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95,100 | Harmonic superspace is given by the product of ordinary extended superspace (with 8 real fermionic coordinatates) with S2 with the nontrivial U(1)R bundle over it. The product is somewhat twisted in that the fermionic coordinates are also charged under U(1)R. This charge is given by θ ± α = u i ± θ i α {\displaystyle \... | Harmonic superspace | 0.778858 |
95,101 | ( u + i ) ∗ = u i − {\displaystyle \left(u^{+i}\right)^{*}=u_{i}^{-}} .The redundancy in the coordinates is given by u + i u i − = 1 {\displaystyle u^{+i}u_{i}^{-}=1} .Everything can be interpreted in terms of algebraic geometry. The projection is given by the "gauge transformation" u ± i → e ± i ϕ u ± i {\displaystyle... | Harmonic superspace | 0.778858 |
95,102 | {\displaystyle G=G_{1}\supseteq G_{2}\supseteq \cdots .} In particular, the quotient group Gk/Gk+1 is a finitely generated abelian group. The Bass–Guivarc'h formula states that the order of polynomial growth of G is d ( G ) = ∑ k ≥ 1 k rank ( G k / G k + 1 ) {\displaystyle d(G)=\sum _{k\geq 1}k\operatorname {rank} (G... | Bass's theorem | 0.778857 |
95,103 | There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolf showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h and independently Hyman Bass (with different proofs) computed the exact order of polynomial growt... | Bass's theorem | 0.778857 |
95,104 | The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length (relative to a symmetric generating set) at most n is bounded above by a polynomial function p(n). The order of growth is then the least degre... | Bass's theorem | 0.778857 |
95,105 | Beyond Gromov's theorem one can ask whether there exists a gap in the growth spectrum for finitely generated group just above polynomial growth, separating virtually nilpotent groups from others. Formally, this means that there would exist a function f: N → N {\displaystyle f:\mathbb {N} \to \mathbb {N} } such that a f... | Bass's theorem | 0.778857 |
95,106 | Later, Terence Tao and Yehuda Shalom modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds. Gromov's theorem also follows from the classification of approximate groups obtained by Breuillard, Green and Tao. A simple and concise proof based on functiona... | Bass's theorem | 0.778857 |
95,107 | In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called the Gromov–Hausdorff convergence, is currently widely used in geometry. A relatively simple proof of the theorem was found by Bruce Kleiner. | Bass's theorem | 0.778857 |
95,108 | In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index. | Bass's theorem | 0.778857 |
95,109 | Submarine commanders were still expected to be able to calculate a firing solution by hand as a backup against mechanical failure, and because many submarines existed at the start of the war were not equipped with a TDC; most could keep the "picture" in their heads and do much of the calculations (simple trigonometry) ... | Torpedo | 0.778856 |
95,110 | In quantum mechanics, the u-bit or ubit is a proposed theoretical entity which arises in attempts to reformulate wave functions using only real numbers instead of the complex numbers conventionally used. | U-bit | 0.778856 |
95,111 | In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells. A geometric honeycomb is a space-filling of p... | Cubic-square tiling honeycomb | 0.778855 |
95,112 | Kolff gave a set of blueprints for his hemodialysis machine to George Thorn at the Peter Bent Brigham Hospital in Boston. This led to the manufacture of the next generation of Kolff's dialyzer, a stainless steel Kolff-Brigham dialysis machine. According to McKellar (1999), a significant contribution to renal therapies ... | Hemodialysis | 0.778855 |
95,113 | However, they are in general more hydrophobic which leads to increased adsorption of proteins to the membrane surface which in turn can lead to complement system activation. Synthetic membranes can be made in either low- or high-flux configuration, but most are high-flux. Nanotechnology is being used in some of the mos... | Hemodialysis | 0.778854 |
95,114 | In mice, it is thought that the lymphatic components form through an angiogenic process. But, there is evidence from bird embryos that gives rise to the idea that lymphatic vessels arise in the embryos through a vasculogenesis-like process from the lymphangioblastic endothelial precursor cells.The development of the ly... | Lymph sac | 0.778854 |
