A.__________________________________________________________________________________________________________________________________________________________________________________________________

abacus.

A counting frame to aid arithmetic computationhsuan-pan in which each wire has two markers in the upper section and five in the lower; and the Japanese soroban, in which the upper section has one marker per wire and the lower has four per wires.

abelian.

Named for the great Norweigan mathematician, Niels Abel (1802-1829), this adjective applies to any system whose primary operation is commutative (PL). Thus, a group (PL) with a commutative group operation (PL) is an abelian group (PL).

abelian field.

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

abelian group.

As noted under abelian, an abelian group is one whose group operation is commutive. The prototype is the permutation (a.k.a. symmetric) group on n members, which is commutative under the group operation of concatenation (PL). Every finite group is equivalent to a subgroup of a permutation group.

abridge.

In multiplication, after multiplying by a digit of the multiplier, to drop those digits not affecting the desired degree of accuracy.

abscissa.

The horizontal coordinate in a 2-D Cartesian coordinate system on the Euclidean plane, usually denoted by x.

absolute error.

the difference between value of quantity obtained by measurement or inference and its "actual" value.

absolute number.

One represented, in algebra, by numerals, not letters.

abstract algebra.

An algebraic repertory (PL) of which numerical algebra (PL) as prototype can generate, by permuting labels, various theories or systems of mathematics, such as groups, rings, modules, fields, numbers, and categories, as well as linear algebra, representations, and Galois theory.

abundant number.

An integer, n, which is not a pefect number (PL) and for which the abundance, s(n) s(n) - n > n, where s() is the divisor function (PL). The first five abundant numbers are 12, 18, 20, 24, 30

accumulation factor.

The term, as in a binomial or comound interets, which is exponentiated..

accumulation point.

A point P, of a point-set, S, s. t. at least one point of S exists distinctly from P in any of neighborhood (PL) of it.

accumulator.

In a computing device , an adder or counter which augments its stored number by each successive number received.

activithm

"[W]hat counts as reality ... as a glass of water or a book or a table ... is a matter of what categories we impose on the world .... Our concept of reality is a matter of our [linguistic] categories."^[19] The "Activithm" Strategy declares that knowledge comes only from risky action in order (sometimes) to create patterns-in-patterns-in-.... To do so: (1) impose a pattern P upon "whatever"; (2) see if pattern P has a subpattern S. (3) If so, see if subpattern S is conserved under a group (PL) G of transformations. (4) If so, then explicate P as the pattern conserved by group G (PL, also colored multiplication patterns as example of activithm). Perhaps the most frequent use of activithm, to date, is imposition of physical dimensions (PL dimension), making possible mathematical physics.

acute angle.

An angle (PL) subtended for less than 90° or p/2.

acute triangle.

A triangle each of whose angles is an acute angle (PL). This is possible because the sum of angles of a triangule equals 180° or p. A special case of the acute triangle is the regular (a.k.a. symmetric) triangle, all of whose angles subtend 60° or p/3.

acyclic.

Describes an ordering which is not "circular". Every finite group (PL) of even order (number of members) (PL) is cyclic, in that the successor of any member of the ordering is a power of that member. Any other finite group is acyclic.

acyclic graph.

A graph (PL) without a circuit (PL). (PL also forest.)

add.

The active verbal form for the arithmetical operation of addition (PL). Our daily language provokes subliminal confusion by "mispeaking" the term "add" instead of "adjoin". Thus, "I adjoin my shirt to the suitcase", but I can't "add" it thereby.

addend.

An addend is an operand (PL) of the operation denoted addition. Default addition is binary (PL) -- extended to n-ary addends by asssociativity of addition (PL) -- hence, properly, an addend is one of a pair of operands in the operation of addition.

addition.

Repeated counting can be tedious -- as in counting thumb and finger, then the other three fingers, then all fingers: "one, two -- one, two, three -- one, two, three, four, five". Hence, as a shortcut, the successor function (PL) of recursion (PL) is invoked to define recursively the (binary) operation of addition of natural numbers (PL): S(a + b) = a + S(b). Addition is shown to be a total operation of the natural number system -- hence, of successive systems -- always yielding a natural number. And commutative and associative laws for addition can be derived, and it can be shown to distritute (PL) with multiplication (PL). Since it is welldefined (PL) and commutative, addition has a single inverse (PL), namely, subtraction (PL).

addition of vectors.

