Scalar value

A scalar-valued function/(/4) of one symmetric second-order tensor A is said to be symmetric if... 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

Analogous results are available for scalar-valued functions of more than two tensor variables, see, e.g., [20]. 

These conditions govern the flow and are therefore of crucial importance. For each condition (see Table 11.2) the flow value and the scalar values are discussed separately. The table contains volumetric sources, which are nor strictly speaking boundary conditions in a mathematical sense. For the CFD engineer they nevertheless define the problem and are therefore included in this table. The problem must also not be overspecified. 

Associated scalar values include temperature, turbulence quantities, contaminant concentrations. 

As shown in Fig. 5.4, the flow domain can be denoted by 2 with inlet streams at Ain boundaries denoted by 3 2, (/el,..., Ain). In many scalar mixing problems, the initial conditions in the flow domain are uniform, i.e., cc(x, 0) = 40). Likewise, the scalar values at the inlet streams are often constant so that cc(x e 3 2, t) = c(f for all / e 1,..., Nm. Under these assumptions,38 the principle of linear superposition leads to the following relationship ... 

Matrix elements are scalar-valued matrix functions of the exponent matrices Lk- Therefore, the appropriate mathematical tool for finding derivatives is the matrix differential calculus [116, 118]. Using this, the derivations are nontrivial but straightforward. We will only present the final results of the derivations. The reader wishing to derive these formulas, or other matrix derivatives, is referred to the Ref. 116 and references therein. 

The scalar values for the vectorial velocities for the four polymer films that account for the heat transfer and dissipative melting may be calculated as follows ... 

For our purpose, it is convenient to classify the measurements according to the format of the data produced. Sensors provide scalar valued quantities of the bulk fluid i. e. density p(t), refractive index n(t), viscosity dielectric constant e(t) and speed of sound Vj(t). Spectrometers provide vector valued quantities of the bulk fluid. Good examples include absorption spectra A t) associated with (1) far-, mid- and near-infrared FIR, MIR, NIR, (2) ultraviolet and visible UV-VIS, (3) nuclear magnetic resonance NMR, (4) electron paramagnetic resonance EPR, (5) vibrational circular dichroism VCD and (6) electronic circular dichroism ECD. Vector valued quantities are also obtained from fluorescence I t) and the Raman effect /(t). Some spectrometers produce matrix valued quantities M(t) of the bulk fluid. Here 2D-NMR spectra, 2D-EPR and 2D-flourescence spectra are noteworthy. A schematic representation of a very general experimental configuration is shown in Figure 4.1 where r is the recycle time for the system. 

Figure 4.1 Schematic diagram of a general-purpose CSTR recycle system for in situ investigations of liquid-phase homogeneous catalyzed reactions. The blocks represent in-line instruments and their signals, namely, (i) sets of scalar valued measurements (sensors), (ii) sets of vector valued measurements (ID spectroscopies) and (iii) sets of matrix valued measurements (2D spectroscopies). The recycle time for the system is given by t.

In principle, one can fashion a p-box that represents the best possible limits on the distribution of a variable given any specific state of knowledge about the variable (Person 2002). Such optimal p-boxes have already been worked out for cases in which the following information is available. (Note that the values can be specified as precise scalar values or as interval bounds.)... 

The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimen-sional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms mt + m2 is R(m,) + R(m2). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. 

Another class of problems requiring iteration is minimization or maximization of a nonlinear scalar valued function g which depends on one or ( variables x (ref. 4). A value r of the independent variables is a local minimum point if g(r) < g(x) for all x in a neighborhood of r. ... 

Note that, although up to this point we have described tlie expectation value of A as though it were a scalar value, it is also possible that A is a function of some experimentally (and computationally) accessible variable, in which case we may legitimately ask about its expectation value at various points along the axis of its independent variable. A good... 

Although the linearity of the chain-rule differential expressions (10.5) confers primitive affine-type spatial structure on thermodynamic variables, it does not yet provide a sense of distance or metric on the space (other than what might be displayed in an arbitrarily chosen axis system). In order to bring intrinsic geometrical structure to the thermodynamic space, we need to define the scalar product (R RJ) [(9.29)] that dictates the spatial metric on Ms- The metric on Ms should reflect intrinsic physical properties of the thermodynamic responses, not merely generic chain rule-type mathematical properties of their differential representation. At the same time, we must exhibit how the space Ms is explicitly connected to the physical measurements of thermodynamic responses. Because such measurements assign scalar values to physical properties, it is natural to associate each scalar product of Ms with the scalar value of an experimental measurement. How can this be done ... 

Some readers will recognize this development as the calculus of variations [7]. A functional is a function of a function in this case, T takes the function (r,t0) and maps it to a scalar value that is numerically equal to the total free energy of the system. 

Secondly, this seemingly innocent relationship between input and scattered wavevec-tors has a significant effect on the scattered (diffracted) beam. Because in all realistic cases the scalar value of k0 will greatly exceed that of K, if one plots the wavevector diagrams one will see that k+ and ko form the two sides of a very nearly flattened triangle, and for... 

Of the three quantities D, E, and p, only one is independent according to Equation (14.44). In the case of the parallel-plate condenser and an isotropic medium, all of the vectors are parallel and normal to the plates of the condenser. We are primarily concerned with their scalar values however, we continue to use the vector symbols for clarity. 

For the system that we consider here, the vectors H, B, and m are all parallel, and scalar values could be used for these quantities. However, we continue to use the vector symbolism for clarity. 

Velocity is frequently, and incorrectly, interchanged with speed. Speed is not a vector value, but rather a scalar value, and does not specify any particular direction of motion. Average speed is defined as ... 

Zadeh, L.A., "Optimality and Non-Scalar Valued Performance Criteria" IEEE Transactions 1963, AC8, 59. 

Using the hydrophobicity scale as an example, the amino acid residues can be represented by the real-numbered scalar value. They can also be classified with respect to their side chains as polar, nonpolar, or amphipathic, depending on the range of the hydrophobicity in the scale, such as in... 

Encoding Method Residue/ Window Vector Size Vector or Scalar Value ... 

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