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79,600 | SuperFabric is a cut and abrasion resistant material, and a registered trademark of Higher Dimension Materials, Inc. As a technical fabric, SuperFabric is created with a base fabric such as nylon, polyester, neoprene, crepe, etc. and is overlaid with tiny, hard guard plates in a specific pattern. Spacings between the g... | SuperFabric | 0.785297 |
79,601 | Semi-amplitude means half of the peak-to-peak amplitude. The majority of scientific literature employs the term amplitude or peak amplitude to mean semi-amplitude. It is the most widely used measure of orbital wobble in astronomy and the measurement of small radial velocity semi-amplitudes of nearby stars is important ... | Peak-to-peak amplitude | 0.785296 |
79,602 | In 1838, Jöns Jacob Berzelius wrote that quercitannate is used to dissolve morphine.In 1865 in the fifth volume of "A dictionary of chemistry", Henry Watts wrote: It exhibits with ferric salts the same reactions as gallotannic acid. It differs however from the latter in not being convertible into gallic acid, and not y... | Tannic acid | 0.785295 |
79,603 | In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas o... | Voigt notation | 0.785295 |
79,604 | Feature engineering can be a time-consuming and error-prone process, as it requires domain expertise and often involves trial and error. Deep learning algorithms may be used to process a large raw dataset without having to resort to feature engineering. However, it's important to note that deep learning algorithms stil... | Feature engineering | 0.785295 |
79,605 | There are a number of open-source libraries and tools that automate feature engineering on relational data and time series: featuretools is a Python library for transforming time series and relational data into feature matrices for machine learning. OneBM or One-Button Machine combines feature transformations and featu... | Feature engineering | 0.785295 |
79,606 | Automation of feature engineering is a research topic that dates back to the 1990s. Machine learning software that incorporates automated feature engineering has been commercially available since 2016. Related academic literature can be roughly separated into two types: Multi-relational decision tree learning (MRDTL) u... | Feature engineering | 0.785295 |
79,607 | The Feature Store is where the features are stored and organized for the explicit purpose of being used to either train models (by data scientists) or make predictions (by applications that have a trained model). It is a central location where you can either create or update groups of features created from multiple dif... | Feature engineering | 0.785295 |
79,608 | Feature engineering or feature extraction or feature discovery is the process of extracting features (characteristics, properties, attributes) from raw data. Due to deep learning networks, such as convolutional neural networks, that are able to learn it by itself, domain-specific- based feature engineering has become o... | Feature engineering | 0.785295 |
79,609 | Consequently, rocket engines and jet engines for supersonic flight incorporate a C-D nozzle which permits further expansion against the inside of the nozzle. However, unlike the fixed convergent-divergent nozzle used on a conventional rocket motor, those on turbojet engines must have heavy and expensive variable geomet... | Propulsive nozzle | 0.785293 |
79,610 | At the other extreme, some high bypass ratio civil turbofans control the fan working line by using a convergent-divergent nozzle with an extremely low (less than 1.01) area ratio on the bypass (or mixed exhaust) stream. At low airspeeds, such a setup causes the nozzle to act as if it had variable geometry by preventing... | Propulsive nozzle | 0.785293 |
79,611 | The afterburners on combat aircraft require a bigger nozzle to prevent adversely affecting the operation of the engine. The variable area iris nozzle consists of a series of moving, overlapping petals with a nearly circular nozzle cross-section and is convergent to control the operation of the engine. If the aircraft i... | Propulsive nozzle | 0.785293 |
79,612 | Nozzles for vectored thrust include fixed geometry Bristol Siddeley Pegasus and variable geometry F119 (F-22). | Propulsive nozzle | 0.785293 |
