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13,100 | Green Chemistry is a monthly peer-reviewed scientific journal covering every aspect of sustainable chemistry and its implementation in chemical engineering. It is published by the Royal Society of Chemistry and was established in 1999 by James Clark (University of York). Articles published in this journal are intended ... | Green Chem | 0.816521 |
13,101 | Research papers (which contain original scientific work that has not been published previously) Communications (original scientific work that is of an urgent nature and that has not been published previously) Green Chemistry news (an easy-to-read magazine style section) | Green Chem | 0.816521 |
13,102 | Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB. It then is argued that angle β has larger measure than angle ... | Triangular inequality | 0.816519 |
13,103 | The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (th... | Triangular inequality | 0.816519 |
13,104 | {\displaystyle {\overline {\mathrm {AC} }}>{\overline {\mathrm {AD} }}\ .} A similar construction shows AC > DC, establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B: (i) as depicted (which is to be proved), or (ii) B coincident with... | Triangular inequality | 0.816519 |
13,105 | In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the... | Triangular inequality | 0.816519 |
13,106 | In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confuse... | Multinomial probit | 0.816517 |
13,107 | The following are equivalent: ( X , T ) {\displaystyle (X,T)} has the small-boundary property. mdim ( X , T ) = 0 {\displaystyle \operatorname {mdim} (X,T)=0} , where mdim {\displaystyle \operatorname {mdim} } denotes mean dimension. For every ε > 0 {\displaystyle \varepsilon >0} , x ≠ y ∈ X {\displaystyle x\neq y\in... | Small boundary property | 0.816516 |
13,108 | It is called infinite if | X | = ∞ {\displaystyle |X|=\infty } . Lindenstrauss introduced SBP and proved:Theorem: Let ( X , T ) {\displaystyle (X,T)} be an extension of an infinite minimal system. | Small boundary property | 0.816516 |
13,109 | The field of p-adic (or nonarchimedean) dynamics is the study of classical dynamical questions over a field K that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of p-adic rationals Qp and the completion of its algebraic closure Cp. The metric on K and the standard de... | Arithmetic dynamics | 0.816512 |
13,110 | Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Douglas Northcott says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The uniform boundedness conjecture for preperiodic points of Patrick Morton and Jose... | Arithmetic dynamics | 0.816512 |
13,111 | The Journal of Statistics Education is a double-blind peer-reviewed scientific journal. It was established in 1992 at North Carolina State University by E. Jacquelin Dietz, and became an official publication of the American Statistical Association beginning in 1999. It is published online three times a year, covering s... | Journal of Statistics Education | 0.816511 |
13,112 | The journal covers curricular reform in statistics, efficacy of group learning and joint projects, innovative pedagogical techniques, case studies, general understanding of probability and statistics, statistical literacy, research on the teaching of statistics, attitudes and beliefs about statistics, the use of comput... | Journal of Statistics Education | 0.816511 |
13,113 | Gundissalinus defined natural science as "the science considering only things unabstracted and with motion," as opposed to mathematics and sciences that rely on mathematics. Following Al-Farabi, he then separated the sciences into eight parts, including physics, cosmology, meteorology, minerals science, and plant and a... | Natural scientist | 0.816511 |
13,114 | In the Byzantine Empire, John Philoponus, an Alexandrian Aristotelian commentator and Christian theologian was the first who questioned Aristotle's teaching of physics. Unlike Aristotle who based his physics on verbal argument, Philoponus instead relied on observation and argued for observation rather than resorting to... | Natural scientist | 0.816511 |
13,115 | According to a famous 1923 textbook, Thermodynamics and the Free Energy of Chemical Substances, by the American chemist Gilbert N. Lewis and the American physical chemist Merle Randall, the natural sciences contain three great branches: Aside from the logical and mathematical sciences, there are three great branches of... | Natural scientist | 0.816511 |
