This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing. The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.
This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.
This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts. Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.
This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing. The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.
Storia: A Journey Through Life and Places By: Alberto Rizzotti Thirty years in the making, Storia is much more than a simple memoir. Author Alberto Rizzotti’s intention is to represent a lifetime of experiences in the context of his emotions. From birth to semi-retirement, he has much to share with regard to adventure, mischief, and achievements, but it’s his open door into his thinking and his most personal feelings that make this book stand out. Rizzotti has struggled with exposing himself to such a degree, opening up to the most intimate of disclosures, but ultimately, he wanted to be true, especially to himself. The writing of Storia is undoubtedly the most effective way for him to do just that, to state emphatically, Like it or not, this is who I am.
Power is an issue that is attracting increased interest among philosophers, theologians and social scientists. The gospel of Mark, especially in 10:32-45, contains teachings attributed to Jesus about the use and abuse of power. This book applies a combination of different methods and approaches: mainly orality, criticism, literary criticism and a sensitivity for the social and cultural environment of the text, showing the centrality of Jesus's message on the issue of power both for the plot and for the theology of Mark. This message is a call to practice leadership in a way that is subversive toward the networks of power of the empire.
The Committee Firenze 2016, on the occasion of the 50th anniversary of the tragic 1966 flood, invited six engineers and scientists to form an International Technical Scientific Committee (ITSC) to assess the current status of flood protection for the city of Florence and identify steps to reduce the risk of flooding facing the city. In this final Report, ITSC concludes that Florence remains at risk to significant flooding and this risk grows each day. It is not a question of whether a flood of the magnitude of 1966 or greater will occur, but when. In fact, the level of protection that exists in Florence now is not on a level appropriate to the citizens and treasures that rest within the city. If, under current conditions, a 1966-like flood occurred, the consequences to human lives, treasures, properties and community infrastructure could be much more catastrophic than they were in 1966.
Fleeing a Hollywood that spurned him, Orson Welles arrived in Italy in 1947 to begin his career anew. Far from being welcomed as the celebrity who directed and starred in Citizen Kane, his six-year exile in Italy was riddled with controversy, financial struggles, disastrous love affairs, and failed projects. Alberto Anile's book depicts the artist's life and work in Italy, including his reception by the Italian press, his contentious interactions with key political figures, and his artistic output, which culminated in the filming of Othello. Drawing on revelatory new material on the artist's personal and professional life abroad, Orson Welles in Italy also chronicles Italian cinema's transition from the social concerns of neorealism to the alienated characters in films such as Federico Fellini's La Dolce Vita, amid the cultural politics of postwar Europe and the beginnings of the cold war.
In this book, two well-known scholars of critical educational studies provide a compelling introduction to the thoughts of Brazilian educator Paulo Freire and German critical theorist Jurgen Habermas. The book compares their theories in-depth and situates their thinking in relation to other social theories and philosophies of education. The authors demonstrate that, despite their differences, these philosophers share crucial views on science, society, critical social psychology, and educational praxis that are mutually illuminating and offer a new point of departure for a critical theory of education.
This book charts ideas European intellectuals (mostly from Great Britain, France, Germany and Italy) put forward to solve the problem of war during the first half of the twentieth century: a period that began with the Anglo-Boer war and that ended with the explosion of the atomic bombs on Hiroshima and Nagasaki. Such ideas do not belong to a homogeneous tradition of thought, but can be understood as a unique discourse that takes different characteristics according to the point of view of each author and of the specific historical situation.
Monsignor Luigi Giussani (1922–2005) was the founder of the Catholic lay movement Communion and Liberation in Italy, which has hundreds of thousands of adherents around the globe. In The Life of Luigi Giussani Alberto Savorana, who spent an important part of his life working and studying with Giussani, draws on many unpublished documents to recount who the priest was and how he lived. Giussani’s life story is particularly significant because it shares many of the same challenges, risks, and paths toward enlightenment that are described in his numerous and influential publications. Savorana demonstrates that the circumstances Giussani experienced and the people he encountered played a crucial role in defining his vocation. Illuminating details are shared about Giussani’s parents, professors, and friends in the seminary, the things he read, his priesthood, his experience teaching, misunderstandings and moments of recognition, and illness. Luigi Giussani considered Christianity to be a fact, a real event in human life, which takes the form of an encounter, inviting anyone and everyone to verify its relevance to life’s needs. This is what happened for so many people all over the world who recognized in this priest and leader, with his rough and captivating voice, not only a teacher to learn from, but above all a man to compare oneself with – a companion for the journey who could be trusted to answer the question: how can we live? In addition to providing the first chronological reconstruction of the life of the founder of Communion and Liberation, The Life of Luigi Giussani provides a detailed account of his legacy and what his life’s work meant to individual people and the Church.
Most books on the biotechnology industry focus on scientific and technological challenges, ignoring the entrepreneurial and managerial complexities faced bio-entrepreneurs. The Business Models for Life Science Firms aims to fill this gap by offering managers in this rapid growth industry the tools needed to design and implement an effective business model customized for the unique needs of research intensive organizations. Onetti and Zucchella begin by unpacking the often-used ‘business model’ term, examining key elements of business model conceptualization and offering a three tier approach with a clear separation between the business model and strategy: focus, exploring the different activities carried out by the organization; locus, evaluating where organizational activities are centered; and modus, testing the execution of the organization’s activities. The business model thus defines the unique way in which a company delivers on its promise to its customers. The theory and applications adopt a global approach, offering business cases from a variety of biotech companies around the world.
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