Sharing storage using dirty vectors

Consider a computation F with n inputs (independent variables) and m outputs (dependent variables) and suppose that we wish to evaluate the Jacobian of F. Automatic differentiation commonly performs this evaluation by associating vector storage either with the program variables (in the case of forward-mode automatic differentiation) or with the adjoint variables (in the case of reverse). Each vector component contains a partial derivative with respect to an independent variable, or a partial derivative of a dependent variable, respectively. The vectors may be full vectors, or they may be dynamically managed sparse data structures. In either case, many of these vectors will be scalar multiples of one another. For example, any intermediate variable produced by a unary operation in the forward mode will have a derivative vector that is a multiple of the derivative for the argument. Any computational graph node that is read just once during its lifetime will have an adjoint vector that is a multiple of the adjoint of the node that reads it. It is frequently wasteful to perform component multiplications explicitly. A scalar multiple of another vector can be replaced by a single multiplicative "scale factor" together with a pointer to the other vector. Automated use of this "dirty vector" technique can save considerable memory management overhead and dramatically reduce the number of floating-point operations required. In particular, dirty vectors often allow shared threads of computation to be reverse-accumulated cheaply. The mechanism permits a number of generalizations, some of which give efficient techniques for preaccumulation

Averages and moments associated to class numbers of imaginary quadratic fields

For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_\ell(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geq 3$.Comment: 26 pages; minor edits to exposition and notation, to agree with published versio

Simultaneous Integer Values of Pairs of Quadratic Forms

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.Comment: 63 page

Investigation of integrating sphere measurement parameters

Directional and bidirectional reflectance of MgO sphere wall coatings, and directional characteristics of photomultiplier tub

Comment on "Summing One-Loop Graphs at Multi-Particle Threshold"

The propagator of a virtual $\phi$-field with emission of $n$ on-mass-shell particles all being exactly at rest is calculated at the tree-level in $\lambda \phi^4$ theory by directly solving recursion equations for the sum of Feynman graphs. It is shown that the generating function for these propagators is equivalent to a Fourier transform of the recently found Green's function within the background-field technique for summing graphs at threshold suggested by Lowell Brown. Also the derivation of the result that the tree-level on-mass-shell scattering amplitudes of the processes $2 \to n$ are exactly vanishing at threshold for $n > 4$ is thus given in the more conventional Feynman diagram technique.Comment: 7 pages, LaTeX, TPI-MINN-92/46-

Exploring the PcycP_{cyc} vs ProtP_{rot} relation with flux transport dynamo models of solar-like stars

Aims: To understand stellar magnetism and to test the validity of the Babcock-Leighton flux transport mean field dynamo models with stellar activity observations Methods: 2-D mean field dynamo models at various rotation rates are computed with the STELEM code to study the sensitivity of the activity cycle period and butterfly diagram to parameter changes and are compared to observational data. The novelty is that these 2-D mean field dynamo models incorporate scaling laws deduced from 3-D hydrodynamical simulations for the influence of rotation rate on the amplitude and profile of the meridional circulation. These models make also use of observational scaling laws for the variation of differential rotation with rotation rate. Results: We find that Babcock-Leighton flux transport dynamo models are able to reproduce the change in topology of the magnetic field (i.e. toward being more toroidal with increasing rotation rate) but seem to have difficulty reproducing the cycle period vs activity period correlation observed in solar-like stars if a monolithic single cell meridional flow is assumed. It may however be possible to recover the $P_{cyc}$ vs $P_{rot}$ relation with more complex meridional flows, if the profile changes in a particular assumed manner with rotation rate. Conclusions: The Babcock-Leighton flux transport dynamo model based on single cell meridional circulation does not reproduce the $P_{cyc}$ vs $P_{rot}$ relation unless the amplitude of the meridional circulation is assumed to increase with rotation rate which seems to be in contradiction with recent results obtained with 3-D global simulations.Comment: 12 pages, 8 figures, accepted for publication by A&A 1: AIM, CEA/DSM-CNRS-Univ. Paris 7, IRFU/SAp, France, 2: D.A.M.T.P., Centre for Mathematical Sciences, Univ. of Cambridge, UK, 3: JILA and Department of Astrophysical and Planetary Sciences, Univ. of Colorado, US

Coulomb blockade in a Si channel gated by an Al single-electron transistor

We incorporate an Al-AlO_x-Al single-electron transistor as the gate of a narrow (~100 nm) metal-oxide-semiconductor field-effect transistor (MOSFET). Near the MOSFET channel conductance threshold, we observe oscillations in the conductance associated with Coulomb blockade in the channel, revealing the formation of a Si single-electron transistor. Abrupt steps present in sweeps of the Al transistor conductance versus gate voltage are correlated with single-electron charging events in the Si transistor, and vice versa. Analysis of these correlations using a simple electrostatic model demonstrates that the two single-electron transistor islands are closely aligned, with an inter-island capacitance approximately equal to 1/3 of the total capacitance of the Si transistor island, indicating that the Si transistor is strongly coupled to the Al transistor.Comment: 3 pages, 4 figures, 1 table; typos corrected, minor clarifications added; published in AP

Attitudes in Physics Education: An Alternative Approach to Teaching Physics to Non-Science College Students

In this article, we present an alternative way of teaching conceptual physics for non-science majors by depicting the role of physics in today\u27s technology. The goal of this approach is to increase in the minds of non-science students the acceptance of physics as a useful component in general education, and as a major tool in comprehending the present-day technological world experienced by students outside the classroom