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11,400 | OCLC 636972310. Reprint by the American Institute of Physics and Tomash Publishing:—Whittaker, E. T. (1987). | A History of the Theories of Aether and Electricity | 0.818022 |
11,401 | Roberto Torretti states, in his 1983 book Relativity and Geometry, "Whittaker's views on the origin of special relativity have been rejected by the great majority of scholars", citing Max Born, Gerald Holton, Charles Scribner, Stanley Goldberg, Elie Zahar, Tetu Hirosige, Kenneth F. Schaffner, and Arthur I. Miller. Whil... | A History of the Theories of Aether and Electricity | 0.818022 |
11,402 | In the second volume, a chapter titled "The Relativity Theory of Poincaré and Lorentz" credits Henri Poincaré and Hendrik Lorentz for developing special relativity, and especially alluded to Lorentz's 1904 paper (dated by Whittaker as 1903), Poincaré's St. Louis speech (The Principles of Mathematical Physics) of Septem... | A History of the Theories of Aether and Electricity | 0.818022 |
11,403 | He goes on to tell his readers to consult both "authoritative" volumes for a "detailed account of the fascinating history".In a 1988 Isis review of a combined reprint of the second edition, including both the first and second volumes bound together, published in New York by the American Institute of Physics and Tomash ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,404 | In a fourth obituary the work is said to be "brilliant" and a "colossal undertaking involving wide reading and accurate understanding".The book was included in a curated 1958 list of "important books on science" in a Science article by Ivy Kellerman Reed and Alexander Gode, where the volumes are said to be the "first e... | A History of the Theories of Aether and Electricity | 0.818022 |
11,405 | The theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypo... | Second incompleteness theorem | 0.818022 |
11,406 | A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is recursively enumerable (Franzén 2005, p. 112). This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not th... | Second incompleteness theorem | 0.818022 |
11,407 | Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC. In 1973, Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first se... | Second incompleteness theorem | 0.818022 |
11,408 | Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics. The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): Th... | Second incompleteness theorem | 0.818022 |
11,409 | There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here,... | Second incompleteness theorem | 0.818022 |
11,410 | {\displaystyle N(wz)=N(w)N(z).} For R , {\displaystyle \mathbb {R} ,} C , {\displaystyle \mathbb {C} ,} H , {\displaystyle \mathbb {H} ,} and O the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In other composition algebras the norm is an isotr... | Equivalent norm | 0.81802 |
11,411 | When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous. In signal processing and statistics, David Donoho referred to the zero "norm" with qu... | Equivalent norm | 0.81802 |
11,412 | In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distan... | Equivalent norm | 0.81802 |
11,413 | The concept of norm N ( z ) {\displaystyle N(z)} in composition algebras does not share the usual properties of a norm as it may be negative or zero for z ≠ 0. {\displaystyle z\neq 0.} A composition algebra ( A , ∗ , N ) {\displaystyle (A,{}^{*},N)} consists of an algebra over a field A , {\displaystyle A,} an involuti... | Equivalent norm | 0.81802 |
11,414 | Let E {\displaystyle E} be a finite extension of a field k {\displaystyle k} of inseparable degree p μ , {\displaystyle p^{\mu },} and let k {\displaystyle k} have algebraic closure K . {\displaystyle K.} If the distinct embeddings of E {\displaystyle E} are { σ j } j , {\displaystyle \left\{\sigma _{j}\right\}_{j},} t... | Equivalent norm | 0.81802 |
11,415 | Each CU contains two sets of an SIMD32 vector unit, an SISD scalar unit, textures units, and a stack of various caches. New low precision data types like INT4 and INT8 are new supported data types for RDNA 2 CUs.The RDNA 2 graphics pipeline has been reconfigured and reordered for greater performance-per-watt and more e... | RDNA 2 | 0.818018 |
11,416 | In queueing theory, a discipline within the mathematical theory of probability, a fluid limit, fluid approximation or fluid analysis of a stochastic model is a deterministic real-valued process which approximates the evolution of a given stochastic process, usually subject to some scaling or limiting criteria. Fluid li... | Fluid limit | 0.818018 |
11,417 | The cladistic idea of strictly using the topology of the phylogenetic tree to define groups necessitates other definitions than crown groups to adequately define commonly discussed fossil groups. Thus, a host of prefixes have been defined to describe various branches of the phylogenetic tree relative to extant organism... | Stem group | 0.818014 |
