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12,000 | Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an n {\displaystyle n} -dimensional vector space is spanned by a basis of 2 n {\di... | Geometric algebra | 0.817466 |
12,001 | Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure. | Geometric algebra | 0.817466 |
12,002 | In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product of k {\displaystyle k} vectors). By way of example, e 1 ∧ e 2 + e 3 ∧ e 4 {\displaystyle e_{1}\wedge e_{2}+e_{3}\wedge e_{4}} in G ( 4 , 0 ) {\displaystyle {\mathcal {G}}(4,0)} cannot be factored; typically,... | Geometric algebra | 0.817466 |
12,003 | Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. If the canonical basis elements are { B i ∣ i ∈ S } {\displaystyle \{B_{i}\mid i\in S\}} with S {\displaystyle S} being an index set, then the geometric product of any two multivectors is ( ∑ i α i B i )... | Geometric algebra | 0.817466 |
12,004 | The set of all possible products of n {\displaystyle n} orthogonal basis vectors with indices in increasing order, including 1 {\displaystyle 1} as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra G ( 3 , ... | Geometric algebra | 0.817466 |
12,005 | (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra G ( p , q ) {\displaystyle {\mathcal {G}}(p,q)} . For example, G ( 3 , 0 ) {\displaystyle {\mathcal {G}}(3,0)} models three-dimensional Euclidean space, G ( 1 , 3 ) {\d... | Geometric algebra | 0.817466 |
12,006 | A multivector that is the exterior product of r {\displaystyle r} linearly independent vectors is called a blade, and is said to be of grade r {\displaystyle r} . A multivector that is the sum of blades of grade r {\displaystyle r} is called a (homogeneous) multivector of grade r {\displaystyle r} . From the axioms, wi... | Geometric algebra | 0.817466 |
12,007 | Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general rotoreflections and rotations. The reflection c ′ {\displaystyle c'} of a vector c {\displaystyle c} along a vector m {\displaystyle m} , or equivalently in the ... | Geometric algebra | 0.817466 |
12,008 | {\displaystyle a=\sum _{i}a_{i}e^{i}\ .} A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing J = ( j 1 , … , j n ) , {\displaystyle J=(j_{... | Geometric algebra | 0.817466 |
12,009 | C ∨ D := ( ( C I − 1 ) ∧ ( D I − 1 ) ) I {\displaystyle C\vee D:=((CI^{-1})\wedge (DI^{-1}))I} with I {\displaystyle I} the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.The inner product on vectors can also be generalized, but in more than one non-equivalent way. T... | Geometric algebra | 0.817466 |
12,010 | Here, the i = γ 0 γ 1 γ 2 γ 3 {\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} is the unit pseudoscalar (or four-dimensional volume element), γ 0 {\displaystyle \gamma _{0}} is the unit vector in time direction, and E {\displaystyle E} and B {\displaystyle B} are the classic electric and magnetic field ve... | Geometric algebra | 0.817466 |
12,011 | In physics, the main applications are the geometric algebra of Minkowski 3+1 spacetime, G ( 1 , 3 ) {\displaystyle {\mathcal {G}}(1,3)} , called spacetime algebra (STA), or less commonly, G ( 3 , 0 ) {\displaystyle {\mathcal {G}}(3,0)} , interpreted the algebra of physical space (APS). While in STA, points of spacetime... | Geometric algebra | 0.817466 |
12,012 | So the torque, the rate of change of work W {\displaystyle W} , due to a force F {\displaystyle F} , is τ = d W d θ = F ⋅ d r d θ = F ⋅ ( r i ) . {\displaystyle \tau ={\frac {dW}{d\theta }}=F\cdot {\frac {dr}{d\theta }}=F\cdot (\mathbf {r} i).} Unlike the cross product description of torque, τ = r × F {\displaystyle \t... | Geometric algebra | 0.817466 |
12,013 | The mathematical description of rotational forces such as torque and angular momentum often makes use of the cross product of vector calculus in three dimensions with a convention of orientation (handedness). The cross product can be viewed in terms of the exterior product allowing a more natural geometric interpretati... | Geometric algebra | 0.817466 |
12,014 | (2010), Geometric Algebra Computing in Engineering and Computer Science, Springer, ISBN 9781849961080 Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory Goldman, Ron (2010), Rethinking Quaternions: Theory and Computation, Morgan & Clayp... | Geometric algebra | 0.817466 |
