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Influid mechanics,hydrostatic equilibrium, also calledhydrostatic balanceandhydrostasy, is the condition of afluidorplasticsolid at rest, which occurs when external forces, such asgravity, are balanced by apressure-gradient force.[1]In the planetary physics of Earth, the pressure-gradient force prevents gravity from co... | https://en.wikipedia.org/wiki/Hydrostatic_equilibrium |
In mathematics, themethod of steepest descentorsaddle-point methodis an extension ofLaplace's methodfor approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point... | https://en.wikipedia.org/wiki/Saddle-point_method |
Inmathematics,Laplace's method, named afterPierre-Simon Laplace, is a technique used to approximateintegralsof the form
wheref{\displaystyle f}is a twice-differentiablefunction,M{\displaystyle M}is a largenumber, and the endpointsa{\displaystyle a}andb{\displaystyle b}could be infinite. This technique was originally p... | https://en.wikipedia.org/wiki/Laplace%27s_method |
Inmathematical analysis, themaximumandminimum[a]of afunctionare, respectively, the greatest and least value taken by the function. Known generically asextremum,[b]they may be defined either within a givenrange(thelocalorrelativeextrema) or on the entiredomain(theglobalorabsoluteextrema) of a function.[1][2][3]Pierre de... | https://en.wikipedia.org/wiki/Maximum_and_minimum |
In the study ofdynamical systems, ahyperbolic equilibrium pointorhyperbolic fixed pointis afixed pointthat does not have anycenter manifolds. Near ahyperbolicpoint the orbits of a two-dimensional,non-dissipativesystem resemble hyperbolas. This fails to hold in general.Strogatznotes that "hyperbolic is an unfortunate n... | https://en.wikipedia.org/wiki/Hyperbolic_equilibrium_point |
Inmathematics,hyperbolic geometry(also calledLobachevskian geometryorBolyai–Lobachevskiangeometry) is anon-Euclidean geometry. Theparallel postulateofEuclidean geometryis replaced with:
(Compare the above withPlayfair's axiom, the modern version ofEuclid'sparallel postulate.)
Thehyperbolic planeis aplanewhere every p... | https://en.wikipedia.org/wiki/Hyperbolic_geometry |
In the mathematical area ofgame theoryand ofconvex optimization, aminimax theoremis a theorem that claims that
under certain conditions on the setsX{\displaystyle X}andY{\displaystyle Y}and on the functionf{\displaystyle f}.[1]It is always true that the left-hand side is at most the right-hand side (max–min inequality... | https://en.wikipedia.org/wiki/Minimax_theorem |
In mathematics, themax–min inequalityis as follows:
When equality holds one says thatf,W, andZsatisfies a strong max–min property (or asaddle-pointproperty). The example functionf(z,w)=sin(z+w){\displaystyle \ f(z,w)=\sin(z+w)\ }illustrates that the equality does not hold for every function.
A theorem giving conditi... | https://en.wikipedia.org/wiki/Max%E2%80%93min_inequality |
Themountain pass theoremis anexistence theoremfrom thecalculus of variations, originally due toAntonio AmbrosettiandPaul Rabinowitz.[1][2]Given certain conditions on a function, the theorem demonstrates the existence of asaddle point. The theorem is unusual in that there are many other theorems regarding the existence... | https://en.wikipedia.org/wiki/Mountain_pass_theorem |
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