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5,400 | In comparison, an element like phosphorus transfers in mineral form, as it is contained in rock material. These cycles also greatly vary in mobility, solubility, and the rate at which they move through their natural cycles. Together, they drive all of the processes of soil chemistry. | Soil chemistry | 0.826056 |
5,401 | A knowledge of environmental soil chemistry is paramount to predicting the fate of contaminants, as well as the processes by which they are initially released into the soil. Once a chemical is exposed to the soil environment, myriad chemical reactions can occur that may increase or decrease contaminant toxicity. These ... | Soil chemistry | 0.826056 |
5,402 | Though there are many other variables, such as texture, soils that lack sufficient organic matter content are susceptible to soil degradation and drying, as there is nothing supporting the soil structure. This often leads to a decline in soil fertility and an increase in erodibility. Other associated concepts: Anion an... | Soil chemistry | 0.826056 |
5,403 | Soil chemistry is the study of the chemical characteristics of soil. Soil chemistry is affected by mineral composition, organic matter and environmental factors. In the early 1870s a consulting chemist to the Royal Agricultural Society in England, named J. Thomas Way, performed many experiments on how soils exchange io... | Soil chemistry | 0.826056 |
5,404 | One can draw conclusions or frame new hypotheses on similar reactions in different soils with diverse textures, organic matter contents, types of clay minerals and oxides, pH, and drainage condition. Laboratory studies have the disadvantage that they lose some of the realism and heterogeneity of undisturbed soil in the... | Soil chemistry | 0.826056 |
5,405 | as occurs when the soil is shaken or leached with an extracting soil solution. These methods usually use surface spectroscopic techniques, such as Fourier transform infrared spectroscopy, nuclear magnetic resonance, Mössbauer spectroscopy, and X-ray spectroscopy. These approaches aim to obtain information on the chemic... | Soil chemistry | 0.826056 |
5,406 | Renewed interest in recent decades has led many soil chemists to maintain soil samples in a field-moist condition and stored at 4 °C (39 °F) under aerobic conditions before and during investigations.Two approaches are frequently used in laboratory investigations in soil chemistry. The first is known as batch equilibrat... | Soil chemistry | 0.826056 |
5,407 | New knowledge about the chemistry of soils often comes from studies in the laboratory, in which soil samples taken from undisturbed soil horizons in the field are used in experiments that include replicated treatments and controls. In many cases, the soil samples are air dried at ambient temperatures (e.g., 25 °C (77 °... | Soil chemistry | 0.826056 |
5,408 | In a class of algebras defined as all the models of a set of equations, it is usually the case that some algebras of the class satisfy more equations than just those needed to qualify them for the class. The class of Boolean algebras is unusual in that, with a single exception, every Boolean algebra satisfies exactly t... | Boolean algebras canonically defined | 0.826046 |
5,409 | Furthermore, every complemented distributive lattice forms a Boolean algebra, and conversely every Boolean algebra forms a complemented distributive lattice. This provides an alternative definition of a Boolean algebra, namely as any complemented distributive lattice. Each of these three properties can be axiomatized w... | Boolean algebras canonically defined | 0.826046 |
5,410 | The lattice is said to be distributive when x∧(y∨z) = (x∧y)∨(x∧z), or equivalently when x∨(y∧z) = (x∨y)∧(x∨z), since either law implies the other in a lattice. These are laws of Boolean algebra whence the underlying poset of a Boolean algebra forms a distributive lattice. Given a lattice with a bottom element 0 and a t... | Boolean algebras canonically defined | 0.826046 |
5,411 | Any poset (B,≤) such that every pair x,y of elements has both a sup and an inf is called a lattice. We write x∨y for the sup and x∧y for the inf. The underlying poset of a Boolean algebra always forms a lattice. | Boolean algebras canonically defined | 0.826046 |
5,412 | The empty sup is 0 (the bottom element) and the empty inf is 1 (top). It follows that every finite set has both a sup and an inf. Infinite subsets of a Boolean algebra may or may not have a sup and/or an inf; in a power set algebra they always do. | Boolean algebras canonically defined | 0.826046 |
5,413 | This is not a restriction because atoms of this form include all the ordinary atoms, namely the constants 0 and 1, which arise here as the n-ary operations nf0 and nf−1 for each n (abbreviating 22n−1 to −1), and the variables x0,...,xn-1 as can be seen from the truth tables where x0 appears as both the unary operation ... | Boolean algebras canonically defined | 0.826046 |
