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26,200 | It is also possible to prove Stallings' theorem for finitely presented groups using Riemannian geometry techniques of minimal surfaces, where one first realizes a finitely presented group as the fundamental group of a compact 4 {\displaystyle 4} -manifold (see, for example, a sketch of this argument in the survey artic... | Stallings theorem about ends of groups | 0.807254 |
26,201 | Niblo obtained a proof of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of H {\displaystyle H} in G {\displaystyle G} ) ... | Stallings theorem about ends of groups | 0.807254 |
26,202 | The case of semi-splittings of finitely generated groups with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson.A number of new proofs of Stallings' theorem have been obtained by others after Stallings' original proof. Dunwoody gave a proof based on the ideas of ... | Stallings theorem about ends of groups | 0.807254 |
26,203 | In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with H {\displaystyle H} , such as for the case where H {\displaystyle H} is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few except... | Stallings theorem about ends of groups | 0.807254 |
26,204 | The case where e ( G , H ) > 1 {\displaystyle e(G,H)>1} is called a semi-splitting of G {\displaystyle G} over H {\displaystyle H} . Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott, Swarup, and others. The work of Sageev and Gerasimov in the 1990s showed that for ... | Stallings theorem about ends of groups | 0.807254 |
26,205 | Using Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if G {\displaystyle G} is a finitely presented group with asymptotic dimension 1 then G {\displaystyle G} is virtually free one can show that for a finitely presented word-hyperbolic group G {\displaystyle G} the hyperbolic... | Stallings theorem about ends of groups | 0.807254 |
26,206 | Linnell showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as Bestvina-Feighn accessibility of finitely presented groups (where the so-... | Stallings theorem about ends of groups | 0.807254 |
26,207 | Among the immediate applications of Stallings' theorem was a proof by Stallings of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free virtually free group is free. Stallings' theorem also implies that the property of having a nontrivial spli... | Stallings theorem about ends of groups | 0.807254 |
26,208 | Let G {\displaystyle G} be a finitely generated group. Then e ( G ) > 1 {\displaystyle e(G)>1} if and only if one of the following holds: The group G {\displaystyle G} admits a splitting G = H ∗ C K {\displaystyle G=H*_{C}K} as a free product with amalgamation where C {\displaystyle C} is a finite group such that C ≠ H... | Stallings theorem about ends of groups | 0.807254 |
26,209 | In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G {\displaystyle G} has more than one end if and only if the group G {\displaystyle G} admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgrou... | Stallings theorem about ends of groups | 0.807254 |
26,210 | In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are ... | Electronic band structure | 0.807254 |
26,211 | The closest band above the band gap is called the conduction band, and the closest band beneath the band gap is called the valence band. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the valence orbitals. In a metal or semimetal, th... | Electronic band structure | 0.807254 |
26,212 | In the nearly free electron approximation, interactions between electrons are completely ignored. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring recipro... | Electronic band structure | 0.807254 |
26,213 | Molecular phylogenetics has revealed that every clade of bacteria with genome sizes under 2 Mb was derived from ancestors with much larger genomes, thus refuting the hypothesis that bacteria evolved by the successive doubling of small-genomed ancestors. Recent studies performed by Nilsson et al. examined the rates of b... | Bacterial genome | 0.807253 |
26,214 | For example, he made the first known reference to the phenomenon that the mineral tourmaline attracts straws and bits of wood when heated, now known to be caused by pyroelectricity. Pliny the Elder makes clear references to his use of the work in his Natural History, while updating and making much new information avail... | Greek science | 0.807252 |
26,215 | Although the author insists that epilepsy has a natural cause, when it comes to explain what that cause is and what the proper treatment would be, the explanation is as short on specific evidence and the treatment as vague as that of his rivals. Nonetheless, observations of natural phenomena continued to be compiled in... | Greek science | 0.807252 |
