id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
19,900 | Ionized gas plasma antennas can eliminate or reduce co-site interference. Smart ionized gas plasma antennas use plasma physics to shape and steer the antenna beams without the need of phased arrays. | Plasma antenna | 0.811228 |
19,901 | An emulsion reaction in fluorinated cyclohexane takes 36 hours and the neat reaction takes even longer (48 hours). Other reactions with apolar reactants such as Ene reactions and Diels–Alder reactions also exhibit rate accelerations. An explanation is not available but it involves hydrogen bonding and the presence of a... | On water reaction | 0.811225 |
19,902 | The set membership relation, ε = "is an element of", satisfies these properties so ε is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.In terms of the calculus of relations, sufficient conditions for a contact relation include where ∋ {\displaystyle \ni } is the conv... | Injective relation | 0.811222 |
19,903 | A common technique in physics is to integrate a vector field along a curve, also called determining its line integral. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where... | Vector fields | 0.81122 |
19,904 | The accessible surface area (ASA) or solvent-accessible surface area (SASA) is the surface area of a biomolecule that is accessible to a solvent. Measurement of ASA is usually described in units of square angstroms (a standard unit of measurement in molecular biology). ASA was first described by Lee & Richards in 1971 ... | Solvent Accessible Surface | 0.811219 |
19,905 | This is a list of algebraic geometry topics, by Wikipedia page. | List of algebraic geometry topics | 0.811217 |
19,906 | Grothendieck topology Topos Derived category Descent (category theory) Grothendieck's Galois theory Algebraic stack Gerbe Étale cohomology Motive (algebraic geometry) Motivic cohomology A¹ homotopy theory Homotopical algebra | List of algebraic geometry topics | 0.811217 |
19,907 | Enriques–Kodaira classification List of algebraic surfaces Ruled surface Cubic surface Veronese surface Del Pezzo surface Rational surface Enriques surface K3 surface Hodge index theorem Elliptic surface Surface of general type Zariski surface | List of algebraic geometry topics | 0.811217 |
19,908 | Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of an algebraic variety Hilbert's Nullstellensatz Complete variety Elimination theory Gröbner basis Projective variety Quasiprojective variety Canonical bundle Complete intersection Serre duality Spaltenstein variety Arithmetic genus, geometric genus... | List of algebraic geometry topics | 0.811217 |
19,909 | Affine space Projective space Projective line, cross-ratio Projective plane Line at infinity Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective pla... | List of algebraic geometry topics | 0.811217 |
19,910 | Niels Henrik Abel Carl Gustav Jacob Jacobi Jakob Steiner Julius Plücker Arthur Cayley Bernhard Riemann Max Noether William Kingdon Clifford David Hilbert Italian school of algebraic geometry Guido Castelnuovo Federigo Enriques Francesco Severi Solomon Lefschetz Oscar Zariski W. V. D. Hodge Sir Michael Atiyah Kunihiko K... | List of algebraic geometry topics | 0.811217 |
19,911 | Kähler manifold Calabi–Yau manifold Stein manifold Hodge theory Hodge cycle Hodge conjecture Algebraic geometry and analytic geometry Mirror symmetry | List of algebraic geometry topics | 0.811217 |
19,912 | Conics, Pascal's theorem, Brianchon's theorem Twisted cubic Elliptic curve, cubic curve Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions Elliptic integral Complex multiplication Weil pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function Modular group S... | List of algebraic geometry topics | 0.811217 |
19,913 | Prime ideal Valuation (algebra) Krull dimension Regular local ring Regular sequence Cohen–Macaulay ring Gorenstein ring Koszul complex Spectrum of a ring Zariski topology Kähler differential Generic flatness Irrelevant ideal | List of algebraic geometry topics | 0.811217 |
19,914 | Locally ringed space Coherent sheaf Invertible sheaf Sheaf cohomology Coherent sheaf cohomology Hirzebruch–Riemann–Roch theorem Grothendieck–Riemann–Roch theorem Coherent duality Dévissage | List of algebraic geometry topics | 0.811217 |
19,915 | Linear algebraic group Additive group Multiplicative group Algebraic torus Reductive group Borel subgroup Parabolic subgroup Radical of an algebraic group Unipotent radical Lie-Kolchin theorem Haboush's theorem (also known as the Mumford conjecture) Group scheme Abelian variety Theta function Grassmannian Flag manifold... | List of algebraic geometry topics | 0.811217 |
