Under the standard duality of plane projective geometry ( where points correspond to lines and collinearity of points corresponds to concurrency of lines ), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa.

Under Poincaré duality, the fundamental class is dual to the bottom class of a connected manifold ( a generator of ): in the closed case, Poincaré duality is the statement that the cap product with the fundamental class yields an isomorphism.

Under Pope Benedict XVI's motu proprio Summorum Pontificum, Roman Catholic bishops, priests, and deacons are again permitted to use the 1962 edition of the Roman Breviary, promulgated by Pope John XXIII to satisfy their obligation to recite the Divine Office every day.

Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number.

Under Dylan the programmer can simply place every function call in the " Private " or " Development " interface, and collect up publicly accessible functions in " Public ".

Under the reforms that instituted the Julian calendar, Intercalaris was abolished, leap years occurred regularly every fourth year, and in leap years February gained a 29th day.

Under such conditions every formula of the form, where ( T ) is a string of quantifiers containing all quantifiers in ( S ) and ( S ') interleaved among themselves in any fashion, but maintaining the relative order inside ( S ) and ( S '), will be equivalent to the original formula Φ '( this is yet another basic result in first-order predicate calculus that we rely on ).

Under Unix, every running program has an associated user id, which normally corresponds to the user that started the process.

Under Cronje's captaincy, South Africa won 27 Tests and lost 11, completing series victories against every team except Australia.

Under the Electoral Act 1997 a Constituency Commission is established following the publication of census figures every five years.

Under this system, every health condition can be assigned to a unique category and given a code, up to six characters long.

Under the 1995 constitution, the president, who is both chief of state and head of the government, is chosen through universal direct suffrage every 5 years.

Under optimal conditions bacteria can grow extremely rapidly and can double as quickly as every 10 minutes.

Under the 1999 Constitution he is allowed to do once every two years, but he must call parliamentary elections with three months.

Under gadaa, every eight years, the Oromo would hold a popular assembly called the Gumi Gayo, where laws were established for the following eight years.

Under the command economy system, every state owned enterprise was required to have a party committee.

Under United Nations authority, humankind has colonized every habitable planet and moon in the solar system.

Under Omani Arabs Zanzibar became East Africa's main slave port, with as many as 50, 000 enslaved Africans passing through every year during the 19th century.

Under international treaty, it may be used " in time of war as in time of peace, by every vessel of commerce or of war, without distinction of flag.

; 1870 – 1891: Under the Elementary Education Act 1870, basic State Education becomes free for every child under the age of 10.

Under SPI, S & T started including a new game in every issue of the magazine, which along with the regular games SPI was publishing vastly increased the number of wargames available.

Under this DS1 pricing model, a provider charges the same price in every geography it services.

Under English law, the burden of proving ( on balance of probability ) the literal truth of every disparaging statement is on the defendant.

Under the concession the NMELT was to construct a tramway and provide electricity within the municipalities of Essendon and Flemington for 30 years, it also mandated a service at least every 20 minutes and had provisions for the undertaking to become property of the municipalities involved earlier than the prescribed 30 years.

Under Spanish law, every corpse can provide organs unless the deceased person had expressly rejected it.

Under the classification-system concept of genre, placing a text into a genre is vital, since " every text is read according to a genre which governs its interpretation " ( Schauber 401 ).

Under the empire, the consuls would change as often as every two months, but only the two consuls at the beginning of the year would lend their names to that year.

In 1981, Lee began using the compact, headless Steinberger bass, which he occasionally used on the supporting tour for Signals and for several tracks on Grace Under Pressure.

Under normalized Ricci flow compact manifolds with this geometry converge to R < sup > 2 </ sup > with the flat metric.

Under normalized Ricci flow compact manifolds with this geometry converge ( rather slowly ) to R < sup > 1 </ sup >.

Under this doctrine The First Nations, Inuit, and Métis are though of a sovereign nations who are in a compact with Canada, rather than minority groups within the Canadian nation state.

Under the random matrix model developed by Nick Katz and Peter Sarnak, there is a conjectural correspondence between ( unitarized ) characteristic polynomials of Frobenius elements and conjugacy classes in the compact Lie group USp ( 2n ) = Sp ( n ).

Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points.

Under the provisions of the compact, the authority is legally incorporated in the District of Columbia, where WMATA maintains its headquarters.

Under Japanese exterior dimension regulations, this series was no longer regarded as a " compact car ".

Under the weak topology the bounded sets coincide with the relatively compact sets which leads to the important Bourbaki – Alaoglu theorem.

Under this assumption the space group must be Af and the following are the positions of the atoms in the unit cell.

Under meager point-set hypotheses, namely sobriety, this is completely accurate — it is possible to recover a sober space from its associated site.

Under this definition, plane tilings and space tilings ( honeycombs ) are considered to be polytopes, and are sometimes classed as apeirotopes because they have infinitely many cells ; tilings of hyperbolic spaces are also included under this definition.

Under both schemes the old rural intersection just outside the Potsdam Gate, and the Octagon ( Leipziger Platz ) just inside, were to be joined together to create a long rectangular space, with a gargantuan edifice standing in the middle of it.

Under liberal democratic systems, privacy creates a space separate from political life, and allows personal autonomy, while ensuring democratic freedoms of association and expression.

Under space law, the only internationally recognized authority for naming celestial bodies is the International Astronomical Union ( IAU ).

Under his leadership, the Examiner devoted 24 percent of its space to crime, presenting the stories as morality plays, and sprinkled adultery and " nudity " ( by 19th century standards ) on the front page.

Under the frequentist interpretation of probability, Bayes ' rule is a general relationship between and, for any events, and in the same event space.

Under the new regulations, smoking rooms are allowed, but are subjected to very strict conditions: they may occupy at most 20 % of the total floor space of the establishment and their size may not be more than 35 m² ; they need to be equipped with separate ventilation which replaces the full volume of air ten times per hour ; the air pressure of the smoking room must constantly be lower than the pressure in the contiguous rooms ; they have doors that close automatically ; no service can be provided in the smoking rooms ; cleaning and maintenance personnel may enter the room only one hour after it was last used for smoking.

Under Project Horizon in the late 1950s, use of spent missile stages for building a manned orbital space station had been proposed, and this general concept was picked up under AAP.

Under the EUL, St. John Properties has entered into a long-term ground lease of federally owned property for the development of commercial office space and research-and-development ( R & D ) space.

Under the current Russian space program Baikonur remains a busy space port, with numerous commercial, military and scientific missions being launched annually.

For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes dot product of vectors.

More space was needed to meet the academic demand, and so in 1948, Mr F. Bray, Under Secretary of the Ministry of Education, opened the converted Downings Warehouse.

" Under Florida law, a convicted murderer could not share cell space with another prisoner before his execution, but another convicted murderer was already awaiting execution at Raiford.

Under this concept, it has become a world leader in the development of low-noise receivers ; large parabolic-dish antennas ; tracking, telemetry, and command systems ; digital signal processing ; and deep space navigation.

Under normal conditions, medial and vestibular nuclei, cortical, thalamic, and cerebellar centers coordinate things such as head and eye movement, and orientation in space.

Under a common house rule, a player who lands on this space wins a jackpot consisting of a collection of fines and taxes that have been placed in the center of the board ( rather than having been paid to the Banker, as they would under the official rules ).

Under this definition, C is called a covering space of X.

Under normal conditions, one can thus calculate the volume of phase space occupied by the gas, by dividing Equation 1 by N !.

In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary ( or antiunitary ) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.