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

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11
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275 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
8
votes
0answers
232 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
8
votes
0answers
309 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at ...
7
votes
0answers
74 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 ...
6
votes
0answers
129 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
106 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
6
votes
0answers
180 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
4
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81 views

$\operatorname{Ext}^n$: computation verification

I would like someone to verify my computation of $\operatorname{Ext}^n$. Problem: Let $p$ be a prime, $k$ a field of characteristic $p$, $G = \langle x \mid x^p = 1 \rangle$, $B = kG$, $S = k(1 ...
4
votes
0answers
101 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
4
votes
0answers
100 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r},$$ which can be proved combinatorically whether one particular element (among the $n$) ...
4
votes
0answers
93 views

Sign of some permutation.

I trying to find the sign of this permutation: $\left(\begin{array}{cccccccccc} 1 & 2 & 3 & \cdots & \cdots & \cdots & \cdots & \cdots & n-1 & n\\ 2 & 4 & ...
4
votes
0answers
286 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
4
votes
0answers
510 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
4
votes
0answers
384 views

Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
3
votes
0answers
58 views

Are there less trivial necessary and sufficient conditions?

Given an infinite set $X$ with the finite-complement topology, find a necessary and sufficient condition for a map $f:X\to X$ to be continuous. I came up with the condition that $\lvert ...
3
votes
0answers
96 views

Should a certain entire function be a polynomial?

Assume $f$ is an entire function such that $$\lim_{z\to\infty}\frac{|f'(z)|}{1+|f(z)|^2}=0,$$ then should $f$ be a polynomial? Picard's Theorem proves this instantly; which states: Let $f$ be a ...
3
votes
0answers
53 views

A Theorem On Compact Connected Metric Spaces by Stadje

I recently came across a surprising theorem, due to Wolfgang Stadje, a special case of which states that: Let $(X,d)$ be a compact connected metric space. Then there exists a unique real number ...
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 ...
3
votes
0answers
45 views

Proving a simple claim concerning order without using LaGrange's Theorem

For whatever reason, I am having trouble proving the following claim without using LaGrange's Theorem. Claim: Let $G$ be a group of order $n < \infty$. Then, $x^{n}=1$, where $1$ is the identity, ...
3
votes
0answers
108 views

Banach Alaoglu different proofs

While trying to prove Banach-Alaoglu theorem I noticed the differrent equivalent definitions of compactness. When I tried to find a proof of Banach-Alaoglu I found a proof in Pedersen Analysis Now and ...
3
votes
0answers
188 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
3
votes
0answers
55 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
3
votes
0answers
84 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
3
votes
0answers
137 views

Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
2
votes
0answers
58 views
+50

Minima and maxima of the 6th degree polynomial are not expressible in radicals.

Question: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima are not expressible in radicals. I have the following proof: the minima and maxima of a 6th ...
2
votes
0answers
49 views

proof about triangles with sides in arithmetic progression

$\require{cancel}$ The sides of a triangle are in an arithmetic progression with $k>0$ and $a>b>c$. Prove that ...
2
votes
0answers
27 views

The proof of the integral test using the contradiction method.

I am currently writing a short note about the proof techniques. I found a random theorem and wanted to write a proof by contradiction as an example. The theorem says The integral ...
2
votes
0answers
59 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 $0,5<\gamma< 0,6.$ ...
2
votes
0answers
45 views

A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
2
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0answers
58 views

Proof Verification for Putnam Problem [Alternate Solution] 1997 A4

I have come across an interesting problem from the Putnam 1997 test, question A4: Problem: Let $G$ be a group with identity $e$ and $\phi: G \to G$ a mapping such that $\phi(g_1)\phi(g_2)\phi(g_3) = ...
2
votes
0answers
33 views

Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...
2
votes
0answers
72 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...
2
votes
0answers
40 views

Let $p\gt 3$ be a prime number and: $\sum_{j=1}^{p-1}\frac{(-1)^{j}}{j} \binom{p-1}{j} =\frac{a}{b}\Rightarrow p^2\mid a$

I want to prove the following statement: Let $p\gt 3$ be a prime number and let: $$\sum_{j=1}^{p-1}\frac{(-1)^{j}}{j} \binom{p-1}{j} =\frac{a}{b}$$ Which $a,b\in \mathbb Z$ and $\gcd(a,b)=1$. ...
2
votes
0answers
49 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
2
votes
0answers
74 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
2
votes
0answers
93 views

Alternative proof that Harmonic sum is not an integer

In the post “Is there an elementary proof that ∑k=1/k is never an integer?” there is a simple and elegant proof by Bill Dubuque, who uses the prime 2 as a basis for his proof. I wondered whether a ...
2
votes
0answers
61 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 ...
2
votes
0answers
133 views

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
2
votes
0answers
57 views

Is this irrationality proof correct?

Consider a non-square integer $n$. If its square root was rational, then we would have $$\sqrt n=\frac{a}{b}$$ for some $a,b\in\mathbb{Z}$ and so $a^2=nb^2$. But this is impossible, because $n$ is ...
2
votes
0answers
35 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
2
votes
0answers
75 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
2
votes
0answers
143 views

Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
2
votes
0answers
44 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
2
votes
0answers
150 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
2
votes
0answers
214 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
2
votes
0answers
114 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
votes
0answers
101 views

Proof for the distributivity of multiplication over addition for a Binary Field

For the standard binary field $\mathbb{F}_{2} = \{0, 1\}$. Where the operations of addition and multiplication exist, and multiplication is equivalent to logical and, and addition is equivalent to ...
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0answers
21 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
1
vote
0answers
28 views

How to show that a function is continuous in the topology of weak convergence

Let $\Omega$ be compact, and let $\omega^* \in \Omega$ be arbitrary. Let $\Delta (\Omega)$ denote the set of all probability measures over $\Omega$, and endow $\Delta ( \Omega)$ with the topology of ...
1
vote
0answers
30 views

Solutions to the heat equation given certain growth conditions

Let $u(x, t)$ be the solution of the following Cauchy problem for the heat equation given by \begin{align*} u_{t}(x, t) - u_{xx}(x, t) &= 0 \quad (x, t) \in \mathbb{R} \times (0, T)\\ u(x, 0) ...