# All Questions

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### About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing. Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph. I would like to understand what is the ...
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### Is there a standard notation for off-diagonal transpose?

Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$. But is there a conventional way of notating the matrix ...
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### Example of torsion in orientable manifolds?

An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples ...
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### Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
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### Bar Construction Model of Ring Spectrum Quotient

Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism ...
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### For a mathematician that English is not the native language, does he/she think in english or graph or native language? [on hold]

For example, if you are a mathematician with Chinese the native language. During your research you find most of the books or papers are in English, of course when you read them, you probably will ...
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### Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$. We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...
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### Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
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### A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a connected finite dimensional Lie group $G$ so that ...
931 views

### Categorical proof subgroups of free groups are free?

This is a crossport of this question from MSE. Is there a categorical proof that subgroups of free groups are free? How about the result that subgroups of free abelian groups are free abelian? ...
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### Iterative Calculation? [on hold]

Apologies, as I do not know how to phrase this question in the correct terms; however, I will try my best. I have an equation that looks like this: D = A - B - C However, C = ( [A - B] * X ) Is ...
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### Why is “The Higman Rope Trick” thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma: If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...
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### Besse p134 Riemann tensor in dimension 4

Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that $$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4)$$ because we are in dimension ...
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### Improper integral calculation - limit at infinity [on hold]

Will you please help me prove the following limit is zero ? $lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} cos(x\cdot u) du$ Thanks in advance
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### Alexander polynomial in branched covers

Suppose I am given a homology sphere as a double branched cover over a link (of determinant one). Let a knot in this space be given as a lift of an arc with endpoints on the link. Is there a way to ...
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### Decidability of first order theory of subclasses of posets

Is the first order theory of finite posets known to be undecidable? Does anyone know a survey about such results?
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### p-divisibility of the connected component of the Picard group [on hold]

Let $X$ to be a smooth projective variety over a field of positive characteristic $p>0$, then can one claim $Pic^0(X)$ is p-divisible.
279 views

### Find an integrable, positive, unbounded, analytic function

Is there a standard example of a function $f \in L^1( \mathbb R)$ which is analytic, positive, integrable but not bounded? An example which comes immediately to mind is to take the series of narrower ...
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### Schur covering group for S4

It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). ...
119 views

### Hilbert vs Chow in nice cases

I'm trying to understand the relationship between the Hilbert schemes and Chow varieties in situations where everything is simple. Suppose that $X$ is a smooth projective variety over $\mathbb C$, ...
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### Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e. ...
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### Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...
96 views

### Help in understanding “Local well-posedness for the Maxwell-Schrodinger system”

Is there someone who knows the following paper "Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada. I'm trying to study it but I've some doubts. In particular I'm not ...
336 views

### Can ZFC prove it cannot derive an inconsistency in $n$ steps?

Let $Con(\mathtt{ZFC}, n)$ denote the statement "$\mathtt{ZFC}$ cannot prove the contradiction within $n$ steps (or better within $n$ symbols) within a given proof system (say a natural deduction to ...
238 views

### Who first talked about “holes” in homology?

The question Why do the homology groups capture holes in a space better than the homotopy groups? and many others here use the idea that homology counts the holes'' in a space. The comments on this ...
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### distances-based dispersion measuring approach

Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ? thanks.
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### Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?

Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial. For $m\in \mathbb{Z}$, we also define ...
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### What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and $$L^1(G:H)=\{f\in L^1(G): R_h f=f\;(a.e)\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in \hat{G}:\xi|_H=1\}$($\hat{G}$ ...
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### Isomorphisms of Positive and Negative Spinor Bundles

Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf, 1998): 2.6 Spin Bundles and the Dirac Operator ...
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### Order-preserving image of a complete lattice

If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving map, does this imply that $P$ is a (complete) lattice?
182 views

### Set with small internal radius, small perimeter and prescribed area

Given a regular set $E\subset \mathbb R^2$ define $$R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\}$$ to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
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### positively invariant set respec to fractional system

In my research I need to show that the set $$M := \{X \in \mathbb{R}^4,X≥0\}$$ where $$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$ is positively invariant with respect the following system of ...
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### Signatures of latin squares: what about the extremal cases?

For a latin square (LS) of order $n$, we will define a cut (or maybe general transversal, I don't know whether there is an entrenched name for this) as a collection of $n$ cells such that no two share ...
I've read somewhere that the cut rule in sequent calculus $$\frac{A \vdash \mathbf{C}, B \qquad A',\mathbf{C} \vdash B'}{A,A' \vdash B,B'} (\text{cut})$$ states that the $\mathbf{C}$ on the right is ...