# All Questions

149,977
questions

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### Alternative construction for the infinite loop space (?)

There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...

1
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0
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14
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### Injective hulls of metric spaces

In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...

0
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0
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4
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### Reference request for dinatural transformations arising from free Cartesian closed categories

Let $g_n$ be a discrete graph with $n$ nodes and $\operatorname{F}$ the free functor of the adjunction between the category of graphs and the category of Cartesian closed categories and functors, as ...

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0
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12
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### Module formulation of chain complex in covering

Given a (regular 2D) CW complex $K$, with cellular chain complex $(C(K,R),\partial )$, ($R$ a field), let $K_H\to K$ be a regular cover with group of deck transformation $G=\pi_1/H$. The cellular ...

3
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34
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### Linear equation among divisors of a positive integer

Let $a,b,c$ be co-prime positive integers and let $\theta \in (1/4, 1/3)$ be a real number. For each positive integer $k$, does there exist a positive integer $N$ such that the linear diophantine ...

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0
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30
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### Local systems on $\mathbb P^1$ and on the formal punctured disc

Consider the projective curve $\mathbb P^1$ over a finite field $k$.
Consider $\ell$-adic local systems $E$ on $\mathbb P^1\backslash \{0,\infty\}$ such that
a) $E$ is tame at $\infty$
b) The ...

0
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0
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12
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### Smooth surface with boundary equal to $N$ distinct lines

In a paper I am reading the following obvious remark is made: if $p_1$,...,$p_N$ are distinct points on $\mathbb{S}^1$ and if $M\subset \overline{B}_1$ is a smooth embedded $1$-dimensional manifold ...

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29
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### 3-functoriality of the lax Gray tensor product

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...

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81
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### Kan extensions in Grothendieck school

Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...

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0
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23
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### Knot invariants in WZW CFT via Holographic Principle

In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...

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72
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### Is the Rado graph the unique countable graph that has all finite graphs as induced subgraphs?

I understand that since the Rado graph is the Fraïsse limit of the class of finite graphs, it is the unique homogeneous graph with this property. Is there another graph not isomorphic to the Rado ...

2
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50
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### Unitary representations of $\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})$

Does the left, unitary action of $\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{C})$ on $\mathcal{L}^2(\text{H}^{+}_{3})$ integrate to its Lie group, i.e $\text{SL}(2,\mathbb{C})\times \...

1
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1
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53
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### How to generate a random function with conditions?

The background is as follows:
I consider the following differential equation
$$\phi_{xx}+u\phi=\lambda \phi,\ \ \lambda=-k^2$$
where $u=u(x),\ \phi=\phi(x,\lambda)$, $\lambda$ is the spectral ...

2
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0
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61
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### Is there a category theoretic definition of a cryptographic commitment scheme?

I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...

0
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0
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15
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### From average degree to a highly connected subhypergraph

I'm looking for a result in $k$-uniform hypergraphs analogous to the following result for graphs, due to Mader:
Every graph of average degree $4r$ has a $r$-connected subgraph.

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53
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### A question about cohomology with local coefficient

Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...

0
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0
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117
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### Hilbert irreducibility and the inverse Galois problem?

A few years ago I wrote a note about Hilbert irreducibility and the Galois problem, which I recently re-uploaded here. It has not been peer-reviewed, so the following should be taken with a caution ...

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1
answer

50
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### derived completion and flat base change

Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings.
We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete.
For a ...

0
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0
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27
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### Kernel perfection in some powers of cycles

Suppose I orient the edges of the power of cycle graph $G=C_n^k$ where $n=16$ and $k=4$ in such a way that all the generated cycles by the elements $1,2,3,4$ are given the standard lexical orientation....

2
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1
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90
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### Three-dimensional analogues of Hirzebruch surfaces

There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....

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43
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### Random walk with same directions and different step sizes

Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$.
Consider the following two random ...

0
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2
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259
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### Groups $G$ with only non-trivial quotient isomorphic to $G$

If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we ...

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45
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### Expansion with Laguerre polynomials

Let $L_{m}^{(\alpha)}\left(x^{2}\right)$ be the Laguerre polynomial of degree $m$ and order $\alpha$. Put $\varphi_{m}^{(\alpha)}(x)=e^{-\frac{x^{2}}{2}} L_{m}^{(\alpha)}\left(x^{2}\right) $.
It's ...

2
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0
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28
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### Truncating the high degree part of a positive boolean function doesn't change the distance to positive functions too much

Given $n\in\mathbb{Z}^{+}$, suppose $f:\{-1,1\}^n\to[0,1], $ then $f$ has a Fourier expansion: $f(x)=\sum_{S\subseteq[n]} \tilde{f}(S)x^S,$ where $x^S=\Pi_{i\in S}x_i$ , $\tilde{f}(S)\in\mathbb{R}$. (...

