Algebraically, these units can be thought of as the squares of the corresponding length units.

Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point.

Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.

Second, let G = G ( s ) be the external force per unit length acting on a small segment of a chain as a function of s. The forces acting on the segment of the chain between s and s + Δs are the force of tension T ( s + Δs ) at one end of the segment, the nearly opposite force − T ( s ) at the other end, and the external force acting on the segment which is

If an equation can be put into the form f ( x ) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x < sub > 1 </ sub > in the basin of attraction of x, and let x < sub > n + 1 </ sub > = f ( x < sub > n </ sub >) for n ≥ 1, and the sequence

For a discrete probability function f ( y ), let y < sub > i </ sub >, i = 1 to n, be the points with non-zero probabilities indexed in increasing order ( y < sub > i </ sub > < y < sub > i + 1 </ sub >).

As an example for English vowels, the pair " let " + " lit " can be used to demonstrate that the phones ( in let ) and ( in lit ) do in fact represent distinct phonemes and.

If we let − V < sub > i </ sub > + 2u =

If n = 2k is even, then let W ′ be an isotropic subspace complementary to W. If n = 2k + 1 is odd let W ′ be a maximal isotropic subspace with W ∩ W ′

In an execution where such a linked list is being shared between multiple processes, two processes may attempt to remove two different nodes simultaneously resulting in the following problem: let nodes i and i + 1 be the nodes to be removed ; furthermore, let neither of them be the head nor the tail ; the next pointer of node i-1 will be changed to point to node i + 1 and the next pointer of node i will be changed to point to node i + 2.

For another method, let the two linear functions: a < sub > 1 </ sub > x + b < sub > 1 </ sub > y + c < sub > 1 </ sub >

Philidor visited England in 1747 and decisively beat the Syrian Phillip Stamma in a match + 8 = 1 − 1, despite the fact Philidor let Stamma have White in every game, and scored all draws as wins for Stamma.

This influence is reflected mainly in the retention of certain lexical items ( cruds or cruddled milk ( cottage cheese ), hap ( comforter ), jag ( to tease or annoy ), jag around ( to fool around or act foolishly ), jagger ( a thorn or burr ), jagoff ( an annoying or irritating person ), neb / nebby / neb-nose ( nosy ), redd up ( to clean ), slippy ( slippery ), yinz / yunz / you ’ uns ( second-person plural ), " punctual " whenever and possibly " positive " anymore and reversed usage of leave and let, but also in the like, need, or want + past participle grammatical constructions i. e. ` the yard needs mowed ' and the discourse marker ‘ n ’ at i. e. ` and so forth / etc.

Proceeding inductively, for each k, let F < sub > k + 1 </ sub > be.

Let M be a monoid with identity element e and let A be the set of all subsets of M. For two such subsets S and T, let S + T be the union of S and T and set ST =

Let vectors < u > a </ u >, < u > b </ u >, < u > c </ u > and < u > h </ u > determine the position of each of the four orthocentric points and let < u > n </ u > = (< u > a </ u > + < u > b </ u > + < u > c </ u > + < u > h </ u >) / 4 be the position vector of N, the common nine-point center.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

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 ).

For example, let n be significantly less than 2 < sup > b </ sup >.

In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.

A bijection g ( t ) from T to ( 0, 1 ) is defined by: If t is the n < sup > th </ sup > string in sequence b, let g ( t ) be the n < sup > th </ sup > number in sequence a ; otherwise, let g ( t ) = 0. t.

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Let K be the triangle's area and let a, b and c, be the lengths of its sides.

The simplest method of this type is to let the interpolating function be a constant function ( a polynomial of degree zero ) which passes through the point (( a + b )/ 2, f (( a + b )/ 2 )).

Let f: D → R be a function defined on a subset D of the real line R. Let I = b be a closed interval contained in D, and let P =

For a let us assume A < sub > 1 </ sub >+ A < sub > 1 </ sub >+ A < sub > 1 </ sub > ... ; for b, A < sub > 2 </ sub >+ A < sub > 2 </ sub >+ A < sub > 2 </ sub > ...; and so on for the rest. But now what relation can there be among these several As, which will restore to us the unity of our original A or substance?

Let x, y, z be complex numbers, and let a, b be real numbers.

Let ( a, L ) and ( b, M ) be two Poincaré transformations, and let us denote their group product by ( a, L ).

In one such case, it is possible to do something which will ( a ) save the five and kill four ( passengers of one or more trolleys and / or the hammock-sleeper ), ( b ) save the five and kill three, ( c ) save the five and kill two, ( d ) save the five and kill one, or ( e ) do nothing and let five die.

He'd be an idiot to let them stay he thought, but he couldn't send them on, either.

If we let them go, they won't stay away, they'll find men to ride with them and they'll be back.

As John T. Westbrook says in his article, `` Twilight Of Southern Regionalism '' ( Southwest Review, Winter 1957 ): `` The miasmal mausoleum where an Old South, already too minutely autopsied in prose and poetry, should be left to rest in peace, forever dead and ( let us fervently hope ) forever done with ''.

let them learn to `` swing '' freely, to let go, to become authentically themselves, and then perhaps civilization will be saved.

If I now risk some comparisons with Sons And Lovers let it be clear that I am not comparing the two works or judging their merits ; ;

It was symbolized ( at least for those of us who recognized ourselves in the image ) by that self-consuming, elegiac candle of Edna St. Vincent Millay's, that candle which from the quatrain where she ensconced it became a beacon to us, but which in point of fact would have had to be as tall as a funeral taper to last even the evening, let alone the night.

An idea, let us say, may be roughly defined as a theme or topic with which our reflection may be concerned.

-- But let him be ready to invent whatever falsehood -- to assail whatever character -- and to prostitute his paper to whatever ends -- and they hug him to their heart.

As a proud man, his prestige would suffer if he let Pike dictate to him through the governor's office, but to lower his prices would be tantamount to an admission that they had been too high in the first place.

or to let the ethical be superseded by the purposive.

As to our action, let us align ourselves with the purpose expressed by Jesus in the Lord's Prayer: `` Thy kingdom come, Thy will be done, on earth as it is in heaven ''.

if he wanted to leave me something let it be a trinket, nothing else.

Those famous lines of the Greek Anthology with which a fading beauty dedicates her mirror at the shrine of a goddess reveal a wise attitude: `` Venus, take my votive glass, Since I am not what I was, What from this day I shall be, Venus, let me never see ''.

That is how the real routine of resistance goes on, and its strength is directly proportionate to the number of insignificant people who can let themselves be taken to pieces, piece by piece, without quitting.

But because of the peculiar nature of the military situation in Laos, the Soviet leader must be tempted to let things ride -- a course that would appear to cost him little on the spot, but would bog Washington in a tactical mess.

Jesus replied, ' let it be so for the present ; ;

The Macmillan government might be willing to let him go, but he has been dead seventy-eight years and even the Soviet morticians could not make him look presentable.

To avoid suspicion of bigotry, let the hand of vengeance be stayed until the meat-wagon has picked up the twenty-five thousandth corpse ; ;

but let the twenty-five thousand and first butchery be the signal for the arrest of the 50 state highway commissioners.

Then let the whole lot be hanged in a public mass execution on July 4, 1963.

Before memory, too, clouds over, let us make a note or two of what could be seen.

and it is still very far from certain how valid the party's claim is that in `` a growing number of kolkhozes '' the peasants are finding it more profitable, to surrender their private plots to the kolkhoz and to let the latter be turned into something increasingly like a state farm.

To be on the safe side I never let Eileen get out of my sight day or night.