79,613 | Practical lessons, over the years, had led to the definition, and establishment, of procedures and tools. Eventually, the tools became systems to manage software changes. Industry-wide practices were offered as solutions, either in an open or proprietary manner (such as Revision Control System). With the growing use of... | Software configuration management | 0.785293 |
79,614 | The history of software configuration management (SCM) in computing can be traced back as early as the 1950s, when CM (configuration management), originally for hardware development and production control, was being applied to software development. Early software had a physical footprint, such as cards, tapes, and othe... | Software configuration management | 0.785293 |
79,615 | In software engineering, software configuration management (SCM or S/W CM) is the task of tracking and controlling changes in the software, part of the larger cross-disciplinary field of configuration management. SCM practices include revision control and the establishment of baselines. If something goes wrong, SCM can... | Software configuration management | 0.785293 |
79,616 | When combined with encryption and digital signatures, digital certificates provide individuals and organizations with a means of privately sharing information so that each party is confident that the individual or organization with which they are communicating is in fact who it claims to be. Hand Geometry Authenticatio... | E-commerce identification and identification types | 0.785293 |
79,617 | A way to moderate the reaction was required, and the method developed was to react the hydrocarbon with cobalt(III) fluoride, rather than fluorine itself. After World War II, much of the technology that had been kept secret was released into the public domain. The March 1947 issue of Industrial and Engineering Chemistr... | Fowler process | 0.785292 |
79,618 | Most cancers, if not all, could cause chromosome abnormalities, with either the formation of hybrid genes and fusion proteins, deregulation of genes and overexpression of proteins, or loss of tumor suppressor genes (see the "Mitelman Database" and the Atlas of Genetics and Cytogenetics in Oncology and Haematology,). Fu... | Chromosomal aberrations | 0.785291 |
79,619 | That same year, Lattice replaced several members of its leadership team, including bringing in a new president and CEO, Jim Anderson, who previously worked at Advanced Micro Devices. Under the new leadership, Lattice shifted the company's focus entirely to low-power field-programmable gate arrays (FPGAs). Lattice acqui... | Lattice Semiconductor | 0.785291 |
79,620 | Lattice primarily focuses on small, efficient low-power field-programmable gate arrays (FPGAs). It also sells programmable mixed-signal and interconnect products, related software and intellectual property (IP), for applications from edge computing to cloud computing. Lattice's main products are the ECP and Certus-NX s... | Lattice Semiconductor | 0.785291 |
79,621 | The facility was designed and constructed to test both nuclear and non-nuclear space hardware in a simulated Low-Earth-Orbiting environment. Although the facility was designed for testing nuclear hardware, only non-nuclear tests have been performed throughout its history. Some of the test programs that have been perfor... | Space Power Facility | 0.785291 |
79,622 | Polyunsaturated fat can be found mostly in nuts, seeds, fish, seed oils, and oysters. "Unsaturated" refers to the fact that the molecules contain less than the maximum amount of hydrogen (if there were no double bonds). These materials exist as cis or trans isomers depending on the geometry of the double bond. | Polyunsaturated fatty acid | 0.78529 |
79,623 | The institute also contributes to the undergraduate programs and university studies as well as continuing vocational trainings. Academic staff is involved in several master programs and coordinates two of them. The institute is also engaged in an Erasmus Mundus master, «EACH» (Excellence in Analytical Chemistry), and t... | Institute of Analytical Sciences | 0.78529 |
79,624 | The Institute brings together over 200 technical and scientific staff with a panel of expertise ranging from theoretical chemistry to physics, through modeling, biochemistry, and analytical chemistry. The ISA also has a tip and global instrumentation equipment: NMR spectrometer (1Ghz), hosted by the «Very High Fields N... | Institute of Analytical Sciences | 0.785289 |
