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10,500 | loc. for an exotic example with 4 parts of type A3.The classification roughly reduces a given hypothetical example to a Radford biproduct of the (coradical-) group and the (connected-) part, which contains the Nichols algebra, by taking the corresponding "graded object" (killing all linkings). With the knowledge from t... | Nichols algebra | 0.818871 |
10,501 | The Nichols algebra appears as quantum Borel part in the classification of finite-dimensional pointed Hopf algebras (without small primes) by Nicolas Andruskiewitsch and Hans-Jürgen Schneider, especially Quantum groups. For example, U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} and their well known truncations for q... | Nichols algebra | 0.818871 |
10,502 | The Nichols algebra B ( V ) {\displaystyle {\mathfrak {B}}(V)} is the unique Hopf algebra in the braided category H H Y D {\displaystyle {}_{H}^{H}{\mathcal {YD}}} generated by the given V ∈ H H Y D {\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}} , such that V ⊂ B ( V ) {\displaystyle V\subset {\mathfrak {B}}(V)} are th... | Nichols algebra | 0.818871 |
10,503 | Particular examples include the Nichols algebra associated to the conjugacy class(es) of reflections in a Coxeter group, they are related to the Fomin Kirilov algebras. It is known these Nichols algebras are finite dimensional for S 3 , S 4 , S 5 , D 4 {\displaystyle \mathbb {S} _{3},\mathbb {S} _{4},\mathbb {S} _{5},\... | Nichols algebra | 0.818871 |
10,504 | Only a handful of finite-dimensional Nichols algebras over k = C are known so far. It is known that in this case each irreducible Yetter–Drinfeld module O χ {\displaystyle {\mathcal {O}}_{}^{\chi }} corresponds to Conjugacy class of the group (together with an irreducible representation of the centralizer of g). An ar... | Nichols algebra | 0.818871 |
10,505 | Nichols algebras are probably best known for being the Borel part of the quantum groups and their generalizations. More precisely let V := x 1 C ⊕ x 2 C ⊕ ⋯ ⊕ x n C {\displaystyle V:=x_{1}\mathbb {C} \oplus x_{2}\mathbb {C} \oplus \cdots \oplus x_{n}\mathbb {C} } be the diagonal Yetter-Drinfel'd module over an abelian ... | Nichols algebra | 0.818871 |
10,506 | This is not a group homomorphism, but Matsumoto's theorem (group theory) tells us that the action of any s ( σ ) {\displaystyle s(\sigma )} on V ⊗ n {\displaystyle V^{\otimes n}} is well-defined independently of the choice of a reduced expression. Finally the Nichols algebra is then W n := ∑ σ ∈ S n s ( σ ): V ⊗ n → V ... | Nichols algebra | 0.818871 |
10,507 | Order statistics First-order statistics, e.g., arithmetic mean, median, quantiles Second-order statistics, e.g., correlation, power spectrum, variance Higher-order statistics, e.g., bispectrum, kurtosis, skewness | Order (algebra) | 0.818868 |
10,508 | Order (journal), an academic journal on order theory Dense order, a total order wherein between any unequal pair of elements there is always an intervening element in the order Glossary of order theory Lexicographical order, an ordering method on sequences analogous to alphabetical order on words List of order topics, ... | Order (algebra) | 0.818868 |
10,509 | Order (group theory), the cardinality of a group or period of an element Order of a polynomial (disambiguation) Order of a square matrix, its dimension Order (ring theory), an algebraic structure Ordered group Ordered field | Order (algebra) | 0.818868 |
10,510 | Dispatchable generation refers to sources of electricity that can be programmed on demand at the request of power grid operators, according to market needs. Dispatchable generators may adjust their power output according to an order. Non-dispatchable renewable energy sources such as wind power and solar photovoltaic (P... | Dispatchable generation | 0.818867 |
10,511 | A 2018 study suggested a new classification of energy generation sources, which accounts for fast increase in penetration of variable renewable energy sources, which result in high energy prices during periods of low availability: "Fuel saving" variable renewable energy, which have near zero variable costs and zero fue... | Dispatchable generation | 0.818867 |
10,512 | There are three Givens rotations in dimension 3: R X ( θ ) = . {\displaystyle R_{X}(\theta )={\begin{bmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{bmatrix}}.} R Y ( θ ) = {\displaystyle {\begin{aligned}\\R_{Y}(\theta )={\begin{bmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&... | Givens rotation | 0.818865 |
10,513 | A Givens rotation is represented by a matrix of the form G ( i , j , θ ) = , {\displaystyle G(i,j,\theta )={\begin{bmatrix}1&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &&\vdots &&\vdots \\0&\cdots &c&\cdots &-s&\cdots &0\\\vdots &&\vdots &\ddots &\vdots &&\vdots \\0&\cdots &s&\cdots &c&\cdots &0\\\vdots &&\... | Givens rotation | 0.818865 |
