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Galileo's paradox
.
Italian mathematician, Galileo Galilei (1564-1642), discovered that a proper part or subset of a set can be corresponded one-one with all of the set. This cannot happen with a finite set. (Try to put part of the fingers of your hands one-one with all of them.) Galileo's discovery: the fact that you can always double a natural or counting number -- resulting in an even number -- means that the even numbers correspond one-one with all the counting numbers, even and odd. And the evens constitute a proper part or subset of all the numbers. Modern mathematicians are not shocked by this, and use it as a definition of an infinite set, that is, a set which can one-one correspond to a proper subset.
Galois correspondence
.
A correspondemce between lattices (PL), each a partial ordering (poset) (PL) s. t. every element has a least upper bound (labeled "join"), , and a greatest lower bound ("meet"). . Axiomatically there is assigned a (not operationally attained) single least element, usually labeled its "0"; also assigned, a (not operationally attained) single greatest element , usually labeled its "1". (In this Dictionary, perhaps uniquely, an indicator table (of zeros and ones) (PL) is assigned to a lattice, conflicting with the standard "0" and "1" labels just cited, hence, these are herein labeled "MIN" and "MAX", to distingish from each sublattice min and max.) Ranging from its 0 or MIN to its 1 or MAX, a lattice has a metric (PL) known as "rank". Every lattice has an associated dual lattice invoked by interchanging meet and join, so that any increase in a lattice is associated with a decrease in its dual lattice, inducing a dual isomorphism (PL fundamental theorem of Galois theory ), so that an inclusion in one lattice corresponds uniquely to an included in the other lattice. This "dual isomorphism" is the Galois correspondence. (When the above partial ordering is changed to a simple or total ordering (PL), the Galois correspondence becomes an antitone (PL), which may be inherent in every physical process, PL antitonic hypothesis.)
Galois extension field
.
Given field , F, and a splitting field for collection of separable polynomials, K, these conditions are equivalent: (a) when K is a finite extension, only one separable polynomial is necessary; (b) the field automorphisms (PL) of K that fix F do not fix any intermediate field between them; (c) every irreducible polynomial (PL) over F having a root in (separable) K factors into linear factor in K; (d) given F of field F, a field automorphism, s: F F for which s(K) = K must fix K . An extension is not a Galois extension if, either it is not normal (PL), or it is not separable.
Galois field
.
A finite field (PL) whose order (PL) is prime or power of a prime. For each of latter, there exists exactly one (up to an isomorphism, PL) finite field, GF (Pn). GF(p), the prime field of order p, is the field of residue classes modulo p, of elements denoted 0, 1, 2, ..., p - 1. Having multiplicative inverses, this type of field is comparable to the rational numbers. (Galois fields are used for economical coding of signals between satellites and earth.)
Galois group
.
Given L as field extension of L, denoted L/K , and G as the set of automorphisms of L/K -- i.e., the set of automorphisms, s of L s.t. s(x) = x -- so that K is fixed. Then G is a group of transformations of L, known as the Galois group of L/K. Thus, The Galois group of complex numbers as extension of the reals, (C/R), consists only of the identity and complex conjugation (PL), both sending a real number into itself.
Galois indicator-signal
.
Galois' great "trick" was using the solvable group (PL) as indicator (PL) input for the solvable equation output. Group-failure explained the equation-failure (nonsolvability by radicals). An indicator under linguistic and physical control is a signal (PL). Development of Galois theory provides linguistic control; attaining algorithmic and programmable facility provides physical control. (PL Lie indicator-signal.)
Galois theory
.
Given 1-1 correspondence between subgroups (PL) and subfields (PL) s. t. G(E(G')) = G', E(G(E')) = E', then E has a Galois theory.
game
.
A conflict between two or more opponents who proceed by formal rules. Pl game theory.
game theory
.
a branch of logico- mathematics dealing with analysis of games.
gamma (statistical) distribution
.
Given a Poisson (statistical) distribution (PL) with exchange rate l, the distribution function for the waiting times until the kth event is D(x) = P(X x) = 1 - D(X > x) = 1 - [G(h, xl )/G(h), for x e [0, ), where G(x) is a complete gamma function and G( a, x) is an imcomplete gamma function. Differentiating for the probability distribution function, P(x) = D'(x) = [g (gx)h-1/(h - 1)!]e-hx. Setting a h (not necessarily integral and q 1/l for time between changes. Then the above p.d.f. becomes P(x) = [xa - 1ex/q]/G(a)q a. The characteristic function (PL) is f(t) = (1 - itq)-a . The mean is m = a q; the variance is s2 = aq2.
