Highest scored questions
159,029 questions
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Definition of tautological vector bundle [closed]
Could you please give a detailed definition (or construction)of tautological vector bundle of Grassmannian over arbitrary base scheme? Thank you in advance!
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votes
2
answers
303
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Constructing a new manifold with a germ of manifold [closed]
Given a germ of manifolds and compatible Riemannian metrics, can we construct a new Hausdorff manifold using the exponential map?
A germ of manifolds at a point $m$ is a series of manifolds $U_i$ ...
user176431
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votes
1
answer
203
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Is there a procedure to derive models from axiomatic systems? [closed]
Is there a systematic procedure to construct a model of an axiomatic system from the system itself?
For example given the abstract postulates of a ring we can show that the integers satisfies them ...
- 4,862
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1
answer
147
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Finitely generated module [closed]
If a finitely generated module $M$ injected in a free module, then would the image of $M$ be a free finitely generated module?
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1
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structure of finite polycyclic groups [closed]
We know that every finite nilpotent group is written as a direct product of its Sylow subgroups.
My question is : can we write finite polycyclic groups as a direct product of some subgroups?
if the ...
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Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?
In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case ...
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1
answer
97
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How to solve a Diophantine equation in six variables? [closed]
Find all the integer solutions $a, b, c, d, e, f$ satisfying the equation $a^2b^3 + c^2d^3 = e^2f^3$.
Note that if we prove that there are no such solutions with the condition $\text{gcd}(ab, cd, ef) ...
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1
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Reference request in optimal stopping [closed]
I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
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1
answer
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How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently? [closed]
I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where
$y$ is a $n \times 1$ vector.
$A$ is a $m \times n$ matrix where $n \gg m$.
$B$ is a $m \times m$ symmetric positive definite matrix; the ...
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Isomorphic functors and their images [closed]
Is the following a true statement?
Let $F:\bf{A}\to\bf{B}$ be a functor, and let $\bf{C}$ be a full subcategory of $\bf{B}$ with inclusion functor $I:\bf{C}\to\bf{B}$. If $G:\bf{C}\to {A}$ is a ...
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categorification of the $\Gamma$ function? [closed]
Is there a combinatorial or information-theoretical meaning to $\Gamma(\frac{1}{2})=\sqrt{\pi}$ ?
The identity $\Gamma(n+1)=\int_0^\infty x^{n+1} e^{-x} \, dx = n!$ suggests something is being ...
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Expression for a complex summation involving factorial [closed]
It is known that $\sum_{k = 0}^{n } {n \choose k}(k!) = \lfloor e \cdot n! \rfloor $ But is it known what $\sum_{p = 0}^{n} \sum_{q = 0}^{n - p} {n \choose p}{{n - p} \choose q} p! \cdot q! \cdot (n-p-...
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The difference between Hilbert Scheme and Chow Scheme
I am confused by Hilbert Scheme and Chow Scheme. Whenever you have a point in hilbert scheme, take its fiber in the universal family and take its fumdamental class, we get a point in Chow Scheme; and ...
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patitions of the number n [closed]
I'm having difficult with the following question :
A. Show that the number of partitions of n where in each one of them the even numbers appears at most once equals to the number of partitions of n ...
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Integral closure of an ideal [closed]
Let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. Does exist a finitely generated ideal $J$, such that $J\subset I$ and $a_i\in J^i$ for all $i=1,\...
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About planar graphs? [closed]
Can any non-planar graph with n minimum crossing points be 'drawn' on a sphere so the vertice and edge sets are the same and it has a connected subset A with minimum r crossing points and a disjoint ...
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on the solvable groups of order $p^aq^b$ [closed]
We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...
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Is the product of closed subgroups in a locally compact group locally compact? [closed]
Let $A$ and $H$ be closed subgroups of a $\sigma$-compact locally compact group $G$. Assume further that $A$ is abelian. Is the group $AH$ locally compact subgroup in the subspace topology?
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Is this a conclusion group for a new fundamental geometry problem? [closed]
Let σ(n) represent all possible values of the types of different lengths of segments connected to each other among n points in the definition,such as in the plane,σ(3)=(1,2,3),σ(3)min=1,in the ...
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0
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212
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Can a mathematics research paper have just propositions & corollaries? [closed]
I am writing a research paper, in which I am proving some properties of new convolution operation $\star$ for some transform, like linearity, associativity, commutativity,distributivity, shift ...
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Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
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Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
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Divisors of n and n + 1
Suppose $a$ is a proper divisor of $n$ (where $n$ is a positive integer), and $b$ a proper divisor of $n + 1$.
Is there a general criterion (or general property of $n$) which enables one to conclude ...
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Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
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Can ZFC be interpreted in this infinitary logic theory?
Working in language $\mathcal L_{\Omega^+,\Omega^+}$ where $\Omega$ is the first strongly inaccessible cardinal. If we add a primitive partial $\Omega$-ary function $F$ and a primitive constant $\...