95,115 | A software map represents static, dynamic, and evolutionary information of software systems and their software development processes by means of 2D or 3D map-oriented information visualization. It constitutes a fundamental concept and tool in software visualization, software analytics, and software diagnosis. Its prima... | Software map | 0.778854 |
95,116 | Software maps methods and techniques belong to the scientific discipline of software visualization and information visualization. They form a key concept and technique within the fields of software diagnosis. They also have applications in software mining and software analytics. Software maps have been extensively deve... | Software map | 0.778854 |
95,117 | Software maps can be used, in particular, as analysis and presentation tool of business intelligence systems, specialized in the analysis of software related data. Furthermore, software maps "serve as recommendation systems for software engineering".Software maps are not limited by software-related information: They ca... | Software map | 0.778854 |
95,118 | Software maps are applied in the context of software engineering: Complex, long-term software development projects are commonly faced by manifold difficulties such as the friction between completing system features and, at the same time, obtaining a high degree of code quality and software quality to ensure software ma... | Software map | 0.778854 |
95,119 | The Journal of Biomedical Nanotechnology is a monthly peer-reviewed scientific journal covering fundamental and applied research pertaining to nanotechnology applications in all fields of the life sciences. It was established in 2005 and is published by American Scientific Publishers, a company identified by Jeffrey Be... | Journal of Biomedical Nanotechnology | 0.778852 |
95,120 | To detect NPPD a 0.1% methanol solution of naphthoresorcinol is added to a swab of the item in question and then concentrated hydrochloric acid is added. The observation of a red color indicates a positive test. Napthoresorcinol was frequently used in early organic chemistry investigations to determine the presence of ... | C11H9NO3 | 0.778851 |
95,121 | Biofuels have a limited ability to replace fossil fuels and should not be regarded as a 'silver bullet' to deal with transport emissions. Biofuels on their own cannot deliver a sustainable transport system and so must be developed as part of an integrated approach, which promotes other renewable energy options and ener... | Sustainable biofuel | 0.778851 |
95,122 | Pongamia pinnata is a legume native to Australia, India, Florida (USA) and most tropical regions, and is now being invested in as an alternative to Jatropha for areas such as Northern Australia, where Jatropha is classed as a noxious weed. Commonly known as simply 'Pongamia', this tree is currently being commercialised... | Sustainable biofuel | 0.778851 |
95,123 | The Brazilian agency Embrapa estimates that there is enough agricultural land available to increase at least 30 times the existing sugarcane plantation without endangering sensible ecosystems or taking land destined for food crops. Most future growth is expected to take place on abandoned pasture lands, as it has been ... | Sustainable biofuel | 0.778851 |
95,124 | Polar Research is a biannual peer-reviewed scientific journal covering natural and social scientific research on the polar regions. It is published by the Norwegian Polar Institute. It covers a wide range of fields from biology to oceanography, including socio-economic and management topics. According to the Journal Ci... | Polar Research | 0.77885 |
95,125 | Research from a wide range of disciplines including psychology, cognitive science, neuroscience, and economics, suggest that humans have limited cognitive resources that can be used at any given time, when resources are allocated to one task, the resources available for other tasks will be limited. Given that attention... | Attention economy | 0.77885 |
95,126 | In physics, a free surface flow is the surface of a fluid flowing that is subjected to both zero perpendicular normal stress and parallel shear stress. This can be the boundary between two homogeneous fluids, like water in an open container and the air in the Earth's atmosphere that form a boundary at the open face of ... | Computational methods for free surface flow | 0.778847 |
95,127 | Membranes are pliant and low mass, which can strengthen coupling with the wave's energy. Their pliancy allows large changes in the geometry of the working surface, which can be used to tune the converter for specific wave conditions and to protect it from excessive loads in extreme conditions. A submerged converter may... | Wave energy | 0.778846 |
