Newtonian analogues of the trigonometric and exponential functions.
Newtonian analogues of trigonometric and exponential functions studied by series
NEWTON THEORY↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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The dynamics of photoinduced defect creation in amorphous chalcogenides: The origin of the stretched exponential function
The article discusses the dynamics of photoinduced defect creations (PDC) in amorphous chalcogenides, which is described by the stretched exponential function (SEF), while the well known photodarkening (PD) and photoinduced volume expansion (PVE) are governed only by the exponential function. It is shown that the exponential distribution of the thermal activation barrier produces the SEF in PDC, suggesting that thermal energy, as well as photon energy, is incorporated in PDC mechanisms. The differences in dynamics among three major photoinduced effects (PD, PVE, and PDC) in amorphous chalcogenides are now well understood.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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.↗
Calculating the divided differences of the exponential function by addition and removal of inputs
We introduce a method for calculating the divided differences of the exponential function by means of addition and removal of items from the input list to the function. Our technique exploits a new identity related to divided differences recently derived by F. Zivcovich. We show that upon adding an item to or removing an item from the input list of an already evaluated exponential, the re-evaluation of the divided differences can be done with only O(sn) floating point operations and O(sn) bytes of memory, where [z 0 ,...,z n ] are the inputs and s ∝ max i,j |z i – z j |. We demonstrate our algorithm’s ability to deal with input lists that are orders-of-magnitude longer than the maximal capacities of the current state-of-the-art. Here, we discuss in detail one practical application of our method: the efficient calculation of weights in the off-diagonal series expansion quantum Monte Carlo algorithm.
97 MATHEMATICS AND COMPUTING↗
Small Deviation Probabilities of a Sum of Independent Positive Random Variables, the Common Distribution of Which Decreases at Zero Not Faster Than Exponential Function
We investigate small deviation probabilities of the cumulative sum of independent positive random variables, the common distribution of which decreases at zero not faster than exponential function.
97 MATHEMATICS AND COMPUTING↗
Curve fitting of aeroelastic transient response data with exponential functions
The extraction of frequency, damping, amplitude, and phase information from unforced transient response data is considered. These quantities are obtained from the parameters determined by fitting the digitized time-history data in a least-squares sense with complex exponential functions. The highlights of the method are described, and the results of several test cases are presented. The effects of noise are considered both by using analytical examples with random noise and by estimating the standard deviation of the parameters from maximum-likelihood theory.
Bennett, R. M.↗
Recovery of Spaceflight-induced Bone Loss: Bone Mineral Density after Long-Duration Missions as Fitted with an Exponential Function
The loss of bone mineral in NASA astronauts during spaceflight has been investigated throughout the more than 40 years of space travel. Consequently, it is a medical requirement at NASA Johnson Space Center (JSC) that changes in bone mass be monitored in crew members by measuring bone mineral density (BMD) with dual-energy x-ray absorptiometry (DXA) before and after flight on astronauts who serve on long-duration missions (4-6 months). We evaluated this repository of medical data to track whether there is recovery of bone mineral that was lost during spaceflight. Our analysis was supplemented by BMD data from cosmonauts ( by convention, a space traveler formally employed by the Russia Aviation and Space Agency or by the previous Soviet Union) who had also flown on long-duration missions. Data from a total of 45 individual crew members -- a small number of whom flew on more than one mission -- were used in this analysis. Changes in BMD (between 56 different sets of pre- and postflight measurements) were plotted as a function of time (days after landing). Plotted BMD changes were fitted to an exponential mathematical function that estimated: i) BMD change on landing day (day 0) and ii) the number of days after landing when 50% of the lost bone would be recovered ("50% recovery time") in the lumbar spine, trochanter, pelvis, femoral neck and calcaneus. In sum, averaged losses of bone mineral after long-duration spaceflight ranged between 2-9% across all sites with our recovery model predicting a 50% restoration of bone loss for all sites to be within 9 months.
