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6,100 | Example areas of study are radio waves, usually emitted by cold objects such as interstellar gas and dust clouds; the cosmic microwave background radiation which is the redshifted light from the Big Bang; pulsars, which were first detected at microwave frequencies. The study of these waves requires very large radio tel... | Stellar physics | 0.824908 |
6,101 | Observational astronomy is a division of the astronomical science that is concerned with recording and interpreting data, in contrast with theoretical astrophysics, which is mainly concerned with finding out the measurable implications of physical models. It is the practice of observing celestial objects by using teles... | Stellar physics | 0.824908 |
6,102 | Custom peptide synthesis is the commercial production of peptides for use in biochemistry, biology, biotechnology, pharmacology and molecular medicine. Custom peptide synthesis provides synthetic peptides as valuable tools to biomedical laboratories. Synthetic oligopeptides are used extensively in research for structur... | Custom peptide synthesis | 0.824891 |
6,103 | Large scale custom peptide synthesis can be carried out either in a liquid solution or in solid phase. In general, peptides shorter than 8 amino acids are prepared more economically by solution chemistry. Peptides larger than 8 residues are generally assembled by solid phase chemistry. Solid phase peptide synthesis (SP... | Custom peptide synthesis | 0.824891 |
6,104 | Its adduct with ammonia, which was known earlier, is very shock-sensitive: it can be set off by the touch of a feather, shifting air currents, or even alpha particles. For this reason, small amounts of nitrogen triiodide are sometimes synthesised as a demonstration to high school chemistry students or as an act of "che... | Nitrogen compounds | 0.82488 |
6,105 | The chemical element nitrogen is one of the most abundant elements in the universe and can form many compounds. It can take several oxidation states; but the most oxidation states are -3 and +3. Nitrogen can form nitride and nitrate ions. It also forms a part of nitric acid and nitrate salts. Nitrogen compounds also ha... | Nitrogen compounds | 0.82488 |
6,106 | Nitrogen is one of the most important elements in organic chemistry. Many organic functional groups involve a carbon–nitrogen bond, such as amides (RCONR2), amines (R3N), imines (RC(=NR)R), imides (RCO)2NR, azides (RN3), azo compounds (RN2R), cyanates and isocyanates (ROCN or RCNO), nitrates (RONO2), nitriles and isoni... | Nitrogen compounds | 0.82488 |
6,107 | In particular, since the B–N unit is isoelectronic to C–C, and carbon is essentially intermediate in size between boron and nitrogen, much of organic chemistry finds an echo in boron–nitrogen chemistry, such as in borazine ("inorganic benzene"). Nevertheless, the analogy is not exact due to the ease of nucleophilic att... | Nitrogen compounds | 0.82488 |
6,108 | → NaNO3 + NaOH + NaHCO3Despite its limited chemistry, the orthonitrate anion is interesting from a structural point of view due to its regular tetrahedral shape and the short N–O bond lengths, implying significant polar character to the bonding. | Nitrogen compounds | 0.82488 |
6,109 | It can only be made in the solid state, because upon melting it spontaneously decomposes to nitrogen dioxide, and liquid nitric acid undergoes self-ionisation to a larger extent than any other covalent liquid as follows: 2 HNO3 ⇌ H2NO+3 + NO−3 ⇌ H2O + + + −Two hydrates, HNO3·H2O and HNO3·3H2O, are known that can be cry... | Nitrogen compounds | 0.82488 |
6,110 | mathematical physics, mathematics pertaining to physical problems. medical physics, the application of physics in medicine to prevention, diagnosis, and treatment. physical chemistry, dealing with physical processes and their relations in the science of physical chemistry. physical oceanography, is the study of physica... | Branches of physics | 0.824876 |
6,111 | engineering physics, the combined discipline of physics and engineering. geophysics, the sciences of physical relations on our planet. | Branches of physics | 0.824876 |
6,112 | To the interdisciplinary fields, which define partially sciences of their own, belong e.g. the agrophysics is a branch of science bordering on agronomy and physics astrophysics, the physics in the universe, including the properties and interactions of celestial bodies in astronomy. space physics is the study of plasmas... | Branches of physics | 0.824876 |
