Henstock-Kurzweil integration on euclidean spaces [[electronic resource] /] / Lee Tuo Yeong

• Record Nr.: UNINA9910464534003321
• Autore: Lee Tuo Yeong <1967->
• Pubbl/distr/stampa: Singapore ; ; Hackensack, N.J., : World Scientific, c2011
• ISBN: 1-283-23477-7; 9786613234773; 981-4324-59-0
• Descrizione fisica: 1 online resource (325 p.)
• Collana: Series in real analysis ; ; v. 12
• Disciplina: 515.43
• Soggetti: Henstock-Kurzweil integral; Lebesgue integral; Calculus, Integral; Electronic books.
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and indexes.
• Nota di contenuto: Preface; Contents; 1. The one-dimensional Henstock-Kurzweil integral; 1.1 Introduction and Cousin's Lemma; 1.2 Definition of the Henstock-Kurzweil integral; 1.3 Simple properties; 1.4 Saks-Henstock Lemma; 1.5 Notes and Remarks; 2. The multiple Henstock-Kurzweil integral; 2.1 Preliminaries; 2.2 The Henstock-Kurzweil integral; 2.3 Simple properties; 2.4 Saks-Henstock Lemma; 2.5 Fubini's Theorem; 2.6 Notes and Remarks; 3. Lebesgue integrable functions; 3.1 Introduction; 3.2 Some convergence theorems for Lebesgue integrals; 3.3 μm-measurable sets; 3.4 A characterization of μm-measurable sets; 3.5 μm-measurable functions3.6 Vitali Covering Theorem; 3.7 Further properties of Lebesgue integrable functions; 3.8 The Lp spaces; 3.9 Lebesgue's criterion for Riemann integrability; 3.10 Some characterizations of Lebesgue integrable functions; 3.11 Some results concerning one-dimensional Lebesgue integral; 3.12 Notes and Remarks; 4. Further properties of Henstock-Kurzweil integrable functions; 4.1 A necessary condition for Henstock-Kurzweil integrability; 4.2 A result of Kurzweil and Jarn ́ık; 4.3 Some necessary and su cient conditions for Henstock- Kurzweil integrability; 4.4 Harnack extension for one-dimensional Henstock-Kurzweil integrals4.5 Other results concerning one-dimensional Henstock- Kurzweil integral; 4.6 Notes and Remarks; 5. The Henstock variational measure; 5.1 Lebesgue outer measure; 5.2 Basic properties of the Henstock variational measure; 5.3 Another characterization of Lebesgue integrable functions; 5.4 A result of Kurzweil and Jarn ́ık revisited; 5.5 A measure-theoretic characterization of the Henstock- Kurzweil integral; 5.6 Product variational measures; 5.7 Notes and Remarks; 6. Multipliers for the Henstock-Kurzweil integral; 6.1 One-dimensional integration by parts6.2 On functions of bounded variation in the sense of Vitali; 6.3 The m-dimensional Riemann-Stieltjes integral; 6.4 A multiple integration by parts for the Henstock-Kurzweil integral; 6.5 Kurzweil's multiple integration by parts formula for the Henstock-Kurzweil integral; 6.6 Riesz Representation Theorems; 6.7 Characterization of multipliers for the Henstock-Kurzweil integral; 6.8 A Banach-Steinhaus Theorem for the space of Henstock- Kurzweil integrable functions; 6.9 Notes and Remarks; 7. Some selected topics in trigonometric series; 7.1 A generalized Dirichlet test7.2 Fourier series; 7.3 Some examples of Fourier series; 7.4 Some Lebesgue integrability theorems for trigonometric series; 7.5 Boas' results; 7.6 On a result of Hardy and Littlewood concerning Fourier series; 7.7 Notes and Remarks; 8. Some applications of the Henstock-Kurzweil integral to double trigonometric series; 8.1 Regularly convergent double series; 8.2 Double Fourier series; 8.3 Some examples of double Fourier series; 8.4 A Lebesgue integrability theorem for double cosine series; 8.5 A Lebesgue integrability theorem for double sine series; 8.6 A convergence theorem for Henstock-Kurzweil integrals
• Sommario/riassunto: The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Per

Bridging the Gap : The Measure of Urban Resilience

• Record Nr.: UNINA9910557489403321
• Autore: Brunetta Grazia
• Pubbl/distr/stampa: Basel, Switzerland, : MDPI - Multidisciplinary Digital Publishing Institute, 2021
• Descrizione fisica: 1 online resource (266 p.)
• Soggetti: Economics, finance, business & management
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Sommario/riassunto: The concept of resilience has arisen as a "new way of thinking", becoming a response to both the causes and effects of ongoing global challenges. As it strongly stresses cities' transformative potential, resilience's final purpose is to prevent and manage unforeseen events and improve communities' environmental and social quality. Although the resilience theory has been investigated in depth, several methodological challenges remain, mainly related to the concept's practical sphere. As a matter of fact, resilience is commonly criticised for being too ambiguous and empty of meaning. At the same time, turning resilience into practice is not easy to do. This will arguably be one of the most impactful global issues for future research on resilience. The Special Issue "Bridging the Gap: The Measure of Urban Resilience" falls under this heading, and it seeks to synthesise state-of-the-art knowledge of theories and practices on measuring resilience. The Special Issue collected 11 papers that address the following questions: "What are the theoretical perspectives of measuring urban resilience? What are the existing methods for measuring urban resilience? What are the main features that a technique for measuring urban resilience needs to have? What is the role of measuring urban resilience in operationalising cities' ability to adapt, recover and benefit from shocks?"