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1answer
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: ...
0
votes
1answer
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
1answer
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 ...
0
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0answers
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
votes
0answers
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 $n \leq 60$.
0
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0answers
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 ...
2
votes
0answers
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 ...
4
votes
0answers
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 ...
2
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
1
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0answers
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 ...
0
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0answers
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 ...
5
votes
0answers
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 ...
2
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0answers
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 $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...
3
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0answers
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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1answer
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 ...
0
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0answers
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 \...
4
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0answers
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,.....
4
votes
1answer
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}^{\...
0
votes
0answers
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
1answer
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 $n/(n+1)$ ...
1
vote
0answers
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(...
2
votes
1answer
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 ...
2
votes
0answers
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 ...
-4
votes
0answers
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
votes
0answers
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 ...
1
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0answers
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 ...
1
vote
1answer
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 $\...
1
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0answers
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 $\mathcal{O}_S$-...
4
votes
0answers
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 ...
0
votes
0answers
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 (\...
0
votes
0answers
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 \...
4
votes
1answer
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 ...
0
votes
0answers
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 $...
1
vote
0answers
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 ...
2
votes
1answer
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(...
4
votes
0answers
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 ...
2
votes
0answers
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
1answer
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 ...
0
votes
0answers
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 ...
1
vote
2answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
6
votes
0answers
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 $2s$...
0
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0answers
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]\...
2
votes
0answers
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 $...
1
vote
0answers
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 ...
2
votes
0answers
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. ...
2
votes
0answers
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 ...

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