When to rebuild or when to adjust scorecards

Data-based scorecards, such as those used in credit scoring, age with time and need to be rebuilt or readjusted. Unlike the huge literature on modelling the replacement and maintenance of equipment there have been hardly any models that deal with this problem for scorecards. This paper identifies an effective way of describing the predictive ability of the scorecard and from this describes a simple model for how its predictive ability will develop. Using a dynamic programming approach one is then able to find when it is optimal to rebuild and when to readjust a scorecard. Failing to readjust or rebuild a scorecard when they aged was one of the defects in credit scoring identified in the investigations into the sub-prime mortgage crisis.     

When bad things happen to good trees*

When the edges in a tree or rooted tree fail with a certain fixed probability, the (greedoid) rank may drop. We compute the expected rank as a polynomial in p and as a real number under the assumption of uniform distribution. We obtain several different expressions for this expected rank polynomial for both trees and rooted trees, one of which is especially simple in each case. We also prove two extremal theorems that determine both the largest and smallest values for the expected rank of a (rooted or unrooted) tree, and precisely when these extreme bounds are achieved. We conclude with directions for further study. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 79–99, 2001     

When almost multiplicative morphisms are close to homomorphisms

It is shown that approximately multiplicative contractive positive morphisms from (with dim ) into a simple -algebra of real rank zero and of stable rank one are close to homomorphisms, provided that certain -theoretical obstacles vanish. As a corollary we show that a homomorphism is approximated by homomorphisms with finite dimensional range, if gives no -theoretical obstacle.

     

When Are Covers Relative to Two Classes Isomorphic

Let ?₁ ? ?₂ be classes of right R-modules, we introduce ?₁-covering modules and projective modules relative to ?₂ to characterize the relations between the existences of ?₁-covers, ?₂-covers and the classes ?₁, ?₂. As corollaries, every module in ? _n is an ?-covering module if and only if every flat cover is an ? _n -cover for each right module if and only if R is a von Neumann regular ring whenever wD(R) < ∞; every flat right R-module is projective relative to 𝒫 if and only if every flat cover is a projective cover for each right module if and only if R is right perfect.     

When nonautonomous equations are equivalent to autonomopus ones

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit constant solutions to which there correspond non-constant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.     

When to refinance a mortgage: A dynamic programming approach

When should one refinance a mortgage loan? It is one of the most common finance questions in today's world. There have been surprisingly few attempts to answer this question in a structured manner, however. Moreover, the existing guidelines for refinancing consist of a short list of very simple rules that have a limited application. This article addresses the question through a dynamic programming model coupled with an analysis of historical interest rates. The analysis reveals a more complex set of rules for an optional refinance decision––oftentimes conflicting with the conventionally accepted idea that rate differences must be greater than two percent.     

When Polarizations Generate

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. From elements of the invariant algebra C[V]^G we obtain, by polarization, elements of C[kV]^G, where k ≥ 1 and kV denotes the direct sum of k copies of V. For G simple, our main result is the classification of the G-modules V and integers k ≥ 2 such that polarizations generate C[kV]^G.     

When K+(n-4)L fails to be nef

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. We study polarized pairs (X,L) for which K+(n−3)L is nef but K+(n−4)L fails to be nef. Supported by MURST funds     

Dynamic Programming in Tennis—When to Use a Fast Serve

Dynamic programming is applied to the problem in tennis of whether to serve fast or slow on either or both serves at each stage in a game, and a simple policy is found.     

When is an Abelian group isomorphic to its endomorphism group?

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.     

When slower is faster

The slower is faster (SIF) effect occurs when a system performs worse as its components try to do better. Thus, a moderate individual efficiency actually leads to a better systemic performance. The SIF effect takes place in a variety of phenomena. We review studies and examples of the SIF effect in pedestrian dynamics, vehicle traffic, traffic light control, logistics, public transport, social dynamics, ecological systems, and adaptation. Drawing on these examples, we generalize common features of the SIF effect and suggest possible future lines of research. © 2015 Wiley Periodicals, Inc. Complexity 21: 9–15, 2015     

When Ramification points meet

Summary This paper contains three applications of the technique of limit series (our [1986]) to the theory of ramification of linear series on smooth curves, and curves of compact type, overC.Let {L _t t|<}, be a family of linear series on a smooth family of smooth curves {C _t }, and letp ₁(t),p ₂(t)C _t be sections of the family which coincide (only) att=0. Setp=p ₁())=p ₂(0)C ₀.We first give a condition related to the Schubert calculus which must be satisfied by the ramification series and the . We then take up the converse problem: In what ways can a given ramification point arise as a limit? We show that if the ramification point isdimensionally proper in the sense of our [1986], then families of every kind allowed by the Schubert calculus condition can actually be constructed. Finally, we prove that dimensional propriety is in a strong sense an open condition, so that ramification points constructed as above are again dimensionally proper.In the body of the paper we work not with pairs of points, as above, but with arbitrary finite collections of points approaching (possibly) several points of the limit curve. Further, by their nature, the results are valid for families of curves of compact type.     

When Teachers Drop Out

Abstract

Like other small liberal arts colleges across the country, Williams College in rural Massachusetts has had to think about the hard questions confronting educators in the coming decade. Decreasing college age population, escalating costs, questions about faculty tenure and salary are common themes to many higher education institutions.

Williams has been known for the innovative planning and thought that has gone into meeting past challenges. As early as 1962, the school had received a Ford Foundation grant for the express purpose of studying the future of Williams College.

In 1969 a second study, this time using sophisticated computer techniques, led to some far reaching decisions for the college. The 1969 study went through several planning stages. Eventually the school administration adopted a plan which allowed the college to grow from 1,200 to 1,800 students by adding 600 women. The addition of women to the student body was a major decision which also affected faculty size. The 1969 plan recommended that the faculty grow by the number which would have accomodated 1,500 students under earlier planning ratios.

The editors of CHANGE felt that the Williams 1980 report would be of interest to other colleges and universities grappling with the same problems. The sections excerpted here are a condensation of the introduction, summary and conclusions of a much longer report.

Copies of the entire report are available from the Office of the President, Williams College, Williamstown, Massachusetts 01267.