Highest scored questions
159,034 questions
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2
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Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
-3
votes
1
answer
237
views
L. Gegenbauer's proof of Infinitude of Primes [closed]
I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that
L. Gegenbauer proved Infinitude of Primes by ...
- 233
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votes
1
answer
181
views
How to find the content of a sphone [closed]
I need to know how to find the contents of a sphone; however I have not been able to find an equation for it online. I noted that the equation for a cone is 1/3(h)(A base). So I thought that perhaps ...
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votes
1
answer
148
views
A proposition about power series
Is this proposition established?
Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series
$$p(x)=\sum_{n=0}^\infty a_nx^n,$$
$$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+...
- 13
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votes
1
answer
534
views
Inside the construction of the Frey curve
Consider the frey curve $E\mathrel: y^2=x(x-a^{p})(x+b^{p})$ with conductor $N =2\prod_{p|(abc)^{2p}}p $. Frey assume that $p$ does not divide $(abc)^{2p} $ so the level of the cusp form predict by ...
- 135
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votes
1
answer
101
views
Orbit size of an element [closed]
Let $H$ be a normal subgroup of $G$ and assume that $G$ is acting over a set $X$. Let $c$ be some element of $X$, is there any relationship among the size of the orbit of $c$ under the action of $H$ ...
- 195
-3
votes
1
answer
227
views
Is this sequence convergent? [closed]
suppose $\exists S \subset \mathbb{R}$ and a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x_0 \in S $ the sequence $x_{n+1} = f(x_n)$ converge to $x \in S$
now, let $\alpha \...
- 159
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votes
1
answer
234
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
- 42k
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votes
1
answer
269
views
Negative Dirichlet Pigeonhole Principle [closed]
From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...
- 13.9k
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votes
1
answer
63
views
How to show $\lambda_i \in \sigma_A(x)$?
Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
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1
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341
views
What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]
This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
- 11.3k
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1
answer
339
views
How to prove the combinatorial equality? [closed]
Please, help me to understand following convolution (or give a reference):
$$
\sum_{R=0}^N \binom{R}{r} \binom{N-R}{n-r} = \binom{N+1}{n+1}
$$
Why is it true?
Thank you!
-3
votes
1
answer
124
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How to find closed form expression for $\int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$?
I am badly stuck in some integration here and will appreciate any help out of it.
$$\int^\infty_0f(r) dr = \int^\infty_0 \frac{Ar}{1+Cr^\alpha} e^{-Br^2} dr$$
If I let $u = Br^2$, then I get
$$ = \...
- 113
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votes
1
answer
117
views
How much information is required to determine integers x,y,z [closed]
what is x+y+z is x,y and z are integers and xy-1 is divisible by z, yz-1 is divisible by x and xz-1 is divisible by y.
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votes
1
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195
views
Matching in a bipartite graph that saturates all vertices
Let $G=(S,T;E)$ be a bipartite graph without isolated vertices.
For every edge $e\in E$, e $=$ $st$ $($ s $\in S$, $ t\in T$) happens the inequality $dG(s)$ $>=$ $dG(t)$.
Prove that in $G$ ...
-3
votes
1
answer
333
views
product distinct prime factors of prime(n)-1 and prime(n)+1
The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with
distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are
such primes common? Can a value of ...
- 145
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votes
1
answer
230
views
Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]
Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.
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votes
1
answer
76
views
If a g-module is sum of irreps then is direct sum of irreps
In Infinite dimensional Lie algebras book by Victor G Kac, In prop.3.6 He proves that, any integrable $g(A)$ - module $V$ is direct sum of finite dimensional, irreducible, $h$ - invariant $g_{(i)}$ ...
- 1,269
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1
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246
views
Encyclopedia of Mathematics?(non-Alphabetical) [closed]
Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level.
And what's the difference between say, ...
- 109
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votes
1
answer
320
views
Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]
I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb R)$...
-3
votes
1
answer
237
views
Without the use of a calculator, how to calculate the logarithm of 2 and 3 in base 10 [closed]
Without the use of calculator how to calculate $\log_{10} ~2$, $\log_{10} ~3$?
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votes
1
answer
166
views
Relations between ordinary functor categories and higher categories [closed]
Definitions of ordinary functor categories and higher categories are considered with very similar algebraic and geometric methods such as graph structures and simplicial sets. I know the differences ...
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votes
1
answer
166
views
Decidable theorem or result that is not weaker than Tarski's theorem
I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem.
Could any one give reference or a simple introduction about such result known in their ...
- 3,104
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votes
2
answers
546
views
Hexagon Formed by connecting Trisections of triangle sides [closed]
Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
-3
votes
1
answer
124
views
Suppose I know $\int h(t) dt = H(t)$, is there a way to find $\int h(t)^N dt$?
I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum_{i=1}^N X_i}]$
Suppose the probability mass function is $f_X(x) = (1 - p)^{x - 1} p$
The moment ...
