id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
15,400 | A symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n, one has P(Xσ(1), Xσ(2), ..., Xσ(n)) = P(X1, X2, ..., Xn). Symmetri... | Symmetry in mathematics | 0.814446 |
15,401 | In quantum mechanics, bosons have representatives that are symmetric under permutation operators, and fermions have antisymmetric representatives. This implies the Pauli exclusion principle for fermions. In fact, the Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring th... | Symmetry in mathematics | 0.814446 |
15,402 | In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\dots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots ,v_{\sigma r})} for every permutation σ of the symbols {1,2,...,r}. Alterna... | Symmetry in mathematics | 0.814446 |
15,403 | {\displaystyle \left(VD^{\frac {1}{2}}V^{-1}\right)^{2}=VD^{\frac {1}{2}}\left(V^{-1}V\right)D^{\frac {1}{2}}V^{-1}=VDV^{-1}=A~.} For example, the matrix A = ( 33 24 48 57 ) {\displaystyle A=\left({\begin{smallmatrix}33&24\\48&57\end{smallmatrix}}\right)} can be diagonalized as VDV−1, where V = ( 1 1 2 − 1 ) {\displays... | Square root of a matrix | 0.814435 |
15,404 | The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of A is the same as the rank of A.The operation of taking the principal square root is continuous on this set of matrices. These properties are consequences of the holomorphic functional c... | Square root of a matrix | 0.814435 |
15,405 | Square roots, and the unitary freedom of square roots, have applications throughout functional analysis and linear algebra. | Square root of a matrix | 0.814435 |
15,406 | This method of defining or computing A 1 2 {\displaystyle A^{\frac {1}{2}}} is especially useful in the case where A {\displaystyle A} is positive semi-definite. In that case, we have ‖ I − A ‖ A ‖ ‖ ≤ 1 {\textstyle \left\|I-{\frac {A}{\|A\|}}\right\|\leq 1} and therefore ‖ ( I − A ‖ A ‖ ) n ‖ ≤ ‖ I − A ‖ A ‖ ‖ n ≤ 1 {... | Square root of a matrix | 0.814435 |
15,407 | The Euler characteristic of the n-sphere Sn is: χ ( S n ) = 1 + ( − 1 ) n = { 2 n even 0 n odd . {\displaystyle \chi (\mathbf {S} ^{n})=1+(-1)^{n}={\begin{cases}2&n{\text{ even}}\\0&n{\text{ odd}}.\end{cases}}} Thus, there is no non-vanishing section of the tangent bundle of even spheres (this is known as the Hairy bal... | Euler class | 0.814431 |
15,408 | In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characte... | Euler class | 0.814431 |
15,409 | In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintai... | Mountain climbing problem | 0.814428 |
15,410 | program in computational biology with the University of Pittsburgh, which was founded as the Joint CMU-Pitt Ph.D. Program in Computational Biology in 2005. This program is currently receiving training support through a National Institutes of Health T32 Training Grant. | Carnegie Mellon University Computational Biology Department | 0.814427 |
15,411 | RECOMB 2014 Great Lakes Bioinformatics Conference 2013 (joint with University of Pittsburgh) Automated Personal Genome Analysis (APGA) 2013 Bioimage Informatics 2010 | Carnegie Mellon University Computational Biology Department | 0.814427 |
15,412 | CBD is the home of the B.S. in Computational Biology, one of the four B.S. degree programs within Carnegie Mellon School of Computer Science. The Computational Biology undergraduate program is ranked as one of the top 3 programs by US News.CBD is the home of an NIH Center for the HuBMAP Integration, Visualization & Eng... | Carnegie Mellon University Computational Biology Department | 0.814427 |
15,413 | The Computational Biology Department (CBD) is a division within the School of Computer Science at Carnegie Mellon University in Pittsburgh, Pennsylvania, United States. It is located in the Gates-Hillman Center. Established in 2007 by Robert F. Murphy as the Lane Center for Computational Biology with funding from Raymo... | Carnegie Mellon University Computational Biology Department | 0.814427 |
15,414 | Joint CMU-Pitt Ph.D. Program in Computational Biology (with University of Pittsburgh) M.S. in Computational Biology (joint with the Department of Biological Sciences) M.S. in Biotechnology Innovation and Computation (joint with Language Technologies Institute) B.S. in Computational Biology Minor in Computational Biolog... | Carnegie Mellon University Computational Biology Department | 0.814427 |