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24
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### Understanding simple point processes

Background
I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes.
A simple point process is a random variable whose ...

0
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0
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21
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### Dimension inequality for primary groups

Let $p$ be a prime number and $G$ an abelian group.
The group $G$ is said to be $\textbf{primary}$ if every element of $G$ has order power of $p$.
For every natural number $n$, we define
$$\ker(p^n)=\{...

6
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2
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149
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### Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation

$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...

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73
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### Zero loci of sections of wedge product of bundles

Let $V$ be a $\mathbf{C}$-vector space of dimension $n$, and consider the Grassmannian $G:=Gr(2, V)$ of 2-dim subspaces of $V$. Then we have the tautological subbundle $E\subset V\otimes \mathcal{O}_G$...

1
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1
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91
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### Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...

1
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0
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96
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### Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$

$\DeclareMathOperator\GL{GL}$In this question, representations means finite-dimensional complex representations.
Fix some $n,m \geq 2$ and some prime $p$. I'm interesting in representations $V$ of $\...

5
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91
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### Is there a sharper Golden–Thompson inequality?

For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...

10
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0
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192
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### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...

0
votes

1
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27
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### How to handle the evaluation of functions on staggered ghost nodes?

I have a convection-diffusion-reaction steady state PDE in the form
$$
\frac{\partial C}{\partial x} = \frac{1}{u_0(x)}\left(\frac{\partial}{\partial z} \left( \mathcal{D}(z) \frac{\partial C}{\...

0
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0
answers

45
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### Moser iteration epsilon-regularity for non-linear system in general dimension

I am attempting to prove the following result in general dimension $n$. Given $(M^n,g)$ a Riemannian manifold with $\mathrm{Ric}_g \geq -(n-1)$ and $\mathrm{Vol}_g(B_1(x)) \geq v > 0$ for all $x \...

4
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4
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506
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### Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...

0
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0
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31
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### Sum of Skellam-distributed number of random variables

Suppose $X_i$ are i.i.d, and $N \sim \text{Skellam}(\mu_1$, $\mu_2$). Is it possible to find a closed form for the p.d.f of $S_N$, defined by $S_N = X_1 + \cdots X_N$ when $N \ge 0$, and $S_{-N} = -...

2
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0
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44
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### Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer

In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...

-1
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0
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### $\{X_{n}\} $ simple random walk, find $P(\max({X_{1} \dotsc X_{n}})=4)$ [closed]

The obvious decision is using total probability. We count the probabilities of all possible variants. We can achieve point 4 the first time on the 4th step, which means:
$$P(X_{4}=4)P(X_{5}\le4,\dotsc,...

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0
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206
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### Is this a known pythagorean triple generator? $a=r^2+ rs , b= s^2/2 + rs , c = s^2/2 +r^2 + rs$? [closed]

This produces all pythagorean triples when $rs$ is an integer.
$r=k_1^{1/2}$
$s=(2k_2)^{1/2}$
for some $k_1, k_2 \in \mathbb{Z^+}$.
(9, 12, 15) can be generated when $r= 3^{1/2}$ and $s = 12^{1/2}$
...

3
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0
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56
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### Are there fast rank and unrank algorithms for integer vectors under the action of a permutation group?

We are distributing $m$ indistinguishable balls in $k$ numbered boxes $S=\{1,2,\ldots,k\}$. A distribution is a tuple of nonnegative integers $a=(a_1,\ldots,a_k)$ whose sum is $m$. We also have a ...

1
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0
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63
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### Conformal laplacian on asymptotically flat manifolds with boundary

Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...

1
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1
answer

75
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### Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...

3
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1
answer

77
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### What are the internal adjunctions in the bicategory $\mathsf{Span}$?

Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...

2
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1
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108
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### Pontryagin product on the homology of cyclic groups

Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...

1
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0
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29
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### Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...

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0
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29
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### Define this set of power curves bounded above by a given geometric curve and below by the x-axis

Let $f(x) = 1/(1-x)$, with $x$ a real number in $[0, 1]$ and $f(x)$ a real number in $[1, \infty]$. This is clearly part of a geometric curve, as well as part of a branch of a hyperbola with ...

2
votes

2
answers

143
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### Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$

Question:
is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$?
I'm convinced it must be true, but can't remember having seen ...

1
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0
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60
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### Adjunction correspondence for Blow up of double point

Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$.
Why holds for ...

5
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2
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301
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### Countable chain condition in topology

A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...

4
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0
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95
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### Elementary equivalence for rings

Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...