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32,900 | A fundamental question in the study of SDR is whether or not an SDR exists. Hall's marriage theorem gives necessary and sufficient conditions for a finite collection of sets, some possibly overlapping, to have a transversal. The condition is that, for every integer k, every collection of k sets must contain in common a... | Transversal (group theory) | 0.803681 |
32,901 | Recombinant inbred strains are now widely used in systems genetics and to study gene–environment interactions. It is possible to accumulate extensive genetic and phenotype data for each member of a family of recombinant inbred strains under several different conditions (e.g., baseline environment versus stressful envir... | Recombinant inbred strain | 0.80368 |
32,902 | (2014). "On Klein's So-called Non-Euclidean geometry". In Ji, L.; Papadopoulos, A. | Cayley metric | 0.80368 |
32,903 | doi:10.1007/s10998-008-8197-5. S2CID 31045705. Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley–Klein approach", Journal of Geometry, 89 (1): 151–170, doi:10.1007/s00022-010-0053-z, ISSN 0047-2468, MR 2739193, S2CID 123015988 A’Campo, N.; Papadopoulos, A. | Cayley metric | 0.80368 |
32,904 | doi:10.1090/S0002-9904-1930-04885-5. Littlewood, J. E. (1986) , Littlewood's miscellany, Cambridge University Press, ISBN 978-0-521-33058-9, MR 0872858 Harvey Lipkin (1985) Metrical Geometry from Georgia Institute of Technology Struve, Horst; Struve, Rolf (2004), "Projective spaces with Cayley–Klein metrics", Journal o... | Cayley metric | 0.80368 |
32,905 | Projective geometry. Boston: Ginn. Liebmann, H. | Cayley metric | 0.80368 |
32,906 | Leipzig: Teubner. Bertrand Russell (1898) An Essay on the Foundations of Geometry, re-issued 1956 by Dover Publications, Inc. Alfred North Whitehead (1898) Universal Algebra, Book VI Chapter 1: Theory of Distance, pp 347–70, especially Section 199 Cayley's Theory of Distance. Hausdorff, F. | Cayley metric | 0.80368 |
32,907 | (eds.). Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. pp. | Cayley metric | 0.80368 |
32,908 | "Circle geometry in affine Cayley-Klein planes". Periodica Mathematica Hungarica. 57 (2): 197–206. | Cayley metric | 0.80368 |
32,909 | "Non-euclidean geometry, a retrospect" (PDF). Bulletin of the American Mathematical Society. 36 (2): 66–76. | Cayley metric | 0.80368 |
32,910 | In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail ... | Cayley metric | 0.80368 |
32,911 | In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent. Friedri... | Friedrichs' inequality | 0.80368 |
32,912 | Julia M. Klein of Johns Hopkins Magazine wrote, "There's nothing small about Johns Hopkins physicist Sean Carroll's latest undertaking. The Biggest Ideas in the Universe: Space, Time, and Motion is the first volume in an ambitious trilogy that seeks to explain physics to a popular audience—one willing to grapple with t... | The Biggest Ideas in the Universe: Space, Time, and Motion | 0.803678 |
32,913 | Although GEEN has higher efficiency than many other methods in reverse genetics, it is still not highly efficient; in many cases less than half of the treated populations obtain the desired changes. For example, when one is planning to use the cell's NHEJ to create a mutation, the cell's HDR systems will also be at wor... | Genome editing with engineered nucleases | 0.803677 |
32,914 | The triplets RA0 = (RA10, RA20, RA30) satisfying these conditions, enter the theory as a given set of real constants. Following Biedenharn and Louck, we introduce an orthonormal body-fixed frame, the Eckart frame, F → = { f → 1 , f → 2 , f → 3 } {\displaystyle {\vec {\mathbf {F} }}=\{{\vec {f}}_{1},{\vec {f}}_{2},{\vec... | Eckart conditions | 0.803675 |
32,915 | It can be shown that the invariance of V under simultaneous translation over t of all nuclei implies that vectors T = (t, ..., t) are in the kernel of H. From the invariance of V under an infinitesimal rotation of all nuclei around s, it can be shown that also the vectors S = (s x R10, ..., s x RN0) are in the kernel o... | Eckart conditions | 0.803675 |
32,916 | In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvatur... | Cheng's eigenvalue comparison theorem | 0.803674 |
32,917 | {\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\geq \lambda _{1}\left(B_{M}(p,r)\right).} S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem. As a special case, if k = −1 and inj(p) = ∞, Cheng’s inequality becomes λ*(N) ≥ λ*(H n(−1)) which is McKean’s inequality. | Cheng's eigenvalue comparison theorem | 0.803674 |