95,128 | A wave farm (wave power farm or wave energy park) is a group of colocated wave energy devices. The devices interact hydrodynamically and electrically, according to the number of machines, spacing and layout, wave climate, coastal and benthic geometry, and control strategies. The design process is a multi-optimization p... | Wave energy | 0.778845 |
95,129 | The Borden Base Line is a historic survey line (7.42 miles, 39,009.73 feet (11,890.17 m) long) running north/south through Hatfield and South Deerfield, Massachusetts. It was completed in 1831. It was designated a National Historic Civil Engineering Landmark by the American Society of Civil Engineers in 1981. The basel... | Borden Base Line | 0.778844 |
95,130 | International cooperation in astronomy started with a first meeting of 15 European astronomers in Gotha in 1798. The society was founded two years in Lilienthal, near Bremen, to organize the compilation of ecliptic star charts, with more than half of the members originated from non-German speaking countries. In the sam... | Vereinigte Astronomische Gesellschaft | 0.778844 |
95,131 | Silicene is a two-dimensional allotrope of silicon, with a hexagonal honeycomb structure similar to that of graphene. Contrary to graphene, silicene is not flat, but has a periodically buckled topology; the coupling between layers in silicene is much stronger than in multilayered graphene; and the oxidized form of sili... | Silicene | 0.778842 |
95,132 | Thus, allows for highly diversified economic functions over an extensive geographical area. This forms a kind of urban mosaic in which the city's spatial geometry is made up of differing nuclei that are no longer organised around a single centre, where the CBD still acts as a functional and important nucleus to the cit... | Morphology (architecture and engineering) | 0.77884 |
95,133 | Recent contemporary advances in analytic and cross platform tools such as 3d printing, virtual reality, and building information modeling make the current contemporary typology formally difficult to pinpoint into one holistic definition. Advances in the study of Architectural (formal) morphology have the potential to i... | Morphology (architecture and engineering) | 0.77884 |
95,134 | Riddles was born in 1892 and entered the Crewe Works of the London and North Western Railway as a premium apprentice in 1909, completing his apprenticeship in 1913. While attending the Mechanics Institute classes he took a course in electrical engineering, feeling there would be a future for electric traction. During t... | Robert Riddles | 0.778835 |
95,135 | Upon the creation of the Railway Executive in 1947, in preparation for the nationalisation of the railways in 1948, he was appointed Member of the Railway Executive for Mechanical and Electrical Engineering. He had two principal assistants, both of whom were also former LMS men: Roland C. Bond, Chief Officer (Locomotiv... | Robert Riddles | 0.778835 |
95,136 | Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions. It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions. However, the loop- (or string-) or membrane-like excitations are extended objects that can have frac... | Anyonic statistics | 0.778834 |
95,137 | Thus, elementary particles are either fermions, whose phase factor is − 1 {\displaystyle -1} , or bosons, whose phase factor is 1 {\displaystyle 1} . These two types have different statistical behaviour. Fermions obey Fermi–Dirac statistics, while bosons obey Bose–Einstein statistics. | Anyonic statistics | 0.778834 |
95,138 | In quantum mechanics, and some classical stochastic systems, indistinguishable particles have the property that exchanging the states of particle i with particle j (symbolically ψ i ↔ ψ j for i ≠ j {\displaystyle \psi _{i}\leftrightarrow \psi _{j}{\text{ for }}i\neq j} ) does not lead to a measurably different many-bod... | Anyonic statistics | 0.778834 |
95,139 | In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the permutation group (SN of N indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic ... | Anyonic statistics | 0.778834 |
95,140 | In the case θ = π we recover the Fermi–Dirac statistics (eiπ = −1) and in the case θ = 0 (or θ = 2π) the Bose–Einstein statistics (e2πi = 1). In between we have something different. Frank Wilczek in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any... | Anyonic statistics | 0.778834 |
