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8,800 | In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but th... | Groups of Lie type | 0.820947 |
8,801 | {\displaystyle F\mapsto (x\mapsto e^{i}(\nabla _{e_{i}}F)(x)).} This operator is independent of the choice of frame, and can thus be used to define what in geometric calculus is called the vector derivative: ∇ = e i ∂ i . | Geometric calculus | 0.820942 |
8,802 | {\displaystyle {\begin{aligned}\int _{V}{\dot {\mathsf {L}}}\left({\dot {\nabla }}dX;x\right)&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,dX\,I^{-1}\rangle \\&=\int _{V}\langle {\dot {F}}(x){\dot {\nabla }}\,|dX|\rangle \\&=\int _{V}\nabla \cdot F(x)\,|dX|.\end{aligned}}} Likewise, ∮ ∂ V L ( d S ; x ) = ∮ ∂ V ⟨ F ... | Geometric calculus | 0.820942 |
8,803 | The reason for defining the vector derivative and integral as above is that they allow a strong generalization of Stokes' theorem. Let L ( A ; x ) {\displaystyle {\mathsf {L}}(A;x)} be a multivector-valued function of r {\displaystyle r} -grade input A {\displaystyle A} and general position x {\displaystyle x} , linear... | Geometric calculus | 0.820942 |
8,804 | Following is a diagram summarizing the history of geometric calculus. | Geometric calculus | 0.820942 |
8,805 | Macdonald, Alan (2012). Vector and Geometric Calculus. Charleston: CreateSpace. ISBN 9781480132450. OCLC 829395829. | Geometric calculus | 0.820942 |
8,806 | Let { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} be a set of basis vectors that span an n {\displaystyle n} -dimensional vector space. From geometric algebra, we interpret the pseudoscalar e 1 ∧ e 2 ∧ ⋯ ∧ e n {\displaystyle e_{1}\wedge e_{2}\wedge \cdots \wedge e_{n}} to be the signed volume of the n {\dis... | Geometric calculus | 0.820942 |
8,807 | With a geometric algebra given, let a {\displaystyle a} and b {\displaystyle b} be vectors and let F {\displaystyle F} be a multivector-valued function of a vector. The directional derivative of F {\displaystyle F} along b {\displaystyle b} at a {\displaystyle a} is defined as ( ∇ b F ) ( a ) = lim ϵ → 0 F ( a + ϵ b ) ... | Geometric calculus | 0.820942 |
8,808 | The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the multivector derivative. Let F {\displaystyle F} be a multivector-valued function of a multivector. The directional derivative of F {\displaystyle F} with respect to X {\display... | Geometric calculus | 0.820942 |
8,809 | On a manifold, locally we may assign a tangent surface spanned by a set of basis vectors { e i } {\displaystyle \{e_{i}\}} . We can associate the components of a metric tensor, the Christoffel symbols, and the Riemann curvature tensor as follows: g i j = e i ⋅ e j , {\displaystyle g_{ij}=e_{i}\cdot e_{j},} Γ i j k = ( ... | Geometric calculus | 0.820942 |
8,810 | In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including vector calculus, differential geometry, and differential forms. | Geometric calculus | 0.820942 |
8,811 | A Key Selection Vector (KSV) is a numerical identifier associated with a Device Key Set which is distributed by a Licensor or its designee to Adopters and is used to support authentication of Licensed Products and Revocation as part of the HDCP copy protection system. The KSV is used to generate confidential keys, spec... | Key selection vector | 0.820932 |
8,812 | In physics, deflection is a change in a moving object's velocity, hence its trajectory, as a consequence of contact (collision) with a surface or the influence of a non-contact force field. Examples of the former include a ball bouncing off the ground or a bat; examples of the latter include a beam of electrons used to... | Deflection (physics) | 0.820924 |
8,813 | The procedure to select the U.S. Physics Team consists of two exams: the "F=ma" and the "USA Physics Olympiad" (USAPhO). Approximately the top 20 finishers are invited to the U.S. Physics Team training camp. (In some previous years, there was also an intermediate, quarterfinal exam.) | United States Physics Olympiad | 0.820923 |
8,814 | Approximately the top 400 students from the F=ma exam are invited to take a free-response, calculus-based exam covering all topics in introductory physics, including mechanics, electricity and magnetism, thermodynamics, fluids, relativity, waves, and nuclear and atomic physics. There are two parts in the exam, each all... | United States Physics Olympiad | 0.820923 |
8,815 | Approximately the top 20 students in the nation are selected for the U.S. Physics Team, and invited to a training camp at the University of Maryland. There, students receive intensive theoretical and experimental preparation and take a series of selection tests. The top five students travel to the International Physics... | United States Physics Olympiad | 0.820923 |
