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Quasispecies Model
The quasispecies model is a description of the process of the Darwinian evolution of certain self-replicating entities within the framework of physical chemistry. A quasispecies is a large group or "cloud" of related genotypes that exist in an environment of high mutation rate (at stationary state), where a large fraction of offspring are expected to contain one or more mutations relative to the parent. This is in contrast to a species, which from an evolutionary perspective is a more-or-less stable single genotype, most of the offspring of which will be genetically accurate copies. It is useful mainly in providing a qualitative understanding of the evolutionary processes of self-replicating macromolecules such as RNA or DNA or simple asexual organisms such as bacteria or viruses (see also viral quasispecies), and is helpful in explaining something of the early stages of the origin of life. Quantitative predictions based on this model are difficult because the parameters that ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation tends to exist within any given population as a result of genetic mutation and recombination. Evolution occurs when evolutionary processes such as natural selection (including sexual selection) and genetic drift act on this variation, resulting in certain characteristics becoming more common or more rare within a population. The evolutionary pressures that determine whether a characteristic is common or rare within a population constantly change, resulting in a change in heritable characteristics arising over successive generations. It is this process of evolution that has given rise to biodiversity at every level of biological organisation, including the levels of species, individual organisms, and molecules. The theory of evolution by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cytosine
Cytosine () ( symbol C or Cyt) is one of the four nucleobases found in DNA and RNA, along with adenine, guanine, and thymine (uracil in RNA). It is a pyrimidine derivative, with a heterocyclic aromatic ring and two substituents attached (an amine group at position 4 and a keto group at position 2). The nucleoside of cytosine is cytidine. In Watson-Crick base pairing, it forms three hydrogen bonds with guanine. History Cytosine was discovered and named by Albrecht Kossel and Albert Neumann in 1894 when it was hydrolyzed from calf thymus tissues. A structure was proposed in 1903, and was synthesized (and thus confirmed) in the laboratory in the same year. In 1998, cytosine was used in an early demonstration of quantum information processing when Oxford University researchers implemented the Deutsch-Jozsa algorithm on a two qubit nuclear magnetic resonance quantum computer (NMRQC). In March 2015, NASA scientists reported the formation of cytosine, along with uracil and thym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron–Frobenius Theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems ( subshifts of finite type); to economics ( Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines (PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducibility (mathematics)
In mathematics, the concept of irreducibility is used in several ways. * A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. * In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. * In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module. * Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as ''irreducible over an algebraic closure''. * In commutative algebra, a commutative ring ''R'' is irreducible if its prime spectrum, that is, the topological space Spec ''R'', is an irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Matrix
Primitive may refer to: Mathematics * Primitive element (field theory) * Primitive element (finite field) * Primitive cell (crystallography) * Primitive notion, axiomatic systems * Primitive polynomial (other), one of two concepts * Primitive function or antiderivative, ''F''′ = ''f'' * Primitive permutation group * Primitive root of unity; See Root of unity * Primitive triangle, an integer triangle whose sides have no common prime factor Sciences * Primitive (phylogenetics), characteristic of an early stage of development or evolution * Primitive equations, a set of nonlinear differential equations that are used to approximate atmospheric flow * Primitive change, a general term encompassing a number of basic molecular alterations in the course of a chemical reaction Computing * Cryptographic primitives, low-level cryptographic algorithms frequently used to build computer security systems * Geometric primitive, the simplest kinds of figures in computer graphics * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonalizable Matrix
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) For a finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation T = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, A is represented by Diagonalization is the process of finding the above P and Diagonalizable matrices and maps are especially easy for computations, once their eigenvalues and eigenvectors are known. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power, and the determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A Equation solving, solution to the system above is given by the Tuple, ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Threshold (evolution)
In evolutionary biology and population genetics, the error threshold (or critical mutation rate) is a limit on the number of base pairs a self-replicating molecule may have before mutation will destroy the information in subsequent generations of the molecule. The error threshold is crucial to understanding "Eigen's paradox". The error threshold is a concept in the origins of life (abiogenesis), in particular of very early life, before the advent of DNA. It is postulated that the first self-replicating molecules might have been small ribozyme-like RNA molecules. These molecules consist of strings of base pairs or "digits", and their order is a code that directs how the molecule interacts with its environment. All replication is subject to mutation error. During the replication process, each digit has a certain probability of being replaced by some other digit, which changes the way the molecule interacts with its environment, and may increase or decrease its fitness, or ability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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