(1) as line segments, head of second vector is placed at tail of first vector, and sum drawn as segment of vector from tail of second vector to head of first vector; (2) as components of orderedform, the ith component of the sum is the sum of the ith component of the addends.

additive.

Adjectival form of verbal form "add" (PL).

additive function.

A function, f, s.t. f(x + y) = f(x) + f(y), definitively .

additive inverse.

The additive inverse in arithmetic is subtraction (PL), as described in the discussion of addition.

additive set function.

A set function, f, such that f(A B) = f(A) + f(B), for any two disjoint (PL) sets A, B on which f is defined..

ad infinitum.

Latin for "to infinity", often used in describing a limit (PL) process.

adjacency matrix.

Matrix (PL), m_rc, representing a graph (PL) s.t. matrix contains entry 1 iff vertex, v(r), is adjacent (PL standard dictionary) to vertex, v(c).

adjacent.

Labeling two vertices (PL) sharing an edge (PL), or two distinct edges with at least one common vertex.

adjacent angles.

Two angles with common side and vertex, lying on opposite sides of their common side.

adjoint.

Given a 2nd order differential equation, L(u) p(x_₀D²_xxu(x) + p(x)₁D_x u(u) + p(x)₂u(x), the adjoint operator is L(u) D_xx[p₀(x)u(x)] - D_x[p₁(x)U(x)] + p₂(x)u(x). (Older texts use dagger notation to denote an adjoint operator. Relative to a Hermitian operator, the term "Hermitian conjugate" is sometimes used.

adjoint of a matrix.

Given M, a matrix (PL), form the matrix of cofactors (PL) of M and transpose (PL) them.

adjoint operator.

Given the 2nd order differential equation, a(x)D_xxu(x) + b(x)D_xu(x) + c(x)u(x). Its adjoint operator is D_xx(a(x)u(x)) + D_x (b(x)u(x)) + c(x)u(x).

admissible.

Said of term (PL) in a string (PL) or word (PL) iff term appears in given sequence (PL). Thus, all terms of m, mo, moom are definitively admissible, but term mm is inadmissible.

agstymie.

When Socrates was condemned to death ("for corrupting Athenian youth"), his pupil Plato sought Pythagorean sanctuary and learning. Repatriated, Plato renounced Pythagoreanism, canonizing Eleatic dogma historically. (PL Zeno's paradoxes.) The motto of Plato's Academy was "Let no one ignorant of geometry enter here!". The incommensurability (PL) of the diagonal of a square with a side was thought to imply that "Geometry is incompatible with arithmetic." Since Platonists further argued that"Motion is geometry set to time", it was concluded that "Motion cannot be arithmetized!" (Velocity, acceleration, force, momentum, energy, etc., seemed meaningless.) Hence, the "AGstymie", since its argument of the incompatibility of arithmetic, "A", and geometry, "G", stymied the development of theoretical mechanics. For, ancient applied mechanics was not rudimentary. Inheriting from the great Alexandrian Period -- which boasted robots and steam cars! -- imperial Roman engineers developed efficient windmills and water mills (For example, in the first century B.C., the Roman engineer Vitrivius described the undershot water wheel, later so useful in medieval and post-medieval Europe, but rarely used in Roman times.) However, believing that theoretical extensions were impossible, no improvements replaced the vast slave power. (Compassion didn't end European slavery -- only disproof of cost-effectiveness. Good reason, the Greco-Roman-Hebraic heritage of the West was one of slavery. That "thou-shalt-not" Book, The Bible nowhere says, "Thou shalt not enslave!" So, the Mathematical Establishment stood aside until evidence of noncosteffectiveness ended widespread slavery -- still existent in "pockets" today.) We glean the efficiency of this machinery from Gimpel²⁶. To produce food, materials for clothing, etc., yet leave time for prayer and meditation, 12th century monks were permitted to read "pagan" Roman manuals about labor-saving devices. Soon thousands of windmills and water mills covered Europe and Britain. This Mechanical Industrial Revolution waned with "The Black Death" (bubonic plague), peaking in 1348. (The Black Death was caused by widespread killing of cats, which had controlled rats with their contagious fleas.) In some regions, 50% of the population died in "The Black Death". Many regions lacked blacksmiths, coopers, wheelrights, other craftsmen. Training had been oral. The Church now allowed more general reading of "pagan" books, initiating The Renaissance. However, the gravest charge against Galileo (1564-1642) was not his support of the argument of Johann Kepler (1571-1620) that the earth revolves around the sun, but rather that Galileo was a "Pythagorean" in promoting a "law of falling bodies". Thus did the Platonists delay the "Industrial Revolution" and prolong slavery! (Today, Platonism -- PL -- is the dominant mathematical philosophy in mathematics departments around the world.)

