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Topic: Multivalued function

	Function
		The mathematical notion of function is not limited to computations using single numbers, or even numbers at all - a function may be any of a wide variety of mappings, maps or transformations.
		As a mathematical term, "function" was coined by Leibniz, in 1694, to describe a quantity related to a curve; such as a curve's slope or a specific point of said curve.
		Functions related to curves are nowaday called differentiable functions and are still the most frequently type of functions encounted by non-mathematicians.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Preimage.html (1779 words)

	math lessons - Function (mathematics)
		In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
		A very common type of function occurs when the argument and the function value are both numbers, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula.
		The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
	www.mathdaily.com /lessons/Function_(mathematics) (2825 words)

	Multivalued function - Wikipedia, the free encyclopedia
		Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic.
		The natural logarithm function from the positive reals to the reals is single-valued, but its generalization to complex numbers (excluding 0) is multiple-valued, because the natural exponential function exp(z) (evaluated at complex arguments z) is periodic with period 2πi.
		For the nth root and logarithm functions, 0 is a branch point; for the arctangent functions, the imaginary units i and −i are branch points.
	en.wikipedia.org /wiki/Multivalued_function (434 words)

	Wikinfo \| Function
		The most familiar kind of function is that where the argument and the function's value are both numbers, and the functional relationship is expressed by a formula, and the value of the function is obtained from the arguments by direct substitution.
		For such kind of functions, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and they are the basics of calculus.
		Those functions, first thought as purely imaginary and called collectively "monsters" as late as the turn of the 20th century, were later found to be important in the modelling of physical phenomena such as Brownian motion.
	www.wikinfo.org /wiki.php?title=Function (2162 words)

	Function - Questionz.net , answers to all your questions (Site not responding. Last check: 2007-10-14)
		History As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve; such as a curve's slope or a specific point of said curve.
		Graph of a functions The graph of a function f is the collection of all points(x, f(x)), for all x in set X. In the example of the discrete function, the graph of f is {(1,a),(2,d),(3,c)}.
		Thus one obtains a function gÊoÊf: XÊ→ÊZ defined by (gÊoÊf)(x)Ê:= g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x).
	www.questionz.net /Family/Function.html (2514 words)

	Partial function
		In mathematics and computer science, a partial function, from the domain X to the codomain Y is a binary relation, over X and Y, which is functional, that is, associates with every element in set X with, at most, one element in set Y.
		If a partial function associates with every element in its domain precisely one element of its codomain, then it is a "total function".
		Partial functions are often used in theoretical computer science: the behavior of a Turing machine for instance can be described by a partial function relating its inputs to its outputs.
	www.ebroadcast.com.au /lookup/encyclopedia/pa/Partial_function.html (169 words)

	Function -- from Wolfram MathWorld
		A function is a relation that uniquely associates members of one set with members of another set.
		Unfortunately, the term "function" is also used to refer to relations that map single points in the domain to possibly multiple points in the range.
		These "functions" are called multivalued functions (or multiple-valued functions), and arise prominently in the theory of complex functions, where the presence of multiple values engenders the use of so-called branch cuts.
	mathworld.wolfram.com /Function.html (310 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		To use this multivalued function in a SPARK problem you can write three atomic classes, each having a single inverse function that calls solveMyModel(x, y, u, v, w).
		The first, qpwl, is a class representing a quadratic function and a 3-part piecewise linear function.
		These example solve functions are written in C/C++, but in principle they could be expressed in any language for which you have a compiler that produces object-level compatibility with Microsoft Visual C/C++, provided that standard argument passing conventions are used.
	www.ecs.fullerton.edu /pub/research/sowell/spark/readmeMVO.txt (742 words)

	Branches of Complex Functions
		In our definition of a function in Section 2.1, we specified that each value of the independent variable in the domain is mapped onto one and only one value in the range.
		Associated with the branch of a function is the
		is a construct useful for visualizing a multivalued function.
	math.fullerton.edu /mathews/c2003/ComplexFunBranchMod.html (634 words)

	Britain.tv Wikipedia - Exponent
		is the inverse function to the natural logarithm.
		When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication.
		A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function.
	www.britain.tv /wikipedia.php?title=Exponent (2661 words)

	Folse6
		The theory of complex functions is one area of mathematics where multivalued functions arise.
		However, for each function the origin point is a "singular point" resulting in the consequence that when a closed curve is traced around the origin, such that it returns to the same value of the independent variable, the value of the function now differs by a constant factor.
		The details are lacking, but Bohr seems to have intended to correlate the relationship between the independent complex variable and the different values of a multivalued function with a descriptive term and the different references the term could have.
	members.tripod.com /~Glove_r/Folse6.html (3273 words)

