Symbolic Powers of Monomial Ideals

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M = (x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.Comment: 15 pages. Fixed typ

Will Renunciation of a Bequest or Failure to Claim a Statutory Share Constitute a Taxable Gift

In 1948 the 80th Congress amended the Internal Revenue Code in an effort to eliminate the discrimination theretofore enjoyed by residents of states which had adopted the community property system.\u27 Substantial equalization in the estate and gift tax fields is expected to follow from the marital deduction and gift tax splitting privileges. Moreover, these changes have focused attention upon a problem which caused considerable concern to conservative tax advisors even before the 1948 Tax Amendment. It has long been a doubtful question whether the renunciation of testamentary benefits would be held to constitute a taxable gift. The Act and the proposed regulations make it reasonably clear that a widow will not incur a gift tax as a result of renouncing a bequest or failing to claim her statutory share. But what of a child who (1) renounces a legacy, or (2) fails to claim a statutory share where such right exists because of his parent\u27s failure to mention him in the will? Will he be deemed to have made a taxable gift? The purpose of this note is to analyze the probable basis of gift tax liability in these two latter situations

Resumming the large-N approximation for time evolving quantum systems

In this paper we discuss two methods of resumming the leading and next to leading order in 1/N diagrams for the quartic O(N) model. These two approaches have the property that they preserve both boundedness and positivity for expectation values of operators in our numerical simulations. These approximations can be understood either in terms of a truncation to the infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a particular two-particle irreducible vacuum energy graph in the effective action of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the case of quantum mechanics where the Lagrangian is $L(x,\dot{x}) = (1/2) \sum_{i=1}^{N} \dot{x}_i^2 - (g/8N) [ \sum_{i=1}^{N} x_i^2 - r_0^2 ]^{2}$. The key to these approximations is to treat both the $x$ propagator and the $x^2$ propagator on similar footing which leads to a theory whose graphs have the same topology as QED with the $x^2$ propagator playing the role of the photon. The bare vertex approximation is obtained by replacing the exact vertex function by the bare one in the exact Schwinger-Dyson equations for the one and two point functions. The second approximation, which we call the dynamic Debye screening approximation, makes the further approximation of replacing the exact $x^2$ propagator by its value at leading order in the 1/N expansion. These two approximations are compared with exact numerical simulations for the quantum roll problem. The bare vertex approximation captures the physics at large and modest $N$ better than the dynamic Debye screening approximation.Comment: 30 pages, 12 figures. The color version of a few figures are separately liste

Chaos in Time Dependent Variational Approximations to Quantum Dynamics

Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first show that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large N (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schrodinger equation.Comment: 10 pages (12 figures), RevTeX (plus macro), uses epsf, minor typos correcte

Improved fidelity of triggered entangled photons from single quantum dots

We demonstrate the on-demand emission of polarisation-entangled photon pairs from the biexciton cascade of a single InAs quantum dot embedded in a GaAs/AlAs planar microcavity. Improvements in the sample design blue shifts the wetting layer to reduce the contribution of background light in the measurements. Results presented show that >70% of the detected photon pairs are entangled. The high fidelity of the (|HxxHx>+|VxxVx>)/2^0.5 state that we determine is sufficient to satisfy numerous tests for entanglement. The improved quality of entanglement represents a significant step towards the realisation of a practical quantum dot source compatible with applications in quantum information.Comment: 9 pages. Paper is available free of charge at http://www.iop.org/EJ/abstract/1367-2630/8/2/029/, see also 'A semiconductor source of triggered entangled photon pairs', R. M. Stevenson et al., Nature 439, 179 (2006

Primary Cardiac Synovial Sarcoma: A Case Report and Brief Review of the Literature

Synovial sarcoma comprises approximately 10% of all soft tissue sarcoma diagnoses; a primary synovial sarcoma of the myocardium is exceedingly rare. There have been very few cases reported in the literature thus far. With the identification of the characteristic and diagnostic chromosomal abnormality t(X;18), this may become an increasingly recognized entity. Our report adds to the limited published cases of primary cardiac synovial sarcoma with the characteristic t(X;18). Further elucidation of the effects of this translocation on the cell cycle may lead to directed therapies in the future