A-integrable.

Generalization of Lebesque integral. A measurable function (PL), f(x) is A-integrable over the closed interval, [A, B], if m{x: |f(x)| > n} = O(1/n), where m is the Lebesgue measure, and where the integral        lim_nb^a [f(x)]_ndx exists, s. t. [f(x)]_n = f(x) if |f(x)| n, = 0 if |f(x)| > n.

Airy differential equation.

General form: D_zzy(z) ± k² y(z)z = 0; "conventional form", D_zzy(z) - y(z)z = 0.

Airy function.

The Ai(z), Bi(z) fumctions are the two linearly independent (PL) solutions of the differential equation (PL), D_zzy(z) - y(z)z = 0, with Ai(z) = (z/x p^3/2 K_1/3(2/3 z ^3/2) and Bi(z) = (z/x)^3/2 [I_-1) (2/3 z ^3/2 + I₁) (2/3 z^3/2], where I(z) is a modified Bessel function of the first kind (PL) and K(z) is a modified Bessel of the second kind (PL).

aleph null.

The cardinal number (PL) of the set of positive integers (PL), or of any set in one-to-one correspondence with the positive integers. Written ₀. Also, called "aleph-zero".

alforder.

An order of numbers in increasing degree similar to alphabetic order (PL), that is, from left-to-right. (PL numorder .)

algebra.

Generalization of numerical algebra (PL) in which numbers are replaced by formal symbols, as in algebra of vectors or multivectors, algebra of statements, algebra of logic circuits, etc. (The word "algebrista" appears in Cervantes' Don Quixote, where it is used for a bone-setter, that is, "a restorer". But, more likely, the label "algebra" derived from a book written by the great Islamic astronomer and mathematician, al-Khwarizmi, circa 750-850 -- a book entitled Al-jabr wa'l muqabulah, circa 830 A.D. Although Greek mathematician, Diophantus (200?-284? A.D.) has been called "the Father of Algebra", the title should be given to al-Khwarizmi, bequeather of the tactics of algebra and of the term "algorithm", for a "sure-fire" problem-solver.)

algebra of dimensional analysis.

PL dimensional analysis algebra.

algebra, numerical.

Arithmetic "backwards": Arithmetic operates on numbers to yield a "goal"-result. Numerical algebra begins with a numerical result and inversively operates "backwards" to determine the set of numbers which can achieve that "goal". (PL annuity and the camel problem as protoypical of numerical algebra.) Equivalently, numerical algebra is the arithmetic of sets of numbers or functions (PL) of sets of numbers.

algebraic.

Of or relating to algebra (PL).

algebraic addition.

Combining algebraic terms (PL) by addition or subtraction (additive inverse, (PL) in an algebraic sum (PL)..

algebraically closed.

Any field -- PL -- (such as the complex numbers, PL) in which all polynomial equations (PL) with coefficients in the field have equational roots in the field. A field is algebraically closed iff its only closure (PL) is itself. Equivalently: (a) Every nonconstant polynomial in a polynomial ring (PL) has a root in that ring. (b) Every nonconstant polynomial in a polynomial ring splits over that ring, i.e. can be written as product of linear factors in that ring. (c) Every irreducible polynomial in a polynomial ring has degree 1 (PL). (d) The field has a subfield such that the field is algebraic (PL) over the subfield and every polynomial in the subfield ring splits in the field..

algebraic closure of a field.