19,916 | In logic, the representability of algebras as relational structures is often used to prove the equivalence of algebraic and relational semantics. Examples of this include Stone's representation of Boolean algebras as fields of sets, Esakia's representation of Heyting algebras as Heyting algebras of sets, and the study ... | Representation (mathematics) | 0.811217 |
19,917 | An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A key class of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, more precisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersec... | Representation (mathematics) | 0.811217 |
19,918 | Although the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout mathematics. | Representation (mathematics) | 0.811217 |
19,919 | Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representation theory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces. | Representation (mathematics) | 0.811217 |
19,920 | The large distances between the telescopes enable very high angular resolutions to be achieved, much greater in fact than in any other field of astronomy. At the highest frequencies, synthesised beams less than 1 milliarcsecond are possible. | Millimeter-wave astronomy | 0.81121 |
19,921 | Water vapor interferes with radio astronomy at higher frequencies, which has led to building radio observatories that conduct observations at millimeter wavelengths at very high and dry sites, in order to minimize the water vapor content in the line of sight. Finally, transmitting devices on Earth may cause radio-frequ... | Millimeter-wave astronomy | 0.81121 |
19,922 | The artificial neuron model assumed by Kurzweil and used in many current artificial neural network implementations is simple compared with biological neurons. A brain simulation would likely have to capture the detailed cellular behaviour of biological neurons, presently understood only in broad outline. The overhead i... | Hard AI | 0.811208 |
19,923 | Writing in The Guardian, roboticist Alan Winfield claimed the gulf between modern computing and human-level artificial intelligence is as wide as the gulf between current space flight and practical faster-than-light spaceflight.Most AI researchers believe strong AI can be achieved in the future, but some thinkers, like... | Hard AI | 0.811208 |
19,924 | Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of superposition ... has no analogy in classical physics". According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory ." Though reasoning by ... | Superposition principle | 0.811208 |
19,925 | In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition ... | Superposition principle | 0.811208 |
19,926 | Generally, a semi-infinite set is bounded in one direction, and unbounded in another. For instance, the natural numbers are semi-infinite considered as a subset of the integers; similarly, the intervals ( c , ∞ ) {\displaystyle (c,\infty )} and ( − ∞ , c ) {\displaystyle (-\infty ,c)} and their closed counterparts are ... | Semi-infinite | 0.811206 |
19,927 | Many optimization problems involve some set of variables and some set of constraints. A problem is called semi-infinite if one (but not both) of these sets is finite. The study of such problems is known as semi-infinite programming. == References == | Semi-infinite | 0.811206 |
19,928 | Shapiro does not give a fair reading of the central dogma. Shapiro's reading of the central dogma requires that only random mutations can be the root of evolutionary change. If this reading is correct, then, ignoring the looseness of such an application of the term "random", the several mechanisms identified by Shapiro... | Natural genetic engineering | 0.811205 |
19,929 | In a 1997 Boston Review article, Shapiro lists four categories of discoveries made in molecular biology that, in his estimation, are not adequately accounted for by the Modern Synthesis: genome organization, cellular repair capabilities, mobile genetic elements and cellular information processing. Shapiro concludes: Wh... | Natural genetic engineering | 0.811205 |
19,930 | With the discovery of the nature and roles of nucleic acids in genetics, this model prompted Francis Crick's so-called Central Dogma of Molecular Biology: " information cannot be transferred back from protein to either protein or nucleic acid. "Shapiro points out that multiple cellular systems can affect DNA in respons... | Natural genetic engineering | 0.811205 |
