# All Questions

114,973
questions

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votes

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59 views

### Fixed points of diffeomorphisms of tori isotopic to identity and their traces under isotopies

Suppose $T^n$ is the $n$-dimensional torus ($n\geq 2$) and $f: T^n\to T^n$ is a diffeomorphism isotopic to the identity and fixing points $x_1,\ldots,x_k\in T^n$. Does there exist an isotopy $\{ f_t: ...

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votes

**1**answer

83 views

### How may a largest fixed-point be defined in second order logic?

Adapting from Anil Gupta and & Nuel Belnap, Revision theory of truth, MIT 1993, p. 194, in the context of a second order logic, where $A(x.G)$ is a formula where $G$ only occurs positively, a ...

**6**

votes

**1**answer

256 views

### A variant on Wieferich primes

Recall that a Wieferich prime is a prime number $p$ such that
$2^{p-1} \equiv 1 \bmod p^2.$
It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many ...

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64 views

### Toric ideals are generated by binomials. $V(x)$ gives affine $n-1$ space in affine $n$-space. $x$ is not a binomial, yet affine $n-1$ space is toric?

Proposition 1.1.11 of Cox-Little-Schenk's Toric Varieties states that an ideal $I \subseteq \mathbb{C}[x_1, \dots, x_n]$ is toric iff it is prime and generated by binomials. Setting $I = (x_1) \subset ...

**6**

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68 views

### Characteristic polynomial of the Gcd matrix

Let $A_n$ be the $n \times n$-matrix with entries $Gcd(i,j)$ and $f_n$ the characteristic polynomial of $A_n$.
Question: Is $f_n$ irreducible for all $n$ except $n=8$?
This is true for $n \leq 60$.

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25 views

### Upper bounding the sum with hypergeometric and binomial probabilities

Could you please help me upper bound this tricky expression:
$$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$.
So far I only ...

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104 views

### The space of rearrangements of a plane curve

I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of ...

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81 views

### Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question.
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. Hironaka tells us that ...

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votes

**1**answer

139 views

### A set of objects classically generates the full subcategory of compact objects iff it generates the whole category

Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...

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vote

**1**answer

40 views

### How do you find the Cholesky decomposition of the sum of two positive definite matrices without adding the matrices directly?

If you're given two positive definite matrices ($A_1,A_2$) and the Cholesky Decomposition of those two matrices ($L_1,L_2$ such that $A_1=L_1L_1^T, A_2=L_2L_2^T$). Is there a way to find the Cholesky ...

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vote

**1**answer

30 views

### Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram

A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...

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vote

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127 views

### $X-Y$, where $X$ and $Y$ are sums of Bernoulli random variables

Let $X = x_1 + x_2 + \ldots + x_n$ and $Y = y_1 + y_2 + \ldots + y_n$, where each $x_i$ is an independent Bernoulli random variable with success probability $p_i$ and each $y_i$ is a Bernoulli random ...

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72 views

### Counting quadratic curves in $\mathbb P^1 \times \mathbb P^1$ passing through seven points in general position

Let $p_1,\dots,p_7 \in \mathbb P^1 \times \mathbb P^1$ be 7 points in general position.
What is the number of maps $F=(F_0,F_1):\mathbb P^1 \to \mathbb P^1 \times \mathbb P^1$ modulo domain ...

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77 views

### Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result:
(C-S-T): Let $G$ be a ...

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55 views

### $t$-balanced numbers

Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...

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48 views

### Calabi-Yau structures on dg-categories

A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here)
$$ A^! = A[-n]$$
Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...

**10**

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**1**answer

234 views

### Which is the more popular approach to forcing in the literature?

There are some interesting questions and answers on the site discussing the different approaches to forcing in set theory, and I understand that the two most important are the ones using countable ...

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30 views

### Finding all possible set of functions

Let $\{ h_n(x)\}_{n=1,..,N}$ a set of $2\pi$ periodic functions such that they satisfy the reflection property
\begin{equation}
e^{h_n (x+\pi) + i\bar{h}_n (x+\pi)} = \sum_m C_{nm} e^{h_m (x) + i \...

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54 views

### Invariant theory for the orthogonal group and Clifford algebras

The first fundamental theorem of invariant theory for
the orthogonal group $O_n(k)$ asserts that the
ring of invariants is generated by the scalar products:
a polynomial function of $m$ vectors $v_1,.....

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votes

**1**answer

223 views

+50

### Does every smooth function agree with some formal power series at uncountably many points?

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function. Does there exist an uncountable set $X\subset \mathbb{R}$ and real numbers $a_0, a_1, \dots$ such that for any $x\in X$ the sum $\sum_{i=0}^{\...

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53 views

### When the sum of two ideals is indecomposable

I am looking for a commutative ring $R$ and two ideals $I$ and $J$ of $R$ and two different maximal ideals $m_1$ and $m_2$ of $R$ such that $ann(I)=m_1$ and $ann(J)=m_2$ and $I+J$ is an ...

**11**

votes

**1**answer

351 views

### Possible limit involving the gamma function

Does $$\lim_{n \to \infty} \int_{0}^{1} \Gamma(x)^{n/(n+1)}dx - n$$ exist?
Here's some background. The integral
$$\int_{0}^{1} \Gamma(x) dx$$
diverges rather slowly. Inserting the exponent $n/(n+1)$ ...

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46 views

### Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?

Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...