Gauss-Bonnet formula
.
Taken as a consequence of the Poincaré-Hopf index theorem (PL), the Gauss map of a orientable 3-D surface is half of the Euler characteristic (PL) for the surface: MKdA = 2pc(M) - Sa i - dMk gds, for compact M.
Gauss constant
The reciprocal of the arithmetic-geometric mean (PL) of 1, 2 (the side and diagonal of a unit square): G 1/M(1, 2) = (1/2p)101/[(1 - x2)1/2]dx = (1/2 p) ½p01/[(1 + sin2q )1/2]dq = (2/p K)(1/2) = 1/(2p)3/2[G(1/4)]2 = 0,83462684167....
Gauss cyclotomic formula
For prime , p > 3, 4[(xp - yp)/(x - y)] = R2(x, y) - (-1)½(p - 1)pS2(x, y), where R(x, y), S(x, y) are homogeneous polynomials (PL) with integral coefficients. Gauss gave the coefficients of R, S up to p = 23. (PL cyclotomic equation.)
Gauss digamma function
For f0(p/q) = - g - ln(2q) - ½pcot(p p/q) + 2S [(q/2)-1]k=1 cos(2ppk/2)ln[sin(ppk/2)], 0 < p < q, where g is the Euler-Mascheroni constant (PL). (PL digamma function.)
Gauss hypergeometric function
.
PL hypergeometric function.
Gaussian elimination
.
A method for solving a system of linear equations, written in matrix form as Mx = c. Proceed by writing M in augmented form, [M|c], using that column matrix for distinguishing matrix rows. Next, perform elementary row, column operations (PL) to reduce the augmented matrix to the upper triangular form, with all zeros below diagonal. Solve the kth row for xk, substituting solution back into the equation of the (k = 1)st row to obtain a solution for xk-1, etc., according to the formula (with hyphonated terms from the augmented matrix), xi = 1/a'ii(b'i - Skj=k+1a'ijxj) .
Gaussian integer
.
A complex number , a + b-1, for integers a, b. Gaussian integers belong to the imaginary quadratic field, Q(-1), forming a ring , Z(i), closed under sum, difference, product, and division provided the quotient is integral.
Gaussian polynomial
.
PL q-binomial .
Gaussian quadrature
.
Best numerical estimate for an integral by selecting optimal abscisssas, xi , at which the integrand, f(x). The fundamental theorem of Gaussian quadrature: the optimal abscissas of the m -point Gaussian quadrature formulas are roots of the orthogonal polynomial for the same intgration interval and weight function. (Weights for abscissas are computed by a Lagrange interpolation polynomial (PL).) Gaussian quadrature is optimal because it exactly fits all polynomials up to degree 2m.
Gauss-Jordan elimination
.
Starts out as in Gaussian elmination (PL). But, in the augmented matrix, the identity matrix is inserted instead of the c-column matrx, and elementary row, column operations (PL) are performed to transform the Mij, which transforms the original identity matrix into terms used in solving the system.
..
Gauss map
.
PL Gauss-Bonnet formula .
Gauss mean value theorem
.
Given an analytic function, f(x), |x - a| < R, then f(x) = 1/2pf(x + reiq)dq, 0 < r < R. (PL mean value theorem.)
Gegenbauer (hyperspherical) differential equation
.
The 2nd order ordinary d. e., (1 - x2y'' + 2 (m + 1)xy' + (v - m) (v + m + 1)y = 0, with solution in terms of an associated Legendre polynomials of the first kind and second kind. Among variations on this equation is the "Gegenbauer ultaspherical d. e.", (1 - x2)y'' - (2m + 1)xy' + v(v + 2m)y = 0, with solution also in terms of the Legendre polynomials. However, if m is integral, that cited solution doesn't work, but requires the Gegenbauer polynomials (PL) as solutions.
Gegenbauer polynomials
.
