Riccati matrix

Mixing of two or more polymers of different chemical composition offers a powerfiil way of tailoring performance and economic relationships using existing materials. As a result, the area of polymer blends or alloys has become an important one for both scientific investigation and commercial product development. Fundamental issues that affect the properties of blends include equilibrium phase and interfacial behavior, physical and chemical interactions between the components, phase morphology, and rheology, all of which relate to pragmatic issues of compatibility. One of the most important examples of polymer blends is the judicious incorporation of an elastomeric phase in a rigid matrix to enhance mechanical toughness.  [c.408]

Copolymers are typically manufactured using weU-mixed continuous-stirred tank reactor (cstr) processes, where the lack of composition drift does not cause loss of transparency. SAN copolymers prepared in batch or continuous plug-flow processes, on the other hand, are typically hazy on account of composition drift. SAN copolymers with as Httle as 4% by wt difference in acrylonitrile composition are immiscible (44). SAN is extremely incompatible with PS as Httle as 50 ppm of PS contamination in SAN causes haze. Copolymers with over 30 wt % acrylonitrile are available and have good barrier properties. If the acrylonitrile content of the copolymer is increased to >40 wt %, the copolymer becomes ductile. These copolymers also constitute the rigid matrix phase of the ABS engineering plastics.  [c.507]

In general, for a second-order system, when Q is a diagonal matrix and R is a scalar quantity, the elements of the Riccati matrix P are  [c.280]

In reverse-time, starting with P(A ) = 0 at NT = 20 seconds, compute the state feedback gain matrix K(kT) and Riccati matrix P(kT) using equations (9.29) and (9.30). Aiso in reverse time, use the desired state vector r(/c7 ) to drive the tracking equation (9.53) with the boundary condition s(N) = 0 and hence compute the command vector y kT).  [c.282]

Continue the recursive steps until the solution settles down (when k = 50, or kT = 5 seconds) and hence determine the steady-state value of the feedback matrix K(0) and Riccati matrix P(0). What are the closed-loop eigenvalues  [c.323]

Steady-state value of Riccati matrix 2.3145 1.6310  [c.410]

Riccati matrix, equation  [c.414]

In order to perform this procedure on all elements of matrix A a margin with constant values aj and a ,j and a length of X (left/right) or % (top/bottom) is added on each side of matrix A. The two new matrices A and Ai are combined to form a resulting matrix Ax for the filtering in x-direction  [c.262]

There are two matrix inverses that appear on the right-hand side of these equations. One of these is trivial the  [c.49]

Rost J M 1998 Semiclassical s-matrix theory for atomic fragmentation Phys. Rep. 297 272-344  [c.1003]

A3.13.1). From [38]. The two-level structure (left) has two models I I = const and random signs (upper part), random V.j but V < V.j < (lower part). The right-hand side shows an evolution with initial diagonal density matrix (upper part) and a single trajectory (lower part).  [c.1079]

The ADT matrix for the lowest two electronic states of H3 has recently been obtained [55]. These states display a conical intersection at equilateral triangle geometi ies, but the GP effect can be easily built into the treatment of the reactive scattering equations. Since, for two electronic states, there is only one nonzero first-derivative coupling vector, w5 2 (Rl), we will refer to it in the rest of this  [c.197]

Note that these matrix elements are numbers, in comparison with the term on the right-hand side of Eq. (A. 10), which involves matrix elements of the second  [c.313]

Let us consider a singlet II electronic species (right-hand side of Fig. 1) and first assume that the magnitude for the splitting of the potential surfaces upon bending is negligible, that is, that the electronic state remains degenerate at small-amplitude bending vibrations (this degeneracy is accidental, because it does not follow for symmetry reasons), and that the bending potential is harmonic. The bending potential then has the same form as that presented on the left-hand side of Fig. 1 (top), representing a E electronic state, but it consists of two potential energy surfaces coinciding with each other. In the II state, we also have the electronic angular momentum besides the vibrational state. In the (hypothetical) case we consider (no splitting of the potential surfaces), the presence of the additional electronic angular momentum has no effects on the position of vibronic energy levels. That becomes obvious if we look at matrix (32)—its off-diagonal elements vanish in this case. On the other hand, the number of levels is doubled. Furthermore, the presence of two angular momenta has as a consequence that the vibronic levels have to be classified according to the quantum number corresponding to their sum being the only angular momentum that commutes with the Hamiltonian. A simple bookkeeping shows that the lowest lying (nondegenerate) vibrational level of the S electronic state, characterized with the quantum numbers u = 0, ( = 0, correlates with the (doubly degenerate) 0 = 0, = 1 level of the II state, that the u = 1, / = 1 level  [c.490]