11,418 | The International Academy of Mathematical Chemistry (IAMC) was founded in Dubrovnik, Croatia, in 2005 by Milan Randić. It is an organization for chemistry and mathematics avocation; its predecessors have been around since the 1930s. There are 88 Academy members (as of 2011) from around the world (27 countries), compris... | International Academy of Mathematical Chemistry | 0.818012 |
11,419 | It is now possible to use powerful non-linear numerical analysis programs (or finite element analysis) to formfind and design fabric and cable structures. The programs must allow for large deflections. The final shape, or form, of a fabric structure depends upon: shape, or pattern, of the fabric the geometry of the sup... | Tensile structure | 0.818009 |
11,420 | A similar solution to that above can be derived where: By equilibrium: W = 4 T d L {\displaystyle W={\frac {4Td}{L}}} d = W L 4 T {\displaystyle d={\frac {WL}{4T}}} By geometry: L = S 2 + 4 d 2 = S 2 + 4 ( W L 4 T ) 2 {\displaystyle L={\sqrt {S^{2}+4d^{2}}}={\sqrt {S^{2}+4\left({\frac {WL}{4T}}\right)^{2}}}} This gives... | Tensile structure | 0.818009 |
11,421 | When calculating keyword density, ignore html tags and other embedded tags which will not appear in the text of the page once published. When calculating the density of a keyword phrase, the formula would be ( N k r ∗ N w p / T k n ) ∗ 100 {\displaystyle (Nkr*Nwp/Tkn)*100} , where Nwp is the number of words in the phra... | Keyword density | 0.818007 |
11,422 | Keyword density is the percentage of times a keyword or phrase appears on a web page compared to the total number of words on the page. In the context of search engine optimization, keyword density can be used to determine whether a web page is relevant to a specified keyword or keyword phrase. In the late 1990s, the e... | Keyword density | 0.818007 |
11,423 | The beginning of carbochemistry goes back to the 16th century. At that time, large quantities of charcoal were needed for the smelting of iron ores. Since the production of charcoal required large amounts of slowly-regenerating wood, the use of coal was studied. The use of pure coal was difficult because of the amount ... | Carbochemistry | 0.818006 |
11,424 | Carbochemistry is the branch of chemistry that studies the transformation of coal (bituminous coal, coal tar, anthracite, lignite, graphite, and charcoal) into useful products and raw materials. The processes that are used in carbochemistry include degasification processes such as carbonization and coking, gasification... | Carbochemistry | 0.818006 |
11,425 | For example, if G is any group, then there exists an automorphism σ of G × G that switches the two factors, i.e. σ(g1, g2) = (g2, g1).For another example, the automorphism group of Z × Z is GL(2, Z), the group of all 2 × 2 matrices with integer entries and determinant, ±1. This automorphism group is infinite, but only ... | Direct product of groups | 0.818 |
11,426 | Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on G × H is associative. Identity The direct product has an identity element, namely (1G, 1H), whe... | Direct product of groups | 0.818 |
11,427 | It is possible to take the direct product of more than two groups at once. Given a finite sequence G1, ..., Gn of groups, the direct product ∏ i = 1 n G i = G 1 × G 2 × ⋯ × G n {\displaystyle \prod _{i=1}^{n}G_{i}\;=\;G_{1}\times G_{2}\times \cdots \times G_{n}} is defined as follows: This has many of the same properti... | Direct product of groups | 0.818 |
11,428 | Mechanism is the belief that natural wholes (principally living things) are similar to complicated machines or artifacts, composed of parts lacking any intrinsic relationship to each other. The doctrine of mechanism in philosophy comes in two different flavors. They are both doctrines of metaphysics, but they are diffe... | Mechanical philosophy | 0.817993 |
11,429 | In organic chemistry, the carbohydrates and carboxylic acids are ternary compounds with carbon, oxygen, and hydrogen. Other organic ternary compounds replace oxygen with another atom to form functional groups. The multiplicity of ternary compounds based on {C, H, O} has been noted. For example, C 9 H 10 O 3 {\displayst... | Ternary compound | 0.81799 |
11,430 | Vigorous shaking and stirring may be needed to bring the reaction to completion. This means that the more finely divided a solid or liquid reactant the greater its surface area per unit volume and the more contact it with the other reactant, thus the faster the reaction. To make an analogy, for example, when one starts... | Chemical Kinetics | 0.817989 |
11,431 | Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is different from chemical thermodynamics, which deals with the direction in which a reaction occurs but in itself tells nothing about its rate. Chemical kinet... | Chemical Kinetics | 0.817989 |
11,432 | In some cases, equations are unsolvable analytically, but can be solved using numerical methods if data values are given. There are two different ways to do this, by either using software programmes or mathematical methods such as the Euler method. Examples of software for chemical kinetics are i) Tenua, a Java app whi... | Chemical Kinetics | 0.817989 |