A field (PL), F', is an algebraic closure of a field, F, iff every polyomial (PL) splits (PL algebraic closed) completely in F'. Thus, the complex number field, C constitutes an algebraic closure of R, field of real numbers.

algebraic curve.

A point-set in n-D Euclidean space satisfying a polynomial in the n coordinate functions. Given such a polynomial of degree m, then the curve is said to be of degree m.

algebraic equation.

An equation (PL) formed by equating an algebraic expression (PL) to zero.

algebraic expression.

A symbolic string formed by performing a finite number of algebraic operations (addition, subtraction, multiplication, division, exponentiating to a rational power) on formal symbols

algebraic extension field.

E is an algebraic extension field of F if (1) F is a subfield of E; (2) every element of F is algebraic over E.

algebraic function.

A function (PL) whose value is given by an algebraic expression ({L) of the functand (PL) of the function.

algebraic geometry.

Study of geometries derived from algebra, particularly from ring (PL); classically, the algebra is the ring of polynomials and the geometry is the set of zeros of the polynomials ("varieties").

algebraic language.

Rules of syntax (PL) for writing mathematical expressions and equations.

algebraic manifold.

A smooth (PL) algebraic variety (PL); can be covered (PL) by coordinate charts (PL) whose transition functions (PL) are rational functions. Thus, the (Riemann) sphere, with chart given by the complex field, C, and another chart at , having transition functions given by 1/z.

algebraic number.

Any root (PL) of an algebraic polynomial with coefficients (PL) which are rational numbers (PL).

algebraic number field.

Any extension field which can be represented as a finite vector space (PL) over the rationals.

algebraic number theory.

Applies algebraic (PL) methods to the study of numbers.

algebraic surface.

The point-set in Euclidean 2-D space represented by algebraic functions of two parameters (for 2-D). In an (n-1)-D hyperspace, an algebraic surface is a 2-manifold in the neighborhood of almost all of its points. Also, algebraic hypersurface; algebraic variety of codimension.

algebraic topology.

Studying topological properties of space via methods of abstract algebra (PL), such as theories of homology, cohomolugy, homotopy..

algorithm.

An algorithm is for thehuman what a computer program is for the computer: a set of instructions which, when complied with, lead to a desired result, say, solution to a problem. (PL heurithm.) The division algorithm (PL) is a protoype, and its antitonic nature leads to the conjecture that all algorithms are antitonic..

alias.

A function (PL) is an alias of another function if the two are indistinguishable in having the same values at a finite set of points. The problem often occurs in an undersampled discrete Fourier transform.

aliasing.

PL alias.

aliquot.

The label "aliquot parts" is "archaic language" for "divisors of a number".

almost everywhere.

A property, P, holds almost everywhere iff the set (PL) of points (PL) where it fails has measure zero (PL).

almost periodic (function).

a.p.f. iff, for every e there exists an l(e) > 0 s.t. every interval (PL) [t₀, t + l(e] contains at least one number t for which metric (PL) r[x(t), x(t + t] < e where - < t < .

alternating group.

A normal group ()L) of the group of permutations (PL) consisting of on a set of countability n.

alternating permutations (PL zigzag permutations) .

Arrangement of c₁c₂,...,c_n with no c_i between c_i-1 and c_i+1. Example: 1,3,2.5,4,6. (Determining 1, 2,...,n of sequence is André's problem.) Given Z_n = 2A_n, sequence can be computed for 2a_n = Sa_ra_s sequenced integers s. t. r + s = n - 1, with a ₀ = a₁ - 1 and A_n = n!a_n . The A_n are labeled Euler zigzag numbers, beginning 1,1,2,5,16,61,272. Even A_n are labeled Euler(a.k.a. secant, zig) numbers; odds, tangent (a.k.a.zag) numbers.

alternating series.

A series (PL) wherin cosecutive terms add and subtract.

alternating series test (Leibnitz criterion).

An a.s. converges (PL) iff terms are monotonically (PL) sequential and its limit is zero.

altitude.

The altitudes of a triangle are its cevians (PL) perpendicular to the legs (PL) opposite. The three altitudes of a triangle meet at its orthocenter.

amicable number.

