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6,300 | A nondeterministic algorithm for determining whether a 2-satisfiability instance is not satisfiable, using only a logarithmic amount of writable memory, is easy to describe: simply choose (nondeterministically) a variable v and search (nondeterministically) for a chain of implications leading from v to its negation and... | 2-satisfiability | 0.824473 |
6,301 | Most chemists of the 1830s and later accepted the suggestion that isomerism resulted from the differing arrangements of the same numbers and types of atoms, resulting in distinct substances. The numbers of isomers proliferated rapidly with the development of organic chemistry, especially after the introduction of atomi... | Atomic Theory | 0.824467 |
6,302 | In order to introduce the Ideal gas law and statistical forms of physics, it was necessary to postulate the existence of atoms. In 1738, Swiss physicist and mathematician Daniel Bernoulli postulated that the pressure of gases and heat were both caused by the underlying motion of molecules. In 1860, James Clerk Maxwell,... | Atomic Theory | 0.824467 |
6,303 | For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the... | Unital algebra | 0.824465 |
6,304 | In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an R-module (instead of a K-vector space). | Unital algebra | 0.824465 |
6,305 | An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra. | Unital algebra | 0.824465 |
6,306 | Reaction overpotential is an activation overpotential that specifically relates to chemical reactions that precede electron transfer. Reaction overpotential can be reduced or eliminated with the use of electrocatalysts. The electrochemical reaction rate and related current density is dictated by the kinetics of the ele... | Overpotential | 0.824449 |
6,307 | In electrochemistry, overpotential is the potential difference (voltage) between a half-reaction's thermodynamically-determined reduction potential and the potential at which the redox event is experimentally observed. The term is directly related to a cell's voltage efficiency. In an electrolytic cell the existence of... | Overpotential | 0.824449 |
6,308 | The "gene chip" industry started to grow significantly after the 1995 Science Magazine article by the Ron Davis and Pat Brown labs at Stanford University. With the establishment of companies, such as Affymetrix, Agilent, Applied Microarrays, Arrayjet, Illumina, and others, the technology of DNA microarrays has become t... | Microarray analysis | 0.824445 |
6,309 | Center for Chemistry at the Space-Time Limit or CaSTL Center is a National Science Foundation Center for Chemical Innovation.The CaSTL Center was established through a cooperative agreement between the National Science Foundation and the University of California, Irvine in 2008. Vartkess Ara Apkarian, a Professor of Ch... | Center for Chemistry at the Space-Time Limit | 0.824442 |
6,310 | CaSTL researchers developed experimental & theoretical tools to image electrostatic fields and charge distributions with sub-nanometer spatial resolution. They demonstrated the first single molecule limit in miniaturization of microElectroMechanical Systems (SMEMS). They demonstrated that vibrations of a single tip-att... | Center for Chemistry at the Space-Time Limit | 0.824442 |
6,311 | The CaSTL Center organized several scientific events such as symposia, workshops, summer schools on single molecule chemistry. Noted among these are the 2018 Telluride Workshop on Molecular Videography and a symposium with the theme "Toward Chemistry in Real Space and Time" at the 2019 Fall Meeting of the American Chem... | Center for Chemistry at the Space-Time Limit | 0.824442 |
6,312 | Internal vibrations of molecules determine the structural transformations that determine chemistry such as reactivity. A CaSTL team led by Vartkess Ara Apkarian measured the vibrational normal modes of a single cobalt-tetraphenylporphyrin molecule on a copper surface with atomically confined light. This study used a va... | Center for Chemistry at the Space-Time Limit | 0.824442 |
6,313 | Polymerase chain reaction (PCR) is a molecular biology technique used to amplify specific DNA segments by several orders of magnitude. The specific segments of DNA is amplified over three processes, denaturation, annealing and extension – where the DNA strands are separated by raising the temperature to the optimal fro... | Hot start PCR | 0.824441 |