8,816 | Approximately 6,000 students take this first exam, which consists of 25 multiple choice questions to be solved in 75 minutes, focusing on algebra-based mechanics. In the past, a quarter point was deducted for each incorrect answer. From 2015 onwards, no points were deducted for incorrect answers. Prior to 2018, the exa... | United States Physics Olympiad | 0.820923 |
8,817 | The United States Physics Olympiad (USAPhO) is a high school physics competition run by the American Association of Physics Teachers and the American Institute of Physics to select the team to represent the United States at the International Physics Olympiad (IPhO). The team is selected through a series of exams testin... | United States Physics Olympiad | 0.820923 |
8,818 | At the 1986 London IPhO, the team brought home three bronze medals. Since then, the US Physics Team has regularly placed in the top ten nations at the international competition. It has accumulated 66 Gold Medals, 48 Silver Medals, 29 Bronze Medals, and 11 Honorable Mentions at IPhO as of 2019. | United States Physics Olympiad | 0.820923 |
8,819 | The International Physics Olympiad began in 1967 among Eastern European countries; many western countries soon joined in the 1970s. In 1986, the American Association of Physics Teachers led by Jack Wilson organized the United States Physics Team for the first time. The 1986 team was made up of 20 talented high school p... | United States Physics Olympiad | 0.820923 |
8,820 | Afterward, the exam was changed to be a single day event in early April, to protect exam security. All USAPhO qualifiers receive certificates of recognition for their physics skill. The medal system is modeled on that of the International Physics Olympiad: approximately the top 10-12% scorers in the USAPhO will receive... | United States Physics Olympiad | 0.820923 |
8,821 | The word syzygy came into mathematics with the work of Arthur Cayley. In that paper, Cayley used it in the theory of resultants and discriminants. As the word syzygy was used in astronomy to denote a linear relation between planets, Cayley used it to denote linear relations between minors of a matrix, such as, in the c... | Linear relation | 0.82092 |
8,822 | For proving this, it suffices to apply twice the preceding proposition for getting two decompositions of the module of the relations between the union of the two generating sets. For obtaining a similar result for higher syzygy modules, it remains to prove that, if M is any module, and L is a free module, then M and M ... | Linear relation | 0.82092 |
8,823 | For k > 1, a kth syzygy module of M is a syzygy module of a (k – 1)-th syzygy module. Hilbert's syzygy theorem states that, if R = K {\displaystyle R=K} is a polynomial ring in n indeterminates over a field, then every nth syzygy module is free. The case n = 0 is the fact that every finite dimensional vector space has... | Linear relation | 0.82092 |
8,824 | In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} are elements of a (left) module M over a ring R (the case of a vector space over a fiel... | Linear relation | 0.82092 |
8,825 | . , x k {\displaystyle x_{1},..,x_{k}} because, although the syzygy module depends on the chosen generating set, most of its properties are independent; see § Stable properties, below. If the ring R is Noetherian, or, at least coherent, and if M is finitely generated, then the syzygy module is also finitely generated. ... | Linear relation | 0.82092 |
8,826 | The construction of higher order syzygy modules is generalized as the definition of free resolutions, which allows restating Hilbert's syzygy theorem as a polynomial ring in n indeterminates over a field has global homological dimension n. If a and b are two elements of the commutative ring R, then (b, –a) is a relatio... | Linear relation | 0.82092 |
8,827 | In this case, one can take L n = S n , {\displaystyle L_{n}=S_{n},} and L k = 0 {\displaystyle L_{k}=0} (the zero module) for every k > n. This allows restating Hilbert's syzygy theorem: If R = K {\displaystyle R=K} is a polynomial ring in n indeterminates over a field K, then every free resolution is finite of length... | Linear relation | 0.82092 |
8,828 | Bioelectromagnetics Bioelectrochemistry European Biophysics Journal International Journal of Bioelectromagnetism, ISBEM, 1999–present, (ISSN 1456-7865) BioMagnetic Research and Technology archive (no longer publishing) Biophysics, English version of the Russian "Biofizika" (ISSN 0006-3509) Radiatsionnaya Bioliogiya Rad... | Bioelectromagnetics | 0.820917 |
8,829 | The Bioelectromagnetics Society (BEMS) European BioElectromagnetics Association (EBEA) Society for Physical Regulation in Biology and Medicine (SPRBM) (formerly the Bioelectrical Repair and Growth Society, BRAGS) International Society for Bioelectromagnetism (ISBEM) The Bioelectromagnetics Lab at University College Cor... | Bioelectromagnetics | 0.820917 |