6,113 | The special theory of relativity enjoys a relationship with electromagnetism and mechanics; that is, the principle of relativity and the principle of stationary action in mechanics can be used to derive Maxwell's equations, and vice versa. The theory of special relativity was proposed in 1905 by Albert Einstein in his ... | Branches of physics | 0.824876 |
6,114 | Classical mechanics is a model of the physics of forces acting upon bodies; includes sub-fields to describe the behaviors of solids, gases, and fluids. It is often referred to as "Newtonian mechanics" after Isaac Newton and his laws of motion. It also includes the classical approach as given by Hamiltonian and Lagrange... | Branches of physics | 0.824876 |
6,115 | Physics is a scientific discipline that seeks to construct and experimentally test theories of the physical universe. These theories vary in their scope and can be organized into several distinct branches, which are outlined in this article. | Branches of physics | 0.824876 |
6,116 | Particle physics studies the nature of particles, while nuclear physics studies the atomic nuclei. | Branches of physics | 0.824876 |
6,117 | Optics is the study of light motions including reflection, refraction, diffraction, and interference. Acoustics is the branch of physics involving the study of mechanical waves in different mediums. | Branches of physics | 0.824876 |
6,118 | computational physics, the application of computers and numerical methods to physical systems. econophysics, dealing with physical processes and their relations in the science of economy. environmental physics, the branch of physics concerned with the measurement and analysis of interactions between organisms and their... | Branches of physics | 0.824876 |
6,119 | The first chapter of The Feynman Lectures on Physics is about the existence of atoms, which Feynman considered to be the most compact statement of physics, from which science could easily result even if all other knowledge was lost. By modeling matter as collections of hard spheres, it is possible to describe the kinet... | Branches of physics | 0.824876 |
6,120 | Other developments include quantum statistics, quantum electrodynamics, concerned with interactions between charged particles and electromagnetic fields; and its generalization, quantum field theory. String Theory A possible candidate for the theory of everything, this theory combines the theory of general relativity a... | Branches of physics | 0.824876 |
6,121 | The matrix mechanics of Werner Heisenberg (1925) makes no mention of wave functions or similar concepts but was shown to be mathematically equivalent to Schrödinger's theory. A particularly important discovery of the quantum theory is the uncertainty principle, enunciated by Heisenberg in 1927, which places an absolute... | Branches of physics | 0.824876 |
6,122 | In 1924, Louis de Broglie proposed that not only do light waves sometimes exhibit particle-like properties, but particles may also exhibit wave-like properties. Two different formulations of quantum mechanics were presented following de Broglie's suggestion. The wave mechanics of Erwin Schrödinger (1926) involves the u... | Branches of physics | 0.824876 |
6,123 | Quantum mechanics is the branch of physics treating atomic and subatomic systems and their interaction based on the observation that all forms of energy are released in discrete units or bundles called "quanta". Remarkably, quantum theory typically permits only probable or statistical calculation of the observed featur... | Branches of physics | 0.824876 |
6,124 | Star formation is the process by which dense regions within molecular clouds in interstellar space collapse to form stars. As a branch of astronomy, star formation includes the study of the interstellar medium and giant molecular clouds (GMC) as precursors to the star formation process, and the study of protostars and ... | Solar radiation pressure | 0.824875 |
6,125 | In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism. | P-form electrodynamics | 0.824861 |
6,126 | Knowledge Systems Laboratory (KSL) was an artificial intelligence research laboratory within the Department of Computer Science at Stanford University until 2007, located in the Gates Computer Science Building, Stanford. Work focused on knowledge representation for shareable engineering knowledge bases and systems, com... | Knowledge Systems Laboratory | 0.824852 |
6,127 | The Graded Brauer group is a synonym for the Brauer–Wall group B W ( F ) {\displaystyle BW(F)} classifying finite-dimensional graded central division algebras over the field F. An A {\displaystyle {\mathcal {A}}} -graded category for a category A {\displaystyle {\mathcal {A}}} is a category C {\displaystyle {\mathcal {... | Graded (mathematics) | 0.82485 |
