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tableau
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tabular interpolation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Tait's conjecture
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangent
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangent bundle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangent formula
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangential
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangential component
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangent space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangent vector
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tangle
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A subfield of topology (PL), of interest to children. Example: With drawing instrument (pencil, pen, etc.) and paper, scrawl a curve, perhaps crossing itself many times, to end connected to the initial point: a table. Problem: Given a point "within" the scrawl, is it "inside" the tangle or "outside" it? (Solution procedure.)
tanh
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May be read at http://www.harcourt.com/dictionary /browse/19/
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telegraph equation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ten's complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor analysis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor component
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor contraction
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor differentiation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor field
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tensor product
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tera-
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May be read at http://www.harcourt.com/dictionary /browse/19/
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term
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May be read at http://www.harcourt.com/dictionary /browse/19/
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terminal vertex
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ternary
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tessellation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tesseract
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A 4-D rectrex (PL), constructed from 8 cubes. (For a tesseract house see REFERENCE [30].
test function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tetradic
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tetragon
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tetrahedron
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May be read at http://www.harcourt.com/dictionary /browse/19/
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theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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theory
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May be read at http://www.harcourt.com/dictionary /browse/19/
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theory of equations
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May be read at http://www.harcourt.com/dictionary /browse/19/
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third isomorphism theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Thom's theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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threshold
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Tietze extension theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tile
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May be
tile (pegboard) geometry
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When polygonal structures are drawn on tiles or outline by string or yarn or by other means on a plane divided into square units, the area of this polygon can easily be calculated by the formula: A = i + b/2 - 1 square units, where i denotes number of unit-points interior to polygon and b denotes unit-ponts on boundary of polygon. This subject is also known as Minkowski Geometry. It is easily taught to children.
read at http://www.harcourt.com/dictionary /browse/19/.
tiling the plane
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May be read at http://www.harcourt.com/dictionary /browse/19/
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time
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A recusion of antitonic chronons. That is, a recursion (PL) on that chronon (PL) having antitonic interior structure ("black box"). Open question: What is the nature of "successor function" for chronons? What is the structure in space-time which is homologous to the successor function (PL) in arithmetic?
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time-series analysis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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t-math(ematics)
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A mathematical system recognizing only type (kind), not order (degree) -- in contrast to an o-math(ematics) (PL), which recognizes both type and order. Classical or default set theory does not recognize multiple tokens of an element. Thus, {2, 2, 3} = {2, 3}. The factor theory of "square-free" numbers is a t-math. But factor theory, in general, recognizes multiple tokens -- for example, that, 12 = 2*2*3, while 6 = 2*3, a distinction showing that general factor theory is an o-math. (This is one technical reason why "The New Math", instituted in 1958, confused students, teachers, and parents by not supporting "the old math of arithmetical factoring".)
topological composite structure
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One which can be written as the Cartesian product of two simpler structures of lower dimension, hence factorable. Example, the cylinder as: C X S, for circle, C, and line segment, L -- every point of the circle is matched by a point on the line segment, constructing the cylinder. (PL topological prime structure, topological prime conjecture.)
topological dimension
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological dual
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological mapping or topological transformation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological prime structure
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One which cannot be written as the Cartesian product of two simpler structures of lower dimension as the cylinder can), hence not factorable. Example: möboid (PL), cross-cap (PL), Boy surface (PL), Roman surface (PL). (PL topological composite structure. topological prime conjecture.)
topological prime conjecture
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The various nonorientable surfaces and solids can be shown to be topologically homologous to prime integers. Among these are Möboid (PL), cross-cap (PL), Boy surface (PL), Roman surface (PL), Klein bottle (PL), which might be comparable to primes 2, 3, 5, 7. An essential of this homology would be some notion of ordering these topological primes. The possibility of this arises from the finding by Garrett Birhoff, in his Lattice Theory that topology can be derived from order theory. (PL topological composite structure, topologicalprime.
