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Topic: Greatest fixed point

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	Fixed point (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function.
		In graphical terms, a fixed point means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
		Attractive fixed points are a special case of a wider mathematical concept of attractors.
	en.wikipedia.org /wiki/Fixed_point_(mathematics) (567 words)

	Fixed point math routines
		Fixed point numbers can be assigned, compared, added, subtracted, negated and shifted (for multiplying or dividing by powers of two) using the normal integer operators, but you should take care to use the appropriate conversion routines when mixing fixed point with integer or floating point values.
		The fixed point square root, sin, cos, tan, inverse sin, and inverse cos functions are implemented using lookup tables, which are very fast but not particularly accurate.
		The fixed point functions used to be named with an "f" prefix instead of "fix", eg.
	www.nsac.ns.ca /eng/courses/temp/allegro/alleg029.html (577 words)

	GFP - Wikipedia, the free encyclopedia
		In neuroscience, GFP mice are used to image neurons in vivo or in vitro neurons.
		This page concerning a three-letter acronym or abbreviation is a disambiguation page—a list of articles associated with the same title.
		If an internal link referred you to this page, you may wish to change the link to point directly to the intended article.
	en.wikipedia.org /wiki/GFP (135 words)

	Knaster-Tarski theorem (Site not responding. Last check: 2007-10-10)
		Then the set of fixed pointss of f in L is also a complete lattice.
		Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least (or greatest) fixed point.
		For example, in theoretical computer science, least fixed points of monotone functions are used to compute program semantics.
	www.brainyencyclopedia.com /encyclopedia/k/kn/knaster_tarski_theorem.html (354 words)

	Papers Report (Site not responding. Last check: 2007-10-10)
		An iteration theory is an algebraic theory equipped with a dagger operation satisfying all equations that hold for the least (or greatest) fixed point operation on monotonic functions on complete lattices.
		We prove that all iteration theories of boolean functions equipped with a pointwise dagger operation consist of monotonic functions, and that the dagger operation is either the least, or the greatest fixed point operation.
		It follows form the general result proved in the paper that if a Conway theory is equipped with a pointwise dagger, then the theory obtained by adjoining all constants is a Conway theory in a unique way, and that the same fact holds for iteration theories.
	www.mfcs.sk /mfcs2000/abstracts/AccAbs6.html (204 words)

	[No title]
		ABSTRACT: Fixed point logics and $\mu$-calculi are obtained from previously existing logical or algebraic frameworks by the addition of least and greatest fixed point operators.
		I'll point out the sense for which initial algebras of functors are a generalization of least fixed points and the relationship to inductively constructed sets (such as lists, finite tress, etc.).
		Some familiar examples of fixed points are presented, such at the transitive closure of a set of relations (R*) and the "reachability" relation.
	pages.cpsc.ucalgary.ca /~robin/CPRGLCT/past.html (1331 words)

	Big Bang - Wikipedia, the free encyclopedia
		Understanding this period of the history of the Universe is one of the greatest unsolved problems in physics.
		Grand unification theories predicted point defects in space that would manifest as magnetic monopoles with a density much higher than was consistent with observations, given that searches have never found any monopoles.
		The entropy of the Universe would increase to the point where no organized form of energy could be extracted from it, a scenario known as heat death.
	en.wikipedia.org /wiki/Big_bang (5691 words)

	Allegro Manual: Fixed point math routines (Site not responding. Last check: 2007-10-10)
		Fixed point math is considered "add-on" material and is kept only for backwards compatibility.
		Note that on machines with very good floating point processors using these functions could be slower in real life code due to cache misses: it may be faster to wait a few extra cicles for a floating point sine result rather than wait for the CPU to fetch the precalculated table from main memory.
		On top of that, the Fix class may be slower than using directly the C functions because of implicit internal conversions from one type to another which you otherwise could avoid or minimise.
	www.talula.demon.co.uk /allegro/onlinedocs/en/alleg032.html (2065 words)

