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26,700 | Lie superalgebra supergroup (physics) supercharge supermultiplet supergravity | List of mathematical topics in quantum theory | 0.807048 |
26,701 | quantum mechanics, matrix mechanics, Hamiltonian (quantum mechanics) particle in a box particle in a ring particle in a spherically symmetric potential quantum harmonic oscillator hydrogen atom ring wave guide particle in a one-dimensional lattice (periodic potential) Fock symmetry in theory of hydrogen | List of mathematical topics in quantum theory | 0.807048 |
26,702 | identical particles angular momentum angular momentum operator rotational invariance rotational symmetry rotation operator translational symmetry Lorentz symmetry Parity transformation Noether's theorem Noether charge Spin (physics) isospin Aman matrices scale invariance spontaneous symmetry breaking supersymmetry brea... | List of mathematical topics in quantum theory | 0.807048 |
26,703 | Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: Every computably enumerable set is Diophantine, and the converse.A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventual... | MRDP theorem | 0.807047 |
26,704 | Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable. One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result: Corresponding to any given consistent axiomatization of number theory, one c... | MRDP theorem | 0.807047 |
26,705 | Consider a relation of the form R(x1, …, xn) = 0, where R is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an implicit curve if n = 2 and an implicit surface if n = 3. The implicit equations are the basis of algebraic geometry, whose basic subjects of study are ... | Implicit differentiation | 0.807045 |
26,706 | For example, the equation x = 0 does not imply a function f(x) giving solutions for y at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling th... | Implicit differentiation | 0.807045 |
26,707 | An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an im... | Implicit differentiation | 0.807045 |
26,708 | Larger sample sizes generally lead to increased precision when estimating unknown parameters. For example, if we wish to know the proportion of a certain species of fish that is infected with a pathogen, we would generally have a more precise estimate of this proportion if we sampled and examined 200 rather than 100 fi... | Required sample sizes for hypothesis tests | 0.807043 |
26,709 | The earliest work on intraclass correlations focused on the case of paired measurements, and the first intraclass correlation (ICC) statistics to be proposed were modifications of the interclass correlation (Pearson correlation). Consider a data set consisting of N paired data values (xn,1, xn,2), for n = 1, ..., N. Th... | Intra-class correlation | 0.807042 |
26,710 | In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of ... | Intra-class correlation | 0.807042 |
26,711 | In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory. | Typical subspace | 0.807042 |
26,712 | The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. That being the case, the state vector would not apply to individual cat experiments, but only to the statistics of many similar prepared cat experiments. Proponents of this interpretation state that thi... | Statistical interpretation | 0.807037 |
26,713 | Popper described "propensities" as the correct notion of probability for quantum mechanics. Although several other notable physicists championed the ensemble concept, including John C. Slater, Edwin C. Kemble, and Dmitrii Blokhintsev, Leslie Ballentine's 1970 paper 'The statistical interpretation of quantum mechanics" ... | Statistical interpretation | 0.807037 |
26,714 | Einstein believed quantum mechanics was correct in the same sense that thermodynamics is correct, but that it was insufficient as means of unifying physics.Also in the years around 1936, Karl Popper published philosophical studies countering the work of Heisenberg and Bohr. Popper considered their work as essentially s... | Statistical interpretation | 0.807037 |
26,715 | In his 1926 paper introducing the concept of quantum scattering theory Max Born proposed to view "the motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws", where the causal laws are Schrodinger's equations. As related in his 1954 Nobel Prize in Physic... | Statistical interpretation | 0.807037 |
