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Topic: Harmonic function

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	Harmonic function - Wikipedia, the free encyclopedia
		The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over R: sums, differences and scalar multiples of harmonic functions are again harmonic.
		Harmonic functions satisfy the following maximum principle: if K is any compact subset of U, then f, restricted to K, attains its maximum and minimum on the boundary of K.
		The generalization of the study of harmonic functions is the study of harmonic forms on Riemannian manifolds, and is known as cohomology.
	en.wikipedia.org /wiki/Harmonic_function (519 words)

	Harmonic conjugate - Wikipedia, the free encyclopedia
		In mathematics, the harmonic conjugate of a harmonic real-valued function of two variables u(x,y), is a function v(x,y) such that v is harmonic and u and v satisfy the Cauchy-Riemann equations, that is, the complex-valued function u(x,y)+iv(x,y) = f(z) is analytic.
		The conformal mapping property of analytic functions (at points where the derivative is not zero) gives rise to a geometric property of harmonic conjugates.
		Another formulation of the harmonic conjugate is given by the theory of the Hilbert transform.
	en.wikipedia.org /wiki/Harmonic_conjugate (228 words)

	Schoenberg on Tonal Function
		For example, harmonic meaning or tonal function might be used as a scalar degree and its variations, used as a root of different chords (2); or even it can be associated to tendencies of individual pitches of a chord (3).
		The function of a pitch is defined by being a scalar degree related to a tonic of a tonality, or of a tonal region.
		The function of degrees 4th and 7th of a scale is of extreme importance, and prevents a possible false interpretation of a tonality with its closest neighbors, the tonalities on both sides of the circle of fifths.
	www.rem.ufpr.br /REMv2.1/vol2.1/Schoenberg/Schoenberg_on_Tonal.html (6280 words)

	PlanetMath: harmonic conjugate function (Site not responding. Last check: 2007-11-07)
		The real part and the imaginary part of a holomorphic function are always the harmonic conjugate functions of each other.
		This is version 15 of harmonic conjugate function, born on 2004-10-20, modified 2005-05-27.
		Then the construction of the harmonic conjugate is still true if this open set is also simply connected, and so on...
	planetmath.org /encyclopedia/HarmonicConjugateFunction.html (235 words)

	Chord (music) - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		A chord is then also only the harmonic function of the group of three notes, and it is unnecessary to have all three notes form a simultaneity.
		Each note has a function within the chord, the note the chord is built on is called the root of the chord, the second note a third above it is called the third of the chord, and the third note a third above the second note is called the fifth of the chord.
		The II and IV chords have Subdominant Function, partially due to the fact that they are a fifth away from the Dominant chords of a key, and partially because in their own Tonic keys, their respective Dominant chords are built on the root notes of the stable Tonic function I and VI.
	encyclopedia.learnthis.info /c/ch/chord__music_.html (1696 words)

	[No title]
		For example, the harmonic meaning of chords is often attributed to each diatonic scale degree and their variants, serving as the roots of a variety of chords.(1) Thus we may say that A-flat major functions as III in F minor, as V in D-flat major, and as flat-VI in C major.
		Riemann explained harmonic coherence with a two-pronged approach: function specified the meaning of a chord in relation to its tonic and its key; the interval of root relation, which is independent of position in the key, specified the strength and directness of progressions.
		Such a fully rule-governed theory of harmonic function has proved on one hand to be an elusive goal, and on the other to be somewhat beside the point, since we have more satisfying deterministic explanations of music these days.
	mto.societymusictheory.org /issues/mto.95.1.3/mto.95.1.3.kopp.art (4164 words)

	Robot Motion Planning (Site not responding. Last check: 2007-11-07)
		Harmonic functions satisfy the min-max principle: spontaneous creation of local minima within the region is impossible if laplace's equation is imposed as constraint on the function used.
		Harmonic functions worked out for the cases shown, were free from local minimas.
		Thus during navigation, when the goal point is usually fixed, one can initially(when the goal point is set) access the harmonic function corresponding to that goal point from the database and use it throughout to navigate from any point in the C-space to the goal point using a steepest descent search routine.
	members.rediff.com /mitulsaha/rmp.html (1223 words)

	Harmonic functions
		Harmonic function is a grouping from the harmonic variety having an extreme characteristic.
		Value of continuity of a harmonic connection is a sum of the particular continuities of all bindings (divided by number of bindings).
		Value of impulse of a harmonic connection is a sum of the particular impulses of all bindings (divided by number of bindings).
	www.sweb.cz /vladimir_ladma/english/music/articles/links/mhfunc.htm (409 words)

