id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
6,000 | In reverse chemogenomics, small compounds that perturb the function of an enzyme in the context of an in vitro enzymatic test will be identified. Once the modulators have been identified, the phenotype induced by the molecule is analyzed in a test on cells or on whole organisms. This method will identify or confirm the... | Chemical genomics | 0.825072 |
6,001 | The reason of over-constraint is the unique geometry of linkages in these mechanisms, which the mobility formula does not take into account. This unique geometry gives rise to "redundant constraints", i.e. when multiple joints are constraining the same degrees of freedom. These redundant constraints are the reason of t... | Overconstrained mechanism | 0.825072 |
6,002 | In mechanical engineering, an overconstrained mechanism is a linkage that has more degrees of freedom than is predicted by the mobility formula. The mobility formula evaluates the degree of freedom of a system of rigid bodies that results when constraints are imposed in the form of joints between the links. If the link... | Overconstrained mechanism | 0.825072 |
6,003 | A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R). | Quadratic Lie algebra | 0.825068 |
6,004 | A quadratic Lie algebra is a Lie algebra (g,) together with a non-degenerate symmetric bilinear form ( . , . ): g ⊗ g → R {\displaystyle (.,. )\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} } that is invariant under the adjoint action, i.e. (,Z)+(Y,)=0where X,Y,Z are elements of the Lie algebra g. A local... | Quadratic Lie algebra | 0.825068 |
6,005 | A big group of examples fits into the category of semisimple Lie algebras, i.e. Lie algebras whose adjoint representation is faithful. Examples are sl(n,R) and su(n), as well as direct sums of them. Let thus g be a semi-simple Lie algebra with adjoint representation ad, i.e. a d: g → E n d ( g ): X ↦ ( a d X: Y ↦ ) {\... | Quadratic Lie algebra | 0.825068 |
6,006 | The hierarchical architecture of the biological neural system inspires deep learning architectures for feature learning by stacking multiple layers of learning nodes. These architectures are often designed based on the assumption of distributed representation: observed data is generated by the interactions of many diff... | Representation learning | 0.825059 |
6,007 | An autoencoder consisting of an encoder and a decoder is a paradigm for deep learning architectures. An example is provided by Hinton and Salakhutdinov where the encoder uses raw data (e.g., image) as input and produces feature or representation as output and the decoder uses the extracted feature from the encoder as i... | Representation learning | 0.825059 |
6,008 | A larger portion of negative samples is typically necessary in order to prevent catastrophic collapse, which is when all inputs are mapped to the same representation. Generative representation learning tasks the model with producing the correct data to either match a restricted input or reconstruct the full input from ... | Representation learning | 0.825059 |
6,009 | Self-supervised representation learning is learning features by training on the structure of unlabeled data rather than relying on explicit labels for an information signal. This approach has enabled the combined use of deep neural network architectures and larger unlabeled datasets to produce deep feature representati... | Representation learning | 0.825059 |
6,010 | With appropriately defined network functions, various learning tasks can be performed by minimizing a cost function over the network function (weights). Multilayer neural networks can be used to perform feature learning, since they learn a representation of their input at the hidden layer(s) which is subsequently used ... | Representation learning | 0.825059 |
6,011 | Neural networks are a family of learning algorithms that use a "network" consisting of multiple layers of inter-connected nodes. It is inspired by the animal nervous system, where the nodes are viewed as neurons and edges are viewed as synapses. Each edge has an associated weight, and the network defines computational ... | Representation learning | 0.825059 |
6,012 | A cache can store data that is computed on demand rather than retrieved from a backing store. Memoization is an optimization technique that stores the results of resource-consuming function calls within a lookup table, allowing subsequent calls to reuse the stored results and avoid repeated computation. It is related t... | Hardware cache | 0.825059 |