26,216 | Two Greek philosophers, Leucippus (first half of the 5th century BC) and Democritus came up with the notion that there were two real entities: atoms, which were small indivisible particles of matter, and the void, which was the empty space in which matter was located. Although all the explanations from Thales to Democr... | Greek science | 0.807252 |
26,217 | Euclid's Elements served as the main textbook for the teaching of theoretical mathematics until the early 20th century. Archimedes (287–212 BC), a Sicilian Greek, wrote about a dozen treatises were he communicated many remarkable results, such as the sum of an infinite geometric series in Quadrature of the Parabola, an... | Greek science | 0.807252 |
26,218 | Greek mathematics in the Hellenistic period reached a level of sophistication not matched for several centuries afterward, as much of the work represented by scholars active at this time was of a very advanced level. There is also evidence of combining mathematical knowledge with high levels of technical expertise, as ... | Greek science | 0.807252 |
26,219 | Ptolemy's Mathēmatikē Syntaxis (Ancient Greek: Μαθηματικὴ Σύνταξις), better known as the Almagest, sought to improve on the work of his predecessors by building astronomy not only upon a secure mathematical basis but also by demonstrating the relationship between astronomical observations and the resulting astronomical... | Greek science | 0.807252 |
26,220 | Claudius Ptolemy (c. 100–170 AD), living in or around Alexandria, carried out a scientific program centered on the writing of about a dozen books on astronomy, astrology, cartography, harmonics, and optics. Despite their severe style and high technicality, a great deal of them have survived, in some cases the sole remn... | Greek science | 0.807252 |
26,221 | Advances in mathematical astronomy also took place during the Hellenistic age. Aristarchus of Samos (310–230 BC) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the known universe, with the Earth revolving around the Sun once a year... | Greek science | 0.807252 |
26,222 | Truth is to be found through rational argumentation, analogous to the demonstrations of mathematicians. For instance, Plato recommended that astronomy be studied in terms of abstract geometrical models rather than empirical observations, and proposed that leaders be trained in mathematics in preparation for philosophy.... | Greek science | 0.807252 |
26,223 | A later account has it that Plato had inscribed at the entrance to the Academy the words "Let no man ignorant of geometry enter." Although the story is most likely a myth, it nonetheless testifies to Plato's interest in mathematics, which is alluded to in several of his dialogues.Plato's philosophy maintained that all ... | Greek science | 0.807252 |
26,224 | The first institution of higher learning in Ancient Greece was founded by Plato (c. 427–c. 347 BC), an Athenian who—perhaps under Pythagorean influence—appears to have identified the ordering principle of the universe as one based on number and geometry. | Greek science | 0.807252 |
26,225 | Apart from astronomy, both the Harmonics and the Optics contain (in addition to mathematical analyses of sound and sight, respectively) instructions on how to construct and use experimental instruments to corroborate theory. In retrospect, it is apparent that Ptolemy adjusted some reported measurements to fit his (inco... | Greek science | 0.807252 |
26,226 | After Plato's death, Aristotle left the Academy and traveled widely before returning to Athens to found a school adjacent to the Lyceum. As one of the most prolific natural philosophers of Antiquity, Aristotle wrote and lecture on many topics of scientific interest, including biology, meteorology, psychology, logic, an... | Greek science | 0.807252 |
26,227 | Some of the more widely known figures active in this period include Hippocrates, Aristotle, Euclid, Archimedes, Hipparchus, Galen, and Ptolemy. Their contributions and commentaries spread throughout the Eastern, Islamic, and Latin worlds and contributed to the birth of modern science. Their works covered many different... | Greek science | 0.807252 |
26,228 | Standard atomic weight is nevertheless the most widely published variant of relative atomic mass. Additionally, the continued use of the term "atomic weight" (for any element) as opposed to "relative atomic mass" has attracted considerable controversy since at least the 1960s, mainly due to the technical difference bet... | Average atomic mass | 0.807251 |