79,625 | Located within the Cité lyonnaise de l’environnement et de l’analyse, the Institute of Analytical Sciences (UMR 5280) is a French research, training and expertise center specializing in analytical chemistry. It is a Joint Research Unit (French: Unité mixte de recherche or UMR) affiliated with the National Centre for Sc... | Institute of Analytical Sciences | 0.785289 |
79,626 | A variety of similar research areas are based on gesture recognition, with many projects using motion capture to track the physical motions of a designer and translate them into three-dimensional geometry suitable for digital fabrication. While these share similarities to responsive design through their cyber-physical ... | Responsive computer-aided design | 0.785289 |
79,627 | Responsive computer-aided design is enabled by ubiquitous computing and the Internet of Things, concepts which describe the capacity for everyday objects to contain computing and sensing technologies. It is also enabled by the ability to directly manufacture one-off objects from digital data, using technologies such as... | Responsive computer-aided design | 0.785289 |
79,628 | ∇ and ∇′ define the same families of affinely parametrized geodesics if and only if S(X, Y) = 0.In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two ... | Torsion of connection | 0.785288 |
79,629 | The Steiner–Lehmus theorem, a theorem in elementary geometry, was formulated by C. L. Lehmus and subsequently proved by Jakob Steiner. It states: Every triangle with two angle bisectors of equal lengths is isosceles.The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a... | Steiner–Lehmus theorem | 0.785285 |
79,630 | Thus a = b. But this may not be considered direct as one must first argue about why the other factor cannot be 0. John Conway has argued that there can be no "equality-chasing" proof because the theorem (stated algebraically) does not hold over an arbitrary field, or even when negative real numbers are allowed as param... | Steiner–Lehmus theorem | 0.785285 |
79,631 | The Steiner–Lehmus theorem can be proved using elementary geometry by proving the contrapositive statement: if a triangle is not isosceles, then it does not have two angle bisectors of equal length. There is some controversy over whether a "direct" proof is possible; allegedly "direct" proofs have been published, but n... | Steiner–Lehmus theorem | 0.785285 |
79,632 | 131–140 Pambuccian, Victor; Struve, Horst; Struve, Rolf (2016). "The Steiner–Lehmus theorem and "triangles with congruent medians are isosceles" hold in weak geometries". Beiträge zur Algebra und Geometrie. | Steiner–Lehmus theorem | 0.785285 |
79,633 | S2CID 125461255. Lewin, Mordechai (1974). "On the Steiner-Lehmus Theorem". | Steiner–Lehmus theorem | 0.785285 |
79,634 | "SSA and the Steiner-Lehmus Theorem". The Mathematics Teacher. 85 (5): 381–383. | Steiner–Lehmus theorem | 0.785285 |
79,635 | 154–166 Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, pp. 224–225 Beran, David (1992). | Steiner–Lehmus theorem | 0.785285 |
79,636 | John Horton Conway, Alex Ryba: The Steiner-Lehmus Angle Bisector Theorem. In: Mircea Pitici (Hrsg. ): The Best Writing on Mathematics 2015. Princeton University Press, 2016, ISBN 9781400873371, pp. | Steiner–Lehmus theorem | 0.785285 |
79,637 | In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or be... | Truncated distribution | 0.785285 |
79,638 | Psychoacoustics is the branch of psychophysics involving the scientific study of sound perception and audiology—how human auditory system perceives various sounds. More specifically, it is the branch of science studying the psychological responses associated with sound (including noise, speech, and music). Psychoacoust... | Perceptual coding | 0.785284 |
79,639 | In aqueous solution chemistry and precipitates ageing, the term refers to the growth of larger crystals from those of smaller size which have a higher solubility than the larger ones. In the process, many small crystals formed initially (nuclei) slowly disappear, except for a few that grow larger, at the expense of the... | Ostwald ripening | 0.785282 |
79,640 | One example of Ostwald ripening is the re-crystallization of water within ice cream which gives old ice cream a gritty, crunchy texture. Larger ice crystals grow at the expense of smaller ones within the ice cream, creating a coarser texture.Another gastronomical example is the ouzo effect, where the droplets in the cl... | Ostwald ripening | 0.785282 |
79,641 | He began research into the fluid mechanics of the flotation process and set in train a series of experimental projects into the effect of particle diameter and bubble size on the flotation rate constant. Much of the research was conducted by honours students in chemical engineering. Jameson accepted the challenge of co... | Jameson cell | 0.785281 |
