An extension of Laplace's method

Asymptotic expansions are obtained for contour integrals of the form $$\int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt,$$ in which $z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are analytic functions of $t$, and the positive constants $\mu$ and $\nu$ are related to the local behaviour of the functions $p(t)$ and $r(t)$ near the endpoint $a$. Our main theorem includes as special cases several important asymptotic methods for integrals such as those of Laplace, Watson, Erd\'elyi and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle using formal, non-rigorous methods. The results of the paper also serve to place Dingle's investigations on a rigorous mathematical foundation. The new results have potential applications in the asymptotic theory of special functions in transition regions, and we illustrate this by two examples.Comment: 19 pages, 2 figures, revised version, accepted for publication in Constructive Approximatio

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal-gain cascades (i.e., when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on the rates and length of the cascade, as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. In particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction.Comment: 18 pages, 7 figure

What two models may teach us about duality violations in QCD

Though the operator product expansion is applicable in the calculation of current correlation functions in the Euclidean region, when approaching the Minkowskian domain, violations of quark-hadron duality are expected to occur, due to the presence of bound-state or resonance poles. In QCD finite-energy sum rules, contour integrals in the complex energy plane down to the Minkowskian axis have to be performed, and thus the question arises what the impact of duality violations may be. The structure and possible relevance of duality violations is investigated on the basis of two models: the Coulomb system and a model for light-quark correlators which has already been studied previously. As might yet be naively expected, duality violations are in some sense "maximal" for zero-width bound states and they become weaker for broader resonances whose poles lie further away from the physical axis. Furthermore, to a certain extent, they can be suppressed by choosing appropriate weight functions in the finite-energy sum rules. A simplified Ansatz for including effects of duality violations in phenomenological QCD sum rule analyses is discussed as well.Comment: 17 pages, 6 figures; version to appear in JHE

A Feynman integral in Lifshitz-point and Lorentz-violating theories in R^D ⨁ R<i>^m</i>

We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X&lt;1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations

Yoga vs. physical therapy vs. education for chronic low back pain in predominantly minority populations: Study protocol for a randomized controlled trial

Background: Chronic low back pain causes substantial morbidity and cost to society while disproportionately impacting low-income and minority adults. Several randomized controlled trials show yoga is an effective treatment. However, the comparative effectiveness of yoga and physical therapy, a common mainstream treatment for chronic low back pain, is unknown.Methods/Design: This is a randomized controlled trial for 320 predominantly low-income minority adults with chronic low back pain, comparing yoga, physical therapy, and education. Inclusion criteria are adults 18-64 years old with non-specific low back pain lasting ≥12 weeks and a self-reported average pain intensity of ≥4 on a 0-10 scale. Recruitment takes place at Boston Medical Center, an urban academic safety-net hospital and seven federally qualified community health centers located in diverse neighborhoods. The 52-week study has an initial 12-week Treatment Phase where participants are randomized in a 2:2:1 ratio into i) a standardized weekly hatha yoga class supplemented by home practice; ii) a standardized evidence-based exercise therapy protocol adapted from the Treatment Based Classification method, individually delivered by a physical therapist and supplemented by home practice; and iii) education delivered through a self-care book. Co-primary outcome measures are 12-week pain intensity measured on an 11-point numerical rating scale and back-specific function measured using the modified Roland Morris Disability Questionnaire. In the subsequent 40-week Maintenance Phase, yoga participants are re-randomized in a 1:1 ratio to either structured maintenance yoga classes or home practice only. Physical therapy participants are similarly re-randomized to either five booster sessions or home practice only. Education participants continue to follow recommendations of educational materials. We will also assess cost effectiveness from the perspectives of the individual, insurers, and society using claims databases, electronic medical records, self-report cost data, and study records. Qualitative data from interviews will add subjective detail to complement quantitative data.Trial registration: This trial is registered in ClinicalTrials.gov, with the ID number: NCT01343927. © 2014 Saper et al.; licensee BioMed Central Ltd

Safety and Reactogenicity of Canarypox ALVAC-HIV (vCP1521) and HIV-1 gp120 AIDSVAX B/E Vaccination in an Efficacy Trial in Thailand

A prime-boost vaccination regimen with ALVAC-HIV (vCP1521) administered intramuscularly at 0, 4, 12, and 24 weeks and gp120 AIDSVAX B/E at 12 and 24 weeks demonstrated modest efficacy of 31.2% for prevention of HIV acquisition in HIV-uninfected adults participating in a community-based efficacy trial in Thailand.Reactogenicity was recorded for 3 days following vaccination. Adverse events were monitored every 6 months for 3.5 years, during which pregnancy outcomes were recorded. Of the 16,402 volunteers, 69% of the participants reported an adverse event any time after the first dose. Only 32.9% experienced an AE within 30 days following any vaccination. Overall adverse event rates and attribution of relatedness did not differ between groups. The frequency of serious adverse events was similar in vaccine (14.3%) and placebo (14.9%) recipients (p = 0.33). None of the 160 deaths (85 in vaccine and 75 in placebo recipients, p = 0.43) was assessed as related to vaccine. The most common cause of death was trauma or traffic accident. Approximately 30% of female participants reported a pregnancy during the study. Abnormal pregnancy outcomes were experienced in 17.1% of vaccine and 14.6% (p = 0.13) of placebo recipients. When the conception occurred within 3 months (estimated) of a vaccination, the majority of these abnormal outcomes were spontaneous or elective abortions among 22.2% and 15.3% of vaccine and placebo pregnant recipients, respectively (p = 0.08). Local reactions occurred in 88.0% of vaccine and 61.0% of placebo recipients (p<0.001) and were more frequent after ALVAC-HIV than AIDSVAX B/E vaccination. Systemic reactions were more frequent in vaccine than placebo recipients (77.2% vs. 59.8%, p<0.001). Local and systemic reactions were mostly mild to moderate, resolving within 3 days.The ALVAC-HIV and AIDSVAX B/E vaccine regimen was found to be safe, well tolerated and suitable for potential large-scale use in Thailand.ClinicalTrials.govNCT00223080