	Retrieval Of Ocean Winds Form Ers-1/2 Scatterometer And Sar Data Using Natural Network.
		To illustrate the functionality of the neural network, a set to wind fields were generated by means of Monte Carlo simulations.
		Then, a geophysical model function proposed by ESA (European Space Agency) was used to produce the simulated normalized radar cross section at three pointing antennae of scatterometer according to the ERS -1 configuration.
		In the above, G is known as geophysical model function and g is inverse transfer function which is usually a multivalued function.
	www.gisdevelopment.net /aars/acrs/1997/ps2/ps4011.asp (851 words)

	[No title]
		Replace G.1.2(20) by: The computed results of the mathematically multivalued functions are rendered single-valued by the following conventions, which are meant to imply that the principal branch is an analytic continuation of the corresponding real-valued function in Ada.Numerics.Generic_Elementary_Functions.
		The rules given in the !summary follow from the assumption that the functions on the complex plane are meant to be analytic continuations of the corresponding functions on the real line.
		Arctanh) functions is discontinuous as the parameter X crosses the real axis to the left of -1.0 or the right of 1.0 The real (resp.
	www.ada-auth.org /cgi-bin/cvsweb.cgi/AIs/AI-00185.TXT?rev=1.7 (3102 words)

	Search ScienceWorld
		The inverse sine is the multivalued function sin^(-1)z (Zwillinger 1995, p.
		124), that is the inverse function of the sine.
		The Hurwitz zeta function zeta(s,a) is a generalization of the Riemann zeta function zeta(s) that is also known as the generalized zeta function.
	scienceworld.wolfram.com /search/index.cgi?as_q=Ema (464 words)

	Toward a General Theory of Process
		This was not applied to the multivalued wave function of quantum physics because, even though Post demonstrated there is an infinite number of possible truth-values, any given proposition still can be ascertained to be single-valued.
		The same function space, however, can repeatedly be reconstructed with each order of logical-value permitted by an m-valued calculus of propositions interpreted in relation to the notion of identity transparency.
		Simple-locality and simple-identity would not be inherent properties of entities mapped into this densely stacked function space, which may be called a multivalued reference space (and is the hyparxic domain to which Julian Barbour’s term “Platonia” refers).
	www.geocities.com /moonhoabinh/ithapapers/Gplinks/gplink21.html (688 words)

	The Saigon Papers
		A multi-dimensional function space, called Hilbert space, came to be used, which may not have been quite sufficient for full insight into the transformation implied by conversion from laws of motion to wave function.
		Multivalued exchange units (MEUs) fully implemented within the global economy on borders that are Koch curves, would completely obliterate the distinction between a domestic economy and the international monetary system, while yet maintaining lateral partitions and vertical scale-levels.
		This multivalued quantum property is incorporated into the quantal exchange unit itself, thus allowing each individual unit to directly carry major components of explicit macroeconomic information into every single market transaction in a fashion impossible to atomistic exchange units.
	www.geocities.com /moonhoabinh/sgpapers/m-valuation.html (6364 words)

	Orðasafn: M
		multivalued, marggildur, = many-valued, = multiple-valued, = multivalent 2.
		multivalued function marggilt fall, kvíslað fall, = branched function, = many-valued function, = multifunction, = multiple-valued function, -> multivalued mapping.
		multivalued mapping marggild vörpun, = many-valued mapping, = multiple-valued mapping, -> multivalued function.
	www.hi.is /~mmh/ord/safn/safnM.html (2128 words)

	Trigonometry and Basic Functions - Numericana
		Lambert's W function is used to solve practical transcendental equations.
		Polynomial functions: The value y is obtained from the variable x using only a finite number of additions and/or multiplications involving given constants.
		Alternately, such functions may be construed as univalued (ordinary) functions of a variable whose domain is a so-called Riemann surface for which several points may have the same projection on the complex plane.
	home.att.net /~numericana/answer/functions.htm (3891 words)

	Multivalued Function -- from Wolfram MathWorld
		When considering multivalued functions, it is therefore necessary to refer to usual "functions" as single-valued functions.
		exponential, and integer power functions are all single-valued functions, their inverses are multivalued.
		The discontinuities of multivalued functions in the complex plane are commonly handled through the adoption of branch cuts, but use of Riemann surfaces is another possibility.
	mathworld.wolfram.com /MultivaluedFunction.html (228 words)

