一个包含Smarandache函数S（n）和Z1（n）的方程及其整数解

对任意正整数n,著名的Smarandache函数S（n）定义为S（n）=min{m：m∈N,n｜m!）,而伪Smarandache函数Z1（n）定义为Z1（n）=min{m：m∈N,n｜1^2＋2^2＋…＋m^2）．研究方程Z1（n）＋1=S（n）的可解性,并利用初等方法得到了该方程的所有正整数解,同时也给出了所有解的具体表示形式．     

关于算术数列中三个或多个素数的和

作为圆法的一个应用,考虑算术数列中的素变数方程P1＋P2＋…＋Pk=N,Pi≡gi（modh）,j=1,2,…,k,∑1≤j≤kgj≡N（modh）,k≥3,给出了方程在大模情况下解的个数的渐近公式,即设≥3,H=sup{β：L（β＋iγ,x）=0},ε〉0,1≤h≤N^δ,0〈δ〈1,则∑p1＋p2＋…＋pk=N/pj≤N,pj≡gj（modu）,1≤j≤k（logp1）（logp2）…（loghk）=1/（k-1）!Nk-1y（k,N）＋O（Nk-2＋H＋c）＋O（Nηk＋c）＋O（Nk-2＋λ＋c）,其中η3=5/9,η4=14/5和ηk=0（k≥5）,λ={β^-,若L函数存在例外零点β^-,/0,若L函数不存在例外零点,y（k,N）=h/φ（h）^k∏p×h,p×N（1＋（-1）^k＋1/（p-1）^k）∏p×h,p｜N（1＋（-1）^k/（p-1）^k-1）.     

关于指数Diophantine方程x^2＋2^（2m）=y^n

运用Lucas数本原素因数存在性的结果讨论方程x^2＋2^2m=y^n的正整数解（x,y,m,n）,证明了该方程仅有正整数解（x,y,m,n）=（2^（n-1）/2,2^r（rs-1）/2,s）和（2^3k.11,2^2k.5,3k＋1,3）适合n〉2,其中r和s是适合s〉2的正奇数,k是非负整数.     

二重双随机Dirichlet级数的收敛性和增长性

研究二重双随机Dirichlet级数F（s,t,ω）=∑m=1^∞∑n=1^∞amnXmn（ω）exp（-λm（ω）s-μn（ω）t）在{Xmn}独立且某阶矩一致有界等条件下的收敛性和增长性,得出二重双随机Diriehlet级数F（s,t,ω）=∑m=1^∞∑n=1^∞amnXmn（ω）exp（-λm（ω）s-μn（ω）t）与二重Dirichlet级数F（s,t）=∑m=1^∞∑n=1^∞amnexp（-sEλm-tEμn）a.s.有相同的收敛横坐标和增长级．     

随机删失下半参数回归模型中估计的渐近性质

考虑半参数回归模型Yi=xiβ g(ti) iσiε,i=1,2,…,n,其中2iσ=f(ui).当Yi因受某种随机干扰而被随机右删失时,就删失分布未知的情形,利用所获得的删失数据定义了β与g(t)的估计^βn和^gn(t),在适当的条件下,证明了^nβ的渐近正态性,同时得到了^gn(t)的最优收敛速度.     

一个数论函数与最小素因子函数的混合均值

设n∈N ,Smarandache函数V(1)=1;当n>1时,令n=p11αp22α…prrα是n的标准分解式,V(n)=min1≤i≤r{iα.pi}.利用初等方法研究了一个包含Smarandache函数与最小素因子函数的混合均值,并给出了一个有趣的渐近公式.     

一个新的伪Smarandache函数及其均值

引入了一个新的伪Smarandche函数Z0（n）．当n为偶数时,定义Z0（n）=m,m为最小的正整数,使得竹整除2＋4＋6＋…＋2m=m（m＋1）,即Z0（n）=min{m：m∈N,n｜m（m＋1）};当行为奇数时,m为最小的整数使得n｜1＋3＋5＋…＋（2m-1）=m^2．即Z0（n）=min{m：m∈N,n｜m^2}．利用解析方法以及Perron公式研究函数Z0（2n-1）的均值性质,并给出了一个较强的渐近公式．     

关于正负相间方幂和中因子p的指数

研究一类正负相间方幂和中素数因子p的指数问题,将方幂和中因子指数转化为由n,r,x和d组成代数式中因子个数计算.设n=pam,p|m,素数p>7,给出了正负相间方幂和中因(-1)k(x+dk)r,d=ps+1.子p的指数:D= 2n-1k=0     

关于Smarandache函数的β次混合均值

研究了Smarandache函数与最大素因子函数P（n）之差的β次方的值分布问题.利用初等方法给出了∑n≤x（S（n）-P（n））β的一个较强的渐近公式,其中x≥3,β〉1为任意实数.     