Photoreceptor Inner Segment Morphology in Best Vitelliform Macular Dystrophy

PURPOSE To characterize outer retina structure in best vitelliform macular dystrophy (BVMD) and to determine the effect of macular lesions on overlying and adjacent photoreceptors. METHODS Five individuals with BVMD were followed prospectively with spectral domain optical coherence tomography and confocal and nonconfocal split-detector adaptive optics scanning light ophthalmoscopy (AOSLO). The AOSLO cone photoreceptor mosaic images were obtained within and around retinal lesions. Cone density was measured inside and outside lesions. In 2 subjects, densities were compared with published measurements acquired ∼2.5 years before. One subject was imaged 3 times over a 5-month period. RESULTS The AOSLO imaging demonstrated that photoreceptor morphology within BVMD retinal lesions was highly variable depending on the disease stage, with photoreceptor structure present even in advanced disease. The AOSLO imaging was repeatable even in severe disease over short-time and long-time intervals. Photoreceptor density was normal in retinal areas immediately adjacent to lesions and stable over ∼2.5 years. Mobile disk-like structures possibly representing subretinal macrophages were also observed. CONCLUSION Combined confocal and nonconfocal split-detector AOSLO imaging reveals substantial variability within clinical lesions in all stages of BVMD. Longitudinal cellular photoreceptor imaging could prove a powerful tool for understanding disease progression and monitoring emerging therapeutic treatment response in inherited degenerations such as BVMD

White-tailed Deer Browse Preferences for South Texas and the Edwards Plateau

8 pp., 2 color photos, 4 tables, 2 graphsDeer prefer forbs, but these plants are not always available. Browse plants actually make up the major part of deer diets, and the specific browse plants deer eat depends on plant palatibility and availability. This publication explains deer nutrition and methods of evaluating the quality of deer diets. There are tables of preferred plants and their nutritional quality

Incomplete concerted evolution and reproductive isolation at the rDNA locus uncovers nine cryptic species within Anopheles longirostris from Papua New Guinea

Abstract. Background. Nuclear ribosomal DNA (rDNA) genes and transcribed spacers are highly utilized as taxonomic markers in metazoans despite the lack of a cohesive understanding of their evolution. Here we follow the evolution of the rDNA second internal transcribed spacer (ITS2) and the mitochondrial DNA cytochrome oxidase I subunit in the malaria mosquito Anopheles longirostris from Papua New Guinea (PNG). This morphospecies inhabits a variety of ecological environments indicating that it may comprise a complex of morphologically indistinguishable species. Using collections from over 70 sites in PNG, the mtDNA was assessed via direct DNA sequencing while the ITS2 was assessed at three levels - crude sequence variation through restriction digest, intragenomic copy variant organisation (homogenisation) through heteroduplex analysis and DNA sequencing via cloning. Results. Genetic evaluation of over 300 individuals revealed that A. longirostris comprises eight ITS2 PCR-RFLP genotypes and nine ITS2 heteroduplex genotypes showing distinct copy variant organization profiles after PCR amplification. Seven of these nine genotypes were found to be sympatric with other genotypes. Phylogenetic analysis of cloned ITS2 PCR products and mtDNA COI confirmed all nine clades with evidence of reproductive isolation at the rDNA locus. Compensatory base changes in the ITS2 secondary structure or in pseudoknots were absent when closely related species were assessed. Individuals from each ITS2 genotype showed the same copy variant heteroduplex profile suggesting that the rDNA array is fixed within each genotype. Conclusion. The centromere-proximal position of the rDNA array in Anopheles mosquitoes has probably reduced interchromosomal recombination leaving intrachromosomal events responsible for the observed pattern of concerted evolution we see in these mosquitoes. The stability of these intragenomic ITS2 copy variants within individuals and interbreeding populations suggests that rDNA is moving as a single evolutionary unit through natural populations to fixation and has provided a complementary diagnostic tool to the restriction digest for studying genetic discontinuities and species boundaries. In this, the utility of the ITS2 as a universal taxonomic marker is probably contingent on several factors pertaining to spacer dimensions and the genomic location of the rDNA array with respect to recombination and proximity to regions potentially under selection