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11,000 | If the codomain has a structure of R {\displaystyle \mathbb {R} } -algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real va... | Function of a real variable | 0.818357 |
11,001 | However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers or the qua... | Function of a real variable | 0.818357 |
11,002 | In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. Most r... | Function of a real variable | 0.818357 |
11,003 | Prune and search is a method of solving optimization problems suggested by Nimrod Megiddo in 1983.The basic idea of the method is a recursive procedure in which at each step the input size is reduced ("pruned") by a constant factor 0 < p < 1. As such, it is a form of decrease and conquer algorithm, where at each step t... | Prune and search | 0.818346 |
11,004 | In prune and search algorithms S(n) is typically at least linear (since the whole input must be processed). With this assumption, the recurrence has the solution T(n) = O(S(n)). This can be seen either by applying the master theorem for divide-and-conquer recurrences or by observing that the times for the recursive sub... | Prune and search | 0.818346 |
11,005 | It is shown that information dimension and differential entropy are tightly connected. Let X {\displaystyle X} be a random variable with continuous density f ( x ) {\displaystyle f(x)} . Suppose we divide the range of X {\displaystyle X} into bins of length Δ {\displaystyle \Delta } . By the mean value theorem, there e... | Information dimension | 0.818346 |
11,006 | SciLifeLab is provided with SEK 150 million per year in state funds separate from other national and European grants and infrastructure support in the fields of drug discovery, drug development and fundamental research. Together with the prestigious American journal Science, SciLifeLab awards a young researcher prize. ... | Science for Life Laboratory | 0.818345 |
11,007 | SciLifeLab (Science for Life Laboratory) is a world-leading Swedish national center for large-scale research and one of the largest molecular biology research laboratories in Europe at the forefront of innovation in life sciences research, computational biology, bioinformatics, training and services in molecular biosci... | Science for Life Laboratory | 0.818345 |
11,008 | In the study of the physics of granular materials, a force chain consists of a set of particles within a compressed granular material that are held together and jammed into place by a network of mutual compressive forces.Between these chains are regions of low stress whose grains are shielded for the effects of the gra... | Force chain | 0.818342 |
11,009 | Roger A Johnson (31 August 2007). Advanced Euclidean Geometry. Dover Publications Inc. ISBN 978-0486462370. | Modern triangle geometry | 0.818337 |
11,010 | The triangle geometry has again become an active area of research pursued by a group of dedicated geometers. As epitomizing this revival, one can point out the formulation of the concept of a "triangle centre" and the compilation by Clark Kimberling of an encyclopedia of triangle centers containing a listing of nearly ... | Modern triangle geometry | 0.818337 |
11,011 | A dearth of connections to other fields. Competition with other topics with a strong visual content like tessellations, fractals, graph theory, etc.A further revival of interest was witnessed with the advent of the modern electronic computer. | Modern triangle geometry | 0.818337 |
11,012 | The exhaustion of its methodologic possibilities. The visual complexity of the so-called deeper results of the subject. The downgrading of the visual in favor of the algebraic. | Modern triangle geometry | 0.818337 |
11,013 | The Lester circle is the circle which passes through the circumcenter, the nine-point center and the outer and inner Fermat points. A generalised Lester circle is a circle which passes through at least four triangle centers. Discoverer has discovered several generalized Lester circles.Sava Grozdev, Hiroshi Okumura, Dek... | Modern triangle geometry | 0.818336 |
11,014 | Theorem 9.1. The Yff center of congruence is the internal center of similitude of the incircle and the circumcircle with respect to the pedal triangle of the incenter. | Modern triangle geometry | 0.818336 |
11,015 | The following sample of theorems gives a flavor of the new results discovered by Discoverer. Theorem 6.1 Let P and Q are points, neither lying on a sideline of triangle ABC. If P and Q are isogonal conjugates with respect to ABC, then the Ceva product of their complements lies on the Kiepert hyperbola. | Modern triangle geometry | 0.818336 |
11,016 | The entry of computers had a deciding influence on the course of development in the interest in triangle geometry witnessed during the closing years of the twentieth century and the early years of the current century. Some of the ways in which the computers had influenced this course have been delineated by Philip Davi... | Modern triangle geometry | 0.818336 |
