全國大學(xué)生數(shù)學(xué)模型聯(lián)試題的解課程

1、1992年全國大學(xué)生數(shù)學(xué)模型聯(lián)試題(1992年11月27-29日)A題 施肥效果分析某地區(qū)作物生長所需的營養(yǎng)素主要是氮(N),鉀(K),磷(P)某作物研究所在該地區(qū)對(duì)土豆與生菜做了一定數(shù)量的實(shí)驗(yàn),實(shí)驗(yàn)數(shù)據(jù)如下列表格所示,其中ha表示公頃,t表示噸, kg表示公斤,當(dāng)一個(gè)營養(yǎng)素的施肥量變化時(shí),總將另二個(gè)營養(yǎng)素的施肥量固定在第7個(gè)水平上,例如N做實(shí)驗(yàn)時(shí),P與K 的施肥量分別取為196kg/ha與372kg/ha.土豆:NPK施肥量(kg/ha)產(chǎn)量(t/ha) 施肥量(kg/ha)產(chǎn)量(t/ha)施肥量(kg/ha)產(chǎn)量(t/ha)0 346710113520225933640447115.182

2、1.3625.7232.2934.0339.4543.1543.4640.8430.75024497398147196245294342 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)產(chǎn)量(t/ha) 施肥量(kg/ha)產(chǎn)量(t/ha)施肥量(kg/ha)產(chǎn)量(t/ha)028568411216822428033639211.0212.7014.5616.

3、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試分析施肥量與產(chǎn)量之間關(guān)系,并對(duì)所得結(jié)果從應(yīng)用價(jià)值與如何改進(jìn)等方面作出估價(jià)本題是由北京理工大學(xué)應(yīng)用數(shù)學(xué)系葉其孝建議的,可參看Tony Barnes, Estimating fertilizer roquirements of veg

4、etable crops, Mathematical ModellingA Source Book of Case Studeies,Edited by I.D.Huntley and D.J.G.James, Oxford University Press, 1990,341356.全國一等獎(jiǎng)?wù)撐谋绢}用回歸模型解決,讓我們看一個(gè)例子例 對(duì)8個(gè)學(xué)生調(diào)查其智商iq和課后復(fù)習(xí)某門課時(shí)間t,該門課考試成績g,得下表試建立由智商和復(fù)習(xí)時(shí)間預(yù)測該課程考試成績的公式表 8個(gè)學(xué)生學(xué)習(xí)成績iqtg10510751101279120668116385122169l13087911420981021576這個(gè)問題

5、中,學(xué)生考試成績g受他的智商iq和復(fù)習(xí)某門課時(shí)間t影響,3個(gè)變量g,iq,t間存在密切關(guān)系但是它們的關(guān)系不是確定性關(guān)系而是相關(guān)關(guān)系對(duì)本例我們假設(shè)分?jǐn)?shù)是可連續(xù)取值的,并且認(rèn)為:智商越高,成績越好;復(fù)習(xí)時(shí)間越多,學(xué)習(xí)成績越好;并且假設(shè)智商與復(fù)習(xí)時(shí)間對(duì)考試成績的影響是線性的但每個(gè)人會(huì)遇到其它方面的影響,如精力旺盛與否,學(xué)習(xí)積極性,親友對(duì)該課程知識(shí)的介紹,報(bào)紙雜志對(duì)該課程的介紹等因此g與iq及t的關(guān)系只能是相關(guān)關(guān)系,于是對(duì)一般的學(xué)生成績建立數(shù)學(xué)模型 用計(jì)算機(jī)軟件卻可以方便地完成回歸計(jì)算,SAS的REG,RSREG,ORTHOREG和GLM過程都可以用來作回歸其中REG過程具有許多功能,例如模型選擇回歸

6、診斷等,所以一般情況下總用REG作線性回歸REG過程主要有兩個(gè)語句:PROC REG語句和MODEL語句,其功能如下(1)PROC REG語句用以調(diào)用REG過程,同時(shí)可以加上若干選項(xiàng),其中DATA=用以說明線性回歸所用的數(shù)據(jù)集,如果沒有這一選項(xiàng),就用最新產(chǎn)生的數(shù)據(jù)集作回歸 (2)MODEL語句中有等號(hào),等號(hào)前的變量被指定為響應(yīng)變量,等號(hào)后的變量被指定為自變量 對(duì)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;pro