26,716 | The claim that the wave functional approach fails to apply to single particle experiments cannot be taken as a claim that quantum mechanics fails in describing single-particle phenomena. In fact, it gives correct results within the limits of a probabilistic or stochastic theory. Probability always requires a set of mul... | Statistical interpretation | 0.807037 |
26,717 | As noted by Paul Humphreys, many physical examples show the lack of reciprocal correlation, for example, the propensity for smokers to get lung cancer does not imply lung cancer has a propensity to cause smoking. Propensity closely matches the application of quantum theory: single event probability can be predicted by ... | Statistical interpretation | 0.807037 |
26,718 | The physical scenario is view as weakly causal condition. The weak causation invalidates Bayes' theorem and correlation is no longer symmetric. | Statistical interpretation | 0.807037 |
26,719 | Quantum observations are inherently statistical. For example, the electrons in a low-intensity double slit experiment arrive at random times and seemingly random places and yet eventually show an interference pattern. The theory of quantum mechanics offer only statistical results. Given that we have prepared a system i... | Statistical interpretation | 0.807037 |
26,720 | Although Einstein described quantum mechanics as clearly an ensemble theory he did present a formal definition of an ensemble. Einstein sought a theory of individual entities, which he argued was not quantum mechanics. Ballentine distinguish his particular ensemble interpretation the Statistical Interpretation. | Statistical interpretation | 0.807037 |
26,721 | But one cannot do so by mixing a fragment of the original unsplit beam with component split sub-beams. That is because one photon cannot both go into the unsplit fragment and go into the split component sub-beams. Bohr felt that talk in statistical terms might hide this fact. The physics here is that the effect of the ... | Statistical interpretation | 0.807037 |
26,722 | This yields the following system of equations: S = M + U M = M²x² + 1 U = Sx + MUx²In this system of equations, S, M, and U are functions of x, so one could also write S ( x ) {\displaystyle S(x)} , M ( x ) {\displaystyle M(x)} , and U ( x ) {\displaystyle U(x)} . The equation system can be resolved after S, resulting ... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,723 | The theorem can be used in analytic combinatorics to estimate the number of words of length n generated by a given unambiguous context-free grammar, as n grows large. The following example is given by Gruber, Lee & Shallit (2012): the unambiguous context-free grammar G over the alphabet {0,1} has start symbol S and the... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,724 | In formal language theory, the Chomsky–Schützenberger enumeration theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger about the number of words of a given length generated by an unambiguous context-free grammar. The theorem provides an unexpected link between the theory of formal languages and a... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,725 | The harder part is to show that there does not exist an unambiguous grammar that generates L G {\displaystyle L_{G}} . This can be proved as follows: If g k {\displaystyle g_{k}} denotes the number of words of length k {\displaystyle k} in L G {\displaystyle L_{G}} , then for the associated power series holds G ( x ) =... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,726 | In classical formal language theory, the theorem can be used to prove that certain context-free languages are inherently ambiguous. For example, the Goldstine language L G {\displaystyle L_{G}} over the alphabet { a , b } {\displaystyle \{a,b\}} consists of the words a n 1 b a n 2 b ⋯ a n p b {\displaystyle a^{n_{1}}ba... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,727 | Having established the necessary notions, the theorem is stated as follows. Chomsky–Schützenberger theorem. If L {\displaystyle L} is a context-free language admitting an unambiguous context-free grammar, and a k := | L ∩ Σ k | {\displaystyle a_{k}:=|L\ \cap \Sigma ^{k}|} is the number of words of length k {\displaysty... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,728 | {\displaystyle f(x)\cdot g(x)=\sum _{k=0}^{\infty }\left(\sum _{i=0}^{k}a_{i}b_{k-i}\right)x^{k}.} In particular, we write f 2 = f ( x ) ⋅ f ( x ) {\displaystyle f^{2}=f(x)\cdot f(x)} , f 3 = f ( x ) ⋅ f ( x ) ⋅ f ( x ) {\displaystyle f^{3}=f(x)\cdot f(x)\cdot f(x)} , and so on. In analogy to algebraic numbers, a power... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,729 | In order to state the theorem, a few notions from algebra and formal language theory are needed. Let N {\displaystyle \mathbb {N} } denote the set of nonnegative integers. A power series over N {\displaystyle \mathbb {N} } is an infinite series of the form f = f ( x ) = ∑ k = 0 ∞ a k x k = a 0 + a 1 x 1 + a 2 x 2 + a 3... | Chomsky–Schützenberger enumeration theorem | 0.807036 |
