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Friday, May 24th, 2019
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1:49 am
uknotting a compact manifold in the PL setting

The general position theorem asserts that any $M$ $m$-manifold unknots in $R^n$ provided $n\geq 2m+2$. The general position theorem assumes a smooth setting. Is the unknotting still hold in the PL setting? what is the lower bound on n in that case? what is the argument for unknotting a general $m$-manifold (compact, closed, not necessarily connected ) in the PL setting?

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1:49 am
Grade 12: Data Management; probability/binomial distribution - Help need!

Struggling to figure these out, I have tried on my own a few times and have yet to get an answer that is remotely close to the correct one or what would be correct. In answer please include all steps

1) it is estimated that 17% of cars are black. In a sample of 150 cars, what is the probability that less than 20 will be black?

2)it is estimated that 7% of cars are blue. in a sample of 50 cars, what is the probability that more than 10 are blue?

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1:49 am
How to prove these following identities?

Prove the followings: 1.tan2A x cotA = 1+ sec2A

2.(1+ sin2A)/ (1+cos2A) = 1/2(1+tanA)^2

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1:20 am
The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower

Let $A_0\subseteq A_1\subseteq...$ be an infinite tower of unital inclusions of finite dimensional $C^*$-algebras and $B_0\subseteq B_1\subseteq ...$ be its associated infinite tower of path algebras. Then how to show that these two towers are tower isomorphic ?

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12:45 am
Local well-posedness of the quadratic NLS on the 1D torus

What is the proof of the local well-posedness of the quadratic nonlinear Schrödinger equation $\mathrm{i} \,\partial_t u + \Delta u \pm \left|u\right| u = 0$ on the 1D torus in $H^s$ for $s > 1$ (a good reference would suffice)?

$H^s(\mathbb{T})$ is an algebra, but $\left|u\right| u$ is not of the form $u^2$, $\overline{u}u$ or $\overline{u}^2$ and so the LWP doest not immediately follow from the Banach contraction mapping principle.

The LWP should hold according to Tao's webpage (even for $s > 0$). However, the above problem is not covered by Theorem I in [Bo1993] (reference as on Tao's webpage) and in fact Remark (ii) after Proposition 5.73 states that uniqueness of the solutions is unclear.

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12:15 am
When is the conductor of an elliptic modular curve equal to its level?

Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer:

$$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$

where

$$f(E/\mathbb{Q}_p)=\begin{cases}0 & E\text{ has good reduction mod }p\\1 & E\text{ has multiplicative reduction mod }p\\ 2 & E\text{ has additive reduction mod }p\end{cases}$$

is the "exponent of the conductor of $E/\mathbb{Q}_p$" (see for example Silverman's book Advanced Topics in the Arithmtic of Elliptic Curves, chapter IV, section 10). In the case of $p=2,3$ the exponent $f(E/\mathbb{Q}_p)$ might have extra terms depending on its wild ramification.

How do you prove that $N=M$?

If $E$ had a Weierstrass equation, the proof would be straight forward since $f(E,\mathbb{Q}_p)$ is easy to calculate. You could also use the Ogg-Saito formula if you can calculate the Neron model of $E/\mathbb{Q}_p$, but this also requires a Weierstrass equation.

I know that the function field of $E/\mathbb{C}$ is equal to $\mathbb{C}(j,j_N)$ and that an algebraic relation between $j$ and $j_N$ gives an equation for $E$, but this polynomial is extremely inconvenient for calculations.

Is there a way to calculate a simple equation for $E/\mathbb{Q}$?

Thursday, May 23rd, 2019
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11:01 pm
Spectrum of a Hamiltonian which is a perturbation of Laplacian

Let $\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ be the Laplacian on $\mathbb{R}^3$. Consider an operator $H$ on complex valued functions on $\mathbb{R}^3$ $$H\psi=\Delta\psi(x) +i\sum_{p=1}^3A_p(x)\frac{\partial \psi(x)}{\partial x_p} +B(x)\psi(x),$$ where $A_i,B$ are smooth real valued functions.