	Iterations of the map
		There is a positive jump from zero up to the fixed point value.25 in the x,A attractor diagram, unlike the logistic map case.
		Thus, the slope is greater than unity and the smaller fixed point is always unstable.
		For the largest stable periodic points, it is the tangent to their curves.
	journal-ci.csse.monash.edu.au /ci/vol02/gottlieb/node4.html (636 words)

	Lecture 5, slide 11 (Site not responding. Last check: 2007-10-10)
		We can characterise the greatest fixed point and the least fixed point very easily: the least fixed point as the infimum of all these objects and the greatest fixed point as the supremum of all these objects.
		And in this definition lies the reason of why having a denotational approach in terms of fixed point, be them either least fixed points or greatest fixed points is mathematically very rewarding.
		We have immediately from this very definition we can extract two reasoning principles: induction and definition by recursion from the least fixed point and co-induction and definition by co-recursion from the greatest fixed point.
	www.europeindia.org /cd09/lectures/lect05/11.htm (212 words)

	Sandra J. Peart, David M. Levy, Happiness, Progress and the Vanity of the Philosopher: Library of Economics and Liberty
		The value of children is the greatest of all encouragements to marriage.
		The key point is that in Smith's view of the world, which as we shall see Malthus followed, incentives and foresight explained the differences between America and Europe, not some exogenous, immutable force.
		The term, general good, may be defined as the rearing of the greatest number of individuals in full vigour and health, with all their faculties perfect, under the conditions to which they are subjected.
	www.econlib.org /library/Columns/y2005/PeartLevymalthus.html (3669 words)

	STANDING - Online Information article about STANDING
		Let C be a point whose ordinates are x and y, and let the river at C have the breadth b, the slope i, and the velocity v.
		This is more likely to the highest level at one point of a river is not always simultaneous with the attainment of the highest level at other points; and that the rise of a river in flood is very different in different parts of its course.
		wool was used loosely fixed in a hole at the centre.
	encyclopedia.jrank.org /SOU_STE/STANDING.html (8111 words)

	MainFrame: A theory of fixed points
		Definition of the notion of a bounded monotonic function and of least and greatest fixed points.
		Proofs that "lfp" gives a fixed point and that it is the least fixed point.
		This is the corresponding theorem for greatest fixed point to the "induction" principle for least fixed points.
	www.rbjones.com /rbjpub/pp/x001-m.html (777 words)

	CATHOLIC ENCYCLOPEDIA: Aristotle
		The greatest of heathen Philosophers, born at Stagira, a Grecian colony in the Thracian peninsula Chalcidice, 384 B.C.; died at Chalcis, in Euboea, 322 B.C. His father, Nicomachus, was court physician to King Amyntas of Macedonia.
		A point which should be emphasized in the exposition of this portion of Aristotle's philosophy is the doctrine that all action consists in bringing into actuality what was somehow potentially contained in the material on which the agent works.
		It is of the nature of moral virtues, therefore, to avoid all excess as well as defect; bashfulness, for example, is as much opposed to the virtue of modesty as shamelessness is. The intellectual virtues (understanding, science, wisdom, art, and practical wisdom) are perfections of reason itself, without relation to the lower faculties.
	www.newadvent.org /cathen/01713a.htm (5735 words)

	A Calculated Look at Fixed-Point Arithmetic
		Certainly none of the small 4- and 8-bit microcontrollers support floating point, even though these are precisely the processors that are going to be at the heart of many apparently "real-number" applications.
		In this case, the decimal point is really an illusion-there are always two decimal places, so instead of working with numbers in the range 0.00 to 50.00 we actually use 0 to 5,000 (the values are scaled up by a factor of 100).
		There are a fixed number of digits after the decimal point; the resolution is explicit.
	www.embedded.com /98/9804fe2.htm (2746 words)

