Highest scored questions
159,030 questions
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Equivalence of categeories-variants of definition [closed]
There is a notion of equivalence of categories which is the functor $F:\mathcal{C} \to \mathcal{D}$ such that there is a functor $G:\mathcal{D} \to \mathcal{C}$ such that $FG \cong id_{\mathcal{D}}$ ...
- 9,340
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1
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233
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First research papers in mathematical logic [closed]
Hello I'm a software engineer who just wants to start research in mathematics soon. I'm interested in the foundations and hence I'm picking mathematical logic. As I have never touched undergraduate-...
-5
votes
1
answer
233
views
Is Ham($S^2$) homeomorphic to SO(3)?
Is it true that Ham($S^2$,\omega), the group of Hamiltonian diffeomorphisms of the sphere $S^2$ with its standard symplectic structure $\omega$, is homeomorphic to SO(3)?
- 21
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1
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240
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Are SE's math sites a good place to check a theorem before sending it to an journal? [closed]
Say I prove an identity that's quite important; important enough that an uneducated person like myself could get it published (given the proof is correct, of course). What then is the best course of ...
- 91
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1
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231
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Are equinumerous size preserving models of a theory isomorphic?
If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then:
is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving ...
- 11.3k
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1
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787
views
An olympiad-like inequality [closed]
I found this problem in my old paper :
Let $f(x)$ be a convex function on $(0,\infty)$ such that $\forall x>0$ we have $f(x)>0$ and $n\geq 3$ a natural number then we have :
$$\Big(f(1)^{f(...
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votes
1
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390
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Can we blend ZFC with true arithmetic?
Can we have a consistent theory whose signature is $(=,\in, S, +, \times)$ standing for identity and membership binary relations and the successor total unary function, addition and multiplication ...
- 11.3k
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votes
1
answer
197
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Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]
It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance,
$0^ \left(
\begin{array}{cc}
\frac{1}{2} &...
- 10.1k
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votes
1
answer
515
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Rejection of a paper because of not suitable level of rigor without a single example of a mathematical error/imprecision [closed]
Question 1. A paper was rejected because of 'not suitable level of rigour' without a single example of a mathematical error/imprecision. What can the author do in this situation?
It sounds a matter of ...
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votes
1
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562
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On the Absolute Value of the Riemann Zeta Function on the Critical Line [closed]
Is there any sort of (closed form preferably, though if not, it's fine) function for $|\zeta(\frac12+it)|$ where $\zeta$ is the Riemann zeta function? Anything is welcome, so I can take it from there. ...
- 265
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1
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454
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Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...
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votes
1
answer
2k
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V.I. Arnold's high school problem [closed]
According to his interview to the Notices of the AMS, when Vladimir I. Arnold was 12 years old (in 1949) his teacher I.V. Morozkin, gave to his classroom (apparently 6th grade of a soviet primary ...
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1
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1k
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How to prove that there are no exponential object in a category?
I have proved that my category $\mathbf{Fcd}$ has small products. (Correction, it seems it has no terminal object that is empty product. I have overlooked this earlier in my proof.) No, it indeed has ...
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483
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For every proximity, does there exist a uniformity which generates this proximity?
For every proximity, does there exist a uniformity which generates this proximity?
This question may be generalized for different generalizations of proximities and uniformities. In fact I need it ...
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1
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88
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Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
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1
answer
151
views
On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
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1
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191
views
How do you prove the validity of this formula for $H(n)$? [closed]
I'm looking for a proof of the identity
$$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$
There is a generalization of this formula for $...
user130611
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votes
1
answer
270
views
Calculus based on pdf [closed]
Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
- 201
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1
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177
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Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed]
Below, we interpret divergent integrals as germs of partial integrals at infinity:
$$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$
where $\operatorname{bigpart}$ means taking ...
- 10.1k
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1
answer
112
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Graphs where every vertex can be its own color class
Let $G=(V,E)$ be a finite, simple, undirected graph. We say $G$ has the singleton coloring class property (SCCP) if for all $v_0\in V$ there is a vertex coloring $c:V\to\{1,\ldots,\chi(G)\}$ such ...
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2
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648
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Another question on Øksendal's book
Hi
On page 98 "Stochastic differential equations" of Øksendal, 6th edition,
the author writes that $$\int_{0}^{u}\Big(\int_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR_{s}\Big)dt=\int_{0}^{u}\Big(\...
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1
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79
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Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
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1
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174
views
It is known if $n$ is prime for all $n\leq N$ [closed]
This is more of a curiosity than a research question, but I could not find it answered anywhere. What is the largest $N$ for which the statement in the title is true? I have recently read that the ...
- 417
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1
answer
63
views
What is the pattern in this sequence of fractions? [closed]
1/2, 1/2, 5/8, 5/7, 17/22, 13/16,...
I notice the top numbers are all primes but could not find how that helps. At first I thought maybe it is similar to a Fibonnaci type sequence because of the first ...
