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9,100 | More precisely, they are equivalent for fields, for which every homomorphism is a monomorphism, and for varieties of universal algebra, that is algebraic structures for which operations and axioms (identities) are defined without any restriction (the fields do not form a variety, as the multiplicative inverse is define... | Surjective homomorphism | 0.820662 |
9,101 | For algebraic structures, monomorphisms are commonly defined as injective homomorphisms. : 134: 29 In the more general context of category theory, a monomorphism is defined as a morphism that is left cancelable. This means that a (homo)morphism f: A → B {\displaystyle f:A\to B} is a monomorphism if, for any pair g {\di... | Surjective homomorphism | 0.820662 |
9,102 | Also in this case, it is K {\displaystyle K} , rather than ∼ {\displaystyle \sim } , that is called the kernel of f {\displaystyle f} . The kernels of homomorphisms of a given type of algebraic structure are naturally equipped with some structure. This structure type of the kernels is the same as the considered structu... | Surjective homomorphism | 0.820662 |
9,103 | In that case the image of X {\displaystyle X} in Y {\displaystyle Y} under the homomorphism f {\displaystyle f} is necessarily isomorphic to X / ∼ {\displaystyle X/\!\sim } ; this fact is one of the isomorphism theorems. When the algebraic structure is a group for some operation, the equivalence class K {\displaystyle ... | Surjective homomorphism | 0.820662 |
9,104 | An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to its target. : 135 The endomorphisms of an algebraic structure, or of an object of a category form a monoid under composition. The endomorphisms of a vector space or of a module form a ring. In the... | Surjective homomorphism | 0.820662 |
9,105 | A linear map is a homomorphism of vector spaces; that is, a group homomorphism between vector spaces that preserves the abelian group structure and scalar multiplication. A module homomorphism, also called a linear map between modules, is defined similarly. An algebra homomorphism is a map that preserves the algebra op... | Surjective homomorphism | 0.820662 |
9,106 | A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures. This means a map f: A → B {\displaystyle f:A\to B} between two sets A {\displaystyle A} , B {\displaystyle B} equipped with the same structure such that, if ⋅ {\displays... | Surjective homomorphism | 0.820662 |
9,107 | In model theory, the notion of an algebraic structure is generalized to structures involving both operations and relations. Let L be a signature consisting of function and relation symbols, and A, B be two L-structures. Then a homomorphism from A to B is a mapping h from the domain of A to the domain of B such that h(F... | Surjective homomorphism | 0.820662 |
9,108 | In software engineering, the initialization-on-demand holder (design pattern) idiom is a lazy-loaded singleton. In all versions of Java, the idiom enables a safe, highly concurrent lazy initialization of static fields with good performance. The implementation of the idiom relies on the initialization phase of execution... | Initialization-on-demand holder idiom | 0.820655 |
9,109 | This can happen for at most 2K(x,y)-l = 2k different strings. These strings can be enumerated given k,l and hence x can be specified by its index in this enumeration. The corresponding program for x has size k + O(1). Theorem proved. | Chain rule for Kolmogorov complexity | 0.820654 |
9,110 | The chain rule for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states: H ( X , Y ) = H ( X ) + H ( Y | X ) {\displaystyle H(X,Y)=H(X)+H(Y|X)} That is, the combined randomness of two sequences X and Y is the sum of the randomness of X plus whatever randomness is left in Y once w... | Chain rule for Kolmogorov complexity | 0.820654 |
9,111 | Thus, many-body theoretical physics most often relies on a set of approximations specific to the problem at hand, and ranks among the most computationally intensive fields of science. In many cases, emergent phenomena may arise which bear little resemblance to the underlying elementary laws. Many-body problems play a c... | Quantum many body problem | 0.820654 |
9,112 | In quantum mechanics, however, the many-body-system is in general in a superposition of combinations of single particle states - all the k n {\displaystyle k^{n}} different combinations have to be accounted for. The dimension of the quantum many body system therefore scales exponentially with n {\displaystyle n} , much... | Quantum many body problem | 0.820654 |
9,113 | The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system. Many can be anywhere from three to inf... | Quantum many body problem | 0.820654 |
9,114 | Mean-field theory and extensions (e.g. Hartree–Fock, Random phase approximation) Dynamical mean field theory Many-body perturbation theory and Green's function-based methods Configuration interaction Coupled cluster Various Monte-Carlo approaches Density functional theory Lattice gauge theory Matrix product state Neura... | Quantum many body problem | 0.820654 |