13,116 | Thales of Miletus, an early philosopher who lived from 625 to 546 BC, explained earthquakes by theorizing that the world floated on water and that water was the fundamental element in nature. In the 5th century BC, Leucippus was an early exponent of atomism, the idea that the world is made up of fundamental indivisible... | Natural scientist | 0.816511 |
13,117 | Some scholars trace the origins of natural science as far back as pre-literate human societies, where understanding the natural world was necessary for survival. People observed and built up knowledge about the behavior of animals and the usefulness of plants as food and medicine, which was passed down from generation ... | Natural scientist | 0.816511 |
13,118 | He introduced the theory of impetus. John Philoponus' criticism of Aristotelian principles of physics served as inspiration for Galileo Galilei during the Scientific Revolution.A revival in mathematics and science took place during the time of the Abbasid Caliphate from the 9th century onward, when Muslim scholars expa... | Natural scientist | 0.816511 |
13,119 | Aristotle's works were influential through the 16th century, and he is considered to be the father of biology for his pioneering work in that science. He also presented philosophies about physics, nature, and astronomy using inductive reasoning in his works Physics and Meteorology. While Aristotle considered natural ph... | Natural scientist | 0.816511 |
13,120 | In molecular biology, the calcium-activated potassium channel beta subunit is a family of proteins comprising the beta subunits of calcium-activated potassium channels. The functional diversity of potassium channels can arise through homo- or hetero-associations of alpha subunits or association with auxiliary cytoplasm... | Calcium-activated potassium channel beta subunit | 0.816509 |
13,121 | A control engineer's career starts with a bachelor's degree and can continue through the college process. Control engineer degrees are well paired with an electrical or mechanical engineering degree. Control engineers usually get jobs in technical managing where they typically lead interdisciplinary projects. | Control system engineering | 0.816508 |
13,122 | American Journal of Physics. 64 (6): 694–98. Bibcode:1996AmJPh..64..694D. doi:10.1119/1.18234. | Quantum Electrodynamics | 0.816508 |
13,123 | Therefore, P(A to B) consists of 16 complex numbers, or probability amplitude arrows. : 120–121 There are also some minor changes to do with the quantity j, which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping. Associated with the fact that the... | Quantum Electrodynamics | 0.816508 |
13,124 | Quantum mechanics introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the square modulus of probability amplitudes, which are complex numbers. Feynman avoids ... | Quantum Electrodynamics | 0.816508 |
13,125 | This is also known as the curvature of the gauge field.Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field B μ {\displaystyle B_{\mu }} set to zero for simplicity) L = − 1 4 F μ ν F μ ν + ψ ¯ ( i γ μ ∂ μ − m ) ψ − e j μ A μ {\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\m... | Quantum Electrodynamics | 0.816508 |
13,126 | Statistics and Computing is a peer-reviewed academic journal that deals with statistics and computing. It was established in 1991 and is published by Springer. | Statistics and Computing | 0.816507 |
13,127 | Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space R 3 . {\displaystyle \mathbb {R} ^{3}.} The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calc... | Vector Calculus | 0.816507 |
13,128 | Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator ( ∇ {\displaystyle \nabla } ), also known as "nabla". The three basic vector operators are: Also commonly used are the two Laplace operators: A quantity called the Jacobi... | Vector Calculus | 0.816507 |
13,129 | Vector calculus is particularly useful in studying: Center of mass Field theory Kinematics Maxwell's equations | Vector Calculus | 0.816507 |
13,130 | From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifie... | Vector Calculus | 0.816507 |
13,131 | Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions. The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, a... | Vector Calculus | 0.816507 |
13,132 | The first, geometric algebra, uses k-vector fields instead of vector fields (in 3 or fewer dimensions, every k-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector field... | Vector Calculus | 0.816507 |
13,133 | In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field... | Vector Calculus | 0.816507 |
13,134 | Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by... | Vector Calculus | 0.816507 |
13,135 | For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rn) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values ... | Vector Calculus | 0.816507 |
