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33,800 | In chemical engineering, the principles of thermodynamics are commonly applied to "open systems", i.e. those in which heat, work, and mass flow across the system boundary. Flows of both heat ( Q ˙ {\displaystyle {\dot {Q}}} ) and work, i.e. W ˙ S {\displaystyle {\dot {W}}_{\text{S}}} (shaft work) and P ( d V / d t ) {\... | Entropy | 0.803261 |
33,801 | Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature i... | Entropy | 0.803261 |
33,802 | As the second law of thermodynamics shows, in an isolated system internal portions at different temperatures tend to adjust to a single uniform temperature and thus produce equilibrium. A recently developed educational approach avoids ambiguous terms and describes such spreading out of energy as dispersal, which leads ... | Entropy | 0.803261 |
33,803 | This texture effect as well as the interaction with other interfaces in the PV module is a challenging optical simulation task. A particularly efficient method for modeling and optimization is the OPTOS formalism. | Photoelectric cells | 0.803257 |
33,804 | Incident light rays onto a textured surface do not reflect back out to the air as opposed to rays onto a flat surface. Rather some light rays are bounced back onto the other surface again due to the geometry of the surface. This process significantly improves light to electricity conversion efficiency, due to increased... | Photoelectric cells | 0.803257 |
33,805 | National Renewable Energy Laboratory tests and validates solar technologies. Three reliable groups certify solar equipment: UL and IEEE (both U.S. standards) and IEC. Solar cells are manufactured in volume in Japan, Germany, China, Taiwan, Malaysia and the United States, whereas Europe, China, the U.S., and Japan have ... | Photoelectric cells | 0.803257 |
33,806 | Since most of the patients were young and otherwise healthy and Parkinson's disease tends to afflict people at a much older age, researchers at the hospital began to scrutinize the illicitly synthesized opiates that the patients had ingested. The researchers discovered that the opiates were tainted with MPTP, which is ... | MPP+ | 0.803255 |
33,807 | MPP+ has been known since at least the 1920s, with a synthesis of the compound being published in a German chemistry journal in 1923. Its neurotoxic effects, however, were not known until much later, with the first paper definitively identifying MPP+ as a Parkinson's-inducing poison being published in 1983. This paper ... | MPP+ | 0.803255 |
33,808 | This led to the study of linkages that could generate straight lines, even if only approximately; and inspired the mathematician J. J. Sylvester, who lectured on the Peaucellier linkage, which generates an exact straight line from a rotating crank.The work of Sylvester inspired A. B. Kempe, who showed that linkages for... | Linkage (mechanical) | 0.803252 |
33,809 | Archimedes applied geometry to the study of the lever. Into the 1500s the work of Archimedes and Hero of Alexandria were the primary sources of machine theory. It was Leonardo da Vinci who brought an inventive energy to machines and mechanism.In the mid-1700s the steam engine was of growing importance, and James Watt r... | Linkage (mechanical) | 0.803252 |
33,810 | A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A ... | Linkage (mechanical) | 0.803252 |
33,811 | Linkage systems are widely distributed in animals. The most thorough overview of the different types of linkages in animals has been provided by Mees Muller, who also designed a new classification system which is especially well suited for biological systems. A well-known example is the cruciate ligaments of the knee. ... | Linkage (mechanical) | 0.803252 |
33,812 | The journal was never indexed by the Science Citation Index and therefore didn't get an official impact factor. According to an unofficial calculation in 2007, it reached an impact factor of 20.1. The journal was discontinued in April 2010 and merged with the existing journal BMC Biology. | Journal of Biology | 0.803251 |
33,813 | The Journal of Biology was a peer-reviewed scientific journal published by BioMed Central. It was established in 2002 with the aim to provide an alternative to biology journals with high-impact factor such as Nature, Science, and Cell. Because of stringent selection criteria, it published only a few research articles p... | Journal of Biology | 0.803251 |
