Fractal-dimensional properties of subordinators

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely limδ→0U(δ)N(t,δ)=t , where N(t,δ) is the minimal number of boxes of size at most δ needed to cover a subordinator’s range up to time t, and U(δ) is the subordinator’s renewal function. Our main result is a central limit theorem (CLT) for N(t,δ) , complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N(t,δ) , but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator’s jumps of size greater than δ . This new process can be manipulated with remarkable ease in comparison with N(t,δ) , and allows better understanding of the box-counting dimension of a subordinator’s range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process

Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Diffusion in multiscale spacetimes

We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples and the most general spectral-dimension profile of multifractional spaces is constructed.Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected, references adde

Boundary non-crossings of Brownian pillow

Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \psi(u;h):=P{B_0(s,t)+h(s,t) \le u(s,t), \forall s,t\in [0,1]}. Further we investigate the asymptotic behaviour of $\psi(u;\gamma h)$ with $\gamma$ tending to infinity, and solve a related minimisation problem.Comment: 14 page

Hausdorff dimension of operator semistable L\'evy processes

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process in \rd with exponent $E$, where $E$ is an invertible linear operator on \rd and $X$ is semi-selfsimilar with respect to $E$. By refining arguments given in Meerschaert and Xiao \cite{MX} for the special case of an operator stable (selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we determine the Hausdorff dimension of the partial range $X(B)$ in terms of the real parts of the eigenvalues of $E$ and the Hausdorff dimension of $B$.Comment: 23 page

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Energy Consumption, Carbon Emissions and Global Warming Potential of Wolfberry Production in Jingtai Oasis, Gansu Province, China

During the last decade, China's agro-food production has increased rapidly and been accompanied by the challenge of increasing greenhouse gas (GHG) emissions and other environmental pollutants from fertilizers, pesticides, and intensive energy use. Understanding the energy use and environmental impacts of crop production will help identify environmentally damaging hotspots of agro-production, allowing environmental impacts to be assessed and crop management strategies optimized. Conventional farming has been widely employed in wolfberry (Lycium barbarum) cultivation in China, which is an important cash tree crop not only for the rural economy but also from an ecological standpoint. Energy use and global warming potential (GWP) were investigated in a wolfberry production system in the Yellow River irrigated Jingtai region of Gansu. In total, 52 household farms were randomly selected to conduct the investigation using questionnaires. Total energy input and output were 321,800.73 and 166,888.80 MJ ha−1, respectively, in the production system. The highest share of energy inputs was found to be electricity consumption for lifting irrigation water, accounting for 68.52%, followed by chemical fertilizer application (11.37%). Energy use efficiency was 0.52 when considering both fruit and pruned wood. Nonrenewable energy use (88.52%) was far larger than the renewable energy input. The share of GWP of different inputs were 64.52% electricity, 27.72% nitrogen (N) fertilizer, 5.07% phosphate, 2.32% diesel, and 0.37% potassium, respectively. The highest share was related to electricity consumption for irrigation, followed by N fertilizer use. Total GWP in the wolfberry planting system was 26,018.64 kg CO2 eq ha−1 and the share of CO2, N2O, and CH4 were 99.47%, 0.48%, and negligible respectively with CO2 being dominant. Pathways for reducing energy use and GHG emission mitigation include: conversion to low carbon farming to establish a sustainable and cleaner production system with options of raising water use efficiency by adopting a seasonal gradient water pricing system and advanced irrigation techniques; reducing synthetic fertilizer use; and policy support: smallholder farmland transfer (concentration) for scale production, credit (small- and low-interest credit) and tax breaks

A Long-Life, High-Rate Lithium/Sulfur Cell: A Multifaceted Approach to Enhancing Cell Performance

Lithium/sulfur (Li/S) cells are receiving significant attention as an alternative power source for zero-emission vehicles and advanced electronic devices due to the very high theoretical specific capacity (1675 mA·h/g) of the sulfur cathode. However, the poor cycle life and rate capability have remained a grand challenge, preventing the practical application of this attractive technology. Here, we report that a Li/S cell employing a cetyltrimethyl ammonium bromide (CTAB)-modified sulfur-graphene oxide (S-GO) nanocomposite cathode can be discharged at rates as high as 6C (1C = 1.675 A/g of sulfur) and charged at rates as high as 3C while still maintaining high specific capacity (~ 800 mA·h/g of sulfur at 6C), with a long cycle life exceeding 1500 cycles and an extremely low decay rate (0.039% per cycle), perhaps the best performance demonstrated so far for a Li/S cell. The initial estimated cell-level specific energy of our cell was ~ 500 W·h/kg, which is much higher than that of current Li-ion cells (~ 200 W·h/kg). Even after 1500 cycles, we demonstrate a very high specific capacity (~ 740 mA·h/g of sulfur), which corresponds to ~ 414 mA·h/g of electrode: still higher than state-of-the-art Li-ion cells. Moreover, these Li/S cells with lithium metal electrodes can be cycled with an excellent Coulombic efficiency of 96.3% after 1500 cycles, which was enabled by our new formulation of the ionic liquid-based electrolyte. The performance we demonstrate herein suggests that Li/S cells may already be suitable for high-power applications such as power tools. Li/S cells may now provide a substantial opportunity for the development of zero-emission vehicles with a driving range similar to that of gasoline vehicles