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A question about finite free convolution

Mon, 12/04/2017 - 09:57

For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.

Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets define the polynomials $p_1(x)$ and $p_2(x)$ as follows : $\chi_x(A)=(x-a)p_1(x)$ and $\chi_x(B)=(x-b)p_2(x)$. Then over uniform sampling from the permutation group $S_n$ one can show (quite a non-trivial proof) that ``finite free convolution" (denoted as $\boxplus$) satisfies the following identity,

$$\mathbb{E}_{P \sim S_n} [\chi_x(A + PBP^T)] = (x-(a+b))[p_1(x) \boxplus p_2(x)]$$

Given a $n-$dimensional symmetric matrix $M$ such that, $\chi_x(M) = (x-1)^{\frac {n}{2}}(x+1)^{\frac {n}{2}}$ define a polynomial $p$ such that $\chi_x(M)=(x-1)p(x)$. Now apparently the following identity holds for any positive integer $d$,

$$\underset{P_1,P_2,..,P_d \sim S_n}{\mathbb{E}}[\chi_x(P_1MP_1^T+ P_2MP_2^T+..+P_dMP_d^T)]\\ = (x-d)[p(x)\boxplus p(x) ..(d \text{ times})..\boxplus p(x)]$$

Can someone kindly help derive the second equality from the first?

I believe this is some kind of an induction but I am unable to get it work. As in even at $d=3$ I cant get this explicitly.

Optimizing input of an unknown function

Mon, 12/04/2017 - 01:40

Suppose we have a machine which takes the input $x_{in}$. In this machine the variable $x_{in}$ is converted to $y_{in}$ with the function $f(x)$, $f(x_{in})=y_{in}$. $f(x)$ is a known function, but not very easy to evaluate.

Secondly the machine is externally measured. This gives a measurement $x_{out}$. Assuming the measurement device has no errors, then there is phenomenon that converts $y_{in}$ to $x_{out}$ by a function, which we call $g(y)$. Since we don't know this phenomenon, $g(y)$ is unknown.

The machine works correct when for every $x_{in} > 0$, $x_{in}$ and $x_{out}$ are close together. To accomplish this, it is possible to set two parameters into the machine. Let's call those parameters $a$ and $b$. These paremeters are used in the following way: we take $y_{in}$ and set $y_{new} = a\times y_{in} + b$.

Since we do not know the function $g(y)$, we do not know what effect those parameters have on the measured output $x_{out}$. So in fact the problem here is about minimizing the following:
$||x_{in} - x_{out}|| = ||x_{in} - g(a\times y_{in} + b)||$
over $a$ and $b$, with the unknown function $g(y)$.

At the moment those parameters are set by using trial and error, but it can take up to two days to get the best set of parameters.

Now I've read some things about

  • Simulated Annealing
  • Black box optimization
  • Surrogate modelling

But I'm not sure if I am looking in the right direction. Or if this problem is even solvable without trial and error. If it is solvable is there someopne who can give me some good referecences to this type of problems?

Gromov-Hausdorff relative compactness without curvature restrictions

Sun, 12/03/2017 - 09:31

A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by Burago-Burago-Ivanov lists a few other such compactness theorems, but they are all dependent on some variant of global curvature bounds.

My question is, are there examples of such compactness theorems which do not use assumptions on the curvature? In other words, is it reasonable to expect some kind of compactness theorem under uniform control of a combination of other geometric quantities like volumes of balls, curve lengths, diameter, injectivity radius etc., but not curvature?

This is mainly a reference request, I am trying to get a feel for whether there is a metamathematical principle which tells us that one cannot expect relative compactness in the Gromov-Hausdorff metric without curvature restrictions.

degree of a line bundle corresponding to an effective divisor is positive

Sun, 12/03/2017 - 09:05

Let $M$ be a compact kahler manifold with kahler form $\omega$. For any line bundle $\mathcal{L}$ on $M$,we define the '$\omega - degree$' of $\mathcal{L}$ to be $deg(\mathcal{L}) :=\int_{M}c_{1}(\mathcal{L})\wedge \omega ^{n-1}$.

