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	Critical point (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a critical point (or critical number) is a point on the domain of a function where the derivative is equal to zero.
		For a function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to the exterior derivative being zero.
		Critical points are also sometimes defined to be points where the derivative of a function is not of maximum rank, i.e.
	en.wikipedia.org /wiki/Critical_point_(mathematics) (425 words)

	Critical Point Analysis (Site not responding. Last check: 2007-11-07)
		Based on the observation that prominent features in the optical response arise from critical points in the joint density of states, critical point modeling was developed to gain an understanding of these spectral features in terms of specific critical points in the band structure.
		The authors present a new approach to critical point modeling of the undifferentiated spectra and interpret the model in terms of balanced sets of critical points which describe the interband transition strength arising from individual pairings of valance and conduction bands.
		Critical point models consisting of 1D balanced sets, used to model the interband transitions of a linear polymer, poly(di-n-hexylsilane).
	www.lrsm.upenn.edu /~frenchrh/critical_point_analysis.htm (493 words)

	Critical Point Drier
		Critical Point Drying is so named as it includes, as part of its process, the occurrence known as the continuity of state for which there is no apparent difference between the liquid and gas state of a medium, the surface tension between this interface reducing to zero.
		The EMS 850 Critical Point Drier is designed for use with CO, having first replaced any water in the specimen by a series of dehydration, often in the same fluid such as Acetone, which will also be the intermediate fluid.
		This ensures the critical point is accurately obtained, avoiding excess pressures or temperatures, or the need to rely on pressure relief valves to control pressure during the heating cycle.
	emsdiasum.com /microscopy/products/equipment/critical_point_drier.aspx (434 words)

	Glossary: Critical Point (Site not responding. Last check: 2007-11-07)
		Given a mapping f of a surface into space and a unit vector z, the critical points of the mapping f in the direction z are the singularities of the height function induced by f in the direction z.
		A level set containing a critical point is called a critical level of the surface in the direction z.
		In general, the critical points are the points where the topology of the level sets (for the given direction) change, usually be a change in the number of components in the level set (a component appears or disappears, two components join at a critical point to form one component, or one component breaks into two).
	www.geom.uiuc.edu /docs/research/RP2-handle/Glossary/CriticalPoint.html (191 words)

	Critical Points
		We recall that a critical point of a function of several variables is a point at which the gradient of the function is either the zero vector 0 or is undefined.
		Although every point at which a function takes a local extreme value is a critical point, the converse is not true, just as in the single variable case.
		A critical point of a function of three variables is degenerate if at least one of the eigenvalues of the Hessian determinant is 0, and has a saddle point in the remaining case: at least one eigenvalue is positive, at least one is negative, and none is 0.
	www.math.umd.edu /~jmr/241/crits.htm (1273 words)

	The momentum equation and the critical point (Site not responding. Last check: 2007-11-07)
		This is the critical distance, or the distance of the critical point.
		is the escape velocity at the critical point.
		The point where the RHS of the momentum equation goes to zero is referred to as the Parker point after Eugene Parker, who described and solved the equations for the solar wind in 1958.
	astron.berkeley.edu /~jrg/ay202/node103.html (964 words)

	Denton DCP-1 Critical Point Dryer (Site not responding. Last check: 2007-11-07)
		Critical Point Drying was developed by Anderson in 1952 using liquid carbon dioxide.
		Critical point drying has proved of great value in the preparation of SEM specimens.
		As the pressure increases, the density of the vapor phase increases until at the critical point (critical temperature and pressure) it is equal to that of the liquid.
	www.dentonvacuum.com /equipment/sem/dcp_sem.html (729 words)

	Critical Points
		The next type of critical point is that where f '(x) is undefined.
		Points of discontinuity are potential local extrema and are therefore critical points.
		Note a point at which f(x) is not defined is a point at which f(x) is not continuous, so even though such a point cannot be a local extrema, it is technically a critical point.
	oregonstate.edu /instruct/mth251/cq/Stage7/Lesson/critical.html (712 words)

	Tousimis SAMDRI-PVT-3B Critical Point Drying Apparatus
		The Tousimis SAMDRI-PVT-3B is a critical point drying apparatus used after HF-release of a MEMS device to prevent stiction in the device.
		The chamber is heated to a temperature above the critical temperature of CO While the chamber volume is constant, there is an accompanying rise of pressure to above the critical pressure of CO and the liquid phase is converted to the gaseous phase.
		This transition of the liquid CO at its critical point avoids the distortion from evaporation boundaries and thermal expansion that leads to stiction problems.
	www.enme.umd.edu /mml/SOP/co2.html (1911 words)

	Calculus I (Math 2413) - Applications of Derivatives - Critical Points (Site not responding. Last check: 2007-11-07)
		Critical points will show up through out a majority of this chapter so it makes some sense to start this chapter defining them and working a few examples.
		Notice that we still have t=0 as a critical point. It’s also become much easier to quickly determine where the derivative will be zero. Recall that a rational expression will only be zero if its numerator is zero (and provided the denominator isn’t also zero at that point of course).
		At this point we need to be careful. The numerator doesn’t factor, but that doesn’t mean that there aren’t any critical points where the derivative is zero. We can use the quadratic formula on the numerator to determine if the fraction as a whole is ever zero.
	tutorial.math.lamar.edu /AllBrowsers/2413/CriticalPoints.asp (1214 words)

