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17,000 | In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. | Rate parameter | 0.813262 |
17,001 | By the substitution rule of integral calculus, we then have 1 = ∫ − ∞ ∞ f ( g ( x ) ) g ′ ( x ) d x = ∫ − ∞ ∞ f s ( x ) d x . {\displaystyle 1=\int _{-\infty }^{\infty }f(g(x))g'(x)\,dx=\int _{-\infty }^{\infty }f_{s}(x)\,dx.} So f s {\displaystyle f_{s}} is also properly normalized. | Rate parameter | 0.813262 |
17,002 | In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be consider... | Symbolic reasoning | 0.813259 |
17,003 | Combinatorics of Experimental Design is a textbook on the design of experiments, a subject that connects applications in statistics to the theory of combinatorial mathematics. It was written by mathematician Anne Penfold Street and her daughter, statistician Deborah Street, and published in 1987 by the Oxford Universit... | Combinatorics of Experimental Design | 0.813259 |
17,004 | The book has 15 chapters. Its introductory chapter covers the history and applications of experimental designs, it has five chapters on balanced incomplete block designs and their existence, and three on Latin squares and mutually orthogonal Latin squares. Other chapters cover resolvable block designs, finite geometry,... | Combinatorics of Experimental Design | 0.813259 |
17,005 | Although intended as an advanced undergraduate textbook, this book can also be used as a graduate text, or as a reference for researchers. Its main prerequisites are some knowledge of linear algebra and linear models, but some topics touch on abstract algebra and number theory as well.Although disappointed by the omiss... | Combinatorics of Experimental Design | 0.813259 |
17,006 | Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite. Estimators of shape ofte... | Shape parameter | 0.813256 |
17,007 | In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributions that is neither a location parameter nor a scale parameter (nor a function of these, such as a rate parameter). Such a parameter must affect the sha... | Shape parameter | 0.813256 |
17,008 | The journal is abstracted and indexed in ABOA/Streamline, Elsevier BIOBASE, BIOSIS, CAB Abstracts, Chemical Abstracts, Current Contents (Agriculture, Biology & Environmental Sciences), Science Citation Index, Scopus and TEEAL. | Soil Research | 0.813255 |
17,009 | Soil Research is an international peer-reviewed scientific journal published by CSIRO Publishing. It aims to rapidly publish high-quality, novel research about fundamental and applied aspects of soil science. As well as publishing in traditional aspects of soil biology, soil physics and soil chemistry across terrestria... | Soil Research | 0.813255 |
17,010 | Just as sketch relations define conditions such as tangency, parallelism, and concentricity with respect to sketch geometry, assembly mates define equivalent relations with respect to the individual parts or components, allowing the easy construction of assemblies. SolidWorks also includes additional advanced mating fe... | SolidWorks | 0.813253 |
17,011 | Relations are used to define attributes such as tangency, parallelism, perpendicularity, and concentricity. The parametric nature of SolidWorks means that the dimensions and relations drive the geometry, not the other way around. The dimensions in the sketch can be controlled independently, or by relationships to other... | SolidWorks | 0.813253 |
17,012 | Operation-based features are not sketch-based, and include features such as fillets, chamfers, shells, applying draft to the faces of a part, etc. Building a model in SolidWorks usually starts with a 2D sketch (although 3D sketches are available for power users). The sketch consists of geometry such as points, lines, a... | SolidWorks | 0.813253 |
17,013 | SolidWorks is a solid modeler, and utilizes a parametric feature-based approach which was initially developed by PTC (Creo/Pro-Engineer) to create 3D CAD models and assemblies. The software uses the Parasolid modeling kernel.Parameters refer to constraints whose values determine the shape or geometry of the model or as... | SolidWorks | 0.813253 |
17,014 | According to quantum mechanics, atoms and molecules can only hold certain defined quantities of energy, or exist in specific states. When electromagnetic radiation is absorbed by an atom or molecule, the energy of the radiation changes the state of the atom or molecule from an initial state to a final state. The number... | Absorption bands | 0.813249 |
17,015 | In quantum mechanics, an absorption band is a range of wavelengths, frequencies or energies in the electromagnetic spectrum that are characteristic of a particular transition from initial to final state in a substance. According to quantum mechanics, atoms and molecules can only hold certain defined quantities of energ... | Absorption bands | 0.813249 |
17,016 | The first classification theorem for persistent homology appeared in 1994 via Barannikov's canonical forms. The classification theorem interpreting persistence in the language of commutative algebra appeared in 2005: for a finitely generated persistence module C {\displaystyle C} with field F {\displaystyle F} coeffici... | Topological Data Analysis | 0.813246 |