Solutions for the Gegenbauer d. e. (PL) for m integral and m is integral and l < ½, as generalizations of the associated Legendre polynomials (solutions of the standard Gegenbauer d. e.) for (n + 2)-space. As formulated by Szego, using the ultraspherical polynomials (PL), P(l)n(x) , the G. ps. are: C(l)n (x) = {[G( l + ½)][G (n + ½)]/[G(2l) G(n + l) + ½]}P(l - ½, l - ½)n(x). These can also be stated in terms of the hypergeometric functions, PL.
generalized function
.
PL distribution .
generative arithmetic
.
The arithmetic of natural numbers is generated recursively from zero by the successor function (PL). The arithmetic of integers is generated as vectors of naturals (PL). The arithmetic of rational numbers is generated as vectors of integers (PL). The arithmetic of real numbers is generated as infinite vectors ("decimal numbers") derived as limits of Cauchy sequences of rationals (PL). The arithmetic of complex numbers is generated as a vector of real numbers (PL), providing the basis for the unending sequence of multivectors of the arithmetic of Clifford numbers (PL). (A generative presentation explains the rules of arithmetic, whereas this is given by fiat in an axiomatic one (PL). Also generative arithmetic can be interactively discovered, which is not the case with an axiomatic presentation.)
generative trigonometry
.
  1. set up Cartesian X-Y axes;
  2. plot circle at origin O, of radius r = 1.
  3. draw a radial vector to some point above X-axis, labeling this point as (x,y);
  4. the projection of this point onto the X-axis creates a right triangle (PL) explicating the Pythagorean formula, x2 + y2 = 1, and radial makes angle q with respect to X-axis;
  5. relabel the above point as (cos q, sin q);
  6. identify cos q as right triangle's opposite side over diagonal, and sin q as its opposite side over diagonal;
  7. then x2 + y2 = 1 becomes the fundamental trigonometric identity, cos2q + sin2 q = 1;
  8. all other trigonometric identities follow from this fundamental one.
generator (of group)
.
Member of a cyclic group (PL) whose powers generate the entire group.
generic character
.
Given form (PL), Q, its generic character, ci(Q), consists of the values of ci(m) where (m, 2d) 1 and Q represent m: c1(Q), c2(Q), ..., cr(Q). The characters apply to the class of properly equivalent forms as they represent the same numbers. PL genus (form).
genus (curve)
.
A Plücker characteristic (PL): p ½(n - 1)(n - 2) - (d + k) = ½(m - 1)(m - 2) - (t + I), where n, d , k, m, t, i are, respectively, order, number of nodes, no. of cusps, class, no. of bitangents, no. of inflection points. (PL Riemann curve theorem.)
genus (form)
.
Given forms, Q, whose generic characters (PL), x (Q) equal some designated array of signs, e1 ..., er; ei = 1, -1; Pri=1 ei = 1, There are 2r-1 possible arrays, with r the number of possible divisors of a field discriminant, d, s. t. the number of forms per array is the genus of the forms. A form for which all ei = 1 is a principal genus of forms. Each genus is a collection of equivalence classes (PL). (PL the fundamental theorem of genera, generic character.)
genus (graph)
.
The minimum number of handles needed for a plane to embed a given graph without crossings.
genus (knot)
.
The least geometric genus or surface genus (PL) of a Seifert surface (PL) for a given knot. The unknot is the only knot with genus zero.
genus (surface)
.
PL geometric genus .
genus theorem
.
The Diophantine equation (PL), x2 + y2 = p, as aprime solution iff p = 2 or p 1 (mod 4) -- uniquely, except for sign-change. This theorem connects with the quadratic reciprocity theorem (PL), generalizing to the quartic reciprocity theorem (PL).
geobias
.
Since geometry was one of the first fields of mathematics to be developed, with useful applications in surveying, engineering, etc., and developed impressive axiomatization, geometry acquired a hallowed curricular status. (American mathematician, Edward Kasner said that he found it easier to teach topology to children since "they haven't been brainwashed by geometry".) Geobias acts to the advantage of males, to disadvantage of females, in our society since males are hypothesized as evolving a superior spatial sense, whereas females are seem superior in language, which would make them "biased" for algebra if it were properly taught. Geobias creates bias in standardized tests, to the disadvantage of students (especially girls), disadvantage of mathematized science, disadvantage of society.
geodesic
.
