Exact Solutions to the Sine-Gordon Equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are exponentially asymptotic to integer multiples of $2\pi$ as $x\to\pm\infty.$ The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.Comment: 43 page

Explicit solutions to the Korteweg-de Vries equation on the half line

Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the $xt$-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schr\"odinger equation. In the reflectionless case such solutions reduce to pure $N$-soliton solutions. An illustrative example is provided.Comment: 17 pages, no figure

Properties of Solutions in 2+1 Dimensions

We solve the Einstein equations for the 2+1 dimensions with and without scalar fields. We calculate the entropy, Hawking temperature and the emission probabilities for these cases. We also compute the Newman-Penrose coefficients for different solutions and compare them.Comment: 16 pages, 1 figures, PlainTeX, Dedicated to Prof. Yavuz Nutku on his 60th birthday. References adde

Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.Comment: 60 pages, 18 figure

Inverse Spectral-Scattering Problem with Two Sets of Discrete Spectra for the Radial Schroedinger Equation

The Schroedinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectru

Patterns of analgesic use, pain and self-efficacy: a cross-sectional study of patients attending a hospital rheumatology clinic

Background: Many people attending rheumatology clinics use analgesics and non-steroidal anti-inflammatories for persistent musculoskeletal pain. Guidelines for pain management recommend regular and pre-emptive use of analgesics to reduce the impact of pain. Clinical experience indicates that analgesics are often not used in this way. Studies exploring use of analgesics in arthritis have historically measured adherence to such medication. Here we examine patterns of analgesic use and their relationships to pain, self-efficacy and demographic factors. Methods: Consecutive patients were approached in a hospital rheumatology out-patient clinic. Pattern of analgesic use was assessed by response to statements such as 'I always take my tablets every day.' Pain and self-efficacy (SE) were measured using the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) and Arthritis Self-Efficacy Scale (ASES). Influence of factors on pain level and regularity of analgesic use were investigated using linear regression. Differences in pain between those agreeing and disagreeing with statements regarding analgesic use were assessed using t-tests. Results: 218 patients (85% of attendees) completed the study. Six (2.8%) patients reported no current pain, 26 (12.3%) slight, 100 (47.4%) moderate, 62 (29.4%) severe and 17 (8.1%) extreme pain. In multiple linear regression self efficacy and regularity of analgesic use were significant (p < 0.01) with lower self efficacy and more regular use of analgesics associated with more pain. Low SE was associated with greater pain: 40 (41.7%) people with low SE reported severe pain versus 22 (18.3%) people with high SE, p < 0.001. Patients in greater pain were significantly more likely to take analgesics regularly; 13 (77%) of those in extreme pain reported always taking their analgesics every day, versus 9 (35%) in slight pain. Many patients, including 46% of those in severe pain, adjusted analgesic use to current pain level. In simple linear regression, pain was the only variable significantly associated with regularity of analgesic use: higher levels of pain corresponded to more regular analgesic use (p = 0.003). Conclusion: Our study confirms that there is a strong inverse relationship between self-efficacy and pain severity. Analgesics are often used irregularly by people with arthritis, including some reporting severe pain

One-year efficacy and safety of routine prasugrel in patients with acute coronary syndromes treated with percutaneous coronary intervention

Objective: To investigate 1‑year outcomes with routine prasugrel treatment after acute coronary syndrome (ACS) in a large-scale registry. Methods: The Rijnmond Collective Cardiology Research registry is a prospective, observational study that enrolled 4,258 consecutive ACS patients treated with percutaneous coronary intervention (PCI) with 1‑year follow-up. Patients received prasugrel as first-choice antiplatelet agent, except for increased bleeding risk patients in which clopidogrel was recommended. Events were validated by an independent clinical endpoint committee. Results: A total number of 2,677 patients received prasugrel at discharge after the index event. Eighty-one percent of the target population was on prasugrel treatment at hospital discharge. At 1 year, the primary endpoint, a composite of all-cause mortality and myocardial infarction, occurred in 2.4% of patients receiving prasugrel. All-cause mortality occurred in 1.0%, myocardial infarction in 1.5%, target-vessel revascularisation in 3.1%, stent thrombosis in 0.6%, and stroke in 0.5% of the patients treated with prasugrel. Thrombolysis in Myocardial Infarction defined major bleeding episodes not related to coronary artery bypass grafting were observed in 1.4% of patients receiving prasugrel. Conclusions: In routine practice, a tailored approach of prasugrel prescription in ACS patients undergoing PCI, resulted in low ischaemic and low bleeding rates up to 1 year post PCI