User AndroidFish

7 votes

3 answers

103 views

Prove $\Pi_{k=1}^\infty(1+q^k)=\Pi_{k=1}^\infty\frac{1}{(1-q^{(2k-1)})}$ are equivalent generating series.

combinatorics generating-functions

asked Nov 13, 2016 at 23:44

5 votes

3 answers

102 views

Prove that $\lim_{n \to\infty} \int_0^1 \frac{x^n}{\sqrt{1+x^n}}\, \mathrm dx=0$.

asked Nov 12, 2016 at 20:35

4 votes

1 answer

88 views

Find two bounded sequences, $\{s_n\}_{n \in \mathbb N}$ and $\{t_n\}_{n \in \mathbb N}$, such that $\limsup(s_n+t_n))<\limsup(s_n)+\limsup(t_n)$

sequences-and-series inequality examples-counterexamples limsup-and-liminf

asked Sep 23, 2016 at 3:01

3 votes

1 answer

986 views

What is the probability that the sequence of $n$ integers between $1$ and $n$ contains exactly $n-2$ different integers?

combinatorics

asked Sep 14, 2016 at 4:20

3 votes

2 answers

687 views

Give an example where $UT$ is one-to-one, but $T$ isn't. Give an example where $UT$ is onto, but $T$ isn't.

linear-algebra linear-transformations

asked Sep 23, 2016 at 2:39

3 votes

2 answers

782 views

Suppose G is a finite group. Let N be a normal subgroup of G and A an arbitrary subgroup. Verify that $|AN|=\frac{|A|*|N|}{|A \cap N|}$

abstract-algebra group-theory finite-groups

asked Feb 21, 2017 at 20:33

2 votes

3 answers

300 views

If $\lvert f(x)-f(y) \rvert \le \lvert x-y \rvert ^2$ for all $x$ and $y$ in $\mathbb{R}$ then $f$ is a constant function [duplicate]

real-analysis continuity epsilon-delta

asked Dec 13, 2016 at 22:37

2 votes

3 answers

1k views

Number of ways to choose bagels

combinatorics

asked Sep 11, 2016 at 19:35

1 vote

2 answers

912 views

Prove that if $T:V\rightarrow V$ is additive (that is, $T(x+y)=T(x)+T(y)$ for all $x,y \in V$), then $T$ is linear.

linear-algebra linear-transformations

asked Sep 21, 2016 at 20:30

1 vote

1 answer

25 views

Fix point interval tends to the fix point

calculus sequences-and-series cauchy-sequences

asked Nov 8, 2015 at 23:40

1 vote

1 answer

30 views

Prove for two groups, one less than the other, the smaller is a cyclic subgroup of larger.

group-theory cyclic-groups

asked Mar 13, 2016 at 18:18

1 vote

1 answer

29 views

Find noninvertible $T:P(\mathbb R) \rightarrow P(\mathbb R)$ and $S:P(\mathbb R) \rightarrow P(\mathbb R)$ such that $TS=I_{P(\mathbb R)}$

linear-algebra vector-spaces linear-transformations

asked Sep 29, 2016 at 20:10

1 vote

2 answers

54 views

What is $\bigcap_{n=1}^{\infty}[\frac{1}{3n}, 1+\frac{1}{n})$

general-topology elementary-set-theory

asked Sep 29, 2016 at 23:55

1 vote

3 answers

56 views

The matrix $e^A$ is defined by $e^A=\Sigma_{k=0}^{\infty}\frac {A^k}{k!}$ Suppose M=$\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$. Calculate $e^M$

linear-algebra matrices

asked Oct 23, 2016 at 2:02

1 vote

1 answer

52 views

Prove that if $\{v_1,v_2,\dots,v_n\}$ is orthonormal, then for any $v$, $\lVert v\rVert^2 \ge \Sigma_{j=1}^n \lvert \langle v,v_j \rangle \rvert$

