The global Goursat problem and scattering for nonlinear wave equations

AbstractThe Goursat problem for nonlinear scalar equations on the Einstein Universe M̃, with finite-energy datum, has a unique global solution in the positive-energy, Sobolev-controllable case. Such equations include those of the form □ϑ + H′(ϑ) = 0, where H denotes a hamiltonian that is a fourth-order polynomial, bounded below, in components of the multicomponent scalar section ϑ. In particular, the conformally invariant equation (□ + 1)ϑ + λϑ3 = 0 (λ ⩾ 0) is included. In the higher-dimensional analog R × Sn to the Einstein Universe the same result holds under the stronger conditions on H required for Sobolev controllability. Irrespective of energy positivity, there is a unique local-in-time solution for arbitrary finite-energy Goursat datum, for all n ⩾ 3, establishing evolution from the given lightcone to any sufficiently close lightcone. These results show the existence of wave operators in the sense of scattering theory, and their continuity in the (Einstein) energy metric, for positive-energy equations of the indicated type. They also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in Minkowski space M0 to the Goursat problem in M̃. In particular, any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction Lagrangian on multicomponent scalar sections, having finite Einstein energy at any one time, is asymptotic to solutions of the corresponding multicomponent free wave equation as the Minkowski time x0 → ± ∞. Thus given a finite-Einstein-energy solution of the equation □ƒ + λƒ3 = 0 on M0 (λ ⩾ 0) there exist unique solutions ƒ± of the free wave equation which approach ƒ in the Minkowski energy norm as x0 → ± ∞, and every finite-Einstein-energy solution of the free wave equation is of the form ƒ+ (or ƒ−) for a unique solution ƒ of the nonlinear equation. This generalizes, in part in maximality sharp form, earlier results of Strauss for this equation

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The Coupled Cluster Method in Hamiltonian Lattice Field Theory: SU(2) Glueballs

The glueball spectrum within the Hamiltonian formulation of lattice gauge theory (without fermions) is calculated for the gauge group SU(2) and for two spatial dimensions. The Hilbert space of gauge-invariant functions of the gauge field is generated by its parallel-transporters on closed paths along the links of the spatial lattice. The coupled cluster method is used to determine the spectrum of the Kogut-Susskind Hamiltonian in a truncated basis. The quality of the description is studied by computing results from various truncations, lattice regularisations and with an improved Hamiltonian. We find consistency for the mass ratio predictions within a scaling region where we obtain good agreement with standard lattice Monte Carlo results.Comment: 13 pages, 7 figure

Path Integral Monte Carlo Approach to the U(1) Lattice Gauge Theory in (2+1) Dimensions

Path Integral Monte Carlo simulations have been performed for U(1) lattice gauge theory in (2+1) dimensions on anisotropic lattices. We extractthe static quark potential, the string tension and the low-lying "glueball" spectrum.The Euclidean string tension and mass gap decrease exponentially at weakcoupling in excellent agreement with the predictions of Polyakov and G{\" o}pfert and Mack, but their magnitudes are five times bigger than predicted. Extrapolations are made to the extreme anisotropic or Hamiltonian limit, and comparisons are made with previous estimates obtained in the Hamiltonian formulation.Comment: 12 pages, 16 figure

Hamiltonian Study of Improved U(1U(1 Lattice Gauge Theory in Three Dimensions

A comprehensive analysis of the Symanzik improved anisotropic three-dimensional U(1) lattice gauge theory in the Hamiltonian limit is made. Monte Carlo techniques are used to obtain numerical results for the static potential, ratio of the renormalized and bare anisotropies, the string tension, lowest glueball masses and the mass ratio. Evidence that rotational symmetry is established more accurately for the Symanzik improved anisotropic action is presented. The discretization errors in the static potential and the renormalization of the bare anisotropy are found to be only a few percent compared to errors of about 20-25% for the unimproved gauge action. Evidence of scaling in the string tension, antisymmetric mass gap and the mass ratio is observed in the weak coupling region and the behaviour is tested against analytic and numerical results obtained in various other Hamiltonian studies of the theory. We find that more accurate determination of the scaling coefficients of the string tension and the antisymmetric mass gap has been achieved, and the agreement with various other Hamiltonian studies of the theory is excellent. The improved action is found to give faster convergence to the continuum limit. Very clear evidence is obtained that in the continuum limit the glueball ratio $M_{S}/M_{A}$ approaches exactly 2, as expected in a theory of free, massive bosons.Comment: 13 pages, 15 figures, submitted to Phys. Rev.

