# All Questions

149 views

84 views

### Complement of invariant subspace

Assuming that I have a given vector space $V$ and a subspace $U$, which is invariant under an endomorphism $A\in End(V)$. I want to prove that $U^\perp$ is also invariant on $A$ .
64 views

### Finding limit function

\begin{align} f(x) &= \lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n ((\infty)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n (1 -1)\\& = ...
113 views

35 views

### If $E_1, E_2$ are connected and $A_1\subseteq E_1$ and $A_2\subseteq E_2$ then $(E_1\times E_2)-(A_1\times A_2)$ is connected [duplicate]

Let $E_1$, $E_2$ connected metric spaces. Let $A_1\subseteq E_1$ and $A_2\subseteq E_2$ proper subsets. Show that the complement of $A_1\times A_2$ $$(E_1\times E_2)-(A_1\times A_2)$$ is connected. I ...
59 views

### Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
218 views

### If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$

Show that if $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$. Can I get some help to solve this problem
47 views

### Justify conditional

If there is an interpretation $I$ in wich $A=\forall x (P(x)\rightarrow Q(x))$ is true, then $(P(x)\rightarrow Q(x))$ is not true ands is not false in $I$. I need to justifiy this is false. So, if A ...
137 views

### Set of permutation matrices

I'm stuck in this problem. Prove the set $P$ of $n×n$ permutation matrices spans a subspace of dimension $(n−1)^2+1$
92 views

### Number of combinations when you can choose none or multiple options for a questions.

I would like to know both the formula and the math name for such combination. Simple example: 1 questions with 3 options, you can choose none, one or multiple options. How can I calculate the number ...
55 views

### One partial differential equation

Where can I find information about equation $$\frac{\partial u(x,t)}{\partial t}-\operatorname{div}\left(A(x)\nabla u(x,t)\right)=f(x,t),\text{where } A(x) \text{ is a matrix 2x2} ?$$ I would be ...
200 views

### Can any finite group be realized as the automorphism group of a directed acyclic graph?

We are given a finite group $G$ and wish to find a DAG (directed acyclic graph) $(V,E)$ whose automorphism group is exactly G (a graph automorphism of a graph is a bijective function $f:V\to V$ such ...
395 views

80 views

### How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?

On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary. Yuval Filmus answered with a explanation ...
29 views

### Finitely many non-convergent ultrafilters

I am trying to prove that if a space $X$ has finitely many non-convergent ultrafilters, then every non-convergent ultrafilter $\mathcal U$ contains a set $A$ that is not contained in any of other ...
193 views

### For real numbers $x$ and $y$, show that $\frac{x^2 + y^2}{4} < e^{x+y-2}$

Show that for $x$, $y$ real numbers, $0<x$ , $0<y$ $$\left(\frac{x^2 + y^2}{4}\right) < e^{x+y-2}.$$ Someone can help me with this please...
62 views

### Applied Probability with Four Disjoint Subsets.

I have a rather longwinded question, so please bear with me! A number of employees conduct duties at company X. Each type of employee meets by department 4 times yearly. Occasionally, a ...
696 views

### Families of curves, differential equation problem [closed]

Obtain the D.E. having a solution as the equation representing all circles whose radius 1 and center's on the line $y=x$.
117 views

### Functions whose derivatives can be written as a function of themself?

What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$? If $f$ is given, can $g$ be solved in terms of the ...
121 views

### In how many ways can $3$ balls be tossed into $3$ boxes?

$3$ balls are tossed into $3$ boxes. In how many ways can that be done? Well, I did in the following way. We have $5$ objects: $3$ balls and $2$ walls of boxes. So a configuration, for example: ...
100 views

### Reference request for the law of the stopping time in the gambler's ruin problem

Suppose we have a sequence of independent and identically distributed random variables $(X_n)_{n\ge 1}$ such that $$P(X_n=1)=p,\quad P(X_n=0)=r,\quad P(X_n=-1)=q$$ with $p,q,r\in[0,1]$, $p+q+r=1$, ...
69 views

### Is this function involving matrices convex?

Let $X\in \mathbb{R}^{n \times n}$. Then, is the function $$\text{Tr}\left( (X^T X )^{-1} \right)$$ convex in $X$? ($\text{Tr}$ denotes the trace operator)
419 views

### Two questions on Lebesgue Decomposition of an increasing function?

I come up with the question in doing Stein's Real analysis, Chap3. Ex. 24, which assert that any increasing function $f$ on $[a,b]$ can be decomposed as $$F=F_A+F_C+F_J,$$ with $F_A$ is absolutely ...
This is the total differential $$df=dx\frac {\partial f}{\partial x}+dy\frac {\partial f}{\partial y}.$$ How do we take higher orders of total differential, $d^2 f=$? Suppose I have $f(x,y)$ and I ...
The reference is http://www.math.columbia.edu/~masdeu/files/notes/FallSeminar.pdf, page 9: Let now $C/R$ be a curve over a noetherian ring $R$; this means that $C$ is smooth, connected, integral, ...