15,415 | In theoretical physics, light cone gauge is an approach to remove the ambiguities arising from a gauge symmetry. While the term refers to several situations, a null component of a field A is set to zero (or a simple function of other variables) in all cases.The advantage of light-cone gauge is that fields, e.g. gluons ... | Light cone gauge | 0.814426 |
15,416 | In software engineering, coupling is the degree of interdependence between software modules; a measure of how closely connected two routines or modules are; the strength of the relationships between modules. Coupling is usually contrasted with cohesion. Low coupling often correlates with high cohesion, and vice versa. ... | Control coupling | 0.814425 |
15,417 | Further, explaining the similarity in IQ of closely related persons requires careful study because environmental factors may be correlated with genetic factors. Early twin studies of adult individuals have found a heritability of IQ between 57% and 73%, with some recent studies showing heritability for IQ as high as 80... | Heritability of IQ | 0.814421 |
15,418 | The brain undergoes morphological changes in development which suggests that age-related physical changes could contribute to this effect.A 1994 article in Behavior Genetics based on a study of Swedish monozygotic and dizygotic twins found the heritability of the sample to be as high as 0.80 in general cognitive abilit... | Heritability of IQ | 0.814421 |
15,419 | Vocabulary size, for example, is very substantially heritable (and highly correlated with general intelligence) although every word in an individual's vocabulary is learned. In a society in which plenty of words are available in everyone's environment, especially for individuals who are motivated to seek them out, the ... | Heritability of IQ | 0.814421 |
15,420 | The heritability of IQ increases with the child's age and reaches a plateau at 14-16 years old, continuing at that level well into adulthood. However, poor prenatal environment, malnutrition and disease are known to have lifelong deleterious effects.Although IQ differences between individuals have been shown to have a ... | Heritability of IQ | 0.814421 |
15,421 | Johnson, Wendy (10 June 2010). "Understanding the Genetics of Intelligence". Current Directions in Psychological Science. | Heritability of IQ | 0.814421 |
15,422 | In the same period, the educational achievement disparity also diminished. Reviews by Flynn and Dickens, Mackintosh, and Nisbett et al. all accept the gradual closing of the gap as a fact. Other recent studies have focused on disparities in nutrition and prenatal care, as well as other health-related environmental disp... | Heritability of IQ | 0.814421 |
15,423 | In the US, individuals identifying themselves as Asian generally tend to score higher on IQ tests than Caucasians, who tend to score higher than Hispanics, who tend to score higher than African Americans –– despite the fact that greater variation in IQ scores exists within each ethnic group than between them. Yet, alth... | Heritability of IQ | 0.814421 |
15,424 | Organoiron chemistry is the chemistry of iron compounds containing a carbon-to-iron chemical bond. Organoiron compounds are relevant in organic synthesis as reagents such as iron pentacarbonyl, diiron nonacarbonyl and disodium tetracarbonylferrate. While iron adopts oxidation states from Fe(−II) through to Fe(VII), Fe(... | Organoiron compound | 0.814419 |
15,425 | The rapid growth of organometallic chemistry in the 20th century can be traced to the discovery of ferrocene, a very stable compound which foreshadowed the synthesis of many related sandwich compounds. Ferrocene is formed by reaction of sodium cyclopentadienide with iron(II) chloride: 2 NaC5H5 + FeCl2 → Fe(C5H5)2 + 2 N... | Organoiron compound | 0.814419 |
15,426 | In the area of bioorganometallic chemistry, organoiron species are found at the active sites of the three hydrogenase enzymes as well as carbon monoxide dehydrogenase. | Organoiron compound | 0.814419 |
15,427 | The study of pollination spans many disciplines, such as botany, horticulture, entomology, and ecology. The pollination process as an interaction between flower and pollen vector was first addressed in the 18th century by Christian Konrad Sprengel. It is important in horticulture and agriculture, because fruiting is de... | Cross pollination | 0.814409 |
