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Conjugate (group Theory), Conjugate
Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change of sign of the imaginary part of a complex number *Conjugate (square roots), the change of sign of a square root in an expression *Conjugate element (field theory), a generalization of the preceding conjugations to roots of a polynomial of any degree *Conjugate transpose, the complex conjugate of the transpose of a matrix * Harmonic conjugate in complex analysis * Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph *In group theory, various notions are called conjugation: **Inner automorphism, a type of conjugation homomorphism **Conjugacy class in group theory, related to matrix similarity in linear algebra **Conjugation (group theory), the image of an element und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grammatical Conjugation
In linguistics, conjugation ( ) is the creation of derived forms of a verb from its principal parts by inflection (alteration of form according to rules of grammar). For instance, the verb ''break'' can be conjugated to form the words ''break'', ''breaks'', and ''broke''. While English has a relatively simple conjugation, other languages such as French and Arabic or Spanish are more complex, with each verb having dozens of conjugated forms. Some languages such as Georgian and Basque (some verbs only) have highly complex conjugation systems with hundreds of possible conjugations for every verb. Verbs may inflect for grammatical categories such as person, number, gender, case, tense, aspect, mood, voice, possession, definiteness, politeness, causativity, clusivity, interrogatives, transitivity, valency, polarity, telicity, volition, mirativity, evidentiality, animacy, associativity, pluractionality, and reciprocity. Verbs may also be affected by agreement, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem addres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Quantities
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty relation corresponds to the symplectic form. Also, conjugate variables are related by Noether's theorem, which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved). Conjugate variables in thermodynamics are widely used. Examples There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Examples of canonically conjugate variables include the following: * Time and fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Variables (thermodynamics)
In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy, pressure and volume, or chemical potential and particle number. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power. For a mechanical system, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" that, when unbalanced, cause certain generalized "displacements", and the product of the two is the energy transferred as a result. These forces and their associated displacements are called ''conjugate variables''. The thermodynamic force is always an intensive variable and the displacement is always an extensive variable, yiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugated System
In physical organic chemistry, a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule, which in general lowers the overall energy of the molecule and increases Chemical stability, stability. It is Resonance (chemistry), conventionally represented as having alternating single and multiple covalent bond, bonds. Lone pairs, radical (chemistry), radicals or carbenium ions may be part of the system, which may be Cyclic molecule, cyclic, acyclic, Linear molecular geometry, linear or mixed. The term "conjugated" was coined in 1899 by the German chemist Johannes Thiele (chemist), Johannes Thiele. Conjugation is the orbital overlap, overlap of one p-orbital with another across an adjacent Sigma bond, σ bond (in transition metals, d-orbitals can be involved). A conjugated system has a region of overlapping p-orbitals, bridging the interjacent locations that simple diagrams illustrate as not having a π bond. They allow a delocalization of pi el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate (acid-base Theory)
A conjugate acid, within the Brønsted–Lowry acid–base theory, is a chemical compound formed when an acid gives a proton () to a base—in other words, it is a base with a hydrogen ion added to it, as it loses a hydrogen ion in the reverse reaction. On the other hand, a conjugate base is what remains after an acid has donated a proton during a chemical reaction. Hence, a conjugate base is a substance formed by the removal of a proton from an acid, as it can gain a hydrogen ion in the reverse reaction. Because some acids can give multiple protons, the conjugate base of an acid may itself be acidic. In summary, this can be represented as the following chemical reaction: \text + \text \; \ce \; \text + \text Johannes Nicolaus Brønsted and Martin Lowry introduced the Brønsted–Lowry theory, which said that any compound that can give a proton to another compound is an acid, and the compound that receives the proton is a base. A proton is a subatomic particle in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugation (biochemistry)
Bioconjugation is a chemical strategy to form a stable Covalent bond, covalent link between two molecules, at least one of which is a biomolecule. Methods to conjugate biomolecules are applied in various field, including medicine, diagnostics, biocatalysis and materials. Synthetically modified biomolecules can have diverse functionalities, such as tracking cellular events, revealing enzyme function, determining protein biodistribution, Imaging science, imaging specific biomarkers, and delivering Pharmaceutical drug, drugs to targeted cells. Bioconjugation is a crucial strategy that links these modified biomolecules with different Substrate (biochemistry), substrates. Besides applications in biomedical research, bioconjugation has recently also gained importance in nanotechnology such as bioconjugated quantum dots. The most common types of bioconjugation include coupling of a small molecule (such as biotin or a fluorescent dye) to a protein. Antibody-drug conjugates such as Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Vaccine
A conjugate vaccine is a type of subunit vaccine which combines a weak antigen with a strong antigen as a carrier so that the immune system has a stronger response to the weak antigen. Vaccines are used to prevent diseases by invoking an immune response to an antigen, part of a bacterium or virus that the immune system recognizes. This is usually accomplished with an attenuated or dead version of a pathogenic bacterium or virus in the vaccine, so that the immune system can recognize the antigen later in life. Most vaccines contain a single antigen that the body will recognize. However, the antigen of some pathogens does not elicit a strong response from the immune system, so a vaccination against this weak antigen would not protect the person later in life. In this case, a conjugate vaccine is used in order to invoke an immune system response against the weak antigen. In a conjugate vaccine, the weak antigen is covalently attached to a strong antigen, thereby eliciting a stronge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bacterial Conjugation
Bacterial conjugation is the transfer of genetic material between Bacteria, bacterial cells by direct cell-to-cell contact or by a bridge-like connection between two cells. This takes place through a pilus. It is a parasexual cycle, parasexual mode of reproduction in bacteria. It is a mechanism of horizontal gene transfer as are Transformation (genetics), transformation and Transduction (genetics), transduction although these two other mechanisms do not involve cell-to-cell contact. Classical ''E. coli'' bacterial conjugation is often regarded as the bacterial equivalent of sexual reproduction or mating, since it involves the exchange of genetic material. However, it is not sexual reproduction, since no exchange of gamete occurs, and indeed no biogenesis, generation of a new organism: instead, an existing organism is transformed. During classical ''E. coli'' conjugation, the ''donor'' cell provides a conjugative or mobilizable genetic element that is most often a plasmid or trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sexual Conjugation
Isogamy is a form of sexual reproduction that involves gametes of the same morphology (indistinguishable in shape and size), and is found in most unicellular eukaryotes. Because both gametes look alike, they generally cannot be classified as male or female. Instead, organisms that reproduce through isogamy are said to have different mating types, most commonly noted as "+" and "−" strains. Etymology The etymology of isogamy derives from the Greek adjective ''isos'' (meaning equal) and the Greek verb ''gameo'' (meaning to have sex/to reproduce), eventually meaning "equal reproduction" which refers to a hypothetical initial model of equal contribution of resources by both gametes to a zygote in contrast to a later evolutional stage of anisogamy. The term isogamy was first used in the year 1891. Characteristics of isogamous species Isogamous species often have two mating types ( heterothallism), but sometimes can occur between two haploid individuals that are mitotic d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Prior
In Bayesian probability theory, if, given a likelihood function p(x \mid \theta), the posterior distribution p(\theta \mid x) is in the same probability distribution family as the prior probability distribution p(\theta), the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function p(x \mid \theta). A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may clarify how a likelihood function updates a prior distribution. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.Howard Raiffa and Robert Schlaifer. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961. A similar c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
