http://planetmath.org/?op=enlist&mode=created http://planetmath.org/encyclopedia/OrthogonalityOfLaguerrePolynomials.html pahio We use the definition of Laguerre polynomials ... via their RodriguesFormula ... The polynomials (1) themselves are not orthogonal to each other, but the expressions ... \, ( ... ) are orthogonal on the interval from 0 to infty, i.e. the polynomials are o ... Saturday, 2009-11-21 http://planetmath.org/encyclopedia/MuOperator.html CWoo Let a property on non-negative integers be given. Informally, the mu-operator looks for the smallest number satisfying the given property. The mu-operator is also known as the (unbounded) minimization operator or the (unbounded) search operator. Formally, ... Saturday, 2009-11-21 http://planetmath.org/encyclopedia/ExamplesOfPrimitiveRecursivePredicates.html CWoo In this entry, we give some examples of primitive recursive predicates. In particular, we will show that the predicate ... is primitive recursive. In the examples, we use ... to indicate a predicate (over the natural numbers ... ). First, two very simple ... Tuesday, 2009-11-17 http://planetmath.org/encyclopedia/VectorProjection.html pahio The principle used in the projection of line segment a line, which results a line segment, may be extended to concern the projection of a vector ... on another non-zero vector ... , resulting a vector. This projection vector, the so-called ... vector proj ... Tuesday, 2009-11-17 http://planetmath.org/encyclopedia/BoundedMaximization.html CWoo The proof involved in showing that functions obtained via BoundedMinimization primitive recursive functions are themselves primitive recursive can be used to show the primitive recursiveness of another family of functions derived from existing primitive r ... Sunday, 2009-11-15 http://planetmath.org/encyclopedia/PropertiesOfOrthogonalPolynomials.html pahio A countable ... system of orthogonal polynomials ... on an interval \, ... ,\, where a inner product of two functions ... is defined with respect to a weight function ... , satisfies the OrthogonalVectors ... One also requires that ... Such a system (1) m ... Saturday, 2009-11-14 http://planetmath.org/encyclopedia/KazimierzZarankiewicz.html ZZZTopologist Kazimierz Zarankiewicz (1902-1959) Polish topologist. He taught at the Society for Scientific Courses in Warsaw its first year in 1906. His work confirmed a result by Urysohn regarding hereditarily locally connected continua. The Zarankiewicz problem conc ... Saturday, 2009-11-14 http://planetmath.org/encyclopedia/OrderlyNumber.html PrimeFan An ... orderly number is an integer n such that there exists at least one other integer k such that each divisor of n from 1 to ... (with ... being the number of divisors function) satisfies the congruences ... , ... through ... . For example, 20 is an or ... Friday, 2009-11-13 http://planetmath.org/encyclopedia/GeneratingFunctionOfHermitePolynomials.html pahio We start from the definition of Hermite polynomials via their 11983 ... The consequence ... of 1150 allows to write (1) as the complex integral ... where C is any contour around the point z and the direction is anticlockwise.\, The 11373 \, ... \, here yi ... Thursday, 2009-11-12 http://planetmath.org/encyclopedia/ImportanceOfPrimitiveRecursion.html CWoo In this entry, we show that the operation of primitive recursion is an important ingredient in the definition of primitive recursive functions. Very basic arithmetic functions, such as addition, depend on primitive recursion in an essential way. One can n ... Thursday, 2009-11-12 http://planetmath.org/encyclopedia/BoundedMinimization.html CWoo One useful way of generating more primitive recursive functions from existing ones is through what is known as ... bounded summation and ... bounded product . Given a primitive recursive function ... , define two functions ... as follows: for ... and ... ... Tuesday, 2009-11-10 http://planetmath.org/encyclopedia/AreaBoundedByArcAndTwoLines.html pahio Let\, ... \, be the equation of a continuous curve in polar coordinates and 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 the ... 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 ... To deduce such a formula the easy way, one starts from the generating function of the reciprocal central binomial coefficients and having in ... 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