33,814 | The group H {\displaystyle H} can, however, be given a different topology, in which the distance between two points h 1 , h 2 ∈ H {\displaystyle h_{1},h_{2}\in H} is defined as the length of the shortest path in the group H {\displaystyle H} joining h 1 {\displaystyle h_{1}} to h 2 {\displaystyle h_{2}} . In this topol... | Lie Group | 0.803251 |
33,815 | We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by H = { ( e 2 π i θ 0 0 e 2 π i a θ ): θ ∈ R } ⊂ T 2 = { ( e 2 π i θ 0 0 e 2 π i ϕ ): θ , ϕ ∈ R } , {\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i... | Lie Group | 0.803251 |
33,816 | Conversely, let G be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group G ′ {\displaystyle G'} that is locally isomorphic to G. Then, by a version of the closed subgroup theorem, G ′ {\displaystyle G'} is a real-analytic manifold and then, through the local i... | Lie Group | 0.803251 |
33,817 | Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the follow... | Lie Group | 0.803251 |
33,818 | A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group GL ( n , C ) {\displaystyle \operatorname {GL} ... | Lie Group | 0.803251 |
33,819 | In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by = a b c − a c b − b a c + b c a + c a b − c b a . {\displaystyle =abc-acb-bac+bca+cab-cba.\,} Also called the ternutator or alternating ternary sum, it is a special case of the n-commutator for n = 3, where... | Ternary commutator | 0.80325 |
33,820 | Certification is often used in the professions of software engineering and information technology. CITP (Chartered IT Professional) conferred by British Computer Society, The Institution of Engineering and Technology and by other professional engineering institutions in the UK and commonwealth | Certification exam | 0.80325 |
33,821 | The American Academy of Environmental Engineers board certifies licensed environmental engineers (Board Certified Environmental Engineer—BCEE) and unlicensed environmental engineering practitioners (Board Certified Environmental Engineering Member—BCEEM) for those with a degree in engineering and at least 8 years of pr... | Certification exam | 0.80325 |
33,822 | It follows that, as with LFSRs and linear complexity, any stream cipher whose N-adic complexity is low should not be used for cryptography. FCSRs and LFSRs are special cases of a very general algebraic construction of sequence generators called Algebraic Feedback Shift Registers (AFSRs) in which the integers are replac... | Feedback with Carry Shift Registers | 0.803249 |
33,823 | Associated with the FCSR is a connection integer q = q r N r + ⋯ + q 1 N 1 − 1 {\displaystyle q=q_{r}N^{r}+\dots +q_{1}N^{1}-1} . Associated with the output sequence is the N-adic number a = a 0 + a 1 N + a 2 N 2 + … {\displaystyle a=a_{0}+a_{1}N+a_{2}N^{2}+\dots } The fundamental theorem of FCSRs says that there is an... | Feedback with Carry Shift Registers | 0.803249 |
33,824 | Then the new state is ( a 1 , a 2 , … , a r ; z ′ ) {\displaystyle (a_{1},a_{2},\dots ,a_{r};z')} . By iterating the state change an FCSR generates an infinite, eventually periodic sequence of numbers in S {\displaystyle S} . FCSRs have been used in the design of stream ciphers (such as the F-FCSR generator), in the cr... | Feedback with Carry Shift Registers | 0.803249 |
33,825 | In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators L n , L ¯ n , − ∞ < n < ∞ {\displaystyle L_{n},{\bar {L}}_{n},-\infty 0, which are the lowering generators. Descendants are obtained from the primaries by acting with L n , L ¯ n {\displaystyle L_{n}... | Primary field | 0.803249 |
33,826 | Generally, each superconformal primary representations will include several primaries of the conformal algebra, which arise by acting with the supercharges Q {\displaystyle Q} on the superconformal primary. There exist also special chiral superconformal primary operators, which are primary operators annihilated by some... | Primary field | 0.803249 |
33,827 | In D ≤ 6 {\displaystyle D\leq 6} dimensions, conformal algebra allows graded extensions containing fermionic generators. Quantum field theories invariant with respect to such extended algebras are called superconformal. In superconformal field theories, one considers superconformal primary operators. In D > 2 {\display... | Primary field | 0.803249 |
33,828 | In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the representation theory point of view, a primary is the lowest dimension o... | Primary field | 0.803249 |
33,829 | The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field eq... | Solutions of the Einstein field equations | 0.803249 |
