id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
9,600 | Suppose that a connected Lie group G with Lie algebra g {\displaystyle {\mathfrak {g}}} acts on a differentiable manifold M. Consider the corresponding representation ρ of G on the space of smooth functions on M. Then elements of g {\displaystyle {\mathfrak {g}}} are represented by first order differential operators on... | Casimir element | 0.819995 |
9,601 | Given a representation ρ of g {\displaystyle {\mathfrak {g}}} on a vector space V, possibly infinite-dimensional, the Casimir invariant of ρ is defined to be ρ(Ω), the linear operator on V given by the formula ρ ( Ω ) = ∑ i = 1 n ρ ( X i ) ρ ( X i ) . {\displaystyle \rho (\Omega )=\sum _{i=1}^{n}\rho (X_{i})\rho (X^{i}... | Casimir element | 0.819995 |
9,602 | {\displaystyle {\mathfrak {g}}.} This corresponds to a symmetric homogeneous polynomial c ( m ) = κ i j ⋯ k t i t j ⋯ t k {\displaystyle c_{(m)}=\kappa ^{ij\cdots k}t_{i}t_{j}\cdots t_{k}} in m indeterminate variables t i {\displaystyle t_{i}} in the polynomial algebra K {\displaystyle K} over a field K. The reason fo... | Casimir element | 0.819995 |
9,603 | The article on universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties. All Casimir operators correspond to symmetric homogeneous polynomials in the symmetric algebra of the adjoint representation ad g . {\displaystyle \operatorname {ad} _{\... | Casimir element | 0.819995 |
9,604 | The Lie algebra s l 2 C {\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} } consists of two-by-two complex matrices with zero trace. There are three standard basis elements, e {\displaystyle e} , f {\displaystyle f} , and h {\displaystyle h} , with e = , f = , h = . {\displaystyle {\begin{aligned}e&={\begin{bmatrix}0&1... | Casimir element | 0.819995 |
9,605 | Let { X i } i = 1 n {\displaystyle \{X_{i}\}_{i=1}^{n}} be any basis of g {\displaystyle {\mathfrak {g}}} , and { X i } i = 1 n {\displaystyle \{X^{i}\}_{i=1}^{n}} be the dual basis of g {\displaystyle {\mathfrak {g}}} with respect to B. The Casimir element Ω {\displaystyle \Omega } for B is the element of the universa... | Casimir element | 0.819995 |
9,606 | Suppose that g {\displaystyle {\mathfrak {g}}} is an n {\displaystyle n} -dimensional Lie algebra. Let B be a nondegenerate bilinear form on g {\displaystyle {\mathfrak {g}}} that is invariant under the adjoint action of g {\displaystyle {\mathfrak {g}}} on itself, meaning that B ( ad X Y , Z ) + B ( Y , ad X Z ) =... | Casimir element | 0.819995 |
9,607 | Since for a simple Lie algebra every invariant bilinear form is a multiple of the Killing form, the corresponding Casimir element is uniquely defined up to a constant. For a general semisimple Lie algebra, the space of invariant bilinear forms has one basis vector for each simple component, and hence the same is true f... | Casimir element | 0.819995 |
9,608 | In order to perform well in the International Physics Olympiad, the Korean Physical Society holds the Korean Physical Olympiad, and through the operation of the winter school, KPS discovers gifted students and educates them to contribute to the development of mathematics, science, and engineering in South Korea. Also, ... | Korean Physics Olympiad | 0.819992 |
9,609 | In 2010, KPhO changed to the form of General Physics Communication Education I and II, selecting the candidates for winter school by disguising it as providing AP courses. As of 2018, there exists a general physical course that includes calculus and one that does not. The first one is usually for science students; 55 a... | Korean Physics Olympiad | 0.819992 |
9,610 | General Physics Communication Training Apply & Selection > General Physics 1 Communication Training > General Physics 2 Communication Training > Winter School Apply & Selection > Winter School > National Representatives Selection >National Representatives Training > International Physics Olympiad == References == | Korean Physics Olympiad | 0.819992 |
9,611 | The Korea Physics Olympiad (KPhO) is a physics competition held by the Korean Physical Society for middle and high school students. | Korean Physics Olympiad | 0.819992 |
9,612 | The Physics Analysis Workstation (PAW) is an interactive, scriptable computer software tool for data analysis and graphical presentation in High Energy Physics (HEP). The development of this software tool started at CERN in 1986, it was optimized for the processing of very large amounts of data. It was based on and int... | Physics Analysis Workstation | 0.819984 |