15,428 | The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete spectral lines seen in the emission spectrum and absorption spectrum of isolated atoms of a chemical element, which only absorb and emit light at particular wavelengths. The technique of spectroscopy i... | Discrete spectrum (physics) | 0.814408 |
15,429 | The time delay neural network (TDNN) was introduced in 1987 by Alex Waibel et al. and was one of the first convolutional networks, as it achieved shift invariance. It did so by utilizing weight sharing in combination with backpropagation training. Thus, while also using a pyramidal structure as in the neocognitron, it ... | Convolutional layer | 0.814407 |
15,430 | The feed-forward architecture of convolutional neural networks was extended in the neural abstraction pyramid by lateral and feedback connections. The resulting recurrent convolutional network allows for the flexible incorporation of contextual information to iteratively resolve local ambiguities. In contrast to previo... | Convolutional layer | 0.814407 |
15,431 | Fully connected layers connect every neuron in one layer to every neuron in another layer. It is the same as a traditional multilayer perceptron neural network (MLP). The flattened matrix goes through a fully connected layer to classify the images. | Convolutional layer | 0.814407 |
15,432 | Convolutional deep belief networks (CDBN) have structure very similar to convolutional neural networks and are trained similarly to deep belief networks. Therefore, they exploit the 2D structure of images, like CNNs do, and make use of pre-training like deep belief networks. They provide a generic structure that can be... | Convolutional layer | 0.814407 |
15,433 | The method also significantly improves training speed. This makes the model combination practical, even for deep neural networks. The technique seems to reduce node interactions, leading them to learn more robust features that better generalize to new data. | Convolutional layer | 0.814407 |
15,434 | A convolutional neural network consists of an input layer, hidden layers and an output layer. In a convolutional neural network, the hidden layers include one or more layers that perform convolutions. Typically this includes a layer that performs a dot product of the convolution kernel with the layer's input matrix. | Convolutional layer | 0.814407 |
15,435 | In 2011, Xavier Glorot, Antoine Bordes and Yoshua Bengio found that ReLU enables better training of deeper networks, compared to widely used activation functions prior to 2011. Other functions can also be used to increase nonlinearity, for example the saturating hyperbolic tangent f ( x ) = tanh ( x ) {\displaystyle ... | Convolutional layer | 0.814407 |
15,436 | LeNet-5, a pioneering 7-level convolutional network by LeCun et al. in 1995, that classifies digits, was applied by several banks to recognize hand-written numbers on checks (British English: cheques) digitized in 32x32 pixel images. The ability to process higher-resolution images requires larger and more layers of con... | Convolutional layer | 0.814407 |
15,437 | This ignores locality of reference in data with a grid-topology (such as images), both computationally and semantically. Thus, full connectivity of neurons is wasteful for purposes such as image recognition that are dominated by spatially local input patterns. Convolutional neural networks are variants of multilayer pe... | Convolutional layer | 0.814407 |
15,438 | For example, in CIFAR-10, images are only of size 32×32×3 (32 wide, 32 high, 3 color channels), so a single fully connected neuron in the first hidden layer of a regular neural network would have 32*32*3 = 3,072 weights. A 200×200 image, however, would lead to neurons that have 200*200*3 = 120,000 weights. Also, such n... | Convolutional layer | 0.814407 |
15,439 | In particular it is not expected to be a correctness assertion for a given program implementing the algorithm on a given machine. That would involve such considerations as limitations on computer memory. A deep result in proof theory, the Curry–Howard correspondence, states that a proof of functional correctness in con... | Correctness (computer science) | 0.814406 |
15,440 | For example, successively searching through integers 1, 2, 3, … to see if we can find an example of some phenomenon—say an odd perfect number—it is quite easy to write a partially correct program (see box). But to say this program is totally correct would be to assert something currently not known in number theory. A p... | Correctness (computer science) | 0.814406 |
