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97,700 | However, thinner fibrils can also be detected. Wågberg et al. reported fibril widths of 5–15 nm for a nanocellulose with a charge density of about 0.5 meq./g. The group of Isogai reported fibril widths of 3–5 nm for TEMPO-oxidized cellulose having a charge density of 1.5 meq./g.Pulp chemistry has a significant influenc... | Nanocrystalline cellulose | 0.777627 |
97,701 | An increase in ionic strength induces the arrest of nanocellulose dispersions into attractive glasses. At further increasing ionic strength, nanocelluloses aggregate into hydrogels. The interactions within nanocelluloses are weak and reversible, wherefore nanocellulose suspensions and hydrogels are self-healing and may... | Nanocrystalline cellulose | 0.777627 |
97,702 | In physics, the Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers su... | Spalart–Allmaras turbulence model | 0.777627 |
97,703 | Proofs of the mathematical result that the rational number 22/7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the th... | Proof that 22/7 exceeds π | 0.777626 |
97,704 | In molecular biology the protein SSI is a Subtilisin inhibitor-like which stands for Streptomyces subtilisin inhibitor. This is a protease inhibitor. These are often synthesised as part of a larger precursor protein, either as a prepropeptide. The function of this protein domain is to prevent access of the substrate to... | SSI protease inhibitor | 0.777625 |
97,705 | Satellite formation flying is the coordination of multiple satellites to accomplish the objective of one larger, usually more expensive, satellite. Coordinating smaller satellites has many benefits over single satellites including simpler designs, faster build times, cheaper replacement creating higher redundancy, unpr... | Satellite formation flying | 0.777625 |
97,706 | In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution (Pareto Type II). It is one example of a Tsallis distribution. | Q-Weibull distribution | 0.777625 |
97,707 | In physics, the Moses effect is a phenomenon of deformation of the surface of a diamagnetic liquid by a magnetic field. The effect was named after the biblical figure Moses, inspired by the mythological crossing of the Red Sea in the Old Testament.The rapid progress in the development of neodymium magnets, supplying ma... | Moses Effect | 0.777619 |
97,708 | The atomic number of tin The fifth magic number in nuclear physics | 50 (number) | 0.777617 |
97,709 | One of the most important applications of LOFAR will be to carry out large-sky surveys. Such surveys are well suited to the characteristics of LOFAR and have been designated as one of the key projects that have driven LOFAR since its inception. Such deep LOFAR surveys of the accessible sky at several frequencies will p... | Low-Frequency Array (LOFAR) | 0.777617 |
97,710 | In type theory, a polynomial functor (or container functor) is a kind of endofunctor of a category of types that is intimately related to the concept of inductive and coinductive types. Specifically, all W-types (resp. M-types) are (isomorphic to) initial algebras (resp. final coalgebras) of such functors. Polynomial f... | Polynomial functor (type theory) | 0.777616 |
97,711 | Kumoh National Institute of Technology was founded on December 31, 1979, by Park Chung-hee, the former President of South Korea. In 1987, a graduate school was established for both master and Ph.D. courses such as mechanical engineering, electronic engineering and civil engineering. KIT turned into a national institute... | Kumoh National Institute of Technology | 0.777616 |
97,712 | All jobs at each node belong to a single "class" and jobs follow the same service-time distribution and the same routing mechanism. Consequently, there is no notion of priority in serving the jobs: all jobs at each node are served on a first-come, first-served basis. Jackson networks where a finite population of jobs t... | Jackson's theorem (queueing theory) | 0.777614 |
97,713 | In queueing theory, a discipline within the mathematical theory of probability, a Jackson network (sometimes Jacksonian network) is a class of queueing network where the equilibrium distribution is particularly simple to compute as the network has a product-form solution. It was the first significant development in the... | Jackson's theorem (queueing theory) | 0.777614 |
97,714 | The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rn and E and F are trivial bundles. At this point, it relies primarily on two lemmas: Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x an... | Peetre theorem | 0.777613 |
