The soil moisture velocity equation
NASA Astrophysics Data System (ADS)
Ogden, Fred L.; Allen, Myron B.; Lai, Wencong; Zhu, Jianting; Seo, Mookwon; Douglas, Craig C.; Talbot, Cary A.
2017-06-01
Numerical solution of the one-dimensional Richards' equation is the recommended method for coupling groundwater to the atmosphere through the vadose zone in hyperresolution Earth system models, but requires fine spatial discretization, is computationally expensive, and may not converge due to mathematical degeneracy or when sharp wetting fronts occur. We transformed the one-dimensional Richards' equation into a new equation that describes the velocity of moisture content values in an unsaturated soil under the actions of capillarity and gravity. We call this new equation the Soil Moisture Velocity Equation (SMVE). The SMVE consists of two terms: an advection-like term that accounts for gravity and the integrated capillary drive of the wetting front, and a diffusion-like term that describes the flux due to the shape of the wetting front capillarity profile divided by the vertical gradient of the capillary pressure head. The SMVE advection-like term can be converted to a relatively easy to solve ordinary differential equation (ODE) using the method of lines and solved using a finite moisture-content discretization. Comparing against analytical solutions of Richards' equation shows that the SMVE advection-like term is >99% accurate for calculating infiltration fluxes neglecting the diffusion-like term. The ODE solution of the SMVE advection-like term is accurate, computationally efficient and reliable for calculating one-dimensional vadose zone fluxes in Earth system and large-scale coupled models of land-atmosphere interaction. It is also well suited for use in inverse problems such as when repeat remote sensing observations are used to infer soil hydraulic properties or soil moisture.Plain Language SummarySince its original publication in 1922, the so-called Richards' <span class="hlt">equation</span> has been the only rigorous way to couple groundwater to the land surface through the unsaturated zone that lies between the water table and land</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://pubs.er.usgs.gov/publication/70012994','USGSPUBS'); return false;" href="http://pubs.er.usgs.gov/publication/70012994"><span>Optimization of one-way wave <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="http://pubs.er.usgs.gov/pubs/index.jsp?view=adv">USGS Publications Warehouse</a></p>
<p>Lee, M.W.; Suh, S.Y.</p>
<p>1985-01-01</p>
<p>The theory of wave extrapolation is based on the square-root <span class="hlt">equation</span> or one-way <span class="hlt">equation</span>. The full wave <span class="hlt">equation</span> represents waves which propagate in both directions. On the contrary, the square-root <span class="hlt">equation</span> represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave <span class="hlt">equation</span>. -from Authors</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21202833','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21202833"><span>Linear determining <span class="hlt">equations</span> for differential constraints</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Kaptsov, O V</p>
<p>1998-12-31</p>
<p>A construction of differential constraints compatible with partial differential <span class="hlt">equations</span> is considered. Certain linear determining <span class="hlt">equations</span> with parameters are used to find such differential constraints. They generalize the classical determining <span class="hlt">equations</span> used in the search for admissible Lie operators. As applications of this approach <span class="hlt">equations</span> of an ideal incompressible fluid and non-linear heat <span class="hlt">equations</span> are discussed.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19920002099','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19920002099"><span>Turbulent fluid motion 3: Basic continuum <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Deissler, Robert G.</p>
<p>1991-01-01</p>
<p>A derivation of the continuum <span class="hlt">equations</span> used for the analysis of turbulence is given. These <span class="hlt">equations</span> include the continuity <span class="hlt">equation</span>, the Navier-Stokes <span class="hlt">equations</span>, and the heat transfer or energy <span class="hlt">equation</span>. An experimental justification for using a continuum approach for the study of turbulence is given.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015CoPhC.192..156D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015CoPhC.192..156D"><span>Solving Parker's transport <span class="hlt">equation</span> with stochastic differential <span class="hlt">equations</span> on GPUs</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Dunzlaff, P.; Strauss, R. D.; Potgieter, M. S.</p>
<p>2015-07-01</p>
<p>The numerical solution of transport <span class="hlt">equations</span> for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker's <span class="hlt">equation</span> using Stochastic Differential <span class="hlt">Equations</span> (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker's transport <span class="hlt">equation</span> and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10-60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker's transport <span class="hlt">equation</span>.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012JMP....53j3520V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012JMP....53j3520V"><span>Fredholm's <span class="hlt">equations</span> for subwavelength focusing</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Velázquez-Arcos, J. M.</p>
<p>2012-10-01</p>
<p>Subwavelength focusing (SF) is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves. Many recent works have described a successful alternative procedure using time reversal methods. The advantage is that we do not need devices which require the complicated manufacture of left-hand materials; nevertheless, the theoretical mathematical bases are far from complete because before now we lacked an adequate easy-to-apply frame. In this work we give, for a broad class of discrete systems, a solid support for the theory of electromagnetic SF that can be applied to communications and nanotechnology. The very central procedure is the development of vector-matrix formalism (VMF) based on exploiting both the inhomogeneous and homogeneous Fredholm's integral <span class="hlt">equations</span> in cases where the last two kinds of integral <span class="hlt">equations</span> are applied to some selected discrete systems. To this end, we first establish a generalized Newmann series for the Fourier transform of the Green's function in the inhomogeneous Fredholm's <span class="hlt">equation</span> of the problem. Then we go from an integral operator <span class="hlt">equation</span> to a vector-matrix algebraic one. In this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous <span class="hlt">equation</span>. Thus, on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a SF. On the other hand, we analyze the homogeneous version of the <span class="hlt">equations</span>, finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering, that can be used in a proposed very powerful way in communications. Also we recover quantum mechanical operator relations that are identical for classical electromagnetics. Finally, we prove two theorems that formalize the relation between the theory of Fredholm's integral <span class="hlt">equations</span> and the VMF we present here.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/28297844','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/28297844"><span>Cable <span class="hlt">equation</span> for general geometry.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>López-Sánchez, Erick J; Romero, Juan M</p>
<p>2017-02-01</p>
<p>The cable <span class="hlt">equation</span> describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this <span class="hlt">equation</span> might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable <span class="hlt">equation</span> for a general cable geometry. This generalized <span class="hlt">equation</span> depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable <span class="hlt">equation</span> depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable <span class="hlt">equation</span> as a diffusion <span class="hlt">equation</span> with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhRvE..95b2403L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhRvE..95b2403L"><span>Cable <span class="hlt">equation</span> for general geometry</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>López-Sánchez, Erick J.; Romero, Juan M.</p>
<p>2017-02-01</p>
<p>The cable <span class="hlt">equation</span> describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this <span class="hlt">equation</span> might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable <span class="hlt">equation</span> for a general cable geometry. This generalized <span class="hlt">equation</span> depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable <span class="hlt">equation</span> depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable <span class="hlt">equation</span> as a diffusion <span class="hlt">equation</span> with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA115279','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA115279"><span><span class="hlt">Equation</span> of State of Simple Metals.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1982-05-10</p>
<p>This is the final report of A. L. Ruoff and N. W. Ashcroft on <span class="hlt">Equation</span> of State of Simple Metals. It includes experimental <span class="hlt">equation</span> of state results for potassium and theoretical calculations of its <span class="hlt">equation</span> of state . (Author)</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Maxwell+AND+equation&id=EJ844007','ERIC'); return false;" href="https://eric.ed.gov/?q=Maxwell+AND+equation&id=EJ844007"><span>How to Obtain the Covariant Form of Maxwell's <span class="hlt">Equations</span> from the Continuity <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Heras, Jose A.</p>
<p>2009-01-01</p>
<p>The covariant Maxwell <span class="hlt">equations</span> are derived from the continuity <span class="hlt">equation</span> for the electric charge. This result provides an axiomatic approach to Maxwell's <span class="hlt">equations</span> in which charge conservation is emphasized as the fundamental axiom underlying these <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=Conservation&pg=7&id=EJ844007','ERIC'); return false;" href="http://eric.ed.gov/?q=Conservation&pg=7&id=EJ844007"><span>How to Obtain the Covariant Form of Maxwell's <span class="hlt">Equations</span> from the Continuity <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Heras, Jose A.</p>
<p>2009-01-01</p>
<p>The covariant Maxwell <span class="hlt">equations</span> are derived from the continuity <span class="hlt">equation</span> for the electric charge. This result provides an axiomatic approach to Maxwell's <span class="hlt">equations</span> in which charge conservation is emphasized as the fundamental axiom underlying these <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA470361','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA470361"><span>Spectral Models Based on Boussinesq <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2006-10-03</p>
<p><span class="hlt">equations</span> assume periodic solutions apriori. This, however, also forces the question of which extended Boussinesq model to use. Various one- <span class="hlt">equation</span> ... <span class="hlt">equations</span> of Nwogu (1993), without the traditional reduction to a one- <span class="hlt">equation</span> model. Optimal numerical techniques to solve this system of <span class="hlt">equations</span> are...A. and Madsen, P. A. (2004). "Boussinesq evolution <span class="hlt">equations</span> : numerical efficiency, breaking and amplitude dispersion," Coastal Engineering, 51, 1117</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADD095338','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADD095338"><span>An <span class="hlt">Equation</span> of State for Fluid Ethylene.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>an <span class="hlt">equation</span> of state , vapor pressure <span class="hlt">equation</span>, and <span class="hlt">equation</span> for the ideal gas heat capacity. The coefficients were determined by a least squares fit...of selected experimental data. Comparisons of property values calculated using the <span class="hlt">equation</span> of state with measured values are given. The <span class="hlt">equation</span> of state is...vapor phases for temperatures from the freezing line of 450 K with pressures to 40 MPa are presented. The <span class="hlt">equation</span> of state and its derivative and</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0739592','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0739592"><span>Nonlinear Evolution <span class="hlt">Equations</span> in Banach Spaces.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>relationship to the evolution <span class="hlt">equation</span> is studied. The results obtained extend several known existence theorems and provide generalized solutions of the evolution <span class="hlt">equation</span> in more general cases. (Author)</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21230210','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21230210"><span>Numerical integration of variational <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Skokos, Ch; Gerlach, E</p>
<p>2010-09-01</p>
<p>We present and compare different numerical schemes for the integration of the variational <span class="hlt">equations</span> of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "tangent map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton <span class="hlt">equations</span> of motion by the repeated action of a symplectic map S , while the corresponding tangent map TS is used for the integration of the variational <span class="hlt">equations</span>. A simple and systematic technique to construct TS is also presented.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2007SPIE.6597E..09U','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2007SPIE.6597E..09U"><span>Integration of quantum hydrodynamical <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ulyanova, Vera G.; Sanin, Andrey L.</p>
<p>2007-04-01</p>
<p>Quantum hydrodynamics <span class="hlt">equations</span> describing the dynamics of quantum fluid are a subject of this report (QFD).These <span class="hlt">equations</span> can be used to decide the wide class of problem. But there are the calculated difficulties for the <span class="hlt">equations</span>, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IJMES..47..552L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IJMES..47..552L"><span>Students' understanding of quadratic <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael</p>
<p>2016-05-01</p>
<p>Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic <span class="hlt">equations</span> in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic <span class="hlt">equations</span>. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic <span class="hlt">equations</span> with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.</p>
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<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009LNCS.5720..104W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009LNCS.5720..104W"><span>Differential <span class="hlt">Equations</span> for Morphological Amoebas</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Welk, Martin; Breuß, Michael; Vogel, Oliver</p>
<p></p>
<p>This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential <span class="hlt">equation</span>. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential <span class="hlt">equation</span> related to self-snakes and the well-known (mean) curvature motion <span class="hlt">equation</span>. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JChPh.119.2165S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JChPh.119.2165S"><span>Fractional reaction-diffusion <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Seki, Kazuhiko; Wojcik, Mariusz; Tachiya, M.</p>
<p>2003-07-01</p>
<p>A fractional reaction-diffusion <span class="hlt">equation</span> is derived from a continuous time random walk model when the transport is dispersive. The exit from the encounter distance, which is described by the algebraic waiting time distribution of jump motion, interferes with the reaction at the encounter distance. Therefore, the reaction term has a memory effect. The derived <span class="hlt">equation</span> is applied to the geminate recombination problem. The recombination is shown to depend on the intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem. Phys. 116, 2338 (2002)], which were obtained from the fractional reaction-diffusion <span class="hlt">equation</span> where the diffusion term has a memory effect but the reaction term does not. The reactivity dependence of the recombination probability is confirmed by numerical simulations.</p>
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<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2890680','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2890680"><span>Fractional-calculus diffusion <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p></p>
<p>2010-01-01</p>
<p>Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion <span class="hlt">equation</span> is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's <span class="hlt">equation</span> which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's <span class="hlt">equation</span> is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014PhRvE..90f3307P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014PhRvE..90f3307P"><span>Numerical optimization using flow <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Punk, Matthias</p>
<p>2014-12-01</p>
<p>We develop a method for multidimensional optimization using flow <span class="hlt">equations</span>. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow <span class="hlt">equation</span>. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow <span class="hlt">equation</span> only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25615222','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25615222"><span>Numerical optimization using flow <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Punk, Matthias</p>
<p>2014-12-01</p>
<p>We develop a method for multidimensional optimization using flow <span class="hlt">equations</span>. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow <span class="hlt">equation</span>. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow <span class="hlt">equation</span> only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/17927422','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/17927422"><span>The room acoustic rendering <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Siltanen, Samuel; Lokki, Tapio; Kiminki, Sami; Savioja, Lauri</p>
<p>2007-09-01</p>
<p>An integral <span class="hlt">equation</span> generalizing a variety of known geometrical room acoustics modeling algorithms is presented. The formulation of the room acoustic rendering <span class="hlt">equation</span> is adopted from computer graphics. Based on the room acoustic rendering <span class="hlt">equation</span>, an acoustic radiance transfer method, which can handle both diffuse and nondiffuse reflections, is derived. In a case study, the method is used to predict several acoustic parameters of a room model. The results are compared to measured data of the actual room and to the results given by other acoustics prediction software. It is concluded that the method can predict most acoustic parameters reliably and provides results as accurate as current commercial room acoustic prediction software. Although the presented acoustic radiance transfer method relies on geometrical acoustics, it can be extended to model diffraction and transmission through materials in future.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CMMPh..57..211V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CMMPh..57..211V"><span>Special solutions to Chazy <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Varin, V. P.</p>
<p>2017-02-01</p>
<p>We consider the classical Chazy <span class="hlt">equation</span>, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy <span class="hlt">equation</span> in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy <span class="hlt">equation</span>. The results have applications to analytic number theory.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22525840','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22525840"><span>Explicit integration of Friedmann's <span class="hlt">equation</span> with nonlinear <span class="hlt">equations</span> of state</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong E-mail: gwg1@damtp.cam.ac.uk</p>
<p>2015-05-01</p>
<p>In this paper we study the integrability of the Friedmann <span class="hlt">equations</span>, when the <span class="hlt">equation</span> of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann <span class="hlt">equation</span> is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann <span class="hlt">equations</span> also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009APS..DPPNI3001C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009APS..DPPNI3001C"><span>Transport <span class="hlt">Equations</span> In Tokamak Plasmas</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Callen, J. D.</p>
<p>2009-11-01</p>
<p>Tokamak plasma transport <span class="hlt">equations</span> are usually obtained by flux surface averaging the collisional Braginskii <span class="hlt">equations</span>. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for: neoclassical effects on the parallel Ohm's law (trapped particle effects on resistivity, bootstrap current); fluctuation-induced transport; heating, current-drive and flow sources and sinks; small B field non-axisymmetries; magnetic field transients etc. A set of self-consistent second order in gyroradius fluid-moment-based transport <span class="hlt">equations</span> for nearly axisymmetric tokamak plasmas has been developed recently using a kinetic-based framework. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfv'en waves (Grad-Shafranov equilibrium, ion radial force balance); sound waves (pressure constant along field lines, incompressible flows within a flux surface); and ion collisions (damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on the plasma fluid: 7 ambipolar collision-based ones (classical, neoclassical, etc.) and 8 non-ambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients etc.). The plasma toroidal rotation <span class="hlt">equation</span> [1] results from setting to zero the net radial current induced by the non-ambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the non-ambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport <span class="hlt">equations</span> do not involve an ambipolar constraint and hence are more directly obtained. The resultant transport <span class="hlt">equations</span> will be presented and contrasted with the usual ones. [4pt] [1] J.D. Callen, A.J. Cole, C.C. Hegna, ``Toroidal Rotation In</p>
</li>
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<ol class="result-class" start="181">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21371198','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21371198"><span>Transport <span class="hlt">equations</span> in tokamak plasmas</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Callen, J. D.; Hegna, C. C.; Cole, A. J.</p>
<p>2010-05-15</p>
<p>Tokamak plasma transport <span class="hlt">equations</span> are usually obtained by flux surface averaging the collisional Braginskii <span class="hlt">equations</span>. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport <span class="hlt">equations</span> for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation <span class="hlt">equation</span> results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport <span class="hlt">equations</span> do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017GrCo...23..280D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017GrCo...23..280D"><span>Einstein <span class="hlt">equations</span> with fluctuating volume</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Dzhunushaliev, Vladimir; Quevedo, Hernando</p>
<p>2017-07-01</p>
<p>We develop a simple model to study classical fields on the background of a fluctuating spacetime volume. It is applied to formulate the stochastic Einstein <span class="hlt">equations</span> with a perfect-fluid source. We investigate the particular case of a stochastic Friedmann-Lema\\^itre-Robertson-Walker cosmology, and show that the resulting field <span class="hlt">equations</span> can lead to solutions which avoid the initial big bang singularity. By interpreting the fluctuations as the result of the presence of a quantum spacetime, we conclude that classical singularities can be avoided even within a stochastic model that include quantum effects in a very simple manner.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/6961552','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/6961552"><span>The nuclear <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Kahana, S.</p>
<p>1986-01-01</p>
<p>The role of the nuclear <span class="hlt">equation</span> of state in determining the fate of the collapsing cores of massive stars is examined in light of both recent theoretical advances in this subject and recent experimental measurements with relativistic heavy ions. The difficulties existing in attempts to bring the softer nuclear matter apparently required by the theory of Type II supernovae into consonance with the heavy ion data are discussed. Relativistic mean field theory is introduced as a candidate for derivation of the <span class="hlt">equation</span> of state, and a simple form for the saturation compressibility is obtained. 28 refs., 4 figs., 1 tab.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013PhRvL.111i6101S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013PhRvL.111i6101S"><span>Young's <span class="hlt">Equation</span> at the Nanoscale</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Seveno, David; Blake, Terence D.; De Coninck, Joël</p>
<p>2013-08-01</p>
<p>In 1805, Thomas Young was the first to propose an <span class="hlt">equation</span> to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/945516','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/945516"><span>Tantalum <span class="hlt">equation</span> of state package</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Young, D A; Orlikowski, D</p>
<p>2008-12-04</p>
<p>We provide here the tantalum <span class="hlt">equation</span> of state (EOS) over broad ranges of temperature and pressure (up to 49,900 K and 9.6 Mbar). This EOS was developed by the quotidian <span class="hlt">equation</span> of state methodology, which is a robust EOS model providing EOS tables for high pressure hydrodynamic simulations. The included tables span densities 5-33.4 g/cc with 51 entries and temperatures 116-49,900 K with 50 entries. There are 16 quantities as a function of that density-temperature grid, which are provided in this EOS package and are listed in the Appendix (Table I).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24730966','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24730966"><span>Investigation of the kinetic model <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Liu, Sha; Zhong, Chengwen</p>
<p>2014-03-01</p>
<p>Currently the Boltzmann <span class="hlt">equation</span> and its model <span class="hlt">equations</span> are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann <span class="hlt">equation</span> is the fundamental <span class="hlt">equation</span> in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model <span class="hlt">equations</span> such as the famous Bhatnagar-Gross-Krook (BGK) <span class="hlt">equation</span> are mathematically simple. Since the computational cost of solving model <span class="hlt">equations</span> is much less than that of solving the full Boltzmann <span class="hlt">equation</span>, the model <span class="hlt">equations</span> are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann <span class="hlt">equation</span> and its model <span class="hlt">equations</span> are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann <span class="hlt">equation</span> and its model <span class="hlt">equations</span> using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann <span class="hlt">equation</span> and BGK <span class="hlt">equation</span> that is also derived in this paper. After the analysis of the difference between the Boltzmann <span class="hlt">equation</span> and the BGK <span class="hlt">equation</span>, an attempt at approximating the collision term is proposed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=equations+AND+states&pg=4&id=EJ753914','ERIC'); return false;" href="http://eric.ed.gov/?q=equations+AND+states&pg=4&id=EJ753914"><span>The Forced Soft Spring <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Fay, T. H.</p>
<p>2006-01-01</p>
<p>Through numerical investigations, this paper studies examples of the forced Duffing type spring <span class="hlt">equation</span> with [epsilon] negative. By performing trial-and-error numerical experiments, the existence is demonstrated of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions. Subharmonic boundaries are…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=pendulum&pg=3&id=EJ831639','ERIC'); return false;" href="http://eric.ed.gov/?q=pendulum&pg=3&id=EJ831639"><span>Pendulum Motion and Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Reid, Thomas F.; King, Stephen C.</p>
<p>2009-01-01</p>
<p>A common example of real-world motion that can be modeled by a differential <span class="hlt">equation</span>, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017GeoJI.208.1567L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017GeoJI.208.1567L"><span>Wave-<span class="hlt">equation</span> dispersion inversion</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Li, Jing; Feng, Zongcai; Schuster, Gerard</p>
<p>2017-03-01</p>
<p>We present the theory for wave-<span class="hlt">equation</span> inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-<span class="hlt">equation</span> traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave <span class="hlt">equation</span> and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-<span class="hlt">equation</span> dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/1391','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/1391"><span>Simple, Flexible, Trigonometric Taper <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>Charles E. Thomas; Bernard R. Parresol</p>
<p>1991-01-01</p>
<p>There have been numerous approaches to modeling stem form in recent decades. The majority have concentrated on the simpler coniferous bole form and have become increasingly complex mathematical expressions. Use of trigonometric <span class="hlt">equations</span> provides a simple expression of taper that is flexible enough to fit both coniferous and hard-wood bole forms. As an illustration, we...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/6742124','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/6742124"><span>Empirical <span class="hlt">equation</span> estimates geothermal gradients</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Kutasov, I.M. )</p>
<p>1995-01-02</p>
<p>An empirical <span class="hlt">equation</span> can estimate geothermal (natural) temperature profiles in new exploration areas. These gradients are useful for cement slurry and mud design and for improving electrical and temperature log interpretation. Downhole circulating temperature logs and surface outlet temperatures are used for predicting the geothermal gradients.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19740012057','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19740012057"><span>The solution of transcendental <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Agrawal, K. M.; Outlaw, R.</p>
<p>1973-01-01</p>
<p>Some of the existing methods to globally approximate the roots of transcendental <span class="hlt">equations</span> namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED502442.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED502442.pdf"><span>Renaissance Learning <span class="hlt">Equating</span> Study. Report</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben</p>
<p>2007-01-01</p>
<p>An <span class="hlt">equating</span> study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Stephen+AND+king&pg=7&id=EJ831639','ERIC'); return false;" href="https://eric.ed.gov/?q=Stephen+AND+king&pg=7&id=EJ831639"><span>Pendulum Motion and Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Reid, Thomas F.; King, Stephen C.</p>
<p>2009-01-01</p>
<p>A common example of real-world motion that can be modeled by a differential <span class="hlt">equation</span>, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/5710156','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/5710156"><span>Lithium <span class="hlt">equation</span>-of-state</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Blink, J.A.</p>
<p>1983-09-01</p>
<p>In 1977, Dave Young published an <span class="hlt">equation</span>-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=symbolism&pg=2&id=EJ749932','ERIC'); return false;" href="http://eric.ed.gov/?q=symbolism&pg=2&id=EJ749932"><span>The Symbolism Of Chemical <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Jensen, William B.</p>
<p>2005-01-01</p>
<p>A question about the historical origin of equal sign and double arrow symbolism in balanced chemical <span class="hlt">equation</span> is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/EJ250344.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/EJ250344.pdf"><span>Differential <span class="hlt">Equations</span> via Population Dynamics.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Sofo, Anthony</p>
<p>1981-01-01</p>
<p>Some single species and two species interactions in population models are presented to show how credible examples can be used to teach an underlying, common mathematical structure within apparently different concepts. The models examined consist of differential <span class="hlt">equations</span>, and focus on real-world issues. (MP)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Symbolism&pg=3&id=EJ749932','ERIC'); return false;" href="https://eric.ed.gov/?q=Symbolism&pg=3&id=EJ749932"><span>The Symbolism Of Chemical <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Jensen, William B.</p>
<p>2005-01-01</p>
<p>A question about the historical origin of equal sign and double arrow symbolism in balanced chemical <span class="hlt">equation</span> is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27586766','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27586766"><span>Sonar <span class="hlt">equations</span> for planetary exploration.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Ainslie, Michael A; Leighton, Timothy G</p>
<p>2016-08-01</p>
<p>The set of formulations commonly known as "the sonar <span class="hlt">equations</span>" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar <span class="hlt">equations</span>, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar <span class="hlt">equations</span> have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar <span class="hlt">equations</span> can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19720016564','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19720016564"><span>Optimized solution of Kepler's <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kohout, J. M.; Layton, L.</p>
<p>1972-01-01</p>
<p>A detailed description is presented of KEPLER, an IBM 360 computer program used for the solution of Kepler's <span class="hlt">equation</span> for eccentric anomaly. The program KEPLER employs a second-order Newton-Raphson differential correction process, and it is faster than previously developed programs by an order of magnitude.</p>
</li>
</ol>
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<ol class="result-class" start="201">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016ZaMP...67..135H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016ZaMP...67..135H"><span>On abstract degenerate neutral differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hernández, Eduardo; O'Regan, Donal</p>
<p>2016-10-01</p>
<p>We introduce a new abstract model of functional differential <span class="hlt">equations</span>, which we call abstract degenerate neutral differential <span class="hlt">equations</span>, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential <span class="hlt">equations</span>. Some examples on nonlinear partial neutral differential <span class="hlt">equations</span> are presented.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=direct-writing&pg=3&id=ED262084','ERIC'); return false;" href="http://eric.ed.gov/?q=direct-writing&pg=3&id=ED262084"><span>Statistical <span class="hlt">Equating</span> of Direct Writing Assessment.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Phillips, Gary W.</p>
<p></p>
<p>This paper provides empirical data on two approaches to statistically <span class="hlt">equate</span> scores derived from the direct assessment of writing. These methods are linear <span class="hlt">equating</span> and <span class="hlt">equating</span> based on the general polychotomous form of the Rasch model. Data from the Maryland Functional Writing Test are used to <span class="hlt">equate</span> scores obtained from two prompts given in…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=quantum&pg=5&id=EJ985449','ERIC'); return false;" href="http://eric.ed.gov/?q=quantum&pg=5&id=EJ985449"><span>Simple Derivation of the Lindblad <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Pearle, Philip</p>
<p>2012-01-01</p>
<p>The Lindblad <span class="hlt">equation</span> is an evolution <span class="hlt">equation</span> for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad <span class="hlt">equation</span> are given. The derivation of the Lindblad <span class="hlt">equation</span> presented here is…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=bayesian&pg=4&id=EJ839749','ERIC'); return false;" href="https://eric.ed.gov/?q=bayesian&pg=4&id=EJ839749"><span>A Bayesian Nonparametric Approach to Test <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Karabatsos, George; Walker, Stephen G.</p>
<p>2009-01-01</p>
<p>A Bayesian nonparametric model is introduced for score <span class="hlt">equating</span>. It is applicable to all major <span class="hlt">equating</span> designs, and has advantages over previous <span class="hlt">equating</span> models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous <span class="hlt">equating</span> models are…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED382644.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED382644.pdf"><span>Simulated <span class="hlt">Equating</span> Using Several Item Response Curves.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Boldt, R. F.</p>
<p></p>
