# All Questions

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### Seemingly complex logic/set-theoretic puzzle

I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed. In front of you is an ...
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### Function with all but mixed second partial derivatives twice differentiable?

Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
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### Decidability survives new constants

Let $L$ be a finite first order language and let $M$ be an $L$-structure with universe $\mathbb{N}$ that interprets all $L$-symbols as recursive sets (so $M$ is a recursive $L$-structure). Let ...
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### Six operations passage from $X_0$ to $X$ reference request

Let $X_0$ be a variety over $\mathbb F_q$ and denote by $X$ its basechange to the algebraic closure. Consider the constructible derived categories $D^b_c(X_0,\mathbb E)$ and $D^b_c(X,\mathbb E)$, ...
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### Quotient Surface of A Hyperelliptic Involution

Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J >$ is a Riemann ...
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### Subtlety in the definition of the Kobayashi metric

When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition: A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
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### Properties of the Euler Discretization of a diffusion

Let $X$ be a continuous 1-d diffusion: $$dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x.$$ W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties. Let ...
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### What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set? (This is mostly ...
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### Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ...
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### How to prove H^2(g,J(g)) is nonzero for a semisimple Lie algebra g, where J(g) is the augmentation ideal of g?

Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, ...
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### Variation on Fatou's lemma for Sobolev norms

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$ If I am not ...
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### Degeneration of the Hodge spectral sequence

Let $f\colon X \to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb Q$-scheme), then Deligne has shown: $R^af_*\Omega^b_{X/S}$ is locally free for ...
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### isotopy doesn't make sense (Milnor)

hello, I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots For each $c$ and $p$ in $\mathbb{R}^{n-1}$ such that $\|c-p\| > < r$, there is ...
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### Existence of solution for this parabolic PDE

The parabolic PDE $$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$ has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form ...
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### Extremal point and probability

Let $(X,\mathcal{F},\mathbf{P})$ be a probability space and $f \colon X \mapsto \mathbf{R}^n$ an integrable function. We assume that $f$ takes its values in a closed convex set $C$ of $\mathbf{R}^n$ ...
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### Real analytic function, injective, non surjective and preserving the rationals ?

I'd like to prove the non-existence of a real analytic function, injective, non-surjective that sends rationals to rationals. Is it a classical result ? If not, any hints on how to prove it ? Thanks ...
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### sum of $1/ \phi(n)^2$

Hello, I am reading some material on circle method. Right now I am at its application to the binary Goldbach problem. To obtain a certain bound the fact $\sum_{n> X} 1/ \phi(n)^2 = O(1/X)$ is ...
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### cohomology of torsion sheaves and nilpotent sheaves

Let $X$ be a scheme and $\mathcal{F}$ be a sheaf on $X$ which is torsion $\mathcal{O}_X-$module (i.e., every local section is annihilated by an element of the ring $\mathcal{O}_X(U)$) or nilpotent ...
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### Gosper's Mathematics

Sometimes I bump into more of the astonishing results of Gosper (some examples follow) and I gather that a lot of them come from hypergeometrics and special functions. Have there been any attempts ...
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### Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
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### Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising: $-\Delta v(x)= e^{v(x)}$ in $R^N_+$ $v= - \infty$ $\qquad$ on $\partial R^N_+$ $\qquad$ $v \le 0$ in $R^N_+$. ...
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### counting k-cliques not also (k+1) on random graphs

consider the set of graphs with $n$ vertices and exactly half of all $\binom n 2$ possible edges. looking for a formula that counts the number of these graphs that have a $k$-clique but not a ...
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### Generalizing the Reshitikhin-Turaev construction possible?

OK, I have to ask a dumb question again: Where do Lie groups enter in the construction of the Reshitikhin-Turaev invariant? The parts of the proof I understand are that 6j symbols take care of ...
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### What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?

There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here: ...