17,017 | A comparison between these tools is done by Otter et al. Giotto-tda is a Python package dedicated to integrating TDA in the machine learning workflow by means of a scikit-learn API. An R package TDA is capable of calculating recently invented concepts like landscape and the kernel distance estimator. The Topology Tool... | Topological Data Analysis | 0.813246 |
17,018 | The first algorithm over all fields for persistent homology in algebraic topology setting was described by Barannikov through reduction to the canonical form by upper-triangular matrices. The algorithm for persistent homology over F 2 {\displaystyle F_{2}} was given by Edelsbrunner et al. Zomorodian and Carlsson gave t... | Topological Data Analysis | 0.813246 |
17,019 | Notably, mutual-informations generalize correlation coefficient and covariance to non-linear statistical dependences. These approaches were developed independently and only indirectly related to persistence methods, but may be roughly understood in the simplicial case using Hu Kuo Tin Theorem that establishes one-to-on... | Topological Data Analysis | 0.813246 |
17,020 | Normal persistence homology studies real-valued functions. The circle-valued map might be useful, "persistence theory for circle-valued maps promises to play the role for some vector fields as does the standard persistence theory for scalar fields", as commented in Dan Burghelea et al. The main difference is that Jorda... | Topological Data Analysis | 0.813246 |
17,021 | This is motivated by theoretical work in TDA, since the Reeb graph is related to Morse theory and MAPPER is derived from it. The proof of this theorem relies on the interleaving distance. | Topological Data Analysis | 0.813246 |
17,022 | There are some attempts to loosen the stricter restriction of the function. Please refer to the Categorification and cosheaves and Impact on mathematics sections for more information. It's natural to extend persistence homology to other basic concepts in algebraic topology, such as cohomology and relative homology/coho... | Topological Data Analysis | 0.813246 |
17,023 | In order to apply tools from machine learning, the information obtained from TDA should be represented in vector form. An ongoing and promising attempt is the persistence landscape discussed above. Another attempt uses the concept of persistence images. However, one problem of this method is the loss of stability, sinc... | Topological Data Analysis | 0.813246 |
17,024 | Short persistence has also been found to be useful, despite the common belief that noise is the cause of the phenomena. This is interesting to the mathematical theory.One of the main fields of data analysis today is machine learning. Some examples of machine learning in TDA can be found in Adcock et al. A conference is... | Topological Data Analysis | 0.813246 |
17,025 | However, persistent topology is able to record the birth (appearance) and death (disappearance) of topological features, thus extra geometric information is embedded in it. One evidence in theory is a partially positive result on the uniqueness of reconstruction of curves; two in application are on the quantitative ana... | Topological Data Analysis | 0.813246 |
17,026 | There are several notable interesting features of the recent applications of TDA: Combining tools from several branches of mathematics. Besides the obvious need for algebra and topology, partial differential equations, algebraic geometry, representation theory, statistics, combinatorics, and Riemannian geometry have al... | Topological Data Analysis | 0.813246 |
17,027 | An interval module k I {\displaystyle k_{I}} assigns to each element s ∈ I {\displaystyle s\in I} the vector space k {\displaystyle k} and assigns the zero vector space to elements in R ∖ I {\displaystyle \mathbb {R} \setminus I} . All maps ρ s t {\displaystyle \rho _{s}^{t}} are the zero map, unless s , t ∈ I {\displa... | Topological Data Analysis | 0.813246 |
17,028 | The most notable result is done by Crawley-Boevey, which solved the case of R {\displaystyle \mathbb {R} } . Crawley-Boevey's theorem states that any pointwise finite-dimensional persistence module is a direct sum of interval modules.To understand the definition of his theorem, some concepts need introducing. An interv... | Topological Data Analysis | 0.813246 |
17,029 | However, this argument fails if the indexing set is ( R , ≤ ) {\displaystyle (\mathbb {R} ,\leq )} .In general, not every persistence module can be decomposed into intervals. Many attempts have been made at relaxing the restrictions of the original structure theorem. The case for pointwise finite-dimensional persistenc... | Topological Data Analysis | 0.813246 |
17,030 | The structure theorem is of central importance to TDA; as commented by G. Carlsson, "what makes homology useful as a discriminator between topological spaces is the fact that there is a classification theorem for finitely generated abelian groups." (see the fundamental theorem of finitely generated abelian groups). The... | Topological Data Analysis | 0.813246 |