<p>The comparison of item response theory models for the Test of English as a Foreign Language (TOEFL) was extended to an <span class="hlt">equating</span> context as simulation trials were used to "<span class="hlt">equate</span> the test to itself." <span class="hlt">Equating</span> sample data were generated from administration of identical item sets. <span class="hlt">Equatings</span> that used procedures based on each model (simple…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=bayesian&pg=3&id=EJ839749','ERIC'); return false;" href="http://eric.ed.gov/?q=bayesian&pg=3&id=EJ839749"><span>A Bayesian Nonparametric Approach to Test <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Karabatsos, George; Walker, Stephen G.</p>
<p>2009-01-01</p>
<p>A Bayesian nonparametric model is introduced for score <span class="hlt">equating</span>. It is applicable to all major <span class="hlt">equating</span> designs, and has advantages over previous <span class="hlt">equating</span> models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous <span class="hlt">equating</span> models are…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=quantum&pg=6&id=EJ985449','ERIC'); return false;" href="https://eric.ed.gov/?q=quantum&pg=6&id=EJ985449"><span>Simple Derivation of the Lindblad <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Pearle, Philip</p>
<p>2012-01-01</p>
<p>The Lindblad <span class="hlt">equation</span> is an evolution <span class="hlt">equation</span> for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad <span class="hlt">equation</span> are given. The derivation of the Lindblad <span class="hlt">equation</span> presented here is…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED572625.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED572625.pdf"><span>The Complexity of One-Step <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Ngu, Bing</p>
<p>2014-01-01</p>
<p>An analysis of one-step <span class="hlt">equations</span> from a cognitive load theory perspective uncovers variation within one-step <span class="hlt">equations</span>. The complexity of one-step <span class="hlt">equations</span> arises from the element interactivity across the operational and relational lines. The higher the number of operational and relational lines, the greater the complexity of the <span class="hlt">equations</span>.…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21550034','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21550034"><span>Relativistic <span class="hlt">equations</span> with fractional and pseudodifferential operators</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Babusci, D.; Dattoli, G.; Quattromini, M.</p>
<p>2011-06-15</p>
<p>In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential <span class="hlt">equations</span> involving operators with noninteger exponents. We apply the method to <span class="hlt">equations</span> resembling generalizations of the heat <span class="hlt">equations</span> and discuss the possibility of extending the procedure to the relativistic Schroedinger and Dirac <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/11736050','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/11736050"><span>Interplays between Harper and Mathieu <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Papp, E; Micu, C</p>
<p>2001-11-01</p>
<p>This paper deals with the application of relationships between Harper and Mathieu <span class="hlt">equations</span> to the derivation of energy formulas. Establishing suitable matching conditions, one proceeds by inserting a concrete solution to the Mathieu <span class="hlt">equation</span> into the Harper <span class="hlt">equation</span>. For this purpose, one resorts to the nonlinear oscillations characterizing the Mathieu <span class="hlt">equation</span>. This leads to the derivation of two kinds of energy formulas working in terms of cubic and quadratic algebraic <span class="hlt">equations</span>, respectively. Combining such results yields quadratic <span class="hlt">equations</span> to the energy description of the Harper <span class="hlt">equation</span>, incorporating all parameters needed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..91a3309Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..91a3309Z"><span>Lattice Boltzmann <span class="hlt">equation</span> method for the Cahn-Hilliard <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zheng, Lin; Zheng, Song; Zhai, Qinglan</p>
<p>2015-01-01</p>
<p>In this paper a lattice Boltzmann <span class="hlt">equation</span> (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard <span class="hlt">equation</span> (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010), 10.1016/j.jcp.2010.07.007]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890000291&hterms=equation+state&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dequation%2Bof%2Bstate','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890000291&hterms=equation+state&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D70%26Ntt%3Dequation%2Bof%2Bstate"><span>Isothermal <span class="hlt">Equation</span> Of State For Compressed Solids</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Vinet, Pascal; Ferrante, John</p>
<p>1989-01-01</p>
<p>Same <span class="hlt">equation</span> with three adjustable parameters applies to different materials. Improved <span class="hlt">equation</span> of state describes pressure on solid as function of relative volume at constant temperature. Even though types of interatomic interactions differ from one substance to another, form of <span class="hlt">equation</span> determined primarily by overlap of electron wave functions during compression. Consequently, <span class="hlt">equation</span> universal in sense it applies to variety of substances, including ionic, metallic, covalent, and rare-gas solids. Only three parameters needed to describe <span class="hlt">equation</span> for given material.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890000291&hterms=covalent+ionic&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3Dcovalent%2Bionic','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890000291&hterms=covalent+ionic&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3Dcovalent%2Bionic"><span>Isothermal <span class="hlt">Equation</span> Of State For Compressed Solids</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Vinet, Pascal; Ferrante, John</p>
<p>1989-01-01</p>
<p>Same <span class="hlt">equation</span> with three adjustable parameters applies to different materials. Improved <span class="hlt">equation</span> of state describes pressure on solid as function of relative volume at constant temperature. Even though types of interatomic interactions differ from one substance to another, form of <span class="hlt">equation</span> determined primarily by overlap of electron wave functions during compression. Consequently, <span class="hlt">equation</span> universal in sense it applies to variety of substances, including ionic, metallic, covalent, and rare-gas solids. Only three parameters needed to describe <span class="hlt">equation</span> for given material.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19830021442','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19830021442"><span>Applications of film thickness <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Hamrock, B. J.; Dowson, D.</p>
<p>1983-01-01</p>
<p>A number of applications of elastohydrodynamic film thickness expressions were considered. The motion of a steel ball over steel surfaces presenting varying degrees of conformity was examined. The <span class="hlt">equation</span> for minimum film thickness in elliptical conjunctions under elastohydrodynamic conditions was applied to roller and ball bearings. An involute gear was also introduced, it was again found that the elliptical conjunction expression yielded a conservative estimate of the minimum film thickness. Continuously variable-speed drives like the Perbury gear, which present truly elliptical elastohydrodynamic conjunctions, are favored increasingly in mobile and static machinery. A representative elastohydrodynamic condition for this class of machinery is considered for power transmission equipment. The possibility of elastohydrodynamic films of water or oil forming between locomotive wheels and rails is examined. The important subject of traction on the railways is attracting considerable attention in various countries at the present time. The final example of a synovial joint introduced the <span class="hlt">equation</span> developed for isoviscous-elastic regimes of lubrication.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012PhRvD..85j4048L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012PhRvD..85j4048L"><span>Graviton corrections to Maxwell's <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Leonard, Katie E.; Woodard, R. P.</p>
<p>2012-05-01</p>
<p>We use dimensional regularization to compute the one loop quantum gravitational contribution to the vacuum polarization on flat space background. Adding the appropriate Bogoliubov-Parsiuk-Hepp-Zimmermann counterterm gives a fully renormalized result which we employ to quantum correct Maxwell’s <span class="hlt">equations</span>. These <span class="hlt">equations</span> are solved to show that dynamical photons are unchanged, provided the free state wave functional is appropriately corrected. The response to the instantaneous appearance of a point dipole reveals a perturbative version of the long-conjectured, “smearing of the light cone”. There is no change in the far radiation field produced by an alternating dipole. However, the correction to the static electric field of a point charge shows strengthening at short distances, in contrast to expectations based on the renormalization group. We check for gauge dependence by working out the vacuum polarization in a general 3-parameter family of covariant gauges.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008APS..MAR.P8015L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008APS..MAR.P8015L"><span>A thermodynamic <span class="hlt">equation</span> of jamming</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Lu, Kevin; Pirouz Kavehpour, H.</p>
<p>2008-03-01</p>
<p>Materials ranging from sand to fire-retardant to toothpaste are considered fragile, able to exhibit both solid and fluid-like properties across the jamming transition. Guided by granular flow experiments, our <span class="hlt">equation</span> of jammed states is path-dependent, definable at different athermal equilibrium states. The non-equilibrium thermodynamics based on a structural temperature incorporate physical ageing to address the non-exponential, non-Arrhenious relaxation of granular flows. In short, jamming is simply viewed as a thermodynamic transition that occurs to preserve a positive configurational entropy above absolute zero. Without any free parameters, the proposed <span class="hlt">equation</span>-of-state governs the mechanism of shear-banding and the associated features of shear-softening and thickness-invariance.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016APS..MARS22006K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016APS..MARS22006K"><span>Implementing Parquet <span class="hlt">equations</span> using HPX</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kellar, Samuel; Wagle, Bibek; Yang, Shuxiang; Tam, Ka-Ming; Kaiser, Hartmut; Moreno, Juana; Jarrell, Mark</p>
<p></p>
<p>A new C++ runtime system (HPX) enables simulations of complex systems to run more efficiently on parallel and heterogeneous systems. This increased efficiency allows for solutions to larger simulations of the parquet approximation for a system with impurities. The relevancy of the parquet <span class="hlt">equations</span> depends upon the ability to solve systems which require long runs and large amounts of memory. These limitations, in addition to numerical complications arising from stability of the solutions, necessitate running on large distributed systems. As the computational resources trend towards the exascale and the limitations arising from computational resources vanish efficiency of large scale simulations becomes a focus. HPX facilitates efficient simulations through intelligent overlapping of computation and communication. Simulations such as the parquet <span class="hlt">equations</span> which require the transfer of large amounts of data should benefit from HPX implementations. Supported by the the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013cpgt.book...59S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013cpgt.book...59S"><span>Systems of Inhomogeneous Linear <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Scherer, Philipp O. J.</p>
<p></p>
<p>Many problems in physics and especially computational physics involve systems of linear <span class="hlt">equations</span> which arise e.g. from linearization of a general nonlinear problem or from discretization of differential <span class="hlt">equations</span>. If the dimension of the system is not too large standard methods like Gaussian elimination or QR decomposition are sufficient. Systems with a tridiagonal matrix are important for cubic spline interpolation and numerical second derivatives. They can be solved very efficiently with a specialized Gaussian elimination method. Practical applications often involve very large dimensions and require iterative methods. Convergence of Jacobi and Gauss-Seidel methods is slow and can be improved by relaxation or over-relaxation. An alternative for large systems is the method of conjugate gradients.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19750048365&hterms=Spiegel&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3DSpiegel','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19750048365&hterms=Spiegel&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3DSpiegel"><span>Modal <span class="hlt">equations</span> for cellular convection</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Gough, D. O.; Spiegel, E. A.; Toomre, J.</p>
<p>1975-01-01</p>
<p>We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possible useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of <span class="hlt">equations</span> which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of 'modal <span class="hlt">equations</span>' is analyzed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008EPJB...65..295M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008EPJB...65..295M"><span>Renewal <span class="hlt">equations</span> for option pricing</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Montero, M.</p>
<p>2008-09-01</p>
<p>In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal <span class="hlt">equations</span>, and therefore it enhances the potential use of CTRW techniques in finance. We solve these <span class="hlt">equations</span> for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.</p>
</li>
</ol>
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<ol class="result-class" start="221">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/12059300','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/12059300"><span>Linear superposition in nonlinear <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Khare, Avinash; Sukhatme, Uday</p>
<p>2002-06-17</p>
<p>Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV <span class="hlt">equations</span> and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/15558698','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/15558698"><span>Linear <span class="hlt">equations</span> with random variables.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Tango, Toshiro</p>
<p>2005-10-30</p>
<p>A system of linear <span class="hlt">equations</span> is presented where the unknowns are unobserved values of random variables. A maximum likelihood estimator assuming a multivariate normal distribution and a non-parametric proportional allotment estimator are proposed for the unobserved values of the random variables and for their means. Both estimators can be computed by simple iterative procedures and are shown to perform similarly. The methods are illustrated with data from a national nutrition survey in Japan.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016Fract..2450028S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016Fract..2450028S"><span>Langevin <span class="hlt">Equation</span> on Fractal Curves</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Satin, Seema; Gangal, A. D.</p>
<p>2016-07-01</p>
<p>We analyze random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, plays an important role in this analysis. A Langevin <span class="hlt">equation</span> with a particular model of noise is proposed and solved using techniques of the Fα-Calculus.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1995PhRvD..52.5141O','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1995PhRvD..52.5141O"><span>Instantaneous Bethe-Salpeter <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Olsson, M. G.; Veseli, Siniša.; Williams, Ken</p>
<p>1995-11-01</p>
<p>We present a systematic algebraic and numerical investigation of the instantaneous Beth-Salpeter <span class="hlt">equation</span>. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector-types. We explore the stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions for all quark masses.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/1214628','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/1214628"><span><span class="hlt">Equation</span> of State Project Overview</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Crockett, Scott</p>
<p>2015-09-11</p>
<p>A general overview of the <span class="hlt">Equation</span> of State (EOS) Project will be presented. The goal is to provide the audience with an introduction of what our more advanced methods entail (DFT, QMD, etc.. ) and how these models are being utilized to better constrain the thermodynamic models. These models substantially reduce our regions of interpolation between the various thermodynamic limits. I will also present a variety example of recent EOS work.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19990047906','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19990047906"><span>The Thin Oil Film <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Brown, James L.; Naughton, Jonathan W.</p>
<p>1999-01-01</p>
<p>A thin film of oil on a surface responds primarily to the wall shear stress generated on that surface by a three-dimensional flow. The oil film is also subject to wall pressure gradients, surface tension effects and gravity. The partial differential <span class="hlt">equation</span> governing the oil film flow is shown to be related to Burgers' <span class="hlt">equation</span>. Analytical and numerical methods for solving the thin oil film <span class="hlt">equation</span> are presented. A direct numerical solver is developed where the wall shear stress variation on the surface is known and which solves for the oil film thickness spatial and time variation on the surface. An inverse numerical solver is also developed where the oil film thickness spatial variation over the surface at two discrete times is known and which solves for the wall shear stress variation over the test surface. A One-Time-Level inverse solver is also demonstrated. The inverse numerical solver provides a mathematically rigorous basis for an improved form of a wall shear stress instrument suitable for application to complex three-dimensional flows. To demonstrate the complexity of flows for which these oil film methods are now suitable, extensive examination is accomplished for these analytical and numerical methods as applied to a thin oil film in the vicinity of a three-dimensional saddle of separation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015CMaPh.337.1317K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015CMaPh.337.1317K"><span>Nonlocal <span class="hlt">Equations</span> with Measure Data</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kuusi, Tuomo; Mingione, Giuseppe; Sire, Yannick</p>
<p>2015-08-01</p>
<p>We develop an existence, regularity and potential theory for nonlinear integrodifferential <span class="hlt">equations</span> involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate <span class="hlt">equations</span> established by Boccardo and Gallouët (J Funct Anal 87:149-169, 1989, Partial Differ Equ 17:641-655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591-613, 1992, Acta Math 172:137-161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón-Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321-381, 1965). Our results cover the case of linear nonlocal <span class="hlt">equations</span> with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22308767','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22308767"><span>The complex chemical Langevin <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Schnoerr, David; Sanguinetti, Guido; Grima, Ramon</p>
<p>2014-07-14</p>
<p>The chemical Langevin <span class="hlt">equation</span> (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master <span class="hlt">equation</span>. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master <span class="hlt">equation</span> of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25027995','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25027995"><span>The complex chemical Langevin <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Schnoerr, David; Sanguinetti, Guido; Grima, Ramon</p>
<p>2014-07-14</p>
<p>The chemical Langevin <span class="hlt">equation</span> (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE's main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE's predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master <span class="hlt">equation</span>. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE's accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the "complex CLE" predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master <span class="hlt">equation</span> of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015APS..MAR.V1283S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015APS..MAR.V1283S"><span>Geometric Implications of Maxwell's <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Smith, Felix T.</p>
<p>2015-03-01</p>
<p>Maxwell's synthesis of the varied results of the accumulated knowledge of electricity and magnetism, based largely on the searching insights of Faraday, still provide new issues to explore. A case in point is a well recognized anomaly in the Maxwell <span class="hlt">equations</span>: The laws of electricity and magnetism require two 3-vector and two scalar <span class="hlt">equations</span>, but only six dependent variables are available to be their solutions, the 3-vectors E and B. This leaves an apparent redundancy of two degrees of freedom (J. Rosen, AJP 48, 1071 (1980); Jiang, Wu, Povinelli, J. Comp. Phys. 125, 104 (1996)). The observed self-consistency of the eight <span class="hlt">equations</span> suggests that they contain additional information. This can be sought as a previously unnoticed constraint connecting the space and time variables, r and t. This constraint can be identified. It distorts the otherwise Euclidean 3-space of r with the extremely slight, time dependent curvature k (t) =Rcurv-2 (t) of the 3-space of a hypersphere whose radius has the time dependence dRcurv / dt = +/- c nonrelativistically, or dRcurvLor / dt = +/- ic relativistically. The time dependence is exactly that of the Hubble expansion. Implications of this identification will be explored.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3097485','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3097485"><span>ON THE GENERALISED FANT <span class="hlt">EQUATION</span></span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Howe, M. S.; McGowan, R. S.</p>
<p>2011-01-01</p>
<p>An analysis is made of the fluid-structure interactions involved in the production of voiced speech. It is usual to avoid time consuming numerical simulations of the aeroacoustics of the vocal tract and glottis by the introduction of Fant’s ‘reduced complexity’ <span class="hlt">equation</span> for the glottis volume velocity Q (G. Fant, Acoustic Theory of Speech Production, Mouton, The Hague 1960). A systematic derivation is given of Fant’s <span class="hlt">equation</span> based on the nominally exact <span class="hlt">equations</span> of aerodynamic sound. This can be done with a degree of approximation that depends only on the accuracy with which the time-varying flow geometry and surface-acoustic boundary conditions can be specified, and replaces Fant’s original ‘lumped element’ heuristic approach. The method determines all of the effective ‘source terms’ governing Q. It is illustrated by consideration of a simplified model of the vocal system involving a self-sustaining single-mass model of the vocal folds, that uses free streamline theory to account for surface friction and flow separation within the glottis. Identification is made of a new source term associated with the unsteady vocal fold drag produced by their oscillatory motion transverse to the mean flow. PMID:21603054</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21205250','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21205250"><span>Five-dimensional monopole <span class="hlt">equation</span> with hedgehog ansatz and Abel's differential <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Kihara, Hironobu</p>
<p>2008-06-15</p>
<p>We consider the generalized monopole in the five-dimensional Euclidean space. A numerical solution with the hedgehog ansatz is studied. The Bogomol'nyi <span class="hlt">equation</span> becomes a second-order autonomous nonlinear differential <span class="hlt">equation</span>. The <span class="hlt">equation</span> can be translated into the Abel's differential <span class="hlt">equation</span> of the second kind and is an algebraic differential <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JDE...263...26F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JDE...263...26F"><span>Boundedness of solutions of measure differential <span class="hlt">equations</span> and dynamic <span class="hlt">equations</span> on time scales</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Federson, M.; Grau, R.; Mesquita, J. G.; Toon, E.</p>
<p>2017-07-01</p>
<p>In this paper, we investigate the boundedness results for measure differential <span class="hlt">equations</span>. In order to obtain our results, we use the correspondence between these <span class="hlt">equations</span> and generalized ODEs. Furthermore, we prove our results concerning boundedness of solutions for dynamic <span class="hlt">equations</span> on time scales, using the fact that these <span class="hlt">equations</span> represent a particular case of measure differential <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/772611','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/772611"><span>ADVANCED WAVE-<span class="hlt">EQUATION</span> MIGRATION</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>L. HUANG; M. C. FEHLER</p>
<p>2000-12-01</p>
<p>Wave-<span class="hlt">equation</span> migration methods can more accurately account for complex wave phenomena than ray-tracing-based Kirchhoff methods that are based on the high-frequency asymptotic approximation of waves. With steadily increasing speed of massively parallel computers, wave-<span class="hlt">equation</span> migration methods are becoming more and more feasible and attractive for imaging complex 3D structures. We present an overview of several efficient and accurate wave-<span class="hlt">equation</span>-based migration methods that we have recently developed. The methods are implemented in the frequency-space and frequency-wavenumber domains and hence they are called dual-domain methods. In the methods, we make use of different approximate solutions of the scalar-wave <span class="hlt">equation</span> in heterogeneous media to recursively downward continue wavefields. The approximations used within each extrapolation interval include the Born, quasi-Born, and Rytov approximations. In one of our dual-domain methods, we use an optimized expansion of the square-root operator in the one-way wave <span class="hlt">equation</span> to minimize the phase error for a given model. This leads to a globally optimized Fourier finite-difference method that is a hybrid split-step Fourier and finite-difference scheme. Migration examples demonstrate that our dual-domain migration methods provide more accurate images than those obtained using the split-step Fourier scheme. The Born-based, quasi-Born-based, and Rytov-based methods are suitable for imaging complex structures whose lateral variations are moderate, such as the Marmousi model. For this model, the computational cost of the Born-based method is almost the same as the split-step Fourier scheme, while other methods takes approximately 15-50% more computational time. The globally optimized Fourier finite-difference method significantly improves the accuracy of the split-step Fourier method for imaging structures having strong lateral velocity variations, such as the SEG/EAGE salt model, at an approximately 30% greater</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015ResPh...5..125A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015ResPh...5..125A"><span>Exact solutions to the Benney-Luke <span class="hlt">equation</span> and the Phi-4 <span class="hlt">equations</span> by using modified simple <span class="hlt">equation</span> method</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Akter, Jesmin; Ali Akbar, M.</p>
<p></p>
<p>The modified simple <span class="hlt">equation</span> (MSE) method is a competent and highly effective mathematical tool for extracting exact traveling wave solutions to nonlinear evolution <span class="hlt">equations</span> (NLEEs) arising in science, engineering and mathematical physics. In this article, we implement the MSE method to find the exact solutions involving parameters to NLEEs via the Benney-Luke <span class="hlt">equation</span> and the Phi-4 <span class="hlt">equations</span>. The solitary wave solutions are derived from the exact traveling wave solutions when the parameters receive their special values.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=equations&pg=2&id=EJ865980','ERIC'); return false;" href="http://eric.ed.gov/?q=equations&pg=2&id=EJ865980"><span>On the Inclusion of Difference <span class="hlt">Equation</span> Problems and Z Transform Methods in Sophomore Differential <span class="hlt">Equation</span> Classes</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Savoye, Philippe</p>
<p>2009-01-01</p>
<p>In recent years, I started covering difference <span class="hlt">equations</span> and z transform methods in my introductory differential <span class="hlt">equations</span> course. This allowed my students to extend the "classical" methods for (ordinary differential <span class="hlt">equation</span>) ODE's to discrete time problems arising in many applications.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=differential&pg=3&id=EJ865980','ERIC'); return false;" href="https://eric.ed.gov/?q=differential&pg=3&id=EJ865980"><span>On the Inclusion of Difference <span class="hlt">Equation</span> Problems and Z Transform Methods in Sophomore Differential <span class="hlt">Equation</span> Classes</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Savoye, Philippe</p>
<p>2009-01-01</p>
<p>In recent years, I started covering difference <span class="hlt">equations</span> and z transform methods in my introductory differential <span class="hlt">equations</span> course. This allowed my students to extend the "classical" methods for (ordinary differential <span class="hlt">equation</span>) ODE's to discrete time problems arising in many applications.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26602880','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26602880"><span>[Dosing adjustment and renal function: Which <span class="hlt">equation(s</span>)?].</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Delanaye, Pierre; Flamant, Martin; Cavalier, Étienne; Guerber, Fabrice; Vallotton, Thomas; Moranne, Olivier; Pottel, Hans; Boffa, Jean-Jacques; Mariat, Christophe</p>
<p>2016-02-01</p>
<p>While the CKD-EPI (for Chronic Kidney Disease Epidemiology) <span class="hlt">equation</span> is now implemented worldwide, utilization of the Cockcroft formula is still advocated by some physicians for drug dosage adjustment. Justifications for this recommendation are that the Cockcroft formula was preferentially used to determine dose adjustments according to renal function during the development of many drugs, better predicts drugs-related adverse events and decreases the risk of drug overexposure in the elderly. In this opinion paper, we discuss the weaknesses of the rationale supporting the Cockcroft formula and endorse the French HAS (Haute Autorité de santé) recommendation regarding the preferential use of the CKD-EPI <span class="hlt">equation</span>. When glomerular filtration rate (GFR) is estimated in order to adjust drug dosage, the CKD-EPI value should be re-expressed for the individual body surface area (BSA). Given the difficulty to accurately estimate GFR in the elderly and in individuals with extra-normal BSA, we recommend to prescribe in priority monitorable drugs in those populations or to determine their "true" GFR using a direct measurement method.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=stratification&pg=7&id=EJ895495','ERIC'); return false;" href="http://eric.ed.gov/?q=stratification&pg=7&id=EJ895495"><span>New <span class="hlt">Equating</span> Methods and Their Relationships with Levine Observed Score Linear <span class="hlt">Equating</span> under the Kernel <span class="hlt">Equating</span> Framework</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Chen, Haiwen; Holland, Paul</p>
<p>2010-01-01</p>
<p>In this paper, we develop a new curvilinear <span class="hlt">equating</span> for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score <span class="hlt">equating</span>. In fact, by applying both the kernel <span class="hlt">equating</span> framework and the mean preserving linear transformation of…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25544787','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25544787"><span>Generalized Ordinary Differential <span class="hlt">Equation</span> Models.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Miao, Hongyu; Wu, Hulin; Xue, Hongqi</p>
<p>2014-10-01</p>
<p>Existing estimation methods for ordinary differential <span class="hlt">equation</span> (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.</p>
</li>
</ol>
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<ol class="result-class" start="241">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JPCM...28m5001M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JPCM...28m5001M"><span>Young’s <span class="hlt">equation</span> revisited</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Makkonen, Lasse</p>
<p>2016-04-01</p>
<p>Young’s construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young’s <span class="hlt">equation</span> can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/12688877','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/12688877"><span>Discrete Boltzmann <span class="hlt">equation</span> for microfluidics.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Li, Baoming; Kwok, Daniel Y</p>
<p>2003-03-28</p>
<p>We propose a discrete Boltzmann model for microfluidics based on the Boltzmann <span class="hlt">equation</span> with external forces using a single relaxation time collision model. Considering the electrostatic interactions in microfluidics systems, we introduce an equilibrium distribution function that differs from the Maxwell-Boltzmann distribution by an exponential factor to represent the action of an external force field. A statistical mechanical approach is applied to derive the equivalent external acceleration force exerting on the lattice particles based on a mean-field approximation, resulting from the electro-static potential energy and intermolecular potential energy between fluid-fluid and fluid-substrate interactions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015AmJPh..83..935P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015AmJPh..83..935P"><span>Advanced lab on Fresnel <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Petrova-Mayor, Anna; Gimbal, Scott</p>
<p>2015-11-01</p>
<p>This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel <span class="hlt">equations</span> for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/240383','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/240383"><span>Asymptotics of radial wave <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Morehead, J.J.</p>
<p>1995-10-01</p>
<p>The Langer modification is an improvement in the WKB analysis of the radial Schroedinger <span class="hlt">equation</span>. We derive a generalization of the Langer modification to any radial operator. For differential operators we write the modified classical symbols explicitly and show that the WKB wavefunctions with the modification have the exact limiting behavior for small radius. Unlike in the Schroedinger case, generally the modified radial analysis is not equivalent to the WKB analysis of the full problem before reduction by the spherical symmetry. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/1238605','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/1238605"><span>Germanium multiphase <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Crockett, Scott D.; Lorenzi-Venneri, Giulia De; Kress, Joel D.; Rudin, Sven P.</p>
<p>2014-05-07</p>
<p>A new SESAME multiphase germanium <span class="hlt">equation</span> of state (EOS) has been developed using the best available experimental data and density functional theory (DFT) calculations. The equilibrium EOS includes the Ge I (diamond), the Ge II (β-Sn) and the liquid phases. The foundation of the EOS is based on density functional theory calculations which are used to determine the cold curve and the Debye temperature. Results are compared to Hugoniot data through the solid-solid and solid-liquid transitions. We propose some experiments to better understand the dynamics of this element</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013APS..SHK.V2005C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013APS..SHK.V2005C"><span>Germanium Multiphase <span class="hlt">Equation</span> of State</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Crockett, Scott; Kress, Joel; Rudin, Sven; de Lorenzi-Venneri, Giulia</p>
<p>2013-06-01</p>
<p>A new SESAME multiphase Germanium <span class="hlt">equation</span> of state (EOS) has been developed utilizing the best experimental data and theoretical calculations. The equilibrium EOS includes the GeI (diamond), GeII (beta-Sn) and liquid phases. We will also explore the meta-stable GeIII (tetragonal) phase of germanium. The theoretical calculations used in constraining the EOS are based on quantum molecular dynamics and density functional theory phonon calculations. We propose some physics rich experiments to better understand the dynamics of this element.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/pages/biblio/1238605-germanium-multiphase-equation-state','SCIGOV-DOEP'); return false;" href="https://www.osti.gov/pages/biblio/1238605-germanium-multiphase-equation-state"><span>Germanium multiphase <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/pages">DOE PAGES</a></p>
<p>Crockett, Scott D.; Lorenzi-Venneri, Giulia De; Kress, Joel D.; ...</p>
<p>2014-05-07</p>
<p>A new SESAME multiphase germanium <span class="hlt">equation</span> of state (EOS) has been developed using the best available experimental data and density functional theory (DFT) calculations. The equilibrium EOS includes the Ge I (diamond), the Ge II (β-Sn) and the liquid phases. The foundation of the EOS is based on density functional theory calculations which are used to determine the cold curve and the Debye temperature. Results are compared to Hugoniot data through the solid-solid and solid-liquid transitions. We propose some experiments to better understand the dynamics of this element</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JPhCS.500c2006C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JPhCS.500c2006C"><span>Germanium multiphase <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Crockett, S. D.; De Lorenzi-Venneri, G.; Kress, J. D.; Rudin, S. P.</p>
