-Let fX i ; Y k g denote a fixed basis of the Lie algebra n of a connected and simply connected nilpotent Lie group N of step two. Under a technical assumption on fX i ; Y k g, we prove that the Lie algebra w of vector fields V on N that satisfy [V ; X i ] l i X i is finite dimensional, a property that we refer to as rigidity. Our proof allows the explicit count of dim w.

Low-rank problems are nonlinear minimization problems in which the objective function, by means of a suitable linear transformation of the variables, depends on very few variables. These problems often arise in quantitative management science applications, for example, in location models, transportation problems, production planning, data envelopment analysis and multiobjective programs. They are usually approached by means of outer approximation, branch and bound, branch and select and optimal level solution methods. The paper studies, from both a theoretical and an algorithmic point of view, a class of large-dimension rank-two nonconvex problems having a polyhedral feasible region and $f(x)=\phi (c^Tx+c_0,d^Tx+d_0)$ as the objective function. The proposed solution algorithm unifies a new partitioning method, an outer approximation approach and a mixed method. The results of a computational test are provided to compare these three approaches with the optimal level solutions method. In particular, the new partitioning method performs very well in solving large problems.

The aim of this book is to provide a practical working tool for students in\ud Engineering, Mathematics, and Physics, or in any other field where rigorous\ud Calculus is needed. The emphasis is thus on problems that enhance students’ skill in\ud solving standard exercises with a careful attitude, encouraging them to devote an\ud attentive eye to what may or may not be done in manipulating formulae or deriving\ud correct conclusions, while maintaining, whenever possible, a fresh approach, that is,\ud seeking guiding ideas.\ud Every chapter starts with a summary of the main results that should be kept\ud in mind and used for the exercises of that chapter; this is followed by a selection of\ud guided exercises. The theoretical preamble is meant to recapitulate the main definitions\ud and results and should also offer a bird’s-eye view on the topic treated in the\ud chapter. Hence, the student can quickly review the main theoretical facts and then,\ud most importantly, “learn by examples,” becoming acquainted with the specific\ud techniques by seeing them applied directly to the problems. Each exercise ends with\ud a short comment which underlines the main issues of that specific exercise, the\ud leading ideas, and the main techniques. A selection of problems closes each\ud chapter, the answers to which are all listed in Solutions. The reader is urged to try to\ud solve some of these problems, which are similar, but not always trivially analogous,\ud to those that have been presented in detail

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