Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique

In this paper, a powerfully analytical technique is proposed for predicting and generating the steady state solution of the fractional differential system based on the method of harmonic balance. The zeroth-order approximation using just one Fourier term is applied to predict the parametric function for the boundary between oscillatory and non-oscillatory regions of the fractional van der Pol oscillator. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy of the solutions successively. The highly accurate solutions to the angular frequency and limit cycle of fractional van der Pol oscillator are obtained and compared. The results reveal that the technique described in this paper is very effective and simple for obtaining asymptotic solution of nonlinear system having fractional order derivative.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Bruceines K and L from the Ripe Fruits of Brucea javanica

Bruceine K ( 1 ), a pentacyclic C₂₀‐quassinoid bearing a unique 12,20‐epoxy moiety, and bruceine L ( 2 ), along with the ten known compounds (6S,7E)‐6,9,10‐trihydroxy‐ and (6S,7E)‐6,9‐dihydroxymegastigma‐4,7‐dien‐3‐one ( 3 and 4 , resp.), cleomiscosins A–C, luteoline, quercetine, bruceantinol, pinoresinol, and thevetiaflavone, were isolated from the ripe fruits of Brucea javanica. Bruceines K ( 1 ) and L ( 2 ) were determined to be (1β,2α,11β,12β,14ξ,15β)‐12,20‐epoxy‐1,2,11,13,14,15‐hexahydroxypicras‐3‐en‐16‐one and (1β,2α,11β,12β,15β)‐13,20‐epoxy‐1,2,11,12‐tetrahydroxy‐16‐oxo‐15‐(senecioyloxy)picras‐3‐en‐21‐oic acid methyl ester (senecioic acid=3‐methylbut‐2‐enoic acid), respectively, on the basis of NMR (¹H‐ and ¹³C‐NMR, DEPT, ¹H,¹H‐COSY, NOESY, HMQC, and HMBC) and ESI‐MS data. Among the known compounds, (6S,7E)‐6,9,10‐trihydroxy‐ and (6S,7E)‐6,9‐dihydroxymegastigma‐4,7‐dien‐3‐one ( 3 and 4 , resp.), cleomiscosin C, luteoline, quercetine, and thevetiaflavone were isolated for the first time from the Brucea plants.     

A natural product-like inhibitor of NEDD8-activating enzyme

The natural product-like 6,6″-biapigenin has been identified as only the second inhibitor of NEDD8-activating enzyme using virtual screening. This compound was active in enzyme and cell-based assays, with potency in the micromolar range.     

Some addition reactions of bisgermavinylidene

The reaction of bisgermavinylidene [(Me₃SiN?PPh₂)₂C?Ge→Ge?C(PPh₂?NSiMe₃)₂] ( 1 ) with AdNCO (Ad = Adamantyl) afforded the [2 + 2] cycloadditon product [(Me₃SiN?PPh₂)₂CGeC(O) NAd] ( 2 ). Similar reaction of 1 with Ph₃SiOH in tetrahydrofuran (THF) yielded the base‐stabilized germanium(II) triphenylsiloxide [H₂C(PPh₂?NSiMe₃)₂Ge(OSiPh₃)₂] ( 3 ). The results suggested that reactive germavinylidene may exist in solution and is capable of forming addition reaction products. The X‐ray structures of 2 and 3 were determined. Copyright © 2007 John Wiley & Sons, Ltd.     

How Pythagoras' theorem is taught in Czech Republic,Hong Kong and Shanghai: A case study

This paper attempts to explore certain characteristics of the mathematics classroom by investigating how teachers from three different cultures, namely, the Czech Republic, Hong Kong and Shanghai, handle Pythagoras' theorem at eighth grade. Based on a fine-grained analysis of one lesson, from each of the three places, some features in terms of the ways of handling the same topic were revealed, as follows: the Hong Kong teacher and the Shanghai teacher emphasized exploring Pythagoras' theorem, the Shanghai teacher seemed to emphasize mathematical proofs, while the Czech teacher and the Hong Kong teacher tended to verify the theorem visually. It was found that the Czech teacher and the Hong Kong teacher put stress on demonstrating with some degree of student input in the process of learning. On the other hand, the Shanghai teacher demonstrated a constructive learning scenario: students were actively involved in the process of learning under the teacher's control through a series of deliberate activities. Regarding the classroom exercises, the Shanghai teacher tended to vary problems implicitly within a mathematical context, while the teachers in the other two places preferred varying problems explicitly within both mathematical and daily life contexts.     

A stochastic programming approach for multi-site aggregate production planning

Production planning problems play a vital role in the supply chain management area, by which decision makers can determine the production loading plan—consisting of the quantity of production and the workforce level at each production plant—to fulfil market demand. This paper addresses the production planning problem with additional constraints, such as production plant preference selection. To deal with the uncertain demand data, a stochastic programming approach is proposed to determine optimal medium-term production loading plans under an uncertain environment. A set of data from a multinational lingerie company in Hong Kong is used to demonstrate the robustness and effectiveness of the proposed model. An analysis of the probability distribution of economic demand assumptions is performed. The impact of unit shortage costs on the total cost is also analysed.     

Adatom concentration on GaAs(001) during MBE annealing

We study the concentration of adatoms on GaAs(001) during annealing under MBE conditions. By rapidly cooling the sample from typical growth temperatures and typical As overpressures, the thermal concentration of adatoms can be frozen into small islands on the terraces. The area of the resulting islands is measured with STM far from terrace steps, giving an estimate of the concentration of adatoms during equilibrium. We find that a surprisingly large concentration of adatoms is present for typical growth temperatures, e.g. 0.18 monolayer at 600°C. Possible consequences for current growth models are discussed.     

Driven nonequilibrium lattice systems with shifted periodic boundary conditions

We present the first study of a driven nonequilibrium lattice system in the two-phase region, withshifted periodic boundary conditions, forcing steps into the interface. When the shift corresponds to small angles with respect to the driving field, we find nonanalytic behavior in the (internal) energy of the system, supporting numerical evidence that interface roughness is suppressed by the field. For larger shifts, the competition between the driving field and the boundary induces the breakup of a single strip with tilted interfaces into many narrower strips with aligned interfaces. The size and temperature dependences of the critical angles of such breakup transitions are studied.