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

learn more… | top users | synonyms

4
votes
1answer
88 views

Wedge product of a direct sum and the Yoneda Lemma

In a comment to http://math.stackexchange.com/a/344851/58601, Martin Brandenburg suggests that one may prove the existence of the canonical isomorphism $\wedge^n(W_1 \oplus W_2) \to \bigoplus_{p+q=n} ...
5
votes
2answers
142 views

Difficulties understanding a proof of $\int_0^{\infty} \frac{\sin(x)}{x} \, dx = \frac{\pi}{2}$

I got a homework and I've trying to do this problem about 2 days, but I "lost my fight". So I turn to you. I have to prove that $$\int _0^\infty \frac{\sin (x)}{x} \, dx = \frac{\pi}{2}.$$ I can't use ...
2
votes
1answer
57 views

An integer square matrix of prime order has size at least $(p-1)\times (p-1)$

There's$\let\geq\geqslant\DeclareMathOperator{\GL}{GL}$ this exercise in my algebra course book: Let $p$ be a prime and $A\neq I$ an $n\times n$ matrix over $\mathbb Z$ such that $A^p=I$. Prove ...
1
vote
2answers
68 views

How can I prove this using number theory only

So this book I'm reading has this question: show that if $(a,n)=(b,n)=1$ the the equation $$ax+by\equiv c(mod( n))$$ has exactly $n$ different solutions. I was only able to prove it using ...
0
votes
0answers
27 views

Is the proof rigorous enough?

Proposition Let $n$ be a Natural Number and let $P(n)$ be a property pertaining to the Natural Numbers such that whenever $P(m{++})$ is true, $P(m)$ is true. Suppose that $P(n)$ is true. ...
2
votes
4answers
88 views

Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$

Show that $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$ Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$. ...
7
votes
0answers
69 views

Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)*(1-x)$ = the area of a ...
1
vote
3answers
67 views

Check proof that operator in unbounded please

I have to show that $f:\mathcal{C}'[a,b]\rightarrow \mathbb{R}$ with $f(x)=x'(\frac{a+b}{2})$ is unbounded. Here $\mathcal{C}'[a,b]$ (the space of continuously differentiable functions) is to be ...
1
vote
2answers
92 views

Show that $[2x]+[2y] \geq [x]+[y]+[x+y]$

Prove that $[2x]+[2y] \geq [x]+[y]+[x+y]$ whenever $x$ and $y$ are real numbers. The $[]$ symbol is the greatest integer or floor function. I have proved this fact by cases, but I stumbled upon what ...
6
votes
3answers
95 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
5
votes
1answer
84 views

A combinatorial proof of Wilson's Theorem

I am looking for a combinatorial proof of Wilson's Theorem. Something along the lines of this kind of proof. $\textbf{Combinatorial proof of Fermat's Little Theorem}$ First consider a $p$ -tuple and ...
2
votes
1answer
25 views

Point in a rectangle

$ABCD$ is a rectangle and $P$ is a point in the same plane. If the perpendicular through $C$ to $AP$ and the perpendicular through $B$ to $DP$ intersect at $Q$, prove that $PQ \parallel AD$. ...
18
votes
1answer
424 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an ...
2
votes
0answers
58 views

Alternative proof of de l'Hospital theorem

I see that the usual proof of de l'Hospital theorem involves the mean value theorem and is carried on in a case-by-case fashion. Could you point out some other proofs of de l'Hospital that are ...
10
votes
4answers
446 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
2
votes
3answers
183 views

Commutative artinian ring is noetherian

Suppose R is a commutative Artinian ring then R is Noetherian. I am aware of the proof which uses the idea of filtration. But I would like to prove this fact without that idea but haven't got far ...
2
votes
2answers
45 views

Disk/Washer method proof checking

This is a homework question, but i am just checking if what i am saying is correct, The question in the book states that A sphere of radius $r$ is cut by a plane of $h$ ($h < r$) units above the ...
2
votes
5answers
689 views

How to prove that there are infinitely many primes without using contradiction

How can I prove that there are infinitely many primes without using contradiction? I know the proof that is (not) by Euclid saying there are infinitely many primes. It assumes that there is a ...
1
vote
1answer
29 views

Identity Tranformation Proof- Is this enough to prove this statement?

Let {v$_1$,...,v$_n$} be a basis for a vector space V and let T:V$\to$V be a linear transformation. Prove that if T(v$_1$)= v$_1$,...,T(v$_n$)= v$_n$, then T is the identity tranformation on V. I'm ...
1
vote
3answers
44 views

Concise induction step for proving $\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$

I recently got a book on number theory and am working through some of the basic proofs. I was able to prove that $$\sum_{i=1}^n i^3 = \left( \sum_{i=1}^ni\right)^2$$ with the help of the identity ...
2
votes
1answer
92 views

A matrix-free way to find a fan basis of $V$?

