# All Questions

139,132
questions

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### Finite domination and Poincaré duality spaces

Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...

0
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0
answers

8
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### Fourier-Mukai kernels for Fano threefolds

Let $Y_1$ and $Y_1'$ be index two degree one Fano threefolds. Suppose we have a Fourier-Mukai equivalence $\Phi_P : \mathrm{D}^b(Y_1) \to \mathrm{D}^b(Y_1')$. Can anything be said about the kernel $P$,...

-1
votes

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3
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### Choice of approximate posterior in variational inference with positive support

I have a simple probabilistic graphical model: $z \longrightarrow x$ where $z_i \sim Exp\left(\lambda_i\right)$ where subscript $i$ denotes the $i$th dimension and $x|z \sim \mathcal{N}\left(f\left(z\...

1
vote

1
answer

19
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### Before focal point, the locally distance function is smooth

I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question: In the proof of Lemma 7.4, He let $d$ be the ...

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15
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### Under what conditions a continues linear map maps a closed subspace to a closed subspace

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$.
It is obviously satisfied if $W$ is ...

1
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0
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17
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### Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group

Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...

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1
answer

29
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### Approximation from above of subharmonic functions

Let $u$ be a subharmonic function on $\mathbb{C}$. It is known that the convolution with standard mollifier gives a sequence $u_{\epsilon}$of subharmonic functions with the property: $u_{\epsilon}\in ...

-5
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24
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### How would I be able to work this question out? [closed]

A stuffed toy in the shape of an ice cream cone consists of a hemisphere attached to the base if t=a cone in such that both the hemisphere and the cone have the same radius. the ratio of the volume of ...

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52
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### Integral over $\Bbb C$ [closed]

Is this correct? Let $a\in\Bbb C^*$ and $f\in L^1(\Bbb C^2)$
$$\int_{\Bbb C^2}|f(a(z,w))|dz dw={1\over |a|^2}\int_{\Bbb C^2}|f((z',w'))|dz' dw'$$
where $dz$ is the usual measure Lebesgue on $\Bbb C$
...

-4
votes

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answers

72
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### Does the permanent equals the determinant over $\mathbb{F}_{2^n}$? [closed]

It is known that the permanent equals the determinant modulo $2$.
We got numerical evidence that this generalizes to $\mathbb{F}_{2^n}$.
Is it true that for all square matrices with entries from $\...

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votes

1
answer

33
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### Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$

Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...

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0
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31
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### Can we have such an infinite descending sequence of functions with prior ones inside their successors?

Let $M$ be some non-well-founded model of $\sf ZF$, can we have a sequence $(S_n)_{n \in \mathbb N}$ of nonempty sets in $M$, where each $S_n \subset \mathcal P(S_{n+1})$; and such that there exists ...

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17
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### Subordinated non-deterministic Gaussian process is non-deterministic

Let $X = \{ X(k), k \in \mathbb{Z} \}$ be a strictly stationary, Gaussian time series whose spectral density $f_X$ exists.
Furthermore, let $X$ be non-deterministic, i.e.
$$
\mathbb{E}\big[ \vert X(n +...

2
votes

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answers

15
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### Topology of the Malcev-Neumann group ring

For a ring $R$ and a group $G$ the group ring $R[G]$ consist of maps from $G$ to $R$ with finite support.
It was shown that if the group is fully ordered them this ring can be embedded in a division ...

2
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29
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### Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?

Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...

2
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26
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### How to get Bakry Emery Criterion $ \Phi'(t)=\frac{d}{dt}\int \Gamma_1(P_t f)d\pi=-\int\Gamma_2(P_tf)d\pi? $

I am reading Bakry Emery Criterion https://terrytao.wordpress.com/2013/02/05/some-notes-on-bakry-emery-theory/.
Let a function $H\in C^2(R)$ and define the infinitesimal generator:
$$
Lf=\Delta f-\...

4
votes

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answers

37
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### Univalence for weakly Tarski universes

In Martin-Löf type theory, a weakly Tarski universe is a type $\mathcal{U}$ with a type family $\mathcal{T}(A)$ indexed by terms $A:\mathcal{U}$, which is closed under identity types, dependent ...

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14
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### Sum over lattice points in homogeneously expanding domains

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...

12
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150
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### A new (?) way of composing monads

By composition of monads, I mean given two monads $S$ and $T$, making their composite $S T$ into a monad. Or more generally, given two monoid $X$ and $Y$ in a non-symetric monoidal category, making $X ...

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24
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### Matrix optimization to find ideal embedding

Basically, I am trying to find the embeddings so I can approximate $K \approx M(\vec{\phi})$. The embeddings are for each one of my samples $\vec{\phi}(x_i) \in \mathbb{R}^D$ so I thought it should ...

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17
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### Netting edges of a graph

Consider a weighted directed graph.
Is there any known algorithm to compute the associated graph with "netted edges"?
By edge netting I mean a function such that the (sub-) branch
...

2
votes

0
answers

51
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### Definition of Radon measure on Takesaki's first volume

Consider the following theorem from Takesaki's first volume "Theory of operator algebras":
In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...

1
vote

1
answer

54
views

### Existence of a family of sets with some properties

Is it possible to find an example of a family $\mathcal{F}$ of $n$ finite distinct non-empty sets, a universe of maximum size $n/4$, with at least $\lfloor \frac{2}{3}{n \choose 2} \rfloor$ unordered ...

5
votes

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76
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### What is the Balmer spectrum of the p-complete stable homotopy category?

When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...

0
votes

1
answer

27
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### Method for (binary) optimization under constraints

I would like to know if there is a method to solve the Problem.
Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...

