B
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Baire category theorem
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A nonempty metric space (PL) cannot be a ,I>union (PL) of countable (PL) family of nowhere dense spaces (PL).
Baire space
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Any topological space which is homeomorphic to a complete metric space (PL).
balanced integer.
An integer (PL) is p-balanced for prime p iff, for all binomial coefficients (PL), C(n,k), k = 0, 1, ..., n (mod p), the number of quadratic residues (mod p) equals the number of nonresidues.
Baker-Campbell-Hausdorff series
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The series Z = (exeyew), for noncommuting x, y, w
ball
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Interior of a sphere.
Banach algebra
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A field (PL) whose norm makes it a Banach space (PL), with continuous multiplication (PL).
Banach space
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A complete (PL) vector space (PL) whose topology (PL) is determined by its norm, with continuous operations.
Banach-Steinhaus theorem
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A pointwise-bounded (PL) family of continuous (PL) linear operators (PL) from a Banach space (PL) to a normed space (PL) is uniformly bounded.
Banach-Tarski paradox
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According to the Axiom of Choice (PL choice function), a geometric structure the size of the moon can be dissected into as few as five parts, then reassembled to about the size of a pea which could be put in a shirt pocket. (For comment on this, see fable.
bandwidth
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The range or difference between the limiting frequencies of a continuous frequency band. In graph theory (PL), givem L(G) as the set of all distinct integral labelings of the vertices of graph G, the bandwidth of graph G is minmax|l(u) - l(v)|, for the length of all edges of the graph.
Barnette's conjecture
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Every 3-connected (PL) bipartite cubic planar graph is Hamiltonian.
barrel
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A solid of revolution (PL) with parallel circular top and bottom, common axis, side of a curve symmetrical about the midplane.
Barth sextic
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In complex 3-D projective space having the maximum possible number of ordinary double points (PL). Of these, 20 nodes (PL) are at the vertices of a a regular dodecahedron (PL) of side length 2/f, and 30 are at the midpoints of the edges of a concentric dodecahedron (PL) of side length 2/f2, for golden ration (PL), f. Discovered by W. Barth, with implicit equations: 4(f2x2 - y 2)(f2y2 - z2)( f2z2 - x2) - (1 + 2f)(x2 + y2 + z2 - w2)2w2 = 0. The B. s. is invariant under the icosahedral group (PL). Under the map, (x, y, z, w) (x2, y2, z2, w2) , the B. s. is the eightfold cover of the Cayley cubic (PL).
barycenter
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In complex 3-D projective space (PL) with the maximum number of double points (PL). Of these, 20 nodes (PL) are at the vertices of a regular dodecahedron (PL) of side length, 2/f, and 30 are at the midpoints of the edges of a concentric dodecahedron of side length, 2/f 2, for the golden ration (PL),
barycenter
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The centroid (PL) in a baracentric coordinate system (PL).
barycentric coordinates
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Number triples for masses at vertices of a reference triangle determining centroid of the three masses.
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base
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The number if digits available for denoting a number system; base n provides n digits: 0, 1, 2, ...,n-1. The decimal or denary base uses ten digits (0,1,2,3,4,5,6,7,8,9). The binary base uses two digits (0,1). Also known as radix.
base of a logarithm
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The arithmetic operation of exponentiation (bp = x) has two distinct inverse operations (since it is not commutative): logarithm and root extraction (PL); bp = x -> logbx = p. The operand b is the logarithmic base. The most familiar bases are the exponential number e, in "natural logarithms"; the number 10 for "common logarithms"; and the number 2 for "information bits".
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base of a number system
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The number whose powersplace value (PL) in a positional representation (PL) of a number system (PL).
basic polynomial sequence
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A polynomial sequence, pn(x) is a b. p. s. for a delta operator (PL) Q if: p0(x) = t; pn(0) = 0, n > 0; Qpnp(x) = npn-1 (x). A b. p. s. for a delta operator is a binomial-type sequence (PL) of polynomials, and conversely.
basis
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A set of linearly independent (PL) ectors in a vector space (PL) such that each vector in the space can be represented as a finite linear combination of vectors from the basis set. Vectors in the basis are called base vectors. The number of base vectors is the dimension of the vector space and may be infinite.
bayesian
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Treating a probability (PL) as a degree of belief modifieable by experience in accordance with Bayes' rule (PL). This contrasts with the frequentist (PL) approach, represents a probability as the relative frequency of occurrence.
bei function
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The imaginary part of Jn(ze±3pi/4), where Jn is the nth Bessel function. PL ber function.
ber function
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The real part of Jn(ze±3pi/4), where Jn is the nth Bessel function. PL bei function.
