Search NASA⌕ Search

SEARCH · Search NASA

Results for “EXPONENTIAL FUNCTION”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 127 records · Page 7

Spiking Neurons for Analysis of Patterns

Artificial neural networks comprising spiking neurons of a novel type have been conceived as improved pattern-analysis and pattern-recognition computational systems. These neurons are represented by a mathematical model denoted the state-variable model (SVM), which among other things, exploits a computational parallelism inherent in spiking-neuron geometry. Networks of SVM neurons offer advantages of speed and computational efficiency, relative to traditional artificial neural networks. The SVM also overcomes some of the limitations of prior spiking-neuron models. There are numerous potential pattern-recognition, tracking, and data-reduction (data preprocessing) applications for these SVM neural networks on Earth and in exploration of remote planets. Spiking neurons imitate biological neurons more closely than do the neurons of traditional artificial neural networks. A spiking neuron includes a central cell body (soma) surrounded by a tree-like interconnection network (dendrites). Spiking neurons are so named because they generate trains of output pulses (spikes) in response to inputs received from sensors or from other neurons. They gain their speed advantage over traditional neural networks by using the timing of individual spikes for computation, whereas traditional artificial neurons use averages of activity levels over time. Moreover, spiking neurons use the delays inherent in dendritic processing in order to efficiently encode the information content of incoming signals. Because traditional artificial neurons fail to capture this encoding, they have less processing capability, and so it is necessary to use more gates when implementing traditional artificial neurons in electronic circuitry. Such higher-order functions as dynamic tasking are effected by use of pools (collections) of spiking neurons interconnected by spike-transmitting fibers. The SVM includes adaptive thresholds and submodels of transport of ions (in imitation of such transport in biological neurons). These features enable the neurons to adapt their responses to high-rate inputs from sensors, and to adapt their firing thresholds to mitigate noise or effects of potential sensor failure. The mathematical derivation of the SVM starts from a prior model, known in the art as the point soma model, which captures all of the salient properties of neuronal response while keeping the computational cost low. The point-soma latency time is modified to be an exponentially decaying function of the strength of the applied potential. Choosing computational efficiency over biological fidelity, the dendrites surrounding a neuron are represented by simplified compartmental submodels and there are no dendritic spines. Updates to the dendritic potential, calcium-ion concentrations and conductances, and potassium-ion conductances are done by use of equations similar to those of the point soma. Diffusion processes in dendrites are modeled by averaging among nearest-neighbor compartments. Inputs to each of the dendritic compartments come from sensors. Alternatively or in addition, when an affected neuron is part of a pool, inputs can come from other spiking neurons. At present, SVM neural networks are implemented by computational simulation, using algorithms that encode the SVM and its submodels. However, it should be possible to implement these neural networks in hardware: The differential equations for the dendritic and cellular processes in the SVM model of spiking neurons map to equivalent circuits that can be implemented directly in analog very-large-scale integrated (VLSI) circuits.

Huntsberger, Terrance↗

The Area Coverage of Geophysical Fields as a Function of Sensor Field-of View

In many remote sensing studies of geophysical fields such as clouds, land cover, or sea ice characteristics, the fractional area coverage of the field in an image is estimated as the proportion of pixels that have the characteristic of interest (i.e., are part of the field) as determined by some thresholding operation. The effect of sensor field-of-view on this estimate is examined by modeling the unknown distribution of subpixel area fraction with the beta distribution, whose two parameters depend upon the true fractional area coverage, the pixel size, and the spatial structure of the geophysical field. Since it is often not possible to relate digital number, reflectance, or temperature to subpixel area fraction, the statistical models described are used to determine the effect of pixel size and thresholding operations on the estimate of area fraction for hypothetical geophysical fields. Examples are given for simulated cumuliform clouds and linear openings in sea ice, whose spatial structures are described by an exponential autocovariance function. It is shown that the rate and direction of change in total area fraction with changing pixel size depends on the true area fraction, the spatial structure, and the thresholding operation used.

Key, Jeffrey R.↗

X-ray study of solar plage regions and a small flare.

The O VII and Ne IX resonance line and 5 keV continuum emission of a small solar flare are interpreted on the basis of a model in which the emission measure of the region is an exponentially decreasing function of temperature. Good agreement with all three sets of data is obtained - a result not possible on the basis of an isothermal approximation. The change in the emission of the flare over a two minute interval appears to be consistent with a simple cooling of the region. Computation of the contribution function versus temperature for each emission indicates that each predominantly comes from a physically separate volume within the feature. The emission measure distribution for the flare is distinctly flatter than that of five plage regions which tend to have similar slopes and which are in turn flatter than the general corona.

Acton, L. W.↗

Computer-controlled environmental test systems - Criteria for selection, installation, and maintenance.

Applications for presently marketed, new computer-controlled environmental test systems are suggested. It is shown that capital costs of these systems follow an exponential cost function curve that levels out as additional applications are implemented. Some test laboratory organization changes are recommended in terms of new personnel requirements, and facility modification are considered in support of a computer-controlled test system. Software for computer-controlled test systems are discussed, and control loop speed constraints are defined for real-time control functions. Suitable input and output devices and memory storage device tradeoffs are also considered.

Chapman, C. P.↗

A nonlinear theory of cosmic ray pitch angle diffusion in homogeneous magnetostatic turbulence

A plasma strong turbulence, weak coupling, theory is applied to the problem of cosmic ray pitch angle scattering in magnetostatic turbulence. The theory used is a rigorous generalization of Weinstock's resonance-broadening theory and contains no ad hoc approximations. A detailed calculation is presented for a model of slab turbulence with an exponential correlation function. The results agree well with numerical simulations. The rigidity dependence of the pitch angle scattering coefficient differs from that found by previous researchers. The differences result from an inadequate treatment of particle trajectories near 90 deg pitch angle in earlier work.

Goldstein, M. L.↗

A nonlinear theory of cosmic-ray pitch-angle diffusion in homogeneous magnetostatic turbulence

A plasma strong turbulence, weak coupling, theory is applied to the problem of cosmic-ray pitch-angle scattering in magnetostatic turbulence. The theory used is a rigorous generalization of Weinstock's 'resonance broadening' theory and contains no ad hoc approximations. A detailed calculation is presented for a model of 'slab' turbulence with an exponential correlation function. The results agree well with numerical simulations. The rigidity dependence of the pitch-angle scattering coefficient differs from that found by previous researchers. The differences result from an inadequate treatment of particle trajectories near 90 deg pitch angle in earlier work.

Goldstein, M. L.↗

Some MACSYMA program for solving difference equations

A set of MACSYMA programs are described for finding closed form solutions to linear recurrence relations in equations having either constant or variable coefficients. In the homogenous case, a polymonial equation is obtained and the solution to the recurrence relation can be written as a linear combination of the roots of the polynomial. Exponential generating functions are used to solve variable coefficient relations. Taking successive derivatives and using the recurrence relation, an ordinary differential equation is obtained. Expanding the solution to the differential equation in a Taylor series, shows that the nth term of the series is the solution to the recurrence relation. For second order recurrences, a check is made for those that can be solved in terms of Bessel functions.

Ivie, J.↗