Relevant bibliographies by topics / TOV Equation

Academic literature on the topic 'TOV Equation'

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Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "TOV Equation"

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Bhatti, M. Z., Z. Yousaf, and Zarnoor. "Stability of charged neutron star in Palatini f(R) gravity." Modern Physics Letters A 34, no. 31 (October 7, 2019): 1950252. http://dx.doi.org/10.1142/s0217732319502523.

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In this work, we discuss the stability of charged neutron star in the background of [Formula: see text] gravity and construct the generalized Tolman–Oppenheimer–Volkoff (TOV) equations. For this, we consider static spherically symmetric geometry to construct the hydrostatic equilibrium equation and deduce TOV equations from modified field equations with electromagnetic effects. We conclude that the generalized TOV equation depicts the stable stars configuration independent of the generic function of the modified gravity if the condition of uniform entropy and chemical composition is assumed.
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Alloy, Marcelo D., Débora P. Menezes, and Manuel Malheiro. "Ansatz for Dense Matter Equation of State." International Journal of Modern Physics: Conference Series 45 (January 2017): 1760049. http://dx.doi.org/10.1142/s2010194517600497.

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The aim of the present work is to try to find an equation of state (EoS) directly from the solution of the Tolman-Oppenheimer-Volkoff (TOV) equations subject to known observational constraints to the maximum mass and corresponding radius and baryonic mass. Hence, instead of solving the TOV equations with an EoS that enters as input, we obtain an EoS as output.
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Carvalho, G. A., S. I. Dos Santos, P. H. R. S. Moraes, and M. Malheiro. "Strange stars in energy–momentum-conserved f(R,T) gravity." International Journal of Modern Physics D 29, no. 10 (July 2020): 2050075. http://dx.doi.org/10.1142/s0218271820500753.

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For the accurate understanding of compact astrophysical objects, the Tolmann–Oppenheimer–Volkoff (TOV) equation has proved to be of great use. Nowadays, it has been derived in many alternative gravity theories, yielding the prediction of different macroscopic features for such compact objects. In this work, we apply the TOV equation of the energy–momentum–conserved version of the [Formula: see text] gravity theory to strange quark stars. The [Formula: see text] theory, with [Formula: see text] being a generic function of the Ricci scalar [Formula: see text] and trace of the energy–momentum tensor [Formula: see text] to replace [Formula: see text] in the Einstein–Hilbert gravitational action, has shown to provide a very interesting alternative to the cosmological constant [Formula: see text] in a cosmological scenario, particularly in the energy–momentum conserved case (a general [Formula: see text] function does not conserve the energy–momentum tensor). Here, we impose the condition [Formula: see text] to the astrophysical case, particularly the hydrostatic equilibrium of strange stars. We solve the TOV equation by taking into account linear equations of state to describe matter inside strange stars, such as [Formula: see text] and [Formula: see text], known as the MIT bag model, with [Formula: see text] the pressure and [Formula: see text] the energy density of the star, [Formula: see text] constant and [Formula: see text] the bag constant.
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Rather, Ishfaq A., Asloob A. Rather, Ilídio Lopes, V. Dexheimer, A. A. Usmani, and S. K. Patra. "Magnetic-field Induced Deformation in Hybrid Stars." Astrophysical Journal 943, no. 1 (January 1, 2023): 52. http://dx.doi.org/10.3847/1538-4357/aca85c.

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Abstract The effects of strong magnetic fields on the deconfinement phase transition expected to take place in the interior of massive neutron stars are studied in detail for the first time. For hadronic matter, the very general density-dependent relativistic mean field model is employed, while the simple, but effective vector-enhanced bag model is used to study quark matter. Magnetic-field effects are incorporated into the matter equation of state and in the general-relativity solutions, which also satisfy Maxwell’s equations. We find that for large values of magnetic dipole moment, the maximum mass, canonical mass radius, and dimensionless tidal deformability obtained for stars using spherically symmetric Tolman–Oppenheimer–Volkoff (TOV) equations and axisymmetric solutions attained through the LORENE library differ considerably. The deviations depend on the stiffness of the equation of state and on the star mass being analyzed. This points to the fact that, unlike what was assumed previously in the literature, magnetic field thresholds for the approximation of isotropic stars and the acceptable use of TOV equations depend on the matter composition and interactions.
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Riazi, Nematollah, S. Sedigheh Hashemi, S. Naseh Sajadi, and Shahrokh Assyyaee. "A new class of anisotropic solutions of the generalized TOV equation." Canadian Journal of Physics 94, no. 10 (October 2016): 1093–101. http://dx.doi.org/10.1139/cjp-2016-0365.

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We present gravitating relativistic spheres composed of an anisotropic, barotropic fluid. We assume a bi-polytropic equation of state that has both linear and power-law terms. The generalized Tolman–Oppenheimer–Volkoff (TOV) equation, which describes the hydrostatic equilibrium, is used and the full system of equations is solved for solutions that are regular at the origin and asymptotically flat. Conditions for the appearance of horizon and a basic treatment of stability are also presented.
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Albino, M. B., F. S. Navarra, R. Fariello, and G. Lugones. "The nature of the quark-hadron phase transition in hybrid stars and the mass-radius diagram." Journal of Physics: Conference Series 2340, no. 1 (September 1, 2022): 012015. http://dx.doi.org/10.1088/1742-6596/2340/1/012015.

