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21,200 | Sometimes, the time taken to undertake optimization therein itself may be an issue. Optimizing existing code usually does not add new features, and worse, it might add new bugs in previously working code (as any change might). Because manually optimized code might sometimes have less "readability" than unoptimized code... | Premature optimization | 0.81041 |
21,201 | Optimization may include finding a bottleneck in a system – a component that is the limiting factor on performance. In terms of code, this will often be a hot spot – a critical part of the code that is the primary consumer of the needed resource – though it can be another factor, such as I/O latency or network bandwidt... | Premature optimization | 0.81041 |
21,202 | The study of shape and statistics in populations are local theories, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape is then the study of diffeomorphic correspondences relative to the template. A core operation is the generation of templates from populations, estimati... | Computational Anatomy | 0.810408 |
21,203 | {\displaystyle (g\circ g^{\prime })\cdot m=g\cdot (g^{\prime }\cdot m)\in {\mathcal {M}}.} The orbit M {\displaystyle {\mathcal {M}}} of the template becomes the space of all shapes, M ≐ { m = g ⋅ m t e m p , g ∈ G } {\displaystyle {\mathcal {M}}\doteq \{m=g\cdot m_{\mathrm {temp} },g\in {\mathcal {G}}\}} , being homog... | Computational Anatomy | 0.810408 |
21,204 | The model of human anatomy is a deformable template, an orbit of exemplars under group action. Deformable template models have been central to Grenander's metric pattern theory, accounting for typicality via templates, and accounting for variability via transformation of the template. An orbit under group action as the... | Computational Anatomy | 0.810408 |
21,205 | The group of diffeomorphisms is very big. To ensure smooth flows of diffeomorphisms avoiding shock-like solutions for the inverse, the vector fields must be at least 1-time continuously differentiable in space. For diffeomorphisms on R 3 {\displaystyle {\mathbb {R} }^{3}} , vector fields are modelled as elements of the... | Computational Anatomy | 0.810408 |
21,206 | In the following, denote square-root ODF ( ODF {\displaystyle {\sqrt {\text{ODF}}}} ) as ψ ( s ) {\displaystyle \psi ({\bf {s}})} , where ψ ( s ) {\displaystyle \psi ({\bf {s}})} is non-negative to ensure uniqueness and ∫ s ∈ S 2 ψ 2 ( s ) d s = 1 {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}... | Computational Anatomy | 0.810408 |
21,207 | Dense LDDMM ODF matching takes the HARDI data as ODF at each voxel and solves the LDDMM variational problem in the space of ODF. In the field of information geometry, the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. For the purpose of LDDMM ODF mapping, the square-root representation is chosen b... | Computational Anatomy | 0.810408 |
21,208 | For the study of rigid body kinematics, the low-dimensional matrix Lie groups have been the central focus. The matrix groups are low-dimensional mappings, which are diffeomorphisms that provide one-to-one correspondences between coordinate systems, with a smooth inverse. The matrix group of rotations and scales can be ... | Computational Anatomy | 0.810408 |
21,209 | Dense image matching has a long history now with the earliest efforts exploiting a small deformation framework. Large deformations began in the early 1990s, with the first existence to solutions to the variational problem for flows of diffeomorphisms for dense image matching established in. Beg solved via one of the ea... | Computational Anatomy | 0.810408 |
21,210 | The original idea of doing differential geometry, curvature and geodesics on infinite dimensional manifolds goes back to Bernhard Riemann's Habilitation (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen); the key modern book laying the foundations of such ideas in global analysis are from Michor.The applicat... | Computational Anatomy | 0.810408 |
21,211 | Computational anatomy is the study of shape and form at the morphome or gross anatomy millimeter, or morphology scale, focusing on the study of sub-manifolds of R 3 , {\displaystyle {\mathbb {R} }^{3},} points, curves surfaces and subvolumes of human anatomy. An early modern computational neuro-anatomist was David Van ... | Computational Anatomy | 0.810408 |