	Critical Point of Benzene
		When the critical point is reached, the meniscus disappears and only diffraction irregularities are present due to temperature gradients.
		As the temperature decreases, the meniscus reappears at the critical temperature.
		The critical temperature is the temperature above which the distinction between liquid and gas no longer applies, so as the benzene is heated above its critical temperature the difference between gas and liquid disappears and the benzene becomes one fluid (usually called gas), and conversely as the benzene is cooled.
	jchemed.chem.wisc.edu /JCESoft/CCA/CCA2/MAIN/BENZENE/CD2R1.HTM (406 words)

	Critical Point Drying Principles
		It may be apparent that we an utilize the Critical Phenomena as a drying technique, as it achieves a phase change from liquid to dry gas without the effects of surface tension and is therefore suitable for delicate biological specimens.
		Therefore, CO remains the most common medium for which to apply the C.P.D. procedure, and is termed the Transitional Fluid', however, it is not miscible with water and we have to replace the water in the specimen with another fluid which is miscible with CO, this is termed the Intermediate Fluid'.
		The conditions for which the Critical Point Passage is obtained, (for CO, 31.1°C and 1072 p.s.i.).
	www.emsdiasum.com /microscopy/products/equipment/critical_drying.aspx (301 words)

	Critical Fluid Thermal Equilibrium Experiment (Site not responding. Last check: 2007-11-07)
		The feverish activity of the 1970's in fluid critical phenomena slowed to a near standstill, partly because of gravitational limitations on the acquisition of further experimental data closer to the critical temperature.
		At the critical point, it is unclear whether a fluid is liquid or vapor, and the fluid is highly compressible.
		a constatn volume sample loaded on to he critical density can not maintain a large portion of the sample at the critical point, because the weight of the fluid is enough to compress half of the sample to a density abouve the critical density.
	zeta.lerc.nasa.gov /expr2/cft.htm (987 words)

	A function with a single critical point, which is a local minimum but not a global minimum (Site not responding. Last check: 2007-11-07)
		A function with a single critical point, which is a local minimum but not a global minimum
		For a function of a single variable f(x), if f is continuous on an interval I, has only one critical point in I, and that critical point is a local minimum, then it is the absolute (or global) minimum.
		The only point solving both of these equations is (0,1), and the second derivative test shows that this point is a local minimum for f.
	www.math.tamu.edu /~tom.vogel/gallery/node16.html (214 words)

	The Mandelbrot set
		A6c: Zero is the critical point of z^2+c, that is, a point where d/dz (z^2+c) = 0.
		Critical points are important because by a result of Fatou: every attracting cycle for a polynomial or rational function attracts at least one critical point.
		The proof is based on the study of the bifurcation of parabolic periodic points.
	www.faqs.org /faqs/fractal-faq/section-6.html (954 words)

	Hysteresis Critical Point
		We found a critical point in our model, separating smooth hysteresis loops from loops with a macroscopic jump or burst.
		This is a simulation of a two-dimensional somewhat above the critical point, where none of the avalanches are very large.
		The critical point is where an infinite avalanche first occurs.
	www.lassp.cornell.edu /sethna/hysteresis/HysteresisCriticalPoint.html (354 words)

	Mathwords: Critical Point
		A point (x, y) on the graph of a function at which the derivative is either 0 or undefined.
		A critical point will often be a minimum or maximum, but it may be neither.
		Note: Finding critical points is an important step in the process of curve sketching.
	www.mathwords.com /c/critical_point.htm (57 words)

	Critical Phenomena
		The isotherms below the critical temperature, for example, temperature equals 0.9, are peculiar to the van der Waals equation of state and are not physically realistic.
		A point where a decreasing function becomes flat before continuing to decrease is called a point of inflection.
		The mathematical characteristic of an inflection point is that the first and second derivatives are zero at that point.
	www.chem.arizona.edu /~salzmanr/480a/480ants/CRITICAL/critical.html (789 words)

	critical point
		the point at which a substance in one phase, as the liquid, has the same density, pressure, and temperature as in another phase, as the gaseous.
		(of a function of a single variable) a point at which the derivative of the function is zero.
		(of a function of several variables) a point at which all partial derivatives of the function are zero.
	www.infoplease.com /dictionary/critical+point (79 words)

	World Web Math: Vector Calculus: Critical Point Test (Site not responding. Last check: 2007-11-07)
		We notice that the point x=1 is a minimum for the first equation and the point y=2 is a minimum for the second equation.
		The critical point of (-1,2) is neither a minimum nor a maximum point for the surface.
		Below are images of a minimum, a maximum, and a saddle point critical point for a two-variable function.
	web.mit.edu /wwmath/vectorc/minmax/critical.html (205 words)

