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Lagrange's four-square theorem
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Also known as Bachet's conjecture, this was stated but not proven by Diophantus (c. 25AD) -- that every positive integer can be written as the sum g(n) of at most four squares. Although proven by Fermat (1601-1665) using infinite descent, the proof was suppressed. Euler (1707-1783) was unable to prove the theorem and the first published proof was given by Lagrange (1736-1813) in 1770, using of Euler's four-square identity (PL). Lagrange proved that g(2) = 4. Legendre (1752-1833) proved that this number is reducible to 3 except for numbers of the form 4n(8k + 7).
Lagrangian multiplier
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Lagrange's Principle derived from the Antitonic Principle (Hays)
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Given the antitone (PL), S * U = CONSTANT , take the logarithm: log S + log U = 0. We assign: log S T, log U V, with another constant: T + V = CONSTANT. for kinetic energy , T and potential energy, V. Result: Langrnage's Principle, derived from The Antitonic Principle. PL Hamilton's Principle derived from the Antitonic Principle (PL), and d'Alembert's Principle derived from Newton's Law , with antitonic form.
Lagrangian multiplier
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Laguerre's differential equation
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lambda calculus
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lambda expression
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Lamé equation
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Landau symbols
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language theory
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Laplace-Beltrami operator
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Laplace's equation
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Laplace's expansion
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larutan
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A set of operations and a set of their operands such that the operands form a monoid under each operation. The natural numbers form a larutan. (PL terms.)
lateral area
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lateral face
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Latin rectangle
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lattice
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latus rectum
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law of cosines
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law of the iterated logarithm (probability)
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Let Sn denote the number of successes in nBernoulli trials (PL) with probability p of success, so that b(k; n, p) is the probability of the event that Sn = k. In approximating this probability between two assigned numbers, the De Moivre-Laplace Limit Theorem (PL) is developed. This limit theorem can take on a simpler form if Sn is replaced by Sn* = (Sn - np)/(npq)&frqac12; , q = 1 - p, that is, measuring deviations of Sn from np in units of (npq) &frqac12;, for the reduced number of successes in n trials. By the De Moivre-Laplace Limit Theorem, for every particular value of n, it is improbable to have a large number of reduced successes (Sn*), but, in a prolonged sequence of trials, Sn* will assume arbitrarily large values. In proving the strong law of large number (PL), one finds (with probability one) that Sn* < 2(log n)½, for all sufficiently large number of trials, n: an upper bound for the fluctuations of reduced successes, Sn*. From this follows the law of iterated logarithm of Russian probabilist, A. Khinchine. Theorem: With probability one, limn supSn*/(2 log log n)½ = 1.
law of the iterated logarithm (number-theoretic).
Let x denote a real number in the interval 0 x < 1, x = 0.a1a2. Such decimal expansions connect with Bernoulli trials (PL) with probability p = 1/10, such that digit zero represents success and all ofther digits represent failure . Thus, in the sample space (PL) of Bernoulli trials, the event "success at nth trial" is represent by all real numbers x whose n th decimal is zero. Thus, all limit theorems for Bernoulli trials for p = 1/10 translate into theorems about decimal expansions, so that "with probability one" translates into "for almost all x" or "almost everywhere". In measure theory (PL) language, the weak law of large number (PL) asserts that S n(x)/n 1/10 in measure, while the strong law of large numbers asserts that Sn(x)/n 1/10 almost everywhere. Then, Kinchine theorem asserts lim sup (Sn(x) - n/10)/(n log log n)½ = (0.3)2, for almost all x. (This answers a problem formulated in a series of papers initiated by Hausdorff in 1913 and by Hardy and Littlewood in 1914.)
law of the iterated logarithm (VERIBILITY).
The verdibility (PL) measure has the same relation to logical statements that probabiity has to events. It follows that the probability law of the iterated logarithm translates into a similar law for verdibility.
law of exponents
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May be read at http://www.harcourt.com/dictionary /browse/19/
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law of signs
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PL signs, law of.
law of sines
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law of tangents
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leading coefficient
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leading zeros
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least common denominator
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The least common multiple (LCM) of the denominators (PL) of several rationals ("fractions") (PL) to be operated upon.
least common multiple (LCM)
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Given two or more factors (PL), of the infinite number of their common multiple, one is the least. It is the number which contains each prime factor to the tokenage in the one with the greatest tokenage. The standard LCM algorithm: (1) find product, P, of the given candidates; (2) use the Euclidean algorithm (PL) to determine the greatest common divisor (GCD) (PL) of the candidates; (3) divide P by GCD to find the given LCM. LCM (similarly GCD) is an operation, since it is a function whose codomain (output) is a subset of its domain (input). But it is a many-one function, hence, it is not welldefined ("cancellative") (PL), so (in contrast to addition, multiplication), LCM (also GCD) cannot have an inverse. A further (but ignored) consequence (PL frinteger) is that, for example, the lattice of a "square-free" number -- one containing each prime exactly once -- has a free lattice that extends membership, apparently violating "The Fundamental Theorem of Arithmetic: only one way of prime factoring, ignoring order", a condition induced by the inversive nature of multiplication, bypassed by noninversive LCM (and GCD). Thus, the complemented disributive lattice (PL) of 30 = 2*3*5 has 23 = 8 members, whereas its free lattice (PL) has 18 members. (Is there a formula in the literature yielding as ouput the number of free lattice members for an input of the number of members in a complemented distributive lattice?) PL repertory, which explains that LCM is homologous to union in set theory, join in lattice theory, disjunction in statement logic, "boolean addition" in "Boolean algebra"
least upper bound
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Lebesgue identity
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(a2 + b2 + c2 + d2)2 = (a2 + b2 - c2 - d2) + (2ac + 2bd)2 + (2ad - 2bc)2 .
