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Topic: Interval (mathematics)

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	Interval (mathematics) - Wikipedia, the free encyclopedia
		Intervals using the round brackets (or) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set.
		Intervals using both square and round brackets [ and) or (and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.
		Interval arithmetic, also called interval mathematics, interval analysis, and interval computation, is being developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors in mathematical computation and thus obtaining very reliable results.
	en.wikipedia.org /wiki/Interval_(mathematics) (910 words)

	PlanetMath: interval
		Loosely speaking, an interval is a part of the real numbers that start at one number and stops at another number.
		The logic is here that a empty circle represent a point not belonging to the interval, while a filled circle represents a point belonging to the interval.
		This is version 11 of interval, born on 2003-07-14, modified 2006-03-24.
	planetmath.org /encyclopedia/Interval.html (431 words)

	ipedia.com: Interval (mathematics) Article (Site not responding. Last check: 2007-10-21)
		Interval notation is where the permitted values for...
		Interval notation is where the permitted values for a variable are expressed as ranging over an interval; for example, "5
		In higher mathematics, a formal definition is the following: An interval is a subset S of a totally ordered set T with the property that whenever x and y are in S and x
	www.ipedia.com /interval__mathematics_.html (752 words)

	Presentation of Modal Interval Analysis
		From an interval point of view the common alternative of computing with single numbers is able to provide only an indication of the true result that could be reached using interval calculus: only a single value pointing somewhere inside the complete interval result.
		For a rational function f(x1,...,xn) a rational interval extension fR(X1,...,Xn) is defined through the sintactic tree of f, with their numerical arguments x1,...,xn replaced by the interval arguments X1,...,Xn and with the numerical operations of f replaced by their corresponding interval operations, which in case of approximate computations must be externally rounded.
		where the "intervals" [2,1] and [6,3] are the dual of the intervals [1,2] and [3,6].
	ima.udg.es /SIGLA/X/mod_interval (1326 words)

	Defintions in Interval Mathematics (Site not responding. Last check: 2007-10-21)
		Interval analysis was originally described as a way to calculate bounding errors in finite computer mathematical calculations inInterval Analysis (Moore, 1966).
		An interval vector can be physically thought of as a n-space geometrical figure, referred to as a box.
		Mathematical operations describing interval analysis are given by Hansen and Neumaier and are decribed below-of particular interest are the interval Newton methods.
	www.nd.edu /~whaynes/interval/node1.html (251 words)

	C H A P T E R 2 - C++ Interval Arithmetic Library Reference (Site not responding. Last check: 2007-10-21)
		In a SRIC containing two interval endpoints, the first number is the infimum or lower bound, and the second is the supremum or upper bound.
		Formulas for computing the endpoints of interval arithmetic operations on finite floating-point intervals are motivated by the requirement to produce the narrowest interval that is guaranteed to contain the set of all possible point results.
		The interval hull is the smallest interval that contains all the elements of the operand intervals.
	docs.sun.com /source/817-6705/iapgCreference.html (2420 words)

	Amazon.com: Interval Methods for Systems of Equations (Encyclopedia of Mathematics and its Applications): Books: A. ... (Site not responding. Last check: 2007-10-21)
		Interval analysis consists of the tools and methods needed to solve linear and nonlinear systems of equations in the presence of data uncertainties.
		Interval arithmetic is an elegant tool for practical work with inequalities, approximate numbers, error bounds, and more generally with certain convex and bounded sets.
		Interval mathematics 1980: Proceedings of an International Symposium on Interval Mathematics, held at the Institut für Angewandte Mathematik, Universität Freiburg i.
	www.amazon.com /exec/obidos/tg/detail/-/052133196X?v=glance (709 words)

	CU-Denver Department of Mathematics Events
		Interval Tournaments A directed graph is an interval digraph if to each vertex v there corresponds an ordered pair of intervals (S(v),T(v)) such that u beats v if and only if the intersection of S(u) and T(v) is nonempty.
		We characterize the tournaments that are interval digraphs via the existence of a large transitive subtournament and by forbidden subtournaments.
		A bipartite graph is an interval bigraph if to each vertex there corresponds an interval such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex belongs to a different partite set.
	www-math.cudenver.edu /events/QueryEvent.php?eid=315 (148 words)

	ABC Mathematics - Words to think about numbers, space and algebra
		Morris Kline's Mathematics in Western Culture, 1953 (Pelican 1972) tells the history of mathematics and relates it to the general history of human thought and to philosophy, art, and science.
		Catchup mathematics is an opportunity for people who may have missed out on the chance to appreciate and enjoy mathematics, the first time around.
		It allows people to reclaim mathematics, in areas that interest them, at a speed they find congenial, and in an environment where they can voice any difficulties and feelings they may have experienced with mathematics previously.
	www.mdx.ac.uk /www/study/glonum.htm (2494 words)

	f95 Interval Reference (Site not responding. Last check: 2007-10-21)
		In this case, an internal interval approximation is constructed that is guaranteed to contain the single decimal literal constant's external value.
		intervals are motivated by the requirement to produce the narrowest interval that is guaranteed to contain the set of all possible point results.
		Results for general set (or interval) inputs are the union of the results of the single-point results as they range over the input sets (or intervals).
	docs.sun.com /source/806-3595/iapg_ref_man.html (3332 words)