9,613 | Coalescent theory relates genetic diversity in a sample to demographic history of the population from which it was taken. It normally assumes neutrality, and so sequences from more neutrally evolving portions of genomes are therefore selected for such analyses. It can be used to infer the relationships between species ... | Evolutionary genetics | 0.819983 |
9,614 | In the extreme case of an asexual population, linkage is complete, and population genetic equations can be derived and solved in terms of a travelling wave of genotype frequencies along a simple fitness landscape. Most microbes, such as bacteria, are asexual. The population genetics of their adaptation have two contras... | Evolutionary genetics | 0.819983 |
9,615 | Traditionally a highly mathematical discipline, modern population genetics encompasses theoretical, laboratory, and field work. Population genetic models are used both for statistical inference from DNA sequence data and for proof/disproof of concept.What sets population genetics apart from newer, more phenotypic appro... | Evolutionary genetics | 0.819983 |
9,616 | Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and population structure.Population genetics was a vital ingredient in the emergence ... | Evolutionary genetics | 0.819983 |
9,617 | In 1932 Wright introduced the concept of an adaptive landscape and argued that genetic drift and inbreeding could drive a small, isolated sub-population away from an adaptive peak, allowing natural selection to drive it towards different adaptive peaks.The work of Fisher, Haldane and Wright founded the discipline of po... | Evolutionary genetics | 0.819983 |
9,618 | Population genetics began as a reconciliation of Mendelian inheritance and biostatistics models. Natural selection will only cause evolution if there is enough genetic variation in a population. Before the discovery of Mendelian genetics, one common hypothesis was blending inheritance. But with blending inheritance, ge... | Evolutionary genetics | 0.819983 |
9,619 | de.NBI develops and supplies about 100 bioinformatics tools for the German and global life sciences community, e.g. Galaxy (computational biology)/useGalaxy.eu (Workflow engine for all Freiburg RNA Tools), EDGAR (Comparative Genome Analyses Plattform), KNIME (Workflow engine), OpenMS (Open-source software C++ library f... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,620 | In addition, online training was introduced on the de.NBI website in 2016. Since 2017, online training formats like hackathons or webinars are offered by all service centers. The de.NBI training program started in 2015 with 17 f2f training courses with 329 participants, steadily increasing to 79 courses with 1586 Parti... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,621 | The de.NBI network offers a large portfolio of resources for the German and international life science community. It mainly includes databases, bioinformatics tools and hardware based on a federated cloud system. | German Network for Bioinformatics Infrastructure | 0.819974 |
9,622 | In addition, as an outreach activity de.NBI supported the Summer School BioByte 2019 at University of Halle addressed at high school students which offers an ideal opportunity to get to know the diversity of bioinformatics. | German Network for Bioinformatics Infrastructure | 0.819974 |
9,623 | In May 2013, the announcement of funding guidelines for a German Network for Bioinformatics Infrastructure (de.NBI) was published by the German Federal Ministry of Education and Research (BMBF). The aim of this announcement was to establish an infrastructure in Germany that will provide solutions to the ‘Big Data Probl... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,624 | The de.NBI network is open for further partners which like to take part in the network. These partners are called Associated Partners and are divided into two groups: Associated Service and Training Partners or Associated Training Partners. Associated Service und Training Partners offer at least one service and trainin... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,625 | The 'German Network for Bioinformatics Infrastructure – de.NBI' is a national, academic and non-profit infrastructure initiated by the Federal Ministry of Education and Research funding 2015-2021. The network provides bioinformatics services to users in life sciences research and biomedicine in Germany and Europe. The ... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,626 | Next to the training courses de.NBI offers annual summer schools to cover a distinct topic in more detail. The first Summer School was held in 2015 by the Service Centers Bielefeld-Gießen (BiGi) Center for Microbial Bioinformatics, RBC and de.NBI-SysBio and was focused in the workflow from genome assembly to genome and... | German Network for Bioinformatics Infrastructure | 0.819974 |
