Direct and inverse scattering problems for operator of order 4 on the line

Abstract. We study both the direct and the inverse scattering problems for a differential operator of order 4 on the line. In the direct scattering problem we start by forming a differential equation from our operator. By applying the fundamental solution of the differential operator we turn the differential equation into an integral equation, and proceed to solve it. Having found the solution to the integral equation we study the asymptotic behaviour of the solution. We conclude the study of the direct scattering problem by defining so called transmission and reflection coefficients, that are needed to solve the inverse scattering problem. In the inverse scattering problem we simplify the operator by choosing its coefficients suitably. It turns out that the operator includes one interesting potential function V. The inverse problem is formulated as follows: find and construct the jumps and singularities of the potential V. By using the reflection coefficient defined previously we define the so called inverse Born approximation V_B. We prove that the difference V-V_B is a continuous function. This means that the jumps and singularities of potential V can be found by calculating V_B.Suora ja käänteinen sirontaongelma neljännen kertaluvun operaattorille. Tiivistelmä. Työssä tutkitaan sekä suoraa että käänteistä sirontaongelmaa neljännen kertaluvun differentiaalioperaattorille. Suorassa sirontaongelmassa käytämme operaattoria muodostaaksemme differentiaaliyhtälön. Soveltaen operaattorin perusratkaisua, voimme muuntaa differentiaaliyhtälön integraaliyhtälöksi ja ratkaista sen. Kun integraaliyhtälön ratkaisu on löydetty, tutkimme sen asymptoottista käyttäytymistä. Päätämme suoran sirontaongelman tarkastelun määrittelemällä asymptoottien avulla ns. välitys- ja heijastuskertoimet, joita tarvitaan käänteisen sirontaongelman ratkaisussa. Käänteisessä sirontaongelmassa yksinkertaistamme operaattoria hieman. Käy ilmi, että operaattori sisältää tällöin yhden kiinnostavan potentiaalifunktion V. Käänteisen sirontaongelman asettelu on seuraava: etsi potentiaalin V mahdolliset hyppyepäjatkuvuudet ja singulariteetit, kun heijastuskerroin on tunnettu. Heijastuskertoimen avulla voimme määritellä ns. käänteisen Bornin approksimaation V_B. Osoitamme, että erotus V-V_B on jatkuva funktio. Tällöin potentiaalin V hypyt ja singulariteetit voidaan löytää laskemalla V_B

Tetrazole as a Replacement of the Electrophilic Group in Characteristic Prolyl Oligopeptidase Inhibitors

4-Phenylbutanoyl-aminoacyl-2(S)-tetrazolylpyrrolidines were studied as prolyl oligopeptidase inhibitors. The compounds were more potent than expected from the assumption that the tetrazole would also here be a bioisostere of the carboxylic acid group and the corresponding carboxylic acids are at their best only weak inhibitors. The aminoacyl groups L-prolyl and L-alanyl gave potent inhibitors with IC50 values of 12 and 129 nM, respectively. This was in line with typical prolyl oligopeptidase inhibitors; however, we did observe a difference with N-methyl-L-alanyl, which gave potent inhibitors in typical prolyl oligopeptidase inhibitors but not in our novel compound series. Furthermore, all studied 4-phenylbutanoyl-aminoacyl-2(S)-tetrazolylpyrrolidines decreased alpha-synuclein dimerization at the concentration of 10 mu M, also when they were only weak inhibitors of the proteolytic activity of the enzyme with an IC50 value of 205 mu M. Molecular docking studies revealed that the compounds are likely to bind differently to the enzyme compared to typical prolyl oligopeptidase inhibitors represented in this study by 4-phenylbutanoyl-aminoacyl-2(S)-cyanopyrrolidines.Peer reviewe

Inverse medium problem for a singular contrast

We consider an inverse medium problem in two- and three-dimensional cases. Namely, we investigate the problem of reconstruction of unknown compactly supported refractive index (contrast) from L-2 with a fixed positive wave number. The proof is based on the new estimates for the Green-Faddeev function in L-infinity space. The main goal of this work is to prove a uniqueness result in the two- and three-dimensional cases and to discuss some possible constructive methods for solving the problem. Finally, we present some numerical examples to demonstrate the results in two dimensions. Published under license by AIP Publishing.Peer reviewe