13,136 | Vector calculus is initially defined for Euclidean 3-space, R 3 , {\displaystyle \mathbb {R} ^{3},} which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of angle, and ... | Vector Calculus | 0.816507 |
13,137 | Vector calculus can also be generalized to other 3-manifolds and higher dimension spaces. | Vector Calculus | 0.816507 |
13,138 | The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions: In two dimensions, the divergence and curl theorems reduce to the Green's theorem: | Vector Calculus | 0.816507 |
13,139 | The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of: Also commonly used are the two triple products: | Vector Calculus | 0.816507 |
13,140 | In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7 (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (gen... | Vector Calculus | 0.816507 |
13,141 | Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while cu... | Vector Calculus | 0.816507 |
13,142 | They have applications in: image and video recognition, recommender systems, image classification, image segmentation, medical image analysis, natural language processing, brain–computer interfaces, and financial time series.CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Networks (SIANN), b... | Pooling (neural networks) | 0.816501 |
13,143 | Convolutional neural network (CNN) is a regularized type of feed-forward neural network that learns feature engineering by itself via filters (or kernel) optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over few... | Pooling (neural networks) | 0.816501 |
13,144 | Theano: The reference deep-learning library for Python with an API largely compatible with the popular NumPy library. Allows user to write symbolic mathematical expressions, then automatically generates their derivatives, saving the user from having to code gradients or backpropagation. These symbolic expressions are a... | Pooling (neural networks) | 0.816501 |
13,145 | Microsoft Cognitive Toolkit: A deep learning toolkit written by Microsoft with several unique features enhancing scalability over multiple nodes. It supports full-fledged interfaces for training in C++ and Python and with additional support for model inference in C# and Java. TensorFlow: Apache 2.0-licensed Theano-like... | Pooling (neural networks) | 0.816501 |
13,146 | Allows the creation of custom layers. Integrates with Hadoop and Kafka. Dlib: A toolkit for making real world machine learning and data analysis applications in C++. | Pooling (neural networks) | 0.816501 |
13,147 | CNNs have been used in computer Go. In December 2014, Clark and Storkey published a paper showing that a CNN trained by supervised learning from a database of human professional games could outperform GNU Go and win some games against Monte Carlo tree search Fuego 1.1 in a fraction of the time it took Fuego to play. La... | Pooling (neural networks) | 0.816501 |
13,148 | It makes the weight vectors sparse during optimization. In other words, neurons with L1 regularization end up using only a sparse subset of their most important inputs and become nearly invariant to the noisy inputs. L1 with L2 regularization can be combined; this is called elastic net regularization. | Pooling (neural networks) | 0.816501 |
13,149 | Deeplearning4j: Deep learning in Java and Scala on multi-GPU-enabled Spark. A general-purpose deep learning library for the JVM production stack running on a C++ scientific computing engine. | Pooling (neural networks) | 0.816501 |
13,150 | Caffe: A library for convolutional neural networks. Created by the Berkeley Vision and Learning Center (BVLC). It supports both CPU and GPU. Developed in C++, and has Python and MATLAB wrappers. | Pooling (neural networks) | 0.816501 |
13,151 | In neural networks, each neuron receives input from some number of locations in the previous layer. In a convolutional layer, each neuron receives input from only a restricted area of the previous layer called the neuron's receptive field. Typically the area is a square (e.g. 5 by 5 neurons). Whereas, in a fully connec... | Pooling (neural networks) | 0.816501 |
13,152 | A well studied complex is tris(glycinato)cobalt(III). It is produced by the reaction of glycine with sodium tris(carbonato)cobalt(III). Similar synthetic methods apply to the preparation of tris(chelates) of other amino acids.Commonly amino acid complexes are prepared by ligand displacement reactions of metal aquo comp... | Transition metal amino acid complexes | 0.816501 |
13,153 | Mixing simple metal salts with solutions of amino acids near neutral or elevated pH often affords bis- or tris complexes. For metal ions that prefer octahedral coordination, these complexes often adopt the stoichiometry M(aa)3 (aa = amino carboxylate, such as glycinate, H2NCH2CO2−). Complexes of the 3:1 stoichiometry h... | Transition metal amino acid complexes | 0.816501 |
