; Algebraically closed field: A field F is algebraically closed if every polynomial in F has a root in F ; equivalently: every polynomial in F is a product of linear factors.

Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point.

Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions.

Algebraically, the projectivization of this vector space is Proj of its symmetric algebra, that is,

; Algebraically independent elements: Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F.

Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.

Algebraically, this dilution factor will give us the physiologic dead space as calculated by the Bohr equation:

Algebraically, these units can be thought of as the squares of the corresponding length units.

Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras.

-- The crystallization of copolymers comprising Af units interspersed with a minor percentage of Af is limited by the inability of the crystal lattice characteristic of the former to accommodate the bulky side group of the latter.

Amethyst is a purple variety of quartz ( SiO < sub > 2 </ sub >) and owes its violet color to irradiation, iron impurities ( in some cases in conjunction with transition element impurities ), and the presence of trace elements, which result in complex crystal lattice substitutions.

A perfect crystal is one in which the internal lattice structure extends uninterrupted in all directions.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

He ordained further that some should be called " Abbreviators of the Upper Bar " ( Abbreviatores de Parco Majori ; the name derived from a space in the chancery, surrounded by a grating, in which the officials sat, which is called higher or lower ( major or minor ) according to the proximity of the seats to that of the vice-chancellor ), the others of the Lower Bar ( Abbreviatores de Parco Minori ); that the former should sit upon a slightly raised portion of the chamber, separated from the rest of the hall or chamber by lattice work, assist the Cardinal Vice-Chancellor, subscribe the letters and have the principal part in examining, revising, and expediting the apostolic letters to be issued with the leaden seal ; that the latter, however, should sit among the apostolic writers upon benches in the lower part of the chamber, and their duty was to carry the signed schedules or supplications to the prelates of the upper bar.

The Funkturm Berlin is a tall lattice radio tower at the fair area, built between 1924 and 1926.

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.

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

In many superconductors, the attractive interaction between electrons ( necessary for pairing ) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice ( the phonons ).

In most materials ( in low temperature superconductors ), this attraction is brought about indirectly by the coupling of electrons to the crystal lattice ( as explained above ).

This is incorporated into the BCS theory, where lattice vibrations yield the binding energy of electrons in a Cooper pair.

Half or more of the trench is filled with " green " ( unfired ) bricks which are stacked in an open lattice pattern to allow airflow.

The lattice is capped with a roofing layer of finished brick.

Also called the " caesium chloride structure ", this structural motif is composed of a primitive cubic lattice with a two-atom basis, each with an eightfold coordination ; the chloride atoms lie upon the lattice points at the edges of the cube, while the caesium atoms lie in the holes in the center of the cubes.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

However, as the temperature of a semiconductor rises above absolute zero, there is more energy in the semiconductor to spend on lattice vibration and on exciting electrons into the conduction band.

The longheld view is that the carbon atoms fit into octahedral interstices in a close packed metal lattice when the metal atom radius is greater than approximately 135 pm:

the body centred cubic structure adopted by vanadium, niobium, tantalum, chromium, molybdenum and tungsten is not a close packed lattice.

The simple view that the lattice of the pure metal " absorbs " carbon atoms can be seen to be untrue as the packing of the metal atom lattice in the carbides is different from the packing in the pure metal, although it is technically correct that the carbon atoms fit into the octahedral interstices of a close-packed metal lattice.

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention.

For example, the operations of set union and set intersection are both idempotent, as are logical conjunction and logical disjunction, and, in general, the meet and join operations of a lattice.

A non-empty subset F of a lattice ( P ,≤) is a filter, if and only if it is an upper set that is closed under finite meets ( infima ), i. e., for all x, y in F, we find that x ∧ y is also in F.

In both of these models the electrons are seen as a gas traveling through the lattice of the solid with an energy that is essentially isotropic in that it depends on the square of the magnitude, not the direction of the momentum vector k. In three-dimensional k-space, the set of points of the highest filled levels ( the Fermi surface ) should therefore be a sphere.

This approach uses a discrete set of space-time points ( called the lattice ) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation which is then carried out on supercomputers like the QCDOC which was constructed for precisely this purpose.

Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).

This partially ordered set is always a distributive lattice.

* Tamari lattice, a partially ordered set in which the elements can be defined as binary trees and the ordering between elements is defined by tree rotation.

In this model, a given reflection is associated with a set of evenly spaced sheets running through the crystal, usually passing through the centers of the atoms of the crystal lattice.

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ( join ) and an infimum ( meet ).

A partially ordered set ( L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound ( the infimum, also called the meet ) and a least upper bound ( the supremum, also called the join ) in ( L, ≤).

Indeed, a totally ordered set ( with its order topology ) is compact as a topological space if it is complete as a lattice.

More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Since the supremum and infimum of an unbounded set of real numbers may not exist ( the reals are not a complete lattice ), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set, which is a complete lattice.

The unit interval is a totally ordered set and a complete lattice ( every subset of the unit interval has a supremum and an infimum ).

A French window ( when hinged French door ) is a large door-sized lattice light, typically set in pairs or multiples thereof.

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.