9,627 | The action of testosterone on hair follicles is mainly due to the more potent androgen, dihydrotestosterone, DHT. With androgen therapy, genetics primarily determines how much hair will develop (and where) as well as whether male pattern baldness will develop. Testosterone is converted (within the cells of the hair fol... | Masculinizing hormone therapy | 0.819968 |
9,628 | In general, the output of an online algorithm is compared to the solution of a corresponding offline algorithm which is necessarily always optimal and knows the entire input in advance (competitive analysis). In many situations, present decisions (for example, resources allocation) must be made with incomplete knowledg... | Online optimization | 0.819962 |
9,629 | Online optimization is a field of optimization theory, more popular in computer science and operations research, that deals with optimization problems having no or incomplete knowledge of the future (online). These kind of problems are denoted as online problems and are seen as opposed to the classical optimization pro... | Online optimization | 0.819962 |
9,630 | The astronomical unit of mass is the solar mass. The symbol M☉ is often used to refer to this unit. The solar mass (M☉), 1.98892×1030 kg, is a standard way to express mass in astronomy, used to describe the masses of other stars and galaxies. It is equal to the mass of the Sun, about 333000 times the mass of the Earth ... | Astronomical units of length | 0.819961 |
9,631 | The astronomical unit of time is the day, defined as 86400 seconds. 365.25 days make up one Julian year. The symbol D is used in astronomy to refer to this unit. | Astronomical units of length | 0.819961 |
9,632 | Machine learning is commonly separated into three main learning paradigms, supervised learning, unsupervised learning and reinforcement learning. Each corresponds to a particular learning task. | Neural network models | 0.819961 |
9,633 | An unreadable table that a useful machine could read would still be well worth having. Biological brains use both shallow and deep circuits as reported by brain anatomy, displaying a wide variety of invariance. Weng argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade ... | Neural network models | 0.819961 |
9,634 | Technology writer Roger Bridgman commented: Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probabilit... | Neural network models | 0.819961 |
9,635 | In 1997, Alexander Dewdney commented that, as a result, artificial neural networks have a "something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are. No human hand (or mind) intervenes; solutions are found as if by magic;... | Neural network models | 0.819961 |
9,636 | A central claim of ANNs is that they embody new and powerful general principles for processing information. These principles are ill-defined. It is often claimed that they are emergent from the network itself. This allows simple statistical association (the basic function of artificial neural networks) to be described ... | Neural network models | 0.819961 |
9,637 | Stochastic neural networks originating from Sherrington–Kirkpatrick models are a type of artificial neural network built by introducing random variations into the network, either by giving the network's artificial neurons stochastic transfer functions, or by giving them stochastic weights. This makes them useful tools ... | Neural network models | 0.819961 |
9,638 | Johnson likens this to a "mosaic" view of development. According to the second, the Interactive Specialization (IS) hypothesis, development is not a unidirectional maturational process, but rather a set of complex, dynamic and back-propagated interactions between genetics, brain, body and environment. Development is no... | Interactive specialization | 0.819957 |
9,639 | Interactive Specialization is a theory of brain development proposed by the British developmental cognitive neuroscientist Mark Johnson, formerly head of the Centre for Brain and Cognitive Development at Birkbeck, University of London, London and who is now Head of Psychology at the University of Cambridge. In his book... | Interactive specialization | 0.819957 |
9,640 | The UKM Medical Molecular Biology Institute, usually referred to as UMBI, is a biomedicine and cancer research institute located in Bandar Tun Razak, Kuala Lumpur, Malaysia. The institute is one of research institute in National University of Malaysia (UKM). UMBI was established in 2003. The institute has been recogniz... | UKM Medical Molecular Biology Institute | 0.819953 |
