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Topic: Equidistributed

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	SINUM Volume 29 Issue 4
		The polynomial interpolation on a quasi-equidistributed node set which is a union of several rotations around the origin for an equidistributed node set on the unit circle is investigated.
		A scheme for generating a sequence of quasi-equidistributed node sets for polynomial interpolation on the unit disk is proposed, in which each set is contained in its successor and the average increased rate of number of nodes at each step is less than 2, in fact, may be arbitrarily close to 1.
		Our interpolation process is as stable as the ordinary process on equidistributed node sets, since the Lebesgue constant of the corresponding interpolation has a logarithmic rate of growth and the (spectral) condition number of the finite dimensional linear transformation, from the sample values to the coefficients of the interpolation polynomial, is bounded.
	locus.siam.org /SINUM/volume-29/art_0729070.html (242 words)

	Clarke, Steingrimsson, Zeng, SLC35c (Site not responding. Last check: 2007-10-14)
		We define or redefine new Mahonian permutation statistics, called "mad," "mak" and "env." Of these, env is shown to equal the classical "inv," that is the number of inversions, while "mak" has been defined in a slightly different way by Foata and Zeilberger.
		It is shown that the triple statistics (des,mak,mad) and (exc,den,env) are equidistributed over the symmetric group.
		In particular, this implies the equidistribution of (exc,inv) and (des,mad).
	www.maths.tcd.ie /EMIS/journals/SLC/wpapers/s35clarke.html (161 words)

	Equiprobable bins (Site not responding. Last check: 2007-10-14)
		The equidistributed interior bin boundaries are located at the x-values for which the accumulated probabilities are
		In that figure suppose that B is the equidistribution point we are looking for, and suppose we know that B lies between the two library x-values A and C.
		Therefore, the quadratic equation for the equidistribution point has two roots, B and D, but the root D is physically impossible.
	nuclear.llnl.gov /CNP/mcapm/mcf/node58.html (365 words)

	On the Solution of Convection-Diffusion Boundary Value Problems Using Equidistributed Grids
		The grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc-length of the solution.
		In particular, a parameter-dependent monitor function is introduced which incorporates fixed meshes, approximate arc-length equidistribution, and equidistribution of the absolute value of the solution, in a single framework.
		It is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that, furthermore, the spurious solutions can be stable for an appropriate moving mesh method.
	epubs.siam.org /sam-bin/dbq/article/28045 (264 words)

	[No title]
		A second pair, (exc, \textsc{den}), was shown to be equidistributed with (des, \textsc{maj}) by Foata and Zeilberger in 1989.
		This left open the problem of finding a natural Eulerian statistic z such that (z, \textsc{inv}) is equidistributed with (des, \textsc{maj}) and (exc, \textsc{den}).
		We present such a statistic z, along with a simple bijective proof that the pairs of statistics are equidistributed.
	maven.smith.edu /~rhaas/cone/abs/skandera.html (125 words)

	Numerical results (Site not responding. Last check: 2007-10-14)
		Although it is difficult to state what would be an optimal mesh, it seems natural to look for meshes such that the error be almost equidistributed in some norm.
		Figure 1-(a) and 1-(b) below show the size of the elements as a function of their distance from the singularity for the described adaptive procedures.
		As it can be seen from these graphs, in both cases, the graduation of the mesh corresponds to that needed for equidistribution of the error.
	cabmec1.cnea.gov.ar /darie/ddpsiam/node5.html (1060 words)

	Correlated angle-energy equidistribution (Site not responding. Last check: 2007-10-14)
		For each of these equidistributed cosines we need to create equidistributed secondary energy bins.
		The third step is the equidistribution of these energies.
		The only new step here is the second step, that of making an energy distribution at one of the edges of an equiprobable cosine bin.
	nuclear.llnl.gov /CNP/mcapm/mcf/node59.html (504 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		----------------------------------------------------- 14.3.04, 13:00 Barbara Schapira Title: Equidistribution of the horocycles of a geometrically finite surface Abstract: Furstenberg proved that the horocycle flow of a compact hyperbolic surface is uniquely ergodic; in particular, all orbits become equidistributed to the unique invariant measure: the Liouville measure.
		When the surface has finite volume, the equidistribution result is essentially the same: all non periodic orbits are equidistributed to the Liouville measure (Dani-Smillie).
		In the case of a geometrically finite surface with infinite volume and variable negative curvature, there is no more natural definition of the horocycle flow, but it is still possible to define averages on the horocycles.
	www.math.technion.ac.il /~techm/20040314130020040314sch (361 words)

