Newest Questions
159,065 questions
1
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Trans-universality for finite-dimensional Banach space
In addition to a specific problem Trans-universality for finitely generated groups, I posted also its general form. It should not hurt to provide another special case:
QUESTION: does there exist a ...
- 4,786
0
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0
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181
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Can a model of "true computation" exist? What would be its consequences?
Analogous to the model of True Arithmetic, the model of "True Computation" is defined to be the set of all true first-order statements about Turing machines i.e. answers to elementary ...
3
votes
0
answers
125
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Does symmetric product functor preserve fibrations?
I know that the symmetric product is a functor, cf: https://en.wikipedia.org/wiki/Symmetric_product_(topology)#Functioriality.
My question is, does it preserve fibrations in the category of ...
- 506
9
votes
1
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227
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Does $\infty$-categorical localization commute with taking directed fibered products?
Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ ...
- 1,349
1
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0
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127
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an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
1
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155
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Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 501
1
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0
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84
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Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 2,178
11
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2
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1k
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When does a group act effectively and holomorphically on some Riemann surface?
Given a Riemann surface $X$, we have some fairly standard methods for identifying which groups $G$ admit an effective and holomorphic action $G \times X \to X$. For instance, some fairly elementary ...
- 609
2
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2
answers
1k
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Existence of Brownian motion using Kolmogorov's extension theorem
When wishing to prove existence of Brownian motion, most authors take the route of defining the desired finite dimensional distributions, showing the family of defined finite dimensional distributions ...
- 123
12
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1
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902
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Positive 4-form
Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 45k
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104
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Lower bound for couples of disjoint sets in some partitions of the power set
Originally posted on MathStackExchange but without answers.
Consider partitions $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_n} \}$ of the powerset without the empty set element $Q = \mathcal{P}([n])...
- 700
3
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0
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71
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Question on the Tits cone of an irreducible affine Coxeter group
Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be the system of simple roots in the standard geometric realization, so the $\{\...
- 565
2
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0
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275
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Is there any relation between Berkovich spaces over $\Bbb Z$ and Arakelov theory?
As I understand it, both Arakelov geometry and Berkovich geometry over $\Bbb Z$ (or $\mathcal O_K$) consider geometric objects that contain in some sense information about both archimdean and ...
- 804
1
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1
answer
198
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An identity for the higher form Levi-Civita connection
Take $M$ a Riemannian manifold and $\Lambda^1$ its space of one forms. The LCC (Levi-Civita connection) $\nabla:\Lambda^1 \to \Lambda^1 \otimes \Lambda^1$ is well known to satisfy the identity $m \...
5
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1
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326
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Does Cesaro convergence along all arithmetic progressions imply convergence on a full density subsequence?
Suppose $\{x_n\}_{n \geq 1}$ is a real valued sequence such that for every $a, r \in \mathbb Z_+$, we have that
$$\lim_{N \to \infty} \frac{1}{N} \sum_{i = 0}^{N-1} x_{a + ir}$$
exists and equals $L$ ...
- 6,313
0
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1
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238
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Trans-universality for finitely generated groups
QUESTION: does there exist a group U such that three conditions hold:
(a) every finitely generated group is isomorphic to a subgroup of U;
(b) for every group G that is not finitely generated there ...
- 4,786
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0
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71
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Space of non-archimedean characters is nonempty
Let $k$ be an algebraically closed complete non-archimedean field. Let $\mathcal{O}_k$ be its ring of integers. Suppose that $A$ is a $k$-Banach algebra, and $B$ is its closed unitary ball. Note that $...
- 1,485
2
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1
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218
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How to do LU factorization efficiently based on the factorized result added with a low-rank matrix?
Suppose a square $n\times n$, dense matrix $A^{\text{old}}$ has been factorized into $L^{\text{old}}$ and $U^{\text{old}}$ components by performing a LU decomposition $A^{\text{old}} = L^{\text{old}}U^...
- 21
2
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1
answer
185
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Zariski openness of Zariski dense representations
Let $\Gamma$ be a finitely generated group and let $G$ be an almost simple algebraic group defined over $\overline{\mathbb{F}_p}$. Consider the representation variety $R$ for $\Gamma$ in $G$. Namely, ...
- 527
2
votes
1
answer
265
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Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
- 1,071
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160
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On the invariance of the Kaledin class
In Formality of DG algebras (after Kaledin), Lunts introduces an $A_\infty$-Hochschild cohomology class, called the Kaledin class, controlling formality of an $A_\infty$-algebra up to a certain order. ...
- 6,777
5
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2
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396
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Algebra with three anti-commutator relations
Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations
$$u a^2 = bc + cb$$
$$v b^2 = ac + ca$$
$$w c^2 = ab + ba$$
Is $V$ generated by ...