Bernoulli differential equation
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A linear (PL) differential equation (PL) of the form: Dxy + yf(x) = yng(x).
Bernoulli's lemniscate
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PL lemniscate.
Bertrand's postulate
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For n > 3, there exists at least one prime (PL) between n and 2n - 2.
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Bessel differential equation
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x2Dxxy + xDxy + (x2 - m2) = 0 , with regular singularity (PL) at 0, irregular singularity (PL) at . Solutions are Bessel functions (PL).
Bessel function
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Solutions of Bessel differential equation (PL). For n a positive or negative integer , the nth Bessel function, Jn (x), is the coefficient of tn in expansion of ex[t - 1/t]/2 in powers of t, 1/t.
Betti number
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A topological invariant , the maximum numbers that do not divide a surface. The n th Betti number is the rank (PL) of the nth homology group (PL).
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biconditional
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Statment logic connective (PL): for statments P, Q, if P, then Q, and if Q, then P; symbolized P <-> Q; Q is a necessary and sufficient condition for P, and vice versa. Also labeled statement equivalence.
biconnected graph
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A graph with no vertex such that removal of its node would disconnect the graph.
bicorn
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Curve with equation: y2(a2 - x2 = (x2 + 2ay - a2)2 .
biharmonic function
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A solution of 4 = 0.
bijection
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A one-one corespondence between two setsl equivalently, a function that is both an injection (into) and a surjection (onto) -- PL both.
bilateral Laplace transform
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A Laplace transform between positive and negative infinity.
bilinear expression
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A functional expression that is separately linear (PL) in each of its two functands (PL).
bilinear function
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PLbilinear expression.
bilinear map
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PL bilinear expression
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billion
A number written 1,000,000,000 = 109.
bimodul (hays)
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A number system which bifurcates into two subsystems each of which forms a modul (PL), i.e., closed under subtraction. Whereas each system of the naturals, integers, rational, and reals forms monolithically a modul -- allowing the basic vector structure of the latter three to be hidden by signs -- by contrast, in the complex number system, the "imaginary" part forms a modul which is subtractively closed off from the "real" part, hence, the vector structure cannot be hidden by mere signs. A consequence is that a complex number also provides the simplest form of a spinor (PL) and the basis of a recursive generation of "the arithmetic of Clifford numbers" or multivectors (PL). Find a further consequence of bimodulity in Quanta, by P. W. Atkins, p. 81, showing that a complex wavefunction actually manifests itself as two separate wavefunctions, one for the real component and one for the imaginary component preceding it in time; the higher the energy, the faster the wavefunction oscillates between real and imaginary.
binary number
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A number in the binary number system.
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binary number system
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A number system using two as the base (PL), written with only the digits 0, 1.
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binary operation
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Rule for combining two elements of a set to yield a third element of the set. (Addition, subtraction, multiplication, division, exponentiation, logarithm, root extraction.)
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binary system
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PL binary number.
binary-to-decimal conversion
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Transformation (PL) from a base 2 (binary, PL) number representation (PL) to a base 10 (decimal. PL) representation.
binary tree
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A tree with two branches at each fork and one or two leaves at each branch.
binomial
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A polynomial (PL) consisting of two terms (PL).
binomial differential equation
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(Dxy)m = 0.
binomial equation
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PL quadratic equation.
binomial surd
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A surd is a root of a number. A binomial surd is a binomial (PL) of the form ac1/m + bd1/n or a + bd1/n, where m, n are integers greater than 1 and the indicated roots are irrational numbers, while conjugate binomial surds are pairs of the form a + bd 1/n and a - bd1/n or ac1/m + bd1/n and ac1/m - bd1/n.
binormal
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The binormal to a curve at a point P in 3-D Euclidean space is the vector through P that is normal to the osculating (PL) plane of the curve at P. The direction of the vector is chosen so that, together with the positive tangent and principal normal to the curve at P, it forms a right-handed Cartesian system (PL).
bipartite cubic graph (bicubic graph, Tutti conjecture)
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Tutti (1971) conjectured that all 3-connected (PL) bicubic graphs are Hamiltonian.