Two naturals or integers are amicable iff sum of divisors of each equals the other one. Example:

amplitude.

The magnitude of an oscillation (PL).

analysis, analytic.

Any mathematical subject or procedure which is dependent upon the limit (PL) process. Protypes are anaylytic geometry and differential. integral calculus (PL).

analytic continuation (continuation).

Means for extending domain of definitin of a complex function. Usually applied when complex analytic function is determined near a point z₀ by means of a power series, f(z) = S _k=0 a_k (z - z₀)^k, which is usually valid only within its radius of convergence. Frequently the function has a larger domain of definition and can be so extended. (Thus, extensions of the trigonometric, exponential, logarithmic, power, and hyperbolic functions from real line R to entire complex plane, C.) Similarly, extending values of a function across a branch cut in the complex plane. Given functions, f₁, f₂, defined respectively on domains, W₁, W₂, with nonepty intersection W₁ W₂, with f₁ = f₂ on this intersection, then f₂ is the continuation of f₂ on W₂, and vice versa. (Existence is unique existence.) Via analytic continuation, initiating with representation of one function by a power series, any number of power series can combine to define the function at all points of the domain. Also, any point can be reached from a point without passing through a singularity of the function, and the combination thereby constitutes the analytic expression of the function. This can result in a multivalued function, such as (z)^1/2 .

analytic function.

A complex function (PL) is analytic in region r iff complex differentiable (PL) at every point of r. If analytic, the ordering of any legitimate limit process applied does not matter. Terms holomorphic function, differential function, complex differentiable function, regular function may interchange .

analytic geometry.

Representation of geometric via coordinates, equations, algebraic manipulation. Strategy:

1. points are intrinsically indistinguishable, whereas numbers are distinguishable, so point become the figure against the ground of number;
2. numbers invoke arithmetic and algebraic tactics.

AND function.

Possible reference for operation of conjunction (PL) in statement and predicate logic.

André's problem.

PL alternating permutation.

angle.

In the plane (PL), a measure of rotation about a fixed axis stated in degrees, radians, revolutions (PL). Also the region between two rays (PL) with a common vertex swept out by such a rotation. In higher dimensions, the angle between two hyperplanes is defined between the normals to the hyperplanes. Label may refer to a geometric figure, numeric quantity, signed algebraic quantity.

annihilator.

variously and widely used, most commonly for set, S, of all functions satisfying a given condition set, C, which is nullified on each member of a given set, G, that is, S is the annihilator on G.

annuity.

Any set of equal payments made at equal intervals of time. Typical xamples: interest payments to bond investors; premiums paid by insurees on insurance policies, payments on installment purchases. The great French-British mathematician, Abraham De Moivre (1767-1854) -- the creator of "the normal distribution curve" usually credited to Gauss -- created the first annuity, in linear form, later modified. Elsewhere (algfront.htm) we note that "algebra is arithmetic backwards". The determination of an annuity is a typical example, of working backward from total payment to equal installments. The development of these financial devices transformed our society into what American economist, John Kenneth Galbraith, calls a "fiduciary society", since so much American wealth is invested in workers' pensions.

ansatz.

This term adopted from German is a procedure in mathematical physics which is only quasi-mathematical. Approximately, it means, "Let's look at it this way, and see what this accomplishes". This is the premise of hypothesis of axiomatic reasoning in mathematics; but it goes forward, at best, to "empirical proof", not logico-mathematical proof. But it calls for the discipline of logico-mathematics to be use most effectively.

antichain.

Subsystem of noncomparable (PL) elements of a poset (PL); forming "row" of poset, just as a chain (PL) forms a "column" of it.

anthropomorphic decimal numeration system.

PL decimal numeration system.

anticlastic.

When Gaussian curvature (PL) is negative, the given saddle-shape (PL) surface is anticlastic.

anticommutator.

For operators (PL), O, P, their anticommutator is [O, P] OP + PO.

anticommute.

Two operators (PL), A, B, iff AB = - BA.

antilogarithm of a number.