97,715 | Let M be a smooth manifold and let E and F be two vector bundles on M. Let Γ ∞ ( E ) , and Γ ∞ ( F ) {\displaystyle \Gamma ^{\infty }(E),\ {\hbox{and}}\ \Gamma ^{\infty }(F)} be the spaces of smooth sections of E and F. An operator D: Γ ∞ ( E ) → Γ ∞ ( F ) {\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\inft... | Peetre theorem | 0.777613 |
97,716 | In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite... | Peetre theorem | 0.777613 |
97,717 | Rescale s by a factor of 2C/g. Then if g is non-zero, by the linearity of D, |Ds(y)|=2C>C, which is impossible by Lemma 1. This proves the theorem in the punctured neighborhood V\{x}.Now, we must continue the differential operator to the central point x in the punctured neighborhood. | Peetre theorem | 0.777613 |
97,718 | Let M be a compact smooth manifold (possibly with boundary), and E and F be finite dimensional vector bundles on M. Let Γ ∞ ( E ) {\displaystyle \Gamma ^{\infty }(E)} be the collection of smooth sections of E. An operator D: Γ ∞ ( E ) → Γ ∞ ( F ) {\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\infty }(F)} is... | Peetre theorem | 0.777613 |
97,719 | Consider the following operator: ( L f ) ( x 0 ) = lim r → 0 2 d r 2 1 | S r | ∫ S r ( f ( x ) − f ( x 0 ) ) d x {\displaystyle (Lf)(x_{0})=\lim _{r\to 0}{\frac {2d}{r^{2}}}{\frac {1}{|S_{r}|}}\int _{S_{r}}(f(x)-f(x_{0}))dx} where f ∈ C ∞ ( R d ) {\displaystyle f\in C^{\infty }(\mathbb {R} ^{d})} and S r {\displaystyle... | Peetre theorem | 0.777613 |
97,720 | The reaction takes its name from its inventor, Eric Jacobsen, with Tsutomu Katsuki sometimes being included. Chiral-directing catalysts are useful to organic chemists trying to control the stereochemistry of biologically active compounds and develop enantiopure drugs. Several improved procedures have been developed.A g... | Jacobsen epoxidation | 0.777611 |
97,721 | Israel Kleiner is a Canadian mathematician and historian of mathematics. Kleiner earned an MA at Yale University (1963) and a PhD at McGill University (1967) under Joachim Lambek with a thesis Lie modules and rings of quotients. Before his retirement as professor emeritus, he spent his career as a mathematics professor... | Israel Kleiner (mathematician) | 0.77761 |
97,722 | Modern research suggests that this policy was misguided, and that a certain level of wildfires is a natural and important part of forest ecology. Side effects of climate engineering to counter global warming could involve even further warming as a consequence of reflectivity-reducing afforestation or crop yield reducti... | Unintended consequence | 0.77761 |
97,723 | In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta fu... | Zeta function | 0.777609 |
97,724 | Saliency estimation may be viewed as an instance of image segmentation. In computer vision, image segmentation is the process of partitioning a digital image into multiple segments (sets of pixels, also known as superpixels). The goal of segmentation is to simplify and/or change the representation of an image into some... | Saliency map | 0.777608 |
97,725 | These algorithms generate a set of bounding boxes of where an object may lie in an image.In addition to classic approaches, neural-network-based are also popular. There are examples of neural networks for motion saliency estimation: TASED-Net: It consists of two building blocks. First, the encoder network extracts low-... | Saliency map | 0.777608 |
97,726 | There are three forms of classic saliency estimation algorithms implemented in OpenCV: Static saliency: Relies on image features and statistics to localize the regions of interest of an image. Motion saliency: Relies on motion in a video, detected by optical flow. Objects that move are considered salient. Objectness: O... | Saliency map | 0.777608 |
97,727 | In computer vision, a saliency map is an image that highlights the region on which people's eyes focus first. The goal of a saliency map is to reflect the degree of importance of a pixel to the human visual system. For example, in this image, a person first looks at the fort and light clouds, so they should be highligh... | Saliency map | 0.777608 |
97,728 | In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral χ 2 {\displaystyle \chi ^{2}} distribution) is a noncentral generalization of the chi-squared distribution. It often arises in the power analysis of statistical tests in which the null distri... | Non-central chi-squared distribution | 0.777606 |
97,729 | In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to... | Dyakis dodecahedron | 0.777606 |