6,013 | In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae (formulae that do not contain any logical connectives). In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusio... | Relevant logic | 0.825056 |
6,014 | Eugene Wigner showed that a symmetry operation S of a Hamiltonian is represented in quantum mechanics either by a unitary operator, S = U, or an antiunitary one, S = UK where U is unitary, and K denotes complex conjugation. Antiunitary operators arise in quantum mechanics due to the time reversal operator If the set of... | Corepresentations of unitary and antiunitary groups | 0.825051 |
6,015 | However, some operators of physical importance such as time reversal are antilinear, and including these in the symmetry group leads to groups including both unitary and antiunitary operators. This article is about corepresentation theory, the equivalent of representation theory for these groups. It is mainly used in t... | Corepresentations of unitary and antiunitary groups | 0.825051 |
6,016 | In quantum mechanics, symmetry operations are of importance in giving information about solutions to a system. Typically these operations form a mathematical group, such as the rotation group SO(3) for spherically symmetric potentials. The representation theory of these groups leads to irreducible representations, whic... | Corepresentations of unitary and antiunitary groups | 0.825051 |
6,017 | For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of... | Pointed unary system | 0.82505 |
6,018 | Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In this way, ev... | Pointed unary system | 0.82505 |
6,019 | Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. Boolean algebra: a complemented distributive lattice. Either of meet or join can be defi... | Pointed unary system | 0.82505 |
6,020 | Simple structures: no binary operation: Set: a degenerate algebraic structure S having no operations.Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. Group: a monoid with a unary op... | Pointed unary system | 0.82505 |
6,021 | The differential forms on a differentiable manifold form an alternating algebra. The exterior algebra is an alternating algebra. The cohomology ring of a topological space is an alternating algebra. | Alternating algebra | 0.825049 |
6,022 | In mathematics, an alternating algebra is a Z-graded algebra for which xy = (−1)deg(x)deg(y)yx for all nonzero homogeneous elements x and y (i.e. it is an anticommutative algebra) and has the further property that x2 = 0 for every homogeneous element x of odd degree. | Alternating algebra | 0.825049 |
6,023 | Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols. | Notation in probability | 0.825048 |
6,024 | Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters). A tilde (~) denotes "has the probability distribution of". Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an estimator of it, e.g., θ ^ {\displaystyle {\widehat {\theta }}} is an estima... | Notation in probability | 0.825048 |
6,025 | The regression would find the functional relationship between voltage and current to be R {\displaystyle R} , such that V = I R {\displaystyle V=IR} Classification problems are those for which the output will be an element from a discrete set of labels. Classification is very common for machine learning applications. I... | Statistical learning theory | 0.825046 |
6,026 | In machine learning problems, a major problem that arises is that of overfitting. Because learning is a prediction problem, the goal is not to find a function that most closely fits the (previously observed) data, but to find one that will most accurately predict output from future input. Empirical risk minimization ru... | Statistical learning theory | 0.825046 |
6,027 | Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields... | Statistical learning theory | 0.825046 |
6,028 | Classical Mechanics is a well-established textbook written by Thomas Walter Bannerman Kibble and Frank Berkshire of the Imperial College Mathematics Department. The book provides a thorough coverage of the fundamental principles and techniques of classical mechanics, a long-standing subject which is at the base of all ... | Classical Mechanics (Kibble and Berkshire) | 0.825042 |
6,029 | Preface Useful Constants and Units Chapter 1: Introduction Chapter 2: Linear motion Chapter 3: Energy and Angular momentum Chapter 4: Central Conservative Forces Chapter 5: Rotating Frames Chapter 6: Potential Theory Chapter 7: The Two-Body Problem Chapter 8: Many-Body Systems Chapter 9: Rigid Bodies Chapter 10: Lagran... | Classical Mechanics (Kibble and Berkshire) | 0.825042 |
6,030 | The various editions are held in 1789 libraries. In comparison, the various (2011) editions of Herbert Goldstein's Classical Mechanics are held in 1772. librariesThe original edition was reviewed in Current Science. The fourth edition was reviewed by C. Isenberg in 1997 in the European Journal of Physics, and the fifth... | Classical Mechanics (Kibble and Berkshire) | 0.825042 |
6,031 | Harmonicity of u is identical to the closedness of the differential 1-form ⋆ d u , {\displaystyle \star du,} defined using the Hodge star operator ⋆ {\displaystyle \star } associated to the Riemannian metric. The Poincaré lemma thus implies the existence of a function v on U with d v = ⋆ d u . {\displaystyle dv=\star d... | Isothermal coordinates | 0.825038 |
6,032 | The existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations. In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to th... | Isothermal coordinates | 0.825038 |
6,033 | The existence of isothermal coordinates can be proved by applying known existence theorems for the Beltrami equation, which rely on Lp estimates for singular integral operators of Calderón and Zygmund. A simpler approach to the Beltrami equation has been given more recently by Adrien Douady.If the Riemannian metric is ... | Isothermal coordinates | 0.825038 |
6,034 | By the 1950s, expositions of the ideas of Korn and Lichtenstein were put into the language of complex derivatives and the Beltrami equation by Lipman Bers and Shiing-shen Chern, among others. In this context, it is natural to investigate the existence of generalized solutions, which satisfy the relevant partial differe... | Isothermal coordinates | 0.825038 |
6,035 | In 1822, Carl Friedrich Gauss proved the existence of isothermal coordinates on an arbitrary surface with a real-analytic Riemannian metric, following earlier results of Joseph Lagrange in the special case of surfaces of revolution. The construction used by Gauss made use of the Cauchy–Kowalevski theorem, so that his m... | Isothermal coordinates | 0.825038 |
6,036 | In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form g = φ ( d x 1 2 + ⋯ + d x n 2 ) , {\displaystyle g=\va... | Isothermal coordinates | 0.825038 |
6,037 | The pathwidth of any n-vertex cubic graph is at most n/6. The best known lower bound on the pathwidth of cubic graphs is 0.082n. It is not known how to reduce this gap between this lower bound and the n/6 upper bound.It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the first paper on g... | Cubic graphs | 0.825038 |
6,038 | By Kőnig's line coloring theorem every bicubic graph has a Tait coloring. The bridgeless cubic graphs that do not have a Tait coloring are known as snarks. They include the Petersen graph, Tietze's graph, the Blanuša snarks, the flower snark, the double-star snark, the Szekeres snark and the Watkins snark. There is an ... | Cubic graphs | 0.825038 |
6,039 | In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. | Cubic graphs | 0.825038 |
6,040 | According to Brooks' theorem every connected cubic graph other than the complete graph K4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color... | Cubic graphs | 0.825038 |
6,041 | A body contact or body fault is a conductive connection between body and active parts of electrical equipment resulting from a fault.A body, short or ground fault is: Perfect (direct) contactIf there is no fault resistance in the circuit Imperfect (indirect) contactIf a fault resistance is present in the circuit (e.g. ... | Body contact (electricity) | 0.825036 |
6,042 | Historically, this, as well as the Chu–Vandermonde identity, ( x + y ) n = ∑ k = 0 n ( n k ) ( x ) n − k ( y ) k , {\displaystyle (x+y)_{n}=\sum _{k=0}^{n}{\binom {n}{k}}(x)_{n-k}\,(y)_{k},} (following from it, and corresponding to the binomial theorem), are included in the observations that matured to the system of um... | Finite difference | 0.825035 |
6,043 | is the binomial coefficient, and ( x ) k = x ( x − 1 ) ( x − 2 ) ⋯ ( x − k + 1 ) {\displaystyle (x)_{k}=x(x-1)(x-2)\cdots (x-k+1)} is the "falling factorial" or "lower factorial", while the empty product (x)0 is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values... | Finite difference | 0.825035 |
6,044 | Kim studied industrial engineering at KAIST in Korea, obtaining a master's degree, and then completed her Ph.D. in computer science in 2010 at Royal Holloway, University of London. Her dissertation was supervised by Gregory Gutin.After postdoctoral research in Montpellier, France, at the Laboratoire d'Informatique, de ... | Eun Jung Kim (parameterized complexity) | 0.82503 |
