Eigenvalues of the k‐th power of a graph

The k‐th power of a graph G, denoted by , is a graph with the same set of vertices as G such that two vertices are adjacent in if and only if their distance in G is at most k. In this paper, we give the bounds on the spectral radius of and . The Nordhaus–Gaddum‐type inequality for the spectral radius of the graph is also presented. Moreover, we obtain an upper bound on the energy of the second power of graphs.     

Spectrality of certain self‐affine measures on the generalized spatial Sierpinski gasket

The self‐affine measure corresponding to a upper or lower triangle expanding matrix M and the digit set in the space is supported on the generalized spatial Sierpinski gasket, where are the standard basis of unit column vectors in . We consider in this paper the existence of orthogonal exponentials on the Hilbert space , i.e., the spectrality of . Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non‐spectrality of . As an application, the spectral property of a class of such self‐affine measures are clarified. The results here generalize the corresponding results in a simple manner.     

Solving unsteady Korteweg–de Vries equation and its two alternatives

This paper aims to present the generalized Kudryashov method to find the exact traveling wave solutions transmutable to the solitary wave solutions of the ubiquitous unsteady Korteweg–de Vries equation and its two famed alternatives, namely, the regularized long‐wave equation and the time regularized long‐wave equation. The exact analytic solutions of the studied equations are constructed explicitly in three forms, namely, hyperbolic, trigonometric, and rational function. The validity of our solutions is verified with MAPLE by putting them back into the original equation and found correct. Moreover, it has shown that the generalized Kudryashov method is an easy and reliable technique over the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Relativistic treatment of the spin-zero particles subject to the second Poschl-Teller-like potential

The three-dimensional Klein-Gordon equation is solved for the case of equal vector and scalar second Poschl-Teller potential by proper approximation of the centrifugal term within the framework of the asymptotic iteration method. Energy eigenvalues and the corresponding wave function are obtained analytically. Eigenvalues are computed numerically for some values of n and It is found that the results are in good agreement with the findings of other methods for short-range potential.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

面外应变对1-3型垂直异质P(VDF-TrFE)基复合薄膜电热性能的调控

考虑1-3型垂直异质铁电P（VDF-TrFE）基复合薄膜结构,利用非线性的热力学理论分析和讨论了平面外应变对复合薄膜电热性能的调控作用. 结果表明,在施加的垂直电场下,平面外应变可以有效地调控电极化、热释电系数、绝热温差等铁电、电热性能. 通过合理的调控平面外应变可以在很大的温度区域范围内获得比纯平面外延薄膜结构更高的绝热温差. 研究结果预示着垂直异质P（VDF-TrFE）基复合薄膜结构在一定的工作温度范围内具有优异的电热性能,在微电源、光通信二极管、红外传感器等微型元件方面有着广泛的应用前景. 关键词： 电热效应 平面外应变 P(VDF-TrFE)复合薄膜 绝热温差     

A method for the numerical solution of the one-dimensional inverse Stefan problem

Summary In this paper we suggest the use of complete families of solutions of the heat equation for the numerical solution of the inverse Stefan problem. Our approach leads to linear optimization problems which can be established and solved easily. Convergence results are proved. In a final section the method is applied to some examples.     

On updating triangular products of Householder reflections

Summary In this paper it is shown how a triangular product of Householder reflections can be updated with an arbitrary elementary orthogonal transformation. The amount of work required is 5/2n ² multiplications for absorbing an arbitrary reflection and 3n ² multiplications for absorbing a plane rotation on subsequent indices.     

On automorphisms of infinite graphs with forbidden subgraphs

LexX be anm-connected infinite graph without subgraphs homeomorphic toKm, n, for somen, and let α be an automorphism ofX with at least one cycle of infinite length. We characterize the structure of α and use this characterization to extend a known result about orientation-preserving automorphisms of finite plane graphs to infinite plane graphs. In the last section we investigate the action of α on the ends ofX and show that α fixes at most two ends (Theorem 3.2).     

Geometric and operational optimization of 20-kHz probe-type sonoreactor for enhancing sonochemical activity

The use of a 20-kHz probe-type sonicator irradiating downward in a 500 mL vessel was optimized for the enhancement of the sonochemical activity in terms of the geometric and operational factors. These factors included the probe immersion depth (the vertical position of the probe), input power, height of the liquid from the bottom, horizontal position of the probe, and thickness of bottom plate The sonochemical oxidation reactions were investigated both quantitatively and qualitatively using calorimetry, KI dosimetry, and luminol (Sonochemiluminescence, SCL) techniques. The sonochemical activity was very positively affected by the vertical boundaries. The highest sonochemical activity was obtained when the probe was placed close to the bottom of the vessel (immersion depth of 60 mm), with a high input power (input power of 75%), and optimal liquid height condition (liquid height of 70 mm). The SCL image analysis showed that the cavitational activity zone gradually expanded around the probe body and changed into a circular shape as the experimental conditions were optimized, and consequently the sonochemical activity increased. The formation of a large bright circular-shaped activity zone could be attributed to the strong reflections of the ultrasound firstly, at the vessel bottom and secondly, at the liquid surface. On the other hand, the cavitational activity zone and the sonochemical activity were negatively affected by the horizontal boundaries when the probe was placed close to the side wall of the vessel. In addition, it was found that the sonochemical activity was also significantly affected by the thickness of the support plate owing to the reflection and transmission of the ultrasound at the boundary between the liquid and the solid media.