The spectrum of an element x ∈ A, denoted by, consists of all those complex scalars λ such that x − λ1 is not invertible in A.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields " yes " ( or true ) ( more accurately the number b in location B is greater than or equal to the number a in location A ) THEN the algorithm specifies B ← B − A ( meaning the number b − a replaces the old b ).

And to " measure " is to place a shorter measuring length s successively ( q times ) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q * s, q being the quotient, or remainder r is the " modulus ", the integer-fractional part left over after the division.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

Again the estimated population variance of 30 is computed correctly by Algorithm II, but the naive algorithm now computes it as − 170. 66666666666666.

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

By international agreement, absolute zero is defined as on the Kelvin scale and as − 273. 15 ° on the Celsius scale.

So a quasar at an absolute magnitude of − 25. 5 is 100 times brighter than our galaxy ( because ( 10 < sup > 0. 4 </ sup >)< sup >(− 20. 5 -(− 25. 5 ))</ sup >

For example, if the array has five elements, indexed 1 through 5, and the base address B is replaced by B − 30c, then the indices of those same elements will be 31 to 35.

And since there are n − 1 links in any tree, the amortized cost is found to be 2 ×( n − 1 )/ n, or approximately 2.

If the balance factor becomes − 2 or + 2 then the subtree is unbalanced and needs to be rotated to fix it.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

For a general n × n invertible matrix A, i. e., one with nonzero determinant, A < sup >− 1 </ sup > can thus be written as an ( n − 1 )- th order polynomial expression in A: As indicated, the Cayley – Hamilton theorem amounts to the identity

Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator.

On the other hand 0 is in the spectrum because the operator R − 0 ( i. e. R itself ) is not invertible: it is not surjective since any vector with non-zero first component is not in its range.

If λI − T is invertible then that inverse is linear ( this follows immediately from the linearity of λI − T ), and by the bounded inverse theorem is bounded.

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. e., if there exists an invertible matrix P such that P < sup > − 1 </ sup > AP is a diagonal matrix.

If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients λ < sub > i </ sub > can be made to be 0, 1, and − 1.

If S is allowed to be any invertible matrix then B can be made to have only 0, 1, and − 1 on the diagonal, and the number of the entries of each type ( n < sub > 0 </ sub > for 0, n < sub >+</ sub > for 1, and n < sub >−</ sub > for − 1 ) depends only on A.

The condition ad − bc ≠ 0 is equivalent to the condition that the determinant of above matrix be nonzero, i. e. that the matrix be invertible.

There is a well-defined function Δ on X, which is the absolute value of the determinant of a matrix — this is constant on the cosets, since an invertible integer matrix has determinant 1 or − 1.

and invertible, then its inverse is a pseudo-differential operator of order − m, and its symbol can be calculated.

If T − λ is invertible ( i. e., if it is one-to-one and onto ), then its inverse is bounded ; this follows directly from the open mapping theorem.

: Proof: r − 1 ( with r as above ) is in the Jacobson radical so r is invertible.

± s ( because a is invertible modulo p ), so r = s because they are both in the range 1 ≤ r ≤ ( p − 1 )/ 2.

This is because we arrange things such that for every integer, there is a distinct odd integer: ... − 2 → − 3, − 1 → − 1, 0 → 1, 1 → 3, 2 → 5, ...; or, more generally, n → 2n + 1.

Similarly, because meters per second, the expression f ′( g ′( 10 )) represents the change in pressure at a height of − 98 meters per second, which is also nonsense.

He implemented these measures because he wanted to resolve serious economic problems and political inertia that clearly threatened to put the Soviet Union into a state of long − term stagnation.

In all four categories and for all three encyclopaedias, the four average grades fell between B − and B +, chiefly because none of the encyclopaedias had an article on sexual harassment in 1994.

The final nonzero remainder r < sub > N − 1 </ sub > is the greatest common divisor of a and b. The number N cannot be infinite because there are only a finite number of nonnegative integers between the initial remainder r < sub > 0 </ sub > and zero.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

We must also prevent the decoder from using the last code in the upper block, 2 < sup > n + 1 </ sup > − 1, because when the decoder fills that slot, it will increase the code width.

Two ’ s complement arithmetic is convenient because there is a perfect one-to-one correspondence between representations and values ( in particular, no separate + 0 and − 0 ), and because addition, subtraction and multiplication do not need to distinguish between signed and unsigned types.

# Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracket of any two derivations is a derivation.

This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as − 1 on the tangent space T < sub > e </ sub >.

because 38 − 14 = 24, which is a multiple of 12.

For example, the square root of 2 is irrational and not transcendental ( because it is a solution of the polynomial equation x < sup > 2 </ sup > − 2 = 0 ).

It was found that for silver atoms, the beam was split in two — the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with L < sub > z </ sub > = − 1, 0, and + 1.

This heuristic argument is actually somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other.

For instance, if m is odd then n − m is also odd, and if m is even, then n − m is even, a non-trivial relation because ( besides 2 ) only odd numbers can be prime.