21,212 | This separates the discipline from the case of incompressible fluids for which momentum is a pointwise function of velocity. Computational anatomy intersects the study of Riemannian manifolds and nonlinear global analysis, where groups of diffeomorphisms are the central focus. Emerging high-dimensional theories of shap... | Computational Anatomy | 0.810408 |
21,213 | Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures. The field is br... | Computational Anatomy | 0.810408 |
21,214 | The spirit of this discipline shares strong overlap with areas such as computer vision and kinematics of rigid bodies, where objects are studied by analysing the groups responsible for the movement in question. Computational anatomy departs from computer vision with its focus on rigid motions, as the infinite-dimension... | Computational Anatomy | 0.810408 |
21,215 | At computational anatomy's heart is the comparison of shape by recognizing in one shape the other. This connects it to D'Arcy Wentworth Thompson's developments On Growth and Form which has led to scientific explanations of morphogenesis, the process by which patterns are formed in biology. Albrecht Durer's Four Books o... | Computational Anatomy | 0.810408 |
21,216 | The successive flow of the Euler equation induces the random space of shapes and forms E x p i d ( v 0 ) ⋅ m ∈ M {\displaystyle Exp_{id}(v_{0})\cdot m\in {\mathcal {M}}} . Performing empirical statistics on this tangent space at the identity is the natural way for inducing probability laws on the statistics of shape. S... | Computational Anatomy | 0.810408 |
21,217 | Shape in computational anatomy is a local theory, indexing shapes and structures to templates to which they are bijectively mapped. Statistical shape in computational anatomy is the empirical study of diffeomorphic correspondences between populations and common template coordinate systems. This is a strong departure fr... | Computational Anatomy | 0.810408 |
21,218 | Computing the flow v 0 {\displaystyle v_{0}} onto coordinates Riemannian logarithm, mapping L o g i d ( ⋅ ): D i f f V → V {\displaystyle Log_{\rm {id}}(\cdot ):Diff_{V}\to V} at identity from φ {\displaystyle \varphi } to vector field v 0 ∈ V {\displaystyle v_{0}\in V} ; L o g i d ( φ ) = v 0 initial condition of EL g... | Computational Anatomy | 0.810408 |
21,219 | More evidence for this being the case is the way that good human players find it much easier to recall positions from genuine chess games, breaking them down into a small number of recognizable sub-positions, rather than completely random arrangements of the same pieces. In contrast, poor players have the same level of... | Computer Chess | 0.810407 |
21,220 | Since the era of mechanical machines that played rook and king endings and electrical machines that played other games like hex in the early years of the 20th century, scientists and theoreticians have sought to develop a procedural representation of how humans learn, remember, think and apply knowledge, and the game o... | Computer Chess | 0.810407 |
21,221 | In mechanical engineering, a rolling-element bearing, also known as a rolling bearing, is a bearing which carries a load by placing rolling elements (such as balls or rollers) between two concentric, grooved rings called races. The relative motion of the races causes the rolling elements to roll with very little rollin... | Rolling-element bearings | 0.810405 |
21,222 | The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences... | Binomial type | 0.810405 |
21,223 | In consequence of this definition the binomial theorem can be stated by saying that the sequence { x n: n = 0 , 1 , 2 , … } {\displaystyle \{x^{n}:n=0,1,2,\ldots \}} is of binomial type. The sequence of "lower factorials" is defined by(In the theory of special functions, this same notation denotes upper factorials, but... | Binomial type | 0.810405 |
21,224 | Here is the main result of this section: Theorem: All polynomial sequences of binomial type are of this form. A result in Mullin and Rota, repeated in Rota, Kahaner, and Odlyzko (see References below) states that every polynomial sequence { pn(x) }n of binomial type is determined by the sequence { pn′(0) }n, but those ... | Binomial type | 0.810405 |
21,225 | The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other fields. | Binomial type | 0.810405 |
21,226 | A photon is massless, has no electric charge, and is a stable particle. In a vacuum, a photon has three possible polarization states. The photon is the gauge boson for electromagnetism,: 29–30 and therefore all other quantum numbers of the photon (such as lepton number, baryon number, and flavour quantum numbers) are z... | Gamma quantum | 0.810404 |