	Myofascial pain / trigger points /nerve root pain/satellite trigger points (Site not responding. Last check: 2007-10-10)
		They were points which became spontaneously tender, and were detected by palpation when the troublesome part of the body was examined.
		These satellite trigger points start exerting pain in their own area of radiation and are quite likely to reactivate the primary trigger point, the cause of the original problem.
		The detection of trigger points should not be difficult as the pressure on the point produces what has been described as the ‘jump sign’.
	www.positivehealth.com /permit/Articles/Bodywork/halfd16.htm (2063 words)

	Mathematical Programming Glossary - C (Site not responding. Last check: 2007-10-10)
		A differentiable curve where each point is the analytic center of a polyhedron associated with a linear program.
		This is repeated until a fixed point is reached.
		In general, it is possible for such a fixed point not to be an optimum (even locally) because a simultaneous change in variables could result in an improvement.
	carbon.cudenver.edu /~hgreenbe/glossary/C.html (6173 words)

	Summary of Recursive Types
		is defined to be a fixed point of the function mapping the type
		Although many such fixed points may exist in general, we can uniquely determine one by choosing either the least fixed point or the greatest fixed point.
		This "circular" definition of lists does not uniquely define the set of possible lists however; it is compatible both with the conclusion that all lists are finite (the least fixed point) and with the conclusion that lists may be finite or infinite (the greatest fixed point).
	www.cs.hmc.edu /~stone/recursive.html (724 words)

	Mth 351 Note - Intel 80x87 FP Data Types IEEE Std. 754
		The largest floating point integer N such that N-1 is also a floating point integer (that is, exactly representable) is called the largest fp integer with a predecessor.
		The largest fp integer with a predecessor and the unit round are convenient measures of the precision of the floating point representation.
		Since the binary point is not actually stored we have to have some agreement about where it falls.
	oregonstate.edu /~peterseb/mth351/docs/351s2001_fp80x87.html (967 words)

	[No title]
		Such a function σ has a least fixed point denoted μy.σ(y), and a greatest fixed point denoted νy.σ(y).
		Thus, EFp is fixed point of the function σ(y)=p \/ EXy.
		Given an LTS and a property p of its states, we identify p with the set of states having the property p.
	www.cs.technion.ac.il /~myoeli/pub/ver_toc/lot_ch15.htm (542 words)

	EMail Msg <9007301428.AA24473@fb14vax.cs.uni-sb.de> (Site not responding. Last check: 2007-10-10)
		In any case, least fixed points are counter-intuitive in a terminological logic where you have only (and C D), (all R C), and (exists R) since concepts that use (exists R) and are defined circularly over R have an empty denotation.
		Nevertheless, it looks unnatural and prohibits the use of such constructions which seems to be useful sometimes.] >It is also not surprising that >defintion of human is no different from that of dog or cat or any other animal.
		There is one VERY BIG problem though: Least and greatest fixpoints have the property to be very rare.
	www-ksl.stanford.edu /email-archives/interlingua.messages/23.html (607 words)

	TAPL exercises for Lecture 6 (2004-10-06) (Site not responding. Last check: 2007-10-10)
		Formulate a type NatTree of binary trees with natural number leaves and show and implementation of sum computing the sum of all leaf elements in a tree.
		A complete lattice is a partial order (S, <=) such that each subset T of S has a least upper bound and greatest lower bound wrt.
		Argue that both lfp F (least fixed point of F) and gfp F (greatest fixed point of F) are well-defined.
	www.diku.dk /undervisning/2004e/224/ex2004-10-06.html (356 words)

	Data Type Propagation (Fixed-Point Blockset)
		Use the number of bits from the Ref1 reference signal, or use the number of bits from widest reference signal.
		Use the range from the Ref2 reference signal, or use the range of the reference signal with the greatest range.
		Use a bias of zero, regardless of the biases used by the reference signals.
	www-rohan.sdsu.edu /doc/matlab/toolbox/fixpoint/datatypepropagation.html (1678 words)