11,433 | It gives rise to a reduced scheme, when we take it over a field K, if and only if the characteristic of K does not divide n. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example μp over a finite field with p elements for any prime n... | Multiplicative group | 0.817981 |
11,434 | In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), wh... | Multiplicative group | 0.817981 |
11,435 | Neutral point clamped (NPC) inverters are widely used topology of multilevel inverters in high-power applications. This kind of inverters are able to be used for up to several megawatts applications. See links for more information. | Neutral Point Clamped | 0.817976 |
11,436 | In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most modern equations of state are formulated in the Helmholtz free energy. Equatio... | State equation | 0.817974 |
11,437 | Consider the argument A: Either it is hot or it is cold It is not hot Therefore it is cold This argument is of the form: Either P or Q Not P Therefore Q or (using standard symbols of propositional calculus): P ∨ {\displaystyle \lor } Q ¬ {\displaystyle \neg } P ____________ Q The corresponding conditional C is: IF ((P ... | Corresponding conditional | 0.817968 |
11,438 | In case of the difficulty in trying to derive a contradiction, one should proceed as follows. From the negation of the corresponding conditional derive a theorem in conjunctive normal form in the methodical fashions described in text books. If, and only if, the original argument was valid will the theorem in conjunctiv... | Corresponding conditional | 0.817968 |
11,439 | The National Cancer Institute’s Genomic and Bioinformatics Group are currently researching microarray gene-expression patterns. Findings in this area will further the ability to analyze cancer-related data. | Compute Against Cancer | 0.817967 |
11,440 | In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time-average probabilities. A Palm probability or Palm expectation, often denoted P 0 ( ⋅ ) {\displaystyle P^{0}(\cdot )} or E 0... | Palm calculus | 0.817966 |
11,441 | So ⟨ T ¯ n , ⟩ = ∫ x n d μ ( x ) . {\displaystyle \langle {\overline {T}}^{n},\rangle =\int x^{n}d\mu (x).} On the other hand, ⟨ T ¯ n , ⟩ = ⟨ T n , ⟩ = m n . {\displaystyle \langle {\overline {T}}^{n},\rangle =\langle T^{n},\rangle =m_{n}.} For an alternative proof of the existence that only uses Stieltjes integ... | Hamburger moment problem | 0.817966 |
11,442 | (Early literature on the relative efficiency is somewhat misleading, as it assumed that square roots were much more time-consuming than divisions,: 69 while on 21st-century computers they are only slightly more expensive.) Efficient algorithms for the Kalman prediction and update steps in the square root form were deve... | Kalman gain | 0.817958 |
11,443 | The filtering method is named for Hungarian émigré Rudolf E. Kálmán, although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was... | Kalman gain | 0.817958 |
11,444 | A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is a concept much applied in time series analysis used for topics such as signal processing and econometrics. Kalman filtering is also one of the m... | Kalman gain | 0.817958 |
11,445 | For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a... | Kalman gain | 0.817958 |
11,446 | In the absence of reliable statistics or the true values of noise covariance matrices Q k {\displaystyle \mathbf {Q} _{k}} and R k {\displaystyle \mathbf {R} _{k}} , the expression P k ∣ k = ( I − K k H k ) P k ∣ k − 1 ( I − K k H k ) T + K k R k K k T {\displaystyle \mathbf {P} _{k\mid k}=\left(\mathbf {I} -\mathbf {K... | Kalman gain | 0.817958 |
11,447 | The Kalman filtering equations provide an estimate of the state x ^ k ∣ k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} and its error covariance P k ∣ k {\displaystyle \mathbf {P} _{k\mid k}} recursively. The estimate and its quality depend on the system parameters and the noise statistics fed as inputs to the estima... | Kalman gain | 0.817958 |
11,448 | Another example provided is the relevance of mathematical notation in the proof of the impossibility of squaring the circle which is used to illustrate the idea that the same procedure being represented in different ways can reveal non-obvious connections within mathematics. The chapter also considers the importance of... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,449 | Colyvan argues that mathematical theory can be inconsistent whilst still being useful, pointing to naive set theory and early infinitesimal calculus as examples of mathematical theories that were later proved to be inconsistent but were fruitfully worked on by mathematicians. He also proposes that the logic used by mat... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,450 | According to this account, mathematical models act as maps of physical systems by abstracting away from particular details to more structural features of the system. In this way, the abstract structures of mathematics can be used to represent physical systems via similarity relations. Colyvan presents the case study of... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,451 | He then distinguishes between intra-mathematical explanations, mathematical explanations of mathematical facts, and extra-mathematical explanations, mathematical explanations of non-mathematical, empirical facts. He uses the unification account of explanation to attempt to explain the difference between mathematical pr... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,452 | It discusses the Löwenheim–Skolem theorem and its connection with Cantor's theorem, including a proof of Cantor's theorem and an explanation of why the two theorems are not contradictory. It also discusses Gödel's incompleteness theorems and Gödel and Cohen's work on the independence of the continuum hypothesis. These ... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,453 | The book has eight chapters and an epilogue with each chapter ending with a list of discussion questions and further readings. Chapter 1 briefly covers what Colyvan calls the "big isms" which dominated early 20th century philosophy of mathematics: logicism, formalism and intuitionism. It then turns to the philosophical... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,454 | Next-generation sequencing (NGS) allows for the rapid sequencing of large amounts of DNA, significantly advancing the study of genetics, and replacing older methods such as Sanger sequencing. This technology is starting to become more common in healthcare and research not only because it is a reliable method of determi... | Exome | 0.817953 |
11,455 | Hunacek said that he was "puzzled" by the inclusion of the epilogue which he said contained items which "seemed to have no particular philosophical significance". Wagner praised the praised the epilogue, calling it "excellent" whilst suggesting some changes to its presentation of the theorems covered. Weber praised the... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,456 | As an introduction to the field, the choice of topics proposed is entirely justified. "Pettigrew said of the epilogue that he would have preferred if it covered fewer theorems and more proofs and applications. Nonetheless, he said that "it is certainly a valuable resource for a student entering the philosophy of mathem... | An Introduction to the Philosophy of Mathematics | 0.817953 |
11,457 | Astronomer T. J. Cornwell writes, "The impact of CLEAN on radio astronomy has been immense", both directly in enabling greater speed and efficiency in observations, and indirectly by encouraging "a wave of innovation in synthesis processing that continues to this day." It has also been applied in other areas of astrono... | CLEAN (algorithm) | 0.81795 |
11,458 | The CLEAN algorithm is a computational algorithm to perform a deconvolution on images created in radio astronomy. It was published by Jan Högbom in 1974 and several variations have been proposed since then.The algorithm assumes that the image consists of a number of point sources. It will iteratively find the highest v... | CLEAN (algorithm) | 0.81795 |
11,459 | We consider a charge density in tube with axis along a magnetic field embedded in an electron gas where r {\displaystyle r} is the distance from the guiding center, L B {\displaystyle L_{B}} is the width of the material in the direction of the magnetic field where the cyclotron frequency is (Gaussian units) and is the ... | Static forces and virtual-particle exchange | 0.817942 |
11,460 | In perturbation theory, forces are generated by the exchange of virtual particles. The mechanics of virtual-particle exchange is best described with the path integral formulation of quantum mechanics. There are insights that can be obtained, however, without going into the machinery of path integrals, such as why class... | Static forces and virtual-particle exchange | 0.817942 |
11,461 | In these cases, the physical interpretation must be re-examined. As an example, the calculations of atomic structure in atomic physics or of molecular structure in quantum chemistry could not easily be repeated, if at all, using the "force-mediating particle" picture.Use of the "force-mediating particle" picture (FMPP)... | Static forces and virtual-particle exchange | 0.817942 |
11,462 | Split-intein circular ligation of peptides and proteins (SICLOPPS) is a biotechnology technique that permits the creation of cyclic peptides. These peptides are produced by ribosomal protein synthesis, followed by an intein-like event that splices the protein into a loop. By contrast with the nonribosomal peptide synth... | Split-intein circular ligation of peptides and proteins | 0.817935 |
11,463 | New rules, called Lorentz transformations, were given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred. : 19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that ... | Quantum fields | 0.817935 |
