The x-ray data did not permit Douglass to determine uniquely the space group, but a negative test for piezoelectricity led him to assume a center of symmetry.

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

However, the possible absence of a center of symmetry not only moves the hydrogen atom off Af, but also allows the oxygen atoms to become nonequivalent, with Af at Af and Af at Af ( space group Af ), where Af represents the oxygens on one side of the Af layers and Af those on the other side.

At the very first, then, Brumidi was required, by the classically pyramidal shape of his central group, to fill in the triangular space above the seated girl on Liberty's right, before starting on the allegorical figures themselves.

Cell dimensions, number of formula units per cell, space group, and specific gravity are given for all substances.

Gray arsenic ( α-As, space group Rm No. 166 ) adopts a double-layered structure consisting of many interlocked ruffled six-membered rings.

Metallic antimony adopts a layered structure ( space group Rm No. 166 ) in which layers consist of fused ruffled six-membered rings.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

To practice architecture means to offer or render services in connection with the design and construction of a building, or group of buildings and the space within the site surrounding the buildings, that have as their principal purpose human occupancy or use.

Communities would come together for an entire day of singing in a large building where they sat in four distinct areas surrounding an open space, one member directing the group as a whole.

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

At ambient conditions, berkelium assumes its most stable α form which has a hexagonal symmetry, space group P6 < sub > 3 </ sub >/ mmc, lattice parameters of 341 pm and 1107 pm.

When compressed at room temperature to 7 GPa, α-berkelium transforms to the beta modification, which has a face-centered cubic ( fcc ) symmetry and space group Fmm.

Upon heating, α-berkelium transforms into another phase with an fcc lattice ( but slightly different from β-berkelium ), space group Fmm and the lattice constant of 500 pm ; this fcc structure is equivalent to the closest packing with the sequence ABC.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.

After the substituents of a stereocenter have been assigned their priorities, the molecule is so oriented in space that the group with the lowest priority is pointed away from the observer.

Following the discovery, starting in the 1990s, of other Kuiper belt objects, Pluto began to be seen not as a planet orbiting alone at 40 AU, but as the largest of a group of icy bodies in that region of space.

In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy.

Communication requires a sender, a message, and a recipient, although the receiver need not be present or aware of the sender's intent to communicate at the time of communication ; thus communication can occur across vast distances in time and space.

Extendible hashing uses a dynamic hash function that requires space proportional to n to compute the hash function, and it becomes a function of the previous keys that have been inserted.

Merge sort requires Ω ( n ) auxiliary space, but heapsort requires only a constant amount.

The precise definition requires that the dimension " d " so defined is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

* In-place ; i. e., only requires a constant amount O ( 1 ) of additional memory space

Communication requires a sender, a message, and an intended recipient, although the receiver need not be present or aware of the sender's intent to communicate at the time of communication ; thus communication can occur across vast distances in time and space.

* It is currently engaged in a further UK space policy review aimed at determining whether the UK requires a dedicated space agency.

However, since s is an unphysical parameter, physical states must be left invariant by " s-evolution ", and so the physical state space is the kernel of H − E ( this requires the use of a rigged Hilbert space and a renormalization of the norm ).

The analysis so far has been two dimensional ; it turns out that an unperturbed orbit is two-dimensional in a plane fixed in space, and thus the extension to three dimensions requires simply rotating the two-dimensional plane into the required angle relative to the poles of the planetary body involved.

» ( in French, each " double punctuation ", as the guillemet, requires a non-breaking space ; in Russian it does not ).

Targeting to the outer membrane, intermembrane space, and inner membrane often requires another signal sequence in addition to the matrix targeting sequence.

The persistent version of red – black trees requires O ( log n ) space for each insertion or deletion, in addition to time.

Tracing garbage collection cycles are triggered too often if the set of live objects fills most of the available memory ; it requires extra space to be efficient.

Also, less importantly, reference counting requires every memory-managed object to reserve space for a reference count.

Using a space requires clear handwriting to prevent confusion.

This can be useful in functional programming, and requires amortized O ( log n ) space per update.