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bipartite graph
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A set of vertices disjoined into two sets s. t. no two vertices in same set are adjacent.
biquadratic
. A polynomial (Pl) of degree 4. Also "quartic"..
biquinary abacus
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Chinese-style abacus with markers separated into 2-part and 5-part sections.
biquinary notation
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Mixed-based notation of numbers as represented on a biquinary abacus (PL). Digits are grouped in pairs, the first of which indicates 0 or 1 unit of 5, and the second 0, 1, 2, 3, or 4 units of 1.
birectangular
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A geometric structure (not necessarily planar) with two right angles (PL).
bisector
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A line or hyperplane dividing a given angle into two equal angles. Also, a point, line, or plane passing through the midpoint of a line segment.
bit
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Place value in a binary number, representing 0 or 1. The basic unit of information in a digital computing system. (Acronym for binary digit.)
body of revolution
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The 3-D structure formed by rotating a planar curve or planar region about a line (axis of revolution) in the same plane. Given a closed curve which does not intersect the axis of revolution, the body of revolution is an annular solid.
boltzmannian theory
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A subsystem of occupancy theory (PL), in turn a subsytem of combinatorics (PL). If b number of boltzmannians in set B are assigned to a c number of cells in set C, then the number of possible assignments is cb. This is homologous to the number of functions from set B to set C. PL fermions, bosons.
Bolzano's theorem
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If f is real-valued and continuous on the real closed interval [a, b] and f(a) > 0 and f(b) < 0, then there exists some number x0 in the open interval, (a,b) for which f(x0) = 0.
Bolzano-Weierstrass theorem
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For bounded infinite point set (PL), S, there exists a point x as limit point (PL) of S. (Usually credited to Karl (1815-1897) Weierstrass but proved by Bernard Bolzano (1781-1848) in 1817, and apparently was known to Augustin Cauchy (1789-1857).
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Boolean algebra
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A misnomer and distortion[5] of the work of George Boole (1815-1864). Boole intended to develop structure homologous to sets with multiple tokens and with signs (positive and negative). But his untimely death left his work to the distortions of W. S. Jevons. The system described by this name is a set of elements a, b, c, ... with binary operators ("join"), ("meet") obeying an idempotent law: a a = a, a a = a; commutative laws: a b = b a, a b = b a; associative laws: a (b c) = (a b) c, a (b c) = (a b) c; distributive laws; a (b c) = (a b) (a c), a (b c) = (a b) (a c); absorption laws: a (a b) = a (a b) = a; with univeral bounds O, I such that O a = O, O a = a, I a = a, I a = I; and a unary operation a -> a' such that a a' = O, a a' = I. Other instances of "Boolean algebras": the subsets of a set under appropriate set-theoretic operations; a statement logic under its appropriate operations; a complemented distributive lattice under its appropriate operations. (This shows that a "Boolean algebra" has some of the general aspects of a repertory -- PL -- without suggestions of educational possibilities and no reference to semiotic transformations.)
Boolean function
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Elements of a Boolean algebra (PL) with unique representation (up to order) as complete products (PL), for total membership of 2nn.
Boolean ring
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A ring (PL) with unit s.t. every element is idempotent -- applies to Boolean algebra (PL).
Borel algebra
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A sigma algebra (PL) generated by a collection of open sets (PL) (or closed sets, PL).
Borel (probabilty) measure
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A measure (PL) on a Borel algebra (PL): all contiuous functions herein are measurable.
bordant
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Two manifolds forming boundary of a third manifold.
bordism
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Term no replacing cobordism. PL bordant.
Borel set
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Element of a Borel algebra .
boson theory
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A subsystem of occupancy theory (PL), in turn a subsytem of combinatorics (PL). The members of a set B, containing b in number, is assigned to a set C of cells, c in number, no restrictions on occupancy. (This differs from boltzmannian theory in that boltzmannians are distinguishable.) The number of possible assignments is the negative binomial or Pascal distribution: (b - 1)!/(b - c)!(c - 1)!, where n! = n(n-1)(n-1)...(2)(1), the factorial function (PL). Example: Let B denote number of indistinguishable passengers on a bus, and let C denote number of bus stops. (PL fermions.) In quantum physics, bosons are the carriers of force or interaction between particles.
bottlecap (figurate, Pythagorean) geometry
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Pythagoras created a figurate or discrete geometry of structures outlined by points (represented by stones, etc.). This is easily taught to children via bottlecaps from soft drinks. (PL gnomon.)
boundary
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The set of all boundary points (PL) of a given subset of a topological space (PL).
