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More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than Planck's constant ħ, so the " deformation parameter " ħ / S can be effectively taken to be zero.
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More and rigorously
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,
More rigorously, if a surface S is a union of finitely many pieces S < sub > 1 </ sub >, …, S < sub > r </ sub > which do not overlap except at their boundaries then
More rigorously, an interpretation of quantum mechanics satisfies CFD if it includes in the statistical population of measurement results, those measurements which are counterfactual by virtue of their being excluded by the quantum mechanical prohibition on simultaneous measurement of certain pairs of properties.
More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of X.
More rigorously, metonymy and synecdoche may be considered as sub-species of metaphor, intending metaphor as a type of conceptual substitution ( as Quintilian does in Institutio oratoria Book VIII ).
More rigorously, since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal.
More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value.
More rigorously, they are said to be isomorphic or “ structure preserving ”.
More serious consultation with a range of internationally adopted people from various professional and community-work based backgrounds needs to be included before the field of adoption study is more truly representative and rigorously informed.
More rigorously the effective temperature corresponds to the temperature at the radius that is defined by the Rosseland optical depth.
The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.
More rigorously
More rigorously, Pickover's " biomorphs " encompass the class of organismic morphologies created by small changes to traditional convergence tests in the field of " Julia set " theory.
More and mathematical
More complex mathematical modeling continues to be done today.
More formally, it is " the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.
More readably and idiomatically, one might use the following equivalent, which demonstrates a simple procedure definition and the use of the mathematical operators and:
More generally, square roots can be considered in any context in which a notion of " squaring " of some mathematical objects is defined ( including algebras of matrices, endomorphism rings, etc.
More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time.
More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics.
More recently ( 2001 and 2002 ), Bailey and Richard Crandall showed that the existence of this and similar formulas has implications for the long-standing question of " normality " – whether and why the digits of certain mathematical constants ( including pi ) appear " random " in a particular sense.
More complex systems arising in biology, medicine, the humanities, management sciences, and similar fields often remained intractable to conventional mathematical and analytical methods.
# More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object.
More than 180 named mathematical facts have so benefited from formal codification.
More recent work has restored rigor to infinitesimals, making them once more a legitimate mathematical topic ( see non-standard analysis ).
More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
More recently, the mathematician Underwood Dudley has written a series of books on mathematical cranks, including The Trisectors, Mathematical Cranks, and Numerology: Or, What Pythagoras Wrought.
More recently, Yaneer Bar-Yam has claimed that the gene-centered view ( and thus Ronald Fisher's treatment of evolution ) relies upon a mathematical approximation that is not generally valid.
" More formal mathematical descriptions of this relation are often called the term structure of interest rates.
More general affine connections were then studied around 1920, by Hermann Weyl, who developed a detailed mathematical foundation for general relativity, and Élie Cartan, who made the link with the geometrical ideas coming from surface theory.
More generally a stack refers to any category acting more or less like a moduli space with a universal family ( analogous to a classifying space ) parametrizing a family of related mathematical objects such as schemes or topological spaces, especially when the members of these families have nontrivial automorphisms.
More recently, Christopher Isham, Jeremy Butterfield, and their students have continued to develop quantum geometrodynamics to take account of recent work toward a quantum theory of gravity and further developments in the very extensive mathematical theory of initial value formulations of general relativity.
More importantly, his mathematical and graphical calculation methods improved turbine design and engineering.
More recently Paul Ernest has proposed a social constructivist account of mathematical knowledge, drawing on the works of both of these sociologists.
More recently, attempts have been made to construct mathematical models of the stereolithography process and design algorithms to determine whether a proposed object may be constructed by the process.
More significantly, he became interested in developing a linguistic theory using a non-taxonomic approach and based on mathematical formalism, and this line of inquiry represented a decisive break with the Bloomfieldian taxonomic structuralist tradition of linguistic analysis.
More critical or optimized designs now routinely use mathematical modeling on a computer.
More and operation
More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S × S to S:
More simple hydrofoils such as the Russian designed ones and those by Rodriquez in Italy are more competitive in price and are still being produced today for operation on ferry routes.
More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H. This is usually represented notationally by H ≤ G, read as " H is a subgroup of G ".
More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then ( A, m, e, inv ) is a group object in the category of groups ( or monoids ).
More powerful PLCs can operate on a group of internal memory locations and execute an operation on a range of addresses, for example, to simulate a physical sequential drum controller or a finite state machine.
More recently, decentralized IP-based CCTV cameras, some equipped with megapixel sensors, support recording directly to network-attached storage devices, or internal flash for completely stand-alone operation.
More generally, the general linear group of degree n over any field F ( such as the complex numbers ), or a ring R ( such as the ring of integers ), is the set of n × n invertible matrices with entries from F ( or R ), again with matrix multiplication as the group operation.
More complicated switch systems, such as double slips are restricted to low-speed operation.
* More common than hardware failure is incorrect or unstable operation, which may require the computer to be restarted frequently and can cause loss of data, which may be relatively small or can destroy the entire filesystem.
More than 79, 000 people were killed at Majdanek ( 59, 000 of them Polish Jews ) during the 34 months of its operation .</ ref > The last major, widely publicized prosecution of 16 SS members from Majdanek ( Majdanek-Prozess in German ) took place from 1975 to 1981 in West Germany.
More than 200 firefighters were involved in the rescue operation.
More recently, companies have offered their services to act as a sort of agent between the writer and a small printing operation.
More specifically, hot swapping describes replacing components without significant interruption to the system, while hot plugging describes the addition of components that would expand the system without significant interruption to the operation of the system.
More complex implementations may recommend but do not require that the component be shut down, with sufficient redundancy in the system to allow operation to continue if a component is removed without being shut down.
More than fifty schools are in operation today.
More complex synchronization mechanisms can be created from this primitive operation.
More than one million Poles, living mostly in the disputed territories, remained in the SU, systematically persecuted by Soviet authorities because of political, economical and religious reasons ( see the Polish operation of the NKVD ).
* More than half of these ancient Israeli structures have now been threatened or obliterated by modern construction projects, including Tumulus # 4, which was excavated hastily in a salvage operation.
More than 4 forest resorts are still on operation.
More than 8, 600 of the division's troops deployed during this operation.
More than 2 W continuous-wave ( CW ) operation at-10 degrees Celsius heat-sink temperature was also reported in 1998 from a VCSEL array consisting of 1, 000 elements, corresponding to a power density of 30 W / cm < sup > 2 </ sup >.
More commonly nowadays, it has become an obstacle to be avoided for normal operation of particle accelerators, vacuum electronics, radars, satellite communication devices, and so forth.
More precisely, such structures generalize well-known spaces of functions in a way that the operation " take a value at this point " may not be defined.
More peaceful operation carried out by the squadron included survey flights, casualty evacuation and carrying mail, while the squadron carried out a long-distance return trip from Aden to Cairo and back in 1932.
2.379 seconds.