A note on some Ostrowski-like type inequalities

AbstractError estimate plays an important role for numerical integration. Some Ostrowski-like type inequalities concerning a new type of quadrature formula are established recently by Huy and Ngô (2009, 2010) [6,7]. In this note, improvement and extension to these inequalities are given

Chebyshev-Grüss- and Ostrowski-type Inequalities

Chebyshev-Grüss- und Ostrowski-typ Ungleichungen Mein Promotionsvorhaben "Chebyshev-Grüss- and Ostrowski-type Inequalities" befasst sich mit Chebyshev-Grüss- und Ostrowski-Typ-Ungleichungen im univariaten und bivariaten Fall. Derartige Ungleichungen haben in den letzten Jahren, auch aufgrund ihrer Anwendungen, viel Aufmerksamkeit auf sich gezogen. Die klassische Form der Grüss-Ungleichung, die zum ersten Mal von G. Grüss im Jahre 1935 publiziert wurde, gibt eine Abschätzung für die Differenz zwischen dem Integral des Produktes und dem Produkt der Integrale zweier Funktionen in C[a,b] an. In den folgenden Jahren erschienen in der Literatur viele Varianten dieser Ungleichung. Die vorgelegte Dissertation besteht aus fünf Kapiteln. Der erste Abschnitt beinhaltet Hilfsmittel, die im weiteren Verlauf benötigt werden. Wesentlich dabei sind: Stetigkeitsmodule, das K- Funktional und seine Beziehung zu den Modulen, positive und nicht notwendig positive lineare Operatoren. Im zweiten Kapitel sind Chebyshev-Grüss-Typ-Ungleichungen im eindimensionalen Fall von Interesse. Zunächst werden einige Hilfsergebnisse und Anwendungen derselben angegeben; ebenso wird auf die Historie verwiesen. Einige Bemerkungen und Ergebnisse zur Ungleichung von Chebyshev werden ebenfalls vorgestellt. (Pre-) Chebyshev-Grüss-Typ-Abschätzungen werden anschließ end eingeführt, und zwar mit Hilfe von zweiten Momenten, ersten absoluten Momenten und Größen, die Differenzen von zweiten und ersten Momenten enthalten. Die wichtigsten Ergebnisse betreffen (positive) lineare Operatoren. Für die entsprechenden Anwendungen sind Oszillationen, die durch die kleinste konkave Majorante des ersten Moduls der Ordnung 1 ausgedrückt werden, das zentrale Hilfsmittel. Die Verwendung solcher Oszillationen umfasst alle Punkte im betrachteten Intervall; dies ist die Motivation für einen weiteren Ansatz, der weniger Punkte betrachtet. Derartige diskrete Oszillationen in Chebyshev-Grüss-Typ-Ungleichungen für mehr als zwei Funktionen werden am Ende dieses Kapitels ebenfalls betrachtet. Der dritte Abschnitt überträgt die Ergebnisse des univariaten auf den bivariaten Fall. Hier wird das Verfahren der parametrischen Erweiterungen verwendet. Hilfs-und historische Ergebnisse werden im ersten Teil dargestellt. Sodann werden für ausgewählte Operatoren deren Tensorprodukte betrachtet. Anwendungen werden sowohl für den Ansatz mit der kleinsten konkaven Majorante als auch für den via diskreter Oszillationen gegeben. Der Zweck des vierten und fünften Kapitels ist es, diese Studie zu vervollständigen in dem Sinne, dass univariate und bivariate Ostrowski-Typ-Ungleichungen dargestellt werden. Hier werden zunächst einige historische Betrachtungen angestellt und die entsprechenden Ergebnisse anschließ end modifiziert und teilweise verbessert. Das letzte Kapitel stellt zwei Beispiele von Ostrowski-Typ-Ungleichungen im bivariaten Fall vor. Die beiden Anwendungen, die hier angegeben wurden, betreffen Produkte von Bernstein-Stancu und Bernstein-Durrmeyer-Operatoren mit Jacobi-Gewichten. In beiden Fällen werden Ostrowski-Typ-Ungleichungen mit oder ohne Beteiligung der Iterierten der Operatoren erhalten. Der Grenzwert der Iterierten von positiven linearen Operatoren wird ebenfalls untersucht. Es gibt eine Verbindung zwischen Ostrowski- und Grüss-Ungleichungen, die den Begriff "Ostrowski-Grüss-Typ-Ungleichungen", der häufig in der Literatur verwendet wird, erklärt. Aus Gründen der Klarheit wird betont, dass der Begriff in dieser Arbeit ausschließlich dann verwendet wird, wenn die untere Schranke der Fehlerterm in der einfachsten Quadraturformel ist, während die obere Schranke eine Differenz der oberen und unteren Grenze der Funktion enthält, so wie dies auch in der Arbeit von G. Grüss aus dem Jahr 1935 der Fall ist. Hierbei sei daran erinnert, dass der Begriff "Ostrowski-Grüss-Typ-Ungleichung" zuerst von Dragomir et al. in einem Artikel aus dem Jahr 1997 geprägt wurde.My PhD thesis deals with Chebyshev-Grüss- and Ostrowski-type inequalities in the univariate and bivariate case. Such inequalities have drawn much attention in recent years due to their applications. The classical form of Grüss' inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions in C[a,b]. In the following years a lot of variants of this inequality appeared in the literature. This talk consists of five parts. The first part includes a motivation, containing some introductory instruments that are further used to obtain the results. In the second section, Chebyshev-Grüss-type inequalities in the one-dimensional case are of interest. The results are introduced with the help of second moments, first absolute moments and quantities over differences of second and first moments. They are applied to (positive) linear operators. Oscillations which are expressed by the least concave majorant of the first order modulus are used in the first place. The use of such oscillations includes all points in the considered interval, and that is the reason why a new approach arises looking at fewer points. When talking about discrete oscillations, Chebyshev-Grüss-type inequalities for more than two functions are obtained at the end of this section. The third section extends the results from the univariate to the bivariate case. The method of parametric extensions by means of product of two compact metric spaces is used. Applications are given for both the approach with the least concave majorant and also for the one via discrete oscillations. The purpose of the fourth and fifth sections is to complete this study, in the sense that univariate and bivariate Ostrowski-type inequalities are also considered. Some applications are specified and Ostrowski type inequalities are obtained, with or without the participation of the iterates of the operators. The limit of the iterates of positive linear operators is also studied

