If you already have a proof for some result, but want to ask for a different proof (using different methods).

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3
votes
3answers
255 views

Finding the shortest distance between two Parabolas

Recently, a problem asked me to find the minimum distance between the parabolas $y=x^2$ and $y=-x^2-16x-65$. I proceeded with the problem as thus. Let $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
1
vote
0answers
27 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
1
vote
1answer
33 views

Prove $ f(c)\int_{a}^{b}g(x)dx=\int_{a}^{b}g(x)f(x)dx$

Assume that $f:[a,b]\rightarrow\mathbb{R}$ is continuous on $[a,b]$ and $g:[a,b]\rightarrow\mathbb{R}$ is integrable and $g(x)\geq0$ for all $x\in[a,b]$. Then there exists a $c\in(a,b)$ such that ...
0
votes
1answer
18 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ...
4
votes
1answer
197 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
15
votes
5answers
2k views

Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
1
vote
0answers
18 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ...
0
votes
0answers
15 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
0
votes
1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
3
votes
4answers
583 views

Prove that the symmetric group $S_n$, $n \geq 3$, has trivial center.

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let ...
1
vote
2answers
27 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
3
votes
3answers
35 views

How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ ...
2
votes
2answers
49 views

Solving the recurrence $A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k}$

Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and $$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$ There is an explicit formula for ...
0
votes
2answers
19 views

Decreasing sequence and prove by contradiction

I have "solved" the following question using prove by contradiction. But it seems a bit off to me: Let {$x_k$} be a sequence satisfying $x_{k+1}\le(1-\beta)x_k$ for $0\lt\beta\lt 1$ , and $x_0\le C$. ...
0
votes
0answers
25 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
4
votes
5answers
159 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
1
vote
1answer
49 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
2
votes
4answers
45 views

Show that $6^n/n! \le 6^5/5! \times 6/n$

I want to show that $$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$ without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this?
2
votes
1answer
67 views

Proving that product of transpose matrix and the matrix is inversible

I need to prove that $A^T$$A$ is an invertible matrix. $$ A= \begin{bmatrix} \vec{a_1} & \vec{a_2} & \ldots & \vec{a_n} \\ \end{bmatrix} $$ Can I prove this using ...
0
votes
1answer
32 views

Alternative proof of $\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it}$

Let $t\in \mathbb{R}$. I want an alternative proof of the following identity $$\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it} \quad(\star)$$ I've came up with this identity observing ...
3
votes
1answer
40 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem ...
2
votes
4answers
78 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
1
vote
2answers
41 views

Simple Inequality of Complex Numbers, $\left| \frac{a-b}{1-\overline{a}b} \right| <1$

Exercise from Ahlfor's Complex: Given $a,b \in \mathbb C$, with $|a| <1$, $|b|<1$, prove: $$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$ My argument: Lemma: If $\alpha, \beta \in ...
2
votes
1answer
32 views

Show that a piecewise function of two solutions to an ODE is a solution to the ODE

Let $x=x(t)$. If the first order ODE $x'=f(t,x) (*)$ is satisfied by $u$ and $v$, each over $I = (a,b)$, show that $$w(t) := u1_{(a,t_0)} + v1_{[t_0,b)}$$ satisfies $(*)$, where $u(t_0) = v(t_0)$ ...
0
votes
1answer
14 views

Show code $C$ is a 1 error correcting

Let $C=\{(0,0,0,0,0),(1,1,1,0,0),(0,0,1,1,1),(1,1,0,1,1)\}\in\mathbb{F}_2$. Show $C$ is $1$ error correcting. Definition: a code $C\subseteq\mathbb{F}^n$ is t error correcting , if for any two ...
1
vote
2answers
24 views

Proof using pumping lemma that $\{0^m1^n \mid m \neq n \}$ is not regular

First of all, there's already a question very similar to this, but in my case I just wanted to show my attempt with the hope that if there are any erros or it's wrong completely you can help me in the ...
1
vote
2answers
2k views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
1
vote
1answer
60 views

For all $x$, $f(x)=\int_{0}^{x}f(t)dt$. Prove that $f(x)=0$ for all $x$.

Suppose that the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $$f(x)=\int_{0}^{x}f(t)dt\qquad\text{for all $x$}$$ Prove that $f(x)=0$ for all $x$. Attempt Since ...
4
votes
0answers
48 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
3
votes
2answers
61 views

Alternative proof: G group of order $p^2$, p prime $\Rightarrow$ $H$ is normal in $G$.

