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	Gibbs Phenomenon
		One shortcoming of Fourier series today known as the Gibbs phenomenon was first observed by H. Wilbraham in 1848 and then analyzed in detail by Josiah W. Gibbs (1839-1903).
		Gibbs got interested to the behavior of the sequence of Fourier partial sums around this point.
		Since the functions are odd, we will only focus on the behavior to the right of 0.
	www.sosmath.com /fourier/fourier3/gibbs.html (333 words)

	Gibbs phenomenon - Wikipedia, the free encyclopedia
		Informally, it reflects the difficulty inherent in approximating a discontinuous function by a series of continuous sine and cosine waves.
		By the same token, it is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction.
		In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as FejÃ©r summation or Riesz summation, or by using sigma-approximation.
	en.wikipedia.org /wiki/Gibbs_phenomenon (727 words)

	Free energy - Wikipedia, the free encyclopedia
		Otherwise, Helmholtz free energy is useful, since it is related to the partition function of a canonical ensemble in statistical mechanics (hence its utility to gas-phase chemists, and physicists).
		Gibbs free energy is H - TS, where H is the enthalpy, T is the temperature, and S is the energy.
		Recently a compromise notation has become common, using A for the Helmholtz function, G for the Gibbs function, and avoiding F entirely.
	en.wikipedia.org /wiki/Free_energy (277 words)

	The Lag Spectrum, Gibbs Phenomenon and Hanning Smoothing (Site not responding. Last check: 2007-11-05)
		The Gibbs phenomenon is the ringing at the edges of the frequency spectrum that results from the truncation of the temporal (lag) cross-correlation spectrum.
		This truncation corresponds to a convolution of the complex bandpass with a sinc function, generating a ripple at the low frequency side of the spectrum (high frequency for U-band; both sides for band width code 9).
		Gibbs phenomenon is described in Chapters 4 and 18 of SIRA as well.
	www.vla.nrao.edu /astro/guides/sline/current/node11.html (721 words)

	Encyclopedia: Free energy (Site not responding. Last check: 2007-11-05)
		The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals.
		In thermodynamics the Gibbs free energy is a state function of any system defined as G = H − T·S where G is the Gibbs free energy, measured in joules H is the enthalpy, measured in joules T is the temperature, measured in kelvins S is the entropy, measured in joules...
		In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium.
	www.nationmaster.com /encyclopedia/Free-energy (746 words)

	Free energy - Open Encyclopedia (Site not responding. Last check: 2007-11-05)
		In the isothermal-isobaric ensemble, the partition function is related to the Gibbs free energy.
		In chemistry, this quantity is called the Helmholtz function or the work content and is denoted A after the German word Arbeit, meaning work.
		The functions are then referred to as the Helmholtz free energy and Gibbs free energy.
	www.open-encyclopedia.com /Free_energy (330 words)

	Gibbs phenomenon Details, Meaning Gibbs phenomenon Article and Explanation Guide (Site not responding. Last check: 2007-11-05)
		More precisely, this is the function f which equals between and and between and for every integer n ; thus this square wave has a jump discontinuity of height at every integer multiple of.
		In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation.
		Let be a piecewise continuously differentiable function which is periodic with some period.
	www.e-paranoids.com /g/gi/gibbs_phenomenon.html (708 words)

	[No title]
		A logistic activation function is used for the output, with scaling factors determined so that the range encountered in the training set is mapped to a restricted part of the logistic functions (0,1) range (e.g.
		The use of the sum-squared network error function means that points lying far from the others have a disproportionate influence on the position of the hyperplanes used in regression.
		Thus, the reliability function (commonly denoted as R(t)) is the complement to the cumulative distribution function (i.e., R(t)=1-F(t)); the reliability function is also sometimes referred to as the survivorship or survival function (since it describes the probability of not failing or surviving until a certain time t; e.g., see Lee, 1992).
	www.statsoft.com /textbook/glosr.html (4576 words)

	THERMODYNAMICS OF AGING (Site not responding. Last check: 2007-11-05)
		According to the thermodynaic theory the specific value of the Gibbs function of the formation of supramolecular structures of the organism tends to a minimum.
		The value of the specific Gibbs function of the formation of the supramolecular structure is an important quantitative measure of the “gerontological value” of a natural foodstuff.
		That optimal stability (estimated by the value of the specific Gibbs function of the formation of the supramolecular structure) facilitates normal metabolism, slows down aging and improves the quality of life.
	www.endeav.org /evolut/text/ta/ta.htm (5766 words)

	FuelCellStore.com Thermodynamic demonstration device (Site not responding. Last check: 2007-11-05)
		This reinforces the notion that the terms in Gibbs' Function are signed quantities.
		The standard Gibbs free energy change is the maximum reversible work that can be obtained from a mole of ideal gas at standard temperature against the resistance of standard atmospheric pressure.
		We are now ready to examine this situation in the light of Gibbs' Phase Rule and to understand why there must be a pressure plateau in a metal hydride isotherm.
	www.fuelcellstore.com /products/hci/toy.html (2425 words)

