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Bernstein Algebra
In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by . In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see , and . Baric algebras Baric algebras (or weighted algebras) were introduced by . A baric algebra over a field ''K'' is a possibly non-associative algebra over ''K'' together with a homomorphism ''w'', called the weight, from the algebra to ''K''. Ber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-associative Algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if it is a vector space over ''K'' and is equipped with a ''K''- bilinear binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zygote
A zygote (, ) is a eukaryotic cell formed by a fertilization event between two gametes. The zygote's genome is a combination of the DNA in each gamete, and contains all of the genetic information of a new individual organism. In multicellular organisms, the zygote is the earliest developmental stage. In humans and most other anisogamous organisms, a zygote is formed when an egg cell and sperm cell come together to create a new unique organism. In single-celled organisms, the zygote can divide asexually by mitosis to produce identical offspring. German zoologists Oscar and Richard Hertwig made some of the first discoveries on animal zygote formation in the late 19th century. Humans In human fertilization, a released ovum (a haploid secondary oocyte with replicate chromosome copies) and a haploid sperm cell (male gamete) combine to form a single diploid cell called the zygote. Once the single sperm fuses with the oocyte, the latter completes the division of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamete
A gamete (; , ultimately ) is a haploid cell that fuses with another haploid cell during fertilization in organisms that reproduce sexually. Gametes are an organism's reproductive cells, also referred to as sex cells. In species that produce two morphologically distinct types of gametes, and in which each individual produces only one type, a female is any individual that produces the larger type of gamete—called an ovum— and a male produces the smaller type—called a sperm. Sperm cells or spermatozoa are small and motile due to the flagellum, a tail-shaped structure that allows the cell to propel and move. In contrast, each egg cell or ovum is relatively large and non-motile. In short a gamete is an egg cell (female gamete) or a sperm (male gamete). In animals, ova mature in the ovaries of females and sperm develop in the testes of males. During fertilization, a spermatozoon and ovum unite to form a new diploid organism. Gametes carry half the genetic information of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure Constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles. Recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket or commutator. Definition Given a set of basis vectors \ for the underlying vector space of the algebra, the structure constants or structure coefficients c_^ express the multiplication \cdot of pairs of vectors as a linear combination: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peirce Decomposition
In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements. The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decomposition for Jordan algebras was introduced by . Peirce decomposition for associative algebras If ''e'' is a commuting idempotent (''e''2 = ''e'' and ''e'' is in the center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ... of ''A'') in an associative algebra ''A'', then the two-sided Peirce decomposition writes ''A'' as the direct sum of ''eAe'', ''eA''(1 − ''e''), (1 − ''e'')''Ae'', and (1 − ''e'')''A''(1 − ''e''). There are also left and right Peirce decompositions, where the left decomposition writes ''A'' as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexible Algebra
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: : a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a for any two elements ''a'' and ''b'' of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, American Journal of Mathematics 76: 435 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power-associative
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is performed an operation * by itself several times, it doesn't matter in which order the operations are carried out, so for instance x*(x*(x*x)) = (x*(x*x))*x = (x*x)*(x*x). Examples and properties Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions and Okubo algebras. Any algebra whose elements are idempotent is also power-associative. Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and population structure. Population genetics was a vital ingredient in the emergence of the modern evolutionary synthesis. Its primary founders were Sewall Wright, J. B. S. Haldane and Ronald Fisher, who also laid the foundations for the related discipline of quantitative genetics. Traditionally a highly mathematical discipline, modern population genetics encompasses theoretical, laboratory, and field work. Population genetic models are used both for statistical inference from DNA sequence data and for proof/disproof of concept. What sets population genetics apart from newer, more phenotypic approaches to modelling evolution, such as evolutionary game theory and adaptive dynamics, is its emphasis on such genetic phenomena as dominance, epi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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