Questions tagged [alternative-proof]
Looking for a proof different from the standard proof(s) of a result
78
questions
1
vote
1answer
81 views
Generalizing Bottema's theorem
Can you provide another proof for the claim given below?
Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
54
votes
9answers
5k views
Proofs of theorems that proved more or deeper results than what was first supposed or stated as the corresponding theorem
Recently, I figured out that a colleague of mine has had published during recent years a proof of a theorem in which he was actually proving a deeper result which we both thought to be still open. ...
23
votes
2answers
866 views
Slick proof of Stirling's Formula?
In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$. This can ...
12
votes
6answers
2k views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
17
votes
0answers
497 views
Picture of Lambert's proof that $\pi$ is irrational?
With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
2
votes
0answers
250 views
Theorems with many proofs
Q. What are the characteristics of theorems that seem to invite (or possess) several or even many distinct proofs?
What I have in mind are examples such as these:
Proofs that there are infinitely ...
4
votes
2answers
354 views
Katz's proof of Cartier's (descent) theorem
I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
1
vote
0answers
59 views
What are some interesting applications of the Archimedean Property?
So a wile back I managed to prove the The Remainder Theorem starting from the Archimidean property and since then I've thought what could be other results which can be proved using it. But I haven't ...
15
votes
1answer
842 views
Topological proof that a Vitali set is not Borel
This question is purely out of curiosity, and well outside my field — apologies if there is a trivial answer. Recall that a Vitali set is a subset $V$ of $[0,1]$ such that the restriction to $V$ of ...
22
votes
4answers
2k views
What's the use of countable ordinals? (prompted by a remark of Tim Gowers)
In a typically lucid and helpful page of notes for students, A beginner’s guide to countable ordinals, Tim Gowers explains how the countable ordinals can be “constructed rigorously in a way that ...
-1
votes
1answer
162 views
Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$
When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $ Re(s)=\frac12 $ and $t >0$ ...
9
votes
0answers
382 views
Including alternative proofs
Suppose I have found two or even more proofs of a theorem and I prepare a paper on it. Is it considered to be a good practice to write down all of them? Or is it considered to be my job as an author ...
3
votes
0answers
294 views
Understanding a part of Friedberg’s Priority Argument Paper
This is Richard Friedberg’s original 1957 proof of the Friedberg-Muchnik Theorem, the origin of the ground-breaking priority argument. (The result proven is that there are two recursively enumerable ...
7
votes
3answers
1k views
Euler's rotation theorem revisited - Elementary geometric proofs
This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE.
Euler's Rotation Theorem, proved ...
2
votes
0answers
167 views
Proof of non-solvability of general equations of one unknown in elementary terms (finite terms)?
This question relates to the solvability of equations of one unknown in elementary terms (finite terms) according to Liouville and Ritt.
Elementary equations and closed-form solutions can be ...
14
votes
0answers
924 views
Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
3
votes
0answers
241 views
A closed set which is the closure of its interior points
This is a cross-post to MSE to this question
https://math.stackexchange.com/questions/3267497/a-set-which-is-the-closure-of-its-interior-points
There is a related question in MO:
Closure of the ...
6
votes
1answer
270 views
Severi Formula for Intersection Multiplicities
I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial.
Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...
4
votes
2answers
587 views
Hard implications that become easy with the right intermediate step
I’m interested in examples of theorems of the form “If $P$ then $Q$“ that were either unsolved or thought to require difficult arguments until someone came up with an $X$ for which “If $P$ then $X$” ...
17
votes
5answers
971 views
Closed-form expression for certain product
$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
9
votes
2answers
414 views
Deuring's result on elliptic curves. Any proof reference
I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\...
4
votes
2answers
309 views
Orthogonal Polynomials and Sturm Liouville operators
Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
0
votes
1answer
269 views
Fibers and Push-forward
Let $f\colon X\longrightarrow Y$ be a surjective morphism of projective varieties. Consider a non-singular irreducible projective curve $T$ and surjective morphism $p_Y\colon Y\longrightarrow T$. Then ...
28
votes
2answers
2k views
Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?
In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...
21
votes
8answers
3k views
Mathematical Writing: Proof Outlines/Overview in a Paper
While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.
I am writing a paper, in which we have a section called "Outline ...
3
votes
2answers
298 views
Alternate descriptions of finite fields
The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...