26,730 | The Hartree atomic unit system uses the following defining constants: e, me, ħ, ke.The Coulomb constant, ke, is generally expressed as 1/4πε0 when working with this system. These units are designed to simplify atomic and molecular physics and chemistry, especially the hydrogen atom, and are widely used in these fields.... | Natural unit system | 0.807035 |
26,731 | This natural unit system, used only in the fields of particle and atomic physics, uses the following defining constants:: 509 c, me, ħ, ε0,where c is the speed of light, me is the electron mass, ħ is the reduced Planck constant, and ε0 is the vacuum permittivity. The vacuum permittivity ε0 is implicitly used as a nondi... | Natural unit system | 0.807035 |
26,732 | In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge e may be used as a natural unit of electric charge, and the speed of light c ... | Natural unit system | 0.807035 |
26,733 | The Helmholtz principle rests mainly on the linearity and superposability of the light field, and it has close analogues in non-electromagnetic linear propagating fields, such as sound. It was discovered before the electromagnetic nature of light became known.The Helmholtz reciprocity theorem has been rigorously proven... | Helmholtz reciprocity | 0.807034 |
26,734 | In the case of inelastic scattering with small energy loss, it can be shown that reciprocity may be used to approximate intensity (rather than wave amplitude). So in very thick samples or samples in which inelastic scattering dominates, the benefits of using reciprocity for the previously mentioned TEM applications are... | Helmholtz reciprocity | 0.807034 |
26,735 | The ring of radical polynomials is a graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement th... | Radical polynomial | 0.807034 |
26,736 | In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if k {\displaystyle k} is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial... | Radical polynomial | 0.807034 |
26,737 | On March 1, 2018, the Defense Advanced Research Projects Agency (DARPA) announced their new Biostasis program under the direction of Dr. Tristan McClure-Begley. The aim of the Biostasis program is to develop new possibilities for extending the golden hour in patients who suffered a traumatic injury by slowing down the ... | Biostasis | 0.807027 |
26,738 | In molecular biology, molecular chaperones are proteins that assist in the folding, unfolding, assembly, or disassembly of other macromolecular structures. Under typical conditions, molecular chaperones facilitate changes in shape (conformational change) of macromolecules in response to changes in environmental factors... | Biostasis | 0.807027 |
26,739 | In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is ... | Cochain complexes | 0.807027 |
26,740 | In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes h... | Cochain complexes | 0.807027 |
26,741 | The decans are 36 groups of stars (small constellations) used in the ancient Egyptian astronomy to conveniently divide the 360 degree ecliptic into 36 parts of 10 degrees. Because a new decan also appears heliacally every ten days (that is, every ten days, a new decanic star group reappears in the eastern sky at dawn r... | Decimal time | 0.807027 |
26,742 | Length of a decimal yearSince there are about 365 days in a year, there are about 365/10 = 36.5 days in a tenth of a year. Hence the year 2020.5 represents the day 2 July 2020. More exactly, a "Julian year" is exactly 365.25 days long, so a tenth of the year is 36.525 days (36 days, 12 hours, 36 minutes). These values,... | Decimal time | 0.807027 |
26,743 | Scientists often record time as decimal. For example, decimal days divide the day into 10 equal parts, and decimal years divide the year into 10 equal parts. Decimals are easier to plot than both (a) minutes and seconds, which uses the sexagesimal numbering system, (b) hours, months and days, which has irregular month ... | Decimal time | 0.807027 |
26,744 | Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to St... | VC theory | 0.807026 |
26,745 | The majority of the arguments of how to bound the empirical process, rely on symmetrization, maximal and concentration inequalities and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the ... | VC theory | 0.807026 |