Let $G$ be a group of order $p^2$ where $p$ is prime. If $H$ is a subgroup of order $p$, show that $H$ is normal in $G$. I would like to prove this with the tools that the book has provided up to ...
1
vote
0answers
32 views

Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
0
votes
1answer
13 views

On a Simple Theorem from Hilbert's *The Foundations of Geometry*

I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ...
1
vote
2answers
37 views

Is there a name for this double summation identity? What is the shortest way to illustrate that it holds?

Say I have the following the expression: $$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$ By enumeration, it is easy to see that: in the case when $j=0$ we have $$\sum\limits_{u=0}^{j} g(u) ...
0
votes
0answers
35 views

Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...
0
votes
2answers
47 views

Can one show that $\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||}$ is always on the range of $\cos \theta $?

A basic property of the dot product of two vectors is that $$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||} = \cos \theta $$ Where $\theta$ is the angle between the two vectors. Since there is ...
1
vote
1answer
55 views

Prove $T$ is invertible

If $T\in L(X,X)$ where $X$ is a Banach space and $L(X,X)$ is denoted as the space of bounded linear maps, and $\|I-T\|<1$ where $I$ is the identity operator, then $T$ is invertible? Here ...
0
votes
0answers
18 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
0
votes
0answers
12 views

Direct proof of the existence of optimal memoryless deterministic policies in MDP

It is well known that (finite-state, finite-action, discrete time) MDPs admit an optimal policy that is memoryless and deterministic (sometimes called pure). The proof of this fact for ...
3
votes
2answers
152 views

Proving finite additivity for this semi-algebra (infinite coin flips)

Background copied and pasted from another one of my questions: Background: Consider flipping a coin $n$ times. Define the sample space as $$ \Omega = \{(r_1,r_2,r_3,\dots); r_i = 0 \text{ or }1\} $$ ...
2
votes
1answer
27 views

Generalisation of Binomial Theorem, Leibniz Formula and similar theorems

Since the beginning of the year, our maths teacher showed us the Binomial Theorem in $\mathbb{R}$\, then in $\mathbb{C}$\, in $M_n(\mathbb{K)}$ with two matrices which commute, and now the Leibniz ...
2
votes
0answers
33 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
4
votes
3answers
78 views

Proving that $\sqrt{a_1^2} + \sqrt{a_2^2} +…+ \sqrt{a_n^2} > \sqrt{a_1^2 + a_2^2 +…+a_n^2}$ using Pythagoras

I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$. I'm interested in whether there's a way to use that proof with Pythagoras to prove the general ...
5
votes
1answer
145 views

How to Prove the Chain Rule for Limits Using a $\varepsilon-\delta$ Argument?

I came across the chain rule for limits the other day and it interested me quite a bit and surprisingly I couldn't find the proof on the internet anywhere. From what I understand the chain rule for ...
1
vote
3answers
131 views

Isomorphism of Non-Symmetric Matrices

$A, B$ are non-symmetric matrices of dimension $m \times n$ where $m=n$ or $m \neq n$. Example: An example of $6 \times 3$ non-symmetric matrix is $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & ...
2
votes
2answers
65 views

Dummit and Foote exercise verification?

I was working on the following problem: Let $\sigma$ be the m-cycle $(1 2...m)$. Show that $\sigma^{i}$ is also an m-cycle iff $\gcd(i,m)=1$ A solution to this problem is given here. But the ...
3
votes
1answer
38 views

Verify proof that if $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian

I have to prove that If $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian We let $S$ be a non-finitely generated submodule of $M\otimes_R ...
3
votes
2answers
170 views

Euler-Mascheroni constant [strategic proof]

I know two proof about the approximation of Euler-Mascheroni constant $\gamma$, but very technical. So I would like to know if someone has a strategic proof to show that $0,5<\gamma< 0,6.$ ...
4
votes
1answer
99 views

Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$

Theorem Let $\{a_n\}$ and $\{b_n\}$ be convergent real sequences. Assume that there exists a $N\in\mathbb{N}$ so $a_n\le b_n$ (eq. 1) for all $n\ge N$. Then $\lim_{n\to\infty}a_n\le ...
9
votes
4answers
85 views

Alternative way to show that a simple group of order $60$ can not have a cyclic subgroup of order $6$

Suppose $G$ is a simple group of order $60$, show that $G$ can not have a subgroup isomorphic to $ \frac {\bf Z}{6 \bf Z}$. Of course, one way to do this is to note that only simple group of ...
7
votes
4answers
142 views

A new approach to find value of $x^2+\frac{1}{x^2}$

When I was teaching in college class ,I write this question on board . if we now $x+\frac{1}{x}=4$ show the value of $x^2+\frac{1}{x^2}=14$ Some student ask me for multi idea to show or prove that ...