32,918 | The second part is a comparison theorem for the Ricci curvature of M: Suppose that the Ricci curvature of M satisfies, for every vector field X, Ric ( X , X ) ≥ k ( n − 1 ) | X | 2 . {\displaystyle \operatorname {Ric} (X,X)\geq k(n-1)|X|^{2}.} Then, with the same notation as above, λ 1 ( B N ( k ) ( r ) ) ≥ λ 1 ( B M... | Cheng's eigenvalue comparison theorem | 0.803674 |
32,919 | Let M be a Riemannian manifold with dimension n, and let BM(p, r) be a geodesic ball centered at p with radius r less than the injectivity radius of p ∈ M. For each real number k, let N(k) denote the simply connected space form of dimension n and constant sectional curvature k. Cheng's eigenvalue comparison theorem com... | Cheng's eigenvalue comparison theorem | 0.803674 |
32,920 | Fixed-effect meta-regression reflects the belief that the studies involved lack substantial difference. An arm-level fixed-effect meta-regression is written as ytk = xtk′β + ɛtk. If only study-level summary statistics are available, the subscript t for treatment assignment can be dropped, yielding yk = xk′β + ɛk. The e... | Meta-regression | 0.803672 |
32,921 | Meta-regression is defined to be a meta-analysis that uses regression analysis to combine, compare, and synthesize research findings from multiple studies while adjusting for the effects of available covariates on a response variable. A meta-regression analysis aims to reconcile conflicting studies or corroborate consi... | Meta-regression | 0.803672 |
32,922 | The most common summary statistic reported in a study is the sample mean and the sample standard deviation, in which case no transformation is needed. It is also possible to derive an aggregate-data model from an underlying individual-participant-data model. For example, if yitk is the binary response either zero or on... | Meta-regression | 0.803672 |
32,923 | However, aggregate data are generally modeled as a normal linear regression ytk = xtk′β + εtk using the central limit theorem and variable transformation, where the subscript k indicates the kth study or trial, t denotes the tth treatment, ytk indicates the response endpoint for the kth study's tth arm, xtk is the arm-... | Meta-regression | 0.803672 |
32,924 | In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimension... | Unit circle | 0.803672 |
32,925 | Thus, by the Pythagorean theorem, x and y satisfy the equation Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant. The interior of ... | Unit circle | 0.803671 |
32,926 | Using the antisymmetry of the electromagnetic field, one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with {λ, μ, ν} being either {1, 2, 3}, {2, 3, 0}, {3, 0, 1}, or {0, 1, 2}. The Faraday–Gauss equation is sometimes written where a semicolon indicates a covariant deri... | Maxwell's equations in curved spacetime | 0.803669 |
32,927 | The electric current is a contravariant vector density, and as such it transforms as follows: J ¯ μ = ∂ x ¯ μ ∂ x α J α det . {\displaystyle {\bar {J}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}J^{\alpha }\det \left.} Verification of this transformation law: So all that remains is to show that ∂... | Maxwell's equations in curved spacetime | 0.803669 |
32,928 | In this view, the Maxwell "equation" d F = 0 {\displaystyle \mathrm {d} \mathbf {F} =0} is a mathematical identity known as the Bianchi identity. The equation d ⋆ F = J {\displaystyle \mathrm {d} \star \mathbf {F} =\mathbf {J} } is the only equation with any physical content in this formulation. This point of view is p... | Maxwell's equations in curved spacetime | 0.803669 |
32,929 | In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwel... | Maxwell's equations in curved spacetime | 0.803669 |
32,930 | A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and ha... | Multi-resolution analysis | 0.803667 |
32,931 | In the history of astronomy, a handful of Solar System bodies other than Jupiter have been counted as the fifth planet from the Sun. | Fifth planet (hypothetical) | 0.803666 |
32,932 | The maximally extended solution does not describe a typical black hole created from the collapse of a star, as the surface of the collapsed star replaces the sector of the solution containing the past-oriented white hole geometry and other universe. While the basic space-like passage of a static black hole cannot be tr... | Penrose diagram | 0.803666 |