The inverse function (PL) of operation of logarithm (PL) s. t., for base, < b , and number, n, we find antilog_b (log_bn) = log_b(antilog_b) = bⁿ, that is, the label distinguishes one of the two inverse functions of exponentiation, the other being root extraction.

antisymmetric matrix.

Is the negative of its transpose (PL).

antisymmetric tensor.

Changes sign for interswitching of indices (PL).

antitone.

The Greek word tonus means "ordering". (Thus, the tones of the musical scale correlate with an ordering of sonic pitches. An antitone is a correlation of two orderings such that one ordering (MAXTONE) increases as the other (MINTONE) decreases, with a bound on one of the orderings. (PL isotone and antitonic algorithm, as well as geometric progression for description of an antitonic proof.)

antitonic algorithm.

Since an antitone coordinates every stage of an increasing order (MAXTONE) with a corresponding stage of a decreasing (MINTONE) ordering, with bounding of one ordering, it follows that this induces a bounding on the other ordering..

antitonic hypothesis .

Hypothesis: all natural processes and modeled antitonically.

antitonic repertory.

An antitone ([lease see above) is both an ordering and a group (PL both) -- the latter, since its correlation achieves both closure and inversion, the neccessaryand sufficient conditions for a group. But it is more than that.

• When the orderings are interpreted as partial ordering, it becomes a Galois correspondence (PL) between a lattice and its antilattic.
• If each element the above lattices holographically compacts a model of the lattice itself, it becomes a holattice (PL) coordination.
• When all correlations of the simple antitone except that between MAXTONE and MINTONE are ignored, it becomes a bypass or conugacy (PL).
• If the simple antitone is embedded in a structure such that one ordering may or may not display one uncoordinated element, it becomes a figure&ground (PL) structure.
• These and any other systems derived from the simple antitone generate the antitonic repertory.

apex.

Standard dictionary usage.

apothem.

Perpendicular distance from midpoint of a chord (P) to associated circle's center; equal to radius (PL) minus the sagitta (PL).

Arabic numerals.

PL Hindu-Arabic numerals.

arboricity.

For graph (PL), G, its arboricity is the minimum number of subgraphs which are line-disjoint (PL) and acyclic (PL) and whose union is G.

arc.

Segment of a curve.

arc length.

Length along a curve. Thus, for a circle of radius, r, the arc length for two points of (radian-measured) angles, q₁ , q₁, the arc length, s is s = r|q₁ - q₂|.

arc function.

PL arc length.

Archimedean.

Attribute of structure or system satisfying Archimedes' axiom PL.

Archimedean spiral.

Spiral with polar equation r = aq.

Archimedean solid.

PL semiragular polyhedra.

Archimedes' axiom.

(Actually due to Eudoxus (z-y).) Axiom: Equivalence a/b = c/d iff one of the following conditions is satisfied for integers m, n:

1. if ma < nb, then mc < nd.
2. if ma = nb, then mc = nd.
3. if ma > nb, then mc > nd.

That is, given two magnitudes with common ratio, there exists a multiple of either esceeding the other. (This was basis of Archimedes' exhaustion procedure for solving area and volume problems, anticipatimg integral calculus.) Any geometry not satisfying this condition is a non-Archimedean geometry, PL horn angle. Pl, also, Eudoxus' axiom

Archimedes' problem.

Cut sphere (PL) by plane (PL) s.t. spherical segments have specified ratio (PL).

area.

Measure of a plane surface..

area-preserving map.

A map (PL), F, Rⁿ Rⁿ is area-preserving iff m(F(A)) = m(A), for each subregion Rⁿ, where m(A) is the n-dimensional measure of A. A linear transformation is area-preserving iff its corresponding determinant equals one.

Argand-Wessel-Gauss diagram.

Standard rectangular representation of the complex plane, with horizontal axis representing the real part of a complex number and the vertical axis representing the complex part. The work of Swiss mathematician, J. R. Argand (1768-1832); similar representations by Norweigan surveyor, Caspar Wessel (1745-1818), and German mathematician, K. F. Gauss (1777-1855).

argument.

Given complex number (PL), z x + iy = x + (-1)y = |x| e^iq, then q is the argument (a.k.a. phase) and is given by arg(x + iy) = arctan (y/x).

arithmetic.