<p>2014-05-01</p>
<p>A new SESAME multiphase germanium <span class="hlt">equation</span> of state (EOS) has been developed utilizing the best available experimental data and density functional theory (DFT) calculations. The equilibrium EOS includes the Ge I (diamond), the Ge II (β-Sn) and the liquid phases. The foundation of the EOS is based on density functional theory calculations which are used to determine the cold curve and the Debye temperature. Results are compared to Hugoniot data through the solid-solid and solid-liquid transitions. We propose some experiments to better understand the dynamics of this element.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JGP...113..206M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JGP...113..206M"><span>On third order integrable vector Hamiltonian <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Meshkov, A. G.; Sokolov, V. V.</p>
<p>2017-03-01</p>
<p>A complete list of third order vector Hamiltonian <span class="hlt">equations</span> with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable <span class="hlt">equation</span> on the sphere is found.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.epa.gov/risk/regional-screening-levels-rsls-equations-may-2016','PESTICIDES'); return false;" href="https://www.epa.gov/risk/regional-screening-levels-rsls-equations-may-2016"><span>Regional Screening Levels (RSLs) - <span class="hlt">Equations</span> (May 2016)</span></a></p>
<p><a target="_blank" href="http://www.epa.gov/pesticides/search.htm">EPA Pesticide Factsheets</a></p>
<p></p>
<p></p>
<p>Regional Screening Level RSL <span class="hlt">equations</span> page provides quick access to the <span class="hlt">equations</span> used in the Chemical Risk Assessment preliminary remediation goal PRG risk based concentration RBC and risk calculator for the assessment of human Health.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.epa.gov/risk/regional-screening-levels-rsls-equations-june-2017','PESTICIDES'); return false;" href="https://www.epa.gov/risk/regional-screening-levels-rsls-equations-june-2017"><span>Regional Screening Levels (RSLs) - <span class="hlt">Equations</span> (June 2017 )</span></a></p>
<p><a target="_blank" href="http://www.epa.gov/pesticides/search.htm">EPA Pesticide Factsheets</a></p>
<p></p>
<p></p>
<p>Regional Screening Level RSL <span class="hlt">equations</span> page provides quick access to the <span class="hlt">equations</span> used in the Chemical Risk Assessment preliminary remediation goal PRG risk based concentration RBC and risk calculator for the assessment of human Health.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/10130','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/10130"><span>A net volume <span class="hlt">equation</span> for Northeastern Minnesota.</span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>Gerhard K. Raile</p>
<p>1980-01-01</p>
<p>Describes a net volume <span class="hlt">equation</span> for northeastern Minnesota developed as part of the 1977 Minnesota Forest Inventory. <span class="hlt">Equation</span> coefficients are presented by species groupings for both cubic foot and board foot volumes for five tree classes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016TMP...188.1172B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016TMP...188.1172B"><span>Bilinear approach to the supersymmetric Gardner <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Babalic, C. N.; Carstea, A. S.</p>
<p>2016-08-01</p>
<p>We study a supersymmetric version of the Gardner <span class="hlt">equation</span> (both focusing and defocusing) using the superbilinear formalism. This <span class="hlt">equation</span> is new and cannot be obtained from the supersymmetric modified Korteweg-de Vries <span class="hlt">equation</span> with a nonzero boundary condition. We construct supersymmetric solitons and then by passing to the long-wave limit in the focusing case obtain rational nonsingular solutions. We also discuss the supersymmetric version of the defocusing <span class="hlt">equation</span> and the dynamics of its solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA344449','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA344449"><span>Systems of Nonlinear Hyperbolic Partial Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1997-12-01</p>
<p>McKinney) Travelling wave solutions of the modified Korteweg - deVries -Burgers <span class="hlt">Equation</span> . J. Differential <span class="hlt">Equations</span> , 116 (1995), 448-467. 4. (with D.G...SUBTITLE Systems of Nonlinear Hyperbolic Partial Differential <span class="hlt">Equations</span> 6. AUTHOR’S) Michael Shearer PERFORMING ORGANIZATION NAMES(S) AND...DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This project concerns properties of wave propagation in partial differential <span class="hlt">equations</span> that are nonlinear</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1992TMP....92..697S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1992TMP....92..697S"><span>Symmetry algebras of linear differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Shapovalov, A. V.; Shirokov, I. V.</p>
<p>1992-07-01</p>
<p>The local symmetries of linear differential <span class="hlt">equations</span> are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential <span class="hlt">equations</span>. For a nonparabolic second-order <span class="hlt">equation</span>, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami <span class="hlt">equation</span>, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JDE...263..285C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JDE...263..285C"><span>Periodic solutions of Fokker-Planck <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Chen, Feng; Han, Yuecai; Li, Yong; Yang, Xue</p>
<p>2017-07-01</p>
<p>In this paper, the existence of periodic solutions of Fokker-Planck <span class="hlt">equations</span> is obtained by discussing the existence of periodic solutions in distribution for some stochastic differential <span class="hlt">equations</span>. To prove the existence of periodic solutions in distribution for stochastic differential <span class="hlt">equations</span>, a new criterion analogous to Halanay's criterion is given. Actually, the criterion is similar to a law of large numbers. Based on this criterion, the existence of periodic solutions in distribution for stochastic (functional) differential <span class="hlt">equations</span> is established by Lyapunov's method.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21501353','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21501353"><span>Wave <span class="hlt">equation</span> on spherically symmetric Lorentzian metrics</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Bokhari, Ashfaque H.; Al-Dweik, Ahmad Y.; Zaman, F. D.; Kara, A. H.; Karim, M.</p>
<p>2011-06-15</p>
<p>Wave <span class="hlt">equation</span> on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the <span class="hlt">equation</span> in terms of explicit functions of {theta} and {phi} are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave <span class="hlt">equation</span> are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave <span class="hlt">equation</span> with variable coefficients to the one with constant coefficients.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20100010899','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20100010899"><span>Solving <span class="hlt">Equations</span> of Multibody Dynamics</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Jain, Abhinandan; Lim, Christopher</p>
<p>2007-01-01</p>
<p>Darts++ is a computer program for solving the <span class="hlt">equations</span> of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical <span class="hlt">equations</span> of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/542113','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/542113"><span>Non-markovian boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Kremp, D.; Bonitz, M.; Kraeft, W.D.; Schlanges, M.</p>
<p>1997-08-01</p>
<p>A quantum kinetic <span class="hlt">equation</span> for strongly interacting particles (generalized binary collision approximation, ladder or T-matrix approximation) is derived in the framework of the density operator technique. In contrast to conventional kinetic theory, which is valid on large time scales as compared to the collision (correlation) time only, our approach retains the full time dependencies, especially also on short time scales. This means retardation and memory effects resulting from the dynamics of binary correlations and initial correlations are included. Furthermore, the resulting kinetic <span class="hlt">equation</span> conserves total energy (the sum of kinetic and potential energy). The second aspect of generalization is the inclusion of many-body effects, such as self-energy, i.e., renormalization of single-particle energies and damping. To this end we introduce an improved closure relation to the Bogolyubov{endash}Born{endash}Green{endash}Kirkwood{endash}Yvon hierarchy. Furthermore, in order to express the collision integrals in terms of familiar scattering quantities (Mo/ller operator, T-matrix), we generalize the methods of quantum scattering theory by the inclusion of medium effects. To illustrate the effects of memory and damping, the results of numerical simulations are presented. {copyright} 1997 Academic Press, Inc.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19740002093','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19740002093"><span>Shaped cassegrain reflector antenna. [design <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Rao, B. L. J.</p>
<p>1973-01-01</p>
<p>Design <span class="hlt">equations</span> are developed to compute the reflector surfaces required to produce uniform illumination on the main reflector of a cassegrain system when the feed pattern is specified. The final <span class="hlt">equations</span> are somewhat simple and straightforward to solve (using a computer) compared to the ones which exist already in the literature. Step by step procedure for solving the design <span class="hlt">equations</span> is discussed in detail.</p>
</li>
</ol>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=relative+AND+measurement+AND+error&id=EJ1030020','ERIC'); return false;" href="http://eric.ed.gov/?q=relative+AND+measurement+AND+error&id=EJ1030020"><span>Local Observed-Score Kernel <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Wiberg, Marie; van der Linden, Wim J.; von Davier, Alina A.</p>
<p>2014-01-01</p>
<p>Three local observed-score kernel <span class="hlt">equating</span> methods that integrate methods from the local <span class="hlt">equating</span> and kernel <span class="hlt">equating</span> frameworks are proposed. The new methods were compared with their earlier counterparts with respect to such measures as bias--as defined by Lord's criterion of equity--and percent relative error. The local kernel item response…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=spline&pg=3&id=EJ314651','ERIC'); return false;" href="http://eric.ed.gov/?q=spline&pg=3&id=EJ314651"><span>Effectiveness of Analytic Smoothing in Equipercentile <span class="hlt">Equating</span>.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Kolen, Michael J.</p>
<p>1984-01-01</p>
<p>An analytic procedure for smoothing in equipercentile <span class="hlt">equating</span> using cubic smoothing splines is described and illustrated. The effectiveness of the procedure is judged by comparing the results from smoothed equipercentile <span class="hlt">equating</span> with those from other <span class="hlt">equating</span> methods using multiple cross-validations for a variety of sample sizes. (Author/JKS)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2000CNSNS...5...64Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2000CNSNS...5...64Y"><span>Lattice Boltzmann solver of Rossler <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yan, Guangwu; Ruan, Li</p>
<p>2000-06-01</p>
<p>We proposed a lattice Boltzmann model for the Rossler <span class="hlt">equation</span>. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler <span class="hlt">equation</span> on the mesescopic scale. The numerical results show the method can be used to simulate Rossler <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/EJ1112187.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/EJ1112187.pdf"><span>Students' <span class="hlt">Equation</span> Understanding and Solving in Iran</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Barahmand, Ali; Shahvarani, Ahmad</p>
<p>2014-01-01</p>
<p>The purpose of the present article is to investigate how 15-year-old Iranian students interpret the concept of <span class="hlt">equation</span>, its solution, and studying the relation between the students' <span class="hlt">equation</span> understanding and solving. Data from two <span class="hlt">equation</span>-solving exercises are reported. Data analysis shows that there is a significant relationship between…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3928957','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3928957"><span>Multi-time <span class="hlt">equations</span>, classical and quantum</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Petrat, Sören; Tumulka, Roderich</p>
<p>2014-01-01</p>
<p>Multi-time <span class="hlt">equations</span> are evolution <span class="hlt">equations</span> involving several time variables, one for each particle. Such <span class="hlt">equations</span> have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics. PMID:24711721</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=stocking&pg=7&id=EJ450921','ERIC'); return false;" href="https://eric.ed.gov/?q=stocking&pg=7&id=EJ450921"><span><span class="hlt">Equating</span> Tests under the Graded Response Model.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Baker, Frank B.</p>
<p>1992-01-01</p>
<p>The procedure of M.L. Stocking and F.M. Lord (1983) for computing <span class="hlt">equating</span> coefficients for tests having dichotomously scored items is extended to the case of graded response items. A system of <span class="hlt">equations</span> for obtaining the <span class="hlt">equating</span> coefficients under the graded response model is derived. (SLD)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED445011.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED445011.pdf"><span>Nonequivalent Group <span class="hlt">Equating</span> via 1-P HGLLM.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Chu, Kwang-lee; Kamata, Akihito</p>
<p></p>
<p>The quality of nonequivalent group <span class="hlt">equating</span> by the one-parameter hierarchical generalized linear logistic model (1-P HGLLM) was examined by comparing it with: (1) traditional concurrent <span class="hlt">equating</span>; (2) Stocking-Lord's method; and (3) multiple-group concurrent <span class="hlt">equating</span>. Root mean squared errors (RMSEs) for item parameters indicated that there was no…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016FBS....57..265B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016FBS....57..265B"><span>On Fractional Duffin-Kemmer-Petiau <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bouzid, N.; Merad, M.; Baleanu, D.</p>
<p>2016-04-01</p>
<p>In this paper we treat a fractional bosonic, scalar and vectorial, time <span class="hlt">equation</span> namely Duffin-Kemmer-Petiau <span class="hlt">Equation</span>. The fractional variational principle was used, the fractional Euler-Lagrange <span class="hlt">equations</span> were presented. The wave functions were determined and expressed in terms of Mittag-Leffler function.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017EJPh...38a5602R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017EJPh...38a5602R"><span>Are Maxwell's <span class="hlt">equations</span> Lorentz-covariant?</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Redžić, D. V.</p>
<p>2017-01-01</p>
<p>It is stated in many textbooks that Maxwell's <span class="hlt">equations</span> are manifestly covariant when written down in tensorial form. We recall that tensorial form of Maxwell's <span class="hlt">equations</span> does not secure their tensorial contents; they become covariant by postulating certain transformation properties of field functions. That fact should be stressed when teaching about the covariance of Maxwell's <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/EJ1053715.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/EJ1053715.pdf"><span>On a <span class="hlt">Equation</span> in Finite Algebraically Structures</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Valcan, Dumitru</p>
<p>2013-01-01</p>
<p>Solving <span class="hlt">equations</span> in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial <span class="hlt">equation</span> of the form (1). We specify that this <span class="hlt">equation</span> has been…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=conditioning&pg=5&id=EJ1027949','ERIC'); return false;" href="http://eric.ed.gov/?q=conditioning&pg=5&id=EJ1027949"><span>More Issues in Observed-Score <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>van der Linden, Wim J.</p>
<p>2013-01-01</p>
<p>This article is a response to the commentaries on the position paper on observed-score <span class="hlt">equating</span> by van der Linden (this issue). The response focuses on the more general issues in these commentaries, such as the nature of the observed scores that are <span class="hlt">equated</span>, the importance of test-theory assumptions in <span class="hlt">equating</span>, the necessity to use multiple…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2007CoTPh..47..995Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2007CoTPh..47..995Y"><span>Symmetry Breaking for Black-Scholes <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yang, Xuan-Liu; Zhang, Shun-Li; Qu, Chang-Zheng</p>
<p>2007-06-01</p>
<p>Black-Scholes <span class="hlt">equation</span> is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes <span class="hlt">equation</span> and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting <span class="hlt">equations</span> are obtained.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=3pl+AND+logistics&id=EJ893357','ERIC'); return false;" href="https://eric.ed.gov/?q=3pl+AND+logistics&id=EJ893357"><span>The Effect of Repeaters on <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Kim, HeeKyoung; Kolen, Michael J.</p>
<p>2010-01-01</p>
<p>Test <span class="hlt">equating</span> might be affected by including in the <span class="hlt">equating</span> analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) <span class="hlt">equating</span> using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Cubic+AND+Spline&id=EJ314651','ERIC'); return false;" href="https://eric.ed.gov/?q=Cubic+AND+Spline&id=EJ314651"><span>Effectiveness of Analytic Smoothing in Equipercentile <span class="hlt">Equating</span>.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Kolen, Michael J.</p>
<p>1984-01-01</p>
<p>An analytic procedure for smoothing in equipercentile <span class="hlt">equating</span> using cubic smoothing splines is described and illustrated. The effectiveness of the procedure is judged by comparing the results from smoothed equipercentile <span class="hlt">equating</span> with those from other <span class="hlt">equating</span> methods using multiple cross-validations for a variety of sample sizes. (Author/JKS)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=Medical+AND+College&pg=5&id=EJ893357','ERIC'); return false;" href="http://eric.ed.gov/?q=Medical+AND+College&pg=5&id=EJ893357"><span>The Effect of Repeaters on <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Kim, HeeKyoung; Kolen, Michael J.</p>
<p>2010-01-01</p>
<p>Test <span class="hlt">equating</span> might be affected by including in the <span class="hlt">equating</span> analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) <span class="hlt">equating</span> using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3637690','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3637690"><span>Sparse dynamics for partial differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley</p>
<p>2013-01-01</p>
<p>We investigate the approximate dynamics of several differential <span class="hlt">equations</span> when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential <span class="hlt">equations</span>, which promote sparsity. We find that our method successfully reduces the dynamics of convection <span class="hlt">equations</span>, diffusion <span class="hlt">equations</span>, weak shocks, and vorticity <span class="hlt">equations</span> with high-frequency source terms. PMID:23533273</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JMP....58h1507B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JMP....58h1507B"><span>Linearizability for third order evolution <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Basarab-Horwath, P.; Güngör, F.</p>
<p>2017-08-01</p>
<p>The problem of linearization for third order evolution <span class="hlt">equations</span> is considered. Criteria for testing <span class="hlt">equations</span> for linearity are presented. A class of linearizable <span class="hlt">equations</span> depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to <span class="hlt">equations</span> within the same class by differential substitutions and connection with KdV and mKdV <span class="hlt">equations</span> is also reviewed in this framework.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1992EM%26P...59..211K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1992EM%26P...59..211K"><span>Poynting-Robertson effect. II - Perturbation <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Klacka, J.</p>
<p>1992-12-01</p>
<p>The paper addresses the problem of the complete set of perturbation <span class="hlt">equations</span> of celestial mechanics as applied to the Poynting-Robertson effect. Differential <span class="hlt">equations</span> and initial conditions for them are justified. The sudden beginning of the operation of the Poynting-Robertson effect (e.g., sudden release of dust particles from a comet) is taken into account. Two sets of differential <span class="hlt">equations</span> and initial conditions for them are obtained. Both of them are completely equivalent to Newton's <span class="hlt">equation</span> of motion. It is stressed that the transformation mu yields mu(1-beta) must be made in perturbation <span class="hlt">equations</span> of celestial mechanics.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015CNSNS..20..674S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015CNSNS..20..674S"><span>Lax integrable nonlinear partial difference <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Sahadevan, R.; Nagavigneshwari, G.</p>
<p>2015-03-01</p>
<p>A systematic investigation to derive nonlinear lattice <span class="hlt">equations</span> governed by partial difference <span class="hlt">equations</span> admitting specific Lax representation is presented. Further whether or not the identified lattice <span class="hlt">equations</span> possess other characteristics of integrability namely Consistency Around the Cube (CAC) property and linearizability through a global transformation is analyzed. Also it is presented that how to derive higher order ordinary difference <span class="hlt">equations</span> or mappings from the obtained lattice <span class="hlt">equations</span> through periodic reduction and investigated whether they are measure preserving or linearizable and admit sufficient number of integrals leading to their integrability.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016ArRMA.222..731Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016ArRMA.222..731Y"><span>Spectrum Analysis of Some Kinetic <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yang, Tong; Yu, Hongjun</p>
<p>2016-11-01</p>
<p>We analyze the spectrum structure of some kinetic <span class="hlt">equations</span> qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann <span class="hlt">equation</span> for hard potentials with or without angular cutoff and the Landau <span class="hlt">equation</span> with {γ≥q-2}. As an application, we show that the solutions to these two fundamental <span class="hlt">equations</span> are asymptotically equivalent (mod time decay rate {t^{-5/4}}) as {tto∞} to that of the compressible Navier-Stokes <span class="hlt">equations</span> for initial data around an equilibrium state.</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26745463','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26745463"><span>The Specific Analysis of Structural <span class="hlt">Equation</span> Models.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>McDonald, Roderick P</p>
<p>2004-10-01</p>
<p>Conventional structural <span class="hlt">equation</span> modeling fits a covariance structure implied by the <span class="hlt">equations</span> of the model. This treatment of the model often gives misleading results because overall goodness of fit tests do not focus on the specific constraints implied by the model. An alternative treatment arising from Pearl's directed acyclic graph theory checks identifiability and lists and tests the implied constraints. This approach is complete for Markov models, but has remained incomplete for models with correlated disturbances. Some new algebraic results overcome the limitations of DAG theory and give a specific form of structural <span class="hlt">equation</span> analysis that checks identifiability, tests the implied constraints, <span class="hlt">equation</span> by <span class="hlt">equation</span>, and gives consistent estimators of the parameters in closed form from the <span class="hlt">equations</span>. At present the method is limited to recursive models subject to exclusion conditions. With further work, specific structural <span class="hlt">equation</span> modeling may yield a complete alternative to the present, rather unsatisfactory, global covariance structure analysis.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010MeSol..45..712S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010MeSol..45..712S"><span>Bending <span class="hlt">equation</span> for a quasianisotropic plate</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Shachnev, V. A.</p>
<p>2010-10-01</p>
<p>In the framework of the linear theory of elasticity, an exact bending <span class="hlt">equation</span> is obtained for the median plane of a plate whose material is a monoclinic system with the axis of symmetry perpendicular to the plate plane. As an example, the <span class="hlt">equation</span> of the median plane of an isotropic plate is considered; the operator of this <span class="hlt">equation</span> coincides with the operator of Sophie Germain's approximate <span class="hlt">equation</span>. As the plate thickness tends to zero, the right-hand side of the <span class="hlt">equation</span> is asymptotically equivalent to the right-hand side of the approximate <span class="hlt">equation</span>. In addition, <span class="hlt">equations</span> relating the median plane transverse stresses and the total stresses in the plate boundary planes to the median plane deflexions are obtained.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhyA..429..103W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhyA..429..103W"><span>Exact solution to fractional logistic <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>West, Bruce J.</p>
<p>2015-07-01</p>
<p>The logistic <span class="hlt">equation</span> is one of the most familiar nonlinear differential <span class="hlt">equations</span> in the biological and social sciences. Herein we provide an exact solution to an extension of this <span class="hlt">equation</span> to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic <span class="hlt">equation</span> (FLE) is obtained using the Carleman embedding technique that allows the nonlinear <span class="hlt">equation</span> to be replaced by an infinite-order set of linear <span class="hlt">equations</span>, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009NewA...14..347B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009NewA...14..347B"><span>The <span class="hlt">equations</span> of medieval cosmology</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Buonanno, Roberto; Quercellini, Claudia</p>
<p>2009-04-01</p>
<p>In Dantean cosmography the Universe is described as a series of concentric spheres with all the known planets embedded in their rotation motion, the Earth located at the centre and Lucifer at the centre of the Earth. Beyond these "celestial spheres", Dante represents the "angelic choirs" as other nine spheres surrounding God. The rotation velocity increases with decreasing distance from God, that is with increasing Power (Virtù). We show that, adding Power as an additional fourth dimension to space, the modern <span class="hlt">equations</span> governing the expansion of a closed Universe (i.e. with the density parameter Ω0 > 1) in the space-time, can be applied to the medieval Universe as imaged by Dante in his Divine Comedy. In this representation, the Cosmos acquires a unique description and Lucifer is not located at the centre of the hyperspheres.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4935996','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4935996"><span>Evolution <span class="hlt">equation</span> for quantum coherence</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Hu, Ming-Liang; Fan, Heng</p>
<p>2016-01-01</p>
<p>The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution <span class="hlt">equation</span> of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1994JMP....35.5035M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1994JMP....35.5035M"><span>Interpolation and partial differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Maligranda, Lech; Persson, Lars Erik; Wyller, John</p>
<p>1994-09-01</p>
<p>One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential <span class="hlt">equations</span> (PDEs). Nowadays interpolation theory has been developed in an almost unbelievable way {see the bibliography of Maligranda [Interpolation of Operators and Applications (1926-1990), 2nd ed. (Luleå University, Luleå, 1993), p. 154]}. In this article some model examples are presented which display how powerful this theory is when dealing with PDEs. One main aim is to point out when it suffices to use classical interpolation theory and also to give concrete examples of situations when nonlinear interpolation theory has to be applied. Some historical remarks are also included and the relations to similar results are pointed out.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1348210','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1348210"><span>An <span class="hlt">equation</span> for behavioral contrast.</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Williams, B A; Wixted, J T</p>
<p>1986-01-01</p>
<p>Pigeons were trained on a three-component multiple schedule in which the rates of reinforcement in the various components were systematically varied. Response rates were described by an <span class="hlt">equation</span> that posits that the response-strengthening effects of reinforcement are inversely related to the context of reinforcement in which it occurs, and that the context is calculated as the weighted average of the various sources of reinforcement in the situation. The quality of fits was comparable to that found with previous quantitative analyses of concurrent schedules, especially for relative response rates, with over 90% of the variance accounted for in every case. As with previous research, reinforcements in the component that was to follow received greater weights in determining the context than did reinforcements in the preceding component. PMID:3950534</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21409730','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21409730"><span>Entropic corrections to Friedmann <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Sheykhi, Ahmad</p>
<p>2010-05-15</p>
<p>Recently, Verlinde discussed that gravity can be understood as an entropic force caused by changes in the information associated with the positions of material bodies. In Verlinde's argument, the area law of the black hole entropy plays a crucial role. However, the entropy-area relation can be modified from the inclusion of quantum effects, motivated from the loop quantum gravity. In this note, by employing this modified entropy-area relation, we derive corrections to Newton's law of gravitation as well as modified Friedmann <span class="hlt">equations</span> by adopting the viewpoint that gravity can be emerged as an entropic force. Our study further supports the universality of the log correction and provides a strong consistency check on Verlinde's model.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=spline&id=EJ1027954','ERIC'); return false;" href="http://eric.ed.gov/?q=spline&id=EJ1027954"><span>Adjoined Piecewise Linear Approximations (APLAs) for <span class="hlt">Equating</span>: Accuracy Evaluations of a Postsmoothing <span class="hlt">Equating</span> Method</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Moses, Tim</p>
<p>2013-01-01</p>
<p>The purpose of this study was to evaluate the use of adjoined and piecewise linear approximations (APLAs) of raw equipercentile <span class="hlt">equating</span> functions as a postsmoothing <span class="hlt">equating</span> method. APLAs are less familiar than other postsmoothing <span class="hlt">equating</span> methods (i.e., cubic splines), but their use has been described in historical <span class="hlt">equating</span> practices of…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=kernel&pg=4&id=EJ819612','ERIC'); return false;" href="http://eric.ed.gov/?q=kernel&pg=4&id=EJ819612"><span>A Comparison of the Kernel <span class="hlt">Equating</span> Method with Traditional <span class="hlt">Equating</span> Methods Using SAT[R] Data</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Liu, Jinghua; Low, Albert C.</p>
<p>2008-01-01</p>
<p>This study applied kernel <span class="hlt">equating</span> (KE) in two scenarios: <span class="hlt">equating</span> to a very similar population and <span class="hlt">equating</span> to a very different population, referred to as a distant population, using SAT[R] data. The KE results were compared to the results obtained from analogous traditional <span class="hlt">equating</span> methods in both scenarios. The results indicate that KE results…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Cubic+AND+Spline&id=EJ1027954','ERIC'); return false;" href="https://eric.ed.gov/?q=Cubic+AND+Spline&id=EJ1027954"><span>Adjoined Piecewise Linear Approximations (APLAs) for <span class="hlt">Equating</span>: Accuracy Evaluations of a Postsmoothing <span class="hlt">Equating</span> Method</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Moses, Tim</p>
<p>2013-01-01</p>
<p>The purpose of this study was to evaluate the use of adjoined and piecewise linear approximations (APLAs) of raw equipercentile <span class="hlt">equating</span> functions as a postsmoothing <span class="hlt">equating</span> method. APLAs are less familiar than other postsmoothing <span class="hlt">equating</span> methods (i.e., cubic splines), but their use has been described in historical <span class="hlt">equating</span> practices of…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4701001','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=4701001"><span>Inferring Mathematical <span class="hlt">Equations</span> Using Crowdsourcing</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Wasik, Szymon</p>
<p>2015-01-01</p>
<p>Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game—so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical <span class="hlt">equations</span> that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an <span class="hlt">equation</span> that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players. PMID:26713846</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26713846','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26713846"><span>Inferring Mathematical <span class="hlt">Equations</span> Using Crowdsourcing.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Wasik, Szymon; Fratczak, Filip; Krzyskow, Jakub; Wulnikowski, Jaroslaw</p>
<p>2015-01-01</p>
<p>Crowdsourcing, understood as outsourcing work to a large network of people in the form of an open call, has been utilized successfully many times, including a very interesting concept involving the implementation of computer games with the objective of solving a scientific problem by employing users to play a game-so-called crowdsourced serious games. Our main objective was to verify whether such an approach could be successfully applied to the discovery of mathematical <span class="hlt">equations</span> that explain experimental data gathered during the observation of a given dynamic system. Moreover, we wanted to compare it with an approach based on artificial intelligence that uses symbolic regression to find such formulae automatically. To achieve this, we designed and implemented an Internet game in which players attempt to design a spaceship representing an <span class="hlt">equation</span> that models the observed system. The game was designed while considering that it should be easy to use for people without strong mathematical backgrounds. Moreover, we tried to make use of the collective intelligence observed in crowdsourced systems by enabling many players to collaborate on a single solution. The idea was tested on several hundred players playing almost 10,000 games and conducting a user opinion survey. The results prove that the proposed solution has very high potential. The function generated during weeklong tests was almost as precise as the analytical solution of the model of the system and, up to a certain complexity level of the formulae, it explained data better than the solution generated automatically by Eureqa, the leading software application for the implementation of symbolic regression. Moreover, we observed benefits of using crowdsourcing; the chain of consecutive solutions that led to the best solution was obtained by the continuous collaboration of several players.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23509385','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23509385"><span>Complex PT-symmetric nonlinear Schrödinger <span class="hlt">equation</span> and Burgers <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Yan, Zhenya</p>
<p>2013-04-28</p>
<p>The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) <span class="hlt">equation</span> were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger <span class="hlt">equation</span> and Burgers <span class="hlt">equation</span>. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger <span class="hlt">equations</span> (known as the Gross-Pitaevskii <span class="hlt">equation</span> in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave <span class="hlt">equations</span> starting from both -symmetric (e.g. the KdV <span class="hlt">equation</span>) and non- -symmetric (e.g. the Burgers <span class="hlt">equation</span>) nonlinear wave <span class="hlt">equations</span>. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers <span class="hlt">equation</span> in detail.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014CoTPh..61..203Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014CoTPh..61..203Z"><span>Generation of Nonlinear Evolution <span class="hlt">Equations</span> by Reductions of the Self-Dual Yang—Mills <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zhang, Yu-Feng; Hon-Wah, Tam</p>
