Recursive state estimation for hybrid systems

The paper deals with recursive state estimation for hybrid systems. An unobservable state of such systems is changed both in a continuous and a discrete way. Fast and efficient online estimation of hybrid system state is desired in many application areas. The presented paper proposes to look at this problem via Bayesian filtering in the factorized (decomposed) form. General recursive solution is proposed as the probability density function, updated entry-wise. The paper summarizes general factorized filter specialized for (i) normal state-space models; (ii) multinomial state-space models with discrete observations; and (iii) hybrid systems. Illustrative experiments and comparison with one of the counterparts are provided.     

Asymptotic approximation of functions and their derivatives by Müller’s Gamma operators

We obtain the complete asymptotic expansion of the image functions of Müller’s Gamma operators and of their derivatives. All expansion coefficients are explicitly calculated. Moreover, we study linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations of which special cases were recently introduced and investigated by other authors.     

Probing structural integrity of single walled carbon nanotubes by dynamic and static compression

We report on a first study of single walled carbon nanotubes (SWCNTs) after application of dynamic (shock) compression. The experiments were conducted at 19 GPa and 36 GPa in a recovery assembly. For comparison, an experiment at a static pressure of 36 GPa was performed on the material from the same batch in a diamond anvil cell (DAC). After the high pressure treatment the samples were characterized by Raman spectroscopy and transmission electron microscopy (TEM). After exposure to 19 GPa of shock compression the CNT material exhibited substantial structural damage such as CNT wall disruption, opening of the tube along its axis (“unzipping”) and tube shortening (“cutting”). Dynamic compression to 36 GPa resulted in essentially complete CNT destruction whereas at least a fraction of the nanotubes was recovered after 36 GPa of static compression though severely damaged. The results of these shock wave experiments underline the prospect of using SWCNTs as reinforcing units in material WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)     

A Nonholonomic Model of the Paul Trap

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.     

Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

We consider two models for directed polymers in space‐time independent random media (the O'Connell‐Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ^1/3, converges to the GUE Tracy‐Widom distribution. We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar‐Parisi‐Zhang equation. The Laplace transform formula we prove can be inverted to give the one‐point probability distribution of the solution to these stochastic PDEs for the class of initial data. © 2014 Wiley Periodicals, Inc.     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.