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Topic: Quartic

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quartic equations

Klein quartic

quartic function

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	Quartic equation - Wikipedia, the free encyclopedia
		In mathematics, a quartic equation is the result of setting a quartic function equal to zero.
		According to the fundamental theorem of algebra, a quartic equation always has four solutions (roots).
		To begin, the quartic must first be converted to a depressed quartic.
	en.wikipedia.org /wiki/Quartic_equation (1529 words)

	Quartic equation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a quartic equation is the result of setting a (additional info and facts about quartic function) quartic function equal to zero.
		As the (additional info and facts about fundamental theorem of algebra) fundamental theorem of algebra tells us, a quartic equation always has four solutions (((botany) the usually underground organ that lacks buds or leaves or nodes; absorbs water and mineral salts; usually it anchors the plant to the ground) roots).
		Hence it would be useful to have a general formula or algorithm (such as the (An equation in which the highest power of an unknown quantity is a square) quadratic equation which solves all quadratics).
	www.absoluteastronomy.com /encyclopedia/q/qu/quartic_equation.htm (1788 words)

	PlanetMath: quartic formula
		The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer.
		A practical algorithm for solving quartic equations by radicals is given in the concluding paragraph of the Galois-theoretic derivation of the quartic formula.
		This is version 3 of quartic formula, born on 2002-01-22, modified 2005-07-07.
	planetmath.org /encyclopedia/QuarticFormula.html (115 words)

	Quartic function - Wikipedia, the free encyclopedia
		A quartic function is a function of the form
		Since a quartic function is a polynomial of even degree, it has the same limit when the argument goes to positive or negative infinity.
		The derivative of a quartic function is a cubic function.
	en.wikipedia.org /wiki/Quartic_function (158 words)

	Solving Cubics and Quartics (Site not responding. Last check: 2007-10-22)
		The cubics derived in the algorithms by Descartes, by Neumark, by Ferrari, by Brown, Yacoub & Fraidenraich, and by Christianson are examined.
		Quartics are the highest degree polynomials which can be solved analytically in general by the method of radicals i.e.: operating on the coefficients with a sequence of operators from the set: sum, difference, product, quotient, and the extraction of an integral order root.
		The signs of each of the quartic coefficients 'a', 'b', 'c', 'd' and the root of the cubic 'y', may be positive or negative, giving 32 possible combinations of signs.
	linus.socs.uts.edu.au /~don/pubs/solving.html (3623 words)

	What is a quartic law?
		One famous example of a quartic law is Ludwig Boltzmann's (and Josef Stefan's) "fourth power radiation law"--to find how intensely a hot object GLOWS you quart the absolute temperature.
		Quartic is how temperature relates to the brightness of shining.
		It doesn't matter here whether or not you remember Boltzmann's (and Stefan's) quartic radiation law--the important thing to notice is that some laws do involve squaring a number twice in succession or, in other words, quarting it.
	www.planck.com /quartic.htm (877 words)

	Pure and Applied Geometry - Polyhedral Models of Klein's Quartic (Site not responding. Last check: 2007-10-22)
		Felix Klein's quartic, also called Klein's curve, Klein's regular map or Klein's group PSL (2,7) is one of the most famous mathematical objects, or, as A.M. Macbeath formulated ([L], p.
		This book was issued on the occasion of the installation of a nice geometric model of Klein's quartic made of Carrara marble by the artist H. Ferguson and put up at the campus of Berkeley.
		The 3-dimensional one comes from the fact that Klein's quartic can be realized as a Riemannian manifold or as a regular map on an oriented 2-manifold of genus 3 and with octahedral symmetry ([L] p.
	www.math.uni-siegen.de /wills/klein (1589 words)

	The "Quartic Formula"
		Shortly after the discovery of a method to solve the cubic equation, Lodovico Ferraria (1522-1565), a student of Cardano, found a way to solve the quartic equation.
		First, the quartic equation is "depressed"; then one reduces the problem to solving a related cubic equation.
		One word of caution: A quartic equation may have four complex roots; so you should expect complex numbers to play a much bigger role in general than in my concrete example.
	www.sosmath.com /algebra/factor/fac12/fac12.html (527 words)

	Read This: The Eightfold Way
		The second part of the article addresses the relevance of Klein's Quartic curve to Kenku's recent proof of the Stark-Heegner theorem on imaginary quadratic number fields of the class number one.
		The Klein Quartic is the plane curve of lowest degree that attains this upper bound.
		The proof that Klein's quartic is a plane curve of genus 3 uses Euler's formula for a triangulation and is arguably simpler than using Hurwitz's formula for branched coverings.
	www.maa.org /reviews/eightfold.html (1298 words)

