# Math Overflow Recent Questions

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### Why is the definition of entropy solution necessary to prove uniqueness for hyperbolic conservation laws?

Mon, 03/12/2018 - 13:30

I'm aware that there are a lot of counterexamples to show that distributional solutions for hyperbolic (scalar) conservation laws are not unique.

1. Conceptually, at which point of a proof of uniqueness is the definition of distributional solution not enough to go on?

2. Why is the definition of entropy solution useful in the proof of uniqueness for hyperbolic conservation laws?

### Coupled Sylvester equations

Mon, 03/12/2018 - 12:33

Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero.

I an looking for two matrices $T_1$ and $T_2$ belonging to $\mathbb{R}^{n \times n}$ such that: \begin{eqnarray} \alpha A T_1 + T_1 B &=& \gamma C T_2 \\ \beta A T_2 + T_2 B &=& \delta C T_1 \end{eqnarray}

1. Is there any known result on when these coupled Sylvester equations admit a solution?
2. If there is a solution, when is it unique (as a function of the scalars $\alpha,\beta,\gamma,\delta$) ?

Edit: as was mentioned by Carlo Beenakker, the equations admit the trivial solution $T_1 = T_2 = 0$. What if now, we assume both $T_1$ and $T_2$ non-singular?

### Integral domain over which any non-constant, one variable, irreducible polynomial has degree 1

Mon, 03/12/2018 - 08:42

Let $R$ be an integral domain such that every non-constant, irreducible polynomial $f(X) \in R[X]$ has degree $1$.

Q. is it true that $R$ is a field?

If $0 \ne a \in R$ , then $X^2-a$ is reducible in $R[X]$. Since this polynomial has content $1$ , we must have a factorization into one degree polynomials $X^2-a=(cX+d)(eX+g)=ecX^2+(de+cg)X+dg$. So $ec=1, de+cg=0, dg=-a$. So $d=-c^2g$, hence $a=-dg=(cg)^2$ . So every element of $R$ is a perfect square in $R$ . So if $R$ is a factorization domain with only irreducible non-constant polynomials in $R[X]$ being of degree $1$, then $R$ has no irreducible elements , hence $R$ must be a field.

I have no idea about what happens if $R$ is not a factorization domain.

### Decomposition into Weyl modules

Mon, 03/12/2018 - 08:26

Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (*), an ascending chain $$0 = V_0 \subset V_1 \subset V_2 \subset \dots$$ of submodules of a $G$-module $V$ is called a Weyl filtration of $V$ if $V = \bigcup_{i \geq 0} V_i$ and if each $V_i/V_{i-1}$ is isomorphic to some Weyl module $V(\lambda_i)$ of highest weight $\lambda_i$.

It seems that if a module admits such a Weyl filtration, we have $V \cong \bigoplus_{i} V(\lambda_i)$. Indeed, according to remark II.4.19 in Jantzen, we can order the $\lambda_i$ such that $\lambda_i < \lambda_j$ implies $i > j$. The decomposition $V \cong \bigoplus_{i} V(\lambda_i)$ then follows by induction from $\operatorname{Ext}^1_G(V(\lambda),V)=0$ if $V$ has no weight $\mu > \lambda$ (see Jantzen II.2.14 Remark 2).

In particular I would like to use that we can write the symmetric square of a Weyl module as a direct sum of Weyl modules and that we can derive the terms from Weyl's character formula. This should follow from Proposition II.4.21 in Jantzen which basically says (the dual of) if $V$ admits a Weyl filtration then so does $V \otimes V$.

However, this results seems quite standard and I am quite surprised that Jantzen doesn't mention it (as far as I know). So my question is whether there is a more direct approach towards this problem and whether anyone knows a good reference.

EDIT. The reasoning in the second paragraph is not correct. We would want the $\lambda_i$ to be ordered such that if $\lambda_i < \lambda_j$ then $i < j$. I am not sure whether this is possible. Probably it is not, which explains why this isn't in Jantzen.