12,015 | (eds. ), Applications of Geometric Algebra in Computer Science and Engineering, Birkhäuser, pp. 35–46, doi:10.1007/978-1-4612-0089-5_2, ISBN 978-1-4612-0089-5 Doran, Chris J. L.; Lasenby, Anthony N. | Geometric algebra | 0.817466 |
12,016 | Projective geometric algebra (PGA), also known as the homogeneous model, provides a complete algebra containing representations of all Euclidean isometries and the linear subspaces on which they operate. In this model, a single degenerate dimension is added to the n ordinary dimensions of space to form the algebra G ( ... | Geometric algebra | 0.817466 |
12,017 | In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory. David Hestenes reinterpreted the Pauli and Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been ... | Geometric algebra | 0.817466 |
12,018 | Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Élie Cartan, Hermann Weyl and Claude Chevalley. The geometrical approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Arti... | Geometric algebra | 0.817466 |
12,019 | Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and... | Geometric algebra | 0.817466 |
12,020 | Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space... | Geometric algebra | 0.817466 |
12,021 | The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in ... | Geometric algebra | 0.817466 |
12,022 | Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derive... | Geometric algebra | 0.817466 |
12,023 | The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chos... | Geometric algebra | 0.817466 |
12,024 | Given a canonical basis consisting of orthonormal vectors e i {\displaystyle e_{i}} of V {\displaystyle V} , the set of all 2 {\displaystyle 2} -vectors is spanned by { e 3 e 2 , e 1 e 3 , e 2 e 1 } . {\displaystyle \{e_{3}e_{2},e_{1}e_{3},e_{2}e_{1}\}.} Labelling these i {\displaystyle i} , j {\displaystyle j} and k {... | Geometric algebra | 0.817466 |
12,025 | Such units arise from one of the many quantities in the real algebra that square to − 1 {\displaystyle -1} , and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. In G ( 3 , 0 ) {\displaystyle {\mathcal {G}}(3,0)} , a further familiar case occurs. | Geometric algebra | 0.817466 |
12,026 | Geometric calculus extends the formalism to include differentiation and integration including differential geometry and differential forms.Essentially, the vector derivative is defined so that the GA version of Green's theorem is true, ∫ A d A ∇ f = ∮ ∂ A d x f {\displaystyle \int _{A}dA\,\nabla f=\oint _{\partial A}dx... | Geometric algebra | 0.817466 |
12,027 | For vectors a {\displaystyle a} and b {\displaystyle b} , we may write the geometric product of any two vectors a {\displaystyle a} and b {\displaystyle b} as the sum of a symmetric product and an antisymmetric product: a b = 1 2 ( a b + b a ) + 1 2 ( a b − b a ) {\displaystyle ab={\frac {1}{2}}(ab+ba)+{\frac {1}{2}}(a... | Geometric algebra | 0.817466 |
12,028 | 256. ISBN 978-3-030-01756-9. Wilmot, G.P. (2023). "The Algebra Of Geometry". | Geometric algebra | 0.817466 |
12,029 | doi:10.1007/s00006-016-0664-z. S2CID 124014198. Dorst, Leo (2016), 3D Oriented Projective Geometry Through Versors of R 3 , 3 {\displaystyle \mathbf {R} ^{3,3}} , Springer, ISBN 9783658076184 Vaz, Jayme; da Rocha, Roldão (2016), An Introduction to Clifford Algebras and Spinors, Oxford University Press, Bibcode:2016icas... | Geometric algebra | 0.817466 |
12,030 | "The Genesis of Geometric Algebra:A Personal Retrospective". Advances in Applied Clifford Algebras. 27 (1): 351–379. | Geometric algebra | 0.817466 |
12,031 | Chapter 1 as PDF Vince, John A. (2008), Geometric Algebra for Computer Graphics, Springer, ISBN 978-1-84628-996-5 Lundholm, Douglas; Svensson, Lars (2009), "Clifford Algebra, Geometric Algebra and Applications", arXiv:0907.5356v1 Perwass, Christian (2009), Geometric Algebra with Applications in Engineering, Geometry a... | Geometric algebra | 0.817466 |
12,032 | (2006), "Spacetime structuralism: §5 Manifolds vs. geometric algebra", in Dennis Dieks (ed. ), The ontology of spacetime, Elsevier, p. 54 ff, ISBN 978-0-444-52768-4 Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007), Geometric algebra for computer science: an object-oriented approach to geometry, Elsevier, ISBN 978-0-1... | Geometric algebra | 0.817466 |