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Abstract In this work, we use a hybrid equation of state that allows us to choose the smoothness of the quark-hadron phase transition, by choosing the value of a continuous parameter μc . To describe the hadron phase, we use an equation of state (EoS) based on a chiral effective field theory (cEFT), and for the quark phase we use the equation of state of the MFTQCD (Mean Field Theory of QCD). We solve simultaneously the TOV equations and the tidal deformability equations and contruct the mass-radius and deformability-mass diagrams for several values of the parameter μc . We find that the curves in these two diagrams are almost insensitive to the smoothness of the phase transition.
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Abbas, G., and M. R. Shahzad. "Quintessence compact stars with Vaidya–Tikekar type grr for anisotropic fluid." Canadian Journal of Physics 98, no. 9 (September 2020): 869–76. http://dx.doi.org/10.1139/cjp-2019-0596.

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The present study provides a new solution to the Einstein field equations for anisotropic matter configuration in static and spherically symmetric space–time. By taking benefit from the conformal Killing vector (CKV) technique and quintessence field specified by a parameter ωq as –1 < ωq < –1/3, we generate an exact solution to the field equations. For this investigation, we have used a specific form of metric potential taken fromVaidya–Tikekar (J. Astrophys. Astron. 3, 325 (1982)) geometry. To canvass the physical plausibility of the presented solution, we explored some analytical expressions such as energy conditions, the TOV equation, stability analysis, and equation of state parameters. We present graphical analysis of the necessary analytical expressions that revealed that our solution satisfies the necessary physical conditions.
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Zhang, Z., X. Wang, H. Zhang, and J. Shi. "Entropy of nonsingular self-gravitating polytropes and their TOV equation." Il Nuovo Cimento B 106, no. 11 (November 1991): 1189–94. http://dx.doi.org/10.1007/bf02728656.

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MIRZA, BABUR M. "THE EQUILIBRIUM STRUCTURE OF CHARGED ROTATING RELATIVISTIC STARS." International Journal of Modern Physics D 17, no. 12 (November 2008): 2291–304. http://dx.doi.org/10.1142/s021827180801387x.

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General relativistic equilibrium conditions imply that an electrically charged compact star, in a spherically symmetric configuration, can sustain a huge amount of electric charge (up to 1020 C). The equilibrium, however, is reached under very critical conditions such that a perturbation to the stellar structure can cause these systems to collapse. We study the effects of rotation in charged compact stars and obtain conditions, the modified Tolman–Oppenheimer–Volkoff (TOV) equations, under which such stars form a stable gravitational system against Coulomb repulsion. We assume the star to be rotating slowly. We also assume that the charge density is proportional to the mass density everywhere inside the star. The modified TOV equations for hydrostatic equilibrium are integrated numerically for the general equation of state for a polytrope. The detailed numerical study shows that the centrifugal force adds to the Coulomb pressure in the star. In the stable equilibrium configurations, therefore, a loss in stellar mass (energy) density occurs for higher values of the angular frequency. The additional energy is radiated in the form of electrical energy. The stellar radius is also decreased so that the star does not necessarily becomes more compact.
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Nayak, S. N., P. K. Parida, and P. K. Panda. "Effects of the cosmological constant on compact star in quark-meson coupling model." International Journal of Modern Physics E 24, no. 10 (October 2015): 1550068. http://dx.doi.org/10.1142/s0218301315500688.

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We study effect of the cosmological constant on compact star with equation of state provided by quark-meson coupling (QMC) model. In this model, baryons are described as a system of nonoverlapping bags interacting through the scalar and vector mesons. We derive the Tolman–Oppenheimer–Volkoff (TOV) equation taking into account the cosmological constant in static and spherically symmetric metric. Using the equation of state given by QMC model, the mass–radius relationship of the compact star has been computed for various values of the cosmological constant.
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Dissertations / Theses on the topic "TOV Equation"

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Rutstam, Nils. "Study of equations for Tippe Top and related rigid bodies." Licentiate thesis, Linköpings universitet, Tillämpad matematik, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-60835.

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The Tippe Top consist of a small truncated sphere with a peg as a handle. When it is spun fast enough on its spherical part it starts to turn upside down and ends up spinning on the peg. This counterintuitive behaviour, called inversion, is a curious feature of this dynamical system that has been studied for some time, but obtaining a complete description of the dynamics of inversion has proved to be a difficult problem. The existing results are either numerical simulations of the equations of motion or asymptotic analysis that shows that the inverted position is the only attractive and stable position under certain conditions. This thesis will present methods to analyze the equations of motion of the Tippe Top, which we study in three equivalent forms that each helps us to understand different aspects of the inversion phenomenon. Our study of the Tippe Top also focuses on the role of the underlying assumptions in the standard model for the external force, and what consequences these assumptions have, in particular for the asymptotic cases. We define two dynamical systems as an aid to understand the dynamics of the Tippe Top, the gliding heavy symmetric top and the gliding eccentric cylinder. The gliding heavy symmetric top is a natural non-integrable generalization of the well-known heavy symmetric top. Equations of motion and asymptotics for this system are derived, but we also show that equations for the gliding heavy symmetric top can be obtained as a limit of the equations for the Tippe Top. The equations for the gliding eccentric cylinder can be interpreted as a special case of the equations for the Tippe Top, and since it is a simpler system, properties of the Tippe Top equations are easier to study. In particular, asymptotic analysis of the gliding eccentric cylinder reveals that the standard model seems to have inconsistencies that need to be addressed.
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Måhl, Anna. "Separation of variables for ordinary differential equations." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5620.