32,933 | The eight floor of the faculty houses an ISO 8, class 100,000 cleanroom for the development of micro satellites. The facility is used both by staff and by graduate students from the space department of the faculty. The cleanroom is used for space related research and for the production of TU Delft's micro satellites, o... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,934 | Current areas of research include novel aerospace materials, Particle Image Velocimetry, CubeSat, Airborne Wind Energy and several others. Currently ten research chairs are grouped under four major departments: Flow Physics and Technology (FPT) Control and Operations (C&O) Aerospace Structures and Materials (ASM) Space... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,935 | The Faculty of Aerospace Engineering at the Delft University of Technology in the Netherlands is the merger of two interrelated disciplines, aeronautical engineering and astronautical engineering. Aeronautical engineering works specifically with aircraft or aeronautics. Astronautical engineering works specifically with... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,936 | In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. It can be defined directly from the action and does not require the canonical coordinates and their canonical momenta to be defined in advance.The bracket {\displaystyle } is defined as D A ( B ) − D B ( A ) {\displaystyle... | Peierls bracket | 0.803665 |
32,937 | In 2023 the TU Delft reached the 3rd place in the "Mechanical, aerospace and Manufacturing Engineering" category of the QS World University Rankings. In 2013 this category got extended to "Mechanical, Aeronautical & Manufacturing Engineering" and the TU Delft jumped to the 18th position worldwide (6th place in Europe).... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,938 | The Faculty is one of the largest of the eight faculties at TU Delft and one of the largest faculties devoted entirely to aerospace engineering in northern Europe. It is the only institute carrying out research and education directly related to aerospace engineering in the Netherlands. Through the years, the Faculty ha... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,939 | The Delft Aerospace Structures and Materials laboratory is one of the largest facilities of the faculty of aerospace engineering with a footprint of over 3600 square meters. The laboratory is split up in multiple smaller laboratories which allow for a wide variety of research and educational activities. Amongst others ... | TU Delft Faculty of Aerospace Engineering | 0.803665 |
32,940 | Air pollutants impact essential ecosystem services such as air and water purification, decomposition and detoxification of waste materials, and climate regulation. When deposition is greater than the critical load of a pollutant for a particular location, it is considered a critical load exceedance, meaning the biota a... | Critical load | 0.803662 |
32,941 | Energy is defined via work, so the SI unit of energy is the same as the unit of work – the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. In slightly more fundamental terms, 1 joule is equal to 1 newton metre and, in terms of SI base units 1 J = 1 k g ( m s ... | Unit of energy | 0.803661 |
32,942 | The calorie is defined as the amount of thermal energy necessary to raise the temperature of one gram of water by 1 Celsius degree, from a temperature of 14.5 °C, at a pressure of 1 atm. For thermochemistry a calorie of 4.184 J is used, but other calories have also been defined, such as the International Steam Table ca... | Unit of energy | 0.803661 |
32,943 | The rotating-vane VGT was first developed under Garrett and patented in 1953.One of the first production cars to use these turbochargers was the 1988 Honda Legend; it used a water-cooled VGT installed on its 2.0-litre V6 engine. The limited-production 1989 Shelby CSX-VNT, with only 500 examples produced, was equipped w... | Variable geometry turbocharger | 0.803661 |
32,944 | Several companies manufacture and supply rotating-vane variable-geometry turbochargers, including Garrett, BorgWarner, and Mitsubishi Heavy Industries. This design is mostly limited to small engines and light-duty applications (passenger cars, race cars and light commercial vehicles). The main supplier of sliding-vane ... | Variable geometry turbocharger | 0.803661 |
32,945 | {\displaystyle X\Vdash A\iff A\in X.} This definition obeys the conditions on satisfaction of →: on one hand, if X ⊩ A → B {\displaystyle X\Vdash A\to B} , and Y ∈ W {\displaystyle Y\in W} is such that Y ⊇ X {\displaystyle Y\supseteq X} and Y ⊩ A {\displaystyle Y\Vdash A} , then Y ⊩ B {\displaystyle Y\Vdash B} by modus... | Lambda elimination | 0.80366 |
