An integral transform with a Whittaker function kernel Interim report, 20 Jun. - 19 Aug. 1968
Integral transformation with Whittaker kernel functions
Wimp, J.↗
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Charged-Particle Bound States in Periodic Boxes
We consider the binding energy of a two-body system with a repulsive Coulomb interaction in a finite periodic volume. We define the finite-volume Coulomb potential as the usual Coulomb potential, except that the distance is defined the shortest separation between the two bodies in the periodic volume. We investigate this problem in one and three-dimensional periodic boxes and derive the asymptotic behavior of the volume dependence for bound states with zero angular momentum in terms of Whittaker functions. We benchmark our results against numerical calculations and show how the method can be used to extract asymptotic normalization coefficients for charged-particle bound states. Furthermore, the results we derive here have immediate applications for calculations of atomic nuclei in finite periodic volumes for the case where the leading finite-volume correction is associated with two charged clusters.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Longitudinal dynamic stability of a shuttle vehicle.
Analytical study of the longitudinal dynamic stability of a nonrolling, lifting vehicle gliding at hypersonic speeds. The analysis applies to shuttle vehicles designed for operating up to the rim of a planetary atmosphere. A general nondimensional time transformation is introduced to derive a unified second-order linear differential equation for the angle of attack, valid for all types of reentry of a general type of vehicle. The stability of motion is discussed for two fundamental regimes of flight that are based on widely different assumptions. For near ballistic entry along a straight line trajectory, the equation reduces to a confluent hypergeometric equation, the solution of which can be expressed in terms of Whittaker's function. Using a theorem in the theory of stability of differential equations, criteria for damped oscillations are derived. It is shown that the aerodynamic criteria for stability are the same as for the case of ballistic entry. In addition, for each vehicle configuration, and specified planetary atmosphere, there exists an altitude range where the angle of attack frequency is nearly equal to the orbital frequency causing instability in pitch. This resonance instability is due to the ellipticity of the orbit. Criteria for eccentricity instability are derived.
Vinh, N. X.↗
Acoustic resonances and sound scattering by a shear layer
The reflection and transmission characteristics of plane waves scattered by a finite-thickness shear layer having a linear velocity profile and bounded by two otherwise uniform parallel flows is examined using the pressure perturbation equation solutions in the shear layer shown previously to be in terms of Whittaker M functions. In addition to the angle of plane wave incidence and the relative Mach number of the flows bounding the shear layer, it is found that the scattering properties of the shear layer depend crucially upon a parameter tau in such a manner that the case tau approaching zero characterizes the long wavelength properties of the layer and the case tau approaching infinity characterizes the short wavelength properties of the layer. In contrast to the region of ordinary reflection in the cases where the corresponding vortex sheet does not have a Brewster angle, the values of the reflection coefficient up to tau of 2 follow those of the vortex sheet; for the case for which the corresponding vortex sheet has a Brewster angle, the magnitude of the reflection coefficient may be sensitive to even small changes in tau in certain cases.
Koutsoyannis, S. P.↗
Characterization of acoustic disturbances in linearly sheared flows
The equation describing the plane wave propagation, the stability, or the rectangular duct mode characteristics in a compressible inviscid linearly sheared parallel, but otherwise homogeneous flow, is shown to be reducible to Whittaker's equation. The resulting solutions, which are real, viewed as functions of two variables, depend on a parameter and an argument, the values of which have precise physical meanings depending on the problem. The exact solutions in terms of Whittaker functions are used to obtain a number of known results of plane wave propagation and stability in linearly sheared flows as limiting cases in which the speed of sound goes to infinity (incompressible limit) or the shear layer thickness, or wave number, goes to zero (vortex sheet limit). The usefulness of the exact solutions is then discussed in connection with the problems of plane wave propagation and the stability of a finite thickness layer with a linear velocity profile.
Koutsoyannis, S. P.↗
Features of sound propagation through and stability of a finite shear layer
The plane wave propagation, the stability and the rectangular duct mode problems of a compressible inviscid linearly sheared parallel, but otherwise homogeneous flow, are shown to be governed by Whittaker's equation. The exact solutions for the perturbation quantities are essentially Whittaker M-functions. A number of known results are obtained as limiting cases of exact solutions. For the compressible finite thickness shear layer it is shown that no resonances and no critical angles exist for all Mach numbers, frequencies and shear layer velocity profile slopes except in the singular case of the vortex sheet.