A locally length-minimizing curve; the path of a nonaccelerating particle. Planar: straight lines. Spherical: great circles. Generally: Riemann metrical (PL). For surface, given parametrically as x = x(u,v), y = y(u,v), z = z(u,v), the geodesic derives
from minimizing the arc length L du = (dx2 + dy2 + dz2)½. Rewriting, u' Dvu, v' D uv, P (Dux)2 + (Duy)2 + (Duz)2, Q DuxDvx + Du yDvy + DuzDvz, R (Dvx) 2 + (Dvy)2 + (Dvz)2, L = (P + 2Qv' + Rv'2)½du = (P + 2Qu' + Ru'2) ½dv.
geometric dual graph
.
PL dual graph .
geometric genus
.
A topological invariant (PL) of a surface (PL): maximum number of nonintersecting simple closed curves which can be drawn on a surface without separating it; equivalently, number of holes in the surface. Denoted g, the g. g. is related to the Euler characteristic (PL) for a surfaceeby x = 2 - 2g .
geometric mean
.
Given a set, S, of numbers of cardinality, c: (1) multiply the numbers of S; (2) compute the cth root of this product; (3) declare this root to be the geometric mean of S. In a geometric progression (PL), its "middle" is the geometric mean of the set. The bypass (PL) for this computation is to compute the arithmetic mean of the logarithm (PL) of each number in S; the cologarithm (PL) is the geometric mean of S. Any extensive measure whose "growth" (as in "the law of cooling') is similar to that of compound interest (PL) is represented by the geometric mean.
geometric progression
Given an initial term, a, and a (nonzero) multiplier, g, you recursively multiply for a geometric sequence: a, ag, ag2, ag3, ag4, ..., agn; and add terms to obtain a geometric progression: a + ag + ag2 + ag2 + ag3 + ... + agn. By an antitonic (PL) proof, the sum of such a geometric progression is given as S = (a - gn+1)/(1 - g). For a = 1 this becomes a, ag, ag2, ag3, ag4, ..., agn; and add terms to obtain a geometric progression: 1 + g + g2 + g2 + g3 + ... + gn with sum, S = (1 - gn+1)/(1 - g) = 1/1 - g + gn+1/(1 - g). For 0 < g < 1, the term gn+1/(1 - g) monotonically decreases, approaching zero, so that the sum monotonically approaches S = 1. (This provides a solution for the ancient Zeno paradox of "Achilles and The Tortoise" -- PL.) This modeled the notion of limit (PL) when the notion of geometric sequence (above) is generalized to Cauchy sequence (PL). (As noted in real numbers, adjoining the transfinitary notion of limit operator to the finitary operators of arithmetic results in the real number system and its arithmetic).
geometric ratio.
May be read at http://www.harcourt.com/dictionary /browse/19/
.
geometry
.
A topology (PL) with a metric (PL) whose various forms yield a Euclidean geometry (PL) or a non-Euclidean geometry (PL), etc. (Was geometry initiated by prehistoric women?
Gibbs' phase rule
.
D + P - C = 2, where D denotes number of degrees of freedom of thermodynamic system, P denotes number of phases of thermodynamic system, C denotes number of components of thermodynamic system . (This is, PL, homologous to Euler's polyhedral formula: V + F - E = 2, where V denotes number of polyhedral vertices, F denotes numberof polyhedral faces, E denotes number of polyhedral edges. HOMOLOGY of Euler to Gibbs: V : D :: F : P :: E : C, that is, V D, F P , E C, and EULER'S RULE GIBB'S RULE. ASSIGNMENT: POINCARÉ generalized Euler's formula for topology to allow for "holes" and such. Is this homologous to any thermodynamic aspect?)
Gibbs phenomenon
.
Overshoot of Fourier series (PL) and other eigenfunctional series (PL) at simple discontinuities, removable by the Lanzcos sigma factor (PL).
girth
.
Given a cycle (PL) in a graph (PL), girth is the length of its longest cycle. (An acyclic graph is considered to have infinite girth.)
gnomon
.