12,033 | (2003), Geometric Algebra for Physicists (PDF), Cambridge University Press, ISBN 978-0-521-71595-9, archived (PDF) from the original on 2009-01-06 Hestenes, David (2003), "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics" (PDF), Am. J. Phys., 71 (2): 104–121, Bibcode:2003AmJPh..71..104H, CiteS... | Geometric algebra | 0.817466 |
12,034 | By the Cartan–Dieudonné theorem we have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors. In group terms, for a real, non-degenerate G ( p , q ) {\displaystyle {\mathcal {G}}(p,q)} , having identified... | Geometric algebra | 0.817466 |
12,035 | A k {\displaystyle k} -versor is a multivector that can be expressed as the geometric product of k {\displaystyle k} invertible vectors. Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refe... | Geometric algebra | 0.817466 |
12,036 | All objects in projective geometric algebra are homogeneous, including magnitudes that would simply be scalars in nonprojective settings. Multiplying any point, line, plane, motor, or flector by a nonzero scalar has no effect on its meaning. An object is projected into 3D space through a process called unitization, whi... | Geometric algebra | 0.817466 |
12,037 | This gives rise not only to a fundamental duality in the geometric objects of the algebra, but also the operations on those objects. Every product in the algebra (wedge product, inner product, geometric product, interior product) has a matching "antiproduct" that performs the same operation on the duals of its operands... | Geometric algebra | 0.817466 |
12,038 | Proper Euclidean isometries, which are always screw motions in 3D space, are represented by motors consisting of eight even-graded components, and improper Euclidean isometries, which contain a reflection, are represented by flectors consisting of eight odd-graded components. The motor algebra is the correct generaliza... | Geometric algebra | 0.817466 |
12,039 | Lundholm defines the Pin {\displaystyle \operatorname {Pin} } , Spin {\displaystyle \operatorname {Spin} } , and Spin + {\displaystyle \operatorname {Spin} ^{+}} subgroups, generated by unit vectors, and in the case of Spin {\displaystyle \operatorname {Spin} } and Spin + {\displaystyle \operatorname {Spin} ^{+}} , onl... | Geometric algebra | 0.817466 |
12,040 | It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above mentioned grade projection operator: C ∧ D := ∑ r , s ⟨ ⟨ C ⟩ r ⟨ D ⟩ s ⟩ r + s {\displaystyle C\wedge D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r+s}} (the... | Geometric algebra | 0.817466 |
12,041 | In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional ob... | Geometric algebra | 0.817466 |
12,042 | In general, information retrieval research and technology can be divided into two broad categories: semantic and statistical. Information retrieval systems that fall into the semantic category will attempt to implement some degree of syntactic and semantic analysis of the natural language text that a human user would p... | Concept search | 0.817457 |
12,043 | Some of the most powerful approaches to semantic processing are based on the use of mathematical transform techniques. Matrix decomposition techniques have been the most successful. Some widely used matrix decomposition techniques include the following: Independent component analysis Semi-discrete decomposition Non-neg... | Concept search | 0.817457 |
12,044 | A variety of techniques based on artificial intelligence (AI) and natural language processing (NLP) have been applied to semantic processing, and most of them have relied on the use of auxiliary structures such as controlled vocabularies and ontologies. Controlled vocabularies (dictionaries and thesauri), and ontologie... | Concept search | 0.817457 |
12,045 | In particle physics, a radiative process refers to one elementary particle emitting another and continuing to exist. This typically happens when a fermion emits a boson such as a gluon or photon. | Radiative process | 0.817455 |
12,046 | According to Pfenex, a clinical-stage biotechnology company, the proposed terms in the Trans-Pacific Partnership, a trade agreement between twelve Pacific Rim countries, meant that all participating countries had to adopt the United States' lengthy drug patent exclusivity protection period of 12 years for biologics and... | Specialty drugs | 0.817453 |
