Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

This theory states that an animal's embryonic development follows exactly the same sequence as the sequence of its evolutionary ancestors.

The principle of superposition states that a sedimentary rock layer in a tectonically undisturbed sequence is younger than the one beneath it and older than the one above it.

In a state-machine, the state transition logic can be designed to catch unused states and trigger a reset sequence or other error recovery routine.

International standard ISO 1155 states that a longitudinal redundancy check for a sequence of bytes may be computed in software by the following algorithm:

It states that a logic 0 is represented by a High-Low signal sequence and a logic 1 is represented by a Low-High signal sequence.

The period of a PRNG is defined as the maximum over all starting states of the length of the repetition-free prefix of the sequence.

* K4 — It should be impossible, for all practical purposes, for an attacker to calculate, or guess from an inner state of the generator, any previous numbers in the sequence or any previous inner generator states.

If the Thue – Morse sequence members are used in order to select program states:

Chromosomal regions can adopt stable and heritable alternative states resulting in bistable gene expression without changes to the DNA sequence.

The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence.

The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence.

At the coarsest level, it appears that in transitioning to the native state, a given amino acid sequence takes on roughly the same route and proceeds through roughly the same intermediates and transition states.

For instance, the concrete semantics of an imperative programming language may associate to each program the set of execution traces it may produce – an execution trace being a sequence of possible consecutive states of the execution of the program ; a state typically consists of the value of the program counter and the memory locations ( globals, stack and heap ).

A Markov chain is a sequence of random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, X < sub > 3 </ sub >, ... with the Markov property, namely that, given the present state, the future and past states are independent.

LR ( 1 ) parsers have the requirement that each rule should be expressed in a complete LR ( 1 ) manner, i. e. a sequence of two states with a specific lookahead.

Therefore the sequence of tokens generated by an HMM gives some information about the sequence of states.

and Dirichlet's theorem states that this sequence contains infinitely many prime numbers.

Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run.

However, it is important to note that the notion of the blasé is actually not the central or final point of the essay, but is part of a description of a sequence of states in an irreversible transformation of the mind.

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent.

< li > A finite table ( occasionally called an action table or transition function ) of instructions ( usually quintuples: q < sub > i </ sub > a < sub > j </ sub >→ q < sub > i1 </ sub > a < sub > j1 </ sub > d < sub > k </ sub >, but sometimes 4-tuples ) that, given the state ( q < sub > i </ sub >) the machine is currently in and the symbol ( a < sub > j </ sub >) it is reading on the tape ( symbol currently under the head ) tells the machine to do the following in sequence ( for the 5-tuple models ):

Given a short exact sequence with maps q and r:

The use of the letters ⟨ q ⟩ and ⟨ w ⟩ are very rare, and up to 2006, the ⟨ v ⟩ and ⟨ w ⟩ were often combined in the collating sequence under ⟨ v ⟩.

A sequence of pseudo-random numbers of length q − 1 will be produced by the expansion.

All these subgroups are conjugate, where conjugation is given by translation from p to q ( which is uniquely defined ), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the short exact sequence

# the circles associated with the fractions r / s that are the neighbours of p / q in some Farey sequence, or

As there is no precomposed character to combine the acute accent with the letter q, striking and then is likely to result in ´ q, with the accent and letter as separate characters, or in some systems the invalid typing sequence may be discarded silently.

It is very common for n = p + q to be another natural index in the spectral sequence.

This spectral sequence is doubly graded by the filtration degree p and the complementary degree q = n − p. ( The complementary degree is often a more convenient index than the total degree n. For example, this is true of the spectral sequence of a double complex, explained below.

Notker learned how to fit the separate syllables of a Latin text to the tones of this jubilation ; this poem was called the sequence ( q. v.

The Arnoldi iteration uses the stabilized Gram – Schmidt process to produce a sequence of orthonormal vectors, q < sub > 1 </ sub >, q < sub > 2 </ sub >, q < sub > 3 </ sub >, …, called the Arnoldi vectors, such that for every n, the vectors q < sub > 1 </ sub >, …, q < sub > n </ sub > span the Krylov subspace.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

On systems with 32-bit or larger words, it is possible to speed up execution of this cipher by combining the < tt > SubBytes </ tt > and < tt > ShiftRows </ tt > steps with the < tt > MixColumns </ tt > step by transforming them into a sequence of table lookups.

Two results which follow from the axiom are that " no set is an element of itself ," and that " there is no infinite sequence ( a < sub > n </ sub >) such that a < sub > i + 1 </ sub > is an element of a < sub > i </ sub > for all i. "

By sweeping this surface through R < sup > 3 </ sup > as a function of the ion sequence input data, such as via ion-ordering, a volume is generated onto which positions the 2D detector positions can be computed and placed three-dimensional space.

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

If the convention B < sub > 1 </ sub >=− is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS ); with the convention B < sub > 1 </ sub >=+ is known as the second Bernoulli numbers ( / in OEIS ).

As an example of this process, in 2008 Danish researchers first revealed the sequence of events when oxygen and hydrogen combine on the surface of titanium dioxide ( TiO < sub > 2 </ sub >, or titania ) to produce water.

In this way, Ω < sub > F </ sub > represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string ( of some finite length ) that is in the domain of F. It is for this reason that Ω < sub > F </ sub > is called a halting probability.

* There is a computable sequence of rational numbers converging to such that < math >| q_i-q_

* If is the norm ( usually noted as ) defined in the square-summable sequence space ℓ < sup > 2 </ sup > ( which also matches the usual distance in a continuous and isotropic cartesian space ), then

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

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.

The sequence defined by x < sub > n </ sub > = is Cauchy, but does not have a limit in the given space.

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if F < sub > n </ sub > is closed and non-empty, for every n, and diam ( F < sub > n </ sub >) → 0, then there is a point x ∈ X common to all sets F < sub > n </ sub >.