A unit ("building-block") of a generated pattern, arising in the (digital) geometric theory of numbers created 2500 years ago by Pythagoras (c. 580-496 BC). (PL bottle-cap geometry.) The notion of recursion -- PL -- is implicit in gnomon. Thus, the odd number is a gnomon of a triangular number pattern. (The term also referred to a vertical metal triangle or pin on a sundial, whose projected shadow is an indicator of the time of day. The term also involves knowledge, as in the suffix, -gnomy: for example, physiognomy.) Using gnomon as a "building-block" of numbers, Pythagoras developed the geometric theory of numbers, giving us such labels as "squares" for 4, 9, 16, 25, etc., and "cubes" for 8, 27, 64, 125, etc., because he created these numbers as 2-D squares of dots or 3-D squares of dots. Thus, Pythagoras showed that the odd number is also the gnomon of the square, since recursive addition of odd numbers creates the sequence of square numbers: 1 = 1 x 1, 4 = 2 x 2, 9 = 3 x 3, 16 = 4 x 4, 25 = 5 x 5, etc., -- constructed by adding the successive odd numbers, 1, 3, 5, 7, 9, etc. Behold: 1; 1 + 3 = 4 = 2 x 2; 1 + 3 + 5 = 9 = 3 x 3; 1+ 3 + 5 + 7 = 16 = 4 x 4; 1 + 3 + 5 + 7 + 9 = 25 = 5 x 5; etc. Children can easily be shown that, given a square of equal rows and columns of dots or bottle-caps or marbles or blocks, the adjunction of one more row and one more column, together with their corner closure, results in the next larger square. Once children have been taught this method of additive construction of number-squares, the "Kierkegard kickback" of recursive inversion-by-subtraction of odd numbers can be applied to extract square roots. Thus, given 36 = 6 x 6: 36 - 1 = 35; 35 - 3 = 32; 32 - 5 = 27; 27 - 7 = 20; 20 - 9 = 11; 11 - 11 = 0. Question: How many odd numbers were used to reduce 36 to 0? Answer: Odd numbers, 1, 3, 5, 7, 9, 11 -- 6 odd numbers. Hence, the square root of 36 is 6. (There's an easy trick that simplifies the work for any "whole number". Given 1225 = 35 x 35, instead of peforming 35 subtractions to discover the square of 1225, this trick requires only 3 + 5 = 8 subtractions to elicit the answer, 35. And determine the square root of 2 to 5 decimal places (1.41421) by 1 + 4 + 1 + 4 + 2 + 1 = 13 subtractions.) As to the loconek (PL) of the sun-dial, Pythagoras built geometric models of numbers whose gnomon is 1 or 2 or 3 or 4, etc. That is, the numbers 1, 2, 3, 4, etc., are built from initiator 1 by the gnomon 1: 1; 1 + 1 = 2; 2 + 1 = 3; 3 + 1 = 4; etc. The numbers 1, 3, 5, 7, etc., are built from initiator 1 by the gnomon 2: 1; 1 + 2 = 3; 3 + 2 = 5; 5 + 2 = 7; etc. The numbers 1, 4, 7, 10, 13, etc., are built from initator 1 by the gnomon 3: 1; 1 + 3 = 4; 4 + 3 = 7; 7 + 3 = 10; 10 + 3 = 13; etc. The numbers 1, 5, 9, 14, 19, 24, etc., are built from initiator 1 by the gnomon 4: 1; 1 + 4 = 5; 5 + 4 = 9; 9 + 4 = 13; etc. And so on for other such constructions. Pythagoras modeled such numbers by dots fanning out in a triangular segment. The model for 1, 2, 3, 4, etc. -- with gnomon of 1 -- occupies a single triangular segment. The model for 1, 3, 5, 7, etc. -- with gnomon of 2 -- occupies a triangular segment bisected into 2 triangular subsegments. The model for 1, 4, 7, 10, etc. -- with gnomon of 3 -- occupies a triangular segment trisected into 3 triangular subsegments. Etc. When combined, these begin to sweep out a circular sector resembling the face of a sun-dial. This was the precursor of sequences of numbers built by successively adding or subtracting the same number (arithmetic progressions), giving gave rise to that form of average we know as the arithmetic mean -- the one most people call "THE AVERAGE", although there are many such. These progressions appear in Euclid's Elements of Geometry as line segments, the "analogic" version the Pythagorean "digital" form. And the basic idea implicit in such constructions invoked the most powerful of all forms of proof, namely,mathematical induction. (Warning! This should be called proof by recursion -- connecting with definition by recursion. This "misnaming" not only disconnects it from the recursive process, but gives rise to confusion with the use of "induction" in logic, a sometimes questionable form of reasoning, which generalizes from a few cases.) An important hiconek -- PL-- of gnomonics occurs in the science of crystallography, which explains beautiful jewelry, the molecular structure of pharmaceuticals and other chemical achievements, the creation of synthetic insulin, and the transistor (as noted in the gnomon-table.) However, the foundational philosophy of Platonism (PL) bans those gnomons known as "atoms") from set theory (PL), since their presence precludes the two most powerful nonconstructive proofs, namely, proof-by-contradiction and the axiom of choice (PL). By so doing, Platonists preclude generative competition with the axiomatics they support.