10,514 | In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. | Givens rotation | 0.818865 |
10,515 | In Clifford algebras and its child structures like geometric algebra rotations are represented by bivectors. Givens rotations are represented by the exterior product of the basis vectors. Given any pair of basis vectors e i , e j {\displaystyle \mathbf {e} _{i},\mathbf {e} _{j}} Givens rotations bivectors are: B i j = ... | Givens rotation | 0.818865 |
10,516 | MEMOCS publishes articles from diverse scientific fields with a specific emphasis on mechanics. Its contents rely on the application or development of rigorous mathematical methods. The journal also publishes original research in related areas of mathematics of well-established applicability, such as variational method... | Mathematics and Mechanics of Complex Systems | 0.818862 |
10,517 | Perhaps one of the main unsolved problems in modern neuroscience is the so-called "cell types" problem which refers to the categorization, definition, and identification of all neuronal/astrocytic cell types in an organism. Usually, this refers to the mouse brain since an understanding of the mouse brain is seen as a s... | Brain science | 0.818858 |
10,518 | The specific topics that form the main focus of research change over time, driven by an ever-expanding base of knowledge and the availability of increasingly sophisticated technical methods. Improvements in technology have been the primary drivers of progress. Developments in electron microscopy, computer science, elec... | Brain science | 0.818858 |
10,519 | The scientific study of the nervous system increased significantly during the second half of the twentieth century, principally due to advances in molecular biology, electrophysiology, and computational neuroscience. This has allowed neuroscientists to study the nervous system in all its aspects: how it is structured, ... | Brain science | 0.818858 |
10,520 | Other major organizations devoted to neuroscience include the International Brain Research Organization (IBRO), which holds its meetings in a country from a different part of the world each year, and the Federation of European Neuroscience Societies (FENS), which holds a meeting in a different European city every two y... | Brain science | 0.818858 |
10,521 | The largest professional neuroscience organization is the Society for Neuroscience (SFN), which is based in the United States but includes many members from other countries. Since its founding in 1969 the SFN has grown steadily: as of 2010 it recorded 40,290 members from 83 countries. Annual meetings, held each year in... | Brain science | 0.818858 |
10,522 | Moreover, the neuroscientific identification of multiple memory systems related to different brain areas has challenged the idea of memory as a literal reproduction of the past, supporting a view of memory as a generative, constructive and dynamic process.Neuroscience is also allied with the social and behavioral scien... | Brain science | 0.818858 |
10,523 | Cognitive neuroscience addresses the questions of how psychological functions are produced by neural circuitry. The emergence of powerful new measurement techniques such as neuroimaging (e.g., fMRI, PET, SPECT), EEG, MEG, electrophysiology, optogenetics and human genetic analysis combined with sophisticated experimenta... | Brain science | 0.818858 |
10,524 | Questions in systems neuroscience include how neural circuits are formed and used anatomically and physiologically to produce functions such as reflexes, multisensory integration, motor coordination, circadian rhythms, emotional responses, learning, and memory. In other words, they address how these neural circuits fun... | Brain science | 0.818858 |
10,525 | Success in computational modeling of neurons have led to the development of stereochemical models that accurately predict acetylcholine receptor-based synapses operating at microsecond time scales.Ultrafine nanoneedles for cellular manipulations are thinner than the smallest single walled carbon nanotubes. Computationa... | Brain science | 0.818858 |
10,526 | Neuromorphic engineering is a branch of neuroscience that deals with creating functional physical models of neurons for the purposes of useful computation. The emergent computational properties of neuromorphic computers are fundamentally different from conventional computers in the sense that they are a complex system,... | Brain science | 0.818858 |
