Sinc-function discrete variable

The MINLP model for the synthesis problem consists of minimizing the objective function in (29) subject to the feasible space defined by eqs. (21)-(28). The continuous variables (/. q, ( hu, qcu, dt, dtcu. discrete variables c, ecu, zhu are O-I. The advantage of this model is that the constraints (21)-(28) are all linear. The nonlinearities have all been placed in the objective function (29). However, it should be noted that since these terms are nonconvex, the MINLP may lead to local optimal solutions. 

Since the design/retrofit problem embeds batch plant scheduling, it systematically includes the determination of the production sequence and the equipment sizes based on a performance criterion. Equipment sizes are considered either as continuous variables or as discrete ones and so the problem can involve either discrete variables or a set of mixed ones. Most of the existing literature has focused on single objectives involving a cost criterion typically based on capital investment. This criterion is generally expressed as a non-linear function of the size of the equipment, following the six-tenths rule. 

The Chebyshev expansion method [430] is applied to the propagation of the nuclear wavepacket subject to the above Hamiltonian for both the two-and three-state models. Nuclear wavefimctions, the potential functions, the matrix to represent the first and second nuclear derivatives arising from T of Eq. (6.70) are all expressed in the sine discrete variable representation (sinc-DVR) [92]. The time length for one-step integration of nuclear wavepacket is set to 0.02 fs. The 1200 DVR grid points are employed within a range from —3 to 14 Bohrs. For a practical reason, the potential function is cut off in the range shorter than 1.2 Bohrs. 

Forming the matrix representation of the Hamiltonian operator and manipulating the Hamiltonian matrix to obtain the observable of interest can be computationally intensive. A discrete variable representation [53-55] (DVR) can ameliorate both of these difficulties. That is, the construction of the Hamiltonian matrix is particularly simple in a DVR because no multidimensional integrals involving the potential function are required. Also, the resulting matrix is sparse because the potential is diagonal, which expedites an iterative solution [37, 38]. In the present research we use a sinc-function based DVR (vide infra) first developed by Colbert and Miller [56] for use in the 5—matrix version of the Kohn variational principle [57, 58], and used subsequently for 5—matrix calculations [37, 38] in addition to N(E) calculations [23]. This is a uniform grid DVR which is constructed from an infinite set of points. It is... 

As is well known, the steady-state behavior of (spherical and disc) microelectrodes enables the generation of a unique current-potential relationship since the response is independent of the time or frequency variables [43]. This feature allows us to obtain identical I-E responses, independently of the electrochemical technique, when a voltammogram is generated by applying a linear sweep or a sequence of discrete potential steps, or a periodic potential. From the above, it can also be expected that the same behavior will be obtained under chronopotentiometric conditions when any current time function I(t) is applied, i.e., the steady-state I(t) —E curve (with E being the measured potential) will be identical to the voltammogram obtained under controlled potential-time conditions [44, 45]. 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Indeed, since the macroscopic states of a protein are discrete, they are described by discrete surfaces in the phase space of considered variables (Pfeil and Privalov, 1976c). The small globular proteins, or individual cooperative domains, which have only two stable macroscopic states, the native (N) and denatured (D), are described by two surfaces in the phase space, corresponding to their extensive thermodynamic functions. The transition between these states is determined by the differences of... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

Another test for the presence of quantum chaos is to investigate the power spectrum of the auto-correlation function 9 t). It is immediately obvious from (1.4.8) that the power spectrum of a bounded system is a countable sum of S functions. Since no chaos is present in (1.4.8), one may conjecture that for chaos to be present it is necessary that the power spectrum of 9 t) contains a continuous component. Other tests for quantum chaos are to compute the power spectrum of expectation values of dynamical variables, for instance the position x(t) = ip t) x ip t)). Again, a purely discrete power spectrum indicates regular time evolution of 0) whereas a continuous component indicates the presence of chaos. Obviously, a continuous component in the power spectrum is only a necessary condition for chaos. It is not sufficient since unbounded systems, for instance scattering systems, may show a continuous component without any sign of chaos in their dynamics. This state of affairs was known more than 20 years ago (Lebowitz and Penrose (1973), Hogg and Huberman (1982), Wunner (1989)). 

The alternative procedure has the advantage of enabling the solution of an equation on the flux form, whereas the temperature equation is formulated on the advective form. Moreover, the enthalpy quantity is often representing a well behaved smooth function whereas the temperature variable might oscillate and represent steep gradients. However, the method has the drawback that in many cases the enthalpy variable has to be converted into temperature at a sufficient number of discretization points and for every time step in the solution process, since the boundary conditions used are normally expressed in terms of temperature. Besides, the transformation formulas for non-ideal reactive flow systems can be rather complex. 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

Since input and/or output variables typically fail in a discrete manner, the evaluation of any single-point failure effects on the software program, the computer hardware, and/or the system can be accomplished through the software FHA. The drawback here is that the specific effects of any such failure may be somewhat difficult to define since they are typically a function of the actual operational state of the computer at the exact time that the failure occurs. Hence, even evaluation of hypothetical scenarios could be a monumental task since so many possible variables are bound to exist. Also, the software FHA can be an extremely lengthy and quite tedious process that may or may not yield any significant results. A decision to proceed with such an effort must therefore be weighed against the anticipated benefits that are expected on its completion. 

The knowledge of these second-order correlations is sufficient for calculation of all dynamic observables in equilibrium since all higher order correlation functions can be expressed via these functions due to independence of modes and Cartesian components. Below we list expressions for most common observables. In the Appendix, we present an alternative solution of Rouse model in continuous variables, which is valid for long enough chains. In the limit N—yoo, two formalisms give same results for but the exact discrete solution of this... 

For some values of 7, this expression will yield MOs that are not normalized.) Equation (15-6) tells us that MO continuous function [by replacing n — )/6 with the continuous variable 0] and then locate the places on this coefficient wave that correspond to the discrete points of interest. Sketches for this coefficient wave when 7 = 3 and 7 = 9 are shown in Fig. 15-6. The special points of interest, where actual coefficient values are given... 

Uniform Distribution n A probability distribution where the probability in the case of a discrete random variable or the density function in the case of a continuous random variable, X, with values, x, are constant (equal) over an interval, a,b), where x is greater than or equal to a and X less than or equal to b and x is zero outside the interval. The uniform distribution is sometimes referred to as the rectangular distribution since, a plot of its probability or density function resembles a rectangle. When the random variable, X, is discrete, the uniform distribution is referred to as the discrete uniform distribution and has the probability, P(x. ), with the form of ... 

---