Homomorphisms on infinite direct product algebras, especially Lie algebras

We study surjective homomorphisms f:\prod_I A_i\to B of not-necessarily-associative algebras over a commutative ring k, for I a generally infinite set; especially when k is a field and B is countable-dimensional over k. Our results have the following consequences when k is an infinite field, the algebras are Lie algebras, and B is finite-dimensional: If all the Lie algebras A_i are solvable, then so is B. If all the Lie algebras A_i are nilpotent, then so is B. If k is not of characteristic 2 or 3, and all the Lie algebras A_i are finite-dimensional and are direct products of simple algebras, then (i) so is B, (ii) f splits, and (iii) under a weak cardinality bound on I, f is continuous in the pro-discrete topology. A key fact used in getting (i)-(iii) is that over any such field, every finite-dimensional simple Lie algebra L can be written L=[x_1,L]+[x_2,L] for some x_1, x_2\in L, which we prove from a recent result of J.M.Bois. The general technique of the paper involves studying conditions under which a homomorphism on \prod_I A_i must factor through the direct product of finitely many ultraproducts of the A_i. Several examples are given, and open questions noted.Comment: 33 pages. The lemma in section 12.1 of the previous version was incorrect, and has been removed. (Nothing else depended on it.) Other changes are improvements in wording, et

The spectrum of tachyons in AdS/CFT

We analyze the spectrum of open strings stretched between a D-brane and an anti-D-brane in planar AdS/CFT using various tools. We focus on open strings ending on two giant gravitons with different orientation in $AdS_5 \times S^5$ and study the spectrum of string excitations using the following approaches: open spin-chain, boundary asymptotic Bethe ansatz and boundary thermodynamic Bethe ansatz (BTBA). We find agreement between a perturbative high order diagrammatic calculation in ${\cal N}=4$ SYM and the leading finite-size boundary Luscher correction. We study the ground state energy of the system at finite coupling by deriving and numerically solving a set of BTBA equations. While the numerics give reasonable results at small coupling, they break down at finite coupling when the total energy of the string gets close to zero, possibly indicating that the state turns tachyonic. The location of the breakdown is also predicted analytically.Comment: 40 pages, lots of figures, v2: typos corrected, accepted for publication in JHE

On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach

A {log-optimal} portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor's wealth. This maximization problem typically assumes that the information of the true distribution of returns is known to the trader in advance. However, in practice, the return distributions are indeed {ambiguous}; i.e., the true distribution is unknown to the trader or it is partially known at best. To this end, a {distributional robust log-optimal portfolio problem} formulation arises naturally. While the problem formulation takes into account the ambiguity on return distributions, the problem needs not to be tractable in general. To address this, in this paper, we propose a {supporting hyperplane approximation} approach that allows us to reformulate a class of distributional robust log-optimal portfolio problems into a linear program, which can be solved very efficiently. Our framework is flexible enough to allow {transaction costs}, {leverage and shorting}, {survival trades}, and {diversification considerations}. In addition, given an acceptable approximation error, an efficient algorithm for rapidly calculating the optimal number of hyperplanes is provided. Some empirical studies using historical stock price data are also provided to support our theory.Comment: submitted for possible publicatio

Tropical Monte Carlo quadrature for Feynman integrals

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.Comment: 6 figures, see http://github.com/michibo/tropical-feynman-quadrature for the referenced program code; v2: typos corrected, version accepted for publication in Annales de l'Institut Henri Poincar\'e

Modeling the Anisotropic Resolution and Noise Properties of Digital Breast Tomosynthesis