	University of Michigan Combinatorics Seminar (Site not responding. Last check: 2007-10-14)
		A second pair, (exc, DEN), was shown to be equidistributed with (des, MAJ) by Foata and Zeilberger in 1989.
		This left open the problem of finding a natural Eulerian statistic z such that (z, INV) is equidistributed with (des, MAJ) and (exc, DEN).
		We will define such a statistic z, and give a simple bijective proof that the pairs of statistics are equidistributed.
	www.math.lsa.umich.edu /seminars/combin/winter00/feb04.html (125 words)

	Sobol (Site not responding. Last check: 2007-10-14)
		On the other hand [J] presents evidence that the Sobol sequence keeps up with the uniform sequence at any number N of points even in high dimensions if the initialization numbers are chosen properly.
		If the sequence is not equidistributed we do not know wether the Monte Carlo integral converges to the true integral as the number N of points inreases to infinity.
		Low discrepancy sequences on the other hand are equidistributed in every dimension and so the Monte Carlo integral is guarenteed to converge to the true value of the integral.
	martingale.berlios.de /javadoc/QuasiRandom/Sobol.html (421 words)

	rng::Taus2 class Reference
		This is a maximally equidistributed combined Tausworthe generator.
		From: P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe Generators", Mathematics of Computation, 65, 213 (1996), 203--213.
		There is an erratum in the paper "Tables of Maximally Equidistributed Combined LFSR Generators", Mathematics of Computation, 68, 225 (1999), 261--269: http://www.iro.umontreal.ca/~lecuyer/myftp/papers/tausme2.ps
	simdata-csp.sourceforge.net /docs/simdata-docs-0.4/classrng_1_1Taus2.html (485 words)

	List of pseudorandom number generators (Site not responding. Last check: 2007-10-14)
		Generator: gsl_rng_taus Generator: gsl_rng_taus2 This is a maximally equidistributed combined Tausworthe generator by L'Ecuyer.
		L'Ecuyer, "Tables of Maximally Equidistributed Combined LFSR Generators", Mathematics of Computation, 68, 225 (1999), 261--269 The generator gsl_rng_taus2 should now be used in preference to gsl_rng_taus.
		The gfsr4 generator is like a lagged-fibonacci generator, and produces each number as an xor'd sum of four previous values.
	www.worldhistory.com /wiki/L/List-of-pseudorandom-number-generators.htm (2699 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		There is a remarkable equidistribution phenomena of complex rational maps: the iterated preimages of every (non-exceptional) point are equidistributed according to a certain measure, often called equilibrium measure (Brolin, Lyubich, Freire-Lopez-Mañé).
		On the other hand, more recently a series of equidistribution of small points phenomena have been discovered, that have several geometric and arithmetic implications.
		We will prove a result in the intersection of these topics: given a rational map with algebraic coefficients, the points of small canonical height are equidistributed with respect to the equilibrium measure.
	www.impan.gov.pl /~feliksp/str/040903.html (190 words)

	Gregory Buck, Celestial Mechanics
		In this paper a large class of approximate solutions to the n-body problem is described: the masses are equidistributed along closed space curves and move with uniform speed along the curves.
		I consider the uniform motion of equidistributed masses on closed space curves, in the limit when the number of particles tends to infinity.
		In this situation, the gravitational force on each particle is proportional to the local curvature, and may be calculated using an integral closely related to the Biot-Savart integral.
	www.anselm.edu /homepage/gbuck/CM3.html (871 words)

	Evaluating Hash Functions
		We want the variable-length hash to be an equidistributed function, that is, it should map an equal number of keys to each output.
		The only equidistributed mixing functions are the permutations of the internal state
		given a fixed internal state, the hash will not be equidistributed even for one-bucket keys.
	burtleburtle.net /bob/hash/birthday.html (740 words)

	Not all (possibly) "random" sequences are created equal -- Pincus and Kalman 94 (8): 3513 -- Proceedings of the ...
		equidistributed sequences), for sequences with a finite state
		refine the notion of limiting equidistribution, or normality.
		of e are not remarkably equidistributed, with a relative ranking
	www.pnas.org /cgi/content/full/94/8/3513 (5039 words)

	Citations: Tables of maximally equidistributed combined LFSR generators - L'Ecuyer (ResearchIndex)
		Citations: Tables of maximally equidistributed combined LFSR generators - L'Ecuyer (ResearchIndex)
		Construction of Equidistributed Generators - Based On Linear
		L'Ecuyer, P. Tables of maximally equidistributed combined LFSR generators.
	citeseer.ist.psu.edu /context/2052413/0 (399 words)

	Construction of Equidistributed Generators (ResearchIndex)
		Low-discrepancy point sets for quasi-Monte Carlo integration can also be constructed based on these linear recurrences.
		The quality of these generators or point sets is usually measured by certain equidistribution criteria.
		3 Increasing the orders of equidistribution of the leading bit..
	citeseer.ist.psu.edu /663255.html (410 words)

	[No title]
		MT has the following merits: It is designed with consideration on the flaws of various existing generators.
		Far longer period and far higher order of equidistribution than any other implemented generators.
		(It is proved that the period is 2^19937-1, and 623-dimensional equidistribution property is assured.) Fast generation.
	www.rksolution.cz /delphi/dtip0008.txt (397 words)

	Please see PDF version
		These equidistribution theorems are the the lower bounds of FejesT6th (1940) and the up
		Equidistribution in the MST, rectangle R C [0, 111,
		that S(Ln) is asymptotically equidistributed boils down problem to resolve these conjo~ctures.
	www-stat.wharton.upenn.edu /~steele/Papers/HTML/EoWPSf.html (2018 words)

	Tausworthe (CILib API) (Site not responding. Last check: 2007-10-14)
		This is a maximally equidistributed combined Tausworthe generator by L'Ecuyer.
		P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe Generators", `Mathematics of Computation', 65, 213 (1996), 203-213.
		P. L'Ecuyer, "Tables of Maximally Equidistributed Combined LFSR Generators", `Mathematics of Computation', 68, 225 (1999), 261-269
	cilib.sourceforge.net /api/net/sourceforge/cilib/Random/Tausworthe.html (74 words)

	Rudin's Theorem of Complex Analysis.
		Two functions f and g with values in a complex plane are called equidistributed, if for any measurable set A of this plane the sets
		Of course, from this theorem it follows that all such inner functions must be equidistributed with the function I(z)=z.
		Since I is a measure-preserving function, all inner funtions f with f(0)=0 must be measure-preserving.
	www.literka.addr.com /mathcountry/innerf.htm (651 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Abstract: "Equidistributed sequences in the unit interval are those which can be used to define the Riemann integral.
		The idea can be extended to apply to many topological measure spaces.
		By studying the algebra of equivalence classes of subsets of $\Bbb N$ modulo sets with asymptotic density, I can show that many Radon measures have equidistributed sequences".
	www.math.toronto.edu /appmath/AppMath_02_03/fremlin.html (67 words)

	DC MetaData for: The Performance of Equidistributed Sequences in Nonparametric Regression Based on a Quasi Least ... (Site not responding. Last check: 2007-10-14)
		DC MetaData for: The Performance of Equidistributed Sequences in Nonparametric Regression Based on a Quasi Least Squares Method
		The Performance of Equidistributed Sequences in Nonparametric Regression Based on a Quasi Least Squares Method
		Abstract: In this paper a method of generating experimental designs for estimating a response function is proposed.
	www.math.fu-berlin.de /publ/preprints/1998/Ab-A-98-12.html (221 words)

	Abstract from Pacific Journal of Mathematics - 216-1-7 - Paul Alton Hagelstein (Site not responding. Last check: 2007-10-14)
		It is well known that if $f$ and $\tilde{f}$ are equidistributed functions supported on $Q = [0,1] \times [0,1]$, then $\int_{Q}M_{\mathrm{HL}}f \sim \int_{Q}M_{\mathrm{HL}}\tilde{f}$.
		This article examines the relationships between $\int_{Q}M_{y}\tilde{f}$ and $\int_{Q}M_{y}f$, $\int_{Q}M_{y}M_{x}\tilde{f}$ and $\int_{Q}M_{y}M_{x}f$, and $\int_{Q}M_{\mathrm{S}}\tilde{f}$ and $\int_{Q}M_{\mathrm{S}}f$ in the scenario in which $\tilde{f}$ and $f$ are \emph{horizontal rearrangements} of one another, meaning that $\tilde{f}\left(\cdot, y\right)$ and $f\left(\cdot, y\right)$ are equidistributed on $\left[0,1\right]$ for any value of $y$.
		The rearrangement results provided are not only of intrinsic interest, but also yield tools for more detailed examinations involving the local integrablility of maximal functions.
	nyjm.albany.edu:8000 /PacJ/2004/216-1-7nf.htm (185 words)

	EMPIRICAL RESULTS
		As a first example of such a test, we would like to check if the bits in the binary representation of pseudorandom number s
		The following test design is due to Leeb (1995), who has also contributed Figures 3 and 4.
		Then, for a given sample size N, we compute 32 values of the -theoretically equidistributed- upper tail probability of the M-tuple test.
	random.mat.sbg.ac.at /generators/wsc95/inversive/node4.html (783 words)

	GNU Scientific Library -- Reference Manual
		The original generator used a default seed of 4357 and choosing
		Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator".
		Note that the algorithm relies on the properties of 32-bit unsigned integers and has been implemented using a bitmask of
	theoryx5.uwinnipeg.ca /gnu/gsl/Random-number-generator-algorithms.html (906 words)

	On the uniform equidistribution of long closed horocycles, Andreas Strömbergsson
		It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length l becomes asymptotically equidistributed as l→∞.
		We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to l.
		[St] A. Strömbergsson, ``Some results on the uniform equidistribution of long closed horocycles'' in Studies in the Analytic and Spectral Theory of Automorphic Forms, Ph.D. thesis, Uppsala University, Uppsala, Sweden, 2001, 137--226.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1086957715 (729 words)

	Quantum Ergodicity - Basic Ideas and Recent Developments (Site not responding. Last check: 2007-10-14)
		The quantum mechanical counterpart to classical ergodicity is that eigenfunctions of the quantized Hamiltonian (stationary states) should be equidistributed in a certain sense.
		If a dynamical system is classically ergodic, Schnirelman's theorem asserts that most stationary states are equidistributed.
		However, subsequences of exceptional eigenfunctions (scars) cannot be ruled out.
	www.matfys.lth.se /quantum.ergodicity/Talks/Par_Kurlberg (268 words)

	Math 18.785: Exam description (Site not responding. Last check: 2007-10-14)
		Proof that {n x} and {n^2 x} are equidistributed mod 1, if x is irrational.
		Idea of the proof that {n^k x} is equidistributed mod 1 for any positive integer k and x irrational (What this means: you don't need to know Van der Corput's inequality, but if I gave you Van Der corput's inequality you should be able to prove {n^k x} equidistributed using that.)
		Kronecker's theorem (on equidistribution of ({nx}, {ny}, {nz},...) if x,y,z...
	www-math.mit.edu /~akshayv/785/know.html (683 words)

	MathLinks Math Forum :: View topic - irrational (Site not responding. Last check: 2007-10-14)
		I think nobody is saying anything not because it's tough, but because stuff like this has been discussed on the forum lots of times.
		I even believe Kent showed Weyl's Theorem, namely that that sequence is equidistributed in
		Try searching the forum for stuff like "dense" and "uniformly distributed".
	www.mathlinks.ro /Forum/post-171749.html (688 words)

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