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Discrete Mathematics and Optimization

Let F be an ordered topological vector space (over R) whose positive cone F+ is weakly closed, and let E⊆ F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-∗ dense in the topological dual wedge E+′. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces., Optimization

Let F be an ordered topological vector space (over R) whose positive cone F+ is weakly closed, and let E⊆ F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-∗ dense in the topological dual wedge E+′. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces., Discrete Mathematics and Optimization

We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points.

Получен ряд вспомогательных утверждений, необходимых для решения задачи глобального управления показателями Ляпунова трехмерных линейных систем обыкновенных дифференциальных уравнений с локально-интегрируемыми и интегрально ограниченными коэффициентами. Основополагающую роль среди этих утверждений для решения такой задачи играет теорема, устанавливающая существование для трехмерной линейной равномерно вполне управляемой системы с вышеуказанными коэффициентами трех измеримых и ограниченных векторных управлений таких, что решения (траектории движения) системы с этими управлениями на произвольном отрезке времени фиксированной длины лежат в трех выпуклых круговых конусах достаточно малой угловой меры и образуют базис трехмерного евклидова векторного пространства достаточно большого объема. = The number of auxiliary statements necessary for solving the problem of global Liapunov indicators management trivariate linearly systems of ordinary equations with locally-integrated and integrally limited quotients has been obtained. The essential role among these statements for solving these task plays the theorem that establishes the existence for trivariate linearly completely controllable system with the abovementioned quotients of three measurable and limited vector control of such kind that the solutions (motion trajectories) of the system with these controls in the arbitrary time interval of the fixed length lie in three convex circular cone of sufficiently small angular measure and form the basis of trivariate Euclidean vector space of sufficiently large volume.

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the 'three-fold way'. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly 'complex' representations), those that are self-dual thanks to a symmetric bilinear pairing (which are 'real', in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are 'quaternionic', in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds-real, complex and quaternionic-can be seen as Hilbert spaces of the other kinds, equipped with extra structure. © 2011 Springer Science+Business Media, LLC.

The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax > 0. A natural condition measure associated with this algorithm is the Euclidean width T of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/T^2, see Rosenblatt 1962. Dunagan and Vempala have developed a re-scaled version of the perceptron algorithm with an improved complexity of O(n ln(1/T)) iterations (with high probability), which is theoretically efficient in T, and in particular is polynomial-time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax is an element of a set int K where K is a regular convex cone. We provide a conic extension of the re-scaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the re-scaled perceptron algorithm is itself theoretically efficient; this includes the cases when K is the cross-product of half-spaces, second-order cones, and the positive semi-definite cone.

The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax > 0. A natural condition measure associated with this algorithm is the Euclidean width T of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/T^2, see Rosenblatt 1962. Dunagan and Vempala have developed a re-scaled version of the perceptron algorithm with an improved complexity of O(n ln(1/T)) iterations (with high probability), which is theoretically efficient in T, and in particular is polynomial-time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax is an element of a set int K where K is a regular convex cone. We provide a conic extension of the re-scaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the re-scaled perceptron algorithm is itself theoretically efficient; this includes the cases when K is the cross-product of half-spaces, second-order cones, and the positive semi-definite cone.

This paper provides a test of convexity of a regression function. This test is based on the least squares splines. The test statistic is shown to be asymptotically of size equal to the nominal level, while diverging to infinity if the convexity is misspecified. Therefore, the test is consistent against all deviations from the null hypothesis.

This paper provides a test of convexity of a regression function. This test is based on the least squares splines. The test statistic is shown to be asymptotically of size equal to the nominal level, while diverging to infinity if the convexity is misspecified. Therefore, the test is consistent against all deviations from the null hypothesis.

We prove that if the second-order sufficient condition and constraint regularity hold at a local minimizer of a nonlinear programming problem, then for sufficiently smooth perturbations of the constraints and objective function the set of local stationary points is nonempty and continuous; further, if certain polyhedrality assumptions hold (as is usually the case in applications) then the local minimizers, the stationary points and the multipliers all obey a type of Lipschitz condition. Through the use of generalized equations, these results are obtained with a minimum of notational complexity., See also Part 1, ADA049495.

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