In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

An equivalent definition is the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with an angular frequency of radians per day ; or that length such that, when used to describe the positions of the objects in the Solar System, the heliocentric gravitational constant ( the product GM < sub >☉</ sub >) is equal to ()< sup > 2 </ sup > AU < sup > 3 </ sup >/ d < sup > 2 </ sup >.

It is irreversibly converted to acetyl-CoA, giving off one carbon atom as the waste product carbon dioxide, generating another reducing equivalent as NADH.

Under the Curry – Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem, as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication.

It is not truly distillation, but a recrystallization where the product is the mother liquor, and does not produce products equivalent to distillation.

Lutz goes on to say that when advertisers state that their product is “ good ", it is equivalent in meaning to saying that their product is the best.

The Euclidean topology turns out to be equivalent to the product topology on R < sup > n </ sup > considered as a product of n copies of the real line R ( with its standard topology ).

) only one isomer was ever found, implying that all six carbons are equivalent, so that substitution on any carbon gives only a single possible product.

A ' smoked ' ham must have been smoked by hanging over burning wood chips in a smokehouse or an atomized spray of liquid smoke such that the product appearance is equivalent ; a " hickory-smoked " ham must have been smoked using only hickory.

His estimated net worth, if calculated as a fraction of the U. S. gross domestic product at the time, would have been equivalent to $ 110. 1 billion in 2006 U. S. dollars, making him the fourth richest person in American history .< ref name = forbes >

It was marketed as a product which could perform as well as its Intel Pentium II equivalent but at a significantly lower price.

In another order of ideas, constructing 3-manifolds, it is known that a solid Klein bottle is topologically equivalent with the Cartesian product:, the Mobius band times an interval.

Any mathematical object is considered to be a product of a construction of a mind, and therefore, the existence of an object is equivalent to the possibility of its construction.

* A function's product integral ( as a continuous equivalent to the product of a sequence or the multiplicative version of the ( normal / standard / additive ) integral.

At the time, the country's $ 800 million external foreign debt was equivalent to almost 175 percent of its gross domestic product.

Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row k, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.

Over the real numbers this is equivalent to being able to define a symmetric scalar product, u. v = ½ ( uv + vu ) that can be used to orthogonalise the quadratic form, to give a set of bases

In the context of the transport of heat, the Peclet number is equivalent to the product of the Reynolds number and the Prandtl number.

Substituting ω for W into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:

The tourist industry and governments, however, focus more on the product aspect, treating ecotourism as equivalent to any sort of tourism based in nature.

A very small number of producers, such as Guillon Painturaud and Moyet, do not blend their final product from different ages of eaux-de-vie to produce a ' purer ' flavour ( a practice roughly equivalent to the production of a single-cask Scotch whisky ).

The outer product u ⊗ v is equivalent to a matrix multiplication uv < sup > T </ sup >, provided that u is represented as a m × 1 column vector and v as a n × 1 column vector ( which makes v < sup > T </ sup > a row vector ).

As it was the custom of that alert colony to take over the property of persons asking for protection, this was an act roughly equivalent to throwing open the door to a pack of wolves and saying `` Come and get it ''.

But, in spite of this, I, at present a man 31 years of age and a College Professor, have been recalled `` by direction of the President '' to report on November 25th to Fort Devens, Massachusetts, for another twelve months of Active Duty as an Sp 4 ( the equivalent of a PFC ).

For grant to the Government of India under subsection ( E ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may from time to time be mutually agreed.

For loan to the Government of India under subsection ( G ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may be mutually agreed.

In the event the total of rupees accruing to the Government of the United States of America as a consequence of sales made pursuant to this Agreement is different from the rupee equivalent of $1,276 million, the amounts available for the purposes specified in paragraph 1, Article 2, will be adjusted proportionately.

The Government of India will provide facilities for the conversions of the rupee equivalent of $4 million ( up to a maximum of $1 million per year ) accruing under the subject agreement for agricultural market development purposes into currencies other than United States dollars on request of the Government of the United States of America.

The rupee equivalent of $63.8 million, but not more than 5 percent of the currencies received under the Agreement will be used for loans to be made by the Export-Import Bank of Washington under Section 104 ( E ) of the Agricultural Trade Development and Assistance Act, as amended ( hereinafter referred to as the Act ), and for administrative expenses of the Export-Import Bank of Washington in India incident thereto.