6,314 | In particle physics the little hierarchy problem in the Minimal Supersymmetric Standard Model (MSSM) is a refinement of the hierarchy problem. According to quantum field theory, the mass of the Higgs boson must be rather light for the electroweak theory to work. However, the loop corrections to the mass are naturally m... | Little hierarchy problem | 0.824431 |
6,315 | Sports handicapping lends itself to information theory extremely well because of the availability of statistics. For many years noted economists have tested different mathematical theories using sports as their laboratory, with vastly differing results. One theory regarding sports betting is that it is a random walk. | Gambling and information theory | 0.824426 |
6,316 | These authors found the heterozygote state was indistinguishable from the noncarrier state. Another theory for the prevalence of the CF mutation is that it provides resistance to tuberculosis. Tuberculosis was responsible for 20% of all European deaths between 1600 and 1900, so even partial protection against the disea... | Heterozygote advantage | 0.824423 |
6,317 | In this instance, the state of the organism's environment will provide selection, with a net effect either favoring or working in opposition to the gene, until an environmentally determined equilibrium is reached. Heterozygote advantage is a major underlying mechanism for heterosis, or "hybrid vigor", which is the impr... | Heterozygote advantage | 0.824423 |
6,318 | A heterozygote advantage describes the case in which the heterozygous genotype has a higher relative fitness than either the homozygous dominant or homozygous recessive genotype. Loci exhibiting heterozygote advantage are a small minority of loci. The specific case of heterozygote advantage due to a single locus is kno... | Heterozygote advantage | 0.824423 |
6,319 | One of the biggest challenges in bioinformatics is documenting and sharing scientific workflows in such a way that the computation and its results can be peer-reviewed or reliably reproduced. Bioinformatic pipelines typically use multiple pieces of software, each of which typically has multiple versions available, mult... | BioCompute Object | 0.824423 |
6,320 | The BioCompute Object (BCO) Project is a community-driven initiative to build a framework for standardizing and sharing computations and analyses generated from High-throughput sequencing (HTS -- also referred to as next-generation sequencing or massively parallel sequencing). The project has since been standardized as... | BioCompute Object | 0.824423 |
6,321 | Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations. Actuarial science (ruin theory) Computational electromagnetics Bo... | Integral equation | 0.824419 |
6,322 | Consider now a continuous map f: K → L {\displaystyle f:{\mathcal {K}}\rightarrow {\mathcal {L}}} . A simplicial map g: K → L {\displaystyle g:{\mathcal {K}}\rightarrow {\mathcal {L}}} is said to be a simplicial approximation of f {\displaystyle f} if and only if each x ∈ K {\displaystyle x\in {\mathcal {K}}} is mapped... | Triangulable space | 0.824416 |
6,323 | Giving spaces the structure of a simplicial structure might help to understand maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem: | Triangulable space | 0.824416 |
6,324 | Allen Hatcher: Algebraic Topology, Cambridge University Press, Cambridge/New York/Melbourne 2006, ISBN 0-521-79160-X James R. Munkres: . Band 1984. Addison Wesley, Menlo Park, California 1984, ISBN 0-201-04586-9 Marshall M. Cohen: A course in Simple-Homotopy Theory . In: Graduate Texts in Mathematics. 1973, ISSN 0072-5... | Triangulable space | 0.824416 |
6,325 | In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, f... | Triangulable space | 0.824416 |
6,326 | The lab frame in some experiments is an inertial frame, but it is not required to be (for example the laboratory on the surface of the Earth in many physics experiments is not inertial). In particle physics experiments, it is often useful to transform energies and momenta of particles from the lab frame where they are ... | Reference frames | 0.824416 |
6,327 | A further aspect of a frame of reference is the role of the measurement apparatus (for example, clocks and rods) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relation between observer and measurement is stil... | Reference frames | 0.824416 |
