* Scanning: If a is the next symbol in the input stream, for every state in S ( k ) of the form ( X → α • a β, j ), add ( X → α a • β, j ) to S ( k + 1 ).

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

If β < sub > 1 </ sub > and β < sub > 2 </ sub > are high enough ( hundreds ), this relation can be approximated with:

If A < sub > OL </ sub > >> 1, then A < sub > fb </ sub > ≈ 1 / β and the effective amplification ( or closed-loop gain ) A < sub > fb </ sub > is set by the feedback constant β, and hence set by the feedback network, usually a simple reproducible network, thus making linearizing and stabilizing the amplification characteristics straightforward.

If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β < α such that x < sub > ι </ sub > is in U for all ι ≥ β.

If α and β are chosen so that the right hand side yields ê < sub > 1 </ sub > or ê < sub > n </ sub >, then the quantity in the parentheses will fulfill the definition of the n < sup > th </ sup > forward or backward vector, respectively.

If the geometry of the body is fixed and in case of symmetric flight ( β = 0 and Q = 0 ), pressure and friction coefficients are functions depending on:

If E is an elliptic curve over a finite field with q elements, then the number of points of E defined over the field with q < sup > m </ sup > elements is 1 − α < sup > m </ sup >− β < sup > m </ sup > + q < sup > m </ sup >,

If α is a limit ordinal, an α-inaccessible is a fixed point of every ψ < sub > β </ sub > for β < α ( the value ψ < sub > α </ sub >( λ ) is the λ < sup > th </ sup > such cardinal ).

# ( Integrality condition ) If α and β are roots in Φ, then the projection of β onto the line through α is a half-integral multiple of α.

:: Note: If β ≠ 0 then α + 2y ≠ 0.

If the V < sub > CB </ sub > of Q < sub > 2 </ sub > increases, so does the output transistor β because of the Early effect: β

If one takes L to be the splitting field of X < sup > 3 </ sup > − a over Q, where a is not a cube in the rational numbers, then L contains a subfield K with three cube roots of 1 ; that is because if α and β are roots of the cubic polynomial, we shall have ( α / β )< sup > 3 </ sup > = 1 and the cubic is a separable polynomial.

: If α and β are algebraic numbers with α ≠ 0, 1 and if β is not a rational number, then any value of α < sup > β </ sup > = exp ( β log α ) is a transcendental number.

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

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If the ratio a '/ a be sufficiently constant, as is often the case, the above relation reduces to the condition of Airy, i. e. tan w '/ tan w = a constant.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* If the operation is commutative, ab = ba, then the value depends only on

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete ( since otherwise NP = co-NP ).

If the user pressed keys 1 + 2 = 3 simultaneously the letter " c " appeared.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

If the distribution of the 71 items were wholly concordant in the two families, the distance would of course be 0.

If ΔS and / or T are small, the condition ΔG < 0 may imply that ΔH < 0, which would indicate an exothermic reaction.

If the color were fully green, its RGBA would be ( 0, 1, 0, 0. 5 ).

If the valid element indices begin at 0, the constant B is simply the address of the first element of the array.

If the numbering does not start at 0, the constant B may not be the address of any element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If Acorn had thought to include this small modification in the original Electron design it is likely the machine would have had a much greater impact as it would have nearly doubled the amount of motion possible in games and saved modes 0 – 3 ( including the only 16 colour mode ) from being nearly useless due to contended memory timings.

If no address is provided, one is picked at random from the " base subnet ", 0.

If the balance factor becomes 0 then the height of the subtree has decreased by one and the retracing needs to continue.

If the rotation leaves the subtree's balance factor at 0 then the retracing towards the root must continue since the height of this subtree has decreased by one.

If the maximum gain is 0 dB, the 3 dB gain is the range where the gain is more than-3dB, or the attenuation is less than + 3dB.

If a DFS ROM is installed, will cause the computer to search for and load a file called on the floppy disk in drive 0.

If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0. 5 and 0. 5 ≤ t ≤ 1, and the same procedure is applied recursively to each half.

If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1 / g ′( f ( 0 )).

If we set, then η is continuous at 0.

If sender0 has code ( 1, – 1 ) and data ( 1, 0, 1, 1 ), and sender1 has code ( 1, 1 ) and data ( 0, 0, 1, 1 ), and both senders transmit simultaneously, then this table describes the coding steps:

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.