9,115 | Biocatalysis refers to the use of living (biological) systems or their parts to speed up (catalyze) chemical reactions. In biocatalytic processes, natural catalysts, such as enzymes, perform chemical transformations on organic compounds. Both enzymes that have been more or less isolated and enzymes still residing insid... | Biocatalysis | 0.820648 |
9,116 | Recently, photoredox catalysis has been applied to biocatalysis, enabling unique, previously inaccessible transformations. Photoredox chemistry relies upon light to generate free radical intermediates. These radical intermediates are achiral thus racemic mixtures of product are obtained when no external chiral environm... | Biocatalysis | 0.820648 |
9,117 | In particle physics, the X charge (or simply X) is a conserved quantum number associated with the SO(10) grand unification theory. It is thought to be conserved in strong, weak, electromagnetic, gravitational, and Higgs interactions. Because the X charge is related to the weak hypercharge, it varies depending on the he... | X (charge) | 0.820645 |
9,118 | The 1928 edition included a clear statement of the Entscheidungsproblem (decision problem) for FOL, and also asked whether that logic was complete (i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules). The former problem was answered in the negative first by Alonzo Church and ... | Principles of Mathematical Logic | 0.820633 |
9,119 | The concept of chemical similarity can be expanded to consider chemical similarity network theory, where descriptive network properties and graph theory can be applied to analyze large chemical space, estimate chemical diversity and predict drug target. Recently, 3D chemical similarity networks based on 3D ligand confo... | Chemical similarity | 0.82062 |
9,120 | A generalization of the concept of Pythagorean triples is the search for triples of positive integers a, b, and c, such that an + bn = cn, for some n strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem because it took longer than a... | Pythagorean triangle | 0.820617 |
9,121 | A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} ; thus, Pythagorean triple... | Pythagorean triangle | 0.820617 |
9,122 | Primitive Pythagorean triples have been used in cryptography as random sequences and for the generation of keys. | Pythagorean triangle | 0.820617 |
9,123 | Multimodal learning, in context of machine learning, is deep learning from a combination of various modalities of data, often arising in real-world applications. An example of multi-modal data is data that combines text (typically represented as feature vector) with imaging data consisting of pixel intensities and anno... | Multimodal learning | 0.820614 |
9,124 | Self Supervised Learning brings a more interesting and powerful model for multimodality. OpenAI developed CLIP and DALL-E models that revolutionized multimodality. Multimodal deep learning is used for cancer screening – at least one system under development integrates such different types of data. | Multimodal learning | 0.820614 |
9,125 | Exact maximum likelihood learning in this model is intractable, but approximate learning of DBMs can be carried out by using a variational approach, where mean-field inference is used to estimate data-dependent expectations and an MCMC based stochastic approximation procedure is used to approximate the model’s expected... | Multimodal learning | 0.820614 |
9,126 | A Boltzmann machine is a type of stochastic neural network invented by Geoffrey Hinton and Terry Sejnowski in 1985. Boltzmann machines can be seen as the stochastic, generative counterpart of Hopfield nets. They are named after the Boltzmann distribution in statistical mechanics. The units in Boltzmann machines are div... | Multimodal learning | 0.820614 |
9,127 | Generally speaking, as the number of compartments increases, it is challenging both to find the algebraic and numerical solutions to the model. However, there are special cases of models, which rarely exist in nature, when the topologies exhibit certain regularities that the solutions become easier to find. The model c... | Multi-compartment model | 0.820612 |
9,128 | The program can both simulate and fit models to data, returning optimal parameter estimates and associated statistics. It was developed by scientists working on metabolism and hormones kinetics (e.g., glucose, lipids, or insulin). It was then used for tracer studies and pharmacokinetics. Albeit a multi-compartment mode... | Multi-compartment model | 0.820612 |
9,129 | Simulation Analysis and Modeling 2 SAAM II is a software system designed specifically to aid in the development and testing of multi-compartment models. It has a user-friendly graphical user interface wherein compartmental models are constructed by creating a visual representation of the model. From this model, the pro... | Multi-compartment model | 0.820612 |
9,130 | Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936, and by others since then. ... But these meta-mathematical proofs cannot be represented within the arithmetical calculus; and, since they are not finitistic, the... | Hilbert's second problem | 0.820611 |