13,154 | In discrete-time Markov Decision Processes, decisions are made at discrete time intervals. However, for continuous-time Markov decision processes, decisions can be made at any time the decision maker chooses. In comparison to discrete-time Markov decision processes, continuous-time Markov decision processes can better ... | Markov Decision Process | 0.8165 |
13,155 | Another application of MDP process in machine learning theory is called learning automata. This is also one type of reinforcement learning if the environment is stochastic. The first detail learning automata paper is surveyed by Narendra and Thathachar (1974), which were originally described explicitly as finite state ... | Markov Decision Process | 0.8165 |
13,156 | They are used in many disciplines, including robotics, automatic control, economics and manufacturing. The name of MDPs comes from the Russian mathematician Andrey Markov as they are an extension of Markov chains. At each time step, the process is in some state s {\displaystyle s} , and the decision maker may choose an... | Markov Decision Process | 0.8165 |
13,157 | In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via ... | Markov Decision Process | 0.8165 |
13,158 | Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical comm... | Minimum Sudoku problem | 0.816489 |
13,159 | Gary McGuire proved a minimum uniquely solvable Sudoku requires 17 clues. Symbolic validation (via computer algebra) of conjectures to motivate the search for an analytical proof Solutions to a special case of the quantum three-body problem known as the hydrogen molecule-ion were found standard quantum chemistry basis ... | Minimum Sudoku problem | 0.816489 |
13,160 | Various proofs of the four colour theorem. Clement Lam's proof of the non-existence of a finite projective plane of order 10. | Minimum Sudoku problem | 0.816489 |
13,161 | Mathematicians have always practiced experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing result... | Minimum Sudoku problem | 0.816489 |
13,162 | It is fundamental in chemical reactions of the form A+B→ C+D that two products are necessarily generated though product C may have been the desired one. That being the case, D is considered a byproduct. As it is a significant goal of green chemistry to maximize the efficiency of the reactants and minimize the productio... | Atom efficiency | 0.816487 |
13,163 | The concept of molecular gastronomy was perhaps presaged by Marie-Antoine Carême, one of the most famous French chefs, who said in the early 19th century that when making a food stock "the broth must come to a boil very slowly, otherwise the albumin coagulates, hardens; the water, not having time to penetrate the meat,... | Molecular cuisine | 0.816485 |
13,164 | Though rarely credited, the origins of the Erice workshops (originally entitled "Science and Gastronomy") can be traced back to cooking teacher Elizabeth Cawdry Thomas, who studied at Le Cordon Bleu in London and ran a cooking school in Berkeley, California. The one-time wife of a physicist, Thomas had many friends in ... | Molecular cuisine | 0.816485 |
13,165 | In 1932, Belle Lowe, then the professor of Food and Nutrition at Iowa State College, published a book titled Experimental Cookery: From The Chemical And Physical Standpoint which became a standard textbook for home economics courses across the United States. The book is an exhaustively researched look into the science ... | Molecular cuisine | 0.816485 |
13,166 | French writer Raymond Roussel, in his 1914 story "L'Allée aux lucioles" ("The Alley of Fireflies"), introduces a fictionalized version of French chemist Antoine de Lavoisier who, in the story, creates an apparently edible semi-permeable coating ("invol ...") that he uses to encase a tiny frozen sculpture made from one ... | Molecular cuisine | 0.816485 |
13,167 | In the late 1990s and early 2000s, the term started to be used to describe a new style of cooking in which some chefs began to explore new possibilities in the kitchen by embracing science, research, technological advances in equipment and various natural gums and hydrocolloids produced by the commercial food processin... | Molecular cuisine | 0.816485 |
13,168 | In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or bin... | Binary exponentiation | 0.816484 |
13,169 | A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In a mathematical sense, these wave functions are the basis set of... | Linear combination of atomic orbitals | 0.816479 |