11,017 | Given any line l, let P, Q, R be the feet of perpendiculars from the vertices A, B, C of triangle ABC to l. The lines through P. Q, R perpendicular respectively to the sides BC, CA, AB are concurrent and the point of concurrence is the orthopole of the line l with respect to the triangle ABC. In modern triangle geometr... | Modern triangle geometry | 0.818336 |
11,018 | For a given triangle ABC with centroid G, the symmedian through the vertex is the reflection of the line AG in the bisector of the angle A. There are three symmedians for a triangle one passing through each vertex. The three symmedians are concurrent and the point of concurrency, commonly denoted by K, is called the Le... | Modern triangle geometry | 0.818336 |
11,019 | Another unifying notion of contemporary modern triangle geometry is that of a central line. This concept unifies the several special straight lines associated with a triangle. The notion of a central line is also related to the notion of a triangle center. | Modern triangle geometry | 0.818336 |
11,020 | A quick glance into the world of modern triangle geometry as it existed during the peak of interest in triangle geometry subsequent to the publication of Lemoine's paper is presented below. This presentation is largely based on the topics discussed in William Gallatly's book published in 1910 and Roger A Johnsons' book... | Modern triangle geometry | 0.818336 |
11,021 | One of the most significant ideas that has emerged during the revival of interest in triangle geometry during the closing years of twentieth century is the notion of triangle center. This concept introduced by Clark Kimberling in 1994 unified in one notion the very many special and remarkable points associated with a t... | Modern triangle geometry | 0.818336 |
11,022 | An introductory account of certain modern ideas and methods in plane analytical geometry. London: Macmillan and Co. Retrieved 10 January 2022. | Modern triangle geometry | 0.818336 |
11,023 | : 1–295. Paul Yiu (December 2012). Introduction to the Geometry of the Triangle (PDF). | Modern triangle geometry | 0.818336 |
11,024 | ISBN 0883856190. Altshiller-Court, Nathan (1952). College geometry; an introduction to the modern geometry of the triangle and the circle. | Modern triangle geometry | 0.818336 |
11,025 | H S M Coexter (5 September 1996). Geometry Revisited. Mathematical Association of America. | Modern triangle geometry | 0.818336 |
11,026 | Ross Honsberger (1995). Episodes in Nineteenth and Twentienth Century Euclidean Geometry. Mathematical Association of America. | Modern triangle geometry | 0.818336 |
11,027 | The Modern Geometry of the Triangle. London: Francis Hodgson. Retrieved 4 January 2022. | Modern triangle geometry | 0.818336 |
11,028 | (The Development of Mathematics, p. 323) Philip Davis has suggested several reasons for the decline of interest in triangle geometry. These include: The feeling that the subject is elementary and of low professional status. | Modern triangle geometry | 0.818336 |
11,029 | (See the conference paper titled "Teaching new geometrical methods with an ancient figure in the nineteenth and twentieth centuries: the new triangle geometry in textbooks in Europe and USA (1888–1952)" by Pauline Romera-Lebret presented in 2009.) However, this escalation of interest soon collapsed and triangle geometr... | Modern triangle geometry | 0.818336 |
11,030 | Later the theory of correspondences which was an offshoot of the theory of geometric transformations was developed to give coherence to the various isolated results. With its development, the expression "new triangle geometry" indicated not only the many remarkable objects associated with a triangle but also the method... | Modern triangle geometry | 0.818336 |
11,031 | The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the p... | Modern triangle geometry | 0.818336 |
11,032 | In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the ti... | Modern triangle geometry | 0.818336 |
11,033 | Applications in artificial intelligence include fault diagnosis, belief revision, and automated planning. The most direct application of abduction is that of automatically detecting faults in systems: given a theory relating faults with their effects and a set of observed effects, abduction can be used to derive sets o... | Inference to the Best Explanation | 0.818319 |
11,034 | Given the many possible explanations for the movement of the eight ball, our abduction does not leave us certain that the cue ball in fact struck the eight ball, but our abduction, still useful, can serve to orient us in our surroundings. Despite many possible explanations for any physical process that we observe, we t... | Inference to the Best Explanation | 0.818319 |
11,035 | Abduction can lead to false conclusions if other rules that might explain the observation are not taken into account—e.g. the grass could be wet from dew. This remains the common use of the term "abduction" in the social sciences and in artificial intelligence. | Inference to the Best Explanation | 0.818319 |