21,227 | Secondary antibodies can be conjugated to enzymes such as horseradish peroxidase (HRP) or alkaline phosphatase (AP); or fluorescent dyes such as fluorescein isothiocyanate (FITC), rhodamine derivatives, Alexa Fluor dyes; or other molecules to be used in various applications. Secondary antibodies are used in many bioche... | Primary and secondary antibodies | 0.810401 |
21,228 | A principal benefit of modeling is the ability to explicitly compare models: Rather than simply returning a value for each component, the modeler can compute confidence intervals on parameters, but, crucially, can drop and add paths and test the effect via statistics such as the AIC. Thus, for instance to test for pred... | Equal environments assumption | 0.810398 |
21,229 | Twin studies are studies conducted on identical or fraternal twins. They aim to reveal the importance of environmental and genetic influences for traits, phenotypes, and disorders. Twin research is considered a key tool in behavioral genetics and in related fields, from biology to psychology. Twin studies are part of t... | Equal environments assumption | 0.810398 |
21,230 | The revolution in molecular genetics has provided more effective tools for describing the genome, and many researchers are pursuing molecular genetics in order to directly assess the influence of alleles and environments on traits. An initial limitation of the twin design is that it does not afford an opportunity to co... | Equal environments assumption | 0.810398 |
21,231 | The basic or classical twin-design contains only identical and fraternal twins raised in their biological family. This represents only a sub-set of the possible genetic and environmental relationships. It is fair to say, therefore, that the heritability estimates from twin designs represent a first step in understandin... | Equal environments assumption | 0.810398 |
21,232 | Several academic bodies exist to support behavior genetic research, including the Behavior Genetics Association, the International Society for Twin Studies, and the International Behavioural and Neural Genetics Society. Behavior genetic work also features prominently in several more general societies, for instance the ... | Equal environments assumption | 0.810398 |
21,233 | Like all behavior genetic research, the classical twin study begins by assessing the variance of behavior (called a phenotype by geneticists) in a large group, and attempts to estimate how much of this is due to: genetic effects (heritability) shared environment – events that happen to both twins, affecting them in the... | Equal environments assumption | 0.810398 |
21,234 | For instance, up to 40% of SARS-CoV-2 infections may be asymptomatic, suggesting that many people are naturally protected from disease. Large genetic studies have defined risk factors for severe SARS-CoV-2 infections, and genome sequences from 659 patients with severe COVID-19 revealed genetic variants that appear to b... | Infection | 0.810395 |
21,235 | A Study of 18-point Mirror Cell Optimization Using Varying Forces Jeff Anderson-Lee, January 2003 David Lewis, Automated Mirror Cell Optimization Mechanical design of the 110 cm cruxis telescope Stellafane links page | PLate OPtimizer | 0.810394 |
21,236 | In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign. The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale calculus. A summation equation compares to a diff... | Summation equation | 0.810391 |
21,237 | A commonly used convention for selecting frames of reference in robotics applications is the Denavit and Hartenberg (D–H) convention which was introduced by Jacques Denavit and Richard S. Hartenberg. In this convention, coordinate frames are attached to the joints between two links such that one transformation is assoc... | Denavit-Hartenberg parameters | 0.810382 |
21,238 | In the horizontal coordinate system, used in celestial navigation, azimuth is one of the two coordinates. The other is altitude, sometimes called elevation above the horizon. It is also used for satellite dish installation (see also: sat finder). In modern astronomy azimuth is nearly always measured from the north. | Flight azimuth | 0.810382 |
21,239 | Multiple nomenclatures (around 12) for linker histone variants have been proposed and used in publications previously, greatly complicating comparison across studies. In 1994 Parseghian et al. have attempted to create a system in which variant designations were applied uniformly to orthologs across mammalian species, h... | Linker histone H1 variants | 0.810381 |
