Individual Subspaces Research Articles

Constructing projections on sums and intersections

In this note, we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of C n in terms of the projections on the individual subspaces.

Quantitative analysis of solitary lesions of bone.

To quantitate the radiographic features of common solitary lesions of bone using vector analysis to optimize their radiographic description. Plain radiographs of 709 solitary bone lesions were reviewed. Relevant demographic, anatomic, and structural data were cataloged. These data were used to approximate the vector space defined by all possible solitary lesions of bone. Vector space analysis on the entire set of lesions was performed to determine similarities among and between lesion types. The 709 lesions generated 614 separate vectors within the appearance vector space. Only 10 vectors were represented more than 3 times, indicating great variability among and between most lesion types. Osteochondromas (with two vectors repeated 10 and 15 times, respectively) and peripheral chondrosarcomas (with one vector repeated 5 times) were relatively uniform in appearance. Gender bias was present for multiple types of lesions. Larger lesions that involved more than one anatomic center were more likely to be malignant. Lesions occurring in the femoral diaphysis and pelvic flat bones were statistically more likely to be malignant. Analysis shows that most lesions have highly varied vectors in the appearances within the vector space. More in-depth analysis of each lesion type using this quantitative technique is required to better define individual lesion subspaces and hence their radiographic appearances with respect to other bone lesions.

Experimental investigations of flow, turbulence and subspace formation on cold models of gas‐stirred reactors with a view to describing mixing processes by mathematical models

AbstractTheoretical and experimental investigations were performed with cold water models of bottom‐blown metallurgical ladles in order to develop a mathematical description of the mixing processes in such reactors. This analytical model is based on the subdivision of the reactor into interacting subspaces, whereby each subspace is regarded as an ideal mixer. The convective mass transfer between the subspaces is represented by constant coupling coefficients, and the mass balance for the material to be mixed in provides a homogeneous system of differential equations to determine the time dependence of concentration in the individual subspaces. Different combinations of interacting subspaces are represented. The flow field and the distribution of turbulence intensity measured by LDA in water models and the concentration profiles in mercury models measured by EMF cells show that a subdivision into five subspaces is adequate. Size, position and interaction coefficients are also determined by experiments. The calculated mixing times agree well with those measured.

Matrix elements of shell model Hamiltonians in multiple angular momentum coupling schemes.

Coefficients of fractional parentage (CFP's) in an L-S (or J-T) coupling scheme have recently been computed by using a new algorithm extending two procedures introduced many years ago: a factorization procedure introduced by Jahn and a combination of numerical and analytic methods introduced by Bayman and Lande. As a consequence of the factorization, coefficients of fractional parentage between states of arbitrary permutational symmetry must be computed within each angular momentum subspace separately. The new algorithm is extended here to calculate Hamiltonian matrix elements in the second quantized formalism. The matrix elements of the creation and annihilation operators, within each individual subspace, between states of arbitrary symmetry, are defined to be proportional to the corresponding coefficients of fractional parentage. The matrix elements for the one- and two-body operators in the Hamiltonian are constructed by combining the matrix elements in the individual subspaces using a new sum-over-path-overlaps technique. This procedure is extremely efficient for quark shell model calculations, where the quarks carry multiple quantum numbers of angular momentum, or more general, type (L, S, color, and flavor).

Reply to Comment by Paul R. Motyka and G. Warren Hall

WE wish to thank P. R. Motyka and G. W. Hall for their comments on our decoupling paper. They have correctly explained why we arrived at a result different from theirs when solving their problem. When the calculations presented in Ref. 2 were done, we were at a loss to explain the nonagreement of results. It was discovered that their solution did not give the prescribed eigenvalues and we incorrectly assumed that was the explanation. As Motyka and Hall have correctly pointed out, we had, in fact, assigned the eigenvalues to different eigenvectors. It is not true, however, that this is an inherent shortcoming of the geometric approach. As was pointed out, the geometric theory provides a method for placing the poles that was not used in Ref. 2. In fact, for the lateral motions case, it is a simple matter to assign eigenvalues independently to the individual controllability subspaces. It is true that the heuristic Penalty Function Method described in Ref. 2 must be used with considerable care, and we are indebted to Motyka and Hall for making this point. Next, we consider the comments relating to exact model following. In order to minimize confusion, we shall try to be somewhat precise. In the notation of Ref. 3, the given system is described by

Level density for some partially random Hamiltonians

We study the level density of a random HamiltonianH such that the total configuration space can be split into irreducible subspaces using for example invariance properties. The level density is supposed to be known for each individual subspace and the matrix elements connecting the various subspaces are treated as random variables. A similar problem arises when the Hamiltonian may be written in the formH0 +V, whereH0 has a known level density andV, the residual interaction, is a random matrix. In both cases, we start from the relation of the level density to the average of the resolvent. We expand the latter in powers of the interaction, and by scaling considerations we get the dominant contributions. The final result is easily obtained when the matrix elements of the residual interaction are statistically independent. Some simple algebraic examples are given; in particular, we recover the « semi-circle » law and indicate some generalizations.