# All Questions

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### Generators of Ideals in Integers in a Number Field

Let $R$ be the ring of integers in a number field $K$. It is known that each ideal of $R$ can be generated by two elements. In fact if $I$ is an ideal of $R$ and $a\in I$ is a nonzero element, then ...
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### Cut locus, conjugate points and smoothness of distance function

I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance ...
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### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
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### Finite Subgroups Of $GL(2,\mathbb{R})$ [migrated]

I have the following question: Is it true that every finite subgroup of odd order in $GL(2,\mathbb{R})$ is cyclic? Thanks!
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### how to project a collision between a pair of polygons under rotation?

I am trying to create a physically plausible 2d physics engine. I have read many documents about detection of collisions, contact resolving, interpenetrations, projection, separating axis theorem ...
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### problem computing block inverse of factorized matix

I'm running into trouble when verifying the inverse of the following block matrix using the Schur complement: My matrix is given by: $$K = USU^T,$$ where U are its eigenvectors and S the ...
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### Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects? What are some applications? For ...
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### Colorful model theory

There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance http://math.univ-lyon1.fr/~wagner/nijmegen.pdf) - which are colorfully named: ...
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### Group of homomorphisms with real coefficients and circle coefficients

Do you know some groups $G$ that have the same group of homomorphisms to $R$ and to $S^1$; i.e. $$Hom(G,R) = Hom(G,S^1)??$$ Is there any special property for $G$ in order to satisfy the last relation? ...
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### Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...
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### Classical and Quantum Chern-Simons Theory

Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway. Let $\Sigma$ be a two-manifold and $M$ a ...
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### Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field) Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...
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### Block Covariance Matrix - Positive Definite? (Quadratic Optimization)

I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form: [ C C ] [ C C ] However, the quadratic solver ...
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### Holomorphic map near the point (0:0:0:1)

Let T := C/ be a complex torus and let : C → T be the quotient map. Let } be the Weierstrass }-function for the lattice and let g2, g3 ∈ C be such that (}′)2 = 4}3 −g2}−g3. Show that the map : T ...
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### Do Hermitian metrics also split on the Riemann sphere?

Maybe this is well known, but i could not find a pointer to some literature: Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that ...
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### General inverse of matrix [on hold]

I'm try to understand one problem. For example I have a Laplacian matrix L, it's symmetric positive semi definite and singular, am I right? In many software (e.g. matlab ) is solve that equation: v = ...
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### Newlander-Nirenberg in dimension 2

What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions ...
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### Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials

By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex polytopes. If $P$ is an integral polytope, the counting function for the number of lattice points ...
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### What do we know about the generlized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$. Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem ...
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### Making Hironaka's theorem explicit for hypersurfaces

Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, ...
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### Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal. The original proof is in German, and I've been ...
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### Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
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### Functors and coverings

A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...
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### The tallest possible lattice?

Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
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### commutative diagram with Yoneda pairing, Weil pairing and edge morphism

$\require{AMScd}$ Why does the following diagram commute? \begin{CD} H^0(X,\mathscr{A}) @VVV \times \mathrm{Ext}^2_X(\mathscr{A},\mu_{\ell^n}) @>>> H^2(X,\mu_{\ell^n}) @| \\ ...
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### expectation of log(x+a) when X follows a beta distribution

Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?
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### intersection of finitely many maximal ideals

For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero? Obviously it holds for Dedekind domains ...
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### Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated. The same question for the Weil pairing ...
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### $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational

We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is ...
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### Normalization of a curve and push forward of vector bundles

Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. ...
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### p-adic valuation of a sum

Let $n_1$, ..., $n_k$ denote positive integers, and let us write $$n_i=\prod_{j=1}^m p_j^{\alpha_{ij}}$$ for $1\le i\le k$, where the $p_j$'s are distinct prime numbers, and $\alpha_{ij}\ge 0$ for ...
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### Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...
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### What are all the stationary and pointwise independent random processes?

In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the ...
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### What is the best result of asymmetric sparse connector of depth 2 so far?

What is the best result of asymmetric sparse connector of depth 2 so far? The sparse connector problem can be represented by a digraph. Given $n, N \in \aleph \left(n\leq N \right)$, construct a ...