(*) Jantzen, Representations of Algebraic Groups (AMS, 2003)

### Simplification of integral on the sphere

Mon, 03/12/2018 - 08:22

In the article: https://arxiv.org/abs/0906.3217 the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, if $h : \Bbb{S}^2 \to \Bbb{R}$ is a $C^2$ function, $Hess(h)(X,Y) = \langle \nabla_X \nabla h,Y\rangle$ is the Hessian of $H$ and $H(h)$ is the determinant of the Hessian, then $$\int_{\Bbb{S}^2} H(h) dA = \frac{1}{2} \int_{\Bbb{S}^2} |\nabla h|^2 dA$$ where $dA$ is the area element on $\Bbb{S}^2$. This formula is helpful and simplifies some aspects of my numerical computations. Here and in the following, $\nabla$ and $\Delta$ represent the tangential gradient and Laplace-Beltrami operator on $\Bbb{S}^2$.

The above formula is proved in connection with bodies of constant width, but in the proof in the article they don't seem to use this fact. In my numerical experiments the formula gives the expected result in the general case. However, I work in the case where $H(h)>0$ (I don't know if this is relevant or not...)

I was wondering if it is possible to obtain a similar simplification for the integral $\int_{\Bbb{S}^2} h H(h) dA$? More precisely, is it possible to obtain something of the form $$\int_{\Bbb{S}^2} h H(h) dA = \int_{\Bbb{S}^2} \mathcal{F}(h,\nabla h,\Delta h)dA$$ where $\mathcal{F}$ is "simple" (polynomial)?

### Equivalence of finiteness of $spliG$ and periodicity isomorphisms being induced by cup product

Mon, 03/12/2018 - 07:04

I am trying to prove that the following are equivalent for a group $G$ with periodic cohomology with period $q$ after $k$ steps:

$(i)\ spliG<\infty$ (where $spliG$ is the supremum of injective length of $\mathbb{Z}G$ - projective modules)

$(ii)$ There is an element $g\in H^q(G,\mathbb{Z})$ such that the cup (Yoneda) product $\ \_\bigcup g:H^i(G,\_)\to H^{i+q}(G,\_)$ is an isomorphism for every $i>k$

It was easy to prove that $(i)\Longrightarrow (ii)$, since $(i)$ is equivalent to the following:

There is a $q$-extension of the form $0\to\mathbb{Z}\to X\to P_{q-2}\to\ldots\to P_0\to\mathbb{Z}\to0$ where $P_i$ is a projective $\mathbb{Z}G$-module for every $i=0,1,\ldots,q-2$ and $pd_{\mathbb{Z}G}X<\infty$.

and $(ii)$ is equivalent to the following:

There is a $\mathbb{Z}$-split, $\mathbb{Z}G$-exact sequence $0\to\mathbb{Z}\to X$, where $X$ is $\mathbb{Z}$-free and $pd_{\mathbb{Z}G}X<\infty$.

However, I can't figure out how to prove $(ii)\Longrightarrow(i)$.

Any help or suggestions would be greatly appreciated.

Thank you.

Edit: There is a mistake in the original question. $(i)$ and $(ii)$ have been interchanged. Actually, the fact that the periodicity isomorphisms are induced by cup product with an element $g\in H^q(G,\mathbb{Z}$ is equivalent to the existence of a $q$-extension $0\to\mathbb{Z}\to X\to P_{q-2}\to\ldots\to P_0\to\mathbb{Z}\to0$ where $P_i$ is a projective $\mathbb{Z}G$-module for every $i=0,1,\ldots,q-2$ and $pd_{\mathbb{Z}G}X<\infty$

while

$spliG<\infty$ is equivalent to the existence of a $\mathbb{Z}$-split, $\mathbb{Z}G$-exact sequence $0\to\mathbb{Z}\to X$, where $X$ is $\mathbb{Z}$-free and $pd_{\mathbb{Z}G}X<\infty$.

