Math Overflow Recent Questions


General result for the N-point correlation of the Poisson process (and its derivative)?

Tue, 03/20/2018 - 14:30

I am specifically interested in computing: $$\mathbb{E}[S_p S_q S_s S_t]$$ where $S_t=\frac{dN_t}{dt}$ and $N_t$ is a Poisson process (so $S_t$ is a "Poisson pulse train"): $$\mathbb{P}(N_t=n)=\frac{(\lambda t)^n}{n!}e^{-\lambda t} \; , \; t>0$$

Here is my attempt:

Assuming $t>s>p>q$, I tried to compute it from the process $N_t$ rewriting it with the independent random variables $N_t-N_s, N_s-N_p, N_p-N_q, N_p$, that we can rename $N_j$, $j=1,2,3,4$: $$\mathbb{E}[N_p N_q N_s N_t]=\mathbb{E}[\prod_{k=1}^4\sum_{j=1}^k N_j]$$

This unfolds a combinatorics which somewhat looks like the one we obtain for normally distributed variables using the Isserlis / Wick theorem.

Is there an analogous theorem about such correlation function which applies to the Poisson process?

This looks simpler for the pulse train since the only term which does not vanish when we take the derivative is: $$\mathbb{E}[S_p S_q S_s S_t]=\frac{d^4}{dpdqdsdt}\mathbb{E}[N_p N_q N_s N_t]=\frac{d^4}{dpdqdsdt}\big(\mathbb{E}[N_1]\mathbb{E}[N_2]\mathbb{E}[N_3]\mathbb{E}[N_4]\Big)=\lambda^4$$ However, this is without considering coincidences among $t,s,p,q$. From here, I see (p.10): $$\mathbb{E}[S_s S_t]=\lambda^2+\lambda \delta(s-t)$$

I deduce that we should have something like $$\mathbb{E}[S_p S_q S_s S_t]=\lambda^4+\lambda^3\sum_{i\neq j}\delta(t_i-t_j)+\lambda^2\sum_{i\neq j,\; k\neq l }\delta(t_i-t_j)\delta(t_k-t_l)+\lambda\sum_{i\neq j,\; k\neq l,\; m\neq n }\delta(t_i-t_j)\delta(t_k-t_l)\delta(t_m-t_n)$$

Is this correct (except the sloppily specified summation indices)? Do you know any related general result?

Reflecting Brownian motions on horns are not Feller?

Tue, 03/20/2018 - 06:20

I have a question about Feller property of reflecting Brownian motions.

Let $D \subset \mathbb{R}^2$ be a domain. Assume $D$ is represented as \begin{equation*} D=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid |y|<H(|x|)\}. \end{equation*} Here, $H$ denotes a smooth function on $[0,\infty)$.

Let $X_t$ be the reflecting Brownian motion on $\overline{D}$ and $X_t^1$ be the first coordinate of $X_t$. According to Pinsky's heuristic argument (see p.4 of 1), the behavior of $\rho(t)=|X_t^1|$ over the long run should be like the behavior of the one dimensional diffusion generated by \begin{equation*} \frac{1}{2}\frac{d^2}{d \rho^2}+\frac{C H'(\rho)}{H(\rho)(H'(\rho)^2+1)}\frac{d}{d \rho}, \end{equation*} where $C$ is some positive constant. Hence, if $H'(\rho) \to 0$ as $\rho \to \infty$, the above diffusion is not much different from the one generated by \begin{equation*} \frac{1}{2}\frac{d^2}{d \rho^2}+\frac{C H'(\rho)}{H(\rho)}\frac{d}{d \rho}. \end{equation*}

My question

Let $H=\exp(-x^4)$. If we believe the above heuristic argument, the inner drift of $|X_t^1|$ becomes stronger as $X_t$ goes to infinity. Hence, I think $X_t$ is not Feller process by means of $p_{t}(C_{\infty}(\overline{D})) \subset C_{\infty}(\overline{D})$. But I couldn't prove this claim which is seemingly correct.

Do you know how to prove this?

Variants of reflection principle

Mon, 03/19/2018 - 07:12

This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.

Let us state the first definition of reflection.

Definition(Reflection): A stationary set $S$ in $[H_\lambda]^\omega$ reflects at $X\subseteq H_\lambda$ if $S\cap [X]^\omega$ is stationary in $[X]^\omega$.

The following is the definition of reflection principle which widely is used in literature. See Jech's book, for example.

