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Topic: Monotonic function

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Monotonicity

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	Monotonic function Totally Explained
		In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order.
		These properties are the reason why monotonic functions are useful in technical work in analysis.
		Monotonicity of entailment is a property of many logic systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions.
	monotonic_function.totallyexplained.com (1151 words)

	Monotonic function - Wikipedia, the free encyclopedia
		Functions that are strictly increasing or decreasing are one-to-one (because a < b implies
		An important application of monotonic functions is in probability theory.
		Monotonicity of entailment is a property of many logic systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions.
	en.wikipedia.org /wiki/Monotonic_function (961 words)

	Monotonicity criterion - Wikipedia, the free encyclopedia
		The monotonicity criterion is a voting system criterion used to analyze both single and multiple winner voting systems.
		Furthermore, although all voting systems are vulnerable to tactical voting, systems which fail the monotonicity criterion suffer an unusual form, where voters might try to elect their candidate by voting against that candidate.
		The single-winner methods of range voting and approval voting are also monotonic as one can never help a candidate by reducing or removing support for them, but these require a slightly different definition of monotonicity as they are not preferential systems.
	en.wikipedia.org /wiki/Monotonicity_criterion (616 words)

	Monotonic function
		The function need not be differentiable; if it is, equivalent definitions can be given in terms of the sign of f'.
		Ordinarily, if a function is simply referred to as being monotonic, that means the function is either monotonic increasing or else monotonic decreasing.
		That is, the definition I gave for "monotonic non-decreasing" they use to define "mono- tonic increasing" and what I gave for "monotonic non-increasing" they use to define "monotonic decreasing".
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH136.HTM (191 words)

	Chapter 3: Fun with Functions
		The inverse of a linear function is linear.
		The exponential is the function whose slope is proportional to its height.
		The logarthmic function is the inverse of the exponential.
	psych.fullerton.edu /mbirnbaum/psych520t/functions.htm (852 words)

	PlanetMath: inverse function theorem
		The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.
		Cross-references: variable, dimension, implicit function theorem, strictly monotonic function, monotonic, strictly, neighbourhood, open interval, theorem, function, Jacobian, open set, mapping, vector-valued function, continuously differentiable
		This is version 6 of inverse function theorem, born on 2002-08-24, modified 2002-12-28.
	planetmath.org /encyclopedia/InverseFunctionTheorem.html (118 words)

	Digitally programmable gain control circuit Number:7,212,592 from the United States Patent and Trademark Office (PTO) ...
		The transfer function segments are separated by an offset value 301.sub.--i to 307.sub.--i and 301.sub.--d to 307.sub.--d, where "i" represents traversing the transfer function in the direction of increasing gain and "d" represents traversing the transfer function in the direction of decreasing gain.
		The corresponding offset values are used to move the PGA 12 from one transfer function segment to a subsequent transfer function segment during operation.
		Note that the plurality of transfer function segments 301 308 are overlapping; therefore, there are no gaps or unwanted separation in between the transfer function segments during operation of the PGA 12 as compared to a conventional programmable gain amplifier.
	www.linkgrinder.com /Patents/Digitally_progr_7212592.html (5509 words)

	The Generic Connectionist Architecture
		The output function of the neuron is a non-linear function of its input and is expressed in Equation 15.
		Consequently, the most common function used is the logistic or sigmoid function which compresses the range of the net input so that the output signal lies between 0 and 1.
		Furthermore, radial basis function networks will also be introduced as they use the same general framework but are differentiated by the net input function that they calculate and the activation function used within the processing units.
	neuron-ai.tuke.sk /NCS/VOL1/P3_html/node30.html (837 words)

	Rotello & Heit (1999). Two-process models of recognition memory: Evidence for recall to reject?
		The Monotonic Function predicts that as the response deadline is increased
		Notice first that the denominator of this equation is exactly the same as the denominator of the monotonic function.
		In the model fitting the monotonic function fit the data as well as did the non-monotonic function.
	www.uark.edu /misc/lampinen/read/rotello99.html (1308 words)