17,031 | The proof of the structure theorem relies on the base domain being field, so not many attempts have been made on persistence homology with torsion. Frosini defined a pseudometric on this specific module and proved its stability. One of its novelty is that it doesn't depend on some classification theory to define the me... | Topological Data Analysis | 0.813246 |
17,032 | Persistent homology has been applied to many types of data across many fields. Moreover, its mathematical foundation is also of theoretical importance. The unique features of TDA make it a promising bridge between topology and geometry. | Topological Data Analysis | 0.813246 |
17,033 | TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. | Topological Data Analysis | 0.813246 |
17,034 | In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is i... | Topological Data Analysis | 0.813246 |
17,035 | This restriction means that the output is in the form of a complex network. Because the topology of a finite point cloud is trivial, clustering methods (such as single linkage) are used to produce the analogue of connected sets in the preimage f − 1 ( U ) {\displaystyle f^{-1}(U)} when MAPPER is applied to actual data.... | Topological Data Analysis | 0.813246 |
17,036 | The basic workflow in TDA is: If X {\displaystyle X} is a point cloud, replace X {\displaystyle X} with a nested family of simplicial complexes X r {\displaystyle X_{r}} (such as the Čech or Vietoris-Rips complex). This process converts the point cloud into a filtration of simplicial complexes. Taking the homology of e... | Topological Data Analysis | 0.813246 |
17,037 | And let f , g: X → Y {\textstyle f,g:X\to Y} (not necessarily continuous) be functions, and F , G {\textstyle F,G} to be the corresponding persistence diagram. Then d I ( F , G ) ≤ d ∞ ( f , g ) := sup x ∈ X d Y ( f ( x ) , g ( x ) ) {\displaystyle d_{I}(F,G)\leq d_{\infty }(f,g):=\sup _{x\in X}d_{Y}(f(x),g(x))} .These... | Topological Data Analysis | 0.813246 |
17,038 | The isometry theorem asserts that the interleaving distance d I {\displaystyle d_{I}} is equal to the bottleneck distance. Bubenik et al. have abstracted the definition to that between functors F , G: P → D {\displaystyle F,G:P\to D} when P {\textstyle P} is equipped with a sublinear projection or superlinear family, i... | Topological Data Analysis | 0.813246 |
17,039 | Stability is of central importance to data analysis, since real data carry noises. By usage of category theory, Bubenik et al. have distinguished between soft and hard stability theorems, and proved that soft cases are formal. Specifically, general workflow of TDA is The soft stability theorem asserts that H F {\displa... | Topological Data Analysis | 0.813246 |
17,040 | Random variables are considered as partitions of the n atomic probabilities (seen as a probability (n-1)-simplex, | Ω | = n {\displaystyle |\Omega |=n} ) on the lattice of partitions ( Π n {\displaystyle \Pi _{n}} ). The random variables or modules of measurable functions provide the cochain complexes while the cobound... | Topological Data Analysis | 0.813246 |
17,041 | The study of random simplicial complexes offers some insight into statistical topology. K. Turner et al. offers a summary of work in this vein. A second way is to study probability distributions on the persistence space. | Topological Data Analysis | 0.813246 |
17,042 | Real data is always finite, and so its study requires us to take stochasticity into account. Statistical analysis gives us the ability to separate true features of the data from artifacts introduced by random noise. Persistent homology has no inherent mechanism to distinguish between low-probability features and high-p... | Topological Data Analysis | 0.813246 |
17,043 | Precursors to the full concept of persistent homology appeared gradually over time. In 1990, Patrizio Frosini introduced a pseudo-distance between submanifolds, and later the size function, which on 1-dim curves is equivalent to the 0th persistent homology. Nearly a decade later, Vanessa Robins studied the images of ho... | Topological Data Analysis | 0.813246 |
17,044 | Roughly speaking, sheaves are the mathematical tool for understanding how local information determines global information. Justin Curry regards level set persistence as the study of fibers of continuous functions. The objects that he studies are very similar to those by MAPPER, but with sheaf theory as the theoretical ... | Topological Data Analysis | 0.813246 |
17,045 | The essential relationship between Cech and Rips complexes can be seen much more clearly in categorical language.The language of category theory also helps cast results in terms recognizable to the broader mathematical community. Bottleneck distance is widely used in TDA because of the results on stability with respect... | Topological Data Analysis | 0.813246 |
