On the spectral radius of ‡-shape trees

Let A(G) be the adjacency matrix of G. The characteristic polynomial of the adjacency matrix A is called the characteristic polynomial of the graph G and is denoted by φ(G, λ) or simply φ(G). The spectrum of G consists of the roots (together with their multiplicities) λ ₁(G) ? λ ₂(G) ? … ? λ _n (G) of the equation φ(G, λ) = 0. The largest root λ ₁(G) is referred to as the spectral radius of G. A ?-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by G(l ₁, l ₂, … l ₇) (l ₁ ? 0, l _i ? 1, i = 2, 3, …, 7) a ?-shape tree such that $G\left( {l_1 ,l_2 , \ldots l_7 } \right) - u - v = P_{l_1 } \cup P_{l_2 } \cup \ldots P_{l_7 }$ , where u and v are the vertices of degree 4. In this paper we prove that ${{3\sqrt 2 } \mathord{\left/ {\vphantom {{3\sqrt 2 } 2}} \right. \kern-0em} 2} < \lambda _1 \left( {G\left( {l_1 ,l_2 , \ldots l_7 } \right)} \right) < {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$ .     

Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process

In this paper, the stochastic harvesting problem is regarded as a mathematical formulation of finding the maximum sustained yield and the corresponding best sustainable harvesting strategies under uncertainty. We use a new method to solve this problem, and prove the equivalency between this method and previous methods. This paper is the first attempt to apply the ergodic theory on the optimal harvesting problem, to the best of our knowledge.     

Connectivity and minimal distance spectral radius of graphs

In this article, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with k cut vertices (respectively, k cut edges) with the minimal distance spectral radius.     

Soybean-Oil-Based CO2-Switchable Surfactants with Multiple Heads

Oligomeric surfactants display the novel properties of low surface activity, low critical micellar concentration and enhanced viscosity, but no CO₂ switchable oligomeric surfactants have been developed so far. The introduction of CO₂ can convert tertiary amine reversibly to quaternary ammonium salt, which causes switchable surface activity. In this study, epoxidized soybean oil was selected as a raw material to synthesize a CO₂-responsive oligomeric surfactant. After addition and removal of CO₂, the conductivity analyzing proves that the oligomeric surfactant had a good response to CO₂ stimulation. The viscosity of the oligomeric surfactant solution increased obviously after sparging CO₂, but returned to its initial low viscosity in the absence of CO₂. This work is expected to open a new window for the study of bio-based CO₂-stimulated oligomeric surfactants.     

The protection zone for biological population in random environment

A biomathematical model is described by stochastic differential equations with Markovian switching. The long‐time dynamical properties are studied both theoretically and numerically. Results show that both the persistence and extinction have close relationship with environmental noises (white and color noises). These results are of important biological significance for biological conservation. Copyright © 2012 John Wiley & Sons, Ltd.     

The catalytic activity of alkali metal alkoxides and titanium alkoxides in the hydrosilylation of unfunctionalized olefins

The catalytic activities of titanium alkoxides and alkali metal alkoxides for hydrosilylation of unfunctionalized olefins have been studied. Titanium(IV) alkoxides showed excellent catalytic activity, while alkali metal alkoxides have low catalytic activity for the hydrosilylation of olefins. However, by using titanocene dichloride as an additive, alkali metal alkoxides showed also excellent catalytic property for hydrosilylation. In comparison with titanium alkoxides, no α-adduct was obtained by using alkali metal alkoxides/Cp₂TiCl₂ as catalysts.     

The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field

Based on the extended Huygens–Fresnel principle, the closed-form propagation equation of partially coherent Sinh-Gaussian (ShG) beams through the turbulent atmosphere has been derived by using the quadratic approximation of the Rytov's phase structure function. The influence of turbulence on propagation properties of partially coherent ShG beams and their beam quality in the far field are studied both analytically and numerical. The fully coherent ShG beam has been treated as a special case of the partially coherent ShG beam when the degree of spatial coherence α=∞. It is shown that in comparison with the free-space propagation the turbulence accelerates the evolution of three stages that partially coherent ShG beams undergo. The smaller α is, the less partially coherent ShG beams are affected by the turbulence. In particular, we find that the β parameter decreases as the decentered parameter δ increases, but the dependence of the Strehl ratio on δ is not monotonous. There exists an optimal δ_opt, the influence of turbulence on the maximum intensity of the corresponding partially coherent ShG beam with δ_opt is the smallest. Therefore, a suitable choice of δ may reduce the influence of turbulence on the beam quality in the far field in practice.     

Spreading of spatially partially coherent polychromatic beams in atmospheric turbulence

Taking the polychromatic Gaussian Schell-model (GSM) beam as a typical example of spatially partially coherent polychromatic beams, the spreading of polychromatic GSM beams in atmospheric turbulence is studied. The mean-squared width of polychromatic GSM beams in turbulence is derived by using the effective source and the strong fluctuation models. It is shown that the same result is obtained using both the models. The diffraction, atmospheric turbulence and beam polychroism result in a spreading of polychromatic GSM beams. If the scaling law fails, the spreading of polychromatic GSM beams increases with increasing bandwidth Γ, but the influence of Γ on the spreading of polychromatic GSM beams becomes small as the structure constant C_n² of the refractive index and spatial correlation parameter α increase. The spreading of polychromatic GSM beams increases as C_n² increases and α decreases. Spatially partially coherent polychromatic beams are less sensitive to the effects of atmospheric turbulence than spatially fully coherent polychromatic beams.