26,229 | In mathematics, Tarski's theorem, proved by Alfred Tarski (1924), states that in ZF the theorem "For every infinite set A {\displaystyle A} , there is a bijective map between the sets A {\displaystyle A} and A × A {\displaystyle A\times A} " implies the axiom of choice. The opposite direction was already known, thus th... | Tarski's theorem about choice | 0.807247 |
26,230 | The goal is to prove that the axiom of choice is implied by the statement "for every infinite set A: {\displaystyle A:} | A | = | A × A | {\displaystyle |A|=|A\times A|} ". It is known that the well-ordering theorem is equivalent to the axiom of choice; thus it is enough to show that the statement implies that for ever... | Tarski's theorem about choice | 0.807247 |
26,231 | In electromagnetism, a branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers, and vector calculus. These representations are for a homogeneous medium, an approximation in an inhomogeneous medium. A matrix representat... | Matrix Representation of Maxwell Equations | 0.807247 |
26,232 | This representation uses a single matrix equation, instead of a pair of matrix equations. In this representation, using 8 × 8 matrices, it has been possible to separate the dependence of the coupling between the upper components (Ψ+) and the lower components (Ψ−) through the two laboratory functions. Moreover, the exac... | Matrix Representation of Maxwell Equations | 0.807247 |
26,233 | The derivation of the Helmholtz equation from the Maxwell's equations is an approximation as one neglects the spatial and temporal derivatives of the permittivity and permeability of the medium. A new formalism of light beam optics has been developed, starting with the Maxwell's equations in a matrix form: a single ent... | Matrix Representation of Maxwell Equations | 0.807247 |
26,234 | In chemistry, an ionic compound is a chemical compound composed of ions held together by electrostatic forces termed ionic bonding. The compound is neutral overall, but consists of positively charged ions called cations and negatively charged ions called anions. These can be simple ions such as the sodium (Na+) and chl... | Ionic solid | 0.807246 |
26,235 | Since 1801 pyrotechnicians have described and widely used metal-containing ionic compounds as sources of colour in fireworks. Under intense heat, the electrons in the metal ions or small molecules can be excited. These electrons later return to lower energy states, and release light with a colour spectrum characteristi... | Ionic solid | 0.807246 |
26,236 | The last four results combine to give the following theorem: If c m {\displaystyle c_{m}} is the largest value of dim Aut P {\displaystyle \dim \operatorname {Aut} {\mathcal {P}}} , where P {\displaystyle {\mathcal {P}}} is a non-classical compact 2m-dimensional topological projective plane, then dim Aut P ≤ c ... | Smooth projective plane | 0.807245 |
26,237 | In geometry, smooth projective planes are special projective planes. The most prominent example of a smooth projective plane is the real projective plane E {\displaystyle {\mathcal {E}}} . Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous ... | Smooth projective plane | 0.807245 |
26,238 | The latter can be characterized as follows: Σ {\displaystyle \Sigma } leaves some classical complex subplane C {\displaystyle {\mathcal {C}}} invariant and induces on C {\displaystyle {\mathcal {C}}} the connected component of its full automorphism group. The Hughes planes are not smooth. This yields a result similar t... | Smooth projective plane | 0.807245 |
26,239 | These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane. More generally, all non-classical compact 2-dimensional planes P {\displaystyle {\mathcal {P}}} such that dim Aut P ≥ 3 {\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\geq ... | Smooth projective plane | 0.807245 |
26,240 | We have:Theorem. If P = ( P , L ) {\displaystyle {\mathcal {P}}=(P,{\mathfrak {L}})} is a smooth plane, then each continuous collineation of P {\displaystyle {\mathcal {P}}} is smooth; in other words, the group of automorphisms of a smooth plane P {\displaystyle {\mathcal {P}}} coincides with Aut P {\displaystyle \op... | Smooth projective plane | 0.807245 |
26,241 | Automorphisms play a crucial role in the study of smooth planes. A bijection of the point set of a projective plane is called a collineation, if it maps lines onto lines. The continuous collineations of a compact projective plane P {\displaystyle {\mathcal {P}}} form the group Aut P {\displaystyle \operatorname {Aut}... | Smooth projective plane | 0.807245 |