6,045 | The split-ubiquitin membrane yeast two-hybrid system uses transcriptional reporters to identify yeast transformants that encode pairs of interacting proteins. In 2006, random forest, an example of a supervised technique, was found to be the most-effective machine learning method for protein interaction prediction. Such... | Protein–protein interaction | 0.825029 |
6,046 | Prediction models using machine learning techniques can be broadly classified into two main groups: supervised and unsupervised, based on the labeling of input variables according to the expected outcome.In 2005, integral membrane proteins of Saccharomyces cerevisiae were analyzed using the mating-based ubiquitin syste... | Protein–protein interaction | 0.825029 |
6,047 | Many computational methods have been suggested and reviewed for predicting protein–protein interactions. Prediction approaches can be grouped into categories based on predictive evidence: protein sequence, comparative genomics, protein domains, protein tertiary structure, and interaction network topology. The construct... | Protein–protein interaction | 0.825029 |
6,048 | It has even been mandated in the curriculum of some regions.Computer algebra systems have been extensively used in higher education. Many universities offer either specific courses on developing their use, or they implicitly expect students to use them for their course work. The companies that develop computer algebra ... | Symbolic algebra | 0.825028 |
6,049 | There have been many advocates for increasing the use of computer algebra systems in primary and secondary-school classrooms. The primary reason for such advocacy is that computer algebra systems represent real-world math more than do paper-and-pencil or hand calculator based mathematics. This push for increasing compu... | Symbolic algebra | 0.825028 |
6,050 | Many also include: a programming language, allowing users to implement their own algorithms arbitrary-precision numeric operations exact integer arithmetic and number theory functionality Editing of mathematical expressions in two-dimensional form plotting graphs and parametric plots of functions in two and three dimen... | Symbolic algebra | 0.825028 |
6,051 | Thus, through effects of canonical neurotransmitter systems such as catecholamines, social stressors are able to penetrate the nucleus of various cell types and alter the gene transcription profiles within these cells. Other transcription factors that have been known to respond to social factors include some factors br... | Social genomics | 0.825024 |
6,052 | Metacomputing, as a computing of computing, includes: the organization of large computer networks, choice of the design criteria (for example: peer-to-peer or centralized solution) and metacomputing software (middleware, metaprogramming) development where, in the specific domains, the concept metacomputing is used as a... | Metacomputing | 0.825007 |
6,053 | It describes how far away our assumptions ( F {\displaystyle {\mathcal {F}}} ) are from the truth ( F ∗ {\displaystyle {\mathcal {F}}^{*}} ). ( b ) {\displaystyle (b)} will be strictly greater than 0 if we make assumptions that are too strong ( F {\displaystyle {\mathcal {F}}} too small). On the other hand, failing to ... | Learnable function class | 0.825006 |
6,054 | If the true relationship between y {\displaystyle y} and x {\displaystyle x} is y ∼ f ∗ ( x ) {\displaystyle y\sim f^{*}(x)} , then by selecting the appropriate loss function, f ∗ {\displaystyle f^{*}} can always be expressed as the minimizer of the expected loss across all possible functions. That is, f ∗ = arg min ... | Learnable function class | 0.825006 |
6,055 | Interplay between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in statistics literature is often known as the bias-variance tradeoff. However, note that in the authors gave an example of stochastic convex optimization for General Setting of Learning where learnability is not equivalent with uniform convergen... | Learnable function class | 0.825006 |
6,056 | Part ( a ) {\displaystyle (a)} in (2) is closely linked to empirical process theory in statistics, where the empirical risk { ∑ i = 1 n L ( y i , f ( x i ) ) , f ∈ F } {\displaystyle \{\sum _{i=1}^{n}L(y_{i},f(x_{i})),f\in {\mathcal {F}}\}} are known as empirical processes. In this field, the function class F {\display... | Learnable function class | 0.825006 |
