The Minkowski ?(x) function and Salem's problem

R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 427-439) asked whether the Fourier-Stieltjes transform of the Minkowski question mark function ?(x) vanishes at infinity. In this note we present several possible approaches towards the solution. For example, we show that this transform satisfies integral and discrete functional equations. Thus, we expect the affirmative answer to Salem's problem. In the end of this note we show that recent attempt to settle this question (S. Yakubovich, C. R. Acad. Sci. Paris, Ser. I 349 (11-12) (2011) 633-636) is fallacious.Comment: 4 pages. C. R. Acad. Sci. Paris, Ser. I. (2012

MODELING AND ANALYSIS OF INTERACTIONS BETWEEN A PULSATILE PNEUMATIC VENTRICULAR ASSIST DEVICE AND THE LEFT VENTRICLE

The use of a ventricular assist device (VAD) is a promising option for the treatment of end-stage heart failure. In many cases VADs provide not only temporary support, but contribute to the recovery of the native ventricle. Many studies have reported incidences where the native ventricle has recovered function, leading to device explantation and eliminating the need for heart transplantation. Despite strong interest in the subject for many years, the determinants of the recovery process are poorly understood and number of patients successfully weaned from chronic support remains low.A mathematical model was developed to gain an understanding of the complex mechanical interactions between a pneumatic, pulsatile VAD and the left ventricle. The VAD model was verified in-vitro using a mock circulatory loop. Over a wide range of experimental conditions, it correctly described observed dynamic behaviors and was accurate in predicting both VAD stroke volume and fill-to-empty rate within 6% error. This validated VAD model was coupled to a simple, lumped parameter cardiovascular model. The coupled model qualitatively reproduced the temporal patterns of various hemodynamic variables observed in clinical data. A concept of VAD characteristic frequency (fc) was developed to facilitate the analysis of VAD-ventricle synchrony. Characteristic frequency, defined as VAD rate in the absence of ventricular contraction, was essentially independent of cardiovascular parameters. For a given set of VAD parameters, synchrony was found to occur over a range of native heart rates. While the lower bound was determined by fc alone, the upper bound was a function of various cardiovascular parameters (e.g., left ventricular contractility, EMAX and systemic vascular resistance, SVR). In the case of synchronous behavior, the VAD and native heart have matched rates and counter-pulse, resulting in reduced ventricular loading. A decrease in EMAX or an increase in SVR increases asynchrony, resulting in frequent occurrences of co-pulsed beats (i.e., high ventricular loading). In conclusion, we found that VAD-ventricle synchrony is determined by a complex interaction between VAD and cardiovascular parameters. Our model-based analysis of VAD-ventricle interaction may be useful for optimizing the VAD operation, characterizing native ventricular contractility, and better understanding of the recovery process

Global existence, singular solutions, and ill-posedness for the Muskat problem

The Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher viscosity fluid expands into the lower viscosity fluid, we show global in time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher viscosity fluid contracts, we construct singular solutions that start off with smooth initial data, but develop a point of infinite curvature at finite time

Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

We obtain optimal trigonometric polynomials of a given degree $N$ that majorize, minorize and approximate in $L^1(\mathbb{R}/\mathbb{Z})$ the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann that generalize a paper of J. Vaaler. As applications we provide the corresponding Erd\"{o}s-Tur\'{a}n-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.Comment: 14 pages. Accepted for publication in the J. Approx. Theory. V2 has additional references and some typos correcte

Symmetric Differentiation on Time Scales

We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can be symmetric differentiable.Comment: This is a preprint of a paper whose final and definite form will be published in Applied Mathematics Letters. Submitted 30-Jul-2012; revised 07-Sept-2012; accepted 10-Sept-201