33,830 | The development of high-speed and high-sensitivity pump–probe imaging techniques has enabled applications in several fields, such as materials science, biology, and art. | Pump-probe microscopy | 0.803249 |
33,831 | Pump–probe microscopy is a non-linear optical imaging modality used in femtochemistry to study chemical reactions. It generates high-contrast images from endogenous non-fluorescent targets. It has numerous applications, including materials science, medicine, and art restoration. | Pump-probe microscopy | 0.803249 |
33,832 | William Thurston proved many knots are hyperbolic knots, meaning that the knot complement (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of hyperbolic geometry. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure... | Knot diagram | 0.803247 |
33,833 | The notion of a knot has further generalisations in mathematics, see: Knot (mathematics), isotopy classification of embeddings. Every knot in the n-sphere S n {\displaystyle \mathbb {S} ^{n}} is the link of a real-algebraic set with isolated singularity in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} (Akbulut & King 1981... | Knot diagram | 0.803247 |
33,834 | This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory terms for basic terminology | List of graph theory topics | 0.803245 |
33,835 | Ant colony algorithm Breadth-first search Depth-first search Depth-limited search FKT algorithm Flood fill Graph exploration algorithm Matching (graph theory) Max flow min cut theorem Maximum-cardinality search Shortest path Dijkstra's algorithm Bellman–Ford algorithm A* algorithm Floyd–Warshall algorithm Topological s... | List of graph theory topics | 0.803245 |
33,836 | Node Child node Parent node Leaf node Root node Root (graph theory) | List of graph theory topics | 0.803245 |
33,837 | The way it is done is by passing to the modern interpretation. Dirac's idea is more directly applicable to solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though t... | Hole theory | 0.803244 |
33,838 | Power system automation enables rapid diagnosis of and precise solutions to specific grid disruptions or outages. These technologies rely on and contribute to each of the other four key areas. Three technology categories for advanced control methods are distributed intelligent agents (control systems), analytical tools... | Smart Grid | 0.803243 |
33,839 | Many in the power systems engineering community believe that the Northeast blackout of 2003 could have been contained to a much smaller area if a wide area phasor measurement network had been in place. Distributed power flow control: power flow control devices clamp onto existing transmission lines to control the flow ... | Smart Grid | 0.803243 |
33,840 | Power from photovoltaic cells (and to lesser extent wind turbines) has also, significantly, called into question the imperative for large, centralised power stations. The rapidly falling costs point to a major change from the centralised grid topology to one that is highly distributed, with power being both generated a... | Smart Grid | 0.803243 |
33,841 | Since the early 21st century, opportunities to take advantage of improvements in electronic communication technology to resolve the limitations and costs of the electrical grid have become apparent. Technological limitations on metering no longer force peak power prices to be averaged out and passed on to all consumers... | Smart Grid | 0.803243 |
33,842 | In order for grids to operate economically and reliably, demand forecasting is essential, because it is used to predict the amount of power that will be consumed by the load. This is dependent on weather conditions, type of day, random events, incidents, etc. For non-linear loads though, the load profile isn't smooth a... | Smart Grid | 0.803243 |
33,843 | Electric power systems can be classified in multiple different ways: non-linear, dynamic, discrete, or random. Artificial Neural Networks (ANNs) attempt to solve the most difficult of these problems, the non-linear problems. Demand ForecastingOne application of ANNs is in demand forecasting. | Smart Grid | 0.803243 |
33,844 | Smart Grid Communication NetworkNetwork Simulators are used to simulate/emulate network communication effects. This typically involves setting up a lab with the smart grid devices, applications etc. with the virtual network being provided by the network simulator. Neural networksNeural networks have been considered for... | Smart Grid | 0.803243 |
33,845 | Many different concepts have been used to model intelligent power grids. They are generally studied within the framework of complex systems. In a recent brainstorming session, the power grid was considered within the context of optimal control, ecology, human cognition, glassy dynamics, information theory, microphysics... | Smart Grid | 0.803243 |
33,846 | J ( x ) ⊢ K {\displaystyle \Lambda x\in C.J(x)\vdash K} . An implementation of the LF logical framework is provided by the Twelf system at Carnegie Mellon University. Twelf includes a logic programming engine meta-theoretic reasoning about logic programs (termination, coverage, etc.) an inductive meta-logical theorem p... | LF (logical framework) | 0.80324 |
33,847 | In the case of the LF logical framework, the meta-language is the λΠ-calculus. This is a system of first-order dependent function types which are related by the propositions as types principle to first-order minimal logic. The key features of the λΠ-calculus are that it consists of entities of three levels: objects, ty... | LF (logical framework) | 0.80324 |
33,848 | In quantum mechanics, and especially quantum information processing, the entropy exchange of a quantum operation ϕ {\displaystyle \phi \,} acting on the density matrix ρ Q {\displaystyle \rho _{Q}\,} of a system Q {\displaystyle Q\,} is defined as S ( ρ , ϕ ) ≡ S = S ( ρ Q R ′ ) {\displaystyle S(\rho ,\phi )\equiv S=S... | Entropy exchange | 0.803239 |
33,849 | Multiple cache replacement policies have attempted to use perceptrons, markov chains or other types of machine learning to predict which line to evict. There are also learning augmented algorithms for the cache replacement problem. | Cache replacement policy | 0.803239 |
33,850 | The algorithm was proposed by Jiapu Pan and Willis J. Tompkins in 1985, in the journal IEEE Transactions on Biomedical Engineering. The performance of the method was tested on an annotated arrhythmia database (MIT/BIH) and evaluated also in presence of noise. Pan and Tompkins reported that the 99.3 percent of QRS compl... | Pan-Tompkins algorithm | 0.803235 |
33,851 | Plate notation has been implemented in various TeX/LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab. == References == | Plate notation | 0.803234 |
33,852 | Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at... | Cycle index | 0.803234 |
33,853 | In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used i... | Cycle index | 0.803234 |
33,854 | {\displaystyle \left({\begin{matrix}1&2&3&4&5&6\\2&1&3&5&6&4\end{matrix}}\right)=(12)(3)(456).} This permutation is the product of three cycles, one of length two, one of length three and a fixed point. The elements in these cycles are disjoint subsets of X and form a partition of X. The cycle structure of a permutatio... | Cycle index | 0.803234 |
33,855 | As polynomials they may also be formally added, subtracted, differentiated and integrated. The area of symbolic combinatorics provides combinatorial interpretations of the results of these formal operations. The question of what the cycle structure of a random permutation looks like is an important question in the anal... | Cycle index | 0.803233 |
33,856 | {\displaystyle \sum _{k=0}^{n}F_{k}t^{k}/k!=Z(G;1+t,1,1,\ldots ,1).} Let G be a group acting on the set X and h a function from X to Y. For any g in G, h(xg) is also a function from X to Y. Thus, G induces an action on the set YX of all functions from X to Y. The number of orbits of this action is Z(G; b, b, ...,b) whe... | Cycle index | 0.803233 |
33,857 | A transitive permutation group is regular (or sometimes referred to as sharply transitive) if the only permutation in the group that has fixed points is the identity permutation. A finite transitive permutation group G on the set X is regular if and only if |G| = |X|. Cayley's theorem states that every abstract group h... | Cycle index | 0.803233 |
33,858 | The group homomorphism can be thought of as a means for permitting the group G to "act" on the set X (using the permutations associated with the elements of G). Such a group homomorphism is formally called a group action and the image of the homomorphism is a permutation representation of G. A given group can have many... | Cycle index | 0.803233 |
33,859 | In photonics, band gaps or stop bands are ranges of photon frequencies where, if tunneling effects are neglected, no photons can be transmitted through a material. A material exhibiting this behaviour is known as a photonic crystal. The concept of hyperuniformity has broadened the range of photonic band gap materials, ... | Band gap engineering | 0.803231 |