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In case of the PDE's the concept of solving by separation of variables

has a well defined meaning. One seeks a solution in a form of a

product or sum and tries to build the general solution out of these

particular solutions. There are also known systems of second order

ODE's describing potential motions and certain rigid bodies that are

considered to be separable. However, in those cases, the concept of

separation of variables is more elusive; no general definition is

given.

In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

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Heavilin, Justin. "The Red Top Model: A Landscape-Scale Integrodifference Equation Model of the Mountain Pine Beetle-Lodgepole Pine Forest Interaction." DigitalCommons@USU, 2007. https://digitalcommons.usu.edu/etd/7137.

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Under normative conditions the mountain pine beetle (Dendroctonus ponderosae Hopkins) has played a regulating role in healthy lodgepole pine (Pinus contorta) forests. However, recently eruptive outbreaks that result from large pine beetle populations have destroyed vast tracts of valuable forest. The outbreaks in North America have received a great deal of attention from both the timber industry and government agencies as well as biologists and ecologists. In this dissertation we develop a landscape-scaled integrodifference equation model describing the mountain pine beetle and its effect on a lodgepole pine forest. The model is built upon a stage-structured model of a healthy lodgepole pine forest with the addition of beetle pressure in the form of an infected tree class. These infected trees are produced by successful beetle attack, modelled by response functions. Different response functions reflect different probabilities for various densities. This feature of the model allows us to test hypotheses regarding density-dependent beetle attacks. To capture the spatial aspect of beetle dispersal from infected trees we employ dispersal kernels. These provide a probabilistic model for finding given beetle densities at some distance from infected trees. Just as varied response functions model different attack dynamics, the choice of kernel can model different dispersal behavior. The modular nature of the Red Top Model yields multiple model candidates. These models allow discrimination between broad possibilities at the land scape scale: whether or not beetles are subject to a threshold effect at the lands cape scale and whether or not host selection is random or directed. We fit the model using estimating functions to two distinct types of data: aerial damage survey data and remote sensing imagery. Having constructed multiple models, we introduce a novel model selection methodology for spatial models based on facial recognition technology. Because the regions and years of aerial damage survey and remote sensing data in the Sawtooth National Recreation Area overlap, we can compare the results from data sets to address the question of whether remote sensing data actually provides insight to the system that coarser scale but less expensive and more readily available aerial damage survey data does not.
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Yang, Ronghua. "Studies on value distribution of solutions of complex linear differential equations /." Joensuu : Joensuun yliopistopaino, 2006. http://www.loc.gov/catdir/toc/fy0706/2006421381.html.

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Dengiz, Suat. "3+1 Orthogonal And Conformal Decomposition Of The Einstein Equation And The Adm Formalism For General Relativity." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12612949/index.pdf.

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In this work, two particular orthogonal and conformal decompositions of the 3+1 dimensional Einstein equation and Arnowitt-Deser-Misner (ADM) formalism for general relativity are obtained. In order to do these, the 3+1 foliation of the four-dimensional spacetime, the fundamental conformal transformations and the Hamiltonian form of general relativity that leads to the ADM formalism, de
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Schulz, Stephan. "Leaning search control knowlledge for equational deduction /." Berlin : AKA, 2000. http://www.loc.gov/catdir/toc/fy0804/2007440965.html.

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Häggström, Johan. "Teaching systems of linear equations in Sweden and China : what is made possible to learn? /." Göteborg : Göteborgs universitet, 2008. http://www.loc.gov/catdir/toc/fy0805/2008380731.html.

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Bernroider, Edward, and Patrick Schmöllerl. "A technological, organisational, and environmental analysis of decision making methodologies and satisfaction in the context of IT induced business transformations." Elsevier, 2013. http://dx.doi.org/10.1016/j.ejor.2012.07.025.

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Although Operational Research (OR) has successfully provided many methodologies to address complex decision problems, in particular based on the rationality principle, there has been too little discussion regarding their limited consideration in IT evaluation practice and associated decision making satisfaction levels in an organisational context. The aim of this paper is to address these issues through providing a current account of diffusion and infusion of OR methodologies in IT decision making practice, and by analysing factors affecting decision making satisfaction from a Technological, Organisational, and Environmental (TOE) framework in the context of IT induced business transformations. We developed a structural equation model and conducted an empirical survey, which supported four out of five developed research hypotheses. Our results show that while Decision Support Systems (DSS), holistic IT evaluation methods, and management support seem to positively affect individual satisfaction, legislative regulation has an adverse effect. Results also revealed a persistent methodology diffusion and infusion gap. The paper discusses implications in each of these aspects and presents opportunities for future work. (authors' abstract)
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Rippeyoung, Phyllis Love Farley. "Is it too late baby? pinpointing the emergence of a black-white test score gap in infancy." Diss., University of Iowa, 2006. http://ir.uiowa.edu/etd/80.

