全国大学生数学模型联试题的解课程.doc

1992年全国大学生数学模型联试题(1992年11月27-29日)A题 施肥效果分析某地区作物生长所需的营养素主要是氮(N),钾(K),磷(P)某作物研究所在该地区对土豆与生菜做了一定数量的实验,实验数据如下列表格所示,其中ha表示公顷,t表示吨, kg表示公斤,当一个营养素的施肥量变化时,总将另二个营养素的施肥量固定在第7个水平上,例如N做实验时,P与K 的施肥量分别取为196kg/ha与372kg/ha.土豆:NPK施肥量(kg/ha)产量(t/ha) 施肥量(kg/ha)产量(t/ha)施肥量(kg/ha)产量(t/ha)0 346710113520225933640447115.1821.3625.7232.2934.0339.4543.1543.4640.8430.75024497398147196245294342 33.4632.4736.0637.9641.0440.0941.2642.1740.3642.730479314018627937246525825118.9827.3534.8638.52 38.4439.7338.4343.8742.7765.22生菜:NPK施肥量(kg/ha)产量(t/ha) 施肥量(kg/ha)产量(t/ha)施肥量(kg/ha)产量(t/ha)028568411216822428033639211.0212.7014.5616.2717.7522.5921.6319.3416.1214.1104998147196294391489587685 6.399.4812.4614.3317.1021.9422.6421.3422.0724.53 0479314018627937246555865115.7516.7616.8916.2417.5619.2017.9715.8420.1119.40试分析施肥量与产量之间关系,并对所得结果从应用价值与如何改进等方面作出估价本题是由北京理工大学应用数学系叶其孝建议的,可参看Tony Barnes, Estimating fertilizer roquirements of vegetable crops, Mathematical ModellingA Source Book of Case Studeies,Edited by I.D.Huntley and D.J.G.James, Oxford University Press, 1990,341356.全国一等奖论文本题用回归模型解决,让我们看一个例子例 对8个学生调查其智商iq和课后复习某门课时间t,该门课考试成绩g,得下表试建立由智商和复习时间预测该课程考试成绩的公式表 8个学生学习成绩iqtg10510751101279120668116385122169l13087911420981021576这个问题中,学生考试成绩g受他的智商iq和复习某门课时间t影响,3个变量g,iq,t间存在密切关系但是它们的关系不是确定性关系而是相关关系对本例我们假设分数是可连续取值的,并且认为:智商越高,成绩越好;复习时间越多,学习成绩越好;并且假设智商与复习时间对考试成绩的影响是线性的但每个人会遇到其它方面的影响,如精力旺盛与否,学习积极性,亲友对该课程知识的介绍,报纸杂志对该课程的介绍等因此g与iq及t的关系只能是相关关系,于是对一般的学生成绩建立数学模型 用计算机软件却可以方便地完成回归计算,SAS的REG,RSREG,ORTHOREG和GLM过程都可以用来作回归其中REG过程具有许多功能,例如模型选择回归诊断等,所以一般情况下总用REG作线性回归REG过程主要有两个语句:PROC REG语句和MODEL语句,其功能如下(1)PROC REG语句用以调用REG过程,同时可以加上若干选项,其中DATA=用以说明线性回归所用的数据集,如果没有这一选项,就用最新产生的数据集作回归 (2)MODEL语句中有等号,等号前的变量被指定为响应变量,等号后的变量被指定为自变量 对于上例可以采用如下SAS程序data score;input iq t g;cards;105 10 75110 12 79120 6 68116 13 85122 16 91130 8 79114 20 98102 15 76;proc reg data=score;/*调用reg过程*/model g=iq t;/*解释变量是iq和t,应变量是g*/run;执行程序后计算机打出2个数表:方差分析表(表头Analysis of Variance),参数估计表(表头Parameter Estimate)以下分别介绍这2个表所反映的信息 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr F Model 2 596.11512 298.05756 32.57 0.0014 Error 5 45.75988 9.15198 Corrected Total 7 641.87500 Root MSE 3.02522 R-Square 0.9287 Dependent Mean 81.37500 Adj R-Sq 0.9002 Coeff Var 3.71763表的上半部分是方差分析表,即表,SourceDFSum of SquaresMean SquareF ValuePrFModel2596.11512298.0575632.570.0014Error545.759889.15198.Corrected Total7641.87500.第1列指出各行平方和来源:第2行是回归平方和;第3行是残差平方和;第4行是前两行之和第2列(DF)表示自由度,分别是2,5和2+5=7;第3列是平方和:SSR=596.11512,SSE=45.75988,SST=SSR+SSE=641.87500第4列是平均平方和298.05756=596.11512/2,9.15198=45.75988/5第5列是F值:32.57=298.05756/9.15198第6列是自由度为2,5的F分布随机变量大于32.57的概率,这概率小于0.01等价于F大于0.99分位数点,因而线性关系是显著的表的下半部分给出 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr |t| Intercept 1 0.73655 16.26280 0.05 0.9656 iq 1 0.47308 0.12998 3.64 0.0149 t 1 2.10344 0.26418 7.96 0.0005上表为参数估计表,即VariableDFParameter EstimateStandard Errort ValuePr|t|Intercept10.7365516.262800.050.9656iq10.473080.129983.640.0149t12.103440.264187.960.0005各列各行含义如下:第1列为变量,从中可见第2行是(intercept), 第3行是(iq的系数),第4行是(t的系数)第2列为自由度,各变量自由度都是1第3列为参数估计值:=0.73655,=0.47308,=2.10344第4列为标准误,第5列为t值:,第6列为n-m-1=5个自由度t分布随机变量大于这些t值的概率:P(T0.05)=0.9656,P(T3.64)=0.0149),P(T7.96)=0.0005概率小于0.05表明变量的作用显著由此可见智商和复习时间对得分的作用是显著的;而常数的作用是不显著的常数反映教师的作用,由检验可以看出教师的教学效果是不好的 