10,527 | Basic questions addressed in molecular neuroscience include the mechanisms by which neurons express and respond to molecular signals and how axons form complex connectivity patterns. At this level, tools from molecular biology and genetics are used to understand how neurons develop and how genetic changes affect biolog... | Brain science | 0.818858 |
10,528 | These questions include how signals are processed by neurites and somas and how neurotransmitters and electrical signals are used to process information in a neuron. Neurites are thin extensions from a neuronal cell body, consisting of dendrites (specialized to receive synaptic inputs from other neurons) and axons (spe... | Brain science | 0.818858 |
10,529 | Furthermore, production of nanomedicine for drug trials is economically consuming, hindering progress in their implementation. Computational models in nanoneuroscience provide alternatives to study the efficacy of nanotechnology-based medicines in neurological disorders while mitigating potential side effects and devel... | Brain science | 0.818858 |
10,530 | Questions in computational neuroscience can span a wide range of levels of traditional analysis, such as development, structure, and cognitive functions of the brain. Research in this field utilizes mathematical models, theoretical analysis, and computer simulation to describe and verify biologically plausible neurons ... | Brain science | 0.818858 |
10,531 | SIAM Journal on Discrete Mathematics is a peer-reviewed mathematics journal published quarterly by the Society for Industrial and Applied Mathematics (SIAM). The journal includes articles on pure and applied discrete mathematics. It was established in 1988, along with the SIAM Journal on Matrix Analysis and Application... | SIAM Journal on Discrete Mathematics | 0.818857 |
10,532 | The idea of general finite differences also means that FPM is not based on a weak formulation like Galerkin's approach. Rather, FPM is a strong formulation which models differential equations by direct approximation of the occurring differential operators. The method used is a moving least squares idea which was especi... | Finite pointset method | 0.818855 |
10,533 | In order to overcome the disadvantages of the classical methods many approaches have been developed to simulate such flows (Hansbo 92, Harlow et al. 1965, Hirt et al. 1981, Kelecy et al. 1997, Kothe at el. 1992, Maronnier et al. 1999, Tiwari et al. 2000). A classical grid free Lagrangian method is Smoothed Particle Hyd... | Finite pointset method | 0.818855 |
10,534 | (1980). "Model-making in physics". Contemporary Physics. | Mathematical Modeling | 0.818843 |
10,535 | Recall that a prestack (of groupoids) on a category C {\displaystyle {\mathcal {C}}} , also known as a 2-presheaf, is a 2-functor X: C opp → G r p {\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} } , where G r p {\displaystyle \mathrm {Grp} } is the 2-category of (set-theoretical) groupoids, their morphism... | Differentiable stack | 0.818842 |
10,536 | Given a Lie group action a: M × G → M {\displaystyle a:M\times G\to M} on M {\displaystyle M} , its quotient (differentiable) stack is the differential counterpart of the quotient (algebraic) stack in algebraic geometry. It is defined as the stack {\displaystyle } associating to any manifold X {\displaystyle X} the ca... | Differentiable stack | 0.818842 |
10,537 | Every Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} gives rise to the differentiable stack B G: M f d opp → G r p {\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} } , which sends any manifold N {\displaystyle N} to the category of G {\displaystyle G} -torsors on N {\displaystyle N} (i.e. G {\dis... | Differentiable stack | 0.818842 |
10,538 | An epimorphism between differentiable stacks G → X {\displaystyle G\to X} is called a gerbe over X {\displaystyle X} if G → G × X G {\displaystyle G\to G\times _{X}G} is also an epimorphism. For example, if X {\displaystyle X} is a stack, B S 1 × X → X {\displaystyle BS^{1}\times X\to X} is a gerbe. A theorem of Giraud... | Differentiable stack | 0.818842 |
10,539 | A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.Differentiable stacks are particularly useful to handle spaces with singul... | Differentiable stack | 0.818842 |
10,540 | A differentiable stack X {\displaystyle X} may be equipped with Grothendieck topology in a certain way (see the reference). This gives the notion of a sheaf over X {\displaystyle X} . For example, the sheaf Ω X p {\displaystyle \Omega _{X}^{p}} of differential p {\displaystyle p} -forms over X {\displaystyle X} is give... | Differentiable stack | 0.818842 |
10,541 | In differential geometry, the torsion of a curve — a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve. | Third derivative | 0.818841 |
