Newest Questions
159,056 questions
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Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
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2
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1k
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Natural density of the set of simple numbers
Let us call $n>1$ simple if every prime power $q$ with $q-1 \mid n-1$ is a prime number. (Please let me know if there is already an established name for these numbers.) The simple numbers $\leq 100$...
- 63.1k
4
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1
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152
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How much can we "shrink" intersecting families
Motivation. An intersecting family is a collection of subsets ${\cal S}\subseteq {\cal P}(X)$ of a set $X\neq \emptyset$ such that $A\cap B\neq \emptyset$ for all $A,B\in {\cal S}$. The intersections ...
- 48.9k
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242
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Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,413
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58
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Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
3
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2
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428
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Inconsistency in determinability of the solution of a linear first order PDE
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \label{1}\tag{1}$$
with $u(x,0)=f(x)$. The solution of \eqref{1}, ...
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1
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Alternate definitions of compact and weak mixing extensions
In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition ...
- 6,313
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189
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Tangent space to the moduli space of abelian varieties
Letting $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. I saw without reference that the tangent space of $\mathcal{A}_g$ at a point $t$ could be canonically identified ...
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2
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1k
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On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 12.6k
2
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2
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276
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The complex $K$-theory of the Thom spectrum $MU$
The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
- 21
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99
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Relation of geometric and polyhedral convergence
By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
- 61
1
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67
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Volume comparison with integral curvature bounds which gives lower bound on volume on Riemannian manifold
In Riemannian geometry, the Bishop-Gromov theorem states that for an $n$-dimensional manifold $M$ with nonnegative Ricci curvature $Rc\geq 0$, then the volume quotient
$$
\frac{\text{Vol} B(x,r)}{\...
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74
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Distribution of an unordered set of random variables
Suppose we have a set of deterministic points $y_{1}, \dots, y_{m} \in \mathbb{R}^{n}$.
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let
$T : \mathbb{R}^{n} \times \Omega \to \...
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0
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91
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Definition of stable solution of elliptic PDE and the classification of the solution (as the critical points of energy functional)
My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here.
For example, for $$
-\Delta u=f(u) \quad \text { in } \Omega,
$$
we call a solution is ...
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167
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Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
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1
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74
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Conditions for absorption
Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $h \colon S \to [0,1]$ be a sub-harmonic or super-harmonic function. Assume that for all $\varepsilon >0$ ...
user514580
1
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126
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Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width? [closed]
Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have?
Thank you!
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365
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Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
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384
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What is the cardinality of liners of rank 4? Is it always equal 27?
Definition 1. A binary operation $\cdot:X\times X\to X$, $\cdot:(xy)\mapsto xy$, on a set $X$ will be called a line operation if
$$xx=x,\quad xy=yx,\quad (xy)x=y$$
for every $x,y\in X$.
Remark 1. ...
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178
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Ext for commutative Gorenstein algebras
Let $A$ be a finite dimensional commutative Gorenstein $K$-algebra over a field $K$.
Question 1: Is there an easy example of $A$-modules $M$ and $N$ such that $\mathrm{Ext}_A^1(M,N)=0$ but $\mathrm{...
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58
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Does uniform convergence on compacts of drifts in rough differential equation imply convergence of solutions?
Consider the RDE
$$dY^n=b_n(Y^n) \, dt+\sigma(Y^n) \, d\mathbf X$$
where $\mathbf X$ is a rough path, $\sigma$ is as smooth as you'd like and $b_n$ are Lipschitz. If $b_n\to b$ uniformly then Friz-...
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113
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What properties do graphs avoiding large regular subgraphs have?
Fix a positive integer $r$ and real $\delta \in (0,1)$.
Let $G$ be an undirected graph on $n$ vertices. Suppose that $G$ does not contain an $r$-regular subgraph on at least $\delta n$ vertices (i.e., ...
- 557
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104
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Are sequences going to +infty along a ultrafilter on $\omega$ essentialy increasing?
The question is essentially in the title: suppose you have a non-principal ultrafilter $p$ on $\omega$ and a sequence $(u_n)_{n\in \omega}$ of elements of $\omega$ such that the $p$-limit of $(u_n)$ ...
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205
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Number fields with prescriped prime decomposition
Pick your favorite prime $p$, as well as three positive integers $e,f,g$.
For each such choice, does there exist at least one Galois number field $K/\mathbf{Q}$ of degree $n=efg$ in which $p$ has ...
- 1,422
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141
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What is the maximal number of $(n-2)$-linear spaces lying on a degree $d$ hypersurface in $\mathbb{A}^n$?