topological product
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological property
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topological vector space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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topology
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More general than GEOMETRY, TOPOLOGY is simply the study of CONNECTIONS (while geometry is the study of CONNECTED systems with specific SHAPE and SIZE). TOPOLOGY IS NECESSARY FOR GEOMETRY, BUT GEOMETRY IS NOT NECESSARY FOR TOPOLOGY. Infants and small children daily grapple with topology. A simple trick illustrates topology. A man can remove his vest without taking off his coat, because (topologically) the vest is outside the coat in the same sense that a paper in a waste basket is outside the basket. How do it? Put arm through, say left sleeve of vest; pull the coat through this vest-hole, leaving the vest hanging on the right arm; then pull the vest out the right sleeve. A triangle, a square, a circle, a rectangle are all equivalent in topology! Because each figure is connected within the plane in the same way. Each figure separates the plane into one inside region and one outside region. Topologists have a special name for any figure separating the plane into one inside and one outside region: A JORDAN CURVE (named for the French mathematician, Camille Jordan (1838-1922), who first gave an enlightening discussion of this subject). However, the figure-eight topologically differs from the square, circle, or rectangle. The figure-eight is not a Jordan curve because the figure-eight connects differently within the plane by comparison to the Jordan curve. The figure-eight separates the plane into TWO distinct INSIDE regions and one outside region. However, a single cut on the figure-eight transforms it into an equivalent of the Jordan curve. Transforming a figure by a cut provides us a classifier for plane figures, just as plants or animals can be biologically classified as of this genus (generative pattern). We define the Jordan curve (triangle, circle, rectangle, etc.) as of genus zero because the Jordan curve is the simplest of figures -- connects most simply in the plane. A triangle requires zero cuts to transform it into the simplest form; similarly, the square, etc. Since a single cut transforms a figure-eight into a Jordan curve (genus zero structure), the topologists says the figure-eight is of genus one. A figure requiring two cuts to become equivalent to a genus-zero figure is of genus two. Etc. Kids can be taught to understand the topology of tangles (PL) they draw on paper. Two structures are topologiclly equivalent iff each transforms into the other tearing or ripping. Why treat the triangle, circle, square, rectangle as the same? PL topophysics. An important part of topology is the mathematical theory of knots and braids, for understanding molecular chemistry, especially for pharmaceuticals. The unravelling of knots mirrors the solution of algebraic equations. But topology is not taught in our schools, or tested in standardized tests. Teachers, students, parents, citizens alike are ignorant of its presence in our daily lives and the great promise of its resources if we use them. Topology has been called "rubber-sheet geometry", since any figure drawn on a rubber sheet and stretched is topologically unchanged. Using putty or playdough, kids could experiment with topological structures -- if only teachers knew enough or cared enough to encourage them to do so. Topology is the creation of the great Swiss mathematician, Leonhard Euler (1707-83). PL topology, set-theoretic; topology, general; topology, algebraic; topology, differential. "The difference between algebraic topology, differential topology, general topology, etc. is the measure we choose for the manifold [PL]."[28]
topometrics
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This is a better label for the subject traditionally labeled "geometry" (from "measurement of earth", for surveying fields). But a geometry is simply a topology (PL) adjoined to a metric (PL). Most of our daily concerns are topological rather than topometric (geometric)
topophysics
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This concerns mathematical physics dependent upon topology. Particle physicists have begun to discuss topological conservation laws, in contrast to the conservation laws (emanating in the past), which are metrical (PL). But "standard physics" can only be stated TOPOLOGICALLY, that is, "within proper systems". For, topophysics at least dates to Newton's treatment of inertia and his "first law if motion": "A particle in uniform motion remains in motion, or at rest remains at rest, unless acted upon by a force". This is a topological statement about what happens within a simply closed volume of space: an inertial system. Outside an inertial system, "things become more complicated". When physicists speak of a "Coriolis force", this is within a geophysical system", and the Coriolis force is not observed outside the earth, another topological statement. Similarly, in electromagnetics. Outside a moving system containing an electric charge, a compass jerks at the induced magnetism, not observed within the system (another topological statement). Without realizing, we observe topologogy in electric circuits. Connect two "electric" wires to poles of a battery and to leads of a small light bulb, with a switch in between. With switch closed (dividing the space into an "inside" and "outisde", electricity flows through wires, switch and bulb, to light up the bulb. But what shape does the circuit have to be? A circle or triangle or square or rectangle or any genus-zero plane figure (PL topology) achieve the same results -- all providing the connectivity required for an electric circuit. But a genus-one figure-eight circuit could induce capacitance effects across the touching wires. (Connectivity-along-the-line becomes Neighbor-connectivity.) Similarly, the two-pole light switch on walls of your "living space". Physics students study two "Kirchoff circuitry laws", which Gustav Kirchoff (1824-87) derived by topological reasoning he learned from his mathematician friend, August Möbius (1790-1868, PL möboid). The widespread ignoring of topophysics may explain the famous Einstein-Bohr debates. Also, PL cartesian product for its application in topology. (Students of biophysics should the topology in their field -- for example, what is different "inside the cell" from its "outside".)