So, the correct question is how to prove that:

If $G$ has periodic cohomology with period $q$ after $k$ steps and $spliG<\infty$ then there exists a $q$-extension of the form $0\to\mathbb{Z}\to X\to P_{q-2}\to\ldots\to P_0\to\mathbb{Z}\to0$ where $P_i$ is a projective $\mathbb{Z}G$-module for every $i=0,1,\ldots,q-2$ and $pd_{\mathbb{Z}G}X<\infty$.

I apologize for the mix-up.

### What is the consistency strength of Z+ Accessibility?

Mon, 03/12/2018 - 06:03

Informally the axiom schema of accessibility states that for each one place function $F$ that is definable over the whole universe of discourse "in the language of set theory", like the powerset function $P$, or the singleton function $\iota$, or the set Union function $\bigcup$, etc.. for each such a function $F$ we can coin an accessibility notion [dependent on $F$] whereby a set $x$ is said to be reached (i.e. is accessible) from a subset $\alpha$ if and only if there is a set $\beta$ that is hereditarily strictly subnumerous to $x$ such that $F(\beta)$ is supernumerous to $x$ and $\alpha \subseteq \beta$ [or otherwise $x$ is subnumerous to $\alpha$]. Now the scheme states that for every set there exists a set of all sets that are hereditarily $F$ accessible from it. The real intention is to have a theory that can prove existence of large cardinals. Infinity is provable since all hereditarily finite sets are seen as hereditarily power accessible from the set $\emptyset$. All $\text{ZF}$ sets are seen as hereditarily Power accessible from $\omega_0$, and so a model of $\text{ZF}$ is definable in this theory. To get to that, we only need to add this axiom scheme on top of axioms of Zermelo set theory minus infinity and add an axiom of Transitivity that asserts that every set is a subset of some transitive set, this way we can define transitive closures and define the hereditary notions.

Formal workup:

$\text{Axiom schema of Accessibility:}$ if $F$ is a one place function symbol that is definable in the language of set theory over the whole universe of discourse, and if $\phi(Y)$ is a formula in which $X$ doesn't occur free and $Y$ occurs free and only free, then all closures of:

$$[\exists \alpha \forall Y (\phi(Y) \to Y \ ..ACC^F \ \alpha)] \to \exists X \forall Y (Y \in X \leftrightarrow \phi(Y))$$ are axioms.

Where $ACC^F$ is defined as:

$$Y \ ACC^F \ \alpha \iff Y\ ..\leq \ \alpha \lor \exists \beta \ [\beta \ ..< \ Y \wedge \alpha \subseteq \beta \wedge F(\beta) \geq Y]$$

Where generally $..R"$ denotes "hereditarily $R$" relation defined as:

$$X \ ..R \ \ Y \iff X \ R \ Y \wedge \forall m \in TC(X) [m \ R \ Y]$$

Where $TC(X)$ is defined in the customary manner as the minimal transitive superset of $X$.

Where: $x < y \iff \exists f (f:x\to y \wedge f \text{ is an injection}) \wedge \not \exists g (g: y \to x \wedge g \text{ is an injection})$

and: $x \geq y \iff \exists f (f: y \to x \wedge f \text { is an injection} )$;

and: $x \leq y \iff y \geq x$

Now the question is:

What is the consistency strength of the theory whose axioms are the axioms of $[\text{Z} - \text{INF.}] + \text{Transitivity} + \text{Accessibility}?$

Where Axiom of Transitivity is the axiom stating that every set is a subset of some transitive set.

### Dirichlet series of the k-th divisor of n^2

Mon, 03/12/2018 - 05:40

I know that the Dirichlet series of the $k$-th divisor of n is $$\sum_{n=0}^\infty \frac{d_k(n)}{n^s}= \zeta^k(s).$$ where $d_k(n)$ is the number of $k$-tuples of integers $(a_1,a_2,\cdots,a_k)$ with product $n$, i.e. $$d_k(n)= \sum_{a_1\cdots a_k=n}1,$$ is the number of divisors of $n$.

I would like to know how or where to find a formula for the Dirichlet series of $$\sum_{n=0}^\infty \frac{d_k(n^2)}{n^s}.$$

I computed when $k=2$, which is $$\sum_{n=0}^\infty \frac{d_2(n^2)}{n^s}=\frac{\zeta^3(s)}{\zeta(2s)}.$$

For $k\ge 2,$ I used the Euler product and stopped with

$$\sum_{n=0}^\infty \frac{d_k(n^2)}{n^s} = \zeta^k(s) \prod_P \frac{1}{2} \left( \left(1-P^{-s/2}\right)^k+\left(1+P^{-s/2}\right)^k\right)$$

### Berry-Esseen type theorem for Monotonic independence

Sat, 03/10/2018 - 20:05

The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative result, because it also specifies the rate at which this convergence takes place by giving a bound on the maximal error of approximation between the normal distribution and the true distribution of the scaled sample mean.

In free probability, analogue of Central Limit Theorem is known where Wigner's semi-circle law plays the role of Normal distribution. Berry–Esseen type theorem for free random variables is due to Vladislav Kargin 2007.

In monotone probability, analogue of Central Limit theorem is also known where arc-sine law plays the role of normal distribution.

But the Berry-Esseen type theorem seems to be missing.. Is it just an open problem or are there known reasons for which such a result won't be possible....thanks....

### A sheaf is a presheaf that preserves small limits

Sat, 03/10/2018 - 03:52

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.

However when reading Lurie's DAGVII he goes on to define a sheaf of spectra on an $\infty$-topos $\mathfrak X$ as a presheaf $\mathcal O:{\mathfrak X}^{op}\to \mathsf {Sp}$ which preserves small limits.

Why can the higher analogue of sheaves of rings be defined like this? My guess is that, because it is higher, it sorts out whatever problems you get when defining a normal sheaf like that. But I am seriously clueless on this matter and I would love for some helpful explanations.

### Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem

Sat, 03/10/2018 - 03:36

This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without directly invoking energy considerations. But I think it is interesting in its own right.

Suppose one has a linear, second-order partial differential operator $P$ on $\mathbb{R}^{n+1}_{(t,x)}$, strictly hyperbolic with respect to the level sets of $t$. Let $\Sigma := \{t=0\}$ and $u_0, u_1 \in \mathscr{E}'(\Sigma)$. Then there is a distributional solution $u$ of $Pu=0$ such that $u$ and $\partial_t u$ (respectively) have $u_0$ and $u_1$ as their restrictions to $\Sigma$. Now let $\xi$ be a non-zero covector at $p \in \Sigma$ such that $(p, \xi) \in WF^s(u_0) \cup WF^{s-1}(u_1)$ for some $s \in \mathbb{R}$ (as in the sister question, $WF^s$ denotes the $H^s$ wave front set of a distribution).

By strict hyperbolicity, there exist precisely two non-zero $\mathbb{R}^{n+1}$-covectors at $p$, characteristic for $P$, which give $\xi$ when pulled back by the embedding $\Sigma \hookrightarrow \mathbb{R}^{n+1}$. Call these two covectors $\eta_+$ and $\eta_-$; in co-ordinates, we may write $\eta_\pm = (\tau_\pm,\xi)$. Does it follow that either $(p, \eta_+)$ or $(p,\eta_-)$ must belong to $WF^s(u)$?

Of course, for the $C^\infty$ wave front set this would be completely standard. It is also known (I think) that $H^s$ microlocal smoothness propagates: if $u_0$ is microlocally in $H^s$ at $(p,\xi)$ and $u_1$ is microlocally in $H^{s-1}$ at $(p,\xi)$ then the resulting solution $u$ is microlocally in $H^s$ at $(p,\eta_\pm)$.

### All polynomials are the sum of three others, each of which has only real roots

Sat, 03/10/2018 - 03:32

It was asked at the Bulletin of the American Mathematical Society Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative real parts) can be divided into the arithmetic sum of two or three polynomials, each of which has positive coefficients and only nonpositive real roots.

I would like to know if the following problem is known and how it can be solved:

Can any polynomial with complex coefficients and degree $n$ can be divided into the arithmetic sum of three complex polynomials, each of which has degree at most $n$ and only real roots?