Definition(Reflection principle) We say ${\rm RP(\lambda)}$ holds if every stationary set of $[H_\lambda]^\omega$ reflects at a set of size $\aleph_1$. We also let ${\rm RP}$ denote ${\rm RP}(\lambda)$ for all regular $\lambda\geq\aleph_2$.

Now we define another definition of reflection:

Definition(Reflection$^*$): A stationary set $S$ in $[H_\lambda]^\omega$ reflects if there is a continuous $\in$-chain $\langle M_\alpha:\alpha<\omega_1\rangle$ of elementary substructures of $H_\lambda$ such that $\{\alpha<\omega_1:M_\alpha\in S\}$ is stationary in $\omega_1$. Now using this definition we can build a principle:

Definition(Reflection principle$^*$) We say ${\rm RP}^*(\lambda)$ holds if every stationary subset of $[H_\lambda]^\omega$ reflects. We also let $RP^*$ denote ${\rm RP}^*(\lambda)$ for all $\lambda\geq\aleph_2$.

The following theorem is easy.

Theorem: ${\rm RP}^*$ implies ${\rm RP}$.

Both of these definitions are used and known in literature as reflection principle. Note that ${\rm SRP}$ (Strong reflection principle) implies ${\rm RP}^*$. I would like to know if they are not equivalent.

Question: Is there any model of ${\rm RP}+\neg {\rm RP}^* $?

Can we estimate the error $\left| \frac{1}{N^2} \sum f ( \{ \sqrt{2} m + \sqrt{3} n \} ) - \int_0^1 f(x) \, dx \right|$?

Mon, 03/19/2018 - 06:47

As a computer experiment I did a few Riemannian sums to see if I could quantify the density statement $\overline{\mathbb{Q}(\sqrt{2}, \sqrt{3})} = \mathbb{R}$ :

$$ \Big| \frac{1}{N^2} \sum_{0 \leq m,n \leq N} \{ \sqrt{2} m + \sqrt{3} n \}^5 - \frac{1}{6} \Big| \stackrel{?}{<} \frac{1}{N^2} $$

A log-plot shows the correct exponent is a bit less than $2$. Is it a Hausdorff dimension of some kind?

The general quantitative statement looks like some error term to an average:

$$ \Big| \frac{1}{N^2} \sum_{0 \leq m,n \leq N} f \big( \{ \sqrt{2} m + \sqrt{3} n \} \big) - \int_0^1 f(x) \, dx \Big| \stackrel{?}{\ll} \frac{1}{N^2} $$

This is certainly false... what might the a good exponent be? The statement could be more general - I have some kind of totally real number field - but then we get a worse exponent.

Relative version of Whitney's embedding theorem

Mon, 03/19/2018 - 05:53

In Milnor's lectures on H-cobordism theorem, there is a theorem without a proof. There are references, but I don't have access to them.

Theorem: Let $f \colon M \to N$ be a smooth map which is an embedding on a closed subset $A \subseteq M$. Provided $\dim N > 2 \dim M$ there exists a smooth embedding $g \colon M \to N$ homotopic to $f$ and such that $g|_A = f|_A$.

Is it possible to prove this theorem "by hand", i.e. without function spaces?

Calculate integral with help the Euler's integrals [migrated]

Mon, 03/19/2018 - 04:49

It is my first question. In advance please sorry for my bad English!

I need to calculate this integral with help the Euler's integrals: $$ \int_0^{+\infty} \frac{1}{1+x^5} $$

I have tried decompose integrand in Taylor Series but I did not get anything. Also I tried use partial fractions and I got crazy expression.

I'm here to get an elegant solution of this question. Thank you for help in advance.

Vanishing sums of roots of unity

Mon, 03/19/2018 - 04:45

Lets say I have a multivariate polynomial $P(x_1, \ldots, x_n)$ of degree at most $d = d(n) = n^c$ for a fixed constant $c$. I know that the Kronecker map ($x_i \to x^{d^i}$) preserves zeroness/nonzeroness of a polynomial, that is, $P(x_1, \ldots, x_n) \neq 0$ iff $P(x^{d^1}, \ldots, x^{d^n}) \neq 0$ .

Instead of this univariate map, if I substitute $x_i = \omega_N^{d^i}$ where $\omega_N$ is the $N$-th ($N = d^{n^2}$) root of unity, is the zeroness/nonzeroness still preseved? If yes, what is the best lower bound known on this non-vanishing sum of roots of unity?