19,931 | Shapiro first laid out his ideas of natural genetic engineering in 1992 and has continued to develop them in both the primary scientific literature and in work directed to wider audiences, culminating in the 2011 publication of Evolution: A View from the 21st Century (second edition in 2022.). Natural genetic engineeri... | Natural genetic engineering | 0.811205 |
19,932 | One of the many standard responses to these arguments is that biology can be sufficiently explained without invoking higher causes. Shapiro's view differs significantly from that of creationists, not the least because his higher causes exist only at the level of cellular machinery. However, to a critic unpersuaded of t... | Natural genetic engineering | 0.811204 |
19,933 | While Shapiro developed NGE in the peer-reviewed literature, the idea attracted far more attention when he summarized his work in his book Evolution: A View from the 21st Century. In part due to its discussion of the Intelligent Design movement, the book was widely and critically reviewed. Criticism falls into two main... | Natural genetic engineering | 0.811204 |
19,934 | Natural genetic engineering (NGE) is a class of process proposed by molecular biologist James A. Shapiro to account for novelty created in the course of biological evolution. Shapiro developed this work in several peer-reviewed publications from 1992 onwards, and later in his 2011 book Evolution: A View from the 21st C... | Natural genetic engineering | 0.811204 |
19,935 | In December 2015, the Department of Biotechnology launched the National Biotechnology Development Strategy 2015–2020 programme. The stated aim of the programme is to intensify research in the fields of vaccines, humane genome, infectious and chronic diseases, crop science, animal agriculture and aquaculture, food and n... | Department of Biotechnology | 0.811202 |
19,936 | Autonomous InstitutesCenter of Innovative and Applied Bioprocessing, Mohali Centre for DNA Fingerprinting and Diagnostics, Hyderabad National Institute of Animal Biotechnology, Hyderabad National Institute of Biomedical Genomics, Kalyani National Centre for Cell Science, Pune National Brain Research Centre, Manesar Kal... | Department of Biotechnology | 0.811202 |
19,937 | The Department of Biotechnology (DBT) is an Indian government department, under the Ministry of Science and Technology responsible for administrating development and commercialisation in the field of modern biology and biotechnology in India. It was set up in 1986. | Department of Biotechnology | 0.811202 |
19,938 | In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function f: M → N {\displaystyle f:M\to N} is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R, f ( x + y ) = f ( x ) + f ... | Additive relation | 0.811202 |
19,939 | A module homomorphism is called a module isomorphism if it admits an inverse homomorphism; in particular, it is a bijection. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism if and only if it is... | Additive relation | 0.811202 |
19,940 | Hom R ( M , R ) {\displaystyle \operatorname {Hom} _{R}(M,R)} is called the dual module of M; it is a left (resp. right) module if M is a right (resp. left) module over R with the module structure coming from the R-action on R. It is denoted by M ∗ {\displaystyle M^{*}} . Given a ring homomorphism R → S of commutativ... | Additive relation | 0.811202 |
19,941 | In quantum mechanics a free particle has as state function a plane wave function, which is a non-square-integrable function of well-defined momentum. The kinetic energy of this particle can take any positive value. The position of the COM is uniformly probable everywhere, in agreement with the Heisenberg uncertainty pr... | Electronic molecular Hamiltonian | 0.811196 |
19,942 | See Ref. for a thorough discussion of the mathematical properties of the Coulomb Hamiltonian. Also it is discussed in this paper whether one can arrive a priori at the concept of a molecule (as a stable system of electrons and nuclei with a well-defined geometry) from the properties of the Coulomb Hamiltonian alone. | Electronic molecular Hamiltonian | 0.811196 |
19,943 | They measured some of these properties using a 500-ton hydraulic press that applied pressures of up to 1.2 gigapascals (GPa). They concluded that the Earth's mantle had a different composition than the crust, perhaps ferromagnesian silicates, and the core was some combination of iron and nickel. They estimated the pres... | Mineral physics | 0.811195 |
