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Topic: Eta function

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	PlanetMath: modular discriminant
		The Dedekind eta function is defined to be
		The Dedekind eta function should not be confused with the Weierstrass eta function,
		-invariant, as a function of lattices, is defined to be:
	planetmath.org /encyclopedia/DedekindEtaFunction.html (102 words)

	Dedekind eta function - Wikipedia, the free encyclopedia
		The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.
		Because of these functional equations the eta function is a modular form of weight 1/2, and can be used to define other modular forms.
		Because the eta function is easy to compute, it is often helpful to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
	en.wikipedia.org /wiki/Dedekind_eta_function (311 words)

	Dirichlet eta function - Wikipedia, the free encyclopedia
		Hardy gave a simple proof of the functional equation for the eta function, which is
		From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
		Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function.
	en.wikipedia.org /wiki/Dirichlet_eta_function (216 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		This function does not print out the all_leaders field, since that field is not used as part of the graph.;#dbg_class Debug class which much be active node to be displayed.
		The memory allocated to the ETAs and STAs of the schedulable object along with the memory allocated to the schedulable object itself is released.
		The function returns the number of schedulable objects that satisfied the criteria and have been placed in the list.;#list[] The list of schedulable objects to be examined against the criteria.
	www.lcdf.org /ctas/data/dp/function-info (9254 words)

	Dedekind eta function -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The Dedekind eta function is a function defined on the (additional info and facts about upper half plane) upper half plane of complex numbers whose imaginary part is positive.
		The eta function is (additional info and facts about holomorphic) holomorphic on the upper half plane but cannot be continued analytically beyond it.
		Because of these functional equations the eta function is a (additional info and facts about modular form) modular form of weight 1/2, and can be used to define other modular forms.
	www.absoluteastronomy.com /encyclopedia/d/de/dedekind_eta_function.htm (392 words)

	[No title]
		\eta \in \loopone \rbrace $ are {\it periodic} points of the Hamiltonian flow $ \F^t_{a_0} $, then $ \F_y (\eta) = \eta $ for all $ \eta \in \loopone $.
		\eta \in \loopone \rbrace $ are periodic is a particular example in which the hypothesis $ \F_y (\eta) = \eta $ is satisfied.
		According to hypothesis $ (H_5)$, $ (\nabla_\xi a_0) (y, \eta) \not = 0 $ for all $ \eta \in \Slyinsupp $.
	mpej.unige.ch /mp_arc/p/00-213 (12879 words)

	CTAS DP Function Database Browser: O-S (Site not responding. Last check: 2007-10-08)
		This function is defined to return a const char * because the caller of this function is not supposed to write anything into the resulting string.
		This function will be removed when all of CTAS uses the new data structures and the lib function used to read the aircraft types has been updated.
		function for maintaining the ssc list it compares the time of the second list entry (if it exists) to see if it has reached current time, and if so removes the first list entry which is no longer necessary.
	www.lcdf.org /ctas/report/dp/f3.html (5074 words)

	[No title]
		The function $F,$% \begin{equation} F=\frac{1}{2}(4ab)^{n}\equiv \frac{1}{2}(v^{2}-u^{2})^{n}~, \label{reaction001} \end{equation} is called the reaction term or reaction front, and we are interested in describing the asymptotic behavior of the function $F$ for large times.
		\label{that} \end{equation} \subsection{The function $V\label{sectionv}$} The function $\widetilde{\mu},$ $\widetilde{\mu}(y)=\mu_{1}(y)/y$ is strictly decreasing on $\mathbf{R}_{+},$ and therefore $\mu_{1}(y)/y\geq\mu _{1}(t^{\gamma}y)/(t^{\gamma}y)$ for $t\geq1.$ Furthermore, the functions $% \eta$ and $\mu_{1}$ are strictly positive and $\mu_{1}$ is strictly increasing.
		\subsubsection{Existence of the function $\protect\varphi_{2}$} The equation (\ref{h}) for $h$ is linear.
	www.ma.utexas.edu /mp_arc/papers/99-116 (5501 words)

	Eta function - TheBestLinks.com - Number theory, Dedekind eta function, Dirichlet eta function, ... (Site not responding. Last check: 2007-10-08)
		Eta function - TheBestLinks.com - Number theory, Dedekind eta function, Dirichlet eta function,...
		There are two distinct functions called the 'eta function in number theory.
		The first is the Dirichlet eta function, and the second is the Dedekind eta function.
	www.thebestlinks.com /Eta_function.html (96 words)