toral subalgebra
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May be read at http://www.harcourt.com/dictionary /browse/19/
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toric surface
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May be read at http://www.harcourt.com/dictionary /browse/19/
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toroid
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May be read at http://www.harcourt.com/dictionary /browse/19/
torsion element
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May be read at http://www.harcourt.com/dictionary /browse/19/
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torsion-free
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May be read at http://www.harcourt.com/dictionary /browse/19/
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torsion group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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torsion module
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May be read at http://www.harcourt.com/dictionary /browse/19/
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torsion of a curve
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May be read at http://www.harcourt.com/dictionary /browse/19/
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torus
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May be read at http://www.harcourt.com/dictionary /browse/19/
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totally bounded
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May be read at http://www.harcourt.com/dictionary /browse/19/
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totally disconnected
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May be read at http://www.harcourt.com/dictionary /browse/19/
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total order(ing)
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Since any ordering relation {PL} must be transitive (PL), a total ordering is an ordering O that is also
  • nonreflexive: for any a, a O a never holds,
  • and asymmetric: if a O b then b O a does not hold .
The best known formal total ordering is the set of natural or counting numbers. (PL total ordering.)
totally ordered set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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total inverse operation
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An inverse operation (PL) is total on a number system (PL) iff the result of the operation exists within the system, equivalently, the system is closed under the operation ("all-in-the-family"), otherwise it is partial. Rendering total the inverses of arithmetic extends the number systems beyond the naturals.
totient (formula) function (Euler's phi-function)
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The label derives from Latin, toties, meaning "as many". Denoted f(n), for number, n, for which another number is a totitive (PL) iff they are mutually coprime (have only unity as common factor). This formula computes the number of totitives of a given by reference to its prime factors, pi, i = 1, 2, ...,r: f(n) = n(1 - 1/p1)(1 - 1/p1) ... (1 - 1/pr).
totitive
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Given natural number or integer, n, then the natural or integral number, m -- or they are mututally totitive -- iff they have only unity as a common factor. Any such number (prime or not) is a totitive of a prime if it is less than the prime. Thus, 1, 2, 3, 4, 5, 6 of prime 7 .
tough
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tournament
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trace
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tractrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trail
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trailing zero
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transcendental field extension
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transcendental function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transfilter induction
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transfinite number
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transform
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transformation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transformation group
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A group of elements interpretable as transformations (PL) on some space; the group operation is interpreted as composition (PL) of functions.
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transition probability
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transitive relation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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translation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transport
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transpose of a matrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transposition
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May be read at http://www.harcourt.com/dictionary /browse/19/
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transversal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trapdoor fumction
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f: {0,1}I(n) X {0,1}n {0,1}m(n) s. t. it is a one-way-functionfixed public key (PL) y e {0,1}I(n), f(x,y) fy(x) acts as map n m(n) in bits (PL). If so, an algorithm exists s. t. {y, fy(x), z} x' s. t. fy(x') = fy(x) for some trapdoor key (PL), z e {0,1}k(n). A hash function (PL) is a one-way hash function if, also, given messages M, f(M), it is difficult to discover a message M' M s. t. (by transmission) f(M') = f(M. Open problem: Can a trapdoor one-way function be constructed from any one-way function? Example of trapdoor one-way function: the factoring of the product of two large primes , as in RSA encryption (PL), conjectured to be trapfoor one-way.