Any help would be appreciated.

### A problem with induction method [on hold]

Sat, 03/10/2018 - 03:14

Nowadays, I study maths after 10 years and I have many problems. I have a problem with induction method.

I review an example and I do not understand this comparative:

$$1^3+2^3+\dots+n^3=(1+2+\dots+n)^2$$

and this is equal:

$$(1+2+\dots+n)^2=\left(\frac{n(n+1)}2\right)^2$$

then:

$$1^3+2^3+\dots+n^3=\left(\frac{n(n+1)}2\right)^2$$

please, who can explain me this example?

Thank you very much!

### Combinatorial Optimization with "Homogenic" Rational Objective Function

Sat, 03/10/2018 - 02:30

This is a followup question to Unconstrained Rational Combinatorial Optimization, where I asked for a solution of $$\min_{\alpha \in \lbrace0,1\rbrace^n}\frac{\alpha^T w}{\alpha^T m},\quad w \in \mathbb{R}^n, \ m\in\mathbb{N}^n,\ \|\alpha\|\ne 0$$

I had accepted the Nimrod Megiddo's publication regarding combinatorial optimization with rational objective function; there is however a little difference in the kind of problem that is solved, namely that w.l.o.g.: $\ \alpha_0 = 1\$is known.

In my problem I don't however know à priori, which of the $\alpha_i$ can assumed to be $1$.

Question:
how can the "homogenic" rational objective function turned into an "inhomogenic" one, from which the solution to the original optimization problem can be retrieved?

I haven't seen that question addressed in any of the articles related to optimization problems with rational objective functions, that I checked.

### Do commutator functor and intersection commute?

Sat, 03/10/2018 - 01:17

For two subgroups $A, B$ in $G$,

$[A,A] \cap [B,B] = [A\cap B, A \cap B]$?

At least, if $G$ is free, is the left contained in the right?

### Moduli space M(0,H,1)

Fri, 03/09/2018 - 23:51

Let $X$ be a K3 surface with $Pic(X) = \mathbf{Z}H$ and $H^2 > 0$. Let $M = M_X(0,H,1)$ be the moduli space of sheaves on $X$ with Mukai vector $v = (0,H,1)$. Then how can we see that $M$ is smooth? I would like to see an argument showing that $ob(E) \in Ext^2(E,E)$ is zero.

The problem is that trace map $tr : Ext^2(E,E) \to k$ is zero. But I don't know how to show this. I think it should follow from the fact that $tr : End(E) \to O_X$ is zero?

### Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process

Fri, 03/09/2018 - 22:24

I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\mathbf{\theta} - \mathbf{S}_t)dt + \mathbf{\sigma}d\mathbf{W}_t, \end{equation*} using the vector autoregression: \begin{equation*} \mathbf{S}_t = \mathbf{A} + \mathbf{BS}_{t-1} + \mathbf{\epsilon}_t. \end{equation*}

Recall that the ideal co-integration factor corresponds to the eigenvector of $\kappa = \mathbf{I-B}$ with the largest corresponding eigenvalue. However, I am encountering an empirical situation in which $\kappa$ is not diagonalizable over $\mathbb{R}$ and has only complex eigenvalues and eigenvectors.

This situation does not seem to be treated in any of the literature, even though there appears to be no reason, a priori, that it should not arise. What are best practices for handling this situation, i.e., computing the co-integration factor when $\kappa$ is not diagonalizable?

### A generating set for injective envelope

Fri, 03/09/2018 - 22:18

Let $m$ be a maximal ideal of a commutative ring $R$ with $1$. Can we construct a generating set $\{x_i\}_{i\in I}$ for the injective envelope $E(R/m)$ of $R/m$ such that $R/m\not\subseteq\langle x_i\rangle$ for each $i\in I$?

Or is there description for a set of generation set of $E(R/m)$ ?

### Reflecting Brownian motion, Feller property

Fri, 03/09/2018 - 21:41

I have a question about reflecting Brownian motions.