9,641 | The UKM Medical Molecular Biology Institute (UMBI) was founded as one of the Centre of Excellence in UKM after the approval from the National University of Malaysia senate meeting. UMBI was officially established in July 2003 with the operating budget allocated to this new institute of RM 25 thousand. Professor Datuk D... | UKM Medical Molecular Biology Institute | 0.819953 |
9,642 | Most artificial neural networks bear only some resemblance to their more complex biological counterparts, but are very effective at their intended tasks (e.g. classification or segmentation). Some artificial neural networks are adaptive systems and are used for example to model populations and environments, which const... | Types of artificial neural networks | 0.819937 |
9,643 | Deep neural networks can be potentially improved by deepening and parameter reduction, while maintaining trainability. While training extremely deep (e.g., 1 million layers) neural networks might not be practical, CPU-like architectures such as pointer networks and neural random-access machines overcome this limitation... | Types of artificial neural networks | 0.819937 |
9,644 | Limiting the degree of freedom reduces the number of parameters to learn, facilitating learning of new classes from few examples. Hierarchical Bayesian (HB) models allow learning from few examples, for example for computer vision, statistics and cognitive science. Compound HD architectures aim to integrate characterist... | Types of artificial neural networks | 0.819937 |
9,645 | Compositional pattern-producing networks (CPPNs) are a variation of artificial neural networks which differ in their set of activation functions and how they are applied. While typical artificial neural networks often contain only sigmoid functions (and sometimes Gaussian functions), CPPNs can include both types of fun... | Types of artificial neural networks | 0.819937 |
9,646 | Biological studies have shown that the human brain operates as a collection of small networks. This realization gave birth to the concept of modular neural networks, in which several small networks cooperate or compete to solve problems. | Types of artificial neural networks | 0.819937 |
9,647 | A neuro-fuzzy network is a fuzzy inference system in the body of an artificial neural network. Depending on the FIS type, several layers simulate the processes involved in a fuzzy inference-like fuzzification, inference, aggregation and defuzzification. Embedding an FIS in a general structure of an ANN has the benefit ... | Types of artificial neural networks | 0.819937 |
9,648 | The Boltzmann machine can be thought of as a noisy Hopfield network. It is one of the first neural networks to demonstrate learning of latent variables (hidden units). Boltzmann machine learning was at first slow to simulate, but the contrastive divergence algorithm speeds up training for Boltzmann machines and Product... | Types of artificial neural networks | 0.819937 |
9,649 | Regulatory feedback networks started as a model to explain brain phenomena found during recognition including network-wide bursting and difficulty with similarity found universally in sensory recognition. A mechanism to perform optimization during recognition is created using inhibitory feedback connections back to the... | Types of artificial neural networks | 0.819937 |
9,650 | Learning vector quantization (LVQ) can be interpreted as a neural network architecture. Prototypical representatives of the classes parameterize, together with an appropriate distance measure, in a distance-based classification scheme. | Types of artificial neural networks | 0.819937 |
9,651 | There are many types of artificial neural networks (ANN). Artificial neural networks are computational models inspired by biological neural networks, and are used to approximate functions that are generally unknown. Particularly, they are inspired by the behaviour of neurons and the electrical signals they convey betwe... | Types of artificial neural networks | 0.819937 |
9,652 | The self-organizing map (SOM) uses unsupervised learning. A set of neurons learn to map points in an input space to coordinates in an output space. The input space can have different dimensions and topology from the output space, and SOM attempts to preserve these. | Types of artificial neural networks | 0.819937 |
9,653 | The feedforward neural network was the first and simplest type. In this network the information moves only from the input layer directly through any hidden layers to the output layer without cycles/loops. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the si... | Types of artificial neural networks | 0.819937 |
