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obelisk
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May be read at http://www.harcourt.com/dictionary /browse/19/
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object
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May be read at http://www.harcourt.com/dictionary /browse/19/
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object function
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oblate spheroid
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oblique
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oblique angle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oblique coordinates
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oblique coordinate system
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oblique lines
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oblique section
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oblique triangle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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obtuse
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obtuse angle
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obtuse triangle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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octagon
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May be read at http://www.harcourt.com/dictionary /browse/19/
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octahedron
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May be read at http://www.harcourt.com/dictionary /browse/19/
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octal number system
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May be read at http://www.harcourt.com/dictionary /browse/19/
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octant
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May be read at http://www.harcourt.com/dictionary /browse/19/
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octillion
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May be read at http://www.harcourt.com/dictionary /browse/19/
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occupancy theory
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A susbsystem of combinatorics (a.k.a. combinatorial algebra, PL). In this theory, elements (distinguishable or indistinguishable) are assigned by rule to cells (distingishable by ordering), with or without restrictions on occupancy. PL boltzmannians, fermions, bosons, multons.
odd function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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odd number
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May be read at http://www.harcourt.com/dictionary /browse/19/
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o-intersection
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Extenstion of the set-theoretic operation of intersection (t-intersection, PL) to recognize multiple tokens of type (Pl typon). Thus, given o-sets -- o-{a, a, a, b, c} and o-{a, a, b, b, c, c}, their o-intersection is:
o-{a, a, a, b, c} o o-{a, a, b, b, c, c} ~o o-{a, a, b, c} -- elements of same cardinality common to both sets.
o-join
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Lattice homologue of o-union(PL), as with homologues of o-sets given there: 120 900 ~o 12000. Note that it is equivalent to LCM (least common factor).
o-math(ematics)
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Mathematics which recognizes both kind or type and also degree or order. The protypes are factor theory and distributive lattice theory (PL). But it can be extended to set theory, statement logic, combination theory, indicator theory, diagram theory. (Until this extension takes hold, we survive the agstymie (PL) only to be stymied by the tostymie.)
o-meet
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Lattice homologue of o-intersection (PL), as with homologues of o-sets given there: 120 900 ~o 60. Note that it is equivalent to GCF (greatest common factor), which t-meet (PL) is not, being (herein) 30.
one-parameter group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one-parameter subgroup
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one-point compactification
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one's complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one-sided limit
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one-to-one
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May be read at http://www.harcourt.com/dictionary /browse/19/
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one-to-one function or one-to-one mapping
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May be read at http://www.harcourt.com/dictionary /browse/19/
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onto
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May be read at http://www.harcourt.com/dictionary /browse/19/
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open ball
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May be read at http://www.harcourt.com/dictionary /browse/19/
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open interval
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May be read at http://www.harcourt.com/dictionary /browse/19/
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open map
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May be read at http://www.harcourt.com/dictionary /browse/19/
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open mapping theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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open set
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May be read at http://www.harcourt.com/dictionary /browse/19/
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operand
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May be read at http://www.harcourt.com/dictionary /browse/19/
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operation
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A function (PL) such that the codomain (output set) is a subset of the domain (input set). An operation has an inverse (PL) iff it is a one-one function. The positive case is well-defined (cancellative), as exemplified by numerical addition. The negative case is nonwell-defined (nonconcellative), as exemplified by least common multiple (LCM) or greatest common divisor (GCD). (PL these terms.)
operational calculu
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May be read at http://www.harcourt.com/dictionary /browse/19/
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operations research
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May be read at http://www.harcourt.com/dictionary /browse/19/
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operator
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May be read at http://www.harcourt.com/dictionary /browse/19/
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operator theory
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May be read at http://www.harcourt.com/dictionary /browse/19/
optimal number system
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This would be a more enlightening label for the real number system (PL), since these are no more "real" than any other numbers, rather all are merely linguistic. But these numbers are optimal for specific conditions. Thus, the circle (with circumference, C, and diameter, d) is the plane figure providing maximum area with minimum perimeter (optimality), encapsulated in the irrational real number, p = C/d. Again, the square is the polygon providing maximum area with minimum perimeter, and its diagonal (segment equally subdividing figure, homologous to diameter of circle) is a multiple of the square root of two, historically the first irrational real number to be discovered. Another such irrational real is the golden mean (PL), praised in many female forms, observed in Nature, and applied in architecture, sculpture, painting, etc. Both Eudoxus (c. 408-355 BC) and Richard Dedekind (1831-1916) showed that a "slot" exists between any consecutive pair of fractions. Students should be taught that the optimal number system reveals to them some of the "secrets" of "Nature" -- "slots" to "squeeze in" any "just right" beauty or utility -- motivating students to seek optimality in their individual and public lives. (PL Bellman on this Homepage.)
optimal policy
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May be read at http://www.harcourt.com/dictionary /browse/19/
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optimal system
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May be read at http://www.harcourt.com/dictionary /browse/19/
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optimization
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May be read at http://www.harcourt.com/dictionary /browse/19/
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optimization theory
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May be read at http://www.harcourt.com/dictionary /browse/19/
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OR
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orbit
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order
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A procedure for assigning element to a sequence (PL). For example:
  • The order of a derivative or differential operator refers to how many times it is applied.
