数学问题解答

20 0 2年 8月号问题解答(解答由问题提供人给出 )1 3 86　试证明 :有两边及一个角的角平分线对应相等的两个三角形全等 .(湖北宜昌市十一中学　是海松　443 0 0 3 )证明　设△ＡＢＣ和△Ａ′Ｂ′Ｃ′的三边分别记为ａ、ｂ、ｃ和ａ′、ｂ′、ｃ′,三条角平分线分别记为ｔａ、ｔｂ、ｔｃ和ｔａ′、ｔｂ′、ｔｃ′,半周长分别记为ｐ和ｐ′.当有两边及它们的夹角的平分线对应相等时 ,不妨设ｂ=ｂ′,ｃ =ｃ′,ｔａ =ｔａ′.由ｔａ =2ｂ+ｃ ｂｃｐ(ｐ -ａ) ,ｔａ′ =2ｂ′+ｃ′ ｂ′ｃ′ｐ′(ｐ′ -ａ′)得 :2ｂ+ｃ ｂｃｐ(ｐ -ａ) =2ｂ+ｃ ｂｃｐ…     

Diastereoselective [4+2] annulation of vinyl carbodiimides with N-alkyl imines. Asymmetric synthetic access to the batzelladine alkaloids

A diastereoselective [4+2] annulation of vinyl carbodiimides with chiral N-alkyl imines has been developed to access the stereochemically rich tricyclic core of the batzelladine alkaloids. Its application to the asymmetric synthesis of batzelladine D permitted the use of long-range, directed hydrogenation and stereoselective intramolecular iodoamination as additional key steps to establish the remaining stereocenters within the natural product with excellent stereocontrol.     

Inductively coupled plasma compared with direct current arc spectrometry for analysis of minor elements in aluminium baths

A validation study was carried out in order to evaluate the efficiency of inductively coupled plasma-optical emission spectrometry (ICP-OES) for the analysis of minor elements (manganese, chromium, copper, iron, and titanium) in aluminium alloys. Aluminium casting samples were obtained by adding compressed powder compacts of each alloying element and aluminium (minitablets) to aluminum baths in a laboratory crucible furnace. Digestion of solid samples was performed using concentrated HCI and H202 35% (v/v) previous to analysis by ICP-OES without any matrix separation. This solution-based method was validated considering direct current arc spectrometry as the reference method based on direct analysis without any pretreatment of the solid samples considered. Univariate statistical procedures were carried out, for which precision <3% and trueness of the analytical results were taken into account.     

Atomic layer deposition in nanometer-level replication of cellulosic substances and preparation of photocatalytic TiO2/cellulose composites

TiO2 replicas of filter paper with nanometer-level accuracy were prepared by atomic layer deposition of thin conformal TiO2 coating, followed by a removal of the paper by air-anneal at 450 degrees C. Photocatalytic anatase TiO2/cellulose composites were also made by leaving the paper intact. The TiO2 films were deposited from Ti(OMe)4 and H2O at 150-250 degrees C. The photocatalytic activity of the TiO2/cellulose composite was verified by photocatalytic reduction of Ag(I) from an aqueous solution to Ag nanoparticles on the TiO2 surface. The TiO2/cellulose composites are mechanically more stable than the free-standing TiO2 replicas and are therefore potentially suitable as lightweight, high surface area photocatalysts.     

Self-assembled peptide tubelets with 7 A pores

Here we report the preparation and structural characteristics of self-assembling peptide tubelets composed of 32-membered rings formed of alternating alpha-amino acids and cis-3-aminocyclohexanecarboxylic acids. The tubelets possess a partial hydrophobic core environment, provided by the projection of the cyclohexane C2 methylene moiety into the lumen, and a Van der Waals pore diameter of about 7 A.