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Topic: Newton polygon

	Newton polygon - Wikipedia, the free encyclopedia
		After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory.
		Newton polygons have also been useful in the study of elliptic curves.
		Newton polygons provide one technique for the study of the behaviour of the roots.
	en.wikipedia.org /wiki/Newton_polygon (480 words)

	Isaac Newton - Wikipedia, the free encyclopedia
		Newton was born in Woolsthorpe-by-Colsterworth (at Woolsthorpe Manor), a hamlet in the county of Lincolnshire.
		Newton was born prematurely, and no one expected him to live; indeed, his mother, Hannah Ayscough Newton, is reported to have said that his body at that time could have fit inside a quart mug (Bell, 1937).
		Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but his only recorded comments were to complain about a cold draft in the chamber and request that the window be closed.
	en.wikipedia.org /wiki/Sir_Isaac_Newton (5361 words)

	Michiel Hazewinkel : Book review
		The boundary of Newton's polygon or polyhedron can be interpreted as one of the possible generalizations of the degree of polynomial (or the order of the formal power series) of one variable to the general case.
		It is well-known that the Newton's polygon and polyhedron play an important role in the theory of small and large solutions of nonlinear operator equations, theory of perturbations of linear operators, the bifurcation and singularities theory, analytical theory of ordinary differential equations and so on.
		However the applications of Newton's polygon and polyhedron in other branches of analysis are known a few and this book is a pleasant exception It turns out that the Newton's polygon and polyhedron are an important and convinient approach to investigating of some difficult problems in the partial differential equations theory.
	homepages.cwi.nl /~mich/reviews/AAA_1147.html (604 words)

	[No title]
		(1.2) c 0 k=1 c c the image polygon may be unbounded; permitted angles lie in the c range -3.le.betam(k).le.1.
		more c precisely, errest is an approximate bound for how far c the true vertices of the image polygon may be from those c computed by numerical integration using the c numerically determined prevertices z(k).
		it is safest to pick wc to be as central as c possible in the polygon in the sense that as few parts c of the polygon as possible are shielded from wc by c reentrant edges.
	www.netlib.org /conformal/scpack (1494 words)

	Newton Polygons
		The newton polygon will have points (i, v_i) where i is the exponent of a term of f and v_i is the valuation of the coefficient of the ith term.
		The newton polygon of the polynomial f where the place p of an algebraic function field is the prime used for determining the valuations of the coefficients of f.
		The Newton polygon that is the compact convex hull of the set or sequence V of points of the form < a, b > where a, b are integers or rational numbers.
	magma.maths.usyd.edu.au /magma/htmlhelp/text739.htm (1502 words)

	Search Results for Newton
		Newton explained a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun.
		Newton did not answer this directly but explained his own idea that the rotation of the Earth could be proved from the fact that an object dropped from the top of a tower should have a greater tangential velocity than one dropped near the foot of the tower.
		Newton argued that inertial motion was relative to absolute space but instead Mach argued that inertial motion was relative to the average of all the mass in the universe.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?BIOGS=1&TOPICS=1&CURVES=1&REFS=1&BIBLI=1&SOCIETIES=1"=1&CHRON=1&WORD=Newton&CONTEXT=1 (16673 words)

	Introduction
		Any Newton polygon is contained in some virtual cartesian product of the rational field with itself (virtual since this plane is not a structure that is intended to be accessible to the user or which is characteristic of the polygon).
		The standard Newton polygon of a polynomial f=f(u, v) is, by definition, the convex hull of the points < a, b >, so-called Newton points, ranging over monomials u^av^b having nonzero coefficient in f together with the points +Infinity on the two axes.
		This is the standard Newton polygon associated with the polynomial f.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text801.htm (696 words)

	Newton Polygons
		More generally, the Newton polygon of L at the place (p) is the Newton polygon at t=0 after rewriting L as a differential operator L' in a local parameter t of (p), such that derivation of L' is of the form t.d/dt.
		This means that for the computation of the Newton polygon L may have had to be rewritten as a differential operator L' over a differential Laurent series ring C((t)), say, such that L' has derivation t.d/dt.
		The definition of the Newton polynomial of a face that is used by magma, is given in Section 3 of [vH97].
	www.math.lsu.edu /magma/text911.htm (358 words)

	References for Newton
		E J Aiton, The contributions of Newton, Bernoulli and Euler to the theory of the tides, Ann.
		Z Bechler, Newton's law of forces which are inversely as the mass : a suggested interpretation of his later efforts to normalise a mechanistic model of optical dispersion, Centaurus 18 (1973/74), 184-222.
		C B Waff, Isaac Newton, the motion of the lunar apogee, and the establishment of the inverse square law, Vistas Astronom.
	www-groups.dcs.st-and.ac.uk /~history/References/Newton.html (3888 words)