- 63.1k
1
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answers
33
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Unusual parameterization of the ring Dupin cyclide
I discovered the following by playing with the formulas given in the paper Sculptures in $S^3$ by Schleimer and Segerman.
First, define the following parameterization of the Clifford torus:
$$
p(\...
- 2,560
1
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1
answer
294
views
Temporal evolution of a globally hyperbolic spacetime
Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal).
For ...
- 612
4
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1
answer
403
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When are the transition densities of an SDE symmetric?
We fix $T>0$. Let $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $\sigma:[0, T] \times \mathbb{R}^d \rightarrow \mathcal{M}^\text{sym}_{d \times d}(\mathbb{R})$ be measurable and ...
- 825
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133
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Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 311
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350
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Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
- 730
3
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0
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116
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Conformal welding and Jordan loop consequences?
In the similar context as Conformal welding of rectifiable curves
In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ ...
- 5,474
23
votes
4
answers
2k
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Lower bounding $|1+z+\cdots + z^{n-1}|$ when $z\approx 1$
I am trying to lower bound $|1+z+\cdots + z^{n-1}|$ when $z$ is a complex number close to $1$ (and $n$ is sufficiently large). My main concern occurs in the case $z = 1 + it$, where $t$ is small. In ...
- 347
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0
answers
120
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Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
5
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2
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411
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Gaussian curvature of a holomorphic curve in complex 2-space
Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$.
Each point of $M$ has ...
- 2,858
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0
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212
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When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
- 293
5
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1
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282
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Questions about algorithms for permutation groups
Let $G < S_n$ be a permutation group of degree $n$, $\mathcal{P(n)}$
denote the set of all partitions of $n$, and $c: G \rightarrow
\mathcal{P}(n)$, where $c(g)$ is the partition given by the ...
- 4,646
3
votes
1
answer
402
views
The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction
$$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$
descends ...
- 64k
1
vote
0
answers
511
views
How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?
Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically
\begin{equation}
\sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
- 311
2
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0
answers
109
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Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
- 295
1
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1
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166
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Derivation of the Dirichlet L-series of order $1$
I would like to know how the Dirichlet L-series(of order $1$) were derived. I independently found sequences analogous to the Dirichlet L-series using a property from: https://math.stackexchange.com/q/...
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390
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A question on the proof of pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes
Here is a question about proving the pullback bundles by homotopic maps are isomorphic in Prof. Ralph Cohen notes Bundles, Homotopy, and Manifolds. The proof is in page 73 of the notes. For me, ...
- 1,173
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votes
1
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624
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Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$
Note that rescaling $2^{j} s\mapsto s$ shows that
$$J(j)\leq 2^{-j(1+a)}\int_{0}^...
- 852
1
vote
1
answer
191
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Concentration inequality for square roots
Given a sequence of (not-necessarily-iid) real-valued random variables $X_n$ that converge to $a\in\mathbb{R}$ in probability, suppose we have an exponential concentration inequality of the form
$$
P(|...
- 13
2
votes
1
answer
236
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Zero sets for entire functions $f$ with $|f(z)| \leq C_f e^{c|z|}$
Let $c>0$ and $X$ the collection of all entire functions $f$ for which there exists $C_f > 0$ s.t.
$$
|f(z)| \leq C_f e^{c|z|}.
$$
Given a sequence $(z_n) \subset (0,\infty)$ s.t.
$$
\frac {z_n}{...
- 127
3
votes
2
answers
429
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Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 4,074
1
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0
answers
106
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
3
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0
answers
146
views
A Hartogs analogue?
Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$.
For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
- 31
2
votes
1
answer
83
views
Borel sets in Vietoris topology
Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel ...
1
vote
1
answer
115
views
What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?
$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
- 515
1
vote
1
answer
259
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On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 563
5
votes
0
answers
214
views
Sperner property of a distributive lattice associated to a divisor poset and the free distributive lattice
Let $P_n$ denote the poset with elements $P_n=\{1,...,n\}$ ordered by divisibility and let $L_n$ denote the distributive lattice of order ideals of $P_n$, whose elements should correspond to primitive ...
- 26.5k
0
votes
0
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25
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Maximum sizes of independent sets in (non-uniform) hypergraphs
It is a very well understood problem to compute the size of the maximum independent set in a uniform hypergraph (in terms of extra conditions).
My question is the following: what is known for ...
- 1,561
5
votes
1
answer
97
views
How to bulge out the curved edges of the stereographic tesseract?
You probably already saw such a representation of the tesseract:
I did something similar on my blog for the truncated tesseract:
The vertices in 3D are the stereographic projections of the original ...
- 2,560