15,441 | In theoretical computer science, an algorithm is correct with respect to a specification if it behaves as specified. Best explored is functional correctness, which refers to the input-output behavior of the algorithm (i.e., for each input it produces an output satisfying the specification).Within the latter notion, par... | Correctness (computer science) | 0.814406 |
15,442 | Its speed in vacuum, 299792458 m/s, is one of the fundamental constants of nature. Like all types of electromagnetic radiation, visible light propagates by massless elementary particles called photons that represents the quanta of electromagnetic field, and can be analyzed as both waves and particles. The study of ligh... | Electromagnetic theory of light | 0.814402 |
15,443 | Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahertz, between the infrared (with longer wavelengths) and the ultraviolet (with shorter ... | Electromagnetic theory of light | 0.814402 |
15,444 | The proposal's vision of the computing future were then unusual or non-existent, but elements of it are now commonplace, such as visualization, workstations, high-speed I/O, data storage, software engineering, and close collaboration with the multi-disciplinary user community. Modern readers of the Black Proposal may g... | National Center for Supercomputing Applications | 0.814401 |
15,445 | Initially, NCSA's administrative offices were in the Water Resources Building and employees were scattered across the campus. NCSA is now headquartered within its own building directly north of the Siebel Center for Computer Science, on the site of a former baseball field, Illini Field. NCSA's supercomputers are at the... | National Center for Supercomputing Applications | 0.814401 |
15,446 | The resolution rule is a single rule of inference that, together with unification, is sound and complete for first-order logic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisfiable. Resolution is commonly used in automated theorem proving. The resolution method w... | Classical predicate logic | 0.814401 |
15,447 | For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum. Since the real line is infinite, any theory satisfied by the real line is also satisfied by some nonstandard models. When the Löwenheim–... | Classical predicate logic | 0.814401 |
15,448 | The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-o... | Classical predicate logic | 0.814401 |
15,449 | In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming. The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constrai... | Revised simplex algorithm | 0.814399 |
15,450 | Corresponding to the constrained optimization problem one can form the Lagrangian function where The Karush–Kuhn–Tucker theorem then states the following. Since the idea of this approach is to find a supporting hyperplane on the feasible set Γ = { x ∈ X: g i ( x ) ≤ 0 , i = 1 , … , m } {\displaystyle \mathbf {\Gamma } ... | KKT conditions | 0.814399 |
15,451 | In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequali... | KKT conditions | 0.814399 |
15,452 | Consider the following nonlinear optimization problem in standard form: minimize f ( x ) {\displaystyle f(\mathbf {x} )} subject to g i ( x ) ≤ 0 , {\displaystyle g_{i}(\mathbf {x} )\leq 0,} h j ( x ) = 0. {\displaystyle h_{j}(\mathbf {x} )=0.} where x ∈ X {\displaystyle \mathbf {x} \in \mathbf {X} } is the optimizatio... | KKT conditions | 0.814399 |
15,453 | In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A. | Cellular algebra | 0.814397 |
15,454 | The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer. However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as coherent algebras. | Cellular algebra | 0.814397 |
15,455 | i {\displaystyle i} . Tits' deformation theorem for cellular algebras: Let A {\displaystyle A} be a cellular R {\displaystyle R} -algebra. Also let R → k {\displaystyle R\to k} be a unitary homomorphism into a field k {\displaystyle k} and K := Quot ( R ) {\displaystyle K:=\operatorname {Quot} (R)} the quotient field... | Cellular algebra | 0.814397 |