  • An n square matrix is said to be of order n.
  • A polynomial with n as largest exponent of a term is of order (or degree) n.
Etc. (PL order relation.)
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ordered field
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ordered pair
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ordered ring
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ordering
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a differential equation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a graph
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a group
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of an element
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of an elliptic function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a pole
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a tensor
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of a zero
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order of magnitude
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order-preserving
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May be read at http://www.harcourt.com/dictionary /browse/19/
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order relation
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A binary relation which is transitive (PL). (PL also total ordering and partial ordering.)
ordinal (extollent, PL) equivalence
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Equivalence in o-mathematics, PL -- arithmetic factor theory, sets, lattices, etc. -- which recognizes multiple tokens of a type (PL typon). It may be denoted as ~o. (PL t-union, o-union, t-intersection, o-intersection, t-join, o-join, t-meet, o-meet.)
ordinal number
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The ordinal number c is the transitive set (PL) of all ordinals preceding c. (PL cardinal number and the paradox of counting.) Children can easily be taught ordinality via a brownbag model.)
ordon
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A gnomon (PL) of the ordinal segment of measurement scales (PL).
outerproduct (multivector)
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xxxxxx (PL innerproduct and multiproduct.)
ordinary differential equation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ordinary point
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May be read at http://www.harcourt.com/dictionary /browse/19/
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ordinate
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May be read at http://www.harcourt.com/dictionary /browse/19/
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Ore's theorem
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orientable manifold
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orientation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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origin
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthocenter
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal
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In elementary geometry: perpendicular. For two vectors: zero inner product, a condition which extends to any inner product space whereby two subspaces are orthogonal iff every element of one is orthogonal to every element of the other subspace. Similar definitions for any symmetric or alternating bilinear form, any Hermitian form.
orthogonal basis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal complement
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal functions
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal group
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May be read at http://www.harcourt.com/dictionary /browse/19/
orthogonality
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PL oethogonal. The equations for rotation of coordinate axes (PL), can be rewritten, by defining, x = x>/sub>1, y = x2, a11 = cos q, a 12 = sin q, a21 = - sin q, a 22 = cos q, to become: x'1 = a11x1 + a12x2, x'2 = a21 x1 + a22x2. The coefficient, aij can be interpreted as one or more direction cosines (PL): the cosine of the angle between x'i and x'j, that is, a12 = cos(x'1, x 2) = sin q, a21 = cos(x'2, x1 ) = cos (q + p/2) = - sin q. Given this new notation, we can rewrite the previous coordinate transformation in summation form: S j=2j=1aijxj, i = 1, 2, which easily extends to any number of dimensions. Now, the sine, cosine functions satisfy orthogonality conditions : 1/L L -L cos(jpL) cos(kpL)dx = 1/L L-Lsin(j pL) sin(kpL)dx = d jk<, where djk is the Kronecker symbol (PL), s. t. djk = 1, j = k and djk = 0, j k. djk = 1, j = k. The orthogonality of the sine, cosine functions is inherited by their "encodments": aijajk = djk. Hence, orthogonality is in implicit in the rotation of coordinate axes equations (PL), hence, also in the test for vectorhood (PL) and in the inner product for vectors (PL) or in general inner product spaces (PL).
orthogonalization
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonally equivalent matrices
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal matrix
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal polynomials
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal projection
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal system
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal trajectory
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal transformation
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthogonal vectors
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthographic projection
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthonormal basis
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthonormal coordinates
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthonormal functions
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthonormal vectors
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May be read at http://www.harcourt.com/dictionary /browse/19/
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orthoptic
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oscillating function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oscillation of a function
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May be read at http://www.harcourt.com/dictionary /browse/19/
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osculating circle
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May be read at http://www.harcourt.com/dictionary /browse/19/
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osculating plane
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May be read at http://www.harcourt.com/dictionary /browse/19/
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o-union
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Union of o-sets (PL), recognizing multiple tokenage of types (PL typons). Thus, given two o-sets -- o-{a, a, b, c, c} and {a, b, b, c} -- their o-union is: o-{a, a, b, c, c} o {a, b, b, c} ~o o-{a, a, a, b, b, b, c, c, c}. Note that, for the factor homologues of these o-sets -- respectively, 900, 90, 27000 -- it is their LCM (least common mulltiple), while the t-union (PL) of these sets would be homologous only to 30.
outer automorphism
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May be read at http://www.harcourt.com/dictionary /browse/19/
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outerproduct (of multivectors, PL)
. Given two multivectors of the same grade, their outerproduct raises this one grade. It is the same as exterior product of differential forms theory. Thus, for basis vectors , e1 = [1, 0] and e 2 = [0, 1], their outerproduct is: e1 Ùe2 = i = -1. Then, for vectors, a = [a1, a2], and b = [b1, b2], their outerproduct is a = (a1b2 - a2b1 ) e1Ùe1 i = (a 1b2 - a2b1) -1, equivalent to a multiple of the "imaginary vector", hence, a subplane of the plane containing the component vectors. It is easily transformed into the standard cross or vector product of vectors, projecting from the plane of the component vectors.
oval
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May be read at http://www.harcourt.com/dictionary /browse/19/
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oval of Cassini
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May be read at http://www.harcourt.com/dictionary /browse/19/
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