Koutsoyannis, S. P.↗
Features of sound propagation through and stability of a finite shear layer
The plane wave propagation, the stability, and the rectangular duct mode problems of a compressible, inviscid, linearly sheared, parallel, homogeneous flow are shown to be governed by Whittaker's equation. The exact solutions for the perturbation quantities are essentially the Whittaker M-functions where the nondimensional quantities have precise physical meanings. A number of known results are obtained as limiting cases of the exact solutions. For the compressible finite thickness shear layer it is shown that no resonances and no critical angles exist for all Mach numbers, frequencies, and shear layer velocity profile slopes except in the singular case of the vortex sheet.
Koutsoyannis, S. P.↗
Multigrid-sinc methods
A Galerkin method using Whittaker cardinal or 'sinc' functions as basis functions is described for the solution of boundary-value problems. When the solution is analytic in the interior of the domain, the error of approximation using 2N + 1 points is O(e exp /-gamma sq rt N/) even if derivatives of the solution are singular at the boundaries. A multigrid method with overall complexity O(N log N) is used to solve the discrete equations. This paper contains a description of the multigrid-sinc algorithm along with some preliminary numerical results for two-point boundary-value problems.
Schaffer, Steve↗
Sampling Theorem in Terms of the Bandwidth and Sampling Interval
An approach has been developed for interpolating non-uniformly sampled data, with applications in signal and image reconstruction. This innovation generalizes the Whittaker-Shannon sampling theorem by emphasizing two assumptions explicitly (definition of a band-limited function and construction by periodic extension). The Whittaker- Shannon sampling theorem is thus expressed in terms of two fundamental length scales that are derived from these assumptions. The result is more general than what is usually reported, and contains the Whittaker- Shannon form as a special case corresponding to Nyquist-sampled data. The approach also shows that the preferred basis set for interpolation is found by varying the frequency component of the basis functions in an optimal way.
Dean, Bruce H.↗
Water-Searchers: A Reconfigurable and Self Sustaining Army of Subsurface Exploration Robots Searching for Water/Ice Using Multiple Sensors
Perhaps the most promising site for extant life on Mars today is where subsurface water has been maintained. Therefore, searching for underground water will provide a good chance to find evidence of life on Mars. The following are scientific/engineering questions that we want to answer using our approach: (1) Is there subsurface water/ice? How deep is it? How much is there? Is it frozen? (2) What kinds of underground layers exist in the Martian crust? (3) What is the density of Martian soil or regolith? Can we dig into it? Should we drill into it? (4) Can a sudden release of underground water occur if a big asteroid hits Mars? Our approach provides essential information to answer these questions. Moreover, dependence on the water content and depth in soil, not only resultant scientific conclusions but also proper digging/drilling methods, are suggested. 'How much water is in the Martian soil?' There can be several possibilities: (1) high water content that is enough to form permafrost; (2) low water content that is not enough to form permafrost; or (3) different layers with different moisture contents. 'How deep should a rover dig into soil to find water/ice?' The exact size-frequency distribution has not been measured for the soil particles. On-board sensors can provide not only the water content but also the density (or porosity) of Martian soil as a function of depth.
Youk, G. U.↗
Sampling and Reconstruction of the Pupil and Electric Field for Phase Retrieval
This technology is based on sampling considerations for a band-limited function, which has application to optical estimation generally, and to phase retrieval specifically. The analysis begins with the observation that the Fourier transform of an optical aperture function (pupil) can be implemented with minimal aliasing for Q values down to Q = 1. The sampling ratio, Q, is defined as the ratio of the sampling frequency to the band-limited cut-off frequency. The analytical results are given using a 1-d aperture function, and with the electric field defined by the band-limited sinc(x) function. Perfect reconstruction of the Fourier transform (electric field) is derived using the Whittaker-Shannon sampling theorem for 1<Q<2. The Fourier transform is constructed by periodic extension, i.e., by spacing copies of the transform in a definite way, recognizing that no aliasing occurs for values of the sampling ratio such that 1<Q<2, which can be used to advantage in the application of phase retrieval estimation. A method was developed for propagating the electro magnetic field with no aliasing, which has been extended to 2-d optical apertures.