17,046 | The understanding of the correspondence between interleaving and matching is of huge importance, since matching has been the method used in the beginning (modified from Morse theory). A summary of works can be found in Vin de Silva et al. Many theorems can be proved much more easily in a more intuitive setting. | Topological Data Analysis | 0.813246 |
17,047 | One advantage of category theory is its ability to lift concrete results to a higher level, showing relationships between seemingly unconnected objects. Bubenik et al. offers a short introduction of category theory fitted for TDA. Category theory is the language of modern algebra, and has been widely used in the study ... | Topological Data Analysis | 0.813246 |
17,048 | It provides a stable and complete representation of PBNs. An ongoing work by Carlsson et al. is trying to give geometric interpretation of persistent homology, which might provide insights on how to combine machine learning theory with topological data analysis.The first practical algorithm to compute multidimensional ... | Topological Data Analysis | 0.813246 |
17,049 | , n ) {\displaystyle s_{i}\leq t_{i},i=1,...,n)} maps satisfy ρ r t = ρ s t ∘ ρ r s {\displaystyle \rho _{r}^{t}=\rho _{s}^{t}\circ \rho _{r}^{s}} for all r ≤ s ≤ t {\displaystyle r\leq s\leq t} It might be worth noting that there are controversies on the definition of multidimensional persistence.One of the advantages... | Topological Data Analysis | 0.813246 |
17,050 | Their definition presents multidimensional persistence with n parameters as a Z n {\displaystyle \mathbb {Z} ^{n}} graded module over a polynomial ring in n variables. Tools from commutative and homological algebra are applied to the study of multidimensional persistence in work of Harrington-Otter-Schenck-Tillman. The... | Topological Data Analysis | 0.813246 |
17,051 | Multidimensional persistence is important to TDA. The concept arises in both theory and practice. The first investigation of multidimensional persistence was early in the development of TDA,. Carlsson-Zomorodian introduced the theory of multidimensional persistence in and in collaboration with Singh introduced the use ... | Topological Data Analysis | 0.813246 |
17,052 | More than one way exists to classify the applications of TDA. Perhaps the most natural way is by field. A very incomplete list of successful applications includes data skeletonization, shape study, graph reconstruction, image analysis, material, progression analysis of disease, sensor network, signal analysis, cosmic w... | Topological Data Analysis | 0.813246 |
17,053 | Probability logic is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false... | Logical deduction | 0.813245 |
17,054 | If the constraints of an optimization problem are mutually contradictory, there are no points that satisfy all the constraints and thus the feasible region is the empty set. In this case the problem has no solution and is said to be infeasible. | Feasible solution | 0.813243 |
17,055 | In optimization and other branches of mathematics, and in search algorithms (a topic in computer science), a candidate solution is a member of the set of possible solutions in the feasible region of a given problem. A candidate solution does not have to be a likely or reasonable solution to the problem—it is simply in ... | Feasible solution | 0.813243 |
17,056 | His 1951 paper used the simpler Brouwer fixed-point theorem for the same purpose.Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many solution concepts ('refinements' of Nash equilibria) designed to rule out im... | Nash Equilibria | 0.81324 |
17,057 | Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. | Nash Equilibria | 0.81324 |
17,058 | In 1971, Robert Wilson came up with the Oddness Theorem, which says that "almost all" finite games have a finite and odd number of Nash equilibria. In 1993, Harsanyi published an alternative proof of the result. "Almost all" here means that any game with an infinite or even number of equilibria is very special in the s... | Nash Equilibria | 0.813239 |
17,059 | Let (M, g) be a compact symmetric Riemannian space of rank greater than one. If F is a symmetric C 2 {\displaystyle C^{2}} Finsler metric whose geodesics coincide with geodesics of the Riemannian metric g, then (M, g) is a symmetric Finsler space. The analogue of this theorem for rank-one symmetric spaces has not been ... | Hilbert's fourth problem | 0.813239 |
17,060 | The multi-dimensional case of the Fourth Hilbert problem was studied by Szabo. In 1986, he proved, as he wrote, the generalized Pogorelov theorem. Theorem. | Hilbert's fourth problem | 0.813239 |
17,061 | Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry ...a geometry in which all the axioms of ordinary euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the... | Hilbert's fourth problem | 0.813239 |
17,062 | Georg Hamel was first to contribute to the solution of Hilbert's fourth problem. He proved the following statement. Theorem. A regular Finsler metric F ( x , y ) = F ( x 1 , … , x n , y 1 , … , y n ) {\displaystyle F(x,y)=F(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})} is flat if and only if it satisfies the conditions: ∂ ... | Hilbert's fourth problem | 0.813239 |
