Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m. We also prove that for each m from 1 to n, there is a nonempty open set of compact, mean convex regions K in R^(n+1) with smooth boundary for which the resulting mean curvature flow has a shrinking S^m x R^(n-m) singularity.Comment: 19 pages. This version includes a new section proving that certain kinds of mean curvature flow singularities persist under arbitrary small perturbations of the initial surface. Newest update (Oct 2013) fixes some bibliographic reference

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

Investigating Off-shell Stability of Anti-de Sitter Space in String Theory

We propose an investigation of stability of vacua in string theory by studying their stability with respect to a (suitable) world-sheet renormalization group (RG) flow. We prove geometric stability of (Euclidean) anti-de Sitter (AdS) space (i.e., $\mathbf{H}^n$) with respect to the simplest RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point of Ricci flow. We therefore choose an appropriate flow for which it is a fixed point, prove a linear stability result for AdS space with respect to this flow, and then show this implies its geometric stability with respect to Ricci flow. The techniques used can be generalized to RG flows involving other fields. We also discuss tools from the mathematics of geometric flows that can be used to study stability of string vacua.Comment: 29 pages, references added in this version to appear in Classical and Quantum Gravit

Gradient flows and instantons at a Lifshitz point

I provide a broad framework to embed gradient flow equations in non-relativistic field theory models that exhibit anisotropic scaling. The prime example is the heat equation arising from a Lifshitz scalar field theory; other examples include the Allen-Cahn equation that models the evolution of phase boundaries. Then, I review recent results reported in arXiv:1002.0062 describing instantons of Horava-Lifshitz gravity as eternal solutions of certain geometric flow equations on 3-manifolds. These instanton solutions are in general chiral when the anisotropic scaling exponent is z=3. Some general connections with the Onsager-Machlup theory of non-equilibrium processes are also briefly discussed in this context. Thus, theories of Lifshitz type in d+1 dimensions can be used as off-shell toy models for dynamical vacuum selection of relativistic field theories in d dimensions.Comment: 19 pages, 1 figure, contribution to conference proceedings (NEB14); minor typos corrected in v

Term structure information and bond strategies

We examine term structure theories by using a novel approach. We form bond investment strategies based on different theories of the term structure in order to determine which strategy performs best. When using a manipulation-proof performance measure, we find that consistent with prior literature, an active strategy that is based on time varying term premiums can indeed form the basis of a successful bond strategy that outperforms an unbiased expectation inspired passive bond buy and hold strategy. This is true, however, for an earlier time period when the literature first made this claim. In a later time period, we find that the passive buy and hold strategy is significantly superior to all active strategies. This result is confirmed by statistical tests and it suggests that once it became known that an active strategy based on time varying term premiums could outperform a passive buy and hold strategy, the markets adjusted and arbitraged away this opportunity. Overall, it appears that the unbiased expectation hypothesis is the most likely explanation of the behaviour of the term structure during more recent times. This is because economically and statistically significant superior performance cannot be achieved if one uses information from the forward curve or the term structure as a guide to adjusting bond portfolios in response to changes in the term premium.This work was supported by Junta de Comunidades de Castilla-La Mancha [grant number PEII11-0031-6939]; Ministerio de Ciencia e Innovación [grant number ECO2011-28134] and partially supported by Fondo Europeo de Desarrollo Regional (FEDER) funds.

Breaking into the blackbox: Trend following, stop losses and the frequency of trading - The case of the S&P500

In this article, we compare a variety of technical trading rules in the context of investing in the S&P500 index. These rules are increasingly popular, both among retail investors and CTAs and similar investment funds. We find that a range of fairly simple rules, including the popular 200-day moving average (MA) trading rule, dominate the long-only, passive investment in the index. In particular, using the latter rule we find that popular stop-loss rules do not add value and that monthly end-of-month investment decision rules are superior to those which trade more frequently: this adds to the growing view that trading can damage your wealth. Finally, we compare the MA rule with a variety of simple fundamental metrics and find the latter far inferior to the technical rules over the last 60 years of investing

Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed

Critical behavior of collapsing surfaces

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the Type II singularity is accurately modeled by the rotationally symmetric translating soliton.Comment: 23 pages, 10 figure

Can Sustainable Withdrawal Rates be Enhanced by Trend Following?

We examine the consequences of alternative popular investment strategies for the decumulation of funds invested for retirement through a defined contribution pension scheme. We examine in detail the viability of specific ‘safe’ withdrawal rates including the ‘4%-rule’ of Bengen (1994). We find two powerful conclusions. First that smoothing the returns on individual assets by simple trend following techniques is a potent tool to enhance withdrawal rates. Second, we show that while diversification across asset classes does lead to higher withdrawal rates than simple equity/bond portfolios, “smoothing” returns in itself is far more powerful a tool for raising withdrawal rates. In fact, smoothing the popular equity/bond portfolios (such as the 60/40 portfolio) is in itself an excellent and simple solution to constructing a retirement portfolio. Alternatively, trend following enables portfolios to contain more risky assets, and the greater upside they offer, for the same level of overall risk compared to standard portfolios