Humans found tedious the corresponding of candidate countables with standard countables (fingers, toes, scratches on bone, etc.), so syntactically formalized countables via recursion (PL) by successor function (PL), achieving closure (PL) in counting numbers or natural numbers (PL). Tedium of concatenated counting motivated syntactic formalization of addition operation via recursion derived from successor-recursion , also found total for natural numbers. Tedium of concatenated addition motivated syntactic formalization of multiplication operation via recursion derived from addditive-recursion, also found total for natural numbers. Tedium of concatenated multiplication motivated syntactic formalization of exponentiation operation via recursion derived from multiplicative-recursion, also found total for natural numbers. Need of inverses for each operation ("We can only understand life backwards, but it must be lived forwards", Sören Kierkegaard) motivated subtraction , defined in terms of addition; of division, defined in terms of multiplication; of logarithm and root extraction, defined in terms of exponentiation. (Note: The commutativity of addition, multiplication yields one distinct inverseeach. Noncommutativity of exponentiation yields two distinct inverses.) Each partial inverse is totalized, first, by integers as vectors of naturals with defined natural-difference rules; by rationals as vectors of integers with defined integral-quotient rules; by real numbers as infinite "Cauchy" vectors of rationals with appropriate rules; by complex numbers as (Hamilton) vectors of reals with appropriate rules. Up to last case, vector form can be syntactically "hidden". But inducing of bimoduls (PL) in last case forces vector form: Clifford Number Arithmetic (PL). Label "arithmetic" often used only rationals, ignoring exponentiation.

arithmetic, intrinsic.

PL intrinsic arithmetic.

arithmetical.

Of or relating to the study of arithmetic (PL)..

arithmetic mean.

Given a set, S, of numbers of cardinality, c: (1) add the numbers of S; (2) divide this sum by c; (3) declare the quotient to be the arithmetic mean of S. The arithmetic mean is "the middle" of an arithmetic progression (PL), hence, its average (representative). The average of a set of extensive measurements (PL) approaches the arithmetic mean, as the set size increases -- explaining why so many people think "the average" means "arithmetic mean". (PL geometric mean, harmonic mean).

arithmetic progression.

Sequence of numbers each equal to preceding term and a constant: f, f + d, f + 2d, f + 3d + ... + f + (n - 1)d, for first term, f; difference, d; total number of terms, n. (PL arithmetic mean.)

arithmetic series.

An unlimited sum of terms having finite subsections that form an arithmetic progression.

arithmetization.

Studying a mathematilcal system using arithmetical operations. ("Analysis has been arithmetized", Henri Poincaré.)

Artinian group.

A group (PL) s.t. any decreasing chain of subgroups terminates finitely.

Artinian module.

A module (PL) fulfilling descending chain condition (PL): every decreasing sequence of submodules(PL) approaches constancy.

Artinian ring.

A noncommutative (PL) semisimple ring satisfying descending chain condition (PL).

ascending chain condition.

All increasing sequences in a partially ordered set approach constancy.

associative algebra.

An algebra satisfying associativity (PL) for its operators (PL).

associative law.

PL associativity .

associative ring.

A ring (PL) satisfying associativity (PL) for its operators (PL).

associativity.

Given an operation, o, and any three operands of a system, a, b, c, then the operation is associative iff (a o b) o c = a o (b o c). Thus, (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c). "What good is associativity? Why study it?" Answer: Associativity extends a binary operation to any number of operands. The operation of vector (cross) product (PL) on vectors is an example of a nonassociative operation. To see that associativity is simply a variation on a familiar pattern, PL commutativity; that, behinds this, is a not generally known strategy, PL bypassing, in turn connected with conservation laws.

asymptote.

A curve, which (arbitrarily closely) approaches a given curve.

asymptotic curve.

PL asymptote..

asymptotic prime number theorem.