79,642 | Jameson's research into flotation began when he was at Imperial College London, in 1969. A colleague, Dr J. A. Kitchener of the Royal School of Mines, pointed out that many of the new mineral deposits being found around the world required fine grinding to separate the valuable particles from the rock in which they were... | Jameson cell | 0.785281 |
79,643 | While older techniques were only able to stain cells with a constricted range of colors, often utilizing bi- and tri-color transgenic mice to unveil limited information in regards to neuronal structures, Brainbow is much more flexible in that it has the capacity to fluorescently label individual neurons with up to appr... | Brainbow | 0.78528 |
79,644 | The complexity of the Drosophila brain, consisting of about 100,000 neurons, makes it an excellent candidate for implementing neurophysiology and neuroscience techniques like Brainbow. In fact, Stefanie Hampel et al. (2011) combined Brainbow in conjunction with genetic targeting tools to identify individual neurons wit... | Brainbow | 0.78528 |
79,645 | The Open Geospatial Consortium (OGC) defines an explicit XML-based exchange format for 3D city models, CityGML, which supports not only geometric descriptions of 3D city model components but also the specification of semantics and topology information. | 3D city models | 0.785276 |
79,646 | Providing high quality visualization of massive 3D city models in a scalable, fast, and cost efficient manner is still a challenging task due to the complexity in terms of 3D geometry and textures of 3D city models. Real-time rendering provides a large number of specialized 3D rendering techniques for 3D city models. E... | 3D city models | 0.785276 |
79,647 | In order to analytically find the stability of fluid flows, it is useful to note that hydrodynamic stability has a lot in common with stability in other fields, such as magnetohydrodynamics, plasma physics and elasticity; although the physics is different in each case, the mathematics and the techniques used are simila... | Dynamic instability (fluid mechanics) | 0.785275 |
79,648 | In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. Since every rank level is itself an antichain, the Spern... | Sperner property of a partially ordered set | 0.785274 |
79,649 | Lemma. The set Z ( X ) ⊂ Gr ( n − k , n ) {\displaystyle Z(X)\subset \operatorname {Gr} (n-k,n)} parametrizing the subspaces of P n − 1 {\displaystyle \mathbb {P} ^{n-1}} which intersect X non-trivially is an irreducible hypersurface of degree d. As a consequence, there exists a degree d form R X {\displaystyle R_{X}... | Chow quotient | 0.785272 |
79,650 | An effective algebraic cycle in P n − 1 {\displaystyle \mathbb {P} ^{n-1}} of codimension 1 and degree d can be defined by the vanishing of a single degree d polynomial in n-many variables, and this polynomial is unique up to rescaling. Letting V d , n {\displaystyle V_{d,n}} denote the vector space of degree d polynom... | Chow quotient | 0.785272 |
79,651 | A Chow quotient parametrizes closures of generic orbits. It is constructed as a closed subvariety of a Chow variety. Kapranov's theorem says that the moduli space M ¯ 0 , n {\displaystyle {\overline {M}}_{0,n}} of stable genus-zero curves with n marked points is the Chow quotient of Grassmannian Gr ( 2 , C n ) {\disp... | Chow quotient | 0.785272 |
79,652 | However, as there is no consensus as to the ‘best’ basis for V k , d , n {\displaystyle V_{k,d,n}} , this term can be ambiguous. From a foundational perspective, the above theorem is usually used as the definition of Gr ( k , d , n ) {\displaystyle \operatorname {Gr} (k,d,n)} . That is, the Chow variety is usually de... | Chow quotient | 0.785272 |
79,653 | The map Gr ( k , d , n ) ↪ P V k , d , n {\displaystyle \operatorname {Gr} (k,d,n)\hookrightarrow \mathbb {P} V_{k,d,n}} which sends X ↦ R X {\displaystyle X\mapsto R_{X}} is a closed embedding of varieties. In particular, an effective algebraic cycle X is determined by its Chow form R X {\displaystyle R_{X}} . If a ... | Chow quotient | 0.785272 |
79,654 | To each degree d algebraic cycle, this associates a degree d form R X {\displaystyle R_{X}} on Gr ( n − k , n ) {\displaystyle \operatorname {Gr} (n-k,n)} , called the Chow form of X, which is well-defined up to scaling. Let V k , d , n {\displaystyle V_{k,d,n}} denote the vector space of degree d forms on Gr ( n −... | Chow quotient | 0.785272 |
