A family of solutions of certain nonautonomous differential equations by series of exponential functions
Solution of differential equations by series of exponential functions
Proctor, T. G.↗
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Exponential estimates and the saddle point property for neutral functional differential equations
Exponential estimates and saddle point property for neutral functional differential equations
Cruz, M. A.↗
Example on how to (intelligently) augment the nuclear-data pipeline with machine learning [Slides]
The presentation discusses how machine learning has helped the Los Alamos National Laboratory (LANL) nuclear-data pipeline. It also discusses the strengths of machine learning as it finds trends in large amounts of data where human brains are overwhelmed and that this information may be crucial to improve our nuclear data. It does stress, however, that machine learning is no "silver bullet" and that it is critical to feed it expert knowledge and use physics intuition to interpret the results. The presentation discusses the need to develop infrastructure and tools to provide data in an easily readable and unambiguously interpretable format (e.g., EXFOR format), to develop experimental data and theory to solve physics questions, and that statisticians and nuclear-data experts must be brought together to correctly interpret the results. The presentation concludes by stating that machine learning is a great tool and that LANL needs to use the algorithms along with developing physics data, tools and infrastructure.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
R-Function Relationships for Application in the Fractional Calculus
The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, e(t), and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, el, in terms of the R-function are developed. Also, some approximations for the R-function are developed.
Lorenzo, Carl F.↗
A Parametric Study of Fine-scale Turbulence Mixing Noise
The present paper is a study of aerodynamic noise spectra from model functions that describe the source. The study is motivated by the need to improve the spectral shape of the MGBK jet noise prediction methodology at high frequency. The predicted spectral shape usually appears less broadband than measurements and faster decaying at high frequency. Theoretical representation of the source is based on Lilley's equation. Numerical simulations of high-speed subsonic jets as well as some recent turbulence measurements reveal a number of interesting statistical properties of turbulence correlation functions that may have a bearing on radiated noise. These studies indicate that an exponential spatial function may be a more appropriate representation of a two-point correlation compared to its Gaussian counterpart. The effect of source non-compactness on spectral shape is discussed. It is shown that source non-compactness could well be the differentiating factor between the Gaussian and exponential model functions. In particular, the fall-off of the noise spectra at high frequency is studied and it is shown that a non-compact source with an exponential model function results in a broader spectrum and better agreement with data. An alternate source model that represents the source as a covariance of the convective derivative of fine-scale turbulence kinetic energy is also examined.
Khavaran, Abbas↗
Finite-analytic numerical method for unsteady two-dimensional Navier-Stokes equations
A finite analytic (FA) numerical solution is developed for unsteady two-dimensional Navier-Stokes equations. The FA method utilizes the analytic solution in a small local element to formulate the algebraic representation of partial differential equations. The combination of linear and exponential functions that satisfy the governing equation is adopted as the boundary function, thereby improving the accuracy of the finite analytic solution. Two flows, one a starting cavity flow and the other a vortex shedding flow behind a rectangular block, are solved by the FA method. The starting square cavity flow is solved for Reynolds number of 400, 1000, and 2000 to show the accuracy and stability of the FA solution. The FA solution for flow over a rectangular block (H x H/4) predicts the Strouhal number for Reynolds numbers of 100 and 500 to be 0.156 and 0.125. Details of the flow patterns are given. In addition to streamlines and vorticity distribution, rest-streamlines are given to illustrate the vortex motion downstream of the block.
Chen, C.-J.↗
Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
Pratt, D. T.↗
A method for the numerical integration of coupled first-order differential equations with greatly different time constants.