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El-Kafri, Manal M. Lutfi. "Symmetry methods applied to Richard's equations and problems of infiltration." Thesis, University of South Wales, 2006. https://pure.southwales.ac.uk/en/studentthesis/symmetry-methods-applied-to-richards-equations-and-problems-of-infiltration(e94a3a66-f16b-46cd-a9c8-192ac6b995bc).html.

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Water resources development around the world has taken many different forms and directions since the dawn of civilization. Water shortage in arid and semiarid regions has encouraged the search for additional sources currently not exploited intensively. Hence, knowledge of the infiltration process is a requirement for understanding water management. The main aim here is to solve the one-dimensional nonlinear time-dependent Richard's equation for water flow in an unsaturated uniform soil. The main theory of soil infiltration is introduced using a mathematical-physical approach to describe water movement in unsaturated soils. This gives rise to Richard's flow equation; which is presented for both unsaturated and also saturated soil. Methods for solving Richard's equation by both analytical and numerical techniques are then introduced. This gives rise to a discussion of the similarity methods first used by Philip to determine analytical solutions of Richard's equation in an unsaturated soil. This is then generalised to determine a broader class of solutions using the Lie (classical) symmetry approach. The non-classical symmetries of Bluman and Cole are also determined. Although these group methods provide the most widely applicable technique to find solutions of ordinary and partial differential equations, a large number of tedious calculations are involved. With the help of computer algebra it is shown that the determining equations for the non-classical case lead to four new highly non-linear equations which are solved in five particular cases. Each case of classical and non-classical solutions is then reduced to an ordinary differential equation and explicit solutions are produced when possible. The potential classical and non-classical method, first suggested by Bluman, Reid and Kumei, is also discussed and presented. The potential non-classical method produced new results, which the potential classical method did not. The solution is useful as a tool by which to judge the quality of numerical methods. A practical solution of classical (Lie/ potential) and non-classical symmetry of Richard's equation is presented. Finally, conclusions and suggestions for further work are discussed.
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Books on the topic "TOV Equation"

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Orlik, Lyubov', and Galina Zhukova. Operator equation and related questions of stability of differential equations. ru: INFRA-M Academic Publishing LLC., 2020. http://dx.doi.org/10.12737/1061676.

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The monograph is devoted to the application of methods of functional analysis to the problems of qualitative theory of differential equations. Describes an algorithm to bring the differential boundary value problem to an operator equation. The research of solutions to operator equations of special kind in the spaces polutoratonny with a cone, where the limitations of the elements of these spaces is understood as the comparability them with a fixed scale element of exponential type. Found representations of the solutions of operator equations in the form of contour integrals, theorems of existence and uniqueness of such solutions. The spectral criteria for boundedness of solutions of operator equations and, as a consequence, sufficient spectral features boundedness of solutions of differential and differential-difference equations in Banach space. The results obtained for operator equations with operators and work of Volterra operators, allowed to extend to some systems of partial differential equations known spectral stability criteria for solutions of A. M. Lyapunov and also to generalize theorems on the exponential characteristic. The results of the monograph may be useful in the study of linear mechanical and electrical systems, in problems of diffraction of electromagnetic waves, theory of automatic control, etc. It is intended for researchers, graduate students functional analysis and its applications to operator and differential equations.
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The human equation. [Edmonton, Alta.]: Human Equation Inc., 2004.

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Escudier, Marcel. Basic equations of viscous-fluid flow. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198719878.003.0015.

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In this chapter it is shown that application of the momentum-conservation equation (Newton’s second law of motion) to an infinitesimal cube of fluid leads to Cauchy’s partial differential equations, which govern the flow of any fluid satisfying the continuum hypothesis. Any fluid flow must also satisfy the continuity equation, another partial differential equation, which is derived from the mass-conservation equation. It is shown that distortion of a flowing fluid can be split into elongational distortion and angular distortion or shear strain. For a Newtonian fluid, the normal and shear stresses in Cauchy’s equations are related to the elongational and shear-strain rates through Stokes’ constitutive equations. Substitution of these constitutive equations into Cauchy’s equations leads to the Navier-Stokes equations, which govern steady or unsteady flow of a fluid. A minor modification of the constitutive equations for a Newtonian fluid allows consideration of generalised Newtonian fluids, for which the viscosity depends upon the shear-strain rates. The boundary conditions for the tangential and normal velocity components are discussed briefly.
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Mann, Peter. Wave Mechanics & Elements of Mathematical Physics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0005.

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This chapter presents an in-depth look at classical wave mechanics and mathematical physics, containing key examples directly relevant to molecular physics. The separation of variables is used to construct the Helmholtz equation from the one-dimensional wave equation before considering the three-dimensional wave equation. From this, equations for the temporal, radial, azimuth and angular components are developed and solutions using the Bessel equations and Legendre polynomials are found. Boundary conditions are explained and the Rayleigh plane wave expansion as the general solution to the Helmholtz equation is reconstructed. Both the Hermite equation and the Legendre equation are derived using the series solution method, and the Laplace equation is discussed.
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Escudier, Marcel. Laminar boundary layers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198719878.003.0017.

