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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 be the probability that a certain coin lands heads up ( H ) when tossed.

Let G and H be groups, and let φ: G → H 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 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??

Let and Hilbert

Let be a Hilbert space and is a vector in.

Let and be the system's and bath's Hilbert spaces, respectively.

Let H be a Hilbert space with an orthonormal basis

Let G = R, and let the complex Hilbert space V be L < sup > 2 </ sup >( R ).

Let G be a topological group and H a complex Hilbert space.

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Let π be a *- representation of a C *- algebra A on the Hilbert space H with cyclic vector ξ having norm 1.

Let M be an algebra of bounded operators on a Hilbert space H, containing the identity operator and closed under taking adjoints.

Let H be a Hilbert space and L ( H ) the bounded operators on H. Consider a self-adjoint subalgebra M of L ( H ).

Let be a weak *- closed operator algebra contained in B ( H ), the set of all bounded operators on a Hilbert space H and for T any operator in B ( H ), let

Let F be any sampling method, i. e. a linear map from the Hilbert space of square-integrable functions to complex space.

Let is a basis of in the Hilbert space sense ; for instance, one could use the eikonal

Let X be an arbitrary set and H a Hilbert space of complex-valued functions on X.

Let H and G be Hilbert spaces of dimension n and m respectively, and Φ be a quantum operation taking the density matrices acting on H to those acting on G. Then there are matrices

Let Φ be a ( not necessarily trace preserving ) quantum operation on a finite dimensional Hilbert space H with two representing sequences of Kraus matrices

Let be a complex separable Hilbert space, be a one-parametric group of unitary operators on and be a statistical operator on.

Let H be the one-particle Hilbert space.

Theorem ( Fuglede ) Let T and N be bounded operators on a complex Hilbert space with N being normal.

Theorem ( Calvin Richard Putnam ) Let T, M, N be linear operators on a complex Hilbert space, and suppose that M and N are normal and MT

Let T ( H ) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map

Let and be the state spaces ( finite-dimensional Hilbert spaces ) of the sending and receiving ends, respectively, of a channel.

Let H be a Hilbert space and let X and Y be its linear subspaces.

Let G be a σ-compact, locally compact topological group and π: G U ( H ) a unitary representation of G on a ( complex ) Hilbert space H. If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an ( ε, K )- invariant vector if π ( g ) ξ-ξ < ε for all g in K.

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