linear-algebra inner-products

asked Nov 13, 2016 at 23:19

1 vote

1 answer

2k views

Assume that $f$ is continuous at $0$ and $g$ is discontinuous at $0$. Prove that if $f(0) \neq 0$, then $f*g$ is not continuous at $0$

real-analysis continuity

asked Oct 10, 2016 at 23:25

1 vote

1 answer

72 views

lucas numbers Prove that $l_0^2+l_1^2+...+l_n^2=l_n*l_n+1+2$ for $n \ge 0$

induction lucas-numbers

asked Nov 2, 2016 at 3:28

1 vote

2 answers

293 views

Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$

abstract-algebra permutations

asked Feb 13, 2017 at 19:29

1 vote

1 answer

27 views

Let $V,W$ be complex inner product spaces. Fix $v \in V$ and $w \in W$ and define $T(u)=\langle u,v\rangle_V w$. Find $T^:W \rightarrow V$

linear-algebra adjoint-operators

asked Nov 14, 2016 at 2:21

1 vote

1 answer

277 views

Find isomorphism between $S_4/H$ and $S_3$ where $H={e,(12)(34),(13)(24),(14)(23)}$ using isomorphism theorem

abstract-algebra group-theory

asked Feb 21, 2017 at 23:24

1 vote

1 answer

49 views

Elementary Divisor Form Uncertainty

group-theory

asked Mar 7, 2017 at 16:10

1 vote

0 answers

459 views

A finite abelian group $G$ has invariant factors $(m_1, m_2, ..., m_k)$. Show that $G$ has an element of order $s$ if and only if $s$ divides $m_k$.

abstract-algebra group-theory finite-groups abelian-groups

asked Mar 16, 2017 at 1:56

1 vote

1 answer

41 views

Suppose $g,h,a∈G$ such that $h=aga^{−1}$. Prove that the formula $x↦ax$ defines a bijection of sets $Fix_X(g)→Fix_X(h)$

abstract-algebra group-theory group-actions

asked Apr 4, 2017 at 2:41

0 votes

1 answer

52 views

Determining the strength of information to vote for candidates.

probability statistics

asked Apr 11, 2017 at 2:40

0 votes

2 answers

78 views

Let $E \subseteq \mathbb{R}^n$ be compact. Let $D = sup\{d(x, y) : x, y ∈ E\}$. Prove that there exists $x_0, y_0$ in $E$ such that $d(x_0, y_0) = D$.

real-analysis

asked Nov 3, 2016 at 18:06

0 votes

0 answers

647 views

Prove that the rank of a nonzero $m \times n$ matrix $A$ is the largest number $r$ such that $A$ has an $r \times r$ minor with nonzero determinant.

linear-algebra matrix-rank

asked Nov 8, 2016 at 21:25

0 votes

2 answers

285 views

Assume that $f:(0,∞)→ℝ$ is twice differentiable with $f(x)>0$ and $f'(x)<0$ for all $x \in (0, ∞)$. Prove that $f''(x)$ cannot always be negative.

real-analysis derivatives

asked Nov 9, 2016 at 22:40

0 votes

1 answer

140 views

Characteristic polynomial of unique matrix [duplicate]

eigenvalues-eigenvectors determinant

asked Oct 15, 2016 at 22:31

0 votes

1 answer

64 views

Prove that $h:S \rightarrow \mathbb R^2$ defined by $h(s)=(f(s),g(s))$ is uniformly continuous on $S$.

real-analysis metric-spaces uniform-continuity

asked Oct 27, 2016 at 20:57

0 votes

1 answer

252 views

Prove that a continuous function from $B(0,1) \subset \mathbb R^2$ to $\mathbb R$ can not be one-to-one.

real-analysis general-topology

asked Oct 27, 2016 at 21:12

Stack Exchange Network

current community

your communities

more stack exchange communities

45 Questions

Prove $\Pi_{k=1}^\infty(1+q^k)=\Pi_{k=1}^\infty\frac{1}{(1-q^{(2k-1)})}$ are equivalent generating series.

Prove that $\lim_{n \to\infty} \int_0^1 \frac{x^n}{\sqrt{1+x^n}}\, \mathrm dx=0$.

