[Home Squat University Mathematics magazine] NO. 393 Zhongshan University 2015-year calculation Mathematics comprehensive examination test questions recall version

The question has 6 big questions, the choice is 4 questions, the following recollection is one of the 4 questions.

1. ($ "$") (1). Trial: $$\bex x,y>0,\ x\neq y\ra (x+y) \ln \frac{x+y}{2}<x\ln x+y\ln y. \eex$$ (2). Trial Certificate: $$\bex 0<e-\sex{1+\frac{1}{n}}^n<\frac{3}{n},\quad n=1,2,\cdots. \eex$$ (3). Test surface $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a}\ (a>0) $ tangent plane The sum of the intercept on the axis is constant.

2. ($ "$") (1). Set $\al_1,\cdots,\al_m\in\bbr^n\ (M\leq N) $. Test: $\sed{\al_1,\cdots,\al_m}$ linear Independent is equivalent to $$\bex \sed{\al_1,\al_1+\al_2,\cdots,\al_1+\al_2+\cdots+\al_m} \eex$$ linear Independent. (2). Set $A $ for $n $ order reversible Phalanx, $\al,\beta$ are $n $ willi vector, trial: $$\bex | a+\al\beta^t|=| a| (1+\beta^ta^{-1}\al). \eex$$

3. ($ $) Set $A $ for $n $ order Positive definite matrix, $\omega>0$, $b $ for constant, trial iteration format (probably so) $$\bex x^{(k+1)}=x^{(k)}-\omega \sex{a\cdot \frac{x^{( k+1)}-x^{(k)}}{2}-b} \eex$$ to $\forall\ x^{(0)}$ in solution Equation $Ax =b$ is both convergent.

4. ($ $) Set $$\bex f (x) =\sedd{\ba{ll} 1+x,&-1\leq x<0,\\ 1-x,&0\leq x<1,\\ 0,&|x|>1. \ea} \eex$$ (1) . Try $\hat F (\xi) $, where $$\bex \hat f (\xi) =\int_{\bbr} f (x) E^{-ix\xi}\rd x. \eex$$ (2). Set $$\bex f (x) =\sum_n C (n) E^{-inx}, \eex$$ try $c (n) $. (3). Set $$\bex h (\xi) =\frac{1}{2}\sum_n C (n) e^{-in\xi}, \eex$$ test: $$\bex H (\XI) +h (\XI+\PI) =1. \eex$$

[Home Squat University Mathematics magazine] NO. 393 Zhongshan University 2015-year calculation Mathematics comprehensive examination test questions recall version