3
votes
2answers
266 views
A direct proof that every $r$-colored complete graph on $n=(r+1)m-(r-1)$ vertices has a monochromatic matching of size $m$?
Cockayne and Lorimer ("The Ramsey number for stripes" 1975) prove that in every $r$-colored complete graph on $n=\sum_{i=1}^rm_i+m_1-(r-1)$ vertices, where $m_1\geq \dots\geq m_r\geq 1$, has a ...
7
votes
1answer
249 views
If $M \otimes -$ is continuous, why is $M$ f.g. projective? Alternative proof
Let $R$ be a commutative ring and $M$ be some $R$-module such that $M \otimes -$ is continuous (i.e. preserves all limits). Then one can show that $M$ is f.g. projective.
One way to prove this is to ...
11
votes
1answer
425 views
What was Smith's proof of Smith's theorem on Hamilton cycles in cubic graphs?
In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.
He attributed the result to his friend CAB ...
0
votes
0answers
133 views
Simple proof for Riemann/Hurwitz ζ functional equation
For the purpose of formalisation, I am looking for a simple proof of the function equation of the Riemann ζ function or the generalisation thereof, the Hurwitz's formula for the Hurwitz ζ function.
...
0
votes
1answer
213 views
How can i justify this multinomial coefficient identity? [closed]
$$\sum_{ k\ge 1 } \sum_{ (s_1,...,s_k) } \binom h{ s_1 ,..., s_k }=\binom{m-1}{h-1}$$
where the second summation is taken over all choices of the numbers
$${ s }_{ 1 },...,{ s }_{ k-1 }\ge 0,\quad {s ...
53
votes
12answers
4k views
Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
26
votes
4answers
5k views
Where to publish a new proof of an old theorem?
A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it ...
11
votes
1answer
586 views
On Selberg's Proof of the Selberg Integral Formula
I'm attempting to read through Mehta's write-up of Selberg's proof of the formula for the Selberg integral formula given below:
$$
\begin{align}
\operatorname{S}_n(\alpha,\ \beta,\ \gamma) & = \...
3
votes
0answers
111 views
Colimits of algebras of an endofunctor
I try to understand a proof in Adamek-Rosicky's book "Locally presentable and accessible categories", Cambridge University Press 1994. In Corollary 2.75 (p. 121) it is proven that the category $\...
21
votes
5answers
3k views
Is Cauchy induction used for proofs other than for AM–GM?
The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.
-2
votes
1answer
241 views
Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [closed]
Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?
A professor said it to me a long time ago, but I don't have any references.
...
8
votes
2answers
825 views
Link between the hairy ball theorem and the fundamental theorem of algebra
I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
21
votes
3answers
2k views
Simplicity of alternating group $A_n$
I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "...
21
votes
2answers
848 views
Evaluating an integral using real methods
This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:
$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$
...
1
vote
0answers
132 views
Another proof of the comparison principle for PDEs by the theory of viscosity solutions
In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions.
So I am now at the end of my tether becasuse I can not ...
4
votes
0answers
145 views
Undecidability of the existential theory
Do you know if I can find the proof that the existential theory of $\mathbb{Z}$ with the structure of addition , divisibility and the relation $(\exists s \in \mathbb{Z})m=np^s$ is undecidable, ...
11
votes
2answers
1k views
Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?
This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\...
8
votes
1answer
353 views
Does the following $ C^{*} $-algebraic result have a purely algebraic proof?
While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * $-...
4
votes
1answer
281 views
Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$
This is related to a question on Math Stack Exchange.
Given a rational number $a>0$ and an $n\in\mathbb{N}$ such that $x^n - a$ is irreducible over $\mathbb{Q}$, it is known that every subfield of ...
8
votes
3answers
821 views
How to prove that two univariate polynomials are always algebraically dependent?
How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...
27
votes
7answers
4k views
Conceptual algebraic proof that Grassmannian is closed in Plucker embedding
I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...
2
votes
0answers
136 views
seek another proof of a result in Fourier analysis
It was proved on page 26 of this note the following result:
Let $\xi$ be an algebraic number that is not a root of unity, then there exists an $n_0\geq 0$ with the property that $$\beta=\sum_{j=-n^2}^...
14
votes
1answer
757 views
Gabber's proof of Br' = Br for quasiprojective schemes
In a note by deJong showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...
6
votes
2answers
337 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 originally posted this on M.SE, but I think it is probably a better fit here.)
I know that Choice ...