8,830 | Superradiant emission using instruments such as FEL are used in materials science, chemical technology and surface analysis. Using the superradiant emission from the instruments, proteins molecules can be imaged. Another use of these instruments in the analysis is the vibrational energy transfer in molecule. High power... | Super radiant emission | 0.820913 |
8,831 | In physics, a number of noted theories of the motion of objects have developed. Among the best known are: Classical mechanics Newton's laws of motion Euler's laws of motion Cauchy's equations of motion Kepler's laws of planetary motion General relativity Special relativity Quantum mechanics | Laws of Motion | 0.82091 |
8,832 | In the modern terminology of differential geometry, polar coordinates provide coordinate charts for the differentiable manifold R2 \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclidean metric tensor is given byThis can be seen via the change of variables formula for the metric tensor, or by computi... | Polar coordinates | 0.820906 |
8,833 | These problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere. The counterparts of a circle in other dimensions can never be packed with complete effici... | Packing problems | 0.820905 |
8,834 | Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packagin... | Packing problems | 0.820904 |
8,835 | In tiling or tessellation problems, there are to be no gaps, nor overlaps. Many of the puzzles of this type involve packing rectangles or polyominoes into a larger rectangle or other square-like shape. There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps: An... | Packing problems | 0.820904 |
8,836 | The total angular momentum satisfies the same commutation relations as orbital angular momentum, namely from which follows where Ji stand for Jx, Jy, and Jz. The quantum numbers describing the system, which are constant over time, are now j and mj, defined through the action of J on the wavefunction Ψ {\displaystyle \P... | Orbital quantum number | 0.820899 |
8,837 | In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being... | Orbital quantum number | 0.820899 |
8,838 | The singleton set consisting of the identity function on R {\displaystyle \mathbb {R} } separates the points of R . {\displaystyle \mathbb {R} .} If X {\displaystyle X} is a T1 normal topological space, then Urysohn's lemma states that the set C ( X ) {\displaystyle C(X)} of continuous functions on X {\displaystyle X} ... | Separating set | 0.820898 |
8,839 | In mathematics, a set S {\displaystyle S} of functions with domain D {\displaystyle D} is called a separating set for D {\displaystyle D} and is said to separate the points of D {\displaystyle D} (or just to separate points) if for any two distinct elements x {\displaystyle x} and y {\displaystyle y} of D , {\displayst... | Separating set | 0.820898 |
8,840 | Ribosome-nascent chain complex (RNC) refers to the collection of molecules that constitute a ribosome attached to the polypeptide (protein) that it is synthesising. The synthesis of the nascent polypeptide may be stalled by one of several methods. RNCs are produced and purified in laboratories to study the dynamics, bi... | Ribosome-nascent chain complex | 0.820895 |
8,841 | However, the joint mutual information I ( X 1 , X 2 ; Y ) = 1 {\displaystyle I(X_{1},X_{2};Y)=1} bit. There is information about the target that can only be extracted from the joint state of the inputs considered jointly, and not any others. There is, at yet, no universal agreement on how synergy can best be quantified... | Drug synergy | 0.820892 |
8,842 | In the natural world, synergistic phenomena are ubiquitous, ranging from physics (for example, the different combinations of quarks that produce protons and neutrons) to chemistry (a popular example is water, a compound of hydrogen and oxygen), to the cooperative interactions among the genes in genomes, the division of... | Drug synergy | 0.820892 |
8,843 | This appears in Karel Čapek's R.U.R., the films A.I. Artificial Intelligence and Ex Machina, as well as the novel Do Androids Dream of Electric Sheep?, by Philip K. Dick. Dick considers the idea that our understanding of human subjectivity is altered by technology created with artificial intelligence. | Machine intelligence | 0.820892 |
8,844 | Thought-capable artificial beings have appeared as storytelling devices since antiquity, and have been a persistent theme in science fiction.A common trope in these works began with Mary Shelley's Frankenstein, where a human creation becomes a threat to its masters. This includes such works as Arthur C. Clarke's and St... | Machine intelligence | 0.820892 |
8,845 | Most EU member states had released national AI strategies, as had Canada, China, India, Japan, Mauritius, the Russian Federation, Saudi Arabia, United Arab Emirates, US and Vietnam. Others were in the process of elaborating their own AI strategy, including Bangladesh, Malaysia and Tunisia. The Global Partnership on Art... | Machine intelligence | 0.820892 |