6,057 | In statistical learning theory, a learnable function class is a set of functions for which an algorithm can be devised to asymptotically minimize the expected risk, uniformly over all probability distributions. The concept of learnable classes are closely related to regularization in machine learning, and provides larg... | Learnable function class | 0.825006 |
6,058 | We can make the condition given in the above equation stronger by requiring that the convergence is uniform for all probability distributions. That is: The intuition behind the more strict requirement is as such: the rate at which sequence { f ^ n } {\displaystyle \{{\hat {f}}_{n}\}} converges to the minimizer of the e... | Learnable function class | 0.825006 |
6,059 | This example shows that classifying spaces may be complicated. In relation with differential geometry (Chern–Weil theory) and the theory of Grassmannians, a much more hands-on approach to the theory is possible for cases such as the unitary groups that are of greatest interest. The construction of the Thom complex MG s... | Classifying space | 0.825003 |
6,060 | Such constructions make evident the connection with group cohomology. Specifically, let EG be the weak simplicial complex whose n- simplices are the ordered (n+1)-tuples {\displaystyle } of elements of G. Such an n-simplex attaches to the (n−1) simplices {\displaystyle } in the same way a standard simplex attaches to... | Classifying space | 0.825003 |
6,061 | This still leaves the question of doing effective calculations with BG; for example, the theory of characteristic classes is essentially the same as computing the cohomology groups of BG, at least within the restrictive terms of homotopy theory, for interesting groups G such as Lie groups (H. Cartan's theorem). As was ... | Classifying space | 0.825003 |
6,062 | Statistical mechanics, also known as statistical thermodynamics, emerged with the development of atomic and molecular theories in the late 19th century and early 20th century, and supplemented classical thermodynamics with an interpretation of the microscopic interactions between individual particles or quantum-mechani... | Thermodynamics | 0.824994 |
6,063 | The modern view is that centrifugal force is indeed an indicator of rotation, but relative to those frames of reference that exhibit the simplest laws of physics. Thus, for example, if we wonder how rapidly our galaxy is rotating, we can make a model of the galaxy in which its rotation plays a part. The rate of rotatio... | History of centrifugal and centripetal forces | 0.824987 |
6,064 | That is, these ideas flowed from operational definitions of measurement coupled with the experimental confirmation of the constancy of the speed of light. Later the general theory of relativity further generalized the idea of frame independence of the laws of physics, and abolished the special position of inertial fram... | History of centrifugal and centripetal forces | 0.824987 |
6,065 | In physics, the history of centrifugal and centripetal forces illustrates a long and complex evolution of thought about the nature of forces, relativity, and the nature of physical laws. | History of centrifugal and centripetal forces | 0.824987 |
6,066 | A DTA curve can be used only as a finger print for identification purposes but usually the applications of this method are the determination of phase diagrams, heat change measurements and decomposition in various atmospheres. DTA is widely used in the pharmaceutical and food industries.DTA may be used in cement chemis... | Differential thermal analysis | 0.824984 |
6,067 | In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are ordinals... | Isomorphism class | 0.824979 |
6,068 | G. A. Barnard (1958) "Studies in the History of Probability and Statistics: IX. Thomas Bayes's Essay Towards Solving a Problem in the Doctrine of Chances", Biometrika 45:293–295. (biographical remarks) Stephen M. Stigler (1982). "Thomas Bayes's Bayesian Inference," Journal of the Royal Statistical Society, Series A, 14... | An Essay towards solving a Problem in the Doctrine of Chances | 0.824977 |
6,069 | Barnard, G. A (1958). "Studies in the History of Probability and Statistics: Ix. Thomas Bayes's Essay Towards Solving a Problem in the Doctrine of Chances". | An Essay towards solving a Problem in the Doctrine of Chances | 0.824977 |
6,070 | Bayes's preliminary results in conditional probability (especially Propositions 3, 4 and 5) imply the truth of the theorem that is named for him. He states:"If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event ... | An Essay towards solving a Problem in the Doctrine of Chances | 0.824977 |