<p>2014-02-01</p>
<p>With the help of some reductions of the self-dual Yang Mills (briefly written as sdYM) <span class="hlt">equations</span>, we introduce a Lax pair whose compatibility condition leads to a set of (2 + 1)-dimensional <span class="hlt">equations</span>. Its first reduction gives rise to a generalized variable-coefficient Burgers <span class="hlt">equation</span> with a forced term. Furthermore, the Burgers <span class="hlt">equation</span> again reduces to a forced Burgers <span class="hlt">equation</span> with constant coefficients, the standard Burgers <span class="hlt">equation</span>, the heat <span class="hlt">equation</span>, the Fisher <span class="hlt">equation</span>, and the Huxley <span class="hlt">equation</span>, respectively. The second reduction generates a few new (2 + 1)-dimensional nonlinear integrable systems, in particular, obtains a kind of (2 + 1)-dimensional integrable couplings of a new (2 + 1)-dimensional integrable nonlinear <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/482311','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/482311"><span>Exact and explicit solitary wave solutions to some nonlinear <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Jiefang Zhang</p>
<p>1996-08-01</p>
<p>Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave <span class="hlt">equations</span> in physics and other fields by using a special transformation. These <span class="hlt">equations</span> include the KdV-Burgers <span class="hlt">equation</span>, the MKdV-Burgers <span class="hlt">equation</span>, the combined KdV-MKdV <span class="hlt">equation</span>, the Newell-Whitehead <span class="hlt">equation</span>, the dissipative {Phi}{sup 4}-model <span class="hlt">equation</span>, the generalized Fisher <span class="hlt">equation</span>, and the elastic-medium wave <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014acm..conf..205H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014acm..conf..205H"><span>Dust levitation about Itokawa's <span class="hlt">equator</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hartzell, C.; Zimmerman, M.; Takahashi, Y.</p>
<p>2014-07-01</p>
<p> levitation about Itokawa, we must include accurate plasma and gravity models. We use a 2D PIC code (described in [8]) to model the plasma environment about Itokawa's <span class="hlt">equator</span>. The plasma model includes photoemission and shadowing. Thus, we model the plasma environment for various solar incidence angles. The plasma model gives us the 2D electric field components and the plasma potential. We model the gravity field around the equatorial cross-section using an Interior Gravity model [9]. The gravity model is based on the shape model acquired by the Hayabusa mission team and, unlike other models, is quick and accurate close to the surface of the body. Due to the nonspherical shape of Itokawa, the electrostatic force and the gravity may not be collinear. Given our accurate plasma and gravity environments, we are able to simulate the trajectories of dust grains about the <span class="hlt">equator</span> of Itokawa. When modeling the trajectories of the grains, the current to the grains is calculated using Nitter et al.'s formulation [10] with the plasma sheath parameters provided by our PIC model (i.e., the potential minimum, the potential at the surface, and the sheath type). Additionally, we are able to numerically locate the equilibria about which dust grains may levitate. Interestingly, we observe that equilibria exist for grains up to 20 microns in radius about Itokawa's <span class="hlt">equator</span> when the Sun is illuminating Itokawa's 'otter tail'. This grain size is significantly larger than the stably levitating grains we observed using our 1D plasma and gravity models. Conclusions and Future Work: The possibility of dust levitation above asteroids has implications both for our understanding of their evolution and for the design of future missions to these bodies. Using detailed gravity and plasma models, we are above to propagate the trajectories of dust particles about Itokawa's <span class="hlt">equator</span> and identify the equilibria about which these grains will levitate. Using these simulations, we see that grains up to 20 microns</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010JPhA...43e5203M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010JPhA...43e5203M"><span>A fractional Dirac <span class="hlt">equation</span> and its solution</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Muslih, Sami I.; Agrawal, Om P.; Baleanu, Dumitru</p>
<p>2010-02-01</p>
<p>This paper presents a fractional Dirac <span class="hlt">equation</span> and its solution. The fractional Dirac <span class="hlt">equation</span> may be obtained using a fractional variational principle and a fractional Klein-Gordon <span class="hlt">equation</span>; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange <span class="hlt">equations</span> of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac <span class="hlt">equation</span> of order α. We also use a fractional Klein-Gordon <span class="hlt">equation</span> to obtain the fractional Dirac <span class="hlt">equation</span> which is the same as that obtained using the fractional variational principle. Eigensolutions of this <span class="hlt">equation</span> are presented which follow the same approach as that for the solution of the standard Dirac <span class="hlt">equation</span>. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac <span class="hlt">equation</span> and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/5733762','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/5733762"><span>Remarks on the Kuramoto-Sivashinsky <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Nicolaenko, B.; Scheurer, B.</p>
<p>1983-01-01</p>
<p>We report here a joint work in progress on the Kuramoto-Sivashinsky <span class="hlt">equation</span>. The question we address is the analytical study of a fourth order nonlinear evolution <span class="hlt">equation</span>. This <span class="hlt">equation</span> has been obtained by Sivashinsky in the context of combustion and independently by Kuramoto in the context of reaction diffusion-systems. Both were motivated by (nonlinear) stability of travelling waves. Numerical calculations have been done on this <span class="hlt">equation</span>. All the results seem to indicate a chaotic behavior of the solution. Therefore, the analytical study is of interest in analogy with the Burger's and Navier-Stokes <span class="hlt">equations</span>. Here we give some existence and uniqueness results for the <span class="hlt">equation</span> in space dimension one, and we also study a fractional step method of numerical resolution. In a forthcoming joint paper with R. Temam, we will study the asymptotic behavior, as t approaches infinity, of the solution of (0.1) and give an estimate on the number of determining modes.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21371054','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21371054"><span>Darboux transformation for the NLS <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Aktosun, Tuncay; Mee, Cornelis van der</p>
<p>2010-03-08</p>
<p>We analyze a certain class of integral <span class="hlt">equations</span> associated with Marchenko <span class="hlt">equations</span> and Gel'fand-Levitan <span class="hlt">equations</span>. Such integral <span class="hlt">equations</span> arise through a Fourier transformation on various ordinary differential <span class="hlt">equations</span> involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schroedinger <span class="hlt">equation</span>.</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015AIPC.1682e0007G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015AIPC.1682e0007G"><span>Stochastic differential <span class="hlt">equation</span> model to Prendiville processes</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Granita, Bahar, Arifah</p>
<p>2015-10-01</p>
<p>The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential <span class="hlt">equation</span> (SDE). This paper discusses the stochastic differential <span class="hlt">equation</span> of Prendiville process. The work started with the forward Kolmogorov <span class="hlt">equation</span> in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck <span class="hlt">equation</span> in relation to the stochastic differential <span class="hlt">equation</span> of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential <span class="hlt">equation</span>. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22492505','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22492505"><span>Stochastic differential <span class="hlt">equation</span> model to Prendiville processes</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Granita; Bahar, Arifah</p>
<p>2015-10-22</p>
<p>The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential <span class="hlt">equation</span> (SDE). This paper discusses the stochastic differential <span class="hlt">equation</span> of Prendiville process. The work started with the forward Kolmogorov <span class="hlt">equation</span> in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck <span class="hlt">equation</span> in relation to the stochastic differential <span class="hlt">equation</span> of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential <span class="hlt">equation</span>. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JAP....93.8966B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JAP....93.8966B"><span><span class="hlt">Equation</span> of state of polytetrafluoroethylene</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bourne, N. K.; Gray, G. T.</p>
<p>2003-06-01</p>
<p>The present drive to make munitions as safe as is feasible and to develop predictive models describing their constitutive response, has led to the development and production of plastic bonded explosives and propellants. There is a range of elastomers used as binder materials with the energetic components. One of these is known as Kel-F-800™ (poly-chloro-trifluroethylene) whose structure is in some ways analogous to that of poly-tetrafluoroethylene (PTFE or Teflon). Thus, it is of interest to assess the mechanical behavior of Teflon and to compare the response of five different production Teflon materials, two of which were produced in pedigree form, one as-received product, and two from previous in-depth literature studies. The <span class="hlt">equations</span> of state of these variants were quantified by conducting a series of shock impact experiments in which both pressure-particle velocity and shock velocity-particle velocity dependencies were measured. The compressive behavior of Teflon, based upon the results of this study, appears to be independent of the production route and additives introduced.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1793e0013B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1793e0013B"><span>Silicon nitride <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Brown, Robert C.; Swaminathan, Pazhayannur K.</p>
<p>2017-01-01</p>
<p>This report presents the development of a global, multi-phase <span class="hlt">equation</span> of state (EOS) for the ceramic silicon nitride (Si3N4).1 Structural forms include amorphous silicon nitride normally used as a thin film and three crystalline polymorphs. Crystalline phases include hexagonal α-Si3N4, hexagonal β-Si3N4, and the cubic spinel c-Si3N4. Decomposition at about 1900 °C results in a liquid silicon phase and gas phase products such as molecular nitrogen, atomic nitrogen, and atomic silicon. The silicon nitride EOS was developed using EOSPro which is a new and extended version of the PANDA II code. Both codes are valuable tools and have been used successfully for a variety of material classes. Both PANDA II and EOSPro can generate a tabular EOS that can be used in conjunction with hydrocodes. The paper describes the development efforts for the component solid phases and presents results obtained using the EOSPro phase transition model to investigate the solid-solid phase transitions in relation to the available shock data that have indicated a complex and slow time dependent phase change to the c-Si3N4 phase. Furthermore, the EOSPro mixture model is used to develop a model for the decomposition products; however, the need for a kinetic approach is suggested to combine with the single component solid models to simulate and further investigate the global phase coexistences.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015APS..SHK.W1018S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015APS..SHK.W1018S"><span>Silicon Nitride <span class="hlt">Equation</span> of State</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Swaminathan, Pazhayannur; Brown, Robert</p>
<p>2015-06-01</p>
<p>This report presents the development a global, multi-phase <span class="hlt">equation</span> of state (EOS) for the ceramic silicon nitride (Si3N4) . Structural forms include amorphous silicon nitride normally used as a thin film and three crystalline polymorphs. Crystalline phases include hexagonal α-Si3N4, hexagonalβ-Si3N4, and the cubic spinel c-Si3N4. Decomposition at about 1900 °C results in a liquid silicon phase and gas phase products such as molecular nitrogen, atomic nitrogen, and atomic silicon. The silicon nitride EOS was developed using EOSPro which is a new and extended version of the PANDA II code. Both codes are valuable tools and have been used successfully for a variety of material classes. Both PANDA II and EOSPro can generate a tabular EOS that can be used in conjunction with hydrocodes. The paper describes the development efforts for the component solid phases and presents results obtained using the EOSPro phase transition model to investigate the solid-solid phase transitions in relation to the available shock data. Furthermore, the EOSPro mixture model is used to develop a model for the decomposition products and then combined with the single component solid models to study the global phase diagram. Sponsored by the NASA Goddard Space Flight Center Living With a Star program office.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015CoPhC.197..169E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015CoPhC.197..169E"><span>Solving <span class="hlt">equations</span> through particle dynamics</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Edvardsson, S.; Neuman, M.; Edström, P.; Olin, H.</p>
<p>2015-12-01</p>
<p>The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear <span class="hlt">equations</span>. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm's merits and also various performance comparisons with existent numerical algorithms are provided.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016WRR....52.1070B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016WRR....52.1070B"><span>Stability analysis of ecomorphodynamic <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bärenbold, F.; Crouzy, B.; Perona, P.</p>
<p>2016-02-01</p>
<p>In order to shed light on the influence of riverbed vegetation on river morphodynamics, we perform a linear stability analysis on a minimal model of vegetation dynamics coupled with classical one- and two-dimensional Saint-Venant-Exner <span class="hlt">equations</span> of morphodynamics. Vegetation is modeled as a density field of rigid, nonsubmerged cylinders and affects flow via a roughness change. Furthermore, vegetation is assumed to develop following a logistic dependence and may be uprooted by flow. First, we perform the stability analysis of the reduced one-dimensional framework. As a result of the competitive interaction between vegetation growth and removal through uprooting, we find a domain in the parameter space where originally straight rivers are unstable toward periodic longitudinal patterns. For realistic values of the sediment transport parameter, the dominant longitudinal wavelength is determined by the parameters of the vegetation model. Bed topography is found to adjust to the spatial pattern fixed by vegetation. Subsequently, the stability analysis is repeated for the two-dimensional framework, where the system may evolve toward alternate or multiple bars. On a fixed bed, we find instability toward alternate bars due to flow-vegetation interaction, but no multiple bars. Both alternate and multiple bars are present on a movable, vegetated bed. Finally, we find that the addition of vegetation to a previously unvegetated riverbed favors instability toward alternate bars and thus the development of a single course rather than braiding.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/139096','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/139096"><span>LINPACK. Simultaneous Linear Algebraic <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Miller, M.A.</p>
<p>1990-05-01</p>
<p>LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic <span class="hlt">equations</span>. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/145553','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/145553"><span>LINPACK. Simultaneous Linear Algebraic <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Dongarra, J.J.</p>
<p>1982-05-02</p>
<p>LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic <span class="hlt">equations</span>. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015JDE...259.1542C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015JDE...259.1542C"><span>Wei-Norman <span class="hlt">equations</span> for classical groups</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Charzyński, Szymon; Kuś, Marek</p>
<p>2015-08-01</p>
<p>We show that the nonlinear autonomous Wei-Norman <span class="hlt">equations</span>, expressing the solution of a linear system of non-autonomous <span class="hlt">equations</span> on a Lie algebra, can be reduced to the hierarchy of matrix Riccati <span class="hlt">equations</span> in the case of all classical simple Lie algebras. The result generalizes our previous one concerning the complex Lie algebra of the special linear group. We show that it cannot be extended to all simple Lie algebras, in particular to the exceptional G2 algebra.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19850010342','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19850010342"><span>Chandrasekhar <span class="hlt">equations</span> for infinite dimensional systems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Ito, K.; Powers, R. K.</p>
<p>1985-01-01</p>
<p>Chandrasekhar <span class="hlt">equations</span> are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati <span class="hlt">equation</span> is strongly differentiable in time and one can define a strong solution of the Riccati differential <span class="hlt">equation</span>. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860058247&hterms=Hereditary&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3DHereditary','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860058247&hterms=Hereditary&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3DHereditary"><span>Chandrasekhar <span class="hlt">equations</span> for infinite dimensional systems</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Ito, K.; Powers, R.</p>
<p>1985-01-01</p>
<p>The existence of Chandrasekhar <span class="hlt">equations</span> for linear time-invariant systems defined on Hilbert spaces is investigated. An important consequence is that the solution to the evolutional Riccati <span class="hlt">equation</span> is strongly differentiable in time, and that a strong solution of the Riccati differential <span class="hlt">equation</span> can be defined. A discussion of the linear-quadratic optimal-control problem for hereditary differential systems is also included.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/279712','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/279712"><span>The Husimi representation and the Vlasov <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>LEplattenier, P.; Suraud, E.; Reinhard, P.G.</p>
<p>1995-12-01</p>
<p>We investigate the {ital h} expansion of the Time-Dependent Hartree Fock <span class="hlt">equation</span> in the Wigner and Husimi representations. Both lead formally to the Vlasov <span class="hlt">equation</span> in lowest order. The Husimi representation delivers a more stable expansion in particular when the self-interaction in the mean field is considered. The test particle solution of the Vlasov <span class="hlt">equation</span> turns out to be closely related to the Husimi representation. Copyright {copyright} 1995 Academic Press, Inc.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/1184176','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/1184176"><span>The Boltzmann <span class="hlt">equation</span> in the difference formulation</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Szoke, Abraham; Brooks III, Eugene D.</p>
<p>2015-05-06</p>
<p>First we recall the assumptions that are needed for the validity of the Boltzmann <span class="hlt">equation</span> and for the validity of the compressible Euler <span class="hlt">equations</span>. We then present the difference formulation of these <span class="hlt">equations</span> and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/19830268','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/19830268"><span>Discrete Surface Modelling Using Partial Differential <span class="hlt">Equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Xu, Guoliang; Pan, Qing; Bajaj, Chandrajit L</p>
<p>2006-02-01</p>
<p>We use various nonlinear partial differential <span class="hlt">equations</span> to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear <span class="hlt">equations</span> used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear <span class="hlt">equations</span> are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD1037543','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD1037543"><span>Stochastic Evolution <span class="hlt">Equations</span> Driven by Fractional Noises</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2016-11-28</p>
<p>Stochastic Evolution <span class="hlt">Equations</span> Driven by Fractional Noises We have introduced a modification of the classical Euler numerical scheme for stochastic...of Papers published in peer-reviewed journals: Final Report: Stochastic Evolution <span class="hlt">Equations</span> Driven by Fractional Noises Report Title We have introduced...case the evolution form of the <span class="hlt">equation</span> will involve a Stratonovich integral (or path-wise Young integral). The product can also be interpreted as a</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016CNSNS..36..378L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016CNSNS..36..378L"><span>Exact solutions of population balance <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Lin, Fubiao; Flood, Adrian E.; Meleshko, Sergey V.</p>
<p>2016-07-01</p>
<p>Population balance <span class="hlt">equations</span> have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics, but the lack of analytical solutions necessitates the use of numerical techniques. The one-dimensional homogeneous population balance <span class="hlt">equation</span> with time dependent but size independent growth rate and time dependent nucleation rate is investigated. The corresponding system of <span class="hlt">equations</span> is solved analytically in this paper.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017CNSNS..45..220Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017CNSNS..45..220Z"><span>Lie symmetry analysis of the Heisenberg <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zhao, Zhonglong; Han, Bo</p>
<p>2017-04-01</p>
<p>The Lie symmetry analysis is performed on the Heisenberg <span class="hlt">equation</span> from the statistical physics. Its Lie point symmetries and optimal system of one-dimensional subalgebras are determined. The similarity reductions and invariant solutions are obtained. Using the multipliers, some conservation laws are obtained. We prove that this <span class="hlt">equation</span> is nonlinearly self-adjoint. The conservation laws associated with symmetries of this <span class="hlt">equation</span> are constructed by means of Ibragimov's method.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/11088406','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/11088406"><span>Material <span class="hlt">equations</span> for electromagnetism with toroidal polarizations.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Dubovik, V M; Martsenyuk, M A; Saha, B</p>
<p>2000-06-01</p>
<p>With regard to the toroid contributions, a modified system of <span class="hlt">equations</span> of electrodynamics moving continuous media has been obtained. Alternative formalisms to introduce the toroid moment contributions in the <span class="hlt">equations</span> of electromagnetism has been worked out. The two four-potential formalism has been developed. Lorentz transformation laws for the toroid polarizations has been given. Covariant form of <span class="hlt">equations</span> of electrodynamics of continuous media with toroid polarizations has been written.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19730012763&hterms=boltzmann&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dboltzmann','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19730012763&hterms=boltzmann&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dboltzmann"><span>Approximation method for the kinetic Boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Shakhov, Y. M.</p>
<p>1972-01-01</p>
<p>The further development of a method for approximating the Boltzmann <span class="hlt">equation</span> is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann <span class="hlt">equation</span> to a series of approximating <span class="hlt">equations</span>.</p>
</li>
</ol>
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<div class="col-sm-12">
<ol class="result-class" start="321">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890000628&hterms=packing&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dpacking','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890000628&hterms=packing&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dpacking"><span>Partitioning And Packing <span class="hlt">Equations</span> For Parallel Processing</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Arpasi, Dale J.; Milner, Edward J.</p>
<p>1989-01-01</p>
<p>Algorithm developed to identify parallelism in set of coupled ordinary differential <span class="hlt">equations</span> that describe physical system and to divide set into parallel computational paths, along with parts of solution proceeds independently of others during at least part of time. Path-identifying algorithm creates number of paths consisting of <span class="hlt">equations</span> that must be computed serially and table that gives dependent and independent arguments and "can start," "can end," and "must end" times of each <span class="hlt">equation</span>. "Must end" time used subsequently by packing algorithm.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19890000628&hterms=packing+algorithm&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dpacking%2Balgorithm','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19890000628&hterms=packing+algorithm&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dpacking%2Balgorithm"><span>Partitioning And Packing <span class="hlt">Equations</span> For Parallel Processing</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Arpasi, Dale J.; Milner, Edward J.</p>
<p>1989-01-01</p>
<p>Algorithm developed to identify parallelism in set of coupled ordinary differential <span class="hlt">equations</span> that describe physical system and to divide set into parallel computational paths, along with parts of solution proceeds independently of others during at least part of time. Path-identifying algorithm creates number of paths consisting of <span class="hlt">equations</span> that must be computed serially and table that gives dependent and independent arguments and "can start," "can end," and "must end" times of each <span class="hlt">equation</span>. "Must end" time used subsequently by packing algorithm.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008PhyA..387.6505T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008PhyA..387.6505T"><span>Fokker Planck <span class="hlt">equation</span> with fractional coordinate derivatives</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Tarasov, Vasily E.; Zaslavsky, George M.</p>
<p>2008-11-01</p>
<p>Using the generalized Kolmogorov-Feller <span class="hlt">equation</span> with long-range interaction, we obtain kinetic <span class="hlt">equations</span> with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck <span class="hlt">equation</span> with fractional coordinate derivative of order 1<α<2.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1983PhRvB..27.4475K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1983PhRvB..27.4475K"><span>Renormalization group and linear integral <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Klein, W.</p>
<p>1983-04-01</p>
<p>We develop a position-space renormalization-group transformation which can be employed to study general linear integral <span class="hlt">equations</span>. In this Brief Report we employ our method to study one class of such <span class="hlt">equations</span> pertinent to the equilibrium properties of fluids. The results of applying our method are in excellent agreement with known numerical calculations where they can be compared. We also obtain information about the singular behavior of this type of <span class="hlt">equation</span> which could not be obtained numerically.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/783765','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/783765"><span>Some remarks on unilateral matrix <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Cerchiai, Bianca L.; Zumino, Bruno</p>
<p>2001-02-01</p>
<p>We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix <span class="hlt">equations</span>. These are algebraic <span class="hlt">equations</span> where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such <span class="hlt">equations</span> have appeared in a discussion of generalized Born-Infeld theories. In particular, two <span class="hlt">equations</span>, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017InJPh..91.1089L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017InJPh..91.1089L"><span>About vortex <span class="hlt">equations</span> of two dimensional flows</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Lee, S.; Ryi, S.; Lim, H.</p>
<p>2017-09-01</p>
<p>A method to obtain a time-independent vortex solution of a nonlinear differential <span class="hlt">equation</span> describing two-dimensional flow is investigated. In the usual way, starting from the Navier-Stokes <span class="hlt">equation</span> the vortex <span class="hlt">equation</span> is derived by taking a curl operation. After rearranging the <span class="hlt">equation</span> of the vortex, we get a continuity <span class="hlt">equation</span> or a divergence-free <span class="hlt">equation</span>: partial _1V_1+partial _2V_2=0. Additional irrotationality of V_1 and V_2 leads us to the Cauchy-Riemann condition satisfied by a newly introduced stream function Ψ and velocity potential Φ. As a result, if we know V_1 and V_2 or a combination of two, the differential <span class="hlt">equation</span> is mapped to a lower-order partial differential <span class="hlt">equation</span>. This differential <span class="hlt">equation</span> is the one satisfied by the stream function ψ where the vorticity vector ω is given by -(partial _1^2+partial _2^2) ψ. A simple solution is discussed for the two different limits of viscosity.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvD..91h5024H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvD..91h5024H"><span>Analytic solutions of the relativistic Boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hatta, Yoshitaka; Martinez, Mauricio; Xiao, Bo-Wen</p>
<p>2015-04-01</p>
<p>We present new analytic solutions to the relativistic Boltzmann <span class="hlt">equation</span> within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart <span class="hlt">equation</span> in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic <span class="hlt">equations</span> at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann <span class="hlt">equation</span> which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second-order hydrodynamics and fluid-gravity correspondence.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/20776862','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/20776862"><span>Analytical solution of tt dilepton <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Sonnenschein, Lars</p>
<p>2006-03-01</p>
<p>The top quark antiquark production system in the dilepton decay channel is described by a set of <span class="hlt">equations</span> which is nonlinear in the unknown neutrino momenta. Its most precise and least time consuming solution is of major importance for measurements of top quark properties like the top quark mass and tt spin correlations. The initial system of <span class="hlt">equations</span> can be transformed into two polynomial <span class="hlt">equations</span> with two unknowns by means of elementary algebraic operations. These two polynomials of multidegree two can be reduced to one univariate polynomial of degree four by means of resultants. The obtained quartic <span class="hlt">equation</span> is solved analytically.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0649174','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0649174"><span>ON A NONHOMOGENEOUS RANDOM DIFFUSION <span class="hlt">EQUATION</span>.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>CORRELATION TECHNIQUES, FUNCTIONS(MATHEMATICS)), (*STOCHASTIC PROCESSES, <span class="hlt">EQUATIONS</span>), STATISTICAL FUNCTIONS, PROBABILITY, HILBERT SPACE, GREEN’S FUNCTION, SERIES(MATHEMATICS), ALLOYS, DIFFUSION , INTEGRALS</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/76406','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/76406"><span>Gibbs adsorption and the compressibility <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Aranovich, G.L.; Donohue, M.D.</p>
<p>1995-08-08</p>
<p>A new approach for deriving the <span class="hlt">equation</span> of state is developed. It is shown that the integral in the compressibility <span class="hlt">equation</span> is identical to the isotherm for Gibbs adsorption in radial coordinates. The Henry, Langmuir, and Frumkin adsorption isotherms are converted into <span class="hlt">equations</span> of state. It is shown that using Henry`s law gives an expression for the second virial coefficient that is identical to the result from statistical mechanics. Using the Langmuir isotherm leads to a new analytic expression for the hard-sphere <span class="hlt">equation</span> of state which can be explicit in either pressure or density. The Frumkin isotherm results in a new <span class="hlt">equation</span> of state for the square-well potential fluid. Conversely, new adsorption isotherms can be derived from <span class="hlt">equations</span> of state using the compressibility <span class="hlt">equation</span>. It is shown that the van der Waals <span class="hlt">equation</span> gives an adsorption isotherm <span class="hlt">equation</span> that describes both polymolecular adsorption and the unusual adsorption behavior observed for supercritical fluids. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19750033973&hterms=Nonlinear+partial+differential+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DNonlinear%2Bpartial%2Bdifferential%2Bequations','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19750033973&hterms=Nonlinear+partial+differential+equations&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D10%26Ntt%3DNonlinear%2Bpartial%2Bdifferential%2Bequations"><span>Prolongation structures of nonlinear evolution <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Wahlquist, H. D.; Estabrook, F. B.</p>
<p>1975-01-01</p>
<p>A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential <span class="hlt">equations</span> in two independent variables. When this is applied to the Korteweg-de Vries <span class="hlt">equation</span>, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential <span class="hlt">equations</span> for the potential functions, linear 'inverse scattering' <span class="hlt">equations</span> for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1999NCimB.114.1239Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1999NCimB.114.1239Z"><span>Scalar field in standard cosmology: time <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zecca, A.</p>
<p>1999-11-01</p>
<p>The separated time <span class="hlt">equation</span> relative to the generalized Klein-Gordon <span class="hlt">equation</span> in the Robertson-Walker space-time is integrated in the background of the standard cosmology. The solutions are given in terms of series that are obtained by the usual integration method of differential <span class="hlt">equations</span> with regular singularity. The normalization of the solutions implied by the requirement of second quantization of the scalar field is performed. The result exhausts the requirement of providing an explicit complete set of normal mode solutions of the scalar field <span class="hlt">equation</span> in standard cosmology.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22617267','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22617267"><span>Optimal Control for Stochastic Delay Evolution <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Meng, Qingxin; Shen, Yang</p>
<p>2016-08-15</p>
<p>In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state <span class="hlt">equation</span> is given by a stochastic delay evolution <span class="hlt">equation</span> with random coefficients, and the corresponding adjoint <span class="hlt">equation</span> is given by an anticipated backward stochastic evolution <span class="hlt">equation</span>. We first prove the continuous dependence theorems for stochastic delay evolution <span class="hlt">equations</span> and anticipated backward stochastic evolution <span class="hlt">equations</span>, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution <span class="hlt">equations</span>. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential <span class="hlt">equation</span> with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential <span class="hlt">equation</span> and an optimal harvesting problem are also considered.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhCS.788a2025K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhCS.788a2025K"><span>Spinor representation of Maxwell’s <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kulyabov, D. S.; Korolkova, A. V.; Sevastianov, L. A.</p>
<p>2017-01-01</p>
<p>Spinors are more special objects than tensor. Therefore possess more properties than the more generic objects such as tensors. Thus, the group of Lorentz two-spinors is the covering group of the Lorentz group. Since the Lorentz group is a symmetry group of Maxwell’s <span class="hlt">equations</span>, it is assumed to reasonable to use when writing the Maxwell <span class="hlt">equations</span> Lorentz two-spinors and not tensors. We describe in detail the representation of the Maxwell’s <span class="hlt">equations</span> in the form of Lorentz two-spinors. This representation of Maxwell’s <span class="hlt">equations</span> can be of considerable theoretical interest.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/12513557','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/12513557"><span>Fractional Schrödinger <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Laskin, Nick</p>
