Similarly to other " bounded error " probabilistic classes the choice of 1 / 3 in the definition is arbitrary.

In what follows, c < sub > 1 </ sub > and c < sub > 2 </ sub > denote arbitrary complex numbers, c * denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

The exponent r can be generalized to an arbitrary real number as follows: if x > − 1, then

where A is a 2 × 2 orthogonal matrix and b = ( b < sub > 1 </ sub >, b < sub > 2 </ sub >) is an arbitrary ordered pair of numbers ; that is,

A multiplier ( which is a nonlinear device ) will generate ideally only the sum and difference frequencies, whereas an arbitrary nonlinear block would generate also signals at e. g. 2 · f < sub > 1 </ sub >- 3 · f < sub > 2 </ sub >, etc.

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups SU ( 3 ) and SU ( 2 ) which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry U ( 1 ) which in principle allows for arbitrary charge assignments.

Some of those algorithms will map arbitrary long string data z, with any typical real-world distribution — no matter how non-uniform and dependent — to a 32-bit or 64-bit string, from which one can extract a hash value in 0 through n − 1.

This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring ; it is not necessarily true in an arbitrary abelian category ( see Roos ' " Derived functors of inverse limits revisited " for examples of abelian categories in which lim ^ n, on diagrams indexed by a countable set, is nonzero for n > 1 ).

The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U ( 1 ).

Also, exponential map is not surjective nor injective for infinite-dimensional ( see below ) Lie groups modelled on C < sup >∞</ sup > Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction ( AND-ing over an empty set of operands ) is often defined as having the result 1.

While there have been many claims through the years of companies achieving " perfect compression " where an arbitrary number N of random bits can always be compressed to N − 1 bits, these kinds of claims can be safely discarded without even looking at any further details regarding the purported compression scheme.

Only in dimension d = 2 can one construct entities where is replaced by an arbitrary complex number with magnitude 1 ( -> anyons ).

In the application's statement, Caleb Bradham describes the trademark as an " arbitrary hyphenated word " PEPSI-COLA ", and indicated that the mark was in continuous use for his business since August 1, 1901.

This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties.

However, the 1 to 5 scale can be arbitrary and need not be on a linear scale.

Again, the 1 to 5 scale can be arbitrary or non-linear depending on decisions by subject-matter experts.

Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat " truth ", also represented as " T " or " 1 ", as an arbitrary constant.

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

Typically an n-bit CRC applied to a data block of arbitrary length will detect any single error burst not longer than n bits and will detect a fraction 1 − 2 < sup >− n </ sup > of all longer error bursts.

A natural generalization of the inverse semigroup is to define an ( arbitrary ) unary operation ° such that ( a °)°= a for all a in S ; this endows S with a type < 2, 1 > algebra.

These must sum to 1, but are otherwise arbitrary.

: Precise instructions ( in language understood by " the computer ") for a fast, efficient, " good " process that specifies the " moves " of " the computer " ( machine or human, equipped with the necessary internally contained information and capabilities ) to find, decode, and then process arbitrary input integers / symbols m and n, symbols + and =

Bliss-characters can now be used in a creative way to create many new arbitrary concepts, by surrounding the invented words with special Bliss indicators ( similar to punctuation ), something which was not possible in the ISO-IR / 169 encoding.

BDB stores arbitrary key / data pairs as byte arrays, and supports multiple data items for a single key.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

( Strictly speaking, monochrome video does use the full spectrum, if arbitrary and unconstrained movement of subjects and / or cameras is permitted.

These applications typically used an arbitrary markup language to define the codes / tags.

The Shannon switching game involves two players coloring the edges of an arbitrary graph, each attempting to connect two distinguished vertices with edges of his / her color.

Substance terms like " water " which are frequently used as mass nouns, can be used as count nouns to denote arbitrary units of a substance (" Two waters please ") or of several types / varieties (" waters of the world ").

If arbitrary values and / or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction.

Derrida contributed to " the understanding of certain deeply hidden philosophical presuppositions and prejudices in Western culture ", arguing that the whole philosophical tradition rests on arbitrary dichotomous categories ( such as sacred / profane, signifier / signified, mind / body ), and that any text contains implicit hierarchies, " by which an order is imposed on reality and by which a subtle repression is exercised, as these hierarchies exclude, subordinate, and hide the various potential meanings.

Modern GPUs are designed primarily for 3d and add the ability to modify bitmaps in mathematically advanced ways, allowing arbitrary image transformations, texture decompression and filtering, shading for illumination models, alpha blend compositing operations, and depth-buffer comparison / update.

where again C ( and C < nowiki >'</ nowiki > = C / 2 ) is an arbitrary constant of integration.

Semaphores which allow an arbitrary resource count are called counting semaphores, while semaphores which are restricted to the values 0 and 1 ( or locked / unlocked, unavailable / available ) are called binary semaphores ( same functionality that mutexes have ).

The Blahut – Arimoto algorithm, co-invented by Richard Blahut, is an elegant iterative technique for numerically obtaining rate – distortion functions of arbitrary finite input / output alphabet sources and much work has been done to extend it to more general problem instances.

Note that, while the overall timing of " Deathmask / Into the Void " is approximately correct, the apportioning of time of " Deathmask " to " Into the Void " may be arbitrary, as the 3: 08 mark of the song falls during " Into the Void "' s middle-8 vocal (" Freedom fighters sent off to the sun ...").

The keyword / value pairs provide information such as size, origin, coordinates, binary data format, free-form comments, history of the data, and anything else the creator desires: while many keywords are reserved for FITS use, the standard allows arbitrary use of the rest of the name-space.

The difference between C macros on one side, and Lisp-like macros and C ++ template metaprogramming on the other side, is that the latter tools allow performing arbitrary computations at compile-time / parse-time, while expansion of C macros does not perform any computation, and relies on the optimizer ability to perform it.

The PMC standard defines which connector pins are used for which PCI signals ; in addition it defines the optional 64 " P4 " connector pins for use of arbitrary I / O signals.

Once the existence of a universal cover C of X is established, if H ≤ π < sub > 1 </ sub >( X, x ) is an arbitrary subgroup, the space C / H is the covering of X corresponding to H. One also needs to check that two covers of C corresponding to the same ( conjugacy class of ) subgroup of π < sub > 1 </ sub >( X, x ) are equivalent.

Let us very roughly consider the line between the " leftmost " and " rightmost " two points of the k selected points ( for some arbitrary left / right axis: we can choose top and bottom for the exceptional vertical case ).