Gödel number
.
A coding of terms in logic used in the proof of Gödel's incompleteness theorem (PL). Each essential term is assigned a power of a prime number s.t. concatenation of terms yields the product of the coded prime powers. This appeals to the fundamental theorem of arithmetic, whereby an integer can be factored into prime factors uniquely. (PL frinteger.)
.
Gödel's incompleteness theorem
.
First Incompleteness Theorem: All consistent axiomatic formulations of number theory include undecidable propositions. Second Incompleteness Theorem: If number theory is conistent, then proof of this is does not exist via methods of first-order predicate logic.
golden mean
.
Given a line segment, of length 1 + f with the point f (the "golden mean") in the segment chosen s. t. we have the golden ratio, f/(1 + f) = 1/f. This yields the quadratic equation: f2 - f - 1 = 0, with solution, f = (1 ± 5)/2 = 1.618033988.... This is supposedly used in many works of art and architecture. Given the regular (even-sided) pentagon; connect each vertex to its two opposite vertices; these segments intersect at the golden mean.
golden ratio
.
PL golden mean.
golden rectangle
.
One with sides 1:f (PL golden mean).
googol
.
A googol is a large number, written as 1 followed by 100 zeros: 10100. Both concept and label are due to the grandson of eminent American mathematician, Edward Kasner (x-y). There isn't a googol of anything in the universe! (Experts estimate that the number of protons in the universe approximate 1050+, so the others would limit the total to less than 1060. However, it is easy to specify a supergoogol of independent choices.
graceful graph
.
Without loops (PL) or multiple edges, its edges are labeled according to the absolute difference between node values.
gradian (grade, gon)
.
Assign 400 gradians to a circle's circumgerence, so a right angle measures 100 gradians.
graded algebra
.
A graded module (PL), M, has a degree=preserving map, f:M X M M, s.t. (M, f) is a graded algebra.
graded module
.
The decomposition of a module (PL) into a direct-sum (PL) of (PL) whose index set (PL) is "the grading set". Graded modules arise naturally in homology theory (PL). Thus, for every integer, i, exists an ith homology group of a space, Hi(X) s.t. the direct sum of these for all i yields the total homology of the space, making this a module graded over the integers.
graded ring
.
A graded algebra (PL) over the integers (PL). The cohomology (PL) of space is a graded ring.
gradient
.
The gradient derivative is derived as an inner product (PL) involving unit vectors (PL) and direction cosines (PL) under the listing of directional derivative (PL). It usually appears as a vector operator , denoted ("del" or "nabla"), often applied to a 3-fold function (PL), f(u1, u1, u1), thus: f grad(f). In cartesian coordinates: f = Dx fx + Dyfy + Dzfz. The direction of f is the orientation in which the directional derivative is maximal and |f| is its value. Also, if nonull, the gradient is perpendicular to the level curve (PL) through {x0, y0 if z = f(x, y) but perpendicular to the level curve through (x0, y0, z0) if F(x, y,z) = 0. In general, the gradient is perpendicular to the surface of its application . In mathematical physics, the gradient of a scalar function connects a force field to a potential field: FORCE = - grad(potential), modeling gravitational fields, electrostatic fields, etc. (The negative sign above is necessary to model, for example, water running downhill.) A force has a potential iff it is irrotational, that is, the work done by the force around a closed loop is zero, as with gravitational and electrostic force fields. Thus, the gradient and the potential indicate a conservative field (PL). Hermann von Helmholtz (x-y) proved that every vector can be factored into an irrorational component and a (rotational) curl (PL). In multivector theory (PL) (a.k.a. geometric algebra, Clifford algebra, Arithmetic of Clifford Numbers), the inner product (PL) and outerproduct (PL) combine in (add up to) a single multiproduct whose derivative combines grad and curl in a single operator.
