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15,100 | It is a type of polyphase system and is the most common method used by electrical grids worldwide to transfer power. It is also used to power large motors and other heavy loads. TorqueIn physics and mechanics, torque is the rotational equivalent of linear force. | Glossary of engineering: M–Z | 0.814638 |
15,101 | QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. Quantum mechanicsIs a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. :... | Glossary of engineering: M–Z | 0.814638 |
15,102 | Quantum electrodynamicsIn particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically desc... | Glossary of engineering: M–Z | 0.814638 |
15,103 | NanoengineeringIs the practice of engineering on the nanoscale. It derives its name from the nanometre, a unit of measurement equalling one billionth of a meter. Nanoengineering is largely a synonym for nanotechnology, but emphasizes the engineering rather than the pure science aspects of the field. | Glossary of engineering: M–Z | 0.814638 |
15,104 | Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle. Weighted arithmetic meanThe weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equall... | Glossary of engineering: M–Z | 0.814638 |
15,105 | Uncertainty principleIn quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as... | Glossary of engineering: M–Z | 0.814638 |
15,106 | Gas, steam, and water turbines have a casing around the blades that contains and controls the working fluid. TurbomachineryTurbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a flui... | Glossary of engineering: M–Z | 0.814638 |
15,107 | TrigonometryIs a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India creat... | Glossary of engineering: M–Z | 0.814638 |
15,108 | Trigonometric functionsIn mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navi... | Glossary of engineering: M–Z | 0.814638 |
15,109 | For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass. In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence ( f k ( x ) ) k ∈ N {\displaystyle (f^{k}(x))... | Glossary of engineering: M–Z | 0.814638 |
15,110 | Both reduce the comfort. ToughnessIn materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. | Glossary of engineering: M–Z | 0.814638 |
15,111 | The theorem also applies to frequency domain AC circuits consisting of reactive and resistive impedances. It means the theorem applies for AC in an exactly same way to DC except that resistances are generalized to impedances. Three-phase electric powerIs a common method of alternating current electric power generation,... | Glossary of engineering: M–Z | 0.814638 |
15,112 | If terminals A and B are connected to one another, the current flowing from A to B will be Vth/Rth. This means that Rth could alternatively be calculated as Vth divided by the short-circuit current between A and B when they are connected together. In circuit theory terms, the theorem allows any one-port network to be r... | Glossary of engineering: M–Z | 0.814638 |
15,113 | Thévenin's theoremAs originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that "For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source Vth in a series ... | Glossary of engineering: M–Z | 0.814638 |
15,114 | Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. | Glossary of engineering: M–Z | 0.814638 |
15,115 | The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and ... | Glossary of engineering: M–Z | 0.814638 |
15,116 | All matter with a temperature greater than absolute zero emits thermal radiation. Particle motion results in charge-acceleration or dipole oscillation which produces electromagnetic radiation. ThermodynamicsIs a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and ... | Glossary of engineering: M–Z | 0.814638 |
15,117 | In brittle materials the ultimate tensile strength is close to the yield point, whereas in ductile materials the ultimate tensile strength can be higher. Tensile testingTensile testing, also known as tension testing, is a fundamental materials science and engineering test in which a sample is subjected to a controlled ... | Glossary of engineering: M–Z | 0.814638 |
15,118 | Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed: E r o t a t i o n a l = 1 2 I ω 2 {\displaystyle E_{\mathrm {rotational} }={\frac {1}{2}}I\omega ^{2}} where ω {\displaystyle \omega \ } is the angular velocity I {\displa... | Glossary of engineering: M–Z | 0.814638 |
15,119 | In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in the order of thousands of km/h, even though the average velocity of its molecules is zero. Rotational energyRotational energy or angular kinetic energy is kinetic energy due to the rotati... | Glossary of engineering: M–Z | 0.814638 |