11,464 | In 1924, Louis de Broglie proposed the hypothesis of wave–particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contribu... | Quantum fields | 0.817935 |
11,465 | Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harm... | Quantum fields | 0.817935 |
11,466 | Quantum field theory results from the combination of classical field theory, quantum mechanics, and special relativity. : xi A brief overview of these theoretical precursors follows. The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absen... | Quantum fields | 0.817935 |
11,467 | In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. : xi QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. Q... | Quantum fields | 0.817935 |
11,468 | The K-groups of finite fields are one of the few cases where the K-theory is known completely: for n ≥ 1 {\displaystyle n\geq 1} , K n ( F q ) = π n ( B G L ( F q ) + ) ≃ { Z / ( q i − 1 ) , if n = 2 i − 1 0 , if n is even {\displaystyle K_{n}(\mathbb {F} _{q})=\pi _{n}(BGL(\mathbb {F} _{q})^{+})\simeq {\begin{cases}\m... | K-groups of a field | 0.817925 |
11,469 | Suslin (1983) showed that the torsion in K-theory is insensitive to extensions of algebraically closed fields. This statement is known as Suslin rigidity. | K-groups of a field | 0.817925 |
11,470 | Namely, they exhibited arithmetic hyperbolic Riemann surfaces with systole behaving as a constant times log ( g ) {\displaystyle \log(g)} . Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times ( log g ) 2 g {\displaystyle {\tfrac {(\log g)^{2}}{g}}} . The... | Systoles of surfaces | 0.817921 |
11,471 | In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The s... | Systoles of surfaces | 0.817921 |
11,472 | This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data is a useful tool for assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role. | Mathematical models | 0.817919 |
11,473 | While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from ... | Mathematical models | 0.817919 |
11,474 | Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks whi... | Mathematical models | 0.817919 |
11,475 | In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint... | Inverse kinematics | 0.817913 |
11,476 | In robotics, inverse kinematics makes use of the kinematics equations to determine the joint parameters that provide a desired configuration (position and rotation) for each of the robot's end-effectors. This is important because robot tasks are performed with the end effectors, while control effort applies to the join... | Inverse kinematics | 0.817913 |
11,477 | Electric and magnetic fields store energy. The (volumetric) energy density is given by u = ε 2 E 2 + 1 2 μ B 2 {\displaystyle u={\frac {\varepsilon }{2}}\mathbf {E} ^{2}+{\frac {1}{2\mu }}\mathbf {B} ^{2}} where E is the electric field, B is the magnetic field, and ε and µ are the permittivity and permeability of the s... | Energies per unit mass | 0.81791 |
11,478 | In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or gravimetric energy density. Often only the useful or extractable energy is measured, which is to say that inacces... | Energies per unit mass | 0.81791 |
11,479 | For instance, for a single bosonic quantum harmonic oscillator we have that the thermal expectation value of the number operator is simply the Bose–Einstein distribution ⟨ b ^ † b ^ ⟩ = T r ( e − β ω b ^ † b ^ b ^ † b ^ ) T r ( e − β ω b ^ † b ^ ) = 1 e β ω − 1 {\displaystyle \langle {\hat {b}}^{\dagger }{\hat {b}}\ran... | Normal order | 0.817909 |
11,480 | However, for this to be a useful procedure one demands that the normal ordered product of any combination of fields has zero expectation value ⟨: ϕ 1 ( x 1 ) ϕ 2 ( x 2 ) … ϕ n ( x n ): ⟩ = 0 {\displaystyle \langle :\phi _{1}(x_{1})\phi _{2}(x_{2})\ldots \phi _{n}(x_{n}):\rangle =0} It is also important for practical ca... | Normal order | 0.817909 |
11,481 | Fermions are particles which satisfy Fermi–Dirac statistics. We will now examine the normal ordering of fermionic creation and annihilation operator products. | Normal order | 0.817909 |
11,482 | This equation is used in defining the contractions used in Wick's theorem. 3. | Normal order | 0.817909 |
11,483 | This equation, which is in the same form as the bosonic case above, is used in defining the contractions used in Wick's theorem. 2. The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice. For example:: f ^ f ^ † f ^ f ^ †: = − f ^ †... | Normal order | 0.817909 |
11,484 | Wick's theorem states the relationship between the time ordered product of n {\displaystyle n} fields and a sum of normal ordered products. This may be expressed for n {\displaystyle n} even as T =: ϕ ( x 1 ) ⋯ ϕ ( x n ): + ∑ perm ⟨ 0 | T | 0 ⟩: ϕ ( x 3 ) ⋯ ϕ ( x n ): + ∑ perm ⟨ 0 | T | 0 ⟩ ⟨ 0 | T | 0 ⟩: ϕ ( x 5 )... | Normal order | 0.817909 |
11,485 | Bosons are particles which satisfy Bose–Einstein statistics. We will now examine the normal ordering of bosonic creation and annihilation operator products. | Normal order | 0.817909 |
