Questions tagged [alternative-proof]
Looking for a proof different from the standard proof(s) of a result
7
votes
1answer
140 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 $\...
3
votes
2answers
144 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
227 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 ...
29
votes
2answers
1k 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 $(\...
3
votes
2answers
168 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
226 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 ...
9
votes
0answers
158 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
112 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
184 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
$${ s }_{ 1 },...,{ s }_{ k-1 }\ge 0,\quad {s ...
52
votes
12answers
4k views
Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
22
votes
3answers
3k views
Where to publish a new proof of an old theorem?
A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it ...
11
votes
1answer
398 views
On Selberg's Proof of the Selberg Integral Formula
I'm attempting to read through Mehta's write-up of Selberg's proof of the formula for the Selberg integral formula given below:
$$
\begin{align}
\operatorname{S}_n(\alpha,\ \beta,\ \gamma) & = \...
3
votes
0answers
101 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 $\...
-2
votes
1answer
233 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
594 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
2answers
722 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
111 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
130 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
341 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
263 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
622 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 ...
23
votes
7answers
3k 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
133 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
634 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 ...
5
votes
2answers
261 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 ...
2
votes
2answers
316 views
Stronger theorem not resulting from proof analysis
Suppose that we proved $\varphi$ from a theory $T$. Often we ask whether or not we could have proved $\varphi$ with a weaker theory, to find out we usually analyze the proof and try to figure out ...
4
votes
1answer
249 views
Alternative proof for counting problem in graphs
Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.
Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of ...
22
votes
12answers
2k views
Instances where an existence result precedes the constructive version
The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. ...
11
votes
1answer
1k views
What is the protocol for making modifications to someone else's proof to prove something slightly stronger?
I have a need to modify Erdős' proof of the Sylvester-Schur Theorem to prove something stronger. See my working document at http://math.rudytoody.us/ or http://math.rudytoody.us/OppermannTheorem.pdf
...
29
votes
6answers
8k views
What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by ...
22
votes
3answers
3k views
Riemann mapping theorem for homeomorphisms
How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
0
votes
1answer
2k views
The Hexagonal Property of Pascal's Triangle
Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that:
the product of non-adjacent vertices is constant.
the greatest common ...
1
vote
1answer
265 views
Synthetic Proof for Ratio of Volumes of Concentric Spheres?
Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following.
$(1)$ Show that $$ \lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}...
8
votes
1answer
2k views
Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory
There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...
23
votes
9answers
12k views
Direct proof of irrationality?
There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form:
Suppose $a/b=\sqrt{2}$ ...
23
votes
3answers
4k views
Different way to view action of fundamental group on higher homotopy groups
There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
8
votes
5answers
2k views
Alternative proof of unique factorization for ideals in a Dedekind ring
I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
24
votes
3answers
7k views
Quick proof of the fact that the ring of integers of Q(\zeta_n) is Z[\zeta_n]?
I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of ...