21,240 | In molecular biology, the linker histone H1 is a protein family forming a critical component of eukaryotic chromatin. H1 histones bind to the linker DNA exiting from the nucleosome core particle, while the core histones (H2A, H2B, H3 and H4) form the octamer core of the nucleosome around which the DNA is wrapped.H1 for... | Linker histone H1 variants | 0.810381 |
21,241 | RIPE is led by the University of Illinois at the Carl R. Woese Institute for Genomic Biology. The project's partner institutions include the Australian National University, Chinese Academy of Sciences, Commonwealth Scientific and Industrial Research Organisation, Lancaster University, Louisiana State University, Univer... | Realizing Increased Photosynthetic Efficiency | 0.81038 |
21,242 | A number of electrical laws apply to all linear resistive networks. These include: Kirchhoff's current law: The sum of all currents entering a node is equal to the sum of all currents leaving the node.Kirchhoff's voltage law: The directed sum of the electrical potential differences around a loop must be zero.Ohm's law:... | Complex circuit | 0.810375 |
21,243 | A Grothendieck topos is a category C which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.) There is a small category D and an inclusion C ↪ Presh(D) that admits a finite-limit-preserving left adjoint. C is the category of sheaves on a ... | Elementary topoi | 0.810374 |
21,244 | Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras. To a scheme and even a stack one may associate an étale topos, an fppf topos, or a Nisnevich topos. Another important example of a topos is from the crystalline site. In the case of the étale topos, these form the ... | Elementary topoi | 0.810374 |
21,245 | More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of a scheme. For each scheme X {\displaystyle X} there is a site Open ( X ) {\displaystyle {\text{Open}}(X)} (of objects given by open subsets and morphisms given by inclu... | Elementary topoi | 0.810374 |
21,246 | However, Harkleroad also notes that some idiosyncrasies of the translation, such as its use of pre-decimal British currency, have since become quaint and old-fashioned. And similarly, although W. W. Sawyer in reviewing the original 1955 publication calls its inclusion of topics from graph theory and topology "truly mod... | Playing with Infinity | 0.810372 |
21,247 | Playing with Infinity presents a broad panorama of mathematics for a popular audience. It is divided into three parts, the first of which concerns counting, arithmetic, and connections from numbers to geometry both through visual proofs of results in arithmetic like the sum of finite arithmetic series, and in the other... | Playing with Infinity | 0.810372 |
21,248 | Let Q(x) denote the number of square-free integers between 1 and x (OEIS: A013928 shifting index by 1). For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these ratios satisfy the multiplicative property (this follows from Chines... | Square-free number | 0.810371 |
21,249 | Most index sets are non-computable, aside from two trivial exceptions. This is stated in Rice's theorem: Let C {\displaystyle {\mathcal {C}}} be a class of partial computable functions with its index set C {\displaystyle C} . Then C {\displaystyle C} is computable if and only if C {\displaystyle C} is empty, or C {\dis... | Index set (computability) | 0.81037 |
21,250 | Materials Chemistry and Physics (including Materials Science Communications) is a peer-reviewed scientific journal published 18 times per year by Elsevier. The focus of the journal is interrelationships among structure, properties, processing and performance of materials. It covers conventional and advanced materials. ... | Materials Chemistry and Physics | 0.810369 |
21,251 | In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers.The notion was introduced in 1974 by Frank Harar... | Integral graph | 0.810367 |
21,252 | Servo motors and stepper motors are specialist types of fractional-horsepower electric motors usually intended for high-precision or robotics applications. Usually running from a DC supply, when combined with a planetary gearbox can offer accuracies less than 8 arc-minutes (2/15ths of a degree, or approx. 2.3 milliradi... | Fractional-horsepower motor | 0.810366 |
21,253 | This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named; in other words there is a first-order sen... | Minimal model (set theory) | 0.810366 |