6,071 | An Essay towards solving a Problem in the Doctrine of Chances is a work on the mathematical theory of probability by Thomas Bayes, published in 1763, two years after its author's death, and containing multiple amendments and additions due to his friend Richard Price. The title comes from the contemporary use of the phr... | An Essay towards solving a Problem in the Doctrine of Chances | 0.824977 |
6,072 | Richard Price discovered Bayes's essay and its now-famous theorem in Bayes's papers after Bayes's death. He believed that Bayes's Theorem helped prove the existence of God ("the Deity") and wrote the following in his introduction to the essay: "The purpose I mean is, to show what reason we have for believing that there... | An Essay towards solving a Problem in the Doctrine of Chances | 0.824977 |
6,073 | When a relatively high energy is applied to a particle with relatively low rest mass, it can be approximated as E ≃ p {\displaystyle E\simeq p} in high-energy physics such that an applied energy in units of eV conveniently results in an approximately equivalent change of momentum in units of eV/c. The dimensions of mom... | Electron volts | 0.824971 |
6,074 | It is a common unit of energy within physics, widely used in solid state, atomic, nuclear, and particle physics, and high-energy astrophysics. It is commonly used with SI prefixes milli-, kilo-, mega-, giga-, tera-, peta- or exa- (meV, keV, MeV, GeV, TeV, PeV and EeV respectively). In some older documents, and in the n... | Electron volts | 0.824971 |
6,075 | In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. When used as a unit of energy, the numerical value of 1 eV in joules (sym... | Electron volts | 0.824971 |
6,076 | Dynamic combinatorial chemistry has been used as a method to develop ligands for biomolecules and receptors for small molecules.Ligands that can recognize biomolecules are being identified by preparing libraries of potential ligands in the presence of a target biomacromolecule. This is relevant for application as biose... | Systems chemistry | 0.824969 |
6,077 | While multicomponent reactions have been studied for centuries, the idea of deliberately analyzing mixtures and reaction networks is more recent. The first mentions of systems chemistry as a field date from 2005. Early adopters focused on prebiotic chemistry combined with supramolecular chemistry, before it was general... | Systems chemistry | 0.824969 |
6,078 | Such homeostatic (open) systems are far-from-equilibrium and are dissipative: they need energy to maintain themselves. In dissipative controlled systems the continuous supply of energy allows a continuous transition between different supramolecular states, where systems with unexpected properties may be discovered. One... | Systems chemistry | 0.824969 |
6,079 | A fundamental difference exists between chemistry as it is performed in most laboratories and chemistry as it occurs in life. Laboratory processes are mostly designed such that the (closed) system goes thermodynamically downhill; i.e. the product state is of lower Gibbs free energy, yielding stable molecules that can b... | Systems chemistry | 0.824969 |
6,080 | Systems chemistry is the science of studying networks of interacting molecules, to create new functions from a set (or library) of molecules with different hierarchical levels and emergent properties.Systems chemistry is also related to the origin of life (abiogenesis). | Systems chemistry | 0.824969 |
6,081 | Systems chemistry is a relatively young sub-discipline of chemistry, where the focus does not lie on the individual chemical components but rather on the overall network of interacting molecules and on their emergent properties. Hence, it combines the classical knowledge of chemistry (structure, reactions and interacti... | Systems chemistry | 0.824969 |
6,082 | An especially distinctive property is that a cross product can be defined only in three or seven dimensions. This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were ... | Seven-dimensional space | 0.824964 |
6,083 | A second of arc, arcsecond (arcsec), or arc second, denoted by the symbol ″, is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 (about 1/206264.8) of a radian. These units originated in Babylonian astronomy as sexagesimal (base 60) subdivisions of the degree; they are used in fields that inv... | Minutes and seconds of arc | 0.824964 |