32,946 | The K and S combinators correspond to the axioms AK: A → (B → A), AS: (A → (B → C)) → ((A → B) → (A → C)),and function application corresponds to the detachment (modus ponens) rule MP: from A and A → B infer B.The calculus consisting of AK, AS, and MP is complete for the implicational fragment of the intuitionistic log... | Lambda elimination | 0.80366 |
32,947 | The Curry–Howard isomorphism implies a connection between logic and programming: every proof of a theorem of intuitionistic logic corresponds to a reduction of a typed lambda term, and conversely. Moreover, theorems can be identified with function type signatures. Specifically, a typed combinatory logic corresponds to ... | Lambda elimination | 0.80366 |
32,948 | A predicate is a combinator that, when applied, returns either T or F. A predicate N is nontrivial if there are two arguments A and B such that N A = T and N B = F. A combinator N is complete if NM has a normal form for every argument M. The analogue of Rice's theorem then says that every complete predicate is trivial.... | Lambda elimination | 0.80366 |
32,949 | But if (S I I Z) does not have a normal form, the foregoing reduces as follows: (N (S I I Z) Ω I) = (K I Ω I) (definition of N) = (I I) = Iwhich means that the normal form of (S I I Z) is simply I, another contradiction. Therefore, the hypothetical normal-form combinator N cannot exist. The combinatory logic analogue o... | Lambda elimination | 0.80366 |
32,950 | The proof of Shannon's channel coding theorem uses random coding to show the existence of a code that achieves channel capacity. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,951 | The weak halfspace theorem for minimal surfaces states that any complete minimal surface of bounded curvature which is not a plane is not contained in any halfspace. This theorem is proved using a coupling between Brownian motions on minimal surfaces. A non-probabilistic proof was available earlier. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,952 | A number of theorems stating existence of graphs (and other discrete structures) with desired properties are proved by the probabilistic method. Non-probabilistic proofs are available for a few of them. The maximum-minimums identity admits a probabilistic proof. Crossing number inequality which is a lower bound on the ... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,953 | A number of other less well-known identities can be deduced in a similar manner.The Picard theorem can be proved using the winding properties of planar Brownian motion. The fact that every Lipschitz function on the real line is differentiable almost everywhere can be proved using martingale convergence. Multidimensiona... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,954 | Non-tangential boundary values of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness. This result (Privalov's theorem), and several results of this kind, are deduced from martingale convergence. Non-probabilistic proofs were available earlier. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,955 | Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century.The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem. A probabilistic proof via n-dimensional Brownian motion is well known. Non-probabilistic proofs were avail... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,956 | Existence of a nowhere differentiable continuous function follows easily from properties of Wiener process. A non-probabilistic proof was available earlier. Stirling's formula was first discovered by Abraham de Moivre in his `The Doctrine of Chances' (with a constant identified later by Stirling) in order to be used in... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,957 | Every continuous function on a compact interval can be uniformly approximated by polynomials, which is the Weierstrass approximation theorem. A probabilistic proof uses the weak law of large numbers. Non-probabilistic proofs were available earlier. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,958 | No deterministic construction is known. The original proof that the Hausdorff–Young inequality cannot be extended to p > 2 {\displaystyle p>2} is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,959 | Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies. The diameter of the Banach–Mazur compactum was calculated using a probabilistic construction. | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,960 | Normal numbers exist. Moreover, computable normal numbers exist. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the strong law of large numbers); (b) some probabilistic inequalities behin... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,961 | Non-commutative dynamics (called also quantum dynamics) is formulated in terms of Von Neumann algebras and continuous tensor products of Hilbert spaces. Several results (for example, a continuum of mutually non-isomorphic models) are obtained by probabilistic means (random compact sets and Brownian motion). One part of... | List of probabilistic proofs of non-probabilistic theorems | 0.803659 |