Gräffe's method
.
For finding roots of univalent polynomial equations, independently invented by Dandelin and Lobachevsky, improved by G. Malajovich and J. P. Zubelli. Method replaces polynomial equation by one whose roots are a 2kth power of the original roots. If roots are real s. t. |r1| > |r2| > ... > |rn|, then k can be maximized so that the ratio of r12k to the coefficient of the next to highest term sufficiently approximates unity, similarly the ratio of coefficient of the third highest term sufficiently approximates unity, etc. In these formulations, |r1|, |r2|, ..., |rn| can be determined. For complex roots, a variation of the method suffices.
Gram matrix
.
Given m with n=dimensional coordinates, vi , consider an m x n matrix, M whose jth column consists of the coordinates of vector v j, j = 1, ..., m. Define the m x m Gram matrix of dot products, mij = vi · v j as G = MTM, where M T is the transpose of M. This Gram matrix determines vectors, vi, up to isometry (PL).
graph
.
Structure composed of point as vertices (nodes) and possibly conected segments as edges -- a binary relation on vertices. For symmetric relation, graph is undirected, otherwise a directed graph. When at most one edge connects two nodes, graph is simple. A self-connected vertex is a loop. The edges may be assigned labels or values for a labeled graph. A one-dimensional comple, s.t. the number of odd nodes is even, sum, differences, powers, unions, products, matrices, and eigenvalues are definable.
graphical analysis
.
PL analytic geometry.
graph theory
.
PL graph.
Grassmann algebra
.
PL exterior algebra.
Grassmann manifold
.
A special case of a flag manifold, PL.
Gray (reflected) code
.
A labeling of numerals s. t. numerals adjacent in the numeration ordering have a single digit differing by one, especially used in binary numeration for encoding bits in computing. (Named "Gray" because it was devised by F. Gray in 1953, it is a Hamiltonian path, PL, on an n-dimensional hypercube, PL, including direction reversals.) Conversion: starting with right-most digit, dn, if d n-1 = 1, rename dn = 1 - dn, otherwise leave unchanged and apply method to dn-1, continuing up to left-most digit, d1, left unchanged since its left-neighbor is zero by definition. Result: g1g2... gn, the reflected Gray code. To convert reflected Gray code into binary numeration, start with right-most digit, gn and compute cn = Si=1n-1 gi (mod 2). If it is one, replace gn by 1 - g, otherwise leave unchanged. Continue this method up to left-most digit, resulting in resoration of binary numeration.
gray list
.
Introduction.
great circle
.
The circumference (PL) of a sphere (PL). It models the longitudinal lines in navigation. An arc (PL) of a great circle is a geodesic (minimal path) in Riemann (elliptic) geometry.
greatest common divisor (GCD)
.
Of the infinite number of divisors of two or more numbers, exactly one is the greatest, the GCD. It is the common divisor containing the prime factors of the candidates to the least allowable tokenage. As explained under least common multiple (LCM), GCD shares with LCM the property of not being welldefined (PL), hence, is noninversive. GCD can be determined by the Euclidean algorithm (PL), and GCD can be used to find the LCM of the candidate numbers. PL repertory, which explains that GCD is homologous to intersection in set theory, meet in lattice theory, conjunction in statement logic, "boolean product" in "Boolean algebra", etc.
greatest integer function
.
PL floor function.
greatest lower bound
.
PL infinum.
greedy algorithm
.
A characterization of many an algorithm (PL) in the literature, it recursively constructs set members by fewest constituent parts.
Green's function
.
