Questions tagged [alternative-proof]

Looking for a proof different from the standard proof(s) of a result

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First known proof of the $2 \cdot n-2$ Theorem for the planar generalization of the Nine dots problem

Reading the Wikipedia page about the well-know Nine dots puzzle, I have just seen that the planar generalization of this problem would have been proven in 1956 (see Wikipedia: Nine dots puzzle), while ...
Marco Ripà's user avatar
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3 votes
1 answer
157 views

Question regarding proof of Littlewood-Paley

I posted this question on Math.SE where I unfortunately received no answers even after a bounty. As such, I am putting it here, in hopes to receive a response. For the proof of Theorem 6.1.6 in ...
newbie's user avatar
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1 vote
1 answer
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Proof of the Dunford-Pettis theorem in the context of probability spaces

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
rfloc's user avatar
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1 vote
1 answer
222 views

Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$

Throughout, $p$ will denote a prime integer, and $k$ an arbitrary integer. I have worked through V. Lebesgue's proof of quadratic reciprocity outlined by Keith Conrad in this MO thread, and I feel ...
D.R.'s user avatar
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3 votes
0 answers
191 views

Hodge symmetry without $\mathbb{C}$ [duplicate]

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that $$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...
Antoine Labelle's user avatar
3 votes
3 answers
475 views

Solving interval problems without outer measure

Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ? Problem 1 If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
Ross Ure Anderson's user avatar
-3 votes
1 answer
584 views

Analysis I, simpler proof of Tao's construction of the integers [closed]

In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers: In the language of set theory, what we are doing here is starting with the ...
HJE's user avatar
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12 votes
0 answers
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Proofs of Serre's theorem on simply-connected finite CW complexes

A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than ...
homotopy-enthusiast's user avatar
8 votes
1 answer
665 views

Measure without measurable sets

This question is a little on the softer and speculative side, so bear with me. Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
Amir Sagiv's user avatar
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4 votes
1 answer
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Are the irrotational and solenoidal parts of a smooth vector field linearly independent?

Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using Helmholtz decomposition that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = ...
MrPie 's user avatar
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-1 votes
1 answer
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Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]

I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer... I've been recently playing around with the linear recurrence sequences. Consider ...
Lab's user avatar
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27 votes
1 answer
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A simple proof of the fundamental theorem of Galois theory

Update. It's now on the arXiv. Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...
Martin Brandenburg's user avatar
3 votes
0 answers
103 views

Examples of "proof by generalising" [duplicate]

In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?). Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...
Chris Sangwin's user avatar
0 votes
1 answer
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Alternative proofs of Euclid-Euler theorem

What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
Ross Ure Anderson's user avatar
9 votes
1 answer
670 views

Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT). Yanofsky [0] has demonstrated several applications of LFPT to ...
jpt4's user avatar
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1 vote
1 answer
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Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below). It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be written as an ...
Tereza Tizkova's user avatar
34 votes
17 answers
5k views

Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
0 votes
0 answers
89 views

Verification of a certain computation of VC dimension

Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
dohmatob's user avatar
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9 votes
1 answer
291 views

Quadrisecants of knots

Recall that a quadrisecant of a knot is a line that passes thru four points on it. If the points appear on the line in the order $a$, $b$, $c$, $d$ and on the knot in the order $a$, $c$, $b$, $d$, ...
Anton Petrunin's user avatar
15 votes
4 answers
3k views

Collecting alternative proofs for the oddity of Catalan

Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
T. Amdeberhan's user avatar
40 votes
11 answers
4k views

Results with short, advanced proofs or long, elementary proofs

Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander ...
2 votes
1 answer
113 views

A question on the applicability Chebyshev inequality for sequence of random quantities

Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function. ...
dohmatob's user avatar
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4 votes
1 answer
321 views

Which step is wrong in the following simplification of Silver's forcing?

Theorem: If M is a countable transitive model of ZFC, and $\kappa$ is a supercompact cardinal in M, and $2^\kappa=\kappa^+$. Then there exists a forcing extension M[G] such that $\kappa$ becomes a ...
Reflecting_Ordinal's user avatar
7 votes
1 answer
1k views

An alternate definition of Sobolev space $W^{1,p}(\Omega)$ when $1<p\leq\infty$ and consequences

Suppose that we define the Sobolev space $W^{1,p}(\Omega)$ with $1<p\leq \infty$, where $\Omega\subset\mathbb{R}^d$ ($d\geq 1$) is an open set (not necessarily bounded), in the following manner. ...
UserA's user avatar
  • 597
1 vote
1 answer
143 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}...
Pedja's user avatar
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55 votes
9 answers
6k 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. ...
25 votes
3 answers
2k 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 ...
Franz Lemmermeyer's user avatar
16 votes
6 answers
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 ...
T. Amdeberhan's user avatar
17 votes
0 answers
713 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 ...
Timothy Chow's user avatar
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3 votes
1 answer
400 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
2 answers
802 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 ...
clarkkent's user avatar
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1 vote
0 answers
82 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 ...
fisura filozofica's user avatar
17 votes
1 answer
2k 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 ...
abx's user avatar
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24 votes
5 answers
3k 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 ...
Peter Smith's user avatar
  • 1,599
-1 votes
1 answer
175 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 $t >0$ ...
zeraoulia rafik's user avatar
11 votes
1 answer
564 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 the proofs? Or is it considered to be my job as an ...
3 votes
0 answers
344 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 ...
Keshav Srinivasan's user avatar
6 votes
3 answers
2k 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 ...
Ross Ure Anderson's user avatar
2 votes
0 answers
182 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 ...
Jürgen Will's user avatar
13 votes
0 answers
1k 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 ...
display llvll's user avatar
4 votes
0 answers
1k 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 ...
Curiosity's user avatar
  • 293
6 votes
1 answer
319 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\...
Vincenzo Zaccaro's user avatar
4 votes
2 answers
611 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
5 answers
1k 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(\...
clathratus's user avatar
11 votes
2 answers
1k 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 $\...
quantum's user avatar
  • 489
4 votes
2 answers
583 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 $...
Amir Sagiv's user avatar
  • 3,554
0 votes
1 answer
408 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 ...
Vincenzo Zaccaro's user avatar
29 votes
2 answers
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 $(\...
Hans-Peter Stricker's user avatar
3 votes
2 answers
354 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 ...
Christopher King's user avatar
3 votes
2 answers
330 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 ...
Louis D's user avatar
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