21,254 | The value of the Gauss sum is an algebraic integer in the pth cyclotomic field Q ( ζ p ) {\displaystyle \mathbb {Q} (\zeta _{p})} . The evaluation of the Gauss sum for an integer a not divisible by a prime p > 2 can be reduced to the case a = 1: g ( a ; p ) = ( a p ) g ( 1 ; p ) . {\displaystyle g(a;p)=\left({\tfrac {a... | Quadratic Gauss sum | 0.810361 |
21,255 | In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These obje... | Quadratic Gauss sum | 0.810361 |
21,256 | Experiments with olfaction, taking quantum mechanics into consideration, suggest that ultimately both theories might work in harmony - first the scent molecules need to fit, as in the docking theory of olfaction model, but then the molecular vibrations of the chemical/atom bonds take over. So in essence your sense of s... | Vibration theory | 0.810359 |
21,257 | Algorithms to locate transition state geometries fall into two main categories: local methods and semi-global methods. Local methods are suitable when the starting point for the optimization is very close to the true transition state (very close will be defined shortly) and semi-global methods find application when it ... | Energy minimization | 0.810358 |
21,258 | The string method uses splines connecting the points, ri, to measure and enforce distance constraints between the points and to calculate the tangent at each point. In each step of an optimization procedure, the points might be moved according to the force acting on them perpendicular to the path, and then, if the equi... | Energy minimization | 0.810358 |
21,259 | There are many variations on the NEB method, such including the climbing image NEB, in which the point with the highest energy is pushed upwards during the optimization procedure so as to (hopefully) give a geometry which is even closer to that of the transition state. There have also been extensions to include Gaussia... | Energy minimization | 0.810358 |
21,260 | Often this constraint is achieved by projecting out components of the force on each bead ri, or alternatively the movement of each bead during optimization, that are tangential to the reaction path. For example, if for convenience, it is defined that gi = ∂E/∂ri, then the energy gradient at each bead minus the componen... | Energy minimization | 0.810358 |
21,261 | Very commonly, these points are referred to as beads due to an analogy of a set of beads connected by strings or springs, which connect the reactant and products. The series of beads is often initially created by interpolating between rreactant and rproduct, for example, for a series of N + 1 beads, bead i might be giv... | Energy minimization | 0.810358 |
21,262 | Chain-of-state methods can be used to find the approximate geometry of the transition state based on the geometries of the reactant and product. The generated approximate geometry can then serve as a starting point for refinement via a local search, which was described above. Chain-of-state methods use a series of vect... | Energy minimization | 0.810358 |
21,263 | Payne et al., "Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients", Reviews of Modern Physics 64 (4), pp. 1045–1097. (1992) (abstract) Stich et al., "Conjugate gradient minimization of the energy functional: A new method for electronic structure calcula... | Energy minimization | 0.810358 |
21,264 | In discrete mathematics, especially combinatorics and finite probability theory, n-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length n. n-tuples whose entries come from a set of m elements are also called arrangements with repetition, permutations of a m... | Empty tuple | 0.810356 |
21,265 | Most typed functional programming languages implement tuples directly as product types, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. A few pr... | Empty tuple | 0.810356 |
21,266 | Braces "{ }" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term tuple can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. | Empty tuple | 0.810356 |
21,267 | This is 'approximately' the circumference of a circle whose diameter is 20,000. Approximating π to four decimal places: π ≈ 62832⁄20000 = 3.1416, Aryabhata stated that his result "approximately" (āsanna "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of... | Computing π | 0.810353 |
21,268 | Many computer families introduced in the 1950s and 1960s include execute instructions: the IBM 709 and IBM 7090 (op code mnemonic: XEC), the IBM 7030 Stretch (EX, EXIC), the PDP-1/-4/-7/-9/-15 (XCT), the UNIVAC 1100/2200 (EXRI), the CDC 924 (XEC), the PDP-6/-10 (XCT), the IBM System/360 (EX), the GE-600/Honeywell 6000 ... | Execute instruction | 0.810352 |