12,047 | : 10 It was consolidated with their pharmacy benefit management company PharmaCare in 2002. In their 2001 annual report, CVS anticipated that the "$16 billion specialty pharmacy market" would grow at "an even faster rate than traditional pharmacy due in large part to the robust pipeline of biotechnology drugs". By 2014... | Specialty drugs | 0.817453 |
12,048 | Gene-based and cellular biologics, for example, often are at the forefront of biomedical research, and may be used to treat a variety of medical conditions for which no other treatments are available." "In contrast to most drugs that are chemically synthesized and their structure is known, most biologics are complex mi... | Specialty drugs | 0.817453 |
12,049 | Specialty drugs are more complex to manufacture. They are "highly complex medications, typically biology-based, that structurally mimic compounds found within the body". Specialty drugs are often biologics—"drugs derived from living cells"—but biologics are "not always deemed to be specialty drugs". Biologics "may be p... | Specialty drugs | 0.817453 |
12,050 | According to the National Center for Biotechnology Information, by the mid-1990s, there were fewer than 30 specialty drugs on the market, but by 2008 that number had increased to 200.Specialty drugs may also be designated as orphan drugs or ultra-orphan drugs under the U. S. Orphan Drug Act of 1983. This was enacted to... | Specialty drugs | 0.817453 |
12,051 | In the left diagram, which expresses the first isomorphism theorem, commutativity of the triangle means that f = f ~ ∘ π {\displaystyle f={\tilde {f}}\circ \pi } . In the right diagram, commutativity of the square means h ∘ f = k ∘ g {\displaystyle h\circ f=k\circ g} . | Commutative diagram | 0.817445 |
12,052 | Most of the computation is taken by the evaluation of ϕ ( P ) {\displaystyle \phi (P)} and ϕ 2 ( P ) {\displaystyle \phi ^{2}(P)} , for each prime l {\displaystyle l} , that is computing x q {\displaystyle x^{q}} , y q {\displaystyle y^{q}} , x q 2 {\displaystyle x^{q^{2}}} , y q 2 {\displaystyle y^{q^{2}}} for each pr... | Schoof's algorithm | 0.817443 |
12,053 | Given t ( mod l i ) {\displaystyle t{\pmod {l_{i}}}} for all l i ∈ S {\displaystyle l_{i}\in S} , the Chinese remainder theorem allows us to compute t ( mod N ) {\displaystyle t{\pmod {N}}} . In order to compute t ( mod l ) {\displaystyle t{\pmod {l}}} for a prime l ≠ p {\displaystyle l\neq p} , we make use of the theo... | Schoof's algorithm | 0.817443 |
12,054 | Hasse's theorem states that if E / F q {\displaystyle E/\mathbb {F} _{q}} is an elliptic curve over the finite field F q {\displaystyle \mathbb {F} _{q}} , then # E ( F q ) {\displaystyle \#E(\mathbb {F} _{q})} satisfies ∣ q + 1 − # E ( F q ) ∣≤ 2 q . {\displaystyle \mid q+1-\#E(\mathbb {F} _{q})\mid \leq 2{\sqrt {q}}.... | Schoof's algorithm | 0.817443 |
12,055 | Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the number of points to judge the difficulty of solving the discrete logarithm problem in the group of points on an elliptic cur... | Schoof's algorithm | 0.817443 |
12,056 | Using the group law on elliptic curves restricted to this set one can see that this set E ( F q ) {\displaystyle E(\mathbb {F} _{q})} forms an abelian group, with O {\displaystyle O} acting as the zero element. In order to count points on an elliptic curve, we compute the cardinality of E ( F q ) {\displaystyle E(\math... | Schoof's algorithm | 0.817443 |
12,057 | The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled. Isotope fractionation occurs during a phase transition, the ratio of light to heavy isotopes in the involved molecules changes. When water vapor condenses (an equilibrium fractionation), the heavier wat... | Second order phase transition | 0.817439 |
12,058 | Other phase changes include: Transition to a mesophase between solid and liquid, such as one of the "liquid crystal" phases. The dependence of the adsorption geometry on coverage and temperature, such as for hydrogen on iron (110). The emergence of superconductivity in certain metals and ceramics when cooled below a cr... | Second order phase transition | 0.817439 |
