|
2d |
accepted |
Vector valued Mean value theorem: Norm for the gradient |
|
2d |
accepted |
Show $5z^n=e^z$ has a finite number of zero in $\{a<\Im z < v\}$ and $\{a < \Re z < b \}$ |
|
|
accepted |
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$ |
|
|
accepted |
Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$ |
|
|
accepted |
Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$ |
|
|
accepted |
Show $\dbinom{n}{2}^{-1} \sum_{i < j} X_i X_j \xrightarrow{p} \mu^2$, when $X_i$ are i.i.d. with mean $\mu$ and finite variance |
|
|
accepted |
$X_n \sim \text{Exponential}(\lambda_n)$, independent, $\sum 1/\lambda_n = \infty$, then, $\sum X_n=\infty$ a.s. |
|
|
accepted |
Strict inequality in Reverse Fatou lemma: $\varlimsup \int f_n\le \int \varlimsup f_n$ |
|
|
accepted |
Show $\det \left[T\right]_\beta=-1$, for any basis $\beta$ when $Tx=x-2(x,u)u$, $u$ unit vector |
|
|
accepted |
Is the set $A=\{1,2,\ldots,\omega\}$ internal, when $\omega$ is infinite |
|
|
accepted |
Show $x\sqrt{n} - n \ln\left(1+\frac{x}{\sqrt{n}}\right) \to \frac{x^2}{2}$ |
|
|
accepted |
Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$ |
|
|
accepted |
If $f_n,g_n \in L^1$ and $f_n,g_n \to 0$, show $\int_A (2f_n g_n)/(1+f_n^2+g_n^2)\to 0$, when $A$ has finite measure |
|
|
accepted |
If $L=\{B : BA = 0 \}$ and $R=\{C : AC = 0 \}$, what is the dimension of $L$, and $R$? $L,R,\{A\} \subset \mathbb R_{n \times n}$ |
|
|
accepted |
Find the common limit of $\frac{2}{1/a_n+1/b_n}$ and $\sqrt{a_n b_n}$ |
|
|
accepted |
If $\dim(A+B)=\dim(A\cap B)+1$, then $A \subset B$ or $B \subset A$ |
|
|
accepted |
Harmonic mean: show $\max\{ax,by\} \ge \frac{1}{a+b}(x+y)$, $a,b>1$, $x,y\ge 0$ |
|
|
accepted |
Compound Poisson process: distribution of stopped local time $L_T$ |
|
|
accepted |
$X_i \sim N(\theta,1), \theta \in \Bbb Z$: $T=\left\lfloor \bar X_n \right \rfloor$ not consistent for $\theta$ |
|
|
accepted |
Show the $l^2$ ellipsoid is close : $\left\{(x_n) \in l^2 : \sum\frac{x_n^2}{a_n^2} \le 1\right\}, a_n \to 0$ |
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