6,128 | A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super Z 2 {\displaystyle \mathbb {Z} _{2}} -gradation. A differential graded Lie algebra is a graded vector space over a... | Graded (mathematics) | 0.82485 |
6,129 | for homogeneous x,y, where | a | {\displaystyle |a|} represents the "parity" of a {\displaystyle a} , i.e. 0 or 1 depending on the component in which it lies. CDGA may refer to the category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space by a... | Graded (mathematics) | 0.82485 |
6,130 | A graded-commutative superalgebra satisfies the "supercommutative" law y x = ( − 1 ) | x | | y | x y . {\displaystyle yx=(-1)^{|x||y|}xy.} | Graded (mathematics) | 0.82485 |
6,131 | A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra). A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. | Graded (mathematics) | 0.82485 |
6,132 | The graded Leibniz rule for a map d: A → A {\displaystyle d\colon A\to A} on a graded algebra A {\displaystyle A} specifies that d ( a ⋅ b ) = ( d a ) ⋅ b + ( − 1 ) | a | a ⋅ ( d b ) {\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)} . A differential graded algebra, DG-algebra or DGAlgebra is a graded algebr... | Graded (mathematics) | 0.82485 |
6,133 | The associated graded module of an R {\displaystyle R} -module M {\displaystyle M} with respect to a proper ideal I {\displaystyle I} is gr I M = ⨁ n ∈ N I n M / I n + 1 M {\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displa... | Graded (mathematics) | 0.82485 |
6,134 | An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). A I {\displaystyle I} -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = ⨁ i ∈ I V i {\textstyle V... | Graded (mathematics) | 0.82485 |
6,135 | In mathematics, the term "graded" has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: An algebraic structure X {\displaystyle X} is said to be I {\displaystyle I} -graded for an index set I {\displaystyle I} if it has a gradation or grading, i.e. a decomposition into a dire... | Graded (mathematics) | 0.82485 |
6,136 | Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability, and had only a superficial resemblance to Gödel's work (van Heijenoort 1967, p. 328). | Incompleteness of formalized arithmetic | 0.824829 |
6,137 | Paul Finsler (1926) used a version of Richard's paradox to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). | Incompleteness of formalized arithmetic | 0.824829 |
6,138 | There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of F as a formula in the language of F. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons(F) from the second incompleteness theore... | Incompleteness of formalized arithmetic | 0.824829 |
6,139 | Large populations between which there is much migration, for example, tend to be little differentiated whereas small populations between which there is little migration tend to be greatly differentiated. FST is a convenient measure of this differentiation, and as a result FST and related statistics are among the most w... | Genetic similarity | 0.824824 |
6,140 | The number of bits needed for the exponent of the extended precision format follows from the requirement that the product of two double-precision numbers should not overflow when computed using the extended format. The largest possible exponent of a double-precision value is 1023 so the exponent of the largest possible... | IEEE double extended precision | 0.824821 |
6,141 | Gödel showed that the classical paradoxes of self-reference, such as "This statement is false", can be recast as self-referential formal sentences of arithmetic. Informally, the sentence employed to prove Gödel's first incompleteness theorem says "This statement is not provable." The fact that such self-reference can b... | On Formally Undecidable Propositions of Principia Mathematica and Related Systems | 0.824819 |
6,142 | The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field of mathematical logic. These appear as theorems VI and XI, respectively, in the paper. In order to prove these results, Gödel introduced a method now known as Gödel numbering. In this method... | On Formally Undecidable Propositions of Principia Mathematica and Related Systems | 0.824819 |
6,143 | "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of Monatshe... | On Formally Undecidable Propositions of Principia Mathematica and Related Systems | 0.824819 |
6,144 | The dependent variable is the event expected to change when the independent variable is manipulated.In data mining tools (for multivariate statistics and machine learning), the dependent variable is assigned a role as target variable (or in some tools as label attribute), while an independent variable may be assigned a... | Extraneous variables | 0.824814 |
