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

Cosmic Neutrinos and the Energy Budget of Galactic and Extragalactic Cosmic Rays

Although kilometer-scale neutrino detectors such as IceCube are discovery instruments, their conceptual design is very much anchored to the observational fact that Nature produces protons and photons with energies in excess of 10^{20} eV and 10^{13} eV, respectively. The puzzle of where and how Nature accelerates the highest energy cosmic particles is unresolved almost a century after their discovery. We will discuss how the cosmic ray connection sets the scale of the anticipated cosmic neutrino fluxes. In this context, we discuss the first results of the completed AMANDA detector and the science reach of its extension, IceCube.Comment: 13 pages, Latex2e, 3 postscript figures included. Talk presented at the International Workshop on Energy Budget in the High Energy Universe, Kashiwa, Japan, February 200

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem--Mahler--Lech theorem, the ${\bf ab}$-series of a level poset is shown to be a rational generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$. In the case the poset is also Eulerian, the analogous result holds for the ${\bf cd}$-series. Using coalgebraic techniques a method is developed to recognize the ${\bf cd}$-series matrix of a level Eulerian poset

High Energy Neutrino Astronomy: Towards Kilometer-Scale Detectors

Of all high-energy particles, only neutrinos can directly convey astronomical information from the edge of the universe---and from deep inside the most cataclysmic high-energy processes. Copiously produced in high-energy collisions, travelling at the velocity of light, and not deflected by magnetic fields, neutrinos meet the basic requirements for astronomy. Their unique advantage arises from a fundamental property: they are affected only by the weakest of nature's forces (but for gravity) and are therefore essentially unabsorbed as they travel cosmological distances between their origin and us. Many of the outstanding mysteries of astrophysics may be hidden from our sight at all wavelengths of the electromagnetic spectrum because of absorption by matter and radiation between us and the source. For example, the hot dense regions that form the central engines of stars and galaxies are opaque to photons. In other cases, such as supernova remnants, gamma ray bursters, and active galaxies, all of which may involve compact objects or black holes at their cores, the precise origin of the high-energy photons emerging from their surface regions is uncertain. Therefore, data obtained through a variety of observational windows---and especially through direct observations with neutrinos---may be of cardinal importance. In this talk, the scientific goals of high energy neutrino astronomy and the technical aspects of water and ice Cherenkov detectors are examined, and future experimental possibilities, including a kilometer-square deep ice neutrino telescope, are explored.Comment: 13 pages, Latex, 6 postscript figures, uses aipproc.sty and epsf.sty. Talk presented at the International Symposium on High Energy Gamma Ray Astronomy, Heidelberg, June 200

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

Cosmic Neutrinos from the Sources of Galactic and Extragalactic Cosmic Rays

Although kilometer-scale neutrino detectors such as IceCube are discovery instruments, their conceptual design is very much anchored to the observational fact that Nature produces protons and photons with energies in excess of 10^20 eV and 10^13 eV, respectively. The puzzle of where and how Nature accelerates the highest energy cosmic particles is unresolved almost a century after their discovery. From energetics considerations we anticipate order 10~100 neutrino events per kilometer squared per year pointing back at the source(s) of both galactic and extragalactic cosmic rays. In this context, we discuss the results of the AMANDA and IceCube neutrino telescopes which will deliver a kilometer-square-year of data over the next 3 years.Comment: 8 pages, 4 figure

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

A new growth chart for preterm babies: Babson and Benda's chart updated with recent data and a new format

BACKGROUND: The Babson and Benda 1976 "fetal-infant growth graph" for preterm infants is commonly used in neonatal intensive care. Its limits include the small sample size which provides low confidence in the extremes of the data, the 26 weeks start and the 500 gram graph increments. The purpose of this study was to develop an updated growth chart beginning at 22 weeks based on a meta-analysis of published reference studies. METHODS: The literature was searched from 1980 to 2002 for more recent data to complete the pre and post term sections of the chart. Data were selected from population studies with large sample sizes. Comparisons were made between the new chart and the Babson and Benda graph. To validate the growth chart the growth results from the National Institute of Child Health and Human Development Neonatal Research Network (NICHD) were superimposed on the new chart. RESULTS: The new data produced curves that generally followed patterns similar to the old growth graph. Mean differences between the curves of the two charts reached statistical significance after term. Babson's 10(th )percentiles fell between the new data percentiles: the 5th to 17th for weight, the 5th and 15th for head circumference, and the 6th and 16th for length. The growth patterns of the NICHD infants deviated away from the curves of the chart in the first weeks after birth. When the infants reached an average weight of 2 kilograms, those with a birthweight in the range of 700 to 1000 grams had achieved greater than the 10(th )percentile on average for head growth, but remained below the 3(rd )percentile for weight and length. CONCLUSION: The updated growth chart allows a comparison of an infant's growth first with the fetus as early as 22 weeks and then with the term infant to 10 weeks. Comparison of the size of the NICHD infants at a weight of 2 kilograms provides evidence that on average preterm infants are growth retarded with respect to weight and length while their head size has caught up to birth percentiles. As with all meta-analyses, the validity of this growth chart is limited by the heterogeneity of the data sources. Further validation is needed to illustrate the growth patterns of preterm infants to older ages