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This would accelerate performance on any platform , but the additional performance gains afforded by adapting code for linear algebra and DSP functions to use the AltiVec processor — the 128-bit vector processing unit available on G4 and G5 chips that permits math operations to be done in parallel — resulted in an unprecedented level of performance . In the second part of the book , various real life applications are presented , most of them formulated as mixed integer linear or nonlinear programming problems . In an ideal homogeneous atmosphere with minimal complications from gradients or unresolved features , photospheric Stokes Spectra should have a straightforward pattern that is symmetric in wavelength for linear polarization and anti-symmetric for circular . Abstract : This paper presents an integer linear programming approach with delayed column generation for the NP Multiple-Depot Vehicle Scheduling Problem ( MDVSP ) in public mass transit . The direct solution of these linear systems using Gaussian elimination is prohibitively expensive in terms of both time and storage , so iterative methods are used . We formulate the problem as a mixed integer linear programming problem and develop a cutting plane algorithm as well as several heuristics to solve it . The problem is formulated as an integer linear programming model where 0-1 variables represent different paths . In infinite dimensional problems , the PCG method may be just formally replaced by any Galerkin method such as FEM for linear elliptic problems . The study of these linear systems started historically in the 1960 's and required a Ph.D. in math as a necessary prerequisite . We formulate the problem as a mixed integer linear programming problem and present a cutting plane algorithm as well as several heuristics for its solution . Instead of seeing ILS systems which deliver highly customized , diagnostic and prescriptive instruction , we see systems which move students along relatively crude learning paths in primarily linear fashion with little adaptation to match individual learning styles and preferences , extremely limited corrective interventions and very little branching to meet individual needs . From the point of first sight to directly underneath the structure is about another 117 linear meters ( 384 linear feet ) .