Digital breast tomosynthesis (DBT) is a 3D imaging modality in which a reconstruction of the breast is generated from various x-ray projections. Due to the newness of this technology, the development of an analytical model of image quality has been on-going. In this thesis, a more complete model is developed by addressing the limitations found in the previous linear systems (LS) model [Zhao, Med. Phys. 2008, 35(12): 5219-32]. A central assumption of the LS model is that the angle of x-ray incidence is approximately normal to the detector in each projection. To model the effect of oblique x-ray incidence, this thesis generalizes Swank\u27s calculations of the transfer functions of x-ray fluorescent screens to arbitrary incident angles. In the LS model, it is also assumed that the pixelation in the reconstruction grid is the same as the detector; hence, the highest frequency that can be resolved is the detector alias frequency. This thesis considers reconstruction grids with smaller pixelation to investigate super-resolution, or visibility of higher frequencies. A sine plate is introduced as a conceptual test object to analyze super-resolution. By orienting the long axis of the sine plate at various angles, the feasibility of oblique reconstruction planes is also investigated. This formulation differs from the LS model in which reconstruction planes are parallel to the breast support. It is shown that the transfer functions for arbitrary angles of x-ray incidence can be modeled in closed form. The high frequency modulation transfer function (MTF) and detective quantum efficiency (DQE) are degraded due to oblique x-ray incidence. In addition, using the sine plate, it is demonstrated that a reconstruction can resolve frequencies exceeding the detector alias frequency. Experimental images of bar patterns verified the existence of super-resolution. Anecdotal clinical examples showed that super-resolution improves the visibility of microcalcifications. The feasibility of oblique reconstructions was established theoretically with the sine plate and was validated experimentally with bar patterns. This thesis develops a more complete model of image quality in DBT by addressing the limitations of the LS model. In future studies, this model can be used as a tool for optimizing DBT

Muon g-2: Review of Theory and Experiment

A review of the experimental and theoretical determinations of the anomalous magnetic moment of the muon is given. The anomaly is defined by a=(g-2)/2, where the Land\'e g-factor is the proportionality constant that relates the spin to the magnetic moment. For the muon, as well as for the electron and tauon, the anomaly a differs slightly from zero (of order 10^{-3}) because of radiative corrections. In the Standard Model, contributions to the anomaly come from virtual `loops' containing photons and the known massive particles. The relative contribution from heavy particles scales as the square of the lepton mass over the heavy mass, leading to small differences in the anomaly for e, \mu, and \tau. If there are heavy new particles outside the Standard Model which couple to photons and/or leptons, the relative effect on the muon anomaly will be \sim (m_\mu/ m_e)^2 \approx 43\times 10^3 larger compared with the electron anomaly. Because both the theoretical and experimental values of the muon anomaly are determined to high precision, it is an excellent place to search for the effects of new physics, or to constrain speculative extensions to the Standard Model. Details of the current theoretical evaluation, and of the series of experiments that culminates with E821 at the Brookhaven National Laboratory are given. At present the theoretical and the experimental values are known with a similar relative precision of 0.5 ppm. There is, however, a 3.4 standard deviation difference between the two, strongly suggesting the need for continued experimental and theoretical studyComment: 103 pages, 57 figures, submitted to Reports on Progress in Physics Final version as published, several minor clarifications to text and a number of references were correcte

Clustering-Based Robot Navigation and Control

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths, capturing the connectivity of the underlying space. This dissertation introduces the use of clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in generally complex-shaped and high-dimensional configuration spaces of robotic systems. We demonstrate some potential applications of such clustering tools to the problem of feedback motion planning and control. The first part of the dissertation presents the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. We reinterpret this classical method for unsupervised learning as an abstract formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions, by relating the continuous space of configurations to the combinatorial space of trees. Based on this new abstraction and a careful topological characterization of the associated hierarchical structure, a provably correct, computationally efficient hierarchical navigation framework is proposed for collision-free coordinated motion design towards a designated multirobot configuration via a sequence of hierarchy-preserving local controllers. The second part of the dissertation introduces a new, robot-centric application of Voronoi diagrams to identify a collision-free neighborhood of a robot configuration that captures the local geometric structure of a configuration space around the robot’s instantaneous position. Based on robot-centric Voronoi diagrams, a provably correct, collision-free coverage and congestion control algorithm is proposed for distributed mobile sensing applications of heterogeneous disk-shaped robots; and a sensor-based reactive navigation algorithm is proposed for exact navigation of a disk-shaped robot in forest-like cluttered environments. These results strongly suggest that clustering is, indeed, an effective approach for automatically extracting intrinsic structures in configuration spaces and that it might play a key role in the design of computationally efficient, provably correct motion planners in complex, high-dimensional configuration spaces