32,962 | In linguistics and semiotics, a notation system is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, a notation is a collection of related symbols that are each given an... | Notation system | 0.803659 |
32,963 | Bra–ket notation, or Dirac notation, is an alternative representation of probability distributions in quantum mechanics. Tensor index notation is used when formulating physics (particularly continuum mechanics, electromagnetism, relativistic quantum mechanics and field theory, and general relativity) in the language of... | Notation system | 0.803659 |
32,964 | Baseball scorekeeping, to represent a game of baseball Aresti Catalogue, to represent aerobatic manoeuvres Chess notation, to represent moves in a game of chess Algebraic notation Portable Game Notation Descriptive notation Forsyth–Edwards Notation Siteswap notation represents a juggling pattern as a sequence of number... | Notation system | 0.803659 |
32,965 | Nucleic acid notation Systems Biology Graphical Notation (SBGN) Sequence motif pattern-description notations Cytogenetic notation Energy Systems Language | Notation system | 0.803659 |
32,966 | Mathematical notation is used to represent various kinds of mathematical ideas. All types of notation in probability Cartesian coordinate system, for representing position and other spatial concepts in analytic geometry Notation for differentiation, common representations of the derivative in calculus Big O notation, u... | Notation system | 0.803659 |
32,967 | In 2017, the Transformer network enabled advancements in generative models, leading to the first generative pre-trained transformer (GPT) in 2018. This was followed in 2019 by GPT-2 which demonstrated the ability to generalize unsupervised to many different tasks as a Foundation model.In 2021, the release of DALL-E, a ... | Generative artificial intelligence | 0.803654 |
32,968 | Beginning in the late 2000s, the emergence of deep learning drove progress and research in image classification, speech recognition, natural language processing and other tasks. Neural networks in this era were typically trained as discriminative models, due to the difficulty of generative modeling.In 2014, advancement... | Generative artificial intelligence | 0.803654 |
32,969 | By the early 1970s, Harold Cohen was creating and exhibiting generative AI works created by AARON, the computer program Cohen created to generate paintings.Markov chains have long been used to model natural languages. Markov himself analyzed the pattern of vowels and consonants in Eugeny Onegin using Markov chains. Onc... | Generative artificial intelligence | 0.803654 |
32,970 | Generative artificial intelligence (also generative AI or GenAI) is artificial intelligence capable of generating text, images, or other media, using generative models. Generative AI models learn the patterns and structure of their input training data and then generate new data that has similar characteristics.In the e... | Generative artificial intelligence | 0.803654 |
32,971 | Generative AI systems trained on words or word tokens include GPT-3, LaMDA, LLaMA, BLOOM, GPT-4, and others (see List of large language models). They are capable of natural language processing, machine translation, and natural language generation and can be used as foundation models for other tasks. Data sets include B... | Generative artificial intelligence | 0.803654 |
32,972 | From the early days of the development of AI there have been arguments put forward by ELIZA creator Joseph Weizenbaum and others about whether tasks that can be done by computers actually should be done by them, given the difference between computers and humans, and between quantitative calculations and qualitative, va... | Generative artificial intelligence | 0.803654 |
