If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If the conditions of the law of large numbers hold, s < sup > 2 </ sup > is a consistent estimator of σ < sup > 2 </ sup >.

If the field K has an absolute value, then the weak topology σ ( X, F ) is induced by the family of

If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X < sub > i </ sub > / σ ( X < sub > i </ sub >) for i = 1, ..., n. This applies to both the matrix of population correlations ( in which case " σ " is the population standard deviation ), and to the matrix of sample correlations ( in which case " σ " denotes the sample standard deviation ).

If we assume that P < sub > X </ sub >( x ) is Gaussian with variance σ < sup > 2 </ sup >, and if we assume that successive samples of the signal X are stochastically independent ( or, if you like, the source is memoryless, or the signal is uncorrelated ), we find the following analytical expression for the rate – distortion function:

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 the observed data X < sub > 1 </ sub >, ..., X < sub > n </ sub > are ( i ) uncorrelated, ( ii ) have a common mean μ, and ( iii ) have a common variance σ < sup > 2 </ sup >, then the sample average < span style =" text-decoration: overline "> X </ span > has mean μ and variance σ < sup > 2 </ sup > / n. If our null hypothesis is that the mean value of the population is a given number μ < sub > 0 </ sub >, we can use < span style =" text-decoration: overline "> X </ span > − μ < sub > 0 </ sub > as a test-statistic, rejecting the null hypothesis if < span style =" text-decoration: overline "> X </ span > − μ < sub > 0 </ sub > is large.

If the sample size is moderate or large, we can substitute the sample variance for σ < sup > 2 </ sup >, giving a plug-in test.

If σ

If σ, ..., σ are the distinct K-linear field embeddings of L into an algebraically closed field containing K ( where n is the degree of the extension L / K ), then

If we assume a normally distributed population with mean μ and standard deviation σ, and choose individuals independently, then we have

If X < sub > 1 </ sub >, ..., X < sub > n </ sub > are independent normally distributed random variables with mean μ and standard deviation σ

If the errors are independent and normally distributed with expected value 0 and variance σ < sup > 2 </ sup >, then the probability distribution of the ith externally studentized residual is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

If ( A, σ ) is an associative algebra with an ( anti -) involution σ, then if σ ( x )= x and σ ( y )= y it follows that

If N is a substructure of M, then both N and M can be interpreted as structures in the signature σ < sub > N </ sub > consisting of σ together with a new constant symbol for every element of N. N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σ < sub > N </ sub >- structures.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

If a correct signature was located on the cartridge, the 7800 would operate in 7800 mode, granting the game access to MARIA and other features.

If a signature was not located, the 7800 remained in 2600 mode and MARIA was unavailable.

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.

The first stage of boot loaders located on fixed disks and removable drives must fit into the first 446 bytes of the Master Boot Record in order to leave room for the default 64-byte partition table with four partition entries and the two-byte boot signature, which the BIOS requires for a proper boot loader — or even less, when additional features like more than four partition entries ( up to 16 with 16 bytes each ), a disk signature ( 6 bytes ), a disk timestamp ( 6 bytes ), an Advanced Active Partition ( 18 bytes ) or special multi-boot loaders If an active partition is found, the MBR code loads the boot sector code from that partition, known as Volume Boot Record ( VBR ), and executes it.

If the Governor fails to return a bill to the legislative house in which it originated within a specified number of days after it was presented to him or her, it becomes a law without their signature.

If Theia had been a separate proto-planet, it probably would have had a different oxygen isotopic signature than Earth, as would the ejected mixed material.

If the smart card is stolen, the thief will still need the PIN code to generate a digital signature.

If the Congress overrides the veto by a two-thirds majority in each house, it becomes law without the President's signature.

If this is not observed, any text following the signature will be ignored, or the entire will may be invalidated if what comes after the signature is so material that ignoring it would defeat the testator's intentions.

If, however, the award was presented posthumously, an image of the Disney Legends statue is engraved instead of the traditional hand prints and signature.

If the king fails to act within six months of the bill being presented to him, it becomes law without his signature.

If M is connected, then the signature of q < sub > m </ sub > does not depend on m.

* If g has signature ( n, 0 ), then g is a Riemannian metric, and M is called a Riemannian manifold.

* If M is four-dimensional with signature ( 1, 3 ) or ( 3, 1 ), then the metric is called Lorentzian.

* If M is 2n-dimensional and g has signature ( n, n ), then the metric is called ultrahyperbolic.

If the tempo is slow or slowing, or if the time signature is compound, a conductor will sometimes indicate " subdivisions " of the beats.

The congregation responds " Amen " to each blessing, and " Baruch Hu Uvaruch Shemo " (" blessed is He and blessed is His Name ") when the chazzan invokes God's name in the signature " Blessed are You, O Lord ..." If there are not six members of the minyan responding " Amen ," the chazzan's blessing is considered in vain.

If, under these circumstances, the president withholds his signature, the bill dies instead of becoming law.

( If the president neither signs nor vetoes a bill when Congress is in session, the bill becomes law without his signature after 10 days.

He and lyricist Carl Sigman wrote " If You Could See Me Now " for Sarah Vaughan and it became one of her first signature songs.

If a document has a digital signature on it, no one else is able to edit the information without being detected.