让我们回到1992试题查文献可知:土豆产量是N,P,K产量的二次多项式于是建立回归模型用土豆-N,P,K数据代入,得SAS程序data npk;input n p k w;nn=n*n;pp=p*p;kk=k*k;np=n*p;nk=n*k;pk=p*k;cards; 0 196 372 15.18 34 196 372 21.36 67 196 372 25.72101 196 372 32.29135 196 372 34.03202 196 372 39.45259 196 372 43.15336 196 372 43.36404 196 372 40.83471 196 372 30.75259 0 372 33.46259 24 372 32.47259 49 372 36.06259 73 372 37.96259 98 372 41.04259 147 372 40.09259 196 372 41.26259 245 372 42.17259 294 372 40.36259 342 372 42.73259 196 0 18.98259 196 47 27.35259 196 93 34.86259 196 140 38.52259 196 186 38.44259 196 279 37.73259 196 372 38.43259 196 465 43.87259 196 558 42.77259 196 651 46.22;proc reg data=npk;model w=n p k nn pp kk np nk pk;run;执行後出现“Model is not full rank. Least-squares solutions for the parameters are not unique”说明模型不满秩,可用逐步回归筛选主要因子采用程序data npk;input n p k w;nn=n*n;pp=p*p;kk=k*k;np=n*p;nk=n*k;pk=p*k;cards; 0 196 372 15.18 34 196 372 21.36 67 196 372 25.72101 196 372 32.29135 196 372 34.03202 196 372 39.45259 196 372 43.15336 196 372 43.36404 196 372 40.83471 196 372 30.75259 0 372 33.46259 24 372 32.47259 49 372 36.06259 73 372 37.96259 98 372 41.04259 147 372 40.09259 196 372 41.26259 245 372 42.17259 294 372 40.36259 342 372 42.73259 196 0 18.98259 196 47 27.35259 196 93 34.86259 196 140 38.52259 196 186 38.44259 196 279 37.73259 196 372 38.43259 196 465 43.87259 196 558 42.77259 196 651 46.22;proc reg data=npk;model w=n p k nn pp kk np nk pk/selection=stepwise;run;得到输出 Stepwise Selection: Step 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr F Model 5 1588.02719 317.60544 52.46 F Intercept 34.21183 1.51322 3094.73186 511.15 .0001 nn -0.00032100 0.00003116 642.51487 106.12 .0001 pp -0.00020621 0.00004702 116.45852 19.24 0.0002 kk -0.00007747 0.00001300 215.00773 35.51 .0001 np 0.00037276 0.00005818 248.56345 41.05 .0001 nk 0.00030883 0.00003063 615.31969 101.63 .0001 Bounds on condition number: 10.342, 202.73由此可得回归方程若再加上市场价格就能寻求最佳施肥方案练习题为了制造猪饲料,采用4种辅料,使用量分别是X1-X4,相应的猪饲料产量是y,试验16次,得到结果如下表找出X1-X4的最好二次多项式,用来预报Y方案序号X1X2X3X4y11012756.363104851010.30410724511.56520126108.6662024755.39720484515.508207251019.53930125512.0810302441013.131130487108.031230726512.4513501241013.491450245510.77155048659.8016507271016.54AcknowledgmentsThe authors would like to thank Johns Hopkins University for the TC-1 cells. This work was supportedby a National Health Research Institutes intramural grant (IV-103-PP-22) and grants from the NationalScience Council, which were awarded to Y.C. Song (NSC 99-2321-B-400-004-MY3) and S.J. Liu (NSC103-2321-B-400-008).Author ContributionsY.C.S. and S.J.L. designed the studies. Y.C.S. performed the research and analyzed the data. Y.C.S. andS.J.L. wrote the manuscript.Additional InformationC57BL/6 mice were immunized subcutaneously (s.c.)once with 1 g of peptide mixed with or without 10 g of CpG adjuvant. After one week, splenocytes wereharvested, and the response of IFN- -secreting cells was determined by ELISPOT after 48 h of peptidestimulation. Briefly, 2 105 splenocytes were incubated with 1 g/ml irrelevant peptide or RAH peptidein an anti-IFN- -coated polyvinylidene fluoride (PVDF) plate for 48 h. After incubation, the cells wereremoved, and a biotinylated anti-IFN- Ab (eBioscience, San