Lattice (group)In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
Integer latticeIn mathematics, the n-dimensional integer lattice (or cubic lattice), denoted \mathbb{Z}^n, is the lattice in the Euclidean space \mathbb{R}^n whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. \mathbb{Z}^n is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2n n!.
Lattice (order)A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection.
Complete latticeIn mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice. Specifically, every non-empty finite lattice is complete. Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.
Zintl phaseIn chemistry, a Zintl phase is a product of a reaction between a group 1 (alkali metal) or group 2 (alkaline earth metal) and main group metal or metalloid (from groups 13, 14, 15, or 16). It is characterized by intermediate metallic/ionic bonding. Zintl phases are a subgroup of brittle, high-melting intermetallic compounds that are diamagnetic or exhibit temperature-independent paramagnetism and are poor conductors or semiconductors. This type of solid is named after German chemist Eduard Zintl who investigated them in the 1930s.
Period 4 elementA period 4 element is one of the chemical elements in the fourth row (or period) of the periodic table of the chemical elements. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behaviour of the elements as their atomic number increases: a new row is begun when chemical behaviour begins to repeat, meaning that elements with similar behaviour fall into the same vertical columns. The fourth period contains 18 elements beginning with potassium and ending with krypton – one element for each of the eighteen groups.
MetalloidA metalloid is a type of chemical element which has a preponderance of properties in between, or that are a mixture of, those of metals and nonmetals. There is no standard definition of a metalloid and no complete agreement on which elements are metalloids. Despite the lack of specificity, the term remains in use in the literature of chemistry. The six commonly recognised metalloids are boron, silicon, germanium, arsenic, antimony and tellurium. Five elements are less frequently so classified: carbon, aluminium, selenium, polonium and astatine.
Period 5 elementA period 5 element is one of the chemical elements in the fifth row (or period) of the periodic table of the chemical elements. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behaviour of the elements as their atomic number increases: a new row is begun when chemical behaviour begins to repeat, meaning that elements with similar behaviour fall into the same vertical columns. The fifth period contains 18 elements, beginning with rubidium and ending with xenon.
Electron configurationIn atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom is 1s2 2s2 2p6, meaning that the 1s, 2s and 2p subshells are occupied by 2, 2 and 6 electrons respectively. Electronic configurations describe each electron as moving independently in an orbital, in an average field created by all other orbitals.
Organogermanium chemistryOrganogermanium chemistry is the science of chemical species containing one or more C–Ge bonds. Germanium shares group 14 in the periodic table with carbon, silicon, tin and lead. Historically, organogermanes are considered as nucleophiles and the reactivity of them is between that of organosilicon and organotin compounds. Some organogermanes have enhanced reactivity compared with their organosilicon and organoboron analogues in some cross-coupling reactions.
Geometry of numbersGeometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by . The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.
DefinitionA definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.
Sphere packingIn geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
Lexical definitionThe lexical definition of a term, also known as the dictionary definition, is the definition closely matching the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictionary. A lexical definition is usually the type expected from a request for definition, and it is generally expected that such a definition will be stated as simply as possible in order to convey information to the widest audience.
X-ray tubeAn X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast to other sources of ionizing radiation, X-rays are only produced as long as the X-ray tube is energized. X-ray tubes are also used in CT scanners, airport luggage scanners, X-ray crystallography, material and structure analysis, and for industrial inspection.
SiliconSilicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic table: carbon is above it; and germanium, tin, lead, and flerovium are below it. It is relatively unreactive. Because of its high chemical affinity for oxygen, it was not until 1823 that Jöns Jakob Berzelius was first able to prepare it and characterize it in pure form.
Orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.
RadonRadon is a chemical element with the symbol Rn and atomic number 86. It is a radioactive, colourless, odourless, tasteless noble gas. It occurs naturally in minute quantities as an intermediate step in the normal radioactive decay chains through which thorium and uranium slowly decay into various short-lived radioactive elements and eventually into stable lead. Radon itself is the immediate decay product of radium. Its most stable isotope, 222Rn, has a half-life of only 3.8 days, making it one of the rarest elements.
Circular definitionA circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic. Circular definitions are related to Circular reasoning in that they both involve a self-referential approach. Circular definitions may be unhelpful if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself.
Gröbner basisIn mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K[x1, ..., xn] over a field K. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite.