11,036 | Peirce long treated abduction in terms of induction from characters or traits (weighed, not counted like objects), explicitly so in his influential 1883 "A theory of probable inference", in which he returns to involving probability in the hypothetical conclusion. Like "Deduction, Induction, and Hypothesis" in 1878, it ... | Inference to the Best Explanation | 0.818319 |
11,037 | A group scheme G over a noetherian scheme S is finite and flat if and only if OG is a locally free OS-module of finite rank. The rank is a locally constant function on S, and is called the order of G. The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves we... | Multiplicative group scheme | 0.818318 |
11,038 | The general linear group GLn is an affine algebraic variety that can be viewed as the multiplicative group of the n by n matrix ring variety. As a functor, it sends an S-scheme T to the group of invertible n by n matrices whose entries are global sections of T. Over an affine base, one can construct it as a quotient of... | Multiplicative group scheme | 0.818318 |
11,039 | The multiplicative group Gm has the punctured affine line as its underlying scheme, and as a functor, it sends an S-scheme T to the multiplicative group of invertible global sections of the structure sheaf. It can be described as the diagonalizable group D(Z) associated to the integers. Over an affine base such as Spec... | Multiplicative group scheme | 0.818318 |
11,040 | Complete group varieties are called abelian varieties. This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers. They are automatically projective, and they have many applications, e.g., in... | Multiplicative group scheme | 0.818318 |
11,041 | Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups. Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in parti... | Multiplicative group scheme | 0.818318 |
11,042 | The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme G, the ring of global sections also has ... | Multiplicative group scheme | 0.818318 |
11,043 | Suppose that G is a group scheme of finite type over a field k. Let G0 be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G0. G has a unique maximal reduced subscheme Gred, and if k is perfect, then Gred is a smooth group var... | Multiplicative group scheme | 0.818318 |
11,044 | In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, s... | Multiplicative group scheme | 0.818318 |
11,045 | This book is intended for advanced undergraduates who have already seen some abstract algebra and real analysis. Reviewer Thomas Shemanske suggests using it as the basis of a senior seminar, as a quick path to many important topics and an interesting example of how these seemingly-separate topics join forces in this ap... | Elementary Number Theory, Group Theory and Ramanujan Graphs | 0.818317 |
11,046 | Several constructions of low-degree Ramanujan graphs are now known, the first of which were by Lubotzky, Phillips & Sarnak (1988) and Margulis (1988). Reviewer Jürgen Elstrod writes that "while the description of these graphs is elementary, the proof that they have the desired properties is not". Elementary Number Theo... | Elementary Number Theory, Group Theory and Ramanujan Graphs | 0.818317 |
11,047 | Elementary Number Theory, Group Theory and Ramanujan Graphs is a book in mathematics whose goal is to make the construction of Ramanujan graphs accessible to undergraduate-level mathematics students. In order to do so, it covers several other significant topics in graph theory, number theory, and group theory. It was w... | Elementary Number Theory, Group Theory and Ramanujan Graphs | 0.818317 |
11,048 | Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existe... | Elementary Number Theory, Group Theory and Ramanujan Graphs | 0.818317 |
11,049 | In graph theory, expander graphs are undirected graphs with high connectivity: every small-enough subset of vertices has many edges connecting it to the remaining parts of the graph. Sparse expander graphs have many important applications in computer science, including the development of error correcting codes, the des... | Elementary Number Theory, Group Theory and Ramanujan Graphs | 0.818317 |
11,050 | Additive Manufacturing (3D printing) technology is a component of the design process for many Newlab residents. Prototyping shops are a distinguishing feature of the hardware-centric facility. Newlab leverages partnerships with firms like AutoDesk, Stratasys, BigRep, Haas, Ultimaker, and others to provide and maintain ... | New Lab | 0.818316 |
11,051 | Newlab opened in June 2016, as a multi-disciplinary technology center. Housed in Building 128 of the Brooklyn Navy Yard, the $35 million project serves as a hardware-focused shared workspace, research lab, and hatchery for socially-oriented tech manufacturing.Using the MIT Media Lab as a model, the impetus for the inde... | New Lab | 0.818316 |
11,052 | The following quote from a paper by R. H. Eddy and R. Fritsch is enough testimony to establish the importance of the Kiepert conics in the study of triangle geometry: "If a visitor from Mars desired to learn the geometry of the triangle but could stay in the earth's relatively dense atmosphere only long enough for a si... | Kiepert parabola | 0.818312 |