Sibonga, J. D.↗
Time-resolved photoluminescence spectroscopy of localized exciton magnetic polarons in Cd{sub 0.70}Mn{sub 0.30}Te spin glass compound
We have investigated dynamics of different localized exciton magnetic polarons (LEMPs) in Cd{sub 0.70}Mn{sub 0.30}Te spin glass (SG) compound below the freezing temperature T{sub f} in the crystal regions, where various microscopic magnetic spin states (MMSSs), namely, “loose” spins, finite, and infinite clusters, are formed. It was shown that there is a broad distribution of the LEMPs lifetimes. The presence of the long-lived LEMPs is caused by the admixture of the optically active bright exciton states to the dark exciton states, i.e., the “brightening” of the dark LEMPs which exist along with the bright LEMPs. The lifetimes of the dark LEMPs correspond to hundreds of nanoseconds. It was found that the time decay of photoluminescence band intensity is approximated by the sum of two functions: a single exponential function and the Kohlrausch–Williams–Watts stretched exponential function. The stretched exponential function describes the recombination processes of the LEMPs formed in the crystal regions of the finite clusters as well as the infinite cluster. This reflects the appearance of spatially heterogeneous dynamics in Cd{sub 0.70}Mn{sub 0.30}Te SG compound below T{sub f} which is due to the disorder in the spin distribution caused by the formation of different MMSSs.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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.↗
SU-E-T-259: Particle Swarm Optimization in Radial Dose Function Fitting for a Novel Iodine-125 Seed
Purpose: To determine the coefficients of bi- and tri-exponential functions for the best fit of radial dose functions of the new iodine brachytherapy source: Iodine-125 Seed AgX-100. Methods: The particle swarm optimization (PSO) method was used to search for the coefficients of the biand tri-exponential functions that yield the best fit to data published for a few selected radial distances from the source. The coefficients were encoded into particles, and these particles move through the search space by following their local and global best-known positions. In each generation, particles were evaluated through their fitness function and their positions were changed through their velocities. This procedure was repeated until the convergence criterion was met or the maximum generation was reached. All best particles were found in less than 1,500 generations. Results: For the I-125 seed AgX-100 considered as a point source, the maximum deviation from the published data is less than 2.9% for bi-exponential fitting function and 0.2% for tri-exponential fitting function. For its line source, the maximum deviation is less than 1.1% for bi-exponential fitting function and 0.08% for tri-exponential fitting function. Conclusion: PSO is a powerful method in searching coefficients for bi-exponential and tri-exponential fitting functions. The bi- and tri-exponential models of Iodine-125 seed AgX-100 point and line sources obtained with PSO optimization provide accurate analytical forms of the radial dose function. The tri-exponential fitting function is more accurate than the bi-exponential function.
60 APPLIED LIFE SCIENCES↗
Transconvolution and the virtual positron emission tomograph-A new method for cross calibration in quantitative PET/CT imaging
Purpose: Positron emission tomography (PET)/computed tomography (CT) measurements on small lesions are impaired by the partial volume effect, which is intrinsically tied to the point spread function of the actual imaging system, including the reconstruction algorithms. The variability resulting from different point spread functions hinders the assessment of quantitative measurements in clinical routine and especially degrades comparability within multicenter trials. To improve quantitative comparability there is a need for methods to match different PET/CT systems through elimination of this systemic variability. Consequently, a new method was developed and tested that transforms the image of an object as produced by one tomograph to another image of the same object as it would have been seen by a different tomograph. The proposed new method, termed Transconvolution, compensates for differing imaging properties of different tomographs and particularly aims at quantitative comparability of PET/CT in the context of multicenter trials. Methods: To solve the problem of image normalization, the theory of Transconvolution was mathematically established together with new methods to handle point spread functions of different PET/CT systems. Knowing the point spread functions of two different imaging systems allows determining a Transconvolution function to convert one image into the other. This function is calculated by convolving one point spread function with the inverse of the other point spread function which, when adhering to certain boundary conditions such as the use of linear acquisition and image reconstruction methods, is a numerically accessible operation. For reliable measurement of such point spread functions characterizing different PET/CT systems, a dedicated solid-state phantom incorporating {sup 68}Ge/{sup 68}Ga filled spheres was developed. To iteratively determine and represent such point spread functions, exponential density functions in combination with a Gaussian distribution were introduced. Furthermore, simulation of a virtual PET system provided a standard imaging system with clearly defined properties to which the real PET systems were to be matched. A Hann window served as the modulation transfer function for the virtual PET. The Hann's apodization properties suppressed high spatial frequencies above a certain critical frequency, thereby fulfilling the above-mentioned boundary conditions. The determined point spread functions were subsequently used by the novel Transconvolution algorithm to match different PET/CT systems onto the virtual PET system. Finally, the theoretically elaborated Transconvolution method was validated transforming phantom images acquired on two different PET systems to nearly identical data sets, as they would be imaged by the virtual PET system. Results: The proposed Transconvolution method matched different PET/CT-systems for an improved and reproducible determination of a normalized activity concentration. The highest difference in measured activity concentration between the two different PET systems of 18.2% was found in spheres of 2 ml volume. Transconvolution reduced this difference down to 1.6%. In addition to reestablishing comparability the new method with its parameterization of point spread functions allowed a full characterization of imaging properties of the examined tomographs. Conclusions: By matching different tomographs to a virtual standardized imaging system, Transconvolution opens a new comprehensive method for cross calibration in quantitative PET imaging. The use of a virtual PET system restores comparability between data sets from different PET systems by exerting a common, reproducible, and defined partial volume effect.