10,542 | In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function y = f ( x ) {\displaystyle y=f(x)} can be denoted by d 3 y d x 3 , f ‴ ( x ) , or d 3 d x 3 . {\d... | Third derivative | 0.818841 |
10,543 | In physics, particularly kinematics, jerk is defined as the third derivative of the position function of an object. It is, essentially, the rate at which acceleration changes. In mathematical terms: j ( t ) = d 3 r d t 3 {\displaystyle \mathbf {j} (t)={\frac {d^{3}\mathbf {r} }{dt^{3}}}} where j(t) is the jerk function... | Third derivative | 0.818841 |
10,544 | The plasma-surface interface is a complex and dynamic system that involves multiple scales, from atomic to macroscopic, and a wide range of physical and chemical processes. The optimization of the plasma-surface interaction regimes and the development of advanced PFCs require the development of accurate models and diag... | Plasma surface interaction | 0.818841 |
10,545 | The interactions between plasma and PFCs are strongly influenced by the plasma parameters, such as density, temperature, and particle flux, as well as the material properties of PFCs, such as composition, surface area, and geometry. In magnetic confinement fusion devices, such as tokamaks and stellarators, the plasma-s... | Plasma surface interaction | 0.818841 |
10,546 | The third power of the trinomial a + b + c is given by ( a + b + c ) 3 = a 3 + b 3 + c 3 + 3 a 2 b + 3 a 2 c + 3 b 2 a + 3 b 2 c + 3 c 2 a + 3 c 2 b + 6 a b c . {\displaystyle (a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3b^{2}a+3b^{2}c+3c^{2}a+3c^{2}b+6abc.} This can be computed by hand using the distributive propert... | Multinomial coefficients | 0.818823 |
10,547 | The numbers ( n k 1 , k 2 , … , k m ) {\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}} appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials: ( n k 1 , k 2 , … , k m ) = n ! k 1 ! | Multinomial coefficients | 0.818823 |
10,548 | One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients. | Multinomial coefficients | 0.818823 |
10,549 | This function drives the genetic selection process.Many chromosomes make up the population; therefore, many different neural networks are evolved until a stopping criterion is satisfied. A common stopping scheme is: When the neural network has learnt a certain percentage of the training data or When the minimum value o... | Recurrent neural networks | 0.818822 |
10,550 | Arbitrary global optimization techniques may then be used to minimize this target function. The most common global optimization method for training RNNs is genetic algorithms, especially in unstructured networks.Initially, the genetic algorithm is encoded with the neural network weights in a predefined manner where one... | Recurrent neural networks | 0.818822 |
10,551 | Training the weights in a neural network can be modeled as a non-linear global optimization problem. A target function can be formed to evaluate the fitness or error of a particular weight vector as follows: First, the weights in the network are set according to the weight vector. Next, the network is evaluated against... | Recurrent neural networks | 0.818822 |
10,552 | Hierarchical recurrent neural networks (HRNN) connect their neurons in various ways to decompose hierarchical behavior into useful subprograms. Such hierarchical structures of cognition are present in theories of memory presented by philosopher Henri Bergson, whose philosophical views have inspired hierarchical models. | Recurrent neural networks | 0.818822 |
10,553 | In this sense, the dynamics of a memristive circuit has the advantage compared to a Resistor-Capacitor network to have a more interesting non-linear behavior. From this point of view, engineering an analog memristive networks accounts to a peculiar type of neuromorphic engineering in which the device behavior depends o... | Recurrent neural networks | 0.818822 |
10,554 | Greg Snider of HP Labs describes a system of cortical computing with memristive nanodevices. The memristors (memory resistors) are implemented by thin film materials in which the resistance is electrically tuned via the transport of ions or oxygen vacancies within the film. DARPA's SyNAPSE project has funded IBM Resear... | Recurrent neural networks | 0.818822 |
10,555 | One approach to the computation of gradient information in RNNs with arbitrary architectures is based on signal-flow graphs diagrammatic derivation. It uses the BPTT batch algorithm, based on Lee's theorem for network sensitivity calculations. It was proposed by Wan and Beaufays, while its fast online version was propo... | Recurrent neural networks | 0.818822 |