5,414 | Example 1. The Boolean prototype is a Boolean algebra, since trivially it satisfies its own laws. It is thus the prototypical Boolean algebra. We did not call it that initially in order to avoid any appearance of circularity in the definition. | Boolean algebras canonically defined | 0.826046 |
5,415 | The laws of Boolean algebra therefore constitute the equational theory of the Boolean prototype. A model of a theory is an algebra interpreting the operation symbols in the language of the theory and satisfying the equations of the theory. A Boolean algebra is any model of the laws of Boolean algebra.That is, a Boolean... | Boolean algebras canonically defined | 0.826046 |
5,416 | The laws of Boolean algebra are the equations in the language of Boolean algebra satisfied by the prototype. The first three of the above examples are Boolean laws, but not the fourth since 1∧0 ≠ 1. The equational theory of an algebra is the set of all equations satisfied by the algebra. | Boolean algebras canonically defined | 0.826046 |
5,417 | It is convenient to consider the i-th n-ary symbol to be nfi as done below in the section on truth tables. An equational theory in a given language consists of equations between terms built up from variables using symbols of that language. Typical equations in the language of Boolean algebra are x∧y = y∧x, x∧x = x, x∧¬... | Boolean algebras canonically defined | 0.826046 |
5,418 | The prototype therefore has 22n n-ary operation symbols, called the Boolean operation symbols and forming the language of Boolean algebra. Only a few operations have conventional symbols, such as ¬ for negation, ∧ for conjunction, and ∨ for disjunction. | Boolean algebras canonically defined | 0.826046 |
5,419 | The operation indexed by each symbol is called the denotation or interpretation of that symbol. Each operation symbol specifies the arity of its interpretation, whence all possible interpretations of a symbol have the same arity. In general it is possible for an algebra to interpret distinct symbols with the same opera... | Boolean algebras canonically defined | 0.826046 |
5,420 | The number of (n+1)-ary operations in the prototype is the square of the number of n-ary operations, so there are 162 = 256 ternary operations, 2562 = 65,536 quaternary operations, and so on. A family is indexed by an index set. In the case of a family of operations forming an algebra, the indices are called operation ... | Boolean algebras canonically defined | 0.826046 |
5,421 | An algebra is finitary when each of its operations takes only finitely many arguments. For the prototype each argument of an operation is either 0 or 1, as is the result of the operation. The maximal such algebra consists of all finitary operations on {0,1}. | Boolean algebras canonically defined | 0.826046 |
5,422 | Boolean algebra treats the equational theory of the maximal two-element finitary algebra, called the Boolean prototype, and the models of that theory, called Boolean algebras. These terms are defined as follows. An algebra is a family of operations on a set, called the underlying set of the algebra. We take the underly... | Boolean algebras canonically defined | 0.826046 |
5,423 | The elements greater than or equal to an atom always form an ultrafilter, but so do many other sets; for example, in the Boolean algebra of finite and cofinite sets of integers, the cofinite sets form an ultrafilter even though none of them are atoms. Likewise, the powerset of the integers has among its ultrafilters th... | Boolean algebras canonically defined | 0.826046 |
5,424 | The subset of B consisting of the former is called an ultrafilter of B. When B is finite its ultrafilters pair up with its atoms; one atom is mapped to 1 and the rest to 0. Each ultrafilter of B thus consists of an atom of B and all the elements above it; hence exactly half the elements of B are in the ultrafilter, and... | Boolean algebras canonically defined | 0.826046 |
5,425 | In the other direction, there may exist many homomorphisms from a Boolean algebra B to 2. Any such homomorphism partitions B into those elements mapped to 1 and those to 0. | Boolean algebras canonically defined | 0.826046 |
5,426 | A Boolean homomorphism is a function h: A→B between Boolean algebras A,B such that for every Boolean operation mfi: The category Bool of Boolean algebras has as objects all Boolean algebras and as morphisms the Boolean homomorphisms between them. There exists a unique homomorphism from the two-element Boolean algebra 2... | Boolean algebras canonically defined | 0.826046 |
5,427 | Birkhoff's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. Now Birkhoff's HSP theorem for varieties can be stated as, every class of models of the equational theory of a class C of algebras is the Homomorphic image of a Subalgebra of a direct Product of a... | Boolean algebras canonically defined | 0.826046 |
5,428 | To go beyond power set algebras we need another construct. A subalgebra of an algebra A is any subset of A closed under the operations of A. Every subalgebra of a Boolean algebra A must still satisfy the equations holding of A, since any violation would constitute a violation for A itself. Hence every subalgebra of a B... | Boolean algebras canonically defined | 0.826046 |