Backing the vehicle requires same technique as backing an ordinary truck / full trailer combination, though the dolly / semi setup is probably longer, thus requiring more space for maneuvering.

Like MRI and CT scans, another test, called a myelogram, uses radiographs and requires a contrast medium to be injected into the subarachnoid space.

In order to analyze how much time and space a given algorithm requires, computer scientists express the time or space required to solve the problem as a function of the size of the input problem.

After about 379, 000 years the electrons and nuclei combined into atoms ( mostly hydrogen ); hence the radiation decoupled from matter and continued through space largely unimpeded.

For example, NASA's workhorse space shuttle used cryogenic hydrogen / oxygen propellant as its primary means of getting into orbit, and all of the rockets built for the Soviet space program by Sergei Korolev used liquid oxygen as their oxidiser.

In the 1960s, Pratt and Whitney licensed Bacon's U. S. patents for use in the U. S. space program to supply electricity and drinking water ( hydrogen and oxygen being readily available from the spacecraft tanks ).

This first generation of stars re-ionized the surrounding neutral hydrogen, creating expanding bubbles of space through which light could readily travel.

The solution of the Schrödinger equation ( wave equations ) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic ( it is radially symmetric in space and only depends on the distance to the nucleus ).

The escape of hydrogen from Earth's atmosphere into space may have occurred at only one percent of the rate previously believed based on revised estimates of the upper atmosphere's temperature.

Eventually, it would reach 3000K ( corresponding to a typical photon energy of 0. 3 eV and so a frequency of 7. 5 × 10 < sup > 13 </ sup > Hz ), and the photons would begin to be absorbed by the hydrogen plasma filling most of the universe, rendering outer space opaque.

In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.

A very common atomic element in space is hydrogen.

For example NASA's space shuttle fires its engines for around 8. 5 minutes, consuming 1, 000 tonnes of solid propellant ( containing 16 % aluminium ) and an additional 2, 000, 000 litres of liquid propellant ( 106, 261 kg of liquid hydrogen fuel ) to lift the 100, 000 kg vehicle ( including the 25, 000 kg payload ) to an altitude of 111 km and an orbital velocity of 30, 000 km / h.

For at least a portion of its life, a star shines due to thermonuclear fusion of hydrogen into helium in its core, releasing energy that traverses the star's interior and then radiates into outer space.

In May, astronauts voiced concerns over the purging of the stage's hydrogen tank in space.

Outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average .< ref name = tadokoro > This source estimates a density of for the Local Group.

A concept to provide low Earth orbit ( LEO ) propellant depots that could be used as way-stations for other spacecraft to stop and refuel on the way to beyond-LEO missions has proposed that waste gaseous hydrogen — an inevitable byproduct of long-term liquid hydrogen storage in the radiative heat environment of space — would be usable as a monopropellant in a solar-thermal propulsion system.

The ACES Integrated Vehicle Fluids option eliminates all hydrazine and helium from the space vehicle — normally used for attitude control and station keeping — and depends instead on solar-thermal monoprop thrusters using waste hydrogen.

Still, the lower thrust-to-weight ratio of nuclear thermal rockets versus chemical rockets ( which have thrust-to-weight ratios of 70: 1 ) and the large tanks necessary for liquid hydrogen storage mean that solid-core engines are best used in upper stages where vehicle velocity is already near orbital, in space " tugs " used to take payloads between gravity wells, or in launches from a lower gravity planet, moon or minor planet where the required thrust is lower.

Interstellar space contains very small amounts of hydrogen.

Interstellar space contains an average of 10 < sup >− 21 </ sup > kg of mass per cubic meter of space, primarily in the form of non-ionized and ionized hydrogen, with smaller amounts of helium, and no significant amounts of other gasses.

This means that the ramjet scoop must sweep 10 < sup > 21 </ sup > cubic meters of space ( approximately the volume of the Earth ) to collect one kilogram of hydrogen.

Such an ion scoop will use electromagnetic funnels, or electrostatic fields to collect ionized hydrogen gas from space for use as propellant by ramjet propulsion systems ( since much of the hydrogen is not ionized, some versions of a scoop propose ionizing the hydrogen, perhaps with a laser, ahead of the ship.