The Maslov Index and the Spectral Flow - revisited

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.Comment: 19 page

HOMOCLINIC SOLUTIONS FOR A CLASS OF AUTONOMOUS SECOND ORDER HAMILTONIAN SYSTEMS WITH A SUPERQUADRATIC POTENTIAL

Abstract. We will prove the existence of a nontrivial homoclinic solution for an autonomous second order Hamiltonian systemq + ∇V (q) = 0, where q ∈ R n , a potential V : R n → R is of the form V (q) = −K(q) + W (q), K and W are C 1 -maps, K satisfies the pinching condition, W grows at a superquadratic rate, as |q| → ∞ and W (q) = o(|q| 2 ), as |q| → 0. A homoclinic solution will be obtained as a weak limit in the Sobolev space W 1,2 (R, R n ) of a sequence of almost critical points of the corresponding action functional. Before passing to a weak limit with a sequence of almost critical points each element of this sequence has to be appropriately shifted

The Equivariant Spectral Flow and Bifurcation for Functionals with Symmetries -- Part I

We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a recent construction of an equivariant spectral flow by the authors, we obtain a bifurcation theorem that generalises well-established results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and Recht. Finally, we discuss some elementary examples of strongly indefinite systems of PDEs and Hamiltonian systems where the mentioned classical approaches fail but an invariance under an orthogonal action of $\mathbb{Z}_2$ makes our methods applicable and yields the existence of bifurcation

Gender differences in hypertension awareness, diagnosis and treatment

Introduction: Hypertension is most important risk factor for cardiovascular disease (CD) and stroke, two of the leading causes of death in adults worldwide. Even though over last decade there has been improvement in increasing the awareness, treatment, and control of hypertension, undiagnosed and uncontrolled hypertension remain one of main public health challenges. In order to make progress in dealing with this problem it important to understand in which areas of the treatment strategy appropriate changes have not yet been implemented. First step to do this is to improve our understanding of the factors that affect the treatment and control of hypertension. International Guidelines for hypertension recommend the same hypertension management for both sexes, however studies reveal that there are actual differences in hypertension management. Purpose of the study: The aim of this study was to conduct a paper review to provide most comprehensive contemporary estimates of sex differences in the awareness, treatment, and control of hypertension. Material and method: A literature review was conducted in English language databases, based only on original articles, regarding awareness, control and treatment of hypertension, using key words: ‘hypertension’, ‘sex- differences’, ‘hypertension treatment’, ‘gender gap’, ‘public health’. Results and conclusions: Based on researched studies we can state that woman have higher awareness, treatment and control rates for hypertension than men worldwide [4,5,6,7,8,9,10]. This suggests that, the guidelines for the treatment and management of hypertension should be gender specific

On the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $q̈(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t$ $\varepsilon$ $\Bbb R$, $q$ $\varepsilon$ $\Bbb R^n$, $a:\Bbb R \longrightarrow \Bbb R^n$ is a continuous positive bounded function, $G: \Bbb R^n \longrightarrow \Bbb R$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f:\Bbb R \longrightarrow \Bbb R^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations

Pomiar parametrów torów telewizyjnych metodą linii kontrolnej. Część II. Przegląd Zagadnień Łączności, 1967, nr 10 (73)

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On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$, $a\colon\mathbb{R}\to\mathbb{R}$ is a continuous positive bounded function, $G\colon\mathbb{R}^n\to\mathbb{R}$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb{R}\to\mathbb{R}^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.Comment: 15 pages, 9 figure

Dealing with the most common side effect of statins

Statins are one of the key drugs in the treatment of cardiology patients. They lower LDL cholesterol levels, thereby reducing the risk of acute cardiovascular events. Statins are among the best studied drugs, and have been proven to be effective and safe in the treatment of dyslipidemia. Data presented in studies on the prevalence of statin intolerance (SI) varies widely, according to some studies, statin tolerance is the same as placebo, however, this is a common problem in clinical practice. The most common reported symptoms of SI include muscle pain. Patients are often afraid of starting satin treatment because of possible side effects. The appearance of myopathy in patients results in their discontinuation of therapy, which results in increased cardiovascular risk. In order to minimize the risk of SI, particularly statin-associated muscular pain (SI), a key role is played by discussion with the patient, exclusion of nocebo effects, identification of risk factors, dose modification and possible discontinuation of the drug. The purpose of this paper is to review the problem of the most common of the adverse effects of statins- myopathy, in terms of incidence, probable cause, and recommended treatment of patients