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Topic: Infinite sequence

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Linguistics

	Sequence Summary
		A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.
		Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science.
		Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0,1}).
	www.bookrags.com /Sequence (2442 words)

	Sequence - Biocrawler (Site not responding. Last check: 2007-10-04)
		An infinite sequence in S is a function from {1,2,...} (the set of natural numbers) to S.
		A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.
		If the terms of the sequence are a subset of a ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
	www.biocrawler.com /encyclopedia/Infinite_sequence (631 words)

	Wikinfo \| Sequence
		In mathematics, a sequence is a list of objects (or events) which have been ordered in a numerical (and sequential) fashion; such that each member either comes before, or after, every other member.
		A sequence is a function with a domain equal to the set of positive integers.
		Infinite sequences are given by listing the first few terms, followed by an ellipsis.
	www.wikinfo.org /wiki.php?title=sequence (352 words)

	Researchers predict infinite genomes
		Armed with the powerful tools of comparative genomics and mathematics, TIGR scientists have concluded that researchers might never fully describe some bacteria and viruses--because their genomes are infinite.
		Sequence one strain of the species, and scientists will find significant new genes.
		To interpret this infinite view of microbial genomes, Tettelin and colleagues propose describing a species by its "pan-genome": the sum of a core genome, containing genes present in all strains, and a dispensable genome, with genes absent from one or more strains and genes unique to each strain.
	www.eurekalert.org /pub_releases/2005-09/tifg-rpi092205.php (660 words)

	Sequence : Infinite sequence (Site not responding. Last check: 2007-10-04)
		A sequence may have a finite or infinite number of terms; thus, it is called either finite or infinite.
		Infinite sequences are given by listing the first few terms, followed by an ellipsis.
		In mediæval Latin literature, a sequence (Latin sequentia) is a poem written in a non-classical metre that uses rhyme and an accentual (stress based) rather than quantititive (vowel length based) verse form.
	www.fastload.org /in/Infinite_sequence.html (293 words)

	Infinite Series \| World of Mathematics
		Thus, in general, the sum of an infinite series is the limit of the sequence of partial sums as n grows without bound, provided that such a limit exists.
		If the sequence of partial sums continues to grow without bound as n grows without bound, then the infinite series does not have a sum and is said to diverge.
		Infinite series may be used to derive a number of remarkable mathematical results.
	www.bookrags.com /research/infinite-series-wom (873 words)

	Infinite sequence (Site not responding. Last check: 2007-10-04)
		cpnDB is a curated collection of chaperonin sequence data collected from public databases or generated by a network of collaborators exploiting the cpn60 target in clinical, phylogenetic and microbial ecology studies.
		The Protein Sequence Database (PSD), a functionally annotated database of protein sequences, is located at PIR.
		Swiss-Prot is a curated database of protein sequences that are highly annotated and have a minimal level of redundancy.
	www.serebella.com /encyclopedia/article-Infinite_sequence.html (1161 words)

	Infinite Series (Site not responding. Last check: 2007-10-04)
		The series that we are often most concern with are infinite series, that is the sum of an infinite sequence.
		In this case, you could look at the sequence and see that for all but the first few terms, the sum of any two sequential terms is larger than the term preceding it in the sequence.
		Following this logic, for any term in the sequence there is a sum of higher index terms that are greater than that term in the sequence, and the sequence will not converge.
	charon.kean.edu /~djoiner/book_1603/node91.html (336 words)

	SparkNotes: Sequences and Series: General Sequences and Series
		Sequences are not typically written as ordered pairs, or drawn as graphs; a sequence is most often represented by a list of its terms starting with the first term, followed by the second, and so on.
		An infinite series is the sum of the terms in an infinite sequence.
		A recursive sequence is a sequence in which the nth term can be expressed as a function of the previous term.
	www.sparknotes.com /math/precalc/sequencesandseries/section1.html (689 words)