	MRC Biostatistics Unit - Research (Site not responding. Last check: 2007-11-05)
		ARS works by constructing an envelope function of the log of the target density, which is then used in rejection sampling (see, for example, Ripley, 1987).
		Firstly, each call to the ARMS function will require an additional point-evaluation of the log-density (to prime the system for Metropolis); and secondly, when log- concavity is assured, squeezing functions can be constructed which may save a function evaluation at each ARS-rejection- step.
		As the envelope function used by both methods progressively adapts to the shape of the target density, sampling becomes progressively more efficient as more points are sampled from the target density.
	www.mrc-bsu.cam.ac.uk /BSUsite/Research/ars.shtml (1549 words)

	Free energy (Site not responding. Last check: 2007-11-05)
		In the isothermal-isobaric ensemble, the partition function is relatedto the Gibbs free energy.
		In chemistry, thisquantity is called the Helmholtz function or the work content and is denoted A after the German word Arbeit, meaning work.
		The functions are thenreferred to as the Helmholtz free energy and Gibbs free energy.
	www.therfcc.org /free-energy-67068.html (338 words)

	[No title]
		The phase stability analysis problem is among the most challenging, because its formulation, either as a Gibbs energy minimization or as a tangent plane distance function minimization, requires robust and reliable numerical techniques to determine the global solution.
		McDonald and Floudas were the first to address the fundamental problem of minimization of the Gibbs free energy and the tangent plane stability criterion that arise in phase and chemical reaction equilibrium as a global optimization problem.
		The objective function and the new method The objective function The stability analysis is based on the tangent plane criterion [3, 4], but uses a different objective function [14 ( 17]: (2) where i = 1,2,..., N (2a) EMBED Word.Picture.8 i = 1,2,..., N (2b) and is set to .
	www.inf.u-szeged.hu /~csendes/stabn14.doc (2170 words)

	R: Performs Gibbs sampling for the Bayesian model used in analyze climate model experiments
		The default values for the Gibbs sampler are the same as those used in the analysis of the paper.
		The function implements a Gibbs sampler for the Gaussian-based statistical model for present and future model biases as described in Quantifying Uncertainty in Projections of Regional Climate Change: A Bayesian Approach to the Analysis of Multimodel Ensembles (2004), Tebaldi, Smith Nychka and Mearns.
		Another detail of the sampling is that a burn in period is required for the Gibbs time series to move away from transient effects from particular initial conditions.
	www.cgd.ucar.edu /~nychka/REA/REA.Gibbs.html (785 words)

	Gibbs Effect (Site not responding. Last check: 2007-11-05)
		Abnormalities of Hemorheological, Endothelial, and Platelet Function in Patients...
		Characterization of GABAA receptor function in human temporal cortical neurons -...
		The Lag Spectrum, Gibbs Phenomenon and Hanning Smoothing...
	www.scienceoxygen.com /math/231.html (236 words)

	Free energy: the Gibbs function
		The Gibbs free energy is the maximum useful work (excluding PV work associated with volume changes of the system) that a system can do on the surroundings when the process occurs reversibly at constant temperature and pressure.
		Willard Gibbs is considered the father of modern thermodynamics and the most brilliant American-born scientist of the 19th century.
		A remarkably rich account of Gibbs' seemingly gray life and his unusual powers of visualization was written by a noted American poet: Rukeyser, M., Willard Gibbs.
	www.chem1.com /acad/webtext/thermeq/TE4.html (3213 words)

	The Gibbs Free Energy (Site not responding. Last check: 2007-11-05)
		It would be preferable to have a state function that depended only on the system, in order to determine whether a process is spontaneous, reversible or nonspontaneous.
		Such a state function exists and is given the name free energy.
		At constant volume and temperature, there is a state function called the Helmholtz free energy, and at constant temperature and pressure (the most important for chemical reactions), there is the Gibbs free energy, which we will focus on below.
	www.nyu.edu /classes/tuckerman/honors.chem/lectures/lecture_18/node4.html (150 words)

	CH101A : INTRODUCTION TO CHEMICAL THERMODYNAMICS (Site not responding. Last check: 2007-11-05)
		Define the Gibbs function and deduce and use it as criteria for spontaneous change.
		Derive an expression showing the dependence of G on p and T. Derive an expression for the pressure dependence of the Gibbs function and of the chemical potential of a perfect gas.
		Define the standard Gibbs functions for a reaction and determine it from Gibbs functions of formation.
	www.uz.ac.zw /science/chemistry/ilove/web-101out.htm (850 words)

	R: GLMMs By Gibbs Sampling (Site not responding. Last check: 2007-11-05)
		A description of the error distribution and link function to be used.
		The number of steps of Gibbs sampling taken before storage of the parameter values begins.
		The number of steps of Gibbs sampling taken once storage of the parameter values has begun.
	www.maths.lth.se /help/R/.R/library/GLMMGibbs/html/glmm.html (429 words)