15,120 | In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data. Root-mean-square speedIn the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is ca... | Glossary of engineering: M–Z | 0.814638 |
15,121 | The goal of robotics is to design machines that can help and assist humans. Robotics integrates fields of mechanical engineering, electrical engineering, information engineering, mechatronics, electronics, bioengineering, computer engineering, control engineering, software engineering, among others. Root mean squareIn ... | Glossary of engineering: M–Z | 0.814638 |
15,122 | One advantage of using the computerized method is that the surgeon does not have to be present, leading to the possibility for remote surgery. RoboticsIs an interdisciplinary field that integrates computer science and engineering. Robotics involves design, construction, operation, and use of robots. | Glossary of engineering: M–Z | 0.814638 |
15,123 | A rigid body is usually considered as a continuous distribution of mass. In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics, a rigid body is usually thought of as a collection of poin... | Glossary of engineering: M–Z | 0.814638 |
15,124 | Rheology is a branch of physics, and it is the science that deals with the deformation and flow of materials, both solids and liquids. Rigid bodyIn physics, a rigid body (also known as a rigid object ) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given point... | Glossary of engineering: M–Z | 0.814638 |
15,125 | Relative velocityThe relative velocity v → B ∣ A {\displaystyle {\vec {v}}_{B\mid A}} (also v → B A {\displaystyle {\vec {v}}_{BA}} or v → B rel A {\displaystyle {\vec {v}}_{B\operatorname {rel} A}} ) is the velocity of an object or observer B in the rest frame of another object or observer A. Reliability engineering... | Glossary of engineering: M–Z | 0.814638 |
15,126 | As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation.... | Glossary of engineering: M–Z | 0.814638 |
15,127 | Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. | Glossary of engineering: M–Z | 0.814638 |
15,128 | Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include dis... | Glossary of engineering: M–Z | 0.814638 |
15,129 | For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). Examples of random phenomena include the weather condition in a future date, the heig... | Glossary of engineering: M–Z | 0.814638 |
15,130 | Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). Probability distributionIn pr... | Glossary of engineering: M–Z | 0.814638 |
15,131 | Power is a scalar quantity. Power factorIn electrical engineering, the power factor of an AC power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit, and is a dimensionless number in the closed interval of −1 to 1. A power factor of less than one indicate... | Glossary of engineering: M–Z | 0.814638 |
15,132 | Power (physics)In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. | Glossary of engineering: M–Z | 0.814638 |
15,133 | PneumaticsThe control of mechanical force and movement, generated by the application of compressed gas. Point estimationIn statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best... | Glossary of engineering: M–Z | 0.814638 |
15,134 | PlasticityIn physics and materials science, plasticity, also known as plastic deformation, is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity a... | Glossary of engineering: M–Z | 0.814638 |
15,135 | The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value h {\displaystyle h} = 6.62607015×10−34 J⋅Hz−1. Plasma (physics)Is one of the four fundamental states of matter, first systematically studied by Irving Langmuir in the 1920s. It consists of a gas of ion... | Glossary of engineering: M–Z | 0.814638 |
15,136 | PhysicsIs the natural science that studies matter, its motion and behavior through space and time, and the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves. Planck constantThe Planck constant, or Planck's con... | Glossary of engineering: M–Z | 0.814638 |
15,137 | Physical chemistryIs the study of macroscopic, and particulate phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mechanics, analytical dynamics and chemical equilibrium. Physical quantityA phys... | Glossary of engineering: M–Z | 0.814638 |
15,138 | It is denoted ϕ ( t ) {\displaystyle \phi (t)} and expressed in such a scale that it varies by one full turn as the variable t {\displaystyle t} goes through each period (and F ( t ) {\displaystyle F(t)} goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasin... | Glossary of engineering: M–Z | 0.814638 |
15,139 | In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air is a third phase over the ice and water. The glass of the jar is another separate phase. (See state of matter § Glass) Phase (waves)In physics and mathematics, the phase of a periodic func... | Glossary of engineering: M–Z | 0.814638 |