10,556 | An online hybrid between BPTT and RTRL with intermediate complexity exists, along with variants for continuous time.A major problem with gradient descent for standard RNN architectures is that error gradients vanish exponentially quickly with the size of the time lag between important events. LSTM combined with a BPTT/... | Recurrent neural networks | 0.818822 |
10,557 | In this context, local in space means that a unit's weight vector can be updated using only information stored in the connected units and the unit itself such that update complexity of a single unit is linear in the dimensionality of the weight vector. Local in time means that the updates take place continually (on-lin... | Recurrent neural networks | 0.818822 |
10,558 | Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function. In neural networks, it can be used to minimize the error term by changing each weight in proportion to the derivative of the error with respect to that weight, provided the non-linear activation functions are diffe... | Recurrent neural networks | 0.818822 |
10,559 | Given a time series x of length sequence_length. In the recurrent neural network, there is a loop that processes all entries of the time series x through the layers neural_network one after another. These have as return value in each time step i both the prediction y_pred and an updated hidden state hidden, which has t... | Recurrent neural networks | 0.818822 |
10,560 | Fully recurrent neural networks (FRNN) connect the outputs of all neurons to the inputs of all neurons. This is the most general neural network topology because all other topologies can be represented by setting some connection weights to zero to simulate the lack of connections between those neurons. The illustration ... | Recurrent neural networks | 0.818822 |
10,561 | A continuous-time recurrent neural network (CTRNN) uses a system of ordinary differential equations to model the effects on a neuron of the incoming inputs. For a neuron i {\displaystyle i} in the network with activation y i {\displaystyle y_{i}} , the rate of change of activation is given by: τ i y ˙ i = − y i + ∑ j =... | Recurrent neural networks | 0.818822 |
10,562 | The Ising model (1925) by Wilhelm Lenz and Ernst Ising was a first RNN architecture that did not learn. Shun'ichi Amari made it adaptive in 1972. This was also called the Hopfield network (1982). See also David Rumelhart's work in 1986. In 1993, a neural history compressor system solved a "Very Deep Learning" task that... | Recurrent neural networks | 0.818822 |
10,563 | The Independently recurrent neural network (IndRNN) addresses the gradient vanishing and exploding problems in the traditional fully connected RNN. Each neuron in one layer only receives its own past state as context information (instead of full connectivity to all other neurons in this layer) and thus neurons are inde... | Recurrent neural networks | 0.818822 |
10,564 | RNNs may behave chaotically. In such cases, dynamical systems theory may be used for analysis. They are in fact recursive neural networks with a particular structure: that of a linear chain. Whereas recursive neural networks operate on any hierarchical structure, combining child representations into parent representati... | Recurrent neural networks | 0.818822 |
10,565 | Neural network pushdown automata (NNPDA) are similar to NTMs, but tapes are replaced by analogue stacks that are differentiable and that are trained. In this way, they are similar in complexity to recognizers of context free grammars (CFGs). | Recurrent neural networks | 0.818822 |
10,566 | Applications of recurrent neural networks include: Machine translation Robot control Time series prediction Speech recognition Speech synthesis Brain–computer interfaces Time series anomaly detection Text-to-Video model Rhythm learning Music composition Grammar learning Handwriting recognition Human action recognition ... | Recurrent neural networks | 0.818822 |
10,567 | Now, consider a case where the mass M {\displaystyle M} is not a point mass, but is distributed in a spherically-symmetric distribution about the center, with an average mass density of ρ {\displaystyle \rho } , ρ = 3 M 4 π R 3 {\displaystyle \rho ={\frac {3M}{4\pi R^{3}}}} ,where the volume of a sphere is: ( 4 / 3 ) π... | Free-fall time | 0.818809 |
10,568 | In this scenario, if we analyze what are his personal preferences and beliefs we will be able to predict which he might choose. (e.g. if someone prioritizes their social life more than academic results, they will go out with their friends). Assuming that the decisions of a person are rational, according to this theorem... | Expected utility | 0.818807 |
10,569 | In 1926, Frank Ramsey introduced the Ramsey's Representation Theorem. This representation theorem for expected utility assumed that preferences are defined over set of bets where each option has a different yield. Ramsey believed that we always choose decisions to receive the best expected outcome according to our pers... | Expected utility | 0.818807 |
10,570 | In the early days of the calculus of probability, classic utilitarians believed that the option which has the greatest utility will produce more pleasure or happiness for the agent and therefore must be chosen The main problem with the expected value theory is that there might not be a unique correct way to quantify ut... | Expected utility | 0.818807 |
10,571 | Cognitive processes and other psychological aspects of decision making matter only to the extent that they have directly measurable implications on choice. The theory of subjective expected utility combines two concepts: first, a personal utility function, and second a personal probability distribution (usually based o... | Expected utility | 0.818807 |
10,572 | In his book, The Foundations of Statistics, Savage integrated a normative account of decision making under risk (when probabilities are known) and under uncertainty (when probabilities are not objectively known). Savage concluded that people have neutral attitudes towards uncertainty and that observation is enough to p... | Expected utility | 0.818807 |
10,573 | In biological organisms, morphogenesis (the development of well-defined shapes and functional structures) is achieved by the interactions between cells guided by the genetic program encoded in the organism's DNA. Inspired by this idea, amorphous computing aims at engineering well-defined shapes and patterns, or coheren... | Natural Computation | 0.818795 |
10,574 | Other approaches to describing accurately and succinctly protein–protein interactions include the use of textual bio-calculus or pi-calculus enriched with stochastic features.Transport networks refer to the separation and transport of substances mediated by lipid membranes. Some lipids can self-assemble into biological... | Natural Computation | 0.818795 |
10,575 | Computational systems biology (or simply systems biology) is an integrative and qualitative approach that investigates the complex communications and interactions taking place in biological systems. Thus, in systems biology, the focus of the study is the interaction networks themselves and the properties of biological ... | Natural Computation | 0.818795 |
10,576 | Conjugation of two ciliates consists of the exchange of their micronuclear genetic information, leading to the formation of two new micronuclei, followed by each ciliate re-assembling the information from its new micronucleus to construct a new functional macronucleus. The latter process is called gene assembly, or gen... | Natural Computation | 0.818795 |
10,577 | These include distinguishing between self and nonself, neutralization of nonself pathogens (viruses, bacteria, fungi, and parasites), learning, memory, associative retrieval, self-regulation, and fault-tolerance. Artificial immune systems are abstractions of the natural immune system, emphasizing these computational as... | Natural Computation | 0.818795 |
10,578 | An artificial neuron A is equipped with a function f A {\displaystyle f_{A}} , receives n real-valued inputs x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} with respective weights w 1 , w 2 , … , w n {\displaystyle w_{1},w_{2},\ldots ,w_{n}} , and it outputs f A ( w 1 x 1 + w 2 x 2 + … + w n x n ) {\disp... | Natural Computation | 0.818795 |
10,579 | Neural computation is the field of research that emerged from the comparison between computing machines and the human nervous system. This field aims both to understand how the brain of living organisms works (brain theory or computational neuroscience), and to design efficient algorithms based on the principles of how... | Natural Computation | 0.818795 |
10,580 | Natural computing, also called natural computation, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration from nature for the development of novel problem-solving techniques; 2) those that are based on the use of computers to synthesize natural phenomena; and 3) those that em... | Natural Computation | 0.818795 |
10,581 | The computation by a membrane system starts with an initial configuration, where the number (multiplicity) of each object is set to some value for each region (multiset of objects). It proceeds by choosing, nondeterministically and in a maximally parallel manner, which rules are applied to which objects. The output of ... | Natural Computation | 0.818795 |
10,582 | The carpenter's rule problem is a discrete geometry problem, which can be stated in the following manner: Can a simple planar polygon be moved continuously to a position where all its vertices are in convex position, so that the edge lengths and simplicity are preserved along the way? A closely related problem is to sh... | Carpenter's rule problem | 0.818794 |