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This chapter starts by introducing the concept of a boundary layer and the associated boundary-layer approximations. The laminar boundary-layer equations are then derived from the Navier-Stokes equations. The assumption of velocity-profile similarity is shown to reduce the partial differential boundary-layer equations to ordinary differential equations. The results of numerical solutions to these equations are discussed: Blasius’ equation, for zero-pressure gradient, and the Falkner-Skan equation for wedge flows. Von Kármán’s momentum-integral equation is derived and used to obtain useful results for the zero-pressure-gradient boundary layer. Pohlhausen’s quartic-profile method is then discussed, followed by the approximate method of Thwaites. The chapter concludes with a qualitative account of the way in which aerodynamic lift is generated.
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Rajeev, S. G. Euler’s Equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0002.

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Euler derived the fundamental equations of an ideal fluid, that is, in the absence of friction (viscosity). They describe the conservation of momentum. We can derive from it the equation for the evolution of vorticity (Helmholtz equation). Euler’s equations have to be supplemented by the conservation of mass and by an equation of state (which relates density to pressure). Of special interest is the case of incompressible flow; when the fluid velocity is small compared to the speed of sound, the density may be treated as a constant. In this limit, Euler’s equations have scale invariance in addition to rotation and translation invariance. d’Alembert’s paradox points out the limitation of Euler’s equation: friction cannot be ignored near the boundary, nomatter how small the viscosity.
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Rajeev, S. G. Integrable Models. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0009.

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Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.
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Deruelle, Nathalie, and Jean-Philippe Uzan. Kinetic theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0010.

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This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.
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Cantor, Brian. The Equations of Materials. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198851875.001.0001.

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This book describes some of the important equations of materials and the scientists who derived them. It is aimed at anyone interested in the manufacture, structure, properties and engineering application of materials such as metals, polymers, ceramics, semiconductors and composites. It is meant to be readable and enjoyable, a primer rather than a textbook, covering only a limited number of topics and not trying to be comprehensive. It is pitched at the level of a final year school student or a first year undergraduate who has been studying the physical sciences and is thinking of specialising into materials science and/or materials engineering, but it should also appeal to many other scientists at other stages of their career. It requires a working knowledge of school maths, mainly algebra and simple calculus, but nothing more complex. It is dedicated to a number of propositions, as follows: 1. The most important equations are often simple and easily explained; 2. The most important equations are often experimental, confirmed time and again; 3. The most important equations have been derived by remarkable scientists who lived interesting lives. Each chapter covers a single equation and materials subject. Each chapter is structured in three sections: first, a description of the equation itself; second, a short biography of the scientist after whom it is named; and third, a discussion of some of the ramifications and applications of the equation. The biographical sections intertwine the personal and professional life of the scientist with contemporary political and scientific developments. The topics included are: Bravais lattices and crystals; Bragg’s law and diffraction; the Gibbs phase rule and phases; Boltzmann’s equation and thermodynamics; the Arrhenius equation and reactions; the Gibbs-Thomson equation and surfaces; Fick’s laws and diffusion; the Scheil equation and solidification; the Avrami equation and phase transformations; Hooke’s law and elasticity; the Burgers vector and plasticity; Griffith’s equation and fracture; and the Fermi level and electrical properties.
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Rajeev, S. G. Hamiltonian Systems Based on a Lie Algebra. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0010.

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There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.
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Book chapters on the topic "TOV Equation"

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Debussche, Arnaud, Berenger Hug, and Etienne Mémin. "Modeling Under Location Uncertainty: A Convergent Large-Scale Representation of the Navier-Stokes Equations." In Mathematics of Planet Earth, 15–26. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-18988-3_2.

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AbstractWe construct martingale solutions for the stochastic Navier-Stokes equations in the framework of the modelling under location uncertainty (LU). These solutions are pathwise and unique when the spatial dimension is 2D. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier-Stokes equation. This warrants that the LU Navier-Stokes equations can be interpreted as a large-scale model of the deterministic Navier-Stokes equation.
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Seifert, Christian, Sascha Trostorff, and Marcus Waurick. "The Fourier–Laplace Transformation and Material Law Operators." In Evolutionary Equations, 67–83. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89397-2_5.

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AbstractIn this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’s equation, the involved material parameters, such as heat conductivity or permeability of the underlying medium, are incorporated within these operators. Hence, these operators are also called “material law operators”. We start our chapter by defining the Fourier transformation and proving Plancherel’s theorem in the Hilbert space-valued case, which states that the Fourier transformation defines a unitary operator on $$L_2(\mathbb {R};H)$$ L 2 ( ℝ ; H ) .
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Deville, Michel O. "Boundary Layer." In An Introduction to the Mechanics of Incompressible Fluids, 175–95. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-04683-4_7.

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AbstractThe Prandtl’s equations for laminar boundary layer are obtained via dimensional analysis. The case of the flat plate is treated as a suitable example for the development of the boundary layer on a simple geometry. Various thicknesses are introduced. The integration of Prandtl’s equation across the boundary layer produces the von Kármán integral equation which allows the elaboration of the approximate von Kármán-Pohlhausen method where the velocity profile is given as a polynomial. The use of a third degree polynomial for the flat plate demonstrates the feasibility of the approach.
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Breda, Dimitri, Jung Kyu Canci, and Raffaele D’Ambrosio. "An Invitation to Stochastic Differential Equations in Healthcare." In Quantitative Models in Life Science Business, 97–110. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-11814-2_6.