Equations (169) and (171), together with Eqs. (170), fomi the basic equations that enable the calculation of the non-adiabatic coupling matrix. As is noticed, this set of equations creates a hierarchy of approximations starting with the assumption that the cross-products on the right-hand side of Eq. (171) have small values because at any point in configuration space at least one of the multipliers in the product is small [115].  [c.698]

Here, tm( >) = 1,2,..., A/ are the eigenvalues of T((p) and A(p is the angular grid size. The order of the multiplication in Eq. (191) is such that the k = 0 term is the first term from the right-hand side in the product. With these definitions the matrix D is defined as D( /) = A(


Equation (B.IO) stands for the j,k) matrix element of the left-hand side of Eq. (B.7). Next, we consider the (j,k) element of the first term on the right-hand side of Eq. (B.7), namely.  [c.720]

Methods for simulating rigid bodies typically rely on introduction of some set of generalized coordinates describing the position and orientation of each body. There are several popular parameterizations in use, including (1) quaternions, (2) cartesian (particle) models, and (3) orientation matrix description. Once a set of variables is chosen, the equations of motion are recast in those variables, and the resulting differential equations are then discretized in time using some numerical timestepping method.  [c.350]

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]  [c.356]

Furfural can be classified as a reactive solvent. It resiniftes in the presence of strong acid the reaction is accelerated by heat. Furfural is an excellent solvent for many organic materials, especially resins and polymers. On catalyzation and curing of such a solution, a hard rigid matrix results, which does not soften on heating and is not affected by most solvents and corrosive chemicals.  [c.75]

An example of such a polycycHc compound is l,2 5,6-dibenzacridine [226-36-8], C22H23N, which, in a rigid matrix, absorbs uv radiation to form a triplet state absorbing strongly in the visible with a maximum at approximately 550 nm.  [c.163]

For the case of a double-D coil we multiply each matrix element with an element shifted by a constant distance of the same line. This is done in x- and y-direction. The distance between the two elements is the correlation length X for filtering in x-direction and a second correlation length for the movement in y-direction. Thus one gets two new matrices Ax and Ax for the filtering from the left to the right (positiv x-direction) and vice versa (negativ x-direction).  [c.261]

Thus, one can solve for the eigenvalue iteratively, by guessing 1, evaluatmg the right-hand side of equation (A 1.1.145), using the resulting value as the next guess and continuing in this manner until convergence is achieved. However, this is not a satisfactory method for solving the Scln-ddinger equation, because the problem of diagonalizing a matrix of dimension A -t 1 is replaced by an iterative procedure in which a matrix of dimension A must be inverted for each successive improvement in the guessed eigenvalue. This is an even more computationally intensive problem than the straightforward diagonalization approach associated with the linear variational principle.  [c.48]

Each temi on the right-hand side of the equation involves matrix produets drat eontain v a speeifie number of times, eitlier explieitly or implieitly (for the temis that involve A i,). Reeognizing that is a zeroth-order quantity, it is straightforward to make the assoeiations  [c.50]