62 RADIOLOGY AND NUCLEAR MEDICINE↗
M. Riesz-Schur-type inequalities for entire functions of exponential type
We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type.
97 MATHEMATICS AND COMPUTING↗
Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type
Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm-Liouville problem, which are exact for entire functions of exponential type, are established. They generalize quadrature formulae involving zeros of Bessel functions, which were first designed by Frappier and Olivier. Bessel quadratures correspond to the Fourier-Hankel integral transform. Some other examples, connected with the Jacobi integral transform, Fourier series in Jacobi orthogonal polynomials and the general Sturm-Liouville problem with regular weight are also given. Bibliography: 39 titles.
97 MATHEMATICS AND COMPUTING↗
A Spectral Lyapunov Function for Exponentially Stable LTV Systems
This paper presents the formulation of a Lyapunov function for an exponentially stable linear timevarying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.
Zhu, J. Jim↗
Experimental observations of bifurcated power decay lengths in the near Scrape-Off Layer of ST40 High Field Spherical Tokamak
The scrape-off layer parallel heat flux decay lengths measured at ST40, a high field, low aspect ratio spherical tokamak, have been observed to bifurcate into two groups. The wide group follows established H-mode scalings (ranging between 2 to 8 mm) while the narrow group falls up to 10 times below these scalings (between 0.2 and 0.8 mm), being comparable to the ion total Larmor radius rather than the ion poloidal Larmor radius. The heat flux profiles of the latter group can only be described by a multi-exponential function, rather than the single exponential function convoluted with a Gaussian. The onset of the narrow scrape-off layer width is observed to be associated with suppressed magnetic fluctuations, suggesting reduced electromagnetic turbulence levels in the SOL.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A novel conformity index for intensity modulated radiation therapy plan evaluation
Purpose: Intensity modulated radiation therapy (IMRT) has gained popularity in the treatment of cancers. Manual evaluation of IMRT plans for head-and-neck cancers has been especially challenging necessitating efficient and objective assessment tools. In this work, the authors address this issue by developing a personalized conformity index (CI) for comparison of IMRT plans for head-and-neck cancers and evaluating its plan quality discerning power in comparison with other widely used CIs. Methods: A two-dimensional CI with dose and distance incorporated (CI{sub DD}) was developed using the MATLAB program language, to quantify the planning target volume (PTV) coverage. Valuable information contained in the digital imaging and communication in medicine (DICOM) RT objects were harvested for computation of each of the CI{sub DD} components. Apart from the dose penalty factor, a distance-based exponential function was employed by varying the penalty weight associated with the location of cold spots within the PTV. With the goal of deriving a customized penalty factor, the distances between individual pixel and its nearest PTV boundary was found. Using the exponential function, the impact of distance penalty was substantially larger for cold spots closer to the PTV centroid but petered out quickly wherever they were situated in the vicinity of PTV border. In order to evaluate the CI{sub DD} scoring system, three CT image data sets of nasopharyngeal carcinoma (NPC) patients were collected. Ten IMRT plans with degrading qualities were generated from each dataset and were ranked based on CI{sub DD} and other existing indices. The coefficient of variance was calculated for each dataset to compare the degree of variation. Results: The CI{sub DD} scoring system that considered spatial importance of each voxel within the PTV was successfully developed. The results demonstrated that the CI{sub DD} including four discrete factors could provide accurate rankings of plan quality by examining the relative importance of each cold spot within the PTVs. Apart from the dose penalty factor, a distance-based exponential function was employed taking the specific tumor geometry into account. Compared with other commonly used CIs, the CI{sub DD} resulted in the largest coefficient of variance among the ten IMRT plans for each dataset, indicating that its discerning power was the best among the CIs being compared. Conclusions: The CI{sub DD} scoring system was successfully developed to incorporate patient-specific spatial dose information and provide a geometry-based physical index for comparison of IMRT plans for head-and-neck cancers. By taking individual tumor geometry into account, the superiority of CI{sub DD} in plan discerning power was demonstrated. The use of CI{sub DD} could provide an effective means of benchmarking performance, reducing treatment plan variability, and advancing the quality of current IMRT planning.
