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### Can a hypergeometric function have two differently sized matrices as input?

I'm currently trying to understand how to calculate the densities of certain Gaussian random matrix ensembles and I'm having trouble with Theorem 7.2.1 from [1]. As far as I can tell, the ...
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### Mean Value Theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
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### Proof of the projection formula (for cohomology of $\mathbf{P}V$)

Let $V\to X$ be a vector bundle (over say a scheme). Then the cohomology of its projectivisation is $$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$ as an algebra, ...
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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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1 vote
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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 ...
1 vote
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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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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 ...
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 ...