32,973 | A generative AI system is constructed by applying unsupervised or self-supervised machine learning to a data set. The capabilities of a generative AI system depend on the modality or type of the data set used. Generative AI can be either unimodal or multimodal; unimodal systems take only one type of input, whereas mult... | Generative artificial intelligence | 0.803654 |
32,974 | This was followed in 2019 by GPT-2 which demonstrated the ability to generalize unsupervised to many different tasks as a Foundation model.In 2021, the release of DALL-E, a transformer-based pixel generative model, followed by Midjourney and Stable Diffusion marked the emergence of practical high-quality artificial int... | Generative artificial intelligence | 0.803654 |
32,975 | By the early 1970s, Harold Cohen was creating and exhibiting generative AI works created by AARON, the computer program Cohen created to generate paintings.The field of machine learning often uses statistical models, including generative models, to model and predict data. Beginning in the late 2000s, the emergence of d... | Generative artificial intelligence | 0.803654 |
32,976 | The academic discipline of artificial intelligence was founded at a research workshop at Dartmouth College in 1956, and has experienced several waves of advancement and optimism in the decades since. Since its founding, researchers in the field have raised philosophical and ethical arguments about the nature of the hum... | Generative artificial intelligence | 0.803654 |
32,977 | In the European Union, the proposed Artificial Intelligence Act includes requirements to disclose copyrighted material used to train generative AI systems, and to label any AI-generated output as such.In the United States, a group of companies including OpenAI, Alphabet, and Meta signed a voluntary agreement with the W... | Generative artificial intelligence | 0.803654 |
32,978 | Weight initialization is another approach that has been proposed to reduce the vanishing gradient problem in deep networks. Kumar suggested that the distribution of initial weights should vary according to activation function used and proposed to initialize the weights in networks with the logistic activation function ... | Vanishing gradient problem | 0.803654 |
32,979 | This section is based on the paper On the difficulty of training Recurrent Neural Networks by Pascanu, Mikolov, and Bengio. | Vanishing gradient problem | 0.803654 |
32,980 | As one example of the problem cause, traditional activation functions such as the hyperbolic tangent function have gradients in the range (0,1], and backpropagation computes gradients by the chain rule. This has the effect of multiplying n of these small numbers to compute gradients of the early layers in an n-layer ne... | Vanishing gradient problem | 0.803654 |
32,981 | In machine learning, the vanishing gradient problem is encountered when training artificial neural networks with gradient-based learning methods and backpropagation. In such methods, during each iteration of training each of the neural networks weights receives an update proportional to the partial derivative of the er... | Vanishing gradient problem | 0.803654 |
32,982 | Behnke relied only on the sign of the gradient (Rprop) when training his Neural Abstraction Pyramid to solve problems like image reconstruction and face localization.Neural networks can also be optimized by using a universal search algorithm on the space of neural network's weights, e.g., random guess or more systemati... | Vanishing gradient problem | 0.803654 |
32,983 | Another technique particularly used for recurrent neural networks is the long short-term memory (LSTM) network of 1997 by Hochreiter & Schmidhuber. In 2009, deep multidimensional LSTM networks demonstrated the power of deep learning with many nonlinear layers, by winning three ICDAR 2009 competitions in connected handw... | Vanishing gradient problem | 0.803654 |
32,984 | Skip connections are a critical component of what allowed successful training of deeper neural networks. ResNets yielded lower training error (and test error) than their shallower counterparts simply by reintroducing outputs from shallower layers in the network to compensate for the vanishing data. Note that ResNets ar... | Vanishing gradient problem | 0.803654 |
32,985 | One of the newest and most effective ways to resolve the vanishing gradient problem is with residual neural networks, or ResNets (not to be confused with recurrent neural networks). ResNets refer to neural networks where skip connections or residual connections are part of the network architecture. These skip connectio... | Vanishing gradient problem | 0.803654 |
