Abstract:
Dirichlet Process Mixtures (DPMs) are a popular class of statistical models to perform density estimation and clustering. However, when the data available have a distribution evolving over time, such models are inadequate. We introduce here a class of time-varying DPMs which ensures that at each time step the random distribution follows a DPM model. Our model relies on an intuitive and simple generalized Polya urn scheme. Inference is performed using Markov chain Monte Carlo and Sequential Monte Carlo. We demonstrate our model on various applications.

Abstract:
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with some numerical examples.

Abstract:
Let $f,g:{\Bbb R}\to{\Bbb R}$ be integrable functions, $f$ nowhere zero, and $\phi (u)=\int du/f(u)$ be invertible. An exact solution to the generalized nonhomogeneous inviscid Burgers' equation $u_t+g(u).u_x=f(u)$ is given, by quadratures.

Abstract:
Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilities P(x_1, ... ,x_N;t|y_1, ... ,y_N;0) of finding N particles on lattice sites x_1, ... ,x_N at time t with initial occupation y_1, ... ,y_N at time t=0.

Abstract:
We present an exact solution for an asymmetric exclusion process on a ring with three classes of particles and vacancies. Using a matrix product Ansatz, we find explicit expressions for the weights of the configurations in the stationary state. The solution involves tensor products of quadratic algebras.

Abstract:
In a recent paper (Beyer and Hennig, 2012 [9]), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub-NUT solutions. Here we derive a three-parametric family of exact solutions within this class, which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a "true spike" in analogy to previously known Gowdy symmetric solutions with spatial T3-topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain "false spikes".

Abstract:
In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulas for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.

Abstract:
We consider a generalized two-color Polya urn (black and withe balls) first introduced by Hill, Lane, Sudderth where the urn composition evolves as follows: let $\pi:\left[0,1\right]\rightarrow\left[0,1\right]$, and denote by $x_{n}$ the fraction of black balls after step $n$, then at step $n+1$ a black ball is added with probability $\pi\left(x_{n}\right)$ and a white ball is added with probability $1-\pi\left(x_{n}\right)$. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology. In this work we discuss large deviations for a wide class of continuous urn functions $\pi$. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit $\phi\left(s\right)=\lim_{n\rightarrow\infty}{\textstyle \frac{1}{n}}\log\mathbb{P}\left(\left\{ nx_{n}=\left\lfloor sn\right\rfloor \right\} \right),\, s\in\left[0,1\right]$, where $nx_{n}$ is the number of black balls at time $n$, and use it to give some insight on the shape of $\phi\left(s\right)$. Under suitable assumptions on $\pi$ we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi-Pal Model, giving the exact implicit expression for $\phi$ in terms of the Cumulant Generating Function.

Abstract:
We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.

Abstract:
We study quenches in integrable spin-1/2 chains in which we evolve the ground state of the antiferromagnetic Ising model with the anisotropic Heisenberg Hamiltonian. For this nontrivially interacting situation, an application of the first-principles-based quench action method allows us to give an exact description of the postquench steady state in the thermodynamic limit. We show that a generalized Gibbs ensemble, implemented using all known local conserved charges, fails to reproduce the exact quench action steady state and to correctly predict postquench equilibrium expectation values of physical observables. This is supported by numerical linked-cluster calculations within the diagonal ensemble in the thermodynamic limit.