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Topic: Quasiperiodic function

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	Almost periodic function - Wikipedia, the free encyclopedia
		A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in.
		In speech processing, audio signal processing, and music synthesis, a quasiperiodic signal, sometimes called a quasiharmonic signal, is a waveform that is virtually periodic microscopically, but not necessarily periodic macroscopically.
		This is the case for musical tones (after the initial attack transient) where all partials or overtones are harmonic (that is all overtones are at frequencies that are an integer multiple of a fundamental frequency of the tone).
	en.wikipedia.org /wiki/Almost_periodic_function (700 words)

	PlanetMath: quasiperiodic function
		Except for the special case of periodicity noted above, the notion of quasiperiodicity is somewhat loose and fuzzy.
		Note that the every complex number can be said to be a quasiperiod of the exponential function.
		This is version 7 of quasiperiodic function, born on 2004-10-03, modified 2006-07-29.
	planetmath.org /encyclopedia/PeriodicFunction.html (175 words)

	PlanetMath: almost periodic function (classical definition)
		A function is called almost periodic iff set of its translates is pre-compact (compact after completion).
		The notion of an almost periodic function should not be confused with the notion of quasiperiodic function.
		This is version 8 of almost periodic function (classical definition), born on 2004-12-12, modified 2005-07-10.
	planetmath.org /encyclopedia/AlmostPeriodicFunction.html (436 words)

	Quasiperiodic motion - Wikipedia, the free encyclopedia
		In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing two incommensurable frequencies.
		(NB the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice is something distinct from this.)
		The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.
	en.wikipedia.org /wiki/Quasiperiodic_motion (200 words)

	Encyclopedia :: encyclopedia : Calculus (Site not responding. Last check: 2007-11-01)
		Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the graph of the function by tangent lines.
		The rigorous foundation of calculus is based on the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum.
		Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
	www.hallencyclopedia.com /Calculus (2145 words)

	Nonlinear Dynamics and Complex Systems Theory (Glossary) (Site not responding. Last check: 2007-11-01)
		If f is a nonlinear function or an operator, and x is a system input (either a function or variable), then the effect of adding two inputs, x1 and x2, first and then operating on their sum is, in general, not equivalent to operating on two inputs separately and then adding the outputs together; i.e.
		Quasiperiodicity often results when nonlinear dynamical systems are driven by periodic driving forces with periods that are incommensurate with (i.e.
		The variation of the Cost Function can be imagined to be a landscape of potential solutions to a problem where the height of each feature represents its cost.
	www.cna.org /isaac/GlossB.htm (8566 words)

	Attractor - Wikipedia, the free encyclopedia
		A fixed point is a point that a system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass.
		It corresponds to a fixed point of the evolution function that is also attracting.
		A limit cycle is a periodic orbit of the system that is isolated.
	en.wikipedia.org /wiki/Attractor (978 words)

	[No title]
		In the case of circularly polarized applied magnetic field, the equation may have periodic and quasiperiodic solutions while chaotic dynamics is precluded.
		The absence of chaos, the existence of pure time-harmonic magnetization modes with no generation of higher-order harmonics, and the existence of quasiperiodic magnetization modes with spontaneous breaking of the rotational symmetry are proven.
		One major difficulty is the lack of convexity of the energy functional whose minimizers are assumed to be equilibrium configurations for the magnetization vector.
	cage.rug.ac.be /~cimo/research/list.bib (4062 words)

	AN AADF PITCH ESTIMATOR
		Quasiperiodic signals Dissimilarity of signals Adaptive Autodissimilarity Aperiodicity, voicing and quasiperiod functions Relationships to other methods for pitch estimation.
		Signals are real functions of variable called time, t, t being either real (analogical signal) or integer (discrete or sampled signals); so signals take real values (so positive or negative), and are bounded and continuous or almost-continuous (with a finite number of discontinuities in any bounded time interval when analogical).
		Thus they are positive continuous functions concentrated around instant 0, that is they are not null in the vicinity of 0 and null otherwise.
	www.uam.es /personal_pdi/filoyletras/jsango/an_aadf_pitch_estimator.htm (3529 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		\begin{defi} A function $f$ is a quasiperiodic function with vector of basic frequencies $\omega=(\omega_1,\ldots,\omega_r)$ if $f(t)=F(\theta_1,\ldots,\theta_r)$, where $F$ is $2\pi$ periodic in all its arguments and $\theta_j=\omega_jt$ for $j=1,\ldots,r$.
		We denote by $\f\_\rho$ the norm $$ \f\_\rho=\sum_{k\in\z^r}{f_ke^{k\rho}}, $$ and it is not difficult to check that it is well defined for any analytical quasiperiodic function defined on a strip of width $\rho$.
		Finally, to define an analytic quasiperiodic matrix, we note that all these definitions hold when $f$ is a matrix-valued function.
	www.ma.utexas.edu /mp_arc/papers/95-14 (2488 words)

