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

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2
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4answers
56 views

Proof verification : every algebraic set is the union of finitely many irreducible algebraic subsets

I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have $I=...
1
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1answer
15 views

Proof improvement for $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ given $(a+ib)(c+id)(e+if)(g+ih) = A + iB$

If $(a+ib)(c+id)(e+if)(g+ih) = A + iB$, prove that $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ My approach is pretty straightforward: $$(a+ib)(c+id)(e+if)(g+ih)$$ $$((ac-bd)+i(ad+bc))(e+if)(...
0
votes
1answer
39 views

If $A$ Is an Upper Triangle Matrix, the Adjoint Is Also Upper Triangular

I already proved it, but it was really laborious. I am wondering if any one has a shorter proof? Write $A = [a_{ik}]$ and let $\overline{A}_{rs} = [c_{ik}]$ denote the minor with row $r$ and column $...
4
votes
3answers
375 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
52 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
41 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 $$ f(...
0
votes
1answer
23 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 \end{...
1
vote
1answer
27 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
1answer
39 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 $C^...
4
votes
1answer
239 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 $$3<\pi&...
1
vote
2answers
31 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 + a_n)^{1 / ...
2
votes
2answers
58 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
20 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 $x_0\le C$. ...
3
votes
3answers
40 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 $$X \setminus V = U \cap X $$ ...
1
vote
1answer
56 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
59 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?
5
votes
5answers
181 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 ...
0
votes
1answer
33 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
47 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
1answer
79 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 ...
2
votes
4answers
84 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
42 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
42 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
20 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 ...
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2answers
41 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
1answer
70 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 $f(x)=\int_{0}^...
4
votes
0answers
54 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
68 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 ...
0
votes
1answer
23 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
41 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
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0answers
33 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 - \...
0
votes
0answers
47 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
49 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
58 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
24 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 ...
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0answers
39 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 ...
1
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0answers
39 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 discounting-...
2
votes
0answers
36 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
88 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 $...
2
votes
1answer
46 views

Generalisation of Binomial Theorem, Leibniz Formula and similar theorems [duplicate]

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 ...
3
votes
2answers
161 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
2answers
74 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
45 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 ...
1
vote
3answers
261 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 & ...
9
votes
4answers
155 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 order $...
0
votes
2answers
77 views

Proof for $ \frac{2}{\pi}x \lt \sin{x} $ for $ x \in (0,\frac{\pi}2) $

The following is part of exercise 6.26.21 from Tom Apostol's Calculus Volume 1. I wonder if my proof is correct and if there is a simpler alternative proof. Prove the following by examining the ...
4
votes
1answer
103 views

A Scalar times the Zero Vector

I'm reading Linear Algebra Done Right by Sheldon Axler and the proof given in the book is the same as the one in the answer provided for this question. I tried to solve this before looking at the ...
6
votes
5answers
139 views

Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$

It's required to prove that $$\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$$ I managed to go about out it two ways: Assume it holds: $$\frac{1+\cos x + \sin x}{1 - \...
4
votes
0answers
113 views

A real symmetric matrix $A$ positive definite if all its eigenvalues are positive

Let $A\in \mathbb R^{n \times n},\ A^T=A$ and the eigenvalues $\lambda_i>0$. Then $v^TAv>0$ for every nonzero vector $v$. I know how to prove the above statement by using the fact that if $A$ ...
3
votes
1answer
53 views

Verify proof of $\sum_{k=1}^{n}{k^{-1/2}}<2\sqrt{n}$ for every $n\ge1$

The following is exercise 2.6.20 (c) from Tom Apostol's Calculus Volume 1, I'd like someone to verify my proof. I'm also interested in simpler alternative proofs. Determine if the following is ...