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Topic: Jacobian

Related Topics

Jacobian determinant

Ordinary differential equation

Nonlinear programming

Metric tensor

Hessian matrix

Determinant

Partial derivative

Gradient

Jacobian variety

Vector field

Linear equation

Eigenvalue

Positive definite

Least squares

Differential geometry of curves

	PlanetMath: Jacobian matrix
		, it is easy to show that the effect of a change of coordinates on volume forms is a local scaling of the volume form by the determinant of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the inverse Jacobian.
		The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.
		This is version 14 of Jacobian matrix, born on 2001-11-14, modified 2007-01-14.
	planetmath.org /encyclopedia/JacobianMatrix.html (278 words)

	Jacobian - Wikipedia, the free encyclopedia
		In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.
		The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
		In this sense, the Jacobian is akin to a derivative of a multivariate function.
	en.wikipedia.org /wiki/Jacobian (640 words)

	Jacobian variety - Wikipedia, the free encyclopedia
		In mathematics, the Jacobian variety of a non-singular algebraic curve C of genus g ≥ 1 is a particular abelian variety J, of dimension g.
		The curve C is a subvariety of J, and generates J as a group.
		An important theorem regarding Jacobian varieties is Abel's theorem.
	en.wikipedia.org /wiki/Jacobian_variety (118 words)

	Intercomparison of forward and Jacobian radiative transfer models for HIRS and AMSU channels
		The Jacobian of a RTM expresses the derivative of brightness temperature (Tb) with respect to model state variables influencing the RTE, such as temperature, T, specific humidity, Q, surface skin temperature, Ts, and pressure, Ps.
		Hence the temperature Jacobian refers to the Tb change due to a 1 K local increase in T and the water vapor and ozone Jacobians refer to the Tb change due to a 10% decrease in concentration.
		Jacobians at the bottom/top levels should be representative of the lower/upper half of the lowest/highest layer.
	collaboration.cmc.ec.gc.ca /science/arma/intercomparison (1336 words)

	IDeA Lab - Software - Jacobian Image Analysis
		Jacobian images can be computed from the high-dimensional warps of one image onto another.
		A 0 or negative Jacobian value would indicate a singularity (folding or tear) in the warp and is avoided by prohibiting any such local volume alterations during the warp computation.
		Log-transformed Jacobian images may be displayed so that contractions are shown to the same scale as expansions.
	neuroscience.ucdavis.edu /idealab/software/jacobian.php (234 words)

	Points on the Jacobian
		The point on the Jacobian J defined by the polynomials a and b and the positive integer d; if not specified then d is taken to be deg(a).
		The point on the Jacobian J (of a hyperelliptic curve C) associated to the divisor D on C. If D does not have degree 0, then a suitable multiple of the divisor at infinity is subtracted.
		Given a point P on the Jacobian J of a hyperelliptic curve, the function returns firstly, a sequence containing the two defining polynomials for the divisor associated to P and secondly, the degree of the divisor.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1238.htm (2209 words)

	Things of interest to number theorists
		The Selmer group is for an isogeny, over a number field, from an abelian variety to the Jacobian of a curve where the kernel of the isogeny is killed by a power of a prime.
		Roughly speaking, the Jacobian of an elliptic curve is itself.
		The rational part of the Jacobian is a finitely generated abelian group called the Mordell-Weil group.
	math.scu.edu /~eschaefe/nt.html (1525 words)

	Points on the Jacobian
		The point on the Jacobian J defined by the polynomials a and b and the positive integer d; if not specified then d is taken to be deg(a).
		Given a point P on a Jacobian J', construct the image of P on J, where J is a base extension of J'.
		Given a point P on the Jacobian J of a hyperelliptic curve, the function returns firstly, a sequence containing the two defining polynomials for the divisor associated to P and secondly, the degree of the divisor.
	www.math.lsu.edu /magma/text1238.htm (1458 words)

	A summary of the definitions and properties of the Jacobian elliptic functions
		The 12 Jacobian elliptic functions are denoted by two letters taken from the quartet s, c, d, n and may be classified into four groups, each with three members, according to the second letter of the function's name.
		The most popular Jacobian elliptic functions is a copolar trio of sine amplitude elliptic function - sn(x,k), cosine amplitude elliptic function - cn(x,k), and delta amplitude elliptic function - dn(x,k).
		It is also worth noting that all Jacobian elliptic functions reduce to a trigonometric function, or to unity, when k=0.
	www.cmmp.ucl.ac.uk /~jlg/Elliptic/appendix.html (352 words)