5,429 | It can be shown that every finite Boolean algebra is isomorphic to some power set algebra. Hence the cardinality (number of elements) of a finite Boolean algebra is a power of 2, namely one of 1,2,4,8,...,2n,... This is called a representation theorem as it gives insight into the nature of finite Boolean algebras by gi... | Boolean algebras canonically defined | 0.826046 |
5,430 | The topology of finite Hausdorff spaces is always both discrete and compact, whereas for infinite spaces "discrete"' and "compact" are mutually exclusive. Thus when generalizing finite algebras (of any kind, not just Boolean) to infinite ones, "discrete" and "compact" part company, and one must choose which one to reta... | Boolean algebras canonically defined | 0.826046 |
5,431 | Thus if one starts with the finite sets and asks how these generalize to infinite objects, there are two ways: "adding" them gives ordinary or inductive sets while "multiplying" them gives Stone spaces or profinite sets. The same choice exists for finite power set algebras as the duals of finite sets: addition yields B... | Boolean algebras canonically defined | 0.826046 |
5,432 | This definition is fleshed out by the next definition.Johnstone (1982) A Boolean algebra is a filtered colimit of finite Boolean algebras. (The circularity in this definition can be removed by replacing "finite Boolean algebra" by "finite power set" equipped with the Boolean operations standardly interpreted for power ... | Boolean algebras canonically defined | 0.826046 |
5,433 | It is no limitation to require the space to be a totally disconnected compact Hausdorff space, or Stone space, that is, every Boolean algebra arises in this way, up to isomorphism. Moreover, if the two Boolean algebras formed as the clopen sets of two Stone spaces are isomorphic, so are the Stone spaces themselves, whi... | Boolean algebras canonically defined | 0.826046 |
5,434 | We have already encountered several definitions of Boolean algebra, as a model of the equational theory of the two-element algebra, as a complemented distributive lattice, as a Boolean ring, and as a product-preserving functor from a certain category (Lawvere). Two more definitions worth mentioning are:. Stone (1936) A... | Boolean algebras canonically defined | 0.826046 |
5,435 | Underlying every Boolean algebra B is a partially ordered set or poset (B,≤). The partial order relation is defined by x ≤ y just when x = x∧y, or equivalently when y = x∨y. Given a set X of elements of a Boolean algebra, an upper bound on X is an element y such that for every element x of X, x ≤ y, while a lower bound... | Boolean algebras canonically defined | 0.826046 |
5,436 | Then to prove s = t in the general case when t may be an application, use the fact that if s = t is an identity then s and t must evaluate to the same atom, call it u. So first prove s = u and t = u as above, that is, evaluate s and t using A1, R1, and R3, and then invoke R2 to infer s = t. In A1, if we view the number... | Boolean algebras canonically defined | 0.826046 |
5,437 | We treat R1 as an inference rule, even though it is like an axiom in having no premises, because it is a domain-independent rule along with R2 and R3 common to all equational axiomatizations, whether of groups, rings, or any other variety. The only entity specific to Boolean algebras is axiom schema A1. In this way whe... | Boolean algebras canonically defined | 0.826046 |
5,438 | The technique we just used to prove an identity of Boolean algebra can be generalized to all identities in a systematic way that can be taken as a sound and complete axiomatization of, or axiomatic system for, the equational laws of Boolean logic. The customary formulation of an axiom system consists of a set of axioms... | Boolean algebras canonically defined | 0.826046 |
5,439 | This ternary operation incidentally formed the basis for Grau's ternary Boolean algebras, which he axiomatized in terms of this operation and negation. The operation is symmetric, meaning that its value is independent of any of the 3! = 6 permutations of its arguments. | Boolean algebras canonically defined | 0.826046 |
5,440 | We have thus calculated, with a dozen or so logical operations on bytes, that the two ternary operations ( x ∧ y ) ∨ ( y ∧ z ) ∨ ( z ∧ x ) {\displaystyle (x\land y)\lor (y\land z)\lor (z\land x)} and ( x ∨ y ) ∧ ( y ∨ z ) ∧ ( z ∨ x ) {\displaystyle (x\lor y)\land (y\lor z)\land (z\lor x)} are actually the same operatio... | Boolean algebras canonically defined | 0.826046 |
5,441 | The n-ary Boolean operations themselves constitute a power set algebra 2W, namely when W is taken to be the set of 2n valuations of the n inputs. In terms of the naming system of operations nfi where i in binary is a column of a truth table, the columns can be combined with Boolean operations of any arity to produce ot... | Boolean algebras canonically defined | 0.826046 |