How does one show that if $D$ is an effective cartier divisor on $M$ and $\mathcal{O}(D)$ be the corresponding line bundle, then $deg \mathcal{O}(D)\geq0$ ? The book I'm reading (Kobayashi's differential geometry on complex vector bundles) states that the integral above actually becomes $\int_{D} \omega^{n-1}$ and thus concludes from there, but I can't figure out why it's true. Thanks in advance!

Frobenius automorphisms of cohomology of a variety

Sun, 12/03/2017 - 09:03

Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are related.

  • If $p$ is a prime of good reduction for $X$, then $H^n_{dR}(X)\otimes\mathbb{Q}_p$ depends functorially on the special fiber of an integral model of $X$. The Frobenius endomorphism of the special fiber induces an automorphism $F_p\in GL\big(H^n_{dR}(X)\otimes\mathbb{Q}_p\big)$.

  • Fix a prime $\ell$ and an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$. Then $H^n_{et}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})$ comes with an action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ that is unramified for all but finitely many primes $p$. For unramified $p$, there is a well-defined conjugacy class $\Phi_p\subset GL\big(H^n_{et}(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})\big)$ coming from the conjugacy class of the Frobenius at $p$ in $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$.

What is the relationship between $F_p$ and $\Phi_p$?

Infinite projective plane with small edges

Sun, 12/03/2017 - 08:40

Let $\kappa$ be an infinite cardinal. We say $E\subseteq {\cal P}(\kappa)$ is an infinite projective plane on $\kappa$ if

  1. $e_1\neq e_2\in E$ implies $|e_1\cap e_2| = 1$, and
  2. whenever $n\neq m\in \kappa$, there is $e\in E$ with $\{n,m\}\subseteq e$.

Is it possible to find an infinite projective plane $E$ on an infinite cardinal $\kappa$ such that for all members of $E$ we have $|e|<\kappa$?

Does there exist homoegeneous polynomial $F$ with its partial derivatives satisfying the following inequalities?

Sun, 12/03/2017 - 08:36

Let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d > 1$ and $n \geq 2$. I am interested in finding an example (or maybe that such $F$ doesn't exist?) with the following properties.

1) $W = \{ \mathbf{x} \in \mathbb{R}^n : F(\mathbf{x}) = 0, \nabla F \not = \mathbf{0} \} \cap (\mathbb{R}_{>0})^n \not = \emptyset$.

2) For all $\mathbf{z} \in W$, we have $\frac{\partial F}{\partial x_i} \cdot \frac{\partial^2 F}{\partial x_i^2} \leq 0$ for every $i \in \{1, ..., n\}$.

I would greatly appreciate if someone could provide me an example of such polynomial, or an argument on why such polynomials do not exist. My guess is that ``most'' random choice of polynomial with non-empty $W$ will not satisfy 2) but I wasn't really sure... Any helpful comments are appreciated as well. Thank you very much.

Map between homology of spectra

Sun, 12/03/2017 - 08:17

Let $X$ be a suspension spectra whose $BP$-homology is infinitely generated ($BP_*(X) = \Sigma^d BP_*/I$, where $I$ has the form $I=(v_0^{i_0}, \dots , v_n^{i_n})$ such that the homology is a $BP_*(BP)$ coalgebra).

Let $C_nX$ the fiber of the map $ X \to L_nX$ and let $\Sigma C_n X$ be its cofiber.

What can be said about the natural map $$BP_*(\Sigma C_n X) \to BP_*(\Sigma C_{n-1} X) ? $$

My guess would be that it's always injective, but i'm not entirely sure about it.

