Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0

We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R-separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases

Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian

Lie theory and separation of variables. 6. The equation iUt + ∆2U = 0

This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method

Do the Barker Codes End?

A Barker code is a binary code with k^th autocorrelation <= 1 for all nonzero k. At the workshop, the Barker code group split into four non-disjoint subgroups: - An "algebra group", who explored symmetries of the search space that preserve the autocorrelations' magnitude. - A "computing group", who explored methods for quickly finding binary codes with very good autocorrelation properties. - A "statistics group", who explored ways to quantify what has been empirically observed about autocorrelation in the search space S_2^N. - A "continuous group", who explored a non-discrete analogue of the problem of finding sequences with good autocorrelations

The Effectiveness of Maintenance Pharmacotherapies for Non-Small Cell Lung Cancer

Although current recommendations for the treatment of advanced non-small cell lung cancer (NSCLC) include a maximum of six cycles of platinum-based combination therapy as a first-line approach, most patients experience progression within 3–4 months. Therefore, a new treatment strategy, maintenance therapy, has been proposed, and several large randomized prospective controlled trials have shown benefits with maintenance therapy. Maintenance therapy can be classified as either continuation maintenance, which is defined as a prolongation of a part of the first-line chemotherapy or molecularly targeted agent until progression, or switch-maintenance, which is defined as the administration of a different cytotoxic chemotherapy or molecularly targeted agent immediately after induction therapy. In this article, recent results from large randomized phase III trials regarding maintenance therapy are reviewed in order to evaluate the role of maintenance therapy in NSCLC

Effects of genuine dimension-six Higgs operators

We systematically discuss the consequences of genuine dimension-six Higgs operators. These operators are not subject to stringent constraints from electroweak precision data. However, they can modify the couplings of the Higgs boson to electroweak gauge bosons and, in particular, the Higgs self-interactions. We study the sensitivity to which those couplings can be probed at future \ee linear colliders in the sub-TeV and in the multi-TeV range. We find that for $\sqrt s=500$ GeV with a luminosity of 1 ab$^{-1}$ the anomalous $WWH$ and $ZZH$ couplings may be probed to about the 0.01 level, and the anomalous $HHH$ coupling to about the 0.1 level.Comment: 21 pages, 17 figures; typos corrected and references adde

Aspects of CP violation in the HZZ coupling at the LHC

We examine the CP-conserving (CPC) and CP-violating (CPV) effects of a general HZZ coupling through a study of the process H -> ZZ* -> 4 leptons at the LHC. We construct asymmetries that directly probe these couplings. Further, we present complete analytical formulae for the angular distributions of the decay leptons and for some of the asymmetries. Using these we have been able to identify new observables which can provide enhanced sensitivity to the CPV $H ZZ$ coupling. We also explore probing CP violation through shapes of distributions in different kinematic variables, which can be used for Higgs bosons with mH < 2 mZ.Comment: 36 pages, 17 figures, LaTeX, version accepted for publicatio

Ethical Acceptability of Postrandomization Consent in Pragmatic Clinical Trials.

Importance: Pragmatic clinical trials that seek informed consent after randomization (ie, postrandomization consent) are increasingly used, but debate on ethics persists because control arm patients are not specifically informed about the trials and randomization occurs before consent for the trials. The public's attitude toward postrandomization consent trials is unknown, but the way the trials are described could bias people's views. Objectives: To assess the attitudes of the US general public toward postrandomization informed consent for pragmatic trials and to measure potential framing and other factors associated with those attitudes. Design, Setting, and Participants: An online, 2 × 2 experimental survey (fielded between February 23 and April 3, 2018) portraying 4 scenarios of postrandomization informed consent (with prior broad consent for medical record use) was conducted. These scenarios included traditional randomized clinical trial language framing vs alternative framing in a high-stakes trial (ie, survival in leukemia) or low-stakes trial (ie, blood glucose level in diabetes). A total of 3793 individuals invited to participate were part of an existing panel representative of the US general public (GfK KnowledgePanel). Main Outcomes and Measures: The proportion of participants who would recommend that an ethics review board approve a postrandomization consent pragmatic trial. Results: A total of 2042 of 3739 invitees (54.6%) responded; after exclusion of 38 incomplete surveys, 2004 participants were included in the analysis. Of these, 997 (49.8%) were women, 1440 (71.9%) were white non-Hispanic, 199 (9.9%) were black non-Hispanic, and 233 (11.6%) were Hispanic. Mean (SD) age was 47.5 (17.4) years. Across scenarios, weighted data showed that 75.4% of the participants would recommend approval of the postrandomization consent pragmatic trial, 20.4% would probably not recommend approval, and 4.2% would definitely not recommend approval. Approval was not sensitive to framing language (traditional vs new framing in high-stakes scenario, 74.3% vs 76.8%, P = .40; in low-stakes scenario, 77.7% vs 72.9%, P = .10) or to the stakes (low vs high stakes in traditional framing, 77.7% vs 74.3%, P = .25; in new framing, 72.9% vs 76.8%, P = .18). Better understanding of the postrandomization consent design was associated with higher rate of approval (78.1% vs 65.0%, P = .002 for high-stakes scenario; 77.2% vs 64.9%, P = .004 for low-stakes scenario), especially among those with less education. However, opinions about personal involvement in the control arm were more cautious (range depending on scenario, 45.6%-59.7%) and sensitive to stakes but not to framing. Conclusions and Relevance: The public's generally high rate of approval of the ethics of postrandomization informed consent for pragmatic trial designs does not appear to be affected by whether postrandomization consent design is framed using traditional randomized clinical trial terminology, regardless of the stakes of the trial. Promoting better understanding of the design may increase its acceptance by the public

Net Fisher information measure versus ionization potential and dipole polarizability in atoms

The net Fisher information measure, defined as the product of position and momentum Fisher information measures and derived from the non-relativistic Hartree-Fock wave functions for atoms with Z=1-102, is found to correlate well with the inverse of the experimental ionization potential. Strong direct correlations of the net Fisher information are also reported for the static dipole polarizability of atoms with Z=1-88. The complexity measure, defined as the ratio of the net Onicescu information measure and net Fisher information, exhibits clearly marked regions corresponding to the periodicity of the atomic shell structure. The reported correlations highlight the need for using the net information measures in addition to either the position or momentum space analogues. With reference to the correlation of the experimental properties considered here, the net Fisher information measure is found to be superior than the net Shannon information entropy.Comment: 16 pages, 6 figure