Numerical integration of coupled first order ODE OF greatly differing time constant
DIFFERENTIAL EQUATION↗
Parametric Modeling and Economic Analysis of a 2MW th 3-Stream sCO 2 Heat Exchanger
Here, this paper presents the design and cost optimization of a novel 2MW th 3-stream sCO2 plate-fin heat exchanger. This heat exchanger design is unique in that it uses reduced metal oxide particle-to-sCO2 heat exchanger for cost-effective energy storage applications. The design uses low velocity, laminar air as the re-oxidizing reactant to transfer the heat of the re-oxidizing reaction to a sCO2 power loop. The design of the heat exchanger is based on a 2-D, 3-fluid plate/fin heat transfer model. The model parameterizes the size, shape, and number of passages of the heat exchanger to calculate the temperature profile, pressure drop, and fluid velocities of all three fluids. Global heat exchanger parameters such as the effectiveness and total heat transferred to the sCO2 are then calculated for overall performance. Due to the value and increased use of sCO2 heat exchangers in power cycles, a cost model of the system based on the unique high temperature/high pressure operating conditions was created using quotes from reference projects and market analysis. These quoted air-to-sCO2 heat exchangers are then processed using multiple weighting factors pertinent to heat exchanger design, including heat exchanger type, maximum temperature, differential pressures, fluids, duty, and more. These factors are then used in an exponential function in order to generate a parameterized cost curve. The design and cost of the heat exchanger are then optimized using the SMPSO genetic algorithm in Python. The optimization objectives for the system are to maximize the overall system effectiveness, including an air recuperator for preheating, and to minimize unit costs. Additional constraints are added to the system for the sCO2 and air pressure drops, air velocity to reduce particle entrainment, and the length and volume of the heat exchanger.
Cost Model↗
Functional methods for waves in random media
Some basic ideas in functional methods for waves in random media are illustrated through a simple random differential equation. These methods are then generalized to solve certain random parabolic equations via an exponential representation given by the Feynman-Kac formula. It is shown that these functional methods are applicable to a number of problems in random wave propagation. They include the forward-scattering approximation in Gaussian white-noise media; the solution of the optical beam propagation problem by a phase-integral method; the high-frequency scattering by bounded random media, and a derivation of approximate moment equations from the functional integral representation.
Chow, P. L.↗
Functional methods for waves in random media
Some basic ideas in functional methods for waves in random media are illustrated through a simple random differential equation. These methods are then generalized to solve certain random parabolic equations via an exponential representation given by the Feynman-Kac formula. It is shown that these functional methods are applicable to a number of problems in random wave propagation. They include the forward-scattering approximation in Gaussian white-noise media; the solution of the optical beam propagation problem by a phase-integral method; the high-frequency scattering by bounded random media; and a derivation of approximate moment equations from the functional integral representation.
Chow, P. L.↗
X-ray Diffraction of Water in Polyvinylpyrrolidone
PVP is a hydrophilic polymer commonly used as an excipient in pharmaceutical formulations. Here we have performed time-resolved high-energy X-ray scattering experiments on pellets of PVP at different humidity conditions for 1-2 days. A two-phase exponential decay in water sorption is found with a peak in the differential pair distribution function at 2.85 Å, which is attributed to the average (hydrogen bonded) carbonyl oxygen-water oxygen distance. Additional scattering measurements on powders with fixed compositions ranging from 2 to 12.3 wt % H 2 O were modeled with Empirical Potential Structure Refinement (EPSR). Further, the models reveal approximately linear relations between the carbonyl oxygen-water oxygen coordination number (nO C - O W ) and the water oxygen-water oxygen coordination number (nO W - O W ) versus water content in PVP. A stronger preference for water-water hydrogen bonding over carbonyl-water bonding is found. At all the concentrations studied the majority of water molecules were found to be randomly isolated, but a wide distribution of coordination environments of water molecules is found within the PVP polymer strands at the highest concentrations. Overall, the EPSR models indicate a continuous evolution in structure versus water content with nO W - O W =1 occurring at similar to 12 wt % H 2 O, i.e., the composition where, on average, each watermolecule is surrounded by one other water molecule.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
New Mathematical Functions for Vacuum System Analysis
A new bivariate function has been found that provides solutions of integrals having the form u (sup minus eta) e (sup u) du which arise when developing predictions for the behavior of pressure within a rigid volume under high vacuum conditions in the presence of venting as well as sources characterized by power law transient decay over the range [0,1] for eta and for u greater than or equal to 0. A few properties of the new function are explored in this work. For instance the eta equals 1/2 case reproduces the Dawson function. In addition, a slight variation of the solution technique reproduces the exponential integral for eta equals 1. The technique used to generate these functions leads to an approach for solving a more general class of nonlinear ordinary differential equations, with the potential for identifying other new functions that solve other integrals.