97,730 | The deltoidal icositetrahedron is a crystal habit often formed by the mineral analcime and occasionally garnet. The shape is often called a trapezohedron in mineral contexts, although in solid geometry the name trapezohedron has another meaning. In Guardians of The Galaxy Vol. 3, the device containing the files about t... | Dyakis dodecahedron | 0.777606 |
97,731 | In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K 5 {\displaystyle K_{5}} (the complete graph on five vertices) or o... | Kuratowski's theorem | 0.777603 |
97,732 | Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph G {\displaystyle G} has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of G {\displaystyle G} itself. Therefore, a graph that contains a Kuratowski subgrap... | Kuratowski's theorem | 0.777603 |
97,733 | If G {\displaystyle G} is a graph that contains a subgraph H {\displaystyle H} that is a subdivision of K 5 {\displaystyle K_{5}} or K 3 , 3 {\displaystyle K_{3,3}} , then H {\displaystyle H} is known as a Kuratowski subgraph of G {\displaystyle G} . With this notation, Kuratowski's theorem can be expressed succinctly:... | Kuratowski's theorem | 0.777603 |
97,734 | A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are often drawn with straig... | Kuratowski's theorem | 0.777603 |
97,735 | A closely related result, Wagner's theorem, characterizes the planar graphs by their minors in terms of the same two forbidden graphs K 5 {\displaystyle K_{5}} and K 3 , 3 {\displaystyle K_{3,3}} . Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not diff... | Kuratowski's theorem | 0.777603 |
97,736 | Kazimierz Kuratowski published his theorem in 1930. The theorem was independently proved by Orrin Frink and Paul Smith, also in 1930, but their proof was never published. The special case of cubic planar graphs (for which the only minimal forbidden subgraph is K 3 , 3 {\displaystyle K_{3,3}} ) was also independently pr... | Kuratowski's theorem | 0.777603 |
97,737 | The +4 oxidation state dominates titanium chemistry, but compounds in the +3 oxidation state are also numerous. Commonly, titanium adopts an octahedral coordination geometry in its complexes, but tetrahedral TiCl4 is a notable exception. Because of its high oxidation state, titanium(IV) compounds exhibit a high degree ... | Titanium metallurgy | 0.777601 |
97,738 | The Directorate General for Civil Aviation (French: Direction générale de l'aviation civile, DGAC) is the French civil aviation authority. Its headquarters are in the 15th arrondissement of Paris, 50 Henry-Farman. It is subordinate to the Ministry of Ecology, Sustainable Development and Energy. The DGAC levies a civil ... | Centre d'études de la navigation aérienne | 0.777601 |
97,739 | Since a six-member cyclic transition state is involved in metallo-ene reaction, high level of stereoselectivity can be expected due to the conservation of orbital symmetry. Indeed, this happens to be the case in many synthetic applications of this reaction. Felkin et al. have found the cis product to be formed as the p... | Metallo-ene reaction | 0.777597 |
97,740 | These features can be detected by multibeam sonar and then sampled by navigated piston coring. Spec and proprietary multibeam seep mapping and core geochemistry by Texas A&M University's Geochemical & Environmental Research Group8 and later TDI Brooks9 demonstrated thermogenic charge in deepwater Angola and deepwater N... | SeaSeep | 0.777597 |
97,741 | The Survey noted several successes against crucial industrial areas: "The Attack on Oil": This section of the USSBS presents the statistics for the Oil Plan portion of the bombing campaign against petroleum, oil, and lubrication (POL) products, particularly regarding the Leuna complex that produced a notable portion of... | Strategic Bombing Survey | 0.777595 |
97,742 | The use of statistics has become so widespread in the social sciences that many universities such as Harvard, have developed institutes focusing on "quantitative social science." Harvard's Institute for Quantitative Social Science focuses mainly on fields like political science that incorporate the advanced causal stat... | Social statistician | 0.777594 |
97,743 | Social scientists use social statistics for many purposes, including: the evaluation of the quality of services available to a group or organization, analyzing behaviors of groups of people in their environment and special situations, determining the wants of people through statistical sampling evaluation of wage expen... | Social statistician | 0.777594 |