11,053 | In triangle geometry, the Kiepert conics are two special conics associated with the reference triangle. One of them is a hyperbola, called the Kiepert hyperbola and the other is a parabola, called the Kiepert parabola. The Kiepert conics are defined as follows: If the three triangles A ′ B C {\displaystyle A^{\prime }B... | Kiepert parabola | 0.818312 |
11,054 | DNA-based computing and quantum computing are areas of active research for both computing hardware and software, such as the development of quantum algorithms. Potential infrastructure for future technologies includes DNA origami on photolithography and quantum antennae for transferring information between ion traps. B... | Applied computing | 0.818306 |
11,055 | It could also ease the transition to renewable energy source, since it would suffice to power one server farm with renewable energy, rather than millions of homes and offices.However, this centralized computing model poses several challenges, especially in security and privacy. Current legislation does not sufficiently... | Applied computing | 0.818306 |
11,056 | Software development, a widely used and more generic term, does not necessarily subsume the engineering paradigm. The generally accepted concepts of Software Engineering as an engineering discipline have been specified in the Guide to the Software Engineering Body of Knowledge (SWEBOK). The SWEBOK has become an interna... | Applied computing | 0.818306 |
11,057 | For example, programs differ substantially in the amount of programming required." The study of IS bridges business and computer science, using the theoretical foundations of information and computation to study various business models and related algorithmic processes within a computer science discipline. The field of... | Applied computing | 0.818306 |
11,058 | Software engineering (SE) is the application of a systematic, disciplined and quantifiable approach to the design, development, operation, and maintenance of software, and the study of these approaches. That is, the application of engineering to software. It is the act of using insights to conceive, model and scale a s... | Applied computing | 0.818306 |
11,059 | π {\displaystyle \pi } is not a Euclidean number ... and therefore it is impossible to construct, by Euclidean methods a length equal to the circumference of a circle of unit diameter A proof exists to demonstrate that any Euclidean number is an algebraic number — a number that is the solution to some polynomial equati... | Proof of impossibility | 0.818304 |
11,060 | The question "Does any arbitrary Diophantine equation have an integer solution?" is undecidable. That is, it is impossible to answer the question for all cases. Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich-Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can d... | Proof of impossibility | 0.818304 |
11,061 | Fermat's Last Theorem was conjectured by Pierre de Fermat in the 1600s, states the impossibility of finding solutions in positive integers for the equation x n + y n = z n {\displaystyle x^{n}+y^{n}=z^{n}} with n > 2 {\displaystyle n>2} . Fermat himself gave a proof for the n = 4 case using his technique of infinite de... | Proof of impossibility | 0.818304 |
11,062 | In political science, Arrow's impossibility theorem states that it is impossible to devise a voting system that satisfies a set of five specific axioms. This theorem is proved by showing that four of the axioms together imply the opposite of the fifth. Similarly, the Gibbard–Satterthwaite theorem states that no voting ... | Proof of impossibility | 0.818304 |
11,063 | 205ff and reference on p. 401 ibid: Greibach "The undecidability of the ambiguity problem for minimal lineal grammars," Information and Control 6:2, 117–125, also reference on p. 402 ibid: Greibach "A note on undecidable properties of formal languages", Math Systems Theory 2:1, 1–6.) Penrose tiling questions. Questio... | Proof of impossibility | 0.818304 |
11,064 | 303)). Rice's theorem: a generalized formulation of Turing's second theorem (cf Hopcroft and Ullman p. 185ff) Greibach's theorem: undecidability in language theory (cf Hopcroft and Ullman p. | Proof of impossibility | 0.818304 |
11,065 | It decides it is the letter-string of a successful machine, but when it tries to do this machine's (its own) computation it locks in a circle and can't continue. Thus, we have arrived at Richard's paradox. (If you are bewildered see Turing's proof for more).A number of similar undecidability proofs appeared soon before... | Proof of impossibility | 0.818304 |
11,066 | Due to the disordered nature of these proteins, topological approaches have been developed to search for conformational patterns in their dynamics. For instance, circuit topology has been applied to track the dynamics of disordered protein domains. By employing a topological approach, one can categorize motifs accordin... | Intrinsically unstructured proteins | 0.818303 |