6,328 | A decision tree is a decision support hierarchical model that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains conditional control statements. Decision trees are commonly used in op... | Decision rules | 0.824398 |
6,329 | Many molecular processes within a cell are carried out by molecular machines that are built from numerous protein components organized by their PPIs. These physiological interactions make up the so-called interactomics of the organism, while aberrant PPIs are the basis of multiple aggregation-related diseases, such as ... | Protein-protein interactions | 0.824397 |
6,330 | For instance, Cytoscape is an open-source software widely used and many plugins are currently available. Pajek software is advantageous for the visualization and analysis of very large networks.Identification of functional modules in PPI networks is an important challenge in bioinformatics. Functional modules means a s... | Protein-protein interactions | 0.824397 |
6,331 | Machine learning (ML) involves training computer programs through exposure to large data sets and examples to learn from experience and solve problems. Machine learning can be used to generate and analyse data as well as make algorithmic calculations and has been applied to image and speech recognition, translations, t... | Automated decision | 0.824396 |
6,332 | Continuous auditing uses advanced analytical tools to automate auditing processes. It can be utilized in the private sector by business enterprises and in the public sector by governmental organizations and municipalities. As artificial intelligence and machine learning continue to advance, accountants and auditors may... | Automated decision | 0.824396 |
6,333 | Automated decision-making technologies (ADMT) are software-coded digital tools that automate the translation of input data to output data, contributing to the function of automated decision-making systems. There are a wide range of technologies in use across ADM applications and systems. ADMTs involving basic computati... | Automated decision | 0.824396 |
6,334 | Automated decision-making (ADM) involves the use of data, machines and algorithms to make decisions in a range of contexts, including public administration, business, health, education, law, employment, transport, media and entertainment, with varying degrees of human oversight or intervention. ADM involves large-scale... | Automated decision | 0.824396 |
6,335 | SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, an... | SMT solver | 0.82439 |
6,336 | In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, a... | SMT solver | 0.82439 |
6,337 | The first order theory of the natural numbers with addition (but not multiplication), called Presburger arithmetic, is also decidable. Since multiplication by constants can be implemented as nested additions, the arithmetic in many computer programs can be expressed using Presburger arithmetic, resulting in decidable f... | SMT solver | 0.82439 |
6,338 | The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.The development of the Cartesian coordinate sys... | Cartesian plane | 0.824385 |
6,339 | This technique originated with Eng and Yates in 1994 which involves comparing a theoretical peptide fragment spectrum to compare an experimentally derived peptide spectrum to and outputting the most likely matches found. However, in the absence of an established peptide database, Proteogenomics instead compares the exp... | Proteogenomics | 0.824364 |
6,340 | In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corre... | Modes of variation | 0.824358 |
6,341 | 3. The converse of the theorem need not hold. For example, R2 and R3 have isomorphic fundamental groups but are still not homeomorphic. | Induced homomorphism | 0.824356 |
6,342 | A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred fr... | Induced homomorphism | 0.824356 |
6,343 | In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More gene... | Induced homomorphism | 0.824356 |
6,344 | A powerful tool in physics is the concept of dimensional analysis and scaling laws. By examining the physical effects present in a system, we may estimate their size and hence which, for example, might be neglected. In some cases, the system may not have a fixed natural length or time scale, while the solution depends ... | Similarity solution | 0.824351 |