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AbstractAn important problem in finance is the evaluation of the value in the future of assets (e.g., shares in company, currencies, derivatives, patents). The change of the values can be modeled with differential equations. Roughly speaking, a typical differential equation in finance has two components, one deterministic (e.g., rate of interest of bank accounts) and one stochastic (e.g., values of stocks) that is often related to the notion of Brownian motions. The solution of such a differential equation needs the evaluation of Riemann–Stieltjes’s integrals for the deterministic part and Ito’s integrals for the stochastic part. For A few types of such differential equations, it is possible to determine an exact solution, e.g., a geometric Brownian motion. On the other side for almost all stochastic differential equations we can only provide approximations of a solution. We present some numerical methods for solving stochastic differential equations.
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Ayala-Rincón, Mauricio, Maribel Fernández, Daniele Nantes-Sobrinho, and Deivid Vale. "Nominal Equational Problems." In Lecture Notes in Computer Science, 22–41. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-71995-1_2.

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AbstractWe define nominal equational problems of the form $$\exists \overline{W} \forall \overline{Y} : P$$ ∃ W ¯ ∀ Y ¯ : P , where $$P$$ P consists of conjunctions and disjunctions of equations $$s\approx _\alpha t$$ s ≈ α t , freshness constraints $$a\#t$$ a # t and their negations: $$s \not \approx _\alpha t$$ s ≉ α t and "Equation missing", where $$a$$ a is an atom and $$s, t$$ s , t nominal terms. We give a general definition of solution and a set of simplification rules to compute solutions in the nominal ground term algebra. For the latter, we define notions of solved form from which solutions can be easily extracted and show that the simplification rules are sound, preserving, and complete. With a particular strategy for rule application, the simplification process terminates and thus specifies an algorithm to solve nominal equational problems. These results generalise previous results obtained by Comon and Lescanne for first-order languages to languages with binding operators. In particular, we show that the problem of deciding the validity of a first-order equational formula in a language with binding operators (i.e., validity modulo $$\alpha $$ α -equality) is decidable.
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Bart, Harm, Marinus A. Kaashoek, and André C. M. Ran. "Convolution equations and the transport equation." In A State Space Approach to Canonical Factorization with Applications, 115–42. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-7643-8753-2_7.

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Hasanov, Fakhri J., Frederick L. Joutz, Jeyhun I. Mikayilov, and Muhammad Javid. "KGEMM Behavioral Equations and Identities." In SpringerBriefs in Economics, 41–83. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-12275-0_7.

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AbstractThis chapter reports the estimated long-run equations and identities in Sects. 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8, and 7.9 while the estimated short-run equations, i.e., final ECM specifications associated with the long-run equations are reported in Appendix B to save space in the main text. Note that the long-run and ECM equations are estimated till 2019 in the fifth version of KGEMM. Starting years of the estimations range from the 1970s to the 1990s dictated by the data availability. For the readers ease, we describe one of the long-run equations below and the rest equations here follow the same context. As an example, we select the first appeared long-run equation, i.e., Eq. (7.43).
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Deville, Michel O. "Turbulence." In An Introduction to the Mechanics of Incompressible Fluids, 211–56. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-04683-4_9.

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AbstractThe Reynolds decomposition and statistical averaging of velocity and pressure generate the Reynolds averaged Navier–Stokes (RANS) equations. The closure problem is solved by the introduction of a turbulence constitutive equation. Several linear turbulence models are presented in the RANS framework: $$K-\varepsilon , K-\omega $$ K - ε , K - ω . The solution of the RANS equations for the turbulent channel flow is elaborated giving the celebrated logarithmic profile. Non-linear models are built on the anisotropy tensor and the incorporation of the concept of integrity bases. The chapter ends with the theory of large eddy simulations with a few up-to-date models: dynamic model, approximate deconvolution method.
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Crisan, Dan, and Prince Romeo Mensah. "Blow-Up of Strong Solutions of the Thermal Quasi-Geostrophic Equation." In Mathematics of Planet Earth, 1–14. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-18988-3_1.

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AbstractThe Thermal Quasi-Geostrophic (TQG) equation is a coupled system of equations that governs the evolution of the buoyancy and the potential vorticity of a fluid. It has a local in time solution as proved in Crisan et al. (Theoretical and computational analysis of the thermal quasi-geostrophic model. Preprint arXiv:2106.14850, 2021). In this paper, we give a criterion for the blow-up of solutions to the Thermal Quasi-Geostrophic equation, in the spirit of the classical Beale–Kato–Majda blow-up criterion (cf. Beale et al., Comm. Math. Phys. 94(1), 61–66, 1984) for the solution of the Euler equation.
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Hoang, Lê Nguyên. "Quick And Not Too Dirty." In The Equation of Knowledge, 249–67. Boca Raton : C&H/CRC Press, 2020. | Translation of: La formule du savoir : une philosophie unifiée du savoir fondée sur le théorème de Bayes: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9780367855307-14.

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Conference papers on the topic "TOV Equation"

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BOONSERM, PETARPA, MATT VISSER, and SILKE WEINFURTNER. "SOLUTION GENERATING THEOREMS: PERFECT FLUID SPHERES AND THE TOV EQUATION." In Proceedings of the MG11 Meeting on General Relativity. World Scientific Publishing Company, 2008. http://dx.doi.org/10.1142/9789812834300_0388.