19,944 | The field of mineral physics was not named until the 1960s, but its origins date back at least to the early 20th century and the recognition that the outer core is fluid because seismic work by Oldham and Gutenberg showed that it did not allow shear waves to propagate.A landmark in the history of mineral physics was th... | Mineral physics | 0.811195 |
19,945 | This information can be used to provide insights into plate tectonics, mantle convection, the geodynamo and related phenomena. Laboratory work in mineral physics require high pressure measurements. The most common tool is a diamond anvil cell, which uses diamonds to put a small sample under pressure that can approach t... | Mineral physics | 0.811195 |
19,946 | Mineral physics is the science of materials that compose the interior of planets, particularly the Earth. It overlaps with petrophysics, which focuses on whole-rock properties. It provides information that allows interpretation of surface measurements of seismic waves, gravity anomalies, geomagnetic fields and electrom... | Mineral physics | 0.811195 |
19,947 | Many of the pioneering studies in mineral physics involved explosions or projectiles that subject a sample to a shock. For a brief time interval, the sample is under pressure as the shock wave passes through. Pressures as high as any in the Earth have been achieved by this method. However, the method has some disadvant... | Mineral physics | 0.811195 |
19,948 | The latter is one of many Grünheisen parameters that play an important role in high-pressure physics. A more realistic EOS is the Birch–Murnaghan equation of state. : 66–73 | Mineral physics | 0.811195 |
19,949 | A student of his, Francis Birch, led a program to apply high-pressure methods to geophysics. Birch extended the Adams-Williamson equation to include the effects of temperature. In 1952, he published a classic paper, Elasticity and constitution of the Earth's interior, in which he established some basic facts: the mantl... | Mineral physics | 0.811195 |
19,950 | Achieving temperatures found within the interior of the earth is just as important to the study of mineral physics as creating high pressures. Several methods are used to reach these temperatures and measure them. Resistive heating is the most common and simplest to measure. The application of a voltage to a wire heats... | Mineral physics | 0.811195 |
19,951 | In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completel... | Torsion-free abelian group | 0.811193 |
19,952 | A homomorphism (or morphism) of inverse semigroups is defined in exactly the same way as for any other semigroup: for inverse semigroups S and T, a function θ from S to T is a morphism if (sθ)(tθ) = (st)θ, for all s,t in S. The definition of a morphism of inverse semigroups could be augmented by including the condition... | Inverse semigroup | 0.811192 |
19,953 | The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is another way of composing partial transformations, which is more restrictive than that used above: two partial transformations α and β are composed if, and only if, the image of α is equal to the domain of β; ... | Inverse semigroup | 0.811192 |
19,954 | Regular polyhedron Platonic solid: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron Regular spherical polyhedron Dihedron, Hosohedron Kepler–Poinsot polyhedron (Regular star polyhedra) Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron Abstract regular polyhedra (P... | List of polygons, polyhedra and polytopes | 0.811192 |
19,955 | Rectification (geometry) Truncation (geometry) Bitruncation Cantellation Runcination Sterication Omnitruncation Expansion (geometry) Snub (geometry) Alternation (geometry) Dual polyhedron Gyration (geometry) Elongation (geometry) Augmentation (geometry) Diminishment (geometry) Greatening (geometry) Aggrandizement (geom... | List of polygons, polyhedra and polytopes | 0.811192 |
19,956 | One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E and e ≤ s, for some s in S, then s is in E.Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum group congruence σ. Then the following are equivalent: S is E-unitary; σ is idempotent pure; ∼ ... | Inverse semigroup | 0.811192 |
19,957 | Machines are commonplace in many industries, including manufacturing, mining, construction and agriculture, and can be dangerous to workers. Many machines involve moving parts, sharp edges, hot surfaces and other hazards with the potential to crush, burn, cut, shear, stab or otherwise strike or wound workers if used un... | Physical hazards | 0.811191 |
