# All Questions

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### $l$-weights and $l$-character of finite-dimensional highest $l$-weight representation of $L\mathfrak{g}$

I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it. Problem In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study ...
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### Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
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### Upper bound involving random orthogonal projection

Let $R$ be an $n\times N$ random matrix with i.i.d. standard Gaussian entries, $n<N$, and let $M:=(RR^T)^{-1/2}R$. Let $u,v\in \mathbb{R}^N$ non-random and s.t. $u^Tv=0$ and $\|u\|>\|v\|$. I ...
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### Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...
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### Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
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### Historical (personal) examples of teaching-based research

The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...
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### G-delta sets and Lebesgue measure [on hold]

The set S of all subsets in R^n which are of the form G\N, where G is a G-delta set and N a null-set (=outer Lebesgue measure zero) coincides with the set of all Lebesgue measurable sets. How could ...
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### Question on homogeneous measures

Let $\mu$ be a strictly positive measure ($m(a)=0$ iff $a=0$) on a Boolean algebra $B$. $\mu$ is called homogeneous if it have the same Maharam type on every $b\in B$. By additive measure algebra I ...
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### Largest ball with fixed center in a a convex region

Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...
assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$, under what condition, we have ${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow ... 1answer 170 views ### Existence of a “quasi-uniform” probablility distribution on$\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 ... 0answers 66 views ### 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 ... 0answers 20 views ### 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 ... 0answers 20 views ###$\mu$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 ... 1answer 29 views ### 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 ... 1answer 141 views ### 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 ... 0answers 46 views ### 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 ... 1answer 77 views ### 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 ... 2answers 392 views ### 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? ... 1answer 70 views ### 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 ... 3answers 171 views ### 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$\mathbb{P}^3$. 1answer 18 views ### 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 ... 1answer 61 views ### 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, ... 0answers 205 views ### 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 ... 0answers 77 views ### 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 ... 1answer 64 views ### 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 ... 1answer 34 views ### 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 ... 1answer 105 views ### 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 ... 1answer 154 views ### 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 ... 1answer 102 views ### 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 ... 1answer 81 views ### 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 ... 0answers 52 views ### 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 ... 0answers 82 views ### 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 ... 1answer 81 views ### 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 ... 0answers 59 views ### 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 ... 0answers 61 views ### 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 ... 0answers 133 views ### Locus where morphism is étale is open on target 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 $$U:= \{ y\in Y : X_y \to y \hspace{1mm} \text{is ... 0answers 25 views ### Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR Let D^n be the closed unit ball in \mathbb{R}^n. Given a compact, n-dimensional, AR(Absolute Retract) metric space X, must it happen that either X embeds in D^n or D^n embeds in X? ... 1answer 126 views ### NCG with all noncommutativity in a nilpotent ideal While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative ... 0answers 26 views ### Minimize Product of Sums of Squared Distances The Question Given two sets of vectors S_1 and S_2，we want to find a unit vector s such that$$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ... 0answers 22 views ### Global error estimates for numerical solutions of ODEs in Matlab or Mathematica [on hold] 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 ... 0answers 33 views ### Local time for reflected random walk [on hold] Say I have a process starting from 0, and last for 100 steps, each step either moves up or down by one unit, within the boundary -10 and 10. My understanding is that expected hitting time would be ... 0answers 34 views ### Need a calculator to evaluate function at irregular input values [on hold] I'm trying to evaluate the magnitude of an appreciably complex transfer function using a variety of input frequencies. Because I'm lazy, I really don't want to have to scroll around in the function ... 1answer 107 views ### The image of the Hurewicz map for rational loop spaces Let$K$be the rationalization of a simply-connected finite CW complex. Then the Samelson product gives$\pi_*(\Omega K)$the structure of a graded Lie algebra, and the Hurewicz map$h: \pi_*(\Omega ...
Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert }$. Let $A$ be the adjacency matrix of the graph. Then define the quantity \$\phi(S)= ...