26,242 | All compact planes P {\displaystyle {\mathcal {P}}} with a 4-dimensional point space and Aut P ≥ 7 {\displaystyle \operatorname {Aut} {\mathcal {P}}\geq 7} have been classified. Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane. According to Bödi (1996, Chap. 1... | Smooth projective plane | 0.807245 |
26,243 | These four cases will be treated separately below. Theorem. The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold. | Smooth projective plane | 0.807245 |
26,244 | A projective plane is called a translation plane if its automorphism group has a subgroup that fixes each point on some line W {\displaystyle W} and acts sharply transitively on the set of points not on W {\displaystyle W} . Theorem. Every smooth projective translation plane P {\displaystyle {\mathcal {P}}} is isomorph... | Smooth projective plane | 0.807245 |
26,245 | The condition, that the geometric operations of a projective plane are complex analytic, is very restrictive. In fact, it is satisfied only in the classical complex plane.Theorem. Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure. | Smooth projective plane | 0.807245 |
26,246 | Let Σ {\displaystyle \Sigma } denote the automorphism group of a compact 16-dimensional topological projective plane P {\displaystyle {\mathcal {P}}} . Either P {\displaystyle {\mathcal {P}}} is the smooth classical octonion plane or dim Σ ≤ 40 {\displaystyle \dim \Sigma \leq 40} . If dim Σ = 40 {\displaystyle \dim... | Smooth projective plane | 0.807245 |
26,247 | In acronyms both parts are usually capitalized, such as VŠCHT for Vysoká škola chemicko-technologická (University of Chemistry and Technology). However, the practice is not unified when writing initial letters of personal names (first name and surname), for example Jan Chudoba can be abbreviated both J. Ch. or J. CH. | Mixed case | 0.807245 |
26,248 | Since classical algebraic geometry allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set i... | Euclidean simplex | 0.807244 |
26,249 | The Mars Organic Molecule Analyser (MOMA) is a mass spectrometer-based instrument on board the Rosalind Franklin rover to be launched in 2028 to Mars on an astrobiology mission. It will search for organic compounds (carbon-containing molecules) in the collected soil samples. By characterizing the molecular structures o... | Mars Organic Molecule Analyser | 0.807244 |
26,250 | The idea of a correlation between molecular geometry and number of valence electron pairs (both shared and unshared pairs) was originally proposed in 1939 by Ryutaro Tsuchida in Japan, and was independently presented in a Bakerian Lecture in 1940 by Nevil Sidgwick and Herbert Powell of the University of Oxford. In 1957... | Valence Shell Electron Pair Repulsion Theory | 0.807238 |
26,251 | : 410–417 The total number of X and E is known as the steric number. For example in a molecule AX3E2, the atom A has a steric number of 5. When the substituent (X) atoms are not all the same, the geometry is still approximately valid, but the bond angles may be slightly different from the ones where all the outside ato... | Valence Shell Electron Pair Repulsion Theory | 0.807238 |
26,252 | Some AX6E1 molecules, e.g. xenon hexafluoride (XeF6) and the Te(IV) and Bi(III) anions, TeCl2−6, TeBr2−6, BiCl3−6, BiBr3−6 and BiI3−6, are octahedral, rather than pentagonal pyramids, and the lone pair does not affect the geometry to the degree predicted by VSEPR. Similarly, the octafluoroxenate ion (XeF2−8) in nitroso... | Valence Shell Electron Pair Repulsion Theory | 0.807238 |
26,253 | As of 5 February 2018, Academic Family Tree contained 689,800 people with 606,600 connections among them. Academic Family Tree encompasses a broad range of disciplines. As of 5 February 2018, there were 55 disciplines spanning science (e.g., human genetics, microbiology, and psychology), mathematics and philosophy, eng... | Academic Tree | 0.807238 |
26,254 | Neurotree and Academic Family Tree have been used as sources of information for the history and prospects of various academic fields such as psychology, meteorology, organizational communication, and neuroscience. Neurotree has also been used to address infometrics and to research issues of scientific methodology.Neuro... | Academic Tree | 0.807238 |
26,255 | Boolean algebras also meet this definition of Kleene algebra. The simplest Kleene algebra that is not Boolean is Kleene's three-valued logic K3. K3 made its first appearance in Kleene's On notation for ordinal numbers (1938). The algebra was named after Kleene by Brignole and Monteiro. | Kleene algebra (with involution) | 0.807236 |
