The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

WAVE BREAKING FOR NONLINEAR NONLOCAL SHALLOW WATER EQUATIONS 233 THEOREM 2.1. Let T>O and vE C 1 ([0, T); H 2(R)). Then for every t~ [0, T) there exists at least one point ~(t)ER with ,~(t) := in~ Ivy(t, x)] = ~ ( t , ~(t)), and the function m is almost everywhere differentiable on (0, T) with dm dt (t)=vt~(t,~(t)) a.e. on (O,T). Proof. Let c>0 stand for a generic constant. Fix te[0, T) and define m(t):=infxcR[v~(t,x)]. If m(t))O we have that v(t , . ) is nondecreasing on R and therefore v(t,. ) 0 (recall v(t,. )cL2(R)) , so that we may assume re(t)<0. Since vx(t,. ) c H I ( R ) we see that limlxl~ ~ Vx(t, x)=0 so that there exists at least a ~(t) e R with re(t) =v~(t, ~(t)). Let now s, tC [0, T) be fixed. If re(t) <~rn(s) we have 0 < re(s) .~(t) = i n f [~x (~, x ) ] ~ ( t , ~(t)) <. ~=(~, ~(t)) -~x(t, ~(t)), and by the Sobolev embedding HI(R) C L ~ ( R ) we conclude that Im(8)-.~(t)l ~< Ivx(t)-v~(s)lL~(~) < c Iv~(t)-v~(8)l.l(R). Hence the mean-value theorem for functions with values in Banach spaces-Hi(R) in the present case--yields (see [12]) jm(t)-m(s)l<~clt-s j m a x [IVt~(T)JHI(R)], t, se[O,T). O~T~max{s,t} Since vt~cC([O,T), Hi(R)) , we see that m is locally Lipschitz on [0, T) and therefore Rademacher's theorem (cf. [14]) implies that m is almost everywhere differentiable on (0,T). Fix tC(0, T). We have that v~(t+h)-vx(t)h vt~(t) Hl(R) ---~0 as h--*O, and therefore vx( t+h ,y ) -vx ( t , y ) sup vtx(t,y) --~0 as h---~O, (2.1) ycl~ h in view of the continuous embedding H 1 ( R ) c L ~ (R). 234 A. C O N S T A N T I N AND J. E S C H E R By the definition of m, m(t+h) = v~(t+h, ((t+h)) <. v~(t+h, ((t)). Consequently, given h>0 , we obtain m( t+h) -m( t ) <<. h Letting h--~O + and using (2.1), we find lim sup m(t+h) -m( t ) h~_~0 + h

/pdf/wave-breaking-for-nonlinear-nonlocal-shallow-water-equations-2y25ekrd18.pdf

Wave breaking for nonlinear nonlocal shallow water equations

https://digital.csic.es/bitstream/10261/15682/1/blanes.pdf

The Magnus expansion and some of its applications

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, e, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, e → 0, δ → 0, independently. Retaining terms O(e), O(δ2) and O(eδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

Camassa-Holm, Korteweg-de Vries and related models for water waves

http://www.cs.biu.ac.il/~schiff/Papers/pap19.pdf

The Camassa Holm equation: conserved quantities and the initial value problem

/pdf/the-camassa-holm-equation-a-loop-group-approach-1ihnoinxtc.pdf

The Camassa-Holm equation: a loop group approach

https://academicworks.cuny.edu/cgi/viewcontent.cgi?article=1443&context=cc_pubs

Kähler-Chern-Simons theory and symmetries of anti-self-dual gauge fields

https://academicworks.cuny.edu/cgi/viewcontent.cgi?article=1407&context=cc_pubs

A Kahler-{Chern-Simons} Theory and Quantization of Instanton Moduli Spaces

This paper introduces a new class of methods, which we call Mobius schemes, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmannian of m-dimensional subspaces of an (n+m)-dimensional vector space. Since the Grassmannians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilities is an artifact of the coordinate system, but since Mobius schemes are based on the natural geometry, they are able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.

/pdf/a-natural-approach-to-the-numerical-integration-of-riccati-1amsjxnn81.pdf

Jeremy Schiff

Papers

The Camassa Holm equation: conserved quantities and the initial value problem

The Camassa-Holm equation: a loop group approach

Kähler-Chern-Simons theory and symmetries of anti-self-dual gauge fields

A Kahler-{Chern-Simons} Theory and Quantization of Instanton Moduli Spaces

A Natural Approach to the Numerical Integration of Riccati Differential Equations