97,744 | It was a confederation of local societies modeled after the British Astronomical Association with its own publication, Amateur Astronomy (AA). Initial members were the MAS, MSIO, and the Madison Astronomical Society. The individual clubs consolidated their own newsletters into AA and traded information on scientific ob... | Milwaukee Astronomical Society | 0.777594 |
97,745 | The Milwaukee Astronomical Society (MAS) is a 501(c)(3) tax-exempt organization serving amateur astronomers in the greater Milwaukee area since 1932, making it one of the oldest continuously running astronomy clubs in the nation. The MAS operates an observatory, one of the largest amateur club observatories in the worl... | Milwaukee Astronomical Society | 0.777594 |
97,746 | Membership in the MAS is open to anyone interested in astronomy without any age restrictions. Membership types are Family, Individual, and Student (under the age of 18). | Milwaukee Astronomical Society | 0.777594 |
97,747 | On the other hand, the secondaries follow a different trend below ≈30 eV: the lower the energy, the longer the mean free path though they are not then able to affect resist development.As they decay, primary photo-electrons and Auger electrons eventually become physically indistinguishable (as in Fermi–Dirac statistics... | X-ray lithography | 0.777591 |
97,748 | Many of the most well-characterized 3FTx proteins exert their toxic effects through binding to nicotinic acetylcholine receptors (nAChRs), a family of ligand-gated ion channels. 3FTx binding interferes with cholinergic intercellular signaling particularly at neuromuscular junctions and causes paralysis. The alpha-neuro... | Three-finger toxin | 0.777588 |
97,749 | Bioinformatics-based surveys of known protein sequences have often identified a number of sequences likely to form a 3FTx protein structure but whose function has not been experimentally characterized. Thus, it is not known whether these "orphan" proteins are in fact toxins or what their cellular targets might be. Geno... | Three-finger toxin | 0.777588 |
97,750 | Thurston never published a complete proof of his theorem for reasons that he explained in (Thurston 1994), though parts of his argument are contained in Thurston (1986, 1998a, 1998b). Wall (1984) and Morgan (1984) gave summaries of Thurston's proof. Otal (1996) gave a proof in the case of manifolds that fiber over the ... | Hyperbolization theorem | 0.777587 |
97,751 | In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. | Hyperbolization theorem | 0.777587 |
97,752 | Otal (1998) and Kapovich (2009) gave proofs of Thurston's theorem for the generic case of manifolds that do not fiber over the circle. The idea of the proof is to cut a Haken manifold M along an incompressible surface, to obtain a new manifold N. By induction one assumes that the interior of N has a hyperbolic structur... | Hyperbolization theorem | 0.777587 |
97,753 | Thurston's original argument for this case was summarized by Sullivan (1981). Otal (1996) gave a proof in the case of manifolds that fiber over the circle. Thurston's geometrization theorem in this special case states that if M is a 3-manifold that fibers over the circle and whose monodromy is a pseudo-Anosov diffeomor... | Hyperbolization theorem | 0.777587 |
97,754 | One form of Thurston's geometrization theorem states: If M is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume. The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyp... | Hyperbolization theorem | 0.777587 |
97,755 | The Mirandola Biomedical District (Italian: Distretto biomedicale di Mirandola) is an industrial district in the biomedical engineering, specialised in the production of medical devices such as equipment and disposable products for therapeutic applications, located in the Mirandola area and in some neighbouring municip... | Mirandola Biomedical District | 0.777587 |
97,756 | In May 2012, a series of earthquakes devastated, along with other towns in Emilia, the northern area of the province of Modena, especially Mirandola and the neighbouring municipalities. The biomedical district suffered very serious damage from 20 May onwards, aggravated by the devastating tremor of magnitude 5.8 on 29 ... | Mirandola Biomedical District | 0.777586 |
97,757 | Let Λ be the lattice in C generated by {1, τ}. It is easy to see that all of the elements of Q(τ) which fix Λ under multiplication form a ring with units, called an order. The other lattices with generators {1, τ′}, associated in like manner to the same order define the algebraic conjugates j(τ′) of j(τ) over Q(τ). Ord... | Klein J-invariant | 0.777585 |
