The Exact Hausdorff Measure of the Zero Set ofFractional Brownian Motion

Let {X(t), t∈ℝN} be a fractional Brownian motion in ℝd of index H. If L(0,I) is the local time of X at 0 on the interval I⊂ℝN, then there exists a positive finite constant c(=c(N,d,H)) such that $$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$$ where $\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}$ , and m φ(E) is the Hausdorff φ-measure ofE. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126-197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ℝ

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

Let {X(t), t is an element of R-N} be a fractional Brownian motion in R-d of index H. If L(0,I) is the local time of X at 0 on the interval I subset of R-N, then there exists a positive finite constant c(=c(N,d,H)) such tha

Dispersive estimates for the Schr\"{o}dinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$H=-\Delta+\sum_{j=1}^N\langle\cdot\,, \varphi_j\rangle \varphi_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1,$$ where each $\varphi_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, \varphi_1,\ldots, \varphi_N)>0$ such that $$\|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0.$$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations ($N=1$). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant $C(N, d, \varphi_1,\ldots, \varphi_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.Comment: 78 page

Effects of extrinsic point defects in phosphorene: B, C, N, O and F Adatoms

Phosphorene is emerging as a promising 2D semiconducting material with a direct band gap and high carrier mobility. In this paper, we examine the role of the extrinsic point defects including surface adatoms in modifying the electronic properties of phosphorene using density functional theory. The surface adatoms considered are B, C, N, O and F with a [He] core electronic configuration. Our calculations show that B and C, with electronegativity close to P, prefer to break the sp3 bonds of phosphorene, and reside at the interstitial sites in the 2D lattice by forming sp2 bonds with the native atoms. On the other hand, N, O and F, which are more electronegative than P, prefer the surface sites by attracting the lone pairs of phosphorene. B, N and F adsorption will also introduce local magnetic moment to the lattice. Moreover, B, C, N and F adatoms will modify the band gap of phosphorene yielding metallic transverse tunneling characters. Oxygen does not modify the band gap of phosphorene, and a diode like tunneling behavior is observed. Our results therefore offer a possible route to tailor the electronic and magnetic properties of phosphorene by the adatom functionalization, and provide the physical insights of the environmental sensitivity of phosphorene, which will be helpful to experimentalists in evaluating the performance and aging effects of phosphorene-based electronic devices

Which way up? Recognition of homologous DNA segments in parallel and antiparallel alignment

Homologous gene shuffling between DNA promotes genetic diversity and is an important pathway for DNA repair. For this to occur, homologous genes need to find and recognize each other. However, despite its central role in homologous recombination, the mechanism of homology recognition is still an unsolved puzzle. While specific proteins are known to play a role at later stages of recombination, an initial coarse grained recognition step has been proposed. This relies on the sequence dependence of the DNA structural parameters, such as twist and rise, mediated by intermolecular interactions, in particular electrostatic ones. In this proposed mechanism, sequences having the same base pair text, or are homologous, have lower interaction energy than those sequences with uncorrelated base pair texts; the difference termed the recognition energy. Here, we probe how the recognition energy changes when one DNA fragment slides past another, and consider, for the first time, homologous sequences in antiparallel alignment. This dependence on sliding was termed the recognition well. We find that there is recognition well for anti-parallel, homologous DNA tracts, but only a very shallow one, so that their interaction will differ little from the interaction between two nonhomologous tracts. This fact may be utilized in single molecule experiments specially targeted to test the theory. As well as this, we test previous theoretical approximations in calculating the recognition well for parallel molecules against MC simulations, and consider more rigorously the optimization of the orientations of the fragments about their long axes. The more rigorous treatment affects the recognition energy a little, when the molecules are considered rigid. However when torsional flexibility of the DNA molecules is introduced, we find excellent agreement between analytical approximation and simulation.Comment: Paper with supplemental material attached. 41 pages in all, 4 figures in main text, 3 figures in supplmental. To be submitted to Journa

On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element

A large-deformation model for thin shells composed of elasto-plastic material is presented in this work, Formulation of the shell model, equivalent to the two-dimensional Cosserat continuum, is developed from the three-dimensional continuum by employing standard assumptions on the distribution of the displacement held in the shell body, A model for thin shells is obtained by an approximation of terms describing the shell geometry. Finite rotations of the director field are described by a rotation vector formulation. An elasto-plastic constitutive model is developed based on the von Mises yield criterion and isotropic hardening. In this work, attention is restricted to problems where strains remain small allowing for all aspects of material identification and associated computational treatment, developed for small-strain elastoplastic models, to be transferred easily to the present elasto-plastic thin-shell model. A finite element formulation is based on the four-noded isoparametric element. A particular attention is devoted to the consistent linearization of the shell kinematics and elasto-plastic material model, in order to achieve quadratic rate of asymptotic convergence typical for the Newton-Raphson-based solution procedures. To illustrate the main objective of the present approach-namely the simulation of failures of thin elastoplastic shells typically associated with buckling-type instabilities and/or bending-dominated shell problems resulting in formation of plastic hinges-several numerical examples are presented, Numerical results are compared with the available experimental results and representative numerical simulations