An integrating kernel (PL) for solving an inhomogeneous differential equation (PL) with boundary conditions (PL) . Homologous (for ordinary differential equation) to Fourier analysis (in partial differential equations). Write a 1-D differential operator (PL) as: L Dn + an-1(t)Dn-1 + ... + a1(t)D + a0(t), with ai, i = 0, 1, ..., n-1 continuous on interval, I, to find solution , y(t) of equation, Ly(t) = h(t), for h(t) a given contnuous function on I . To solve this, seek a function: g: C(I) C(I) s. t. L(g(h)) = h, where y(t) = g(h(t)), a convolution (PL) equation of the form, y = g * h, with solution, y(t) = t t0g(t - x)h(x)dx. The integrating kernel, g(t), here, is the Green's function for L on t, often with h(t) d (t) (the Dirac function, PL), so that we have: y = g * h, with solution, y(t) = tt0g(t - x)d(x)dx, where Lg(t) d(t). But a Green's function is uniquely determined only by initial or boundary conditions (PL). In 3=D, we have: g(r, r') = d(r , r'). Then the solution to Lf = f is f(r) = g(r, r')f(r')d3r' .
Green's identities
.
Three vector/ integral identities, derivable from vector derivative identities: · (yf ) = y2f + (y) · (f)[1] and · (fy) = y2f + ( f) · ( y) [2]. where · is the divergence operator (PL), is the gradient operator (PL), 2 is the Laplacian operator (PL), and _ · _ is inner or dot product. The divergence theorem writes a volume integral as a surface integral involving a normal: V ( · F)dV = S F · dn [3]. Inserting [3] in [2]: V· (f y)[y2f + (f ) · ( y)]dV = S · ( fy) · dn [4], Green's first integral identity. Subtracting [2] from [1]: · (fy - yf) = f 2y - y 2f. So: V(f2y - y2f)dV = S(fy - yf)dn [7], Green's second identity. If u has continuous first partial derivatives and is harmonic inside the region of integration, then, writing C as a circular integral, we have Greem's third identity: u(x, y) = 1/2 p C [ln (1/r) Dnu - u Dn ln (1/r)] ds.
Green's theorem
.
A vector identity equivalent to the curl theorem (PL). Over a region D in the plane, with boundary, b(D), b(D)[f(x, y)dx + g(x, y)dy] = D(Dxy - D yx)dxdy, or b(D)F · ds = D ( X F) · k d A. If the region D is on the left when traveling around b(D), then the area of D can be computed as: A = ½ b(D)[xdy - ydx].
group
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A set of operations and a set of their operands such that the operands are closed under the operations and each operand has an inverse. Equivalently, a monoid such that each operand has an inverse. A vast system can be encapsulated in a single sentence by the group concept (PL activithm). Thus, Euclidean geometry is the set of all properties invariant under the Euclidean group (PL). Or, The Special Theory of Relativity concerns all physical properties invariant under the Einstein-Lorentz group. The group concept can be easily explained to children via the creeping baby group and colored multiplication patterns.
group algebra
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For field (PL), F, and group, G, with operation, *, the group algebra, F[G(_*_)], is the set of all linear combinations of any finity of members of G with coefficients in F, i.e. of the form, c1g1 + c2g3 + ... + cng n, for ci e F, gi e G, i = 1,2,...,n; in general, Sckg, k e G. Then, F[G]is a group algebra over field F for addition Sckg + S dkg = S(ck + dk)g, k e G; for scalar multiple, s Sckg = S(sck)g, k e G; for multiplication, (Sc kg)(Sdkh) = (S(ck dke G. The identity of G is the unit of F[G], which is commutative iff G is an Abelian group. (Replacement of the field by a ring yields a ring algebra.)
group, campanological
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In "cathedral towns" of Great Britain, enthusiasts "ring all the changes on their bells of different pitch". They do not realize that this procedure follows the procedure of recursive generation of a permutation group, which knowledgeable people have labeled a "campanological group", since "campanology" is the art of bell-ringing.
.
groupoid
.
A set of operations and a set of their operands such that the operands are closed under the operations.
group theory
.
PL group.
Gudermannian function
.
From inverse equations for the Mercator map projection (PL). f (y) = gd(y) describes the latitude f in terms of the vertical position, y in the Mercator projection, so the Gudermannian is defined, thus: gd(x) 0xdt/cosh t = 2 arctan [tanh (½x)] = arctan (sinh x) = 2arctan (ex) - ½p. The derivative of the Gudermannian is: Dx gd(x) = sech x, and, of course, it connects with trignometric, hyperbolic and exponential functions.