Diego, CA, USA) was added to each well.The plates were incubated at 37 C for 2 h. Following the addition of the avidin-HRP reagent (eBioscience,CA, USA), the assay was developed using a 3-amine-9-ethyl carbazole (AEC; Sigma-Aldrich, MO,USA) staining solution. The reaction was stopped after 46 min by placing the plate under tap water.The spots were counted using an ELISPOT reader (Cellular Technology Ltd., Shaker Heights, OH, USA).For RAH-specific T cell staining, spleens were harvested seven days after the immunizations, andRAH-specific CD8+ T cells were detected by tetramer staining using a PE-labeled RAH tetramer(Beckman Coulter, CA, USA) and a FITC-labeled anti-CD8 monoclonal antibody (mAb) (eBioscience,CA, USA). The stained RAH-specific CD8+ T cells were analyzed by flow cytometry.Supplementary information accompanies this paper at http:/www.nature.com/srepCompeting financial interests: The authors declare no competing financial interests.How to cite this article: Song, Y.-C. and Liu, S.-J. A TLR9 agonist enhances the anti-tumor immunityof peptide and lipopeptide vaccines via different mechanisms. Sci. Rep. 5, 12578; doi: 10.1038/srep12578 (2015).This work is licensed under a Creative Commons Attribution 4.0 International License.The images or other third party material in this article are included in the articles CreativeCommons license, unless indicated otherwise in the credit line; if the material is not included underthe Creative Commons license, users will need to obtain permission from the license holder toreproduce the material. To view a copy of this license, visit expression of anti-apoptotic molecules such as the BCL-2 familymembers BCL-XL and CASP8 and FADD-like apoptosis regulator(CFLAR, best known as cFLIP), thereby allowing CTLs to surviveand reach neoplastic of various TLR agonists promotes the immunogenicity of DC/malignant cell fusions through the upregulation of IL-12.14,18In this setting, we used a de from Coriolus versicolor (PSK, which operates as a TLR2 agonist)and lyophilized preparations of a low-virulence strain (Su) ofStreptococcus pyogenes (OK-432, which acts as a TLR4 agonist),both of which can be produced as good manufacturing practice(GMP)-grade agents and have been previously used in the clinic asbiological response modifiers.18,19 Of note, DC/cancer cell fusionsactivated in the presence of both TLR2 and TLR4 agonists,but not DC/malignant cell fusions that were left unstimulatedor were exposed to either TLR agonist alone, overcame theimmunosuppressive activity of tumor-derived molecules suchas transforming growth factor 1 (TGF1). In particular,TLR2/4-activated DCs (or the corresponding fusions): (1) exhibitincreased expression levels of MHC class II molecules and CD86on the cell surface; (2) manifest an improved fusion efficacy;(3) produce elevated levels of IL-12; (4) simultaneously activateCD4+ and CD8+ T cells, which secrete high levels of interferon (IFN); (5) potently induce antigen-specific CTL activity;and (6) manifest a superior efficacy in inhibiting the generationof CD4+CD25+FOXP3+ Tregs.20 Nonetheless, when DC/cancercell fusions are generated with neoplastic cells producing extremelyhigh levels of TGF1, they inhibit