson equation for the velocity field with Dirichlet boundary conditions. The perfect large scale Poisson equation is used in a numerical simulation, and is shown to represent the continuum flow exactly. For non-square cross sections we use a numerical iterative procedure to derive flow equations that are approximately perfect.Comment: 25 pages, tex., using epsfig, minor changes, refernces adde

Scattering in a Simple 2-d Lattice Model

L\"uscher has suggested a method to determine phase shifts from the finite volume dependence of the two-particle energy spectrum. We apply this to two models in d=2: (a) the Ising model, (b) a system of two Ising fields with different mass and coupled through a 3-point term, both considered in the symmetric phase. The Monte Carlo simulation makes use of the cluster updating and reduced variance operator techniques. For the Ising system we study in particular O($a^2$) effects in the phase shift of the 2-particle scattering process.Comment: 4 p + 2 PS-figures, UNIGRAZ-UTP-21-10-9

Spinor Bose Condensates in Optical Traps

In an optical trap, the ground state of spin-1 Bosons such as $^{23}$Na, $^{39}$K, and $^{87}$Rb can be either a ferromagnetic or a "polar" state, depending on the scattering lengths in different angular momentum channel. The collective modes of these states have very different spin character and spatial distributions. While ordinary vortices are stable in the polar state, only those with unit circulation are stable in the ferromagnetic state. The ferromagnetic state also has coreless (or Skyrmion) vortices like those of superfluid $^{3}$He-A. Current estimates of scattering lengths suggest that the ground states of $^{23}$Na and $^{87}$Rb condensate are a polar state and a ferromagnetic state respectively.Comment: 11 pages, no figures. email : [email protected]

Bouncing wave packets, Ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box

For a particle in a box, the operator is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it

Distribution of velocities and acceleration for a particle in Brownian correlated disorder: inertial case

We study the motion of an elastic object driven in a disordered environment in presence of both dissipation and inertia. We consider random forces with the statistics of random walks and reduce the problem to a single degree of freedom. It is the extension of the mean field ABBM model in presence of an inertial mass m. While the ABBM model can be solved exactly, its extension to inertia exhibits complicated history dependence due to oscillations and backward motion. The characteristic scales for avalanche motion are studied from numerics and qualitative arguments. To make analytical progress we consider two variants which coincide with the original model whenever the particle moves only forward. Using a combination of analytical and numerical methods together with simulations, we characterize the distributions of instantaneous acceleration and velocity, and compare them in these three models. We show that for large driving velocity, all three models share the same large-deviation function for positive velocities, which is obtained analytically for small and large m, as well as for m =6/25. The effect of small additional thermal and quantum fluctuations can be treated within an approximate method.Comment: 42 page