12,059 | While Galileo's First and Second Letters had been written in response to Scheiner's Tres Epistolae, his Third Letter responded to Accuratior Disquisitio. : 234 Galileo was angry to see that once again Scheiner was making claims about the moons of Jupiter, since he regarded them as his own discovery. To demonstrate the ... | Letters on Sunspots | 0.817438 |
12,060 | The cosmology of Galileo's time, based on Aristotle's Physics, held that the Sun was 'perfect' and unflawed. Only with the invention of the telescope was it possible for sunspots to be systematically observed. Many who had never seen them found the idea of them morally and philosophically repugnant. Those who could see... | Letters on Sunspots | 0.817438 |
12,061 | In 1632 Galileo published Dialogo sopra i due Massimi Sistemi del Mondo (Dialogue Concerning the Two Chief World Systems), a fictitious four day-long discussion about natural philosophy between the characters Salviati (who argued for Copernican ideas and was effectively a mouthpiece of Galileo), Sagredo, who represente... | Letters on Sunspots | 0.817438 |
12,062 | In the Letters on Sunspots Galileo responded to claims by Scheiner about the phases of Venus, which were an important question in the astronomy of the time. There were different schools of thought about whether Venus had phases at all – to the naked eye, none were visible. In 1610, using his telescope, Galileo had disc... | Letters on Sunspots | 0.817438 |
12,063 | In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the n... | Algebraic lattices | 0.817438 |
12,064 | The greatest element of Con(A) is the set AxA, which is the congruence corresponding to the constant homomorphism. The smallest congruence is the diagonal of AxA, corresponding to isomorphisms. Con(A) is a complete lattice. The compact elements of Con(A) are exactly the finitely generated congruences. Con(A) is an alge... | Algebraic lattices | 0.817438 |
12,065 | The greatest lower bound of any family of substructures is their intersection (or A if the family is empty). The compact elements of Sub(A) are exactly the finitely generated substructures of A. Every substructure is the union of its finitely generated substructures; hence Sub(A) is an algebraic lattice.Also, a kind of... | Algebraic lattices | 0.817438 |
12,066 | Then: The set Sub(A), ordered by set inclusion, is a lattice. The greatest element of Sub(A) is the set A itself. For any S, T in Sub(A), the greatest lower bound of S and T is the set theoretic intersection of S and T; the smallest upper bound is the subalgebra generated by the union of S and T. The set Sub(A) is even... | Algebraic lattices | 0.817438 |
12,067 | A poset in which every element is the supremum of the compact elements below it is called an algebraic poset. Such posets that are dcpos are much used in domain theory. As an important special case, an algebraic lattice is a complete lattice L where every element x of L is the supremum of the compact elements below x. ... | Algebraic lattices | 0.817438 |
12,068 | A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow sharing of resources and information. When at least one process in one device is able to send or receive data to or from at least one process residing in a ... | Computer use | 0.817432 |
12,069 | Some theories beyond the Standard Model include a hypothetical fifth force, and the search for such a force is an ongoing line of experimental physics research. In supersymmetric theories, some particles acquire their masses only through supersymmetry breaking effects and these particles, known as moduli, can mediate n... | Four fundamental forces | 0.817431 |
12,070 | {\displaystyle p(p-1).} Abel–Ruffini theorem results from this and the fact that there are polynomials with a symmetric Galois group. | Primitive group | 0.817428 |
12,071 | In the same letter in which he introduced the term "primitive", Galois stated the following theorem:If G is a primitive solvable group acting on a finite set X, then the order of X is a power of a prime number p. Further, X may be identified with an affine space over the finite field with p elements, and G acts on X as... | Primitive group | 0.817428 |