9,654 | Cascade correlation is an architecture and supervised learning algorithm. Instead of just adjusting the weights in a network of fixed topology, Cascade-Correlation begins with a minimal network, then automatically trains and adds new hidden units one by one, creating a multi-layer structure. Once a new hidden unit has ... | Types of artificial neural networks | 0.819937 |
9,655 | A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern/summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each cl... | Types of artificial neural networks | 0.819937 |
9,656 | A district from conventional neural networks, stochastic artificial neural network used as an approximation to random functions. | Types of artificial neural networks | 0.819937 |
9,657 | Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution.Unlike other deep arc... | Types of artificial neural networks | 0.819937 |
9,658 | A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Dong. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalizatio... | Types of artificial neural networks | 0.819937 |
9,659 | A committee of machines (CoM) is a collection of different neural networks that together "vote" on a given example. This generally gives a much better result than individual networks. Because neural networks suffer from local minima, starting with the same architecture and training but using randomly different initial ... | Types of artificial neural networks | 0.819937 |
9,660 | Recurrent neural networks (RNN) propagate data forward, but also backwards, from later processing stages to earlier stages. RNN can be used as general sequence processors. | Types of artificial neural networks | 0.819937 |
9,661 | A more straightforward way to use kernel machines for deep learning was developed for spoken language understanding. The main idea is to use a kernel machine to approximate a shallow neural net with an infinite number of hidden units, then use a deep stacking network to splice the output of the kernel machine and the r... | Types of artificial neural networks | 0.819937 |
9,662 | Multilayer kernel machines (MKM) are a way of learning highly nonlinear functions by iterative application of weakly nonlinear kernels. They use kernel principal component analysis (KPCA), as a method for the unsupervised greedy layer-wise pre-training step of deep learning.Layer ℓ + 1 {\displaystyle \ell +1} learns th... | Types of artificial neural networks | 0.819937 |
9,663 | Memory networks incorporate long-term memory. The long-term memory can be read and written to, with the goal of using it for prediction. These models have been applied in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base and the output is a textual response... | Types of artificial neural networks | 0.819937 |
9,664 | When solving dynamic optimization problems by numerical backward induction, the objective function must be computed for each combination of values. This is a significant obstacle when the dimension of the "state variable" is large. | Curse of dimensionality | 0.819932 |
9,665 | Machine learning can be understood as the problem of assigning instances to their respective generative process of origin, with class labels acting as symbolic representations of individual generative processes. The curse's derivation assumes all instances are independent, identical outcomes of a single high dimensiona... | Curse of dimensionality | 0.819932 |
9,666 | For large d, distribution of r 2 / d {\displaystyle r^{2}/d} is close to the normal distribution with the mean 1/3 and the standard deviation 2 / 45 d {\displaystyle 2/{\sqrt {45d}}} according to the central limit theorem. Thus, when uniformly generating points in high dimensions, both the "middle" of the hypercube, an... | Curse of dimensionality | 0.819932 |
9,667 | In machine learning problems that involve learning a "state-of-nature" from a finite number of data samples in a high-dimensional feature space with each feature having a range of possible values, typically an enormous amount of training data is required to ensure that there are several samples with each combination of... | Curse of dimensionality | 0.819932 |
9,668 | The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in... | Curse of dimensionality | 0.819932 |
9,669 | "The blessing of dimensionality and the curse of dimensionality are two sides of the same coin." For example, the typical property of essentially high-dimensional probability distributions in a high-dimensional space is: the squared distance of random points to a selected point is, with high probability, close to the a... | Curse of dimensionality | 0.819932 |