15,456 | Direct products of finitely many cellular R {\displaystyle R} -algebras are cellular.If R {\displaystyle R} is an integral domain then there is a converse to this last point: If ( A , i ) {\displaystyle (A,i)} is a finite-dimensional R {\displaystyle R} -algebra with an involution and A = A 1 ⊕ A 2 {\displaystyle A=A_{... | Cellular algebra | 0.814397 |
15,457 | Tensor products of finitely many cellular R {\displaystyle R} -algebras are cellular. A R {\displaystyle R} -algebra A {\displaystyle A} is cellular if and only if its opposite algebra A op {\displaystyle A^{\text{op}}} is. If A {\displaystyle A} is cellular with cell-datum ( Λ , i , M , C ) {\displaystyle (\Lambda ,i,... | Cellular algebra | 0.814397 |
15,458 | i {\displaystyle i} is defined as a direct decomposition A = ⨁ k = 1 m U k {\displaystyle A=\bigoplus _{k=1}^{m}U_{k}} into free R {\displaystyle R} -submodules such that i ( U k ) = U k {\displaystyle i(U_{k})=U_{k}} J k := ⨁ j = 1 k U j {\displaystyle J_{k}:=\bigoplus _{j=1}^{k}U_{j}} is a two-sided ideal of A {\disp... | Cellular algebra | 0.814397 |
15,459 | Let R {\displaystyle R} be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also A {\displaystyle A} be an R {\displaystyle R} -algebra. | Cellular algebra | 0.814397 |
15,460 | Assume for the rest of this section that the ring R {\displaystyle R} is a field. With the information contained in the invariant bilinear forms one can easily list all simple A {\displaystyle A} -modules: Let Λ 0 := { λ ∈ Λ ∣ ϕ λ ≠ 0 } {\displaystyle \Lambda _{0}:=\lbrace \lambda \in \Lambda \mid \phi _{\lambda }\neq ... | Cellular algebra | 0.814397 |
15,461 | R d × d {\displaystyle R^{\,d\times d}} is cellular. A cell datum is given by i ( A ) = A T {\displaystyle i(A)=A^{T}} and Λ := { 1 } {\displaystyle \Lambda :=\lbrace 1\rbrace } M ( 1 ) := { 1 , … , d } {\displaystyle M(1):=\lbrace 1,\dots ,d\rbrace } For the basis one chooses C s t 1 := E s t {\displaystyle C_{st}^{1}... | Cellular algebra | 0.814397 |
15,462 | Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as T w ↦ T w − 1 {\displaystyle T_{w}\mapsto T_{w^{-1}}} . This includes for example the integral group algebra of the symmetric groups as well as all other finite Weyl groups. A basi... | Cellular algebra | 0.814397 |
15,463 | det ( C A ) = 1 {\displaystyle \det(C_{A})=1} .If A {\displaystyle A} is Morita equivalent to B {\displaystyle B} and the characteristic of R {\displaystyle R} is not two, then B {\displaystyle B} is also cellular w.r.t. a suitable involution. In particular A {\displaystyle A} is cellular (to some involution) if and on... | Cellular algebra | 0.814397 |
15,464 | Let i: A → A {\displaystyle i:A\to A} be an anti-automorphism of R {\displaystyle R} -algebras with i 2 = id {\displaystyle i^{2}=\operatorname {id} } (just called "involution" from now on). A cell ideal of A {\displaystyle A} w.r.t. i {\displaystyle i} is a two-sided ideal J ⊆ A {\displaystyle J\subseteq A} such that ... | Cellular algebra | 0.814397 |
15,465 | A theory of statistical inference was developed by Charles S. Peirce in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883), two publications that emphasized the importance of randomization-based inference in statistics. | Design of Experiments | 0.814397 |
15,466 | As with other branches of statistics, experimental design is pursued using both frequentist and Bayesian approaches: In evaluating statistical procedures like experimental designs, frequentist statistics studies the sampling distribution while Bayesian statistics updates a probability distribution on the parameter spac... | Design of Experiments | 0.814397 |
15,467 | In 1950, Gertrude Mary Cox and William Gemmell Cochran published the book Experimental Designs, which became the major reference work on the design of experiments for statisticians for years afterwards. Developments of the theory of linear models have encompassed and surpassed the cases that concerned early writers. To... | Design of Experiments | 0.814397 |
15,468 | Certain phenomena in quantum mechanics, such as quantum entanglement, might give the superficial impression of allowing communication of information faster than light. According to the no-communication theorem these phenomena do not allow true communication; they only let two observers in different locations see the sa... | Causality problem | 0.814397 |
15,469 | Since the underlying behavior does not violate local causality or allow FTL communication, it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. The uncertainty principle implies that individual photons may travel for short distances at speeds somewhat faster (or slower)... | Causality problem | 0.814397 |