26,746 | For binary classification and the 0/1 loss function we have the following generalization bounds: P ( sup f ∈ F | R ^ n ( f ) − R ( f ) | > ε ) ≤ 8 S ( F , n ) e − n ε 2 / 32 E ≤ 2 log S ( F , n ) + log 2 n {\displaystyle {\begin{aligned}P\left(\sup _{f\in {\mathcal {F}}}\left|{\hat {R}}_{n}(f)-R(f)\right|>\varepsi... | VC theory | 0.807026 |
26,747 | Of course since P X Y {\displaystyle P_{XY}} is unknown in general, one has no access to R ( f ) {\displaystyle R(f)} . However the empirical risk, given by: R ^ n ( f ) = 1 n ∑ i = 1 n I ( f ( X i ) ≠ Y i ) {\displaystyle {\hat {R}}_{n}(f)={\dfrac {1}{n}}\sum _{i=1}^{n}\mathbb {I} (f(X_{i})\neq Y_{i})} can certainly b... | VC theory | 0.807026 |
26,748 | A similar setting is considered, which is more common to machine learning. Let X {\displaystyle {\mathcal {X}}} is a feature space and Y = { 0 , 1 } {\displaystyle {\mathcal {Y}}=\{0,1\}} . A function f: X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} is called a classifier. Let F {\displaystyle {\mathcal {F}}... | VC theory | 0.807026 |
26,749 | A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal ... | VC theory | 0.807026 |
26,750 | These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal... | VC theory | 0.807026 |
26,751 | . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} .uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigar... | VC theory | 0.807026 |
26,752 | MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware. Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki. In 2011, they show... | Memristor | 0.807026 |
26,753 | The devices' resistance would be read with alternating current so that the stored value would not be affected. In May 2012, it was reported that the access time had been improved to 90 nanoseconds, which is nearly one hundred times faster than the contemporaneous Flash memory. At the same time, the energy consumption w... | Memristor | 0.807026 |
26,754 | Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. For more see als... | Elliptic curves | 0.807017 |
26,755 | They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invari... | Elliptic curves | 0.807017 |
26,756 | Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in numb... | Elliptic curves | 0.807017 |
26,757 | If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. Mor... | Elliptic curves | 0.807017 |
26,758 | Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian... | Elliptic curves | 0.807017 |
26,759 | In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be d... | Elliptic curves | 0.807017 |
26,760 | The regulation of science refers to use of law, or other ruling, by academic or governmental bodies to allow or restrict science from performing certain practices, or researching certain scientific areas. Science could be regulated by legislation if areas are seen as harmful, immoral, or dangerous. For these reasons sc... | Regulation of science | 0.807016 |
26,761 | The Scopes Trial had an important impact in the larger creation versus evolution debate. In the following decades, the term "evolution" was omitted in many biology textbooks, even when the text discusses it. These bans on teaching evolution were overturned by a Supreme Court ruling in Epperson v. Arkansas in 1968. | Regulation of science | 0.807016 |
26,762 | Narrower gauge railways usually cost less to build because they are usually lighter in construction, using smaller cars and locomotives (smaller loading gauge), as well as smaller bridges, smaller tunnels (smaller structure gauge). Narrow gauge is thus often used in mountainous terrain, where the savings in civil engin... | Track gauge | 0.807015 |
26,763 | Although Dyalog APL included an implementation of shared variables for communication with the now-deprecated Microsoft Windows Dynamic Data Exchange (DDE), only IBM continued to use Shared Variables as a means to supply new features to their versions of the APL2 language for non-mainframe computers.Nearly all other APL... | Shared variable | 0.807015 |
26,764 | The high friction against the ground while turning also requires high-torque engines to overcome the friction. By comparison, the design of the Mecanum wheel allows for in-place rotation with minimal ground friction and low torque. Youth robotics competitions such as FIRST Tech Challenge and VEX Robotics often use meca... | Mecanum wheel | 0.807013 |
