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More and rigorously
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, 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.
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 divergence
More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface.
More precisely, they were only able to perform the calculation when the theory contains one less flavor of chiral matter than the number of colors in the special unitary gauge group, because in the presence of fewer flavors an unbroken nonabelian gauge symmetry leads to an infrared divergence and in the case of more flavors the contribution in equal to zero.

More and vector
More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.
More generally, if a linear operator on a vector space ( possibly infinite-dimensional ) has finite-dimensional range ( e. g., a finite-rank operator ), then the rank of the operator is defined as the dimension of the range.
More generally yet, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.
More formally, a multivariate random variable is a column vector X = ( X < sub > 1 </ sub >, ..., X < sub > n </ sub >)< sup > T </ sup > ( or its transpose, which is a row vector ) whose components are scalar-valued random variables on the same probability space ( Ω,, P ), where Ω is the sample space, is the sigma-algebra ( the collection of all events ), and P is the probability measure ( a function returning every event's probability ).
More generally, a norm on a vector space is Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.
After a basis for the vector space is chosen, this ring is naturally identified with the ring of n-by-n matrices with entries in K. More generally, the endomorphism algebra of the free module is naturally n-by-n matrices with entries in R.
More generally, every module category over a ring is additive, and so in particular, the category of vector spaces over a field is additive.
More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales.
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
More formally, a support vector machine constructs a hyperplane or set of hyperplanes in a high-or infinite-dimensional space, which can be used for classification, regression, or other tasks.
More precisely, in order to define the covariant derivative of it is necessary first to extend to a continuously differentiable vector field in an open set.
More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle.
More generally, let V be a vector space over a field K, and let
More generally still, the general linear group of a vector space GL ( V ) is the abstract automorphism group, not necessarily written as matrices.
More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.
More precisely, if V is a vector space over &# x211D ;, the inner product is a function
More recently, a number of loop-free distance vector protocols have been developed — notable examples are EIGRP, DSDV and Babel.
More precisely, the vorticity of a flow is a vector field, equal to the curl ( rotational ) of its velocity field v. It can be expressed by the vector analysis formula, where is the nabla operator.

More and field
More broadly, scholars in the field define aesthetics as " critical reflection on art, culture and nature.
More sophisticated theories are valence bond theory which includes orbital hybridization and resonance, and the linear combination of atomic orbitals molecular orbital method which includes ligand field theory.
More precisely, and technically, a Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory.
More central to this field have been:
More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of the electrons in the superconducting band and consequently a longer London penetration depth of external magnetic fields and currents.
More often than not, a woman serving as a courtesan would last in that field only as long as she could prove herself useful to her companion, or companions.
More recently, this inclined geometry for field stars in the LMC has been confirmed via observations of Cepheids, core helium-burning red clump stars and the tip of the red giant branch.
More importantly, each is associated with a different field of study ; deixis is associated with linguistics, while indexicality is associated with philosophy.
More generally with these operations any field of sets is a Boolean ring.
More precise theory shows that during the action of the laser field, the time-averaged effective potential of H < sub > 2 </ sup >< sup >+</ sup > is formed which binds the electron even if very strong electromagnetic field is present in similarity to the hydrogen molecule ion and prevents ionization.
More so than simply sparking improvements in the budding field of home computing and gaming, the Z-80 also sparked a revolution in electronic music, as the first truly programmable polyphonic synthesizers ( as well as their peripherals ) relied heavily on implementations of this CPU.
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 generalized in the field of economics, cost is a metric that is totaling up as a result of a process or as a differential for the result of a decision.

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