	Function (mathematics) - Wikipedia, the free encyclopedia
		A map is usually a (total) function as defined here, especially for authors using function as synonym for partial functions with a domain (of definition) not necessarily containing all points of the source or "departure set".
		) is the composition of the sine function and the squaring function.
		In some formal settings, use of reverse Polish notation, x f, eliminates the need for any parentheses; and, for example, the factorial function is always written n!, even though its generalization, the gamma function, is written Γ(n).
	en.wikipedia.org /wiki/Function_(mathematics) (3522 words)

	Dr. Dobb's \| Math Power \| March 18, 2003 (Site not responding. Last check: 2007-10-14)
		For example the function x/¦x¦ is discontinuous at x=0 but is said to be continuous on the right and on the left by virtue of its being undefined at x=0.
		A simply discontinuous function like x/¦x¦ with value left undefined at x=0 is a joy to behold because it is continuous on both right and left sides of the jump point; both its square and its absolute value are everywhere 1 (see axiom 1); and its reciprocal is just itself.
		There are physical functions that have a dependence on the direction of action—whether by nature or by design: magnetic hysteresis in magnetic materials, backlash in geartrains, toggling in light switches and in some hydraulic valves, differential in thermostats, voltage hysteresis in electronic comparators.
	www.ddj.com /184405306 (3191 words)

	The Complex Logarithm Function
		as the inverse of the exponential function; that is,
		, is a single-valued branch of the logarithm function.
		is similar to the one we presented for the square root function.
	math.fullerton.edu /mathews/c2003/ComplexFunLogarithmMod.html (625 words)

	[No title]
		This requires a multivalued or even real- most of the previous research, positions were valued evaluation function.
		They explained the demonstrated on chess that a two-valued evaluation function pathology, but their explanation involves multiple values performs poorly compared to a multivalued one.
		Benefits of using multivalued threshold at the root and dispersing gradually as we advance functions for minimaxing.
	www.ijcai.org /papers/1223.txt (2330 words)

	CDAM Research Report Series LSE-CDAM-2003-03 (Site not responding. Last check: 2007-10-14)
		that may be considered as generalizations of (linear) threshold functions, or as discretized versions of artificial neurons.
		Such functions have arisen in the context of multiple-valued logic and artificial neural networks.
		For any fixed k, we present two procedures which, given a set of points labelled with the values of some (unknown) multivalued multithreshold function, will produce such a function that achieves the same classifications on the points.
	www.cdam.lse.ac.uk /Reports/Abstracts/cdam-2003-03.html (267 words)

	Math 132 Applet 4 (Site not responding. Last check: 2007-10-14)
		This applet is similar to Applet 2, but displays functions which are multi-valued rather than single-valued.
		One of the values has been designated as the "principal" value of the function; this value is marked by a red ball.
		By making such a choice, we can restrict the multi-valued function to a single-valued function; this function is called a branch of the original function.
	www.math.ucla.edu /~tao/java/Multi.html (364 words)

	[No title]
		The function exp(-t)/(1+zt) is indeed analytic for all complex z except real and negative values.
		The function is not even continuous near any point of the negative real axis.
		This is closely akin to the exponential integral function Ei(z) = -integral_{-z}^infinity exp(-t)dt / t (1) where we take the principal value of this integral to deal with the singularity at t = 0.
	www.math.niu.edu /~rusin/known-math/00_incoming/divseq (882 words)

	Atlas: On the Borel class of multivalued functions of two variables by Grazyna Kwiecinska (Site not responding. Last check: 2007-10-14)
		Obviously each multivalued function of two variables x and y may be treated as a multivalued function of a single variable (x, y).
		The essential difference is the possibility of formulation of hypotheses concerning multivalued functions in terms of its sectionwise properties.
		A multivalued function F from a topological space (X, to topological space (Z, (Z) is said to be lower (resp.
	atlas-conferences.com /cgi-bin/abstract/caeh-57 (318 words)

	Nano and Giga Abstract (Site not responding. Last check: 2007-10-14)
		at parallel alignment it appears to be the multivalued function of the conductance
		We argue that conduction quantization and, hence, the giant conductance fluctuations should survive even at room temperature because of the large exchange splitting of conduction band.
		This multivalued behavior of magnetoresistance should not be treated as experimental errors, and subjected to a conventional procedure of avaraging.
	www.atomicscaledesign.net /moscow/abstracts/tagirov1 (190 words)

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