5,442 | The operations of a direct product are the corresponding operations of the constituent algebras acting within their respective coordinates; in particular operation nfj of the product operates on n I-tuples by applying operation nfj of Bi to the n elements in the i-th coordinate of the n tuples, for all i in I. When all... | Boolean algebras canonically defined | 0.826046 |
5,443 | Negation ¬x becomes ~X, complement relative to W. There is also set difference X\Y = X∩~Y, symmetric difference (X\Y)∪(Y\X), ternary union X∪Y∪Z, and so on. The atoms here are the singletons, those subsets with exactly one element. Examples 2 and 3 are special cases of a general construct of algebra called direct produ... | Boolean algebras canonically defined | 0.826046 |
5,444 | Example 3. The power set algebra, the set 2W of all subsets of a given set W. This is just Example 2 in disguise, with W serving to index the bit positions. Any subset X of W can be viewed as the bit vector having 1's in just those bit positions indexed by elements of X. Thus the all-zero vector is the empty subset of ... | Boolean algebras canonically defined | 0.826046 |
5,445 | In the mathematics of graph theory, two graphs, G and H, are called homomorphically equivalent if there exists a graph homomorphism f: G → H {\displaystyle f\colon G\to H} and a graph homomorphism g: H → G {\displaystyle g\colon H\to G} . An example usage of this notion is that any two cores of a graph are homomorphica... | Homomorphic equivalence | 0.826045 |
5,446 | Although ordinary division is not defined on Z {\displaystyle \mathbb {Z} } , the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < ... | Integer | 0.826037 |
5,447 | The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of i... | Integer | 0.826037 |
5,448 | All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in Z {... | Integer | 0.826037 |
5,449 | Plant Physiology is a monthly peer-reviewed scientific journal that covers research on physiology, biochemistry, cellular and molecular biology, genetics, biophysics, and environmental biology of plants. The journal has been published since 1926 by the American Society of Plant Biologists. The current editor-in-chief i... | Plant Physiology (journal) | 0.82603 |
5,450 | The language of odds, such as the use of phrases like "ten to one" for intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory. Shakespeare wrote: Knew that we ventured on such dangerous seas That if we wrought out life 'twas ten to one The sixteenth-century pol... | Odds | 0.826028 |
5,451 | There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue. In probability theory and statistics, where the variable p is the probability in favor of a binary event, and the probability against the event is therefore 1-p, "the odds" of the event are the quotient of the two, or p 1 − p {\displaystyle {\f... | Odds | 0.826028 |
5,452 | Similar ratios are used elsewhere in statistics; of central importance is the likelihood ratio in likelihoodist statistics, which is used in Bayesian statistics as the Bayes factor. Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on a last spec... | Odds | 0.826028 |
5,453 | In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtrac... | Odds | 0.826028 |
5,454 | Some chess problems, like the eight queens puzzle or the knight's tour problem, have connections to mathematics, especially to graph theory and combinatorics. Many famous mathematicians have studied such problems, including Euler, Legendre, and Gauss. Besides finding a solution to a particular puzzle, mathematicians ar... | Chess puzzle | 0.826024 |
5,455 | In an enclosed system with a certain number of particles, energy is the only macroscopic variable affected by the microphysics. The Monte Carlo simulation of a microcanonical ensemble thus requires sampling different microscopic states with the same energy. When the number of possible microscopic states of thermodynami... | Demon algorithm | 0.826023 |
5,456 | Computer simulations of the full equations of motion for every individual particle to simulate microscopic properties is computationally very expensive. Monte Carlo methods can overcome this problem by sampling microscopic states according to stochastic rules instead of modeling the complete microphysics. The microcano... | Demon algorithm | 0.826023 |
5,457 | Directed edges connect these vertices into triangles, which by the right hand rule, then have a direction called the normal. Each triangle forms a face of the mesh. These are combinatoric in nature and encode the topology of the shape. | Geometry processing | 0.826022 |
5,458 | Shapes in geometry processing are usually represented as triangle meshes, which can be seen as a graph. Each node in the graph is a vertex (usually in R 3 {\displaystyle R^{3}} ), which has a position. This encodes the geometry of the shape. | Geometry processing | 0.826022 |
5,459 | Like any other shape, the shapes used in geometry processing have properties pertaining to their geometry and topology. The geometry of a shape concerns the position of the shape's points in space, tangents, normals, and curvature. It also includes the dimension in which the shape lives (ex. R 2 {\displaystyle R^{2}} o... | Geometry processing | 0.826022 |