	Gibbs's Phenomena
		This seems counter-intuitive, however, as complex exponentials are continuous functions.
		J. Willard Gibbs first explained this phenomenon in 1899, and therefore these discontinuous points are referred to as Gibbs Phenomenon.
		For Fourier series, Gibb's phenomenon peaks have finite height and zero width: The error differs from zero only at isolated points--whenever the periodic signal contains discontinuities--and equals about 9% of the size of the discontinuity.
	cnx.rice.edu /content/m10092/latest (1205 words)

	CEA HISTORY
		Huff and Gordon (1950) published tables of thermodynamic functions for 42 substances for a temperature range of 300 to 6000 K, based on various sources available at the time, as well as computations performed by NACA.
		A general discussion of methods used to calculate the internal partition function of ideal gases was documented in 1963 (McBride, Heimel, Ehlers and Gordon, 1963).
		However, sometime in the 1960's, the switch was made to the minimization of Gibbs function for deriving the chemical equilibrium iteration equations.
	www.grc.nasa.gov /WWW/CEAWeb/ceaHistory.htm (1488 words)

	Gibbs Phenomenon
		which is continuous along with its first derivative except for a finite number of first-order discontinuity points (jumps); the left and right limits for the function and its first derivative at the discontinuity points must exist.
		The Fourier series approximation displays an overshoot in the left-sided interval of the discontinuity (and a symmetric undershoot in the right-sided interval).
		While the convergence of the Fourier series to the sawtooth function improves anywhere else, the height of the overshoot does not decrease with augmenting the number of terms.
	lcavwww.epfl.ch /~prandoni/dsp/gibbs/gibbs.html (301 words)

	Chemistry: Free energy (Site not responding. Last check: 2007-11-05)
		When a system of molecules undergoes change, whether chemical reaction or changes in physical states such as phase changes, there are two tendencies driving the changes: * Free Energy tends to decrease, * Entropy tends to increase.
		If E represents the energy, T the temperature, and S the entropy, these two tendencies can be combined by stating that the expression E - TS, the Helmholtz function tends to decrease.
		If the pressure is constant, as in open containers, the enthalpy H = E + PV (where P represents the pressure and V represents the volume) replaces the energy, and thus the quantity that must be minimized is H - TS = E + PV - TS, the Gibbs function.
	www.theparentingsearch.com /Chemistry/Gibbs_free_energy.shtml (235 words)

	Gibbs and Helmholtz Free Energies, Equilibrium, Maxwell's Relations
		Since both U and H are natural functions of entropy, it is fair to ask how convenient it is to have S as a variable.
		For this reason the Helmholtz free energy is sometimes called the "work function." When a physicist says "free energy" without indicating Helmholtz or Gibbs, he usually means Helmholtz free energy.
		Any spontaneous process in a system at constant T and p must decrease the Gibbs free energy (if the system is away from equilibrium) or leave the Gibbs free energy unchanged (if the system is at equilibrium).
	www.chem.arizona.edu /~salzmanr/480a/480ants/ageq&max/ageq&max.html (1326 words)

	Biochemical Thermodynamics Professor
		Since this change results in a true negative minimum in the Gibbs free energy of reaction, it is clear that a temperature-dependent heat capacity of reaction plays the role of a thermodynamic molecular switch.
		While the latter are usually not altered significantly in micromolecular interactions, it was Benzinger's conjecture that the large-scale and long-range changes of conformation which accompany protein folding or assembly might generate significant energy differences due to the cumulative alteration of their numerous covalent bond structures.
		Application of the thermal work function permits the separation of 0 K energy differences and energy differences associated with heat capacity integrals for a fuller understanding of reaction energies.
	www.med.ufl.edu /biochem/pchun (2924 words)

	18. The Gibbs Free Energy (Site not responding. Last check: 2007-11-05)
		However, eventually this function would have to decrease when the particles all congregated in another corner of the box, millions of universe lifetimes later.
		As with the enthalpy, the standard Gibbs free energy of formation for a chemical element in its standard state is 0.
		That is, the total change in the Gibbs free energy obtained when several chemical reactions are added together is the sum of the Gibbs free energy changes for each of the individual steps.
	scholar.chem.nyu.edu /0109/suppnotes/tuckerman/node19.html (2653 words)

	Equivalence of thermodynamical fundamental equations (Site not responding. Last check: 2007-11-05)
		The Gibbs function, which depends on the intensive variables T and P, is easier to obtain experimentally than any other thermodynamical potential.
		The two fundamental equations (Gibbs function and internal energy) are examined and their output compared.
		The formalism of the Gibbs function is entirely analytical, while that based on the internal energy is, in this case, numerical.
	stacks.iop.org /0143-0807/21/395 (329 words)

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