Are there Hilbert-style Pure Type Systems?

For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements with empty contexts). For intuitionistic logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S. Hilbert-style versions of illative combinatory logic have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two classes. For some PTSs however, including lambda* and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.Comment: Accepted in Logical Methods in Computer Scienc

Ellipsoidal optical reflectors reproduced by electroforming

An accurately dimensioned convex ellipsoidal surface, which will become a master after polishing, is fabricated from 316L stainless steel. When polishing of the master is completed, it is suspended in a modified watt bath for electroforming of nickel reflectors

Conforming polisher for aspheric surface of revolution Patent

Conforming polisher for aspheric surfaces of revolution with inflatable tub

Fixed-Base Simulator Studies of the Ability of the Human Pilot to Provide Energy Management Along Abort and Deep-Space Entry Trajectories

A simulation study has been made to determine a pilot's ability to control a low L/D vehicle to a desired point on the earth with initial conditions ranging from parabolic orbits to abort conditions along the boost phase of a deep-space mission. The program was conducted to develop procedures which would allow the pilot to perform the energy management functions required while avoiding the high deceleration or skipout region and to determine the information display required to aid the pilot in flying these procedures. The abort conditions studied extend from a region of relatively high flight-path angles at suborbital velocities while leaving the atmosphere to a region between orbital and near-escape velocity outside the atmosphere. The conditions studied included guidance from suborbital and superorbital aborts as well as guidance following return from a deepspace mission. In this paper, the role of the human pilot?s ability to combine safe return abort procedures with guidance procedures has been investigated. The range capability from various abort and entry conditions is also presented

Role of Self-Interaction Effects in the Geometry Optimization of Small Metal Clusters

By combining the Self-Interaction Correction (SIC) with pseudopotential perturbation theory, the role of self-interaction errors inherent to the Local Density Approximation (LDA) to Density Functional Theory is estimated in the determination of ground state and low energy isomeric structures of small metallic clusters. Its application to neutral sodium clusters with 8 and 20 atoms shows that the SIC provides sizeable effects in Na_8, leading to a different ordering of the low lying isomeric states compared with ab-initio LDA predictions, whereas for Na_20, the SIC effects are less pronounced, such that a quantitative agreement is achieved between the present method and ab-initio LDA calculations.Comment: RevTeX, 4 pages, 1 figure available from [email protected]

A Bohmian approach to quantum fractals

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory--based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.Comment: 19 pages, 3 figures (revised version

The Tulczyjew triple for classical fields

The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important feature of our approach is that we do not postulate {\it ad hoc} the ingredients of the theory, but obtain them as unavoidable consequences of the variational calculus. This picture of Field Theory is covariant and complete, containing not only the Lagrangian formalism and Euler-Lagrange equations but also the phase space, the phase dynamics and the Hamiltonian formalism. Since the configuration space turns out to be an affine bundle, we have to use affine geometry, in particular the notion of the affine duality. In our formulation, the two maps $\alpha$ and $\beta$ which constitute the Tulczyjew triple are morphisms of double structures of affine-vector bundles. We discuss also the Legendre transformation, i.e. the transition between the Lagrangian and the Hamiltonian formulation of the first-order field theor

Building Middle-Level Mathematics Teachers\u27 Capacities as Teachers and Leaders: The Math in the Middle Institute Partnership

This article describes professional development for middle-level mathematics teachers offered through the Math in the Middle Institute Partnership, a National Science Foundation-funded project to build teachers’ capacities to improve mathematics learning for all students. An overview of the project, including descriptions of its goals and curriculum are provided. Detailed descriptions of two mathematics courses and one pedagogy course are offered. The mathematics courses included here are the introductory course to the Math in the Middle Institute, as well as one of the final math courses of the Institute in which participants apply mathematical knowledge and processes to real-world problems. The pedagogy course features curriculum that enables teachers to acquire an understanding of the nature and purpose of action research, and launches teachers into planning and implementing systematic inquiry in their own mathematics classrooms around topics of their choosing. The varied abilities of teachers, as well as growth in teachers’ mathematical and pedagogical capacities, are represented by several samples of student work provided within the article. In addition, mathematical and pedagogical products of student work are also provided through the project’s URL links. Improving teacher quality is identified as a national need in mathematics education and one many universities and schools across the country are working in partnership to try to address. This article describes a professional development project aimed at improving mathematics teaching and learning in the middle grades. An overview of the project, along with a close look at several of its course offerings, are presented highlighting mathematical and pedagogical goals, challenges, and accomplishments

Imaging the Radio Photospheres of Mira Variables

We have used the VLA at 43 GHz to image the radio continuum emission from o Ceti, R Leo, and W Hya and to precisely locate their SiO maser emission with respect to the star. The radio continuum emission region for all three stars has a diameter close to 5.6 AU. These diameters are similar to those measured at infrared wavelengths in bands containing strong molecular opacity and about twice those measured in line-free regions of the infrared spectrum. Thus, the radio photosphere and the infrared molecular layer appear to be coextensive. The 43 GHz continuum emission is consistent with temperatures near 1600 K and opacity from H-minus free-free interactions. While the continuum image of o Ceti appears nearly circular, both R Leo and W Hya display significant elongations. The SiO masers for all three stars show partial rings with diameters close to 8 AU.Comment: 14 pages; 3 figure