<p>2002-11-01</p>
<p>Some properties of the fractional Schrödinger <span class="hlt">equation</span> are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger <span class="hlt">equation</span> we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An <span class="hlt">equation</span> for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrödinger <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017Nonli..30.3932B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017Nonli..30.3932B"><span>Symmetric solutions of evolutionary partial differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bruell, Gabriele; Ehrnström, Mats; Geyer, Anna; Pei, Long</p>
<p>2017-10-01</p>
<p>We show that for a large class of evolutionary nonlinear and nonlocal partial differential <span class="hlt">equations</span>, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the <span class="hlt">equations</span>. The first principle covers <span class="hlt">equations</span> that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes <span class="hlt">equations</span> that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the <span class="hlt">equation</span> contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such <span class="hlt">equations</span> one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known <span class="hlt">equations</span> and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these <span class="hlt">equations</span> may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from Ehrnström et al (2009 Int. Math. Res. Not. 2009 4578–96) to nonlocal <span class="hlt">equations</span>, systems and higher dimensions, as well as a study of the standing and mixed cases.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012EJPh...33..805P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012EJPh...33..805P"><span>Simple derivation of the Lindblad <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Pearle, Philip</p>
<p>2012-07-01</p>
<p>The Lindblad <span class="hlt">equation</span> is an evolution <span class="hlt">equation</span> for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad <span class="hlt">equation</span> are given. The derivation of the Lindblad <span class="hlt">equation</span> presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19800037736&hterms=golberg&qs=N%3D0%26Ntk%3DAuthor-Name%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dgolberg','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19800037736&hterms=golberg&qs=N%3D0%26Ntk%3DAuthor-Name%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dgolberg"><span>Integral <span class="hlt">equations</span> for flows in wind tunnels</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Fromme, J. A.; Golberg, M. A.</p>
<p>1979-01-01</p>
<p>This paper surveys recent work on the use of integral <span class="hlt">equations</span> for the calculation of wind tunnel interference. Due to the large number of possible physical situations, the discussion is limited to two-dimensional subsonic and transonic flows. In the subsonic case, the governing boundary value problems are shown to reduce to a class of Cauchy singular <span class="hlt">equations</span> generalizing the classical airfoil <span class="hlt">equation</span>. The theory and numerical solution are developed in some detail. For transonic flows nonlinear singular <span class="hlt">equations</span> result, and a brief discussion of the work of Kraft and Kraft and Lo on their numerical solution is given. Some typical numerical results are presented and directions for future research are indicated.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008PhDT........19Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008PhDT........19Y"><span>Growth estimates for Dyson-Schwinger <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yeats, Karen Amanda</p>
<p></p>
<p>Dyson-Schwinger <span class="hlt">equations</span> are integral <span class="hlt">equations</span> in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams. Taken as recursive <span class="hlt">equations</span>, the Dyson-Schwinger <span class="hlt">equations</span> describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger <span class="hlt">equations</span> to the following system of differential <span class="hlt">equations</span>, gr1x =Prx- sign srg r1x2 +j∈R sjg j1x x6xgr 1x where r∈R,R is the set of amplitudes of the theory which need renormalization, gr1 is the anomalous dimension associated to r, Pr( x) is a modified version of the function for the primitive skeletons contributing to r, and x is the coupling constant. Next, we approach the new system of differential <span class="hlt">equations</span> as a system of recursive <span class="hlt">equations</span> by expanding gr1x =Sn≥1gr1,nx n . We obtain the radius of convergence of Sgr1,nxn/n! in terms of that of SPrnx n/n! . In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013AIPC.1570..343G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013AIPC.1570..343G"><span>Far field expansion for Hartree type <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Georgiev, V.; Venkov, G.</p>
<p>2013-12-01</p>
<p>We consider the scalar field <span class="hlt">equation</span> -Δu(x)+(1/|x|*u2(x))u(x)-E2u(x)/|x|+u(x) = 0 where u = u(|x|) is a radial positive solution and * is the convolution operator in R3. This <span class="hlt">equation</span> can be rewritten as ordinary differential <span class="hlt">equation</span> -ru"(r)-2u'(r)+r ∫ r∞(1/s-1/r)u2(s)s2dsu(r)+ru(r) = 0 and this note is concerned with asymptotic behavior at infinity of solutions of this <span class="hlt">equation</span>.</p>
</li>
</ol>
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<ol class="result-class" start="341">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/783741','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/783741"><span>Some remarks on unilateral matrix <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Cerchiai, Bianca L.; Zumino, Bruno</p>
<p>2001-02-01</p>
<p>We briefly review the results of our paper LBNL-46775: We study certain solutions of left-unilateral matrix <span class="hlt">equations</span>. These are algebraic <span class="hlt">equations</span> where the coefficients and the unknown are square matrices of the same order, or, more abstractly, elements of an associative, but possibly noncommutative algebra, and all coefficients are on the left. Recently such <span class="hlt">equations</span> have appeared in a discussion of generalized Born-Infeld theories. In particular, two <span class="hlt">equations</span>, their perturbative solutions and the relation between them are studied, applying a unified approach based on the generalized Bezout theorem for matrix polynomials.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22617365','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22617365"><span>On implicit abstract neutral nonlinear differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Hernández, Eduardo; O’Regan, Donal</p>
<p>2016-04-15</p>
<p>In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential <span class="hlt">equations</span>. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral <span class="hlt">equations</span> and we focus on applications to partial “nonlinear” neutral differential <span class="hlt">equations</span>. Some applications involving partial neutral differential <span class="hlt">equations</span> are presented.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JPhCS.681a2025S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JPhCS.681a2025S"><span>Resonance regions of extended Mathieu <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Semyonov, V. P.; Timofeev, A. V.</p>
<p>2016-02-01</p>
<p>One of the mechanisms of energy transfer between degrees of freedom of dusty plasma system is based on parametric resonance. Initial stage of this process can de described by <span class="hlt">equation</span> similar to Mathieu <span class="hlt">equation</span>. Such <span class="hlt">equation</span> is studied by analytical and numerical approach. The numerical solution of the extended Mathieu <span class="hlt">equation</span> is obtained for a wide range of parameter values. Boundaries of resonance regions, growth rates of amplitudes and times of onset are obtained. The energy transfer between the degrees of freedom of dusty plasma system can occur over a wide range of frequencies.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhyA..471..212R','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhyA..471..212R"><span>Generalized Thomas-Fermi <span class="hlt">equations</span> as the Lampariello class of Emden-Fowler <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Rosu, Haret C.; Mancas, Stefan C.</p>
<p>2017-04-01</p>
<p>A one-parameter family of Emden-Fowler <span class="hlt">equations</span> defined by Lampariello's parameter p which, upon using Thomas-Fermi boundary conditions, turns into a set of generalized Thomas-Fermi <span class="hlt">equations</span> comprising the standard Thomas-Fermi <span class="hlt">equation</span> for p = 1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel <span class="hlt">equations</span> whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of <span class="hlt">equations</span> for the standard Thomas-Fermi <span class="hlt">equation</span> and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015ZNatA..70..122K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015ZNatA..70..122K"><span>Exact Travelling Wave Solutions of the Nonlinear Evolution <span class="hlt">Equations</span> by Auxiliary <span class="hlt">Equation</span> Method</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet</p>
<p>2015-10-01</p>
<p>The auxiliary <span class="hlt">equation</span> method presents wide applicability to handling nonlinear wave <span class="hlt">equations</span>. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron <span class="hlt">equation</span>, coupled Higgs <span class="hlt">equation</span>, and equal width wave <span class="hlt">equation</span>. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave <span class="hlt">equations</span> in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21251526','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21251526"><span>Cylindrical nonlinear Schroedinger <span class="hlt">equation</span> versus cylindrical Korteweg-de Vries <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.</p>
<p>2008-10-15</p>
<p>A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) <span class="hlt">equations</span> and the one of cylindrical Korteweg-de Vries (cKdV) <span class="hlt">equations</span> is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS <span class="hlt">equation</span> and generalized KdV <span class="hlt">equation</span>, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS <span class="hlt">equation</span> can be associated with non-stationary soliton-like solutions of cKdV <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1995AIPC..334..897P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1995AIPC..334..897P"><span>Covariant <span class="hlt">equations</span> for the NN-πNN system</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Phillips, D. R.; Afnan, I. R.</p>
<p>1995-05-01</p>
<p>We explain the deficiencies of the current NN-πNN <span class="hlt">equations</span>, sketch the derivation of a set of covariant NN-πNN <span class="hlt">equations</span> and describe the ways in which these <span class="hlt">equations</span> differ from previous sets of covariant <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006AGUFMNG31C1607C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006AGUFMNG31C1607C"><span>Constitutive <span class="hlt">Equation</span> for Anisotropic Rock</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Cazacu, O.</p>
<p>2006-12-01</p>
<p>In many rocks, due to the existence of well-defined fabric elements such as bedding, layering, foliation or lamination planes, or due to the existence of linear structures, anisotropy can be important. The symmetries most frequently encountered are: transverse isotropy and orthotropy. By adopting both theoretical and experimental approaches, many authors have investigated the effect of the presence within the rock of pronounced anisotropic feature on the mechanical behavior in the elastic regime and on strength properties. Fewer attempts however have been made to capture the anisotropy of rocks in the plastic range. In this paper an elastic/viscoplastic non-associated constitutive <span class="hlt">equation</span> for an initially transversely isotropic material is presented. The model captures the observed dependency of the elastic moduli on the stress state. The limit of the elastic domain is given by an yield function whose expression is a priori unknown and is determined from data. The basic assumption adopted is that the type of anisotropy of the rock does not change during the deformation process. The anisotropy is thus described by a fourth order tensor invariant with respect to any transformation belonging to the symmetry group of the material. This tensor is assumed to be constant: it does not depend on time nor on deformation; A is involved in the expression of the flow rule, of the yield function, and of the failure criterion in the form of a transformed stress tensor. The components of the anisotropic tensor A are determined from the compressive strengths in conjunction with an anisotropic short- term failure The irreversibility is supposed to be due to transient creep, the irreversible stress work per unit volume being considered as hardening parameter. The adequacy of the model is demonstrated by applying it to a stratified sedimentary rock, Tournemire shale.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1995JOSAA..12.1254T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1995JOSAA..12.1254T"><span>Relation between the Rayleigh <span class="hlt">equation</span> in diffraction theory and the <span class="hlt">equation</span> based on Green's formula</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Tatarskii, V. I.</p>
<p>1995-06-01</p>
<p>The steps necessary to produce the Rayleigh <span class="hlt">equation</span> that is based on the Rayleigh hypothesis from the <span class="hlt">equation</span> that is based on the Green's formula are shown. First a definition is given for the scattering amplitude that is true not only in the far zone of diffraction but also near the scattering surface. With this definition the Rayleigh <span class="hlt">equation</span> coincides with the rigorous <span class="hlt">equation</span> for the surface secondary sources that is based on Green's formula. The Rayleigh hypothesis is equivalent to substituting the far-zone expression of the scattering amplitude into this rigorous <span class="hlt">equation</span>. In this case it turns out to be the <span class="hlt">equation</span> not for the sources but directly for the scattering amplitude, which is the main advantage of this method. For comparing the Rayleigh <span class="hlt">equation</span> with the initial rigorous <span class="hlt">equation</span>, the Rayleigh <span class="hlt">equation</span> is represented in terms of secondary sources. The kernel of this <span class="hlt">equation</span> contains an integral that converges for positive and diverges for negative values of some parameter. It is shown that if we regularize this integral, defining it for the negative values of this parameter as an analytical continuation from the domain of positive values, this kernel becomes equal to the kernel of the initial rigorous <span class="hlt">equation</span>. It follows that the formal perturbation series for the scattering amplitude obtained from the Rayleigh <span class="hlt">equation</span> and from Green's <span class="hlt">equation</span> always coincide. This means that convergence of the perturbation series is a sufficient condition</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=omega&pg=2&id=EJ857937','ERIC'); return false;" href="http://eric.ed.gov/?q=omega&pg=2&id=EJ857937"><span>The Forced van der Pol <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Fay, Temple H.</p>
<p>2009-01-01</p>
<p>We report on a study of the forced van der Pol <span class="hlt">equation</span> x + [epsilon](x[superscript 2] - 1)x + x = F cos[omega]t, by solving numerically the differential <span class="hlt">equation</span> for a variety of values of the parameters [epsilon], F and [omega]. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment,…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19770021807&hterms=differential+ordinary+equation&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Ddifferential%2Bordinary%2Bequation','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19770021807&hterms=differential+ordinary+equation&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Ddifferential%2Bordinary%2Bequation"><span>MACSYMA's symbolic ordinary differential <span class="hlt">equation</span> solver</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Golden, J. P.</p>
<p>1977-01-01</p>
<p>The MACSYMA's symbolic ordinary differential <span class="hlt">equation</span> solver ODE2 is described. The code for this routine is delineated, which is of interest because it is written in top-level MACSYMA language, and may serve as a good example of programming in that language. Other symbolic ordinary differential <span class="hlt">equation</span> solvers are mentioned.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19810031719&hterms=differential+ordinary+equation+nonlinear&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Ddifferential%2Bordinary%2Bequation%2Bnonlinear','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19810031719&hterms=differential+ordinary+equation+nonlinear&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Ddifferential%2Bordinary%2Bequation%2Bnonlinear"><span>Singular perturbation <span class="hlt">equations</span> for flexible satellites</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Huang, T. C.; Das, A.</p>
<p>1980-01-01</p>
<p>Force <span class="hlt">equations</span> of motion of the individual flexible elements of a satellite were obtained in a previous paper. Moment <span class="hlt">equations</span> of motion of the composite bodies of a flexible satellite are to be developed using two sets of <span class="hlt">equations</span> which form the basic system for any dynamic model of flexible satellites. This basic system consists of a set of N-coupled, nonlinear, ordinary, or partial differential <span class="hlt">equations</span>, for a flexible satellite with n generalized, structural position coordinates. For single composite body satellites, N is equal to (n + 3); for dual-spin systems, N is equal to (n + 9). These <span class="hlt">equations</span> involve time derivatives up to the second order. The study shows a method of avoiding this linearization by reducing the N <span class="hlt">equations</span> to 3 or 9 nonlinear, coupled, first order, ordinary, differential <span class="hlt">equations</span> involving only the angular velocities of the composite bodies. The solutions for these angular velocities lead to linear <span class="hlt">equations</span> in the n generalized structural position coordinates, which can be solved by known methods.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=cameras+AND+work&pg=4&id=EJ831999','ERIC'); return false;" href="http://eric.ed.gov/?q=cameras+AND+work&pg=4&id=EJ831999"><span>Does the Wave <span class="hlt">Equation</span> Really Work?</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Armstead, Donald C.; Karls, Michael A.</p>
<p>2006-01-01</p>
<p>The wave <span class="hlt">equation</span> is a classic partial differential <span class="hlt">equation</span> that one encounters in an introductory course on boundary value problems or mathematical physics, which can be used to describe the vertical displacement of a vibrating string. Using a video camera and Wave-in-Motion software to record displacement data from a vibrating string or spring,…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=oxidation+AND+number&pg=2&id=EJ321581','ERIC'); return false;" href="http://eric.ed.gov/?q=oxidation+AND+number&pg=2&id=EJ321581"><span>How Should <span class="hlt">Equation</span> Balancing Be Taught?</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Porter, Spencer K.</p>
<p>1985-01-01</p>
<p>Matrix methods and oxidation-number methods are currently advocated and used for balancing <span class="hlt">equations</span>. This article shows how balancing <span class="hlt">equations</span> can be introduced by a third method which is related to a fundamental principle, is easy to learn, and is powerful in its application. (JN)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/18521608','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/18521608"><span>Qualitative permanence of Lotka-Volterra <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Hofbauer, Josef; Kon, Ryusuke; Saito, Yasuhisa</p>
<p>2008-12-01</p>
<p>In this paper, we consider permanence of Lotka-Volterra <span class="hlt">equations</span>. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra <span class="hlt">equation</span> whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22212870','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22212870"><span>Entropy viscosity method applied to Euler <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Delchini, M. O.; Ragusa, J. C.; Berry, R. A.</p>
<p>2013-07-01</p>
<p>The entropy viscosity method [4] has been successfully applied to hyperbolic systems of <span class="hlt">equations</span> such as Burgers <span class="hlt">equation</span> and Euler <span class="hlt">equations</span>. The method consists in adding dissipative terms to the governing <span class="hlt">equations</span>, where a viscosity coefficient modulates the amount of dissipation. The entropy viscosity method has been applied to the 1-D Euler <span class="hlt">equations</span> with variable area using a continuous finite element discretization in the MOOSE framework and our results show that it has the ability to efficiently smooth out oscillations and accurately resolve shocks. Two <span class="hlt">equations</span> of state are considered: Ideal Gas and Stiffened Gas <span class="hlt">Equations</span> Of State. Results are provided for a second-order time implicit schemes (BDF2). Some typical Riemann problems are run with the entropy viscosity method to demonstrate some of its features. Then, a 1-D convergent-divergent nozzle is considered with open boundary conditions. The correct steady-state is reached for the liquid and gas phases with a time implicit scheme. The entropy viscosity method correctly behaves in every problem run. For each test problem, results are shown for both <span class="hlt">equations</span> of state considered here. (authors)</p>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19720048095&hterms=basic+algebra&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Dbasic%2Balgebra','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720048095&hterms=basic+algebra&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D30%26Ntt%3Dbasic%2Balgebra"><span>Lie algebras and linear differential <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Brockett, R. W.; Rahimi, A.</p>
<p>1972-01-01</p>
<p>Certain symmetry properties possessed by the solutions of linear differential <span class="hlt">equations</span> are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential <span class="hlt">equation</span> theory.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=consumer+AND+spending&pg=2&id=EJ231058','ERIC'); return false;" href="https://eric.ed.gov/?q=consumer+AND+spending&pg=2&id=EJ231058"><span>Recent Methodological Advances in Economic <span class="hlt">Equation</span> Systems.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Theil, Henri; Clements, Kenneth W.</p>
<p>1980-01-01</p>
<p>Examines economic <span class="hlt">equation</span> systems by describing the simultaneous <span class="hlt">equation</span> model, its application to the economy as a whole, and a systemwide approach to microeconomics. The systems approach focuses on particular segments of the economy such as consumer spending. (Author/KC)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/6967629','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/6967629"><span>Global existence proof for relativistic Boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Dudynski, M. ); Ekiel-Jezewska, M.L. )</p>
<p>1992-02-01</p>
<p>The existence and causality of solutions to the relativistic Boltzmann <span class="hlt">equation</span> in L[sup 1] and in L[sub loc][sup 1] are proved. The solutions are shown to satisfy physically natural a priori bounds, time-independent in L[sup 1]. The results rely upon new techniques developed for the nonrelativistic Boltzmann <span class="hlt">equation</span> by DiPerna and Lions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=complex+AND+number&pg=2&id=EJ908773','ERIC'); return false;" href="http://eric.ed.gov/?q=complex+AND+number&pg=2&id=EJ908773"><span>Solving Cubic <span class="hlt">Equations</span> by Polynomial Decomposition</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Kulkarni, Raghavendra G.</p>
<p>2011-01-01</p>
<p>Several mathematicians struggled to solve cubic <span class="hlt">equations</span>, and in 1515 Scipione del Ferro reportedly solved the cubic while participating in a local mathematical contest, but did not bother to publish his method. Then it was Cardano (1539) who first published the solution to the general cubic <span class="hlt">equation</span> in his book "The Great Art, or, The Rules of…</p>
</li>
</ol>
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<ol class="result-class" start="361">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19810018240','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19810018240"><span>Symbolic Solution of Linear Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Feinberg, R. B.; Grooms, R. G.</p>
<p>1981-01-01</p>
<p>An algorithm for solving linear constant-coefficient ordinary differential <span class="hlt">equations</span> is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21501240','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21501240"><span>On solvable Dirac <span class="hlt">equation</span> with polynomial potentials</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Stachowiak, Tomasz</p>
<p>2011-01-15</p>
<p>One-dimensional Dirac <span class="hlt">equation</span> is analyzed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the <span class="hlt">equation</span> in question is not solvable.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1998TMP...115..737P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1998TMP...115..737P"><span>Diophantine <span class="hlt">equations</span> related to quasicrystals: A note</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Pelantová, E.; Perelomov, A. M.</p>
<p>1998-06-01</p>
<p>We give the general solution of three Diophantine <span class="hlt">equations</span> in the ring of integer of the algebraic number field ${\\bf Q}[{\\sqr 5}]$. These <span class="hlt">equations</span> are related to the problem of determination of the minimum distance in quasicrystals with fivefold symmetry.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010AGUFM.H23E1241T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010AGUFM.H23E1241T"><span>Approximate Solution to the Generalized Boussinesq <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Telyakovskiy, A. S.; Mortensen, J.</p>
<p>2010-12-01</p>
<p>The traditional Boussinesq <span class="hlt">equation</span> describes motion of water in groundwater flows. It models unconfined groundwater flow under the Dupuit assumption that the equipotential lines are vertical, making the flowlines horizontal. The Boussinesq <span class="hlt">equation</span> is a nonlinear diffusion <span class="hlt">equation</span> with diffusivity depending linearly on water head. Here we analyze a generalization of the Boussinesq <span class="hlt">equation</span>, when the diffusivity is a power law function of water head. For example polytropic gases moving through porous media obey this <span class="hlt">equation</span>. Solving this <span class="hlt">equation</span> usually requires numerical approximations, but for certain classes of initial and boundary conditions an approximate analytical solution can be constructed. This work focuses on the latter approach, using the scaling properties of the <span class="hlt">equation</span>. We consider one-dimensional semi-infinite initially empty aquifer with boundary conditions at the inlet in case of cylindrical symmetry. Such situation represents the case of an injection well. Solutions would propagate with the finite speed. We construct an approximate scaling function, and we compare the approximate solution with the direct numerical solutions obtained by using the scaling properties of the <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=Nonlinear+AND+partial+AND+differential+AND+equations&id=EJ892212','ERIC'); return false;" href="http://eric.ed.gov/?q=Nonlinear+AND+partial+AND+differential+AND+equations&id=EJ892212"><span>Solving Differential <span class="hlt">Equations</span> Using Modified Picard Iteration</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Robin, W. A.</p>
<p>2010-01-01</p>
<p>Many classes of differential <span class="hlt">equations</span> are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential <span class="hlt">equations</span> considered include typical initial value, boundary value and eigenvalue problems arising in physics and…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/20150021165','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/20150021165"><span>Energy <span class="hlt">Equation</span> Approximation in Fluid Mechanics</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Goldstein, Arthur W.</p>
<p>1959-01-01</p>
<p>There is some confusion in the literature of fluid mechanics in regard to the correct form of the energy <span class="hlt">equation</span> for the study of the flow of nearly incompressible fluids. Several forms of the energy <span class="hlt">equation</span> and their use are therefore discussed in this note.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA501428','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA501428"><span><span class="hlt">Equation</span> of State of Ballistic Gelatin</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2008-06-23</p>
<p>We determined the <span class="hlt">equation</span> of state for ballistic gelatin using the Brillouin scattering spectroscopy with a diamond anvil cell by measuring the...0 to 100 deg C between ambient and 12 GPa. We analyzed the Brillouin data using a high temperature Vinet <span class="hlt">equation</span> of state and obtained the bulk</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA546054','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA546054"><span><span class="hlt">Equation</span> of State of Ballistic Gelatin (II)</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>2011-01-03</p>
<p>We determined the <span class="hlt">equation</span> of state of ballistic gelatin (20%) using Brillouin scattering spectroscopy with diamond anvil cells by measuring the...purposes, we also measured the pressure dependence of sound velocity of lamb tissues up to 10 GPa. We analyzed the Brillouin data using the Vinet <span class="hlt">equation</span> of state and</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=Euler%2c+AND+Leonhard&id=EJ440172','ERIC'); return false;" href="https://eric.ed.gov/?q=Euler%2c+AND+Leonhard&id=EJ440172"><span>Euler's Amazing Way to Solve <span class="hlt">Equations</span>.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Flusser, Peter</p>
<p>1992-01-01</p>
<p>Presented is a series of examples that illustrate a method of solving <span class="hlt">equations</span> developed by Leonhard Euler based on an unsubstantiated assumption. The method integrates aspects of recursion relations and sequences of converging ratios and can be extended to polynomial <span class="hlt">equation</span> with infinite exponents. (MDH)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/765639','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/765639"><span>Operational <span class="hlt">equations</span> for data in common arrays</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Silver, G.L.</p>
<p>2000-10-01</p>
<p>A new method for interpolating experimental data by means of the shifting operator was introduced in 1985. This report illustrates new interpolating <span class="hlt">equations</span> for data in the five-point rectangle and diamond configurations, new measures of central tendency, and new <span class="hlt">equations</span> for data at the vertices of a cube.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=standard+AND+addition+AND+method&pg=2&id=EJ1037236','ERIC'); return false;" href="https://eric.ed.gov/?q=standard+AND+addition+AND+method&pg=2&id=EJ1037236"><span>Improving the Bandwidth Selection in Kernel <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Andersson, Björn; von Davier, Alina A.</p>
<p>2014-01-01</p>
<p>We investigate the current bandwidth selection methods in kernel <span class="hlt">equating</span> and propose a method based on Silverman's rule of thumb for selecting the bandwidth parameters. In kernel <span class="hlt">equating</span>, the bandwidth parameters have previously been obtained by minimizing a penalty function. This minimization process has been criticized by practitioners…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JPCRD..43d3102S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JPCRD..43d3102S"><span>A Fundamental <span class="hlt">Equation</span> of State for Ethanol</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Schroeder, J. A.; Penoncello, S. G.; Schroeder, J. S.</p>
<p>2014-12-01</p>
<p>The existing fundamental <span class="hlt">equation</span> for ethanol demonstrates undesirable behavior in several areas and especially in the critical region. In addition, new experimental data have become available in the open literature since the publication of the current correlation. The development of a new fundamental <span class="hlt">equation</span> for ethanol, in the form of Helmholtz energy as a function of temperature and density, is presented. New, nonlinear fitting techniques, along with the new experimental data, are shown to improve the behavior of the fundamental <span class="hlt">equation</span>. Ancillary <span class="hlt">equations</span> are developed, including <span class="hlt">equations</span> for vapor pressure, saturated liquid density, saturated vapor density, and ideal gas heat capacity. Both the fundamental and ancillary <span class="hlt">equations</span> are compared to experimental data. The fundamental <span class="hlt">equation</span> can compute densities to within ±0.2%, heat capacities to within ±1%-2%, and speed of sound to within ±1%. Values of the vapor pressure and saturated vapor densities are represented to within ±1% at temperatures of 300 K and above, while saturated liquid densities are represented to within ±0.3% at temperatures of 200 K and above. The uncertainty of all properties is higher in the critical region and near the triple point. The <span class="hlt">equation</span> is valid for pressures up to 280 MPa and temperatures from 160 to 650 K.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=euler&pg=7&id=EJ440172','ERIC'); return false;" href="http://eric.ed.gov/?q=euler&pg=7&id=EJ440172"><span>Euler's Amazing Way to Solve <span class="hlt">Equations</span>.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Flusser, Peter</p>
<p>1992-01-01</p>
<p>Presented is a series of examples that illustrate a method of solving <span class="hlt">equations</span> developed by Leonhard Euler based on an unsubstantiated assumption. The method integrates aspects of recursion relations and sequences of converging ratios and can be extended to polynomial <span class="hlt">equation</span> with infinite exponents. (MDH)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=theory+AND+graphs&pg=7&id=EJ684039','ERIC'); return false;" href="http://eric.ed.gov/?q=theory+AND+graphs&pg=7&id=EJ684039"><span>The Specific Analysis of Structural <span class="hlt">Equation</span> Models</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>McDonald, Roderick P.</p>
<p>2004-01-01</p>
<p>Conventional structural <span class="hlt">equation</span> modeling fits a covariance structure implied by the <span class="hlt">equations</span> of the model. This treatment of the model often gives misleading results because overall goodness of fit tests do not focus on the specific constraints implied by the model. An alternative treatment arising from Pearl's directed acyclic graph theory…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=flexible+AND+hours+AND+own+AND+business&id=ED413447','ERIC'); return false;" href="http://eric.ed.gov/?q=flexible+AND+hours+AND+own+AND+business&id=ED413447"><span>The New Economic <span class="hlt">Equation</span>. Executive Summary.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Joshi, Pamela; Carre, Francoise; Place, Angela; Rayman, Paula</p>
<p></p>
<p>The New Economic <span class="hlt">Equation</span> Project opened in May 1995 with a 3-day working conference for 50 national leaders. The <span class="hlt">equation</span> was defined as follows: economic well-being = integration of work, family, and community. Conference participants identified key economic, work, and family concerns facing the United States today. Outreach activities in…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED462425.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED462425.pdf"><span>IRT <span class="hlt">Equating</span> of the MCAT. MCAT Monograph.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Hendrickson, Amy B.; Kolen, Michael J.</p>
<p></p>
<p>This study compared various <span class="hlt">equating</span> models and procedures for a sample of data from the Medical College Admission Test(MCAT), considering how item response theory (IRT) <span class="hlt">equating</span> results compare with classical equipercentile results and how the results based on use of various IRT models, observed score versus true score, direct versus linked…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19720048095&hterms=linear+algebra&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dlinear%2Balgebra','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19720048095&hterms=linear+algebra&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dlinear%2Balgebra"><span>Lie algebras and linear differential <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Brockett, R. W.; Rahimi, A.</p>
<p>1972-01-01</p>
<p>Certain symmetry properties possessed by the solutions of linear differential <span class="hlt">equations</span> are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential <span class="hlt">equation</span> theory.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/11377','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/11377"><span>A Local Net Volume <span class="hlt">Equation</span> for Iowa</span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>Jerold T. Hahn</p>
<p>1976-01-01</p>
