Recent MathOverflow Questions

Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball

Math Overflow Recent Questions - Sat, 10/13/2018 - 12:51

Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [-1,1]$ for $n=0,1,\dots$ is a sequence of functions that are $L$-Lipschitz with regards to the max metric on $[-1,1]^n$, i.e. $$|f_n(x_0,\dots,x_{n-1})-f_n(y_0,\dots,y_{n-1})|\leq L \max\{|x_0-y_0|,\dots,|x_{n-1}-y_{n-1}|\}$$ for any $n$ and $x_0,\dots,x_{n-1},y_0,\dots,y_{n-1}\in [-1,1]$. Note that each of these functions interpreted as a function on the $\ell^\infty$ unit ball is weak* continuous and therefore the function $f(x_0,x_1,\dots)=\sup_nf_n(x_0,\dots,x_{n-1})$ is weak* semicontinuous (and in particular measurable). Also note that $f$ is still $L$-Lipschitz with regards to the $\ell^\infty$ norm.

Let $Y=f(X_0,X_1,\dots)$ be the random variable given by applying $f$ to the sequence $X_i$. I want to be able to say that $Y$ cannot be too sharply bimodal with a bound given by $L$. To be more precise I want to say that there is some $r(L)<\frac{1}{2}$ such that $\min\{P(Y\leq\frac{1}{3}),P(Y\geq \frac{2}{3})\}\leq r(L)$ for any sequence of $L$-Lipschitz functions $f_n$.

Does such an estimate exist for every $L$?

Casimir operator of su(2) and relation with its matrix representation

Math Overflow Recent Questions - Sat, 10/13/2018 - 12:47

I'm following Gilmore's recipe to compute the Casimir operator of a given algebra (in this example, I refer to algebra su(2)). My problem is that I obtain different results according to the specific matrix representation that I choose. To be more concrete, look at the difference between these two cases:

1. Consider the defining commutators: $$[J_1,J_2]=iJ_3, \qquad [J_2,J_3]=iJ_1, \qquad [J_3,J_1]=iJ_2$$ The following $3\times 3$ matrices are a representation of this algebra: $$J_1=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \\ \end{array} \right), \qquad J_2=\left( \begin{array}{ccc} 0 & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{array} \right), \qquad J_3= \left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right).$$ Applying Gilmore's method (see pag. 140), one can write matrix $$X= \sum_{i=1}^3a_iJ_i, \qquad a_i\in\mathbb{R}$$ One therefore obtains: $$X=\left( \begin{array}{ccc} 0 & -i a_3 & i a_2 \\ i a_3 & 0 & -i a_1 \\ -i a_2 & i a_1 & 0 \\ \end{array} \right)$$ At this point one computes the characteristic polynomial $$P(\lambda)=\mathrm{det}(X-\lambda\mathbb{I})=-\lambda^3+\lambda(a_1^2+a_2^2+a_3^2)$$ Performing the substitution $a_i\to J_i$ in the coefficient of $\lambda$, as prescribed by the algorithm, one obtains that the algebra's Casimir is $$C=\left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{array} \right)$$ which indeed commutes with $J_i, \quad \forall i$. This scheme therefore gives a correct result.

2. Now consider the following alternative (but equivalent) definition of the algebra su(2):
$$[J_+,J_-]=2 J_3, \qquad [J_3,J_+]=+J_+, \qquad [J_3,J_-]=-J_-.$$ The following $3\times 3$ matrices are a representation of this algebra: $$J_+= \left( \begin{array}{ccc} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 0 & 2 & 0 \\ \end{array} \right), \qquad J_-=\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ -2 & 0 & 0 \\ \end{array} \right), \qquad J_3=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right)$$ Introducing three unknown coefficients $a_+,\,a_-,\,a_3$, one can computes matrix $X$ and the characteristic polynomial $P(\lambda)$ as seen before. One obtains that $$P(\lambda)= -\lambda^3 +\lambda(a_3^2+4a_+a_-)$$ At this point, one has to perform the substitution $a_i\to J_i$. Even if one takes into accunt the need for symmetrization, i.e. $a_+a_-\to (J_+J_-+J_-J_+)/2$, the algorithm retrieves an incorrect result, i.e. the following matrix $$\bar{C}=\left( \begin{array}{ccc} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \\ \end{array} \right)$$ which does not commute with matrices $J_+$ and $J_-$ and so constitutes a wrong result.