33,860 | Intel's C and Fortran compilers enable the DAZ (denormals-are-zero) and FTZ (flush-to-zero) flags for SSE by default for optimization levels higher than -O0. The effect of DAZ is to treat subnormal input arguments to floating-point operations as zero, and the effect of FTZ is to return zero instead of a subnormal float... | Denormal numbers | 0.803231 |
33,861 | This rule was introduced by Amos Storkey in 1997 and is both local and incremental. Storkey also showed that a Hopfield network trained using this rule has a greater capacity than a corresponding network trained using the Hebbian rule. The weight matrix of an attractor neural network is said to follow the Storkey learn... | Hopfield neural network | 0.803229 |
33,862 | (Note that the Hebbian learning rule takes the form w i j = ( 2 V i s − 1 ) ( 2 V j s − 1 ) {\displaystyle w_{ij}=(2V_{i}^{s}-1)(2V_{j}^{s}-1)} when the units assume values in { 0 , 1 } {\displaystyle \{0,1\}} .) Once the network is trained, w i j {\displaystyle w_{ij}} no longer evolve. If a new state of neurons V s ′... | Hopfield neural network | 0.803229 |
33,863 | An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous va... | Hopfield neural network | 0.803229 |
33,864 | Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or con... | Hopfield neural network | 0.803229 |
33,865 | Hopfield nets have a scalar value associated with each state of the network, referred to as the "energy", E, of the network, where: E = − 1 2 ∑ i , j w i j s i s j − ∑ i θ i s i {\displaystyle E=-{\frac {1}{2}}\sum _{i,j}w_{ij}s_{i}s_{j}-\sum _{i}\theta _{i}s_{i}} This quantity is called "energy" because it either decr... | Hopfield neural network | 0.803229 |
33,866 | Minimizing the Hopfield energy function both minimizes the objective function and satisfies the constraints also as the constraints are “embedded” into the synaptic weights of the network. Although including the optimization constraints into the synaptic weights in the best possible way is a challenging task, many diff... | Hopfield neural network | 0.803229 |
33,867 | Hopfield and Tank presented the Hopfield network application in solving the classical traveling-salesman problem in 1985. Since then, the Hopfield network has been widely used for optimization. The idea of using the Hopfield network in optimization problems is straightforward: If a constrained/unconstrained cost functi... | Hopfield neural network | 0.803229 |
33,868 | Deep Convolutional features extracted from a comprehensive labeled dataset, contain substantial representations which could be effectively used in a new domain. Despite the fact that generic features achieved good results in many visual tasks, fine-tuning is required for pretrained deep CNN models to be more effective ... | Hopfield neural network | 0.803229 |
33,869 | Bruck shed light on the behavior of a neuron in the discrete Hopfield network when proving its convergence in his paper in 1990. A subsequent paper further investigated the behavior of any neuron in both discrete-time and continuous-time Hopfield networks when the corresponding energy function is minimized during an op... | Hopfield neural network | 0.803229 |
33,870 | The dynamical equations describing temporal evolution of a given neuron are given by This equation belongs to the class of models called firing rate models in neuroscience. Each neuron I {\displaystyle I} collects the axonal outputs g J {\displaystyle g_{J}} from all the neurons, weights them with the synaptic coeffici... | Hopfield neural network | 0.803229 |
33,871 | Biological neural networks have a large degree of heterogeneity in terms of different cell types. This section describes a mathematical model of a fully connected modern Hopfield network assuming the extreme degree of heterogeneity: every single neuron is different. Specifically, an energy function and the correspondin... | Hopfield neural network | 0.803229 |
33,872 | A Hopfield network (or Amari-Hopfield network, Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described by Shun'ichi Amari in 1972 and by Little in 1974 based on Ernst Ising's work with ... | Hopfield neural network | 0.803229 |
33,873 | The Ising model of a recurrent neural network as a learning memory model was first proposed by Shun'ichi Amari in 1972 and then by William A. Little in 1974, who was acknowledged by Hopfield in his 1982 paper. Networks with continuous dynamics were developed by Hopfield in his 1984 paper. A major advance in memory stor... | Hopfield neural network | 0.803229 |