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In surreal numbers, why log-atomic numbers are not EL-numbers?
In surreal numbers, the log-atomic numbers are those numbers that can be obtained from $\omega$ and its powers via iterated logarithm or exponential function.
At the same time, Timothy Chow's EL-...
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A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
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Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
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Charpit's method and a nonlinear PDE
I have the nonlinear PDE
$$p^2 + 2q = x$$
with the initial condition $u(0, y) = -y^2$, and $y > 0$.
Here's what I have done so far:
I defined the function $F$ to be equal
$$F(x, y, p, q, u) = p^2 + ...
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1
answer
220
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What is the computationally simplest way to universally index the set of simple graphs?
If given a simple, integer-labeled, but not necessarily connected, graph $G := (V,E)$ consisting of at least one vertex, i.e. $\lvert \rvert V \lvert \rvert \geq 1$, then is there a function to ...
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1
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Regular graph such that $2$ distinct vertices have same neighborhood set [closed]
If $G=(V,E)$ is a simple, undirected graph and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$.
Is there an integer $k>1$ and a connected $k$-regular graph $G=(V,E)$ such that there are $v\neq ...
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How many integers between $\left[2^{2^k}, 2^{2^{k+1}}\right]$? [closed]
Suppose $k$ and $n$ are natural numbers such that $2^{2^k} \lt n \lt 2^{2^{k+1}}$. I am curious how many integers are there in the interval $\left[2^{2^k}, 2^{2^{k+1}}\right]$ in terms of $n$.
I need ...
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Existence and uniqueness of solutions for a system of first order PDEs [closed]
Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...
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2
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An interesting, simple, sequence - surprised to find little material. [closed]
I've been considering this sequence:
$$1,2,3,6,12,24,48,96,192,...$$
I've generated the sequence from the rule
$$V_n=\sum_{0\leq i \lt n} V_i$$
$$V_0=1; V_1=2V_0=V_0+V_0$$
What interests me most, ...
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About the definitions of well-foundedness in this extension of NFU that interprets ZFC?
Lets see how the world of sets could look like from the perspective of $\sf NFU$. So, here we work within the first order language of set theory, with the following extra-logical axioms:
1. Quine atom:...
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1
answer
199
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Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
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1
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310
views
Does Goldbach's Conjecture imply a stronger statement about prime pair distributions?
The discussion on F. Brunault's question and extensive calculations suggest:
$$ \qquad f_{n+1} < 4p_n \quad\text{for}\quad n > 0$$
where $f_n$ is the Frobenius number of the numerical semigroup ...
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What are the properties of 3-dimensional split-complex numbers?
I have often encountered claims that 3-dimensional numbers are impossible. But
it seems to me that $\mathbb{R}^3$ with Hadamard multiplication should in fact behave quite similar to split-complex ...
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2
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If $p^k m^2$ is an odd perfect number with special prime $p$, then $p^k < 2am$ for some positive integer $a < m$ [closed]
(Preamble: Andy Putman asserts, in the comments, that MO policy prohibits "requests to check completeness of proofs". I have therefore trimmed down my original question to the bare ...
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Kadison-Singer problem
The Kadison-Singer problem is the following statement:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition $(...
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1
answer
284
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Does the set of Goldbach numbers have positive density?
We say that an integer $n > 1$ is a Goldbach number if $n$ is the sum of two primes. The famous Goldbach conjecture says that every even integer greater than $2$ is Goldbach.
Consider the following ...
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1
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754
views
Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]
The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
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3
answers
4k
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Gaussian curvature and mean curvature. [closed]
Define Gaussian curvature for a nonorientable surface. Can you define mean curvature for a nonorientable surface?
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$23005\cdot (2^n-1)\cdot 2^n +1=p^2$ [closed]
Consider:
$23005\cdot (2^n-1)\cdot 2^n +1=p^2$ , where n is an natural number and p a prime.
I conjecture that $p=9631$ is the only prime satisfying the above equation for $n=6$.
The curious fact is ...
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2
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555
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When did we stop the challenges between two mathematicians? [closed]
In this video(*) of Veritasium, you can see a challenge(**) between two mathematicians : Tartaglia and Fior, during the Renaissance in Italia.
When did we stop the challenges(**) between two ...
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1
answer
472
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Is this a valid definition of Euclidean geometry? [closed]
In trying to understand what actually constitutes a "geometry" I came across many definitions of Euclidean spaces and geometries. Euclidean space is defined as an affine space with an inner ...
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1
answer
1k
views
Does $\sum_n \frac{\sin n}n$ converge absolutely? [closed]
Using Dirichlet's test, one can prove that $\sum_{n\geq 1} \frac{\sin n}n$ converges. Does it converge absolutely?
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1
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Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
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votes
1
answer
313
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Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]
In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
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Is smooth stack separated?
Let $X$ be a smooth algebraic stack. Is it true that $X$ is separated? I was searching on google but could not find the answer. Please provide a reference.
user111251