Thanks

Smooth dependence on parameters for an ODE

Sun, 12/03/2017 - 08:01

We consider the ODE

$$ icv'+v''+v(1-|v|^2)=0,~{in}~\mathbb{R}$$

(from non-linear Schrodinger equation), where $c \in \mathbb{R}.$ In the article, the author claims that :

" By standard results on smooth dependence on the parameters for an ODE, $c\mapsto v=v(c) $ and $c\mapsto 1-|v(c)|^{2}$ are smooth with values into any Sobolev space $W^{s,p}(\mathbb{R}$) and have exponential decay "

What the author is refering to? I don't find/know any theorems/results which can give me that.

http://math.unice.fr/~chiron/fichierspdf/TW1d.pdf (page 6)

Matrix sieve - an alternative for sieve of Eratosthenes? [on hold]

Sun, 12/03/2017 - 07:49

----------I have derived following theorem

Odd integer $N=6p+5$ is a prime number iff neither of two diophantine equations

$6x^2−1+(6x−1)y=p$

$6x^2−1+(6x+1)y=p$

has solution.

Odd integer $N=6p+7$ is a prime number iff neither of two diophantine equations

$6x^2−1−2x+(6x−1)y=p$

$6x^2−1+2x+(6x+1)y=p$

has solution.

$x=1,2,3,..y=0,1,2,...p=0,1,2,..$

Note: All primes (except 2 and 3) are in one of two forms

$6p+5$ or $6p+7$

Proposed theorem can be formulated as "matrix sieve": Positive integers which do not appear in both 2-dimensional arrays:

|5 10 15 20...| P1(i,j)= |23 34 45 56...| |53 70 87 104...| |95 118 141 164...| |149 178 207 236...| |... ... ... ... | | 5 12 19 26 ..| |23 36 49 62...| P2(i,j)= |53 72 91 110...| |95 120 145 170...| |149 180 211 242...| |... ... ... ...|

$P1(i,j)=6i^2-1+(6i-1)(j-1)$

$P2(i,j)=6i^2-1+(6i+1)(j-1)$

are indexes $p$ of primes in the sequence $S1(p)=6p+5$. Positive integers which do not appear in both 2-dimensional arrays

|3 8 13 18 ..| |19 30 41 52...| |47 64 81 98...| P3(i,j)= |87 110 133 156...| |139 168 197 226...| |... ... ... ... | |7 14 21 28 ..| |27 40 53 66...| P4(i,j)= |59 78 97 116..| |103 128 153 178...| |159 190 221 252...| |... ... ... ... |

$P3(i,j)=6i^2-1-2i+(6i-1)(j-1)$

$P4(i,j)=6i^2-1+2i+(6i+1)(j-1)$

are indexes $p$ of primes in the sequence $S2(p)=6p+7$.

Is proposed "matrix sieve" useful for number theory?

Brownian sausage surgery of Poisson point process

Sun, 12/03/2017 - 07:29

Let $\mathcal P$ be a unit intensity Poisson point process on $\mathbb R^d$. Fix $r>0$ and let $W_t = \cup_{s \leq t} \mathbb B(B_t,r)$ be the Brownian sausage around a Brownian motion $B_t$. Run the process until the time $\tau = \inf \{ t \colon W_t \cap \mathcal P \neq \emptyset\}$ that the sausage hits a point in $\mathcal P$.

Now, let $\mathcal P'$ be an independent unit intensity Poisson point process. Define the set $$\mathcal P'' = (\mathcal P - W_\tau) \cup (\mathcal P' \cap W_\tau).$$ So we are taking out the point that $W_\tau$ hit and putting back in $W_\tau \cap \mathcal P'$.

Is $\mathcal P''$ a unit intensity Poison point process?

Finding a PA cut in a nonstandard model of PA

Sun, 12/03/2017 - 06:53

For a certain project I am currently working on, I need to be able to find PA cuts in nonstandard models of PA, in desirable intervals. For example, I wonder if the following is true, where $\newcommand\PA{\text{PA}}\PA_k$ refers to the $\Sigma_k$ fragment of $\PA$.

Question. If $M$ is a model of $\PA$ in which $\PA_{k-1}$ is consistent, but $\PA_k$ is not (so $k$ is nonstandard), then is there a $\PA$ cut in $M$ above $k$ in which $\PA_k$ is consistent?