21,269 | Where the regular unification found in Prolog is linear in the size of terms compared, the extension to faithfully capture equivalence of nominal terms, called nominal unification in the literature, is quadratic (Calvès 2013). Based on an earlier PTIME algorithm for nominal unification, alphaProlog is a Prolog-like log... | Nominal terms (computer science) | 0.810352 |
21,270 | Many interesting calculi, logics and programming languages that are commonly seen in computer science feature name binding constructs. For instance, the universal quantifier from first-order logic, the lambda-binder from the lambda-calculus, and the pi-binder from the pi-calculus are all examples of name-binding constr... | Nominal terms (computer science) | 0.810352 |
21,271 | He entitled the volume Unto This Last, and his central point was this: "It is impossible to conclude, of any given mass of acquired wealth, merely by the fact of its existence, whether it signifies good or evil to the nation in the midst of which it exists. Its real value depends on the moral sign attached to it, just ... | Monetary value | 0.81035 |
21,272 | In quantum chemistry, n-electron valence state perturbation theory (NEVPT) is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known second-order Møller–Plesset perturbation theory to multireference Complete Active Space cases. The theo... | N-Electron Valence state Perturbation Theory | 0.810348 |
21,273 | Liane Gabora, Department of Psychology, University of British Columbia Russell Gray Max Planck Institute for the Science of Human History, Jena, Germany Herb Gintis, Emeritus Professor of Economics, University of Massachusetts & Santa Fe Institute Kevin Laland, School of Biology, University of St. Andrews Ruth Mace, De... | Gene-culture coevolution | 0.810347 |
21,274 | In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or... | Circular notation | 0.810346 |
21,275 | The Secrets of Triangles: A Mathematical Journey is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and Ingmar Lehmann, and published in 2012 by Prometheus Books. | The Secrets of Triangles | 0.810343 |
21,276 | Reviewer Alasdair McAndrew criticizes the book as being too "breathless" in its praise of the geometry it discusses and too superficial to be of interest to professional mathematicians, and Patricia Baggett writes that it little of its content would be of use in high school mathematics education. However, Baggett sugge... | The Secrets of Triangles | 0.810343 |
21,277 | The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension. If E / F {\displaystyle E/F} is a Galois extension, then Gal ( E / F ) {\... | Galois group | 0.810342 |
21,278 | A basic example of a field extension with an infinite group of automorphisms is Aut ( C / Q ) {\displaystyle \operatorname {Aut} (\mathbb {C} /\mathbb {Q} )} , since it contains every algebraic field extension E / Q {\displaystyle E/\mathbb {Q} } . For example, the field extensions Q ( a ) / Q {\displaystyle \mathbb ... | Galois group | 0.810342 |
21,279 | In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory,... | Galois group | 0.810342 |
21,280 | InfiniteGraph is produced and supported by Objectivity, Inc., a company that develops database management technologies for large-scale, distributed data management and relationship analytics. The new InfiniteGraph was released in May 2021. | InfiniteGraph | 0.810337 |
21,281 | Identifiability analysis is a group of methods found in mathematical statistics that are used to determine how well the parameters of a model are estimated by the quantity and quality of experimental data. Therefore, these methods explore not only identifiability of a model, but also the relation of the model to partic... | Identifiability analysis | 0.810334 |
21,282 | Structural identifiability analysis is a particular type of analysis in which the model structure itself is investigated for non-identifiability. Recognized non-identifiabilities may be removed analytically through substitution of the non-identifiable parameters with their combinations or by another way. The model over... | Identifiability analysis | 0.810334 |
21,283 | In mathematics, the either–or topology is a topological structure defined on the closed interval by declaring a set open if it either does not contain {0} or does contain (−1, 1). | Either–or topology | 0.810334 |
21,284 | In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, s... | Prime triplet | 0.810327 |