On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down

Mon, 03/19/2018 - 04:41

Let $R$ be an integral domain with fraction field $K$ such that for every ring $R \subseteq S \subseteq K$ , the ring extension $R \subseteq S$ satisfies Going Down property i.e. for any chain of prime ideals $P_1 \subseteq P_2$ in $R$ and any prime ideal $Q_2$ in $S$ with $Q_2 \cap R=P_2$, there is a prime ideal $Q_1$ of $S$ such that $Q_1 \cap R=P_1$ .

My question is : If $0\ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is principal for every $b\in R$ ?

Are two triangles with equal corresponding medians , congruent?

Mon, 03/19/2018 - 04:38

Is the hyperbolic or spherical analogy of the following Euclidean fact, true?

Two triangles with equal corresponding medians are congruent.

More precisely: Assume that $\Delta ABC$ and $ \Delta A'B'C'$ are two triangles in the hyperbolic space $\mathbb{H}^2$ or elliptic space $\mathbb{S}^2$ such that $$AM_1=A'M_1',\; BM_2=B'M_2',\;CM_3=C'M_3'$$ where $M_i (M_i')$ in $XM_i(X'M_i')$ is the mid point of the edge opposite to the vertex $X(X')$, respectively.

Under this condition, is there an isometry of the corresponding space carrying the first triangle to the second one?

Non-isomorphic graphs with diameter two

Mon, 03/19/2018 - 03:50

Let $n>1$ be an integer and $[n] = \{1,\ldots,n\}$. What is an example of non-isomorphic simple, undirected graphs $G_i = ([n], E_i)$ for $i=1,2$ with the following properties?

  1. both $G_1$ and $G_2$ have diameter two,
  2. $\chi(G_1)=\chi(G_2)$.

Modular functions of the type $\mathfrak f(\cdot)^{k}\mathfrak f(\cdot)^{23nk}$

Mon, 03/19/2018 - 03:39

Let $\eta(\omega)$ be the Dedekind eta function and let $$\mathfrak f(w)=e^{-\pi i/24}\frac{\eta((\omega+1)/2)}{\eta(\omega)}.$$

In his paper On the “gap” in a theorem of Heegner, Stark fills the gap in the Heegner's proof by producing a certain transformation equation for the function $\mathfrak f(\omega)^{24}$. In order to do this, he considers the following set of functions (where $n \geq 1$ is odd and $k \geq1$ divisible by $3$)

$$\mathcal F(n,k)=\bigg\lbrace \mathfrak f(\frac{r\omega+s}{t})^k\mathfrak f(\omega)^{23nk}:r,s,t \in \mathbb Z\bigg\rbrace$$

the numbers $r,s,t$ satisfying $$r>0,\\ 16 \mid s,\\0 \leq s \leq 16t,\\rt=n,\\(r,s,t)=1.$$

Now Stark claims that the set $\mathcal F(n,k)$ is permuted under the transformations $$\omega \mapsto \omega +2,\\\omega \mapsto -1/\omega .$$

These two transformations generate a congruence subgroup $G$ of the modular group of level $2$ and index $3$. We know how the function $\eta(\omega)$ transforms under the modular group, thus we can infer the behavior of $\mathfrak f(\omega)$ under $G$. How can we investigate the properties of the functions $$\mathfrak f \circ \begin{pmatrix} r & s \\ 0 & t \end{pmatrix}$$ under the action of $G$? Stark refers to Section 73 of Weber's Lehrbuch der Algebra but unfortunately I don't speak German. Is there some English reference?

Banach space properties defined by compact operators, strictly singular operators and strictly cosingular operators

Mon, 03/19/2018 - 02:29

Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by the space of all strictly singular operators from $X$ to $Y$ and $SC(X,Y)$ by the space of all strictly cosingular operators from $X$ to $Y$.

It was proved in Theorem 2.4.10 [F. Albiac and N. Kalton, Topics in Banach space theory] that for every Banach space $Y$, one has $\mathcal{K}(c_{0}, Y)=S(c_{0},Y)$.

Definition 1. We say that a Banach space $X$ has the Bessaga-Pełczyński property I (this name is used temporarily here) if $\mathcal{K}(X,Y)=S(X,Y)$ for every Banach space $Y$.

Question 1. Is there any Banach space enjoying the Bessaga-Pełczyński property I besides $c_{0}$?

Question 2. Is the Bessaga-Pełczyński property I interesting?

Definition 2. We say that a Banach space $Y$ has the Bessaga-Pełczyński property II if $\mathcal{K}(X,Y)=SC(X,Y)$ for every Banach space $X$.

It was proved by Pełczyński in 1965 that $l_{1}$ has the Bessaga-Pełczyński property II.

Question 3. Is there any Banach space enjoying the Bessaga-Pełczyński property II besides $l_{1}$?

