The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.Comment: 9 page

Sheila McA and Robert F. Brown to Jim (3 October 1962)

Signed by Sheila McA and Robert F. Brownhttps://egrove.olemiss.edu/mercorr_pro/1165/thumbnail.jp

Lifting classes for the fixed point theory of nn-valued maps

The theory of lifting classes and the Reidemeister number of single-valued maps of a finite polyhedron $X$ is extended to $n$-valued maps by replacing liftings to universal covering spaces by liftings with codomain an orbit configuration space, a structure recently introduced by Xicot\'encatl. The liftings of an $n$-valued map $f$ split into self-maps of the universal covering space of $X$ that we call lift-factors. An equivalence relation is defined on the lift-factors of $f$ and the number of equivalence classes is the Reidemeister number of $f$. The fixed point classes of $f$ are the projections of the fixed point sets of the lift-factors and are the same as those of Schirmer. An equivalence relation is defined on the fundamental group of $X$ such that the number of equivalence classes equals the Reidemeister number. We prove that if $X$ is a manifold of dimension at least three, then algebraically the orbit configuration space approach is the same as one utilizing the universal covering space. The Jiang subgroup is extended to $n$-valued maps as a subgroup of the group of covering transformations of the orbit configuration space and used to find conditions under which the Nielsen number of an $n$-valued map equals its Reidemeister number. If an $n$-valued map splits into $n$ single-valued maps, then its $n$-valued Reidemeister number is the sum of their Reidemeister numbers.Comment: near complete rewrite from previous versio

Non-Cointegration and Econometric Evaluation of Models of Regional Shift and Share

This paper tests for cointegration between regional output of an industry and national output of the same industry. An equilibrium economic theory is presented to argue for the plausibility of cointegration, however, regional economic forecasting using the shift and share framework often acts as if cointegration does not exist. Data analysis on broad industrial sectors for 20 states finds very little evidence for cointegration. Forecasting models with and without imposing cointegration are than constructed and used to forecast out of sample. The simplest, non-cointegrating models are the best.

Formulating Linear and Integer Linear Programs: A Rogues' Gallery

INFORMS Transactions on Education, 7, 2007, pp. 153-159.The article of record as published may be located at http://dx.doi.org/10.1287/ited.7.2.153The art of formulating linear and integer linear programs is, well, an art: It is hard to teach, and even harder to learn. To help demystify this art, we present a set of modeling building blocks that we call “formulettes.” Each formulette consists of a short verbal description that must be expressed in terms of variables and constraints in a linear or integer linear program. These formulettes can better be discussed and analyzed in isolation from the much more complicated models they comprise. Not all models can be built from the formulettes we present. Rather, these are chosen because they are the most frequent sources of mistakes. We also present Naval Postgraduate School (NPS) format; a define-before-use formulation guide we have followed for decades to express a complete formulation

A Method for Distinguishing Between Transiently Accreting Neutron Stars and Black Holes, in Quiescence

We fit hydrogen atmosphere models to the X-ray data for four neutron stars (three from a previous paper, plus 4U 2129+47) and six black hole candidates (A0620-00, GS 2000+25, GS 1124-68, GS 2023+33, GRO J1655-40, and GRO J0422+32). While the neutron stars are similar in their intrinsic X-ray spectra (similar effective temperatures and emission area radii ~10 km), the spectra of two black hole candidates are significantly different, and the spectra of the remaining four are consistent with a very large parameter space that includes the neutron stars. The spectral differences between the neutron stars and black hole candidates favors the interpretation that the quiescent neutron star emission is predominantly thermal emission from the neutron star surface. Our work suggests that an X-ray spectral comparison in quiescence provides an additional means for distinguishing between neutron stars and black holes. The faint X-ray sources in globular clusters are also a class of objects which can be investigated in this manner.Comment: 33 pages, including 3 ps figures, LaTeX. To appear in Ap

Scheduling Coast Guard District Cutters

Interfaces, 26, March - April 1996, pp. 59-72Center for Infrastructure Defense (CID) Paper.United States Coast Guard (USCG) districts schedule cutters 180 feet or less in length to weekly statuses (statuses is USCGjargon for assignments) from which they primarily respond to calls for search and rescue, law enforcement, and pollution control. The First Coast Guard District, based in Boston, has one of the largest scheduling problems: Each of 16 cutters is as- signed weekly to one of six statuses to ensure patrol coverage, enforce equitable distribution of patrols, and honor restrictions on consecutive cutter statuses. When we state this quarterly scheduling problem as an elastic mixed-integer linear program, we obtain face-valid schedules—superior to manually prepared schedules for all measures of effectiveness considered—within a few minutes on a personal computer. Initial acceptance of the model was hampered by disruptive schedule revisions that re- sulted from minor changes in input. Modifications to preserve run-to-run persistence of solutions have brought success