10,583 | In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is a prime number for all positive natural numbers n. This constant is named after William Harold Mills who proved in 19... | Mills' constant | 0.818792 |
10,584 | In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms wit... | Positive current | 0.818791 |
10,585 | Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.Theorem: Let M be a compact complex manifold. Then M does not admit a Kähler structure if and only if M admits a non-zero positive (1,1)-current Θ {\displaystyle \Theta } which is a (1,1)-part of an exact 2-cu... | Positive current | 0.818791 |
10,586 | Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle to tâtonnement) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other... | Economical mathematics | 0.818791 |
10,587 | Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics. These models lacked the inequalities of the next generation of mathematical economics. In the landmark treatise Foundations of Economic Analysis (1947), Paul Samuelson identified a com... | Economical mathematics | 0.818791 |
10,588 | Restricted models of general equilibrium were formulated by John von Neumann in 1937. Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point ... | Economical mathematics | 0.818791 |
10,589 | Extensions to nonlinear optimization with inequality constraints were achieved in 1951 by Albert W. Tucker and Harold Kuhn, who considered the nonlinear optimization problem: Minimize f ( x ) {\displaystyle f(x)} subject to g i ( x ) ≤ 0 {\displaystyle g_{i}(x)\leq 0} and h j ( x ) = 0 {\displaystyle h_{j}(x)=0} where ... | Economical mathematics | 0.818791 |
10,590 | Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.Broad applications include: optimization problems as to goal equilibrium, whether of a household, business firm, or policy... | Economical mathematics | 0.818791 |
10,591 | Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or othe... | Economical mathematics | 0.818791 |
10,592 | In the last decade, with the rise of the internet, mathematical economists and optimization experts and computer scientists have worked on problems of pricing for on-line services --- their contributions using mathematics from cooperative game theory, nondifferentiable optimization, and combinatorial games. Robert M. S... | Economical mathematics | 0.818791 |
10,593 | "Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in mathematical optimization and mathematical statistics and increasingly in theoretical computer science. Mathematical economics and other mathematical sciences have a history in which t... | Economical mathematics | 0.818791 |
10,594 | Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108 and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingen... | Decimal | 0.818789 |
10,595 | from +0.2 V to -1.8 V vs reversible hydrogen electrode (RHE). Hg electrode is particularly well-suited for studying electroreduction reactions. 2) very reproducible electrode surface, since mercury is liquid. 3) very easy cleaning of the electrode surface by making a new drop of mercury from a large Hg pool connected b... | Polarography | 0.818779 |
10,596 | See conjugated systems for electron bond chemistry that causes these molecules to have pigment. Heme/porphyrin-based: chlorophyll, bilirubin, hemocyanin, hemoglobin, myoglobin Light-emitting: luciferin Carotenoids: Hematochromes (algal pigments, mixes of carotenoids and their derivates) Carotenes: alpha and beta carote... | Biological pigments | 0.818778 |
10,597 | Multiple testing: When multiple true null hypothesis tests are conducted at once without adjustment, the overall probability of Type I error is higher than the nominal alpha level. Those making critical decisions based on the results of a hypothesis test are prudent to look at the details rather than the conclusion alo... | Hypothesis Testing | 0.818774 |
10,598 | In nuclear physics, a magic number is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the atomic nucleus. As a result, atomic nuclei with a 'magic' number of protons or neutrons are much more stable than other nuclei. The seven most widely recognized... | Magic number (physics) | 0.818773 |
10,599 | Where a filter is being designed with more than one passband, the m-type topology will result in a filter where each passband has an analogous frequency-domain response. It is possible to generalise the m-type topology for filters with more than one passband using parameters m1, m2, m3 etc., which are not equal to each... | Multiple feedback topology (electronics) | 0.81877 |
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