33,874 | It is important to note that Hopfield's network model utilizes the same learning rule as Hebb's (1949) learning rule, which basically tried to show that learning occurs as a result of the strengthening of the weights by when activity is occurring. Rizzuto and Kahana (2001) were able to show that the neural network mode... | Hopfield neural network | 0.803229 |
33,875 | It is convenient to define these activation functions as derivatives of the Lagrangian functions for the two groups of neurons This way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are ... | Hopfield neural network | 0.803229 |
33,876 | Kate Jones, Chair of Ecology and Biodiversity at University College London, says zoonotic diseases are increasingly linked to environmental change and human behaviour. The disruption of pristine forests driven by logging, mining, road building through remote places, rapid urbanisation and population growth is bringing ... | Zoonotic bacteria | 0.803225 |
33,877 | The three wind tunnels that are part of this system are: 11-by 11-foot Transonic Test Section. A closed-return, variable-density tunnel with a fixed-geometry, ventilated test section with a flexible wall nozzle with a Mach range of 0.20 to 1.45. 9-by 7-foot Supersonic Test Section. | Unitary Plan Wind Tunnel | 0.803223 |
33,878 | The Transonic wind tunnel is a closed-return, variable density tunnel with a fixed test section geometry. The 11-ft is capable of speeds from Mach 0.25 to Mach 1.4. Airflow is produced by a three-stage, axial-flow compressor. Supersonic conditions are achieved by moving a symmetric flexible wall in a nozzle configurati... | Unitary Plan Wind Tunnel | 0.803223 |
33,879 | String theory and M-theory are two examples where n > 4. In string theory, there appears conformal field theories with 1 + 1 spacetime dimensions. de Sitter space can be formulated as a submanifold of generalized Minkowski space as can the model spaces of hyperbolic geometry (see below). | Flat spacetime | 0.80322 |
33,880 | Influential articles that appeared in the journal include Katona's elegant proof of the Erdős–Ko–Rado theorem and a series of papers spanning over 500 pages, appearing from 1983 to 2004, by Neil Robertson and Paul D. Seymour on the topic of graph minors, which together constitute the proof of the graph minors theorem. ... | Journal of Combinatorial Theory | 0.803219 |
33,881 | From 2010 to 2012, NIMBioS, in conjunction with the UT's James R. Cox Endowment Fund, sponsored a Songwriter-in-Residence Program to encourage the creation and production of songs involving ideas of modern biology and the lives of scientists who pursue research in biology. A total of five songwriters were supported. Ea... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,882 | The National Institute for Mathematical and Biological Synthesis is a research institute focused on the science of mathematics and biology, located on the University of Tennessee, Knoxville, campus. Known by its acronym NIMBioS (pronounced NIM-bus), the Institute is a National Science Foundation (NSF) Synthesis Center ... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,883 | The need for the Institute arose out of the significant growth of the field of mathematical biology over the last decade with research becoming more closely linked to observation and experiment. Rather than starting from mathematical abstractions, it is now common for researchers to: 1) Begin with observations; 2) Use ... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,884 | Primary goals of NIMBios are: to address key biological questions using cross-disciplinary approaches in mathematical biology to foster the development of a cadre of researchers who are capable of conceiving and engaging in creative and collaborative connections across disciplines.To achieve its goals, NIMBioS advances... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,885 | NIMBioS organizes a Summer Research Experience for Undergraduates and Teachers program for eight weeks each summer. Participants live on campus and conduct research in teams with UT professors, NIMBioS researchers, and collaborators on projects at the interface of math and biology. NIMBioS also hosts the annual Undergr... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,886 | NIMBioS’ director is Nina H. Fefferman, Professor of Ecology & Evolutionary Biology and Mathematics at the UT. Louis J. Gross, Professor of Ecology & Evolutionary Biology and Mathematics at the UT served as the previous Director. NIMBioS leadership team also includes associate directors and a deputy director. NIMBioS h... | National Institute for Mathematical and Biological Synthesis | 0.803218 |