关于幂等算子代数的注记

给出幂等算子代数的一个刻画.定义了希尔波特空间H的幂等算子代数.设Ω是B（H）上的一个子代数,且满足Ω^1=Ω,Ω^n=Ω^（n-1）Ω＋Ω^（n-2）Ω^2＋…＋ΩΩ^（n-1）,n=1,2,…,当Ω^2=Ω时,Ω是幂等的.经过研究,得出了幂等算子代数的一些重要性质.从而,进一步得到一个算子代数是幂等算子代数的充分条件.如果Ω不含单位元,对Ω中的每一个元A,都存在一个非零复数λA,使得R（A）真包含于N（A-λA）,那么,Ω就是幂等算子代数.     

Reliable identification of grapevine rootstock varieties using RAPD PCR on woody samples

Since grapevine ( Vitis spp .) rootstock material is being traded increasingly as disbudded woody material a lack of distinctive morphological features on such material necessitates an alternative and reliable means of identification. Methods described here were developed for rapid and efficient extraction of DNA from woody samples rich in phenolic compounds and polysaccharides, and for subsequent identification of varieties by RAPD PCR. Using these methods, and with the application of only one selected RAPD primer, we were able to differentiate sixteen rootstock varieties, including the seven varieties most commonly used in Germany. Problems commonly encountered with reproducibility of RAPD patterns were avoided by choosing primers with a dinucleotide sequence and a high G/C content that allowed a rather high annealing temperature of 45°C. Methods described here should also be useful for other horticultural crops, especially those with woody tissues rich in phenolic compounds and polysaccharides.     

An internet compendium of analytical methods and spectroscopic information for monomers and additives used in food packaging plastics

An internet website (http://cpf.jrc.it/smt/) has been produced as a means of dissemination of methods of analysis and supporting spectroscopic information on monomers and additives used for food contact materials (principally packaging). The site which is aimed primarily at assisting food control laboratories in the European Union contains analytical information on monomers, starting substances and additives used in the manufacture of plastics materials. A searchable index is provided giving PM and CAS numbers for each of 255 substances. For each substance a data sheet gives regulatory information, chemical structures, physico-chemical information and background information on the use of the substance in particular plastics, and the food packaging applications. For monomers and starting substances (155 compounds) the infra-red and mass spectra are provided, and for additives (100 compounds); additionally proton NMR are available for about 50% of the entries. Where analytical methods have been developed for determining these substances as residual amounts in plastics or as trace amounts in food simulants these methods are also on the website. All information is provided in portable document file (PDF) format which means that high quality copies can be readily printed, using freely available Adobe Acrobat Reader software. The website will in future be maintained and up-dated by the European Commission's Joint Research Centre (JRC) as new substances are authorized for use by the European Commission (DG-ENTR formerly DGIII). Where analytical laboratories (food control or other) require reference substances these can be obtained free-ofcharge from a reference collection housed at the JRC and maintained in conjunction with this website compendium.     

Aroma characterization of various apricot varieties using headspace–solid phase microextraction combined with gas chromatography–mass spectrometry and gas chromatography–olfactometry

The characterization of the aromatic profile of several apricot cultivars with molecular tracers in order to obtain objective data concerning the aromatic quality of this fruit was undertaken using headspace–solid phase microextraction (HS–SPME). Six apricot cultivars were selected according to their organoleptic characteristics: Iranien, Orangered, Goldrich, Hargrand, Rouge du Roussillon and A4025. The aromatic intensity of these varieties measured by HS–SPME–Olfactometry were defined and classified according to the presence and the intensity of grassy, fruity and apricot like notes. In the six varieties, 23 common volatile compounds were identified by HS–SPME–GC–MS. Finally, 10 compounds, ethyl acetate, hexyl acetate, limonene, β-cyclocitral, γ-decalactone, 6-methyl-5-hepten-2-one, linalool, β-ionone, menthone and (E)-hexen-2-al were recognized by HS–SPME–GC–O as responsible of the aromatic notes involved in apricot aroma and considered as molecular tracers of apricot aromatic quality which could be utilized to discriminate apricot varieties.     