26,256 | If a De Morgan algebra additionally satisfies x ∧ ¬x ≤ y ∨ ¬y, it is called a Kleene algebra. (This notion should not to be confused with the other Kleene algebra generalizing regular expressions.) This notion has also been called a normal i-lattice by Kalman. Examples of Kleene algebras in the sense defined above incl... | Kleene algebra (with involution) | 0.807236 |
26,257 | If the pseudocomplement satisfies the law of the excluded middle, the resulting algebra is also Boolean. However, if only the weaker law ¬x ∨ ¬¬x = 1 is required, this results in Stone algebras. More generally, both De Morgan and Stone algebras are proper subclasses of Ockham algebras. | Kleene algebra (with involution) | 0.807236 |
26,258 | De Morgan algebras are not the only plausible way to generalize Boolean algebras. Another way is to keep ¬x ∧ x = 0 (i.e. the law of noncontradiction) but to drop the law of the excluded middle and the law of double negation. This approach (called semicomplementation) is well-defined even for a (meet) semilattice; if t... | Kleene algebra (with involution) | 0.807236 |
26,259 | (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = (, max(x, y), min(x, y), 0, 1, 1 − x) is a... | Kleene algebra (with involution) | 0.807236 |
26,260 | Thus ¬ is a dual automorphism of (A, ∨, ∧, 0, 1). If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementati... | Kleene algebra (with involution) | 0.807236 |
26,261 | In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies ... | Kleene algebra (with involution) | 0.807236 |
26,262 | Nonetheless, this does avoid the need to handle an exceptional case when computing the length of each row, as it guarantees the formula ROW_INDEX − ROW_INDEX works for any row i. Moreover, the memory cost of this redundant storage is likely insignificant for a sufficiently large matrix. The (old and new) Yale sparse ma... | Dense matrix | 0.807232 |
26,263 | While I find pretty much all math to be at least somewhat interesting, I don't (yet) see as much beauty in the other three theorems. FWIW, there are formulas that are already in the article that I also find less interesting than this subsequence formula. But obviously this isn't about me, and unfortunately I don't have... | Talk:Fibonacci sequence | 0.807227 |
26,264 | –jacobolus (t) 15:23, 12 May 2023 (UTC) Why does it appeal to me? : because I find it interesting that a sequence that is every k-th term of the Fibonacci sequence obeys a linear recurrence (and that the linear recurrence has easily computed coefficients). Of the above, only Theorem 7 is about such a sequence. | Talk:Fibonacci sequence | 0.807227 |
26,265 | doi:10.2307/3619903. JSTOR 3619903.It has Theorems 4–7: L 2 m + n − ( − 1 ) m L n = 5 F m F m + n , F 2 m + n − ( − 1 ) m F n = F m L m + n , L 2 m + n + ( − 1 ) m L n = L m L m + n , F 2 m + n + ( − 1 ) m F n = L m F m + n . {\displaystyle {\begin{aligned}L_{2m+n}-(-1)^{m}L_{n}&=5F_{m}F_{m+n},\\F_{2m+n}-(-1)^{m}F_{n}&... | Talk:Fibonacci sequence | 0.807227 |
26,266 | –jacobolus (t) 05:14, 12 May 2023 (UTC) For example, it's exercise 52–53 in https://doi.org/10.1002/9781118033067.ch5 and theorems 5, 7 in https://doi.org/10.2307/3619903 –jacobolus (t) 09:51, 12 May 2023 (UTC) If we can find discussion in Fibonacci Quarterly, perhaps that will signal noteworthiness as well. Unfortunat... | Talk:Fibonacci sequence | 0.807227 |
26,267 | In all likelihood the only reason it wasn't already used was because someone was adding references of different kinds and didn't want to figure out which references needed which template. Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 25... | Talk:Fibonacci sequence | 0.807227 |
26,268 | If kinematic viscosity tends to zero simultaneously with the resistivity, and if the plasma becomes turbulent (associated with high Reynolds numbers), then Lagrangian trajectories will no longer be unique. The conventional "naïve" flux freezing argument, discussed above, does not apply in general, and stochastic flux f... | Frozen-in flux theorem | 0.807226 |
26,269 | In image processing and computer vision, traditional image segmentation models often assign to one pixel only one exclusive pattern. In fuzzy or soft segmentation, any pattern can have certain "ownership" over any single pixel. If the patterns are Gaussian, fuzzy segmentation naturally results in Gaussian mixtures. Com... | Gaussian mixture model | 0.807226 |
26,270 | While his work was successful in identifying two potentially distinct sub-populations and in demonstrating the flexibility of mixtures as a moment matching tool, the formulation required the solution of a 9th degree (nonic) polynomial which at the time posed a significant computational challenge. Subsequent works focus... | Gaussian mixture model | 0.807226 |
