Letter R

R _________________________________________________________________________________________________

Rademacher function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radial distribution
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radial of a curve
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radian
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The circumference (PL) of a circle has the form C = 2pr, for radius, r, which sweeps out 360 ° in the circle. Hence, we can define: 1 radian = (360 °)/2p.

radical axis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radical ring
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radius
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The semidiameter of a circle.

radius of curvature
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radius vector
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radix approximation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radix complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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radix notation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Radon-Nikodyn theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ramanujan's sums
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ramsey's theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rand
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Computer jargon for operand. (PL rator.)

random function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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random number
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May be read at http://www.harcourt.com/dictionary /browse/19/
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random variable
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May be read at http://www.harcourt.com/dictionary /browse/19/
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random vector
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May be read at http://www.harcourt.com/dictionary /browse/19/
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range
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May be read at http://www.harcourt.com/dictionary /browse/19/
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range space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ratio
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rational
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rational function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rationalize
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rational number ("fraction")
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Vector (or 2-tuple) of integers: [i₁, i₂]; equivalently, vector of vector of natural numbers: [i₁, i₂] = [[n₁₁, n₁₂], [n₂₁, n₂₂]]. Has operational-relational rules derived from closure on defined quotients (PL). The equivalence rule invokes three equivalence classes (for integers, i, k): [ki, i] = [k, 1], integral vector; [i, ki] = [1, k], Egyptian vector; [i,k] (where integers i, k are coprime, q.v.) as rational vector. All three forms are hidden by fractional or nonfractional signs: [k, 1] -> k; [1,k] -> 1/k; [i, k] -> i/k, as shown by a bypass. (Fra Pacioli (1445-1514), who invented debit-credit bookkeeping thought fractions are unBiblical because a fraction less than 1 decreases when squared -- 1/2 x 1/2 = 1/4 < 1/2 -- and "The Bible says, 'Thou shalt multiply and increase.'")

rator
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Computer jargon for operator. (PL rand.)

ray (arithmetic)

An arithmetic system closed under multiplication ^[11].

ray
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Rayleigh-Ritz method
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May be read at http://www.harcourt.com/dictionary /browse/19/
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real axis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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real function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reallification
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May be read at http://www.harcourt.com/dictionary /browse/19/
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real number
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The usual explanation of the need for the real number system is geometric: to render arithmetical the diagonal of a unit square, that is, give meaning to "the square root of two". The arithmetical reason is that the real number system arises to render total one of the two inverses of exponentiation (PL). This is logarithm (usually treated as "something the cat drug in"): log_bx = p iff b^p = x. But logarithm is partial in naturals, integers, rationals. It can be rendered total only via "infinite vectors" of "Cauchy sequences of rationals", as shown in a bypass.
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real orthogonal group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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real part
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May be read at http://www.harcourt.com/dictionary /browse/19/
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real variable
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reciprocal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reciprocity
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reconstruction conjecture
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectangle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectangular
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectangular hyperbola
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectangular parallelepiped
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectifiable curve
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectilinear
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rectilinear generators
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rextrex
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Emulating the simplex, define: R_n , the n + 1 retrices, where R₀ is a point, R₁ is a unit line segment, R₂ is a unit square, R_e is a unit cube, R₄ is a unit tesseract (PL), etc.

recurrent function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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recursion formula
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reduced form
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reduced residue
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reducible configuration
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reducible polynomial
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reductio ad absurdum
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May be read at http://www.harcourt.com/dictionary /browse/19/
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refinement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reflection
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reflex angle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reflexive
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May be read at http://www.harcourt.com/dictionary /browse/19/
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region
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regula falsi
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regular
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regular embedding
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regular ideal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regular representation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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regular topological space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relatand
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That part of a relation which is invoked by a relator (PL).

relation
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Every relation has, as protype, the binary relation (PL), since, for example, the ternary relation of "Mary gave a present to Tom" can be formalized as [gave, [present, Tom] -- binary in a binary.

relation, equivalence
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A binary relation (PL) which is reflexive, symmetric, transitive, thus. (PL all three terms). Some of the best known equivalence relations are "equals in arithmetic", "congruence in geometry", "modular congruence in number theory", and "conjugacy among matrices". An equivalence relation, r, separates a set, S, into equivalence classes -- a condition denoted by S/R, labeled the quotient set -- homologous to species in biology and an antitonic process.

relator
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In this Dictionary, the invoker of a relation is labeled "relator" to match the term "operator". A relator invokes a relatand (to match "operand"). (PL functor and functand, involving function.

relative complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relative error
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relatively closed set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relatively compact
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relatively open set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relatively prime
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relative topography
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May be read at http://www.harcourt.com/dictionary /browse/19/
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relaxation method
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May be read at http://www.harcourt.com/dictionary /browse/19/
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remainder formula
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May be read at http://www.harcourt.com/dictionary /browse/19/
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remainder theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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removable discontinuity
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May be read at http://www.harcourt.com/dictionary /browse/19/
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removable singularity
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Renaissance mathematics
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Much evidence exists that advances in algebra and number theory attributed to Renaissance European mathematicians were discovered earlier by Islamic mathematicians. The mathematical and scientific history of this period needs extensive investigation. For example, Galileo (1564-1648) is widely credited with discovering "the law of falling bodies", whereas it was the discovery of the 13th century "Merton Scholars" of Merton College of Cambridge University, fellow students of Thomas Bradwardine (1290-1349), who extended the Eudoxian theory of proportions to the exponential function (PL) -- two of many historic events not taught in our schools and colleges.

