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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273 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
298 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
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 ...
7
votes
0answers
182 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 ...
6
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128 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
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105 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 ...
5
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178 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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96 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
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0answers
89 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
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91 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
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457 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
364 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
92 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
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0answers
52 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
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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
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0answers
56 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 ...
3
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0answers
44 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
174 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
3
votes
0answers
99 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
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0answers
180 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
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0answers
53 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
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0answers
134 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
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0answers
62 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
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0answers
38 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
33 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
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0answers
63 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
91 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
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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 ...
2
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0answers
132 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
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
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0answers
69 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
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0answers
134 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
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0answers
42 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
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0answers
142 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
183 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
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0answers
112 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
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0answers
100 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 ...
2
votes
0answers
251 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 ...
1
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0answers
27 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
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0answers
28 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) ...
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0answers
52 views

Looking for an alternative solution for the mutilated chessboard problem

Given a mutilated chessboard where two diagonally opposite squares are missing (the unmutilated version of it has $64$ squares), and given $31$ domino pieces, is it possible to cover the entire ...
1
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0answers
30 views

Prove without method of contradiction that there exists a real number less than every positive real number that is positive

This question was asked before for proof by contradiction and which got me into thinking whether i could prove it without using a contradiction Original problem statement is here Prove by ...
1
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0answers
131 views

Partition of unity of Lagrange polynomials

Given a sequence of increasing real numbers $T = \{t_1 < t_2 < ... < t_{d+1}\}$, the $d+1$ Lagrange polynomials $L_i(t)$ of degree $d$ are defined as $$L_i(t) = \prod_{\substack{1\leqslant j ...
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170 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
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36 views

Prove that the maximum in absolute value of any monic real polynomial of n-th degree on [-1, 1] is not less than $\frac{1}{2^{n-1}}$

One solution is: Note that equality holds for a multiple of the n-th Chebyshev polynomial $T_{n}(X)$ The leading coefficient of $T_{n}$ equals $2^{n-1}$, so $C_{n}(X) = \frac{1}{2^{n-1}}T_{n}(X)$ is ...
1
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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?
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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 ...
1
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0answers
30 views

Riemann-Stieltjes Integrals with $n$ discontinuities (Proof Review)

First of all is the proof legitimate? If so, then are there other methods to achieve the same result that are neater than mine? Let $\alpha : [a,b] \to \mathbb{R}$ be a step function with ...
1
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0answers
42 views

Abelian Group (Alternative Proof)

Is there an alternative method to prove $(ab)^{2}=a^{2}b^{2}$ for all elements $a,b \in G \implies$ $(ab)^{-1}=a^{-1}b^{-1}$ for all elements $a,b \in G$. then the one I give below? Let $G$ be ...