	Interaction - Smart Board
		A "relatively compact" set is one in which any infinite sequence of points (or functions) has a Cauchy subsequence; that is, a subset whose points eventually cluster together within an arbitrarily small region (essentially "converging" to a single point).
		Figure 2 shows a sequence of illustrations similar to what one might draw in attempting to find the desired connection; such illustrations are very common in analysis, and they lend themselves to random construction by computer.
		The key to the proof is in utilizing the finiteness of the cover; since the sequence is infinite, it must hit at least one of the e-balls in the cover "infinitely often," which is another common idiom of analysis that can be given a visual representation.
	www.cs.brown.edu /stc/resea/interaction/research_I12a.html (974 words)

	GAF - Grupo de Acción Filosófica
		If this is an infinite sequence of justifications invoked to justify an observational report, it is reasonable to consider that we make use of infinite sequences of justifications all the time and that most cases of justification seem to invoke an infinite regress.
		Infinite sequences of justfications like the one above bear resemblence to what is a commonplace about truth that, in its turn, is related to what Tarski called the material condition for adequacy in a theory of truth.
		Recursive infinite sequences, however, can be more complex and can involve more interesting (or at least more surprising) justifications but they have the advantage of coming to view in a finite number of steps; namely the recursive clauses.
	www.accionfilosofica.com /blog/mensaje.pl?id=94 (2053 words)

	Infinite Descent versus Induction (Site not responding. Last check: 2007-10-04)
		It is often said that the principle of "infinite descent", developed by Fermat in his study of Diophantine analysis, is equivalent to induction.
		Those who argue that infinite descent is equivalent to induction typically refer to a proof such as the following for the fact that the sum of the first n integers is n(n+1)/2: If not, there is a least integer, say N, for which this is false.
		Thus, there is some validity in the claim that Fermat's infinite descent concept is one of the proto-types for reasoning about infintesimals and limiting sequences, which is perhaps not surprising in view of Fermat's own contributions to the foundation of the differential calculus.
	www.mathpages.com /home/kmath144.htm (1225 words)

	Finitism and Hypercomputation
		For a sufficiently fast growing sequence A, a recursive relation has an infinite descending path through n iff it has an infinite descending path through n and then through a natural number less than A(max(n, m)) where m is the length of the given recursive definition of the relation.
		It enumerates potential initial segments (of the length of the longest sequence in the set of sequences) of the truth predicate for the language with a distinguished symbol for a well-ordering of the elements.
		Consider a well-founded tree of such sequences (ordered by extension) such that every non-leaf node can be extended by any natural number, and if a sequence is a leaf node, then it is sufficiently long relative to itself (or relative to its rate of growth; the difference does not affect the expressive power).
	web.mit.edu /dmytro/www/FinitismPaper.htm (3168 words)

	infinite sequence - a Whatis.com definition
		An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integers {1, 2, 3,...}.
		An infinite series is the sum of the values in an infinite sequence of numbers.
		An infinite series is the sum, if defined, of the numbers in a specific infinite sequence.
	searchnetworking.techtarget.com /gDefinition/0,294236,sid44_gci804476,00.html (295 words)

	Peter Suber, "Infinite Sets"
		To show that an infinite set, like the even numbers, can be put into one-to-one correspondence with another, like the odd numbers, we need only produce a rule-governed sequence for each set which runs through the members without omission or repetition, for example, 2, 4, 6...
		The resulting infinite string of "yeses" and "noes" is demonstrably different from every row of the infinite table, for it differs from the first row in the first term, from the second row in the second term, and so on.
		It may therefore be taken as the defining condition of infinite magnitude, and its absence as the defining condition of finitude.
	www.earlham.edu /~peters/writing/infapp.htm (6879 words)

	Composite number for pi satisfies Euler's equation.
		sequence, it cannot be constructed as a line segment by any method whatsoever.
		P sequence is twice the previous term divided by the new differential as shown by steps 30 and 40 below.
		Since the limits of the sequences are in the same proportion as the terms, we can review the formulas for the terms to ascertain the limits of the sequences.
	members.ispwest.com /r-logan/math.html (1291 words)

	Nielsen's Third Theorem of Infinite Sets
		Also note, you are creating an infinite list of finite sets for every decimal position di in D. Since the set of numbers that corespond to each di are finite and countable, then the union of all those sets is also countable.
		For example, 1/3 is not represented unless the binary sequence is infinite.
		and this is not represented until the binary sequence is infinite.
	www.marknielsen.net /Math/Counting/Nielsen_Theorem_Of_Infinite_Sets/Nielsens_Third_Theorem_Of_Infinite_Sets.html (695 words)