9,131 | Gödel's second incompleteness theorem shows that it is not possible for any proof that Peano Arithmetic is consistent to be carried out within Peano arithmetic itself. This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency... | Hilbert's second problem | 0.820611 |
9,132 | While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether (or in what way) these theorems answer Hilbert's second problem. Simpson (1988) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consisten... | Hilbert's second problem | 0.820611 |
9,133 | Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called finitistic because they are completely constructive and do not presuppose a completed infinity of natural numbers. Göde... | Hilbert's second problem | 0.820611 |
9,134 | In one English translation, Hilbert asks: "When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following a... | Hilbert's second problem | 0.820611 |
9,135 | In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in Hilbert (1900), which include a second or... | Hilbert's second problem | 0.820611 |
9,136 | In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex. There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also... | Expanded 6-simplex | 0.820605 |
9,137 | Logic Theorist is a computer program written in 1956 by Allen Newell, Herbert A. Simon, and Cliff Shaw. It was the first program deliberately engineered to perform automated reasoning, and has been described as "the first artificial intelligence program". Logic Theorist proved 38 of the first 52 theorems in chapter two... | Logic Theory Machine | 0.820599 |
9,138 | They called these ad hoc rules "heuristics", using a term introduced by George Pólya in his classic book on mathematical proof, How to Solve It. (Newell had taken courses from Pólya at Stanford). Heuristics would become an important area of research in artificial intelligence and remains an important method to overcome... | Logic Theory Machine | 0.820599 |
9,139 | Heavy metals – Variously-defined group of metals, on the base of their density, atomic number, or toxicity. Heavy atom – term used in computational chemistry to refer to any element other than hydrogen and helium. Minor actinides – Actinides found in significant quantities in nuclear fuel, other than U and Pu: Np, Am, ... | Names for sets of chemical elements | 0.820597 |
9,140 | Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey... | Discrete Mathematics (journal) | 0.820597 |
9,141 | "A proof of Andrews' q-Dyson conjecture". Discrete Mathematics. 54 (2): 201–224. doi:10.1016/0012-365X(85)90081-0. MR 0791661.) | Discrete Mathematics (journal) | 0.820597 |
9,142 | "Combinatorial aspects of continued fractions" (PDF). Discrete Mathematics. 32 (2): 125–161. | Discrete Mathematics (journal) | 0.820597 |
9,143 | "Tough graphs and Hamiltonian circuits". Discrete Mathematics. 5 (3): 215–228. | Discrete Mathematics (journal) | 0.820597 |
9,144 | The 1973 short note "Acyclic orientations of graphs" by Richard Stanley on the study of the chromatic polynomial and its generalizations (Stanley, R. P. (1973). "Acyclic orientations of graphs". Discrete Mathematics. | Discrete Mathematics (journal) | 0.820597 |
9,145 | Let q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time-averaged kinetic energy of the particle is: ⟨ q ⋅ F ⟩ = ⟨ q x d p x d t ⟩ + ⟨ q y d p y d t ⟩ + ⟨ q z d p z d t ⟩ = − ⟨ q x ∂ H... | Ideal gas equation | 0.820595 |
9,146 | Laws that govern our own Universe often seem contrary to common sense." This theme is described by Alan Cholodenko in his article, "The Nutty Universe of Animation".In a Garfield animated short entitled "Secrets of the Animated Cartoon", the characters Orson and Wade give demonstrations of different laws of the cartoon... | Cartoon physics | 0.820594 |
9,147 | Several aspects of cartoon physics were discussed in the film's dialogue, and the concept was a minor plot theme. In 1993, Stephen R. Gould, then a financial training consultant, writing in New Scientist, said that "... these seemingly nonsensical phenomena can be described by logical laws similar to those in our world... | Cartoon physics | 0.820594 |
9,148 | The idea that cartoons behave differently from the real world, but not randomly, is virtually as old as animation. Walt Disney, for example, spoke of the plausible impossible in 1956 on an episode of the Disneyland television program.Warner Brothers' Looney Tunes and Merrie Melodies series had numerous examples of thei... | Cartoon physics | 0.820594 |
9,149 | Cartoon physics or animation physics are terms for a jocular system of laws of physics (and biology) that supersedes the normal laws, used in animation for humorous effect. Many of the most famous American animated films, particularly those from Warner Bros. and Metro-Goldwyn-Mayer studios, indirectly developed a relat... | Cartoon physics | 0.820594 |
9,150 | The Ernest P. Worrell film series often made note of the title character's cartoon-like traits, with Ernest himself remarking in Ernest Rides Again that he would be dead "if I wasn't this close to being an actual cartoon." In a review of one of the Home Alone films, film critic Roger Ebert noted that in the case of liv... | Cartoon physics | 0.820594 |
