In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

A third problem is to minimize the total number of real multiplications and additions, sometimes called the " arithmetic complexity " ( although in this context it is the exact count and not the asymptotic complexity that is being considered ).

** Context-adaptive binary arithmetic coding ( CABAC ), an algorithm to losslessly compress syntax elements in the video stream knowing the probabilities of syntax elements in a given context.

For example, a calculator program would look at an input such as "" and split it into the tokens,,,,,,,,, each of which is a meaningful symbol in the context of an arithmetic expression.

In the context of second-order arithmetic, results such as Post's theorem establish a close link between the complexity of a formula and the ( non ) computability of the set it defines.

Burgess ( 2005 ) discusses predicative and impredicative theories at some length, in the context of Frege's logic, Peano arithmetic, second order arithmetic, and axiomatic set theory.

Perhaps one of the most striking applications of systoles is in the context of the Schottky problem, by P. Buser and P. Sarnak, who distinguished the Jacobians of Riemann surfaces among principally polarized abelian varieties, laying the foundation for systolic arithmetic.

In his 1996 book, I Was Wrong, he admitted that the first time he actually read the Bible all the way through was while he was in prison, and that it made him realize he had taken certain passages out of context — passages which he had used as " proof texts " to back up his prosperity teachings.

In the context of a loyalty review, membership in a listed organization was meant to raise a question, but not to be considered proof of disloyalty.

In this context, it shows that if we assume there is a formal sentence ( X → Y ), where X itself is equivalent to ( X → Y ), then we can prove Y with a formal proof.

Proposition B is a corollary of proposition A if B can readily be deduced from A or is self-evident from its proof, but the meaning of readily or self-evident varies depending upon the author and context.

In the context of Morocco's Truth Commission of 1999 regarding torture and secret detentions, Wu and Livescu state that the fact that someone remained silent is no proof of their ignorance about a specific piece of information.

Kairos is, for Aristotle, the time and space context in which the proof will be delivered.

Its title is an Arabic phrase meaning Book of the Khazars, while the subtitle " The book of refutation and proof on behalf of the most despised religion " shows its purpose and context in medieval Jewish apologetics.

" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

One of the early definitions of the term " proof of concept " was by Bruce Carsten in the context of a " proof of concept prototype " in the column " Carsten's Corner ":

As a result, a proof that relies on category theoretic ideas often seems slightly out of context to those who are not used to such abstraction, sometimes to the extent that it resembles a comical non sequitur.

NEXPTIME often arises in the context of interactive proof systems, where there are two major characterizations of it.

When the term is used in an informal context rather than as a term of U. S. law, any distilled spirit of high alcoholic purity ( e. g., 170 proof or higher ) that does not contain added flavoring may be referred to as neutral alcohol.

Bunched logic is a variety of substructural logic that, like linear logic, has classes of multiplicative and additive operators, but differs from usual proof calculi in having a tree-like context of hypotheses instead of a flat list-like structure ; it is thus a calculus of deep inference.

KerPass uses an exclusive server side password validation technology that makes possible using a KerPass password in the context of zero knowledge password proof algorithm like SPEKE or SRP.

Examinations at the C1 level may be used as proof of the level of language necessary to work in an international legal context or to follow a course of legal study at university level.

Informal procedures will suffice ; in this context, due process does not require a particular order of proof or mode of offering evidence.

However, the term " no-fault " is most commonly used in the context of state / provincial automobile insurance laws in the United States, Canada, and Australia, in which a policyholder ( and his / her passengers ) are not only reimbursed by the policyholder ’ s own insurance company without proof of fault, but also restricted in the right to seek recovery through the civil-justice system for losses caused by other parties.

My father used to tell me that a text without a context becomes a pretext for a proof text, so when I was still quite young I learned to look at the context.

Here is a proof that is valid in this context.

When placed alongside other songs on the Bear Family box set, " Life Ain't Worth Living " and " Give Me A Future " jar somewhat as demonstration recordings within otherwise released material, but placed in their original context, and although the latter is a mere seventy-six seconds in length, it is clear proof of their ability as early as 1956, incorporating some of the most accomplished harmony singing and triple-time guitar playing on the album.

Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity.

" Kuhrt comments that " the purely Babylonian context of the Cylinder provides no proof " of the historicity of Cyrus's return of the Jewish exiles and the rebuilding of the Temple in Jerusalem, though Becking links this with the lack of any references to the Jews in surviving Achaemenid texts – an indication that the Persians seem not to have regarded them as being of any great importance.

The formula holds in this context as well.

The term " deed ", also known in this context as a " specialty ", is common to signed written undertakings not supported by consideration: the seal ( even if not a literal wax seal but only a notional one referred to by the execution formula, " signed, sealed and delivered ", or even merely " executed as a deed ") is deemed to be the consideration necessary to support the obligation.

In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler – Maclaurin formula is

Parmanand Singh cites Pingala's cryptic formula misrau cha (" the two are mixed ") and cites scholars who interpret it in context as saying that the cases for m beats ( F < sub > m + 1 </ sub >) is obtained by adding a to F < sub > m </ sub > cases and to the F < sub > m − 1 </ sub > cases.

They have also said that the revision of the words of consecration of the wine invalidates the sacrament by moving the phrase mysterium fidei ( in the English translation, " the mystery of faith "), from the middle of the formula of consecration of the wine to after it and changing its context from, they say, referring to the transubstantiated Sacrament to the mystery of Christ's death and resurrection.

) has a specific meaning when used in the context of DNA synthesis: whereas an individual phosphoramidite nucleobase to be added to a DNA polymer has protecting groups and has its molecular weight quoted including these groups, the amount of molecular weight that is ultimately added by this nucleobase to a DNA polymer is referred to as the nucleobase's formula weight ( i. e., the molecular weight of this nucleobase within the DNA polymer, minus protecting groups ).

Though the context makes it perfectly obvious that no vows or obligations towards others are implied, there have been many who were misled into believing that by means of this formula all their vows and oaths are annulled.

In a very broad context, the program built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Selberg and others.

Incidentally, the credit for Shannon's entropy formula ( though not for its use in an information theory context ) really belongs to Boltzmann, who derived it much earlier for use in his H-theorem of statistical mechanics.

* 1990: A. Connes and H. Moscovici prove the local index formula in the context of non-commutative geometry.

Modern usages ( especially in the context of computer science with mathematical software such as model checkers, automated theorem provers, interactive theorem provers ) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness, i. e. of the concrete string representation of formulas ( using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention, using Polish or infix notation, etc.

The Lefschetz trace formula holds in this context and reads:

The Clausius – Mossotti relation is named after the Italian physicist Ottaviano-Fabrizio Mossotti, whose 1850 book analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book in the context not of dielectric constants but of indices of refraction.

The same formula also arises in the context of conductivity, in which it is known as Maxwell's formula.

This formulation was first described by Eugene Wigner in 1932 in the context of quantum mechanics and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the Wigner – Ville distribution, as it was shown in that Wigner's formula needed to use the anaytic signal defined in Ville's paper to be useful as a representation and for a practical analysis.

Suspension, in the context of the formula for absolution, refers to a canonical penalty which can be incurred only by clerics ; therefore, it is omitted when absolving a layman.

In this context, a very good approximation can usually be made using the August-Roche-Magnus formula ( usually called the Magnus or Magnus-Tetens approximation, though this is historically inaccurate ):

Thus, landless nobles were created under the formula Edler von XYZ: either the surname or a place-name followed the German preposition von, which, in this context, was taken to denote nobility.

The name is derived from jantar (" instrument "), and Mantar (" formula ", or in this context " calculation ").