My question is: why does the first way give the correct result but the second scheme does not work? Did I miss any hypothesis which is needed by Gilmore's algorithm? I suspect that the problem might be either in the use of $J_\pm$ as basis elements of algebra su(2) or in the substitutions $a^i\to J_i$ (the book uses, in fact, upper and lower indices, a formalism which I am not familiar with). Please, take into account that my target is to understand how to compute the abstract Casimir operator of a certain Lie algebra in an algorithmic way. The matrix representation of linear and quadratic operators is just auxiliary to reach the target but it is not my core business.

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

Math Overflow Recent Questions - Sat, 10/13/2018 - 12:23

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $\mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.$$ The $\cup_1$ is a higher cup product. The $Sq^1 x= x \cup_1 x$. It shall be true that $$\mathcal{P}(x) \mod 2= x \cup x.$$

Question 1: If $x \cup x =0 \mod 2$, and if $x \cup_1 Sq^1 x =0 \mod 2$, is it true that $$\mathcal{P}(x) =0 \mod 4?$$

• If not, please provide some counter examples.

--

Question 2: $\mathcal{P}(x)$ is a well-defined invariant for the cobordism $\Omega^4_{SO}(B^2 \mathbb{Z}_2)=\mathbb{Z}_4$. Is $$\frac{1}{2}(\mathcal{P}(x) -x^2) \mod 2 = x \cup_1 Sq^1 x \mod 2$$ a well-defined invariant of the cobordism $\Omega^4_{Spin}(B^2 \mathbb{Z}_2)=\mathbb{Z}_2$?

• If not, please provide the correct way to write the cobordism generator of $\Omega^4_{Spin}(B^2 \mathbb{Z}_2)=\mathbb{Z}_2$.

Laplacian of an infinite graph and connected components

Math Overflow Recent Questions - Sat, 10/13/2018 - 11:33

For a finite graph with undirected, unweighted edges, a well-known result is that the dimension of the null space of the Laplacian matrix gives the number of connected components. Does this result apply to infinite graphs as well?

The infinite graphs I'm interested are locally finite. That is, the degree of each node is finite. In my case, the number of nodes is countably infinite, there are no self-edges, and the edges are undirected.

At least according to the PDF from this course: http://www.maths.nuigalway.ie/~rquinlan/linearalgebra/section3-1.pdf

the connected components theorem does not assume anything about finite graphs. Can someone provide a reference (a paper or text) where this is explicitly discussed?

Thank You!

What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$

Math Overflow Recent Questions - Sat, 10/13/2018 - 10:19

or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?

The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutter paper, "Generalizations of the Kunen inconsistency", Annals of Pure and Applied Logic, 163 (2012) pp. 1873-1876 (Section 1, "A few metamathematical preliminaries").

In order that this question should not be deemed overbroad, answers must be limited in scope to specific mathematical difficulties (eg. the Erdos-Hajnal theorem used in in proving the Kunen inconsistency in $NGBC$ or $NBC$ + $Choice$ has no known proof in $NGB$ or is known not to be provable without choice--however, those who would say that the Erdos-Hajnal theorem cannot be proven without choice must show how this fact allows for a non-trivial elementary embedding $j$: $V$ $\rightarrow$ $V$ and the existence of the requisite critical point $\kappa$ in $NGB$). Those who are so inclined might make mention of Rupert McCallum's recent attempt in showing that "the existence of a non-trivial elementary embedding $j$: $V_{\lambda + 2}$ $\rightarrow$ $V_{\lambda + 2}$ for some ordinal $\lambda$...is not consistent with $ZF$" (quote from McCallum's withdrawn short preprint, The Choiceless Cardinals are Inconsistent, arXiv:1712.09768, which is discussed on Joel David Hamlin's blog — I would be especially interested in getting a nice explanation of the gap in this argument and the Laver paper which Gabriel Goldberg directed him to).