6,145 | In mathematical modeling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model yi = a + bxi + ei the term yi is the ith value of the dependent variable and xi is the ith value of the independent variable. The term ei is known as the ... | Extraneous variables | 0.824814 |
6,146 | Other mechanisms for course allocation include fair random assignment, stable matching, and optimization based mechanisms. == References == | Course allocation | 0.82481 |
6,147 | The Hahn decomposition theorem states that for every measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} with a signed measure μ , {\displaystyle \mu ,} there is a partition of X {\displaystyle X} into a positive and a negative set; such a partition ( P , N ) {\displaystyle (P,N)} is unique up to μ {\displaystyle \m... | Positive and negative sets | 0.824806 |
6,148 | {\displaystyle \mu .} In the light of Radon–Nikodym theorem, if ν {\displaystyle \nu } is a σ-finite positive measure such that | μ | ≪ ν , {\displaystyle |\mu |\ll \nu ,} a set A {\displaystyle A} is a positive set for μ {\displaystyle \mu } if and only if the Radon–Nikodym derivative d μ / d ν {\displaystyle d\mu /d\... | Positive and negative sets | 0.824806 |
6,149 | Thomas Patrick Coohill, former president of the American Society for Photobiology Harold F. Blum, who explored sunlight-induced skin cancer Paul Bert, 1878 photobiology pioneer | Photobiology | 0.824805 |
6,150 | This area of Photobiology studies the reactivity of a molecule when it absorbs energy that comes from light. It also studies what happens with this energy, it could be given off as heat or fluorescence so the molecule goes back to ground state. There are 3 basic laws of photochemistry: 1) First Law of Photochemistry: T... | Photobiology | 0.824805 |
6,151 | This area of Photobiology focuses on the physical interactions of light and matter. When molecules absorb photons that matches their energy requirements they promote a valence electron from a ground state to an excited state and they become a lot more reactive. This is an extremely fast process, but very important for ... | Photobiology | 0.824805 |
6,152 | Photobiology is the scientific study of the beneficial and harmful interactions of light (technically, non-ionizing radiation) in living organisms. The field includes the study of photophysics, photochemistry, photosynthesis, photomorphogenesis, visual processing, circadian rhythms, photomovement, bioluminescence, and ... | Photobiology | 0.824805 |
6,153 | Target is Vulkan 1.1.72 in Intel ANV and AMD RADV driver. OpenGL 4.6 with spir-V is also main target. Permanent work is possible completion of Features and Optimization of drivers for older hardware like AMD R600/Evergreen, Nvidia Tesla and before, Fermi, Kepler or Intel Sandybridge, Ivybridge, Haswell or Broadwell. | Generic Buffer Management | 0.824797 |
6,154 | In the United States, although efforts to increase the number of women and African Americans in STEM fields have been ongoing, as recently as 2010 the results have been evaluated as "poor". In 2014, one report declared that "traditionally underrepresented groups remain underrepresented", while another article commented... | STEM pipeline | 0.824794 |
6,155 | In statistics, the Page test for multiple comparisons between ordered correlated variables is the counterpart of Spearman's rank correlation coefficient which summarizes the association of continuous variables. It is also known as Page's trend test or Page's L test. It is a repeated measure trend test. The Page test is... | Page's trend test | 0.824787 |
6,156 | The Concorde TSP Solver is a program for solving the travelling salesman problem. It was written by David Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook, in ANSI C, and is freely available for academic use. Concorde has been applied to problems of gene mapping, protein function prediction, vehicle routi... | Concorde TSP Solver | 0.824786 |
6,157 | Assuming the receiver clock can be related to the synchronized clocks on the satellites, the transit time can be found and used to provide the distance to each satellite. Receiver clock error is corrected by combining the data from four satellites.Such techniques vary in accuracy according to the distances over which t... | Range finding | 0.824784 |
6,158 | Laplace's equation in spherical coordinates is: Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation: The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ... | Laplace’s equation | 0.824778 |