97,758 | The j-invariant has many remarkable properties: If τ is any point of the upper half plane whose corresponding elliptic curve has complex multiplication (that is, if τ is any element of an imaginary quadratic field with positive imaginary part, so that j is defined), then j(τ) is an algebraic integer. These special valu... | Klein J-invariant | 0.777585 |
97,759 | 41 (2nd ed. ), doi:10.1007/978-1-4612-0999-7, ISBN 978-0-387-97127-8, MR 1027834 Berndt, Bruce C.; Chan, Heng Huat (1999), "Ramanujan and the modular j-invariant", Canadian Mathematical Bulletin, 42 (4): 427–440, doi:10.4153/CMB-1999-050-1, MR 1727340. Provides a variety of interesting algebraic identities, including t... | Klein J-invariant | 0.777585 |
97,760 | Apostol, Tom M. (1990), Modular functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, vol. | Klein J-invariant | 0.777585 |
97,761 | Apostol, Tom M. (1976), Modular functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, vol. 41, New York: Springer-Verlag, MR 0422157. Provides a very readable introduction and various interesting identities. | Klein J-invariant | 0.777585 |
97,762 | More remarkably, the Fourier coefficients for the positive exponents of q are the dimensions of the graded part of an infinite-dimensional graded algebra representation of the monster group called the moonshine module – specifically, the coefficient of qn is the dimension of grade-n part of the moonshine module, the fi... | Klein J-invariant | 0.777585 |
97,763 | The vehicle was demonstrated live to their online audience on Facebook Live at a media event in National Harbor. On January 2, 2018, Local Motors received a pledge of up to a $1 billion in financing and operational support to customers of Olli from Florida-based Elite Transportation Services (ETS) with additional fundi... | Local Motors | 0.777584 |
97,764 | Local Motors was an American motor vehicle manufacturing company focused on low-volume manufacturing of open-source motor vehicle designs using multiple microfactories. It was founded in 2007 by John B. Rogers Jr and had headquarters in Phoenix, Arizona. The company's lineup before its closure included the Rally Fighte... | Local Motors | 0.777583 |
97,765 | Small changes in DNA in individuals are what leads to the uniqueness of every person as well as differences in looks, abilities, brain functioning, and all the factors that culminate to develop a cohesive personality.Cattell and Eysenck have proposed that genetics have a powerful influence on personality. A large part ... | Personality psychology | 0.777582 |
97,766 | Ever since the Human Genome Project allowed for a much more in depth comprehension of genetics, there has been an ongoing controversy involving heritability, personality traits, and environmental vs. genetic influence on personality. The human genome is known to play a role in the development of personality. Previously... | Personality psychology | 0.777582 |
97,767 | Biology plays a very important role in the development of personality. The study of the biological level in personality psychology focuses primarily on identifying the role of genetic determinants and how they mold individual personalities. Some of the earliest thinking about possible biological bases of personality gr... | Personality psychology | 0.777582 |
97,768 | Psychoanalytic theories explain human behavior in terms of the interaction of various components of personality. Sigmund Freud was the founder of this school of thought. He drew on the physics of his day (thermodynamics) to coin the term psychodynamics. Based on the idea of converting heat into mechanical energy, Freud... | Personality psychology | 0.777582 |
97,769 | The major theories include dispositional (trait) perspective, psychodynamic, humanistic, biological, behaviorist, evolutionary, and social learning perspective. Many researchers and psychologists do not explicitly identify themselves with a certain perspective and instead take an eclectic approach. Research in this are... | Personality psychology | 0.777582 |
97,770 | Heredity (nature) versus environment (nurture) – Personality is thought to be determined largely either by genetics and biology, or by environment and experiences. Contemporary research suggests that most personality traits are based on the joint influence of genetics and environment. One of the forerunners in this are... | Personality psychology | 0.777582 |
97,771 | From the combination of these two types of research, Terkel was able to analyze the mechanisms involved in this social learning to determine that this eating behavior resulted from a combination of ecology and cultural transmission, as the rats could not figure out how to eat the pinecones without being "shown" by matu... | Cultural transmission in animals | 0.777582 |
