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Page "Picard group" ¶ 10
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quotient and V
where v, v < sub > 1 </ sub > and v < sub > 2 </ sub > are vectors from V, while w, w < sub > 1 </ sub >, and w < sub > 2 </ sub > are vectors from W, and c is from the underlying field K. Denoting by R the space generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space
A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T ( V ), and then enforce the fundamental identity by taking a suitable quotient.
and define Cℓ ( V, Q ) as the quotient algebra
If e < sub > 1 </ sub >, ... e < sub > d </ sub > is a basis of V, the unital zero algebra is the quotient of the polynomial ring k ..., E < sub > n </ sub > by the ideal generated by the E < sub > i </ sub > E < sub > j </ sub > for every pair ( i, j ).
For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.
Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.
where each successive quotient has order p. Apartments are defined by fixing a basis ( v < sub > i </ sub >) of V and taking all lattices with basis
One way in which a projective representation can arise is by taking a linear group representation of G on V and applying the quotient map
The projectivization P ( V ) of a vector space V over a field k is defined to be the quotient of by the action of the multiplicative group k < sup >×</ sup >.
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i. e. by constructing certain quotient algebras of T ( V ).
The quotient space of T ( V ) on which it becomes an internal operation is the exterior algebra of V ; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V.
Over the complex numbers, the Jacobian variety can be realized as the quotient space V / L, where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form
More generally, if W is a linear subspace of a ( possibly infinite dimensional ) vector space V then the codimension of W in V is the dimension ( possibly infinite ) of the quotient space V / W, which is more abstractly known as the cokernel of the inclusion.

quotient and )/
The quotient group O ( n )/ SO ( n ) is isomorphic to O ( 1 ), with the projection map choosing or according to the determinant.
The quotient group O ( 1, 3 )/ SO < sup >+</ sup >( 1, 3 ) is isomorphic to the Klein four-group.
De Sitter space can also be defined as the quotient O ( 1, n )/ O ( 1, n − 1 ) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.
In this case, the restriction of the elements of Gal ( E / F ) to E < sup > H </ sup > induces an isomorphism between Gal ( E < sup > H </ sup >/ F ) and the quotient group Gal ( E / F )/ H.

quotient and <
The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.
Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.
The goal of the kth step is to find a quotient q < sub > k </ sub > and remainder r < sub > k </ sub > such that the equation is satisfied
For example, if a < b, the initial quotient q < sub > 0 </ sub > equals zero, and the remainder r < sub > 0 </ sub > is a.
It results in two polynomials, a quotient and a remainder that are characterized by the following property of the polynomials: given two polynomials a and b such that b ≠ 0, there exists a unique pair of polynomials, q, the quotient, and r, the remainder, such that a = b q + r and degree ( r ) < degree ( b ) ( here the polynomial zero is supposed to have a negative degree ).
One particularly useful condition is that the length of the sequence is finite and each quotient module M < sub > i </ sub >/ M < sub > i + 1 </ sub > is simple.
Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.
In the above difference quotient, all the variables except x < sub > i </ sub > are held fixed.
A space is regular if and only if its Kolmogorov quotient is T < sub > 3 </ sub >; and, as mentioned, a space is T < sub > 3 </ sub > if and only if it's both regular and T < sub > 0 </ sub >.

quotient and sup
Consequently, the Roman surface is a quotient of the real projective plane RP < sup > 2 </ sup >
By associating the element a in G with the inner automorphism ƒ ( x ) = x < sup > a </ sup > in Inn ( G ) as above, one obtains an isomorphism between the quotient group G / Z ( G ) ( where Z ( G ) is the center of G ) and the inner automorphism group:
If we add the relation x < sup > 2 </ sup > = 1 to the presentation of Dic < sub > n </ sub > one obtains a presentation of the dihedral group Dih < sub > 2n </ sub >, so the quotient group Dic < sub > n </ sub >/< x < sup > 2 </ sup >> is isomorphic to Dih < sub > n </ sub >.
The quotient R /( Z + αZ ), for some irrational α, is the irrational torus, a diffeological space diffeomorphic to the quotient of the regular 2-torus R < sup > 2 </ sup >/ Z < sup > 2 </ sup > by a line of slope α.

quotient and 0
Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method.
The last expression shows that the difference quotient equals 6 + h when h ≠ 0 and is undefined when h
Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer.
In OBGYN residency programs, the average laparoscopy-to-laparotomy quotient ( LPQ ) is 0. 55.
For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.
In particular, the theorem asserts that integers called the quotient q and remainder r always exist and that they are uniquely determined by the dividend a and divisor d, with d ≠ 0.
In yet a simpler way, we may think of the Jacobson radical of a ring as method to " mod out bad elements " of the ring that is, members of the Jacobson radical act as 0 in the quotient ring, R / J ( R ).
A more precise form is given by the celebrated prime number theorem: the quotient of the two expressions approaches 1. 0 as tends to infinity.
Since it is the halfway point that is of interest, the quotient of steps 1 and 2 is 0. 5.
If we try to use the quotient to compute f '( 0 ), however, an undefined value will result, since | x | is nondifferentiable at x = 0.
: The difference quotient as a derivative needs no explanation, other than to point out that, since P < sub > 0 </ sub > essentially equals P < sub > 1 </ sub >
* The brain to body mass ratio ( as distinct from encephalization quotient ) in some members of the odontocete superfamily Delphinoidea ( dolphins, porpoises, belugas, and narwhals ) is second only to modern humans, and greater than all other mammals ( there is debate whether that of the treeshrew might be second ) In some dolphins, it is less than half that of humans: 0. 9 % versus 2. 1 %.
is obtained as the quotient of Co < sub > 0 </ sub > ( group of automorphisms of the Leech lattice Λ that fix the origin ) by its center, which consists of the scalar matrices ± 1.
) Similarly, any expression of the form a / 0, with a0 ( including a =+∞ and a =-∞), is not an indeterminate since a quotient giving rise to such an expression will always diverge.
If a and d are natural numbers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder.
It can be proved that there exists a unique integer quotient q and a unique real remainder r such that a = qd + r with 0 ≤ r < | d |.
The Jacobian of, denoted, is a quotient group, thus the elements of the Jacobian are not points, they are equivalence classes of divisors of degree 0 under the relation of linear equivalence.
This I. Q ( a play on the term " intelligence quotient ") measures the player's efficiency in capturing cubes as well as the total number captured on a scale of 0 to 999 ( for instance, beating the game without using a continue gives you an I. Q of at least 350 ).

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