Total space

The total space of system coordinates consists of a tagged coordinate Q (conjugate momentum P) and a set of mass-scaled bath coordinates q (conjugate momenta p). The Hamiltonian reads... 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

The ratio of equations 8.3.58 and 8.3.57 gives the relative total space time requirement for a cascade of stirred tank reactors vis a vis a plug flow reactor. 

Sentential calculus can only be used when the total space of all statements is easily numerable, since all properties about objects and relationships must be stated in separate explicit sentences. 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

As remarked above, we may define the unstable manifold W of the fixed point set A E for the total space of a holomorphic line bundle over E. More generally, let X be a surface containing a curve E. By identifying a tubular neighborhood of E with the total space of the normal bundle of E in X, we can define the unstable manifold W. In fact, this can be defined intrinsically as follows. Let tt X "] S "X be the Hilbert-Chow morphism. We define... 

The integral of dfjdE over a total space of E gives the total number Z of electrons in the molecule under consideration,... 

The theory of gravitation, however, need not involve four dimensions information [99] may exist on a two dimensional surface, such as the event horizon of a black hole. If the symmetries relevant to gravitation involve the evolution of a two-dimensional surface, then an SU(2) x SU(2) x SU(3) gauge theory plus gravity would be 11-dimensional, and duality between the two surfaces that construct spacetime would reduce this to nine dimensions. However, the issue of duality with nuclear interactions would still increase the dimensionality required to 12 or 14, and supergravity requires a total space of 11 dimensions. Strings exist at 10 dimensions. 

Averaging is a different kind of operation since it associates with a stochastic variable a non-stochastic or sure number. Alternatively it may be viewed as a projection in the following way. The set of all stochastic variables contains a subset of variables whose probability density is a delta peak. This subset is isomorphic with the sure numbers of the range and may therefore be identified with them. The operation of taking the average is then a projection of the total space of stochastic variables onto this subset. 

Void Volume This is the total space surrounding the gel particles in a packed column. This value is determined by measuring the volume of solvent required to elute a solute that is completely excluded from the gel matrix. Most columns can be calibrated for void volume with a dye, blue dextran, which has an average molecular mass of2,000,000 daltons. 

A morphism f X — S can always be considered as a family of schemes if we view it as the totality of its fibres, if f is flat we call it a fiat family of schemes parametrized bv 5 X is called the total space of the family f If 5 is connected and s e 5 is a k-rational point, f is also called a flat family of deformations of the fibre X(s). When S=Spec(A) with A a local k-algebra with residue field k, the flat morphism f is called a local family of deformations of %(q), the ribre over the closed point 5 if moreover the local k-algebra A is artinian, then f is called an infinitesimal family of deformations, or an infinitesimal deformation, of X(o). 

BertinTs. 11-16, 11-17 Clifford s, 11-19 Hilbert-Burch, 9-11 Macaulay s, 9-7 Schiessinger s, 6-14 Semicontinuity, 3-10 Total space, 2-4 Trivial relation(s), 4-9... 

After coating, the titer-plates are inserted into the reactor to test their activity. The reaction conditions were held constant in the following experiments. The reactor was heated to 475 °C and held at a pressure of 0.2 bar (g). The heat exchanger was operated at 50 °C, and the throughput for a single well was adjusted to 1 ml min-1, resulting in a total space velocity of 9000 h 1. The residence time in the wells was 0.4 s. 

The region of our ne versus T plot in which plasma-assisted CVD reactors function is also shown by the shaded area. Clearly, this only represents a very small region of the total space within which a reactor could operate. In the future, operation of CVD reactors in other regions of this plot may lead to CVD... 

The value of 1.003 atm is slightly larger than that of He, owing to the fact that the larger S02 molecules occupy more total space, which means they will experience more collisions with the container than He atoms. The much higher correction factor for molecular attractions (0.0134 atm) is higher than He because the much larger molecule is more capable of intermolecular interactions than He. 

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