97,772 | The farther two populations of a species are separated from each other, the less genetic traits they will share in common, and this may be one source of variance in culture.Another argument against the "ethnographic method" is that it is impossible to prove that there are absolutely no ecological or genetic factors in ... | Cultural transmission in animals | 0.777582 |
97,773 | In other words, when an animal changes its behavior over its lifespan, this is most often a result of changes in its environment. Furthermore, animal behavior is also influenced by evolved predispositions, or genetics. It is very possible that "correlation between distance between sites and 'cultural difference' might ... | Cultural transmission in animals | 0.777582 |
97,774 | A popular method of approaching the study of animal culture (and its transmission) is the "ethnographic method," which argues that culture causes the geographical differences in the behavioral repertoires of large-brained mammals. Some researchers argue this downplays the roles that ecology and genetics play in influen... | Cultural transmission in animals | 0.777582 |
97,775 | For many purposes, it is only necessary to know that an expansion for Z {\displaystyle Z} in terms of iterated commutators of X {\displaystyle X} and Y {\displaystyle Y} exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorph... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,776 | Thus, if one wants Z to be an actual element of the Lie algebra containing X and Y (as opposed to a formal power series), one has to assume that X and Y are small. Thus, the conclusion that the product operation on a Lie group is determined by the Lie algebra is only a local statement. Indeed, the result cannot be glob... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,777 | Suppose X {\displaystyle X} and Y {\displaystyle Y} are the following matrices in the Lie algebra s l ( 2 ; C ) {\displaystyle {\mathfrak {sl}}(2;\mathbb {C} )} (the space of 2 × 2 {\displaystyle 2\times 2} matrices with trace zero): Then It is then not hard to show that there does not exist a matrix Z {\displaystyle Z... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,778 | There are numerous other expressions for Z {\displaystyle Z} , many of which are used in the physics literature. A popular integral formula is involving the generating function for the Bernoulli numbers, utilized by Poincaré and Hausdorff. | Baker-Campbell-Hausdorff formula | 0.777581 |
97,779 | Now assume that the commutator is a multiple of Y {\displaystyle Y} , so that = s Y {\displaystyle =sY} . Then all iterated commutators will be multiples of Y {\displaystyle Y} , and no quadratic or higher terms in Y {\displaystyle Y} appear. Thus, the O ( Y 2 ) {\displaystyle O\left(Y^{2}\right)} term above vanishes ... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,780 | This is the degenerate case used routinely in quantum mechanics, as illustrated below. In this case, there are no smallness restrictions on X {\displaystyle X} and Y {\displaystyle Y} . This result is behind the "exponentiated commutation relations" that enter into the Stone–von Neumann theorem. | Baker-Campbell-Hausdorff formula | 0.777581 |
97,781 | If X {\displaystyle X} and Y {\displaystyle Y} commute, that is = 0 {\displaystyle =0} , the Baker–Campbell–Hausdorff formula reduces to e X e Y = e X + Y {\displaystyle e^{X}e^{Y}=e^{X+Y}} . Another case assumes that {\displaystyle } commutes with both X {\displaystyle X} and Y {\displaystyle Y} , as for the nilpote... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,782 | A special case of the Baker–Campbell–Hausdorff formula is useful in quantum mechanics and especially quantum optics, where X and Y are Hilbert space operators, generating the Heisenberg Lie algebra. Specifically, the position and momentum operators in quantum mechanics, usually denoted X {\displaystyle X} and P {\displ... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,783 | Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {\displaystyle G} —writt... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,784 | In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {\displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic terms, that i... | Baker-Campbell-Hausdorff formula | 0.777581 |
97,785 | In 2003, Rupert Soar secured funding and formed the freeform construction group at Loughborough University, UK, to explore the potential for up-scaling existing 3D printing techniques for construction applications. In 2005, the group secured funding to build a large-scale construction 3D printing machine using 'off the... | Building printing | 0.777581 |