Dawson Function↗
Measurement of the central exclusive production of charged particle pairs in proton-proton collisions at $\sqrt{s}$ = 200 GeV with the STAR detector at RHIC
We report on the measurement of the Central Exclusive Production of charged particle pairs h+h- (h = π, K, p) with the STAR detector at RHIC in proton-proton collisions at $\sqrt{s}$ = 200 GeV. The charged particle pairs produced in the reaction pp → p' + h + h - + p' are reconstructed from the tracks in the central detector and identified using the specific energy loss and the time of flight method, while the forward-scattered protons are measured in the Roman Pot system. Exclusivity of the event is guaranteed by requiring the transverse momentum balance of all four final-state particles. Differential cross sections are measured as functions of observables related to the central hadronic final state and to the forward-scattered protons. They are measured in a fiducial region corresponding to the acceptance of the STAR detector and determined by the central particles’ transverse momenta and pseudorapidities as well as by the forward-scattered protons’ momenta. This fiducial region roughly corresponds to the square of the four-momentum transfers at the proton vertices in the range 0.04 GeV 2 < -t 1 , -t 2 < 0.2 GeV 2 , invariant masses of the charged particle pairs up to a few GeV and pseudorapidities of the centrally-produced hadrons in the range |η| < 0.7. The measured cross sections are compared to phenomenological predictions based on the Double Pomeron Exchange (DPE) model. Structures observed in the mass spectra of π + π - and K + K - pairs are consistent with the DPE model, while angular distributions of pions suggest a dominant spin-0 contribution to π + π - production. For π + π - production, the fiducial cross section is extrapolated to the Lorentz-invariant region, which allows decomposition of the invariant mass spectrum into continuum and resonant contributions. The extrapolated cross section is well described by the continuum production and at least three resonances, the f 0 (980), f 2 (1270) and f 0 (1500), with a possible small contribution from the f 0 (1370). Fits to the extrapolated differential cross section as a function of t 1 and t 2 enable extraction of the exponential slope parameters in several bins of the invariant mass of π + π - pairs. These parameters are sensitive to the size of the interaction region.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Modeling of aircraft unsteady aerodynamic characteristics. Part 1: Postulated models
A short theoretical study of aircraft aerodynamic model equations with unsteady effects is presented. The aerodynamic forces and moments are expressed in terms of indicial functions or internal state variables. The first representation leads to aircraft integro-differential equations of motion; the second preserves the state-space form of the model equations. The formulations of unsteady aerodynamics is applied in two examples. The first example deals with a one-degree-of-freedom harmonic motion about one of the aircraft body axes. In the second example, the equations for longitudinal short-period motion are developed. In these examples, only linear aerodynamic terms are considered. The indicial functions are postulated as simple exponentials and the internal state variables are governed by linear, time-invariant, first-order differential equations. It is shown that both approaches to the modeling of unsteady aerodynamics lead to identical models.