I am looking for a precise result of the following approximate form: (1) if $A_i$ and $B$ are 'small' then the discrete spectrum of $H$ is non-positive. (2) If $A_i,B$ are 'large' then the discrete spectrum of $H$ contains necessarily a positive element.

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11:01 pm
Can Berthelot-Ogus be used to prove that varieties do not lift?

Given a "nice" scheme $X$ over $F_p$, Berthelot-Ogus comparison relates its crystalline cohomology to algebraic de Rham cohomology of its lift to $\mathbb{Z}_p$. The nice thing about crystalline cohomology is that it can be defined even when there is no lift.

Are there any examples when there are purely cohomological obstructions to the existence of a lift, and thus Berthelot-Ogus can be used to show that there is no lift?

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11:01 pm
Continuity of The Restriction Map Between Function Spaces

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as \begin{align} \rho:&C(X,Y)\rightarrow C(Z,Y)\\ &f\mapsto f|_Z. \end{align}

Is the map $\rho$ continuous?

I see this "type of" operation used all the time in Sheaf theory but I never stopped to wonder if it was indeed continuous?

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11:01 pm
constraints of number of variables on a production mix problem

I have a maximization problem with 10 variables. how can I express a constraint in which the number of variables can only be < = 3?

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9:45 pm
Pads Approximants of Power Series With Natural Boundaries

Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ for some set $V$ of positive integers. I call this the “set-series” of $V$. There is a beautiful theorem due to Otto Szëgo which, for the case of set-series, shows that $\varsigma_{V}\left(z\right)$ is either a rational function whose poles are simple and located at roots of unity, or that $\varsigma_{V}\left(z\right)$ is a transcendental function with the unit circle ($\partial\mathbb{D}$) as a natural boundary.

Natural boundaries generally occur as the result of singularities clustering arbitrarily close to one another. My intuition tells me that in the case where $\varsigma_{V}\left(z\right)$ has a natural boundary (example: $V=\left\{ 2^{n}:n\geq0\right\}$, $V=\left\{ n^{2}:n\geq0\right\}$, etc), the clustering singularities in question are simple poles.

I figure a good way to try to see this would be via Padé approximants. The “rigorous” statement of my intution would then be something along the lines of: for an appropriately chosen sequence of Padé approximants $\left\{ P_{n}\left(z\right)\right\} _{n\geq1}$ of $\varsigma_{V}\left(z\right)$ (where $\varsigma_{V}\left(z\right)$ has a natural boundary on $\partial\mathbb{D})$, for every $\epsilon>0$ and every $\xi\in\partial\mathbb{D}$, there is an $N_{\epsilon,\xi}\geq1$ so that, for all $n\geq N_{\epsilon,\xi}$, any pole $s$ of $P_{n}\left(z\right)$ satisfying $\left|s-\xi\right|<\epsilon$ is necessarily simple.

With the literature on Padé Approximants appears to be quite extensive (while the literature on natural boundaries appears to be comparatively paltry), I was wondering if anyone knew of anything about this question, or something similar. Insight and/or references would be most appreciated.

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9:45 pm
A closed formula for $\det(\partial/\partial U)^p\prod_{i=1}^n\prod_{j=1}^p U_{i\sigma_j(i)}$

This is a continuation of this question. Is there a simple formula for $$I(\sigma_1,\cdots,\sigma_p)=(-1)^\sigma\left(\left(\det\left(\frac{\partial}{\partial U_{ij}}\right)_{i,j=1}^n\right)^p \prod_{i=1}^n\prod_{j=1}^p U_{i\sigma_j(i)}\right)|_{U=0},$$ where all $\sigma_i\in S_n$? Equivalently $$I(\sigma_1,\cdots,\sigma_p)=\sum_{\pi\in C}\sum_{\tau_1,\tau_2\in R}(-1)^{\sigma\pi} [\tau_1\pi\tau_2=\sigma],$$ where $C$ and $R$ are the Young subgroups of $S_{pn}$ $$C=\mathrm{Sym}(1,\dots,n)\times\cdots\times\mathrm{Sym}((p-1)n+1,\dots,pn)\cong S_n^p$$ and $$R=\mathrm{Sym}(1,\cdots,(p-1)n+1)\times\cdots\times\mathrm{Sym}(n,\cdots,pn)\cong S_p^n$$ and $\sigma=\sigma_1\oplus\cdots\oplus\sigma_p\in C$.