	The Newton-Raphson Method
		If the function is monotonic on that interval, we know that the zero is unique and methods such as bisection and regula falsi are sure to find it.
		If the function is not monotonic, it is difficult to ensure that all the zeroes on the interval are found.
		function is not a very efficient way to raise a number to an integer power, since it uses far more multiplications than necessary.
	pathfinder.scar.utoronto.ca /~dyer/csca57/book_P/node35.html (1246 words)

	PlanetMath: increasing/decreasing/monotone function
		increasing function, strictly increasing function, decreasing function, strictly decreasing function, monotone function, monotonic function, strictly monotone function, strictly monotonic function
		Cross-references: almost everywhere, differentiable, countable, discontinuous, interval, open interval, unbounded, bounded, theorem, strictly, function, subset
		This is version 5 of increasing/decreasing/monotone function, born on 2003-04-28, modified 2006-03-06.
	planetmath.org /encyclopedia/IncreasingdecreasingmonotoneFunction.html (116 words)

	A Monotonic Superclass Linearization for Dylan
		With a monotonic linearization, it is easier to understand the behavior of classes when multiple inheritance is used, largely because behavior of instances of a class can be explained in terms of the direct superclasses.
		A domain of a generic function may be sealed, which prohibits the definition of new methods or classes which would change the applicability of methods for the types specified by the domain.
		It should be clear that the Dylan linearization is monotonic, because the merge procedure never reorders the linearizations of superclasses when producing the linearization.
	www.webcom.com /haahr/dylan/linearization-oopsla96.html (5476 words)

	[No title]
		Give an example of monotonic function on [0,1] which is discontinuous at all rationals.
		For each group, invent a monotonic function that is discontinuous at each point of the group.
		The converse is not true: The absolute value function is continuous at 0, but does not have a derivative at 0, because it has a "corner" at 0.
	www.mathnerds.com /mathnerds/best/jumpingfunction/solution.aspx (343 words)

	Karl's Calculus Tutor - Box 6.0: Proof that Exponentials are Continuous
		In order to know that taking such a derivative can even be done, we need to know that the definition of exponential functions is useful and that exponentials are continuous (remember that you can't take the derivative of a function that is not continuous).
		Monotonic means that the function either always increases or always decreases with increasing
		If you want to accept on faith that exponentials are monotonic and cut right to the chase, then click here, but you'll be missing some very pretty algebra logic.
	www.karlscalculus.org /l6_0.html (1282 words)

	Rate Monotonic Analysis
		Rate Monotonic Analysis (RMA) is a collection of quantitative methods and algorithms that allows engineers to specify, understand, analyze, and predict the timing behavior of real-time software systems, thus improving their dependability and evolvability.
		The term "rate monotonic" originated as a name for the optimal task priority assignment in which higher priorities are accorded to tasks that execute at higher rates (that is, as a monotonic function of rate).
		During the 1980s the limitations of the original theory were overcome and the theory was generalized to the point of being practicable for a large range of realistic situations encountered in the design and analysis of real-time systems [Sha 91a].
	www.sei.cmu.edu /str/descriptions/rma_body.html (1446 words)

	Wikinfo \| Spiral
		polar coordinates by saying that r is a continuous monotonic function of θ.
		The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).
		As in the two-dimensional case, r is a continuous monotonic function of θ.
	www.wikinfo.org /wiki.php?title=Spiral (299 words)

	[No title] (Site not responding. Last check: )
		This function's default is m = 1.7, % the authors' compromise.
		The parameter n is the exponent for the weighed % estimate.
		% Obsolete text: % x monotonic: The effect of this on oceanic profiles is that any % inversions will be neutralized, but their data not completely overlooked % since it is used but with a value equal to the next (deeper) value.
	www.nrsc.no /~even/matlab/evenmat/interpwp.m (828 words)

