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If λ < sub > 1 </ sub >, ..., λ < sub > ν </ sub > are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others.
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If and λ
# If V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.
# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.
# If A and B are Lebesgue measurable and A is a subset of B, then λ ( A ) ≤ λ ( B ).
# If A is a Lebesgue measurable set with λ ( A ) = 0 ( a null set ), then every subset of A is also a null set.
If we do not alter the system, then the parameter λ is unchanged from measurement to measurement ; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.
If λ represents wavelength and f represents frequency ( note, λf = c where c is the speed of light ), then z is defined by the equations:
* If the base field is R, then for all real numbers λ.
* If the base field is C, then for all complex numbers λ, where denotes the complex conjugation of λ.
If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync ( add constructively ) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ.
If λ < sub > c </ sub > is the central wavelength, λ < sub > 0 </ sub > is the central wavelength at normal incidence, and n < sup >*</ sup > is the filter effective index of refraction, then:
If n is a positive integer, then λ ( n ) is defined as:
T, and a free variable of T. We say x is bound in M and free in T. If T contains a subterm λ x.
If a vector v of size n happens to be an eigenvector of A with eigenvalue λ, in other words if, then
If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value λ will be the corresponding eigenvalue of the equation A e
No notation is used to describe the set of all λ, which satisfy this condition, but for a subset: If λ-T does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by σ < sub > r </ sub >( T ).
If T is a compact operator, then it can be shown that any nonzero λ in the spectrum is an eigenvalue.
If α is a limit ordinal, an α-inaccessible is a fixed point of every ψ < sub > β </ sub > for β < α ( the value ψ < sub > α </ sub >( λ ) is the λ < sup > th </ sup > such cardinal ).
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 ).
0.197 seconds.