	[No title]
		A semblance spectrum over the nonhyperbolic parameter eta is obtained by scanning the CMP gather of traces along trajec- tories for signal coherence.
		These scans establish an eta function (i.e., eta versus vertical seismic time) that is used in calculating the fourth order NMO corrections prior to the stacking of the CMP traces.
		The eta functions determined from the semblance spectra must be translated into an eta field (usp traces) for use by vtinmo.
	www.freeusp.org /RaceCarWebsite/TheToolkit/Man_alphabetic/vtispec.html (1140 words)

	Basic Functions (Site not responding. Last check: 2007-10-08)
		The function iso is used to either obtain or set the G interaction tensor isotropic value.
		The function eta is used to either obtain or set the G interaction asymmetry.
		The function wG0 (or its equivalent wGoriented) is used to obtain or generate the 1st order G frequency for a chosen orientation in Hz.
	gamma.magnet.fsu.edu /html/modules/rank2/intga2.htm (1492 words)

	[No title]
		Then by definition of ``link function'' the relation between the mean value parameter $\mu_i$ and the linear predictor $\eta_i$ is given by the link function $$ \eta_i = g(\mu_i).
		In general, as is the case in probit regression, the link function $g$ and the function $b'$ that connects the canonical and mean value parameters are unrelated.
		Thus the variance function is $V(\mu) = \mu$, the dispersion parameter is known ($\phi = 1$), and the weight is also unity ($w = 1$).
	www.stat.umn.edu /geyer/5931/mle/glm.Rnw (2541 words)

	Elliptic and Modular Functions
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.math.niu.edu /help/math/magmahelp/text554.html (1242 words)

	E.M. Barrel Uniformity (Site not responding. Last check: 2007-10-08)
		The data analysis at 300 GeV shows the presence of non-negligible disuniformities in the reconstructed energy as a function of eta.
		These are due to a regular pattern with a periodicity of 8 Middle cells along eta, whose origins are under investigation.
		Total energy vs eta, after correcting for the fitted oscillation: the fitted function obtained in previous figure is used to correct cell by cell the average energy.
	www.mi.infn.it /~fanti/atlas/uniformity.html (479 words)

	GNU Scientific Library -- Reference Manual: Radial Functions for Hyperbolic Space
		The following spherical functions are specializations of Legendre functions which give the regular eigenfunctions of the Laplacian on a 3-dimensional hyperbolic space H3d.
		Of particular interest is the flat limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.
		This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_123.html (180 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		function [beta,eta,gamma,r,yesno]=newtoncond(f,fprime,K,a,b,x0) % % function [beta,eta,gamma,r,yesno]=newtoncond(f,fprime,K,a,b,x0) % Computes parameters to determine if Newton's method will converge for a % given initial guess.
		% % Parameters are -- f -- A string representing the function.
		% K -- A bound on the second derivative of the function.
	www.cse.psu.edu /~barlow/cse455/newtoncond.m (127 words)

	Eta Function
		For better numerical stability, Dataplot actually computes ETA(s) - 1.
		Dataplot uses a Fortran translation of a C routine given in "Atlas For Computing Mathematical Functions" (see the Reference section below).
		is a variable or a parameter (depending on what is) where the computed eta - 1 values are stored;
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/eta.htm (105 words)

	GNU Scientific Library -- Reference Manual - Special Functions
		The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code.
		The Debye functions are defined by the integral D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)).
		The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2.
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_7.html (6072 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		This function was adapted * from a routine in lib.
		This function will be removed when all of CTAS * uses the new data structures and the lib function used * to read the aircraft types has been updated.
		If the reason we are waiting for the first ETA is * because the schedulable object is newly active, then we * reset the schedulable object's departure time difference.
	www.lcdf.org /ctas/DP/sched_obj.c (2066 words)

	ETA and ETB receptor function in pancreatitis-associated microcirculatory failure, inflammation, and parenchymal injury ...
		ETA and ETB receptor function in pancreatitis-associated microcirculatory failure, inflammation, and parenchymal injury -- Plusczyk et al.
		Functional capillary density (FCD) was defined as the capillary
		Changes in pancreatic functional capillary density (FCD) after induction of experimental pancreatitis and pretreatment with either a combined ET receptor antagonist in 2 different concentrations (4 and 10 mg/kg body wt) or a selective ET receptor antagonist.
	ajpgi.physiology.org /cgi/content/full/285/1/G145 (4727 words)

	Function and survival of dendritic cells depend on endothelin-1 and endothelin receptor autocrine loops -- Guruli et ...
		Function and survival of dendritic cells depend on endothelin-1 and endothelin receptor autocrine loops -- Guruli et al.
		Function and survival of dendritic cells depend on endothelin-1 and endothelin receptor autocrine loops
		The functional consequence of DC maturation is commonly measured
	www.bloodjournal.org /cgi/content/full/104/7/2107 (5626 words)