Dean, Bruce↗
Design of a walking robot
Carnegie Mellon University's Autonomous Planetary Exploration Program (APEX) is currently building the Daedalus robot; a system capable of performing extended autonomous planetary exploration missions. Extended autonomy is an important capability because the continued exploration of the Moon, Mars and other solid bodies within the solar system will probably be carried out by autonomous robotic systems. There are a number of reasons for this - the most important of which are the high cost of placing a man in space, the high risk associated with human exploration and communication delays that make teleoperation infeasible. The Daedalus robot represents an evolutionary approach to robot mechanism design and software system architecture. Daedalus incorporates key features from a number of predecessor systems. Using previously proven technologies, the Apex project endeavors to encompass all of the capabilities necessary for robust planetary exploration. The Ambler, a six-legged walking machine was developed by CMU for demonstration of technologies required for planetary exploration. In its five years of life, the Ambler project brought major breakthroughs in various areas of robotic technology. Significant progress was made in: mechanism and control, by introducing a novel gait pattern (circulating gait) and use of orthogonal legs; perception, by developing sophisticated algorithms for map building; and planning, by developing and implementing the Task Control Architecture to coordinate tasks and control complex system functions. The APEX project is the successor of the Ambler project.
Whittaker, William↗
Resolving buried optoelectronic features in metal halide perovskites via modulation spectroscopy studies
As research on both bulk and low dimensional metal halide perovskites (MHPs) continues to grow, the tools necessary to gain insights into their exotic and highly convoluted optoelectronic features must also expand. Electroabsorption (EA) is a modulation spectroscopy technique that is exceptionally valuable at deconvoluting overlapping spectral features as well as discerning between different electronic environments in semiconductors. This review outlines the critical shortcomings of traditional optical spectroscopy in determining essential properties such as the bandgap and exciton binding energy. In this report we provide a brief overview of the working principles of EA, focusing on the differences between bound and unbound charge carriers that result in unique behavior when interacting with an applied electric field. We discuss key studies highlighting the unique characterization available when coupling EA spectroscopy with traditional steady-state optical tools to investigate interesting and complex optical features in MHPs. We conclude that EA is a crucial tool for pushing the boundaries of our understanding of the optoelectronic properties of MHP thin films. When other optical characterization methods fall short in the richness of data provided, EA can bridge the knowledge gap to provide a complete optoelectronic characterization of novel MHPs and deepen the understanding of current structures and compositions as a function of their structural and morphological dimensionality.
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
A workflow to assess the efficacy of brine extraction for managing injection-induced seismicity potential using data from a CO 2 injection site near Decatur, Illinois
Injection of CO 2 for storage in a geologic formation increases pore pressure and alters in situ stresses. Depending on the orientation of any existing fault and fracture planes, such as critically stressed planes, this stress alteration will modify normal stresses acting on planes and could result in frictional sliding and release stored energy in the form of seismicity. Brine extraction (BE) is a technique that can be applied prior to, or during, CO 2 injection to reduce pore pressure for increasing storage capacity and, potentially, for reducing the likelihood of frictional sliding. Here a workflow is described to assess the efficacy of BE for mitigating frictional sliding (i.e., seismicity) during injection and entails: site characterization, stress calculations and failure assessment, static and dynamic modeling, and BE operational planning. Site characterization describes the stress field used to calculate the Coulomb Failure Function (CFF) that constrains allowable pore pressure changes and injection rates in the numerical simulation of CO 2 injection scenarios. The inclusion of BE in the workflow allows for determination of the potential need for pressure reduction, and evaluation of the effectiveness of this operation. Example application of the workflow using an injection field dataset near Decatur, IL, provides insight on fracture planes and stresses at the site, formation properties and the impact of variable CO 2 injection-rate targets on whether BE plans are required. The study workflow indicates that BE could enhance CO 2 injection rate by 39% and correspondingly reduce the potential for injection-induced seismicity as indicated by a reduction in CFF.
58 GEOSCIENCES↗
Basic statistical analyses of candidate nickel-hydrogen cells for the Space Station Freedom
Nickel-Hydrogen (Ni/H2) secondary batteries will be implemented as a power source for the Space Station Freedom as well as for other NASA missions. Consequently, characterization tests of Ni/H2 cells from Eagle-Picher, Whittaker-Yardney, and Hughes were completed at the NASA Lewis Research Center. Watt-hour efficiencies of each Ni/H2 cell were measured for regulated charge and discharge cycles as a function of temperature, charge rate, discharge rate, and state of charge. Temperatures ranged from -5 C to 30 C, charge rates ranged from C/10 to 1C, discharge rates ranged from C/10 to 2C, and states of charge ranged from 20 percent to 100 percent. Results from regression analyses and analyses of mean watt-hour efficiencies demonstrated that overall performance was best at temperatures between 10 C and 20 C while the discharge rate correlated most strongly with watt-hour efficiency. In general, the cell with back-to-back electrode arrangement, single stack, 26 percent KOH, and serrated zircar separator and the cell with a recirculating electrode arrangement, unit stack, 31 percent KOH, zircar separators performed best.