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19
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### Turing reaction diffusion equations and neural networks

Suppose you have a certain Turing-type reaction-diffusion equations that describe the formation of a some 2D pattern. I wonder if there are published works that trace some sort of strong connection ...

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80
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### Can we have a proper class of infinitely descending infinite ordinals?

Working in $\sf ZF-Reg.$, can we have a transitive model $M$ of $\sf ZF$ such that there exists a proper class (i.e. a subset of $M$ that is not an element of $M$) of infinitely descending infinite ...

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votes

0
answers

29
views

### Term for degrees realizing least possible first n jumps

Is there a term for (Turing) degrees which realize the least possible jump (in the following sense) for the first n jumps.
That is degrees which satisfy for all $0 < m \leq n$:
$$X^m \equiv_T X \...

0
votes

1
answer

116
views

### Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book

Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...

4
votes

2
answers

357
views

### Do non-zero derivatives imply tangent lines (and vice versa)?

Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$.
We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...

6
votes

0
answers

78
views

### Automorphisms of Lubotzky–Phillips–Sarnak graphs

For the Lubotzky–Phillips–Sarnak (LPS) graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=\mathrm{PSL}_2(\mathbb{F}_q)$, so clearly ...

2
votes

0
answers

41
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### Derivative of anti-self-dual forms on Kähler space

I am puzzled if we can establish differential relations about anti-self-dual 2-forms on the Kähler space similar to ones for self-dual forms?
Let $(\mathcal{M},g,J,\omega = J^{(1)})$ be a Kähler space....

2
votes

0
answers

36
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### Cross-ratio for projective lines over division rings

If one considers a projective line over a field $k$, then the cross-ratio $(w,x;y,z)$ is a well-known geometric tool.
But what if $k$ is not commutative, that is, if $k$ is a division ring ?
Is there ...

0
votes

1
answer

23
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### When a polygonal line become a loop in hyperbolic plane?

Suppose we have a 5 tuple of positive real numbers $(l_1,l_2,m_1,m_2,m_3)$, with $m_i \in (0,\pi)$ for all $i$. Now fix a point $v_1$ in the hyperbolic plane. Then consider a geodesic of length $l_1$ ...

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votes

0
answers

8
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### Dual equivalence of minimum feedback-vertex sets and cycle packing

it is known that the duals of feedback-set problems are set-packing problems; in the context of digraphs the feedback set are either a minimal set of vertices or edges that hit every oriented cycle; ...

0
votes

1
answer

25
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### Skip-free random walks: recurrence and transience

Let us define a one dimensional random walk: for all $n\in\mathbb{N}$
$$
X_n:=\sum_{i=1}^nZ_i
$$
with $Z_i$ i.i.d. random variables taking values in $\{-1,0,1,2,\dots\}$. This process is sometimes ...

4
votes

0
answers

74
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### Is $C^r(M)$ non-isomorphic to $C^s(N)$ for $r\neq s$ and nontrivial manifolds $M,N$?

This is an obvious continuation of an MO question. Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a ...

3
votes

1
answer

116
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### Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder https://arxiv.org/pdf/2207.06477.pdf Lemma 6.18, $D^b(...

5
votes

0
answers

33
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### Generalizations of classical integrals of motion

In classical Hamiltonian mechanics, we are given the Hamiltonian $H\in C^\infty(M)$. $L\in C^\infty(M)$ is called an integral of motion (or a conserved quantity) if the Poisson bracket of $H$ and $L$ ...

0
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0
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26
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### Rates of convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables in $\mathbb {R}^d$ with common distribution $\mu$, and
$\mu_N = \frac 1N \sum_{k=1}^N \delta_{X_k}$, $N \geq 1$, the associated empirical
measures. If $\...

4
votes

1
answer

459
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### Can deleting a random entry from an iid sequence destroy the iid property?

Let $X=(X_1,\ldots,X_n)$ be an iid sequence of random variables, and let $\nu$ be a uniformly random integer in the range $1,\ldots,n$. Then $\xi_\nu$ is a random entry of $X$. Is it always true that ...

1
vote

0
answers

26
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### Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...

6
votes

1
answer

71
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### Onsager-Machlup functional when drift is time-dependent

Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...

2
votes

0
answers

61
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### Embeddings of reductive groups over algebraically closed fields

Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups.
Do there exist split, reductive ...

0
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0
answers

26
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### Variance of sum of combinations of a discrete set

I have a vector of weights $\vec{w}$ (size $n$) and a random boolean vector $\vec{x}$ sampled uniformly from all possible boolean vector of size $n$.
Is there a shortcut to compute the variance of $\...

4
votes

1
answer

145
views

### What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...

3
votes

1
answer

54
views

### Matrix determinant lemma for non-rank-one updates

The well-known matrix determinant lemma states that for an invertible square matrix $A$ and column vectors $u,v$ one has
$$
\det(A + uv^T) = \det(A)(1 + v^T A^{-1} u).
$$
Is there any analogous ...

2
votes

1
answer

150
views

### Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$

Let $X$ be a complex manifold, there is a natural map $f:H^2(X,\mathbb C)\to H^2(X,\mathcal O)$ induced by the inclusion map $\mathbb C\hookrightarrow \mathcal O$ which coincides with the natural ...

3
votes

0
answers

29
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### The embedding of a Banach lattice in an ultrapower

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...

1
vote

0
answers

40
views

### Least squares problem with left and right unknowns

For $i=1,...,n$, let $b_i$ be a scalar and $A_i$ be an $k\times l$ matrix. Is there a closed form solution for the following problem assuming $n>k+l$?
$$\min_{x\in \mathbb{R}^k ,y\in \mathbb{R}^l} \...