<p>As a part of the 1974 Forest Survey of Iowa, the Station''s Forst Resources Evaluatioin Research Staff developed a merchantable tree volume <span class="hlt">equation</span> and tables of coefficients for Iowa. They were developed for both board-foot (International ?-inch rule) and cubic foot volumes, for several species and species groups of growing-stock trees. The <span class="hlt">equation</span> and...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/10341','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/10341"><span>A net volume <span class="hlt">equation</span> for Indiana.</span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>W. Brad Smith; Carol A. Weist</p>
<p>1982-01-01</p>
<p>Describes a Weibull-type volume <span class="hlt">equation</span> for Indiana developed as part of the ongoing Resource Evaluation research in the Central States. <span class="hlt">Equation</span> coefficients are presented by species groupings for both cubic foot and board foot volumes for three tree class categories.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19950010045','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19950010045"><span><span class="hlt">Equation</span> solvers for distributed-memory computers</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Storaasli, Olaf O.</p>
<p>1994-01-01</p>
<p>A large number of scientific and engineering problems require the rapid solution of large systems of simultaneous <span class="hlt">equations</span>. The performance of parallel computers in this area now dwarfs traditional vector computers by nearly an order of magnitude. This talk describes the major issues involved in parallel <span class="hlt">equation</span> solvers with particular emphasis on the Intel Paragon, IBM SP-1 and SP-2 processors.</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=differential&pg=4&id=EJ892212','ERIC'); return false;" href="https://eric.ed.gov/?q=differential&pg=4&id=EJ892212"><span>Solving Differential <span class="hlt">Equations</span> Using Modified Picard Iteration</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Robin, W. A.</p>
<p>2010-01-01</p>
<p>Many classes of differential <span class="hlt">equations</span> are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential <span class="hlt">equations</span> considered include typical initial value, boundary value and eigenvalue problems arising in physics and…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=kernel&id=EJ1037236','ERIC'); return false;" href="http://eric.ed.gov/?q=kernel&id=EJ1037236"><span>Improving the Bandwidth Selection in Kernel <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Andersson, Björn; von Davier, Alina A.</p>
<p>2014-01-01</p>
<p>We investigate the current bandwidth selection methods in kernel <span class="hlt">equating</span> and propose a method based on Silverman's rule of thumb for selecting the bandwidth parameters. In kernel <span class="hlt">equating</span>, the bandwidth parameters have previously been obtained by minimizing a penalty function. This minimization process has been criticized by practitioners…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19860010881','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19860010881"><span>Breakdown of the conservative potential <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Salas, M. D.; Gumbert, C. R.</p>
<p>1986-01-01</p>
<p>The conservative full-potential <span class="hlt">equation</span> is used to study transonic flow over five airfoil sections. The results of the study indicate that once shock are present in the flow, the qualitative approximation is different from that observed with the Euler <span class="hlt">equations</span>. The difference in behavior of the potential eventually leads to multiple solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED336427.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED336427.pdf"><span>Congeneric Models and Levine's Linear <span class="hlt">Equating</span> Procedures.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Brennan, Robert L.</p>
<p></p>
<p>In 1955, R. Levine introduced two linear <span class="hlt">equating</span> procedures for the common-item non-equivalent populations design. His procedures make the same assumptions about true scores; they differ in terms of the nature of the <span class="hlt">equating</span> function used. In this paper, two parameterizations of a classical congeneric model are introduced to model the variables…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/33038','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/33038"><span>Modeling animal movements using stochastic differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>Haiganoush K. Preisler; Alan A. Ager; Bruce K. Johnson; John G. Kie</p>
<p>2004-01-01</p>
<p>We describe the use of bivariate stochastic differential <span class="hlt">equations</span> (SDE) for modeling movements of 216 radiocollared female Rocky Mountain elk at the Starkey Experimental Forest and Range in northeastern Oregon. Spatially and temporally explicit vector fields were estimated using approximating difference <span class="hlt">equations</span> and nonparametric regression techniques. Estimated...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/1304734','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/1304734"><span>xRage <span class="hlt">Equation</span> of State</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Grove, John W.</p>
<p>2016-08-16</p>
<p>The xRage code supports a variety of hydrodynamic <span class="hlt">equation</span> of state (EOS) models. In practice these are generally accessed in the executing code via a pressure-temperature based table look up. This document will describe the various models supported by these codes and provide details on the algorithms used to evaluate the <span class="hlt">equation</span> of state.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.fs.usda.gov/treesearch/pubs/33580','TREESEARCH'); return false;" href="https://www.fs.usda.gov/treesearch/pubs/33580"><span>Compatible taper <span class="hlt">equation</span> for loblolly pine</span></a></p>
<p><a target="_blank" href="http://www.fs.usda.gov/treesearch/">Treesearch</a></p>
<p>J. P. McClure; R. L. Czaplewski</p>
<p>1986-01-01</p>
<p>Cao's compatible, segmented polynomial taper <span class="hlt">equation</span> (Q. V. Cao, H. E. Burkhart, and T. A. Max. For. Sci. 26: 71-80. 1980) is fitted to a large loblolly pine data set from the southeastern United States. <span class="hlt">Equations</span> are presented that predict diameter at a given height, height to a given top diameter, and volume below a given position on the main stem. All...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2009APS..SHK.W3002C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2009APS..SHK.W3002C"><span>A Gallium Multiphase <span class="hlt">Equation</span> of State</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Crockett, Scott; Greeff, Carl</p>
<p>2009-06-01</p>
<p>A new SESAME multiphase gallium <span class="hlt">equation</span> of state (EOS) has been developed. The <span class="hlt">equation</span> of state includes two of the solid phases (Ga I, Ga III) and a fluid phase. The EOS includes consistent latent heat between the phases. We compare the results to the liquid Hugoniot data. We will also explore refreezing via isentropic release and compression.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010JCoAM.233.1596P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010JCoAM.233.1596P"><span>Trotter products and reaction-diffusion <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Popescu, Emil</p>
<p>2010-01-01</p>
<p>In this paper, we study a class of generalized diffusion-reaction <span class="hlt">equations</span> of the form , where A is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference <span class="hlt">equation</span> for numerical solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014EL....10528001D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014EL....10528001D"><span>On the granular stress-geometry <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>DeGiuli, Eric; Schoof, Christian</p>
<p>2014-01-01</p>
<p>Using discrete calculus, we derive the missing stress-geometry <span class="hlt">equation</span> for rigid granular materials in two dimensions, in the mean-field approximation. We show that i) the <span class="hlt">equation</span> imposes that the voids cannot carry stress, ii) stress transmission is generically elliptic and has a quantitative relation to anisotropic elasticity, and iii) the packing fabric plays an essential role.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JPhCS.766a2029E','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JPhCS.766a2029E"><span>A note on Berwald eikonal <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ekici, Cumali; Muradiye, Çimdiker</p>
<p>2016-10-01</p>
<p>In this study, firstly, we generalize Berwald map by introducing the concept of a Riemannian map. After that we find Berwald eikonal <span class="hlt">equation</span> through using the Berwald map. The eikonal <span class="hlt">equation</span> of geometrical optic that examining light reflects, refracts at smooth, plane interfaces is obtained for Berwald condition.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=graph+AND+theory&pg=7&id=EJ684039','ERIC'); return false;" href="https://eric.ed.gov/?q=graph+AND+theory&pg=7&id=EJ684039"><span>The Specific Analysis of Structural <span class="hlt">Equation</span> Models</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>McDonald, Roderick P.</p>
<p>2004-01-01</p>
<p>Conventional structural <span class="hlt">equation</span> modeling fits a covariance structure implied by the <span class="hlt">equations</span> of the model. This treatment of the model often gives misleading results because overall goodness of fit tests do not focus on the specific constraints implied by the model. An alternative treatment arising from Pearl's directed acyclic graph theory…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://files.eric.ed.gov/fulltext/ED507814.pdf','ERIC'); return false;" href="http://files.eric.ed.gov/fulltext/ED507814.pdf"><span>Construction of Chained True Score Equipercentile <span class="hlt">Equatings</span> under the Kernel <span class="hlt">Equating</span> (KE) Framework and Their Relationship to Levine True Score <span class="hlt">Equating</span>. Research Report. ETS RR-09-24</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Chen, Haiwen; Holland, Paul</p>
<p>2009-01-01</p>
<p>In this paper, we develop a new chained equipercentile <span class="hlt">equating</span> procedure for the nonequivalent groups with anchor test (NEAT) design under the assumptions of the classical test theory model. This new <span class="hlt">equating</span> is named chained true score equipercentile <span class="hlt">equating</span>. We also apply the kernel <span class="hlt">equating</span> framework to this <span class="hlt">equating</span> design, resulting in a…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012PhLA..376.2588T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012PhLA..376.2588T"><span>A generalized fractional sub-<span class="hlt">equation</span> method for fractional differential <span class="hlt">equations</span> with variable coefficients</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong</p>
<p>2012-08-01</p>
<p>In this Letter, a generalized fractional sub-<span class="hlt">equation</span> method is proposed for solving fractional differential <span class="hlt">equations</span> with variable coefficients. Being concise and straightforward, this method is applied to the space-time fractional Gardner <span class="hlt">equation</span> with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential <span class="hlt">equations</span> in mathematical physics.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016IJMPB..3040018M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016IJMPB..3040018M"><span>Lump-type solutions to nonlinear differential <span class="hlt">equations</span> derived from generalized bilinear <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ma, Wen-Xiu; Zhou, Yuan; Dougherty, Rachael</p>
<p>2016-08-01</p>
<p>Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential <span class="hlt">equations</span> derived from generalized bilinear differential <span class="hlt">equations</span>. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential <span class="hlt">equations</span> on f are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential <span class="hlt">equations</span> on u, generated from taking a transformation of dependent variables u = 2(ln f)x.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JPhA...39.1151M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JPhA...39.1151M"><span>Third-order integrable difference <span class="hlt">equations</span> generated by a pair of second-order <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Matsukidaira, Junta; Takahashi, Daisuke</p>
<p>2006-02-01</p>
<p>We show that the third-order difference <span class="hlt">equations</span> proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference <span class="hlt">equations</span>. In some cases, the pair of the second-order <span class="hlt">equations</span> are equivalent to the Quispel-Robert-Thomson (QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1863I0005Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1863I0005Y"><span>Discrete fractional solutions of the radial <span class="hlt">equation</span> of the fractional Schrödinger <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yilmazer, Resat; Ozturk, Okkes</p>
<p>2017-07-01</p>
<p>One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus (DFC) has also an important position in the fractional calculus. The nabla operator in DFC is practical for the singular differential <span class="hlt">equations</span>. In this study, we investigated the radial <span class="hlt">equation</span> of the fractional Schrödinger <span class="hlt">equation</span>. The particular solutions of this <span class="hlt">equation</span> was obtained as discrete fractional forms via ∇-discrete fractional operator out of known methods.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19740041251&hterms=burger&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dburger','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19740041251&hterms=burger&qs=N%3D0%26Ntk%3DAll%26Ntx%3Dmode%2Bmatchall%26Ntt%3Dburger"><span>Partial implicitization. [numerical stability of Burger <span class="hlt">equation</span> model for Navier-Stokes <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Graves, R. A., Jr.</p>
<p>1973-01-01</p>
<p>The steady-state solution to the full Navier-Stokes <span class="hlt">equations</span> for complicated flows is generally difficult to obtain. The Burgers (1948) <span class="hlt">equation</span> is used as a model of the Navier-Stokes <span class="hlt">equations</span>. The steady-state solution is obtained by a one-step explicit technique resulting from a partial implicitization of the difference <span class="hlt">equation</span>. Stability analysis shows that the technique is unconditionally stable, and numerical tests show the technique to be accurate.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014JMP....55h3301A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014JMP....55h3301A"><span>Classical non-Markovian Boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Alexanian, Moorad</p>
<p>2014-08-01</p>
<p>The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion <span class="hlt">equations</span> with memory in space and time. The usual Boltzmann <span class="hlt">equation</span>, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann <span class="hlt">equation</span> with resulting transport <span class="hlt">equations</span> for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion <span class="hlt">equations</span> for modeling transport in terms of spatial and temporal fractional derivatives.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19740030149&hterms=differential+ordinary+equation+nonlinear&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Ddifferential%2Bordinary%2Bequation%2Bnonlinear','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19740030149&hterms=differential+ordinary+equation+nonlinear&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3Ddifferential%2Bordinary%2Bequation%2Bnonlinear"><span>Singular perturbation <span class="hlt">equations</span> for flexible satellites</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Huang, T. C.; Das, A.</p>
<p>1973-01-01</p>
<p>The dynamic model of a flexible satellite with n generalized structural position coordinates requires the solution of a set of N coupled nonlinear ordinary or partial differential <span class="hlt">equations</span>. For single composite body satellites, N is equal to (n + 3). For dual-spin systems, N is equal to (n + 9). These <span class="hlt">equations</span> usually involve time derivatives up to the second order. For large values of n, linearization of the system has so far been the only practicable way of solution. The present study shows a method of avoiding this linearization by reducing the N <span class="hlt">equations</span> to three or nine nonlinear, coupled, first-order ordinary differential <span class="hlt">equations</span> involving only the angular velocities of the composite bodies. The solutions for these angular velocities lead to linear <span class="hlt">equations</span> in the n generalized structural position coordinates, which can then be solved by known methods.</p>
</li>
</ol>
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<ol class="result-class" start="401">
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21409048','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21409048"><span><span class="hlt">Equation</span> of state of tracker fields</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Chiba, Takeshi</p>
<p>2010-01-15</p>
<p>We derive the <span class="hlt">equation</span> of state of tracker fields, which are typical examples of freezing quintessence (quintessence with the <span class="hlt">equation</span> of state approaching toward -1), taking into account of the late-time departure from the tracker solution due to the nonzero density parameter of dark energy {Omega}{sub {phi}.} We calculate the <span class="hlt">equation</span> of state as a function of {Omega}{sub {phi}}for constant {Gamma}=VV{sup ''}/(V{sup '}){sup 2} (during matter era) models. The derived <span class="hlt">equation</span> of state contains a single parameter, w{sub (0)}, which parametrizes the <span class="hlt">equation</span> of state during the matter-dominated epoch. We derive observational constraints on w{sub (0)} and find that observational data are consistent with the cosmological constant: -1.11<w{sub (0)}<-0.96(1{sigma}).</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19900001078','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19900001078"><span>Turbulence kinetic energy <span class="hlt">equation</span> for dilute suspensions</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Abou-Arab, T. W.; Roco, M. C.</p>
<p>1989-01-01</p>
<p>A multiphase turbulence closure model is presented which employs one transport <span class="hlt">equation</span>, namely the turbulence kinetic energy <span class="hlt">equation</span>. The proposed form of this <span class="hlt">equation</span> is different from the earlier formulations in some aspects. The power spectrum of the carrier fluid is divided into two regions, which interact in different ways and at different rates with the suspended particles as a function of the particle-eddy size ratio and density ratio. The length scale is described algebraically. A mass/time averaging procedure for the momentum and kinetic energy <span class="hlt">equations</span> is adopted. The resulting turbulence correlations are modeled under less retrictive assumptions comparative to previous work. The closures for the momentum and kinetic energy <span class="hlt">equations</span> are given. Comparisons of the predictions with experimental results on liquid-solid jet and gas-solid pipe flow show satisfactory agreement.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22306199','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22306199"><span>Classical non-Markovian Boltzmann <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Alexanian, Moorad</p>
<p>2014-08-01</p>
<p>The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion <span class="hlt">equations</span> with memory in space and time. The usual Boltzmann <span class="hlt">equation</span>, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann <span class="hlt">equation</span> with resulting transport <span class="hlt">equations</span> for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion <span class="hlt">equations</span> for modeling transport in terms of spatial and temporal fractional derivatives.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012TRACE...5..371Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012TRACE...5..371Y"><span><span class="hlt">Equation</span> of State for Monochloropentafluoroethane (R115)</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yin, Jian-Min; Yada, N.; Watanabe, K.</p>
<p></p>
<p>Based on the available experimental PVT measurements reported in the literature, a modified BWR <span class="hlt">equation</span> of state for refrigerant R115 (C2CIF5) is proposed. The characteristics of derived thermodynamic properties such as the isochoric specific heat capacity, the isobaric specific heat capacity and the speed of sound have been critically examined in developing the present <span class="hlt">equation</span>. The auxiliary saturated liquid density <span class="hlt">equation</span>, which has been used to calculate the saturated thermodynamic properties over a wide range of temperatures, is also developed. The developed <span class="hlt">equation</span> is effective for a range of temperatures from 220K to 450K and of pressures up to 10 MPa which corresponds to the density range up to 1,244 kg/m3. Comparisons with the available <span class="hlt">equations</span> of state are also discussed here.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/19517504','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/19517504"><span>Theory of electrophoresis: fate of one <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Gas, Bohuslav</p>
<p>2009-06-01</p>
<p>Electrophoresis utilizes a difference in movement of charged species in a separation channel or space for their spatial separation. A basic partial differential <span class="hlt">equation</span> that results from the balance laws of continuous processes in separation sciences is the nonlinear conservation law or the continuity <span class="hlt">equation</span>. Attempts at its analytical solution in electrophoresis go back to Kohlrausch's days. The present paper (i) reviews derivation of conservation functions from the conservation law as appeared chronologically, (ii) deals with theory of moving boundary <span class="hlt">equations</span> and, mainly, (iii) presents the linear theory of eigenmobilities. It shows that a basic solution of the linearized continuity <span class="hlt">equations</span> is a set of traveling waves. In particular cases the continuity <span class="hlt">equation</span> can have a resonance solution that leads in practice to schizophrenic dispersion of peaks or a chaotic solution, which causes oscillation of electrolyte solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=426982','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=426982"><span>On Coupled Rate <span class="hlt">Equations</span> with Quadratic Nonlinearities</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Montroll, Elliott W.</p>
<p>1972-01-01</p>
<p>Rate <span class="hlt">equations</span> with quadratic nonlinearities appear in many fields, such as chemical kinetics, population dynamics, transport theory, hydrodynamics, etc. Such <span class="hlt">equations</span>, which may arise from basic principles or which may be phenomenological, are generally solved by linearization and application of perturbation theory. Here, a somewhat different strategy is emphasized. Alternative nonlinear models that can be solved exactly and whose solutions have the qualitative character expected from the original <span class="hlt">equations</span> are first searched for. Then, the original <span class="hlt">equations</span> are treated as perturbations of those of the solvable model. Hence, the function of the perturbation theory is to improve numerical accuracy of solutions, rather than to furnish the basic qualitative behavior of the solutions of the <span class="hlt">equations</span>. PMID:16592013</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/21861561','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/21861561"><span>Ordinary differential <span class="hlt">equation</span> for local accumulation time.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Berezhkovskii, Alexander M</p>
<p>2011-08-21</p>
<p>Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential <span class="hlt">equation</span>. Using this <span class="hlt">equation</span> one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential <span class="hlt">equation</span>, as was done in previous studies. We derive this ordinary differential <span class="hlt">equation</span> together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26026319','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26026319"><span>Riemann <span class="hlt">equation</span> for prime number diffusion.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Chen, Wen; Liang, Yingjie</p>
<p>2015-05-01</p>
<p>This study makes the first attempt to propose the Riemann diffusion <span class="hlt">equation</span> to describe in a manner of partial differential <span class="hlt">equation</span> and interpret in physics of diffusion the classical Riemann method for prime number distribution. The analytical solution of this <span class="hlt">equation</span> is the well-known Riemann representation. The diffusion coefficient is dependent on natural number, a kind of position-dependent diffusivity diffusion. We find that the diffusion coefficient of the Riemann diffusion <span class="hlt">equation</span> is nearly a straight line having a slope 0.99734 in the double-logarithmic axis. Consequently, an approximate solution of the Riemann diffusion <span class="hlt">equation</span> is obtained, which agrees well with the Riemann representation in predicting the prime number distribution. Moreover, we interpret the scale-free property of prime number distribution via a power law function with 1.0169 the scale-free exponent in respect to logarithmic transform of the natural number, and then the fractal characteristic of prime number distribution is disclosed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1984PhRvB..30.1387K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1984PhRvB..30.1387K"><span>Linear integral <span class="hlt">equations</span> and renormalization group</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Klein, W.; Haymet, A. D. J.</p>
<p>1984-08-01</p>
<p>A formulation of the position-space renormalization-group (RG) technique is used to analyze the singular behavior of solutions to a number of integral <span class="hlt">equations</span> used in the theory of the liquid state. In particular, we examine the truncated Kirkwood-Salsburg <span class="hlt">equation</span>, the Ornstein-Zernike <span class="hlt">equation</span>, and a simple nonlinear <span class="hlt">equation</span> used in the mean-field theory of liquids. We discuss the differences in applying the position-space RG to lattice systems and to fluids, and the need for an explicit free-energy rescaling assumption in our formulation of the RG for integral <span class="hlt">equations</span>. Our analysis provides one natural way to define a "fractal" dimension at a phase transition.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19750015139','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19750015139"><span>Almost periodic solutions to difference <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Bayliss, A.</p>
<p>1975-01-01</p>
<p>The theory of Massera and Schaeffer relating the existence of unique almost periodic solutions of an inhomogeneous linear <span class="hlt">equation</span> to an exponential dichotomy for the homogeneous <span class="hlt">equation</span> was completely extended to discretizations by a strongly stable difference scheme. In addition it is shown that the almost periodic sequence solution will converge to the differential <span class="hlt">equation</span> solution. The preceding theory was applied to a class of exponentially stable partial differential <span class="hlt">equations</span> to which one can apply the Hille-Yoshida theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous <span class="hlt">equation</span> (which can be approximated by almost periodic sequences) which are the solutions to appropriate discretizations. Two methods of discretizations are discussed: the strongly stable scheme and the Lax-Wendroff scheme.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=transformation+AND+methods&pg=5&id=EJ1027914','ERIC'); return false;" href="http://eric.ed.gov/?q=transformation+AND+methods&pg=5&id=EJ1027914"><span>Statistical Models and Inference for the True <span class="hlt">Equating</span> Transformation in the Context of Local <span class="hlt">Equating</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>González, B. Jorge; von Davier, Matthias</p>
<p>2013-01-01</p>
<p>Based on Lord's criterion of equity of <span class="hlt">equating</span>, van der Linden (this issue) revisits the so-called local <span class="hlt">equating</span> method and offers alternative as well as new thoughts on several topics including the types of transformations, symmetry, reliability, and population invariance appropriate for <span class="hlt">equating</span>. A remarkable aspect is to define equating…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=terms+AND+examination&pg=2&id=EJ1130875','ERIC'); return false;" href="https://eric.ed.gov/?q=terms+AND+examination&pg=2&id=EJ1130875"><span>A Comparative Analysis of Pre-<span class="hlt">Equating</span> and Post-<span class="hlt">Equating</span> in a Large-Scale Assessment, High Stakes Examination</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Ojerinde, Dibu; Popoola, Omokunmi; Onyeneho, Patrick; Egberongbe, Aminat</p>
<p>2016-01-01</p>
<p>Statistical procedure used in adjusting test score difficulties on test forms is known as "<span class="hlt">equating</span>". <span class="hlt">Equating</span> makes it possible for various test forms to be used interchangeably. In terms of where the <span class="hlt">equating</span> method fits in the assessment cycle, there are pre-<span class="hlt">equating</span> and post-<span class="hlt">equating</span> methods. The major benefits of pre-<span class="hlt">equating</span>, when…</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2013PhDT........11V','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2013PhDT........11V"><span>Solution Methods for Certain Evolution <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Vega-Guzman, Jose Manuel</p>
<p></p>
<p>Solution methods for certain linear and nonlinear evolution <span class="hlt">equations</span> are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential <span class="hlt">equations</span>, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type <span class="hlt">equation</span> on the entire real line. Explicit transformations are used to reduce the <span class="hlt">equations</span> under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic <span class="hlt">equation</span> considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution <span class="hlt">equation</span>. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck <span class="hlt">equation</span>, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type <span class="hlt">equation</span>. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic <span class="hlt">equations</span> is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers <span class="hlt">equation</span> are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008AGUFMED13A0593C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008AGUFMED13A0593C"><span>Turning <span class="hlt">Equations</span> Into Stories: Using "<span class="hlt">Equation</span> Dictionaries" in an Introductory Geophysics Class</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Caplan-Auerbach, J.</p>
<p>2008-12-01</p>
<p>To students with math fear, <span class="hlt">equations</span> can be intimidating and overwhelming. This discomfort is reflected in some of the frequent questions heard in introductory geophysics: "which <span class="hlt">equation</span> should I use?" and "does T stand for travel time or period?" Questions such as these indicate that many students view <span class="hlt">equations</span> as a series of variables and operators rather than as a representation of a physical process. To solve a problem they may simply look for an <span class="hlt">equation</span> with the correct variables and assume that it meets their needs, rather than selecting an <span class="hlt">equation</span> that represents the appropriate physical process. These issues can be addressed by encouraging students to think of <span class="hlt">equations</span> as stories, and to describe them in prose. This is the goal of the <span class="hlt">Equation</span> Dictionary project, used in Western Washington University's introductory geophysics course. Throughout the course, students create personal <span class="hlt">equation</span> dictionaries, adding an entry each time an <span class="hlt">equation</span> is introduced. Entries consist of (a) the <span class="hlt">equation</span> itself, (b) a brief description of <span class="hlt">equation</span> variables, (c) a prose description of the physical process described by the <span class="hlt">equation</span>, and (d) any additional notes that help them understand the <span class="hlt">equation</span>. Thus, rather than simply writing down the <span class="hlt">equations</span> for the velocity of body waves, a student might write "The speed of a seismic body wave is controlled by the material properties of the medium through which it passes." In a study of gravity a student might note that the International Gravity Formula describes "the expected value of g at a given latitude, correcting for Earth's shape and rotation." In writing these definitions students learn that <span class="hlt">equations</span> are simplified descriptions of physical processes, and that understanding the process is more useful than memorizing a sequence of variables. Dictionaries also serve as formula sheets for exams, which encourages students to write definitions that are meaningful to them, and to organize their thoughts clearly. Finally</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhCS.845a2013I','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhCS.845a2013I"><span>The electromagnetic field <span class="hlt">equations</span> for moving media</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ivezić, T.</p>
<p>2017-05-01</p>
<p>In this paper a formulation of the field <span class="hlt">equation</span> for moving media is developed by the generalization of an axiomatic geometric formulation of the electromagnetism in vacuum (Ivezić T 2005 Found. Phys. Lett. 18 401). First, the field <span class="hlt">equations</span> with bivectors F (x) and ℳ(x) are presented and then these <span class="hlt">equations</span> are written with the 4D vectors E(x), B(x), P (x) and M(x). The latter contain both the 4D velocity vector u of a moving medium and the 4D velocity vector v of the observers who measure E and B fields. They do not appear in previous literature. All these <span class="hlt">equations</span> are also written in the standard basis and compared with Maxwell’s <span class="hlt">equations</span> with 3D vectors. In this approach the Ampère-Maxwell law and Gauss’s law are inseparably connected in one law and the same happens with Faraday’s law and the law that expresses the absence of magnetic charge. It is shown that Maxwell’s <span class="hlt">equations</span> with 3D vectors and the field <span class="hlt">equations</span> with 4D geometric quantities are not equivalent in 4D spacetime</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JHEP...10..149D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JHEP...10..149D"><span>General solution of the scattering <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Dolan, Louise; Goddard, Peter</p>
<p>2016-10-01</p>
<p>The scattering <span class="hlt">equations</span>, originally introduced by Fairlie and Roberts in 1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension, have been reformulated in polynomial form. The scattering <span class="hlt">equations</span> for N particles are equivalent to N - 3 polynomial <span class="hlt">equations</span> h m = 0, 1 ≤ m ≤ N - 3, in N - 3 variables, where h m has degree m and is linear in the individual variables. Facilitated by this linearity, elimination theory is used to construct a single variable polynomial <span class="hlt">equation</span>, Δ N = 0, of degree ( N - 3)! determining the solutions. Δ N is the sparse resultant of the system of polynomial scattering <span class="hlt">equations</span> and it can be identified as the hyperdeterminant of a multidimensional matrix of border format within the terminology of Gel'fand, Kapranov and Zelevinsky. Macaulay's Unmixedness Theorem is used to show that the polynomials of the scattering <span class="hlt">equations</span> constitute a regular sequence, enabling the Hilbert series of the variety determined by the scattering <span class="hlt">equations</span> to be calculated, independently showing that they have ( N - 3)! solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006JDE...228..140C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006JDE...228..140C"><span>Stochastic nonhomogeneous incompressible Navier-Stokes <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Cutland, Nigel J.; Enright, Brendan</p>
<p></p>
<p>We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes <span class="hlt">equations</span> with general multiplicative noise. These <span class="hlt">equations</span> model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic <span class="hlt">equations</span> were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the <span class="hlt">equations</span> of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, <span class="hlt">Equations</span> de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992]. The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes <span class="hlt">equations</span> with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes <span class="hlt">equations</span>, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2001PhFl...13..276H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2001PhFl...13..276H"><span>Next-order structure-function <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hill, Reginald J.; Boratav, Olus N.</p>
<p>2001-01-01</p>