6,159 | If a trial form is then the Cauchy–Riemann equations will be satisfied if we set This relation does not determine ψ, but only its increments: The Laplace equation for φ implies that the integrability condition for ψ is satisfied: and thus ψ may be defined by a line integral. The integrability condition and Stokes' theo... | Laplace’s equation | 0.824778 |
6,160 | The Applied Probability Trust as a long history of donating funds to institutions around the world to support the awarding of prizes. The form a prize takes is at the discretion of the host institution, however the award title often comes with small cash sum or funds to purchase books. Current prizes include: Australia... | Mathematical Spectrum | 0.82477 |
6,161 | Synthetic biology aims at engineering synthetic biological components, with the ultimate goal of assembling whole biological systems from their constituent components. The history of synthetic biology can be traced back to the 1960s, when François Jacob and Jacques Monod discovered the mathematical logic in gene regula... | Natural computing | 0.82476 |
6,162 | Through Shor's polynomial algorithm for factoring integers, and Grover's algorithm for quantum database search that has a quadratic time advantage, quantum computers were shown to potentially possess a significant benefit relative to electronic computers. Quantum cryptography is not based on the complexity of the compu... | Natural computing | 0.824759 |
6,163 | The dual aspect of natural computation is that it aims to understand nature by regarding natural phenomena as information processing. Already in the 1960s, Zuse and Fredkin suggested the idea that the entire universe is a computational (information processing) mechanism, modelled as a cellular automaton which continuou... | Natural computing | 0.824759 |
6,164 | Reviewer Ivailo Mladenov notes with approval the book's attention to example-first exposition, and despite pointing to a minor inaccuracy regarding the nationality of Sophus Lie, recommends it to both undergraduate and graduate students. Reviewer Richard Montgomory writes that the book does "an excellent job of leading... | Symmetry in Mechanics | 0.824758 |
6,165 | The book is written as a textbook for undergraduate mathematics and physics students, with many exercises, and it assumes that the students are already familiar with multivariable calculus and linear algebra, a significantly lower level of background material than other books on symplectic geometry in mechanics. It is ... | Symmetry in Mechanics | 0.824758 |
6,166 | The Kepler problem in classical mechanics is a special case of the two-body problem in which two point masses interact by Newton's law of universal gravitation (or by any central force obeying an inverse-square law). The book starts and ends with this problem, the first time in an ad hoc manner that represents the prob... | Symmetry in Mechanics | 0.824758 |
6,167 | Symmetry in Mechanics: A Gentle, Modern Introduction is an undergraduate textbook on mathematics and mathematical physics, centered on the use of symplectic geometry to solve the Kepler problem. It was written by Stephanie Singer, and published by Birkhäuser in 2001. | Symmetry in Mechanics | 0.824758 |
6,168 | In molecular biology, calmodulin binding domain (CaMBD) is a protein domain found in small-conductance calcium-activated potassium channels (SK channels). These channels are independent of voltage and gated solely by intracellular Ca2+. They are heteromeric complexes that comprise pore-forming alpha-subunits and the Ca... | Calmodulin binding domain | 0.824753 |
6,169 | They are used in synthetic biology to construct tissue-like networks of artificial cells, and to fuse synthetic membranes together to initiate biochemical reactions. They are also widely employed in genetic studies and research on chromosome structure and dynamics. In 2003 the techniques of optical tweezers were applie... | Dipole trap | 0.824753 |
6,170 | Optical traps allowed these biophysicists to observe the forces and dynamics of nanoscale motors at the single-molecule level; optical trap force-spectroscopy has since led to greater understanding of the stochastic nature of these force-generating molecules. Optical tweezers have proven useful in other areas of biolog... | Dipole trap | 0.824753 |
6,171 | In the late 1980s, Arthur Ashkin and Joseph M. Dziedzic demonstrated the first application of the technology to the biological sciences, using it to trap an individual tobacco mosaic virus and Escherichia coli bacterium. Throughout the 1990s and afterwards, researchers like Carlos Bustamante, James Spudich, and Steven ... | Dipole trap | 0.824753 |