9,151 | Cartoon physics is not limited to physics: For example, when a character recovers impossibly fast from a serious injury, the laws of biology rather than physics are being altered. It is also not limited to cartoons; in live-action, the physics-defying stunts would fall under the umbrella of slapstick. Live-action shows... | Cartoon physics | 0.820594 |
9,152 | Specific reference to cartoon physics extends back at least to June 1980, when an article "O'Donnell's Laws of Cartoon Motion" appeared in Esquire. A version printed in V.18 No. 7 p. 12, 1994 by the Institute of Electrical and Electronics Engineers in its journal helped spread the word among the technical crowd, which ... | Cartoon physics | 0.820594 |
9,153 | Kent Pitman's Theory of RelativeTV (Soap Opera Physics) | Cartoon physics | 0.820594 |
9,154 | In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms. Double groupoids are often used to capture information about geometrical objects... | Categorical Algebra | 0.820589 |
9,155 | A claim was then made that, with the gauge group SU(2), "the extended TQFT, or ETQFT, gives a theory equivalent to the Ponzano–Regge model of quantum gravity"; similarly, the Turaev–Viro model would be then obtained with representations of SUq(2). Therefore, one can describe the state space of a gauge theory – or many ... | Categorical Algebra | 0.820589 |
9,156 | In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. | Categorical Algebra | 0.820589 |
9,157 | A first step towards defining higher dimensional algebras is the concept of 2-category of higher category theory, followed by the more 'geometric' concept of double category. A higher level concept is thus defined as a category of categories, or super-category, which generalises to higher dimensions the notion of categ... | Categorical Algebra | 0.820589 |
9,158 | Socrates', Plato's and Aristotle's ideas about truth are seen by some as consistent with correspondence theory. In his Metaphysics, Aristotle stated: "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true". The Stanford E... | Truth theory | 0.820583 |
9,159 | This contradicts the principle of bivalence: every sentence must be either true or false. Since this principle is a key premise in deriving the liar paradox, the paradox is dissolved.However, it has been shown by Gödel that self-reference cannot be avoided naively, since propositions about seemingly unrelated objects c... | Truth theory | 0.820582 |
9,160 | The MOS awards prizes in the field of optimization, including the Fulkerson Prize, the Dantzig Prize and the Tucker Prize. | Mathematical Programming Computation | 0.820572 |
9,161 | The International Conference on Stochastic Programming (ICSP) takes place every three years and is devoted to optimization using uncertain input data. The Nordic MOS conference is a biannual meeting of researchers from Scandinavia working in all fields of optimization. At the Université de Montréal, annual seminars on ... | Mathematical Programming Computation | 0.820572 |
9,162 | There are several publications by the Mathematical Optimization Society: The journal Mathematical Programming (series A/B): series A publishes articles from all fields of optimization; each issue of series B is devoted to a particular subject. The journal Mathematical Programming Computation Optima, the newsletter of t... | Mathematical Programming Computation | 0.820571 |
9,163 | The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, is an international association of researchers active in optimization. The MOS encourages the research, development, and use of optimization—including mathematical theory, software implementation, and practical applic... | Mathematical Programming Computation | 0.820571 |
9,164 | Several conferences are organized or co-organized by the Mathematical Optimization Society, for instance: The International Symposium on Mathematical Programming (ISMP), organized every three years, is open to all fields of mathematical programming. The Integer Programming and Combinatorial Optimization (IPCO) conferen... | Mathematical Programming Computation | 0.820571 |
9,165 | For classical logics with negation, it is generally possible to re-express the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular φ is valid if and only if ¬φ is unsatisfiable, which is to sa... | Satisfiability (logics) | 0.820571 |
9,166 | For instance, consider the first-order logic formula obtained as the conjunction of the following sentences, where a 0 {\displaystyle a_{0}} and a 1 {\displaystyle a_{1}} are constants: R ( a 0 , a 1 ) {\displaystyle R(a_{0},a_{1})} ∀ x y ( R ( x , y ) → ∃ z R ( y , z ) ) {\displaystyle \forall xy(R(x,y)\rightarrow \ex... | Satisfiability (logics) | 0.820571 |
9,167 | Numerical constraints often appear in the field of mathematical optimization, where one usually wants to maximize (or minimize) an objective function subject to some constraints. However, leaving aside the objective function, the basic issue of simply deciding whether the constraints are satisfiable can be challenging ... | Satisfiability (logics) | 0.820571 |