6,084 | If RGR specifically is of interest, another approach is to separate size effects from intrinsic growth differences mathematically.Decomposing the RGR ignores the dependency of plant growth rate on plant size (or allometry) and assumes, incorrectly, that plant growth is directly proportional to total plant size (isometr... | Plant growth analysis | 0.82496 |
6,085 | In comparing different treatments, genotypes or species, the simplest type of growth analysis is to evaluate size of plants after a certain period of growth, typically from the time of germination. In plant biology, size is often measured as dry mass of whole plants (M), or the above-ground part of it. In high-throughp... | Plant growth analysis | 0.82496 |
6,086 | Although the ULR is not equal to the rate of photosynthesis per unit leaf area, both values are often well correlated. The LAR can be further subdivided into two other variables that are relevant for plant biology: Specific leaf area (SLA) and Leaf Mass Fraction (LMF). SLA is the leaf area of a plant (or a given leaf) ... | Plant growth analysis | 0.82496 |
6,087 | Molecular Plant is a monthly peer-reviewed scientific journal which publishes both original research and review articles in the field of plant biology, with a particular emphasis on plant cell biology, physiology, biochemistry, molecular biology, genetics, development, plant-microbe interaction, genomics, bioinformatic... | Molecular Plant | 0.824932 |
6,088 | MHD applies to astrophysics, including stars, the interplanetary medium (space between the planets), and possibly within the interstellar medium (space between the stars) and jets. Most astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all th... | Electrically conducting fluid | 0.824931 |
6,089 | Lesion studies are commonly used in cognitive neuroscience research. Lesions can occur naturally through trauma or disease, or they can be surgically induced by researchers. In the study of declarative memory, the hippocampus and the amygdala are two structures frequently examined using this technique. | Explicit memory | 0.824928 |
6,090 | This can be proven using Gauss's lemma. Quadratic Reciprocity (Gauss; Fourth Proof). Let a, b, c, ... be unequal positive odd primes, whose product is n, and let m be the number of them that are ≡ 3 (mod 4); check whether n/a is a residue of a, whether n/b is a residue of b, .... The number of nonresidues found will be... | Quadratic Reciprocity | 0.824926 |
6,091 | (This formally resembles the residue theorem from complex analysis.) The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol. There is no kind of reciproc... | Quadratic Reciprocity | 0.824926 |
6,092 | Legendre lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: He says that since expressions of the form N c − 1 2 mod c , gcd ( N , c ) = 1 {\displaystyle N^{\frac {c-1}{2}}{\bmod {c}},\... | Quadratic Reciprocity | 0.824926 |
6,093 | He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2). Generalizing the reciprocity law to higher powers has been a lead... | Quadratic Reciprocity | 0.824926 |
6,094 | (Art. 151)Privately, Gauss referred to it as the "golden theorem". | Quadratic Reciprocity | 0.824926 |
6,095 | {\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.} This formula only works if it is known in advance that a {\displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadra... | Quadratic Reciprocity | 0.824926 |
6,096 | In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calc... | Quadratic Reciprocity | 0.824926 |
6,097 | The study of the Sun has a special place in observational astrophysics. Due to the tremendous distance of all other stars, the Sun can be observed in a kind of detail unparalleled by any other star. Understanding the Sun serves as a guide to understanding of other stars. The topic of how stars change, or stellar evolut... | Stellar physics | 0.824908 |
6,098 | Ultraviolet, X-ray and gamma ray astronomy study very energetic processes such as binary pulsars, black holes, magnetars, and many others. These kinds of radiation do not penetrate the Earth's atmosphere well. There are two methods in use to observe this part of the electromagnetic spectrum—space-based telescopes and g... | Stellar physics | 0.824908 |
6,099 | Infrared observations are usually made with telescopes similar to the familiar optical telescopes. Objects colder than stars (such as planets) are normally studied at infrared frequencies. Optical astronomy was the earliest kind of astronomy. | Stellar physics | 0.824908 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.