6,172 | This research earned Chu the 1997 Nobel Prize in Physics along with Claude Cohen-Tannoudji and William D. Phillips. In an interview, Steven Chu described how Ashkin had first envisioned optical tweezing as a method for trapping atoms. Ashkin was able to trap larger particles (10 to 10,000 nanometers in diameter) but it... | Dipole trap | 0.824753 |
6,173 | The detection of optical scattering and the gradient forces on micron sized particles was first reported in 1970 by Arthur Ashkin, a scientist working at Bell Labs. Years later, Ashkin and colleagues reported the first observation of what is now commonly referred to as an optical tweezer: a tightly focused beam of ligh... | Dipole trap | 0.824753 |
6,174 | Astrophysics for People in a Hurry is a 2017 popular science book by Neil deGrasse Tyson, centering around a number of basic questions about the universe. Published on May 2, 2017, by W. W. Norton & Company, the book is a collection of Tyson's essays that appeared in Natural History magazine at various times from 1997 ... | Astrophysics for People in a Hurry | 0.824752 |
6,175 | Neil deGrasse Tyson's Astrophysics for People in a Hurry is a popular introduction to the main concepts and issues of modern astrophysics. The author explains the origin and structure of the Universe, the force of gravity, light, dark matter and dark energy, about our place in the Cosmos and how we try to understand it... | Astrophysics for People in a Hurry | 0.824752 |
6,176 | If k = R {\displaystyle k=\mathbb {R} } or C {\displaystyle \mathbb {C} } then this makes G ( k ) {\displaystyle \mathrm {G} (k)} into a Lie group. Not all Lie groups can be obtained via this procedure, for example the universal cover of SL2(R), or the quotient of the Heisenberg group by an infinite normal discrete sub... | Algebraic groups | 0.824749 |
6,177 | If the field k {\displaystyle k} is a local field (for instance the real or complex numbers, or a p-adic field) and G {\displaystyle \mathrm {G} } is a k {\displaystyle k} -group then the group G ( k ) {\displaystyle \mathrm {G} (k)} is endowed with the analytic topology coming from any embedding into a projective spac... | Algebraic groups | 0.824749 |
6,178 | The latter are classified over algebraically closed fields via their Lie algebra. The classification over arbitrary fields is more involved but still well-understood. If can be made very explicit in some cases, for example over the real or p-adic fields, and thereby over number fields via local-global principles. | Algebraic groups | 0.824749 |
6,179 | {\displaystyle _{q}!} ; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element. | Algebraic groups | 0.824749 |
6,180 | There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is n ! {\displaystyle n!} , and the number of elements of the general linear group over a finite field is (up to some factor) the q-factorial q ! | Algebraic groups | 0.824749 |
6,181 | certain semidirect products, for instance Jet groups, or some solvable groups such as that of invertible triangular matrices.Linear algebraic groups can be classified to a certain extent. Levi's theorem states that every such is (essentially) a semidirect product of a unipotent group (its unipotent radical) with a redu... | Algebraic groups | 0.824749 |
6,182 | There are many examples of such groups beyond those given previously: orthogonal and symplectic groups are affine algebraic groups. unipotent groups. algebraic tori. | Algebraic groups | 0.824749 |
6,183 | An algebraic group is said to be affine if its underlying algebraic variety is an affine variety. Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its coordinate ring it can be shown that every affine alge... | Algebraic groups | 0.824749 |
6,184 | Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field k {\displaystyle k} is associated a Lie algebra over k {\displaystyle k} . As a vector space the Lie algebra is isomorphic to the tangent space at the identity element. The Lie bracket can be constructed from its interpretation as... | Algebraic groups | 0.824749 |
6,185 | Not all algebraic groups are linear groups or abelian varieties, for instance some group schemes occurring naturally in arithmetic geometry are neither. Chevalley's structure theorem asserts that every connected algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if K is a... | Algebraic groups | 0.824749 |
6,186 | Since equations (1) through (4) above cannot be satisfied simultaneously when the hidden variable, λ, takes the same value in each equation, GHSZ proceed by allowing λ to take different values in each equation. They define Λ1: the set of all λs such that equation (1) holds, Λ2: the set of all λs such that equation (2) ... | GHZ experiment | 0.824742 |
