Element Pools and Numeric Coprocessors for Linear Programming on Personal Computers

This note discusses the need to use various data-compaction schemes in the implementation of sophisticated large-scale linear programming codes on modest personal computers. It demonstrates a method whereby the overhead of using these schemes is significantly reduced on Intel 80 x 86 processors.     

No-wait scheduling of position dependent jobs and adjustable processing intervals

In this paper, we consider a single machine no-wait scheduling model whereby job processing times are general functions of their position in the job sequence. We assume that the single machine must operate at a certain cycle, which can be determined by the scheduler. Furthermore, exactly one job has to be completed by the end of each cycle. Using different variations of the Linear Assignment Problem formulation, we develop polynomial time algorithms for minimizing the following objectives: makespan, total completion time, maximum earliness and total earliness.     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems that arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Moré and Sorensen, methods that obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS{\tt TRS} and RQS{\tt RQS} as part of the GALAHAD optimization library, and is especially designed for large-scale problems.     

Extensions of dually bipartite regular polytopes

This paper addresses the problem of finding abstract regular polytopes with preassigned facets and preassigned last entry of the Schläfli symbol. Using C-group permutation representation (CPR) graphs we give a solution to this problem for dually bipartite regular polytopes when the last entry of the Schläfli symbol is even. This construction is related to a previous construction by Schulte that solves the problem when the entry of the Schläfli symbol is 6.     

Non‐rainbow colorings of 3‐, 4‐ and 5‐connected plane graphs

We study vertex‐colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 ‐connected plane graph with n vertices, then the number of colors in such a coloring does not exceed . If G is 4 ‐connected, then the number of colors is at most , and for n≡3(mod8), it is at most . Finally, if G is 5 ‐connected, then the number of colors is at most . The bounds for 3 ‐connected and 4 ‐connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129–145, 2010     

Coloring plane graphs with independent crossings

We show that every plane graph with maximum face size four in which all faces of size four are vertex‐disjoint is cyclically 5‐colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 184–205, 2010     

Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential $U=A{q}_1^2+B{q}_2^2+C{q}_1^2{q}_2+D{q}_2^3$, with $(q_1,q_2)=$ standard Cartesian coordinates and $(A,B,C,D)\in {(0,\infty )}^2\times R^2$, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between $C$ and $D$ varies. More precisely, there are two symmetric pitchfork bifurcations along the line $2C?3D=0$, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.