Synchrotron and Inverse Compton Constraints on Lorentz Violations for Electrons

We present a method for constraining Lorentz violation in the electron sector, based on observations of the photons emitted by high-energy astrophysical sources. The most important Lorentz-violating operators at the relevant energies are parameterized by a tensor c^{nu mu) with nine independent components. If c is nonvanishing, then there may be either a maximum electron velocity less than the speed of light or a maximum energy for subluminal electrons; both these quantities will generally depend on the direction of an electron's motion. From synchrotron radiation, we may infer a lower bound on the maximum velocity, and from inverse Compton emission, a lower bound on the maximum subluminal energy. With observational data for both these types of emission from multiple celestial sources, we may then place bounds on all nine of the coefficients that make up c. The most stringent bound, on a certain combination of the coefficients, is at the 6 x 10^(-20) level, and bounds on the coefficients individually range from the 7 x 10^(-15) level to the 2 x 10^(-17) level. For most of the coefficients, these are the most precise bounds available, and with newly available data, we can already improve over previous bounds obtained by the same methods.Comment: 28 page

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge

Sums, series, and products in Diophantine approximation

There is not much that can be said for all $x$ and for all $n$ about the sum $$\sum_{k=1}^n \frac{1}{|\sin k\pi x|}.$$ However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools of continued fractions. We present in detail the appearance of these sums in the singular series for the circle method. One particular interest of the paper is the detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication. This story includes various parts of late 19th century and early 20th century mathematics.Comment: 83 pages; thanks for correspondence with Christoph Aistleitne

Singularities, Supersymmetry and Combinatorial Reciprocity

This work illustrates a method to investigate certain smooth, codimension-two, real submanifolds of spheres of arbitrary odd dimension (with complements that fiber over the circle) using a novel supersymmetric quantum invariant. Algebraic (fibered) links, including Brieskorn-Pham homology spheres with exotic differentiable structure, are examples of said manifolds with a relative diffeomorphism-type that is determined by the corresponding (multivariate) Alexander polynomial.Engineering and Applied Science

Ambient air pollution in Massachusetts: inequality trends, residential infiltration, and childhood weight growth trajectories

Exposure to pollutants of ambient origin contributes significantly to the global disease burden (Cohen et al., 2017). Mounting evidence has demonstrated disproportionately high ambient PM2.5 and NO2 concentrations in the U.S. among nonwhite and low-income populations, potentially contributing to environmental health disparities (Bell and Ebisu, 2012; Clark et al., 2014; Morello-Frosch and Lopez, 2006). There is limited understanding of temporal trends and underlying causes of exposure inequalities (EIs), and whether residential building characteristics modify observed EIs. Further, while ambient pollutants have been linked to cardiometabolic disease in adulthood, few studies have documented the link between early-life ambient air pollution exposure and weight growth trajectories in early childhood- an informative step on the causal pathway between early life exposures and chronic outcomes. Using 1 km2 PM2.5 and NO2 predictions in Massachusetts and Census data, we quantify longitudinal EI between sociodemographic groups over a decade. We estimate AER for all Massachusetts residential parcels using publicly available data and assess whether accounting for AER exacerbates or ameliorates PM2.5 inequalities. We examine associations of weight growth trajectories in early childhood with residential prenatal and postnatal PM2.5 and distance to road (traffic) exposure in the Boston-based Children’s HealthWatch cohort. PM2.5 and NO2 inequalities increased across the study period in urban areas, and EIs were more pronounced for NO2 than PM2.5 and among racial/ethnic groups compared to other population subgroups. Analyzing EI longitudinally revealed that spatio-temporal shifts in air pollution, and not demographic distributions, contributed to exposure disparities. We found substantial variability in estimated AER across the state, and that PM2.5 EIs were magnified when AER was considered. Prenatal PM2.5 >9.5 µg/m3 predicted higher weight growth rates among females, but with an opposite direction of effect in males. This association was modified by birth weight and AER, with a stronger magnitude of effect in low-birthweight and higher-AER females. These findings underscore the importance of considering vulnerable communities and residential characteristics in ambient air pollution reduction strategies. This dissertation provides an opportunity to understand susceptible phenotypes and periods of potential intervention to reduce ambient air pollution impacts on cardiometabolic outcomes.2020-03-17T00:00:00