8,846 | The regulation of artificial intelligence is the development of public sector policies and laws for promoting and regulating artificial intelligence (AI); it is therefore related to the broader regulation of algorithms. The regulatory and policy landscape for AI is an emerging issue in jurisdictions globally. According... | Machine intelligence | 0.820892 |
8,847 | Although his arguments had been ridiculed and ignored when they were first presented, eventually, AI research came to agree.The issue is not resolved: sub-symbolic reasoning can make many of the same inscrutable mistakes that human intuition does, such as algorithmic bias. Critics such as Noam Chomsky argue continuing ... | Machine intelligence | 0.820892 |
8,848 | Symbolic AI (or "GOFAI") simulated the high-level conscious reasoning that people use when they solve puzzles, express legal reasoning and do mathematics. They were highly successful at "intelligent" tasks such as algebra or IQ tests. In the 1960s, Newell and Simon proposed the physical symbol systems hypothesis: "A ph... | Machine intelligence | 0.820892 |
8,849 | Natural language processing (NLP) allows programs to read, write and communicate in human languages such as English. Specific problems include speech recognition, speech synthesis, machine translation, information extraction, information retrieval and question answering.Early work, based on Noam Chomsky's generative gr... | Machine intelligence | 0.820892 |
8,850 | Machine perception is the ability to use input from sensors (such as cameras, microphones, wireless signals, active lidar, sonar, radar, and tactile sensors) to deduce aspects of the world. Computer vision is the ability to analyze visual input. The field includes speech recognition,image classification,facial recognit... | Machine intelligence | 0.820892 |
8,851 | ProPublica claims that the COMPAS-assigned recidivism risk level of black defendants is far more likely to be overestimated than that of white defendants, despite the fact that the program was not told the races of the defendants.Health equity issues may also be exacerbated when many-to-many mapping is done without tak... | Machine intelligence | 0.820892 |
8,852 | Machine learning applications will be biased if they learn from biased data. The developers may not be aware that the bias exists. For example, on June 28, 2015, Google Photos's new image labeling feature mistakenly identified Jacky Alcine and a friend as "gorillas" because they were black. The system was trained on a ... | Machine intelligence | 0.820892 |
8,853 | LCCN 20190474. Rich, Elaine; Knight, Kevin; Nair, Shivashankar B (2010). Artificial Intelligence (3rd ed.). New Delhi: Tata McGraw Hill India. ISBN 978-0070087705.These were the four the most widely used AI textbooks in 2008: | Machine intelligence | 0.820891 |
8,854 | Artificial Intelligence: A Modern Approach (4th ed.). Hoboken: Pearson. ISBN 978-0134610993. | Machine intelligence | 0.820891 |
8,855 | For many specific tasks, other methods were abandoned. Deep learning's success was based on both hardware improvements (faster computers, graphics processing units, cloud computing) and access to large amounts of data (including curated datasets, such as ImageNet). Deep learning's success led to an enormous increase in... | Machine intelligence | 0.820891 |
8,856 | This "narrow" and "formal" focus allowed researchers to produce verifiable results and collaborate with other fields (such as statistics, economics and mathematics). By 2000, solutions developed by AI researchers were being widely used, although in the 1990s they were rarely described as "artificial intelligence".Sever... | Machine intelligence | 0.820891 |
8,857 | Robotics researchers, such as Rodney Brooks, rejected "representation" in general and focussed directly on engineering machines that move and survive.. Judea Pearl, Lofti Zadeh and others developed methods that handled incomplete and uncertain information by making reasonable guesses rather than precise logic. But the ... | Machine intelligence | 0.820891 |
8,858 | and British governments to restore funding for academic research. However, beginning with the collapse of the Lisp Machine market in 1987, AI once again fell into disrepute, and a second, longer-lasting winter began.Many researchers began to doubt that the current practices would be able to imitate all the processes of... | Machine intelligence | 0.820891 |
8,859 | Both the U.S. and British governments cut off exploratory research in response to the criticism of Sir James Lighthill and ongoing pressure from the US Congress to fund more productive projects. Minsky's and Papert's book Perceptrons was understood as proving that artificial neural networks approach would never be usef... | Machine intelligence | 0.820891 |
8,860 | Herbert Simon predicted, "machines will be capable, within twenty years, of doing any work a man can do". Marvin Minsky agreed, writing, "within a generation ... the problem of creating 'artificial intelligence' will substantially be solved".They had, however, underestimated the difficulty of the problem. | Machine intelligence | 0.820891 |
8,861 | The attendees became the leaders of AI research in the 1960s. They and their students produced programs that the press described as "astonishing": computers were learning checkers strategies, solving word problems in algebra, proving logical theorems and speaking English.By the middle of the 1960s, research in the U.S.... | Machine intelligence | 0.820891 |
8,862 | In the late 2010s, graphics processing units (GPUs) that were increasingly designed with AI-specific enhancements and used with specialized TensorFlow software, had replaced previously used central processing unit (CPUs) as the dominant means for large-scale (commercial and academic) machine learning models' training. ... | Machine intelligence | 0.820891 |
8,863 | In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph. | Edge space | 0.820878 |
8,864 | Thus differential diagnosis, in which several possible explanations are compared and contrasted, must be performed. This involves the correlation of various pieces of information followed by the recognition and differentiation of patterns. Occasionally the process is made easy by a sign or symptom (or a group of severa... | Diagnostic medicine | 0.820865 |
8,865 | Nucleic acid sequence-based amplification, commonly referred to as NASBA, is a method in molecular biology which is used to produce multiple copies of single stranded RNA. NASBA is a two-step process that takes RNA and anneals specially designed primers, then utilizes an enzyme cocktail to amplify it. | NASBA (molecular biology) | 0.820862 |
8,866 | In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. The gravitational for... | Acceleration of free fall | 0.820859 |
8,867 | In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies accelerate in vacuum at the same rate, regardless of the masses or compositi... | Acceleration of free fall | 0.820859 |
8,868 | A clinical biologist is a health professional such as a doctor of medicine, pharmacist, chemist or biologist that is specialized in clinical biology, a medical specialty derived from clinical pathology. The concept includes interventional biology, including assisted reproductive technology. These professionals follow a... | Clinical biologist | 0.820857 |
8,869 | The earliest mathematical work of antiquity to come down to our time is On the rotating sphere (Περὶ κινουμένης σφαίρας, Peri kinoumenes sphairas) by Autolycus of Pitane, who lived at the end of the fourth century BC.Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Gree... | Spherical geometry | 0.820856 |
8,870 | Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of f, this exceeds 180°. | Spherical geometry | 0.820856 |
8,871 | If "line" is taken to mean great circle, spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three. Contrary to the fir... | Spherical geometry | 0.820856 |
8,872 | An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-... | Spherical geometry | 0.820856 |
8,873 | Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of a non-Euclidean geometry and is sometimes described as being one. However, spherical geometry was not considered a full-fledged non-Euclidean geometry sufficient to resolve the ancient problem of whether the parallel ... | Spherical geometry | 0.820856 |
8,874 | The Book of Unknown Arcs of a Sphere written by the Islamic mathematician Al-Jayyani is considered to be the first treatise on spherical trigonometry. The book contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.The book On Tria... | Spherical geometry | 0.820856 |
8,875 | Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere or the n-dimensional surface of higher dimensional spheres. Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical tr... | Spherical geometry | 0.820856 |
8,876 | Spherical geometry has the following properties: Any two great circles intersect in two diametrically opposite points, called antipodal points. Any two points that are not antipodal points determine a unique great circle. There is a natural unit of angle measurement (based on a revolution), a natural unit of length (ba... | Spherical geometry | 0.820856 |
8,877 | Leonhard Euler published a series of important memoirs on spherical geometry: L. Euler, Principes de la trigonométrie sphérique tirés de la méthode des plus grands et des plus petits, Mémoires de l'Académie des Sciences de Berlin 9 (1753), 1755, p. 233–257; Opera Omnia, Series 1, vol. XXVII, p. 277–308. | Spherical geometry | 0.820856 |
8,878 | Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions. Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses so... | Spherical geometry | 0.820856 |