	Math Forum: Ask Dr. Math FAQ: Cubic and Quartic Equations
		To try to go backward and come up with a closed form for the Cubic Formula in terms of the original a, b, c, d would be a real pain.
		The quartic in y must factor into two quadratics with real coefficients, since any complex roots must occur in conjugate pairs.
		Once you have factored the quartic into two quadratics, finishing the finding of the roots is simple, using the Quadratic Formula.
	mathforum.org /dr.math/faq/faq.cubic.equations.html (1185 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-22)
		I didn't have much trouble with the Cubic equations, and I'm hoping to be able to use it to add Quartic Splines to my realtime raytracer (computationally, as for 3rd/2nd degree, direct methods should be faster than doing Newton Raphson right?).
		Date: 05/28/98 at 13:40:13 From: Doctor Rob Subject: Re: Solving quartic equations directly For a full discussion of quartic equations, see the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.cubic.equations.html I like this method to solve quartic equations.
		If this occurs, the quartic has nicely reduced itself to a quadratic in (y^2) which can be easily solved by the quadratic formula (followed by simple factoring, solving each factor for 2 roots, and then re-substituting for all four solutions).
	mathforum.org /library/drmath/view/52874.html (543 words)

	quartic root calculator
		The program is operated by entering the coefficients for the quartic polynomial to be solved, selecting the rounding option desired, and then pressing the Calculate button.
		In this case, the quartic polynomial can be reduced to a cubic which cannot be solved using this calculator; try the Cubic Root Calculator.
		A quartic polynomial can have four real zeros, or two real zeros and one pair of complex zeros, or two pairs of complex zeros.
	home.att.net /~srschmitt/script_quartic.html (271 words)

	Compositions of Cubic and Quartic Surfaces
		We can specify any quartic surface centered at the origin of the world coordinate system by the 35 coefficients of the quartic equation that describes it.
		The translation transformation is reflected in the coefficients of the quartic in a similar manner.
		Thus, for those cases we found the second derivative roots--a linear and a quadratic equation for the case of cubics and quartics respectively.
	www.people.fas.harvard.edu /~serban/cs275project.html (1024 words)

	[No title]
		This is not particularly ugly, unless you try to express the polynomials algebraically in terms of the coefficients of the quartic, and leave the quartic in the original general form.
		Frankly, it may be easier in practice simply to find the roots of the derivative of the quartic and see if at any of those (1 or 3) points the quartic is negative; this is equivalent to determining if the quartic has real roots.
		I pursued this because when I realized that all quartics could result from some pair of ellipses, that meant that any approach to determining the existence of intersections would be equivalent to determining the existence of real roots for an arbitrary quartic.
	www.math.niu.edu /~rusin/known-math/95/ellipse.intersect (1528 words)

	ipedia.com: PSL(2,7) Article (Site not responding. Last check: 2007-10-22)
		It is the automorphism group of the Klein quartic,...
		It is the automorphism group of the Klein quartic, and it is the second-smallest nonabelian simple group, next to the alternating group A
		Klein's quartic pops up all over the place in mathematics, not least of which includes representation theory, homology theory, octonion multiplication, Fermat's last theorem, and Stark's theorem on imaginary quadratic number fields of class number 1!
	www.ipedia.com /psl_2_7_.html (485 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-22)
		Quartic vertices provide a window into one of the most important problems in particle physics; the understanding of electroweak symmetry breaking.
		I survey the various processes that have been proposed to study quartic gauge boson couplings at future{ital e}{sup+}{ital e}{sup -},{ital e}{gamma},{gamma}{gamma},{ital e}{sup -}{ital e}{sup -}, and{ital pp} colliders.
		For quartic couplings involving photons,{gamma}{gamma} collisions appear to be the best place to measure these couplings.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=253429 (254 words)

	Code Documentation (Site not responding. Last check: 2007-10-22)
		Quartic (const double and, const double and, const double and, const double and, const double and)
		Basically this function calls other root solving functions, invoking Cubic::Solve() or Quadric::Solve() as appropriate if the degree is lower than four (which means that closed form expressions of the roots exist).
		Otherwise, solve a quartic using the method of Francois Vieta (Circa 1735).
	www710.univ-lyon1.fr /~abarbier/Doc/Math/a00021.html (284 words)

	5.3.4 Quartic (Site not responding. Last check: 2007-10-22)
		The corresponding object of degree 4 is the quartic object, with syntax
		QUARTIC: quartic { < QUARTIC_COEFFICIENTS > [POLY_MODIFIERS] } QUARTIC_COEFFICIENTS: 35 FLOATs separated by commas.
		An example of such a quartic is the equation for the torus (obtained by isolating in the formula for the torus above the term
	www.mat.univie.ac.at /~kriegl/Skripten/CG/node99.html (69 words)

	Karl's Calculus Tutor - Box 5.3b Solving a Quartic (4th degree) Polynomial
		But seeing how to develop a solution to the general quartic is much more interesting than the formula itself, which is what I will do here.
		Once again, this is optional material, placed here for those who have both a curious mind and the extra time needed to spend on topics that won't be on the exam.
		To be honest, I don't think I could have come up with a solution to the quartic on my own even if I had studied the problem for years.
	www.karlscalculus.org /quartic.html (699 words)