26,271 | In ideal magnetohydrodynamics, Alfvén's theorem, or the frozen-in flux theorem, states that electrically conducting fluids and embedded magnetic fields are constrained to move together in the limit of large magnetic Reynolds numbers. It is named after Hannes Alfvén, who put the idea forward in 1943. Alfvén's theorem im... | Frozen-in flux theorem | 0.807226 |
26,272 | As an alternative to the EM algorithm, the mixture model parameters can be deduced using posterior sampling as indicated by Bayes' theorem. This is still regarded as an incomplete data problem whereby membership of data points is the missing data. A two-step iterative procedure known as Gibbs sampling can be used. | Gaussian mixture model | 0.807226 |
26,273 | MEDLINE (Medical Literature Analysis and Retrieval System Online, or MEDLARS Online) is a bibliographic database of life sciences and biomedical information. It includes bibliographic information for articles from academic journals covering medicine, nursing, pharmacy, dentistry, veterinary medicine, and health care. M... | MEDLINE | 0.807219 |
26,274 | Google has signed two contracts with wind developers to power its data center in Finland with 100% renewable energy. Facebook decided to build a data center in Iowa, and has helped drive the local energy provider to scrap plans to build a nuclear power plant, and instead build a $2bn(£1.23bn) wind farm, which has led t... | Internet science | 0.807213 |
26,275 | Internet science is an interdisciplinary science that examines all aspects of the co-evolution in Internet networks and society. It works in the intersection of and in the gaps among a wide range of disciplines that have had to respond to the impact of the Internet on their 'home turf' and/or offer specific conceptual ... | Internet science | 0.807213 |
26,276 | 1976: Queen Elizabeth II sends her first e-mail. 1979: USENET forms to host news and discussion groups. 1981: The National Science Foundation (NSF) provided a grant to demonstrate the Computer Science Network (CSNET) and afterwards to provide networking services to university computer scientists. | Internet science | 0.807213 |
26,277 | Saunders and Gasseling published data in the Journal of Experimental Biology in 1948, showing that reference marks inserted near the rim of the apical border of the wing bud are dispersed throughout the whole forearm of the wing. This led them to believe that the apical ectoderm may play a role in forming parts of the ... | Zone of polarizing activity | 0.807205 |
26,278 | In physics and color science, the Planckian locus or black body locus is the path or locus that the color of an incandescent black body would take in a particular chromaticity space as the blackbody temperature changes. It goes from deep red at low temperatures through orange, yellowish white, white, and finally bluish... | Planckian locus | 0.807205 |
26,279 | Roots, rodents, worms, and freezing-thawing cycles further break the peds into smaller peds of a more or less spherical shape.At a smaller scale, plant roots extend into voids (macropores) and remove water causing macroporosity to increase and microporosity to decrease, thereby decreasing aggregate size. At the same ti... | Physical properties of soil | 0.807203 |
26,280 | One problem in communication networks is understanding how to control congestion and maintain normal and efficient network function.Zonghua Liu et al. (2006) showed that congestion is more likely to occur at the nodes with high degrees in networks, and an efficient approach of selectively enhancing the message-process ... | Gradient network | 0.807203 |
26,281 | The fact that the topology of the substrate network influence the level of network congestion can be illustrated by a simple example: if the network has a star-like structure, then at the central node, the flow would become congested because the central node should handle all the flow from other nodes. However, if the ... | Gradient network | 0.807203 |
26,282 | The concept of a gradient network was first introduced by Toroczkai and Bassler (2004).Generally, real-world networks (such as citation graphs, the Internet, cellular metabolic networks, the worldwide airport network), which often evolve to transport entities such as information, cars, power, water, forces, and so on, ... | Gradient network | 0.807203 |
26,283 | Genome sequencing of endangered species is the application of Next Generation Sequencing (NGS) technologies in the field of conservation biology, with the aim of generating life history, demographic and phylogenetic data of relevance to the management of endangered wildlife. | Genome sequencing of endangered species | 0.807202 |