19,958 | Faulty electrical connections and damaged electrical equipment can lead to an electric shock to workers and to others at or near the workplace.According to the Bureau of Labor Statistics, a total 1,738 accidents occurred due to contact with electric current between 2003 and 2010 and out of that 849, the highest number ... | Physical hazards | 0.811191 |
19,959 | In 2015, around 136 U.S. workers died in fatalities related to confined spaces according to the data collected in response to the annual Census of Fatal Occupational Injuries (CFOI) program of the Bureau of Labor Statistics. Hazards like entrapment and drowning to asphyxiation and toxic chemical exposure results in the... | Physical hazards | 0.811191 |
19,960 | The Underground Area 3 (UA3) experiment was a high-energy physics experiment at the Proton-Antiproton Collider (SppS), a modification of the Super Proton Synchrotron (SPS), at CERN. The experiment ran from 1978 to 1988 with the objective of searching for magnetic monopoles. No evidence for magnetic monopoles was found ... | UA3 experiment | 0.81119 |
19,961 | Structure-based methods require 3D structures of both proteins being considered. They are often slower than sequence-based methods, but are able to detect circular permutations between distantly related proteins with low sequence similarity. Some structural methods are topology independent, meaning that they are also a... | Circular permutation in proteins | 0.811188 |
19,962 | Many sequence alignment and protein structure alignment algorithms have been developed assuming linear data representations and as such are not able to detect circular permutations between proteins. Two examples of frequently used methods that have problems correctly aligning proteins related by circular permutation ar... | Circular permutation in proteins | 0.811188 |
19,963 | Many proteins have their termini located close together in 3D space. Because of this, it is often possible to design circular permutations of proteins. Today, circular permutations are generated routinely in the lab using standard genetics techniques. Although some permutation sites prevent the protein from folding cor... | Circular permutation in proteins | 0.811188 |
19,964 | Under Newtonian physics, an object in an (isolated) two-body system, consisting of the object orbiting a spherical mass, would trace out an ellipse with the center of mass of the system at a focus of the ellipse. The point of closest approach, called the periapsis (or when the central body is the Sun, perihelion), is f... | Tests of general relativity | 0.811186 |
19,965 | This is gravitational-wave astronomy. Gravitational-wave astronomy can test general relativity by verifying that the observed waves are of the form predicted (for example, that they only have two transverse polarizations), and by checking that black holes are the objects described by solutions of the Einstein field equ... | Tests of general relativity | 0.811186 |
19,966 | The σ-algebra H {\displaystyle {\mathcal {H}}} controls the "granularity" of the conditioning. A conditional expectation E ( X ∣ H ) {\displaystyle E(X\mid {\mathcal {H}})} over a finer (larger) σ-algebra H {\displaystyle {\mathcal {H}}} retains information about the probabilities of a larger class of events. A conditi... | Conditional expected value | 0.811184 |
19,967 | The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation.The Hilbert subspace M = { g ( Y ): E ( g ( Y ) 2 ) < ∞ } {\displaystyle M=\{g(Y):\operatorname {E} (g(Y)^{2})<\infty \}} defined above is replaced with... | Conditional expected value | 0.811184 |
19,968 | Consider, in addition to the above, A measurable space ( U , Σ ) {\displaystyle (U,\Sigma )} , and A random variable Y: Ω → U {\displaystyle Y\colon \Omega \to U} .The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y: E := E {\displaystyle \operato... | Conditional expected value | 0.811184 |
19,969 | The Newtonian dynamical system (3) constrained to the configuration manifold M {\displaystyle \displaystyle M} by the constraint equations (6) is described by the differential equations where Γ i j s {\displaystyle \Gamma _{ij}^{s}} are Christoffel symbols of the metric connection produced by the Riemannian metric (11)... | Particle dynamics | 0.811183 |
19,970 | In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion. | Particle dynamics | 0.811183 |
19,971 | These are summed and averaged to obtain an overall representative diffraction template from each grouping. Different orientations can be assigned colors which helps visualize individual grain orientations. With proper tilting and utilizing precession electron diffraction (PED) it is even possible to make 3D tomographic... | 4D Scanning Transmission Electron Microscopy (4D STEM) | 0.811182 |