6,187 | However, GHZ and collaborators also require that the hidden variable argument to functions A(), B(), and C() may take the same value, λ, even in distinct trials, being characterized by distinct experimental contexts. This is the statistical independence assumption (also assumed in Bell's theorem and commonly known as "... | GHZ experiment | 0.824742 |
6,188 | The Greenberger–Horne–Zeilinger experiment or GHZ experiments are a class of physics experiments that may be used to generate starkly contrasting predictions from local hidden-variable theory and quantum mechanical theory, and permit immediate comparison with actual experimental results. A GHZ experiment is similar to ... | GHZ experiment | 0.824742 |
6,189 | Prior to any measurements being made, the polarizations of the photons are indeterminate; If a measurement is made on one of the photons using a two-channel polarizer aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50% probability for each orientation,... | GHZ experiment | 0.824742 |
6,190 | Schwarzschild's equation contains the fundamental physics needed to understand and quantify how increasing greenhouse gases (GHGs) in the atmosphere reduce the flux of thermal infrared radiation to space. If no other fluxes change, the law of conservation of energy demands that the Earth warm (from one steady state to ... | Schwarzschild's equation for radiative transfer | 0.824741 |
6,191 | It is written entirely in C. a factoring applet by Dario Alpern, that uses the quadratic sieve if certain conditions are met. The PARI/GP computer algebra package includes an implementation of the self-initialising multiple polynomial quadratic sieve implementing the large prime variant. It was adapted by Thomas Papani... | Quadratic sieve | 0.82472 |
6,192 | The self initialisation scheme is based on an idea from the thesis of Thomas Sosnowski. A variant of the quadratic sieve is available in the MAGMA computer algebra package. It is based on an implementation of Arjen Lenstra from 1995, used in his "factoring by email" program. | Quadratic sieve | 0.82472 |
6,193 | To summarize, the basic quadratic sieve algorithm has these main steps: Choose a smoothness bound B. The number π(B), denoting the number of prime numbers less than B, will control both the length of the vectors and the number of vectors needed. Use sieving to locate π(B) + 1 numbers ai such that bi = (ai2 mod n) is B-... | Quadratic sieve | 0.824719 |
6,194 | PPMPQS and PPSIQS mpqs SIMPQS is a fast implementation of the self-initialising multiple polynomial quadratic sieve written by William Hart. It provides support for the large prime variant and uses Jason Papadopoulos' block Lanczos code for the linear algebra stage. SIMPQS is accessible as the qsieve command in the Sag... | Quadratic sieve | 0.824719 |
6,195 | From the geometry we easily conclude that: Plugging ω = − Ω sin α {\displaystyle \omega =-\Omega \sin \alpha } into equation (1) we finally get As α {\displaystyle \alpha } adiabatically approaches zero, the angular velocity of the point of contact Ω {\displaystyle \Omega } becomes very large, and one hears a high-fr... | Euler's Disk | 0.824708 |
6,196 | The solutions in this case are known as Bloch states. Since Bloch's theorem applies only to periodic potentials, and since unceasing random movements of atoms in a crystal disrupt periodicity, this use of Bloch's theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without wh... | Solid State Physics | 0.824706 |
6,197 | Here, the electrons are modelled as a Fermi gas, a gas of particles which obey the quantum mechanical Fermi–Dirac statistics. The free electron model gave improved predictions for the heat capacity of metals, however, it was unable to explain the existence of insulators. The nearly free electron model is a modification... | Solid State Physics | 0.824706 |
6,198 | Properties of materials such as electrical conduction and heat capacity are investigated by solid state physics. An early model of electrical conduction was the Drude model, which applied kinetic theory to the electrons in a solid. By assuming that the material contains immobile positive ions and an "electron gas" of c... | Solid State Physics | 0.824706 |
6,199 | In the United States and Europe, solid state became a prominent field through its investigations into semiconductors, superconductivity, nuclear magnetic resonance, and diverse other phenomena. During the early Cold War, research in solid state physics was often not restricted to solids, which led some physicists in th... | Solid State Physics | 0.824706 |
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