# All 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 ...
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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 $P$,...
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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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### 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 $\Bbb C$ ...
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### 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 ...
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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....
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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 ...
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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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### 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; ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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