Lebesgue integral
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Defined in terms of upper and lower bounds via the Lebesgue measure (PL), it uses a Lebesgue sum (PL), Sn = him{Ea}, where hi is the value of the function in interval i and m(Ea } is the Lebesgue measure of the set, Ea, of points for which values are approximately hi. Covering a wider class of functions than the Riemann integral, it is often written as X f dm for measure space X and measure m .
Lebesgue measure
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Extends classical length and area to more complicated structures. An open set S Sk(ak, bk) of disjoint intervals has the measure, m L(S) Sk(bk - ak). A closed set S' [a, b] - Sk(ak, bk has measure mL(S') (b - a)Sk(bk - ak). (A unit line segment has Lebesgue measure of one; the Cantor set (PL) has Lebesgue measure zero; the Minkowski measure (PL) of a bounded closed set is the same as its Lebesgue measure.)
Lebesgue sum
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PL Lebesgue measure .
left-continuous function
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left-hand derivative
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left-handed coordinate system
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left-invariant
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Legendre equation
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Legendre's theorem
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Legendre symbol
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Legendre transformation
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lemma
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lemniscate (of Bernoulli)
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Levi-Civita connection
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Levi-Civita symbol
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Levitsky's theorem
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lexicographic order
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l'Huilier's formula
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Lie algebra
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Lie bracket
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Lie derivative
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Lie group
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Lie indicator-signal
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Following Galois' example (PL Galois indicator-signal), Lie used the group as indicator (PL) input to solvability of differential equation by quadrature as output. (The difference was that Galois groups are discointiuous; Lie groups are continuous.) An indicator under linguistic and physical control is a signal. Development of Lie thory provided linguistic control; algorithmic and programmable facility provided physical control.
lift
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May be read at http://www.harcourt.com/dictionary /browse/19/
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likelihood function
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limaçon
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limit (as antitone)
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For its two forms, discontinuous and continuous, an antitone (PL) is a one-one correspondence between elements ("steps") of an increasing ordering (maxtone) and a decreasing ordering (mintone) s. t., a bound on one ordering induces a bound on the other. (Its partial ordering extension is the more familiar Galois connection, critical to Galois theory, PL.) The continuous antitone explicates a limit more simply (as simply as climbing stairs) than the familiar "epsilon-delta" definition of the limit of a sequence (PL limit (Cauchy). The maxtone explicates the increasing sequence of terms (increasing risers); the mintone explicates distance from limit "L" (top of stairs); choice, for the maxtone, of an epsilon-distance from the limit, induces a delta-bound on the maxtone; specification allow repeated "squeezing" choices on maxtone inducing "squeezes" on mintone, "realizing" the limit. This can be adapted for any limit in analysis (PL). When adjoined, as a transfinitary operator to the finitary operattions of arithmetic (addition, subtraction, multiplication, division), then limit provides for closure (PL) of one of the inverses of exponentiation (PL), namely, logarithm (PL), by constructing real numbers (PL) as infinite vectors of Cauchy sequences (PL) of rational numbers (PL).
limit (Cauchy)
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Augustin Cauchy (1789- 1857) formulated the first satisfactory definition of a limit: The sequence Sn = s1, s2, s3, ... has the real number, L, as a limit iff, for every e > 0, there exists a d > 0 s. t. |sn - L| < e if n d . (PL limit (as antitone). The series S sn has such a limit iff its sequence of partial sums s1, s1 + s2, s1 + s2 + s 1, ... has such a limit.
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limit inferior
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May be read at http://www.harcourt.com/dictionary /browse/19/
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limit number
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limit of a function
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limit point
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Lindelöf space
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Lindelöf theorem
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line
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linear algebra
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linear combination
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linear differential equation
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linear functional
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linear hull
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linear interpolation
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linearity
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linearly dependent
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linearly disjoint
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linearly independent
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linearly ordered set
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linear manifold
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linear order
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linear programming
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linear space
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linear system
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linear transformation
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line element
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line segment
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links
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Liouville's conformability (conic) theorem
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In space (PL), the only conformal mappings (PL) are inversions (PL), similarity transformations (PL), and congruence transformations (PL). Equivalently: every angle- preserving transformation (PL) is a sphere-transformation(PL).
Liouville's theorem
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Lipschitz continuous
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literal notation
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lituus
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local
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local coordinate system
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local distortion
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local extremum
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localization
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locally arcwise connected
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locally compact
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locally connected
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locally convex
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locally Euclidean
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locally finite
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locally one to one
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local maximum
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local minimum
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local ring
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local solution
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local transformation
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locus
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May be read at http://www.harcourt.com/dictionary /browse/19/
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logarithm
This is one of the two operations which are inverse to exponentiation. (PL italicized words.) Thus, logbn = p iff bp = n. Counterexample: log103 = p implies that 10p = 3, but no natural number p or integer p or rational number p satisfies this condition, proving that logarithm is only partial for naturals, integers, rationals. Only by adjoining the transfinitary operation of limit to the finitary operations of arithmetic can we create a (real) number system making total the operation of logarithm. Unlike the finitary integral vector of naturals and the finitary rational vector of integers, this requires infinite vectors derived from Cauchy sequences of rationals known as "decimal numbers".
logarithmic integral
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PL asymptotic prime number theorem.
logarithmically convex function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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logarithmic coordinate paper
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logarithmic coordinates
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logarithmic curve
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logarithmic differentiation
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logarithmic equation
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logarithmic scale
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logarithmic series
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logarithmic spiral
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