12,072 | One of the major uses of signaling games both in economics and biology has been to determine under what conditions honest signaling can be an equilibrium of the game. That is, under what conditions can we expect rational people or animals subject to natural selection to reveal information about their types? If both par... | Signaling game | 0.817426 |
12,073 | When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a differential equation, so Ohm's law (as defined above) does not directly apply since tha... | Ohm’s law | 0.817416 |
12,074 | After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form: A perfect crystal lattice, with low enough thermal motion and no deviations from... | Ohm’s law | 0.817416 |
12,075 | Ohm explained his experimental results by a slightly more complex equation than the modern form above (see § History below). In physics, the term Ohm's law is also used to refer to various generalizations of the law; for example the vector form of the law used in electromagnetics and material science: where J is the cu... | Ohm’s law | 0.817416 |
12,076 | Spontaneous potentials are often measured down boreholes for formation evaluation in the oil and gas industry, and they can also be measured along the Earth's surface for mineral exploration or groundwater investigation. The phenomenon and its application to geology was first recognized by Conrad Schlumberger, Marcel S... | Spontaneous potential | 0.817415 |
12,077 | Oligogenic inheritance (Greek ὀλίγος – ὀligos = few, a little) describes a trait that is influenced by a few genes. Oligogenic inheritance represents an intermediate between monogenic inheritance in which a trait is determined by a single causative gene, and polygenic inheritance, in which a trait is influenced by many... | Oligogenic inheritance | 0.817408 |
12,078 | In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features ... | Regular languages | 0.817404 |
12,079 | Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.As a consequenc... | Nested dissection | 0.817399 |
12,080 | Molecular Biology and Evolution is a monthly peer-reviewed scientific journal published by Oxford University Press on behalf of the Society for Molecular Biology and Evolution. It publishes work in the intersection of molecular biology and evolutionary biology. The founding editors-in-chief were Walter Fitch and Masato... | Molecular Biology and Evolution | 0.817389 |
12,081 | The bulk of identifications using CODIS rely on short tandem repeats (STRs) that are scattered throughout the human genome and on statistics that are used to calculate the rarity of that specific profile in the population. STRs are a type of copy-number variation and comprise a sequence of nucleotide base pairs that is... | Combined DNA Index System | 0.817387 |
12,082 | Upload of these profiles to the national level of CODIS requires at least eight of the core loci to be present as well as a profile rarity of 1 in 10 million (calculated using population statistics).Loci that fall within a gene are named after the gene. For example, TPOX, is named after the human thyroid peroxidase gen... | Combined DNA Index System | 0.817387 |
12,083 | {\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n! }{k_{1}!\,k_{2}!\cdots k_{m}!}}} is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices k1 through km such that the sum of all ki is n. That is, for each term in the expansion, the exponents of the xi must add u... | Multinomial formula | 0.817385 |
12,084 | In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. | Multinomial formula | 0.817385 |
12,085 | A correct reaction mechanism is an important part of accurate predictive modeling. For many combustion and plasma systems, detailed mechanisms are not available or require development. Even when information is available, identifying and assembling the relevant data from a variety of sources, reconciling discrepant valu... | Organic reaction mechanisms | 0.817384 |
12,086 | Photochemistry of CO2 and CO in ionosphere can produce CO2+ and CO+ ions, respectively: CO2 + hν ⟶ CO+2 + e− CO + hν ⟶ CO+ + e−An ion and an electron can recombine and produce electronic-neutral products. The products gain extra kinetic energy due to the Coulomb attraction between ions and electrons. This process is ca... | Mars atmosphere | 0.817384 |