6,345 | In the study of partial differential equations, particularly in fluid dynamics, a self-similar solution is a form of solution which is similar to itself if the independent and dependent variables are appropriately scaled. Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (fo... | Similarity solution | 0.824351 |
6,346 | By definition, the latter says that, for any x ′ ≥ x {\displaystyle x'\geq x} , the function Δ ( − c ) = ∫ T d λ ( t ) , {\displaystyle \Delta (-c)=\int _{T}\,d\lambda (t),} is a single crossing function. For each t {\displaystyle t} , δ ( − c , t ) = u ( Π ( x ′ ; − c , t ) ) − u ( Π ( x ; − c , t ) ) {\displaystyle ... | Monotone comparative statics | 0.824351 |
6,347 | Application (Monopoly problem under uncertainty): A firm faces uncertainty over the demand for its output x {\displaystyle x} and the profit at state t ∈ T ⊂ R {\displaystyle t\in T\subset \mathbb {R} } is given by Π ( x ; − c , t ) = x P ( x ; t ) − c x {\displaystyle \Pi (x;-c,t)=xP(x;t)-cx} , where c {\displaystyle ... | Monotone comparative statics | 0.824351 |
6,348 | Theorem 7: Let ( T , T , μ ) {\displaystyle (T,{\mathcal {T}},\mu )} be a finite measure space and suppose that, for each s ∈ S {\displaystyle s\in S} , f ( s ; t ) {\displaystyle f(s;t)} is a bounded and measurable function of t ∈ T {\displaystyle t\in T} . Then F ( s ) = ∫ T f ( s ; t ) d μ ( t ) {\displaystyle F(s)=... | Monotone comparative statics | 0.824351 |
6,349 | Note that { Π ( ⋅ , − c ) } ( − c ) ∈ R − {\displaystyle \{\Pi (\cdot ,-c)\}_{(-c)\in \mathbb {R} _{-}}} obey single crossing differences. Indeed, take any x ′ ≥ x {\displaystyle x'\geq x} and assume that x ′ P ( x ′ ) − c x ′ ≥ ( > ) x P ( x ) − c x {\displaystyle x'P(x')-cx'\geq (>)\ xP(x)-cx} ; for any c ′ {\display... | Monotone comparative statics | 0.824351 |
6,350 | Theorem 1: Define F Y ( s ) := arg max x ∈ Y f ( x ; s ) {\displaystyle F_{Y}(s):=\arg \max _{x\in Y}f(x;s)} . The family { f ( ⋅ ; s ) } s ∈ S {\displaystyle \{f(\cdot ;s)\}_{s\in S}} obey single crossing differences if and only if for all Y ⊆ X {\displaystyle Y\subseteq X} , we have F Y ( s ′ ) ≥ S S O F Y ( s ) {\... | Monotone comparative statics | 0.824351 |
6,351 | It focuses on the main property underpinning monotone comparative statics, which is a form of complementarity between the endogenous variable and exogenous parameter. Roughly speaking, a maximization problem displays complementarity if a higher value of the exogenous parameter increases the marginal return of the endog... | Monotone comparative statics | 0.824351 |
6,352 | Monotone comparative statics is a sub-field of comparative statics that focuses on the conditions under which endogenous variables undergo monotone changes (that is, either increasing or decreasing) when there is a change in the exogenous parameters. Traditionally, comparative results in economics are obtained using th... | Monotone comparative statics | 0.824351 |
6,353 | Basic techniques – Milgrom and Shannon (1994)., Milgrom (1994), Shannon (1995), Topkis (1998), Edlin and Shannon (1998), Athey (2002), Quah (2007), Quah and Strulovici (2009, 2012), Kukushkin (2013); Production complementarities and their implications – Milgrom and Roberts (1990a, 1995); Topkis (1995); Games with strat... | Monotone comparative statics | 0.824351 |
6,354 | For any x ′ {\displaystyle x'} , x ∈ X {\displaystyle x\in X} , x ′ ≥ x {\displaystyle x'\geq x} , V ( x ′ ) − V ( x ) + ( − p ) ( x ′ − x ) {\displaystyle V(x')-V(x)+(-p)(x'-x)} is increasing in ( − p ) {\displaystyle (-p)} . Hence, { Π ( ⋅ ; p ) } p ∈ R + + l {\displaystyle \{\Pi (\cdot ;p)\}_{p\in \mathbb {R} _{++}^... | Monotone comparative statics | 0.824351 |
6,355 | That is, if function f {\displaystyle f} is quasisupermodular, then so is function g := H ∘ f {\displaystyle g:=H\circ f} , where H {\displaystyle H} is some strictly increasing function. Theorem 3: Let ( X , ≥ X ) {\displaystyle (X,\geq _{X})} is a lattice, ( S , ≥ S ) {\displaystyle (S,\geq _{S})} a partially ordered... | Monotone comparative statics | 0.824351 |
6,356 | Such a family need not obey single crossing differences. A function f: X × S → R {\displaystyle f:X\times S\to \mathbb {R} } is regular if arg max x ∈ f ( x ; s ) {\displaystyle \arg \max _{x\in }f(x;s)} is non-empty for any x ∗ ∗ ≥ x ∗ {\displaystyle x^{**}\geq x^{*}} , where {\displaystyle } denotes the interval ... | Monotone comparative statics | 0.824351 |