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]). Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).
Of the four possible WMEL diagrams for each the and doorway generators, only one encounters the Raman resonance in each case. We start with two parallel horizontal solid lines, togedier representing the energy gap of a Raman resonance. For ket evolution using, we start on the left at the lowest solid line (the ground state, g) and draw a long solid arrow pointing up (+co ), followed just to the right by a shorter solid arrow pointing down (-CO2) to reach the upper solid horizontal line, / The head of the first arrow brings die ket to a virtual state, from which the second arrow carries the ket to the upper of the two levels of the Raman transition. Since the bra is until now unchanged, it remains in g ((g ) this doorway event leaves the density matrix at second order off-diagonal in which is not zero. Thus a Raman coherence has been established. Analogously, the doorway action on the ket side must be short solid arrow down (-012) from g to a virtual ket state, then long arrow up (+C0j) to /from the virtual state. This evolution also produces. Both doorway actions contain the same Raman resonance denominator, but differ in the denominator appearing at the first step the downward action is iidierently anti-resonant ( N for nonresonant) in the first step, the upward action is potentially resonant ( R for resonant) in the first step and is therefore stronger. Accordingly, we distinguish these two doorway events by labels and respectively (see figure BL3.2. In resonance Raman spectroscopy, this first step in is fiilly resonant and overwhelms D-. (The neglect of D- is known as the rotating wave approximation.) It is unnecessary to explore the bra-side version of these doorway actions, for they would appear in the fiilly conjugate version of these doorway events. Each of the doorway steps,  [c.1188]

On the molecular level, spectacular AFM images have been obtained for a niunber of systems. In the case of isotactic polypropylene, for example, Snetivy and Vancso [143] have succeeded in imaging individual methyl groups on the polymer chain, and distinguishing between left- and right-handed helices in the crystalline i-polypropylene matrix (figure B 1.19.31). The same group has also used AFM to image the phenylene groups in poly(/i-phenyleneterephthalamide) fibres, and have used this data to show the existence of a new polymorphic fonn that had previously only been suggested by computer simulations.  [c.1706]

Beratan D N and Hopfield J J 1984 Calculation of electron tunneling matrix elements in rigid systems mixed valence dithiaspirocyclobutane molecules J. Am. Chem. Soc. 106 1584-94  [c.2995]

Note that the exact adiabatic functions are used on the right-hand side, which in practical calculations must be evaluated by the full derivative on the left of Eq. (24) rather than the Hellmann-Feynman forces. This forai has the advantage that the R dependence of the coefficients, c, does not have to be considered. Using the relationship Eq. (78) for the off-diagonal matrix elements of the right-hand side then leads directly to  [c.292]

THE cvcLOBUTADENE-TETRAHEDRANE SYSTEM. A related reaction is the photoisomerization of cyclobutadiene (CBD). It was found that unsubstituted CBD does not react in an argon matrix upon irradiation, while the tri-butyl substituted derivative forms the corresponding tetrahedrane [86,87]. These results may be understood on the basis of a conical intersection enclosed by the loop shown in Figure 37. The analogy with the butadiene loop (Fig. 13) is obvious. The two CBDs and the biradical shown in the figure are the three anchors in this system. With small substituents, the two lobes containing the lone electrons can be far  [c.370]

The bent point of view offers the explanation of one other aspect of the vibronic energy pattern presented. On the left-hand side of the bottom part of Figure 1 are presented the bending levels for two S electronic states having the same potential surface as the components of the fl state considered. This situation can be looked upon as a paiticular case, A = 0, of the matrix representation (25) the coupling between the electronic states vanishes and each of them has its own bending levels with the pattern analogous to that on the left-hand side of Figure 1, top. The difference in the vibronic structure of two E and a fl electronic state is caused by the presence of the off-diagonal elements of the mabix (25) in the latter case. However, even in H elecbonic states the off-diagonal elements vanish for the particular case A = 0 and these vibronic levels belong exclusively to one of the adiabatic elecbonic states. This is indicated symbolically on the right-hand side of Figure 1 by the corresponding energy level lines matching exactly one of the adiabatic potential curves. The +j— symmetry of a A = 0 level is determined by the symmetiy of the adiabatic state  [c.491]

See pages that mention the term Riccati matrix : [c.22]    [c.277]    [c.288]    [c.358]    [c.39]    [c.46]    [c.781]    [c.1187]    [c.2303]    [c.2349]    [c.2496]    [c.3064]    [c.140]    [c.193]    [c.193]    [c.214]    [c.718]    [c.352]   
Advanced control engineering (2001) -- [ c.0 ]