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NARVÁEZ MACARRO, L. "D-MODULES IN DIMENSION 1." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0001.

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CASTRO JIMÉNEZ, FRANCISCO J. "MODULES OVER THE WEYL ALGEBRA." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0002.

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LÊ, DŨNG TRÁNG, and BERNARD TEISSIER. "GEOMETRY OF CHARACTERISTIC VARIETIES." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0003.

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DELABAERE, E. "SINGULAR INTEGRALS AND THE STATIONARY PHASE METHODS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0004.

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JAMBU, MICHEL. "HYPERGEOMETRIC FUNCTIONS AND HYPERPLANE ARRANGEMENTS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0005.

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GRANGER, MICHEL. "BERNSTEIN-SATO POLYNOMIALS AND FUNCTIONAL EQUATIONS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0006.

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MALGRANGE, B. "DIFFERENTIAL ALGEBRAIC GROUPS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0007.

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"FRONT MATTER." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_fmatter.

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Tsourkas, Philippos K., and Boris Rubinsky. "Laplace’s Equation, Genetic Algorithms, and Evolution." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32658.

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With the advent of problems in genetics that are either too difficult or too dangerous to solve experimentally, it is important to have mathematical tools available so that these problems may be solved through modeling and computation. To this end we developed a mathematical experimentation procedure to simulate the evolution of a population of individuals. The procedure employs genetic algorithm methodology to study a ‘species’ that is comprised of solutions to the Laplace equation. The algorithm is applied to the study of a particularly significant and controversial problem: The release of genetically engineered organism in the wild.
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Reports on the topic "TOV Equation"

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Ostashev, Vladimir, Michael Muhlestein, and D. Wilson. Extra-wide-angle parabolic equations in motionless and moving media. Engineer Research and Development Center (U.S.), September 2021. http://dx.doi.org/10.21079/11681/42043.

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Wide-angle parabolic equations (WAPEs) play an important role in physics. They are derived by an expansion of a square-root pseudo-differential operator in one-way wave equations, and then solved by finite-difference techniques. In the present paper, a different approach is suggested. The starting point is an extra-wide-angle parabolic equation (EWAPE) valid for small variations of the refractive index of a medium. This equation is written in an integral form, solved by a perturbation technique, and transformed to the spectral domain. The resulting split-step spectral algorithm for the EWAPE accounts for the propagation angles up to 90° with respect to the nominal direction. This EWAPE is also generalized to large variations in the refractive index. It is shown that WAPEs known in the literature are particular cases of the two EWAPEs. This provides an alternative derivation of the WAPEs, enables a better understanding of the underlying physics and ranges of their applicability, and opens an opportunity for innovative algorithms. Sound propagation in both motionless and moving media is considered. The split-step spectral algorithm is particularly useful in the latter case since complicated partial derivatives of the sound pressure and medium velocity reduce to wave vectors (essentially, propagation angles) in the spectral domain.
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Fujisaki, Masatoshi. Normed Bellman Equation with Degenerate Diffusion Coefficients and Its Application to Differential Equations. Fort Belvoir, VA: Defense Technical Information Center, October 1987. http://dx.doi.org/10.21236/ada190319.

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Over, Thomas, Riki Saito, Andrea Veilleux, Padraic O’Shea, Jennifer Sharpe, David Soong, and Audrey Ishii. Estimation of Peak Discharge Quantiles for Selected Annual Exceedance Probabilities in Northeastern Illinois. Illinois Center for Transportation, June 2016. http://dx.doi.org/10.36501/0197-9191/16-014.

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This report provides two sets of equations for estimating peak discharge quantiles at annual exceedance probabilities (AEPs) of 0.50, 0.20, 0.10, 0.04, 0.02, 0.01, 0.005, and 0.002 (recurrence intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years, respectively) for watersheds in Illinois based on annual maximum peak discharge data from 117 watersheds in and near northeastern Illinois. One set of equations was developed through a temporal analysis with a two-step least squares-quantile regression technique that measures the average effect of changes in the urbanization of the watersheds used in the study. The resulting equations can be used to adjust rural peak discharge quantiles for the effect of urbanization, and in this study the equations also were used to adjust the annual maximum peak discharges from the study watersheds to 2010 urbanization conditions. The other set of equations was developed by a spatial analysis. This analysis used generalized least-squares regression to fit the peak discharge quantiles computed from the urbanization-adjusted annual maximum peak discharges from the study watersheds to drainage-basin characteristics. The peak discharge quantiles were computed by using the Expected Moments Algorithm following the removal of potentially influential low floods defined by a multiple Grubbs-Beck test. To improve the quantile estimates, regional skew coefficients were obtained from a newly developed regional skew model in which the skew increases with the urbanized land use fraction. The skew coefficient values for each streamgage were then computed as the variance-weighted average of at-site and regional skew coefficients. The drainage-basin characteristics used as explanatory variables in the spatial analysis include drainage area, the fraction of developed land, the fraction of land with poorly drained soils or likely water, and the basin slope estimated as the ratio of the basin relief to basin perimeter. This report also provides: (1) examples to illustrate the use of the spatial and urbanization-adjustment equations for estimating peak discharge quantiles at ungaged sites and to improve flood-quantile estimates at and near a gaged site; (2) the urbanization-adjusted annual maximum peak discharges and peak discharge quantile estimates at streamgages from 181 watersheds including the 117 study watersheds and 64 additional watersheds in the study region that were originally considered for use in the study but later deemed to be redundant. The urbanization-adjustment equations, spatial regression equations, and peak discharge quantile estimates developed in this study will be made available in the web-based application StreamStats, which provides automated regression-equation solutions for user-selected stream locations. Figures and tables comparing the observed and urbanization-adjusted peak discharge records by streamgage are provided at http://dx.doi.org/10.3133/sir20165050 for download.
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Hart, Carl, and Gregory Lyons. A tutorial on the rapid distortion theory model for unidirectional, plane shearing of homogeneous turbulence. Engineer Research and Development Center (U.S.), July 2022. http://dx.doi.org/10.21079/11681/44766.