26,765 | In mechanical engineering, a key is a machine element used to connect a rotating machine element to a shaft. The key prevents relative rotation between the two parts and may enable torque transmission. For a key to function, the shaft and rotating machine element must have a keyway and a keyseat, which is a slot and po... | Key (engineering) | 0.807012 |
26,766 | This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra. The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted... | Axiomatic proof | 0.807012 |
26,767 | In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system tha... | Axiomatic proof | 0.807012 |
26,768 | COLMAP: A general-purpose Structure-from-Motion (SfM) and Multi-View Stereo (MVS) pipeline with a graphical and command-line interface. BSD license. Theia: A computer vision library aimed at providing efficient and reliable algorithms for Structure from Motion (SfM). New BSD license. Ames Stereo Pipeline has a tool for... | Bundle adjustment | 0.80701 |
26,769 | g2o: General Graph Optimization (C++) - framework with solvers for sparse graph-based non-linear error functions. LGPL. DGAP: The program DGAP implement the photogrammetric method of bundle adjustment invented by Helmut Schmid and Duane Brown. | Bundle adjustment | 0.80701 |
26,770 | LGPL. OpenCV: Computer Vision library in the Images stitching module. BSD license. | Bundle adjustment | 0.80701 |
26,771 | In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from... | Bundle adjustment | 0.80701 |
26,772 | Bundle adjustment is almost always used as the last step of feature-based 3D reconstruction algorithms. It amounts to an optimization problem on the 3D structure and viewing parameters (i.e., camera pose and possibly intrinsic calibration and radial distortion), to obtain a reconstruction which is optimal under certain... | Bundle adjustment | 0.80701 |
26,773 | Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space, and separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external. Furthermore, it assumes that spac... | Space Time Continuum | 0.807009 |
26,774 | The result OB > OC corresponds again to above. The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion can affect the observer's observations of the object. | Space Time Continuum | 0.807009 |
26,775 | It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.) Analogous logic can be used to demonstrate the origin of gravit... | Space Time Continuum | 0.807009 |
26,776 | In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature. The fact that physical processes don't care where in... | Space Time Continuum | 0.807009 |
26,777 | Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost co... | Almost-complex structure | 0.807006 |
26,778 | If J2 = −1 then (det J)2 = (−1)n. But if M is a real manifold, then det J is a real number – thus n must be even if M has an almost complex structure. One can show that it must be orientable as well. An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. | Almost-complex structure | 0.807006 |
26,779 | The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether S6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For real-analytic ... | Almost-complex structure | 0.807005 |
26,780 | {\displaystyle -(N_{A})_{ij}^{k}=A_{i}^{m}\partial _{m}A_{j}^{k}-A_{j}^{m}\partial _{m}A_{i}^{k}-A_{m}^{k}(\partial _{i}A_{j}^{m}-\partial _{j}A_{i}^{m}).} In terms of the Frölicher–Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor NA is just one-half of . The Newlander–Nirenbe... | Almost-complex structure | 0.807005 |
26,781 | Digital electronics inherently lacks the temperature sensitive drift and noise level issues of analog electronics and the slippage and "lash" issues of mechanical analog systems. For problems that can be expressed as differential equations, a hardware DDA can solve them much faster than a general purpose computer (usin... | Digital Differential Analyzer | 0.807004 |
26,782 | In natural language processing, it is often necessary to compare tree structures (e.g. parse trees) for similarity. Such comparisons can be performed by computing dot products of vectors of features of the trees, but these vectors tend to be very large: NLP techniques have come to a point where a simple dependency rela... | Tree kernel | 0.807004 |
26,783 | In machine learning, tree kernels are the application of the more general concept of positive-definite kernel to tree structures. They find applications in natural language processing, where they can be used for machine-learned parsing or classification of sentences. | Tree kernel | 0.807004 |
