# All Questions

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Can the adjoint of the derivative of the exponential map of a lie group, be understood via the ad operator? If so how, explicitly?
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### how can i calculate the 2x2x2 (doublet tensorproduct) [on hold]

help me the simple question how can I calculate the doublet@doublet@doublet @ : tensor product please teach me
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### Should one post a paper on the arXiv if it is not intended to be published?

A brief description: I have written a paper which contains a new result which I believe is somewhat important but not vital to the field. It is a generalization of an existing proof to get ...
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### Can $SL(d, \mathbb{R})$ be embedded in $sp(n, \mathbb{R})$ for n large enough? [on hold]

my question is that can $SL(d, \mathbb{R})$ be embedded in $sp(n, \mathbb{R})$ for n large enough? And i also ask similar question for the Lie algebra.
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### Existence of a “quasi-uniform” probablility distribution on $\mathbb{Z}$

Does there exist a probability distribution on $\mathbb{Z}$ such that for every integer $n\geq 1$, the probability that a random integer $x$ is divisible by $n$ equals $1/n$? Henry Cohn has an ...
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### Between Tietze's and Dugundji's Extension Theorems

The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen ...
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### What is $T>0$ large enough such that $\mu\left(B\right)<\varepsilon$?

Let $\left(M,\sigma,\mu\right)$ where $\sigma$ is a Borell $\sigma$-algebra and $\mu$ is a probability $f$-invariant. Let $x\in M$, $E\subset M$ measurable and $f:M\rightarrow M$ a measurable ...
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### $\mu$ is a $f$-invariant measure [on hold]

Let $\left(M,\sigma\left(\tau\right),\mu\right)$ a measure space where $\mu$ is a measure finite, $\tau$ is a topology in $M$, i, e, $\sigma\left(\tau\right)$ is a Borel $\sigma-$algebra. Let ...
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### Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as ...
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### Character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$

Is there any reference where I can find the character table of $\mathrm{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$? A simple search in google gave me this paper of Philip C. Kutzko on "The characters of the ...
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### Is Ш a good parameter for the failure of Global-Local principle for abelian varieties?

Comparing to class group cases : we have an isomorphism $Cl(K)\rightarrow \prod \left(K^\times \backslash K_p^\times /O_p^\times \right)$ for a number field $K$. Similarly, for an elliptic curve ...
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### Existence of $\kappa$-Suslin trees above a measurable cardinal

We have learned from Joel David Hamkins and Monroe Eskew that: Answers: Having a measurable cardinal $\delta$ we can force a $\kappa$-Suslin tree for many $\kappa$'s above $\delta$. But is the ...
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### What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner. I know that this is a famous problem, but what is it called? ...
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### Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside simple Lie algebras (of type ABCDEFG) have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h ...
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### Is every algebraic $K3$ surface a quartic surface? [on hold]

Algebraic $K3$ surface means the $K3$ surface admits an ample line bundle. So the question is equivalent to asking whether every algebraic $K3$ surface can be embedded in $\mathbb{P}^3$.
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### A short problem with minimal projections and biprojections

Let $(N \subset M)$ be a finite index irreducible subfactor, $P=P(N \subset M)$ its planar algebra. Notation: For $a,b \in P_{2,+}$ positive operators, then $\langle a,b \rangle$ is the biprojection ...
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### Decay of solutions to Schrodinger equation with local minimum in potential

Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by $$L = - \partial_x^2 + V$$ where $V$ is a potential with the following properties: $V$ is non-negative, ...
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### Why does this fundamental group not have elements of finite order? [duplicate]

Let $X$ be a subset of $\mathbb R^3$ with its induced topology and let $a\in X$ be a point. Then the fundamental group $\pi_1(X,a)$ seems not to have elements of finite order (except the identity of ...
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### How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...
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### Representability of deformation functors via SGA

I'm trying to understand Böckle's proof of Theorem 2.1.1 in his notes on deformation theory. Let's start with some motivation. Let $\Gamma$ be a profinite group (I'm thinking of an absolute Galois ...
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### Is $[0,1]^\kappa$ an affine complete lattice?

A $k$-ary function $f$ on a bounded distributive lattice $L$ is called compatible if for any congruence relation $\theta$ on $L$ and $(a_i, b_i)\in \theta$ for $i=1,\ldots,k$ we always have ...
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### Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected. Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...
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### Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...
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### Embedding of classical into intuitionistic linear logic

Following on from this recent question, there is another construction that is well-known, but I don’t know a good primary source for: the Kolmogorov-style double-negation embedding of classical into ...
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### If $(X_n+Y_n)$ has bounded variance, is the same true for $(X_n)$ and $(Y_n)$? [on hold]

let $(X_n)$ and $(Y_n)$ be two sequences of random variables defined on the same probability space such that the variance of all components $X_n$, $Y_n$ is finite and the sequence of variances of ...
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### When does the integral of a Dolbeault-exact form vanish?

What conditions (if any) can be imposed on a Kahler manifold $M$ so that we get a Dolbeault analogue of Stokes' theorem on a closed manifold, i.e. $\int_M \partial ( ... ) =0$ The trivial solution ...
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### Normal basis with cyclotomic units

Let p be an odd prime integer and let $\zeta$ be a primitive p-th root of unity. Let $\alpha$ be a non-trivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form ...
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### Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact). Question 1. Is it true that every point of $X$ has a ...
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### conjugate operation on vector bundle

Is the conjugate operation on $\overset{\sim}{K}(\mathbb{C}\mathbb{P}^n)$ known? If so, can I get the full formula at least in terms of the basis $\eta^i$? Here $\overset{\sim}{K}(X)$ denotes the ...
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### Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis? Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and ...
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Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is finite, flat and locally of finite presentation. Then I can prove that the set $$U:= \{ y\in Y : X_y \to y \hspace{1mm} \text{is ... 0answers 25 views ### Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR Let D^n be the closed unit ball in \mathbb{R}^n. Given a compact, n-dimensional, AR(Absolute Retract) metric space X, must it happen that either X embeds in D^n or D^n embeds in X? ... 1answer 131 views ### NCG with all noncommutativity in a nilpotent ideal While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ... 0answers 30 views ### Minimize Product of Sums of Squared Distances The Question Given two sets of vectors S_1 and S_2，we want to find a unit vector s such that$$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
I need to find the first zero (smallest positive root) of the solution of the initial value problem $ry''+y'+f(r)y=0, \ \ y(0)=y'(0)=1$ for certain $f \in C^{\infty}(R)$. One can easily use ...