Tests of potential functional barriers for laminated multilayer food packages. Part I: Low molecular weight permeants

The advent of the functional barrier concept in food packaging has brought with it a requirement for fast tests of permeation through potential barrier materials. In such tests it would be convenient for both foodstuffs and materials below the functional barrier (sub-barrier materials) to be represented by standard simulants. By means of inverse gas chromatography, liquid paraffin spiked with appropriate permeants was considered as a potential simulant of sub-barrier materials based on polypropylene (PP) or similar polyolefins. Experiments were performed to characterize the kinetics of the permeation of low molecular weight model permeants (octene, toluene and isopropanol) from liquid paraffin, through a surrogate potential functional barrier (25 μm-thick oriented PP) into the food simulants olive oil and 3% (w/v) acetic acid. These permeation results were interpreted in terms of three permeation kinetic models regarding the solubility of a particular model permeant in the post-barrier medium (i.e. the food simulant). The results obtained justify the development and evaluation of liquid sub-barrier simulants that would allow flexible yet rigorous testing of new laminated multilayer packaging materials.     

Chlorohydrins of bisphenol A diglycidyl ether (BADGE) and of bisphenol F diglycidyl ether (BFDGE) in canned foods and ready-to-drink coffees from the Japanese market

BADGE.2HCl and BFDGE.2HCl were determined in 28 samples of ready-to-drink canned coffee and 18 samples of canned vegetables (10 corn, 5 tomatoes and 3 others), all from the Japanese market. HPLC was used as the principal analytical method and GCMS for confirmation of relevant LC fractions. BADGE.2HCl was found to be present in one canned coffee and five samples of corn, BFDGE.2HCl in four samples of canned tomatoes and in one canned corn. No sample was found which exceeded the 1mg/kg limit of the EU for the BADGE chlorohydrins. However the highest concentration was found for the sum of BFDGE.2HCl and BFDGE.HCl.H₂O at a level of 1.5mg/kg. A Beilstein test confirmed that all cans containing foods contaminated with BADGE.2HCl or BFDGE.2HCl had at lest one part coated with a PVC organosol.     

European priorities for research to support legislation in the area of food contact materials and articles

A strong science base is required to underpin the planning and decision-making process involved in determining future European community legislation on materials and articles in contact with food. Significant progress has been made in the past 5 years in European funded work in this area, with many developments contributing to a much better understanding of the migration process, and better and simpler approaches to food control. In this paper this progress is reviewed against previously identified work-areas (identified in 1994) and conclusions are reached about future requirements for R&D to support legislation on food contact materials and articles over the next 5 or so years.     

Analysis of benzo[a]pyrene in spiked fatty foods by second derivative synchronous spectrofluorimetry after microwave-assisted treatment of samples

A simple, rapid and inexpensive method has been developed for the determination of benzo[a     

The development of reference materials for paralytic shellfish poisoning toxins in lyophilized mussel. II: Certification study

This paper describes the second part of a project undertaken to develop certified mussel reference materials for paralytic shellfish poisoning toxins. In the first part two interlaboratory studies were undertaken to investigate the performance of the analytical methodology for several PSP toxins, in particular saxitoxin and decarbamoyl-saxitoxin in lyophilized mussels, and to set criteria for the acceptance of results to be applied during the certification exercise. Fifteen laboratories participated in this certification study and were asked to measure saxitoxin and decarbamoyl-saxitoxin in rehydrated lyophilized mussel material and in a saxitoxin-enriched mussel material. The participants were allowed to use a method of their choice but with an extraction procedure to be strictly followed. The study included extra experiments to verify the detection limits for both saxitoxin and decarbamoyl-saxitoxin. Most participants (13 of 15) were able to meet all the criteria set for the certification study. Results for saxitoxin.2HCl yielded a certified mass fraction of <0.07 mg/kg in the rehydrated lyophilized mussels. Results obtained for decarbamoyl-saxitoxin.2HCl yielded a certified mass fraction of 1.59+/-0.20 mg/kg. The results for saxitoxin.2HCl in enriched blank mussel yielded a certified mass fraction of 0.48 +/- 0.06 mg/kg. These certified reference materials for paralytic shellfish poisoning toxins in lyophilized mussel material are the first available for laboratories to test their method for accuracy and performance.     

Detection of adulteration of locust bean gum with guar gum by capillary electrophoresis and polarized light microscopy

Capillary electrophoresis (CE) and polarized light microscopy (PLM) were utilized in the detection of the adulteration of locust bean gum with guar gum. For CE analyses, standards of locust bean and guar gums were extracted with 30% CH₃CN, removing the residual proteins from the gum matrix. A 8.75 mM NaH₂PO₄-20.6 mM Na₂B₄O₇ buffer, pH 9, was used to separate these proteins and to identify marker proteins that were present in the guar gum. These markers did not co-migrate with components in the extracts of mechanically processed locust bean gum, and are used as indicators of adulteration. Using PLM with toluidine blue and iodine staining techniques, unadulterated locust bean gum samples were distinguished from mixed samples through the differential staining of components in locust bean versus guar and tara gums. These experiments in the use of CE and PLM provide orthogonal and complementary methods for the verification of 'true' positives and the elimination of 'false' positives.