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The theory of near-surface atmospheric wind noise is largely predicated on assuming turbulence is homogeneous and isotropic. For high turbulent wavenumbers, this is a fairly reasonable approximation, though it can introduce non-negligible errors in shear flows. Recent near-surface measurements of atmospheric turbulence suggest that anisotropic turbulence can be adequately modeled by rapid-distortion theory (RDT), which can serve as a natural extension of wind noise theory. Here, a solution for the RDT equations of unidirectional plane shearing of homogeneous turbulence is reproduced. It is assumed that the time-varying velocity spectral tensor can be made stationary by substituting an eddy-lifetime parameter in place of time. General and particular RDT evolution equations for stochastic increments are derived in detail. Analytical solutions for the RDT evolution equation, with and without an effective eddy viscosity, are given. An alternative expression for the eddy-lifetime parameter is shown. The turbulence kinetic energy budget is examined for RDT. Predictions by RDT are shown for velocity (co)variances, one-dimensional streamwise spectra, length scales, and the second invariant of the anisotropy tensor of the moments of velocity. The RDT prediction of the second invariant for the velocity anisotropy tensor is shown to agree better with direct numerical simulations than previously reported.
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Luc, Brunet. Systematic Equations Handbook : Book 1-Energy. R&D Médiation, May 2015. http://dx.doi.org/10.17601/rd_mediation2015:1.

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The energy equation handbook is the complete collection of physically coherent expression of energy computed using from 2 to 7 physical units among: density(ML-3) energy (ML2T-2) time (T) force (MLT-2) power (ML2T-3) current (I) temperature (Th) quantity (N) mass (M) length (L) candela (J) surface (L2) volume (L3) concentration (ML-3) frequency (T-1) acceleration (LT- 2) speed (LT-1) pressure (ML-1T-2) viscosity (ML-1T-1) luminance (L- 2J) MolarMass (MN-1) MassicEnergy (L2T-2) resistance (ML2T-3I-2) voltage (ML2T-3I-1) Farad (M-1L-2T4I2) Thermal- Conductivity (MLT-3Th-1) SpecificHeat (L2T-2Th-1) MassFlux (MT-1) SurfaceTension (MT-2) Charge (TI) Resistivity (ML3T-3I-2) The complete list of 4196 equations is sorted by number of variable required to obtain an energy in Joules. All the units are in MKSA.
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Kahan, W. To Solve a Real Cubic Equation. Fort Belvoir, VA: Defense Technical Information Center, November 1986. http://dx.doi.org/10.21236/ada206859.

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Holmes, Eleanor, Laurie Gainey, and John Hanna. Upgrades to the Parabolic Equation Model. Fort Belvoir, VA: Defense Technical Information Center, March 1988. http://dx.doi.org/10.21236/ada211899.

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Baader, Franz, and Alexander Okhotin. On Language Equations with One-sided Concatenation. Aachen University of Technology, 2006. http://dx.doi.org/10.25368/2022.154.

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Language equations are equations where both the constants occurring in the equations and the solutions are formal languages. They have first been introduced in formal language theory, but are now also considered in other areas of computer science. In the present paper, we restrict the attention to language equations with one-sided concatenation, but in contrast to previous work on these equations, we allow not just union but all Boolean operations to be used when formulating them. In addition, we are not just interested in deciding solvability of such equations, but also in deciding other properties of the set of solutions, like its cardinality (finite, infinite, uncountable) and whether it contains least/greatest solutions. We show that all these decision problems are ExpTime-complete.
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Robinson, J. R. BASOPS - Missing Link to the Readiness Equation. Fort Belvoir, VA: Defense Technical Information Center, February 1999. http://dx.doi.org/10.21236/ada363889.

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Baader, Franz, Pavlos Marantidis, and Alexander Okhotin. Approximately Solving Set Equations. Technische Universität Dresden, 2016. http://dx.doi.org/10.25368/2022.227.

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Unification with constants modulo the theory ACUI of an associative (A), commutative (C) and idempotent (I) binary function symbol with a unit (U) corresponds to solving a very simple type of set equations. It is well-known that solvability of systems of such equations can be decided in polynomial time by reducing it to satisfiability of propositional Horn formulae. Here we introduce a modified version of this problem by no longer requiring all equations to be completely solved, but allowing for a certain number of violations of the equations. We introduce three different ways of counting the number of violations, and investigate the complexity of the respective decision problem, i.e., the problem of deciding whether there is an assignment that solves the system with at most l violations for a given threshold value l.
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