79,655 | In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety Gr ( k , d , n ) {\displaystyle \operatorname {Gr} (k,d,n)} is the f... | Chow quotient | 0.785272 |
79,656 | This is a direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the d = 1 {\displaystyle d=1} case of Chow varieties. Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective s... | Chow quotient | 0.785272 |
79,657 | Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule. | Tensor multiplication | 0.785271 |
79,658 | Another example: let U be a tensor of type (1, 1) with components U β α , {\displaystyle U_{\beta }^{\alpha },} and let V be a tensor of type ( 1 , 0 ) {\displaystyle (1,0)} with components V γ . {\displaystyle V^{\gamma }.} Then and Tensors equipped with their product operation form an algebra, called the tensor algeb... | Tensor multiplication | 0.785271 |
79,659 | The eigenconfiguration of A {\displaystyle A} consists of n {\displaystyle n} points in P n − 1 , {\displaystyle \mathbb {P} ^{n-1},} provided A {\displaystyle A} is generic and K {\displaystyle K} is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let A = ( a i 1 i 2 ⋯ i d ) {... | Tensor multiplication | 0.785271 |
79,660 | Nonlinear Oscillations is a quarterly peer-reviewed mathematical journal that was established in 1998. It is published by Springer Science+Business Media on behalf of the Institute of Mathematics, National Academy of Sciences of Ukraine. It covers research in the qualitative theory of differential or functional differe... | Nonlinear Oscillations | 0.785269 |
79,661 | If X {\displaystyle X} is a compact topological space, and ( f n ) n ∈ N {\displaystyle (f_{n})_{n\in \mathbb {N} }} is a monotonically increasing sequence (meaning f n ( x ) ≤ f n + 1 ( x ) {\displaystyle f_{n}(x)\leq f_{n+1}(x)} for all n ∈ N {\displaystyle n\in \mathbb {N} } and x ∈ X {\displaystyle x\in X} ) of con... | Dini's theorem | 0.785269 |
79,662 | In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. | Dini's theorem | 0.785269 |
79,663 | The International Astronomy Olympiad (since 1996) | Eurasian Astronomical Society | 0.785268 |
79,664 | Initiatives to spur adoption by clinicians include the Ubiquitous Pharmacogenomics (U-PGx) program in Europe and the Clinical Pharmacogenetics Implementation Consortium (CPIC) in the United States. In a 2017 survey of European clinicians, in the prior year two-thirds had not ordered a pharmacogenetic test.In 2010, Vand... | Ultra-rapid metabolizer | 0.785267 |
79,665 | MnCO3 adopts a structure like calcite, consisting of manganese(II) ions in an octahedral coordination geometry.Treatment of aqueous solutions of manganese(II) nitrate with ammonia and carbon dioxide leads to precipitation of this faintly pink solid. The side product, ammonium nitrate is used as fertilizer. | CMnO3 | 0.785267 |
79,666 | and simply write ℓ ( θ ∣ X , Y ) = ∑ i = 1 m ( y i θ ′ x i − e θ ′ x i ) . {\displaystyle \ell (\theta \mid X,Y)=\sum _{i=1}^{m}\left(y_{i}\theta 'x_{i}-e^{\theta 'x_{i}}\right).} To find a maximum, we need to solve an equation ∂ ℓ ( θ ∣ X , Y ) ∂ θ = 0 {\displaystyle {\frac {\partial \ell (\theta \mid X,Y)}{\partial \... | Poisson regression | 0.785265 |
79,667 | When estimating the parameters for Poisson regression, one typically tries to find values for θ that maximize the likelihood of an expression of the form ∑ i = 1 m log ( p ( y i ; e θ ′ x i ) ) , {\displaystyle \sum _{i=1}^{m}\log(p(y_{i};e^{\theta 'x_{i}})),} where m is the number of examples in the data set, and p ... | Poisson regression | 0.785265 |
79,668 | In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown pa... | Poisson regression | 0.785265 |
79,669 | The second type of failure is from fatigue or corrosion, caused by instability in the structure's geometry, design or material properties. These failures usually begin when cracks form at stress points, such as squared corners or bolt holes too close to the material's edge. | Structural integrity and failure | 0.785265 |
79,670 | In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population. With background allele frequency p the allele frequencies, in sub-populations separated by Wright's FST F, are distributed according to independent draws from B ( 1 − F... | Balding–Nichols model | 0.785265 |
79,671 | Can the "scallop" swim by periodically opening and closing the hinge? No: regardless of how the cycle of opening and closing depends on time, the scallop will always return to its starting point at the end of the cycle. Here originated the striking quote: "Fast or slow, it exactly retraces its trajectory and it's back ... | Microswimmer | 0.785264 |
79,672 | Purcell concluded that only forces that are exerted in the present moment on a microscale body contribute to its propulsion, so a constant energy conversion method is essential.Microorganisms have optimized their metabolism for continuous energy production, while purely artificial microswimmers (microrobots) must obtai... | Microswimmer | 0.785264 |
79,673 | Magnetic actuation is most often included for controlled in vivo guiding, even for microswimmers which rely on a different type of propulsion. In 2020, Koleoso et al. reviewed the use of magnetic small scale robots for biomedical applications and provide details about the various magnetic fields and actuation systems d... | Microswimmer | 0.785264 |
79,674 | As is the case for microtechnology and nanotechnology in general, the history of microswimmer applications arguably starts with Richard Feynman’s famous lecture There's Plenty of Room at the Bottom. In the visionary speech, among other topics, Feynman addressed the idea of microscopic surgeons, saying: "...it would be ... | Microswimmer | 0.785264 |
79,675 | On the contrary, endowing artificial microswimmers with an internal feedback mechanism, which regulates motility in response to stimuli that are decoupled from the source of propulsion, remains an elusive task.A promising route to achieve this goal is to exploit the coupling between particle shape and motility. Efficie... | Microswimmer | 0.785264 |
79,676 | Reconfigurable synthetic or artificial microswimmers need internal feedback Self-propelling microparticles are often proposed as synthetic models for biological microswimmers, yet they lack the internally regulated adaptation of their biological counterparts. Conversely, adaptation can be encoded in larger-scale soft-r... | Microswimmer | 0.785264 |
79,677 | One of the challenges in proposing an artificial swimmer lies in the fact that the proposed movement stroke should not be reciprocal otherwise it cannot propel itself due to the Scallop theorem. In Scallop theorem, Purcell had argued that a swimmer with one-hinge or one degree of freedom is bound to perform reciprocal ... | Microswimmer | 0.785264 |
79,678 | This means the physics of how microswimmers swim is dominated by viscous drag forces, a problem which has been discussed extensively by physicists in the field. This kind of swimming has challenged engineers as it is not commonly experienced in everyday life, but can nonetheless be observed in nature for motile microor... | Microswimmer | 0.785264 |
79,679 | (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and ... | Nonstandard calculus | 0.785263 |
79,680 | Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level. Thus, the epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the found... | Nonstandard calculus | 0.785263 |
79,681 | To calculate the derivative f ′ {\displaystyle f'} of the function y = f ( x ) = x 2 {\displaystyle y=f(x)=x^{2}} at x, both approaches agree on the algebraic manipulations: Δ y Δ x = ( x + Δ x ) 2 − x 2 Δ x = 2 x Δ x + ( Δ x ) 2 Δ x = 2 x + Δ x ≈ 2 x {\displaystyle {\frac {\Delta y}{\Delta x}}={\frac {(x+\Delta x)^{2}... | Nonstandard calculus | 0.785263 |
79,682 | The fact that a continuous function on a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem) admits a succinct hyperreal proof. Let x, y be hyperreals in the natural extension I* of I. Since I is compact, both st(x) and st(y) belong to I. If x and y were infinitely close, then by the trian... | Nonstandard calculus | 0.785263 |
79,683 | Such a sequence of values is called a partition or mesh and sup k ( x k + 1 − x k ) {\displaystyle \sup _{k}(x_{k+1}-x_{k})} the width of the mesh. In the definition of the Riemann integral, the limit of the Riemann sums is taken as the width of the mesh goes to 0. Theorem: Let f be a real-valued function defined on an... | Nonstandard calculus | 0.785263 |
79,684 | Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted. For the second theorem, the Riemann integral is defined as the limit, if it exists, of a directed family of Riemann sums; these are sums of the form ∑ k = 0 n − 1 f ( ξ k ) ( x k... | Nonstandard calculus | 0.785263 |