9,670 | One example of the blessing of dimensionality phenomenon is linear separability of a random point from a large finite random set with high probability even if this set is exponentially large: the number of elements in this random set can grow exponentially with dimension. Moreover, this linear functional can be selecte... | Curse of dimensionality | 0.819932 |
9,671 | The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the derivative of the input function, while others work on every continuous function. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing... | Root-finding algorithms | 0.819914 |
9,672 | The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points. The behavior of general root-finding algorithms is studied in numerical analysis. However, for polynomials, root-finding study belon... | Root-finding algorithms | 0.819914 |
9,673 | A series of studies have shown that gene disrupting spontaneous (de novo) CNVs disrupt genes approximately four times more frequently in autism than in controls and contribute to approximately 5–10% of cases. Inherited variants also contribute to around 5–10% of cases of autism.Structural variations also have its funct... | Structural variations | 0.819911 |
9,674 | When there's a single process starting the election, the algorithm requires 3N-1 sequential messages, in the worst case. Worst case is when the process starting the election is the immediate following to the one with greatest UID: it takes N-1 messages for the election message to reach it, then N messages for it to get... | Chang and Roberts algorithm | 0.819907 |
9,675 | The trial was both double-blind and placebo-controlled.The methodology of clinical trials was further developed by Sir Austin Bradford Hill, who had been involved in the streptomycin trials. From the 1920s, Hill applied statistics to medicine, attending the lectures of renowned mathematician Karl Pearson, among others. | Drug trials | 0.819903 |
9,676 | The protocol also informs the study administrators (often a contract research organization). The format and content of clinical trial protocols sponsored by pharmaceutical, biotechnology or medical device companies in the United States, European Union, or Japan have been standardized to follow Good Clinical Practice gu... | Drug trials | 0.819903 |
9,677 | Winner: Claudia Bonfio (MRC Laboratory of Molecular Biology and University of Cambridge, UK), Project: Unlocking Primitive Chemical MessagesFinalists: Ivana Drienovska (Vrije Universiteit Amsterdam, The Netherlands), Project: New-to-Nature Reactivities in Biocatalysis: A Closer Look at Enzymatic Fluorination Pawel Dydi... | Dream Chemistry Award | 0.819903 |
9,678 | The contest is for scientists who are younger than 38 years of age who have been nominated by respected experts. The winner of the contest receives a financial reward of €10,000. In addition, starting in 2019, the finalists receive a reward of €1,000.The coordinators of the competition are Pavel Jungwirth from IOCB Pra... | Dream Chemistry Award | 0.819903 |
9,679 | The Dream Chemistry Award is an international competition for young scientists organized by the Institute of Organic Chemistry and Biochemistry of the Czech Academy of Sciences (IOCB Prague) and the Institute of Physical Chemistry of the Polish Academy of Sciences.The competition was established in 2013 by Robert Hołys... | Dream Chemistry Award | 0.819903 |
9,680 | The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It ... | Linear assignment problem | 0.819896 |
9,681 | Due to the equipartition theorem, larger molecules re-orient more slowly than do smaller objects and, hence, measurements of the rotational diffusion constants can give insight into the overall mass and its distribution within an object. Quantitatively, the mean square of the angular velocity about each of an object's ... | Rotational diffusion | 0.819894 |
9,682 | Rotational diffusion has multiple applications in chemistry and physics, and is heavily involved in many biology based fields. For example, protein-protein interaction is a vital step in the communication of biological signals. In order to communicate, the proteins must both come into contact with each other and be fac... | Rotational diffusion | 0.819894 |
9,683 | π, the Euler–Mascheroni constant γ, and Khinchin's constant itself, based on numerical evidence, are thought to be among the numbers whose geometric mean of the coefficients ai in their continued fraction expansion tends to Khinchin's constant. However, none of these limits have been rigorously established. It is not k... | Khinchin's constant | 0.819889 |
