Fractal space frames and metamaterials for high mechanical efficiency

A solid slender beam of length $L$, made from a material of Young's modulus $Y$ and subject to a gentle compressive force $F$, requires a volume of material proportional to $L^{3}f^{1/2}$ [where $f\equiv F/(YL^{2})\ll 1$] in order to be stable against Euler buckling. By constructing a hierarchical space frame, we are able to systematically change the scaling of required material with $f$ so that it is proportional to $L^{3}f^{(G+1)/(G+2)}$, through changing the number of hierarchical levels $G$ present in the structure. Based on simple choices for the geometry of the space frames, we provide expressions specifying in detail the optimal structures (in this class) for different values of the loading parameter $f$. These structures may then be used to create effective materials which are elastically isotropic and have the combination of low density and high crush strength. Such a material could be used to make light-weight components of arbitrary shape.Comment: 6 pages, 4 figure

Almost sure exponential stability of numerical solutions for stochastic delay differential equations

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the Borel–Cantelli lemma

Photometric properties and luminosity function of nearby massive early-type galaxies

We perform photometric analyses for a bright early-type galaxy (ETG) sample with 2949 galaxies ($M_{\rm r}<-22.5$ mag) in the redshift range of 0.05 to 0.15, drawn from the SDSS DR7 with morphological classification from Galaxy Zoo 1. We measure the Petrosian and isophotal magnitudes, as well as the corresponding half-light radius for each galaxy. We find that for brightest galaxies ($M_{\rm r}<-23$ mag), our Petrosian magnitudes, and isophotal magnitudes to 25 ${\rm mag/arcsec^2}$ and 1\% of the sky brightness are on average 0.16 mag, 0.20 mag, and 0.26 mag brighter than the SDSS Petrosian values, respectively. In the first case the underestimations are caused by overestimations in the sky background by the SDSS PHOTO algorithm, while the latter two are also due to deeper photometry. Similarly, the typical half-light radii ($r_{50}$) measured by the SDSS algorithm are smaller than our measurements. As a result, the bright-end of the $r$-band luminosity function is found to decline more slowly than previous works. Our measured luminosity densities at the bright end are more than one order of magnitude higher than those of Blanton et al. (2003), and the stellar mass densities at $M_{\ast}\sim 5\times10^{11} M_{\odot}$ and $M_{\ast}\sim 10^{12} M_{\odot}$ are a few tenths and a factor of few higher than those of Bernardi et al. (2010). These results may significantly alleviate the tension in the assembly of massive galaxies between observations and predictions of the hierarchical structure formation model.Comment: 43 pages, 14 figures, version accepted for publication in the Astrophysical Journa

Microlensing of Sub-parsec Massive Binary Black Holes in Lensed QSOs: Light Curves and Size-Wavelength Relation

Sub-parsec binary massive black holes (BBHs) are long anticipated to exist in many QSOs but remain observationally elusive. In this paper, we propose a novel method to probe sub-parsec BBHs through microlensing of lensed QSOs. If a QSO hosts a sub-parsec BBH in its center, it is expected that the BBH is surrounded by a circum-binary disk, each component of the BBH is surrounded by a small accretion disk, and a gap is opened by the secondary component in between the circum-binary disk and the two small disks. Assuming such a BBH structure, we generate mock microlensing light curves for some QSO systems that host BBHs with typical physical parameters. We show that microlensing light curves of a BBH QSO system at the infrared-optical-UV bands can be significantly different from those of corresponding QSO system with a single massive black hole (MBH), mainly because of the existence of the gap and the rotation of the BBH (and its associated small disks) around the center of mass. We estimate the half-light radii of the emission region at different wavelengths from mock light curves and find that the obtained half-light radius vs. wavelength relations of BBH QSO systems can be much flatter than those of single MBH QSO systems at a wavelength range determined by the BBH parameters, such as the total mass, mass ratio, separation, accretion rates, etc. The difference is primarily due to the existence of the gap. Such unique features on the light curves and half-light radius-wavelength relations of BBH QSO systems can be used to select and probe sub-parsec BBHs in a large number of lensed QSOs to be discovered by current and future surveys, including the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), the Large Synoptic Survey telescope (LSST) and Euclid.Comment: 18 pages, 17 figures, accepted for publication in the Astrophysical Journa

Delay-dependent robust stability of stochastic delay systems with Markovian switching

In recent years, stability of hybrid stochastic delay systems, one of the important issues in the study of stochastic systems, has received considerable attention. However, the existing results do not deal with the structure of the diffusion but estimate its upper bound, which induces conservatism. This paper studies delay-dependent robust stability of hybrid stochastic delay systems. A delay-dependent criterion for robust exponential stability of hybrid stochastic delay systems is presented in terms of linear matrix inequalities (LMIs), which exploits the structure of the diffusion. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method

A Rate-Splitting Based Bound-Approaching Transmission Scheme for the Two-User Symmetric Gaussian Interference Channel with Common Messages

This paper is concerned with a rate-splitting based transmission strategy for the two-user symmetric Gaussian interference channel that contains common messages only. Each transmitter encodes its common message into multiple layers by multiple codebooks that drawn from one separate code book, and transmits the superposition of the messages corresponding to these layers; each receiver decodes the messages from all layers of the two users successively. Two schemes are proposed for decoding order and optimal power allocation among layers respectively. With the proposed decoding order scheme, the sum-rate can be increased by rate-splitting, especially at the optimal number of rate-splitting, using average power allocation in moderate and weak interference regime. With the two proposed schemes at the receiver and the transmitter respectively, the sum-rate achieves the inner bound of HK without time-sharing. Numerical results show that the proposed optimal power allocation scheme with the proposed decoding order can achieve significant improvement of the performance over equal power allocation, and achieve the sum-rate within two bits per channel use (bits/channel use) of the sum capacity

Switching and diffusion models for gene regulation networks

We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis