-1
votes
0answers
10 views

Adjoint of Expoenential Map

Can the adjoint of the derivative of the exponential map of a lie group, be understood via the ad operator? If so how, explicitly?
0
votes
0answers
7 views

A solvable Lie algebra

Consider the Lie algebra $\mathfrak{g}$ span by $x_1,\ldots,x_n,y_1,\ldots,y_n$ with the commutator $\left[y_i,x_i\right]=y_i, \left[x_i,x_{i+1}\right]=y_i, \left[y_i,x_{i+1}\right]=-y_i,\quad 1\leq ...
-2
votes
0answers
21 views

What are all the factors of n=p^2q^3r? [on hold]

n=p^2q^3r It has a total of 24 factors. I need to show all of the factors.
0
votes
0answers
11 views

$l$-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 ...
0
votes
0answers
11 views

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 $\nu = \nu_1 + \nu_2 + \cdots$ ...
0
votes
0answers
17 views

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 ...
2
votes
0answers
43 views

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 ...
0
votes
0answers
38 views

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 ...
0
votes
0answers
16 views

Rate-Distortion theory: What is the distribution of distortion on an optimal encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, rate-distortion theory tells us that the lowest mean-squared-error we can achieve from encoding is $\sigma^2 ...
-4
votes
0answers
35 views

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
11
votes
2answers
500 views

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 ...
-2
votes
0answers
53 views

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.
2
votes
2answers
49 views

Moment problem for discrete distributions

Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B ...
17
votes
7answers
406 views

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 ...
0
votes
0answers
35 views

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 ...
0
votes
0answers
36 views

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 ...
0
votes
0answers
43 views

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 ...
0
votes
0answers
25 views

monotonicity alike functions

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 ...
9
votes
1answer
176 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 ...
7
votes
0answers
67 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 ...
0
votes
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 ...
-2
votes
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 ...
1
vote
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 ...
2
votes
1answer
145 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 ...
-1
votes
0answers
47 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 ...
4
votes
1answer
79 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 ...
3
votes
2answers
400 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? ...
0
votes
1answer
72 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 ...
2
votes
3answers
185 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 $\mathbb{P}^3$.
0
votes
1answer
20 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 ...
5
votes
1answer
63 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, ...
4
votes
0answers
207 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 ...
0
votes
0answers
78 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 ...
1
vote
1answer
65 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 ...
3
votes
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 ...
2
votes
1answer
107 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 ...
2
votes
1answer
157 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 ...
4
votes
1answer
104 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 ...
-1
votes
1answer
84 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 ...
0
votes
0answers
53 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 ...
0
votes
0answers
83 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 ...
2
votes
1answer
82 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 ...
0
votes
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 ...
4
votes
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 ...
1
vote
0answers
135 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 ...
1
vote
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 $X$? ...
8
votes
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 ...
1
vote
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, ...
2
votes
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 ...
0
votes
0answers
34 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 ...

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