26,284 | Despite a faster and easier data production and a continuous improvement of sequencing technologies, there is still a marked delay of data analysis and processing techniques. Genome-wide analysis and big genomes studies require advances in bioinformatics and computational biology. At the same time improvements in the s... | Genome sequencing of endangered species | 0.807202 |
26,285 | The development of indirect sequencing methods has to some degree mitigated the lack of efficient DNA sequencing technologies. These techniques allowed researchers to increase scientific knowledge in fields like ecology and evolution. Several genetic markers, more or less well suited for the purpose, were developed hel... | Genome sequencing of endangered species | 0.807202 |
26,286 | In the context of conservation biology, genomic technologies such as the production of large-scale sequencing data sets via DNA sequencing can be used to highlight the relevant aspects of the biology of wildlife species for which management actions may be required. This may involve the estimation of recent demographic ... | Genome sequencing of endangered species | 0.807202 |
26,287 | Miller stellations. In "The Fifty-Nine Icosahedra" Coxeter, Du Val, Flather and Petrie record five rules suggested by Miller. Although these rules refer specifically to the icosahedron's geometry, they have been adapted to work for arbitrary polyhedra. | Compound polygon | 0.8072 |
26,288 | Magnetometers are used extensively in experimental particle physics to measure the magnetic field of pivotal components such as the concentration or focusing beam-magnets. | Magnetometer | 0.8072 |
26,289 | In saturated groundwater aquifers, all available pore spaces are filled with water (volumetric water content = porosity). Above a capillary fringe, pore spaces have air in them too. Most soils have a water content less than porosity, which is the definition of unsaturated conditions, and they make up the subject of vad... | Water content | 0.807199 |
26,290 | Methods designed for hardware implementation generally do not scale to integers with thousands or millions of decimal digits; these frequently occur, for example, in modular reductions in cryptography. For these large integers, more efficient division algorithms transform the problem to use a small number of multiplica... | Newton–Raphson algorithm | 0.807198 |
26,291 | As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective. Toroidal polyhedra, made of wood and used to suppor... | Polyhedral surface | 0.807196 |
26,292 | Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by triangulation). For example, the volume of a regular polyhedron can be computed by dividing it into congruent pyramids, with each pyramid having a face of the polyhedron as its base and the centre of the polyhed... | Polyhedral surface | 0.807195 |
26,293 | Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the regular skew polyhedra and to develop the theory of complex polyhedra first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of g... | Polyhedral surface | 0.807195 |
26,294 | The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. Coxeter's analysis signalled a rebirth of interest in geometry. | Polyhedral surface | 0.807195 |
26,295 | Mathematics in the 20th century dawned with Hilbert's problems, one of which, Hilbert's third problem, concerned polyhedra and their dissections. It was quickly solved by Hilbert's student Max Dehn, introducing the Dehn invariant of polyhedra. Steinitz's theorem, published by Ernst Steinitz in 1992, characterized the g... | Polyhedral surface | 0.807195 |
26,296 | It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's rigidity theorem, flexible polyhedra must be non-convex. The volume of a flexible polyhedron... | Polyhedral surface | 0.807195 |
26,297 | Forward and reverse accumulation are just two (extreme) ways of traversing the chain rule. The problem of computing a full Jacobian of f: Rn → Rm with a minimum number of arithmetic operations is known as the optimal Jacobian accumulation (OJA) problem, which is NP-complete. Central to this proof is the idea that algeb... | Automatic differentiation | 0.807195 |
26,298 | The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, define... | Automatic differentiation | 0.807195 |
26,299 | Multivariate functions can be handled with the same efficiency and mechanisms as univariate functions by adopting a directional derivative operator. That is, if it is sufficient to compute y ′ = ∇ f ( x ) ⋅ x ′ {\displaystyle y'=\nabla f(x)\cdot x'} , the directional derivative y ′ ∈ R m {\displaystyle y'\in \mathbb {R... | Automatic differentiation | 0.807195 |
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