Let $f:V\to V$ be a linear map, $\dim V =n$. A basis $( v_1, \ldots, v_n)$ of $V$ such that for all $j=1, \ldots,n$ the space $\text{span}(v_1,\ldots,v_j)$ is $f$-invariant is called a fan basis of ...
1
vote
0answers
28 views

Gradient points in the direction of greatest change

Can anyone provide me with an alternative, possibly more intuitive proof of this proposition? I'm confused with where $cos\theta$ has come from?
3
votes
1answer
74 views

Proof Verification of Schröder–Bernstein theorem

So I've spent some time studying the Schröder–Bernstein theorem, but I'm trying to do the exercise in "Naive Set Theory" by Paul Halmos regarding the theorem. The exercise is finding an alternative ...
1
vote
3answers
96 views

Evaluate $\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$ [duplicate]

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$ where $n\in\mathbb{N}$ This one is another intriguing question from my worksheet. I'm only allowed to use ...
3
votes
2answers
59 views

Prove every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.)

Let $A$ be a commutative ring with unity. Prove: Proof every maximal ideal of $A$ is a prime ideal (Hint: Use the fact that $J$ is a maximal ideal iff $A/J$ is a field.) In the question before ...
1
vote
1answer
96 views

Cutting chocolate diagonally

Given is chocolate with rectangular pieces of size $a \times b$. If cut diagonally, how many pieces will it be split into? If knife passes exactly by co-catenating we assume there is no damage to ...
1
vote
1answer
72 views

Application of Contraction Principle?

Show that there is a unique solution to the equation \begin{equation} \frac{df}{dx}=(f(x)+x)x \tag{*}\end{equation} for $0 \le x \le 1$ and $f(0)=0$. Clearly, $(*)$ is a first order linear ...
0
votes
2answers
83 views

Proof of Lindelof Theorem

I have been surfing the net to read the proof of the Lindelof Theorem: Let $U\in \mathbb{R}^n$ be open and $U=\bigcup_{\lambda \in \Lambda} U_{\lambda}$where $\Lambda$ is an index set, ...
1
vote
2answers
53 views

Proof of Bézout's identity - Cohn - CA p26

Given two integers $a$ and $b$, there exist integers $u$ and $v$ such that $$au+bv=1$$ if and only if $a$ and $b$ are coprime. Attempt Proof: Assume $a$ and $b$ are not coprime, e.g. $a=kb,k\in ...
0
votes
0answers
52 views

Show that this Hypergeometric Function equal to this gamma function

I have a question related to hypergeometric functions: Show that ...
1
vote
2answers
67 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [closed]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
3
votes
0answers
48 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
0
votes
2answers
49 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
0
votes
1answer
46 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
6
votes
2answers
61 views

$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials

Prove that there exists constant $C>0$ that for all $f \in P_n$ we have: $$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$$ Where $P_n$ is space of polynomials with ...
0
votes
0answers
22 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
7
votes
3answers
107 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
1
vote
1answer
28 views

Looking for a way to improve my inductive proof of a statement derived by Rolle's Theorem

The following problem is 'absolutely' clear: Problem: Let $f$ be continuous on the interval $[a,b]$ and $n$-times differentiable on $(a,b)$ and $f$ vanishes on $n+1$ points $x_0< x_1 < \dots ...
1
vote
1answer
62 views

Accounts of the proof of Fermat's Last Theorem

I would like to collect a set of references to pieces of Wiles' 1995 proof of Fermat's Last Theorem. Has anyone recompiled the proof into another paper? Are there any books or articles that describe ...
5
votes
2answers
378 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
4
votes
3answers
96 views

Prove $-1$ and $1$ are the only units in $\mathbb{Z}$

Prove $\mathbb Z^*=\{-1,1\}.$ I have a proof, which is posted as an answer below. I'm looking for an alternate proof.
1
vote
1answer
74 views

An alternative proof of the Tietze Extension Theorem(s)

Last summer I was working through a lot of Topology. I made it through the sections of my notes that dealt with separation properties, covering properties and continuous functions between spaces ...
3
votes
2answers
38 views

What is $\gcd(x,x+2)$?

Show that $\gcd(x,x+2)$ is $1$ if $x$ is odd and $2$ if $x$ is even. I am looking for a much simpler proof beside the one which I have posted.
1
vote
0answers
36 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
0
votes
0answers
39 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
35 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
2
votes
2answers
81 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
39 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
7
votes
2answers
113 views

An alternative proof for sum of alternating series evaluates to $\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$

How does one prove the given series? $$\sum_{n=0}^\infty\left(\frac{(-1)^n}{4n-a+2}+\frac{(-1)^n}{4n+a+2}\right)=\frac{\pi}{4}\sec\left(\frac{a\pi}{4}\right)$$ This series came up in xpaul's ...
1
vote
1answer
125 views

Proof of easy matching condition for Hall's theorem

I was studying with the recitations provided in the course 6.042 "Mathematics for Computer Science" of MIT OCW and while studying the proof of Hall's marriage problem, I understood the first proof ...