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**1**answer

148 views

### Pell equation and quadratic residues

We say an integer $k$ is Pell if there exist some integers $p,q$ such that
$$
p^2k-q^2=1
$$
In studying a physics system we ended up with two weaker notions of Pell:
We say an integer $k$ is pre-Pell ...

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85 views

### What is the multiplicative order of this number

Let $q, r \in \mathbb{P}$ and $r$ is the next prime to $q$.
What is the multiplicative order of $r$ modulo $\displaystyle\bigg( \prod_{\substack{p \leq q \\\text{p prime}}} p \bigg)$ ?
In other word ...

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29 views

### Uniquesnes of basis of the matrices with degenerate eigenvalues [closed]

How to prove the statement, "when two or more eigenvalues of a matrix, say $M$, are equal, the basis of eigenstates of $M$ is not unique"?

**2**

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35 views

### Link between characters and isotypic components

I am currently studying finite complex reflection groups using the book written by Lehrer and Taylor called "Unitary Reflection Groups" and I am unsure if I understood isotypic components ...

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69 views

### Commutative C*-rings

Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital subring (closed ...

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**1**answer

41 views

### Has the covariant Hom-functor of the category of additive categories a left adjoint?

Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...

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96 views

### Pushforward of the structure sheaf for smooth morphisms

Let $\pi:X\to S=\mathbb{P}^1_{\mathbb{Z}}$ be a smooth morphism. Is there a smooth separated morphism $\pi':X'\to S$ such that $\pi_*\mathcal{O}_X\approx \pi'_*\mathcal{O}_{X'}$ as $\mathcal{O}_S$-...

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78 views

### Introductory text for representations of compact/profinite groups

I am looking for a text on representation theory of topological (at least profinite) groups over fields (allowing the non-algebraically closed case). Reasonably selfcontained (or assuming a standard ...

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67 views

### Linearization Navier-Stokes

I am considering the classical form of the stationary Navier-Stokes equation given by
$$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...

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61 views

### Does this hereditary size based definition of cardinality work under grounds weaker than regularity and choice?

To $\sf ZF - Regularity$ add the following axiom:
Hereditary size: $\forall x \ \exists H_x \ \exists f (f: x \rightarrowtail H_x)$
Where: $H_x= \{y: \forall z \in TC(\{y\})\exists f (f: z \...

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votes

**1**answer

135 views

### The Thom map for the Brown-Peterson cohomology

For a prime number $p$ and the Brown-Peterson spectrum $BP$, let $T:BP\to H\mathbb{Z}_{(p)}$ be the Thom map, and $T':BP\to H\mathbb{Z}_p$ be the mop $p$ reduction of $T$. Tamanoi (1) determined the ...

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14 views

### Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic

I first asked this question on math.stackexchange as I first thought it was somewhat innocent.
I would like if one can solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where
$...

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55 views

### Terminology: Existence + Representation

I'm looking to describe a result in a recent paper of mine, but I don't know if there is a term used for a result which is both an existence theorem and a representation theorem.
Specifically, the ...

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**1**answer

116 views

### Can we show that this transition semigroup preserves a certain Wasserstein space?

Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...

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88 views

### Measurable total order

Under what conditions on a metric space $X$, equipped with the Borel $\sigma$-algebra, does there exist a measurable total ordering of the elements of $X$?
By "measurable total ordering" we ...

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81 views

+50

### Anti-self-dual Ricci-flat and Kähler Ricci-flat manifolds

Let $(M^4,g)$ be a complete Riemannian $4$-manifold with anti-self-dual (i.e. $W_+=0$) and Ricci-flat metric $g$.
Can we find a finite cover $(\tilde M, \tilde g)$ of $(M,g)$ such that $(\tilde M, \...

**3**

votes

**1**answer

100 views

### Renorming of $C[0,1]$ for a strictly convex dual

Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is ...

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23 views

### Barycentric spanner for set of subsets

Let $V=\{1,...,n\}$ be the ground set, and $\mathcal{S} \subseteq 2^V$ be a set of subsets of $V$. Are there algorithms to compute barycentric spanner of $\mathcal{S}$ of size O(n)?
Let us describe ...

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vote

**2**answers

95 views

### When is the inside of a Jordan curve open? [closed]

I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...

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15 views

### Error in second derivative using lagrange quadratic interpolation

Quadratic lagrange interpolation is given by $P(x)=\sum _{k=0}^2 l_k(x) f_k$ with error term $E(x)=\frac{(x-x_0)(x-x_1)(x-x_2) f^{'''}(\xi )}{3!}$.
To calculate, error for second derivative at point ...

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**1**answer

83 views

### Natural candidates for sub-half-exponential which limit to half-exponential function from below

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However sub-half-exponentials (functions whose composition grows ...

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142 views

### When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...

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31 views

### Derivative of matrix argument function with respect to eigenvalues of argument

Let $\mathsf{SPD}_n$ denote the set of all real symmetric and positive definite $n\times n$ matrices. This set is convex so for every $A,B\in\mathsf{SPD}_n$ there exists a smooth path $\varphi:[0,1]\...

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133 views

### Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...

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43 views

### Bruhat-Tits theory: how does the normalizer act on an apartment?

Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...

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73 views

### Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...

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37 views

### Notation for induced subgraphs

For a graph $G=(V,E)$, is there a standard notation for the induced subgraph on $V \setminus \{v,w\}$ where $v,w$ are the endpoints of some edge $e$? I know $G[V \setminus \{v,w\}]$ is an option, but ...