# All Questions

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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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### What do $\Gamma$-sets classify?

The category $\Gamma^{\mathrm{op}}$ is defined to be a skeleton of the category of finite pointed sets (see also this question). Then $\Gamma$-spaces, meaning space-valued presheaves ...
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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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### A combinatorial problem - counting the solutions

Consider a square. There are 16 ways to paint its sides with two colors. For convenience, we will represent one color with a blank side, and the other - with a line drawn from the squares' center to ...
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### In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?

What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?
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### Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
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### Cubic graphs decompositions

There are many interesting computational problems related to connected cubic graph decomposition. For instance, decomposition of cubic graph into a perfect matching and a connected 2-factor ...
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### Comparison between operators

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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### Letter from Grothendieck to Tate on “crystals”

I have downloaded from this link a quite poor quality scan of the letter dating May 1966 that Grothendieck sent to Tate mentioning his ideas about generalizing Monsky-Washnitzer cohomology. I am ...
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### Density of characters

Did Harish-Chandra prove that characters of irreducible representations of a $p$-adic reductive group $G$ span a dense subspace of the space of conjugation-invariant distributions on $G$? What is the ...
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### Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)? I am especially interested in the quaternary case. There exist some ...
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### In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
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### Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
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### Minimal (semi)lattice containing a given poset

For a given poset, (I think that) it is easy to construct the minimal join-semilattice containing that poset. I wonder whether the minimal lattice containing that poset is also easy to construct. I ...
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### Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for the shape that encloses three given volumes in $\mathbb{R}^3$ and has minimal surface area? I know the planar triple bubble conjecture was ...
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### unique types and decidability

Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is ...
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### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
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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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### 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 ...
Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$: $$... 2answers 99 views ### Invariant measures on a compact metric space I'm dealing with a continuous flow on a compact metric space X, and \mu, \nu are two invariant Borel probability measures on X. If I know that \mu(A)=\nu(A) for all the invariant Borel ... 0answers 24 views ### 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 ... 3answers 344 views ### 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 ... 1answer 49 views ### 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 ... 2answers 212 views ### (Homotopy) Y ENR and contractible subset implies Y is a retract I'm trying to solve the following question: Suppose Y \subset R^n is a Euclidean neighborhood retract. I want to prove that if Y is contractible, then it is a retract of R^n. 0answers 42 views ### Generators of Ideals in Integers in a Number Field [migrated] 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 ... 1answer 126 views ### \text{Rep}(D(G)) as representation category of a vertex operator algebra The category of representations \text{Rep}(D(G)) of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ... 0answers 51 views ### 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 ... 1answer 433 views ### Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?) The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ... 0answers 348 views ### Fine and acyclic sheaves on locales Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let (X,\Sigma,\mu) be a measure space, we can define a Grothendieck pretopology on it (and ... 0answers 240 views ### Constructing the Stone Space of a Distributive Lattice Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian logics"? ... 0answers 37 views ### 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! 0answers 27 views ### 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 ...
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 ...