17,063 | Theorem. In the three-dimensional case any complete continuous flat metric is a σ {\displaystyle \sigma } -metric in generalized meaning. Busemann, in his review to Pogorelov’s book "Hilbert’s Fourth Problem" wrote, "In the spirit of the time Hilbert restricted himself to n = 2, 3 and so does Pogorelov. | Hilbert's fourth problem | 0.813239 |
17,064 | For three dimensional case Pogorelov proved the following theorem. Theorem. Any three-dimensional regular complete flat metric is a σ {\displaystyle \sigma } -metric. However, in the three-dimensional case σ {\displaystyle \sigma } -measures can take either positive or negative values. | Hilbert's fourth problem | 0.813239 |
17,065 | Desargues's theorem: If two triangles lie on a plane such that the lines connecting corresponding vertices of the triangles meet at one point, then the three points, at which the prolongations of three pairs of corresponding sides of the triangles intersect, lie on one line. The necessary condition for solving Hilbert'... | Hilbert's fourth problem | 0.813239 |
17,066 | Before 1900, there was known the Cayley–Klein model of Lobachevsky geometry in the unit disk, according to which geodesic lines are chords of the disk and the distance between points is defined as a logarithm of the cross-ratio of a quadruple. For two-dimensional Riemannian metrics, Eugenio Beltrami (1835–1900) proved ... | Hilbert's fourth problem | 0.813239 |
17,067 | In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with th... | Hilbert's fourth problem | 0.813239 |
17,068 | The following theorem was proved by Pogorelov in 1973Theorem. Any two-dimensional continuous complete flat metric is a σ {\displaystyle \sigma } -metric. Thus Hilbert's fourth problem for the two-dimensional case was completely solved. A consequence of this is that you can glue boundary to boundary two copies of the sa... | Hilbert's fourth problem | 0.813239 |
17,069 | In 1966, in his talk at the International Mathematical Congress in Moscow, Herbert Busemann introduced a new class of flat metrics. On a set of lines on the projective plane R P 2 {\displaystyle RP^{2}} he introduced a completely additive non-negative measure σ {\displaystyle \sigma } , which satisfies the following co... | Hilbert's fourth problem | 0.813239 |
17,070 | {\displaystyle F_{U}(x,y)={\frac {1}{2}}\|y\|_{\mathbb {E} }\left({\frac {1}{\|x-x_{+}\|_{\mathbb {E} }}}+{\frac {1}{\|x-x_{-}\|_{\mathbb {E} }}}\right).} The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of the Lobachevsky geometry. If the hypersurface ∂ U {\displaystyle \pa... | Hilbert's fourth problem | 0.813239 |
17,071 | Let U ⊂ ( E n + 1 , ‖ ⋅ ‖ E ) {\displaystyle U\subset (\mathbb {E} ^{n+1},\|\cdot \|_{\mathbb {E} })} be a bounded open convex set with the boundary of class C2 and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface ∂ U {\displaystyle \partial U} is called the absolute of Hilbert's geometr... | Hilbert's fourth problem | 0.813239 |
17,072 | A useful technique for treating the volume integral without discretising the volume is the dual-reciprocity method. The technique approximates part of the integrand using radial basis functions (local interpolating functions) and converts the volume integral into boundary integral after collocating at selected points d... | Boundary element | 0.81323 |
17,073 | The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form), including fluid mechanics, acoustics, electromagnetics (where the technique is known as method of moments or abbreviat... | Boundary element | 0.81323 |
17,074 | The sequential limits on parallel performance dictated by Amdahl's law also do not apply in the same way because data dependencies are implicitly handled by the programmable node interconnect. Therefore, systolic arrays are extremely good at artificial intelligence, image processing, pattern recognition, computer visio... | Multiple instruction, single data | 0.813229 |
17,075 | In physics, drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an average velocity of zero. Applying an electric field adds to this random moti... | Electron velocity | 0.813226 |
17,076 | The Gershgorin circle theorem applies the companion matrix of the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing a root of the polynomial; see Durand–Kerner method § Root inclusion via Gerschgorin's circles for details. If the interpolation ... | Properties of polynomial roots | 0.813217 |
17,077 | The Gauss–Lucas theorem states that the convex hull of the roots of a polynomial contains the roots of the derivative of the polynomial. A sometimes useful corollary is that, if all roots of a polynomial have positive real part, then so do the roots of all derivatives of the polynomial. A related result is Bernstein's ... | Properties of polynomial roots | 0.813217 |
17,078 | Both bounds result from the Gershgorin circle theorem applied to the companion matrix of the polynomial and its transpose. They can also be proved by elementary methods. These bounds are not invariant by scaling. | Properties of polynomial roots | 0.813217 |
