* 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: 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 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 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 ( A < sub > i </ sub >)< sub > i ∈ I </ sub > be a family of groups and suppose we have a family of homomorphisms f < sub > ij </ sub >: A < sub > j </ sub > → A < sub > i </ sub > for all i ≤ j ( note the order ) with the following properties:

With their third consecutive single (" Let Love In ") from the album, the Goo Goo Dolls hit a record 12 Top 10 hits in Adult Top 40 history, beating Matchbox Twenty and Sheryl Crow until Matchbox Twenty's release of Exile on Mainstream and the Goo Goo Dolls ' release of " Before It's Too Late " from the Transformers Soundtrack, which left both groups with 13 Top 10 hits in the Adult Top 40.

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n.

Let no mistake be made: if we let the Ontario minority be crushed, it will soon be the turn of other French groups in English Canada.

Vocalist Joey Ramone was an avid fan of Spector's early work, including albums by many girl groups and Let It Be by The Beatles.

) Let and be the ideal groups of A and B, respectively ( i. e., the sets of fractional ideals.

With hits such as " Hold On ", " My Lovin ' ( You're Never Gonna Get It )", and " Don't Let Go ( Love )", En Vogue has become one of the most successful girl groups in music history.

Let be a topological space, and let be a presheaf of abelian groups on.

Let be additive groups and a multiplicative group, all of prime order.

* Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let q be a prime power, d a positive integer, and p a prime divisor of q − 1 with d ≤ p. Fix some field F of order q and some element z of this field of order p. The Frobenius complement H is the cyclic subgroup generated by the diagonal matrix whose i, ith entry is z < sup > i </ sup >.

Since then there have been two groups both calling themselves the Libertarian Alliance and using the same logo and using the phrase " Let a Thousand Libertarian Alliances Bloom !".

Let S < sup > n </ sup > denote the n-sphere and note that it is ( n − 1 )- connected so that the groups π < sub > n + k </ sub >( S < sup > n </ sup >) stabilize for

Let the concentrations of the two groups be x, y with reproduction rates a > b, respectively ; let Q be the probability of a virus in the first group ( x ) mutating to a member of the second group ( y ) and let R be the probability of a member of the second group returning to the first ( via an unlikely and very specific mutation ).

In recent years Glover has discovered and managed groups including B * Witched, Let Loose, and D-Side