That is, I want to cut $M$ below the first proof of a contradiction in $\PA_k$, but above $k$, and have $\PA+\text{Con}(\PA_k)$.

Alternatively, is there some other $\Sigma_1$ property of $k$, other than $\neg\text{Con}(\PA_k)$, such that I can always find a $\PA$ cut in $M$ between $k$ and the witness of that property? Kameryn Williams suggested that the Paris-Harrington result may provide this, since it is designed to ensure $\PA$ cuts below the corresponding PH-Ramsey number. But I would need, however, that one can always end-extend the model so as to make the $\Sigma_1$ property true. Does the PH construction have both these features?

With the consistency statements, for example, for any nonstandard $k$ in any model $M$ of $\PA$, there is always an end-extension of $M$ to a model of $\PA$ with $\neg\text{Con}(\PA_k)$.

Variation of Pursuit-evasion (Cops and Robbers)

Sun, 12/03/2017 - 06:50

I am considering to investigate on a variation of the cops and robber game where the robber is considered as an "invisible evader" for their location is unknown until one of the cops are at an adjacent node to the robber.

I was just wondering whether this has been previously studied by any individuals and if so, how much. It would be really helpful if I can be directed to that study.

Thank you.

Boundary conditions for trace theorem in Sobolev spaces

Sun, 12/03/2017 - 06:34

The classic boundary condition for trace theorem is $C^m$, (Ref: Adams R A, Fournier J J F. Sobolev spaces[M]. Academic press, 2003.), but in practice, we always encounter polygonal domain, this domain has some non-smooth points.

So is there a trace theorem for strong Lipschitz boundary conditions, or even weaker boundary condition? and can we generalize trace theorem in the smooth manifold with corners?

For Goldbach's conjecture I need to rule of two sequence [on hold]

Sun, 12/03/2017 - 05:05

Consider the sequence in $\Bbb N\times\Bbb N$:

$(0,0),(1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),...,(1,2k-2),(2,2k-3),...,(k-1,k),(k,k-1),...,(2k-3,2),(2k-2,1),(1,2k-1),(2,2k-2),...,(k,k),...(2k-2,2),(2k-1,1),...$

Question $1$: To define an Abelian group structure on $\Bbb N$ that is not a finitely generated Abelian group and be isomorphic to $(\Bbb Q,+)$, I need to know: what is rule of this sequence?

Consider the sequence in $\Bbb N\times\Bbb N$:

$(1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),...,(1,2k-2),(2,2k-3),...,(k-1,k),(k,k-1),...,(2k-3,2),(2k-2,1),(1,2k-1),(2,2k-2),...,(k,k),...(2k-2,2),(2k-1,1),...$

Question $2$: To define an Abelian group structure on $\Bbb N$ that is not a finitely generated Abelian group and be isomorphic to $(\Bbb Q\setminus\{0\},\times)$, I need to know: what is rule of this sequence?

Thanks in advance.

Corporate salesman problem

Sun, 12/03/2017 - 04:16

Asalesman is employed by a large corporation. He has a $n$ cities to visit, connected by roads, forming a graph. But as travel takes a lot of time, he has to pick hotels between visits. He cannot take any hotel he wishes; rather there are precisely $m$ hotels where he may rest.

He has to plan his travel in such way that after visiting $p$ cities, he has to visit a hotel. We may generalise it to say he has to visit $q$ different hotels. (Maybe it will be traveling celebrity problem?)

So basically he has a graph $G(E,V)$, where $E$ are the edges, $V$ the nodes, and two sets of nodes: $C$ (cities) and $H$ (hotels) with $C \cup H = V$ and $|C|=n$, $|H|=m$. Find a path in the graph starting at one of the $C$ nodes, ending at a different $C$ node and forming pattern $(p,q)$, $p$ nodes from set $C$, then $q$ nodes from set $H$, then repeat. The path may not visit every $H$ element and it may visit some of $H$ elements many times, but it has to visit every $C$ node once.