	[No title]
		[5] Probably the central core of functional theory (the part that Agmon wishes to retain), unlike many of Riemann's ideas, was not merely a case of "theory for the sake of theory," but was rather a well-meaning attempt to respond to problems posed by a wide variety of perceptual phenomena.
		The simplified harmonic basis of the passage is shown in example 4a.
		Agmon applies to it as well the functional symbol D--surely a gross overburdening with harmonic significance of one of the most elemental of contrapuntal phenomena: parallel motion in 6/3 chords.
	mto.societymusictheory.org /issues/mto.96.2.1/mto.96.2.1.rothgeb.tlk (2019 words)

	[No title]
		[T] functions but with chords that are not in the root position; and final cadence progression with the 3 harmonic functions [ST] -> [D] -> [T], and with root positions (the root notes appear in the bass).
		Harmonic degree: I IV V7 VI II6 V7 I6 V7 I (rest); I IV V7 VI II6 V7 ; (I) Harm.
		Function: [T][SD][D][T] [SD][D] [T] [D] [T] The chords repertoire in the progression is richer, but the harmonic progression is as above.
	www.cs.cmu.edu /~music/392/course-material/lecture-notes/class10-tonality2/class11-tonality3/class11.scm (673 words)

	[No title]
		An harmonic progression is a sequence of harmonic functions, associated with: * A possibly extended sequence -- by repetition of functions.
		POPULATION: It is recommended that harmonic functions in a progressions in the beginning or ending of a sequence are populated by their primary representative, while within middle harmonic progressions, the functions can be populated by secondary representatives.
		The same rule applies to harmonic functions within an harmonic progressions: Beginning or ending functions should be populated by primary representatives, while middle functions can be populated by secondary representatives.
	www.cs.cmu.edu /~music/392/course-material/homework/Harmonic-progressioons-mini-project.doc (657 words)

	Basic Harmonic Function
		When discussing harmonic function in music, it may be helpful to think of it similarly to the function of grammar in a written and spoken language.
		No, composers have always used harmonic regressions and successions in their music, just as much of the great literature of the world has instances of non-standard grammar.
		However, most of the music of the common practice period follows harmonic progressions, so it is very important to memorize and be able to recognize these functions within their musical context.
	www.smu.edu /totw/function.htm (1265 words)

	[No title]
		I ended the section on harmonic functions from the last lesson rather cryptically, saying there exists a explicit construction of any harmonic.
		A piecewise harmonic function is a continuous function which is harmonic on each of its cells.
		The space of all piecewise harmonic functions which are zero on the boundary points is called the spline space.
	www.math.unl.edu /~bbockelm/dsweb/lesson2/index.php (1731 words)

	PlanetMath: harmonic function (Site not responding. Last check: 2007-11-07)
		Indeed, a holomorphic function is harmonic, and a real harmonic function
		This is version 6 of harmonic function, born on 2002-06-04, modified 2005-03-25.
		A couple of the entries attached to Harmonic Functions should be moved over to the separate entry on harmonic functions on graphs.
	planetmath.org /encyclopedia/HarmonicFunction.html (161 words)

	[No title]
		A recursive function can also eat up lots of memory as it is running, but it doesn't necessarily have to; we'll see more about this later.
		Once we move out of the functional paradigm, we can't rely on the substitution model any longer; because evaluation will involve keeping track of all sorts of additional information, we'll have to use a more complicated model called "the environment model of evaluation", but don't worry about that for now.
		Apply the function definition to the evaluated arguments (i.e., replace the argument place- holders in the new function definition with the corresponding evaluated arguments).
	www.cc.gatech.edu /computing/classes/AY2000/cs1301x/lectureFall2000/lecture5.html (4284 words)

	HARMONIC - Online Information article about HARMONIC
		algebra is such that the reciprocals of the proportionals are in arithmetical proportion; thus, if a, b, c be in harmonic proportion then 1/a, 1/b, 1/c are in arithmetical proportion; this leads to the relation 2/b=ac/(a+c).
		The connexion between acoustical and mathematical harmonicals is most probably to be found in the See also:
		The mathematical investigation of the form of a vibrating string led to such phrases as harmonic See also:
	encyclopedia.jrank.org /HAN_HEG/HARMONIC.html (335 words)

	Fourier Optics
		Harmonic analysis is the expansion of an arbitrary function of time
		If the response of the system to each harmonic function is known, the response to an arbitrary input function is determined by the use of harmonic analysis at the input and superposition at the output
		This harmonic function is the 2-D building block of the theory.
	astron.berkeley.edu /~jrg/ScalarWave/node12.html (298 words)