<p>Kolmogorov's <span class="hlt">equation</span> [Dokl. Akad. Nauk SSSR 32, 16 (1941)] relates the two-point second- and third-order velocity structure functions and the energy dissipation rate. The analogous next higher-order two-point <span class="hlt">equation</span> relates the third- and fourth-order velocity structure functions and the structure function of the product of pressure-gradient difference and two factors of velocity difference, denoted Tijk. The <span class="hlt">equation</span> is simplified on the basis of local isotropy. Laboratory and numerical simulation data are used to evaluate and compare terms in the <span class="hlt">equation</span>, examine the balance of the <span class="hlt">equation</span>, and evaluate components of Tijk. Atmospheric surface-layer data are used to evaluate Tijk in the inertial range. Combined with the random sweeping hypothesis, the <span class="hlt">equation</span> relates components of the fourth-order velocity structure function. Data show the resultant error of this application of random sweeping. The next-order <span class="hlt">equation</span> constrains the relationships that have been suggested among components of the fourth-order velocity structure function. The pressure structure function, pressure-gradient correlation, and mean-squared pressure gradient are related to Tijk. Inertial range formulas are discussed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/7112684','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/7112684"><span>The zero dispersion limits of nonlinear wave <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Tso, T.</p>
<p>1992-01-01</p>
<p>In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries <span class="hlt">equation</span>, the Benjamin-Bona-Mahony <span class="hlt">equation</span>, and the nonlinear Schroedinger <span class="hlt">equation</span> for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV <span class="hlt">equation</span> obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM <span class="hlt">equation</span> as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation <span class="hlt">equations</span> of the KdV <span class="hlt">equation</span>, the BBM <span class="hlt">equation</span>, the Klein-Gordon <span class="hlt">equation</span>, the NLS <span class="hlt">equation</span>, the mKdV <span class="hlt">equation</span>, and the P-system. It is shown that the modulation <span class="hlt">equations</span> of the KdV <span class="hlt">equation</span>, the K-G <span class="hlt">equation</span>, the NLS <span class="hlt">equation</span>, and the mKdV <span class="hlt">equation</span> are hyperbolic but those of the BBM <span class="hlt">equation</span> and the P-system are not hyperbolic. Also, the relations are studied of the KdV <span class="hlt">equation</span> and the mKdV <span class="hlt">equation</span>. Finally, the author studies the complex mKdV <span class="hlt">equation</span> to compare with the NLS <span class="hlt">equation</span>, and then study the complex gKdV <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008AIPC.1084..224T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008AIPC.1084..224T"><span>Exact Pressure Evolution <span class="hlt">Equation</span> for Incompressible Fluids</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Tessarotto, M.; Ellero, M.; Aslan, N.; Mond, M.; Nicolini, P.</p>
<p>2008-12-01</p>
<p>An important aspect of computational fluid dynamics is related to the determination of the fluid pressure in isothermal incompressible fluids. In particular this concerns the construction of an exact evolution <span class="hlt">equation</span> for the fluid pressure which replaces the Poisson <span class="hlt">equation</span> and yields an algorithm which is a Poisson solver, i.e., it permits to time-advance exactly the same fluid pressure without solving the Poisson <span class="hlt">equation</span>. In fact, the incompressible Navier-Stokes <span class="hlt">equations</span> represent a mixture of hyperbolic and elliptic pde's, which are extremely hard to study both analytically and numerically. This amounts to transform an elliptic type fluid <span class="hlt">equation</span> into a suitable hyperbolic <span class="hlt">equation</span>, a result which usually is reached only by means of an asymptotic formulation. Besides being a still unsolved mathematical problem, the issue is relevant for at least two reasons: a) the proliferation of numerical algorithms in computational fluid dynamics which reproduce the behavior of incompressible fluids only in an asymptotic sense (see below); b) the possible verification of conjectures involving the validity of appropriate <span class="hlt">equations</span> of state for the fluid pressure. Another possible motivation is, of course, the ongoing quest for efficient numerical solution methods to be applied for the construction of the fluid fields {ρ,V,p}, solutions of the initial and boundary-value problem associated to the incompressible N-S <span class="hlt">equations</span> (INSE). In this paper we intend to show that an exact solution to this problem can be achieved adopting the approach based on inverse kinetic theory (IKT) recently developed for incompressible fluids by Tessarotto et al. [7, 6, 7, 8, 9]. In particular we intend to prove that the evolution of the fluid fields can be achieved by means of a suitable dynamical system, to be identified with the so-called Navier-Stokes (N-S) dynamical system. As a consequence it is found that the fluid pressure obeys a well-defined evolution <span class="hlt">equation</span>. The result appears</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25973605','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25973605"><span>A generalized simplest <span class="hlt">equation</span> method and its application to the Boussinesq-Burgers <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Sudao, Bilige; Wang, Xiaomin</p>
<p>2015-01-01</p>
<p>In this paper, a generalized simplest <span class="hlt">equation</span> method is proposed to seek exact solutions of nonlinear evolution <span class="hlt">equations</span> (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary <span class="hlt">equation</span>. This method can yield a Bäcklund transformation between NLEEs and a related constraint <span class="hlt">equation</span>. By dealing with the constraint <span class="hlt">equation</span>, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers <span class="hlt">equation</span> by using the generalized simplest <span class="hlt">equation</span> method.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21067443','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21067443"><span>Generalized Directional Gradients, Backward Stochastic Differential <span class="hlt">Equations</span> and Mild Solutions of Semilinear Parabolic <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Fuhrman, Marco Tessitore, Gianmario</p>
<p>2005-05-15</p>
<p>We study a forward-backward system of stochastic differential <span class="hlt">equations</span> in an infinite-dimensional framework and its relationships with a semilinear parabolic differential <span class="hlt">equation</span> on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic <span class="hlt">equation</span> in a suitable class of locally Lipschitz real functions. The parabolic <span class="hlt">equation</span> is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions.The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay <span class="hlt">equations</span> and reaction-diffusion <span class="hlt">equations</span>),where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic <span class="hlt">equations</span> of Black-Scholes or Hamilton-Jacobi-Bellman type.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1332965','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1332965"><span>An analytic comparison of Herrnstein's <span class="hlt">equations</span> and a multivariate rate <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>McDowell, J. J</p>
<p>1980-01-01</p>
<p>Herrnstein's <span class="hlt">equations</span> are approximations of the multivariate rate <span class="hlt">equation</span> at ordinary rates of reinforcement and responding. The rate <span class="hlt">equation</span> is the result of a linear system analysis of variable-interval performance. Rate <span class="hlt">equation</span> matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate <span class="hlt">equation</span> contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate <span class="hlt">equation</span>, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate <span class="hlt">equation</span> asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding. PMID:16812172</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2007EJASP2007...19B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2007EJASP2007...19B"><span>On the Solution of the Rational Matrix <span class="hlt">Equation[InlineEquation</span> not available: see fulltext.</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Benner, Peter; Faßbender, Heike</p>
<p>2007-12-01</p>
<p>We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix <span class="hlt">equation[InlineEquation</span> not available: see fulltext.], where[Inline<span class="hlt">Equation</span> not available: see fulltext.] is symmetric positive definite and[Inline<span class="hlt">Equation</span> not available: see fulltext.] is nonsingular. Such <span class="hlt">equations</span> arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati <span class="hlt">equation</span> (DARE). We discuss how to use the butterfly[Inline<span class="hlt">Equation</span> not available: see fulltext.] algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/6189747','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/6189747"><span>The gBL transport <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Mynick, H.E.</p>
<p>1989-05-01</p>
<p>The transport <span class="hlt">equations</span> arising from the ''generalized Balescu- Lenard'' (gBL) collision operator are obtained, and some of their properties examined. The <span class="hlt">equations</span> contain neoclassical and turbulent transport as two special cases, having the same structure. The resultant theory offers potential explanation for a number of results not well understood, including the anomalous pinch, observed ratios of Q/GAMMAT on TFTR, and numerical reproduction of ASDEX profiles by a model for turbulent transport invoked without derivation, but by analogy to neoclassical theory. The general <span class="hlt">equations</span> are specialized to consideration of a number of particular transport mechanisms of interest. 10 refs.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3619330','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3619330"><span>Schrödinger <span class="hlt">equation</span> revisited</span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Schleich, Wolfgang P.; Greenberger, Daniel M.; Kobe, Donald H.; Scully, Marlan O.</p>
<p>2013-01-01</p>
<p>The time-dependent Schrödinger <span class="hlt">equation</span> is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger <span class="hlt">equation</span> from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton–Jacobi <span class="hlt">equation</span>. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave. PMID:23509260</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA145812','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA145812"><span>Difference Schemes for <span class="hlt">Equations</span> of Schrodinger Type.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1984-06-01</p>
<p>the numerical solution of the <span class="hlt">equation</span> aa = Aus , (1.1) with A w a + il and a_ 0, and its extension to higher dimensions: Alualu, (1.2) where A, at...definition allows the numerical solution to grow with the number of time steps taken. For <span class="hlt">equations</span> the solutions of which are known to be nonincreasing in...Application, of the Spl’t.step Fourier method to the numerical solution of nonlinear and variable coefficient wave <span class="hlt">equation</span> , SIAM Review, 15 (1973), pp. 423</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19750011026','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19750011026"><span>Formulas for precession. [motion of mean <span class="hlt">equator</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Kinoshita, H.</p>
<p>1975-01-01</p>
<p>Literal expressions for the precessional motion of the mean <span class="hlt">equator</span> referred to an arbitrary epoch are constructed. Their numerical representations, based on numerical values recommended at the working meeting of the International Astronomical Union Commission held in Washington in September 1974, are obtained. In constructing the <span class="hlt">equations</span> of motion, the second-order secular perturbation and the secular perturbation due to the long-periodic terms in the motions of the moon and the sun are taken into account. These perturbations contribute more to the motion of the mean <span class="hlt">equator</span> than does the term due to the secular perturbation of the orbital eccentricity of the sun.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23509260','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23509260"><span>Schrödinger <span class="hlt">equation</span> revisited.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Schleich, Wolfgang P; Greenberger, Daniel M; Kobe, Donald H; Scully, Marlan O</p>
<p>2013-04-02</p>
<p>The time-dependent Schrödinger <span class="hlt">equation</span> is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger <span class="hlt">equation</span> from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi <span class="hlt">equation</span>. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA115568','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA115568"><span>Reducibility of Matrix <span class="hlt">Equations</span> Containing Several Parameters.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1981-12-01</p>
<p>AD-AI15 568 AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOO;-ETC EF G 12 1ADA1551 REDUCIBILITY OF MATRIX <span class="hlt">EQUATIONS</span> CONTAINING SEVERAL...PARAMETERS.E U)CA E UNCLASSIFIED AFIT/GE/RA/81D-1 N P11111111II soonhh Eu;o I. ’Trm * a, ~t- NMI 4 i’- 00Nt. met r~ REDUCIBILITY OF MATRIX <span class="hlt">EQUATIONS</span> CONTAINING...1 REDUCIBILITY OF MATRIX <span class="hlt">EQUATIONS</span> CONTAINING SEVERAL PARAMETERS THESIS Presented to the Faculty of the School of Engineering of the Air Force</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21057257','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21057257"><span>Physical Fields Described By Maxwell's <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Ahmetaj, Skender; Veseli, Ahmet; Jashari, Gani</p>
<p>2007-04-23</p>
<p>Fields that satisfy Maxwell's <span class="hlt">equations</span> of motion are analyzed. Investigation carried out in this work, shows that the free electromagnetic field, spinor Dirac's field without mass, spinor Dirac's field with mass, and some other fields are described by the same variational formulation. The conditions that a field be described by Maxwell's <span class="hlt">equations</span> of motion are given in this work, and some solutions of these conditions are also given. The question arises, which physical objects are formulated by the same or analogous <span class="hlt">equations</span> of physics.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2010AJ....139..803Q','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2010AJ....139..803Q"><span>A Symplectic Integrator for Hill's <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Quinn, Thomas; Perrine, Randall P.; Richardson, Derek C.; Barnes, Rory</p>
<p>2010-02-01</p>
<p>Hill's <span class="hlt">equations</span> are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's <span class="hlt">equations</span> based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV, and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's <span class="hlt">equations</span> over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19920032044&hterms=equation+state&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dequation%2Bof%2Bstate','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19920032044&hterms=equation+state&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D50%26Ntt%3Dequation%2Bof%2Bstate"><span>Shock wave <span class="hlt">equation</span> of state of muscovite</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Sekine, Toshimori; Rubin, Allan M.; Ahrens, Thomas J.</p>
<p>1991-01-01</p>
<p>Shock wave data were obtained between 20 and 140 GPa for natural muscovite obtained from Methuen Township (Ontario), in order to provide a shock-wave <span class="hlt">equation</span> of state for this crustal hydrous mineral. The shock <span class="hlt">equation</span> of state data could be fit by a linear shock velocity (Us) versus particle velocity (Up) relation Us = 4.62 + 1.27 Up (km/s). Third-order Birch-Murnaghan <span class="hlt">equation</span> of state parameters were found to be K(OS) = 52 +/-4 GPa and K-prime(OS) = 3.2 +/-0.3 GPa. These parameters are comparable to those of other hydrous minerals such as brucite, serpentine, and tremolite.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19890009887','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19890009887"><span>Transonic flutter calculations using the Euler <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Bendiksen, Oddvar O.; Kousen, Kenneth A.</p>
<p>1989-01-01</p>
<p>In transonic flutter problems where shock motion plays an important part, it is believed that accurate predictions of the flutter boundaries will require the use of codes based on the Euler <span class="hlt">equations</span>. Only Euler codes can obtain the correct shock location and shock strength, and the crucially important shock excursion amplitude and phase lag. The present study is based on the finite volume scheme developed by Jameson and Venkatakrishnan for the 2-D unsteady Euler <span class="hlt">equations</span>. The <span class="hlt">equations</span> are solved in integral form on a moving grid. The variable are pressure, density, Cartesian velocity components, and total energy.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2006tmgm.meet.1440W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2006tmgm.meet.1440W"><span>Path Deviation <span class="hlt">Equations</span> in AP-Geometry</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Wanas, M. I.; Kahil, M. E.</p>
<p>2006-02-01</p>
<p>Recently, it has been shown that Absolute Parallelism (AP) geometry admits paths that are naturally quantized. These paths have been used to describe the motion of spinning particles in a background gravitational field. In case of a weak static gravitational field limits, the paths are applied successfully to interpret the discrepancy in the motion of thermal neutrons in the Earth's gravitational field (COW-experiment). The aim of the present work is to explore the properties of the deviation <span class="hlt">equations</span> corresponding to these paths. In the present work the deviation <span class="hlt">equations</span> are derived and compared to the geodesic deviation <span class="hlt">equation</span> of the Riemannian geometry.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/7246860','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/7246860"><span>Nonlinear gyrokinetic <span class="hlt">equations</span> for tokamak microturbulence</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Hahm, T.S.</p>
<p>1988-05-01</p>
<p>A nonlinear electrostatic gyrokinetic Vlasov <span class="hlt">equation</span>, as well as Poisson <span class="hlt">equation</span>, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present <span class="hlt">equations</span> conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017eqft.proc..379B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017eqft.proc..379B"><span>Minimal String Theory and the Douglas <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Belavin, A. A.; Belavin, V. A.</p>
<p></p>
<p>We use the connection between the Frobenius manifold and the Douglas string <span class="hlt">equation</span> to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string <span class="hlt">equation</span> and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string <span class="hlt">equation</span> has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/22105488','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/22105488"><span>Supersymmetric Ito <span class="hlt">equation</span>: Bosonization and exact solutions</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Ren Bo; Yu Jun; Lin Ji</p>
<p>2013-04-15</p>
<p>Based on the bosonization approach, the N=1 supersymmetric Ito (sIto) system is changed to a system of coupled bosonic <span class="hlt">equations</span>. The approach can effectively avoid difficulties caused by intractable fermionic fields which are anticommuting. By solving the coupled bosonic <span class="hlt">equations</span>, the traveling wave solutions of the sIto system are obtained with the mapping and deformation method. Some novel types of exact solutions for the supersymmetric system are constructed with the solutions and symmetries of the usual Ito <span class="hlt">equation</span>. In the meanwhile, the similarity reduction solutions of the model are also studied with the Lie point symmetry theory.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19920032044&hterms=birch&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dbirch','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19920032044&hterms=birch&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D80%26Ntt%3Dbirch"><span>Shock wave <span class="hlt">equation</span> of state of muscovite</span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Sekine, Toshimori; Rubin, Allan M.; Ahrens, Thomas J.</p>
<p>1991-01-01</p>
<p>Shock wave data were obtained between 20 and 140 GPa for natural muscovite obtained from Methuen Township (Ontario), in order to provide a shock-wave <span class="hlt">equation</span> of state for this crustal hydrous mineral. The shock <span class="hlt">equation</span> of state data could be fit by a linear shock velocity (Us) versus particle velocity (Up) relation Us = 4.62 + 1.27 Up (km/s). Third-order Birch-Murnaghan <span class="hlt">equation</span> of state parameters were found to be K(OS) = 52 +/-4 GPa and K-prime(OS) = 3.2 +/-0.3 GPa. These parameters are comparable to those of other hydrous minerals such as brucite, serpentine, and tremolite.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015ArRMA.216..881D','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015ArRMA.216..881D"><span>Green's Functions of Wave <span class="hlt">Equations</span> in</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Deng, Shijin; Wang, Weike; Yu, Shih-Hsien</p>
<p>2015-06-01</p>
<p>We study the d'Alembert <span class="hlt">equation</span> with a boundary. We introduce the notions of Rayleigh surface wave operators, delayed/advanced mirror images, wave recombinations, and wave cancellations. This allows us to obtain the complete and simple formula of the Green's functions for the wave <span class="hlt">equation</span> with the presence of various boundary conditions. We are able to determine whether a Rayleigh surface wave is active or virtual, and study the lacunas of the wave <span class="hlt">equation</span> in three dimensional with the presence of a boundary in the case of a virtual Rayleigh surface wave.</p>
</li>
</ol>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017ArRMA.tmp...17H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017ArRMA.tmp...17H"><span>Cusp Formation for a Nonlocal Evolution <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hoang, Vu; Radosz, Maria</p>
<p>2017-02-01</p>
<p>Córdoba et al. (Ann Math 162(3):1377-1389, 2005) introduced a nonlocal active scalar <span class="hlt">equation</span> as a one-dimensional analogue of the surface-quasigeostrophic <span class="hlt">equation</span>. It has been conjectured, based on numerical evidence, that the solution forms a cusp-like singularity in finite time. Up until now, no active scalar with nonlocal flux is known for which cusp formation has been rigorously shown. In this paper, we introduce and study a nonlocal active scalar, inspired by the Córdoba-Córdoba-Fontelos <span class="hlt">equation</span>, and prove that either a cusp- or needle-like singularity forms in finite time.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/982435','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/982435"><span>String Field <span class="hlt">Equations</span> from Generalized Sigma Model</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Bardakci, K.; Bernardo, L.M.</p>
<p>1997-01-29</p>
<p>We propose a new approach for deriving the string field <span class="hlt">equations</span> from a general sigma model on the world-sheet. This approach leads to an <span class="hlt">equation</span> which combines some of the attractive features of both the renormalization group method and the covariant beta function treatment of the massless excitations. It has the advantage of being covariant under a very general set of both local and non-local transformations in the field space. We apply it to the tachyon, massless and first massive level, and show that the resulting field <span class="hlt">equations</span> reproduce the correct spectrum of a left-right symmetric closed bosonic string.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PhLB..771..277H','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PhLB..771..277H"><span>Horizon thermodynamics from Einstein's <span class="hlt">equation</span> of state</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Hansen, Devin; Kubizňák, David; Mann, Robert B.</p>
<p>2017-08-01</p>
<p>By regarding the Einstein <span class="hlt">equations</span> as <span class="hlt">equation(s</span>) of state, we demonstrate that a full cohomogeneity horizon first law can be derived in horizon thermodynamics. In this approach both the entropy and the free energy are derived concepts, while the standard (degenerate) horizon first law is recovered by a Legendre projection from the more general one we derive. These results readily generalize to higher curvature gravities where they naturally reproduce a formula for the entropy without introducing Noether charges. Our results thus establish a way of how to formulate consistent black hole thermodynamics without conserved charges.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017PEPI..270...40K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017PEPI..270...40K"><span>Towards constitutive <span class="hlt">equations</span> for the deep Earth</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kennett, B. L. N.</p>
<p>2017-09-01</p>
<p>A new formulation of constitutive <span class="hlt">equations</span> for states of high compression is introduced for isotropic media, exploiting a separation between hydrostatic and deviatoric components in strain energy. The strain energy is represented as functions of strain invariants, with one purely volumetric component and the other which vanishes for purely hydrostatic deformation. This approach preserves the form of familiar <span class="hlt">equations</span> of state through the volumetric component, but allows the addition of volume and pressure dependence of the shear modulus from the deviatoric term. A suitable shear modulus representation to accompany a Keane <span class="hlt">equation</span> of state is demonstrated.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/5993426','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/5993426"><span>Minimal relativistic three-particle <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Lindesay, J.</p>
<p>1981-07-01</p>
<p>A minimal self-consistent set of covariant and unitary three-particle <span class="hlt">equations</span> is presented. Numerical results are obtained for three-particle bound states, elastic scattering and rearrangement of bound pairs with a third particle, and amplitudes for breakup into states of three free particles. The mathematical form of the three-particle bound state <span class="hlt">equations</span> is explored; constraints are set upon the range of eigenvalues and number of eigenstates of these one parameter <span class="hlt">equations</span>. The behavior of the number of eigenstates as the two-body binding energy decreases to zero in a covariant context generalizes results previously obtained non-relativistically by V. Efimov.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/27258860','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/27258860"><span>Entanglement Equilibrium and the Einstein <span class="hlt">Equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Jacobson, Ted</p>
<p>2016-05-20</p>
<p>A link between the semiclassical Einstein <span class="hlt">equation</span> and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein <span class="hlt">equation</span> implies the validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein <span class="hlt">equation</span> holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/23944601','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/23944601"><span>Solutions of the coupled Higgs field <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Talukdar, Benoy; Ghosh, Swapan K; Saha, Aparna; Pal, Debabrata</p>
<p>2013-07-01</p>
<p>By an appropriate choice for the phase of the complex nucleonic field and going over to the traveling coordinate, we reduce the coupled Higgs <span class="hlt">equations</span> to the Hamiltonian form and treat the resulting <span class="hlt">equation</span> using the dynamical system theory. We present a phase-space analysis of its stable points. The results of our study demonstrate that the <span class="hlt">equation</span> can support both traveling- and standing-wave solutions. The traveling-wave solution appears in the form of a soliton and resides in the midst of doubly periodic standing-wave solutions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1863.0087S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1863.0087S"><span>Transformation <span class="hlt">equations</span> for the fifth dimension</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Said, M. Helmy</p>
<p>2017-07-01</p>
<p>Scientists have been arguing for a long time if there are particles faster than the speed of light or not. Those who denied the existence of particles faster than light speed always refer to Lorentz <span class="hlt">equation</span>. This <span class="hlt">equation</span> deals with particles in only four dimensions. In this paper, we show what would happen if we add one more dimension to this <span class="hlt">equation</span> to make it deals with five dimensions instead of four. The addition of this fifth dimension will greatly help us understand the state of particles before, near, at, and above the speed of light.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21301345','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21301345"><span>A SYMPLECTIC INTEGRATOR FOR HILL'S <span class="hlt">EQUATIONS</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Quinn, Thomas; Barnes, Rory; Perrine, Randall P.; Richardson, Derek C.</p>
<p>2010-02-15</p>
<p>Hill's <span class="hlt">equations</span> are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's <span class="hlt">equations</span> based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV, and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's <span class="hlt">equations</span> over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/5935123','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/5935123"><span>Fokker-Planck <span class="hlt">equation</span> in mirror research</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Post, R.F.</p>
<p>1983-08-11</p>
<p>Open confinement systems based on the magnetic mirror principle depend on the maintenance of particle distributions that may deviate substantially from Maxwellian distributions. Mirror research has therefore from the beginning relied on theoretical predictions of non-equilibrium rate processes obtained from solutions to the Fokker-Planck <span class="hlt">equation</span>. The F-P <span class="hlt">equation</span> plays three roles: Design of experiments, creation of classical standards against which to compare experiment, and predictions concerning mirror based fusion power systems. Analytical and computational approaches to solving the F-P <span class="hlt">equation</span> for mirror systems will be reviewed, together with results and examples that apply to specific mirror systems, such as the tandem mirror.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=Separation&pg=3&id=EJ971001','ERIC'); return false;" href="http://eric.ed.gov/?q=Separation&pg=3&id=EJ971001"><span>Connecting Related Rates and Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Brandt, Keith</p>
<p>2012-01-01</p>
<p>This article points out a simple connection between related rates and differential <span class="hlt">equations</span>. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1999CoPhC.121..376B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1999CoPhC.121..376B"><span>Microscopic models of traveling wave <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Brunet, Eric; Derrida, Bernard</p>
<p>1999-09-01</p>
<p>Reaction-diffusion problems are often described at a macroscopic scale by partial derivative <span class="hlt">equations</span> of the type of the Fisher or Kolmogorov-Petrovsky-Piscounov <span class="hlt">equation</span>. These <span class="hlt">equations</span> have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N=1016 particles at the microscopic scale, where particles react and diffuse according to some stochastic rules, we show that a single velocity is selected for the front. This velocity converges logarithmically to the solution of the F-KPP <span class="hlt">equation</span> with minimal velocity when the number N of particles increases. A simple calculation of the effect introduced by the cutoff due to the microscopic scale allows one to understand the origin of the logarithmic correction.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21428659','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21428659"><span>Moyal-Nahm <span class="hlt">equations</span> in seven dimensions</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Martinez Merino, Aldo Aparicio</p>
<p>2010-10-11</p>
<p>We present current research on the connection between the Nahm's approach to self-dual Yang-Mills <span class="hlt">equations</span> defined on a manifold with Spin(7) holonomy and its gravitational counterpart via the Moyal deformation of the Poisson algebra.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012JPhA...45h5202F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012JPhA...45h5202F"><span>An integrable coupled short pulse <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Feng, Bao-Feng</p>
<p>2012-03-01</p>
<p>An integrable coupled short pulse (CSP) <span class="hlt">equation</span> is proposed for the propagation of ultra-short pulses in optical fibers. Based on two sets of bilinear <span class="hlt">equations</span> to a two-dimensional Toda lattice linked by a Bäcklund transformation, and an appropriate hodograph transformation, the proposed CSP <span class="hlt">equation</span> is derived. Meanwhile, its N-soliton solutions are given by the Casorati determinant in a parametric form. The properties of one- and two-soliton solutions are investigated in detail. Same as the short pulse <span class="hlt">equation</span>, the two-soliton solution turns out to be a breather type if the wave numbers are complex conjugate. We also illustrate an example of soliton-breather interaction.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016APS..DPPG10141M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016APS..DPPG10141M"><span>Relativistic Langevin <span class="hlt">equation</span> for runaway electrons</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Mier, J. A.; Martin-Solis, J. R.; Sanchez, R.</p>
<p>2016-10-01</p>
<p>The Langevin approach to the kinetics of a collisional plasma is developed for relativistic electrons such as runaway electrons in tokamak plasmas. In this work, we consider Coulomb collisions between very fast, relativistic electrons and a relatively cool, thermal background plasma. The model is developed using the stochastic equivalence of the Fokker-Planck and Langevin <span class="hlt">equations</span>. The resulting Langevin model <span class="hlt">equation</span> for relativistic electrons is an stochastic differential <span class="hlt">equation</span>, amenable to numerical simulations by means of Monte-Carlo type codes. Results of the simulations will be presented and compared with the non-relativistic Langevin <span class="hlt">equation</span> for RE electrons used in the past. Supported by MINECO (Spain), Projects ENE2012-31753, ENE2015-66444-R.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017InJPh..91..209Z','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017InJPh..91..209Z"><span>Exact solutions for nonlinear foam drainage <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani</p>
<p>2017-02-01</p>
<p>In this paper, the modified simple <span class="hlt">equation</span> method, the exp-function method, the soliton ansatz method, the Riccati <span class="hlt">equation</span> expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage <span class="hlt">equation</span>. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential <span class="hlt">equations</span> in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JDE...263.1323C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JDE...263.1323C"><span>Asymptotically dichotomic almost periodic differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Campos, Juan; Tarallo, Massimo</p>
<p>2017-07-01</p>
<p>Consider a non-linear differential <span class="hlt">equation</span> in RN which asymptotically behaves as a linear <span class="hlt">equation</span> admitting an exponential dichotomy. We wonder if almost periodic solutions exist when we add to the <span class="hlt">equation</span> an almost periodic forcing term, large enough and not vanishing too much. A positive answer has been given in [3] for the scalar case N = 1 and our aim is to extend that result to higher dimensions. We discover that the extension seems to be driven by a new ingredient, namely the type of the exponential dichotomy: besides the pure stable types, the mixed hyperbolic type is now possible and leads to a weaker than expected extension. An example shows that a stronger extension cannot be obtained by the same method. The approach is blended and mixes methods of differential <span class="hlt">equations</span> and functional analysis, especially when estimating norm and spectral radius of some crucial positive but non-compact linear integral operators.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011SPIE.8321E..13F','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011SPIE.8321E..13F"><span>Underwater photogrammetric theoretical <span class="hlt">equations</span> and technique</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Fan, Ya-bing; Huang, Guiping; Qin, Gui-qin; Chen, Zheng</p>
<p>2011-12-01</p>
<p>In order to have a high level of accuracy of measurement in underwater close-range photogrammetry, this article deals with a study of three varieties of model <span class="hlt">equations</span> according to the way of imaging upon the water. First, the paper makes a careful analysis for the two varieties of theoretical <span class="hlt">equations</span> and finds out that there are some serious limitations in practical application and has an in-depth study for the third model <span class="hlt">equation</span>. Second, one special project for this measurement has designed correspondingly. Finally, one rigid antenna has been tested by underwater photogrammetry. The experimental results show that the precision of 3D coordinates measurement is 0.94mm, which validates the availability and operability in practical application with this third <span class="hlt">equation</span>. It can satisfy the measurement requirements of refraction correction, improving levels of accuracy of underwater close-range photogrammetry, as well as strong antijamming and stabilization.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19860033007&hterms=disruption&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Ddisruption','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19860033007&hterms=disruption&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D90%26Ntt%3Ddisruption"><span>Turbulent disruptions from the Strauss <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Dahlburg, J. P.; Montgomery, D.; Matthaeus, W. H.</p>
