Fourier Series Formalization in ACL2(r)

We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework for formally evaluating definite integrals of real-valued, continuous functions using the Second Fundamental Theorem of Calculus. Our extended framework is also applied to functions containing free arguments. Using this framework, we are able to prove the orthogonality relationships between trigonometric functions, which are the essential properties in Fourier series analysis. The sum rule for definite integrals of indexed sums is also formalized by applying the extended framework along with the First Fundamental Theorem of Calculus and the sum rule for differentiation. The Fourier coefficient formulas of periodic functions are then formalized from the orthogonality relations and the sum rule for integration. Consequently, the uniqueness of Fourier sums is a straightforward corollary. We also present our formalization of the sum rule for definite integrals of infinite series in ACL2(r). Part of this task is to prove the Dini Uniform Convergence Theorem and the continuity of a limit function under certain conditions. A key technique in our proofs of these theorems is to apply the overspill principle from non-standard analysis.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

Report on the formal specification and partial verification of the VIPER microprocessor

The formal specification and partial verification of the VIPER microprocessor is reviewed. The VIPER microprocessor was designed by RSRE, Malvern, England, for safety critical computing applications (e.g., aircraft, reactor control, medical instruments, armaments). The VIPER was carefully specified and partially verified in an attempt to provide a microprocessor with completely predictable operating characteristics. The specification of VIPER is divided into several levels of abstraction, from a gate-level description up to an instruction execution model. Although the consistency between certain levels was demonstrated with mechanically-assisted mathematical proof, the formal verification of VIPER was never completed

Texas Forestry Paper No. 22

Some site, stand, and tree effects on diameter growth of east texas pine treeshttps://scholarworks.sfasu.edu/texas_forestry_papers/1010/thumbnail.jp

Effects of rotation and sloping terrain on fronts of density current fronts

The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity f/2 is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán–Benjamin gravity current front at tF=0. It is shown how, on a time-scale of order 1/f, as a result of ageostrophic dynamics, the slope and front speed UF are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to O(NH0), where N=g′/H0−−−−−√ is the buoyancy frequency, and g′=gΔρ/ρ0 is the reduced acceleration due to gravity. Here ρ0 is the density and Δρ and H0 are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope σ on the bottom boundary shows that, without rotation, UF has a maximum value for σ=\upi/6, while with rotation, UF tends to zero on any slope. For the asymptotic stage when ftF≫1, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a ‘balanced’ component satisfying the Margules geostrophic relation and an equally large ‘unbalanced’ component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale f−1, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio \it Bu−−−−−√=LR/R0, where LR=NH0/f is the Rossby deformation radius and R0 is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed UF from halting sharply at ftF∼1. Depending on the initial value of LR/R0, physical arguments show that UF decreases slowly in proportion to (ftF)−1/2, i.e. UF/UF0=F(ftF,\it Bu). Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as ftF→∞) for finite values of LR/R0. However, as LR/R0→0, it reaches this state when ftF∼1. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier–Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency f and the slow growth of the radius, when ftF≥1, are consistent with recent experiments

Multiparameter vision testing apparatus

Compact vision testing apparatus is described for testing a large number of physiological characteristics of the eyes and visual system of a human subject. The head of the subject is inserted into a viewing port at one end of a light-tight housing containing various optical assemblies. Visual acuity and other refractive characteristics and ocular muscle balance characteristics of the eyes of the subject are tested by means of a retractable phoroptor assembly carried near the viewing port and a film cassette unit carried in the rearward portion of the housing (the latter selectively providing a variety of different visual targets which are viewed through the optical system of the phoroptor assembly). The visual dark adaptation characteristics and absolute brightness threshold of the subject are tested by means of a projector assembly which selectively projects one or both of a variable intensity fixation target and a variable intensity adaptation test field onto a viewing screen located near the top of the housing

Management of Severe Asthma

In this summary of therapy for severe asthma there is no mention of etiologic factors. The author presumes that if infection is the primary factor in the progression to severe asthma that this will be recognized and appropriately treated. A chest x-ray must be an initial laboratory study for the recognition of pneumonia or complicating pneumothorax. However, unlike respiratory failure due to emphysema, here the specific therapy of the altered pulmonary physiology is of paramount importance

Non-Ventilator Management of Respiratory Failure: The Ventimask

This paper will detail the conservative management of acute respiratory failure in patients with chronic respiratory failure due to chronic bronchitis and emphysema. It is important to recognize that this is a very specific group of patients. They have had a chronic hypoxia and hypercarbia for months or years preceding their current episode of acute respiratory failure

Solid metabolic waste transport and stowage investigation

The basic Waste Collection System (WCS) design under consideration utilized air flow to separate the stool from the WCS user and to transport the fecal material to a slinger device for subsequent deposition on a storage bowel. The major parameters governing stool separation and transport were found to be the area of the air inlet orifices, the configuration of the air inlet orifice and the transport air flow. Separation force and transport velocity of the stool were studied. The developed inlet orifice configuration was found to be an effective design for providing fecal separation and transport. Simulated urine tests and female user tests in zero gravity established air flow rates between 0.08 and 0.25 cu sm/min (3 and 9 scfm) as satisfactory for entrapment, containment and transport of urine using an urinal. The investigation of air drying of fecal material as a substitute for vacuum drying in a WCS breadboard system showed that using baseline conditions anticipated for the shuttle cabin ambient atmosphere, flow rates of 0.14 cu sm/min (5 cfm) were adequate for drying and maintaining biological stability of the fecal material