A fair sack is a finite set of independent dice, not required to be fair and allowed to have any number of sides, for which all totals are equally likely. These have been studied for over 60 years. Most results restrict the possible orders of dice in such a sack and almost no examples were known. Building on a rather different approach due to Gasarch and Kruskal, we give an explicit construction of all such sacks.

We have conducted Raman spectroscopy experiments on liquid ethane (C2H6) at 300 K, obtaining a large amount of data at very high resolution. This has enabled the observation of Raman peaks expected but not previously observed in liquid ethane and a detailed experimental study of the liquid that was not previously possible. We have observed a transition between rigid and nonrigid liquid states in liquid ethane at ca. 250 MPa corresponding to the recently proposed Frenkel line, a dynamic transition between rigid liquid (liquidlike) and nonrigid liquid (gaslike) states beginning in the subcritical region and extending to arbitrarily high pressure and temperature. The observation of this transition in liquid (subcritical) ethane allows a clear differentiation to be made between the Frenkel line (beginning in the subcritical region at higher density than the boiling line) and the Widom lines (emanating from the critical point and not existing in the subcritical region). Furthermore, we observe a narrow transition at ca. 1000 MPa to a second rigid liquid state. We propose that this corresponds to a state in which orientational order must exist to achieve the expected density and can view the transition in analogy to the transition in the solid state away from the orientationally disordered phase I to the orientationally ordered phases II and III.

The parts-to-totals map sends the distributions of a set of independent random variables on a finite set of probability spaces to the total distribution of their sum on the product space. We study, in special cases modeled by dice, the geometry of the extension to complex pseudoprobabilities of this map, arithmetic questions about the existence of real points in certain fibers, and, when these exist, of strict points, having all coordinates in the unit interval.

The Handbook of Moduli , comprising three volumes, offers a multi-faceted survey of a rapidly developing subject aimed not just at specialists but at a broad community of producers of algebraic geometry, and even at some consumers from cognate areas. The thirty-five articles in the Handbook , written by fifty leading experts, cover nearly the entire range of the field. They reveal the relations between these many threads and explore their connections to other areas of algebraic geometry, number theory, differential geometry, and topology. The goals of the Handbook are to introduce the techniques, examples, and results essential to each topic, and to say enough about recent developments to provide a gateway to the primary sources. Many articles are original treatments commissioned to bridge gaps in the literature and to make important problems accessible to a wide audience for the first time, and many others illustrate yogas and heuristics that experts use privately to guide intuition or simplify calculation, but that do not appear in published work aimed at other specialists. This is the third of three volumes constituting the Handbook of Moduli , and is also available as part of a three volume set.

This paper is an expository survey of results about the effective divisors on moduli spaces, with a focus on what is known about the effective cones of moduli spaces of stable curves and of principally polarized abelian varieties. This version incorporates clarifications and suggestions of Ana-Maria Castravet and the referee, both of whom we thank. To appear in the proceedings of the conference, Joe at 60:A Celebration of Algebraic Geometry edited by Brendan Hassett, James McKernan, Jason Starr and Ravi Vakil.

We give a method for verifying, by a symbolic calculation, the stability or semistability with respect to a linearization of fixed, possibly small, degree $m$, of the Hilbert point of a scheme $X \in {\mathbb P}(V)$ having a suitably large automorphism group. We also implement our method and apply it to analyze the stability of bicanonical models of certain curves. Our examples are very special, but they arise naturally in the log minimal model program for $\bar{\mathcal M}_g$. In some examples, this connection provides a check of our computations; in others, the computations confirm predictions about conjectural stages of the program.

We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus g >= 3 are the moduli space (M) over bar (ps)(g) of pseudostable curves constructed by Schubert in [10] using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps lambda of the m(th) Hilbert points of curves X with elliptic tails. We compute the exact weight with which lambda acts, and not just the leading term in m of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semistable orbits, we are able to again identify it as (M) over bar (ps)(g). Our computations yield, as byproducts, examples of both m-Hilbert unstable and m-Hilbert stable X that are Chow strictly semistable.

Intrigued by the recent assertion that the “ice-on-eyes” test is 100% specific for myasthenia gravis,1–3 we performed this test in a patient presenting with partial ophthalmoplegia and unilateral ptosis due to Miller Fisher syndrome. This was a 24-year-old female who developed a diarrhoeal illness lasting several days. A right ptosis developed 10 days later, and …

This paper reports investigations into the influence of hydrogen on the magnetic properties of the YCo3-H system. We report results on the magnetic structure and magnetic transitions of YCo3 using a combination of neutron powder diffraction measurements and first-principles full potential augmented plane wave + local orbital calculations under the generalized gradient approximation. The ferromagnetic and ferrimagnetic structures are examined on an equal footing. However, we identify that, no matter which structure is used as the starting point, the neutron diffraction data always refines down to the ferrimagnetic structure with the Co-2 atoms having antiparallel spins. In the ab initio calculations, the inclusion of spin-orbit coupling is found to be important in the prediction of the correct magnetic ground state. Here, the results suggest that, for zero external field and sufficiently low temperatures, the spin arrangement of YCo3 is ferrimagnetic rather than ferromagnetic as previously believed. The fixed spin moment calculation technique has been employed to understand the two successive field-induced magnetic transitions observed in previous magnetization measurements under increasing ultrahigh magnetic fields. We find that the magnetic transitions start from the ferrimagnetic phase (0.61 mu(B)/Co) and terminate with the ferromagnetic phase (1.16 mu(B)/Co), while the spin on the Co-2 atoms progressively changes from antiparallel ferrimagnetic to paramagnetic and then to ferromagnetic. Our neutron diffraction measurements, ab initio calculations, and the high field magnetization measurements are thus entirely self-consistent.

