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Induction
Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell types by deliberate epigenetic reprogramming * Cellular differentiation, the process where a cell changes from one cell type to another * Enzyme induction and inhibition, a process in which a molecule induces the expression of an enzyme * Morphogenesis, the biological process that causes an organism to develop its shape * Regulation of gene expression, the means by which a gene product is either induced or inhibited Chemistry * Induction period, the time interval between cause and measurable effect * Inductive cleavage, in organic chemistry * Inductive effect, the redistribution of electron density through molecular sigma bonds * Asymmetric induction, the formation of one specific stereoisomer in the presence of a nearby chiral center ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' reasoning. If the premises are correct, the conclusion of a deductive argument is ''certain''; in contrast, the truth of the conclusion of an inductive argument is '' probable'', based upon the evidence given. Types The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference. Inductive generalization A generalization (more accurately, an ''inductive generalization'') proceeds from a premise about a sample to a conclusion about the population. The observation obtained from this sample is projected onto the broader population. : The proportion Q of the sample has attribute A. : Therefore, the proportion Q of the population has attribute A. For example, say there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backward Induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying what action would be most optimal at that moment. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation (i.e. for every possible information set) at every point in time. Backward induction was first used in 1875 by Arthur Cayley, who uncovered the method while trying to solve the infamous Secretary problem. In the mathematical optimization method of dynamic programming, backward induction is one of the main methods for solving the Bellman equation. In game theory, backward induction is a method used to compute subgame perfect equilibria in sequential games. The only difference is that optimizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon-induction
In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations. Statement The schema is for any given property \psi of sets and states that, if for every set x, the truth of \psi(x) follows from the truth of \psi for all elements of x, then this property \psi holds for all sets. In symbols: :\forall x. \Big(\big(\forall (y \in x). \psi(y)\big)\,\to\,\psi(x)\Big)\,\to\,\forall z. \psi(z) Note that for the "bottom case" where x denotes the empty set \, the subexpression \forall(y\in x).\psi(y) is vacuously true for all propositions and so that implication is proven by just proving \psi(\). In words, if a property is persistent when collecting any sets with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Labor Induction
Labor induction is the process or treatment that stimulates childbirth and delivery. Inducing (starting) labor can be accomplished with pharmaceutical or non-pharmaceutical methods. In Western countries, it is estimated that one-quarter of pregnant women have their labor medically induced with drug treatment. Inductions are most often performed either with prostaglandin drug treatment alone, or with a combination of prostaglandin and intravenous oxytocin treatment. Medical uses Commonly accepted medical reasons for induction include: * Postterm pregnancy, i.e. if the pregnancy has gone past the end of the 42nd week. * Intrauterine fetal growth restriction (IUGR). * There are health risks to the woman in continuing the pregnancy (e.g. she has pre-eclampsia). * Premature rupture of the membranes (PROM); this is when the membranes have ruptured, but labor does not start within a specific amount of time. * Premature termination of the pregnancy (abortion). * Fetal death in utero and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymmetric Induction
In stereochemistry, asymmetric induction (also enantioinduction) describes the preferential formation in a chemical reaction of one enantiomer or diastereoisomer over the other as a result of the influence of a chiral feature present in the substrate, reagent, catalyst or environment. Asymmetric induction is a key element in asymmetric synthesis. Asymmetric induction was introduced by Hermann Emil Fischer based on his work on carbohydrates. Several types of induction exist. Internal asymmetric induction makes use of a chiral center bound to the reactive center through a covalent bond and remains so during the reaction. The starting material is often derived from chiral pool synthesis. In relayed asymmetric induction the chiral information is introduced in a separate step and removed again in a separate chemical reaction. Special synthons are called chiral auxiliaries. In external asymmetric induction chiral information is introduced in the transition state through a catalyst o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatic Induction
Electrostatic induction, also known as "electrostatic influence" or simply "influence" in Europe and Latin America, is a redistribution of electric charge in an object that is caused by the influence of nearby charges. In the presence of a charged body, an insulated conductor develops a positive charge on one end and a negative charge on the other end. Induction was discovered by British scientist John Canton in 1753 and Swedish professor Johan Carl Wilcke in 1762. Electrostatic generators, such as the Wimshurst machine, the Van de Graaff generator and the electrophorus, use this principle. See also Stephen Gray in this context. Due to induction, the electrostatic potential (voltage) is constant at any point throughout a conductor. Electrostatic induction is also responsible for the attraction of light nonconductive objects, such as balloons, paper or styrofoam scraps, to static electric charges. Electrostatic induction laws apply in dynamic situations as far as the quasistat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induction (teachers)
Induction is the support and guidance provided to novice teachers and school administrators in the early stages of their careers. Induction encompasses orientation to the workplace, socialization, mentoring, and guidance through beginning teacher practice. England and Wales Induction is used to refer to a period during which a Newly Qualified Teacher in England or Wales is both supported and assessed to ensure that regulatory standards are met. Although probation periods for new teachers had only been dropped in 1992, the Teaching and Higher Education Act 1998 introduced arrangements by which the Secretary of State for Education could bring about regulations requiring new teachers to serve a period of induction. The following year, the then secretary of state, David Blunkett introduced an induction period under The Education (Induction Arrangements for School Teachers) (England) Regulations 1999. These regulations made it a requirement that all teachers complete an induction peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetic Induction
Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field. Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the direction of the induced field. Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism. Electromagnetic induction has found many applications, including electrical components such as inductors and transformers, and devices such as electric motors and generators. History Electromagnetic induction was discovered by Michael Faraday, published in 1831. It was discovered independently by Joseph Henry in 1832. In Faraday's first experimental demonstration (August 29, 1831), he wrapped two wires around opposite sides of an iron ring or "torus" (an arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Induction
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction. Structural induction is used to prove that some proposition ''P''(''x'') holds for all ''x'' of some sort of recursively defined structure, such as formulas, lists, or trees. A well-founded partial order is defined on the structures ("subformula" for formulas, "sublist" for lists, and "subtree" for trees). The structural induction proof is a proof that the proposition holds for all the minimal structures and that if it holds for the immediate substructures of a certain structure ''S'', then it must hold for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then is also true. Then transfinite induction tells us that is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that is true. * Successor case: Prove that for any , follows from (and, if necessary, for all ). * Limit case: Prove that for any |
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