A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

With this alternate notion of choice function, the axiom of choice can be compactly stated as

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

With eleven keys in a 3 / 4 / 4 arrangement, 43 symbols could be arranged allowing for lowercase text, numbers and a modest number of punctuation symbols to be represented along with a ' shift ' function for accessing uppercase letters.

With most IIR transfer functions there are related transfer functions having a frequency response with the same magnitude but a different phase ; in most case the so-called minimum phase transfer function is preferred.

With such a function one can directly locate the desired entry in a hash table, without any additional searching.

With its four arms attached with a combination of different active agents, for example, iRNA ( for gene silencing ), microRNA ( for gene expression regulation ), aptamer ( for targeting ) and ribozyme ( as catalyst ), the X-shaped RNA can achieve therapeutic and diagnostic functions by regulating gene expression and cellular function, and binding to cancer cells with precision, enhanced by its polyvalent nature and synergistic effects by design.

With northern blotting it is possible to observe cellular control over structure and function by determining the particular gene expression levels during differentiation, morphogenesis, as well as abnormal or diseased conditions.

With these hypotheses ruled out, the best-supported hypothesis for crest function seems to be as a sexual display.

With enough samples, the graph of the histogram will approximate the density function of a chi-squared distribution with one degree of freedom.

With optical imaging devices, for example, it is the Fourier transform of the point spread function ( hence a function of spatial frequency ) i. e. the intensity distribution caused by a point object in the field of view.

With no new designs of nuclear weapons, the warheads in the U. S. stockpile must continue to function far past their original expected lifetimes.

With the aperture function which is 1 for transparent parts of the object plane and 0 otherwise ( i. e. It is 0 if the direct line between source and the point on the screen passes through the blocking circular object.

With reference to mental illness, the function ( or dysfunction ) of the neurotransmitters glutamate and dopamine are thought to be particularly important.

With the judges were associated a body of elders who shared in the decision, but whose exact function is not yet clear.

With the additional operation of function composition, End ( A ) is a ring with multiplicative identity.

With the definition of elliptic functions given above ( which is due to Weierstrass ) the Weierstrass elliptic function is constructed in the most obvious way: given a lattice as above, put

With the right type of fiber it is possible to have a device that does both simultaneously, and can function as an optical add-drop multiplexer.

With permutation matrices the determinant matches the signature, being + 1 or − 1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.

With a single channel ( single-function, one-button ) remote control the presence of a carrier signal can be used to trigger a function.

With the assumptions of the Black – Scholes model, this second order partial differential equation holds for any type of option as long as its price function is twice differentiable with respect to and once with respect to.

With the construction of ramparts and moats the castle became located within the city's fortifications, and the castle lost its function as a stronghold.

With the development of the lead chamber process in 1746 and the Leblanc process, allowing large-scale production of sulfuric acid and sodium carbonate, respectively, chemical reactions became implemented into the industry.

With the increase in access to personal computers in the 1980s, spaced repetition began to be implemented with computer-assisted language learning software-based solutions.

With the two side-by-side, the new ideas of the Protestant Reformation could be quickly implemented, and sermons in the church where held in Swedish starting in 1525 and Latin abolished in 1530.

With the release of SYSTRAN Server 7 in 2010, SYSTRAN implemented a hybrid rule-based / Statistical machine translation ( SMT ) technology which was the first of its kind in the marketplace.

With the rising of militarism during the Shōwa era, the Imperial Japanese Army and the Imperial Japanese Navy implemented swords called Shin guntō and Kai guntō which were worn tachi style ( cutting edge down ).

With the development of the pentagrid converter, these functions were combined inside a single tube which applied the RF signal to the control grid, but also implemented the local oscillator using additional grids.

With this approach, context switching can be done very quickly and, in addition, it can be implemented even on simple kernels which do not support threading.

With only 12 millions of annual income, the Guise implemented, to replenish the state coffers, a policy of draconian austerity that contributed to their unpopularity.

With further sponsorship from BHP, the Foundation implemented the Mungo National Park 60 km long guided vehicle drive in 1990.

With cargo volume that grows year after year following the same economic development rhythm implemented in the State, the Port of Salvador, located in the Bahia de Todos os Santos, holds status as the port with the highest movement of containers of the North / Northeast and the second-leading fruit exporter in Brazil.

With the First World War waging between the United Kingdom and Germany, in 1917 the British government implemented the Military Service Act III, conscripting men into the armed forced who had previously been rejected on medical grounds.

With bipartisan support, and without raising taxes, Barbour implemented a plan called Operation: Streamline to cut the budget deficit in half.

With an ill grace and some shuffling, the companies implemented the Act.

With support from Rae, Peterson's government implemented socially progressive legislation on matters such as pay equity, brought an end to extra-billing by doctors, and established campaign spending limits.

With version 3 they added a more sophisticated overlay called CCL ( Concise Command Language ) that implemented many more commands.

With the Triennial Review in August 2003, the FCC began to rewrite a large portion of the rules implemented by the Telecommunications Act of 1996.

With the war brewing in Europe, Civil Defence schemes were prepared and implemented by district officers.

With the start of World War II in 1939, a group was formed to come up with a solution that could be rapidly implemented.

With respect to Medicaid, the Congressional Budget Office ( CBO ) has estimated that the plan would increase costs for States or those States would reduce enrollment ; with respect to Medicare, the CBO has said the average senior would pay nearly twice what they currently contribute for the same coverage when the plan is fully implemented.

With worsening economic conditions globally Zarlink implemented several workforce reductions while trying to widely diversify the company's product portfolio.

With national expansion of the competition implemented for the 1995 season the NSWRL passed control of the Winfield Cup competition to the ARL.

With woodwork and furnishings designed by Richard Van der Boyen and implemented by Jules Allard and Sons, the music room boasts a gilt coffered ceiling lined with silver and gold, as well as an elliptical ceiling molding which bears the inscription in French of song, music, harmony and melody.

With the formation of ICAB, Duarte Costa implemented a number of reforms of what he saw as problems in the Roman Catholic Church.

With Horace Enea, he designed and implemented Compel an early single assignment language.