One-Electron Spectral Functions of the Attractive Hubbard Model for Intermediate Coupling

We calculate the one-electron spectral function of the attractive (negative-$U$) Hubbard model. We work in the intermediate coupling and low density regime and obtain the self-energy in an approximate analytical form. The excitation spectrum is found to consist of three branches. The results are obtained in a framework, based on the self-consistent T-matrix approximation, which is compatible with the Mermin-Wagner theorem.Comment: 7 pages, revtex, 7 figures in postscript format, submitted to PR

Inverse problems in thermoacoustics

Thermoacoustics is a branch of fluid mechanics, and is as such governed by the conservation laws of mass, momentum, energy and species. While computational fluid dynamics (CFD) has entered the design process of many applications in fluid mechanics, its success in thermoacoustics is limited by the multi-scale, multi-physics nature of the subject. In his influential monograph from 2006, Prof. Fred Culick writes about the role of CFD in thermoacoustic modeling: The main reason that CFD has otherwise been relatively helpless in this subject is that problems of combustion instabilities involve physical and chemical matters that are still not well understood. Moreover, they exist in practical circumstances which are not readily approximated by models suitable to formulation within CFD. Hence, the methods discussed and developed in this book will likely be useful for a long time to come, in both research and practice. [. . . ] It seems to me that eventually the most effective ways of formulating predictions and theoretical interpretations of combustion instabilities in practice will rest on combining methods of the sort discussed in this book with computational fluid dynamics, the whole confirmed by experimental results. Despite advances in CFD and large-eddy simulation (LES) in particular, unsteady simulations for more than a few selected operating points are computationally infeasible. The ‘methods discussed in this book’ refer to reduced-order models of thermoacoustic oscillations. Whether intentional or not, the last sentence anticipates the advent of data-driven methods, and encapsulates the philosophy behind this work. This work brings together two workhorses of the design process: physics-informed reduced-order models and data from higher-fidelity sources such as simulations and experiments. The three building blocks to all our statistical inference frameworks are: (i) a hierarchical view of reduced-order models consisting of states, parameters and governing equations; (ii) probabilistic formulations with random variables and stochastic processes; and (iii) efficient algorithms from statistical learning theory and machine learning. While leveraging advances in statistical and machine learning, we demonstrate the feasibility of Bayes’ rule as a first principle in physics-informed statistical inference. In particular, we discuss two types of inverse problems in thermoacoustics: (i) implicit reduced-order models representative of nonlinear eigenproblems from linear stability analysis; and (ii) time-dependent reduced-order models used to investigate nonlinear dynamics. The outcomes of statistical inference are improved predictions of the state, estimates of the parameters with uncertainty quantification and an assessment of the reduced-order model itself. This work highlights the role that data can play in the future of combustion modeling for thermoacoustics. It is increasingly impractical to store data, particularly as experiments become automated and numerical simulations become more detailed. Rather than store the data itself, the techniques in this work optimally assimilate the data into the parameters of a physics-informed reduced-order model. With data-driven reduced-order models, rapid prototyping of combustion systems can feed into rapid calibration of their reduced-order models and then into gradient-based design optimization. While it has been shown, e.g. in the context of ignition and extinction, that large-eddy simulations become quantitatively predictive when augmented with data, the reduced-order modeling of flame dynamics in turbulent flows remains challenging. For these challenging situations, this work opens up new possibilities for the development of reduced-order models that adaptively change any time that data from experiments or simulations becomes available.Schlumberger Cambridge International Scholarshi

Simulated Maximum Likelihood Estimation for Latent Diffusion Models

In this paper a method is developed and implemented to provide the simulated maximum likelihood estimation of latent diffusions based on discrete data. The method is applicable to diffusions that either have latent elements in the state vector or are only observed at discrete time with a noise. Latent diffusions are very important in practical applications in nancial economics. The proposed approach synthesizes the closed form method of Aït-Sahalia (2008) and the ecient importance sampler of Richard and Zhang (2007). It does not require any inll observations to be introduced and hence is computationally tractable. The Monte Carlo study shows that the method works well in finite sample. The empirical applications illustrate usefulness of the method and find no evidence of infinite variance in the importance sampler.Closed-form approximation; Diusion Model; Ecient importance sampler

Estimating the GARCH Diffusion: Simulated Maximum Likelihood in Continuous Time

A new algorithm is developed to provide a simulated maximum likelihood estimation of the GARCH diffusion model of Nelson (1990) based on return data only. The method combines two accurate approximation procedures, namely, the polynomial expansion of Aït-Sahalia (2008) to approximate the transition probability density of return and volatility, and the Efficient Importance Sampler (EIS) of Richard and Zhang (2007) to integrate out the volatility. The first and second order terms in the polynomial expansion are used to generate a base-line importance density for an EIS algorithm. The higher order terms are included when evaluating the importance weights. Monte Carlo experiments show that the new method works well and the discretization error is well controlled by the polynomial expansion. In the empirical application, we fit the GARCH diffusion to equity data, perform diagnostics on the model fit, and test the finiteness of the importance weights.Ecient importance sampling; GARCH diusion model; Simulated Maximum likelihood; Stochastic volatility

Stimulated Maximum Likelihood Estimation of Continuous Time Stochastic Volatility Models

In this paper we develop and implement a method for maximum simulated likelihood estimation of the continuous time stochastic volatility model with the constant elasticity of volatility. The approach do not require observations on option prices nor volatility. To integrate out latent volatility from the joint density of return and volatility, a modified efficient importance sampling technique is used after the continuous time model is approximated using the Euler-Maruyama scheme. The Monte Carlo studies show that the method works well and the empirical applications illustrate usefulness of the method. Empirical results provide strong evidence against the Heston model.Efficient importance sampler; Constant elasticity of volatility

One-pot nucleophilic substitution–double click reactions of biazides leading to functionalized bis(1,2,3-triazole) derivatives

The nucleophilic substitution of benzylic bromides with sodium azide was combined with a subsequent copper-catalyzed (3 + 2) cycloaddition with terminal alkynes. This one-pot process was developed with a simple model alkyne, but then applied to more complex alkynes bearing enantiopure 1,2-oxazinyl substituents. Hence, the precursor compounds 1,2-, 1,3- or 1,4-bis(bromomethyl)benzene furnished geometrically differing bis(1,2,3-triazole) derivatives. The use of tris[(1-benzyl-1H-1,2,3-triazol-4-yl)methyl]amine (TBTA) as ligand for the click step turned out to be very advantageous. The compounds with 1,2-oxazinyl end groups can potentially serve as precursors of divalent carbohydrate mimetics, but the reductive cleavage of the 1,2-oxazine rings to aminopyran moieties did not proceed cleanly with these compounds

Optional Decomposition and Lagrange Multipliers

Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand ф such that the difference X−ф•S is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption