Counting faces of cubical spheres modulo two

AbstractSeveral recent papers have addressed the problem of characterizing the f-vectors of cubical polytopes. This is largely motivated by the complete characterization of the f-vectors of simplicial polytopes given by Stanley (Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, Vol. 440, 1985, pp. 212–223) and Billera and Lee (Bull. Amer. Math. Soc. 2 (1980) 181–185) in 1980. Along these lines Blind and Blind (Discrete Comput. Geom. 11(3) (1994) 351–356) have shown that unlike in the simplicial case, there are parity restrictions on the f-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes must have an even number of vertices. Here this result is extended to a class of zonotopal complexes which includes simply connected odd dimensional manifolds. This paper then shows that the only modular equations which hold for the f-vectors of all d-dimensional cubical polytopes (and hence spheres) are modulo two. Finally, the question of which mod two equations hold for the f-vectors of PL cubical spheres is reduced to a question about the Euler characteristics of multiple point loci from codimension one PL immersions into the d-sphere. Some results about this topological question are known (Eccles, Lecture Notes in Mathematics, Vol. 788, Springer, Berlin, 1980, pp. 23–38; Herbert, Mem. Amer. Math. Soc. 34 (250) (1981); Lannes, Lecture Notes in Mathematics, Vol. 1051, Springer, Berlin, 1984, pp. 263–270) and Herbert's result we translate into the cubical setting, thereby removing the PL requirement. A central definition in this paper is that of the derivative complex, which captures the correspondence between cubical spheres and codimension one immersions

Five-Torsion in the Homology of the Matching Complex on 14 Vertices

J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex $M_{14}$ on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case $n=14$ is exceptional; for all other $n$, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of $M_n$ when $n \ge 13$ and $n \neq 14$.Comment: 11 page

The pre-WDVV ring of physics and its topology

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex $\Delta_n$ is homotopy equivalent to a wedge of $(n-2)!$ spheres of dimension $n-4$. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.Comment: 13 pages, 4 figures, 2 table

Restricted Colored Permutations and Chebyshev Polynomials

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schr¨oder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

Atmospheric Muon Flux at Sea Level, Underground, and Underwater

The vertical sea-level muon spectrum at energies above 1 GeV and the underground/underwater muon intensities at depths up to 18 km w.e. are calculated. The results are particularly collated with a great body of the ground-level, underground, and underwater muon data. In the hadron-cascade calculations, the growth with energy of inelastic cross sections and pion, kaon, and nucleon generation in pion-nucleus collisions are taken into account. For evaluating the prompt muon contribution to the muon flux, we apply two phenomenological approaches to the charm production problem: the recombination quark-parton model and the quark-gluon string model. To solve the muon transport equation at large depths of homogeneous medium, a semi-analytical method is used. The simple fitting formulas describing our numerical results are given. Our analysis shows that, at depths up to 6-7 km w. e., essentially all underground data on the muon intensity correlate with each other and with predicted depth-intensity relation for conventional muons to within 10%. However, the high-energy sea-level data as well as the data at large depths are contradictory and cannot be quantitatively decribed by a single nuclear-cascade model.Comment: 47 pages, REVTeX, 15 EPS figures included; recent experimental data and references added, typos correcte

High Involvement Management, High Performance Work Systems and Well-being

Studies on the impact of high-performance work systems on employees' well-being are emerging but the underlying theory remains weak. This paper attempts to develop theory of the effects on well-being of four dimensions of high-performance work systems: enriched jobs, high involvement management, employee voice, and motivational supports. Hypothesized associations are tested using multilevel models and data from Britain's Workplace Employment Relations Survey of 2004 (WERS2004). Results show that enriched jobs are positively associated with both measures of well-being: job satisfaction and anxiety–contentment. Voice is positively associated with job satisfaction, and motivational supports with neither measure. The results for high involvement management are not as predicted because it increases anxiety and is independent of job satisfaction

Center of mass, spin supplementary conditions, and the momentum of spinning particles

We discuss the problem of defining the center of mass in general relativity and the so-called spin supplementary condition. The different spin conditions in the literature, their physical significance, and the momentum-velocity relation for each of them are analyzed in depth. The reason for the non-parallelism between the velocity and the momentum, and the concept of "hidden momentum", are dissected. It is argued that the different solutions allowed by the different spin conditions are equally valid descriptions for the motion of a given test body, and their equivalence is shown to dipole order in curved spacetime. These different descriptions are compared in simple examples.Comment: 45 pages, 7 figures. Some minor improvements, typos fixed, signs in some expressions corrected. Matches the published version. Published as part of the book "Equations of Motion in Relativistic Gravity", D. Puetzfeld et al. (eds.), Fundamental Theories of Physics 179, Springer, 201

Detroit's East Side Village Health Worker Partnership: Community-Based Lay Health Advisor Intervention in an Urban Area

In recent years, there have been few reports in the literature of interventions using a lay health advisor approach in an urban area. Consequently, little is known about how implementation of this type of community health worker model, which has been used extensively in rural areas, may differ in an urban area. This article describes the implementation of the East Side Village Health Worker Partnership, a lay health advisor intervention, in Detroit, Michigan, and notes how participatory action research methods and principles for community-based partnership research are being used to guide the intervention. Findings are presented on how the urban context is affecting the design and implementation of this intervention. Implications of the findings for health educators are also presented and include the utility of a participatory action research approach, the importance of considering the context and history of a community in designing a health education intervention, and the importance of recognizing and considering the differences between rural and urban settings when designing a health education intervention.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67390/2/10.1177_109019819802500104.pd