19,972 | In particle physics, the Wess–Zumino gauge is a particular choice of a gauge transformation in a gauge theory with supersymmetry. In this gauge, the supersymmetrized gauge transformation is chosen in such a way that most components of the vector superfield vanish, except for the usual physical ones when the function of... | Wess–Zumino gauge | 0.811174 |
19,973 | Photobiology is the scientific study of the beneficial and harmful interactions of non-ionizing radiation in living organisms, conventionally demarcated around 10 eV, the first ionization energy of oxygen. UV ranges roughly from 3 to 30 eV in energy. Hence photobiology entertains some, but not all, of the UV spectrum. | UV light | 0.811174 |
19,974 | The methods in this category employ the similarity and differences of substrings in a pair of sequences. These algorithms were mostly used for string processing in computer science. | Alignment-free sequence analysis | 0.811171 |
19,975 | In bioinformatics, alignment-free sequence analysis approaches to molecular sequence and structure data provide alternatives over alignment-based approaches.The emergence and need for the analysis of different types of data generated through biological research has given rise to the field of bioinformatics. Molecular s... | Alignment-free sequence analysis | 0.811171 |
19,976 | Genomic rearrangements Molecular phylogenetics Metagenomics Next generation sequence data analysis Epigenomics Barcoding of species Population genetics Horizontal gene transfer Sero/genotyping of viruses Allergenicity prediction SNP discovery Recombination detection Viral Classification Archaea Taxonomic Identification | Alignment-free sequence analysis | 0.811171 |
19,977 | D 2 S {\displaystyle D_{2}^{S}} and D 2 ∗ {\displaystyle D_{2}^{*}} These approachese are variants of the D 2 {\displaystyle D_{2}} statistics that counts the number of k {\displaystyle k} -mer matches between two sequences. They improve the simple D 2 {\displaystyle D_{2}} statistics by taking the background distribut... | Alignment-free sequence analysis | 0.811171 |
19,978 | Alignment-free methods can broadly be classified into five categories: a) methods based on k-mer/word frequency, b) methods based on the length of common substrings, c) methods based on the number of (spaced) word matches, d) methods based on micro-alignments, e) methods based on information theory and f) methods based... | Alignment-free sequence analysis | 0.811171 |
19,979 | Multi-SpaM (MultipleSpaced-word Matches) is an approach to genome-based phylogeny reconstruction that extends the FSWM idea to multiple sequence comparison. Given a binary pattern P of match positions and don't-care positions, the program searches for P-blocks, i.e. local gap-free four-way alignments with matching nucl... | Alignment-free sequence analysis | 0.811171 |
19,980 | Since the origin of bioinformatics, sequence analysis has remained the major area of research with wide range of applications in database searching, genome annotation, comparative genomics, molecular phylogeny and gene prediction. The pioneering approaches for sequence analysis were based on sequence alignment either g... | Alignment-free sequence analysis | 0.811171 |
19,981 | A linear interval system consists of a matrix interval extension ∈ n × m {\displaystyle \in ^{n\times m}} and an interval vector ∈ n {\displaystyle \in ^{n}} . We want the smallest cuboid ∈ m {\displaystyle \in ^{m}} , for all vectors x ∈ R m {\displaystyle \mathbf {x} \in \mathbb {R} ^{m}} which there is a pair ... | Interval mathematics | 0.811169 |
19,982 | The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry should not be regarded as a priori truth. He held that axioms in geometry should be chosen for the resu... | Philosophy of mathematics | 0.811168 |
19,983 | Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof. Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral... | Ancient mathematics | 0.81116 |
19,984 | 214–298 AD), who wrote on polygonal numbers and a work in pre-modern algebra (Arithmetica), Pappus of Alexandria (c. 290–350 AD), who compiled many important results in the Collection, Theon of Alexandria (c. 335–405 AD) and his daughter Hypatia (c. | Ancient mathematics | 0.81116 |
19,985 | 100–170 AD) was of a very advanced level and rarely mastered outside a small circle. There is also evidence of combining mathematical knowledge with technical or practical applications, as found for instance in the work of Menelaus of Alexandria (c. 70–130 AD), who wrote a work dealing with the geometry of the sphere a... | Ancient mathematics | 0.81116 |