Maloney, Thomas M.↗
Establishing Self-Dopant Design Principles from Structure–Function Relationships in Self-n-Doped Perylene Diimide Organic Semiconductors
Self-doping is a particular doping method that has been applied to a wide range of organic semiconductors. However, there is a lack of understanding regarding the relationship between dopant structure and function. A structurally diverse series of self-n-doped perylene diimides (PDIs) is investigated to study the impact of steric encumbrance, counterion selection, and dopant/PDI tether distance on functional parameters such as doping, stability, morphology, and charge-carrier mobility. The studies show that self-n-doping is best enabled by the use of sterically encumbered ammoniums with short tethers and Lewis basic counterions. Additionally, water is found to inhibit doping, which concludes that thermal degradation is merely a phenomenological feature of certain dopants, and that residual solvent evaporation is the primary driver of thermally activated doping. In situ grazing-incidence wide-angle X-ray scattering studies show that sample annealing increases the π–π stacking distance and shrinks grain boundaries for improved long-range ordering. These features are then correlated to contactless carrier-mobility measurements with time-resolved microwave conductivity before and after thermal annealing. In conclusion, the collective relationships between structural features and functionality are finally used to establish explicit self-n-dopant design principles for the future design of materials with improved functionality.
36 MATERIALS SCIENCE↗
Analysis of the molecular determinants for furin cleavage of the spike protein S1/S2 site in defined strains of the prototype coronavirus murine hepatitis virus (MHV)
We analyzed the spike protein S1/S2 cleavage of selected strains of a prototype coronavirus, mouse hepatitis virus (MHV) by the cellular protease furin, in order to understand the structural requirements underlying the sequence selectivity of the scissile segment. The probability of cleavage of selected MHV strains was first evaluated from furin cleavage scores predicted by the ProP computer software, and then cleavage was measured experimentally with a fluorogenic peptide cleavage assay consisting of S1/S2 peptide mimics and purified furin. We found that in vitro cleavability varied across MHV strains in line with predicted results—but with the notable exception of MHV-A59, which was not cleaved despite a high score predicted for its sequence. Using the known X-Ray structure of furin in complex with a substrate-like inhibitor as an initial structural reference, we carried out molecular dynamics (MD) simulations to learn the modes of binding of the peptides in the furin active site, and the suitability of the complex for initiation of the enzymatic cleavage. We identified the 3D structural requirements of the furin active site configuration that enable bound peptides to undergo cleavage, and the way in which the various strains tested experimentally are fulfilling these requirements. We find that despite some flexibility in the organization of the peptide bound to the active site of the enzyme, the presence of a histidine at P2 of MHV-A59 fails to properly orient the sidechain of His194 of the furin catalytic triad and therefore produces a distortion that renders the peptide/complex structural configuration in the active site incompatible with requirements for cleavage initiation. The Ser/Thr in P1 of MHV-2 and MHV-S has a similar effect of distorting the conformation of the furin active site residues produced by the elimination of the canonical salt-bridge formed by arginine in P1 position. This work informs a study of coronavirus infection and pathogenesis with respect to the function of the viral spike protein, and suggests an important process of viral adaptation and evolution within the spike S1/S2 structural loop.
60 APPLIED LIFE SCIENCES↗
Optimal trajectories for hypersonic launch vehicles
In this paper, we derive a near-optimal guidance law for the ascent trajectory from earth surface to earth orbit of a hypersonic, dual-mode propulsion, lifting vehicle. Of interest are both the optical flight path and the optimal operation of the propulsion system. The guidance law is developed from the energy-state approximation of the equations of motion. Because liquid hydrogen fueled hypersonic aircraft are volume sensitive, as well as weight sensitive, the cost functional is a weighted sum of fuel mass and volume; the weighting factor is chosen to minimize gross take-off weight for a given payload mass and volume in orbit.
Ardema, Mark D.↗