	[No title]
		Any function satisfying some simple properties can be written as a weighed sum of harmonic functions (shifted and scaled sine curves), and (F(f))(s), called the Fourier transform or spectrum of f, gives the weight of the harmonic function of frequency s in f.
		The Fourier transform of a comb function is another comb function, thus the Fourier transform of the sampled function is the Fourier transform of the original function, "replicated" infinitely often by the comb, see Figure 6.
		Truncating a function means multiplying it with a box filter, and this is equivalent to convolving the spectrum of the function with a sinc function.
	graphics.cs.ucdavis.edu /~okreylos/PhDStudies/Winter2000/SamplingTheory.html (3840 words)

	[No title]
		If f_n is a sequence of analytic maps which converges uniformly on compact subsets of D to a function f, then f is analytic too.
		An analytic function f is conformal at every point where its derivative f'(z) is different from 0.
		If D is a regular domain in the complex plane and f is a continous function on the boundary of D, then there exists a unique harmonic function f on D such that h(z)=f(z) for all boundary points of D. Dirichlet problem +------------------------------------------------------------ Let K be a compact subset of the complex plane.
	www.math.harvard.edu /~knill/sofia/data/potential.txt (536 words)

	CONGEN - Constraints
		The connection between the harmonic section of the potential and the constant force section is done by an inverted harmonic piece.
		In this equation the two J coupling constants are "joined" together, and the constraint function is computed based on relationships involving the sums and magnitudes of the differences of the two J couplings, which obviates the need for stereospecific assignments.
		The difference function is harmonic where the magnitude of the calculated difference is bigger than the experimental difference.
	www-nmr.cabm.rutgers.edu /NMRsoftware/homologymodel/congen_12.html (4297 words)

	Harmonic Functions
		On the Poisson Representation of a Function Harmonic in the Upper Half-Plane
		Two-dimensional flow in porous strata with conductivity modeling by a harmonic function of the coordinates.
		Harmonic functions and solutions of the wave equation with three independent variables.
	math.fullerton.edu /mathews/n2003/harmonicfun/HarmonicFunBib/Links/HarmonicFunBib_lnk_3.html (1156 words)

	LPR Motion Planning and Control (Site not responding. Last check: 2007-11-07)
		To generate a reach, or to navigate a mobile robot, we use a harmonic potential function defined on a discrete configuration space grid in which obstacles are held at a potential of 1, and goals are held at a potential of 0.
		The function used is called harmonic because its value at any point is the average of values in the immediate neighborhood (Connolly and Grupen 1993).
		This function also has an important probabilistic interpretation for robot control: the value of the function at any point is the hitting probability, i.e.
	www-robotics.cs.umass.edu /Research/Archive/navigation_fnc.html (527 words)

	TimFun: time function definition (Site not responding. Last check: 2007-11-07)
		The user provided time function is an arbitrary function of time involving the input parameters.
		The user provided function must be coded as a routine to be added to the code.
		A plot of the time function will be generated, if requested by the plotting control parameters.
	www.ae.gatech.edu /people/obauchau/manual/TimFun.html (256 words)

	Alexander Teplyaev -- List of Publications (abstracts)
		I considered both nondegenerate and degenerate harmonic structures (where a nonzero harmonic function can be identically zero on an open set).
		In this paper I obtained certain continuity properties of the gradient for a function in the domain of the Laplacian.
		As an appendix, I proved an estimate of the local energy of harmonic functions which was stated by Strichartz as a hypothesis.
	www.math.mcmaster.ca /tepla/atc/atc.html (1206 words)

	University of York Mathematics: Projects Availableprojects-corriganpreverittprojectssudbery
		A function f on the unit disc is called harmonic, if it is twice differentiable and satisfies the Laplace equation.
		Each holomorphic function is harmonic, and also the complex conjugate of each holomorphic is harmonic (although it is in general not holomorphic).
		For example, the restriction of a function f to the circle can be understood as a periodic function g on R by writing g(t) = f(exp(it)).
	www.york.ac.uk /depts/maths/teaching/sp/Web/pott.htm (817 words)

	harmonic functions of real variables
		Figure 8: A harmonic function averages the values of its neighbors.
		An important practical consequence is that such functions never have maxima nor minima (unless they are confined by a boundary).
		This observation is a precursor of the maximum modulus principle, which holds that the critical points of an analytic function are saddle points, and that extreme values of such a function can only occur on the boundary of a region over which it is examined.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node17.html (159 words)

	Home Page of Eugenia Malinnikova
		Local structure of the zero sets of harmonic polynomials of two variables is well understood, at each critical point it is a union of finitely many curves that pass through the point and form equal angles at this point.
		Given two harmonic functions whose level surfaces are orthogonal at each point.
		One of the ways to start the study of this field is to look at monogenic functions with values in a Clifford algebra as at a generalization of complex valued analytic function.
	www.math.ntnu.no /~eugenia/projects.html (586 words)

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