	Dirichlet Eta Function -- from Harry J. Smith (Site not responding. Last check: 2007-10-08)
		The Dirichlet eta function eta(x) = 1 − 2^−x + 3^−x −...
		Eta is defined for all values of x.
		Dirichlet eta function of x > 0, is defined by the series 1 − 1/2^x + 1/3^x − 1/4^x +...
	www.geocities.com /hjsmithh/Numbers/Eta.html (171 words)

	Approximations to via the Dedekind eta function (Site not responding. Last check: 2007-10-08)
		This connection is shown utilizing their {\it alpha}-function which is defined in terms of theta-functions.
		They are able to find $p$-th order iterations based on this function using modular equations for the theta-functions.
		In this paper we construct an infinite family of functions $\alpha_p$.
	www.cecm.sfu.ca /organics/papers/garvan (197 words)

	Modular Equation Databases
		Magma contains several databases of standard equations for modular curves which are used throughout the system for construction of isogenies of elliptic curves, and which are made available to the user.
		The third distinguished class of modular equations are the Atkin modular equations Xi_N(X, Y), satisfying the property that Xi_N(f(tau), j(tau)) = 0 for a modular function f(tau) on X_0(N) invariant under the Atkin-Lehner involution.
		Since the function f(tau) is well-defined as a function on the Atkin-Lehner quotient curve X_0^ + (N), sometimes denoted X_0^ * (N), these equations are also referred to as star modular equations in the literature.
	www.math.niu.edu /help/math/magmahelp/text1042.html (612 words)

	TREASURY INSPECTOR GENERAL
		ETA also strengthened admittance standards to reduce the number of abusive preparers in the e-file program.
		Additionally, the IRS established procedures to ensure that unscrupulous preparers are referred to the Examination function for penalty assessment or prohibition from participation in the e-file program.
		ETA will coordinate with Examination to assume responsibility for monitoring and removal of noncompliant electronic return originators.
	www.ustreas.gov /tigta/auditreports/reports/199940062es.html (613 words)

	[No title]
		The first implements a newton's method c and the second evaluates sin(eta) + eta = f and stores values of c eta and corresponding values of f in another array.
		This should work since f is a c single valued function of eta.
		c Instead of solving previous form of equation to get r as a function c of tau and Rstar we solve for eta as a function of tau and Rstar.
	godel.ph.utexas.edu /Members/mijan/class/examples/novikov/code/newton.f (200 words)

	Modular Polynomial Databases
		Magma contains several databases of standard defining polynomials for modular curves which are used throughout the system for construction of isogenies of elliptic curves, and which are made available to the user.
		The third distinguished class of modular polynomials are the Atkin modular polynomials Xi_N(X, Y), satisfying the property that Xi_N(f(tau), j(tau)) = 0 for a modular function f(tau) on X_0(N) invariant under the Atkin-Lehner involution.
		Since the function f(tau) is well-defined as a function on the Atkin-Lehner quotient curve X_0^ + (N), sometimes denoted X_0^ * (N), these polynomials are also referred to as star modular polynomials in the literature.
	www.math.lsu.edu /magma/text1252.htm (591 words)

	[No title]
		LINK is the link function to use in place of % the canonical link.
		The coefficient of the constant term is % the first element of B. (Do not enter a column of ones directly % into the X matrix.) % % LINK defines a function f that defines the relationship f(mu) = xb % between the distribution parameter mu and the linear combination of % predictors xb.
		This allows the function FN to be either % a user-defined function or one of the pre-defined functions % provided in the GLMFIT function, without contaminating the name % space with those pre-defined functions.
	visionlab.harvard.edu /Members/Anne/Math/glmfitMW.m (493 words)

	[No title]
		If the function $q^+_{\Lambda_2}(\sigma\cdot)$ is continuous at $\omega$ for any $\sigma$, then the function $q^+_{\Lambda_1}(\sigma\cdot)$ is continuous for any $\sigma$ at $\omega'$ such that $\omega'_{\Lambda_2^c}= \omega_{\Lambda_2^c}$.
		\ee By $(H_4)$ the function $a_{j,\Lambda}(\cdot)$ is increasing, and the function $b_{j,\Lambda}(\cdot)$ decreasing.
		Therefore \be \int\gamma_{\Lambda_n}(d\sigma+)a_{j,\Lambda_m}(_j\sigma) \ee is decreasing as a function of $\Lambda_n$, and \be \int\gamma_{\Lambda_n}(d\sigma+)b_{j,\Lambda_m}(_j\sigma) \ee is increasing as a function of $\Lambda_n$.
	www.ma.utexas.edu /mp_arc/papers/94-113 (4486 words)

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