Approximation of Feynman path integrals with non-smooth potentials

We study the convergence in $L^2$ of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here arise from a series expansion of the action. The results are ultimately based on function spaces, tools and strategies which are typical of Harmonic and Time-frequency analysis.Comment: 18 page

On the pointwise convergence of the integral kernels in the Feynman-Trotter formula

We study path integrals in the Trotter-type form for the Schr\"odinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by Albeverio and It\^o in celebrated papers - of Fourier transforms of complex measures. Essentially, $V$ is bounded and has the regularity of a function whose Fourier transform is in $L^1$. Whereas the strong convergence in $L^2$ in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes, that are the integral kernels of the approximation operators. We prove that, for the above class of potentials, such a convergence at the level of the integral kernels in fact occurs, uniformly on compact subsets and for every fixed time, except for certain exceptional time values for which the kernels are in general just distributions. Actually, theorems are stated for potentials in several function spaces arising in Harmonic Analysis, with corresponding convergence results. Proofs rely on Banach algebras techniques for pseudo-differential operators acting on such function spaces.Comment: 26 page

Extruding the vortex lattice: two reacting populations of dislocations

A controllable soft solid is realised in vortex matter in a type II superconductor. The two-dimensional unit cell area can be varied by a factor of $10^4$ in the solid phase, without a change of crystal symmetry offering easy exploration of extreme regimes compared to ordinary materials. The capacity to confine two-dimensional vortex matter to mesoscopic regions provides an arena for the largely unexplored metallurgy of plastic deformation at large density gradients. Our simulations reveal a novel plastic flow mechanism in this driven non-equilibrium system, utilising two distinct, but strongly interacting, populations of dislocations. One population facilitates the relaxation of density; a second aids the relaxation of shear stresses concentrated at the boundaries. The disparity of the bulk and shear moduli in vortex matter ensures the dislocation motion follows the overall continuum flow reflecting density variation

Social Security Reform: Legal Analysis of Social Security Benefit Entitlement Issues

[Excerpt] Calculations indicating that in the long run the Social Security program will not be financially sustainable under the present statutory scheme have fueled the current debate regarding Social Security reform. This report addresses selected legal issues which may be raised regarding entitlement to Social Security benefits as Congress considers possible changes to the Social Security program, and in view of projected long-range shortfalls in the Social Security Trust Funds

Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks

In this paper we consider the application of importance sampling in simulations of Markovian tandem networks in order to estimate the probability of rare events, such as network population overflow. We propose a heuristic methodology to obtain a good approximation to the 'optimal' state-dependent change of measure (importance sampling distribution). Extensive experimental results on 2-node tandem networks are very encouraging, yielding asymptotically efficient estimates (with bounded relative error) where no other state-independent importance sampling techniques are known to be efficient The methodology avoids the costly optimization involved in other recently proposed approaches to approximate the 'optimal' state-dependent change of measure. Moreover, the insight drawn from the heuristic promises its applicability to larger networks and more general topologies

Banking globalization, monetary transmission and the lending channel

The globalization of banking in the United States is influencing the monetary transmission mechanism both domestically and in foreign markets. Using quarterly information from all U.S. banks filing call reports between 1980 and 2005, we find evidence for the lending channel for monetary policy in large banks, but only those banks that are domestically-oriented and without international operations. We show that the large globally-oriented banks rely on internal capital markets with their foreign affiliates to help smooth domestic liquidity shocks. We also show that the existence of such internal capital markets contributes to an international propagation of domestic liquidity shocks to lending by affiliated banks abroad. While these results imply a substantially more active lending channel than documented in the seminal work of Kashyap and Stein (2000), the lending channel within the United States is declining in strength as banking becomes more globalized. --Lending channel,Bank,global,liquidity,transmission,internal capital markets

Information Gains from Cosmological Probes

In light of the growing number of cosmological observations, it is important to develop versatile tools to quantify the constraining power and consistency of cosmological probes. Originally motivated from information theory, we use the relative entropy to compute the information gained by Bayesian updates in units of bits. This measure quantifies both the improvement in precision and the 'surprise', i.e. the tension arising from shifts in central values. Our starting point is a WMAP9 prior which we update with observations of the distance ladder, supernovae (SNe), baryon acoustic oscillations (BAO), and weak lensing as well as the 2015 Planck release. We consider the parameters of the flat $\Lambda$CDM concordance model and some of its extensions which include curvature and Dark Energy equation of state parameter $w$. We find that, relative to WMAP9 and within these model spaces, the probes that have provided the greatest gains are Planck (10 bits), followed by BAO surveys (5.1 bits) and SNe experiments (3.1 bits). The other cosmological probes, including weak lensing (1.7 bits) and {$\rm H_0$} measures (1.7 bits), have contributed information but at a lower level. Furthermore, we do not find any significant surprise when updating the constraints of WMAP9 with any of the other experiments, meaning that they are consistent with WMAP9. However, when we choose Planck15 as the prior, we find that, accounting for the full multi-dimensionality of the parameter space, the weak lensing measurements of CFHTLenS produce a large surprise of 4.4 bits which is statistically significant at the 8 $\sigma$ level. We discuss how the relative entropy provides a versatile and robust framework to compare cosmological probes in the context of current and future surveys.Comment: 26 pages, 5 figure

Isotropic vs. Anisotropic components of BAO data: a tool for model selection

We conduct a selective analysis of the isotropic ($D_V$) and anisotropic ($AP$) components of the most recent Baryon Acoustic Oscillations (BAO) data. We find that these components provide significantly different constraints and could provide strong diagnostics for model selection, also in view of more precise data to arrive. For instance, in the $\Lambda$CDM model, we find a mild tension of $\sim 2 \sigma$ for the $\Omega_m$ estimates obtained using $D_V$ and $AP$ separately. Considering both $\Omega_k$ and $w$ as free parameters, we find that the concordance model is in tension with the best-fit values provided by the BAO data alone at 2.2$\sigma$. We complemented the BAO data with the Supernova Ia (SNIa) and Observational \textit{Hubble} datasets to perform a joint analysis on the $\Lambda$CDM model and its standard extensions. By assuming $\Lambda$CDM scenario, we find that these data provide $H_0 = 69.4 \pm 1.7$ \text{km/s Mpc$^{-1}$} as the best-fit value for the present expansion rate. In the $k\Lambda$CDM scenario we find that the evidence for acceleration using the BAO data alone is more than $\sim 5.8\sigma$, which increases to $8.4 \sigma$ in our joint analysis.Comment: Accepted for publication in JCAP. References update