15,140 | Exploration, by earth scientists, and petroleum engineering are the oil and gas industry's two main subsurface disciplines, which focus on maximizing economic recovery of hydrocarbons from subsurface reservoirs. Petroleum geology and geophysics focus on provision of a static description of the hydrocarbon reservoir roc... | Glossary of engineering: M–Z | 0.814638 |
15,141 | The currently dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model. Thus, modern particle physics generally investigates the Standard Model and its various possible extensions, e.g. to the newest "known" particle, the Higgs boson, or even to the olde... | Glossary of engineering: M–Z | 0.814638 |
15,142 | Particle physicsParticle physics (also known as high energy physics) is a branch of physics that studies the nature of the particles that constitute matter and radiation. Although the word particle can refer to various types of very small objects (e.g. protons, gas particles, or even household dust), particle physics u... | Glossary of engineering: M–Z | 0.814638 |
15,143 | In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation; it is the same as a 180°-rotation. In quantum mechanics, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are... | Glossary of engineering: M–Z | 0.814638 |
15,144 | The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P (in any num... | Glossary of engineering: M–Z | 0.814638 |
15,145 | By contrast, -3, 5, 7, 21 are odd numbers as they leave a remainder of 1 when divided by 2. Parity (physics)In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all t... | Glossary of engineering: M–Z | 0.814638 |
15,146 | This difference in mass arises due to the difference in atomic binding energy between the nuclei before and after the reaction. Fusion is the process that powers active or main sequence stars and other high-magnitude stars, where large amounts of energy are released. Nuclear physicsThe science that describes the compon... | Glossary of engineering: M–Z | 0.814638 |
15,147 | Norton's theoremIn direct-current circuit theory, Norton's theorem (aka Mayer–Norton theorem) is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single re... | Glossary of engineering: M–Z | 0.814638 |
15,148 | NanotechnologyThe technology of systems built with moving parts on the order of a nanometre in size. Navier–Stokes equationsIn physics, the Navier–Stokes equations are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Na... | Glossary of engineering: M–Z | 0.814638 |
15,149 | Researchers at University of Rochester reported in 2023 that significant improvements in perovskite solar cell efficiency can be achieved by utilizing Purcell effect to extend the duration of photon induced electron-hole pairs spontaneous recombination time thus enabling them to reach the cell electrodes.The Purcell ef... | Purcell effect | 0.814634 |
15,150 | Logic: Many logic systems induce information algebras (Wilson & Mengin 1999). Reducts of cylindric algebras (Henkin, Monk & Tarski 1971) or polyadic algebras are information algebras related to predicate logic (Halmos 2000). Module algebras: (Bergstra, Heering & Klint 1990);(de Lavalette 1992). Linear systems: Systems ... | Information algebra | 0.814634 |
15,151 | Here follows an incomplete list of instances of information algebras: Relational algebra: The reduct of a relational algebra with natural join as combination and the usual projection is a labeled information algebra, see Example. Constraint systems: Constraints form an information algebra (Jaffar & Maher 1994). Semirin... | Information algebra | 0.814634 |
15,152 | The axioms of the two-sorted algebra ( Φ , D ) {\displaystyle (\Phi ,D)} , in addition to the axioms of the lattice D {\displaystyle D}: A two-sorted algebra ( Φ , D ) {\displaystyle (\Phi ,D)} satisfying these axioms is called an Information Algebra. | Information algebra | 0.814634 |
15,153 | The axioms for information algebras are derived from the axiom system proposed in (Shenoy and Shafer, 1990), see also (Shafer, 1991). | Information algebra | 0.814634 |
15,154 | More precisely, in the two-sorted algebra ( Φ , D ) {\displaystyle (\Phi ,D)} , the following operations are defined Additionally, in D {\displaystyle D} the usual lattice operations (meet and join) are defined. | Information algebra | 0.814634 |
15,155 | The majority of the glycans are therefore stalled as immature 'high-mannose' glycans not normally present on secreted or cell surface human glycoproteins. The unusual processing and high density means that almost all broadly neutralising antibodies that have so far been identified (from a subset of patients that have b... | V3 loop | 0.814631 |
15,156 | The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which the area of a face is well-defined. The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. ... | Topological polyhedra | 0.814626 |