	Jacobian Engineering - Home
		Founded in 2005, Jacobian Engineering is a San Francisco based innovative engineering design firm with a strong background in aerospace and government contracting.
		For ongoing projects, we are able to work within the framework of existing requirements and customize a solution to meet and often exceed the minimum criteria for a program.
		Because Jacobian Engineering is able to offer such dynamic solutions based upon each individual customer's needs, let us sit with you and work through how we will add value, collaborate, and engineer solutions that meet your programmatic needs on-time and within budget.
	www.jacobianeng.com (206 words)

	JACOBIAN Statement (Site not responding. Last check: 2007-10-23)
		To specify the values of nonzero derivatives, the variables specified in the JACOBIAN statement have to be defined in the left-hand side of algebraic expressions in programming statements.
		The JACOBIAN statement is useful only if more than one objective function is given in the MIN, MAX, or LSQ statement, or if a DATA= input data set specifies more than one function.
		In the case of least-squares minimization, the crossproduct Jacobian is used as an approximative Hessian matrix.
	v8doc.sas.com /sashtml/ormp/chap5/sect15.htm (244 words)

	Amazon.com: Jacobian (Site not responding. Last check: 2007-10-23)
		det (a) is the Jacobian of the transition map from...
		On the Shafarevich-Tate Group of the Jacobian of a Quotient of the Fermat Curve: MSRI 03720-87; March 1987 by W. McCallum (Paperback - 1987)
		Tables of Jacobian Elliptic Functions whose arguments are rational fractions of the Quarter Period by Alan Robert Curtis (Unknown Binding - 1964)
	www.amazon.com /s?ie=UTF8&keywords=Jacobian&tag=lexico&index=blended&link_code=qs&page=1 (533 words)

	Methods for Solving Nonlinear Equations
		However, when close enough to a root, it is possible to leave the Jacobian frozen for a few steps (though this does certainly affect the convergence rate).
		Of course for a simple one-dimensional root, updating the Jacobian is trivial in cost, so holding the update is only of use here to demonstrate the idea.
		For sparse Jacobian matrices with larger problems, the finite difference Newton method will usually be more efficient since the secant method does not take advantage of sparsity in any way.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/Optimization/UnconstrainedOptimization/1.4.html (1261 words)

	Springer Online Reference Works
		» Encyclopaedia of Mathematics » J » Jacobian conjecture
		Yamabe in 1960 is the global asymptotic stability Jacobian conjecture, also called the Markus–Yamabe conjecture.
		H. Bass, E.H. Connell, D. Wright, "The Jacobian conjecture: reduction of degree and formal expansion of the inverse" Bull.
	eom.springer.de /J/j120010.htm (373 words)

	Levenberg-Marquardt in C/C++
		Notice that using finite differences to approximate the Jacobian implies repetitive evaluations of the function to be fitted.
		If \delta<0, the Jacobian is approximated * with central differences which are more accurate (but slower!) compared to the forward differences * employed by default.
		A structure containing pointers to appropriate data structures can be set up and a pointer to it can be passed to the LM function which then passes it uninterpreted to each call of the user-supplied routines.
	www.ics.forth.gr /~lourakis/levmar (2045 words)

	fsolve (Optimization Toolbox)
		Sparsity pattern of the Jacobian for finite differencing.
		However, if the Jacobian of the system is singular, the algorithm might converge to a point that is not a solution of the system of equations (see Limitations and Diagnostics following).
		This algorithm is a subspace trust region method and is based on the interior-reflective Newton method described in [1],[2].
	www.mathworks.com /access/helpdesk_r13/help/toolbox/optim/fsolve.html (1351 words)

	The Jacobian in Thermodynamics - Introduction & Background
		The use of the Jacobian in thermodynamics was first introduced by Dr. Kali Mukerjee at Michigan State University.
		The Jacobian is a useful way to find a relation between two different vector spaces.
		Building off from this idea it is possible to formulate a set of equations that can be applied to a wide variety of thermodynamic problems including multicomponent systems.
	www.egr.msu.edu /classes/msm851/details.html (206 words)

	odeset :: Functions (MATLAB Function Reference)
		The Jacobian for the stiff van der Pol problem example, described in the MATLAB Mathematics documentation, can be coded as
		The solver uses this sparsity pattern to generate a sparse Jacobian matrix numerically.
		Function that evaluates the Jacobian or a cell array of constant values.
	www.mathworks.com /access/helpdesk/help/techdoc/ref/odeset.html (2386 words)