15,470 | Nimtz has also claimed that "evanescent modes are not fully describable by the Maxwell equations and quantum mechanics have to be taken into consideration." Other scientists such as Herbert G. Winful and Robert Helling have argued that in fact there is nothing quantum-mechanical about Nimtz's experiments, and that the ... | Causality problem | 0.814397 |
15,471 | The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's valence electron shell and to complexes of ions like Cu2+ which have a single "vacancy" in the valence shell.In the specif... | Double group | 0.814397 |
15,472 | In magnetochemistry, the need for a double group arises in a very particular circumstance, namely, in the treatment of the paramagnetism of complexes of a metal ion in whose electronic structure there is a single electron (or its equivalent, a single vacancy) in a metal ion's d- or f- shell. This occurs, for example, w... | Double group | 0.814397 |
15,473 | If B {\displaystyle B} is not empty and if Y {\displaystyle Y} is a 1-dimensional vector subspace of span B {\displaystyle \operatorname {span} B} then B ∩ Y {\displaystyle B\cap Y} is either { 0 } {\displaystyle \{0\}} or else it is absorbing in Y . {\displaystyle Y.} for any x ∈ X , {\displaystyle x\in X,} if B ∩ s... | Balanced set | 0.814396 |
15,474 | This theory can be generalized from a theory with U ( 1 ) {\displaystyle U(1)} gauge symmetry containing a scalar field ϕ {\displaystyle \phi } valued in C {\displaystyle \mathbb {C} } coupled to a gauge field A μ {\displaystyle A_{\mu }} to a theory with gauge symmetry under the gauge group G {\displaystyle G} , a Lie... | Scalar electrodynamics | 0.814385 |
15,475 | From the geometric viewpoint, λ {\displaystyle \lambda } is an infinitesimal change of trivialization, which generates the finite change of trivialization e i e λ: R 1 , 3 → U ( 1 ) . {\displaystyle e^{ie\lambda }:\mathbb {R} ^{1,3}\rightarrow {\text{U}}(1).} In physics, it is customary to work under an implicit choice... | Scalar electrodynamics | 0.814385 |
15,476 | In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via ... | Scalar electrodynamics | 0.814385 |
15,477 | Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem. Algorithms, such as Montgomery reductio... | Modular arithmatic | 0.814383 |
15,478 | For her Intel Science Fair project Flannery prepared a demonstration where the same plaintext was enciphered using both RSA and her new Cayley–Purser algorithm and it did indeed show a significant time improvement. Returning to the ESAT Young Scientist and Technology Exhibition in 1999, Flannery formalised Cayley-Purse... | Cayley–Purser algorithm | 0.814382 |
15,479 | The Cayley–Purser algorithm was a public-key cryptography algorithm published in early 1999 by 16-year-old Irishwoman Sarah Flannery, based on an unpublished work by Michael Purser, founder of Baltimore Technologies, a Dublin data security company. Flannery named it for mathematician Arthur Cayley. It has since been fo... | Cayley–Purser algorithm | 0.814382 |
15,480 | In statistics, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results are observed. Thus a ... | Sequential analysis | 0.814382 |
15,481 | (Koch and Silver, 2005) "The evolution of modern cells is arguably the most challenging and important problem the field of Biology has ever faced. In Darwin's day the problem could hardly be imagined. For much of the 20th century it was intractable. | Cellular evolution | 0.814379 |
15,482 | If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the t... | Group ring | 0.814377 |
15,483 | In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its mu... | Group ring | 0.814377 |
15,484 | Assembling these mappings gives an algebra isomorphism ρ ~: C → ⨁ k = 1 m E n d ( V k ) ≅ ⨁ k = 1 m M d k ( C ) , {\displaystyle {\tilde {\rho }}:\mathbb {C} \to \bigoplus _{k=1}^{m}\mathrm {End} (V_{k})\cong \bigoplus _{k=1}^{m}M_{d_{k}}(\mathbb {C} ),} where dk is the dimension of Vk. The subalgebra of C correspondi... | Group ring | 0.814377 |
15,485 | . . , m, these correspond to group homomorphisms ρ k: G → A u t ( V k ) {\displaystyle \rho _{k}:G\to \mathrm {Aut} (V_{k})} and hence to algebra homomorphisms ρ ~ k: C → E n d ( V k ) {\displaystyle {\tilde {\rho }}_{k}:\mathbb {C} \to \mathrm {End} (V_{k})} . | Group ring | 0.814377 |
