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Page "Analysis of variance" ¶ 147
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If and F
If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.
If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.
If F is also surjective, then the Banach space X is called reflexive.
If F. tularensis were used as a weapon, the bacteria would likely be made airborne for exposure by inhalation.
If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.
If F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:
If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
If certain coordinate systems are used, for instance, polar-toroidal coordinates ( common in plasma physics ) using the notation ∇ × F will yield an incorrect result.
If φ is a scalar valued function and F is a vector field, then
If F is clear from context then Ω < sub > F </ sub > may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.
If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.
If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.
If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).
If only layer C changes, we should find a way to avoid re-blending all of the layers when computing F. Without any special considerations, four full-image blends would need to occur.

If and
If n 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.
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 an equation can be put into the form f ( x ) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x < sub > 1 </ sub > in the basin of attraction of x, and let x < sub > n + 1 </ sub > = f ( x < sub > n </ sub >) for n 1, and the sequence
If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β α, the point x < sub > β </ sub > lies in Y.
If Φ is a unital positive map, then for every normal element a in its domain, we have Φ ( a * a ) Φ ( a *) Φ ( a ) and Φ ( a * a ) Φ ( a ) Φ ( a *).
If M is an Hermitian positive-semidefinite matrix, one sometimes writes M 0 and if M is positive-definite one writes M > 0 .< ref > This may be confusing, as sometimes nonnegative matrices are also denoted in this way.
If M N > 0 then N < sup >− 1 </ sup > M < sup >− 1 </ sup > > 0.
< li > If M = ( m < sub > ij </ sub >) 0 then the diagonal entries m < sub > ii </ sub > are real and non-negative.
* If f: R → R is a Lebesgue integrable function and f ( x ) 0 almost everywhere, then
If there exists a K 1 with
If we restrict attention to real-valued W then the relation is defined only for x − 1 / e, and is double-valued on (− 1 / e, 0 ); the additional constraint W − 1 defines a single-valued function W < sub > 0 </ sub >( x ).
If n 2, then the group GL ( n, F ) is not abelian.
If is a monotone sequence of real numbers ( i. e., if a < sub > n </ sub > ≤ a < sub > n + 1 </ sub > or a < sub > n </ sub > a < sub > n + 1 </ sub > for every n 1 ), then this sequence has a finite limit if and only if the sequence is bounded.
If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of R < sup > n </ sup > ( n 0 ) to X such that the following hold:
If the chosen anti-aliasing filter ( a low-pass filter in this case ) has a transition band of 2000 Hz, then the cut-off frequency should be ≤ 20050 Hz to yield a signal with negligible power at frequencies 22050 Hz and complete pass of frequencies ≤ 20 kHz ( within the human hearing range ).
# 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 X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β < α such that x < sub > ι </ sub > is in U for all ι β.
If X is a connected cell complex with homotopy groups π < sub > n </ sub >( X ) = 0 for all n 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K ( G, 1 ) for G = π < sub > 1 </ sub >( X ).
If a function f is concave, and f ( 0 ) 0, then f is subadditive.

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 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 )( h ) for x, y in L < sup > 1 </ sup >( G ).

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