Klein, Vladislav↗
Uplift and Seismicity Driven by Magmatic Inflation at Sierra Negra Volcano, Galápagos Islands
Abstract Although episodes of surface uplift and elevated seismicity precede many volcanic eruptions, their temporal evolution is often complex, and apparently in contradiction to simple trends predicted by mechanical deformation models. Here, we use continuous global positioning system and seismic data recorded at Sierra Negra volcano, Galápagos Islands, to show how the edifice responded to stress changes driven by magma accumulation in a shallow sill. The rate of uplift varied during the 13 years and 6.5 m of inflation before the 2018 eruption. The number of earthquakes per unit of uplift increased exponentially with total uplift as the differential stress increased. Accordingly, the temporal seismicity rate varied in time as a function of both the total uplift and the uplift rate. The Gutenberg‐Richter b ‐value decreased as a function of total uplift. In the final six months before the eruption, a sequence of large ( M > 4) earthquakes regulated the state of stress on the fault, each being followed by 2–3 days of postseismic quiescence, and retarding the increase in seismicity rate. These earthquakes did not affect the overall uplift rate. Subsidence of 8.5 m accompanied the 2‐month eruption. On resumption of uplift, the number of earthquakes per unit of uplift was very low, and the b ‐value high, reflecting the relaxed stress state of the fault system. These observations show that crustal deformation becomes increasingly brittle at higher stress states, and supports theoretical models based on elastic‐brittle mechanics. They suggest that joint interpretation of deformation and seismicity is key for forecasting future eruptions in similar volcanic settings.
Bell, Andrew Forbes↗
Modulation of lens cell adhesion molecules by particle beams
Cell adhesion molecules (CAMs) are proteins which anchor cells to each other and to the extracellular matrix (ECM), but whose functions also include signal transduction, differentiation, and apoptosis. We are testing a hypothesis that particle radiations modulate CAM expression and this contributes to radiation-induced lens opacification. We observed dose-dependent changes in the expression of beta 1-integrin and ICAM-1 in exponentially-growing and confluent cells of a differentiating human lens epithelial cell model after exposure to particle beams. Human lens epithelial (HLE) cells, less than 10 passages after their initial culture from fetal tissue, were grown on bovine corneal endothelial cell-derived ECM in medium containing 15% fetal bovine serum and supplemented with 5 ng/ml basic fibroblast growth factor (FGF-2). Multiple cell populations at three different stages of differentiation were prepared for experiment: cells in exponential growth, and cells at 5 and 10 days post-confluence. The differentiation status of cells was characterized morphologically by digital image analysis, and biochemically by Western blotting using lens epithelial and fiber cell-specific markers. Cultures were irradiated with single doses (4, 8 or 12 Gy) of 55 MeV protons and, along with unirradiated control samples, were fixed using -20 degrees C methanol at 6 hours after exposure. Replicate experiments and similar experiments with helium ions are in progress. The intracellular localization of beta 1-integrin and ICAM-1 was detected by immunofluorescence using monoclonal antibodies specific for each CAM. Cells known to express each CAM were also processed as positive controls. Both exponentially-growing and confluent, differentiating cells demonstrated a dramatic proton-dose-dependent modulation (upregulation for exponential cells, downregulation for confluent cells) and a change in the intracellular distribution of the beta 1-integrin, compared to unirradiated controls. In contrast, there was a dose-dependent increase in ICAM-1 immunofluorescence in confluent, but not exponentially-growing cells. These results suggest that proton irradiation downregulates beta 1-integrin and upregulates ICAM-1, potentially contributing to cell death or to aberrant differentiation via modulation of anchorage and/or signal transduction functions. Quantification of the expression levels of the CAMs by Western analysis is in progress.
NASA Discipline Radiation Health↗
Spectral scheme for atomic structure calculations in density functional theory
In this study, we present a spectral scheme for atomic structure calculations in pseudopotential Kohn-Sham density functional theory. In particular, after applying an exponential transformation of the radial coordinates, we employ global polynomial interpolation on a Chebyshev grid, with derivative operators approximated using the Chebyshev differentiation matrix, and integrations using Clenshaw-Curtis quadrature. We demonstrate the accuracy and efficiency of the scheme through spin-polarized and unpolarized calculations for representative atoms, while considering local, semilocal, and hybrid exchange-correlation functionals. In particular, we find that $\mathcal{O}$(200) grid points are sufficient to achieve an accuracy of 1 microhartree in the eigenvalues for optimized norm conserving Vanderbilt pseudopotentials spanning the periodic table from atomic number Ζ = 1 to 83.
74 ATOMIC AND MOLECULAR PHYSICS↗