Find two bounded sequences, $\{s_n\}_{n \in \mathbb N}$ and $\{t_n\}_{n \in \mathbb N}$, such that $\limsup(s_n+t_n))<\limsup(s_n)+\limsup(t_n)$

What is the probability that the sequence of $n$ integers between $1$ and $n$ contains exactly $n-2$ different integers?

Give an example where $UT$ is one-to-one, but $T$ isn't. Give an example where $UT$ is onto, but $T$ isn't.

Suppose G is a finite group. Let N be a normal subgroup of G and A an arbitrary subgroup. Verify that $|AN|=\frac{|A|*|N|}{|A \cap N|}$

If $\lvert f(x)-f(y) \rvert \le \lvert x-y \rvert ^2$ for all $x$ and $y$ in $\mathbb{R}$ then $f$ is a constant function [duplicate]

Number of ways to choose bagels

Prove that if $T:V\rightarrow V$ is additive (that is, $T(x+y)=T(x)+T(y)$ for all $x,y \in V$), then $T$ is linear.

Fix point interval tends to the fix point

Prove for two groups, one less than the other, the smaller is a cyclic subgroup of larger.

Find noninvertible $T:P(\mathbb R) \rightarrow P(\mathbb R)$ and $S:P(\mathbb R) \rightarrow P(\mathbb R)$ such that $TS=I_{P(\mathbb R)}$

What is $\bigcap_{n=1}^{\infty}[\frac{1}{3n}, 1+\frac{1}{n})$

The matrix $e^A$ is defined by $e^A=\Sigma_{k=0}^{\infty}\frac {A^k}{k!}$ Suppose M=$\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$. Calculate $e^M$

Prove that if $\{v_1,v_2,\dots,v_n\}$ is orthonormal, then for any $v$, $\lVert v\rVert^2 \ge \Sigma_{j=1}^n \lvert \langle v,v_j \rangle \rvert$

Assume that $f$ is continuous at $0$ and $g$ is discontinuous at $0$. Prove that if $f(0) \neq 0$, then $f*g$ is not continuous at $0$

lucas numbers Prove that $l_0^2+l_1^2+...+l_n^2=l_n*l_n+1+2$ for $n \ge 0$

Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$

Let $V,W$ be complex inner product spaces. Fix $v \in V$ and $w \in W$ and define $T(u)=\langle u,v\rangle_V w$. Find $T^:W \rightarrow V$

Find isomorphism between $S_4/H$ and $S_3$ where $H={e,(12)(34),(13)(24),(14)(23)}$ using isomorphism theorem

Elementary Divisor Form Uncertainty

A finite abelian group $G$ has invariant factors $(m_1, m_2, ..., m_k)$. Show that $G$ has an element of order $s$ if and only if $s$ divides $m_k$.

Suppose $g,h,a∈G$ such that $h=aga^{−1}$. Prove that the formula $x↦ax$ defines a bijection of sets $Fix_X(g)→Fix_X(h)$

Determining the strength of information to vote for candidates.

Let $E \subseteq \mathbb{R}^n$ be compact. Let $D = sup\{d(x, y) : x, y ∈ E\}$. Prove that there exists $x_0, y_0$ in $E$ such that $d(x_0, y_0) = D$.

Prove that the rank of a nonzero $m \times n$ matrix $A$ is the largest number $r$ such that $A$ has an $r \times r$ minor with nonzero determinant.

Assume that $f:(0,∞)→ℝ$ is twice differentiable with $f(x)>0$ and $f'(x)<0$ for all $x \in (0, ∞)$. Prove that $f''(x)$ cannot always be negative.

Characteristic polynomial of unique matrix [duplicate]

Prove that $h:S \rightarrow \mathbb R^2$ defined by $h(s)=(f(s),g(s))$ is uniformly continuous on $S$.

Prove that a continuous function from $B(0,1) \subset \mathbb R^2$ to $\mathbb R$ can not be one-to-one.