95,141 | In two-dimensional systems, however, quasiparticles can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977. In the case of two particles this can be expressed as | ψ 1 ψ 2 ⟩ = e... | Anyonic statistics | 0.778834 |
95,142 | {\displaystyle {\begin{aligned}&0!=1.\\&(n+1)!=(n+1)\cdot n!.\end{aligned}}} This definition is valid for each natural number n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function n!, starting from n = 0 and pr... | Recursive definition | 0.778833 |
95,143 | In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A rec... | Recursive definition | 0.778833 |
95,144 | His 1837 Physikalischer Atlas, probably the first thematic atlas ever published, includes several color chorochromatic maps. These included a variety of topics from both physical and human geography: watersheds, geology, agriculture, biogeography, and ethnicity. Much of the data was apparently obtained from Alexander v... | Chorochromatic map | 0.778833 |
95,145 | In model checking, a field of computer science, a difference bound matrix (DBM) is a data structure used to represent some convex polytopes called zones. This structure can be used to efficiently implement some geometrical operations over zones, such as testing emptyness, inclusion, equality, and computing the intersec... | Zone (convex polytope) | 0.778833 |
95,146 | Platoon capability might require buying new vehicles, or it may be something that can be retrofitted. Drivers would probably need a special license endorsement on account of the new skills required and the added responsibility when driving in the lead. Smart cars with artificial intelligence could automatically join an... | Automated highway systems | 0.778832 |
95,147 | With no emissions, the Solar Buses will reduce noxious gases in some of Brighton and Hove's most polluted areas and will power the 52 route between Woodingdean and Brighton on 100% renewable energy. The Solar Bus project was one of 199 different applications to the scheme, 125 of which were shortlisted. These were put ... | Solar bus | 0.778831 |
95,148 | The first Solar Bus in the UK was launched in Brighton in April 2017. Following a marathon six week effort from hundreds of local people, The Big Lemon and Brighton Energy Coop's joint Solar Bus project has won funding from the M&S Community Energy Fund to cover the roof of The Big Lemon’s bus depot in solar panels to ... | Solar bus | 0.778831 |
95,149 | The distinction between a solar-only electric bus and an electro-solar bus is fluid, as the distinction depends on the actual usage: whether the bus is recharged from solar or other power sources. Solar-only bus services involve recharging the bus from solar energy, usually from solar panel-covered bus station canopies... | Solar bus | 0.778831 |
95,150 | In principle, also trolleybuses or other non-autonomous electric buses or alternately powered buses such as fuel cell buses or dual-mode buses could be used for solar bus services, provided the origin of all or most of the energy used for proposing the bus would be solar energy. In practice, however, such systems also ... | Solar bus | 0.778831 |
95,151 | In fact, the 32 nm pitch DRAM by EUV will lengthen up to at least 9F2 cell area, where F=active area half-pitch (traditionally, it had been 6F2). With a 2-D self-aligned double-patterning active area cut, the cell area is still lower at 8.9F2.Aberrations, originating from deviations of optical surfaces from subatomic (... | Extreme ultraviolet lithography | 0.77883 |
95,152 | Largely due to the slit shape, and the presence of residual aberrations, the effectiveness of SMO varies across slit position. At each slit position, there are different aberrations and different azimuthal angles of incidence leading to different shadowing. Consequently, there could be uncorrected variations across sli... | Extreme ultraviolet lithography | 0.77883 |
95,153 | In particle physics, strangeness ("S") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a particle is defined as: where ns represents the number of strange quarks (s) and ns r... | Strangeness | 0.77883 |
95,154 | In statistics and econometrics one often assumes that an observed series of data values is the sum of a series of values generated by a deterministic linear process, depending on certain independent (explanatory) variables, and on a series of random noise values. Then regression analysis is used to infer the parameters... | White noise | 0.778829 |
