But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

( Expressed more technically, in each case the pair ( m, n ) decreases in the lexicographic order on pairs, which is a well-ordering, just like the ordering of single non-negative integers ; this means one cannot go down in the ordering infinitely many times in succession.

A binary relation is the special case of an n-ary relation R ⊆ A < sub > 1 </ sub > × … × A < sub > n </ sub >, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A < sub > j </ sub > of the relation.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n:

Bennett, Bernstein, Brassard, and Vazirani proved in 1996 that a brute-force key search on a quantum computer cannot be faster than roughly 2 < sup > n / 2 </ sup > invocations of the underlying cryptographic algorithm, compared with roughly 2 < sup > n </ sup > in the classical case.

Fortune's Algorithm uses a sweepline technique to achieve O ( n log n ) runtime in the planar case.

For the case of n

If the conjecture were true, it would be a generalization of Fermat's last theorem, which could be seen as the special case n = 2: if, then.

The Earley parser executes in cubic time in the general case, where n is the length of the parsed string, quadratic time for unambiguous grammars, and linear time for almost all LR ( k ) grammars.

This is the use of the factor n-1 in the denominator of the formula, rather than just n. This occurs when the sample mean rather than the population mean is used to centre the data and since the sample mean is a linear combination of the data the residual to the sample mean overcounts the number of degrees of freedom by the number of constraint equations — in this case one.

It is not true that if it always has been the case that a God-like being exists then a God-like being exists and it is always going to be the case that a God-like being exists in ( n ).

It is not the case that a God-like being will always exist in ( n ).

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

In the case where V is of finite dimension n it is common to choose a basis for V and identify GL ( V ) with GL ( n, K ) the group of n-by-n invertible matrices on the field K.

Over 99. 94 % of an atom's mass is concentrated in the nucleus ,< ref group = note > In the case of hydrogen-1, with a single electron and nucleon, the proton is, or 99. 95 % of the total atomic mass.

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.

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

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.

In the first case, ƒ ( x ) has y = c as asymptote when x tends to −∞, and in the second that ƒ ( x ) has y = c as an asymptote as x tends to +∞

Thābit's rule corresponds to the case m = n-1.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

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 ).

3-satisfiability is a special case of k-satisfiability ( k-SAT ) or simply satisfiability ( SAT ), when each clause contains exactly k = 3 literals.

In the case of SWNT, covalent functionalization will break some C = C double bonds, leaving " holes " in the structure on the nanotube and, thus, modifying both its mechanical and electrical properties.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.

One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, K < sub > d </ sub > has a simple physical interpretation: when = K < sub > d </ sub >, = or equivalently /(+)= 1 / 2.

Consider the case when we have four layers to blend to produce the final image: F =

In this case, we might compute T = A *( B * D ) only once and simply blend T * C to produce F, a single operation.

In this case we may compute S = A * B once and save this result.

In type 3, this general relationship is maintained peripherally but not centrally where the pulmonary vein follows a more independent path to the hilum as is the case throughout the lung in type 2.

The indecision did not last long: during May 1963 the CCITT Working Party on the New Telegraph Alphabet proposed to assign lower case characters to columns 6 and 7, and International Organization for Standardization TC 97 SC 2 voted during October to incorporate the change into its draft standard.

The capacitors on each drum were organized into 32 " bands " of 50 ( 30 active bands and 2 spares in case a capacitor failed ), giving the machine a speed of 30 additions / subtractions per second.

In addition to the balancing described above for insertions, if the balance factor for the tree is 2 and that of the left subtree is 0, a right rotation must be performed on P. The mirror of this case is also necessary.

In the Jewish Deuterocanonical book Second Maccabees, Chapter 2, " one finds in the records " that Jeremiah, having received an oracle of the Lord, ordered that the tent and the ark and the altar of incense should follow him to the mountain of God where he sealed them up in a cave, and he told those who followed him in order to mark the way ( but they could not find it ) " The place shall remain unknown until God gathers his people together again and shows his mercy, and then the Lord will disclose these things, and the glory of the Lord and the cloud shall appear, as they were shown in the case of Moses, and as Solomon asked that the place be specially consecrated.

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

In this case, the pinch hitter would receive the plate appearance ( and potential of an at-bat ) unless the original batter is replaced when having 2 strikes against him and the pinch hitter subsequently completes the strikeout.

For example, bacteria catalyse the breakdown of the mineral pyrite ( FeS < sub > 2 </ sub >) by oxidising the sulfur and metal ( in this case ferrous iron, ( Fe < sup > 2 +</ sup >)) using oxygen.

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

SAT is also easier if the number of literals in a clause is limited to 2, in which case the problem is called 2SAT.

In the case of the Creme Brulee, placing the ramekins in a roasting pan and filling the pan with hot water until it is 1 / 2 to 2 / 3 of the way up the sides of the ramekins transfers the heat to the custard gently, which prevents the custard from curdling.

As a special case, the three naturally occurring isotopes of the element hydrogen are often specified as H for < sup > 1 </ sup > H ( protium ), D for < sup > 2 </ sup > H ( deuterium ), and T for < sup > 3 </ sup > H ( tritium ).

In the most famous case of policy failure, Black Wednesday, George Soros arbitraged the pound sterling's relationship to the ECU and ( after making $ 2 billion himself and forcing the UK to spend over $ 8bn defending the pound ) forced it to abandon its policy.

Under the federal Holidays Act, Canada Day is observed on July 1 unless that date falls on a Sunday, in which case July 2 is the statutory holiday, although celebratory events generally take place on July 1 even though it is not the legal holiday.

( Note: ∇< sup > 2 </ sup > F represents, in this case, the vector formed by the individual Laplacian of each component of the vector in question )

Since these stock additions retain the short pistol barrel ( as short as 100 mm ( 4 inches )) they are highly restricted under the NFA unless the shoulder stocks are of original manufacture for the gun and the gun has been " delisted " as outlined in links 1 & 2 in the paragraph above, as is the case with so-equipped " Broomhandle " Mausers and Lugers.

In this case, the ordinals 0, 1,,, and are regular, whereas 2, 3,, and ω < sub > ω · 2 </ sub > are initial ordinals which are not regular.