On the Escape Probability for a Left or Right Continuous Random Walk

A formula for the escape probability in a left or right continuous random walk is derived in a simple manner, under natural conditions.AMS Subject Classification: 60G50.     

Density of states in an electrically biased quantum well

Density of states in a quantum well has been studied in the presence of an electric field applied perpendicular to the growth direction. We have shown that an extra quantization is introduced to the motion of the electron due to the discrete energy levels known as Wannier-Stark ladder states and the nature of density of electronic states changes from quasi two-dimensional to quasi one-dimensional.      

One‐step synthesis of ladder‐type fused poly(benzopentalene) derivatives with tunable energy levels by variable substituents

Novel ladder‐type conjugated polymers, fused poly (benzopentalene) derivatives, were synthesized from the readily accessible 1,4‐dibromo‐2,5‐diethynylbenzene derivatives by the Pd‐catalyzed self‐polycondensation in one‐step with high yields. The low solubility of the ladder structure was suggested when the triisopropylsilyl substituents were selected. However, when longer alkyl chains were introduced into the peripheral moieties, such as the dialkylanilino (DAA) and alkyloxyphenyl groups, a high solubility was achieved and the number‐average molecular weight (M_n) reached 18,000. The UV‐Vis absorption spectral shapes of the polymers were similar to the reported dibenzopentalene derivatives, except for the bathochromically shifted end absorptions. This result suggests an extension of the π‐conjugated systems due to the polymerization. Moreover, the almost defect‐free structure of the ladder‐type polymers was confirmed by the quantitative tetracyanoethylene (TCNE) addition to the DAA‐activated alkynes. The titration experiments of TCNE to the polymers revealed the number of terminal alkynes, which enabled us to calculate the molecular weight of the polymers. The calculated molecular weight was consistent with that determined by GPC. After the TCNE addition, the polymer band gaps reasonably decreased as suggested by the UV‐Vis‐NIR absorption and electrochemical measurements. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2010     

Erds-Ko-Rado Theorem for Ladder Graphs

For a graph G and an integer r≥1,G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star(a family of independent r-sets containing some fixed vertex in G),and G is strictly r-EKR if every extremal intersecting family of independent r-sets is a star.Recently,Hurlbert and Kamat gave a preliminary result about EKR property of ladder graphs.They showed that a ladder graph with n rungs is 3-EKR for all n≥3.The present paper proves that this graph is r-EKR for all 1≤r≤n,and strictly r-EKR except for r=n-1.     

布朗运动的最大值和阶梯期权

在奇异期权定价中经常遇到的具有漂移的布朗运动的最大值问题,我们运用布朗运动的反射原理和G irsanov定理给出了在有限[0,T]区间上的具有漂移的布朗运动的最大值分布及其与终值的联合分布.然后把其应用到阶梯期权,得到了阶梯期权封闭形式的解.     

Heteroleptic Pd(II) dithiocarbamates: synthesis,characterization, packing and in vitro anticancer activity against HeLa cell line

Two new heteroleptic Pd(II) dithiocarbamates (1 and 2) have been synthesized by reaction of equimolar quantities of palladium(II) chloride, sodium 4-(3-methoxyphenyl)piperazine-1-carbodithioate, and appropriate substituted triphenylphosphines. The synthesized complexes have been characterized by their physical, spectral (IR, ¹H, ¹³C, and ³¹P NMR), and X-ray crystallographic data. Complexes 1 and 2 showed square-planar geometry both in solution and solid states. The crystal packing of both complexes revealed similar 3-D-supramolecular networks comprised of 1-D chains. However, the nature and strength of various non-covalent interactions of these networks were slightly different. The DNA interaction studies of the complexes have been carried out by UV–visible spectroscopy to evaluate their anticancer potential. The study suggested intercalative interaction with 2.402 × 10⁴ and 2.713 × 10³ M^?1 binding constants, respectively. The complexes have also been evaluated for their anticancer activity against HeLa cell line. Both complexes showed higher activity with IC₅₀ values much lower (22.176 and 26.166 μM for 1 and 2, respectively) than the standard cisplatin (78.075 μM). Furthermore, the complexes induced stronger DNA fragmentation as investigated by DNA ladder assay for apoptosis. Our findings suggested that the anticancer action of these compounds stems from their interaction with DNA leading to DNA damage and apoptosis. The excellent activity of 1 and 2 deserves to be a focus for further research and in vivo studies.     

Identification of intrinsic gapped behavior in spin-1/2 ladder with staggered dimerization

The zero- and low-temperature behaviors of spin-1/2 two-leg ladder with staggered dimerization are investigated by the Green’s function theory. At zero temperature, the ground state phase diagram is explored, wherein the leg-dimer and rung-singlet phases are revealed, which reflect two different intrinsic gapped behaviors. The former is attributed to the bond alternation along the legs, while the latter is due to the strong rung coupling. It is found that the quantum phase transition from one to another is of the first order, which can be clearly signaled by the rung entanglement entropy. At finite temperatures, the temperature dependence of thermodynamic quantities such as the magnetic susceptibility, specific heat, thermal Drude weight and rung entanglement entropy are calculated to characterize the corresponding quantum phases. It is shown that the magnetic behaviors clearly manifest a typical antiferromagnetism at low temperature, which is in accordance with the experimental results. It is also found that the intrinsic gapped low-lying excitations are responsible for the observed thermodynamic behaviors.     

Stochastic comparisons for bulk queues

We consider two important classes of single-server bulk queueing models: M^(X)/G^(Y)/1 with Poisson arrivals of customer groups, and G^(X)/m^(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G^(s)/1,s 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G^(s)/1,s 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.This research was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

A Rational Approach to Crosslinking of Coordination Polymers Using the Photochemical [2+2] Cycloaddition Reaction

Summary: Polymerization as a result of photocrosslinking is one of the most versatile technologies for producing organic polymers. This article highlights the formation of crosslinking in metal coordination polymers in the solid‐state using a photochemical approach and its importance in material science. Past and current developments of solid‐state photochemical [2+2] cycloaddition in organic and inorganic systems relevant to the topic are reviewed. The challenges for the coordination chemist in utilising this rational approach to orient the coordination polymers and the scope of this work are delineated.

The CC‐containing molecules can be aligned by metal coordination bonds in such a way as to promote crosslinking by a photochemical [2+2] cycloaddition reaction.     

Conformational analysis of siloxane-based enzyme-mimic precursors[1]

The conformational flexibility of six hybrid organodisiloxane oligomers were studied using the Low Mode-Monte Carlo conformational search method with the MM2* force field and the Generalized Born/Surface Area continuum solvent model for water. These systems have enzyme-like properties as synthetic acyltransferases and contain aminopyridine groups in various states of protonation. An ensemble of low energy structures was generated and used to investigate the dependence of molecular shape and flexibility on protonation, which plays an important role in catalyst solubility and self-association. The results as measured by the number of unique conformations, end-to-end or longest intramolecular distance and radius of gyration of the conformational point cloud indicate that the number of protonated pyridines plays a significant role in the overall molecular shape. A similar study was also carried out on various POSS-substitutive organodisiloxane oligomers.