9,684 | Then μ is a probability measure on the σ-algebra of Borel subsets of I. The measure μ is equivalent to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ. Moreover, it can be proved that T is an ergodic transformation of the measurable space I endowed with th... | Khinchin's constant | 0.819889 |
9,685 | The proof presented here was arranged by Czesław Ryll-Nardzewski and is much simpler than Khinchin's original proof which did not use ergodic theory. Since the first coefficient a0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduce... | Khinchin's constant | 0.819889 |
9,686 | In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant. That is, for x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle x... | Khinchin's constant | 0.819889 |
9,687 | The journal is abstracted and indexed in: Current Index to Statistics MEDLINE Science Citation Index Expanded Zentralblatt MATH | Statistical Applications in Genetics and Molecular Biology | 0.819885 |
9,688 | Statistical Applications in Genetics and Molecular Biology is a bimonthly peer-reviewed scientific journal covering the application of statistics to problems in computational biology. It was established in 2002 and is published by de Gruyter. The editor-in-chief is Guido Sanguinetti. According to the Journal Citation R... | Statistical Applications in Genetics and Molecular Biology | 0.819885 |
9,689 | The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countabl... | Number | 0.819878 |
9,690 | A computable number, also known as recursive number, is a real number such that there exists an algorithm which, given a positive number n as input, produces the first n digits of the computable number's decimal representation. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calcul... | Number | 0.819878 |
9,691 | European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). René Descartes called them false root... | Number | 0.819878 |
9,692 | This is in contrast to receptor potentials, whose amplitudes are dependent on the intensity of a stimulus. In both cases, the frequency of action potentials is correlated with the intensity of a stimulus. Despite the classical view of the action potential as a stereotyped, uniform signal having dominated the field of n... | Action potentials | 0.819878 |
9,693 | Mathematical and computational models are essential for understanding the action potential, and offer predictions that may be tested against experimental data, providing a stringent test of a theory. The most important and accurate of the early neural models is the Hodgkin–Huxley model, which describes the action poten... | Action potentials | 0.819878 |
9,694 | A non-linear system of hyperbolic partial differential equations representing a set of conservation laws in one spatial dimension can be written in the form ∂ U ∂ t + ∂ F ( U ) ∂ x = 0. {\displaystyle {\frac {\partial {\boldsymbol {U}}}{\partial t}}+{\frac {\partial {\boldsymbol {F}}({\boldsymbol {U}})}{\partial x}}=0.... | Roe solver | 0.819875 |
9,695 | This torque of 1/3 balances the parabola, which is at a distance 1 from the fulcrum. Hence, the area of the parabola must be 1/3 to give it the opposite torque. This type of method can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of ... | Method of Mechanical Theorems | 0.819874 |
9,696 | To explain Archimedes' method today, it is convenient to make use of a little bit of Cartesian geometry, although this of course was unavailable at the time. His idea is to use the law of the lever to determine the areas of figures from the known center of mass of other figures. The simplest example in modern language ... | Method of Mechanical Theorems | 0.819874 |
9,697 | Archimedes did not admit the method of indivisibles as part of rigorous mathematics, and therefore did not publish his method in the formal treatises that contain the results. In these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer require... | Method of Mechanical Theorems | 0.819874 |
9,698 | Archimedes uses a more elegant method, but in Cartesian language, his method is calculating the integral which can easily be checked nowadays using elementary integral calculus. The idea is to mechanically balance the parabola (the curved region being integrated above) with a certain triangle that is made of the same m... | Method of Mechanical Theorems | 0.819874 |
9,699 | A series of propositions of geometry are proved in the palimpsest by similar arguments. One theorem is that the location of a center of mass of a hemisphere is located 5/8 of the way from the pole to the center of the sphere. This problem is notable, because it is evaluating a cubic integral. | Method of Mechanical Theorems | 0.819874 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.