Let $f$ be a degree $d\ge 2$ polynomial in $n$ variables with coefficients in $\mathbb{C}$, and consider the hypersurface $X=V(f)$ cut by $f$ in $\mathbb{A}^n$. Assume that $X$ contains finitely many ...
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Expected time to absorption for Markov chains
Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $ T = [ z \in S : \ P(z,z) = 1 ]$. Let $\tau = \inf [ k \geq 0 : X_k \in T ]$ and assume that $ \mathbb{E}^x ...
user514580
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0
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90
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Isogenous elliptic curves in characteristic zero and in characteristic $p$
Assume two elliptic curves (with CM), $E_{1}$ and $E_{2}$, are isogenous over a field $K$ of characteristic zero. Are the following two statements true?
(a) Their $V_{p}$ modules are $G_{K}$-...
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308
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Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
Richard Schwartz considered ...
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Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
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Examples of bilimits that aren't 2-limits, and some related questions
Recently I've received an email from Sori Lee about an earlier question I had asked, and we ended up with a number of questions about 2-limits and 2-bilimits which I couldn't quite answer, and decided ...
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181
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Kirby diagram of Enriques surface (as the "(1/2) K3 surface")
Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...
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Basic obstruction to anything like holomophic symmetric functions of infinitely many variables?
The totality of all holomorphic functions on the unit disk forms some sort of infinite-dimensional complex manifold, where the coefficients of the Taylor expansion might serve as coordinates for the ...
- 17.6k
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105
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Definition of term functions, in universal algebra
According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...
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514
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The category of groupoids vs the category of sets
Hopefully this question makes sense. As we know that Kan complexes are the "$\infty$-version" of groupoids for $\infty$-categories as groupoids for categories. On the other hand, the $\infty$...
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How is it possible for PA+¬Con(PA) to be consistent?
I'm having some trouble understanding how a certain first-order theory isn't just straight-up inconsistent.
Let $PA$ be the axioms of (first-order) Peano arithmetic and let $C$ be the following ...
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Chirality of octonion algebras
Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams.
This depicts two ...
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Is every irreducible representation of $Sp(m)$ a representation of $U(2m)$?
If $Sp(m)$ is the group of linear automorphisms of $\mathbb{C}^{2m}$ which fix a complex symplectic form $\omega$ and a quaternionic structure $j$ on $\mathbb{C}^{2m}$, then it is clear that $Sp(m)$ ...
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Is a compact set of extreme points contained in a compact face?
I have run into the following question in convex analysis, which I haven't found answered in the literature:
Suppose that $K$ is a "nice-enough" non-compact convex subset of a Hausdorff ...
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143
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$K_0$ group of an infinite factor
The following question was already posted in this link but I could not understand hints given in this post.
Let $\mathcal{M}$ be an infinite factor and my question is how to prove that $K_0(\mathcal{M}...
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330
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Billiard circuits in pentagons
A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...
- 151k
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268
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Formal series which are always zero
Let $(k, |\cdot|)$ be a complete field with a non-Archimedean norm, not necessarily algebraically closed. Define the Tate algebra as follows:
\begin{align*}
k \langle T_1, \dots, T_n \rangle = \{ \...
- 1,485
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4
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520
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"Upside-down unimodal" sequences in combinatorics
Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
- 24.2k
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98
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How large can a subset of computable reals, whose comparison function is computable, grow?
How large can a subset of computable reals, whose comparison function is computable, grow?
For example, rational numbers are computable reals, and its comparison function is computable. As another ...
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118
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Can you metrize "convergence in probability with respect to every probability measure absolutely continuous"
Let $X_n$ be a sequence of real valued (or more general) random variables and let $\mu$ be some Borel measure on $\mathbb R$ - not necessarily finite.
Suppose for every probability measure $P \ll \mu$ ...
- 1,914
2
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1
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377
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Maximal subgroups of projective general linear group
$\newcommand{\sc}{\mathrm{sc}}$All the groups below are algebraic groups over an algebraically closed field,
From Page $163$ of Malle and Testerman's book "Linear algebraic groups and finite ...
- 501
6
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1
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286
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Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
- 14.2k
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651
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What is the value of $j(2\sqrt{-163})$?
My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...
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2
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797
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Operations on the set of large cardinal axioms
Here's a question from a non-set-theorist, but a sometime-user of large cardinals.
The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages ...
- 64k
1
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0
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124
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Abelian varieties with endomorphism structure
Let me stick to principally polarised abelian varieties $X$ over $\mathbb C$.
I have seen several definitions of what it means for $X$ to have real multiplication by a totally real field $F$:
There ...
3
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0
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73
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Is the discrete logarithm equivalent to solving polynomial discrete logarithms?
Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$.
An interesting observation is that ...
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