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1
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185
views
Two inequalities in $\mathbb{R}$ [closed]
How to prove that for real numbers $a$ and $b$, the following inequalities hold?
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq 2^{2-p}|a-b|^p$,if $p\geq 2$
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq (p-1)\frac{|a-b|^2}{(|...
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votes
2
answers
224
views
Has the equation $p^3-q^2+2=2^3\cdot q$ infinitely many solutions for $p$ and $q$ prime? [closed]
Consider the equation:
$p^3-q^2+2=8\cdot q$
Has this equation infinitely many solutions for $p$ and $q$ prime?
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1
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184
views
a question of definite integral [closed]
1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$
2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$
please help me. I tried to use MATLAB but ...
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1
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315
views
How to find matrix representations of a boolean algebra? [closed]
Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...
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0
answers
85
views
Every smooth function contains a bijection [closed]
Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
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0
answers
83
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Research without context is no research? [closed]
I was wondering what is research in mathematics. Can it be considered research only the research in open problems or it can be considered research also the finding of new formulas, new classes of ...
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0
answers
126
views
Is a quiver variety a moduli stack of quiver representations?
As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
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0
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250
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Can Cardinality Theory capture ZFC?
Cardinality Theory "CT" is a theory of sets of cardinals and links between them, only sets of cardinals can be assigned cardinalities. The links are unordered edges linking cardinals, they ...
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1
answer
94
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optimization problem: find ingredients that add up to given totals [closed]
imagine you have a list of ingredients. Each ingredient has five traits, let's call them A, B, C, D and E. You want a mix of ingredients for which the sum of each trait is known.
for example:
...
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votes
1
answer
149
views
Lottery in O(1) per participant
Goal: implement in $O(1)$ per participant a lottery where each participant has some large number of tickets, and the best (e.g. least) one wins, without actually burning electricity in proportion to ...
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1
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409
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Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes
Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, "Stable Moduli spaces of High Dimensional Manifolds". For each characteristic class of oriented $...
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2
answers
2k
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Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
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2
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1k
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Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]? [closed]
I'm sorry if this isn't an appropriate question for MO. I've been reading here for a while, but I still haven't got a good grasp of what's a good question.Given a field A and the polynomial ring A[x], ...
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1
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614
views
Proof of formula for $\pi$ [closed]
The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
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2
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357
views
Are the zeros of $\zeta'$ exactly the zeros of $\zeta$? [closed]
The Riemann Hypothesis is known to be equivalent to the statement that $\zeta'$ (the derivative of the Riemann zeta function) has no zeros in the region $0< \Re(s) < 1/2$. By the functional ...
-6
votes
1
answer
139
views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by
\begin{equation}
f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))}
\end{equation}
is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
-6
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1
answer
403
views
Is ZFC set theory a satisfactory foundation for mathematics?
The conventional wisdom seems to be that it is, but there are problematic mismatches. Some are well-known: the use of first-order logic, and the many different implementations of the axioms, neither ...
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1
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331
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Numerical evidence that $\pi$ is not normal in base two [closed]
Confusion is possible, but we got numerical evidence against
popular belief about the normality of $\pi$ in base two.
According to wikipedia
a real number is said to be simply normal in an integer ...
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1
answer
175
views
Continuous function $f:\mathbb{R}\to\mathbb{R}$ with fixed size finite fibers [closed]
During a business meeting, I was trying to find a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $|f^{-1}(\{y\})| = 2$ for all $y\in \mathbb{R}$, and after some experimentation I found $$f:\...
- 48.9k
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1
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295
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Is the least ordinal containing all countable ordinals defined by a formula an element of itself? (Out of date, see below)
[UPDATE] The question has been updated and is mostly unrelated to the question above the line. The updated question is below the line.
The following argument supports a "yes" answer; is it convincing?...
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2
answers
554
views
Do degrees determine the chromatic number?
Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}_G(v) = \deg_H(\varphi(v)).$...
- 48.9k
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votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
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votes
1
answer
623
views
Is it true AI achieves silver-medal standard solving IMO problems (2024UK)? [closed]
I am always interested in the development of AI within mathematics, particularly in number theory. I read on this website that AI has achieved a silver-medal level in solving IMO problems, which has ...
-6
votes
1
answer
572
views
(not so) Spectacular $\ 7^3$
Indeed,
Conjecture Let $\ p\ $ be a prime, $\ n>1\, $ -- a natural number, and let the largest prime divisor $\ q\ $ of $\ s(p^n) :=\sum_{k=0}^n\,p^k\ $ satisfy $ q<p.\ $ Then $\ p^n\ =\ 7^3.$
...
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1
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779
views
De-Lifting Lemma, does it hold? [closed]
Let $\sigma$ denote an independent simultaneous substitution. Now I wonder if the following holds:
If $\Gamma \vartriangleright (A\ (\sigma\ \tau))\ \rho$ then there are $\psi$, $\phi$ such that $\...
Countably Infinite