6,357 | The agent chooses x {\displaystyle x} to maximize V ( x ; t ) := ∫ S u ( ( w − x ) R + x s ) λ ( s ; t ) d s . {\displaystyle V(x;t):=\int _{S}u((w-x)R+xs)\lambda (s;t)\,ds.} Note that { u ^ ( ⋅ ; s ) } s ∈ S {\displaystyle \{{\hat {u}}(\cdot ;s)\}_{s\in S}} , where u ^ ( x ; s ) := u ( w R + x ( s − R ) ) {\displaysty... | Monotone comparative statics | 0.824351 |
6,358 | Theorem 6: Suppose { f ( ⋅ ; s ) } s ∈ S {\displaystyle \{f(\cdot ;s)\}_{s\in S}} (for S ⊆ R {\displaystyle S\subseteq \mathbb {R} } ) obeys X. Then the family { F ( ⋅ ; t ) } t ∈ T {\displaystyle \{F(\cdot ;t)\}_{t\in T}} obeys X if { λ ( ⋅ ; t ) } t ∈ T {\displaystyle \{\lambda (\cdot ;t)\}_{t\in T}} is ordered with ... | Monotone comparative statics | 0.824351 |
6,359 | Then, F ( x ′ ; t ) − F ( x ; t ) = ∫ S λ ( s ; t ) d s {\displaystyle F(x';t)-F(x;t)=\int _{S}\lambda (s;t)ds} , or equivalently F ( x ′ ; t ) − F ( x ; t ) = ∫ S ϕ ( s ) λ ( s ; t ) d s {\displaystyle F(x';t)-F(x;t)=\int _{S}\phi (s)\lambda (s;t)ds} . Since { f ( ⋅ ; s ) } s ∈ S {\displaystyle \{f(\cdot ;s)\}_{s\in ... | Monotone comparative statics | 0.824351 |
6,360 | Theorem 5: Suppose { f ( ⋅ ; s ) } s ∈ S {\displaystyle \{f(\cdot ;s)\}_{s\in S}} ( S ⊆ R ) {\displaystyle (S\subseteq \mathbb {R} )} obeys increasing differences. If { λ ( ⋅ ; t ) } t ∈ T {\displaystyle \{\lambda (\cdot ;t)\}_{t\in T}} is ordered with respect to first order stochastic dominance, then { F ( ⋅ ; t ) } t... | Monotone comparative statics | 0.824351 |
6,361 | {\displaystyle F(x;t)=\int _{S}f(x;s)\,\lambda (s;t)\,ds.} For arg max x ∈ X F ( x ; t ) {\displaystyle \arg \max _{x\in X}F(x;t)} to be increasing in t {\displaystyle t} , it suffices (by Theorems 1 and 2) that family { F ( ⋅ ; t ) } t ∈ T {\displaystyle \{F(\cdot ;t)\}_{t\in T}} obey single crossing differences or ... | Monotone comparative statics | 0.824351 |
6,362 | Let X ⊂ R {\displaystyle X\subset \mathbb {R} } , and { f ( ⋅ ; s ) } s ∈ S {\displaystyle \{f(\cdot ;s)\}_{s\in S}} be a family of real-valued functions defined on X {\displaystyle X} that obey single crossing differences or the interval dominance order. Theorem 1 and 3 tell us that arg max x ∈ X f ( x , ; s ) {\dis... | Monotone comparative statics | 0.824351 |
6,363 | Theorem 4: Suppose u: R + + n → R {\displaystyle u:\mathbb {R} _{++}^{n}\rightarrow \mathbb {R} } is supermodular and concave. Then the demand correspondence is normal in the following sense: suppose w ″ > w ′ {\displaystyle w''>w'} , x ″ ∈ D ( p , w ″ ) {\displaystyle x''\in D(p,w'')} and x ′ ∈ D ( p , w ′ ) {\display... | Monotone comparative statics | 0.824351 |
6,364 | A basic property of consumer demand is normality, which means (in the case where demand is unique) that the demand of each good is increasing in wealth. Theorem 3 cannot be straightforwardly applied to obtain conditions for normality, because B ( p , w ′ ) ≱ S S O B ( p , w ) {\displaystyle B(p,w')\not \geq _{SSO}B(p,w... | Monotone comparative statics | 0.824351 |
6,365 | In some important economic applications, the relevant change in the constraint set cannot be easily understood as an increase with respect to the strong set order and so Theorem 3 cannot be easily applied. For example, consider a consumer who maximizes a utility function u: X → R {\displaystyle u:X\to \mathbb {R} } sub... | Monotone comparative statics | 0.824351 |
6,366 | The Buckingham π theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the rank of the dimensional matrix. Furthermore, and most importantly, it provides a method for computing these dimensionless param... | Dimension analysis | 0.824345 |
6,367 | Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same unit as some derived quantity X can be expressed in the general form X = ∏ i = 1 m ( π i ) k i . {\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.} Consequently, every possible commensurate eq... | Dimension analysis | 0.824344 |
6,368 | The AMC 10 and AMC 12 are 25 question, 75-minute multiple choice competitions in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators have not been allowed on the AMC 10/12 since 2008.High scores on the AMC 10 or 12 can qualify the participant for t... | American Mathematics Competitions | 0.824343 |
6,369 | The Orientation-Cum-Selection-Camp (OCSC), Chemistry, consists of rigorous training and testing in theory and experiment. The top four performers here are selected to represent India in the International Chemistry Olympiad. Before the INChO, the selected team undergoes rigorous training in theory and experiments in a P... | Indian National Chemistry Olympiad | 0.824338 |