15,157 | By forgetting the face structure, any polyhedron gives rise to a graph, called its skeleton, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear... | Topological polyhedra | 0.814626 |
15,158 | Developed by the Aachen Computer Graphics Group, RWTH Aachen University. Polyhedronisme Archived 2012-04-25 at the Wayback Machine – Web-based tool for generating polyhedra models using Conway Polyhedron Notation. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files. The 3D files can be opened in CAD so... | Topological polyhedra | 0.814626 |
15,159 | The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.There are several types of highly symmetric polyhedron, classified by which kind of element... | Topological polyhedra | 0.814626 |
15,160 | A convex polyhedron in which all vertices have integer coordinates is called a lattice polyhedron or integral polyhedron. The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polyn... | Topological polyhedra | 0.814626 |
15,161 | Euler's work (together with his earlier solution to the puzzle of the Seven Bridges of Königsberg) became the foundation of the new field of topology. The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by Henri Poincaré, Enrico Betti, Bern... | Topological polyhedra | 0.814626 |
15,162 | René Descartes, in around 1630, wrote his book De solidorum elementis studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was Descartes' theorem on total angular defect, which ... | Topological polyhedra | 0.814626 |
15,163 | This can also be used to identify differences in stereochemistry of peptides based on intermolecular ionic interactions. Some configuration changes of protein could also be observed from the translocation curve. Understanding this also contributes more data to understanding the sequence of the peptide fully in its envi... | Nanopore | 0.814625 |
15,164 | Apart from rapid DNA sequencing, other applications include separation of single stranded and double stranded DNA in solution, and the determination of length of polymers. At this stage, nanopores are making contributions to the understanding of polymer biophysics, single-molecule analysis of DNA-protein interactions, ... | Nanopore | 0.814625 |
15,165 | In contrast to the limitations of solid-state pores, they allow for the optimization of the resistance pulse magnitude relative to the background current by matching the pore-size closely to the particle-size. As detection occurs on a particle by particle basis, the true average and polydispersity distribution can be d... | Nanopore | 0.814625 |
15,166 | Since the discovery of track-etched technology in the late 1960s, filter membranes with needed diameter have found application potential in various fields including food safety, environmental pollution, biology, medicine, fuel cell, and chemistry. These track-etched membranes are typically made in polymer membrane thro... | Nanopore | 0.814625 |
15,167 | In 1954, Sir Edmund Taylor Whittaker, an English mathematician and historian of science, credited Henri Poincaré with the equation E = m c 2 {\displaystyle E=mc^{2}} , and he included a chapter entitled The Relativity Theory of Poincaré and Lorentz in his book A History of the Theories of Aether and Electricity. He cre... | Relativity priority dispute | 0.814619 |
15,168 | Lorentz's 1904 paper contained the transformations bearing his name that appeared in the Annalen der Physik. Some authors claim that Einstein worked in relative isolation and with restricted access to the physics literature in 1905. Others, however, disagree; a personal friend of Einstein, Maurice Solovine, acknowledg... | Relativity priority dispute | 0.814619 |
15,169 | To fill this gap, I introduced the principle of the constancy of the velocity of light, which I borrowed from H. A. Lorentz's theory of the stationary luminiferous ether, and which, like the principle of relativity, contains a physical assumption that seemed to be justified only by the relevant experiments (experiments... | Relativity priority dispute | 0.814619 |
15,170 | Whittaker's claims were criticized by Gerald Holton (1960, 1973). He argued that there are fundamental differences between the theories of Einstein on one hand, and Poincaré and Lorentz on the other hand. Einstein radically reformulated the concepts of space and time, and by that removed "absolute space" and thus the s... | Relativity priority dispute | 0.814619 |
15,171 | 118: "Throughout the paper of 1895, the Fresnel aether is postulated explicitly"; p. 125: "Like Voigt before him, Lorentz regarded the transformation ... only as a convenient mathematical tool for proving a physical theorem ... he proposed to call t the general time and t' the local time. Although he didn't say it expl... | Relativity priority dispute | 0.814619 |
