EXISTENCE AND UNIQUENESS THEOREMS FOR HIGHER-ORDER BOUNDARY VALUE PROBLEMS

An existence and uniqueness of solutions for boundary value problemsandare established, by employing a priori estimates for all solutions and the Schau-der fixed point theorem. Some of these results extend earlier results due toUsmani(1979), Yang(1988), and Pei(1997).     

Counting polynomial subset sums

Let D be a subset of a finite commutative ring R with identity. Let \(f(x)\in R[x]\) be a polynomial of degree d. For a nonnegative integer k, we study the number \(N_f(D,k,b)\) of k-subsets S in D such that

$$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$

In this paper, we establish several bounds for the difference between \(N_f(D,k, b)\) and the expected main term \(\frac{1}{|R|}{|D|\atopwithdelims ()k}\), depending on the nature of the finite ring R and f. For \(R=\mathbb {Z}_n\), let \(p=p(n)\) be the smallest prime divisor of n, \(|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c\) and \(f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]\) with \((a_d, \ldots , a_1, n)=1\). Then

$$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$

answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where \(\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}\) and \(C_d=e^{1.85d}\). Furthermore, if n is a prime power, then \(\delta (n) =1/p\) and one can take \(C_d=4.41\). Similar and stronger bounds are given for two more cases. The first one is when \(R=\mathbb {F}_q\), a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.

     

A strategic r&d planning model with an application to an energy research organization

This paper describes a two-stage strategic planning model for determining the optimal R&D portfolio in the presence of both exploratory and developmental research projects. An example is illustrated in the context of energy research. Research portfolios are analysed using a Bayesian approach in which probability assessments are updated as new information generated by exploratory research becomes available. Useful insights are obtained through sensitivity analysis. It is found that exploratory research tends to exhibit a lumpy characteristic.     

Strong magnetic instability in correlated metallic Bi(2)Ir(2)O(7)

We report an experimental/theoretical study of single-crystal Bi(2)Ir(2)O(7) that possesses a metallic state with strongly exchange-enhanced paramagnetism. The ground state of Bi(2)Ir(2)O(7) is characterized by the following features: (1) a divergent low-temperature magnetic susceptibility that indicates no long-range order down to 50?mK; (2) strongly field-dependent coefficients of the low-temperature T and T(3) terms of the specific heat; (3) a conspicuously large Wilson ratio R(W)?≈?53.5; and (4) unusual temperature and field dependences of the Hall resistivity that abruptly change below 80?K, without any clear correlation with the magnetic behavior. All these unconventional properties suggest the existence of an exotic ground state in Bi(2)Ir(2)O(7).     

Structure elucidation and NMR assignment of two new eudesmane derivatives from Merremia yunnanensis

Two new eudesmane derivatives, 1α,6β,9β-trihydroxy-eudesm-3-ene-1-O-β-d -glucopyranoside ( 1 ) and 1α,6β,9β-trihydroxy-eudesm-3-ene-1-(6-cinnamoyl)-O-β-d -glucopyranoside ( 2 ) were discovered from Merremia yunnanensis. The structures were elucidated by analysis of their spectroscopic data including HR-ESI-MS, 1D, and 2D NMR. It should be noted that this is the first report about structure elucidation and NMR assignment of compounds from M. yunnanensis.     

DCS——偶氮胂与稀土元素显色反应及其在水分析中的应用

本文研究了DCS——偶氮胂与稀土的显色反应,在测定水中稀土时,用PMBP萃淋树脂分离干扰元素。用计算机处理实验数据,相关系数r=0.9992,分析结果令人满意。     

Optical Properties of Fullerenated Poly(epoxy propyl carbazole)

Sincethedevelopmentofmethodslformassproductionoffullerenes,therehasbeengTeatinterestinit-Manyresearchgroups'havepaidgreatattentiontothefiJllerene-dopedpolymersinceWang'discoveredinl992thatfullerene-dopedPVKhasanapparentlyincreasingphotoconductivity.WehavereportedthechendcalattachmentofCcocagestothecarbazolegroupsonpoly(epoxypropylcarbazole)(PEPC)chainsbyFriedel-Craftstypereaction'.Here,theopticalpropertiesoffullerenatedPEPCwereinvestigated,whichwillshowitsgoodphotoconductivepropeFtiesbe…     

A Supramolecularly Activated Radical Cation for Accelerated Catalytic Oxidation

Tuning the activity of radicals is crucial for radical reactions and radical‐based materials. Herein, we report a supramolecular strategy to accelerate the Fenton reaction through the construction of supramolecularly activated radical cations. As a proof of the concept, cucurbit[7]uril (CB[7]) was introduced, through host–guest interactions, onto each side of a derivative of 1,4‐diketopyrrolo[3,4‐c]pyrrole (DPP), a model dye for Fenton oxidation. The DPP radical cation, the key intermediate in the oxidation process, was activated by the electrostatically negative carbonyl groups of CB[7]. The activation induced a drastic decrease in the apparent activation energy and greatly increased the reaction rate. This facile supramolecular strategy is a promising method for promoting radical reactions. It may also open up a new route for the catalytic oxidation of organic pollutants for water purification and widen the realm of supramolecular catalysis.     

Separate and Concentrate Lactic Acid Using Combination of Nanofiltration and Reverse Osmosis Membranes

The processes of lactic acid production include two key stages, which are (a) fermentation and (b) product recovery. In this study, free cell of Bifidobacterium longum was used to produce lactic acid from cheese whey. The produced lactic acid was then separated and purified from the fermentation broth using combination of nanofiltration and reverse osmosis membranes. Nanofiltration membrane with a molecular weight cutoff of 100–400 Da was used to separate lactic acid from lactose and cells in the cheese whey fermentation broth in the first step. The obtained permeate from the above nanofiltration is mainly composed of lactic acid and water, which was then concentrated with a reverse osmosis membrane in the second step. Among the tested nanofiltration membranes, HL membrane from GE Osmonics has the highest lactose retention (97 ± 1%). In the reverse osmosis process, the ADF membrane could retain 100% of lactic acid to obtain permeate with water only. The effect of membrane and pressure on permeate flux and retention of lactose/lactic acid was also reported in this paper.     

The effect of substrate concentration on biohydrogen production by using kinetic models

The effect of substrate concentration ranging from 0 to 300 g/L on fermentative hydrogen production by mixed cultures was investigated in batch tests using glucose as substrate. The experimental results showed that, at 35℃ and initial pH 7.0, during the fermentative hydrogen production, the hydrogen production potential and hydrogen production rate increased with increasing substrate concentration from 0 to 25 g/L. The maximal hydrogen production potential of 426.8 mL and maximal hydrogen pro-duction rate of 15.1 mL/h were obtained at the substrate concentration of 25 g/L. The maximal hydrogen yield and the maximal substrate degradation efficiency were respectively 384.3 mL/g glucose and 97.6%, at the substrate concentration of 2 g/L. The modified Logistic model could be used to describe the progress of cumulative hydrogen production in this study successfully. The Han-Levenspiel model could be used to describe the effect of substrate concentration on fermentative hydrogen production rate.