http://planetmath.org/?op=enlist&mode=created http://planetmath.org/encyclopedia/KurodaNormalForm.html CWoo A formal grammar ... is in ... Kuroda normal form if its productions have one of the following forms: ... , ... , ... . ... where ... and ... . Note: Sometimes the form ... , where ... is added to the list above. However, we may replace a production of th ... Thursday, 2009-7-2 http://planetmath.org/encyclopedia/ProofOfRadonNikodymTheorem.html Ziosilvio The following proof of Radon-Nikodym's theorem is based on the original argument by John von Neumann. We suppose that mu and nu are real, nonnegative, and finite; the extension to the general case is a standard exercise, as is mu-a.e. uniqueness of Radon- ... Thursday, 2009-7-2 http://planetmath.org/encyclopedia/SetMembership.html pahio When one always uses ... sets in mathematics, a set S is thought to consist of certain distinct objects x satisfying some known condition.\, Then we say that x is a ... member or an ... element of the set S (denoted\, ... ).\, In the case, when x does not ... Thursday, 2009-6-25 http://planetmath.org/encyclopedia/EquivalenceOnRegularExpressions.html CWoo Let Sigma be an alphabet, and ... be the set of all regular expressions over Sigma. Two expressions ... are said to be ... equivalent , written ... , if they describe the same language: ... . This relation is clearly an equivalence relation on ... , and t ... Monday, 2009-6-22 http://planetmath.org/encyclopedia/StarHeight.html CWoo The ... star height of a regular expression p over an alphabet Sigma, denoted by ... , is defined recursively as follows: ... for any ... ; ... ; ... . ... Let ... . The following expressions have star height ... : ... Since any regular expression p descr ... Monday, 2009-6-22 http://planetmath.org/encyclopedia/NielsHenrikAbel.html pahio The Norwegian mathematician Niels Henrik Abel (1802--1829) was born in a family with many, where all got the basic education by the father.\, Abel got 1815 into the cathedral school of Kristiania (Oslo), where the mathematics teacher B. M. Holmboe soon sa ... Monday, 2009-6-22 http://planetmath.org/encyclopedia/MarkovAlgorithm.html CWoo ... A ... Markov algorithm is a variant of a rewriting system, invented by mathematician Andrey Andreevich Markov Jr. in 1960. Like a rewriting system, a Markov algorithm consists of an alphabet and a set of productions. Furthermore, rewriting is done by ... Monday, 2009-6-15 http://planetmath.org/encyclopedia/NumerationSystem.html CWoo A ... numeration system is a triple ... , where ... is a positive integer, Sigma is a non-empty alphabet, and d is a one-to-one function from Sigma to the set of non-negative integers ... . Order elements of Sigma so that their values are in increasing or ... Sunday, 2009-6-14 http://planetmath.org/encyclopedia/ArcLengthExample.html pahio The functions ... belong to the few real functions, the arc length of which are expressible in closed form (other ones are mentionned in the entry arc length of parabola). We calculate the arc length of the curve ... By the chain rule, we have ... Hence t ... Sunday, 2009-6-14 http://planetmath.org/encyclopedia/LinearBoundedAutomaton.html CWoo A ... linear bounded automaton , or LBA for short, is a restricted form of a non-deterministic Turing machine with a single tape and a single tape head, such that, given an input word on the tape, the tape head can only scan and rewrite symbols on the cel ... Friday, 2009-6-12 http://planetmath.org/encyclopedia/NearSquarePrime.html PrimeFan A ... near-square prime is a prime number p of the form ... , with n being any integer and ... also an integer. Since for any nonzero real number x it is always the case that ... , it doesn't matter if n is negative. ... |r|r|r|r|r|r|r|r|r|r|r|r|r| 5 & & ... Friday, 2009-6-12 http://planetmath.org/encyclopedia/ResiduesOfTangentAndCotangent.html pahio We will determine the residues of the tangent and the cotangent at their poles, which by the 9074 are SimplePole . By the rule in the entry coefficients of Laurent series, in a simple pole \, ... \, of f one has ... We get first ... All the poles of cotan ... Wednesday, 2009-6-10 http://planetmath.org/encyclopedia/LindenmayerSystem.html CWoo ... Lindenmayer systems, or L-systems for short, are a variant of general rewriting systems. Like a rewriting system, an L-system is also a language generator, where words are generated by applications of finite numbers of rewriting steps to some initial ... Wednesday, 2009-6-10 http://planetmath.org/encyclopedia/ClosedSublattice.html porton A sublattice K of a complete lattice L is a ... closed sublattice of L iff K contains the meet and the join of any its nonempty subset. Examples: Any complete sublattice is a closed sublattice. ... is a closed sublattice of ... . The set of rational numbe ... Tuesday, 2009-6-9 http://planetmath.org/encyclopedia/54RMP53.html milogardner Reading and translating the Rhind Mathematical Papyrus (RMP) and other hieratic texts dating to 2,000 BCE and 1500 BCE requires decoding keys. Following a well known adage, to read any good book, in any era, hieratic texts should be read at least http://t ... Saturday, 2009-6-6 http://planetmath.org/encyclopedia/GeneralizedSequentialMachine.html CWoo ... Generalized sequential machines are generalizations of finite state sequential machines. In a finite state machine ... , the next-state (delta) and output (lambda) functions work independently of one another in the sense that no next states affect the ... Friday, 2009-6-5 http://planetmath.org/encyclopedia/ArcLengthOfParabola.html pahio The parabola is one of the quite few plane curves, the arc length of which is expressible in closed form; other ones are line, Circle , semicubical parabola, catenary, tractrix, cycloid, clothoid, astroid, Nielsen's spiral, logarithmic spiral.\, Determini ... Wednesday, 2009-6-3 http://planetmath.org/encyclopedia/RydiasMathemagicMinute.html PrimeFan ... Rydia's Mathemagic Minute is a mini-game from ... Final Fantasy XIII in which Rydia presents the player a riddle in the form of four numbers, each in the range ... which the player must then employ in conjunction with a selection of basic arithmetic o ... Tuesday, 2009-6-2 http://planetmath.org/encyclopedia/SubgroupsOfFiniteCyclicGroup.html pahio Let n be the order of a finite cyclic group G.\, For every positive Divisibility m of n, there exists one and only one subgroup of order m of G.\, The group G has no other subgroups. ... Proof. \, If g is a generator of G and\, ... ,\, then ... generates ... Tuesday, 2009-6-2 http://planetmath.org/encyclopedia/MetalinearLanguage.html CWoo Recall that a linear grammar is a formal grammar ... whose productions are of the form ... , where A is a terminal symbol, and x is a word over Sigma, with at most one occurrence of a non-terminal symbol. The concept of a linear grammar can be generalized ... Monday, 2009-6-1