	What is infinite sequence? - a definition from Whatis.com - see also: infinite series
		An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integers {1, 2, 3,...}.
		An infinite series is the sum of the values in an infinite sequence of numbers.
		An infinite series is the sum, if defined, of the numbers in a specific infinite sequence.
	whatis.techtarget.com /definition/0,,sid9_gci804476,00.html (324 words)

	Real Numbers as Equivalence Classes of Cauchy Convergent Sequences
		Although it is tempting, and commonly done, to define real numbers as infinite sequences of digits there are insurmountable logical difficulties with that construction.
		The conceptual difficulty in defining real numbers as convergent sequences is that there is no entities to be taken as the limits of convergent sequences.
		Two convergent sequences are equivalent; i.e., belong to the same equivalence class, is their difference is in the equivalence class of zero.
	www.sjsu.edu /faculty/watkins/cauchy.htm (1120 words)

	Introduction
		For directed graphs (digraphs), we show that the infinite terms of the out-distance sequences of vertex-transitive digraphs are nonincreasing, and additionally that any nonincreasing sequence of infinite cardinals is the out-distance sequence of some vertex-transitive digraph.
		Lemma 1.7 For a vertex-transitive digraph of finite or infinite out-degree, the out-distance sequence
		For a sequence of infinite cardinals, log-concavity implies that the sequence is constant except for its first and last terms.
	people.cs.uchicago.edu /~wes/dlocinf/node1.html (432 words)

	College Algebra Tutorial on Sequences
		Sequences of math are a string of numbers that are tied together with some sort of consistent rule, or set of rules, that determines the next number in the sequence.
		An arithmetic sequence is a sequence such that each successive term is obtained from the previous term by addition or subtraction of a fixed number called a difference.
		A geometric sequence is a sequence such that each successive term is obtained from the previous term by multiplying by a fixed number called a ratio.
	www.wtamu.edu /academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54a_seq.htm (1456 words)

	MathComplete.com - Sequence and Series - Tutorial (Site not responding. Last check: 2007-10-04)
		A sequence of numbers is a function defined on the set of positive integer.
		Arithmetic sequence is a sequence of a number each of which, after the first, is obtained by adding to the preceding number a constant number called the common difference.
		Geometric sequence is a sequence of a number each of which, after the first, is obtained by multiplying the preceding number by a constant number called common ratio.
	www.mathcomplete.com /tutorial/sequence/default.asp?pg=1 (283 words)

	Geometric Series (Site not responding. Last check: 2007-10-04)
		The first thing that must be discussed when working with infinite series is the meaning of convergence of an infinite sequence of real numbers.
		A sequence is a function whose domain is the positive integers (sometimes the nonnegative integers).
		On the other hand, if the sequence consists of terms that continue to get smaller and smaller, then it is not clear whether the sum will grow without bound.This is where the need for sequences comes in.
	mathcircle.berkeley.edu /BMC4/Handouts/serie/node2.html (297 words)

	SparkNotes: Sequences and Series: Terms and Formulae
		Finite Sequence - A sequence which is defined only for positive integers less than or equal to a certain given integer.
		Geometric Sequence - A sequence in which the ratio between each term and the previous term is a constant ratio.
		Recursive Sequence - A sequence in which a general term is defined as a function of one or more of the preceding terms.
	www.sparknotes.com /math/precalc/sequencesandseries/terms.html (388 words)

	Miscellaneous Questions - Numericana
		In an infinite sequence of lattice points where the distance [or gap] between two consecutive points is bounded...
		For any sequence S of points (X(n),Y(n)) where two consecutive points are at most D units apart, we may consider the sequence C obtained by removing from the sequence (floor(X(n)/D),floor(Y(n)/D)) any consecutive elements which happen to be equal.
		When this sequence is displayed as a binary tree, each level features the same fractions as in the Stern-Brocot tree, but at different locations: The respective positions [0 = leftmost] of a given fraction are mirror images in binary numeration.
	home.att.net /~numericana/answer/misc.htm (2164 words)

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