12,087 | In early 2016, Stratospheric Observatory for Infrared Astronomy (SOFIA) detected atomic oxygen in the atmosphere of Mars, which has not been found since the Viking and Mariner mission in the 1970s.In 2019, NASA scientists working on the Curiosity rover mission, who have been taking measurements of the gas, discovered t... | Mars atmosphere | 0.817384 |
12,088 | Insurers' business model aims to collect more in premium and investment income than is paid out in losses, and to also offer a competitive price which consumers will accept. Profit can be reduced to a simple equation: Profit = earned premium + investment income – incurred loss – underwriting expenses.Insurers make mone... | Combined ratio | 0.817379 |
12,089 | Exactly how this is supposed to work has been the subject of much debate. Why we should assign probabilities at all to outcomes that are certain to occur in some worlds, and why should the probabilities be given by the Born rule? Several ways to answer these questions in the many-worlds framework have been proposed, bu... | Schrödinger's equation | 0.817376 |
12,090 | The post-measurement wave function generally cannot be known prior to the measurement, but the probabilities for the different possibilities can be calculated using the Born rule. Other, more recent interpretations of quantum mechanics, such as relational quantum mechanics and QBism also give the Schrödinger equation a... | Schrödinger's equation | 0.817376 |
12,091 | The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. The meaning of the Schrödinger equation and how the mathematical entities in it relate to physical reali... | Schrödinger's equation | 0.817376 |
12,092 | : 10 Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl defines the state of a quantum mechanical system to be a vector | ψ ⟩ {\displaystyle |\psi \rangle } belonging to a (separable) Hilbert space H {\displays... | Schrodinger's equation | 0.817376 |
12,093 | Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic calculus, particularly derivatives with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms... | Schrodinger's equation | 0.817376 |
12,094 | Most solar observatories observe optically at visible, UV, and near infrared wavelengths, but other solar phenomena can be observed — albeit not from the Earth's surface due to the absorption of the atmosphere: Solar X-ray astronomy, observations of the Sun in x-rays Multi-spectral solar telescope array (MSSTA), a rock... | Solar telescope | 0.81737 |
12,095 | In the field of amateur astronomy there are many methods used to observe the Sun. Amateurs use everything from simple systems to project the Sun on a piece of white paper, light blocking filters, Herschel wedges which redirect 95% of the light and heat away from the eyepiece, up to hydrogen-alpha filter systems and eve... | Solar telescope | 0.81737 |
12,096 | In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to a sequence of previous logics. The concept was introduced in 1938 by Alan Turing in his PhD dissertation at Princeton in view of Gödel's incompleteness theorems.While Gödel showed that every logic system suffers... | Ordinal logic | 0.817359 |
12,097 | Nongeniculate corallines are of particular significance in the ecology of coral reefs, where they add calcareous material to the structure of the reef, help cement the reef together, and are important sources of primary production. Coralline algae are especially important in reef construction, as they lay down calcium ... | Coralline algae | 0.817354 |
12,098 | Since coralline algae contain calcium carbonate, they fossilize fairly well. They are particularly significant as stratigraphic markers in petroleum geology. Coralline rock was used as building stone since the ancient Greek culture.The calcite crystals composing the cell wall are elongated perpendicular to the cell wal... | Coralline algae | 0.817354 |
12,099 | Coralline algae are red algae in the order Corallinales. They are characterized by a thallus that is hard because of calcareous deposits contained within the cell walls. The colors of these algae are most typically pink, or some other shade of red, but some species can be purple, yellow, blue, white, or gray-green. Cor... | Coralline algae | 0.817354 |
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