17,079 | In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plan... | Properties of polynomial roots | 0.813217 |
17,080 | The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib). It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjug... | Properties of polynomial roots | 0.813217 |
17,081 | The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than the circle), then it was called solid; t... | Geometric construction | 0.813216 |
17,082 | Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837 using field theory, namely trisecting an arbitrary angle and doubling the volume of a cube (see § impossible constructions). Many of these problems are easily solvable provided that other geometric transformations... | Geometric construction | 0.813216 |
17,083 | Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are no... | Geometric construction | 0.813216 |
17,084 | In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, ... | Geometric construction | 0.813216 |
17,085 | Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries. The role of primitive notions, or undefined concepts, was clearly put forward by Alessandro Padoa of the Peano delegation at the 1900 Paris conference: ...when we begin to formulate the theory, we can imagine that... | Classical geometry | 0.813215 |
17,086 | In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2x + 1 (a line), or x2 + y2 = 7 (a circle). Also in the 17th century, Girard Desargues, motivated by the theory of perspective, introduced th... | Classical geometry | 0.813215 |
17,087 | René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra.In this approach, a point on a plane is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, ... | Classical geometry | 0.813215 |
17,088 | Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials science to study molecular systems ranging from small chemical systems to large b... | Molecular Modelling | 0.813209 |
17,089 | In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems. | Symbolic-numeric computation | 0.813206 |
17,090 | 19th-century attempts to algebraize logic, such as the work of Boole (1815–1864) and Venn (1834–1923), typically yielded systems highly influenced by the term-logic tradition. The first predicate logic was that of Frege's landmark Begriffsschrift (1879), little read before 1950, in part because of its eccentric notatio... | Term logic | 0.813201 |
17,091 | Term logic began to decline in Europe during the Renaissance, when logicians like Rodolphus Agricola Phrisius (1444–1485) and Ramus (1515–1572) began to promote place logics. The logical tradition called Port-Royal Logic, or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, ... | Term logic | 0.813201 |
17,092 | There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to ... | Combinatorics on words | 0.813189 |
17,093 | Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer sc... | Combinatorics on words | 0.813189 |
17,094 | His theorem states that if the positive integers are partitioned into k classes, then there exists a class c such that c contains an arithmetic progression of some unknown length. An arithmetic progression is a sequence of numbers in which the difference between adjacent numbers remains constant.When examining unavoida... | Combinatorics on words | 0.813189 |
17,095 | Patterns can be either avoidable patterns, or unavoidable. A significant contributor to the work of unavoidable patterns, or regularities, was Frank Ramsey in 1930. His important theorem states that for integers k, m≥2, there exists a least positive integer R(k,m) such that despite how a complete graph is colored with ... | Combinatorics on words | 0.813189 |
17,096 | Necklaces are constructed from words of circular sequences. They are most frequently used in music and astronomy. Flye Sainte-Marie in 1894 proved there are 22n−1−n binary de Bruijn necklaces of length 2n. A de Bruijn necklace contains factors made of words of length n over a certain number of letters. | Combinatorics on words | 0.813189 |
17,097 | However, μ2 does not equal its identity component because the fiber over every point of Spec(Z) except 2 consists of two discrete points.An algebraic group G over a topological field K admits two natural topologies, the Zariski topology and the topology inherited from K. The identity component of G often changes depend... | Component group | 0.813182 |
17,098 | The Weyl group of a reductive algebraic group G is the components group of the normalizer group of a maximal torus of G. Consider the group scheme μ2 = Spec(Z/(x2 - 1)) of second roots of unity defined over the base scheme Spec(Z). Topologically, μn consists of two copies of the curve Spec(Z) glued together at the poin... | Component group | 0.813182 |
17,099 | The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is ({1,−1},•). Consider the group of units U in the ring of split-complex numbers. In the ordinary topology of the plane {z = x + j y: x, y ∈ R}, U is divided into four components by the lines y = x and y = − x ... | Component group | 0.813182 |
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