renaming rule
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May be read at http://www.harcourt.com/dictionary /browse/19/

renormalization transformation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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repeating decimal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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repertory
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Set or collection of mathematical systems which are syntactically equivalent but semantically distinct. That is, changing the labels of its components (as a girl changes the costumes of a Barbie Doll) results in another mathematical system. Prototype: simple Pecking Order (PL) -> arithmetic factoring of "square-free" numbers -> statement logic -> complemented distributive lattice -> (computer) circuit logic table -> etc. A repertory consists of whatever is invariant under a group of repertorial-semantic transformations. In V. II, The Feynman Lectures on Physics, p. 3-6. ff, and 12-2, ff, existence is shown of a Physics repertory on Conservative Laws.

representation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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representation theory
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May be read at http://www.harcourt.com/dictionary /browse/19/
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residual set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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residual spectrum
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May be read at http://www.harcourt.com/dictionary /browse/19/
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residue
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May be read at http://www.harcourt.com/dictionary /browse/19/
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residue class
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May be read at http://www.harcourt.com/dictionary /browse/19/
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residue theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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resolution of a vector
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May be read at http://www.harcourt.com/dictionary /browse/19/
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resolution of the identity
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May be read at http://www.harcourt.com/dictionary /browse/19/
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resolvent set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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restriction
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May be read at http://www.harcourt.com/dictionary /browse/19/
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resultant
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reticular density
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May be read at http://www.harcourt.com/dictionary /browse/19/
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retract
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May be read at http://www.harcourt.com/dictionary /browse/19/
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reverse curve
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rhomboid
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rhombus
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ribbon
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riccati equation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ricci calculus
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ricci equations
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ricci's lemma
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ricci tensor
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann curvature tensor
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann hypothesis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemannian geometry
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemannian manifold
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann-Lebesgue lemma
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann mapping theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann method
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann sphere
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann-Stieltjes integral
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riemann surface
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May be read at http://www.harcourt.com/dictionary /browse/19/

Riesz-Fischer theorem
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Riesz representation theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riesz-Schauder theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Riesz theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/

right angle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right-continuous function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right-hand derivative
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right-handed coordinate system
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right section
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right strophoid
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May be read at http://www.harcourt.com/dictionary /browse/19/
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right triangle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ring
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ring of quotients
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ring of sets
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ring theory
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Robertson-Seymour theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Rodrigues' formula
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Rolle's theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Roman (Steiner) surface
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A quartic (PL) nonorientable (PL) surface, one of three generated by sewing a Möboid (PL) to a disk (Pl), the others being the cross-cap (PL) and Boy surface (PL), all three homeomorphic (PL) to the real projective plane (PL). The center point of the R. s. is an ordinary triple points: (&plsmn;1, 0, 0) = (0, &plsmn;1, 0) = (0, 0, &plsmn;1), and the six endpoints of the three lines of self-intersection are singlar pinch points (PL) or Whitney singularities (PL). The R. s. is essentially a fusion of six cross-caps (PL), containing a double infinity of conics. The R.s. can be given by the equation: x²y² + x²z² + y²z² = 2kxyz. For a homotopy between the R. s. and the Boy surface PL B. s..

root
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rooted tree
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May be read at http://www.harcourt.com/dictionary /browse/19/

root-finding algorithm
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May be read at http://www.harcourt.com/dictionary /browse/19/

root of an equation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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root of unity
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May be read at http://www.harcourt.com/dictionary /browse/19/
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root system
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May be read at http://www.harcourt.com/dictionary /browse/19/
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root vertex
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rose
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rotation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rotation of coordinate axes
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The point {x, y, z) transforms into the point (x' y', z') as follows: x' = xcosq + ysinq + z, y' = -xsinq + ycosq + z, z = z. This is easily explained by its matrix (PL) form, whose determinant (PL) equals one, hence, "acts as the identity transformation". (PL vector, orthogonal.)

rotation group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Rouché's theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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roulette
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rounding
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May be read at http://www.harcourt.com/dictionary /browse/19/
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rounding error
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May be read at http://www.harcourt.com/dictionary /browse/19/
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row
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May be read at http://www.harcourt.com/dictionary /browse/19/
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row operations
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May be read at http://www.harcourt.com/dictionary /browse/19/
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row rank
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May be read at http://www.harcourt.com/dictionary /browse/19/
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row space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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row vector
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May be read at http://www.harcourt.com/dictionary /browse/19/
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RSA encryption
.
A public-key encryption, using factor of product of two primes, n = pq as the trapdoor one-way function. Define private and public keys, d,e s. t. you have the congruence (PL), de 1(mod f(n)) with public key e and totient function (PL) f(n) coprime(have only unity as common factor). The message-sender converts the message to number, M, makes public n, e and sends E M^e (mod m). In decoding, the receiver (knowing d computes E ^d (M^e)^d M^ed M^{N f(n)+1} M (mod n), for integer N. To "crack" (decode) the code, d must be discovered, requiring factoring of n, since f = (p - 1)(q - 1). The sender should arrange that p ± 1, q ± 1 are divisible by large primes -- else, Pollard p-1 factoring (PL) or Williams p+1 factoring might easily factor n. Also desirable to choose f(f(n)) large, and divisble by large primes. Repeated use of this encription may lead to cracking if a unit of Z/f(n)Z has small order, where Z/sZ is the ring of integers (PL) between 0 and s - 1 under addition and multiplication mod(s. (However, Meijer in 1996 suggested ways of avoiding this.) Under RSA, the sender's identity can be validated w/o revealing his/her private code.

ruled surface
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Runge-Kutta method
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Russell's paradox
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May be read at http://www.harcourt.com/dictionary /browse/19/
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