60 APPLIED LIFE SCIENCES↗
Fracture length data for geothermal applications
Fracture lengths govern permeability and are unknowns in geothermal assessment. Along their lengths, fracture widths vary due to growth by linkage. Under the influence of diagenesis, narrow widths seal, breaking porosity continuity and reducing open length. The largest range of widths and thus susceptibility to fill occurs where fractures are linked by narrow segments. Outcrops of a geothermal target, Cambrian Potsdam quartz arenite, contain opening-mode fractures having lengths spanning five orders of magnitude from 0.082 mm to 17.9 m. Combined lengths measured at a range of scales can be described by power laws, but at a given image resolution, lengths are best fit by exponential functions. Owing to preferential sealing of small fractures, open fractures follow exponential functions, but values depend on rules for designating fractures as continuous. En échelon segments, offset 10 mm, are connected by narrow fractures or microfractures (hard linked) not evident on outcrop 1 m-elevation LiDAR or 30 m-height drone images. A rule that identifies where narrow and likely connected segments are located can yield lengths meaningful for flow simulation. Depending on diagenesis, continuity rules can halve or double average and maximum length values. Length values from outcrop for geothermal applications should be adjusted based on wellsite-specific diagenesis information.
15 GEOTHERMAL ENERGY↗
Translational diffusion of water inside hydrophobic carbon micropores studied by neutron spectroscopy and molecular dynamics simulation
When water molecules are confined to nanoscale spacings, such as in the nanometer-size pores of activated carbon fiber (ACF), their freezing point gets suppressed down to very low temperatures (~150 K), leading to a metastable liquid state with remarkable physical properties. Here we have investigated the ambient pressure diffusive dynamics of water in microporous Kynol ACF-10 (average pore size ~11.6 Å, with primarily slit-like pores) from temperature T = 280 K in its stable liquid state down to T = 230 K into the metastable supercooled phase. The observed characteristic relaxation times and diffusion coefficients are found to be, respectively, higher and lower than those in bulk water, indicating a slowing down of the water mobility with decreasing temperature. The observed temperature-dependent average relaxation time (${{\tau}}$) when compared to previous findings indicate that it is the width of the slit pores-not their curvature-that primarily affects the dynamics of water for pore sizes larger than 10 Å. The experimental observations are compared to complementary molecular dynamics simulations of a model system, in which we studied the diffusion of water within the 11.6 Å gap of two parallel graphene sheets. We find generally a reasonable agreement between the observed and calculated relaxation times at the low momentum transfer Q (Q ≤ 0.9 Å -1) . At high Q, however, where localized dynamics becomes relevant, this ideal system does not satisfactorily reproduce the measurements. Consequently, the simulations are compared to the experiments at low Q, where the two can be best reconciled. The best agreement is obtained for the diffusion parameter D associated with the hydrogen-site when a representative stretched exponential function, rather than the standard bimodal exponential model, is used to parametrize the self-correlation function I (Q,t).
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