97,786 | In a matter of months, many new companies began to emerge. This led to many new endeavors that reached the media, such as, in 2017, the first pedestrian 3d printed bridge and the first cyclist 3d printed bridge, plus an early structural element made with 3d printing in 2016, among many others. Recently, COBOD Internati... | Building printing | 0.777581 |
97,787 | 600 elements of the walls were printed in the shop and assembled at the construction site. After completing the roof structure and interior decoration, the company presented a fully finished 3D building in October 2017.Dutch and Chinese demonstration projects are slowly constructing 3D-printed buildings in China, Dubai... | Building printing | 0.777581 |
97,788 | Robotic bricklaying was conceptualized and explored in the 1950s and related technology development around automated construction began in the 1960s, with pumped concrete and isocyanate foams. Development of automated fabrication of entire buildings using slip forming techniques and robotic assembly of components, akin... | Building printing | 0.777581 |
97,789 | Large-scale, cement-based 3D printing disposes the need for conventional molding by precisely placing, or solidifying, specific volumes of material in sequential layers by a computer controlled positioning process. This 3D printing approach consist of three general stages: data preparation, concrete preparation and com... | Building printing | 0.777581 |
97,790 | The 3D Print Canal House was a construction project.The first residential building in Europe and the CIS, constructed using the 3D printing construction technology, was the home in Yaroslavl (Russia) with the area of 298.5 sq. meters. The walls of the building were printed by the company SPECAVIA in December 2015. | Building printing | 0.777581 |
97,791 | Regardless of the system used for printing (gantry crane or robotic arm), the coordination between the nozzle travel speed and the material flow rate is crucial to the outcome of the printed filament. In some cases, multiple 3D printing robotic arms can be programmed to run simultaneously resulting in decreased constru... | Building printing | 0.777581 |
97,792 | General specifications of wet concrete 3D printing are categorized into four main characteristics: Pumpability: The ease and reliability with which material is moved through the delivery system Printability: The ease and reliability of depositing material through a deposition device Buildability: The resistance of a de... | Building printing | 0.777581 |
97,793 | In this method, each layer consists of a contour line and a filling pattern which can be implemented as honeycomb structures or space-filling curves. Another method is the tangential continuity method which produces 3-dimensional building paths with locally varying thicknesses. This method results in creating constant ... | Building printing | 0.777581 |
97,794 | There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these freque... | Chaotic attractor | 0.77758 |
97,795 | Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global ... | Chaotic attractor | 0.77758 |
97,796 | The Almagest also contains a star catalogue, which is a version of a catalogue created by Hipparchus. Its list of forty-eight constellations is ancestral to the modern system of constellations but, unlike the modern system, they did not cover the whole sky (only what could be seen with the naked eye in the northern hem... | Ptolemy | 0.77758 |
97,797 | Ptolemy's Mathēmatikē Syntaxis (Koine Greek: Μαθηματικὴ Σύνταξις, lit. "Mathematical Systematic Treatise"), better known as the Almagest, is the only surviving comprehensive ancient treatise on astronomy. Although Babylonian astronomers had developed arithmetical techniques for calculating and predicting astronomical p... | Ptolemy | 0.77758 |
97,798 | It was first translated from Arabic into Latin by Plato of Tivoli (Tiburtinus) in 1138, while he was in Spain.Much of the content of the Tetrabiblos was collected from earlier sources; Ptolemy's achievement was to order his material in a systematic way, showing how the subject could, in his view, be rationalized. It is... | Ptolemy | 0.77758 |
97,799 | Although the inscription has not survived, someone in the sixth century transcribed it and manuscript copies preserved it through the Middle Ages. It begins: "To the saviour god, Claudius Ptolemy (dedicates) the first principles and models of astronomy", following by a catalogue of numbers that define a system of celes... | Ptolemy | 0.77758 |
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