Total set

Abstract: Consider solving a linear program in standard form where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is $O(m^2n)$. We propose to reduce that cost by replacing the normal equation matrix, $AD^2A^{\T}$, where $D$ is a diagonal matrix, with a "reduced" version (of same dimension), $A_QD_Q^2A_Q^{\T}$, where $Q$ is an index set including the indices of $M$ most nearly active (or most violated) dual constraints at the current iterate, with $M\geq m$ a prescribed integer. This can result in a speedup of close to $n/|Q|$ at each iteration. Promising numerical results are reported for constraint-reduced versions of a dual-feasible affine-scaling algorithm and of Mehrotra's predictor-corrector method [S. Mehrotra, it SIAM J. Optim., 2 (1992), pp. 575-601]. In particular, while it could be expected that neglecting a large portion of the constraints, especially at early iterations, may result in a significant deterioration of the search direction, it appears that the total number of iterations typically remains essentially constant as the size of the reduced constraint set is decreased down to some threshold. In some cases this threshold is a small fraction of the total set. In the case of the affine-scaling algorithm, global convergence and local quadratic convergence are proved.

Abstract: We describe a class of measurable subsets s in Md such that L2(Q) has an orthogonal basis of frequencies ex(x) = el2nX'x{x e £2) indexed by X e A C iR? . We show that such spectral pairs (I2, A) have a self-similarity which may be used to generate associated fractal measures p with Cantor set support. The Hubert space L2(fi) does not have a total set of orthogonal frequencies, but a harmonic analysis of ii may be built instead from a natural representation of the Cuntz C *algebra which is constructed from a pair of lattices supporting the given spectral pair (I2, A). We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on L2(/i).

Abstract: Under suitable conditions, an infinite direct product ⊗nUn(t) of continuous unitary one‐parameter groups Un(t) is again a continuous unitary one‐parameter group. This question is discussed here in terms of the generators An of Un(t). It is shown that the generator A of ⊗nUn(t) has a total set of product vectors in its domain of definition. As examples, the particle number, energy–momentum, and angular momentum operators in non‐Fock direct product representations of free fields are investigated. The spectra of these operators are determined.

Abstract: The invention provides a traffic signal period division method and system based on two-dimensional clustering. The method comprises steps that traffic flow data of a crossing is acquired; according tothe traffic flow data, a total flow sequence is determined, division processing is carried out based on flow change conditions of the total flow sequence, and the total flow sequence is divided intomultiple sub sequences to acquire multiple time segments which are taken as a first division point set V1; based on the branch directions of the crossing and the traffic flow data, the branch flow vector in each direction at each time segment is calculated, a sum vector angle of the branch flow vector of each time segment is determined, the sum vector angle is compared with a determination threshold, and multiple time segments are determined according to the comparison result and are taken as a second division point set V2; merging processing of the first division point set V1 and the second division point set V2 is carried out to acquire a total set V, and time segment division based on the total set V is carried out. The method is advantaged in that a problem of division failure caused by single parameter extraction is solved.