	IEEE floating-point representations of real numbers
		Only the part of the mantissa that comes after the binary point is actually stored, since the bit to the left of the binary point is completely predictable (it's always 1, since the mantissa is always greater than or equal to one and less than two).
		Mantissas less than one are said to be ``unnormalized'' (because the ``normal form'' is the one in which the mantissa is greater than or equal to one and less than the base of numeration), so an all-zero exponent indicates an unnormalized number.
		that precedes the binary point is once again a ``hidden bit.'' As in single-precision representations, the all-zero exponent is used for unnormalized numbers and (with an all-zero mantissa) for 0, and the all-one exponent is used for the pseudo-numbers positive infinity, negative infinity, and NaN.
	www.math.grin.edu /~stone/courses/fundamentals/IEEE-reals.html (2234 words)

	The mc package: all functions
		points to the original AG formula, and top node of phi points to the original phi.
		This is because we often have to merge paths where the end point of the first is the starting point of the second.
		After the call, it points to an array that contains one entry for every fairness constraint: entry i contains the onionrings for the last EU computation that was performed for fairness constraint i.
	embedded.eecs.berkeley.edu /research/Vis/doc/html/mcAllDet.html (4217 words)

	Common Knowledge
		C is the greatest fixed point of f.
		Barwise's fixed point analysis of common knowledge is favored by those who are especially interested in the applications of common knowledge to problems in logic, while the hierarchical and the partition accounts are favored by those who wish to apply common knowledge in social philosophy and social science.
		Barwise's fixed point account is indeed equivalent to the hierarchical and the partition accounts given the account of knowledge characterized by (K1)-(K4) that most practitioners accept.
	plato.stanford.edu /entries/common-knowledge (14481 words)

	Wizbang
		From the outset, I fixed on the word "fixed" and pegged that as the Achilles heel of the whole "memo scandal." It was the classic tempest in a teapot.
		Many point to Kofi Annan's statment that the war was illegal, but he was referring to the war as a basis of pre-emptive self-defence, regime change, or UN 1441.
		Hence “fixed around the policy” clearly meant putting together a narrative using the facts as they were known to support a particular policy decision, and not to manufacture facts to support a policy.
	wizbangblog.com /archives/006138.php (12353 words)

	MATHS: Category Theory
		Whenever a pair of arrows share a common point: a-mab->b-mbc->c then we are given a way of composing the labels (traditional written o) and an arrow connecting the outer pair a to b.
		A fixed point of a functor is not exactly an object in an category.
		Tarski 55, A Tarski, "A Lattice Theoretical Fixed point Theorem and its
	www.csci.csusb.edu /dick/maths/math_25_Categories.html (3607 words)

	Compiler Optimization Meets Compiler Verification COCV 2004 (Site not responding. Last check: 2007-10-10)
		Formal semantics of programming languages needs to to model the potentially infinite state transition behavior of programs as well as the computation of their final results simultaneously.
		We show that a greatest fixed point interpretation of natural semantics is able to model both aspects equally well.
		Technically, we infer this interpretation of natural semantics based on an easily omprehensible introduction to the dual definition and proof principles of induction and coinduction.
	www.complang.tuwien.ac.at /knoop/COCV2005/Abstract/7.html (123 words)

	[No title]
		The basic idea is to fix an alphabet (which need not be finite) L of labels and to define our evaluation relation -> as a relation on triples of an expression, a label, and another expression.
		The problem is that, of course, F({}) = {} since each of the clauses in F demands that we build a pair of sequences out of a pre-existing pair of sequences in the input.
		The problem is that we were trying to define =~= as the least fixed point of F. If instead, we take =~= to be the *greatest* fixed point of F (assuming one exists and it's unique) then perhaps that would work?
	www.eecs.harvard.edu /~greg/cs256sp2005/lec22.txt (1188 words)

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