	ZeroSystem.Jacobian (JMSL Numerical Library)
		Returns the value of the Jacobian at the given point.
		array which contains the point at which the Jacobian is to be evaluated.
		The contents of this array must not be altered by this function.
	www.vni.com /products/imsl/jmsl/v25/api/com/imsl/math/ZeroSystem.Jacobian.html (68 words)

	Jacobian Conjecture
		One simple and useful result from the Jacobian conjecture is that by a simple reduction argument we know if there is a counter example to the Jacobian conjecture, then there must be a counter example $f, g$ with the degrees of $f, g$ non-divisible by each other.
		Since the kernel of the problem is about the Jacobian of a transformation, we decided to call it the Jacobian Conjecture.
		From mathematical reasoning we were able to show that the condition of the Jacobian being non-zero constant is very restrictive and with the help of a computing program, to deduce that the conjecture is true for polynomials of degrees less than or equal to one hundred.
	www.math.purdue.edu /~ttm/jacobian.html (859 words)

	3.2 Finding the Jacobian
		One approach is to use geometric models, derive an analytical expression for the Jacobian, and calculate a numerical Jacobian at each sampled point
		We use a Jacobian model estimation and updating scheme based on the information obtained more or less for free while performing the task.
		In the Jacobian this derivative is represented in the first column.
	www.md.chalmers.se /~jag/PercAct/thesis/node13.html (573 words)

	Solver Advanced Tutorial - Gradients, Linearity, and Sparsity
		In this case, the Jacobian matrix (sometimes with the objective gradient as an extra row) is often called the LP coefficient matrix.
		In a nonlinear optimization problem, the objective gradient and Jacobian matrix elements are not all constant.
		In a smooth nonlinear optimization problem, the partial derivatives may be variable, but they change continuously as the variables change -- they do not "jump" from one value to another.
	www.solver.com /tutorialadv.htm (811 words)

	3.2 Finding the Jacobian
		One approach is to use geometric models, derive an analytical expression for the Jacobian, and calculate a numerical Jacobian at each sampled point
		We use a Jacobian model estimation and updating scheme based on the information obtained more or less for free while performing the task.
		In the Jacobian this derivative is represented in the first column.
	www.cs.rochester.edu /u/jag/PercAct/thesis/node13.html (582 words)

	Citebase - Jacobian pairs
		Authors: Abhyankar, S. Assi, A. We study meromorphic jacobian pairs, i.e., pairs of polynomials in one variable, with coefficients meromorphic series in a second variable, whose jacobian relative to the two variables depends only on the second variable.
		We pose two meromorphic jacobian conjectures about such pairs, one of which is in terms of an invariant of the pair which we call the beta invariant.
		These conjectures are shown to imply the bivariate algebraic jacobian conjecture which predicts that two bivariate polynomials generate the polynomial ring if their jacobian is a nonzero constant.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0209159 (396 words)

	Jacobian Elliptic Functions
		Only the Jacobian elliptic functions will be discussed here, which are the ones most closely related to the familiar three types of elliptic integrals.
		In practice, values of sn(u) and other Jacobian elliptic functions are found as ratios of theta functions.
		This was one case where we could proceed directly from the integral, but normally bigger guns are necessary, in particular the theory of theta functions.
	www.du.edu /~jcalvert/math/jacobi.htm (2457 words)

	Robotics Institute: Jacobian Images of Super-Resolved Texture Maps for Model-Based Motion Estimation and Tracking
		During tracking, a super-resolved estimate of the texture present on the object or in the scene is obtained.
		A key result is the notion of Jacobian images, which can be viewed as a generalization of traditional gradient images, and represent the crucial computation in the tracking process.
		The approach is illustrated with three sample applications: full 3D tracking of planar surface patches, a projective surface tracker for uncalibrated camera scenarios, and a fast, Kalman filtered version of mosaicking with detection of independently moving objects.
	www.ri.cmu.edu /pubs/pub_489.html (329 words)

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