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.
Hamming codeIn computer science and telecommunication, Hamming codes are a family of linear error-correcting codes. Hamming codes can detect one-bit and two-bit errors, or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three. Richard W.
Periodic trendsIn chemistry, periodic trends are specific patterns that are present in the Periodic table that illustrate different aspects of certain elements when grouped by period and/or group. They were discovered by the Russian chemist Dmitri Mendeleev in the year 1863. Major periodic trends include atomic radius, ionization energy, electron affinity, electronegativity, valency and metallic character. These trends exist because of the similar electron configurations of the elements within their respective groups or periods; they reflect the periodic nature of the elements.
Periodic tableThe periodic table, also known as the periodic table of the elements, arranges the chemical elements into rows ("periods") and columns ("groups"). It is an organizing icon of chemistry and is widely used in physics and other sciences. It is a depiction of the periodic law, which says that when the elements are arranged in order of their atomic numbers an approximate recurrence of their properties is evident. The table is divided into four roughly rectangular areas called blocks.
E8 latticeIn mathematics, the E_8 lattice is a special lattice in R^8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E_8 root system. The norm of the E_8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H.
Block (periodic table)A block of the periodic table is a set of elements unified by the atomic orbitals their valence electrons or vacancies lie in. The term appears to have been first used by Charles Janet. Each block is named after its characteristic orbital: s-block, p-block, d-block, f-block and g-block. The block names (s, p, d, and f) are derived from the spectroscopic notation for the value of an electron's azimuthal quantum number: sharp (0), principal (1), diffuse (2), or fundamental (3).
Modular formIn mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
Euclidean spaceEuclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
Hamming distanceIn information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of substitutions required to change one string into the other, or the minimum number of errors that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences.
Error correction codeIn computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors.
Extended periodic tableAn extended periodic table theorises about chemical elements beyond those currently known in the periodic table and proven. , the element with the highest atomic number known is oganesson (Z = 118), which completes the seventh period (row) in the periodic table. All elements in the eighth period and beyond thus remain purely hypothetical. Elements beyond 118 will be placed in additional periods when discovered, laid out (as with the existing periods) to illustrate periodically recurring trends in the properties of the elements concerned.
Unimodular latticeIn geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E8 lattice and the Leech lattice are two famous examples. A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·). The lattice is integral if (·,·) takes integer values. The dimension of a lattice is the same as its rank (as a Z-module).
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.
Euclidean distanceIn mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
Period (periodic table)A period on the periodic table is a row of chemical elements. All elements in a row have the same number of electron shells. Each next element in a period has one more proton and is less metallic than its predecessor. Arranged this way, elements in the same group (column) have similar chemical and physical properties, reflecting the periodic law. For example, the halogens lie in the second-to-last group (group 17) and share similar properties, such as high reactivity and the tendency to gain one electron to arrive at a noble-gas electronic configuration.
E8 (mathematics)DISPLAYTITLE:E8 (mathematics) In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.
Finite fieldIn mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power.
Leech latticeIn mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1. It is even; i.e.
Pseudo-Euclidean spaceIn mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x. For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic.
Hilbert modular formIn mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m-fold product of upper half-planes satisfying a certain kind of functional equation. Let F be a totally real number field of degree m over the rational field. Let be the real embeddings of F. Through them we have a map Let be the ring of integers of F. The group is called the full Hilbert modular group.