19,986 | The Hellenistic era began in the late 4th century BC, following Alexander the Great's conquest of the Eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholars... | Ancient mathematics | 0.81116 |
19,987 | The Faculty of Biotechnology has adequately equipped classrooms and laboratories situated in the Faculty's buildings in Podgorica, Bar and Bijelo Polje, as well as experimental plots for a part of students’ professional practice and organization of Faculty production. | University of Montenegro Faculty of Biotechnology | 0.81116 |
19,988 | In probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by Edward O. Thorp ... | Proebsting's paradox | 0.811157 |
19,989 | Projected area is the two dimensional area measurement of a three-dimensional object by projecting its shape on to an arbitrary plane. This is often used in mechanical engineering and architectural engineering related fields, especially for hardness testing, axial stress, wind pressures, and terminal velocity. The geom... | Projected area | 0.811156 |
19,990 | In pseudospectral discretization the entire trajectory is represented by a collection of basis functions in the time domain (independent variable). The basis functions need not be polynomials. Pseudospectral discretization is also known as spectral collocation. When used to solve a trajectory optimization problem whose... | Trajectory optimization | 0.811154 |
19,991 | When solving a trajectory optimization problem with an indirect method, you must explicitly construct the adjoint equations and their gradients. This is often difficult to do, but it gives an excellent accuracy metric for the solution. Direct methods are much easier to set up and solve, but do not have a built-in accur... | Trajectory optimization | 0.811154 |
19,992 | The first optimal control approaches grew out of the calculus of variations, based on the research of Gilbert Ames Bliss and Bryson in America, and Pontryagin in Russia. Pontryagin's maximum principle is of particular note. | Trajectory optimization | 0.811154 |
19,993 | Trajectory optimization first showed up in 1697, with the introduction of the Brachystochrone problem: find the shape of a wire such that a bead sliding along it will move between two points in the minimum time. The interesting thing about this problem is that it is optimizing over a curve (the shape of the wire), rath... | Trajectory optimization | 0.811154 |
19,994 | There are many techniques to choose from when solving a trajectory optimization problem. There is no best method, but some methods might do a better job on specific problems. This section provides a rough understanding of the trade-offs between methods. | Trajectory optimization | 0.811154 |
19,995 | There are a wide variety of applications for trajectory optimization, primarily in robotics: industry, manipulation, walking, path-planning, and aerospace. It can also be used for modeling and estimation. | Trajectory optimization | 0.811154 |
19,996 | There are a variety of different applications for trajectory optimization within the field of walking robotics. For example, one paper used trajectory optimization of bipedal gaits on a simple model to show that walking is energetically favorable for moving at a low speed and running is energetically favorable for movi... | Trajectory optimization | 0.811154 |
19,997 | Differential dynamic programming, is a bit different than the other techniques described here. In particular, it does not cleanly separate the transcription and the optimization. Instead, it does a sequence of iterative forward and backward passes along the trajectory. Each forward pass satisfies the system dynamics, a... | Trajectory optimization | 0.811154 |
19,998 | Depending on the configuration, open-chain robotic manipulators require a degree of trajectory optimization. For instance, a robotic arm with 7 joints and 7 links (7-DOF) is a redundant system where one cartesian position of an end-effector can correspond to an infinite number of joint angle positions, thus this redund... | Trajectory optimization | 0.811154 |
19,999 | Decision variables The set of unknowns to be found using optimization.Trajectory optimization problem A special type of optimization problem where the decision variables are functions, rather than real numbers.Parameter optimization Any optimization problem where the decision variables are real numbers.Nonlinear progra... | Trajectory optimization | 0.811154 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.