6,370 | Among the 30,000+ students who sit for the NSEC, only the top 1% are selected for the INChO. About 35 students are selected from the written examination. A total of 30 students are chosen from these to attend the Orientation-Cum-Selection-Camp (OCSC), chemistry, held at HBCSE, Mumbai. Most students qualifying for the I... | Indian National Chemistry Olympiad | 0.824338 |
6,371 | The Indian National Chemistry Olympiad (INChO for short) is an Olympiad in Chemistry held in India. The theory part of the INChO examination is held in end-January/beginning of February of every year. It is conducted by the Indian Association of Chemistry Teachers. School students (usually of standards 11 and 12) first... | Indian National Chemistry Olympiad | 0.824338 |
6,372 | All points are in the same plane, space, etc. (depending on the dimension one chooses to work within). (Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other p... | Ordered geometry | 0.824336 |
6,373 | Axiom of dimensionality: For planar ordered geometry, all points are in one plane. Or If ABC is a plane, then there exists a point D not in the plane ABC. | Ordered geometry | 0.824336 |
6,374 | The only primitive notions in ordered geometry are points A, B, C, ... and the ternary relation of intermediacy which can be read as "B is between A and C". | Ordered geometry | 0.824336 |
6,375 | Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective ... | Ordered geometry | 0.824336 |
6,376 | In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first published in 1993.The idea of the algorithm is based on the concept of contraction of an edge ( u , v ) {\displaystyle (u,v)} in an undirected graph... | Karger's algorithm | 0.824335 |
6,377 | For weighted graphs with positive edge weights w: E → R + {\displaystyle w\colon E\rightarrow \mathbf {R} ^{+}} the weight of the cut is the sum of the weights of edges between vertices in each part w ( S , T ) = ∑ u v ∈ E: u ∈ S , v ∈ T w ( u v ) , {\displaystyle w(S,T)=\sum _{uv\in E\colon u\in S,v\in T}w(uv)\,,} whi... | Karger's algorithm | 0.824335 |
6,378 | In civil engineering applications, one typically considers structures to be in static equilibrium: that is, are either unchanging with time, or are changing slowly enough for viscous stresses to be unimportant (quasi-static). In mechanical and aerospace engineering, however, stress analysis must often be performed on p... | Stress–strain analysis | 0.824334 |
6,379 | In any case, for two- or three-dimensional domains one must solve a system of partial differential equations with specified boundary conditions. Analytical (closed-form) solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. For more... | Stress–strain analysis | 0.824334 |
6,380 | In this case the differential equations that define the stress tensor are also linear. Linear equations are much better understood than non-linear ones; for one thing, their solution (the calculation of stress at any desired point within the structure) will also be a linear function of the applied forces. For small eno... | Stress–strain analysis | 0.824334 |
6,381 | There are a number of experimental methods which may be used: Tensile testing is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure. The results from the test are commonly used to select a material for an application, for quality control, or to predict how a material w... | Stress–strain analysis | 0.824334 |
6,382 | Because of the many important problems in this class, there have been extensive efforts to find polynomial-time algorithms for problems in NP. However, there remain a large number of problems in NP that defy such attempts, seeming to require super-polynomial time. Whether these problems are not decidable in polynomial ... | NP class | 0.82433 |
6,383 | Many computer science problems are contained in NP, like decision versions of many search and optimization problems. | NP class | 0.82433 |
6,384 | In terms of descriptive complexity theory, NP corresponds precisely to the set of languages definable by existential second-order logic (Fagin's theorem). NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial... | NP class | 0.82433 |
6,385 | The phenotypic correlation will be limited by the degree of genetic correlation and also by the heritability of each trait. The expected phenotypic correlation is the bivariate heritability' and can be calculated as the square roots of the heritabilities multiplied by the genetic correlation. (Using a Plomin example, f... | Genetic correlations | 0.824315 |