So it is like finding a Hamiltonian path but with "rests".

  1. Does this problem have a name or it is something new?
  2. In what cases does it have a solution? It probably depends both on the numbers of nodes $p$, $q$, and where on the graph they are located.
  3. How can we find the shortest path, ignoring hotel costs?
  4. What is a way to find an optimal solution (cheapest travel) if every hotel cost is the same?
  5. What is the optimal solution when costs of hotels are different?

about transfer from Hilbert modular forms to Siegel modular forms

Sun, 12/03/2017 - 04:03

Suppose $F$ is a totally real field of degree $d$. Is there an explicit way (like theta series or so) to construct automorphic forms on $G_{sp}(2d)$ from Hilbert modular forms of ${\rm GL}_2(F)$?

If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?

Sun, 12/03/2017 - 02:34

This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$, namely, $f=u(h)$ and $g=v(h)$ for some $u(t),v(t) \in k[t]$; the answer is positive.

Is it possible to replace the field $k$ by an integral domain $D$? Namely: If $f,g \in D[x,y]$ are two algebraically dependent polynomials over an arbitrary integral domain $D$, is it true that there exists a polynomial $h \in D[x,y]$ such that $f,g \in D[h]$?

Denote the field of fractions of $D$ by $Q(D)$. It is clear that if $f,g \in D[x,y] \subset Q(D)[x,y]$ are two algebraically dependent polynomials over $D$, then from the above question there exists a polynomial $h \in Q(D)[x,y]$ such that $f,g \in Q(D)[h]$, namely, $f=u(h)$ and $g=v(h)$ for some $u(t),v(t) \in Q(D)[t]$.

I do not see why, for example, $D[x][y] \ni f=u(h)=u_mh^m+\cdots+u_1h+u_0$ should imply that $h \in D[x,y]$ and $u_j \in D$ (changing variables does not seem to help, namely if the leading term is $cy^l$, with $c \in Q(D)$).

Any comments are welcome.

The two ways Feynman diagrams appear in mathematics

Sun, 12/03/2017 - 02:23

I've heard about two ways mathematician describe Feynman diagrams:

  • They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in monoidal closed category.

  • They are combinatorial tools that allows to give formulas for the asymptotical expansion of integrals of the form:

$$ \int_{\mathbb{R}^n} g(x) e^{-S(x)/\hbar} $$

when $\hbar \rightarrow 0$ in terms of asymptotical expansion for $g$ and $S$ around $0$ (with $S$ having a unique minimum at $0$ and increasing quickly enough at $\infty$ and often with a very simple $g$, like a product of linear forms), as well as some variation of this idea, or for the slightly more subtle ``oscilating integral'' version of it, with $e^{-i S(x)/\hbar}$.

My question is: is there a relation between the two ?

I guess what I would like to see is a "high level" proof of the kind of formula we get in the second point in terms of monoidal categories which explains the link between the terms appearing in the expansion and arrows in a monoidal category... But maybe there is another way to understand it...

Endomorphism algebras of indecomposable quiver representations

Sat, 12/02/2017 - 23:40

Let $Q$ be a wild quiver without oriented cycles and let $V$ be an indecomposable representation of $Q$. Assume that $V_i\neq 0$ for each vertex $i$ of $Q$. The base field $k$ is algebraically closed. If $V$ is not a Schur representation, $\operatorname{End}V$ is a local $k$-algebra different from $k$, so there is a non-trivial nilpotent endomorphism. What I would like to know is the following: does there exist a nilpotent endomorphism $\phi$ such that for each vertex $i$ $\phi$ is not zero at $V_i$?

I have a large body of examples in which this question has positive answer, however I still can't prove it in full generality... am I missing some key example?

This question arised while solving problems for a homework sheet for a course on quiver representations, however it does not help me in any way to solve that problem, which was simply to prove that $V$ is indecomposable iff $End V$ is local.

Edit: forgot the key word, I want $\phi$ to be nilpotent.

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