95,155 | Yang was born in Shanghai, China in 1963. She entered Fudan University in 1980, before transferring to Stony Brook University in the United States in 1983, where she earned her B.A. degree. She earned her M.A. (1985) and Ph.D. (1991) in Biochemistry & Molecular Biophysics from Columbia University. | Wei Yang (biologist) | 0.778829 |
95,156 | Since 1995 she has been a senior scientist in the Laboratory of Molecular Biology at the National Institutes of Health. Her research mainly focuses on DNA mismatch repair, translesion synthesis, and V(D)J recombination. Her lab discovered that DNA synthesis and RNA degradation reactions are propelled by cation traffick... | Wei Yang (biologist) | 0.778829 |
95,157 | A proof of the theorem can be obtained with generating functions. If we note p d ( n ) {\displaystyle p_{d}(n)} the number of partitions with no parts divisible by d and q d ( n ) {\displaystyle q_{d}(n)} the number of partitions with no parts repeated more than d-1 times, then the theorem means that for all n p d ( n ... | Glaisher's theorem | 0.778828 |
95,158 | It states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part is repeated d or more times, which can be written formally as partitions of the form n = λ 1 + ⋯ + λ k {\displaystyle n=\lambda _{1}+\cdots +\la... | Glaisher's theorem | 0.778828 |
95,159 | In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer n {\displaystyle n} into parts not divisible by d {\displaystyle d} is equal to the number of partitions in which no part i... | Glaisher's theorem | 0.778828 |
95,160 | A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. | Stable fixed point | 0.778828 |
95,161 | In molecular biology, the KilA-N domain is a conserved DNA-binding domain found at the N-terminus of the poxvirus D6R/NIR proteins. It is also found in a wide range of proteins of large bacterial and eukaryotic DNA viruses. Putative proteins with homology to the KilA-N domain have also been identified in Maverick trans... | KilA-N domain | 0.778828 |
95,162 | Although based at UCL's campus in Bloomsbury, the LCN includes research in departments of Imperial's South Kensington campus and in King's Strand campus. The LCN's work requires it to draw on the combined skills of multiple departments, including medicine, chemistry, physics, electrical and electronic engineering, bioc... | London Centre for Nanotechnology | 0.778827 |
95,163 | LCN's research is organised around three themes, which it characterizes as follows: • Information Technology: Computing and communications needs continue to grow and underpin all other human endeavours. Current technologies are limited and new nanotechnology-driven paradigms such as quantum computing and spintronics ar... | London Centre for Nanotechnology | 0.778827 |
95,164 | LCN has access to a range of facilities include: • Nano-CAD: techniques to simulate, visualize and design nano-scale structures and devices in the biological and non-biological areas; first principles atomic/molecular level theory, systems modelling and other powerful computational tools. • Nano-characterisation: the f... | London Centre for Nanotechnology | 0.778827 |
95,165 | The London Centre for Nanotechnology was established as a joint venture between UCL and Imperial College London in 2003 following the award of a £13.65m higher education grant under the Science Research Infrastructure Fund. In October 2006 the LCN installed the first monochromated electron microscope in the UK at its s... | London Centre for Nanotechnology | 0.778827 |
95,166 | The London Centre for Nanotechnology is a multidisciplinary research centre in physical and biomedical nanotechnology in London, United Kingdom. It brings together three institutions that are world leaders in nanotechnology, University College London, Imperial College London and King's College London. It was conceived ... | London Centre for Nanotechnology | 0.778827 |
95,167 | Smoky is an 80 node linux cluster used for application development. Research projects are refined and tested on Smoky before running on larger machines such as Titan.In 1989 programmers at the Oak Ridge National Lab wrote the first version of Parallel Virtual Machine (PVM), software that enables distributed computing o... | Oak Ridge National Laboratory | 0.778826 |
95,168 | Throughout the history of the Oak Ridge National Laboratory it has been the site of various supercomputers, home to the fastest on several occasions. In 1953, ORNL partnered with the Argonne National Laboratory to build ORACLE (Oak Ridge Automatic Computer and Logical Engine), a computer to research nuclear physics, ch... | Oak Ridge National Laboratory | 0.778826 |