15,486 | The dimension of the vector space K is just equal to the number of elements in the group. The field K is commonly taken to be the complex numbers C or the reals R, so that one discusses the group algebras C or R. The group algebra C of a finite group over the complex numbers is a semisimple ring. This result, Maschke's... | Group ring | 0.814377 |
15,487 | Thinking of the free vector space as K-valued functions on G, the algebra multiplication is convolution of functions. While the group algebra of a finite group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of finite sums, corresponds... | Group ring | 0.814377 |
15,488 | The group algebra is an algebra over itself; under the correspondence of representations over R and R modules, it is the regular representation of the group. Written as a representation, it is the representation g ↦ ρg with the action given by ρ ( g ) ⋅ e h = e g h {\displaystyle \rho (g)\cdot e_{h}=e_{gh}} , or ρ ( g ... | Group ring | 0.814377 |
15,489 | The group algebra generalizes to the monoid ring and thence to the category algebra, of which another example is the incidence algebra. | Group ring | 0.814377 |
15,490 | The group algebra K has a natural structure of a Hopf algebra. The comultiplication is defined by Δ ( g ) = g ⊗ g {\displaystyle \Delta (g)=g\otimes g} , extended linearly, and the antipode is S ( g ) = g − 1 {\displaystyle S(g)=g^{-1}} , again extended linearly. | Group ring | 0.814377 |
15,491 | Taking K to be an abstract algebra, one may ask for representations of the algebra acting on a K-vector space V of dimension d. Such a representation ρ ~: K → End ( V ) {\displaystyle {\tilde {\rho }}:K\rightarrow {\mbox{End}}(V)} is an algebra homomorphism from the group algebra to the algebra of endomorphisms of V, ... | Group ring | 0.814377 |
15,492 | The Journal of Computational Biology is a monthly peer-reviewed scientific journal covering computational biology and bioinformatics. It was established in 1994 and is published by Mary Ann Liebert The editors-in-chief are Sorin Istrail (Brown University) and Michael S. Waterman (University of Southern California). Acc... | Journal of Computational Biology | 0.814377 |
15,493 | 57, No 3, 171–182 (with discussion). For a concise modern statement see Chapter 10 of "All of Statistics: A Concise Course in Statistical Inference," Springer; 1st Corrected ed. 20 edition (September 17, 2004). Larry Wasserman. | P value | 0.814376 |
15,494 | For the important case in which the data are hypothesized to be a random sample from a normal distribution, depending on the nature of the test statistic and the hypotheses of interest about its distribution, different null hypothesis tests have been developed. Some such tests are the z-test for hypotheses concerning t... | P value | 0.814376 |
15,495 | According to the ASA, there is widespread agreement that p-values are often misused and misinterpreted. One practice that has been particularly criticized is accepting the alternative hypothesis for any p-value nominally less than .05 without other supporting evidence. Although p-values are helpful in assessing how inc... | P value | 0.814376 |
15,496 | The same question was later addressed by Pierre-Simon Laplace, who instead used a parametric test, modeling the number of male births with a binomial distribution: In the 1770s Laplace considered the statistics of almost half a million births. The statistics showed an excess of boys compared to girls. | P value | 0.814376 |
15,497 | In statistics, every conjecture concerning the unknown probability distribution of a collection of random variables representing the observed data X {\displaystyle X} in some study is called a statistical hypothesis. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis i... | P value | 0.814376 |
15,498 | The use of the p-value in statistics was popularized by Ronald Fisher, and it plays a central role in his approach to the subject. In his influential book Statistical Methods for Research Workers (1925), Fisher proposed the level p = 0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical sign... | P value | 0.814376 |
15,499 | 161 in "The Emergence of Mathematical Statistics: A Historical Sketch with Particular Reference to the United States," "On the History of Statistics and Probability," ed. D.B. | P value | 0.814376 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.