	CBofN - Glossary
		Function Approximation The task of finding an instance from a class of functions that is minimally different from an unknown function.
		The sigmoidal activation function of a multilayer perceptron is monotonically increasing.
		Quasiperiodic motion is always composed of multiple but simpler periodic motions.
	mitpress.mit.edu /books/FLAOH/cbnhtml/glossary.html (8347 words)

	[No title]
		An infinite continuous function sin(2 Pi t) would have a power spectrum consisting of a delta function centered on 1, P(f) = c delta(f - 1), where the coefficient c is not necessarily one since there are various normalizations to P(f) that one can apply.
		Their analysis also shows that the transition from quasiperiodic state to chaotic one is continuous, the linewidths grow smoothly with increasing Ra.
		A truncated sinusoidal function, the same signal except all values are zero beyond the observation interval, requires NEW Fourier modes to bring the signal abruptly to zero at the beginning and end of the observation interval.
	www.phy.duke.edu /~hsg/213/lectures/9-29-03.txt (14243 words)

	Nonlinear optics of periodic and quasiperiodic structures
		Till recent times most of the development in nonlinear optics of periodic and quasiperiodic media has been in the theory front, largely due to the lack of proper and comparatively cheap fabrication techniques.
		For example, the system can be composed of periodic (quasiperiodic) arrangement of dielectric slabs with given optical properties and widths or can be a medium with the corresponding variation of, perhaps, refractive index.
		The most interesting property of such structures is the self-similarity of the transmission coefficient as a function of optical width for various generations.
	www.ias.ac.in /currsci/may25/articles17.htm (4674 words)

	SEP: Bell's Theorem
		In these theories the entity supplementing the quantum state (which is a wave function in the position representation) is typically a classical entity, located in a classical phase space and therefore characterized by both position and momentum variables.
		The probability function p will be assumed to be non-negative and to sum to unity when the summation is taken over all allowed values of s and t.
		Bohm's nonlocal model peacefully coexists with relativistic locality for another reason: that the width of the effective wave function which is employed in the guidance equation is not sufficiently controllable to ensure a desired result of measurement of the quantity required to transmit a message.
	plato.stanford.edu /entries/bell-theorem (12693 words)

	Quasiperiodicity and Chaos in Cardiac Fibrillation -- Garfinkel et al. 99 (2): 305 -- Journal of Clinical Investigation
		Several pathways from quasiperiodicity to chaos have been described, such as torus breakdown (21) and torus doubling.
		Unstable quasiperiodicity is also the cause of spiral breakup leading to the fibrillation-like state in the computer simulation.
		The quasiperiodic scenario explains the origin of the ringlike structures seen in the Poincaré plots.
	www.jci.org /cgi/content/full/99/2/305 (5983 words)

	Almost periodic sphere packings (Site not responding. Last check: 2007-11-01)
		Computer assisted searches on those manifolds of quasiperiodic packings allows to construct many packings with explicitely known density.
		The figure shows the piecewise linear function (a1,a2) -> N(2,r,a) which is the minimum of all (n,a) mod 1, where n runs over all nonzero integer lattice points n satisfying n
		This means that on the corresponding finite dimensional manifolds of almost periodic packings, the maximal density is achieved by periodic packings.
	www.math.harvard.edu /~knill/oldinterests/kepler.html (180 words)

	[No title]
		The function f appears to be a very pure harmonic function that oscillates between the values of 0.833 and 2.089 with period near 124.
		The function f(m) seems to match very closely to the form f(m) = A - B sqrt(C - sin(m/k)) Essentially it's a sine wave, but the peaks are a bit narrower than the valleys.
		and f(i) is a quasiperiodic function with the single irrational period 2 log(q) / log(z) = 3.51420405240870236831...
	www.math.niu.edu /~rusin/known-math/95/heron_tri (3366 words)

	[No title]
		Numerical experiments show that the functional equation has analogs expressing T(i+k+1)T(i-k) as a linear combination of T(i)T(i+1) and T(i-1)T(i+2) for all k.
		[wrong: log(q) should be log(1/q) --NDE] and f(i) is a quasiperiodic function with the single irrational period 2 log(q) / log(z) = 3.51420405240870236831...
		The theta function is defined by a rapidly converging sum, so there's no problem.
	www.math.wisc.edu /~propp/somos/elliptic (1503 words)

	PlanetMath:
		partition generating function (in partition function) owned by silverfish
		periodic function (in quasiperiodic function) owned by rspuzio
		Perron function (in Perron family) owned by jirka
	planetmath.org /encyclopedia/P (2857 words)

	Search Results for Dynamics
		In the two essays on dynamics Hamilton first applied the characteristic function V to dynamics just as he had in optics, the characteristic function being the action of the system in moving from its initial to its final point in configuration space.
		In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics.
		These include papers on minimal surfaces, some on the theory of functions of a complex variable where he was particularly interested in applying topological methods, papers on differential topology and on dynamics.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Dynamics&CONTEXT=1 (6230 words)