	Newton Polygons (Site not responding. Last check: 2007-10-09)
		The applet takes a given polynomial f(x) with integer coefficients and a given prime p and constructs the Newton polygon of f(x) with respect to p.
		The Newton polygon of f(x) with respect to p is defined as follows.
		The Newton polygon of f(x) with respect to p is the lower convex hull of the set of all such points.
	www.math.sc.edu /~filaseta/newton/newton.html (204 words)

	Galileo Galilei - Wikipedia, the free encyclopedia
		He was a pioneer, at least in the European tradition, in performing rigorous experiments and insisting on a mathematical description of the laws of nature.
		Galileo also put forward the basic principle of relativity, that the laws of physics are the same in any system that is moving at a constant speed in a straight line, regardless of its particular speed or direction.
		This principle provided the basic framework for Newton's laws of motion and is the infinite speed of light approximation to Einstein's special theory of relativity.
	en.wikipedia.org /wiki/Galileo_Galilei (4233 words)

	Newton polygon: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-09)
		In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation]...
		Then the Newton polygon of is defined to be the convex hull convex hull quick summary:
		In mathematics, the convex hull or convex envelope for an object or a set of objects is the minimal convex set containing the given objects....
	www.absoluteastronomy.com /encyclopedia/n/ne/newton_polygon.htm (1062 words)

	Symmetric function - Wikipedia, the free encyclopedia
		The formulae for doing this are attributed to Isaac Newton.
		They also support the theory of the Newton polygon, part of the theory of ramification.
		in a formal power series ring; here passage to the algebraic closure is the theory of Puiseux expansions in fractional powers, and the Newton polygon is a device for computing the required exponents.
	www.sciencedaily.com /encyclopedia/symmetric_function (547 words)

	Newton's Problem of the Body of Minimal Resistance
		Newton himself provided a solution to this minimization problem in the case where Omega is a ball in R², limiting himself to functions that have radial symmetry.
		Newton's formula for the air resistance (the integral in (1)) is found in many engineering handbooks, and his solution is presented as a simple application of the calculus of variations.
		The screwdriver corresponds to the simplest polygon, with two sides, and it is the optimal one when M is large; as we decrease M, optimality jumps from one regular polygon to the next:
	www.win.tue.nl /~mpeletie/Research/PubsNewton.shtml (1150 words)

	The Homepage of TWIGS (The "What Is...?" Graduate Seminar)
		To a polynomial with rational coefficients and a prime p, we attach in a simple way a polygon in the plane, its p-adic Newton Polygon, which captures a surprisingly large amount of data concerning the roots of f.
		I'll describe the origins of this concept in the work of Newton, its development in the hands of Hensel and Dumas, and its applications to algebraic questions such as factorization and calculation of Galois groups.
		The formula for the number of solutions is given by the "mixed volume" of the Newton polytopes, a classical quantity from convex geometry studied by Minkowski.
	www.math.umass.edu /~hajir/twigsf03.html (802 words)

	Applied Mathematics: Seminar Abstract (Site not responding. Last check: 2007-10-09)
		The underlying problem is to determine the behavior of the roots of a polynomial where the coefficients of the polynomial are smooth functions of a perturbation parameter.
		This will be followed by a short quiz on drawing Newton polygons and applying them to the development of series expansions for the roots of perturbed polynomials (the quiz will not count toward your grade).
		The Newton envelope is derived from the Newton polygon and is a convenient tool for classifying all of the potential splitting behavior of the spectrum for a given deformation of a fixed matrix.
	www.amath.washington.edu /seminars/abstracts/spring98/abst.burke.html (279 words)

	Science News Online - Ivars Peterson's MathLand - 11/2/96
		Then, as I was idly sketching the gallery's layout on a paper napkin, I noticed that this configuration was really just a graph, which is what mathematicians call a set of points (known as vertices) and a set of lines (known as edges) joining pairs of these points.
		I realized I could draw the polygonal art gallery as a graph consisting of 11 vertices (for the corners) and 11 edges (for the walls).
		Polygonal art gallery represented as a graph with 11 vertices and 11 edges (left), and one possible triangulation of that polygon (right).
	www.sciencenews.org /pages/sn_arch/11_2_96/mathland.htm (755 words)

