Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

/pdf/a-kernel-two-sample-test-2mr8hwo4ml.pdf

A kernel two-sample test

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Genetics and analysis of quantitative traits

In the last two lectures, I have given you, I hope, some feeling for the state of the art in the general theory of non-linear wave equations. In this lecture I want to describe some recent work [1] of a more special nature. The theory of propagation of singularities for linear wave equations has had a long development and is well understood. Let’s consider the same problem for the equation 

$$\matrix{{{{\rm{u}}_{{\rm{tt}}}} - {{\rm{u}}_{{\rm{xx}}}} = {\rm{f}}({\rm{x}},{\rm{u}},{{\rm{u}}_{\rm{x}}},{{\rm{u}}_{\rm{t}}})} \cr {{\rm{u}}({\rm{x}},{\rm{o}}) = {{\rm{f}}_1}({\rm{x}})} \cr {{{\rm{u}}_{\rm{t}}}({\rm{x}},{\rm{o}}) = {{\rm{f}}_2}({\rm{x}})} \cr }$$

(1)

in one dimension. By propagation of singularities one means the relation between the smoothness properties of u (x, t) in ℝ2 and the smoothness properties of f1 and f2 on ℝ1 The result is that, as in the linear case, singularities propagate along the boundary of the lightcone, but not into its interior. Of course, since f is non-linear one cannot deal with distributions as one can in the linear case, so we must say what we mean by “singularity”. By a solution of (1) we will mean a once continuously differentiable solution of the corresponding integral equation.

Propagation of Singularities

A completely elementary proof is given for Segal's nonlinear semigroup theorem The purpose of this paper is to give a completely elementary proof of an important theorem of Irving Segal [61 Segal's results give conditions on the nonlinear terms in wave equations so that unique local, global, and smooth solutions exist and these techniques have found wide and important applications in the study of nonlinear equations; for examples, see the work of Strauss [7] or Chadam [11, [2] The theorems which we prove are not exactly those of Segal and the differences are discussed at the end of the paper We restrict our exposition to the Hilbert space case since, due to the discontinuity of the free wave group on the LP energy spaces [41, this is the case which arises in applications Let A be a selfadjoint operator with domain D(A) on a Hilbert space H and let J be a densely defined mapping on H The problem is to give conditions on J that guarantee the existence, uniqueness, and smoothness of a solution to the Cauchy problem (1) +b'(t) = iAb(t) + J(q(t)), O(M) = 00 Theorem 1 (Local existence, uniqueness, smoothness) Let A be a selfadjoint operator on a Hilbert space J and N a positive integer Suppose that J is a densely defined mapping on I so that J: D(AW) D(Ai), for 0 <j<N, and: (Hj) |Aj) J())II < C((III,1 , IA'(kI),|Af)A A/, (Hi ) 11 AiJ(j o J(qj))ll < C(ll011, 11gx11, * ,11A if I, jj1 A ,jj)jj1 0 A iVxll Received by the editors May 6, 1974 AMS (MOS) subject classifications (1970) Primary 35L05, 47D05

https://www.ams.org/proc/1975-051-01/S0002-9939-1975-0377238-0/S0002-9939-1975-0377238-0.pdf

Higher order estimates and smoothness of nonlinear wave equations

On the self-adjointness of quantum fields and hamiltonians

A problem in Quantum Field Theory leads to the study of a representation of the torus, T3, as automorphisms of the infinite dimensional Clifford algebra. It is shown that the irreducible product representations of the Clifford algebra fall into two categories: the discrete representations where the automorphisms are unitarily implementable, and all the others in which the automorphisms are not implementable and which cannot even appear as subrepresentations of larger representations in which the automorphisms are implementable.

/pdf/torus-invariance-for-the-clifford-algebra-i-2y2s8wpngk.pdf

Torus invariance for the Clifford algebra. I

In order to study how the convergence of many variable-response neurons on a single target can sharpen timing information, we investigate the limit as the number of input neurons and the number of incoming spikes required to fire the target both get large with the ratio fixed. We prove that the standard deviation of the firing time of the target cell goes to zero in this limit, and we derive the asymptotic forms of the density and the standard deviation near the limit. We use the theorems to understand the behavior of octopus cells in the mammalian cochlear nucleus.

/pdf/neural-timing-in-highly-convergent-systems-47m6lje0yx.pdf

Michael C. Reed

Papers

Propagation of Singularities

Higher order estimates and smoothness of nonlinear wave equations

On the self-adjointness of quantum fields and hamiltonians

Torus invariance for the Clifford algebra. I

Neural Timing in Highly Convergent Systems