the activity of CTLs in vitro.Therefore, incorporating the simultaneous activation of multipleTLRs and the blockade of immunosuppressive that are intrinsicallyproduced by DC/neoplastic cell fusions may significantly enhancethe therapeutic potential of this approach.Improving the Immunogenicity of Malignant CellsMost, if not all, malignant cells secrete multipleimmunosuppressive mediators such as TGFmolecules normally inhibit the initiation of efficient CTLresponses,21 the microenvironment of malignant cells used forthe generation of DC/cancer cell fusions immunostimulatory. Several strategies to inhibit the production ofimmunosuppressive factors by cancer cells have been developed,including the administration of neutralizing antibodies22 and smallchemical inhibitors,23 as well as the transfection of specific smallinterferingRNAs (siRNAs)24 or constructs coding for a solublevariant of the TGF receptor.25 Also heat-shock proteins (HSPs),which have recently been implicated in the immunogenicity ofapoptotic and necrotic cells, might constitute effective adjuvantfor boosting the efficacy of DC/neoplastic cell fusions.26,27 HSPsgenerally operate as chaperons for a wide panel of peptides,including antigenic peptides, and HSP/peptide complexes notonly can be efficiently taken up by DCs through specific receptors,but also can be presented in molecules the DC surface.28 We have previously reported thatTLR2-stimulated DCs fused with heat-treated cancer cells areimmunogenic, as demonstrated by: (1) the upregulation of multipleHSPs, MHC class I and II molecules, TAAs, CD80, CD86, CD83,and IL-12; (2) their ability producing high levels of IFN; and (3) the capacity to efficientlyelicited antigen-specific CTL activity.26 More recently, we havedemonstrated that the secretion of TGF1, IL-10 and VEGRfrom whole cancer cells is significantly limited upon exposure topharmaceutical grade ethanol, a maneuver that does not reduce theevels of MHC class I molecules and TAAs on the cell surface.27Moreover, ethanol, employed at concentrations that affect tumorgrowth, promoted the upregulation of HSPs. HSPs exposed bycancer cells can be recognized by DCs via TLR4, facilitating theiractivation and promoting antigen processing and presentation.29Of note, malignant cells that undergo immunogenic apoptosisectopically expose the Ca2+-binding chaperone calreticulin (CRT)on the cell surface, allowing TAAs to efficiently traffic to theDC antigen-presenting compartment.30 Moreover, high-mobilitygroup box 1 (HMGB1) passively released from dying neoplasticcells can stimulate antigen processing and presentation in DCs viaa TLR4-dependent signaling pathway.31,32 Therefore, the exposureof CRT and the release of HMGB1 by ethanol-treated malignantcells enhance the immunogenicity of DC/cancer cell fusions.27Importantly, fusions involving DCs and ethanol-treated cancercells activate T cells to produce high levels of IFN, boosting theelicitation of antigen-specific CTL response in vitro.27 In addition,HSP70-peptide complexes derived from DC/cancer cell fusionsappear to possess superior immunogenic properties as comparedwith similar complexes obtain from neoplastic cells.33Synergistic Effects of Fusions Generated