trapezium
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trapezoid
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trapezoidal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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traveling-salesman problem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tree
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May be read at http://www.harcourt.com/dictionary /browse/19/
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triangle
A geometric structure resulting from the intersecting of three line segments. The specification merely of the interecting points is "the 1-D triangle"; the typical triangle is "the 2-D triangle". The triangle, with the interior space filled in is "the 3-D triangle".
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triangle, scalar
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An ordinary triangle, so labeled to distinguish from "vector triangle".
triangle inequality
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May be read at http://www.harcourt.com/dictionary /browse/19/
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triangle, vector
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A triangle each side of which is a vector (PL), such that, given vectors, v1, v2, v3, the "end" of v2 begins at the "arrow" of v1, while the "end" of v3 begins at the "end" of v1 and its "arrow" goes to the "arrow" of v2, "closing the triangle", such v3 is "the sum of the other two vectors". (PL innerproduct.)
triangular
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May be read at http://www.harcourt.com/dictionary /browse/19/
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triangular face
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May be read at http://www.harcourt.com/dictionary /browse/19/
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triangular matrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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triangulation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trichotomy property
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trident of Newton
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May be read at http://www.harcourt.com/dictionary /browse/19/
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tridiagonal matrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trigonometric
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trigonometric functions
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trigonometry
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trihedral
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trillion
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trinomial
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trisect
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trisecting the angle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trisectrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trit
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trivial
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May be read at http://www.harcourt.com/dictionary /browse/19/
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trochoid
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May be read at http://www.harcourt.com/dictionary /browse/19/
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truncate
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May be read at http://www.harcourt.com/dictionary /browse/19/
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truth function
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Assigns to any statement (PL) of statement logic (PL) the "truth value" -- "T" fpr "true", "F" for "false" -- of that statement. Provides for modeling of a simple, complex, comound statement by one or more instances of a truth table.
truth functional
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A simple statement (PL) is truth functional, since it requires no logical connective (PL) to formulate. The connectives of and, or, conditional, biconditional, negation are truth functional because the truth value of any complex or comopound statement constructed by such connectives has a truth value (PL) dependent solely upon the truth values of the given connectives in use. This puts a severe limit upon the use of such logic to model daily language -- with its moods of imperative, interrogative, petitive, etc. -- and even computer programming language, which can be modeled declaratively, except for the imperative of assignment. This is bypassed via vector logic (PL).
truth table
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The truth value (T, F) of any simple/compound statement of statement logic (PL) can be "measured" by entries in a table of "T, F" entries. (Sometimes "1,0" are used in place of "T, F".) Given n independent simple statements (lacking any connective, PL), the subtable for each as 2n entries. ( Example for n = 2.)
truth value
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Statement logic (PL) adheres to the bivalent convention laid down by Chrisippus the Stoic (c. 280-207 BC): a statement is either TRUE or FALSE. (PL truth table.)
T space
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Tukey's lemma
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Turing computable function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Tutte's Hamiltonian theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Tutte's matching theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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two's complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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two-sided ideal
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May be read at http://www.harcourt.com/dictionary /browse/19/
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two-valued logic
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More often labeled bivalent logic, the criterion that a statement is either true or false is attributed to Chrisippus the Stoic (c. 280-207 BC).
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Tychonoff theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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typical fiber
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May be read at http://www.harcourt.com/dictionary /browse/19/
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typological (extensional) equivalence
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The standard equivalence in t-mathematics (PL) -- sets, lattices, etc. -- which recognizes only one token of a type It can be distinguished as ~t . (In contrast, PL ordinal equivalence .)
typon
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Unit of gnomon (PL) of typology (typological measure). It also labels "atom" (PL) of a lattice. In a factor , the prime numbers are atoms>. (PL also ordon.)
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