The Trial Brief

From the chapter Introduction: The object of the preceding chapters is to show the brief maker where to find the material for his brief, how to find it, and how to select out of the mass of material found that which will be suitable for his use.... The present purpose is to outline a course of investigation suitable to the preparation of a case for trial and to suggest methods of making the material collected during the search for authorities readily available. [p.417-418

An overview of Viscosity Solutions of Path-Dependent PDEs

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12]

A Corticothalamic Switch: Controlling the Thalamus with Dynamic Synapses

SummaryCorticothalamic neurons provide massive input to the thalamus. This top-down projection may allow the cortex to regulate sensory processing by modulating the excitability of thalamic cells. Layer 6 corticothalamic neurons monosynaptically excite thalamocortical cells, but also indirectly inhibit them by driving inhibitory cells of the thalamic reticular nucleus. Whether corticothalamic activity generally suppresses or excites the thalamus remains unclear. Here we show that the corticothalamic influence is dynamic, with the excitatory-inhibitory balance shifting in an activity-dependent fashion. During low-frequency activity, corticothalamic effects are mainly suppressive, whereas higher-frequency activity (even a short bout of gamma frequency oscillations) converts the corticothalamic influence to enhancement. The mechanism of this switching depends on distinct forms of short-term synaptic plasticity across multiple corticothalamic circuit components. Our results reveal an activity-dependent mechanism by which corticothalamic neurons can bidirectionally switch the excitability and sensory throughput of the thalamus, possibly to meet changing behavioral demands

Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

A differential method for bounding the ground state energy

For a wide class of Hamiltonians, a novel method to obtain lower and upper bounds for the lowest energy is presented. Unlike perturbative or variational techniques, this method does not involve the computation of any integral (a normalisation factor or a matrix element). It just requires the determination of the absolute minimum and maximum in the whole configuration space of the local energy associated with a normalisable trial function (the calculation of the norm is not needed). After a general introduction, the method is applied to three non-integrable systems: the asymmetric annular billiard, the many-body spinless Coulombian problem, the hydrogen atom in a constant and uniform magnetic field. Being more sensitive than the variational methods to any local perturbation of the trial function, this method can used to systematically improve the energy bounds with a local skilled analysis; an algorithm relying on this method can therefore be constructed and an explicit example for a one-dimensional problem is given.Comment: Accepted for publication in Journal of Physics

Dynamical response of the "GGG" rotor to test the Equivalence Principle: theory, simulation and experiment. Part I: the normal modes

Recent theoretical work suggests that violation of the Equivalence Principle might be revealed in a measurement of the fractional differential acceleration $\eta$ between two test bodies -of different composition, falling in the gravitational field of a source mass- if the measurement is made to the level of $\eta\simeq 10^{-13}$ or better. This being within the reach of ground based experiments, gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in low orbit around the Earth is likely to provide a much better accuracy. We report on the progress made with the "Galileo Galilei on the Ground" (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test. In the present and following paper (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation -in particular its normal modes (Part I) and rejection of common mode effects (Part II)- can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining quantitative agreement with the available experimental data on the frequencies of the normal modes, and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus.Comment: Accepted for publication by "Review of Scientific Instruments" on Jan 16, 2006. 16 2-column pages, 9 figure

Further improvements on TRACE 3-D

TRACE 3-D, an interactive beam-dynamics program that calculates the envelopes of a bunched beam (including linear space-charge forces) through a user-defined transport system, has undergone several upgrades in physics, coding, and capabilities. Recent modifications include centroid tracking (and misalignment capabilities) and an improved beam description that allows study of some nonlinear effects such as wakefields. The Fortran code has been made portable and runs on numerous platforms. It can be used with a variety of graphics packages. The additional beamline elements, new commands, expanded fitting capabilities, improved beam description, and coding modifications have extended TRACE 3-D`s usefulness and applicability to the accelerator community. These changes are documented in the third edition of TRACE 3-D Documentation

Pricing Options in Incomplete Equity Markets via the Instantaneous Sharpe Ratio

We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).Comment: Keywords: Pricing derivative securities, incomplete markets, Sharpe ratio, correlated assets, stochastic volatility, non-linear partial differential equations, good deal bound

Adaptive Markets and the New World Order

In the adaptive markets hypothesis (AMH) intelligent but fallible investors learn from and adapt to changing economic environments. This implies that markets are not always efficient but are usually competitive and adaptive, varying in their degree of efficiency as the environment and investor population change over time. The AMH has several implications, including the possibility of negative risk premiums, alpha converging to beta, and the importance of macro factors and risk budgeting in asset allocation policies