	DIMACS Discrete Mathematics Seminar (Site not responding. Last check: 2007-10-21)
		Interval graphs have been studied extensively both because they have many applications and because many problems which are NP-complete for general graphs, admit polynomial-time solutions when restricted to the class of interval graphs.
		Formally, a graph G is an interval graph if it can be represented as follows: each vertex of G corresponds to a real interval so that two vertices are adjacent in G if and only if their corresponding intervals intersect.
		Clearly the class of unit interval graphs is a subset of the class of proper interval graphs.
	dimacs.rutgers.edu /Events/Titles/1995/Trenk.html (216 words)

	Interval Computations \| Lambda the Ultimate
		But I mention AD not as a competitor for interval arithemtic, but as a neat twist on interval arithmetic that has formal similarity and a similar implementation, but is useful for a slightly different, but related, class of problem.
		And interval arithmetic can be combined with AD to give optimisation algorithms that are both efficient and are guaranteed not to miss optima because you can guarantee, say, that in a certain volume, the derivative of your objective function never reaches zero.
		Interval arithmetic is, quite simply, one of the easiest ways to let your program automatically analyze its own algorithms, and provide truly rigorous results.
	lambda-the-ultimate.org /node/1301#comment-14650 (5288 words)

	Best RELATIVE PLACEMENT
		Two different approaches for solving a BRP problem are discussed which are based on the least-squares method,and interval mathematics.
		One of interesting advantage of using interval mathematics in this case study, is the manoevre field of designers.
		Lastly, interval mathematics, the theory of conjugate method and BRP intiates a new ability for geo-spatial analysis.
	www.geocomputation.org /2000/GC057/Gc057.htm (929 words)

	Sun Studio White Papers
		For a purely software interval implementation, the simple system described in this paper is a reasonable compromise between narrow interval results for special edge cases, and speed for cases that occur more frequently.
		In interval mathematics, the distinction between constants and variables is even more important than it is in point mathematics.
		Following this convention prevents an interval implementation from constructing narrow intervals when, for example, 0.1 is encountered in a double precision interval expression.
	developers.sun.com /prodtech/cc/products/whitepapers/index.html (801 words)

	[No title]
		The details of interval arithmetic are difficult to explain to the untrained, and even some highly trained people simply refuse to listen.
		However, with intervals, the right answer is always captured within a machine-representable interval, he says.
		With intervals, Walster says, "Sun has at least as big an opportunity to gain a technological advantage as any company has ever had in history of science and technology.
	research.sun.com /minds/2004-0527 (1487 words)

	Infolytica » Online Library » Abstract
		The use of interval mathematics in a knowledge-based system for electromagnetic device design is discussed.
		It is shown that, through interval propagation, the expert designer can observe the consequences of imposing interval bounds on parameters so that the nonexpert user can then receive advice regarding the legal range of design specifications.
		Interval techniques have proved to be valuable at various stages of the design process as a knowledge integrity tool and as an aid to end-users of the CAD (computer-aided design) system.
	www.infolytica.com /en/library/abstracts/i009.html (163 words)

	Neli Dimitrova
		The aim of this project was the development of modern methods for mathematical modelling under uncertain data like reliable computing, interval analysis, numerical methods with result verification, controlability of dynamical systems etc.
		In the frames of this project an algorithm with result verification for solving linear systems with intervals in the matrix and in the right-hand side vector is implemented in Maple.
		Theoretical studies in the field of quasilinear spaces of intervals, interval analysis etc. support the construction of the algorithms.
	www.math.bas.bg /~bio/ndim/ndim.html (1311 words)

	A.P.Ershov Institute of Informatics Systems - Andrei Ershov Fifth International Conference - Workshop "Interval ... (Site not responding. Last check: 2007-10-21)
		An interval algorithm for determining soil static pressure on multilined stiff round pipes in a high embankment.
		Interval approach to the solution of the interval constraint satisfaction problem.
		On interval methods for differential problems with data uncertainty in the form of convex sets.
	www.iis.nsk.su /psi03/meeting/index_e.shtml (285 words)

	[No title]
		Academics have the chance to participate in a working environment where their mathematical skills are applied to non-standard but topical industrial problems and opportunities for interactive and collaborative research projects result.
		Date: Wed, 3 Nov 93 09:38:36 CST Subject: Position at University of Kansas Department of Mathematics University of Kansas Applications are invited for a tenure-track position at the assistant professor level and for a visiting position at the assistant professor level (pending funding) beginning August 16, 1994 or as negotiated.
		In addition to strong representation in pure mathematics, the department has an outstanding group of applied mathematicians in the areas of control and optimization, numerical analysis, ordinary and partial differential equations, and probability and stochastic processes.
	www.netlib.org /na-digest/93/v93n41 (4326 words)

