Juan Luis Vázquez
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ...
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This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.Less
This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of physics, chemistry, biology, and engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis. Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this book is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including time decay, smoothing, extinction in finite time, and delayed regularity.
Robert W. Batterman
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195146479
- eISBN:
- 9780199833078
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195146476.001.0001
- Subject:
- Philosophy, Philosophy of Science
This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. ...
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This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. Asymptotic reasoning has received systematic treatment in physics and applied mathematics, but virtually no attention has been paid to it by philosophers of science. I argue that once one understands the role played by asymptotic reasoning in explanatory arguments of scientists, our philosophical conceptions of explanation, reduction, and emergence require significant modification.Less
This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. Asymptotic reasoning has received systematic treatment in physics and applied mathematics, but virtually no attention has been paid to it by philosophers of science. I argue that once one understands the role played by asymptotic reasoning in explanatory arguments of scientists, our philosophical conceptions of explanation, reduction, and emergence require significant modification.
Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven
- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.001.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization
Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a ...
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Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.Less
Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.
Robert W. Batterman
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195146479
- eISBN:
- 9780199833078
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195146476.003.0004
- Subject:
- Philosophy, Philosophy of Science
This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of ...
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This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of asymptotic explanation in general. Two other, superficially quite different, explanatory accounts involving “intermediate asymptotics” are then discussed. It is argued that these different examples exhibit the same general asymptotic explanatory strategy – one that is ubiquitous in physics and applied mathematics. The chapter concludes with a discussion of the importance of stability considerations in the asymptotic explanations.Less
This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of asymptotic explanation in general. Two other, superficially quite different, explanatory accounts involving “intermediate asymptotics” are then discussed. It is argued that these different examples exhibit the same general asymptotic explanatory strategy – one that is ubiquitous in physics and applied mathematics. The chapter concludes with a discussion of the importance of stability considerations in the asymptotic explanations.
Hans Ringström
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0026
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology
In Chapter 26, we formulate criteria ensuring future global existence of solutions (in the spatially homogeneous setting). Moreover, we provide some rough conclusions concerning the asymptotics. ...
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In Chapter 26, we formulate criteria ensuring future global existence of solutions (in the spatially homogeneous setting). Moreover, we provide some rough conclusions concerning the asymptotics. Finally, we discuss the case of isotropic initial data on the 3-sphere.Less
In Chapter 26, we formulate criteria ensuring future global existence of solutions (in the spatially homogeneous setting). Moreover, we provide some rough conclusions concerning the asymptotics. Finally, we discuss the case of isotropic initial data on the 3-sphere.
Hans Ringström
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0027
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology
In Chapter 27, we focus on the case that the scalar field converges to a positive non-degenerate minimum of the potential. In the spatially homogeneous setting, it is then possible to derive detailed ...
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In Chapter 27, we focus on the case that the scalar field converges to a positive non-degenerate minimum of the potential. In the spatially homogeneous setting, it is then possible to derive detailed asymptotics. We first discuss the behaviour of the scalar field and the metric. We then turn to the asymptotics for the distribution function.Less
In Chapter 27, we focus on the case that the scalar field converges to a positive non-degenerate minimum of the potential. In the spatially homogeneous setting, it is then possible to derive detailed asymptotics. We first discuss the behaviour of the scalar field and the metric. We then turn to the asymptotics for the distribution function.
Hans Ringström
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0031
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology
In Chapter 31, we prove future global existence in the case that the initial data are specified on the 3-torus, and assuming the initial data are close to those of de Sitter space. The proof is a ...
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In Chapter 31, we prove future global existence in the case that the initial data are specified on the 3-torus, and assuming the initial data are close to those of de Sitter space. The proof is a bootstrap argument, one important ingredient being a system of differential inequalities. At the end of the chapter, we derive rough conclusions concerning the asymptotics.Less
In Chapter 31, we prove future global existence in the case that the initial data are specified on the 3-torus, and assuming the initial data are close to those of de Sitter space. The proof is a bootstrap argument, one important ingredient being a system of differential inequalities. At the end of the chapter, we derive rough conclusions concerning the asymptotics.
Hans Ringström
- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199680290
- eISBN:
- 9780191760235
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199680290.003.0032
- Subject:
- Mathematics, Mathematical Physics, Geometry / Topology
In Chapter 32, we derive detailed asymptotics of the solutions constructed in Chapter 31. We separate between the two cases that the scalar field is zero and non-zero. The case of zero scalar field ...
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In Chapter 32, we derive detailed asymptotics of the solutions constructed in Chapter 31. We separate between the two cases that the scalar field is zero and non-zero. The case of zero scalar field corresponds to the presence of a positive cosmological constant.Less
In Chapter 32, we derive detailed asymptotics of the solutions constructed in Chapter 31. We separate between the two cases that the scalar field is zero and non-zero. The case of zero scalar field corresponds to the presence of a positive cosmological constant.
Charles L. Epstein and Rafe1 Mazzeo
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0012
- Subject:
- Mathematics, Probability / Statistics
This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components ...
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This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β⁰(P).Less
This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β⁰(P).
Christopher D. Sogge
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.003.0003
- Subject:
- Mathematics, Numerical Analysis
This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) ...
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This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ) = cλⁿ + O(λsuperscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.Less
This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ) = cλⁿ + O(λsuperscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.
Christopher D. Sogge
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.003.0005
- Subject:
- Mathematics, Numerical Analysis
This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral ...
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This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral bands, details and proves a lemma, and gives a related generalization of the Weyl formula from Chapter 3 that involves pseudodifferential operators. The chapter then proves its main result by using a version of the Duistermaat-Guillemin theorem, which allows the use of the Hadamard parametrix and the arguments from Chapter 3. To conclude, the chapter shows that one can improve the sup-norm estimates from Chapter 3 if one assumes a condition on the geodesic flow that is similar to a hypothesis laid out in the Duistermaat-Guillemin theorem.Less
This chapter proves an improved Weyl formula under the assumption that the set of periodic geodesics for (M,g) has measure zero. It then shows trace estimates associated with shrinking spectral bands, details and proves a lemma, and gives a related generalization of the Weyl formula from Chapter 3 that involves pseudodifferential operators. The chapter then proves its main result by using a version of the Duistermaat-Guillemin theorem, which allows the use of the Hadamard parametrix and the arguments from Chapter 3. To conclude, the chapter shows that one can improve the sup-norm estimates from Chapter 3 if one assumes a condition on the geodesic flow that is similar to a hypothesis laid out in the Duistermaat-Guillemin theorem.