Density Matrix Renormalisation Group Approach to the Massive Schwinger Model

The massive Schwinger model is studied, using a density matrix renormalisation group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are almost two orders of magnitude more accurate than previous calculations. Coleman's picture of `half-asymptotic' particles at background field theta = pi is confirmed. The predicted phase transition at finite fermion mass (m/g) is accurately located, and demonstrated to belong in the 2D Ising universality class.Comment: 38 pages, 18 figures, submitted to PR

NMDAR inhibition-independent antidepressant actions of ketamine metabolites

Major depressive disorder afflicts ~16 percent of the world population at some point in their lives. Despite a number of available monoaminergic-based antidepressants, most patients require many weeks, if not months, to respond to these treatments, and many patients never attain sustained remission of their symptoms. The non-competitive glutamatergic N-methyl-D-aspartate receptor (NMDAR) antagonist, (R,S)-ketamine (ketamine), exerts rapid and sustained antidepressant effects following a single dose in depressed patients. Here we show that the metabolism of ketamine to (2S,6S;2R,6R)-hydroxynorketamine (HNK) is essential for its antidepressant effects, and that the (2R,6R)-HNK enantiomer exerts behavioural, electroencephalographic, electrophysiological and cellular antidepressant actions in vivo. Notably, we demonstrate that these antidepressant actions are NMDAR inhibition-independent but they involve early and sustained α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid (AMPA) receptor activation. We also establish that (2R,6R)-HNK lacks ketamine-related side-effects. Our results indicate a novel mechanism underlying ketamine’s unique antidepressant properties, which involves the required activity of a distinct metabolite and is independent of NMDAR inhibition. These findings have relevance for the development of next generation, rapid-acting antidepressants

Prenatal Detection of Aneuploidy and Imbalanced Chromosomal Arrangements by Massively Parallel Sequencing

Fetal chromosomal abnormalities are the most common reasons for invasive prenatal testing. Currently, G-band karyotyping and several molecular genetic methods have been established for diagnosis of chromosomal abnormalities. Although these testing methods are highly reliable, the major limitation remains restricted resolutions or can only achieve limited coverage on the human genome at one time. The massively parallel sequencing (MPS) technologies which can reach single base pair resolution allows detection of genome-wide intragenic deletions and duplication challenging karyotyping and microarrays as the tool for prenatal diagnosis. Here we reported a novel and robust MPS-based method to detect aneuploidy and imbalanced chromosomal arrangements in amniotic fluid (AF) samples. We sequenced 62 AF samples on Illumina GAIIx platform and with averagely 0.01× whole genome sequencing data we detected 13 samples with numerical chromosomal abnormalities by z-test. With up to 2× whole genome sequencing data we were able to detect microdeletion/microduplication (ranged from 1.4 Mb to 37.3 Mb of 5 samples from chorionic villus sampling (CVS) using SeqSeq algorithm. Our work demonstrated MPS is a robust and accurate approach to detect aneuploidy and imbalanced chromosomal arrangements in prenatal samples

Risk factors for positive and negative COVID-19 tests: a cautious and in-depth analysis of UK Biobank data

Background The recent COVID-19 outbreak has generated an unprecedented public health crisis, with millions of infections and hundreds of thousands of deaths worldwide. Using hospital-based or mortality data, several COVID-19 risk factors have been identified, but these may be confounded or biased. Methods Using SARS-CoV-2 infection test data (N=4,509 tests; 1,325 positive) from Public Health England, linked to the UK Biobank study, we explored the contribution of demographic, social, health risk, medical, and environmental factors to COVID-19 risk. We used multivariable and penalised logistic regression models for the risk of (i) being tested, (ii) testing positive/negative in the study population and, adopting a test negative design, (iv) the risk of testing positive within the tested population. Results In the fully adjusted model, variables independently associated with the risk of being tested for COVID-19 with OR >1.05 were: male sex; Black ethnicity; social disadvantage (as measured by education, housing and income); occupation (healthcare worker, retired, unemployed); ever smoker; severely obese; comorbidities; and greater exposure to PM2.5-absorbance. Of these, only male sex, non-White ethnicity, lower educational attainment, and none of the comorbidities or health risk factors, were associated with testing positive among tested individuals. Conclusions We adopted a careful and exhaustive approach within a large population-based cohort, which enabled us to triangulate evidence linking, male sex, lower educational attainment, non-White ethnicity with the risk of COVID-19. The elucidation of the joint and independent effects of these factors is a high-priority area for further research to inform on COVID-19 natural history