Thank you!

Complete intersections in toric varieties

Mon, 03/19/2018 - 02:17

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?

Symmetric C* property

Mon, 03/19/2018 - 01:54

This is a cross-post from

What can be said about elements $a$ of a C*-algebra which fulfil the 'symmetric C* property' $$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$

I'd guess that this is not a general property, but I don't have a good idea which elements (except for self-adjoint and normal ones, of course) should satisfy this.

Sequence problem find its character [on hold]

Mon, 03/19/2018 - 00:23

The sequence $(y_n)$ satisfies the relationship ${y_{n-1}}{y_{n+1}} + y_n = 1$ for all $n \ge 2$. If $y_1= 1$ and $y_2= 2$, What can you say about the sequence? What happens for other starting values?

On cardinality of generating subsets of some submodules

Sun, 03/18/2018 - 12:49

Let $R$ be a commutative ring with unity. Let $\alpha$ be an infinite cardinal . Let $M$ be an $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be such that $rm \in N$, $\mu (N+rM) < \alpha$ and $\mu (N+Rm) < \alpha$ . Then is it true that $\mu (N) < \alpha$ ?

If this is not true in general for all infinite cardinal $\alpha$, can we atleast characterize those $\alpha$ for which it is true ? In particular, is it true for $\alpha= \aleph_0$ ?

NOTE : For an $R$-module $M$, by $\mu (M)$ we denote the minimal no. of generators of $M$ . So $\mu (M) < \alpha$ means $M$ can be generated by a set $S \subseteq M$ such that $|S| < \alpha$ .

Critical points of Dirichlet L functions

Sun, 03/18/2018 - 10:58

Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached through real values of $s$, $1 < s < \infty$ .

For example, let $\chi_4$ be the non-trivial character $(\mathrm{mod}\ 4)$ taking values $(1,0,-1,0)$ at $(1,2,3,4)$. We have $L(1,\chi_4) < 1$ and, by an alternating series argument, $L(s,\chi_4)$ is monotonically increasing for $s > 1$.

Next consider $\chi_7$ $(\mathrm{mod}\ 7)$ taking values $(1,1,-1,1,-1,-1,0)$ at $(1,2,3,4,5,6,7)$. Note that $L(1,\chi_7) > 1$. However, $L(s,\chi_7)$ is not monotone decreasing for $s > 1$, but first increases with a peak near $s = 1.1$ before decreasing. What can one say in general?

Questions: 1) Are there infinitely many $\chi$ (real-valued as above) for which $L(s,\chi)$ has no critical points on the interval $1 < s < \infty$ (i.e. monotonic)?

2) Is it possible for $L(s,\chi)$ to have more than one critical point in $(1,\infty)$? If so, is there any upper bound on the number of such critical points which holds for all $\chi$?

3) Are there any books or papers which display graphs of $L$-functions on the real line?


Inertia of a class of Cayley graphs

Sun, 03/18/2018 - 03:39

Let $H^n_2(d)$ be the Cayley graph with vertex set $\{0,1\}^n$ where two strings form an edge iff they have Hamming distance at least $d$. What is the inertia of these graphs, that is, the numbers of positive, negative and zero eigenvalues? Thank you.

monadic decomposition

Sat, 03/17/2018 - 21:17

Let $\mathrm{F}: \mathcal{C} \rightleftarrows \mathcal{D} : \mathrm{G} $ be an adjunction with associated monad $\mathrm{T} = \mathrm{G} \mathrm{F} .$

If $\mathcal{D} $ admits coequalizers of $\mathrm{G} $-split pairs, then the comparison functor $ \bar{\mathrm{G}}: \mathcal{D} \to \mathrm{Alg}_{ \mathrm{T}} (\mathcal{C}) $ admits a left adjoint $ \bar{\mathrm{F}}.$

So we obtain an adjunction $ \bar{\mathrm{F}}: \bar{ \mathcal{C} }:= \mathrm{Alg}_{ \mathrm{T}} (\mathcal{C}) \rightleftarrows \mathcal{D} : \bar{\mathrm{G}} $ with associated monad $\bar{\mathrm{T}} = \bar{\mathrm{G}} \bar{\mathrm{F}}.$

If $\mathcal{D} $ admits coequalizers of $\bar{\mathrm{G}} $-split pairs, then the comparison functor $ \mathcal{D} \to \mathrm{Alg}_{ \bar{\mathrm{T}}} (\bar{ \mathcal{C} }) $ admits a left adjoint.