9,168 | Example: for nonzero real numbers a and b, the subgroup of R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, meaning that a/b belongs to the rational numbers Q. In geometric group theory, a finitely generated group is viewed as a metric space u... | Commensurability (group theory) | 0.82056 |
9,169 | The abstract commensurator of a group G {\displaystyle G} , denoted Comm ( G ) {\displaystyle (G)} , is the group of equivalence classes of isomorphisms ϕ: H → K {\displaystyle \phi :H\to K} , where H {\displaystyle H} and K {\displaystyle K} are finite index subgroups of G {\displaystyle G} , under composition. Elemen... | Commensurability (group theory) | 0.82056 |
9,170 | Principles of Biology is a college level biology electronic textbook published by Nature Publishing in 2011. The book is not a digitally reformatted version of a paper book. The book, the first in a projected series, is Nature Publishing's first foray into textbook publishing. | Principles of Biology | 0.820552 |
9,171 | As there are now not only 4f but also 5d and 6s subshells at similar energies, competition occurs once again with many irregular configurations; this resulted in some dispute about where exactly the f-block is supposed to begin, but most who study the matter agree that it starts at lanthanum in accordance with the Aufb... | Table of elements | 0.820544 |
9,172 | The periodic table therefore ignores these and considers only idealised configurations.At zinc ( 3d10 4s2), the 3d orbitals are completely filled with a total of ten electrons. Next come the 4p orbitals, completing the row, which are filled progressively by gallium ( 3d10 4s2 4p1) through krypton ( 3d10 4s2 4p6), in a ... | Table of elements | 0.820544 |
9,173 | On chemical grounds Bassett, Werner, and Bury grouped scandium and yttrium with lutetium rather than lanthanum (the former two left an empty space below yttrium as lutetium had not yet been discovered). Hund assumed in 1927 that all the lanthanide atoms had configuration 4f0−145d16s2, on account of their prevailing tri... | Table of elements | 0.820544 |
9,174 | More recently, usage of the term has been extended to the study of human social phenomena. Thus, natural and social sciences are commonly classified as science, whereas the study of classics, languages, literature, music, philosophy, history, religion, and the visual and performing arts are referred to as the humanitie... | Human science | 0.820544 |
9,175 | As a result of the positivist influence, the term science is frequently employed as a synonym for empirical science. Empirical science is knowledge based on the scientific method, a systematic approach to verification of knowledge first developed for dealing with natural physical phenomena and emphasizing the importanc... | Human science | 0.820544 |
9,176 | Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms m: G × G → G (thought of as the "group multiplication") e: 1 → G (thought of as the "inclusion of the identity element") inv... | Group object | 0.820543 |
9,177 | See also Eckmann–Hilton argument. The strict 2-group is the group object in the category of small categories. Given a category C with finite coproducts, a cogroup object is an object G of C together with a "comultiplication" m: G → G ⊕ {\displaystyle \oplus } G, a "coidentity" e: G → 0, and a "coinversion" inv: G → G t... | Group object | 0.820543 |
9,178 | In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. A localic group is a group object in the category of locales. The group objects in the category of groups (or monoids) are the abelian groups. | Group object | 0.820543 |
9,179 | A Lie group is a group object in the category of smooth manifolds with smooth maps. A Lie supergroup is a group object in the category of supermanifolds. An algebraic group is a group object in the category of algebraic varieties. | Group object | 0.820543 |
9,180 | Much of group theory can be formulated in the context of the more general group objects. The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, n... | Group object | 0.820543 |
9,181 | His aim was to train general science graduates who would be scientifically literate, numerate and easily able to communicate across a wide range of disciplines, replacing the traditional Classics training for higher-level government and management careers. Central topics include the evolution of humans, their behaviour... | Human science | 0.820543 |
9,182 | Defining equations do not find out any information about a physical system, they simply re-state one measurement in terms of others. Results, theorems, and laws, on the other hand do provide meaningful information, if only a little, since they represent a calculation for a quantity given other properties of the system,... | Defining equation (physics) | 0.820542 |
9,183 | This is the dispersion relation for wave propagation. The coefficient of restitution for an object colliding is defined using the speeds of separation and approach with respect to the collision point, but depends on the nature of the surfaces in question. Definitions vs. theorems: There is a very important difference b... | Defining equation (physics) | 0.820542 |