In the linked question, Carlo Beenakker guessed and Abdelmalek Abdesselam and David Speyer both proved that $I(\sigma_1,\sigma_2)=2^{\#\mathrm{cyc}(\sigma_1\sigma_2^{-1})}$ when $p=2$. This can also be expressed in the group algebra $\mathbb{C}[S_n]$ as $$\sum_{\sigma\in S_n} I(1,\sigma)\sigma=(2+J_1)\cdots(2+J_n),$$ where the $J_k$ are the Jucys-Murphy elements.

Note that $I$ is symmetric and $I(\sigma_1,\cdots,\sigma_p)=I(\rho\sigma_1,\cdots,\rho\sigma_p)$ for all $\rho$. When $(p,n)\ge(4,3)$ or $(3,5)$, $I$ can be negative. It appears that the prime factors of $I$ are at most $p$.

Here are the values of $I(\sigma_1,\sigma_2,1)/12$ when $p=3$ and $n=2$.

Here are the values of $I(\sigma_1,\sigma_2,1)/24$ when $p=3$ and $n=3$.

Here are the values of $I(\sigma_1,\sigma_2,1)/24$ when $p=3$ and $n=4$.

When $p=3$ and $n=5$, $I/24$ takes the values $-1,0,1,2,4,6,12,36,108,324$.

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9:45 pm
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?

More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of

  • a function $$u_1 \in BV(\mathbb R^2; \mathbb R)$$ with only jump part in the derivative $$Du_1 = D^{jump} u_1$$
  • and of a function with only Cantor part in the derivative: $$u_2 \in BV(\mathbb R^2; \mathbb R)$$ with $$Du_2 = D^{cantor} u_2$$

A related more general question is Heuristic and graphic representation of BV functions and their singularities


Clearly one could take a one dimensional example $f \in BV(\mathbb R)$ and then consider $g(x_1,x_2) := f(x_1)$. However, I'd like to see a "genuinely" two-dimensional example (if it exists).

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9:45 pm
What was Smith's proof of Smith's theorem on Hamilton cycles in cubic graphs?

In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.

He attributed the result to his friend CAB Smith, but explicitly mentioned that the proof was not Smith's proof.

What was Smith's proof?

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9:17 pm
Topological vs algebraic intersection forms

Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in cohomology and intersection theory, where $q_X$ and $q_X'$ are the intersection (bilinear) forms.

Is there an example of $X$ for which the two lattices are not equivalent? What if we restrict $q_X$ to $H^2(X,\mathbb Z)\cap H^{1,1}(X)$?

Also, what can be said about the cycle map $A^1(X)\to H^2(X,\mathbb Z)\cap H^{1,1}(X)\subseteq H^2(X,\mathbb Z)$ in this case? And if we take coefficients in $\mathbb Q$?

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9:17 pm
What is the complexity of finding a third Hamilton Cycle in cubic graph?

According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known that SMITH is in PPA, but it is unknown whether is it PPA-complete.

More details of this problem can be found here: https://kintali.wordpress.com/tag/ppa-completeness/

According to Tutte[1]: Every edge of a cubic graph lies on an even number of Hamilton cycles.Consequently a cubic hamiltonian graph has at least three Hamilton cycles.

My question: Given one Hamilton Cycle, what is the complexity of finding a third Hamilton Cycle in cubic graph?

For the problem that given one Hamilton Cycle in a cubic graph, to find the second one, Thomason[2] gave an exponential time algorithm, which can be converted into a PPA-membership proof of SMITH. How about if given one Hamilton Cycle in a cubic graph, to find the THIRD one? If we still use the algorithm of Thomason, it is no better than exponential time, and it is at least in PPA. But can we do better?

[1] W.T. Tutte, On Hamiltonian circuits, J. London Math. Soc., 21 (1946), 98–101.

