Besides HALT, Minsky's machine includes three assignment ( replacement, substitution ) operations: ZERO ( e. g. the contents of location replaced by 0: L ← 0 ), SUCCESSOR ( e. g. L ← L + 1 ), and DECREMENT ( e. g. L ← L − 1 ).

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

Spassky won the first game as Black in good style, but tenacious, aggressive play from Karpov secured him overall victory by + 4 − 1 = 6.

This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.

variance_n * len ( dataWeightPairs )/( len ( dataWeightPairs ) − 1 )

Below pH 2. 2, the predominant form will have a neutral carboxylic acid group and a positive α-ammonium ion ( net charge + 1 ), and above pH 9. 4, a negative carboxylate and neutral α-amino group ( net charge − 1 ).

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.

This formula requires only k multiplications and k − 1 additions, for any array that can fit in memory.

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

The retracing can stop if the balance factor becomes − 1 or + 1 indicating that the height of that subtree has remained unchanged.

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.

Detailed analysis shows that the complexity class is unchanged by allowing error as high as 1 / 2 − n < sup >− c </ sup > on the one hand, or requiring error as small as 2 < sup >− n < sup > c </ sup ></ sup > on the other hand, where c is any positive constant, and n is the length of input.

Diagram of forces acting on a segment of a catenary from c to r. The forces are the tension T < sub > 0 </ sub > at c, the tension T at r, and the weight of the chain ( 0, − λgs ).

Finally, weight of the chain is represented by ( 0, − λgs ) where λ is the mass per unit length, g is the acceleration of gravity and s is the length of chain between c and r.

The Italian physician Guido da Vigevano ( c. 1280 − 1349 ), planning for a new crusade, made illustrations for a paddle boat and war carriages that were propelled by manually turned compound cranks and gear wheels ( center of image ).

: c < sub > b </ sub > = α f < sub > b </ sub > + ( 1 − α ) b < sub > b </ sub >

Therefore, c divides the initial remainder r < sub > 0 </ sub >, since r < sub > 0 </ sub > = a − q < sub > 0 </ sub > b = mc − q < sub > 0 </ sub > nc = ( m − q < sub > 0 </ sub > n ) c. An analogous argument shows that c also divides the subsequent remainders r < sub > 1 </ sub >, r < sub > 2 </ sub >, etc.

* Any Cartesian product of intervals b and d is Lebesgue measurable, and its Lebesgue measure is ( b − a )( d − c ), the area of the corresponding rectangle.

Pope Sergius III ( c. 860 − 14 April 911 ) was a pope of the Catholic Church from 29 January 904 to 14 April 911, during a period of feudal violence and disorder in central Italy, when the Papacy was a pawn of warring aristocratic factions, often led by prominent women.

Sweyn I Forkbeard ( Old Norse: Sveinn Tjúguskegg ; c. 960 − 3 February 1014 ) was king of Denmark and England, as well as parts of Norway.

In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that ( p ′ − p ) < ( c ln p ) where p ′ denotes the next prime after p. This result was successively improved ; in 1986 Helmut Maier showed that a constant c < 0. 25 can be used.

A simple example is one in which X has a normal distribution with expected value 0 and variance 1, and Y = X if | X | > c and Y = − X if | X | < c, where c > 0.

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.

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

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

:: A ( m, n ) = hyper ( 2, m, n + 3 ) − 3.

:( n = 1 and n = 2 would correspond with A ( m ,− 2 ) = − 1 and A ( m ,− 1 ) = 1, which could logically be added.

An example is ƒ ( x ) = x − 1 / x, which has the oblique asymptote y = x ( m = 1, n = 0 ) as seen in the limits

: r = 9 × 2 < sup > 2n − 1 </ sup > − 1,

: r = ( 2 < sup >( n-m )</ sup >+ 1 )< sup > 2 </ sup > × 2 < sup > m + n </ sup > − 1,

Other exotic baryons have been proposed, such as pentaquarks — baryons made of four quarks and one antiquark ( B = + + + − = 1 ), but their existence is not generally accepted.

Note that t < sup > 0 </ sup > = 1, ( 1 − t )< sup > 0 </ sup > = 1, and that the binomial coefficient,, also expressed as or is:

a − λ1 is not invertible ( because the spectrum of a is not empty ) hence a = λ1: this algebra A is naturally isomorphic to C ( the complex case of the Gelfand-Mazur theorem ).

Some authors use the convention B < sub > 1 </ sub > = − 1 / 2 and state Bernoulli's formula in this way:

This is an elliptic curve with one solution as v < sub > 1 </ sub > = − 31 / 467.

has no real number solution since no real number squared equals − 1.

A common solution is to compute a fixed hash function with a very large range ( say, 0 to 2 < sup > 32 </ sup > − 1 ), divide the result by n, and use the division's remainder.

The converse is not true: not all irrational numbers are transcendental, e. g. the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x < sup > 2 </ sup > − 2

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 ).

The only real solution is − 1, as it factors to

It is clear that the great majority of positions in the puzzle will never be reached when using the shortest possible solution ; indeed, if the priests of the legend are using the longest possible solution ( without re-visiting any position ) it will take them 3 < sup > 64 </ sup > − 1 moves, or more than 10 < sup > 23 </ sup > years.

Acids with a pK < sub > a </ sub > value of less than about − 2 are said to be strong acids ; a strong acid is almost completely dissociated in aqueous solution, to the extent that the concentration of the undissociated acid becomes undetectable.

The solution is to consider the integer number line (..., − 3, − 2, − 1, 0, 1, 2, 3, ...).

It can be used to solve various equations involving exponentials ( e. g. the maxima of the Planck, Bose-Einstein, and Fermi-Dirac distributions ) and also occurs in the solution of delay differential equations, such as y ( t ) = a y ( t − 1 ).

In geometry, a hyperplane of an n-dimensional space V is a " flat " subset of dimension n − 1, or equivalently, of codimension 1 in V ; it may therefore be referred to as an ( n − 1 )- flat of V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly ; in all cases however, any hyperplane can be given in coordinates as the solution of a single ( due to the " codimension 1 " constraint ) algebraic equation of degree 1 ( due to the " flat " constraint ).

When n is an integer, the solution P < sub > n </ sub >( x ) that is regular at x = 1 is also regular at x = − 1, and the series for this solution terminates ( i. e. is a polynomial ).

When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as ( x − r )< sup > 3 </ sup >, with the same root r occurring three times, then the solution would take the form

The solution is a least-squares solution of the original, overdetermined system, minimizing the Euclidean norm || Ax − y ||, a measure of the discrepancy between the two sides in the original system.

# The product log is an implicit function giving the solution for y of the equation x − y e < sup > y </ sup > = 0.

Clearly when the square root was extracted, it was the negative root − 2, rather than the positive root, that was relevant for the particular solution in the problem.