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Page "Inversive geometry" ¶ 18
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If and circles
If the dominant country's influence is felt in social and cultural circles, such as " foreign " music being popular with young people, it may be described as cultural imperialism.
If the parameter " dopefish " is added to the executable, a sample of burping is heard and Scott's Mystical Head is seen spinning in circles on the screen.
If we draw both circles, two new points are created at their intersections.
If you measure the circumferences of circles of steadily larger diameters and divide the former by the latter, all three geometries give the value π for small enough diameters but the ratio departs from π for larger diameters unless Ω = 1:
* If four arbitrary points A, B, C, D are given that do not form an orthocentric system, then the nine-point circles of ABC, BCD, CDA and DAB concur at a point.
* If four points A, B, C, D are given that form a cyclic quadrilateral, then the nine-point circles of ABC, BCD, CDA and DAB concur at the anticenter of the cyclic quadrilateral.
( If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may " reverse orientation "; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori .< ref > Ronald Fintushel, Local S < sup > 1 </ sup > actions on 3-manifolds, Pacific J. o. M. 66 No1 ( 1976 ) 111-118, http :// projecteuclid. org /...) The classification of such ( oriented ) manifolds is given in the article on Seifert fiber spaces.
* If a circle q passes through two distinct points A and A ', inverses with respect to a circle k, then the circles k and q are orthogonal.
If the radical center lies outside of all three circles, then it is the center of the unique circle ( the radical circle ) that intersects the three given circles orthogonally ; the construction of this orthogonal circle corresponds to Monge's problem.
If p / q is between 0 and 1, the Ford circles that are tangent to C can be described variously as
If C and C are tangent Ford circles, then the half-circle joining ( p / q, 0 ) and ( r / s, 0 ) that is perpendicular to the x-axis is a hyperbolic line that also passes through the point where the two circles are tangent to one another.
If M consists of a circle, and N of two circles, M and N together make up the boundary of a pair of pants W ( see the figure at right ).
If you watch the animation above you will see two circles ( one about half way between the edge and center, and the other on the edge itself ) and a straight line bisecting the disk, where the displacement is close to zero.
If a straight line is considered a degenerate circle with zero curvature ( and thus infinite radius ), Descartes ' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line.
If four circles are tangent to each other at six distinct points, and the circles have curvatures k < sub > i </ sub > ( for i = 1, ..., 4 ), Descartes ' theorem says:
If one of the three circles is replaced by a straight line, then one k < sub > i </ sub >, say k < sub > 3 </ sub >, is zero and drops out of equation ( 1 ).
If two circles are replaced by lines, the tangency between the two replaced circles becomes a parallelism between their two replacement lines.
If random sections of the cloth are bound, the result will be a pattern of random circles.
If the cloth is first folded then bound, the resulting circles will be in a pattern depending on the fold used.

If and k
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
If the air mass is colder than the ground below it, it is labeled k. If the air mass is warmer than the ground below it, it is labeled w. While air mass identification was originally used in weather forecasting during the 1950s, climatologists began to establish synoptic climatologies based on this idea in 1973.
If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.
If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,
If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:
If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.
* Scanning: If a is the next symbol in the input stream, for every state in S ( k ) of the form ( X → α • a β, j ), add ( X → α a • β, j ) to S ( k + 1 ).
If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k + 1.
If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity.
If the ranges of the morphisms of the inverse system of abelian groups ( A < sub > i </ sub >, f < sub > ij </ sub >) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j: one says that the system satisfies the Mittag-Leffler condition.
If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.
If the value being sought occurs k times in the list, and all orderings of the list are equally likely, the expected number of comparisons is
If streams > k Do ; print number of execution streams, optionally corrected
If the move was in heap k, we have x < sub > i </ sub > = y < sub > i </ sub > for all i ≠ k, and x < sub > k </ sub > > y < sub > k </ sub >.
If this makes all the heaps of size zero ( in misère play ), the winning move is to take k objects from one of the heaps.

If and q
If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.
If T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor
*( EF1 ) If a and b are in R and b is nonzero, then there are q and r in R such that and either r = 0 or.
This can be done for all m of the p < sub > i </ sub >, showing that m ≤ n. If there were any q < sub > j </ sub > left over we would have
* If q is a prime power, and if F
If X is a positive random variable and q > 0 then for all ε > 0
If we compress data in a manner that assumes q ( X ) is the distribution underlying some data, when, in reality, p ( X ) is the correct distribution, the Kullback – Leibler divergence is the number of average additional bits per datum necessary for compression.
If Alice knows the true distribution p ( x ), while Bob believes ( has a prior ) that the distribution is q ( x ), then Bob will be more surprised than Alice, on average, upon seeing the value of X.
If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force
# If p is an odd prime, then any prime q that divides 2 < sup > p </ sup > − 1 must be 1 plus a multiple of 2p.
#* Proof: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).
# If p is an odd prime, then any prime q that divides must be congruent to ± 1 ( mod 8 ).
If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.
If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with
If we shift the constant term to the right hand side, factor a p and multiply by q < sup > n </ sup >, we get
If we instead shift the leading term to the right hand side and multiply by q < sup > n </ sup >, we get
Sherlock Holmes's straightforward practical principles are generally of the form, " If p, then q ," where " p " stands for some observed evidence and " q " stands for what the evidence indicates.
A proposition such as " If p and q, then p ." is considered to be logical truth because it is true because of the meaning of the symbols and words in it and not because of any facts of any particular world.
If q is the product of that curvature with the circle's radius, signed positive for epi-and negative for hypo -, then the curve: evolute similitude ratio is 1 + 2q.
If H is a subgroup of G, the set of left or right cosets G / H is a topological space when given the quotient topology ( the finest topology on G / H which makes the natural projection q: G → G / H continuous ).

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