So if we assume that $\mathcal{D} $ admits coequalizers (in fact reflexive coequalizers are enough), we can iterate this process starting with an adjunction $\mathrm{F_1}: \mathcal{C}_1 \rightleftarrows \mathcal{D} : \mathrm{G}_1 $ and get a sequence $ (\mathrm{F_\mathrm{i}}: \mathcal{C}_\mathrm{i}\rightleftarrows \mathcal{D} : \mathrm{G}_\mathrm{i})_{ \mathrm{i} \geq 1 } $ of adjunctions, where $\mathrm{G}_\mathrm{i}: \mathcal{D} \to \mathcal{C}_\mathrm{i}:= \mathrm{Alg}_{ \mathrm{T}_{\mathrm{i}-1} } ( \mathcal{C}_{\mathrm{i}-1} ) $ is the comparison functor for the adjunction $ \mathrm{F_{\mathrm{i}-1}}: \mathcal{C}_{\mathrm{i}-1}\rightleftarrows \mathcal{D} : \mathrm{G}_{\mathrm{i}-1} $ with associated monad $\mathrm{T}_{\mathrm{i}-1} $ if $\mathrm{i} > 1.$

Write $\mathcal{C}_{\infty}$ for the limit of the diagram $ ... \to \mathcal{C}_3 \to \mathcal{C}_2 \to \mathcal{C}_1$ so that we obtain a functor $\mathcal{D} \to \mathcal{C}_{\infty}$.

The functor $\mathcal{C}_{\infty} \to \mathcal{C}_1$ is right adjoint and conservative.

If $\mathcal{D}$ and all the categories $ \mathcal{C}_{\mathrm{i}} $ are presentable, all the right adjoint functors $ \mathrm{G}_{\mathrm{i}} $ preserve small limits and are accessible so that $\mathcal{D} \to \mathcal{C}_{\infty}$ preserves small limits and is accessible and so admits a left adjoint by the adjoint functor theorem.

Does the functor $\mathcal{D} \to \mathcal{C}_{\infty}$ admit a left adjoint without the presentability assumption on $\mathcal{D}?$

Denote $\mathrm{T}_\infty$ the monad associated to the adjunction $\mathcal{C}_{\infty} \rightleftarrows \mathcal{D}.$

It is tempting to believe that the monadic forgetful functor $\mathrm{Alg}_{\mathrm{T_\infty}} ( \mathcal{C}_\infty) \to \mathcal{C}_\infty $ is an equivalence. This is equivalent to the condition that $\mathrm{T_\infty}$ is the identity monad or that the left adjoint of the functor $\mathcal{D} \to \mathcal{C}_\infty $ is fully faithul. Is this true?

What can one say more about $\mathcal{C}_{\infty}$ and the functor $\mathcal{D} \to \mathcal{C}_{\infty}$?

Propagation of Singularities

Sat, 03/17/2018 - 14:01

I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - - to prove the Propagation of Singularities Theorem.

Since $WF(u)$ is a closed set and a bicharacteristic curve $\gamma$ is a connected set, the only fact to prove is that $WF(u) \cap \gamma$ is an open subset of $\gamma$. Thus, given a $(x_0,\xi_0) \in WF(u) \subset \operatorname{Char} a$, we want to prove that the bicharacteristic curve $(x(t),\xi(t)) $starting at this point is locally contained in $WF(u)$. We will proceed by contradiction, assuming that this is not true.

I could not see where the contradiction hypothesis -there is a point $(x(s),\xi(s)) \not\in WF(u)$ for an $s$ in the domain of definition of $\gamma$ - is used in the proof (Raymond, p. 83).

Can anybody help me?

Update: I think I understood, it remains to show two things: $\operatorname{supp} c(s) \subset \operatorname{supp}c_0 \subset e^{sH_q}\operatorname{supp}c_0$ and $w(s)=c(s,c,D)v=\chi \in C_0^{\infty}$.

Then, by a reversion in the time we will have that $\tilde{w}(t)=w(t+s)$ is solution of the problem $\partial \tilde{w}(t)-ib(x,D)\tilde{w}(t)=g(t+s)$ with $\tilde{w}(0)=\chi \in C_{0}^{\infty}$. Hence, $\tilde{w} \in \bigcap_{k} C^{0}(I:H^{k})$. In particular, $\tilde{w}(-s)=w(0)=c(x,D)v \in C^\infty$, which proves that $(x_0,\xi_0) \not\in WF(u),$ since $c_0(x_0,\xi_0)\neq 0.$

P.S: I have posted this in Math Stack Exchange but didn't get a reply and so I have posted it here. The page mentioned is in the link below.