32,986 | Similar ideas have been used in feed-forward neural networks for unsupervised pre-training to structure a neural network, making it first learn generally useful feature detectors. Then the network is trained further by supervised backpropagation to classify labeled data. The deep belief network model by Hinton et al. (... | Vanishing gradient problem | 0.803654 |
32,987 | One possible WDM candidate particle with a mass of a few keV comes from introducing two new, zero charge, zero lepton number fermions to the Standard Model of Particle Physics: "keV-mass inert fermions" (keVins) and "GeV-mass inert fermions" (GeVins). keVins are overproduced if they reach thermal equilibrium in the ear... | Warm dark matter | 0.803652 |
32,988 | Scenario-based and multi-mode approaches are essential techniques in embedded-systems, e.g., design space exploration for MPSoCs and reconfigurable systems. Optimization techniques for the generation of schedule graphs supporting such a SBMeS approach have been developed and implemented. SBMeS can promise better perfor... | Meta-scheduling | 0.80365 |
32,989 | Any function f(x1, x2, ..., xn) of n real variables can be considered as a function on Rn (that is, with Rn as its domain). The use of the real n-space, instead of several variables considered separately, can simplify notation and suggest reasonable definitions. Consider, for n = 2, a function composition of the follow... | Real plane | 0.803645 |
32,990 | Euclidean R4 also attracts the attention of mathematicians, for example due to its relation to quaternions, a 4-dimensional real algebra themselves. See rotations in 4-dimensional Euclidean space for some information. In differential geometry, n = 4 is the only case where Rn admits a non-standard differential structure... | Real plane | 0.803645 |
32,991 | The Model C stellarator was the first large-scale stellarator to be built, during the early stages of fusion power research. Planned since 1952, construction began in 1961 at what is today the Princeton Plasma Physics Laboratory (PPPL). The Model C followed the table-top sized Model A, and a series of Model B machines ... | Model C stellarator | 0.803644 |
32,992 | Minkowski's theorem relates the number d(Λ) and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coeffi... | Lattice point | 0.803642 |
32,993 | The TRIAD method is the earliest published algorithm for determining spacecraft attitude, which was first introduced by Harold Black in 1964. Given the knowledge of two vectors in the reference and body coordinates of a satellite, the TRIAD algorithm obtains the direction cosine matrix relating to both frames. Harold B... | Triad Method | 0.80364 |
32,994 | This is because of the simple fact that, in Euclidean geometry, the angle between any two vectors remains invariant to coordinate transformations. Therefore, the determinant d e t ( Γ ) {\displaystyle det\left(\Gamma \right)} is 1 {\displaystyle 1} or − 1 {\displaystyle -1} depending on whether its columns are right-ha... | Triad Method | 0.80364 |
32,995 | Consider the affine line A1(k) = Spec(k) over a field k. If k is algebraically closed, there are exactly two types of prime ideals, namely (t − a), a ∈ k (0), the zero-ideal.The residue fields are k ( t − a ) / ( t − a ) k ( t − a ) ≅ k {\displaystyle k_{(t-a)}/(t-a)k_{(t-a)}\cong k} k ( 0 ) ≅ k ( t ) {\displaystyle... | Residue field | 0.80364 |
32,996 | Solar energy is stored as heat energy in thermal mass such as water, stone, masonry, soil and plant biomass. The second is the biology inside the bioshelter. Earle Barnhart of the New Alchemy Institute has compared a bioshelter to a contained ecosystem. | Bioshelter | 0.80364 |
32,997 | The first is architecture designed to nurture an ecosystem within. A bioshelter structure uses glazing to contain and protect the living biology inside, control air exchange and absorb energy. The building exchanges nutrients, gases and energy with the surrounding environment, produces crops, and recycles waste organic... | Bioshelter | 0.80364 |
32,998 | Suppose that R is a commutative local ring, with maximal ideal m. Then the residue field is the quotient ring R/m. Now suppose that X is a scheme and x is a point of X. By the definition of scheme, we may find an affine neighbourhood U = Spec(A), with A some commutative ring. Considered in the neighbourhood U, the poin... | Residue field | 0.803639 |
32,999 | In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal. This construction is applied in algebraic ge... | Residue field | 0.803639 |
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