Coping strategies, stress, physical activity and sleep in patients with unexplained chest pain

BACKGROUND: The number of patients suffering from unexplained chest pain (UCP) is increasing. Intervention programmes are needed to reduce the chest pain and suffering experienced by these patients and effective preventive strategies are also required to reduce the incidence of these symptoms. The aim of this study was to describe general coping strategies in patients with UCP and examine the relationships between coping strategies, negative life events, sleep problems, physical activity, stress and chest pain intensity. METHOD: The sample consisted of 179 patients younger than 70 years of age, who were evaluated for chest pain at the emergency department daytime Monday through Friday and judged by a physician to have no organic cause for their chest pain. The study had a cross-sectional design. RESULTS: Emotive coping was related to chest pain intensity (r = 0.17, p = 0.02). Women used emotive coping to a greater extent than did men (p = 0.05). In the multivariate analysis was shown that physical activity decreased emotive coping (OR 0.13, p < 0.0001) while sex, age, sleep, mental strain at work and negative life events increased emotive coping. Twenty-seven percent of the patients had sleep problems 8 to14 nights per month or more. Permanent stress at work during the last year was reported by 18% of the patients and stress at home by 7%. Thirty-five percent of the patients were worried often or almost all the time about being rushed at work and 23% were worried about being unable to keep up with their workload. Concerning total life events, 20% reported that a close relative had had a serious illness and 27% had reasons to be worried about a close relative. CONCLUSION: Our results indicated that patients with more intense UCP more often apply emotive coping in dealing with their pain. Given that emotive coping was also found to be related to disturbed sleep, negative life events, mental strain at work and physical activity, it may be of value to help these patients to both verbalise their emotions and to become cognizant of the influence of such factors on their pain experience

Direct and inverse scattering problems for perturbations of the biharmonic operator

Abstract This dissertation is a combination of four articles on the topic of scattering problems for a biharmonic operator. The operator of interest has two coefficients which may be complex-valued and singular. Each of the articles concerns a different aspect of the problem. Namely, the first article discusses the direct scattering problem in higher dimensions and culminates in a proof of Saito's formula, which yields a uniqueness result for the inverse scattering problem. The second paper is about a backscattering problem in two and three dimensions. We prove that the inverse Born approximation can be used to recover the singularities in the coefficients of the operator. The third article fills in an answer to the question about recovering the complex-valued coefficients in three dimensions that was left open in the second article. The final article studies the inverse scattering problem on the line for a quasi-linear operator.Tiivistelmä Väitöskirjatyö koostuu neljästä artikkelista, jotka käsittelevät sirontaongelmia biharmoniselle operaattorille. Työn kohteena olevalla operaattorilla on kaksi kerrointa, jotka voivat olla kompleksiarvoisia ja singulaarisia. Kukin artikkeli käsittelee sirontaongelmaa eri näkökulmasta. Ensimmäinen artikkeli koostuu pääasiassa suorasta sirontateoriasta korkeammissa ulottuvuuksissa huipentuen lopulta Saiton kaavan todistukseen, jonka seurauksena saadaan yksikäsitteisyystulos käänteiselle sirontaongelmalle. Toisen artikkelin aiheena on takaisinsirontaongelma kahdessa ja kolmessa ulottuvuudessa. Todistamme, että käänteistä Bornin approksimaatiota voidaan käyttää paikantamaan kertoimien mahdolliset singulariteetit. Kolmas artikkeli vastaa toisessa artikkelissa avoimeksi jääneeseen kysymykseen kompleksiarvoisien kertoimien rekonstruoimisesta kolmessa ulottuvuudessa. Viimeisessä artikkelissa tutkitaan käänteistä sirontaongelmaa kvasilineaariselle operaattorille yhdessä ulottuvuudessa

Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D

Abstract We consider a backscattering Born approximation for a perturbed biharmonic operator in three space dimensions. Previous results on this approach for biharmonic operator use the fact that the coefficients are real-valued to obtain the reconstruction of singularities in the coefficients. In this text we drop the assumption about real-valued coefficients and also establish the recovery of singularities for complex coefficients. The proof uses mapping properties of the Radon transform

Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line

Abstract We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients