αB-spline: a linear singular blending B-spline

A linear singular blending (LSB) technique can enhance the shape—control capability of the B-spline. This capability is derived from the blending parameters defined at the B-spline control vertices and blends LSB line segments or bilinear surface patches with the B-spline curve or surface. Varying the blending parameters between zero and unity applies tension for reshaping. The reshaped curve or surface retains the same smoothness properties as the original B-spline; it possesses the same strict parametric continuities. This is different from the gif" alt="beta" align="MIDDLE" BORDER="0">-spline, which introduces additional control to the B-spline by imposing geometrical continuities to the joints of curve segments or surface patches. For applications in which strict parametric continuities cannot be compromised, LSB provides an intuitive way to introduce tension to the B-spline.     

Reasoning about action and change

Reasoning about change is a central issue in research on human and robot planning. We study an approach to reasoning about action and change in a dynamic logic setting and provide a solution to problems which are related to the Frame problem. Unlike most work on the frame problem the logic described in this paper is monotonic. It (implicitly) allows for the occurrence of actions of multiple agents by introducing non-stationary notions of waiting and test. The need to state a large number of gif" alt="ldquo" align="MIDDLE" BORDER="0">frame axiomsgif" alt="rdquo" align="MIDDLE" BORDER="0"> is alleviated by introducing a concept of chronological preservation to dynamic logic. As a side effect, this concept permits the encoding of temporal properties in a natural way. We compare the relative merits of our approach and non-monotonic approaches as regards different aspects of the frame problem. Technically, we show that the resulting extended systems of propositional dynamic logic preserve (weak) completeness, finite model property and decidability.     

Algorithms to Separate $\{0,\frac{1}{2}\}$ -Chvátal-Gomory Cuts

Chvátal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or gif" alt="$\frac {1}{2}$" align="middle" border="0"> , such cuts are known as gif" alt="$\{0,\frac {1}{2}\}$" align="middle" border="0"> -cuts. It has been proven by Caprara and Fischetti (Math. Program. 74:221–235, 1996) that separation of gif" alt="$\{0,\frac {1}{2}\}$" align="middle" border="0"> -cuts is gif" alt="$\mathcal {NP}$" align="middle" border="0"> -hard. In this paper, we study ways to separate gif" alt="$\{0,\frac {1}{2}\}$" align="middle" border="0"> -cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated gif" alt="$\{0,\frac {1}{2}\}$" align="middle" border="0"> -cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.     

Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem

This paper presents an algebraic approach to polynomial spectral factorization, an important mathematical tool in signal processing and control. The approach exploits an intriguing relationship between the theory of Gröbner bases and polynomial spectral factorization which can be observed through the sum of roots, and allows us to perform polynomial spectral factorization in the presence of real parameters. It is discussed that parametric polynomial spectral factorization enables us to express quantities such as the optimal cost in terms of parameters and the sum of roots. Furthermore an optimization method over parameters is suggested that makes use of the results from parametric polynomial spectral factorization and also employs two quantifier elimination techniques. This proposed approach is demonstrated in a numerical example of a particular control problem.