	Interval mathematics - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		Start the Interval mathematics article or add a request for it.
		Look for Interval mathematics in Wiktionary, our sister dictionary project.
		Look for Interval mathematics in the Commons, our repository for free images, music, sound, and video.
	www.sciencedaily.com /encyclopedia/interval__mathematics_ (185 words)

	M1
		II International Simposium on Interval Mathematics, Freiburg (FR Germany), May 27-31, 1980.
		III International Simposium on Interval Mathematics, Freiburg (FR Germany), September 23-26, 1985.
		M.S. Petkovic: Introduction to Interval Mathematics (in Serbian).
	mspetkovic.tripod.com (700 words)

	No Title (Site not responding. Last check: 2007-10-21)
		Interval techniques have recently gained notariety because they can be employed to solve systems of non-linear equations for ALL roots, with mathematical and computational certainty.
		This mathematical and computaional certainty is unavailable with other techniques, and is still relatively unknown in the computing community.
		This web page is designed as an introduction to interval mathematics.
	www.nd.edu /~whaynes/interval (119 words)

	Directed Interval Arithmetic
		Directed interval arithmetic is obtained as an extension of the set of normal intervals by improper intervals and a corresponding extension of the definitions of the interval arithmetic operations.
		Using data type INTERVAL which is part of the language core of PASCAL-XSC, exi_ari module supplies all interval arithmetic operations, functions and procedures overloaded to produce correct results for directed intervals, too, as well as many other functions necessary for computations in extended interval arithmetic spaces.
		Mathematica notebook indeterm.ma (40329 bytes ASCII file) contains a case study of the kernel functions on intervals involving the symbol Indeterminate and a detailed demonstration of the directed interval arithmetic on intervals involving the symbol Indeterminate.
	www.math.bas.bg /~epopova/directed.html (856 words)

	References
		In Interval Mathematics: Proceedings of the International Symposium, volume 29 of Lecture Notes In Computer Science, pages 7-18, Berlin, May 1975.
		Based on a conference organized by the Institute of Mathematics and Its Applications, Canterbury, England, 1967.
		Towards an abstract mathematical theory of floating-point arithmetic.
	www.dgp.toronto.edu /~mooncake/thesis/node179.html (432 words)

	Mathematics
		The minimal interval (i.e., point-valued) type of vector interval is exemplified by a vector such as `10 meters due east'.
		#$UnitVectorInterval is a vector interval with a magnitude of 1.
		#$FrameOfReference is a mathematical (and hence intangible) representation of the context in which certain data are to be interpreted.
	www.cyc.com /cycdoc/vocab/math-vocab.html (5058 words)

	[No title]
		During the last decade, Interval methods have received great attention from the Control community and major advances has been made in the application of Interval methods to control problems and many interval methods has been integrated in tools for CACSD.
		Interval analysis provides guaranteed approximations of the set of all the actual solutions of the problem.
		This minisymposium is devoted to the theory and application of interval analysis in validated computing and deterministic global optimization.
	interval.louisiana.edu /reliable_computing.archive/reliable_computing.0401 (5417 words)

	Interval Mathematics 1985
		Karl Nickel (Ed.): Interval Mathemantics 1985: Proceedings of the International Symposium, Freiburg i.
		Klaus D. Schmidt: Embedding Theorems for Cones and Applications to Classes of Convex Sets Occurring in Interval Mathematics.
		Peter Thieler: Technical Calculations by Means of Interval Mathematics.
	www.informatik.uni-trier.de /~ley/db/conf/interval/interval1985.html (279 words)

	Interval Methods in Global Optimization (Site not responding. Last check: 2007-10-21)
		Talks are devoted to development of corresponding mathematical structures, design of computer tools, and applications in a wide range of areas.
		INTERVAL'98 will emphasize the application of interval mathematics and its combination with the interdisciplinary topics from the two preceding meetings in global optimization.
		Especially welcome are contributions which use interval arithmetic tools for solving problems of practical relevance.
	www.csc.fi /math_topics/Mail/NANET97-3/msg00027.html (185 words)

	h51f in fm97
		The aquifer transmissivity was represented by interval and the width of the resulting hydraulic head interval was used as a measure of uncertainty due to imprecise input parameter.
		Analysis of the results indicated that degree of uncertainty associated with the predicted hydraulic head was found to increase as the width of the input parameter interval increased.
		This approach was compared to the stochastic technique and the results indicated that the distribution of the width of the interval along the flow direction is similar to the hydraulic head variance distribution resulted from Monte Carlo simulation approach.
	www.agu.org /cgi-bin/SFgate/SFgate?&listenv=table&multiple=1&range=1&directget=1&application=fm97&database=/data/epubs/wais/indexes/fm97/fm97&maxhits=200&="H51F" (3441 words)

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