Continued fraction solution of Krein's inverse problem

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.Comment: 18 pages, 2 figure

CSF level of beta-amyloid peptide predicts mortality in Alzheimer's disease

Objective Alzheimer’s disease (AD) is the sixth leading cause of death, with an average survival estimated between 5 and 10 years after diagnosis. Despite recent advances in diagnostic criteria of AD, few studies have used biomarker-based diagnostics to determine the prognostic factors of AD. We investigate predictors of death and institutionalization in a population of AD patients with high probability of AD physiopathology process assessed by positivity of three CSF biomarkers. Methods Three hundred twenty-one AD patients with abnormal values for CSF beta-amyloid peptide (Aβ42), tau, and phosphorylated tau levels were recruited from a memory clinic-based registry between 2008 and 2017 (Lariboisiere hospital, Paris, France) and followed during a median period of 3.9 years. We used multivariable Cox models to estimate the hazard ratio (HR) of death and institutionalization for baseline clinical data, genotype of the apolipoprotein E (APOE), and levels of CSF biomarkers. Results A total of 71 (22%) patients were institutionalized and 57 (18%) died during the follow-up. Greater age, male sex, lower MMSE score, and lower CSF Aβ42 level were associated with an increased risk of mortality. One standard deviation lower CSF Aβ42 (135 pg/mL) was associated with a 89% increased risk of death (95% CI = 1.25–2.86; p = 0.002). This association was not modified by age, sex, education, APOE ε4, and disease severity. There was no evidence of an association of tau CSF biomarkers with mortality. None of the CSF biomarkers were associated with institutionalization. Conclusions Lower CSF Aβ42 is a strong prognostic marker of mortality in AD patients, independently of age or severity of the disease. Whether drugs targeting beta-amyloid peptide could have an effect on mortality of AD patients should be investigated in future clinical trials

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.Comment: 17 pages, submitte

Some recent developments in the transmutation operator approach

This is a brief overviewof some recent developments in the transmutation operator approach to practical solution of mathematical physics problems. It introduces basic notions and results of transmutation theory, and gives a brief historical survey with some important references. Mainly applications to linear ordinary and partial differential equations and to related boundary value and spectral problems are discusse

The inverse spectral problem and the factorization

SIGLEAvailable from British Library Document Supply Centre- DSC:D73505/87 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Computing eigenvalues of the string by sampling

AbstractIn this paper, we shall use the Shannon-Whittacker-Kotelnikov sampling theorem to approximate the eigenvalues of the string, y″(t) + μ2w(t)y(t) = 0, where the weight w(t) ≥ 0 is allowed to vanish on subintervals. After a discussion on the truncation error, numerical results are provided