26,784 | In statistics, one-way analysis of variance (abbreviated one-way ANOVA) is a technique that can be used to compare whether two sample's means are significantly different or not (using the F distribution). This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usual... | One-way ANOVA | 0.806999 |
26,785 | The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the central limit theorem.... | One-way ANOVA | 0.806999 |
26,786 | The discovery of Cyclin-dependent kinase inhibitor proteins in 1990 opened the door in how we think about cell cycle control. It has steered to various other fields of study such as developmental biology, cell biology and cancer research. The discovery of the first CKIs in yeast (Far1) and P21 in mammals has led to res... | Cyclin dependent kinase inhibitor | 0.806997 |
26,787 | In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified sampling. Results from probability theory and statistical theory are employed to guide the practice. In business and medical research, sampling is widely used for gathering information about a population. ... | Statistical sampling | 0.806997 |
26,788 | In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of t... | Statistical sampling | 0.806997 |
26,789 | Other pharmaceutical and agrochemical companies gradually followed suit and started outsourcing the procurement of fine chemicals. An example in case is F.I.S., Italy, which partnered with Roche, Switzerland for custom manufacturing precursors of the benzodiazepine class of tranquilizers, such as Librium (chlordiazepox... | Fine chemicals | 0.806993 |
26,790 | The overall emphasis of fine chemical R&D is more on development than on research. The main tasks are (1) designing, respectively duplicating and adapting in case of custom manufacture, and developing laboratory procedures for new products or processes; (2) transferring the processes from the laboratory via pilot plant... | Fine chemicals | 0.806993 |
26,791 | According to experts in the field, 70% of all chemical reactions could be done in microreactors, however only 10-15% are economically justified. With the exception of some stereospecific reactions, particularly biotechnology, mastering these technologies does not represent a distinct competitive advantage. Most reactio... | Fine chemicals | 0.806993 |
26,792 | With the exception of some stereospecific reactions, particularly biotechnology, mastering these technologies does not represent a distinct competitive advantage. Most reactions can be carried out in standard multipurpose plants. The very versatile organometallic reactions (e.g., conversions with lithium aluminum hydri... | Fine chemicals | 0.806993 |
26,793 | Once a new chemical process has been introduced successfully on an industrial scale, process optimization is called upon to improve the economics. As a rule of thumb it should be attempted to reduce the costs of goods sold (COGS) by 10-20%, every time the yearly production quantity has doubled. The task extends from fi... | Fine chemicals | 0.806993 |
26,794 | In case of cGMP fine chemicals also a process validation is required. It consists of the three elements process design, process qualification and continued process verification. Process Optimization. | Fine chemicals | 0.806993 |
26,795 | Depending on the volume requirements, three different types of equipment are used for process research, development and optimization, namely bench-scale laboratories for gram to 100 gram, kilo-labs for kg to 10 kg and pilot plants for 100 kg to ton quantities. Particularities of laboratory processes that have to be eli... | Fine chemicals | 0.806993 |
26,796 | It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other. The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dih... | Facet of a polyhedron | 0.806993 |
26,797 | In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them. The analogous question for polyhedra was the subject of Hilbert's third problem. Max Dehn solved this problem by sh... | Facet of a polyhedron | 0.806993 |
26,798 | Virtual Reality Polyhedra – The Encyclopedia of Polyhedra. Electronic Geometry Models – Contains a peer reviewed selection of polyhedra with unusual properties. Polyhedron Models – Virtual polyhedra. Paper Models of Uniform (and other) Polyhedra Polyhedra Viewer – Web-based tool for visualizing the relationships betwee... | Facet of a polyhedron | 0.806992 |
26,799 | In geometry, a polyhedron (PL: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many', and ἕδρον (-hedron) 'base, seat') is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be co... | Facet of a polyhedron | 0.806992 |
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