Though at first glance the answers to these questions might seem "primarily opinion-based", the mathoverflow community should realize the value of informed opinion and (yes) informed speculation regarding this topic, and use this question as an opportunity to provide these 'clues' (think of the proof of the 4-color theorem) to both the mathoverflow community and the mathematical community as a whole.

Improvement of Chernoff bound in Binomial case

Math Overflow Recent Questions - Sat, 10/13/2018 - 01:44

We know from Chernoff bound $P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where $X$ follows Binomial($N, \frac{1}{2}$).

If I take $N=1000, \epsilon=0.01$, the upper bound is 0.82. However, the actual value is 0.27. Can we improve this Chernoff bound?

Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

Math Overflow Recent Questions - Fri, 10/12/2018 - 20:31

The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, then $\mathcal{C}_1\boxtimes\mathcal{C}_2 =A\otimes B\operatorname{-mod}$ is independent of the choice of $A$ and $B$ (since Morita equivalence of one of the factors will induce Morita equivalence of the tensor product).

Is there some clever name for this operation?

Apologies if I should know this; it's not such an easy thing to Google for.

Lattice points in a square pairwise-separated by integer distances

Math Overflow Recent Questions - Fri, 10/12/2018 - 18:08

Let $S_n$ be an $n \times n$ square of lattice points in $\mathbb{Z}^2$.

Q1. What is the largest subset $A(n)$ of lattice points in $S_n$ that have the property that every pair of points in $A(n)$ are separated by an integer Euclidean distance?

Is it simply that $|A(n)| = n$? And similarly in $\mathbb{Z}^d$ for $d>2$?

$5 \times 5$ lattice square, $5$ collinear points.

Q2. What is the largest subset $B(n)$ of lattice points in $S_n$, not all collinear, that have the property that every pair of points in $B(n)$ are separated by an integer Euclidean distance?

$5 \times 5$ lattice square, $4$ noncollinear points. A $9 \times 9$ example with $5$ noncollinear points, also based on $3{-}4{-}5$ right triangles, is illustrated in the Wikipedia article Erdős–Diophantine graph.

Behaviour of direct limit with matrices

Math Overflow Recent Questions - Fri, 10/12/2018 - 02:02

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:

Let $(A_n,f_n)$ be a direct sequence of $C^*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim_{\rightarrow} M_2(A_n)=M_2(\lim_{\rightarrow} A_n)$$ Where for the system $M_2(A_n)$ the connecting maps are the natural maps obtained using $f_n$ componentwise.

I do feel like the result should be true, but I don’t really have an argument. Any ideas?

Is transcendental Goldbach Conjecture true of the real numbers?

Math Overflow Recent Questions - Fri, 10/12/2018 - 00:59

Let $x0$ be the real number $Pi$, Consider the below sequence of real numbers:

• $s{0}$ = .1415926535897932384626433832795028841971...
• $s{1}$ = .415926535897932384626433832795028841971...
• $s{2}$ = .15926535897932384626433832795028841971...
• $s{3}$ = .5926535897932384626433832795028841971...
• ...

In other words, the real number $s{n+1}$ is formed by shifting the decimal expansion of $s{n}$ to the left, eliminating the first digit after the decimal point of $s{n}$. Naturally, this sequence has a $lub$ called, say, the $Major$ number of $x0$ and denoted by $Major(x0)$. Similarly, $minor(x0)$ can be defined with the $glb$ of the sequence.

In general, we can define $Major(x)$, $minor(x)$ for any real number $x$ since it has a decimal expansion. Similarly, $Major(c)$, $minor(c)$ where $c$ is a complex could also be defined.

Given Champernowne($x$) means $x$ is a Champernowne number, let's now state the, say, transcendental Goldbach Conjecture (akin to Goldbach Conjecture of the natural numbers):