95,169 | The Fusion Energy Division pursues short-term goals to develop components such as high temperature superconductors, high-speed hydrogen pellet injectors and suitable materials for future fusion research. Much research into the behaviour and maintenance of a plasma takes place at the Fusion Energy Division to further th... | Oak Ridge National Laboratory | 0.778826 |
95,170 | The Health Physics Research Reactor built in 1962 was used for radiation exposure experiments leading to more accurate dosage limits and dosimeters, and improved radiation shielding.In 1964 the Molten-Salt Reactor Experiment began with the construction of the reactor. It was operated from 1966 until 1969 (with six mont... | Oak Ridge National Laboratory | 0.778826 |
95,171 | Oak Ridge National Laboratory's biological research covers genomics, computational biology, structural biology and bioinformatics. The BioEnergy Program aims to improve the efficiency of all stages of the biofuel process to improve the energy security of the United States. The program aims to make genetic improvements ... | Oak Ridge National Laboratory | 0.778826 |
95,172 | As is the case for all designated user facilities, the resources of the High Temperature Materials Laboratory are available for free if the results are published; private research is permitted but requires payment.The Center for Nanophase Materials Sciences (CNMS) researches the behaviour and fabrication of nanomateria... | Oak Ridge National Laboratory | 0.778826 |
95,173 | Oak Ridge National Laboratory conducts research into materials science in a range of areas. Between 2002 and 2008 ORNL partnered with Caterpillar Inc. (CAT) to form a new material for their diesel engines that can withstand large temperature fluctuations. The new steel, named CF8C Plus, is based on conventional CF8C st... | Oak Ridge National Laboratory | 0.778826 |
95,174 | Its scientific programs focus on materials, nuclear science, neutron science, energy, high-performance computing, systems biology and national security, sometimes in partnership with the state of Tennessee, universities and other industries. ORNL has several of the world's top supercomputers, including Frontier, ranked... | Oak Ridge National Laboratory | 0.778826 |
95,175 | Electron microscopy – ORNL's electron microscopy program investigates key issues in condensed matter, materials, chemical and nanosciences. Nuclear medicine – The laboratory's nuclear medicine research is focused on the development of improved reactor production and processing methods to provide medical radioisotopes, ... | Oak Ridge National Laboratory | 0.778826 |
95,176 | ORNL conducts research and development activities that span a wide range of scientific disciplines. Many research areas have a significant overlap with each other; researchers often work in two or more of the fields listed here. The laboratory's major research areas are described briefly below. Chemical sciences – ORNL... | Oak Ridge National Laboratory | 0.778826 |
95,177 | This naturally sets the low energy scale of the dark energy.In 2004, when scientists fit the evolution of dark energy with the cosmological data, they found that the equation of state had possibly crossed the cosmological constant boundary (w = −1) from above to below. A no-go theorem has been proved that this scenario... | Dark Energy | 0.778826 |
95,178 | Scalar fields are predicted by the Standard Model of particle physics and string theory, but an analogous problem to the cosmological constant problem (or the problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses. The coincidence... | Dark Energy | 0.778826 |
95,179 | In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the universe does not expand at a constant rate; rather, the universe's expansion is a... | Dark Energy | 0.778826 |
95,180 | The adjoint representation can also be defined for algebraic groups over any field.The co-adjoint representation is the contragredient representation of the adjoint representation. Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philosop... | Adjoint endomorphism | 0.778826 |
95,181 | In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle GL(n,\mathbb {R} )} , the Lie group of real n-by-n inve... | Adjoint endomorphism | 0.778826 |
95,182 | The existence of dark energy, in whatever form, is needed to reconcile the measured geometry of space with the total amount of matter in the universe. Measurements of cosmic microwave background anisotropies indicate that the universe is close to flat. For the shape of the universe to be flat, the mass–energy density o... | Dark Energy | 0.778826 |