	Citebase - Bootstraping the QCD Critical Point
		Authors: Antoniou, N. Kapoyannis, A. It is shown that hadronic matter formed at high temperatures, according to the prescription of the statistical bootstrap principle, develops a critical point at nonzero baryon chemical potential.
		The location of the critical point in the phase diagram, however, depends on the detailed knowledge of the partition function of the deconfined phase, near the critical line.
		In a simplified version of the quark-gluon partition function, the resulting location of the critical point is compared with the solutions of other approaches and in particular with the results of lattice QCD.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-ph/0211392 (172 words)

	Critical point - Wikipedia, the free encyclopedia
		The term critical point can mean any of:
		This is a disambiguation page: a list of articles associated with the same title.
		If an internal link referred you to this page, you may wish to change the link to point directly to the intended article.
	en.wikipedia.org /wiki/Critical_point (81 words)

	Types of Critical Points of Multivariable Functions (Site not responding. Last check: 2007-11-07)
		But there are different kinds of critical points: maxima and minima for example, and there are more.
		Since the contour curves wrap around A, and hence have tangents pointing in every direction, we have that aritrarily near to A there are points at which the gradient is perpendicular to any given direction.
		This nested family of hyperbolas between which two contour curves cross is the visual signature of a type of critical point that we call a saddle point.
	www.math.gatech.edu /~carlen/2507/notes/CritPtType.html (744 words)

	HAZARD ANALYSIS AND CRITICAL CONTROL POINT (HACCP) SYSTEM AND GUIDELINES FOR ITS APPLICATION
		The second section provides general guidance for the application of the system while recognizing that the details of application may vary depending on the circumstances of the food operation[1].
		Critical Control Point (CCP): A step at which control can be applied and is essential to prevent or eliminate a food safety hazard or reduce it to an acceptable level.
		Monitoring is the scheduled measurement or observation of a CCP relative to its critical limits.
	www.fao.org /DOCREP/005/Y1579E/y1579e03.htm (2133 words)

	The stripe critical point for cuprates
		The experimental determination of the quantum critical point (QCP) that triggers the self-organization of charged striped domains in cuprate perovskites is reported.
		The phase diagram of doped cuprate superconductors is determined by a first variable, the hole doping δ, and a second variable, the micro-strain ε of the Cu-O bond length, obtained from the Cu K-edge extended x-ray absorption fine structure.
		The critical charge, orbital and spin fluctuations near this strain QCP provide the interaction for the pairing.
	stacks.iop.org /0953-8984/12/10655 (285 words)

	Victor Davis Hanson on National Review Online
		In all major wars there reaches a critical tipping point when the ultimate outcome of the conflict begins to become clear.
		I don't know at what point Eastern Europe grasped that the Soviet Union was tottering and its planned uprisings were not going to follow the failures of past slaughters in Hungary and Czechoslovakia.
		But after the rise of Solidarity, the deployment of Pershing missiles, the threat of Star Wars, and Ronald Reagan's massive military buildup, the floodgates were unleashed and there was a mad rush not to be the last Soviet Communist in Europe.
	www.nationalreview.com /hanson/hanson200312120835.asp (2330 words)

	ReadWriteThink: Lesson Plan: Critical Literacy: Point of View
		There are a number of approaches to the teaching of critical text analysis, including juxtaposing texts, supplying alternative endings, role-playing, and role reversal.
		Some interesting discussions usually develop at this point and students often want to go back to the website to look at other tales.
		Their tales can be read in front of the class or their point of view can be shared.
	www.readwritethink.org /lessons/lesson_view.asp?id=23 (1692 words)

	Section 3 - THE HAZARD ANALYSIS AND CRITICAL CONTROL POINT (HACCP) SYSTEM
		Determine the points, procedures or operational steps that can be controlled to eliminate the hazard(s) or minimize its (their) likelihood of occurrence.
		The trainees should have the necessary knowledge and abilities to determine critical control points, which should be demonstrated during their reports on the exercise of using the Codex decision tree to determine critical control points for their selected operations.
		The Codex guidelines define a critical control point (CCP) as "a step at which control can be applied and is essential to prevent or eliminate a food safety hazard or reduce it to an acceptable level".
	www.fao.org /docrep/w8088e/w8088e05.htm (13313 words)

	Critical Point Dryers - tousimis Supercritical Point Dryer (Site not responding. Last check: 2007-11-07)
		line of Supercritical Point Dryers is successfully used to increase yield and uniformity of MEMS devices.
		line of Supercritical Point Drying tools used for the MEMS CO2 dry release after wet etching is a most beneficial anti-stiction tool for the MEMS fab.
		Note: Actual delivered model or accessories may vary slightly, as advancements are being applied constantly.
	tousimis.com /critical_point_dryers/MEMS_drying_system.html (153 words)

	Safecom's critical point
		Safecom’s saga may be atypical, but it serves to illustrate that a stable, sturdy management structure is critical to effective cross-agency collaboration.
		“Strong leadership is critical to the success of intergovernmental initiatives,” GAO said in a 2003 report on e-government collaboration.
		For example, Grants.gov and e-Rulemaking have developed a consensus with partner agencies on funding contributions based on agency-unique characteristics, such as the volume of business an agency expects to conduct under the initiative, GAO reported.
	www.gcn.com /print/23_17/26415-1.html (1901 words)

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