21,285 | There is also an algebraic version of incompressibility. Suppose ι: S → M {\displaystyle \iota :S\rightarrow M} is a proper embedding of a compact surface in a 3-manifold. Then S is π1-injective (or algebraically incompressible) if the induced map ι ⋆: π 1 ( S ) → π 1 ( M ) {\displaystyle \iota _{\star }:\pi _{1}(S)\ri... | Incompressible surface | 0.810327 |
21,286 | A Seifert surface S for an oriented link L is an oriented surface whose boundary is L with the same induced orientation. If S is not π1 injective in S3 − N(L), where N(L) is a tubular neighborhood of L, then the loop theorem gives a compressing disk that one may use to compress S along, providing another Seifert surfac... | Incompressible surface | 0.810327 |
21,287 | Let S be a compact surface properly embedded in a smooth or PL 3-manifold M. A compressing disk D is a disk embedded in M such that D ∩ S = ∂ D {\displaystyle D\cap S=\partial D} and the intersection is transverse. If the curve ∂D does not bound a disk inside of S, then D is called a nontrivial compressing disk. If S h... | Incompressible surface | 0.810327 |
21,288 | A strategy for addressing this problem has been suggested; the idea is to study the boundary amplitude, or transition amplitude of a compact region of spacetime, namely a path integral over a finite space-time region, seen as a function of the boundary value of the field. In conventional quantum field theory, this boun... | Gibbons–Hawking–York boundary term | 0.810327 |
21,289 | In pure gravity without cosmological constant, since the Ricci scalar vanishes on solutions of Einstein's equations, the bulk action vanishes and the Hamilton's principal function is given entirely in terms of the boundary term, S = ∫ Σ K a b q a b q d 3 σ {\displaystyle S=\int _{\Sigma }K^{ab}q_{ab}{\sqrt {q}}\;d^{3... | Gibbons–Hawking–York boundary term | 0.810327 |
21,290 | In the quantum theory, the object that corresponds to the Hamilton's principal function is the transition amplitude. Consider gravity defined on a compact region of spacetime, with the topology of a four dimensional ball. The boundary of this region is a three-dimensional space with the topology of a three-sphere, whic... | Gibbons–Hawking–York boundary term | 0.810327 |
21,291 | By the Palatini identity g α β δ R α β = δ V μ ; μ , δ V μ = g α β δ Γ α β μ − g α μ δ Γ α β β . {\displaystyle g^{\alpha \beta }\delta R_{\alpha \beta }=\delta {V^{\mu }}_{;\mu },\qquad \delta V^{\mu }=g^{\alpha \beta }\delta \Gamma _{\alpha \beta }^{\mu }-g^{\alpha \mu }\delta \Gamma _{\alpha \beta }^{\beta }.} We wi... | Gibbons–Hawking–York boundary term | 0.810326 |
21,292 | Integrative Biology is a monthly peer-reviewed scientific journal covering the interface between biology and the fields of physics, chemistry, engineering, imaging, and informatics. It was published by the Royal Society of Chemistry from its launch in 2008 until 2018. Since 2019, it has been published by Oxford Univers... | Integrative Biology | 0.810325 |
21,293 | One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem in the Euclidean setting. Subsequently Hadwiger-type theorems were established in various settings, notably in hermitian geometry, using advanced tools from valuation theory. The more recent meaning of integral geometry is that... | Integral geometry | 0.810321 |
21,294 | The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of the differential forms.A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability... | Integral geometry | 0.810321 |
21,295 | (Note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes—for example Bertrand's paradox.) We can therefore say that integral geometry in this sense is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. | Integral geometry | 0.810321 |
21,296 | Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of interse... | Integral geometry | 0.810321 |
21,297 | In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of f... | Integral geometry | 0.810321 |
21,298 | Since HP carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence Q ↔ H P {\displaystyle Q\leftrightarrow HP} brings this structure to Q. It can be grasped using the notion of hyperbolic motions. Since geodesics in HP are semicircles with centers on the bo... | Hyperbolic coordinates | 0.81032 |
21,299 | In special relativity the focus is on the 3-dimensional hypersurface in the future of spacetime where various velocities arrive after a given proper time. Scott Walter explains that in November 1907 Hermann Minkowski alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathemati... | Hyperbolic coordinates | 0.81032 |
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