Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$ with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=0}^t HF(T/I,i)=\prod_{i=2}^n a_i.$$ I know that $I$ is a complete intersection ideal, so $T/I$ is Artinian and Gorenstein, so there exists $\tau$ s.t. $\dim_k [T/I]_{\tau}=1$ (and of course I can cut this sum after the $\tau$ index), but I can't see why I can pass from this summatory to a product of esponents. T tried some combinatorial proofs without much success.

This question is taken by these two paper (resp. page 8 and 2, both at the bottom), which I link you:

https://www.sciencedirect.com/science/article/pii/S0021869312003730

https://arxiv.org/abs/1110.0745

Can anyone help me? Thanks in advance.

This question is motivated by physics --- trying to understanding the so-called 'accidental' (i.e. non representation-theoretic) degeneracies that occur in the spectrum of the Haldane--Shastry spin chain --- but let me formulate it combinatorially. The resulting introduction is still rather long, my apologies.

**Remark.** I believe that the terminology "motif" (Haldane et al, Phys. Rev. Lett. 69, 2021 (1992)) in this question comes from thinking of the $\mu$ below as certain patterns; it is not related to "motive" in other areas of mathematics.

**Set-up.** A *motif* is a sequence $\mu=\{\mu_i\}_i$ of (strictly positive) integers that increase with step size at least two:
$$ \mu_{i+1} > \mu_i + 1 \, . \tag{*}$$
The empty motif $\mu = \varnothing$ is allowed.

Write $\mathcal{M}_N$ for the set of all motifs with entries bounded by $1\leq\mu_i\leq N-1$. Let's call such motifs *$N$-motifs*. The longest $N$-motif has length $\#\mu = \lfloor N/2\rfloor$. Clearly $\mathcal{M}_{N-1} \subset \mathcal{M}_N$, and moreover $\mathcal{M}_N \setminus \mathcal{M}_{N-1} \cong \mathcal{M}_{N-2}$ with the isomorphism given by appending the entry $N-1$. Therefore we get the recursive construction $\mathcal{M}_N \cong \mathcal{M}_{N-1} \dot\cup \, \mathcal{M}_{N-2}$ (disjoint union), and it follows that the number of $N$-motifs forms a Fibonacci sequence with offset one: $\# \mathcal{M}_N = \text{Fib}_{N-1}$.

Now consider the "energy function" $E_N\,\colon \mathcal{M}_N \longrightarrow \mathbb{R}_{\geq0}$ given by $$E_N(\varnothing)=0 \, , \qquad E_N(\mu) = \sum_{i=1}^{\#\mu} \mu_i \, (N-\mu_i) \, . \tag{#} $$ (In particular, since all $\mu_i$ are positive, this function is related to the $\ell^2$-norm of the sequence $\mu'$ obtained from $\mu$ by subtracting $N/2$ from all its entries: $E_N(\mu) = \#\mu \times N^2/4 - \|\mu'\|_2^{\,2}$.)

As an aside: to get some feeling for this function one may check that $E_N$ is maximized (for $N$ even) by the longest motif, with $E_N(1,3,5,\cdots,N-1) = N\, (N^2+2)/12$.

**Goal.** I would like to understand when $N$-motifs have the same energy.

Let's count. According to Mathematica, the number of tuples consisting of $N$-motifs with the same energy is ($N\geq 2$) $$2, 2, 4, 5, 9, 10, 19, 21, 40, 40, 79, 68, 146, 110, 234, 166, 358, 234, 514, \cdots\,.$$ Unfortunately this sequence is unknown in the OEIS, as are the increasing subsequences obtained by restricting to even or odd $N$.

Here are some examples for low $N$ of all tuples of $N$-motifs with the same energy, where I visualize a motif as a subset $\mu \subset \{1,2,\cdots,N-1\}$:

Here I have ommited the singletons of $N$-motifs with a unique energy. The number of these tuples excluding singletons per $N$ are ($N\geq 2$): $$0, 1, 1, 3, 4, 8, 13, 19, 30, 37, 65, 64, 129, 105, 222, 163, 347, 230, 501,\cdots\,.$$ Again no match with the OEIS, nor for its even/odd-$N$ increasing subsequences.