15,172 | Although he was apparently trying to make a point concerning Whittaker's treatment of the origin of special relativity, Pais' phrasing of that statement was rebuked by at least one notable reviewer of his 1982 book as being "scurrilous" and "lamentable". Also in contrast to Pais' overgeneralized claim, notable scientis... | Relativity priority dispute | 0.814619 |
15,173 | In a 1965 series of articles tracing the history of relativity, Keswani claimed that Poincaré and Lorentz should have the main credit for special relativity – claiming that Poincaré pointedly credited Lorentz multiple times, while Lorentz credited Poincaré and Einstein, refusing to take credit for himself. He also down... | Relativity priority dispute | 0.814619 |
15,174 | In Anatoly Logunov's book about Poincaré's relativity theory, there is an English translation (on p. 113, using modern notations) of the part of Poincaré's 1900 article containing E=mc2. Logunov states that Poincaré's two 1905 papers are superior to Einstein's 1905 paper. According to Logunov, Poincaré was the first sc... | Relativity priority dispute | 0.814619 |
15,175 | 69. "How did Albert Einstein...arrive at his special theory of relativity?...I want only to stress that it is impossible to understand Einstein's discovery (if that is the right word) of special relativity without taking on board the impacts of the quantum in physics." | Relativity priority dispute | 0.814619 |
15,176 | Like Einstein half a decade later, Poincaré wanted new physics, not a reinterpretations or reorganization of existing notions. "Brown denies the idea of other authors and historians, that the major difference between Einstein and his predecessors is Einstein's rejection of the aether, because, it is always possible to ... | Relativity priority dispute | 0.814619 |
15,177 | Poincaré was the first to extend the relativity principle to optics and electrodynamics exactly. Whereas Lorentz, in his theorem of corresponding states, had from 1899 effectively assumed this extension of the relativity principle up to second-order effects, Poincaré took it to hold for all orders. Poincaré was the fir... | Relativity priority dispute | 0.814619 |
15,178 | Harvey R. Brown (2005) (who favors a dynamical view of relativistic effects similar to Lorentz, but "without a hidden aether frame") wrote about the road to special relativity from Michelson to Einstein in section 4: p. 40: "The cradle of special theory of relativity was the combination of Maxwellian electromagnetism a... | Relativity priority dispute | 0.814619 |
15,179 | "I turn now to basic conceptual differences. Einstein completely eliminated the ether, required that the expression of the laws of physics should be the same in any inertial frame, and introduced a "new kinematics" in which the space and time measured in different inertial systems were all on exactly the same footing. ... | Relativity priority dispute | 0.814619 |
15,180 | They both required that every law of physics should be invariant under these transformations. They both gave the relativistic laws of motion. They both recognized that the relativity principle and the energy principle led to paradoxes when conjointly applied to radiation processes. | Relativity priority dispute | 0.814619 |
15,181 | Katzir (2005) argued that "Poincaré's work should not be seen as an attempt to formulate special relativity, but as an independent attempt to resolve questions in electrodynamics." Contrary to Miller and others, Katzir thinks that Poincaré's development of electrodynamics led him to the rejection of the pure electromag... | Relativity priority dispute | 0.814619 |
15,182 | In addition to division rings, there are various other weaker algebraic structures related to fields such as quasifields, near-fields and semifields. There are also proper classes with field structure, which are sometimes called Fields, with a capital F. The surreal numbers form a Field containing the reals, and would ... | Field axioms | 0.814615 |
15,183 | A subset S of a field F is a transcendence basis if it is algebraically independent (do not satisfy any polynomial relations) over E and if F is an algebraic extension of E(S). Any field extension F / E has a transcendence basis. Thus, field extensions can be split into ones of the form E(S) / E (purely transcendental ... | Field axioms | 0.814615 |
15,184 | The above-mentioned field of rational fractions E(X), where X is an indeterminate, is not an algebraic extension of E since there is no polynomial equation with coefficients in E whose zero is X. Elements, such as X, which are not algebraic are called transcendental. Informally speaking, the indeterminate X and its pow... | Field axioms | 0.814615 |