95,183 | In 1998, the High-Z Supernova Search Team published observations of Type Ia ("one-A") supernovae. In 1999, the Supernova Cosmology Project followed by suggesting that the expansion of the universe is accelerating. The 2011 Nobel Prize in Physics was awarded to Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess for th... | Dark Energy | 0.778826 |
95,184 | The theory of large-scale structure, which governs the formation of structures in the universe (stars, quasars, galaxies and galaxy groups and clusters), also suggests that the density of matter in the universe is only 30% of the critical density. A 2011 survey, the WiggleZ galaxy survey of more than 200,000 galaxies, ... | Dark Energy | 0.778826 |
95,185 | In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value... | Residual (statistics) | 0.778824 |
95,186 | In polymer chemistry, free-radical polymerization (FRP) is a method of polymerization by which a polymer forms by the successive addition of free-radical building blocks (repeat units). Free radicals can be formed by a number of different mechanisms, usually involving separate initiator molecules. Following its generat... | Radical polymerisation | 0.778824 |
95,187 | The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties beca... | Hilbert series | 0.778822 |
95,188 | The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. | Hilbert series | 0.778822 |
95,189 | In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filt... | Hilbert series | 0.778822 |
95,190 | The additivity of Hilbert series shows that H S A / ( f ) ( t ) = ( 1 − t ) H S A ( t ) {\displaystyle HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)} . Iterating this a number of times equal to the Krull dimension of A, we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial P(t). | Hilbert series | 0.778822 |
95,191 | {\displaystyle HS_{R/\langle f\rangle }(t)=\left(1-t^{\delta }\right)HS_{R}(t).} Looking on the numerators this proves the following generalization of Bézout's theorem: Theorem - If f is a homogeneous polynomial of degree δ {\displaystyle \delta } , which is not a zero divisor in R, then the degree of the intersection ... | Hilbert series | 0.778822 |
95,192 | Thus R 0 {\displaystyle R_{0}} is an Artinian ring, which is a k-vector space of dimension P(1), and Jordan–Hölder theorem may be used for proving that P(1) is the degree of the algebraic set V. In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.... | Hilbert series | 0.778822 |
95,193 | {\displaystyle HS_{R_{0}}(t)=(1-t)HS_{R_{1}}(t)=P(t).} Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra. | Hilbert series | 0.778822 |
95,194 | Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g1, ..., gn of degree 1, then the map which sends Xi onto gi defines an homomorphism of graded rings from R n = K {\displaystyle R_{n}=K} ont... | Hilbert series | 0.778822 |
95,195 | 41). All these definitions may be extended to finitely generated graded modules over S, with the only difference that a factor tm appears in the Hilbert series, where m is the minimal degree of the generators of the module, which may be negative. The Hilbert function, the Hilbert series and the Hilbert polynomial of a ... | Hilbert series | 0.778822 |
95,196 | Consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that S = ⨁ i ≥ 0 S i {\displaystyle S=\bigoplus _{i\geq 0}S_{i}} and that S 0 = K {\displaystyle S_{0}=K} . The Hilbert function H F S: n ⟼ dim K S n {\displaystyle HF_{S}:... | Hilbert series | 0.778822 |
95,197 | A graded algebra A generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is nilpotent. This implies that the dimension of A as a K-vector space is finite and the Hilbert series of A is a polynomial P(... | Hilbert series | 0.778822 |
95,198 | {\displaystyle V_{0}.} This makes V 0 {\displaystyle V_{0}} an affine algebraic set, which has R 0 = R 1 / ⟨ h d − 1 ⟩ {\displaystyle R_{0}=R_{1}/\langle h_{d}-1\rangle } as its ring of regular functions. The linear polynomial h d − 1 {\displaystyle h_{d}-1} is not a zero divisor in R 1 , {\displaystyle R_{1},} and one... | Hilbert series | 0.778822 |
95,199 | The definition of a regular sequence implies the existence of exact sequences 0 ⟶ ( R / ⟨ h 0 , … , h k − 1 ⟩ ) ⟶ h k R / ⟨ h 1 , … , h k − 1 ⟩ ⟶ R / ⟨ h 1 , … , h k ⟩ ⟶ 0 , {\displaystyle 0\longrightarrow \left(R/\langle h_{0},\ldots ,h_{k-1}\rangle \right)^{}{\stackrel {h_{k}}{\longrightarrow }}R/\langle h_{1},\ldot... | Hilbert series | 0.778822 |
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