Yields asymptotic for prime number counting theorem, PL, p(n) for number of primes n, Carl Friedrich Gauss (1777-1855) suggested p (n) _~ li(x), where li(x) is the logarithmic integral, defined as the Cauchy principal value (PL) of li(x) ₀^xdt/lnt . Using Soldner's constant, m = 1.4513692346... for which li(x) = 0, this can be reformulated: li(x) _m^xdt/ln t, x >m . Proven independently by J. S. Hadmard (1865-1963) and Baron de la Vallëe Poussin (1866-1962) proved this theorem by showing that the Riemann zeta function, PL has no zeros of the form 1 + (-1)^½t, so no deeper values are required. Other mathematicians improved this proof.

atlas.

Collection of coordinate charts (PL) on a manifold (PL) with transition functions which are smooth (PL).

atom (lattice).

An element in a lattice (PL) that is dominated by no other element of the lattice and dominates only the null (empty) element. Standard language designates the singletons (PL) of a lattice as "atoms". In a factor lattice, the prime numbers are atoms. But recognizing multiple tokens of the same element is to introduce an extension of the "atom" concept which is forbidden in standard ("Platonic") treatment, since it allows exceptions to the tertium non datur proof and the axiom of choice, which engender nonconstructive proofs. (PL typon.).

attractor.

Set of states (PL) (points in the phase space, PL), invariant under the dynamics, toward which neighboring states (in a given basin of attraction, PL) approach during dynamic evolution.

aug.

A commutative, associative, distributive operator in o-set theory (PL) achieving the effect of multiplication in arithmetic. Definition: Given o-sets A, B their augment, denoted by A @ B is the set formed by the juxtaposition of the memberships of each set. It has an inverse, coaug, denoted @. Thus, we can formulate a set-theoretic homologue of the arithmetic product 12 * 90 = 2*2*3 * 2*3*3*5 = 1080 = 2*2*2*2*3*3*5, with quotient, 1080 90 = 12. One such homologue is obtained from the two o-sets: o-{a, a, b}, o-{a,b,b,c}, with augment, o-{a, a, b} @ o-{a, b, b, c} ~ o-{a, a, a, b, b, b, c}, and coaugment, coaug, o-{a, a, a, b, b, b, c} @ o-{a, a, b} ~ o-{a, a, b}.

augend.

Initial term of a succession of addends (PL).

automorphic function.

Function, f(z) of complex functand (PL) which is analytic (PL) -- except for poles (PL) -- on a domain and is invariant (PL) under a denumerable infinite (PL) group of linealry fractional transformations (a.k.a. Möbius transformations, PL): z' = (az + b)/(cz + d). Automorphic functions generalize trigonometric and elliptic functions (PL).

automorphic number.

A number, m s.t. nm² has its last digt(s) equal to m is n-automorphic. Examples: 1(5)² = 25; 1(6)² = 36; etc.

automorphism.

A morphism (PL) or transformation which leaves a form unchanged.

autonomous system.

Said of an ordinary differential equation (ODE, PL) or systems of these free of the independent (PL) functand, usually denoted by t (say, for time). An autonomous ODE is independent of the intial condition so that all particles pass through a given point of phase space (PL)..

average.

A number which represents a set of numbers. Various averages exist (PL): mode, median, arithmetic mean, geometric mean, harmonic mean, etc. (The word derives from Latin, havaria, meaning "share". In ancient times, merchants shipping on the Mediterranean lost goods due to piracy, storms, rats, etc. A merchant who lost goods was compensated by receiving a share of goods from other merchants who had made such an agreement -- a primitive form of insurance.)

axial vector (pseudovector).

Does not reverse sign when coordinate axes are reversed (as is case for polar vector. PL). In multivector theory (PL) a polar vector is a 1-vector and an axial vector is a 2-vector or bivector (PL).

axiom.

Any assumption or postulate on which a given mathematical system is based. All other propositions of the theory can be derived from is axioms.

axiom of choice.

PL choice function..

axiom of continuity.

David Hilbert (1862-1943) showed that this must be adjoined to Euclid's axioms to ensure that two circles of radius r will intersect if separation of their centers is less than 2r.

axiom of infinity.

In Zermelo-Fraenkel set theory, this ensures existence of natural numbers.

axis.

A line with respect to which a figure is draw or measured or rotated, etc. Also, a line through a sheaf (PL) of planes (PL).

axis of abscissa.

PL x-axis..

axis of ordinates.

PL y-axis..