Scalar point function

Static pre.s.sure is the pressure of the moving fluid. The static pressure of a gas is the same in all directions and is a scalar point function. It can be measured by drilling a hole in the pipe and keeping a probe flush with the pipe wall. 

Suppose that 4> x,y,z) is a scalar point function, that is, a scalar function that is uniquely defined in a given region. Under a change of coordinate system to, say, x y z, it will take on another form, although its value at any point remains the same. Applying the chain rule (Section 2.12),... 

The static pressure in a flnid has the same vmue in all directions and can be considered as a scalar point function. It is the pressure of a flowing fluid. It is normal to the surface on which it acts and at any given point has the same magnitude irrespective of the orientation of the surface. The static pressure arises because of the random motion in the fluid of the molecules that make up the fluid. In a diffuser or nozzle, there is an increase or decrease in the static pressure due to the change in velocity of the moving fluid. 

The static pressure in a fluid has the same value in all directions and can be considered as a scalar point function. It is the pressure of a flowing fluid. It is normal to the surface on which it acts and at any... 

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. ... 

There have been many surveys on evolutionary techniques for MOO (Fonseca and Fleming, 1995 Coello Coello, 1998 Van Veldhuizen and Lamont, 2000 Tan et al, 2002 Chapter 3 in this book). While conventional methods combined multiple criteria to form a composite scalar objective function, modern approach incorporates the concept of Pareto optimality or modified selection schemes to evolve a family of solutions at multiple points along the tradeoffs simultaneously (Tan et al, 2002). 

On the other hand the so called wedge product of 1-forms a, p, is an example of a 2-form, at any point in phase space it can be viewed as a quadratie form, i.e. a scalar valued function of two vectors which is linear in each argument. It is written a A P and is defined, for vectors , j R by... 

This section collects results obtained with the three exact-decoupling methods within the same implementation and follows the discussion in Ref. [647]. The number of matrix operations necessary for the implementation of different two-component approaches has been collected in Table 14.2. The multiplication of a general matrix with a diagonal matrix requires O(m ) multiplications of floating-point numbers, where m is the dimension of the matrix identical to the number of (scalar) basis functions in this context. The multiplication of two general matrices scales formally as If m is large, the cost of the... 

The function ITxc is a pressure-like scalar memory function of two variables. In practice, ITxc is fully determined by requiring it to reproduce the scalar hnear response of the homogeneous electron gas. Expression (4.47) is clearly non-local in the time-domain but still local in the spatial coordinates. From the previous considerations it is clear that it must violate Galilean invariance. To correct this problem we use a concept borrowed from hydrodynamics. It is assumed that, in the electron liquid, memory resides not with each fixed point r, but rather within each separate fluid element . Thus the element which arrives at location r at time t remembers what happened to it at earlier times t when it was at locations different from its present... 

Let jP be a vector whose components are functions of a scalar variable (e.g. time-dependent position vector of a point F in a three-dimensional domain)... 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

For any matrix A it is convenient to let AK represent the set of all points Ax for which xeK, and to define aK — (al)K for any scalar a. Then a converse to the above theorem, which also holds, can be stated as follows if K is a bounded, dosed, equilibrated, convex body, then the function... 

On taking the scalar product with x° and recalling that for a linear programming problem the values of the objective function of the original problem and of the dual coincide at the solution points, we conclude that whenever xf > 0 we must have j8, = 0. 

The term scalar field is used to describe a region of space in which a scalar function is associated with each point. If there is a vector quantity specified at each point, the points and vectors constitute a vector field. 

In formulating physical problems it is often necessary to associate with every point (x, y, z) of a region R of space some vector a(x, y, z). It is usual to call a(x,y,z) a vector function and to say that a vector field exists in II. If a scalar x, y, z) is defined at every point of R then a scalar field is said to exist in R. 

Consider a function (or scalar field) ()> xl) of the point M (coordinates xl) and defined in the neighbourhood of M. Being a function of a point, the value of does not change when described in a different coordinate system. By the rules of differentiation... 

The concept of potential energy in mechanics is one example of a scalar field, defined by a simple number that represents a single function of space and time. Other examples include the displacement of a string or a membrane from equilibrium the density, pressure and temperature of a fluid electromagnetic, electrochemical, gravitational and chemical potentials. All of these fields have the property of invariance under a transformation of space coordinates. The numerical value of the field at a point is the same, no matter how or in what form the coordinates of the point are expressed. 

The two-point Green function for the /5-scalar field is defined, then, by... 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

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