Let $D \subset \mathbb{R}^d$ be a smooth domain and $X=(X_t,P_x)$ is the reflecting Brownian motion on $\overline{D}$. $X$ is defined as a Hunt process associated the following regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^{2}(\overline{D},dx)$: \begin{equation*} \mathcal{E}(f,g)=\frac{1}{2}\int_{D}(\nabla f,\nabla g)\,dx,\quad f,g \in H^{1}(D). \end{equation*} Here, $H^{1}(D):=\{f \in L^{2}(D,dx) \mid \partial{f}/\partial{x_i} \in L^{2}(D,dx),\ 1 \le i\le d\}$ and $\partial{f}/\partial{x_i}$ is the distributional derivative of $f$, and $\nabla f=(\partial{f}/\partial{x_1},\ldots,\partial{f}/\partial{x_d})$.

My motivation

When $D=\{(x,y)\in \mathbb{R}^2 \mid |y|<1/(1+x^2)\}$, I think the reflecting Brownian motion $X$ on $\overline{D}$ does not satisfy the Feller property by means of $p_{t}(C_{\infty}(\overline{D})) \subset C_{\infty}(\overline{D})$. Here, $C_{\infty}(\overline{D})$ denotes the set of continuous functions on $\overline{D}$ which vanish at infinity. I want to prove $X$ does not have Feller property. However, I couldn't prove it...

My attempt

Now, we can assume $X$ satisfies the strong Feller property by means of $p_{t}(\mathcal{B}_{b}(\overline{D}) \subset C_{b}(\overline{D})$. This property is proved for many smooth unbounded domains (see e.g. enter link description here). Then, Feller proper is equivalent to the following condition: for every $t>0$ and increasing bounded sets $B_n \in \mathcal{B}(\overline{D})$ with $\overline{D}=\bigcup_{n=1}^{\infty}B_n$, \begin{equation*} \lim_{|x| \to \infty}p_{t}(x,B_n)=0,\quad \text{ for any }n \in \mathbb{N}. \end{equation*} Here, $p_{t}(x,B_n)=P_{x}(X_t \in B_n)$. Thus, I think it is important to estimate $\lim_{|x| \to \infty}P_{x}(\tau_{B(x,r) \cap \overline{D}} \le t)$ ($\tau_{\overline{D} \cap B(x,r)}:=\inf \{t>0 \mid X_t^0 \notin \overline{D} \cap B(x,r)\}$). Because we should be able to prove $$(1)\quad\lim_{|x| \to \infty}P_{x}(\tau_{B(x,r) \cap \overline{D}} \le t)=1,\quad t>0,\ r>0$$ if $X$ does not satisfy the Feller property. For the proof of (1), I decided to use the SDE of $X$. Now, $\partial D$ is smooth enough, $X$ satisfies the following Skorohod type SDE: \begin{equation*} (2)\quad X_t-x=B_t+\int_{0}^{t}n(X_s)\,dL_s,\quad t \ge 0. \end{equation*} Here, $n=(n_1,n_2)$ is the inward unit normal vector on $\partial D$ and $L_t$ is the boundary local time of $X$.

If $t<\tau_{B(x,r) \cap \overline{D}}$, by using (2), we have \begin{align*} r&>\left|\int_{0}^{t}n(X_s)\,dL_s \right|-|B_t| \end{align*} Hence, $P_{x}(\tau_{B(x,r) \cap \overline{D}} \le t) \ge P_{x}(r+|B_t| \le \left|\int_{0}^{t}n(X_s)\,dL_s \right|)$.

Is there a nice lower bound of $\left|\int_{0}^{t}n(X_s)\,dL_s \right|$? If $|x|$ is large enough, because $L_t$ is large, I think that $\left|\int_{0}^{t}n(X_s)\,dL_s \right|$ is not too small.

### Calculate the expectation of the maximum of averaged random walks

Fri, 03/09/2018 - 21:10

Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$

Is there any principled approach to calculate or approximate this expectation in the literature? If not, can we still do something for some special distiributions of $X_1$?