1
vote
1answer
8 views

What's the formal name of this matiematical modeling problem

There is a right angle corner with the width of 1 in the both directions. I wonder the shape that can pass this corner with the largest area. I know that this is a famous problem, but what's the ...
-1
votes
0answers
7 views

Equivalence of Lie subalgebras, within a (irreducible) representation

Lie subalgebras inside Lie algebras have been classified up to equivalence, and linear equivalence (by Dynkin et al). How does one classify embeddings of a Lie algebra h inside a Lie algebra g, where ...
-1
votes
2answers
60 views

Is every algebraic $K3$ surface a quartic surface?

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
0answers
5 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 ...
2
votes
1answer
15 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, ...
2
votes
0answers
75 views

Why does this fundamental group do not have elements of finite order?

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
38 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
0answers
27 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
27 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 ...
1
vote
1answer
35 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 ...
0
votes
0answers
34 views

Existence of a projection operator onto subspace of Hilbert space

Let $V \subset H$ be Hilbert spaces with a continuous 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 orthogonal. ...
1
vote
0answers
38 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 ...
0
votes
1answer
55 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 ...
1
vote
0answers
37 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
1answer
50 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 ...
1
vote
1answer
51 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
43 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
37 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 ...
0
votes
0answers
76 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 ...
0
votes
0answers
14 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 $X$? ...
7
votes
0answers
75 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
19 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
18 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
30 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 ...
-5
votes
0answers
29 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 ...
4
votes
1answer
90 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 ...
0
votes
0answers
36 views

About expectation norms on graphs

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)= ...
1
vote
2answers
67 views

Solving $Ax=e_k$ for standard basis vector $e_k$, sparse $A$

Given a sparse matrix $A \in \mathbb{R}^{n \times m}$, are there any efficient methods for determining whether there exists an $x \in \mathbb{R}^m$ such that $Ax=e_k$, the $k^{th}$ standard basis ...
1
vote
0answers
100 views

Avoiding Fibonacci-like sequences

Suppose we are trying to avoid 3-term arithmetic progressions. There are two relevant sequences in the OEIS pertaining to this: A003278: The sequence whose $n^{\text{th}}$ term is the smallest number ...
0
votes
0answers
32 views

On OPNs and SOPNs

(I hope that this question is appropriate for this site. If it is not, please feel free to point it out and I will then cross-post to MSE.) OPNs are odd perfect numbers. SOPNs are spoof odd perfect ...
4
votes
2answers
201 views

Universal coefficient theorem for group homology and cohomology

I've been looking for any kind of universal coefficient theorem for group homology and cohomology, including dual universal coefficient theorems. However, the only things I can find are ones where the ...
2
votes
0answers
82 views

Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type. I find there are plenty of research already been done and ...
1
vote
0answers
45 views

Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...
-3
votes
0answers
43 views

linear transformation question [on hold]

T $\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$ = $\bigl( \begin{smallmatrix} a & -b \\ b & a \end{smallmatrix} \bigr)$ prove that any given matrix on image of ...
-4
votes
0answers
32 views

Integration maths problem between two curves [on hold]

Find the area of the region bounded above by the curve x^2 +y^2 = 2 and below by the curve y = x^2 please help me explain how to do it please
0
votes
0answers
17 views

Schur complement for Square root information matrix [on hold]

Consider a joint information matrix I over X and Y (both vectors). Now, I would like to get the marginal information matrix for X: $I_x$. This can be of course performed via Schur complement. Now ...
0
votes
0answers
37 views

modules over iwasawa ring

Let $G$ be a p-analytic group. Let $M$ ba a finitely generated compact $\Bbb{Z}_p[[G]]$ module. Let $I_{H_i}=Ker (\Bbb{Z}_p[[G]] -> \Bbb{Z}_p[G/G_i]). $ Let $d_i$ denote the free part of the ...
0
votes
0answers
39 views

Projective family of probability spaces

This is a crosspost of this question from MSE. I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions ...
1
vote
0answers
43 views

Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve $$\int_0^\infty\int_\Omega \nabla v\nabla ...
1
vote
1answer
173 views

Infinitely many real roots

Given a non-acyclic quiver without loops with Kac's root system associated. When do we know there are infinitely many real roots?
2
votes
1answer
89 views

Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...
3
votes
0answers
56 views

A construction with homotopy colimits and homotopy pullbacks for descent

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
4
votes
1answer
150 views

Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic

It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already ...
1
vote
0answers
57 views

Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function ...
-4
votes
0answers
44 views

Quardic Equation [on hold]

We know that a quardic equation have two roots.After solving this equation:8x^2-33x-35=0 we get two roots.The first one is:5 and the second one is:-7/8.But the root -7/8 doesn't satisfy the given ...
1
vote
0answers
54 views

Generalize Gauss-Bonnet Formula to non-simple closed curves [migrated]

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense: For a domain $\Omega$ enclosed by an non-simple closed curve ...
3
votes
1answer
106 views

A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any ...
7
votes
0answers
87 views

What is the Turing degree of $\mathbb{C}_{exp}$?

Let $\mathbb{C}_{exp}$ be the theory of the complex numbers in the language of exponential rings. I am interested in the Turing degree of $\mathbb{C}_{exp}$. As the natural numbers are definable in ...
-2
votes
0answers
18 views

Random sum of random variables, not in expectation [on hold]

If $N\geq 1$ is a finite random variable (in this case a binomial Bin(n,p) random variable conditioned to be $\geq 1$) then can we say the following? $$\sum\limits_{i=1}^N \frac{1}{N^2} ...
0
votes
0answers
121 views

Flatness and intersection of fibers

Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ ...

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