The crystalline and magnetic structures of the YCo(3)-H(D) system have been investigated by means of x-ray and neutron diffraction with the objective of understanding the complex magnetic changes that are observed in this system as hydrogen is added. Synchrotron x-ray diffraction (XRD) patterns were first refined to yield the lattice parameters and coordination of Y and Co atoms in the metal and two beta-hydride phases while XRD was used for the gamma phase. In situ neutron powder diffraction measurements of YCo(3)D(x) were then made in all four phases to determine the deuterium site occupancies and magnetic structures. The site occupancies were also rationalized using the Westlake geometric model. The highest hydrogen concentration measured was YCo(3)H(4.6). Using the Westlake model, we conclude that the saturated hydrogen content would be YCo(3)H(5). Our results reported here and in Part I [Phys. Rev. B 76, 184443 (2007)] have enabled us to rationalize the changes in the magnetic structures in terms of changes in the cobalt-cobalt distance caused by the addition of hydrogen. In particular, in the antiferromagnetic gamma phase, we observe Co atomic displacements that enable the structure to adopt a particular antiferromagnetic structure in a manner that is reminiscent of a Peierls distortion as observed in transitions from the conducting to nonconducting hydrides on addition of hydrogen in YH(3).

Real space, high-resolution transmission electron microscopy observations of Xe confined in nanometer size faceted cavities in Al yield information on both the inert gas and the matrix in which it is confined. At room temperature, Xe in such cavities can be liquid or an fcc solid. In larger cavities, Xe within can undergo melting and recrystallization. The Al surface energy can be deduced from the largest Xe nanocrystal at 300 K by setting the corresponding calculated Laplace pressure equal to the equilibrium pressure for melting of Xe, obtained from empirical bulk compression data. These surface energy values are 1.05 J m−2 for {111} facets and 1.10 J m−2 for {200} facets. Because of the weak interactions, these values correspond to the surface tensions for Al at 300 K. At room temperature, fluid Xe confined in small faceted cavities in aluminum has up to three ordered layers of Xe atoms at the Al interface. Conceptually in a three-dimensionally confined system of sufficiently small size, complete three-dimensional ordering of the fluid may occur. Molecular dynamics simulations have revealed that such ordering would result in fluid Xe confined to a small tetragonal volume solidifying as a body-centered cubic phase on compression.

Polymyositis is an inflammatory myopathy of skeletal muscle. Cardiac involvement in the disease was first described in 1899 but has only been studied in detail over the past 20 years. About 10–15% of all patients with polymyositis have a cardiac abnormality as their initial presenting feature, while up to 70% of all those with polymyositis will have some cardiac involvement diagnosed non-invasively during the course of their illness. These can manifest either as cardiac failure; electrocardiographic changes, including nonspecific ST changes, and varying forms of AV block; myocarditis; valve disease and myocardial ischaemia in the presence of normal coronary vasculature. Myasthenia gravis is thought to be an autoimmune disease characterised by the presence of antibodies to the acetylcholine receptor of the motor end plate. Idiopathic forms are rarely associated with polymyositis, but a much stronger correlation exists in the presence of a thymoma, particularly in those with myocardial involvement, when anti-titin antibodies are usually present. However, thymoma is often less than 1 mm in size and therefore not visible on computed tomographic (CT) scanning. We describe a 56 year old woman with polymyositis, who initially presented with myocardial involvement and was also found to have antibodies consistent with myasthenia gravis.

Polymyositis is an inflammatory myopathy of skeletal muscle. Cardiac involvement in the disease was first described in 1899 but has only been studied in detail over the past 20 years. About 10–15% of all patients with polymyositis have a cardiac abnormality as their initial presenting feature, while up to 70% of all those with polymyositis will have some cardiac involvement diagnosed non-invasively during the course of their illness. These can manifest either as cardiac failure; electrocardiographic changes, including nonspecific ST changes, and varying forms of AV block; myocarditis; valve disease and myocardial ischaemia in the presence of normal coronary vasculature. Myasthenia gravis is thought to be an autoimmune disease characterised by the presence of antibodies to the acetylcholine receptor of the motor end plate. Idiopathic forms are rarely associated with polymyositis, but a much stronger correlation exists in the presence of a thymoma, particularly in those with myocardial involvement, when anti-titin antibodies are usually present. However, thymoma is often less than 1 mm in size and therefore not visible on computed tomographic (CT) scanning. We describe a 56 year old woman with polymyositis, who initially presented with myocardial involvement and was also found to have antibodies consistent with myasthenia gravis.

In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \vmgn 0. 2g+n. \to \mgn$ whose image is the locus $\rgn$ of curves with all components rational. Any 1-strata either lies in $\rgn$ or is numerically equivalent to a family $E$ of elliptic tails and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^*(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\mgn$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary. Finally, by more ad-hoc arguments, we prove the nefness of certain special classes.