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1
answer
330
views
Is there a precise definition of "mathematical formula"? [closed]
In the Wikipedia article for Formula (which has no references), it is claimed that:
"The informal use of the term formula in science refers to the general construct of a relationship between given ...
- 3,359
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votes
1
answer
606
views
Simple group of order 504 [closed]
As we know,there are 9 Sylow 2-subgroup in the Simple group of order 504.Can anyone prove it only by Sylow's theorem?
(you can't use knowledge about PSL(2,8))
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votes
1
answer
472
views
singular locus of projective variety
Let $\phi: \mathbb{P}^1 \longrightarrow \mathbb{P}^9_{(Z_0,Z_1,Z_2,Z_3,Z_4,Z_5,Z_6,Z_7,Z_8,Z_9)}$ defined by $(t,s) \longmapsto (t^{18},t^{16}s,t^{14}s^2,t^{12}s^3, t^{10}s^4, t^8 s^5, t^6 s^6, t^4 s^...
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votes
1
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329
views
What do we mean by "Proving an algorithm"? [closed]
Hello,
Thanks in advance for answering my questions :)
The question is: What do we mean by "Proving an algorithm"?
I'm having a problem in where to start (if I want to use contradiction for example)...
-3
votes
1
answer
810
views
epsilon tube around continuous path in open set in R^n [closed]
This may have a simple answer, but I'm not getting anywhere.
If $U$ is an open set in $\mathbb{R}^n$ (usual topology), and $p:[0,1] \to U$ is a continuous path, from $x=p(0)$ to $y=p(1)$, with $x,y \...
- 155
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votes
1
answer
1k
views
An elementary question about the Krull dimension of modules [closed]
Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{...
- 1,207
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votes
0
answers
7
views
Request for Peer Review and Guidance to Refine Mathematical Manuscript
Dear Colleagues and Enthusiasts,
I am seeking peer review and constructive feedback on my manuscript titled "From Chaos to Order: A Study of Balance Chaos Mathematics and Absolute Mathematical ...
-3
votes
0
answers
28
views
Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures [closed]
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
- 1
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votes
0
answers
48
views
Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
- 415
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votes
0
answers
76
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
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votes
0
answers
64
views
Can both conditions about vertex degrees hold true in a planar graph? [closed]
I am working on a problem about planar graphs and trying to understand if two statements can both be true at the same time.
The problem states that for any planar graph with at least 3 or more ...
- 1
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votes
0
answers
157
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
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votes
0
answers
33
views
Bayesian Inference for Parameters Estimation in ARMA Model [closed]
In the usual sense, Maximum Likelihood Estimation is the common method for Parameter estimation in ARMA(p,q) model.
If I am looking to estimate parameters for ARMA(p,q) with Bayesian Inference, how ...
- 1
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votes
0
answers
137
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619
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votes
1
answer
162
views
Amenable non-Hausdorff groupoids
Is there any clear definition of amenable non Hausdorff groupoids? It should be possibly non-separable nuclear C*-algebras? Please let me know if there is any existing literature talking about this.
- 15
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votes
1
answer
116
views
Enumerative number theory term searching [closed]
Given $a,b$ positive numbers such that $gcd(a,b)=1$.Prove that there are infinitely many $n$ positive integers such that $x_n=a+nb$ sequence has many terms such that it is not divisible by any prime'...
-3
votes
1
answer
134
views
SU(2) and entangled particles [closed]
We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...
- 1
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votes
1
answer
201
views
Structure of the automorphism group of an L-rig
This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted.
Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto ...
- 6,982
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votes
1
answer
130
views
Equation $p\cdot q\cdot r=a^3-1+43\cdot (b^2-1)$
$p\cdot q\cdot r=a^3-1+43\cdot (b^2-1)$
p, q, r are primes.
a, b integers>0.
Is this equation a Mordell equation?
Has this equation infinitely many solutions?
-3
votes
1
answer
1k
views
Matrix sieve theorem [closed]
I have formulated the following conjecture:
Odd positive integer $ N=6n-1$ is a prime number iff neither of two diophantine equations
$6x^2+(6x−1)y=n$
$6x^2+(6x+1)y=n$
has solution. $x=1,2,3,..y=0,...
- 11
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votes
1
answer
441
views
Depth or Grade of an ideal
Let $R$$\subset$$ S$ be commutative noetherian rings,and $I$ is an ideal of $S$.
We now that $I$ is a $R-$module.
Do we have $grade_{R}(I)$ $\le$ $ grade_{S}(I)$?
Thank you!
- 359
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votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
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votes
1
answer
141
views
Approximate martingales by truncation
Let $(X,Y)$ be a $\mathbb R-$valued martingale. For any $\varepsilon>0$, is it possible to find another martingale $(X',Y')$ s.t. $X'$ and $Y'$ are supported on a compact set, and
$$
\mathbb E\big[\...
- 1,835
-3
votes
1
answer
262
views
An axiomatic system with a set of constants that form a complete ordered field [closed]
I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
- 201
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votes
1
answer
270
views
Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)
For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
- 101