<p>1985-01-01</p>
<p>Preliminary results are reported from application of a three-dimensional spectral method model to the solution of the Strauss (1976) reduced MHD <span class="hlt">equations</span>. The investigation was focused on describing MHD turbulence in a current-carrying bounded magnetofluid. A cylindrical geometry with a square cross-section was considered, with the walls being rigid perfect conductors with free-slip boundary conditions. A uniform magnetic field and the electric current density both point in the z-direction. Initial conditions are specified which feature small amounts of random noise expressed as Fourier modes. Linearized <span class="hlt">equations</span> are defined for tracing the movement to equlibrium conditions or other temporal development. The model is further refined with nonlinear <span class="hlt">equations</span> to examine the effects of the appearance of disruptions. Comparisons are drawn between solutions obtained with linear and nonlinear <span class="hlt">equations</span>, with an eye to the associated physical realities.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://cfpub.epa.gov/si/si_public_record_report.cfm?dirEntryId=61330&keyword=Kinetic+AND+theory&actType=&TIMSType=+&TIMSSubTypeID=&DEID=&epaNumber=&ntisID=&archiveStatus=Both&ombCat=Any&dateBeginCreated=&dateEndCreated=&dateBeginPublishedPresented=&dateEndPublishedPresented=&dateBeginUpdated=&dateEndUpdated=&dateBeginCompleted=&dateEndCompleted=&personID=&role=Any&journalID=&publisherID=&sortBy=revisionDate&count=50','EPA-EIMS'); return false;" href="http://cfpub.epa.gov/si/si_public_record_report.cfm?dirEntryId=61330&keyword=Kinetic+AND+theory&actType=&TIMSType=+&TIMSSubTypeID=&DEID=&epaNumber=&ntisID=&archiveStatus=Both&ombCat=Any&dateBeginCreated=&dateEndCreated=&dateBeginPublishedPresented=&dateEndPublishedPresented=&dateBeginUpdated=&dateEndUpdated=&dateBeginCompleted=&dateEndCompleted=&personID=&role=Any&journalID=&publisherID=&sortBy=revisionDate&count=50"><span>THE BERNOULLI <span class="hlt">EQUATION</span> AND COMPRESSIBLE FLOW THEORIES</span></a></p>
<p><a target="_blank" href="http://oaspub.epa.gov/eims/query.page">EPA Science Inventory</a></p>
<p></p>
<p></p>
<p>The incompressible Bernoulli <span class="hlt">equation</span> is an analytical relationship between pressure, kinetic energy, and potential energy. As perhaps the simplest and most useful statement for describing laminar flow, it buttresses numerous incompressible flow models that have been developed ...</p>
</li>
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<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/26274137','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/26274137"><span>Approximate probability distributions of the master <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Thomas, Philipp; Grima, Ramon</p>
<p>2015-07-01</p>
<p>Master <span class="hlt">equations</span> are common descriptions of mesoscopic systems. Analytical solutions to these <span class="hlt">equations</span> can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master <span class="hlt">equation</span> using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master <span class="hlt">equation</span>, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=methods+AND+separation&pg=2&id=EJ971001','ERIC'); return false;" href="https://eric.ed.gov/?q=methods+AND+separation&pg=2&id=EJ971001"><span>Connecting Related Rates and Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Brandt, Keith</p>
<p>2012-01-01</p>
<p>This article points out a simple connection between related rates and differential <span class="hlt">equations</span>. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://eric.ed.gov/?q=arrhenius&pg=2&id=EJ301949','ERIC'); return false;" href="https://eric.ed.gov/?q=arrhenius&pg=2&id=EJ301949"><span>The Development of the Arrhenius <span class="hlt">Equation</span>.</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>Laidler, Keith J.</p>
<p>1984-01-01</p>
<p>Traces the development of the Arrhenius <span class="hlt">equation</span> from its beginning, examining the more important alternate proposals and the work that supported them. Aside from its historical interest, this examination affords insight into how scientific progress is made. (JN)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1990ASPC...11...95K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1990ASPC...11...95K"><span><span class="hlt">Equations</span> of state and bump Cepheids</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kanbur, Shashi</p>
<p></p>
<p>The paper presents results of calculations investigating the consequences of a number of recent advances in atomic physics for stellar pulsations, i.e., the Hummer-Mihalas-Dappen (HMD) <span class="hlt">equation</span> of state and opacities generated with new atomic data (Seaton, 1987). The sensitivity of theoretical linear nonadiabatic (LNA) bump Cepheid calculations to the <span class="hlt">equation</span> used in such calculations is examined. LNA periods, growth rates, and period ratios for a specified model grid were calculated using both the HMD and the Saha <span class="hlt">equations</span> of state. The model grid is taken from Simon and Davis (1983), who used the Los Alamos equastion of state in their calculations. A comparison in terms of theoretical linear pulsation results can thus be made between three <span class="hlt">equations</span> of state.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2014OPhy...12..233S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2014OPhy...12..233S"><span>Wong's <span class="hlt">equations</span> in Yang-Mills theory</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Storchak, Sergey</p>
<p>2014-04-01</p>
<p>Wong's <span class="hlt">equations</span> for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The <span class="hlt">equations</span> obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these <span class="hlt">equations</span>, we obtain Wong's <span class="hlt">equations</span> in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0482688','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0482688"><span>AN APPROXIMATE <span class="hlt">EQUATION</span> OF STATE OF SOLIDS.</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>research. By generalizing experimental data and obtaining unified relations describing the thermodynamic properties of solids, and approximate <span class="hlt">equation</span> of state is derived which can be applied to a wide class of materials. (Author)</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/ADA019504','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/ADA019504"><span>Analytic <span class="hlt">Equation</span> of State for Sea Water</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p>1975-12-01</p>
<p>represent the sea water data of Wilson and Bradley. In this paper the thermal expansion data of Bradshaw and Schleicher and the sound velocity data of Wilson have been incorporated to yield a new <span class="hlt">equation</span> of state for sea water.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/25375482','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/25375482"><span>Cattaneo-type subdiffusion-reaction <span class="hlt">equation</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Kosztołowicz, Tadeusz</p>
<p>2014-10-01</p>
<p>Subdiffusion in a system in which mobile particles A can chemically react with static particles B according to the rule A+B→B is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion <span class="hlt">equations</span>. Starting with the difference <span class="hlt">equation</span>, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential <span class="hlt">equation</span> with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction <span class="hlt">equation</span> in the case in which mobile particles of species A and B can chemically react according to a more complicated rule.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2008JCoAM.218..149S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2008JCoAM.218..149S"><span>Systems of fuzzy <span class="hlt">equations</span> in structural mechanics</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Skalna, Iwona; Rama Rao, M. V.; Pownuk, Andrzej</p>
<p>2008-08-01</p>
<p>Systems of linear and nonlinear <span class="hlt">equations</span> with fuzzy parameters are relevant to many practical problems arising in structure mechanics, electrical engineering, finance, economics and physics. In this paper three methods for solving such <span class="hlt">equations</span> are discussed: method for outer interval solution of systems of linear <span class="hlt">equations</span> depending linearly on interval parameters, fuzzy finite element method proposed by Rama Rao and sensitivity analysis method. The performance and advantages of presented methods are described with illustrative examples. Extended version of the present paper can be downloaded from the web page of the UTEP [I. Skalna, M.V. Rama Rao, A. Pownuk, Systems of fuzzy <span class="hlt">equations</span> in structural mechanics, The University of Texas at El Paso, Department of Mathematical Sciences Research Reports Series, <http://www.math.utep.edu/preprints/2007/2007-01.pdf>, Texas Research Report No. 2007-01, 2007].</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2003JPhB...36.4731G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2003JPhB...36.4731G"><span>The stochastic Gross Pitaevskii <span class="hlt">equation</span>: II</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Gardiner, C. W.; Davis, M. J.</p>
<p>2003-12-01</p>
<p>We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii <span class="hlt">equation</span>, as introduced by Gardiner et al (2002 J. Phys. B: At. Mol. Opt. Phys. 35 1555). This derivation does not rely on the concept of local energy and momentum conservation and is based on a quasiclassical Wigner function representation of a 'high temperature' master <span class="hlt">equation</span> for a Bose gas, which includes only modes below an energy cut-off ER that are sufficiently highly occupied (the condensate band). The modes above this cut-off (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provides noise and damping terms in the <span class="hlt">equation</span> of motion for the condensate band, which we call the stochastic Gross-Pitaevskii <span class="hlt">equation</span>. This approach is distinguished by the control of the approximations made in its derivation and by the feasibility of its numerical implementation.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015NaPho...9....2M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015NaPho...9....2M"><span>How Maxwell's <span class="hlt">equations</span> came to light</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Mahon, Basil</p>
<p>2015-01-01</p>
<p>The nineteenth-century Scottish physicist James Clerk Maxwell made groundbreaking contributions to many areas of science including thermodynamics and colour vision. However, he is best known for his <span class="hlt">equations</span> that unified electricity, magnetism and light.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2012PMag...92.3882W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2012PMag...92.3882W"><span>Laplace and the era of differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Weinberger, Peter</p>
<p>2012-11-01</p>
<p>Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential <span class="hlt">equation</span> originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's "fluxionary calculus" finally gave way to a French-type notation of handling differential <span class="hlt">equations</span>. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential <span class="hlt">equations</span>. A remark about Schrödinger and his <span class="hlt">equation</span> for the hydrogen atom finally will lead back to present times.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017IJT....38...59W','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017IJT....38...59W"><span>Interpolation Errors in Thermistor Calibration <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>White, D. R.</p>
<p>2017-04-01</p>
<p>Thermistors are widely used temperature sensors capable of measurement uncertainties approaching those of standard platinum resistance thermometers. However, the extreme nonlinearity of thermistors means that complicated calibration <span class="hlt">equations</span> are required to minimize the effects of interpolation errors and achieve low uncertainties. This study investigates the magnitude of interpolation errors as a function of temperature range and the number of terms in the calibration <span class="hlt">equation</span>. Approximation theory is used to derive an expression for the interpolation error and indicates that the temperature range and the number of terms in the calibration <span class="hlt">equation</span> are the key influence variables. Numerical experiments based on published resistance-temperature data confirm these conclusions and additionally give guidelines on the maximum and minimum interpolation error likely to occur for a given temperature range and number of terms in the calibration <span class="hlt">equation</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://cfpub.epa.gov/si/si_public_record_report.cfm?dirEntryId=61330&keyword=euler&actType=&TIMSType=+&TIMSSubTypeID=&DEID=&epaNumber=&ntisID=&archiveStatus=Both&ombCat=Any&dateBeginCreated=&dateEndCreated=&dateBeginPublishedPresented=&dateEndPublishedPresented=&dateBeginUpdated=&dateEndUpdated=&dateBeginCompleted=&dateEndCompleted=&personID=&role=Any&journalID=&publisherID=&sortBy=revisionDate&count=50&CFID=78784837&CFTOKEN=66032399','EPA-EIMS'); return false;" href="http://cfpub.epa.gov/si/si_public_record_report.cfm?dirEntryId=61330&keyword=euler&actType=&TIMSType=+&TIMSSubTypeID=&DEID=&epaNumber=&ntisID=&archiveStatus=Both&ombCat=Any&dateBeginCreated=&dateEndCreated=&dateBeginPublishedPresented=&dateEndPublishedPresented=&dateBeginUpdated=&dateEndUpdated=&dateBeginCompleted=&dateEndCompleted=&personID=&role=Any&journalID=&publisherID=&sortBy=revisionDate&count=50&CFID=78784837&CFTOKEN=66032399"><span>THE BERNOULLI <span class="hlt">EQUATION</span> AND COMPRESSIBLE FLOW THEORIES</span></a></p>
<p><a target="_blank" href="http://oaspub.epa.gov/eims/query.page">EPA Science Inventory</a></p>
<p></p>
<p></p>
<p>The incompressible Bernoulli <span class="hlt">equation</span> is an analytical relationship between pressure, kinetic energy, and potential energy. As perhaps the simplest and most useful statement for describing laminar flow, it buttresses numerous incompressible flow models that have been developed ...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhA...50g3001K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhA...50g3001K"><span>Geometric aspects of Painlevé <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kajiwara, Kenji; Noumi, Masatoshi; Yamada, Yasuhiko</p>
<p>2017-02-01</p>
<p>In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé <span class="hlt">equations</span>, with a particular emphasis on the discrete Painlevé <span class="hlt">equations</span>. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on {{{P}}}1× {{{P}}}1 and classified according to the degeneration of points. We give a systematic description of the <span class="hlt">equations</span> and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; <span class="hlt">equations</span>, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/960909','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/960909"><span>Finite scale <span class="hlt">equations</span> for compressible fluid flow</span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Margolin, Len G</p>
<p>2008-01-01</p>
<p>Finite-scale <span class="hlt">equations</span> (FSE) describe the evolution of finite volumes of fluid over time. We discuss the FSE for a one-dimensional compressible fluid, whose every point is governed by the Navier-Stokes <span class="hlt">equations</span>. The FSE contain new momentum and internal energy transport terms. These are similar to terms added in numerical simulation for high-speed flows (e.g. artificial viscosity) and for turbulent flows (e.g. subgrid scale models). These similarities suggest that the FSE may provide new insight as a basis for computational fluid dynamics. Our analysis of the FS continuity <span class="hlt">equation</span> leads to a physical interpretation of the new transport terms, and indicates the need to carefully distinguish between volume-averaged and mass-averaged velocities in numerical simulation. We make preliminary connections to the other recent work reformulating Navier-Stokes <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017AIPC.1863K0007A','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017AIPC.1863K0007A"><span>Finite element schemes for Fermi <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Asadzadeh, M.; Beilina, L.; Naseer, M.; Standar, C.</p>
<p>2017-07-01</p>
<p>A priori error estimates are derived for the streamline diffusion (SD) finite element methods for the Fermi pencil-beam <span class="hlt">equation</span>. Two-dimensional numerical examples confirm our theoretical investigations.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.dtic.mil/docs/citations/AD0657639','DTIC-ST'); return false;" href="http://www.dtic.mil/docs/citations/AD0657639"><span>WHAT IS A SATISFACTORY QUADRATIC <span class="hlt">EQUATION</span> SOLVER?</span></a></p>
<p><a target="_blank" href="http://www.dtic.mil/">DTIC Science & Technology</a></p>
<p></p>
<p></p>
<p>The report discusses precise requirements for a satisfactory computer program to solve a quadratic <span class="hlt">equation</span> with floating - point coefficients. The principal practical problem is coping with overflow and underflow.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JPhA...50K5205M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JPhA...50K5205M"><span>On the solution of the Liouville <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Menotti, Pietro</p>
<p>2017-09-01</p>
<p>We give a short and rigorous proof of the existence and uniqueness of the solution of the Liouville <span class="hlt">equation</span> with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://eric.ed.gov/?q=chemical+AND+equation&pg=5&id=EJ121337','ERIC'); return false;" href="http://eric.ed.gov/?q=chemical+AND+equation&pg=5&id=EJ121337"><span>Writing Chemical <span class="hlt">Equations</span>: An Introductory Experiment</span></a></p>
<p><a target="_blank" href="http://www.eric.ed.gov/ERICWebPortal/search/extended.jsp?_pageLabel=advanced">ERIC Educational Resources Information Center</a></p>
<p>LeMay, H. Eugene, Jr.; Kemp, Kenneth C.</p>
<p>1975-01-01</p>
<p>Describes an experiment in which possible products of a series of reactions are tabulated together with properties which may be useful in identifying each substance. The student deduces the products and writes a balanced chemical <span class="hlt">equation</span> for the reaction. (GS)</p>
</li>
</ol>
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<ol class="result-class" start="481">
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015PhRvE..92a2120T','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015PhRvE..92a2120T"><span>Approximate probability distributions of the master <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Thomas, Philipp; Grima, Ramon</p>
<p>2015-07-01</p>
<p>Master <span class="hlt">equations</span> are common descriptions of mesoscopic systems. Analytical solutions to these <span class="hlt">equations</span> can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master <span class="hlt">equation</span> using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master <span class="hlt">equation</span>, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.youtube.com/watch?v=tlwmEll_l6U','SCIGOVIMAGE-NASA'); return false;" href="http://www.youtube.com/watch?v=tlwmEll_l6U"><span>Solar Cycle: Magnetized March to <span class="hlt">Equator</span></span></a></p>
<p><a target="_blank" href="https://images.nasa.gov/">NASA Image and Video Library</a></p>
<p></p>
<p></p>
<p>Bands of magnetized solar material – with alternating south and north polarity – march toward the sun's <span class="hlt">equator</span>. Comparing the evolution of the bands with the sunspot number in each hemisphere over...</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016NLE.....5..219M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016NLE.....5..219M"><span>Linearized Implicit Numerical Method for Burgers' <span class="hlt">Equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Mukundan, Vijitha; Awasthi, Ashish</p>
<p>2016-12-01</p>
<p>In this work, a novel numerical scheme based on method of lines (MOL) is proposed to solve the nonlinear time dependent Burgers' <span class="hlt">equation</span>. The Burgers' <span class="hlt">equation</span> is semi discretized in spatial direction by using MOL to yield system of nonlinear ordinary differential <span class="hlt">equations</span> in time. The resulting system of nonlinear differential <span class="hlt">equations</span> is integrated by an implicit finite difference method. We have not used Cole-Hopf transformation which gives less accurate solution for very small values of kinematic viscosity. Also, we have not considered nonlinear solvers that are computationally costlier and take more running time.In the proposed scheme nonlinearity is tackled by Taylor series and the use of fully discretized scheme is easy and practical. The proposed method is unconditionally stable in the linear sense. Furthermore, efficiency of the proposed scheme is demonstrated using three test problems.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016JDE...260..478B','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016JDE...260..478B"><span>Geometric investigations of a vorticity model <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Bauer, Martin; Kolev, Boris; Preston, Stephen C.</p>
<p>2016-01-01</p>
<p>This article consists of a detailed geometric study of the one-dimensional vorticity model <span class="hlt">equation</span> which is a particular case of the generalized Constantin-Lax-Majda <span class="hlt">equation</span>. Wunsch showed that this <span class="hlt">equation</span> is the Euler-Arnold <span class="hlt">equation</span> on Diff (S1) when the latter is endowed with the right-invariant homogeneous H ˙ 1 / 2-metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-Córdoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to Córdoba-Córdoba.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017JLTP..187..639P','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017JLTP..187..639P"><span>The Hypernetted Chain <span class="hlt">Equations</span> for Periodic Systems</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Panholzer, Martin</p>
<p>2017-06-01</p>
<p>Starting from the general inhomogeneous Fermi hypernetted chain <span class="hlt">equations</span>, the <span class="hlt">equations</span> for periodic systems are derived by simple Fourier transform. It is shown how the symmetry reduces the size of the involved quantities. First results for a one-dimensional (1D) model system are presented. The results allow a reliable estimation of the numerical demand even for realistic 3D systems, such as solids. It is shown that treatment of this systems is feasible with moderate computational resources.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.ncbi.nlm.nih.gov/pubmed/24516624','PUBMED'); return false;" href="https://www.ncbi.nlm.nih.gov/pubmed/24516624"><span>Validating and improving interrill erosion <span class="hlt">equations</span>.</span></a></p>
<p><a target="_blank" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pubmed">PubMed</a></p>
<p>Zhang, Feng-Bao; Wang, Zhan-Li; Yang, Ming-Yi</p>
<p>2014-01-01</p>
<p>Existing interrill erosion <span class="hlt">equations</span> based on mini-plot experiments have largely ignored the effects of slope length and plot size on interrill erosion rate. This paper describes a series of simulated rainfall experiments which were conducted according to a randomized factorial design for five slope lengths (0.4, 0.8, 1.2, 1.6, and 2 m) at a width of 0.4 m, five slope gradients (17%, 27%, 36%, 47%, and 58%), and five rainfall intensities (48, 62.4, 102, 149, and 170 mm h(-1)) to perform a systematic validation of existing interrill erosion <span class="hlt">equations</span> based on mini-plots. The results indicated that the existing interrill erosion <span class="hlt">equations</span> do not adequately describe the relationships between interrill erosion rate and its influencing factors with increasing slope length and rainfall intensity. Univariate analysis of variance showed that runoff rate, rainfall intensity, slope gradient, and slope length had significant effects on interrill erosion rate and that their interactions were significant at p = 0.01. An improved interrill erosion <span class="hlt">equation</span> was constructed by analyzing the relationships of sediment concentration with rainfall intensity, slope length, and slope gradient. In the improved interrill erosion <span class="hlt">equation</span>, the runoff rate and slope factor are the same as in the interrill erosion <span class="hlt">equation</span> in the Water Erosion Prediction Project (WEPP), with the weight of rainfall intensity adjusted by an exponent of 0.22 and a slope length term added with an exponent of -0.25. Using experimental data from WEPP cropland soil field interrill erodibility experiments, it has been shown that the improved interrill erosion <span class="hlt">equation</span> describes the relationship between interrill erosion rate and runoff rate, rainfall intensity, slope gradient, and slope length reasonably well and better than existing interrill erosion <span class="hlt">equations</span>.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2015TePhL..41.1170M','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2015TePhL..41.1170M"><span>An <span class="hlt">equation</span> of turbulent motion in tubes</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Melamed, L. E.</p>
<p>2015-12-01</p>
<p>An <span class="hlt">equation</span> of turbulent motion in a round cylindrical channel, which involves no empirical quantities, is proposed for the first time. A solution of this <span class="hlt">equation</span> describes the average velocity profile in all regions of the channel cross section in the entire range of turbulent Reynolds numbers. The computational model is based on taking into account the internal distributed resistance created by a turbulent "vortex backfill."</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://hdl.handle.net/2060/19790017615','NASA-TRS'); return false;" href="http://hdl.handle.net/2060/19790017615"><span>Program for solution of ordinary differential <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Sloate, H.</p>
<p>1973-01-01</p>
<p>A program for the solution of linear and nonlinear first order ordinary differential <span class="hlt">equations</span> is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential <span class="hlt">equations</span> with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19990008067&hterms=1605&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3D%2526%25231605','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19990008067&hterms=1605&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D40%26Ntt%3D%2526%25231605"><span>Efficient High-Pressure State <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Harstad, Kenneth G.; Miller, Richard S.; Bellan, Josette</p>
<p>1997-01-01</p>
<p>A method is presented for a relatively accurate, noniterative, computationally efficient calculation of high-pressure fluid-mixture <span class="hlt">equations</span> of state, especially targeted to gas turbines and rocket engines. Pressures above I bar and temperatures above 100 K are addressed The method is based on curve fitting an effective reference state relative to departure functions formed using the Peng-Robinson cubic state <span class="hlt">equation</span> Fit parameters for H2, O2, N2, propane, methane, n-heptane, and methanol are given.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017IJGMM..1450013Y','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017IJGMM..1450013Y"><span>(2+1)-dimensional supersymmetric integrable <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Yan, Zhao-Wen; Tala; Chen, Fang; Liu, Tao-Ran; Han, Jing-Min</p>
<p>2017-09-01</p>
<p>By means of two different approaches, we construct the (2+1)-dimensional supersymmetric integrable <span class="hlt">equations</span> based on the super Lie algebra osp(3/2). We relax the constraint condition of homogenous space of super Lie algebra osp(3/2) in the first approach. In another one, the technique of extending the dimension of the systems is used. Furthermore for the (2 + 1)-dimensional supersymmetric integrable <span class="hlt">equations</span>, we also derive their Bäcklund transformations.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19810000370&hterms=weed&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dweed','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19810000370&hterms=weed&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dweed"><span>Numerical Solution for Navier-Stokes <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Warsi, Z. U. A.; Weed, R. A.; Thompson, J. F.</p>
<p>1982-01-01</p>
<p>Carefully selected blend of computational techniques solves complete set of <span class="hlt">equations</span> for viscous, unsteady, hypersonic flow in general curvilinear coordinates. New algorithm has tested computation of axially directed flow about blunt body having shape similar to that of such practical bodies as wide-body aircraft or artillery shells. Method offers significant computational advantages because of conservation-law form of <span class="hlt">equations</span> and because it reduces amount of metric data required.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2016PhLB..756..188L','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2016PhLB..756..188L"><span>Legendre transformations and Clairaut-type <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Lavrov, Peter M.; Merzlikin, Boris S.</p>
<p>2016-05-01</p>
<p>It is noted that the Legendre transformations in the standard formulation of quantum field theory have the form of functional Clairaut-type <span class="hlt">equations</span>. It is shown that in presence of composite fields the Clairaut-type form holds after loop corrections are taken into account. A new solution to the functional Clairaut-type <span class="hlt">equation</span> appearing in field theories with composite fields is found.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://www.osti.gov/scitech/servlets/purl/5245157','SCIGOV-STC'); return false;" href="http://www.osti.gov/scitech/servlets/purl/5245157"><span>Recent developments in the Virasoro master <span class="hlt">equation</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Halpern, M.B.</p>
<p>1991-09-03</p>
<p>The Virasoro master <span class="hlt">equation</span> collects all possible Virasoro constructions which are quadratic in the currents of affine Lie g. The solution space of this system is immense, with generically irrational central charge, and solutions which have so far been observed are generically unitary. Other developments reviewed include the exact C-function, the superconformal master <span class="hlt">equation</span> and partial classification of solutions by graph theory and generalized graph theories. 37 refs., 1 fig., 1 tab.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/1988IJTP...27.1083C','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/1988IJTP...27.1083C"><span>Curvature tensors unified field <span class="hlt">equations</span> on SEXn</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Chung, Kyung Tae; Lee, Il Young</p>
<p>1988-09-01</p>
<p>We study the curvature tensors and field <span class="hlt">equations</span> in the n-dimensional SE manifold SEXn. We obtain several basic properties of the vectors S λ and U λ and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field <span class="hlt">equations</span> is discussed in SEXn and one of its particular solutions is constructed and displayed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3916424','PMC'); return false;" href="https://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=3916424"><span>Validating and Improving Interrill Erosion <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=pmc">PubMed Central</a></p>
<p>Zhang, Feng-Bao; Wang, Zhan-Li; Yang, Ming-Yi</p>
<p>2014-01-01</p>
<p>Existing interrill erosion <span class="hlt">equations</span> based on mini-plot experiments have largely ignored the effects of slope length and plot size on interrill erosion rate. This paper describes a series of simulated rainfall experiments which were conducted according to a randomized factorial design for five slope lengths (0.4, 0.8, 1.2, 1.6, and 2 m) at a width of 0.4 m, five slope gradients (17%, 27%, 36%, 47%, and 58%), and five rainfall intensities (48, 62.4, 102, 149, and 170 mm h−1) to perform a systematic validation of existing interrill erosion <span class="hlt">equations</span> based on mini-plots. The results indicated that the existing interrill erosion <span class="hlt">equations</span> do not adequately describe the relationships between interrill erosion rate and its influencing factors with increasing slope length and rainfall intensity. Univariate analysis of variance showed that runoff rate, rainfall intensity, slope gradient, and slope length had significant effects on interrill erosion rate and that their interactions were significant at p = 0.01. An improved interrill erosion <span class="hlt">equation</span> was constructed by analyzing the relationships of sediment concentration with rainfall intensity, slope length, and slope gradient. In the improved interrill erosion <span class="hlt">equation</span>, the runoff rate and slope factor are the same as in the interrill erosion <span class="hlt">equation</span> in the Water Erosion Prediction Project (WEPP), with the weight of rainfall intensity adjusted by an exponent of 0.22 and a slope length term added with an exponent of −0.25. Using experimental data from WEPP cropland soil field interrill erodibility experiments, it has been shown that the improved interrill erosion <span class="hlt">equation</span> describes the relationship between interrill erosion rate and runoff rate, rainfall intensity, slope gradient, and slope length reasonably well and better than existing interrill erosion <span class="hlt">equations</span>. PMID:24516624</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://ntrs.nasa.gov/search.jsp?R=19810000370&hterms=weeds&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dweeds','NASA-TRS'); return false;" href="https://ntrs.nasa.gov/search.jsp?R=19810000370&hterms=weeds&qs=Ntx%3Dmode%2Bmatchall%26Ntk%3DAll%26N%3D0%26No%3D20%26Ntt%3Dweeds"><span>Numerical Solution for Navier-Stokes <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://ntrs.nasa.gov/search.jsp">NASA Technical Reports Server (NTRS)</a></p>
<p>Warsi, Z. U. A.; Weed, R. A.; Thompson, J. F.</p>
<p>1982-01-01</p>
<p>Carefully selected blend of computational techniques solves complete set of <span class="hlt">equations</span> for viscous, unsteady, hypersonic flow in general curvilinear coordinates. New algorithm has tested computation of axially directed flow about blunt body having shape similar to that of such practical bodies as wide-body aircraft or artillery shells. Method offers significant computational advantages because of conservation-law form of <span class="hlt">equations</span> and because it reduces amount of metric data required.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('https://www.osti.gov/scitech/biblio/21255541','SCIGOV-STC'); return false;" href="https://www.osti.gov/scitech/biblio/21255541"><span>A Chebychev propagator for inhomogeneous Schroedinger <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://www.osti.gov/scitech">SciTech Connect</a></p>
<p>Ndong, Mamadou; Koch, Christiane P.; Tal-Ezer, Hillel; Kosloff, Ronnie</p>
<p>2009-03-28</p>
<p>A propagation scheme for time-dependent inhomogeneous Schroedinger <span class="hlt">equations</span> is presented. Such <span class="hlt">equations</span> occur in time dependent optimal control theory and in reactive scattering. A formal solution based on a polynomial expansion of the inhomogeneous term is derived. It is subjected to an approximation in terms of Chebychev polynomials. Different variants for the inhomogeneous propagator are demonstrated and applied to two examples from optimal control theory. Convergence behavior and numerical efficiency are analyzed.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2017TMP...192..958K','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2017TMP...192..958K"><span>Bifurcations in Kuramoto-Sivashinsky <span class="hlt">equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Kashchenko, S. A.</p>
<p>2017-07-01</p>
<p>We consider the local dynamics of the classical Kuramoto-Sivashinsky <span class="hlt">equation</span> and its generalizations and study the problem of the existence and asymptotic behavior of periodic solutions and tori. The most interesting results are obtained in the so-called infinite-dimensional critical cases. Considering these cases, we construct special nonlinear partial differential <span class="hlt">equations</span> that play the role of normal forms and whose nonlocal dynamics thus determine the behavior of solutions of the original boundary value problem.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2002JDE...178..541S','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2002JDE...178..541S"><span>Normal Forms for Nonautonomous Differential <span class="hlt">Equations</span></span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Siegmund, Stefan</p>
<p>2002-01-01</p>
<p>We extend Henry Poincarés normal form theory for autonomous differential <span class="hlt">equations</span> x=f(x) to nonautonomous differential <span class="hlt">equations</span> x=f(t, x). Poincarés nonresonance condition λj-∑ni=1 ℓiλi≠0 for eigenvalues is generalized to the new nonresonance condition λj∩∑ni=1 ℓiλi=∅ for spectral intervals.</p>
</li>
<li>
<p><a target="_blank" onclick="trackOutboundLink('http://adsabs.harvard.edu/abs/2011JHEP...12..015G','NASAADS'); return false;" href="http://adsabs.harvard.edu/abs/2011JHEP...12..015G"><span>Fisher <span class="hlt">equation</span> for a decaying brane</span></a></p>
<p><a target="_blank" href="http://adsabs.harvard.edu/abstract_service.html">NASA Astrophysics Data System (ADS)</a></p>
<p>Ghoshal, Debashis</p>
<p>2011-12-01</p>
<p>We consider the inhomogeneous decay of an unstable D-brane. The dynamical <span class="hlt">equation</span> that describes this process (in light-cone time) is a variant of the non-linear reaction-diffusion <span class="hlt">equation</span> that first made its appearance in the pioneering work of (Luther and) Fisher and appears in a variety of natural phenomena. We analyze its travelling front solution using singular perturbation theory.</p>
</li>
</ol>
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