6,386 | Genetic correlations are not the same as heritability, as it is about the overlap between the two sets of influences and not their absolute magnitude; two traits could be both highly heritable but not be genetically correlated or have small heritabilities and be completely correlated (as long as the heritabilities are ... | Genetic correlations | 0.824315 |
6,387 | Genetic correlations have been found to be common in non-human genetics and to be broadly similar to their respective phenotypic correlations, and also found extensively in human traits, dubbed the 'phenome'.This finding of widespread pleiotropy has implications for artificial selection in agriculture, interpretation o... | Genetic correlations | 0.824315 |
6,388 | Genetic correlation models were introduced into behavioral genetics in the 1970s–1980s. Genetic correlations have applications in validation of genome-wide association study (GWAS) results, breeding, prediction of traits, and discovering the etiology of traits & diseases. They can be estimated using individual-level da... | Genetic correlations | 0.824315 |
6,389 | In multivariate quantitative genetics, a genetic correlation (denoted r g {\displaystyle r_{g}} or r a {\displaystyle r_{a}} ) is the proportion of variance that two traits share due to genetic causes, the correlation between the genetic influences on a trait and the genetic influences on a different trait estimating t... | Genetic correlations | 0.824315 |
6,390 | (However, if axioms are given for a branch of mathematics, this creates a logical system —Wittgenstein 1921 Tractatus Logico-Philosophicus 5.13; Lakatos claimed that proofs from such a system were tautological, i.e. internally logically true, by rewriting forms, as shown by Poincaré, who demonstrated the technique of t... | Scientific methods | 0.824314 |
6,391 | Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. | Scientific methods | 0.824314 |
6,392 | Building on Pólya's work, Imre Lakatos argued that mathematicians actually use contradiction, criticism, and revision as principles for improving their work. In like manner to science, where truth is sought, but certainty is not found, in Proofs and Refutations, what Lakatos tried to establish was that no theorem of in... | Scientific methods | 0.824314 |
6,393 | The term "infrared" did not appear until late 19th century.Other important dates include: 1830: Leopoldo Nobili made the first thermopile IR detector. 1840: John Herschel produces the first thermal image, called a thermogram. 1860: Gustav Kirchhoff formulated the blackbody theorem E = J ( T , n ) {\displaystyle E=J(T,n... | Infrared rays | 0.824313 |
6,394 | 1970: Willard Boyle and George E. Smith proposed CCD at Bell Labs for picture phone. 1973: Common module program started by NVESD. 1978: Infrared imaging astronomy came of age, observatories planned, IRTF on Mauna Kea opened; 32 × 32 and 64 × 64 arrays produced using InSb, HgCdTe and other materials. 2013: On 14 Februa... | Infrared rays | 0.824313 |
6,395 | 1901: Max Planck published the blackbody equation and theorem. He solved the problem by quantizing the allowable energy transitions. 1905: Albert Einstein developed the theory of the photoelectric effect. | Infrared rays | 0.824313 |
6,396 | In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) + a ∘ ( c ∘ b ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).} Zinbiel algebras were introduced by Jean-Louis Lo... | Zinbiel algebra | 0.824296 |
6,397 | The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.In any Zinbiel algebra, the symmetrised product a ⋆ b = a ∘ b + b ∘ a {\displaystyle a\star b=a\circ b+b\circ a} is associative. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the t... | Zinbiel algebra | 0.824296 |
6,398 | A 2008 review paper written by Philips researcher Clemens J. M. Lasance notes that: "Although there is an analogy between heat flow by conduction (Fourier's law) and the flow of an electric current (Ohm’s law), the corresponding physical properties of thermal conductivity and electrical conductivity conspire to make th... | Thermal resistivity | 0.82429 |
6,399 | However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action. The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right cosets of a subgroup of a group, which ari... | Equivalence classes | 0.824281 |
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