11,067 | Proteasome inhibitors have effective anti-tumor activity in cell culture, inducing apoptosis by disrupting the regulated degradation of pro-growth cell cycle proteins. This approach of selectively inducing apoptosis in tumor cells has proven effective in animal models and human trials. Lactacystin, a natural product sy... | Proteasome | 0.818299 |
11,068 | Thus portfolio optimization can be implemented using a two-moment decision model. Example 3: Suppose that a price-taking, risk-averse firm must commit to producing a quantity of output q before observing the market realization p of the product's price. Its decision problem is to choose q so as to maximize the expected ... | Two-moment decision model | 0.818294 |
11,069 | The two main motivations for the development of abstract algebraic logic are closely connected to (1) and (3) above. With respect to (1), a critical step in the transition was initiated by the work of Rasiowa. Her goal was to abstract results and methods known to hold for the classical propositional calculus and Boolea... | Abstract algebraic logic | 0.818293 |
11,070 | The Atmospheric Chemistry Suite (ACS) is a science payload consisting of three infrared spectrometer channels abord the ExoMars Trace Gas Orbiter (TGO) orbiting Mars since October 2016. The three channels are: the near-infrared channel (NIR), the mid-infrared channel (MIR), and the far infrared channel (FIR, also calle... | Atmospheric Chemistry Suite | 0.818293 |
11,071 | Of particular interest to this astrobiology mission, is the detection and characterization of the atmospheric methane (CH4), as it may be of geological or biological nature. Large differences in the abundances were measured between observations taken in 2003, 2006, and in 2014 NASA reported that the Curiosity rover det... | Atmospheric Chemistry Suite | 0.818293 |
11,072 | These results explain many properties of different classes of logical systems previously explained only on a case-by-case basis or shrouded in mystery. Perhaps the most important achievement of abstract algebraic logic has been the classification of propositional logics in a hierarchy, called the abstract algebraic hie... | Abstract algebraic logic | 0.818293 |
11,073 | (3) owes much to the joint work of Blok and Pigozzi exploring the different forms that the well-known deduction theorem of classical propositional calculus and first-order logic takes on in a wide variety of logical systems. They related these various forms of the deduction theorem to the properties of the algebraic co... | Abstract algebraic logic | 0.818293 |
11,074 | Abstract algebraic logic is a modern subarea of algebraic logic that emerged in Poland during the 1950s and 60s with the work of Helena Rasiowa, Roman Sikorski, Jerzy Łoś, and Roman Suszko (to name but a few). It reached maturity in the 1980s with the seminal publications of the Polish logician Janusz Czelakowski, the ... | Abstract algebraic logic | 0.818293 |
11,075 | Later, Tarski and his American students (whose ranks include Don Pigozzi) went on to discover cylindric algebra, whose representable instances algebraize all of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories. Classical algebraic logic, which compris... | Abstract algebraic logic | 0.818293 |
11,076 | The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 18... | Abstract algebraic logic | 0.818293 |
11,077 | In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems. | Abstract algebraic logic | 0.818293 |
11,078 | As well as the direct sum, another way to combine free abelian groups is to use the tensor product of Z {\displaystyle \mathbb {Z} } -modules. The tensor product of two free abelian groups is always free abelian, with a basis that is the Cartesian product of the bases for the two groups in the product.Many important pr... | Free abelian group | 0.818292 |
11,079 | Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind was a precursor to the analogous Nielsen–Schreier theorem that every subgroup of a free group is free, and is a generalization of the fact that every nontrivial subgroup of the infinite cyclic group is infinite cyclic... | Free abelian group | 0.818292 |
11,080 | Each non-negative x {\displaystyle x} is the sum of x {\displaystyle x} copies of 1 {\displaystyle 1} , and each negative integer x {\displaystyle x} is the sum of − x {\displaystyle -x} copies of − 1 {\displaystyle -1} , so the basis property is also satisfied.An example where the group operation is different from the... | Free abelian group | 0.818292 |
11,081 | In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis B {\displaystyle B} may be described in several equivalent ways. These include formal sums over B {\displaystyle B} , which are expressi... | Free abelian group | 0.818292 |
11,082 | In algebraic topology, a formal sum of k {\displaystyle k} -dimensional simplices is called a k {\displaystyle k} -chain, and the free abelian group having a collection of k {\displaystyle k} -simplices as its basis is called a chain group. The simplices are generally taken from some topological space, for instance as ... | Free abelian group | 0.818292 |
