http://planetmath.org/?op=enlist&mode=created http://planetmath.org/encyclopedia/AreaBoundedByArcAndTwoLines.html pahio Let\, ... \, be the equation of a continuous curve in polar coordinates.\, Let A be the area of the planar region by the curve and the line segments from the origin to two points of the curve corresponding the polar angles alpha and beta ( ... ).\, Then t ... Saturday, 2009-11-7 http://planetmath.org/encyclopedia/GeneratingFunctionForTheReciprocalCatalanNumbers.html juanman The series ... whose coefficients are the reciprocal of the Catalan numbers ... , has as a generating function ... Saturday, 2009-11-7 http://planetmath.org/encyclopedia/GlobalDimensionOfASubring.html joking Let S be a ring with identity and ... a subring, such that R is contained in the center of S. In this case S is a (left) R-module via multiplication. Throughout by modules we will understand left modules and by global dimension we will understand left glo ... Friday, 2009-11-6 http://planetmath.org/encyclopedia/ExamplesOfPrimitiveRecursiveFunctions.html CWoo Starting from the simplest primitive recursive functions, we can build more complicated primitive recursive functions by functional composition and primitive recursion. Below are some examples, starting from the simpler to the more complicated: ... simple ... Thursday, 2009-11-5 http://planetmath.org/encyclopedia/CircularSegment.html pahio A chord of a circle the corresponding disk into two ... circular segments .\, The perimetre of a circular segment consists thus of the chord (c) and a circular arc (a). The magnitude r of the radius of circle and the magnitude alpha of a central angle nat ... Thursday, 2009-11-5 http://planetmath.org/encyclopedia/DerivationOfGeneratingFunctionForTheReciprocalCentralBinomialCoefficients.html rm50 According to the article, the ordinary generating function for ... is ... To see this, let ... , and ... its ordinary generating function. Then ... Thus ... so that ... A little algebra gives ... so that ... and, collecting terms, ... We now have a first- ... Tuesday, 2009-11-3 http://planetmath.org/encyclopedia/ExactSequencesForModulesWithFiniteProjectiveDimension.html joking ... Proposition. Let R be a ring and M be a (left) R-module, such that ... . If ... is an exact sequence of R-modules, such that each ... is projective, then K is projective. ... Proof. Since ... , then there exists exact sequence of R-modules ... Note th ... Tuesday, 2009-11-3 http://planetmath.org/encyclopedia/RecursiveFunctionIsURMComputable.html CWoo ... Every recursive function is URM-computable function. ... The proof can be broken down in several simple steps. ... The zero function, the successor function, and the projection functions are URM-computable. ... The zero function is computed by ... , t ... Tuesday, 2009-11-3 http://planetmath.org/encyclopedia/InverseLaplaceTransformOfMeromorphicFunction.html pahio The complex integral appearing in Mellin's inverse formula is determined usually by utilising Cauchy residue theorem.\, In the applications, one has most often to find the inverse Laplace transform of a function which is analytic in the whole complex plan ... Monday, 2009-11-2 http://planetmath.org/encyclopedia/DivisibilityOfNineNumbers.html pahio We know that 9 is divisible by the prime number 3 and that 99 by another prime number 11.\, If we study the divisibility other ... by primes, we can see that 999 is divisible by a greater prime number 37 and 9999 by 101 which also is a prime, and so on.\, ... Friday, 2009-10-30 http://planetmath.org/encyclopedia/TableOfArithmeticDerivativesFor1N100.html PrimeFan ... |r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r| n & ... & n & ... & n & ... & n & ... & n & ... & n & ... & n & ... & n & ... & n & ... & n & ... 0 & 0 & 10 & 7 & 20 & 24 & 30 & 31 & 40 & 68 & 50 & 45 & 60 & 92 & 70 & 59 & 80 & 176 & 90 & 123 1 & 0 & 11 & 1 ... Thursday, 2009-10-29 http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html milogardner The Rhind Mathematical Papyrus (RMP), a 1650 BCE text, began with 51 2/n table members. The table included no calculations. Ahmes, the RMP scribe, discussed 87 problems. The problems included sparse and fragmented initial, intermediate, final and proof sh ... Thursday, 2009-10-22 http://planetmath.org/encyclopedia/CayleysTheoremForSemigroups.html Ziosilvio Let X be a set. We can define on ... , the set of functions from X to itself, a structure of semigroup by putting ... . Such semigroup is actually a monoid, whose identity element is the identity function of X. ... [Cayley's theorem for semigroups] ... Fo ... Wednesday, 2009-10-21 http://planetmath.org/encyclopedia/ExamplesOfRadicalsOfIdealsInCommutativeRings.html joking Let R be a commutative ring. Recall, that ideals ... in R are called ... coprime iff ... . It can be shown, that if ... are coprime, then ... . Elements ... are called ... pairwise coprime iff ... for ... . It follows by induction, that for pairwise copri ... Tuesday, 2009-10-20 http://planetmath.org/encyclopedia/IdealsWithMaximalRadicalsArePrimary.html joking ... Proposition. Assume that R is a commutative ring and ... is an ideal, such that the radical ... of I is a maximal ideal. Then I is a primary ideal. ... Proof. We will show, that every zero divisor in ... is nilpotent (please, see parent object for det ... Tuesday, 2009-10-20 http://planetmath.org/encyclopedia/CharacetrizationOfPrimaryIdeals.html joking ... Proposition. Let R be a commutative ring and ... an ideal. Then I is primary if and only if every zero divisor in ... is nilpotent. ... Proof. ,,Rightarrow'' Assume, that we have ... such that ... is a zero divisor in ... . In particular ... and there ... Tuesday, 2009-10-20 http://planetmath.org/encyclopedia/EveryPIDIsAUFDAlternativeProof.html joking ... Proposition. If R is a principal ideal domain, then R is a unique factorization domain. ... Proof. Recall, that due to Kaplansky Theorem (see EquivalentDefinitionsForUFD for details) it is enough to show that every nonzero prime ideal in R contains a ... Monday, 2009-10-19 http://planetmath.org/encyclopedia/IdealsContainedInAUnionOfRadicalIdeals.html joking Let R be a commutative ring and ... an ideal. Recall that ... the radical of I is defined as ... It can be shown, that ... is again an ideal and ... . Let ... Of course ... (because I is contained in at least one maximal ideal) and it can be shown, that . ... Monday, 2009-10-19 http://planetmath.org/encyclopedia/UniformAlgebra.html karstenb ... A commutative, unital Banach algebra ... is called ... uniform Banach algebra (for short: uB algebra) if for all ... we have ... In what follows we will show that the Gelfand transform ... of a commutative, unital Banach algebra ... is an isometry if ... Sunday, 2009-10-18 http://planetmath.org/encyclopedia/ExampleOfArtinianModuleWhichIsNotNoetherian.html joking It is well known, that left (right) Artinian ring is left (right) Noetherian (Akizuki-Hopkins-Levitzki theorem). We will show that this no longer holds for modules. Let ... be the ring of integers and ... the field of rationals. Let ... be a prime number ... Sunday, 2009-10-18