8,879 | This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships. Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is more than an analogy; spher... | Spherical geometry | 0.820856 |
8,880 | In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic... | Spherical geometry | 0.820856 |
8,881 | In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are point and great circle. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry. In the extrinsic 3-dimensional picture, a great ci... | Spherical geometry | 0.820856 |
8,882 | A field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds: Each absolutely irreducible variety V {\displaystyle V} defined over K {\displaystyle K} has a K {\displaystyle K} -rational point. For each absolutely irreducible polynomial f ∈ K {\displaystyle ... | Pseudo algebraically closed | 0.820854 |
8,883 | In mathematics, a field K {\displaystyle K} is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967. | Pseudo algebraically closed | 0.820854 |
8,884 | In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantization of plasma oscillations, just like phonons are quantizations of mechani... | Plasmon | 0.820846 |
8,885 | This branch is typically taught in secondary education or in the first year of university. Outline of arithmetic Outline of discrete mathematics List of calculus topics List of geometry topics Outline of geometry List of trigonometry topics Outline of trigonometry List of trigonometric identities List of logarithmic id... | Lists of mathematics topics | 0.820845 |
8,886 | The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science, and engineering. List of algorithm general topics List of computability and complexity topi... | Lists of mathematics topics | 0.820845 |
8,887 | In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The following pages list the integrals of many different functions. Lists of integrals List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions... | Lists of mathematics topics | 0.820845 |
8,888 | Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension. Glossary of differential geometry and to... | Lists of mathematics topics | 0.820845 |
8,889 | Combinatorics concerns the study of discrete (and usually finite) objects. Aspects include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), ... | Lists of mathematics topics | 0.820845 |
8,890 | Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things. List of mathematical examples List of algebraic surfaces List of curves List of complex reflection groups List of compl... | Lists of mathematics topics | 0.820845 |
8,891 | Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Outline of algebra Glossary of fiel... | Lists of mathematics topics | 0.820845 |
8,892 | A mathematical statement amounts to a proposition or assertion of some mathematical fact, formula, or construction. Such statements include axioms and the theorems that may be proved from them, conjectures that may be unproven or even unprovable, and also algorithms for computing the answers to questions that can be ex... | Lists of mathematics topics | 0.820845 |
8,893 | A differential equation is an equation involving an unknown function and its derivatives. In a dynamical system, a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish eac... | Lists of mathematics topics | 0.820845 |
8,894 | Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians. List of algebraic number theory topics List of number theory topics Li... | Lists of mathematics topics | 0.820845 |
8,895 | The branch of mathematics deals with the properties and relationships of numbers, especially positive integers. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the scie... | Lists of mathematics topics | 0.820845 |
8,896 | Probability theory is the formalization and study of the mathematics of uncertain events or knowledge. The related field of mathematical statistics develops statistical theory with mathematics. Statistics, the science concerned with collecting and analyzing data, is an autonomous discipline (and not a subdiscipline of ... | Lists of mathematics topics | 0.820845 |
8,897 | Operations research is the study and use of mathematical models, statistics, and algorithms to aid in decision-making, typically with the goal of improving or optimizing the performance of real-world systems. List of knapsack problems List of network theory topics | Lists of mathematics topics | 0.820845 |
8,898 | Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".1 List of mathematical topics in classical mechanics List of mathematical topics in quantum theory List o... | Lists of mathematics topics | 0.820845 |
8,899 | Signals of interest include sound, images, biological signals such as ECG, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise, compression, and feature extraction. List of algebraic coding theory topics List of information theo... | Lists of mathematics topics | 0.820845 |
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