11,486 | Likewise, the specific crystal structure adopted by a crystalline solid depends on the material involved and on how it was formed. While many common objects, such as an ice cube or a coin, are chemically identical throughout, many other common materials comprise a number of different substances packed together. For exa... | Solidiification point | 0.817908 |
11,487 | The atoms in a solid are bound to each other, either in a regular geometric lattice (crystalline solids, which include metals and ordinary ice), or irregularly (an amorphous solid such as common window glass). Solids cannot be compressed with little pressure whereas gases can be compressed with little pressure because ... | Solidiification point | 0.817908 |
11,488 | Organic chemistry studies the structure, properties, composition, reactions, and preparation by synthesis (or other means) of chemical compounds of carbon and hydrogen, which may contain any number of other elements such as nitrogen, oxygen and the halogens: fluorine, chlorine, bromine and iodine. Some organic compound... | Solidiification point | 0.817908 |
11,489 | One important property of carbon in organic chemistry is that it can form certain compounds, the individual molecules of which are capable of attaching themselves to one another, thereby forming a chain or a network. The process is called polymerization and the chains or networks polymers, while the source compound is ... | Solidiification point | 0.817908 |
11,490 | The study of metallic elements and their alloys makes up a significant portion of the fields of solid-state chemistry, physics, materials science and engineering. Metallic solids are held together by a high density of shared, delocalized electrons, known as "metallic bonding". In a metal, atoms readily lose their outer... | Solidiification point | 0.817908 |
11,491 | Available from SourceForge. MANLAB: Computation of equilibrium, periodic and quasi-periodic solution of differential equations using Fourier series (harmonic balance method) developments of the solution and Taylor series developments (asymptotic numerical method) of the solution branch. Available from LMA Marseille. Bi... | Numerical continuation | 0.817904 |
11,492 | DDEBIFTOOL: Computation of solutions of Delay Differential Equations. A MATLAB package. | Numerical continuation | 0.817904 |
11,493 | A regular solution is a solution at which the Jacobian is full rank ( n ) {\displaystyle (n)} . A singular solution is a solution at which the Jacobian is less than full rank. A regular solution lies on a k-dimensional surface, which can be parameterized by a point in the tangent space (the null space of the Jacobian).... | Numerical continuation | 0.817904 |
11,494 | The constant term in the Taylor series of the scaled bifurcation equation is called the algebraic bifurcation equation, and the implicit function theorem applied the bifurcation equations states that for each isolated solution of the algebraic bifurcation equation there is a branch of solutions of the original problem ... | Numerical continuation | 0.817904 |
11,495 | If the columns of the matrix Φ {\displaystyle \Phi } are an orthonormal basis for the null space of J = {\displaystyle J=\left} and the columns of the matrix Ψ {\displaystyle \Psi } are an orthonormal basis for the left null space of J {\displaystyle J} , then the system F ( x , λ ) = 0 {\displaystyle F(x,\lambda )=0}... | Numerical continuation | 0.817904 |
11,496 | Finding the solution curves leaving a singular point is called branch switching, and uses techniques from bifurcation theory (singularity theory, catastrophe theory). For finite-dimensional systems (as defined above) the Lyapunov-Schmidt decomposition may be used to produce two systems to which the Implicit Function Th... | Numerical continuation | 0.817904 |
11,497 | A regular point of F {\displaystyle F} is a point ( u , λ ) {\displaystyle (\mathbf {u} ,\lambda )} at which the Jacobian of F {\displaystyle F} is full rank ( n ) {\displaystyle (n)} . Near a regular point the solution component is an isolated curve passing through the regular point (the implicit function theorem). In... | Numerical continuation | 0.817904 |
11,498 | Especially in cases where the dynamical system is prone to blow-up at certain parameter values (or combination of values for multiple parameters).It is extremely insightful as to the presence of stable solutions (attracting or repelling) in the study of Nonlinear Partial Differential Equations where time stepping in th... | Numerical continuation | 0.817904 |
11,499 | In some nonlinear systems, parameters are explicit. In others they are implicit, and the system of nonlinear equations is written F ( u ) = 0 {\displaystyle F(\mathbf {u} )=0} where u {\displaystyle \mathbf {u} } is an n-vector, and its image F ( u ) {\displaystyle F(\mathbf {u} )} is an n-1 vector. This formulation, w... | Numerical continuation | 0.817904 |
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