5,460 | This can mean it is consumed by a viewer as a rendered asset in a game or movie, for instance. The end of a shape's life can also be defined by a decision about the shape, like whether or not it satisfies some criteria. Or it can even be fabricated in the real world, through a method such as 3D printing or laser cuttin... | Geometry processing | 0.826021 |
5,461 | Geometry processing involves working with a shape, usually in 2D or 3D, although the shape can live in a space of arbitrary dimensions. The processing of a shape involves three stages, which is known as its life cycle. At its "birth," a shape can be instantiated through one of three methods: a model, a mathematical rep... | Geometry processing | 0.826021 |
5,462 | That means we would like ∇ u = ∇ v ⊥ {\displaystyle \nabla u=\nabla v^{\perp }} . In addition, we would also like the mapping to have proportionally similar sized regions as the original. This results to setting the Jacobian of the u and v coordinate functions to 1. = 1 {\displaystyle {\begin{bmatrix}{\dfrac {\partial ... | Geometry processing | 0.826021 |
5,463 | Occasionally, we need to flatten a 3D surface onto a flat plane. This process is known as parameterization. The goal is to find coordinates u and v onto which we can map the surface so that distortions are minimized. In this manner, parameterization can be seen as an optimization problem. One of the major applications ... | Geometry processing | 0.826021 |
5,464 | The allegations came from a contract with the Naval Oceanographic Office Major Shared Resource Center in Mississippi. On May 25, 2011, Lockheed Martin bought the first Quantum Computing System from D-Wave Systems. Lockheed Martin and D-Wave will collaborate to realize the benefits of a computing platform based upon a q... | Double Helix Methodology | 0.826015 |
5,465 | Gjedrem (1979) showed that selection of Atlantic salmon (Salmo salar) led to an increase in body weight by 30% per generation. A comparative study on the performance of select Atlantic salmon with wild fish was conducted by AKVAFORSK Genetics Centre in Norway. The traits, for which the selection was done included growt... | Selective breeding | 0.826007 |
5,466 | Its first chapter discusses selective breeding and domestication of such animals as pigeons, cats, cattle, and dogs. Darwin used artificial selection as an analogy to propose and explain the theory of natural selection but distinguished the latter from the former as a separate process that is non-directed.The deliberat... | Selective breeding | 0.826007 |
5,467 | Originally, each of the three sections was scored on a scale of 20 to 80 points, adding up to a maximum composite score of 240 points. This paralleled the SAT, which is graded on a scale of 200 to 800 for each section (the narrower range is to distinguish from which test a score comes and to denote less accuracy). Howe... | Preliminary SAT | 0.826003 |
5,468 | Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicul... | Algebra of concepts | 0.825992 |
5,469 | Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings. | Algebra of concepts | 0.825992 |
5,470 | Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library. He called for the creation of an empirical database as a way to further all sciences. His charact... | Algebra of concepts | 0.825992 |
5,471 | Leibniz, for example, proposed the following classification system in the Otivm Hanoveranvm Sive Miscellanea (1737).Leibniz's Idea of Arranging a Narrower Library Theology Jurisprudence Medicine Intellectual Philosophy Philosophy of the Imagination or Mathematics Philosophy of Sensible Things or Physics Philology or La... | Algebra of concepts | 0.825992 |
5,472 | Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition (II... | Algebra of concepts | 0.825992 |
5,473 | In 1693, Leibniz described a design of a machine which could, in theory, integrate differential equations, which he called "integraph".Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined... | Algebra of concepts | 0.825992 |
5,474 | They were not an unambiguous success because they did not fully mechanize the carry operation. Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations. Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher... | Algebra of concepts | 0.825992 |
5,475 | His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1961, Norbert Wiener suggested that Leibniz should be considered the patron saint of cybernetics. Wiener is quoted with "Indeed, the general idea of a computing machine is nothing but a mechanization of Leibniz's Calculus Ratiocinator. | Algebra of concepts | 0.825992 |
5,476 | Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commi... | Algebra of concepts | 0.825992 |
5,477 | Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so... | Algebra of concepts | 0.825992 |
5,478 | They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes. Leibniz was appointed Librarian of the Herzog August Library in Wolfenbüttel... | Algebra of concepts | 0.825992 |
5,479 | The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on calculus in 1674; the earliest evi... | Algebra of concepts | 0.825992 |