13,170 | A strain gauge takes advantage of the physical property of electrical conductance and its dependence on the conductor's geometry. When an electrical conductor is stretched within the limits of its elasticity such that it does not break or permanently deform, it will become narrower and longer, which increases its elect... | Strain gauges | 0.816472 |
13,171 | As alternative to piezo-resistant strain gauges, integrated optical ring resonators may be used to measure strain in microoptoelectromechanical systems (MOEMS). Capacitive strain gauges use a variable capacitor to indicate the level of mechanical deformation. Vibrating wire strain gauges are used in geotechnical and ci... | Strain gauges | 0.816472 |
13,172 | The setting where L = − K X {\displaystyle L=-K_{X}} is ample so that X {\displaystyle X} is a Fano manifold is of particular importance, and in that setting many tools are known to verify the K-stability of Fano varieties. For example using purely algebraic techniques it can be proven that all Fermat hypersurfaces F n... | K-stability | 0.816468 |
13,173 | It has been known since the original work of Pierre Deligne and David Mumford that smooth algebraic curves are asymptotically stable in the sense of geometric invariant theory, and in particular that they are K-stable. In this setting, the Yau–Tian–Donaldson conjecture is equivalent to the uniformization theorem. Namel... | K-stability | 0.816468 |
13,174 | That is, we have ⋃ i ≥ 0 F i R = R {\displaystyle \bigcup _{i\geq 0}F_{i}R=R} .Given a filtration χ {\displaystyle \chi } , its Rees algebra is defined by Rees ( χ ) = ⨁ i ≥ 0 ( F i R ) t i ⊂ R . {\displaystyle \operatorname {Rees} (\chi )=\bigoplus _{i\geq 0}(F_{i}R)t^{i}\subset R.} We say that a filtration is fini... | K-stability | 0.816468 |
13,175 | Define the weight function of the test configuration ( X , L ) {\displaystyle ({\mathcal {X}},{\mathcal {L}})} to be the function w ( k ) {\displaystyle w(k)} where w ( k ) {\displaystyle w(k)} is the total weight of the C ∗ {\displaystyle \mathbb {C} ^{*}} action on the vector space H 0 ( X 0 , L 0 k ) {\displaystyle ... | K-stability | 0.816468 |
13,176 | This definition is differential geometric and is directly related to the existence problems in Kähler geometry. Algebraic definitions were given by using Donaldson-Futaki invariants and CM-weights defined by intersection formula. By definition an action of C ∗ {\displaystyle \mathbb {C} ^{*}} on a polarised scheme come... | K-stability | 0.816468 |
13,177 | Yau suggested this stability condition should be an analogue of slope stability of vector bundles.In 1997, Tian suggested such a stability condition, which he called K-stability after the K-energy functional introduced by Toshiki Mabuchi. The K originally stood for kinetic due to the similarity of the K-energy function... | K-stability | 0.816468 |
13,178 | {\displaystyle \operatorname {YM} (\nabla )=\int _{X}\|F_{\nabla }\|^{2}\,d\operatorname {vol} .} On a Riemann surface such a connection is projectively flat, and its holonomy gives rise to a projective unitary representation of the fundamental group of the Riemann surface, thus recovering the original statement of the... | K-stability | 0.816468 |
13,179 | In 1983, Donaldson produced a new proof of the Narasimhan–Seshadri theorem. As proved by Donaldson, the theorem states that a holomorphic vector bundle over a compact Riemann surface is stable if and only if it corresponds to an irreducible unitary Yang–Mills connection. That is, a unitary connection which is a critica... | K-stability | 0.816468 |
13,180 | In the case where c 1 ( X ) > 0 {\displaystyle c_{1}(X)>0} , that is when X {\displaystyle X} is a Fano manifold, a Kähler–Einstein metric does not always exist. Namely, it was known by work of Yozo Matsushima and André Lichnerowicz that a Kähler manifold with c 1 ( X ) > 0 {\displaystyle c_{1}(X)>0} can only admit a K... | K-stability | 0.816468 |
13,181 | K-stability was originally introduced as an algebro-geometric condition which should characterise the existence of a Kähler–Einstein metric on a Fano manifold. This came to be known as the Yau–Tian–Donaldson conjecture (for Fano manifolds). The conjecture was resolved in the 2010s in works of Xiuxiong Chen, Simon Donal... | K-stability | 0.816468 |
13,182 | K-stability is defined by analogy with the Hilbert–Mumford criterion from finite-dimensional geometric invariant theory. This theory describes the stability of points on polarised varieties, whereas K-stability concerns the stability of the polarised variety itself. The Hilbert–Mumford criterion shows that to test the ... | K-stability | 0.816468 |
