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Page "Quantum operation" ¶ 22
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Let and H
* Let H be a group, and let G be the direct product H × H. Then the subgroups
Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).
Let A be the arithmetic mean and H be the harmonic mean of n positive real numbers.
Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by
Let be the probability that a certain coin lands heads up ( H ) when tossed.
Let G and H be groups, and let φ: GH be a homomorphism.
Let G be a group with identity element e, N a normal subgroup of G ( i. e., N ◁ G ) and H a subgroup of G. The following statements are equivalent:
Let H be the hashing function and m the message:
Let gas 1 be H < sub > 2 </ sub > and gas 2 be O < sub > 2 </ sub >.
Let A and H be groups and Ω a set with H acting on it.
Let us assume that H < sub > f </ sub > is in this time complexity class, and we will attempt to reach a contradiction.
Let p ( n ; H ) be the probability that during this experiment at least one value is chosen more than once.
Let n ( p ; H ) be the smallest number of values we have to choose, such that the probability for finding a collision is at least p. By inverting this expression above, we find the following approximation
Let Q ( H ) be the expected number of values we have to choose before finding the first collision.
Let U be a unitary operator on a Hilbert space H ; more generally, an isometric linear operator ( that is, a not necessarily surjective linear operator satisfying ‖ Ux ‖
Let the coin tosses be represented by a sequence of independent random variables, each of which is equal to H with probability p, and T with probability Let N be time of appearance of the first H ; in other words,, and If the coin never shows H, we write N is itself a random variable because it depends on the random outcomes of the coin tosses.

Let and G
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).
Let f and g be any two elements of G. By virtue of the definition of G, = and =, so that =.
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
Let a, b, and c be elements of G. Then:
Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.
Let R be a ring and G be a monoid.
Let this point be called G. Point G is also the midpoint of line segment FQ:
Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).
Let ( G ,.
Let G be a group.
Let S be a subgroup of G, and let N be a normal subgroup of G. Then:
Let G be a group.
Let N and K be normal subgroups of G, with

Let and be
Let the open enemy to it be regarded as a Pandora with her box opened ; ;
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
`` Let him be now ''!!
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let Af be the null space of Af.
Let N be a linear operator on the vector space V.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let this be denoted by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let not your heart be troubled, neither let it be afraid ''.
The same God who called this world into being when He said: `` Let there be light ''!!
For those who put their trust in Him He still says every day again: `` Let there be light ''!!
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let her out, let her out -- that would be the solution, wouldn't it??

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