79,685 | If f is a real valued function defined on an interval , then the transfer operator applied to f, denoted by *f, is an internal, hyperreal-valued function defined on the hyperreal interval . Theorem: Let f be a real-valued function defined on an interval . Then f is differentiable at a < x < b if and only if for every n... | Nonstandard calculus | 0.785263 |
79,686 | One immediate application is an extension of the standard definitions of differentiation and integration to internal functions on intervals of hyperreal numbers. An internal hyperreal-valued function f on is S-differentiable at x, provided Δ h f = st f ( x + h ) − f ( x ) h {\displaystyle \Delta _{h}f=\operatorname ... | Nonstandard calculus | 0.785263 |
79,687 | In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Non-rigorous calculations with infinitesimals were wide... | Nonstandard calculus | 0.785263 |
79,688 | RSC Advances is an online-only peer-reviewed scientific journal covering research on all aspects of the chemical sciences. It was established in 2011 and is published by the Royal Society of Chemistry. The current editor-in-chiefs are Russell Cox (Leibniz Universität Hannover) and Karen Faulds (University of Strathclyd... | RSC Advances | 0.785261 |
79,689 | X is finitely generated i.e. it is in the final hull of the finite spaces. (This means that there is a final sink fi: Xi → X where each Xi is a finite topological space. )Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov-discrete spaces and their top... | Alexandroff topology | 0.78526 |
79,690 | The closure operator of X distributes over arbitrary unions of subsets. Preorder characterizations: Specialization preorder. T is the finest topology consistent with the specialization preorder of X i.e. the finest topology giving the preorder ≤ satisfying x ≤ y if and only if x is in the closure of {y} in X. Open up-s... | Alexandroff topology | 0.78526 |
79,691 | Interior and closure algebraic characterizations: Interior operator. The interior operator of X distributes over arbitrary intersections of subsets. Closure operator. | Alexandroff topology | 0.78526 |
79,692 | Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. Alexandrov-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topol... | Alexandroff topology | 0.78526 |
79,693 | Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder ≤ on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the upper sets with respect to ≤. | Alexandroff topology | 0.78526 |
79,694 | In topology, an Alexandrov topology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the intersection of every finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known ... | Alexandroff topology | 0.78526 |
79,695 | Hydraulic routing is based on the solution of partial differential equations of unsteady open-channel flow. The equations used are the Saint-Venant equations or the associated dynamic wave equations.The hydraulic models (e.g. dynamic and diffusion wave models) require the gathering of a lot of data related to river geo... | Routing (hydrology) | 0.785256 |
79,696 | A bibliographic database is a database of bibliographic records. This is an organised online collection of references to published written works like journal and newspaper articles, conference proceedings, reports, government and legal publications, patents and books. In contrast to library catalogue entries, a majorit... | Bibliographic databases | 0.785256 |
79,697 | Hydrologic routing uses the continuity equation for hydrology. In its simplest form, inflow to the river reach is equal to the outflow of the river reach plus the change of storage: I = O + Δ S Δ t {\displaystyle I=O+{\frac {\Delta S}{\Delta t}}} , whereI is average inflow to the reach during Δ t {\displaystyle \Delta ... | Routing (hydrology) | 0.785256 |
79,698 | Semi-distributed models such as Muskingum–Cunge family procedures are also available. Simple physical concepts and common river characteristics such as channel geometry, reach length, roughness coefficient, and slope are used to estimate the model parameters without complex and expensive numerical solutions. | Routing (hydrology) | 0.785256 |
79,699 | Morpholinos are used as research tools for reverse genetics by knocking down gene function. This article discusses only the Morpholino antisense oligomers, which are nucleic acid analogs. The word "Morpholino" can occur in other chemical names, referring to chemicals containing a six-membered morpholine ring. | Morpholino | 0.785255 |
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