Not only should every educator above the rank of instructor be expected to be a member of one of the professional organizations, but his first qualification for membership as an educator should be so sharply scrutinized that membership would be equivalent to certification to teach the subject.

It is well to bear in mind that gasoline will cost from $.80 to $.90 for the equivalent of a United States gallon and while you might prefer a familiar Ford, Chevrolet or even a Cadillac, which are available in some countries, it is probably wiser to choose the smaller European makes which average thirty, thirty-five and even forty miles to the gallon.

but if drinking is a gerundial noun modifying water and without drinking water is equivalent to without water for drinking, there will be stronger stress on drinking than on water.

Conventional energy for processing foods is available in the range of at most a few cents per kwhr for electric power and the equivalent of a few mills per kwhr for process steam.

The machine shall contain an opening for the insertion of a thermometer or other equivalent equipment for determining the temperature of the water during the washing and rinsing procedures.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

In addition, disclosures that missile workers were earning sums far in excess of what is paid for equivalent work elsewhere provoked his indignation on behalf of the American taxpayer who was footing the bill.

It is the equivalent of 8 per cent for an unmarried investor with more than $16,000 of income to be taxed, or for a married couple with more than $32,000 of taxed income.

perhaps that is the Comedie's equivalent for thrusting this play's characters into our own time.

This time scale is expressed in the form of tables of differences UTC-UTC ( k ) ( equivalent to TAI-TAI ( k )) for each participating institution k. ( The same circular also gives tables of TAI-TA ( k ), for the various unsynchronised atomic time scales.

Most European domestic power supplies run at 230 V, so the current drawn by a particular European appliance ( in Europe ) will be less than for an equivalent American one ( in the United States ).< ref group =" Note "> The formula for power is given by

When the same sequence is equivalent in meaning to I have instructions which I am to leave, dominant stress is ordinarily on instructions.

The two words may be derived from the same Indo-European form * ṇ-mṛ-to-: immortal ( n-: negative prefix equivalent to the prefix a-in both Greek and Sanskrit ; mṛ: zero grade of * mer-: to die ; and-to-: adjectival suffix ).

Therefore rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.

The usage is roughly equivalent to the term " Commie ", " Red " or " pinko " in the United States during the same period.

For instance, the English declarative sentence " snow is white " refers to the same proposition as the equivalent French declarative sentence " la neige est blanche "; two sentences, one proposition.

No one has given a causal explanation, they argue, of why it would not be possible for a functionally equivalent non-conscious organism ( i. e., a philosophical zombie ) to achieve the very same survival advantages as a conscious organism.

Because each governor held equal imperium to the equivalent magistrate, they were escorted by the same number of lictors ( 12 ) and could only be vetoed by a reigning Consul or Praetor.

Such a definition can be formulated in terms of equivalence classes of smooth functions on M. Informally, we will say that two smooth functions f and g are equivalent at a point x if they have the same first-order behavior near x.

All three computational processes ( recursion, the λ-calculus, and the Turing machine ) were shown to be equivalent — all three approaches define the same class of functions.

Concurrently, until the early 1980s, U. S. Navy and U. S. Coast Guard captains selected for promotion to the rank of rear admiral ( lower half ), would wear the same insignia as rear admiral ( upper half ), i. e., two silver stars for collar insignia or sleeve braid of one wide and one narrow gold stripe, even though they were actually only equivalent to one-star officers.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

They were nicknamed Hundemarken ( the German equivalent of " dog tags ") and compared to a similar identification system instituted for dogs in the Prussian capital city of Berlin at about the same time.

YPLL measurements do not account for how disabled a person is before dying, so the measurement treats a person who dies suddenly and a person who died at the same age after decades of illness as equivalent.

Two ( equivalent ) Zeners in series and in reverse order, in the same package, constitute a transient absorber ( or Transorb, a registered trademark ).

Cropping an image and enlarging to the same size final image as an uncropped image taken under the same conditions is equivalent to using a smaller format under the same conditions, so the cropped image has less DOF.

According to this notation, duodecimal 50 expresses the same quantity as decimal 60 (= five times twelve ), duodecimal 60 is equivalent to decimal 72 (= six times twelve = half a gross ), duodecimal 100 has the same value as decimal 144 (= twelve times twelve

Two elements of the set are considered equivalent ( with respect to the equivalence relation ) if and only if they are elements of the same cell.

Then all elements of X equivalent to each other are also elements of the same equivalence class.

Two such functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.

Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are three equivalent ways of specifying the same thing.