11,083 | Moreover, the sequence d 1 , d 2 , … , d k {\displaystyle d_{1},d_{2},\ldots ,d_{k}} depends only on F {\displaystyle F} and G {\displaystyle G} and not on the basis. A constructive proof of the existence part of the theorem is provided by any algorithm computing the Smith normal form of a matrix of integers. Uniquenes... | Free abelian group | 0.818292 |
11,084 | A homomorphism between two Boolean algebras A and B is a function f: A → B such that for all a, b in A: f(a ∨ b) = f(a) ∨ f(b), f(a ∧ b) = f(a) ∧ f(b), f(0) = 0, f(1) = 1.It then follows that f(¬a) = ¬f(a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory... | Axiomatization of Boolean algebras | 0.818291 |
11,085 | Generalized Boolean lattices are exactly the ideals of Boolean lattices. A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbe... | Axiomatization of Boolean algebras | 0.818291 |
11,086 | Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that a ≤ b, there exists an element x such that a ∧ x = 0 {... | Axiomatization of Boolean algebras | 0.818291 |
11,087 | It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two. Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to ... | Axiomatization of Boolean algebras | 0.818291 |
11,088 | In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very ... | Numerical sign problem | 0.818287 |
11,089 | The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas: Condensed matter physics — It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the Hubbard model. Nuclear physics — It prevents the ab ... | Numerical sign problem | 0.818287 |
11,090 | In physics the sign problem is typically (but not exclusively) encountered in calculations of the properties of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions. Because the particles are strongly interacti... | Numerical sign problem | 0.818287 |
11,091 | The journal covers astronomy, astrophysics, cosmology, planetary science, solar-terrestrial physics, global and regional geophysics, and the history of these subjects. It also publishes thematic articles regarding interdisciplinary research, science policy, news, opinions, correspondence, and book and software reviews.... | Astronomy & Geophysics | 0.818287 |
11,092 | Astronomy & Geophysics (A&G) is a scientific journal and trade magazine published on behalf of the Royal Astronomical Society (RAS) by Oxford University Press. It is distributed bimonthly to members of the RAS. A&G publishes content of interest to professional astronomers and geophysicists, including: news reports, int... | Astronomy & Geophysics | 0.818287 |
11,093 | The Quarterly Journal of the Royal Astronomical Society was formerly indexed in: GeoRef International Aerospace Abstracts Computer & Control Abstracts Electrical & Electronics Abstracts Physics Abstracts. Science Abstracts. Series A Chemical AbstractsAll articles are indexed, with abstracts, in the Astrophysics Data Sy... | Astronomy & Geophysics | 0.818287 |
11,094 | The Quarterly Journal of the Royal Astronomical Society (Q. J. R. Astron. Soc, ISSN 0035-8738) was published by the Royal Astronomical Society from September 1960 to December 1996, and was produced by Blackwell Science in its later period. During that time 37 volumes were issued. The journal included articles reviewing... | Astronomy & Geophysics | 0.818287 |
11,095 | In topology and in calculus, a round function is a scalar function M → R {\displaystyle M\to {\mathbb {R} }} , over a manifold M {\displaystyle M} , whose critical points form one or several connected components, each homeomorphic to the circle S 1 {\displaystyle S^{1}} , also called critical loops. They are special ca... | Round function | 0.818285 |
11,096 | In particular, Z× is finite only for n ∈ {1, 2, 3, 4, 6}. The torsion subgroup of Z× is the group of roots of unity in Q(ζn), which was described in the previous item. Cyclotomic units form an explicit finite-index subgroup of Z×. The Kronecker–Weber theorem states that every finite abelian extension of Q in C is conta... | Cyclotomic fields | 0.818282 |
11,097 | In particular, Q(ζn) / Q is unramified above every prime not dividing n. If n is a power of a prime p, then Q(ζn) / Q is totally ramified above p. If q is a prime not dividing n, then the Frobenius element Frob q ∈ Gal ( Q ( ζ n ) / Q ) {\displaystyle \operatorname {Frob} _{q}\in \operatorname {Gal} (\mathbf {Q} (\ze... | Cyclotomic fields | 0.818282 |
11,098 | In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep... | Cyclotomic fields | 0.818282 |
11,099 | Statistical relational learning (SRL) is a subdiscipline of artificial intelligence and machine learning that is concerned with domain models that exhibit both uncertainty (which can be dealt with using statistical methods) and complex, relational structure. Note that SRL is sometimes called Relational Machine Learning... | Statistical relational learning | 0.818281 |
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