13,183 | In this section we work over the complex numbers C {\displaystyle \mathbb {C} } , but the essential points of the definition apply over any field. A polarised variety is a pair ( X , L ) {\displaystyle (X,L)} where X {\displaystyle X} is a complex algebraic variety and L {\displaystyle L} is an ample line bundle on X {... | K-stability | 0.816468 |
13,184 | {\displaystyle \|({\mathcal {X}},{\mathcal {L}})\|^{2}=c_{0}-{\frac {b_{0}^{2}}{a_{0}}}.} According to the analogy with the Hilbert-Mumford criterion, once one has a notion of deformation (test configuration) and weight on the central fibre (Donaldson-Futaki invariant), one can define a stability condition, called K-st... | K-stability | 0.816468 |
13,185 | In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was... | K-stability | 0.816468 |
13,186 | As discussed, the Yau–Tian–Donaldson conjecture has been resolved in the Fano setting. It was proven by Donaldson in 2009 that the Yau–Tian–Donaldson conjecture holds for toric varieties of complex dimension 2. For arbitrary polarised varieties it was proven by Stoppa, also using work of Arezzo and Pacard, that the exi... | K-stability | 0.816468 |
13,187 | In molecular biology, STRING (Search Tool for the Retrieval of Interacting Genes/Proteins) is a biological database and web resource of known and predicted protein–protein interactions. The STRING database contains information from numerous sources, including experimental data, computational prediction methods and publ... | STRING | 0.816465 |
13,188 | Number theory covers the topics of the discrete quantities as numbers: number systems with their kinds and relations. Geometry studies the issues of spatial magnitudes: straight lines, curved lines, surfaces and solids, all with their respective measurements and relationships. A traditional Aristotelian realist philoso... | Quantity | 0.816458 |
13,189 | Microsatellite enrichment is a method in molecular biology used for enriching the amount of microsatellite sequences in a DNA sample. This can be achieved by designing oligonucleotide probes that hybridize with the repeats in the microsatellites and then pull out the probe/microsatellite complexes from the solution. Th... | Microsatellite enrichment | 0.816452 |
13,190 | The most common choice in optimization of parameters a i {\displaystyle a_{i}} is the root mean square criterion which is also called the autocorrelation criterion. In this method we minimize the expected value of the squared error E {\displaystyle E} , which yields the equation ∑ i = 1 p a i R ( j − i ) = R ( j ) , {... | Linear prediction | 0.816452 |
13,191 | Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples. In digital signal processing, linear prediction is often called linear predictive coding (LPC) and can thus be viewed as a subset of filter theory. In system analysis, a su... | Linear prediction | 0.816452 |
13,192 | The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is the Cartesian product of N intervals each of which is defined by the minimal and maximal value of the corresponding... | Oriented bounding box | 0.816446 |
13,193 | The following concepts and constructs have been used as interpretations of OOP concepts: co algebraic data types recursive types encapsulated state inheritance records are basis for understanding objects if function literals can be stored in fields (like in functional-programming languages), but the actual calculi need... | Object-oriented code | 0.816441 |
13,194 | This is equivalent to U and V being orthogonal as subspaces of H. The sum of the two projections PU and PV is a projection only if U and V are orthogonal to each other, and in that case PU + PV = PU+V. The composite PUPV is generally not a projection; in fact, the composite is a projection if and only if the two projec... | Hilbert space dimension | 0.81644 |
13,195 | The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator). The (geometrical) Hahn... | Hilbert space dimension | 0.81644 |
13,196 | Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous a... | Hilbert space dimension | 0.81644 |
13,197 | If Ω is a suitable domain, then one can define the Sobolev space Hs(Ω) as the space of Bessel potentials; roughly, Here Δ is the Laplacian and (1 − Δ)−s / 2 is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer s, this definition also has pa... | Hilbert space dimension | 0.81644 |
13,198 | Sobolev spaces, denoted by Hs or Ws, 2, are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenie... | Hilbert space dimension | 0.81644 |
13,199 | The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ‖x‖, and to the angle θ between two vectors x and y by means of the formula Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have... | Hilbert space dimension | 0.81644 |
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