boundary condition
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Constraining the solutions of a system of differential equations in terms a specified set of values of the independent functand(s), or one or more values of a difference equation or recurrence relation, in order to initiate the computation.
boundary point
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Member of the closure of a set and of the closure of its complement.
boundary value problem
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Solving a system of differential, integral, or difference equations with its boundary conditions by requiring the the operators of the system to take on specified values along portions of the boundary of the space-time solution domain.
bounded function
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A real function f for which there is a positive real number r, bounding f such that f(x) < r for all x in the domain of f. In general, a function whose range is a metric space is bounded if an open ball (PL) exists that entirely contains the range of the function.
bounded set
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A set of real numbers M is bounded above if there exists a real (upper bounding) number R such that x B for all x in M. M is bounded below if (lower bounding) R x. Also, a subset of a metric space is bounded if it can be contained in some open ball of finite radius.
bounded variation
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Property of a function (PL), f(x) if, for the closed interval (PL), [a, b], there exists an L such that |f(x1) - f(a)| + |f(x2) - f(x1)| + ... + |f(b) - f(xn-1)| < L
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Bowditch curve
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PLLissajous curve .
Boy surface
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A nonorientable surface (PL) and sextic surface (PL), one of three surfaces produced by a Möboid (Möbius strip, M. band) (PL) to a disk (PL); the other two are the cross-cap (PL) and the Roman surface (PL). The B. s. is a model of the projective plane (PL) w/osingularities . In 1986, contradicting Hopf, F. Apéry showed that the B. s. can generated by the general method for nonorientable surfaces: f1(x, y, z) = ½ (2x2 - y2 - z2)(x2 + y2 + z2) + 2yz(y2 - z2) + zx(x2 - z2) + xy(y2 - x 2), f2(x, y, z) = ½(3)-1/2[(y2 - z2) (x2 + y2 + z2) + zx(z2 - x2) + xy(y2 - x2), f3(x, y, z) = 1/8(x + y + z)[(x + y + z)3 +4(y - x)(z - y)(x - z)]. Inserting x = cos u sin v, y = sin u sin v, z = cos v produces the Boy surface. A homotopy (PL) (smooth deformation) between the Roman surface and Boy derives from equations: x(u, v) = [(2) ½) cos (2u) cos2 v + cos u sin (2v)]/[2 - a(2) ½ sin (3u) sin (2v)], y(uv) = [(2)½) sin (2u) cos2 v + sin u sin (2v)]/[2 - a(2)½ sin (3u) sin (2v)], z (uv) = (3 cos 2 v)/[2 - a(2)½ sin (3u) sin (2v)], a = 0 R.s., a = 1 B.s..
boxcar function
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The function (PL), Bc(a, b) = c|H(x - a) - H(x - b)| which is equal to c for a < x < b and zero otherwise, where H(x) is the Heavyside function (PL). The unit rectangle function is B1(-1/2, 1/2).
brachistochrone problem
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To find curve s.t. a bead sliding down it (from rest) does so in the least time. It is the cycloid (PL). (PL tautochrone problem.)
bracket
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PL commutator.
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branch
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A maximal subtree (PL) containing a given point as endpoint.
branch cut
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A line in the complex plane rendering continuous (PL) a multivalued function./DD>
branch of a function
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PL branch cut .
branch point
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PL branch.
Brianchon's theorem
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The dual (PL) of Fermat's theorem (PL).
bridge
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The edges (PL) of a connected graph (PL) whose removal disconects the graph.
Property of a functiOn (PL), f(x) if, for the closed interval (PL), [a, b], there exists an L such that |f(x1) - f(a)| + |f(x2) - f(x1)| + ... + |f(b) - f(xn-1)| < L.
Brouwer fixed-point theorem
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Any contiuous function has a fixed-point (PL) in a unit n -ball (PL).
Brownian motion
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PL Wiener process .
Buffon's needle problem
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The probability (PL) P(l, d) that a tossed needle of length l lands on a grid of equally-spaced lines, a distance d apart. Solved by naturalist Buffon in 1771. P(l, d) = 2l/(d p).
bullet nose
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Curve of equation a2y2 - b2x2 = x2y .
bundle
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PL fiber bundle.
Burali-Forti paradox
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Incompatability of conditions on transfinite ordinal numbers.
butterfly catastrophe
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PL catsastrophe .
bypass
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Melzak: "Homind may have become human by internalizing bypass." Me: Language became mathematics by formalizing bypass.