15,185 | A benefit of the quaternion formulation of the composition of two rotations RB and RA is that it yields directly the rotation axis and angle of the composite rotation RC = RBRA. Let the quaternion associated with a spatial rotation R be constructed from its rotation axis S with the rotation angle φ {\displaystyle \varp... | Quaternions and spatial rotation | 0.814614 |
15,186 | From this all of the rules of quaternion arithmetic follow, such as the rules on multiplication of quaternion basis elements. Using these rules, one can show that: ( a + b i + c j + d k ) ( e + f i + g j + h k ) = ( a e − b f − c g − d h ) + ( a f + b e + c h − d g ) i + ( a g − b h + c e + d f ) j + ( a h + b g − c f ... | Quaternions and spatial rotation | 0.814614 |
15,187 | The complex numbers can be defined by introducing an abstract symbol i which satisfies the usual rules of algebra and additionally the rule i2 = −1. This is sufficient to reproduce all of the rules of complex number arithmetic: for example: ( a + b i ) ( c + d i ) = a c + a d i + b i c + b i d i = a c + a d i + b c i +... | Quaternions and spatial rotation | 0.814614 |
15,188 | It is reported that the existence and continued usage of an alternative quaternion convention in the aerospace and, to a lesser extent, robotics community is incurring a significant and ongoing cost . This alternative convention is proposed by Shuster M.D. in and departs from tradition by reversing the definition for m... | Quaternions and spatial rotation | 0.814614 |
15,189 | Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in... | Quaternions and spatial rotation | 0.814614 |
15,190 | The table below groups applications by their adherence to either quaternion convention: While use of either convention does not impact the capability or correctness of applications thus created, the authors of argued that the Shuster convention should be abandoned because it departs from the much older quaternion multi... | Quaternions and spatial rotation | 0.814614 |
15,191 | The rotated quaternion p' = q p q−1 needs to be differentiated with respect to the rotating quaternion q, when the rotation is estimated from numerical optimization. The estimation of rotation angle is an essential procedure in 3D object registration or camera calibration. For unitary q and pure imaginary p, that is fo... | Quaternions and spatial rotation | 0.814614 |
15,192 | This is a geometric proof that conjugation by q and by −q must produce the same rotational transformation matrix. That fact is confirmed algebraically by noting that the conjugation is quadratic in q, so the sign of q cancels, and does not affect the result. (See 2:1 mapping of SU(2) to SO(3)) If both rotations are a h... | Quaternions and spatial rotation | 0.814614 |
15,193 | In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle θ {\displaystyle \theta } about a fixed axis (called the Euler axis) that runs through the fixed point. The Eu... | Quaternions and spatial rotation | 0.814614 |
15,194 | "Quaternions, Interpolation and Animation" (PDF). Leandra, Vicci (2001). "Quaternions and Rotations in 3-Space: The Algebra and its Geometric Interpretation" (PDF). | Quaternions and spatial rotation | 0.814614 |
15,195 | In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all ... | Infinite-dimensional vector space | 0.81461 |
15,196 | It is commonly believed that a lower word error rate shows superior accuracy in recognition of speech, compared with a higher word error rate. However, at least one study has shown that this may not be true. In a Microsoft Research experiment, it was shown that, if people were trained under "that matches the optimizati... | Word error rate | 0.814608 |
15,197 | If this has happened, the atom(s) which are now adjacent needs to be moved away from the list of mobile atoms, and correspondingly their jump events removed from the list of possible events. Naturally in applying KMC to problems in physics and chemistry, one has to first consider whether the real system follows the ass... | Kinetic Monte Carlo | 0.814608 |
15,198 | Each first-place team member, as well as the winner of the Elizabeth Lowell Putnam Prize, receives $1,000. Sixth through 15th place individuals receive $1,000 and the next ten receive $250. The names of the top 100 students are published in the American Mathematical Monthly, and competition results are published in ear... | William Lowell Putnam Mathematical Competition | 0.814607 |
15,199 | The Putnam competition takes place on the first Saturday in December, and consists of two three-hour sittings separated by a lunch break. The competition is supervised by faculty members at the participating schools. Each one consists of twelve challenging problems. The problems cover a range of advanced material in un... | William Lowell Putnam Mathematical Competition | 0.814607 |
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