	[No title] (Site not responding. Last check: )
		Decidability and halting problem We need approximations of functions which satisfy a given definition (so there is no doubt over their legitimacy).
		Definition: t is continuous if it is monotonic, and for every chain of monotonic functions {f1, f2, f3,...}, we have t[lub{f1, f2, f3,...}] = lub{t[f1], t[f2], t[f3 ],...}.
		The function computed by any recursive program is the least fixed point of the corresponding functional.
	www.ecst.csuchico.edu /~amk/foo/csci351/Chapter8.doc (668 words)

	[No title] (Site not responding. Last check: )
		The output of a neuron is typically a function of the sum of its weighted inputs and a bias value.
		This function, also referred to as an activation function, is typically a sigmoid function.
		That is, a function is minimized (referred to as"the objective function") subject to one or more constraints.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=99/67728.991229&ELEMENT_SET=DECL (8528 words)

	The Shape of a Function
		The derivative of a function f'(x) tells us how fast f(x) is changing with respect to x and gives us a measure of the steepness of the graph of f.
		The first qualitative judgement we will make is whether a function is increasing (the value of the function gets larger as the value of the independent variable gets larger) or decreasing (the function gets smaller as the independent variable gets larger).
		If the function "turns around", then the function is increasing beforehand and is decreasing afterwards or vice versa.
	oregonstate.edu /instruct/mth251/cq/Stage7/Lesson/shape.html (424 words)

	[No title] (Site not responding. Last check: )
		] The integral of the limit of a monotone increasing sequence of nonnegative measurable functions is equal to the limit of the integrals of the functions in the sequence.
		Also known as monotone decreasing function; monotonically nonincreasing function.
		A sequence of real-valued functions, defined on the same domain, that is either monotone-nondecreasing or monotone-nonincreasing.
	www.accessscience.com /Dictionary/M/M35/DictM35.html (2320 words)

	VVELOC
		(1) The coordinate definition function is presumed to be a monotonic decreasing function of the state variables.
		The algorithm does not care what function you use for F so long as F and F_Desired are consistent and F is a monotonic decreasing function over k.
		The algorithm attempts to move mass so that a linearly interpolated value of the coordinate definition function at the layer interfaces is kept close to the desired interfacial value through simple relaxation.
	grads.iges.org /colagmu/poseidon/doc/neptune4.4/M/POSEIDON_MODULE/S/vveloc.html (461 words)

	Calculus II Review Questions Exam 1 (Site not responding. Last check: )
		As with monotonic functions, if the number of subdivisions is big enough, Riemann sums give very good estimates for integrals of non-monotonic functions.
		Even though you could "get lucky" with the integral of a non-monotonic function, the LHS and RHS are guaranteed to bracket or enclose the exact value only for integrals of monotonic functions.
		Even with integrals of monotonic functions, the RHS or LHS could be better than their average, but in most cases, the average of the sums is better than either sum alone, whether the function is monotonic or not.
	www.willamette.edu /~mjaneba/courses/ma142/tfas-ex1.html (825 words)

	Quantum Mechanics 2
		This is typical of monotonically increasing or decreasing functions.
		and its 1st derivative to be continuous, so the oscillating function must join smoothly to the monotonic function at the turning point.
		If the forbidden region is not of infinite extent, then the rising monotonic function does not necessarily violate the conditions of finiteness.
	www.phy.duke.edu /courses/100/lectures/QM2/QM2.html (1109 words)

	The Honest Capitalization Rate (Site not responding. Last check: )
		Logistic growth involves a function that uses the base of the natural log, which is just a number approximately equal to 2.71828, modifying the function to account for growth that occurs at different rates during different time periods.
		The function is an equation having five components.
		The remaining terms in the function are e, the base of the natural log and n, the holding period.
	www.hostsrv.com /webmaa/webm1027/private/entreprenlogic.html (195 words)

	The New Algorithm
		An optimized inhomogeneous energy grid can be obtained by projecting a homogeneous grid from the y-axis of the monotonic function onto the energy axis.
		The integral over a spiked function is a monotonic increasing function.
		Figure (1b) indicates how a homogeneous grid on the abscissa of a monotonic function projects to an inhomogeneous grid on the ordinate.
	www.cfdrc.com /nemo/pubs/iwce_95_html/node3.html (410 words)

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