7、c reg data=score;/*調(diào)用reg過程*/model g=iq t;/*解釋變量是iq和t,應(yīng)變量是g*/run;執(zhí)行程序后計(jì)算機(jī)打出2個(gè)數(shù)表:方差分析表(表頭Analysis of Variance),參數(shù)估計(jì)表(表頭Parameter Estimate)以下分別介紹這2個(gè)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

8、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列指出各行平方和來源:第2行是回歸平方和;第3行是殘差平方和;第4行是前兩行之和第2列(DF)表

9、示自由度,分別是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列是自由度為2,5的F分布隨機(jī)變量大于32.57的概率,這概率小于0.01等價(jià)于F大于0.99分位數(shù)點(diǎn),因而線性關(guān)系是顯著的表的下半部分給出 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value

10、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上表為參數(shù)估計(jì)表,即VariableDFParameter EstimateStandard Errort ValuePr|t|Intercept10.7365516.262800.050.9656iq10.473080.129983.640.0149t12.103440.264187.960.0005各列各行含義如下:第1列為變量,從中可見第2行是(intercept),

11、 第3行是(iq的系數(shù)),第4行是(t的系數(shù))第2列為自由度,各變量自由度都是1第3列為參數(shù)估計(jì)值:=0.73655,=0.47308,=2.10344第4列為標(biāo)準(zhǔn)誤,第5列為t值:,第6列為n-m-1=5個(gè)自由度t分布隨機(jī)變量大于這些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表明變量的作用顯著由此可見智商和復(fù)習(xí)時(shí)間對(duì)得分的作用是顯著的;而常數(shù)的作用是不顯著的常數(shù)反映教師的作用,由檢驗(yàn)可以看出教師的教學(xué)效果是不好的 讓我們回到1992試題查文獻(xiàn)可知:土豆產(chǎn)量是N,P,K產(chǎn)量的二次多項(xiàng)式于是建立回歸模型用土豆-N,

12、P,K數(shù)據(jù)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 36.0

13、6259 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=n

14、pk;model w=n p k nn pp kk np nk pk;run;執(zhí)行後出現(xiàn)“Model is not full rank. Least-squares solutions for the parameters are not unique”說明模型不滿秩,可用逐步回歸篩選主要因子采用程序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

15、 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.

16、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;得到輸出 Stepwise Selection: Step 7 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr F Mo

17、del 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 0.00003

18、063 615.31969 101.63 .0001 Bounds on condition number: 10.342, 202.73由此可得回歸方程若再加上市場價(jià)格就能尋求最佳施肥方案練習(xí)題為了制造豬飼料,采用4種輔料,使用量分別是X1-X4,相應(yīng)的豬飼料產(chǎn)量是y,試驗(yàn)16次,得到結(jié)果如下表找出X1-X4的最好二次多項(xiàng)式,用來預(yù)報(bào)Y方案序號(hào)X1X2X3X4y11012756.363104851010.30410724511.56520126108.6662024755.39720484515.508207251019.53930125512.0810302441013.131130487

19、108.031230726512.4513501241013.491450245510.77155048659.8016507271016.54AcknowledgmentsThe 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 Coun

20、cil, 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 immuniz

21、ed 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 irreleva

22、nt 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 a

23、vidin-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., Shak

24、er 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) (eBioscien

25、ce,CA, USA). The stained RAH-specific CD8+ T cells were analyzed by flow cytometry.Supplementary informationCompeting 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 pepti

26、de 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,

27、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 fami

28、lymembers 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 Co

29、riolus 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 cli

30、nic 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 molecule

31、s 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) simultaneo

32、usly 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 cell

33、s 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 potent

34、ial 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/can

35、cer 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 (siR

36、NAs)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 HSPsgener

37、ally 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

38、 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 red