33,887 | In statistical mechanics (see any book) one derives the so-called distribution functions f {\displaystyle f} for various statistics. In the case of Fermi–Dirac statistics and Bose–Einstein statistics these functions are respectively These functions are derived for (1) a system in dynamic equilibrium (i.e. under steady,... | Bayesian prior | 0.803213 |
33,888 | The LFT analysis is highly dependent on the geometry of the complex, but most explanations begin by describing octahedral complexes, where six ligands coordinate to the metal. Other complexes can be described by reference to crystal field theory. Inverted ligand field theory (ILFT) elaborates on LFT by breaking assumpt... | Ligand Field Theory | 0.80321 |
33,889 | In the Lagrangian formalism, homogeneity in space implies conservation of momentum, and homogeneity in time implies conservation of energy. This is shown, using variational calculus, in standard textbooks like the classical reference text of Landau & Lifshitz. This is a particular application of Noether's theorem. | Homogeneous media | 0.803209 |
33,890 | When A is unital, M(A) = A. It also follows from the definition that for any D containing A as an essential ideal, the multiplier algebra M(A) contains D as a C*-subalgebra. The existence of M(A) can be shown in several ways. A double centralizer of a C*-algebra A is a pair (L, R) of bounded linear maps on A such that ... | Multiplier algebra | 0.803206 |
33,891 | Let X be a locally compact Hausdorff space, A = C0(X), the commutative C*-algebra of continuous functions that vanish at infinity. Then M(A) is Cb(X), the continuous bounded functions on X. By the Gelfand–Naimark theorem, one has the isomorphism of C*-algebras C b ( X ) ≃ C ( Y ) {\displaystyle C_{b}(X)\simeq C(Y)} whe... | Multiplier algebra | 0.803206 |
33,892 | When A is unital, M(A) = A, and the strict topology coincides with the norm topology. For B(H) = M(K(H)), the strict topology is the σ-strong* topology. It follows from above that B(H) is complete in the σ-strong* topology. | Multiplier algebra | 0.803206 |
33,893 | Consider the topology on M(A) specified by the seminorms {la, ra}a ∈ A, where l a ( x ) = ‖ a x ‖ , r a ( x ) = ‖ x a ‖ . {\displaystyle l_{a}(x)=\|ax\|,\;r_{a}(x)=\|xa\|.} The resulting topology is called the strict topology on M(A). A is strictly dense in M(A) . | Multiplier algebra | 0.803206 |
33,894 | K(E)) be the adjointable (resp. compact) operators on E M(A) can be identified via a *-homomorphism of A into B(E). Something similar to the above lemma is true: Lemma. If I is an ideal in a C*-algebra B, then any faithful nondegenerate *-homomorphism π of I into B(E)can be extended uniquely to B. Consequently, if π is... | Multiplier algebra | 0.803206 |
33,895 | If I is an ideal in a C*-algebra B, then any faithful nondegenerate representation π of I can be extended uniquely to B. Now take any faithful nondegenerate representation π of A on a Hilbert space H. The above lemma, together with the universal property of the multiplier algebra, yields that M(A) is isomorphic to the ... | Multiplier algebra | 0.803206 |
33,896 | The set of double centralizers of A can be given a C*-algebra structure. This C*-algebra contains A as an essential ideal and can be identified as the multiplier algebra M(A). For instance, if A is the compact operators K(H) on a separable Hilbert space, then each x ∈ B(H) defines a double centralizer of A by simply mu... | Multiplier algebra | 0.803206 |
33,897 | An ideal I in a C*-algebra B is said to be essential if I ∩ J is non-trivial for every ideal J. An ideal I is essential if and only if I⊥, the "orthogonal complement" of I in the Hilbert C*-module B is {0}. Let A be a C*-algebra. Its multiplier algebra M(A) is any C*-algebra satisfying the following universal property:... | Multiplier algebra | 0.803206 |
33,898 | In mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by Busby (1968). For ... | Multiplier algebra | 0.803206 |
33,899 | The DFD notation draws on graph theory, originally used in operational research to model workflow in organizations. DFD originated from the activity diagram used in the structured analysis and design technique methodology at the end of the 1970s. DFD popularizers include Edward Yourdon, Larry Constantine, Tom DeMarco, ... | Data-flow diagram | 0.803206 |
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