	CRM: Chaire André Aisenstadt 1998-1999 (Site not responding. Last check: 2007-10-09)
		Grothendieck showed that under specialization Newton Polygons "go up"; his conjecture says that, conversely, for a given p-divisible group, and a given "lower" Newton Polygon such a specialization should be possible.
		In proving this conjecture one encounters the problem that the variation of the Newton Polygon under a deformation is very difficult to follow (this is why it took us so long to give a proof for this reasonable conjecture).
		Determine the structure of Newton Polygon strata (dimension, irreducibility, etc.).
	crm.umontreal.ca /act/theme/1999/at_1999_05_14.html (459 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		All of the data provided to Newton by BECO were captured photogrammetrically from aerial photography taken in the spring of 1991.
		It also covers issues involving lines that are too close (i.e., one line should be present rather than two for the mapping scale involved), polygons that are too small for the mapping scale and accuracy, and lines that undershoot or overshoot their point of termination.
		These are created as placeholders for polygons such as ponds and median strips that must have a centroid to preserve topology but are not property parcels.
	www.ci.newton.ma.us /mis/gis/datadictionary.doc (12332 words)

	Operations on Polynomials which use Newton Polygons
		If the coefficient ring of the series ring is a finite field whose characteristic is less than the degree of the polynomial then the denominators computed in the newton polygon part of the algorithm may not be bounded and the function will never return.
		Here it can be seen that the newton polygon part of the algorithm is substantially faster using Duval's method, though the converting of the parametrization to a series is not as fast.
		Therefore an error will be given if there is not enough precision to calculate the part of the root that results from the newton polygon part of the algorithm and the root is not known to be a single root since the multiplicity may not be able to be calculated correctly.
	www.umich.edu /~gpcc/scs/magma/text704.htm (2760 words)

	Adam Parusinski; Abstracts
		Newton Polygon Relative to an Arc, (with T.-C. Kuo), in Real and Complex Singularities (Sao Carlos, 1998), Chapman & Hall Res.
		We define a generalisation of Newton polygon and apply it to study polar curves, {\L}ojasiewicz exponents, singularities at infinity of complex polynomials (Ha Huy Vui's theorem), and $\mu$-constant deformations.
		Newton polygon relative to an arc $\lambda$ exposes $f$ in a horn neighborhood of $\lambda$.
	math.univ-angers.fr /~parus/abstracts.html (1278 words)

	Publikationen (Site not responding. Last check: 2007-10-09)
		In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well-known Agmon-Agranovich-Vishik condition of ellipticity with parameter which guarantees the existence of a ray of minimal growth of the resolvent.
		The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon.
		On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/Denk/publikationen/a08.html (198 words)

	Software - DAEs Solver (Site not responding. Last check: 2007-10-09)
		Similarly to the well known algorithm for algebraic equations in two variables, we will use here the Newton polyhedron, which is a generalisation of the Newton polygon to the higher dimensions.
		The Newton polyhedron of an equation is defined as being the sum of the convex hull of its support and the cone
		The Newton polyhedron is a polyhedral set, that is to say, a intersection of a finite number of hyperplanes.
	www-lmc.imag.fr /lmc-cf/Frederic.Beringer/Software (452 words)

	SINGULAR Examples
		If the Newton Polygon is non-degenerate, then the number of branches can be computed combinatorially from the faces.
		If the Newton Polygon is degenerate and has more than one face, then f can be splitted (modulo analytic equivalence) into a product.
		If the Newton Polygon is degenerate and has only one face, then we use the Puiseux expansion to compute the number of branches.
	www.singular.uni-kl.de /DEMO_HTML/Examples/Genus/genus3.html (94 words)

	Ivars Peterson's MathTrek - Euler Bricks and Perfect Polyhedra
		You can construct a pyramid by drawing a polygon (to serve as the base), then joining each vertex of the polygon to a point not in the plane of the polygon.
		Because the faces of an integer polyhedron must themselves be integer polygons, it's natural to use integer polygons as the building blocks of integer polyhedra, Peterson and Jordan remark.
		In this case, all of the points lying along a line through this circle's center and perpendicular to the plane of the heptagon are equidistant from the polygon's vertices.
	www.maa.org /mathland/mathtrek_10_25_99.html (695 words)

	Atlas: Newton polygon method and solvability of free boundary problems by E. Radkevich
		Our goal is to investigate structural characteristics of new problems, on basis of general methods which became prevalent recently : the localization method (model problems [], []) and the Newton polygon method [], as tools for the investigation of the model problems.
		The existing formalization of the construction process of a classical solution is universal enough and allows to reduce the problem to the investigation of the Fréchet derivative corresponding to the non-linear problem (see for example [], [], []).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carf-19.
	atlas-conferences.com /cgi-bin/abstract/carf-19 (942 words)

	research
		The slopes of the Newton polygon are the valuations of eigenvalues.
		The Newton polygon of a CORG is bounded below by a parabola.
		Numerical experiments suggest that the Newton polygons are also bounded above by a parabola, with the same quadratic term; the difference between the bounding parabola and the Newton polygon grows linearly; and the slopes grow linearly, with a deviation that grows logarithmically.
	www.math.cornell.edu /~lawren/research (2084 words)

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