	[No title]
		The function $\omega \mapsto \epscrit$ is called the {\em critical function} for the standard map.
		It is an open question to understand completely its geometry: it could be an ellipse whose major semi--axis is in the direction of real $\epsilon$ with a length proportional to the square root of the minor semi--axis, or probably a more complicated curve with the nearest singularity to the origin placed on the imaginary axis.
		Studying $\Delta \omega_{p/q}\left(\epsilon\right)$ as a function of $\epsilon$ we are then able to reconstruct the curve $\epsilon_{crit}\left(\Delta\omega_{p/q} \right)$ for a fixed resonance.
	www.ma.utexas.edu /mp_arc/papers/00-117 (3773 words)

	Session Q17 - Nonlinear Phenomenon: General.
		Skewing of the bistable potential function by the DC signal leads to the appearance of lines at even multiples as well.
		We show the different quasiperiodic and chaotic motions the soliton undergo as a time-dependent force pumps energy into the traslational mode of the kink and relate these dynamics with the excitation of the shape modes of the soliton.
		We show that the dependence of R on the pulse duration of the optimal control field may be singular, and that R may display resonances as a function of the field period, for control by a periodic field.
	flux.aps.org /meetings/YR97/BAPSMAR97/abs/S4870.html (1603 words)

	Quasiperiodic Regime
		That is, the root-mean-square displacements from the trap center which are immediately on the ``dense'' side of the heating regime constitute non-heating configurations.
		The stability of crystals will be further investigated after a discussion of the crystalline phase.
		The term ``quasiperiodic'' may be somewhat of a misnomer--- in this closely-coupled cloud phase,
	webphysics.davidson.edu /Projects/SuFischer/node24.html (293 words)

	[No title]
		This statement is, in fact correct: you can not obtain quasiperiodic dynamics from ANY flow with just two variables, no matter how complicated the right side functions.
		The non-crossing theorem then implies that all bounded dynamics are boring: the solution can only go monotonically up or down in value since a reversal in direction (e.g., x(t) going to the right and then going to the left) would constitute a forbidden crossing in phase space.
		So no matter how complex the right side function f(x) for the autonomous ode, if f(x) is smooth so that unique solutions exist, then there can never be periodic or quasiperiodic or chaotic behavior.
	www.phy.duke.edu /~hsg/213/lectures/10-15-03.txt (5668 words)

	Curriculum Vitae (Site not responding. Last check: 2007-11-01)
		Gasparian and G. Nimtz, Propagation of plane waves and waveguide modes in quasiperiodic dielectric heterostructures, Phys.
		Gasparian, and A. Khachatrian, "Resistance's distribution function of a one--dimensional chain of periodically arranged random delta--scatterers'' II.
		Gasparian, and A. Khachatrian, "Resistance's distribution function of a one - dimensional chain of periodically arranged random delta - scatterers'' I. Proceeding of National Academy of Scinces of Armenia, Physics (in Russian),
	www.csub.edu /~vgasparyan/list.html (1540 words)

	Jeremy Osinski's MATH 447 home page (Site not responding. Last check: 2007-11-01)
		Also, a time step indicator is centered at the top of each animated pendulum simulation (problems 2-4).
		Based on Matlab output, the error as a function of h (stepsize) -- E(h) = h^2 for trapezoid method and h^1 for Euler's method.
		The "aforce" parameter is the maximum amplitude of the forcing function.
	mason.gmu.edu /~josinski/p2/proj2.html (769 words)

	Abstracts for the Mars Data Analysis Program
		An internal potential function was created using the averaged vector data released by Mario Acuna for altitudes from 95 to 209 km above the Martian geoid.
		We are calibrating the spectral properties of dust deposited on MA surfaces as a function of dust optical depth using a copy of the Pathfinder MA.
		The amount of diffuse vs direct sunlight (and therefore the color of the total illumination) varies strongly as a function of the lighting geometry due to sun position and the shape of the illuminated surface.
	research.hq.nasa.gov /code_s/nra/current/NRA-00-OSS-01/MDAP_R&A_Abstracts.html (15398 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Also, no %quasi-periodic (almost periodic) nor periodic behavior was observed unless %the initial conditions provided low kinetic and low potential energy.
		When d becomes very %high, we see almost exclusively quasiperiodic behaviors.
		Periodic behaviors can only be found at low %initial energies, and the energies have to be integer multiples of each %other (i.e.
	mason.gmu.edu /~josinski/p2/pend2ode45.m (530 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Kel’manov A.V., Khamidullin S.A. A Posteriori Detection of a Quasiperiodically Recurring Fragment in Numerical Sequences in the Presence of Noise and Data Loss // Siberian Journal of Industrial Mathematics.
		Kel’manov A.V., Khamidullin S.A. A Posteriori Detection of a Quasiperiodically Recurring Fragment in Numerical Sequences in the Presence of Noise and Data Loss // Pattern Recognition and Image Analysis, Vol.
		Kel’manov A.V., Michailova L.V. Joint Detection in the Quasiperiodic Sequence a Given Number of Fragments belonging to the Reference Tuple and Segmentation of this Sequence on Areas Containing Identical Fragments // Siberian Journal of Industrial Mathematics.
	www.math.nsc.ru /~kelmanov/d11_ENG.htm (3743 words)

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