[2] A. G. Thomason, Hamiltonian cycles and uniquely edge colourable graphs, Ann. Discrete. Math. 3 (1978), 259-268.

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8:50 pm
Convexity and integrands. Simple proof desired

The following result is well-known:

Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have

$$H \left( (1 - \lambda)(x_1,y_1) + \lambda (x_2,y_2) \right) \geq H(x_1,y_1)^{1 - \lambda} H(x_2,y_2)^{\lambda},$$

and let $M(y)$ denote the marginal distribution obtained by integrating over $x$

$$M(y) = \int_{\mathbb{R}^m} H(x,y) \, dx.$$

Let $y_1$ $y_2 \in \mathbb R^n$ and $\lambda \in (0,1)$ be given. Then the Prékopa–Leindler inequality applies. It can be written in terms of $M$ as

$$M((1-\lambda) y_1 + \lambda y_2) \geq M(y_1)^{1-\lambda} M(y_2)^\lambda$$ which is the log-concavity for $M$.

Now, I wanted to understand this for a very simple example where $f: \mathbb R^2 \rightarrow \mathbb R:$

$$e^{-g(y)} = \int_{\mathbb R} e^{-f(y,z)} \ dz.$$

Then, I want to prove that $g''\ge 0$ if $f$ satisfies $D^2f > 0$ globally as a matrix. We assume for simplicity that $f$ is such that the above integral is well-defined.

It is easy to see that

$$g''(y) = \langle D_{yy}f \rangle_z - \operatorname{ Var}_z (D_{y}f)$$

where is the expectatio $\langle F \rangle_z(y) := \int_{\mathbb R} F(y,z) e^{-f(y,z)} \ dz / \int_{\mathbb R} e^{-f(y,z)} \ dz$

and $\operatorname{ Var}_z$ is the variance with respect to the above probability measure.

However, it is not at all clear to me from this representation why $g''\ge 0$ holds.

Is there a pedestrian way to see this from the above expression for the second derivative?

I am looking for a more "Calculus" based derivation (using the 2nd derivative) of this inequality than the usual convex-combinatorial arguments.

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8:50 pm
The set that mazimizes a holomorphic mapping on the unit sphere can be made disjoint from a quarter-circle

I am hoping the below is true. If so, I can prove this: Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert for $n=2$. Mention of related ideas/topics is also appreciated.

Let $S= \{(\cos(t),\sin(t)) : t \in [0, \frac{\pi}{2}] \} \subseteq \mathbb{R}^2 \subseteq \mathbb{C}^2$ and let $\partial B$ denote the boundary of the unit ball centered at the origin in $\mathbb{C}^2$. Suppose $F:\mathbb{C}^2 \to \mathbb{C}^2$ is an injective holomorphic mapping s.t. $F(0) = 0$ and $d F(0) = I_2$ and $F$ is not the identity. Let $M = \sup_{z \in \partial B} ||F(z)||$ where $|| \cdot ||$ is the usual Euclidean norm. Then there exists a $2 \times 2$ unitary matrix $U$ s.t. $ \{ z \in \partial B: M = ||F(z)|| \} \cap U(S) = \varnothing$.

(Here $U(S)$ is the image of $S$ after applying $U$.)

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8:50 pm
Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $ 0 \le \sigma \le \frac12$

Based on limited numerical evidence, I suspect this conjecture.

Conjecture: Fix $ 0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$), there is always a zero of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$).

Verification for several random $\sigma$ and $ 0 < t < 30000$ and for a few random intervals didn't show any counterexamples.

For $\sigma > \frac12$ it is false and on the other hand this appears counterintuitive to me.

Counterexamples? (please check for closely spaced zeros that might look like a single local minimum on a large plot).

Does this contradict something?

Even if it is true, a conditional proof probably will be hard yet welcome.

For Siegel $Z$ function on the critical line RH implies this for $t$ large enough.

Maybe can be generalized to $\sigma \le \frac12$.

Plot of a random interval:

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7:30 pm
Need help series n/100n^3+1

I need help with this serie. I need to know if its convergent or divergent.

$$x = {{n} \over n^3+1}.$$

I tried with ratio test but i can't make it.

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