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* If π < sub > 1 </ sub >( M ) is virtually cyclic but not finite then the geometric structure on M is S < sup > 2 </ sup >× R, and M is compact.
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If the phase difference is 180 degrees ( π radians ), then the two oscillators are said to be in antiphase.
If the inclination is zero or 180 degrees ( π radians ), the azimuth is arbitrary.
If, instead of 1 matching entry, there are k matching entries, the same algorithm works but the number of iterations must be π ( N / k )< sup > 1 / 2 </ sup >/ 4 instead of πN < sup > 1 / 2 </ sup >/ 4.
If a line is perpendicular to another as shown, all of the angles created by their intersection are called right angles ( right angles measure π / 2 radians, or 90 °).
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:
:: Example: If the CSPRNG under consideration produces output by computing bits of π in sequence, starting from some unknown point in the binary expansion, it may well satisfy the next-bit test and thus be statistically random, as π appears to be a random sequence.
# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).
* If π < sub > 1 </ sub >( M ) is finite then the geometric structure on M is spherical, and M is compact.
* If π < sub > 1 </ sub >( M ) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
* If π < sub > 1 </ sub >( M ) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
* If π < sub > 1 </ sub >( M ) is virtually solvable but not virtually nilpotent then the geometric structure on M is sol geometry, and M is compact.
* If π < sub > 1 </ sub >( M ) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H < sup > 2 </ sup >× R or the universal cover of SL ( 2, R ).
If it is compact, then the 2 geometries can be distinguished by whether or not π < sub > 1 </ sub >( M ) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else.
* If π < sub > 1 </ sub >( M ) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.
If tan ( φ ) tan ( δ ) > 1, then the sun does not set and the sun is already risen at h = π, so h < sub > o </ sub > = π.
If c < sub > 1 </ sub > ∈ C is another pre-image of x in C then the subgroups p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) and p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c < sub > 1 </ sub >)) are conjugate in π < sub > 1 </ sub >( X, x ) by p-image of a curve in C connecting c to c < sub > 1 </ sub >.
the set of homotopy classes of those closed curves γ based at x whose lifts γ < sub > C </ sub > in C, starting at c, are closed curves at c. If X and C are path-connected, the degree of the cover p ( that is, the cardinality of any fiber of p ) is equal to the index ( π < sub > 1 </ sub >( C, c )) of the subgroup p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) in π < sub > 1 </ sub >( X, x ).
If and <
* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.
If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )
If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.
If ΔS and / or T are small, the condition ΔG < 0 may imply that ΔH < 0, which would indicate an exothermic reaction.
If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.
Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that
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 activated cytotoxic CD8 < sup >+</ sup > T cells recognize them, the T cells begin to secrete various toxins that cause the lysis or apoptosis of the infected cell.
If ADH production is excessive in heart failure, Na < sup >+</ sup > level in the plasma may fall ( hyponatremia ), and this is a sign of increased risk of death in heart failure patients.
If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have
Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.
If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).
If and sub
If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).
If and 1
If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.
If desired, sprinkle with 1 teaspoon drained crushed pineapple.
If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;
If, in a certain part of the range, it starts life 1 foot longer than do any of the other ( relatively large ) giants, and reaches maturity at, let us guess, 18 inches longer than the others, a quadrupling of the maturity length would result in a maximum of ( nearly ) 40 feet.
If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.
If the compiler finds that there are no more electronic switches available for assignment, the warning message `` No More Electronic Switches Available '' will be placed in the object program listing, the table will be altered to show that electronic switches 1 through 30 are available, and assignment will continue as before.
If the resulting four kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit ( i. e. 1 kilobyte ) table by the use of circular rotates.
If more than one player selects a tile in that row, then the player whose tile is in the leftmost column ( closest to 1 ) goes first.
If the color were fully green, its RGBA would be ( 0, 1, 0, 0. 5 ).
* If the balance factor of R is + 1, two different rotations are needed.
* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).
If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).
If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.
If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O ' 1 there is a circular disk of confusion of radius O ' 1R, and in a parallel plane at O ' 2 another one of radius O ' 2R2 ; between these two is situated the disk of least confusion.
If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.
If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.
If there is no number to its left, simply look at the column headed " 1 " in the previous row.
If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.
If the boat floats, the mass of the boat ( plus contents ) as a whole divided by the volume below the waterline is equal to the density of water ( 1 kg / l ).
0.322 seconds.