	Diagonal quartic surfaces (Site not responding. Last check: 2007-10-22)
		The algorithm for computing the Galois groups involved is available separately here, with more comments.
		These are some scripts for the GP interpreter, which implement the various algorithms I wrote to deal with the Picard groups of diagonal quartic surfaces.
		They are diagonal quartic surfaces which are everywhere locally soluble but have no rational point of small height.
	www.boojum.org.uk /maths/quartic-surfaces (451 words)

	Polynomial and Polynomial Functions classification and solutions, Cubic, Quartic, Quintic, nth-degree polynomial
		The described procedure can, on some way, be compared with many of those known methods used to reduce general or standard form of a polynomial.
		Let us compare mentioned reduced forms of the cubic, quartic and quintic polynomial with the basic source forms obtained from the source polynomial (6);
		Thus, the classification defines; three types of the cubic functions, ten types of quartic functions, and hundred and sixteen types of quintic polynomial functions, that means, defined are the necessary and the sufficient conditions for each type.
	www.nabla.hr (337 words)

	Klein's Quartic Curve
		Part of the distortion is caused by embedding the Klein quartic in ordinary 3d Euclidean space.
		If we gave the Klein quartic the metric it inherits from the hyperbolic plane, the edges of the cube would be geodesics.
		The Klein quartic is tiled by 56 triangles.
	math.ucr.edu /home/baez/klein.html (3607 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		S(t) quartic spline interpolation function EMBED Equation.3 quartic B-spline function ci coefficients of spline function f(x) original function Introduction Splines are used to interpolate data with m-splines being the most commonly used.
		The Quartic B-Splines All m-splines can be written in the form EMBED Equation.3 , (2) where the EMBED Equation.3 are the m-B-splines and EMBED Equation.3 are coefficients.
		Results By means of quartic spline interpolation routine developed in this work, we interpolated the experimental data above and plotted the original function and its interpolated graph (see Figure 1).
	www.msu.edu /~gumussoy/cv/project3.doc (1542 words)

	Global Minimization of Normal Quartic Polynomials Based on Global Descent Directions (Site not responding. Last check: 2007-10-22)
		A normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite.
		For a normal quartic polynomial, we present a criterion to find a global descent direction at a noncritical point, a saddle point, or a local maximizer.
		We give sufficient conditions to judge whether a local minimizer is global and give a method for finding a global descent direction at a local, but not global, minimizer.
	epubs.siam.org /sam-bin/dbq/article/42085 (277 words)

	Solving Quartic Equations
		Then the four roots of the quartic equation are:
		Quartic Equation With 2 Real and 2 Complex Roots
		Find the square roots by going to the: Complex Number Calculator.
	www.1728.com /quartic2.htm (464 words)

	QUARTIC EQUATION CALCULATOR
		To see the method for solving quartic equations, click HERE
		If you want to use the "long hand" method for solving quartics, you may find the information in the boxes below very helpful to check your work as you go along.
		Remember, doing this by hand is VERY laborious, time-consuming and the possibility for making errors is HUGE.
	www.1728.com /quartic.htm (225 words)

	- SHOP.COM
		The theory of surfaces has reached a certain stage of completeness and major efforts concentrate on solving concrete questions rather than further developing the formal theory.
		Many of these questions are touched on in this classic volume: such as the classification of quartic surfaces, the description of moduli spaces for abelian surfaces, and the automorphism group of a Kummer surface.
		First printed in 1905 after the untimely death of the author, this work has stood for most of this century as one of the classic reference works in geometry.
	www.shop.com /op/aprod-p26411636 (240 words)

	Subroutines for solving cubic, quartic or quintic equations (Site not responding. Last check: 2007-10-22)
		Subroutines for solving cubic, quartic or quintic equations
		Here you can download subroutines for solving cubic, quartic or quintic equations:
		Calculation of the molar volume for given T and p with a quintic equation of state given in Fluid Phase Equilibria, 162, 115 (1999) :
	van-der-waals.pc.uni-koeln.de /quartic/quartic.html (103 words)

	Quartic Engineering \| Short run machining specialists (Site not responding. Last check: 2007-10-22)
		We can provide a solution for you that will carry our name with pride.
		"Quartic Engineering provides a complete capacity for low to medium engineering tasks where quality and attention to detail are assured."
		We have the tools, including CNC machine centres, CAM software, manual machine craftsmanship, solids CAD design software and straight advice to realise your ideas.
	www.quartic.co.nz (165 words)

	The Quartic Equation, A Solution Utility
		This page contains a routine that solves a Quartic Equation.
		It is written in JavaScript, so make sure that JavaScript is enabled in your browser.
		NOTE: The solutions may include a non-zero imaginary component, as indicated by the "i" field.
	www3.telus.net /thothworks/Quad4Deg.html (116 words)

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