5,480 | Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where Newton accused him of having seen his unpublished work on calculus in advance. This was alleged to be evidence supporting the accusation, made decades later, that he had stolen calculus from Newton... | Algebra of concepts | 0.825992 |
5,481 | He regarded such relations as (real) qualities of things (Leibniz admitted unary predicates only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of De Morgan, Peirce, Schröder and Russell himself, now standard in predicate logic. No... | Algebra of concepts | 0.825992 |
5,482 | Leibniz's relationism, in contrast, describes space and time as systems of relations that exist between objects. The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light. One of Leibniz's projects was to recast Newton's theory as a vortex theory. Ho... | Algebra of concepts | 0.825992 |
5,483 | Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An ... | Algebra of concepts | 0.825992 |
5,484 | Evaluation of effects during spacecraft design includes application of various models of the environment, including radiation belt models, spacecraft-plasma interaction models and atmospheric models to predict drag effects encountered in lower orbits and during reentry. The field often overlaps with the disciplines of ... | Space environment | 0.825983 |
5,485 | Space environment is a branch of astronautics, aerospace engineering and space physics that seeks to understand and address conditions existing in space that affect the design and operation of spacecraft. A related subject, space weather, deals with dynamic processes in the solar-terrestrial system that can give rise t... | Space environment | 0.825983 |
5,486 | These effects corresponded to the deviation from the value −2 for the spectroscopic electron g-factor that are predicted by the Dirac equation. Later, Hans Bethe theoretically calculated those shifts in the hydrogen energy levels due to vacuum polarization on his return train ride from the Shelter Island Conference to ... | Vacuum polarisation | 0.82598 |
5,487 | The tangent at a point (a, b) of the curve is the line of equation ( x − a ) p x ′ ( a , b ) + ( y − b ) p y ′ ( a , b ) = 0 {\displaystyle (x-a)p'_{x}(a,b)+(y-b)p'_{y}(a,b)=0} , like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simple... | Algebraic curves | 0.825967 |
5,488 | Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point a... | Algebraic curves | 0.825967 |
5,489 | Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve m... | Algebraic curves | 0.825967 |
5,490 | If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence. These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must ... | Algebraic curves | 0.825967 |
5,491 | These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is... | Algebraic curves | 0.825967 |
5,492 | In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizin... | Algebraic curves | 0.825967 |
5,493 | Thus, it appears quite possible that much of the real discovery work is buried in unexplained procedures.Lenat claimed that the system had rediscovered both Goldbach's conjecture and the fundamental theorem of arithmetic. Later critics accused Lenat of over-interpreting the output of AM. In his paper Why AM and Eurisko... | Automated Mathematician | 0.825964 |
5,494 | The scale is important — an individual with both parents born in the United Kingdom is not inbred relative to that country's population, but is more inbred than two humans selected from the entire world. This motivates the derivation of Wright's F-statistics (also called "fixation indices"), which measure inbreeding th... | Population structure (genetics) | 0.825963 |
5,495 | For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itself corresponding to the set of all nodes. (In general each node of the Dynkin diagram de... | Borel subgroup | 0.825957 |
5,496 | Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Parabolic subgroups P are also characterized, among algebraic subgroups, by the condition that G/P is a complete variety. Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgro... | Borel subgroup | 0.825957 |
5,497 | In physics, a mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium. While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefore, the oscillating material does not move far from its initial equilibrium posi... | Mechanical waves | 0.825953 |
5,498 | Nucleotides also play a central role in metabolism at a fundamental, cellular level. They provide chemical energy—in the form of the nucleoside triphosphates, adenosine triphosphate (ATP), guanosine triphosphate (GTP), cytidine triphosphate (CTP) and uridine triphosphate (UTP)—throughout the cell for the many cellular ... | Nucleotide | 0.825944 |
5,499 | Variant annotation tools use machine learning algorithms to predict variant annotations. Different annotation tools use different algorithms. Common algorithms include: Interval/Random forest-eg.MutPred, SNPeff Neural networks-eg.SNAP Support Vector Machines-e.g. PhD-SNP, SNPs&GO Bayesian classification-eg.PolyPhen-2 | SNP annotation | 0.825938 |
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