9,184 | Vector algebra There is no way to divide a vector by a vector, so there are no product or quotient forms. Elementary calculus The arithmetic operations are modified to the limiting cases of differentiation and integration. Equations can be expressed in these equivalent and alternative ways.Vector calculus Tensor analys... | Defining equation (physics) | 0.820542 |
9,185 | See the classical mechanics section below for nomenclature and diagrams to the right. Elementary algebra Operations are simply multiplication and division. Equations may be written in a product or quotient form, both of course equivalent. | Defining equation (physics) | 0.820542 |
9,186 | For vector equations, sometimes the defining quantity is in a cross or dot product and cannot be solved for explicitly as a vector, but the components can. ExamplesElectric current density is an example spanning all of these methods, Angular momentum is an example which doesn't require calculus. | Defining equation (physics) | 0.820542 |
9,187 | Often definitions can start from elementary algebra, then modify to vectors, then in the limiting cases calculus may be used. The various levels of maths used typically follows this pattern. Typically definitions are explicit, meaning the defining quantity is the subject of the equation, but sometimes the equation is n... | Defining equation (physics) | 0.820542 |
9,188 | Defining equations are normally formulated in terms of elementary algebra and calculus, vector algebra and calculus, or for the most general applications tensor algebra and calculus, depending on the level of study and presentation, complexity of topic and scope of applicability. Functions may be incorporated into a de... | Defining equation (physics) | 0.820542 |
9,189 | {\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} .\,\!} However, only one of the above equations can be used to define B for the following reason, given that A, r, v, and F have been defined elsewhere unambiguously (most likely mechanics and Euclidean geometry). If the force equation defines ... | Defining equation (physics) | 0.820542 |
9,190 | Much of physics requires definitions to be made for the equations to make sense. Theoretical implications: Definitions are important since they can lead into new insights of a branch of physics. Two such examples occurred in classical physics. When entropy S was defined – the range of thermodynamics was greatly extende... | Defining equation (physics) | 0.820542 |
9,191 | In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units. | Defining equation (physics) | 0.820542 |
9,192 | In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 (Isaacs 1994). In this section only finite groups are considered. A monomial group is ... | Monomial group | 0.820538 |
9,193 | While all these advancements were being made, the community felt the need to have standardised datasets and evaluation metrics so the performances can be compared. This led to the emergence of challenges like the Pascal VOC challenge and the ImageNet challenge. The availability of standard evaluation metrics and the op... | Visual Turing Test | 0.820537 |
9,194 | The late 1990s and early 2000s saw the birth of modern computer vision. One of the reasons this happened was due to the availability of key, feature extraction and representation algorithms. Features along with the already present machine learning algorithms were used to detect, localise and segment objects in Images. | Visual Turing Test | 0.820537 |
9,195 | During this time the neural networks also resurfaced as it was shown that the limitations of the perceptrons can be overcome by Multi-layer perceptrons. Also in the early 1990s convolutional neural networks were born which showed great results on digit recognition but did not scale up well on harder problems. | Visual Turing Test | 0.820537 |
9,196 | These simple neural networks could not live up to their expectations and had certain limitations due to which they were not considered in future research. Later with the availability of the hardware and some processing power the research shifted to image processing which involves pixel-level operations, like finding ed... | Visual Turing Test | 0.820537 |
9,197 | Roughly 50% of the human brain is devoted in processing vision, which indicates that it is a difficult problem. Later there were attempts to solve the problems with models inspired by the human brain. Perceptrons by Frank Rosenblatt, which is a form of the neural networks, was one of the first such approaches. | Visual Turing Test | 0.820537 |
9,198 | Research in computer vision dates back to the 1960s when Seymour Papert first attempted to solve the problem. This unsuccessful attempt was referred to as the Summer Vision Project. The reason why it was not successful was because computer vision is more complicated than what people think. The complexity is in alignmen... | Visual Turing Test | 0.820537 |
9,199 | Companies like Google and Facebook are investing millions of dollars into computer vision research, and are trying to build systems that closely resemble the human visual system. Recently Facebook announced its new platform M, which looks at an image and provides a description of it to help the visually impaired. Such ... | Visual Turing Test | 0.820537 |
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