Of course there are some simple instances of $N$-motifs with the same energy: we certainly have $E_N(\mu)=E_N(\nu)$ when $\nu$ can be obtained from $\mu$ by 'reflecting' one or more of its entries as $\mu_i \longmapsto N-\mu_i$. (In particular this includes pairs of mirror-image motifs.) Let's call such motifs equivalent. Then for low $N$ all tuples (excluding singletons) with the same energy are

where I've picked the representative of an equivalence class with the lowest $\ell^1$-norm. (Reversely, the other $N$-motifs in this class are reconstructed by applying pointwise 'reflections' as above to the representative and checking that the result obeys $(*)$.)

The number of these tuples (excluding singletons) per $N$ is ($N\geq 2$): $$0, 0, 0, 0, 0, 1, 1, 4, 5, 14, 20, 39, 58, 79, 150, 129, 270, 197, 415, \cdots \, .$$ Again no match in the OEIS, nor for the even/odd-$N$ subsequences.

This still contains some further structure: if $\mu,\nu$ have the same energy then so will the pair obtained from $\mu$ and $\nu$ by including the same (up to 'reflection') entry, provided that this is allowed by $(*)$. For example, at $N=9$ the last pair in the table is obtained from the second by including the entry $6$, and at $N=11$ the second triple includes the pair $(4,6)$, $(1,5,9)$ obtained from the first pair listed --- i.e. $(4)$, $(1,9)$ --- by adding the 'reflected' entries $6=11-5$.

Finally, then, my questions are:

**Question 1.** How many tuples (excluding singletons, before or after the identification) of equal-energy $N$-motifs are there?

**Question 2.** Has this combinatorial problem appeared elsewhere in some guise? Is anything known about it?

**Bonus.** Let me just mention that the above setting has a nice generalization, which goes as follows. Given $n\in \mathbb{N}$ define an *$\mathfrak{sl}_n$-motif* to be a strictly increasing sequence $\mu=\{\mu_i\}_i$ of (strictly positive) integers for which $(*)$ is relaxed to

$$ \mu_{i+n-1} > \mu_i + n-1 \, , $$

that is, there can be at most $n-1$ adjacent $\mu_i$ at a time. For $n=2$ we reproduce the above setting.

Analogously to before, write $\mathcal{M}_{n,N}$ for the set of all such motifs with entries $1\leq\mu_i\leq N-1$. Notice the stability under increasing $n$: $\mathcal{M}_{n-1,N} \subset \mathcal{M}_{n,N}$. There is again a construction of $\mathcal{M}_{n,N}$ by recursion in $N$, now in terms of $\mathcal{M}_{n,N'}$ for $N-1\leq N' \leq N-n+1$, and the number of $\mathfrak{sl}_n$-$N$-motifs is given by is a generalized Fibonacci number.

For $n=\infty$ we just get strictly increasing sequences, i.e. subsets of $\{1,\cdots,N-1\}$, so $\mathcal{M}_{\infty,N} \cong 2^{\{1,\cdots,N-1\}}$.

The energy functional is as above (so independent of $n$), and one can ask similar questions to the above. I have not thought about this yet.

**Related problems.** Variations in another direction are obtained by replacing $(\#)$ by another energy function. In particular, for the Frahm--Polychronakos spin chain the energy is just the $\ell^1$-norm:

$$E'_N(\mu) = \sum_{i=1}^{\#\mu} \mu_i = \|\mu\|_1$$

If one then takes $n=\infty$ the above problem is related to integer partitions with distinct parts, whose number (by Euler) is equal to the number of partitions with odd parts, and for which the generating function is known.

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, **connected**, and smoothly bounded.

I am concerned with the subsets $$ A = \ker (H^1(Y)\to H^1(\Sigma)), \quad B=\ker (H^1(Y)\to H^1(\Sigma^*))$$ of the first de Rham cohomology of $Y$. The maps are induced by the restriction maps.

My conjecture is that one of the following three statements must hold true:

- $A=0$ or
- $B=0$ or
- $A=B$ and $\dim A=1$.

Can you help me prove or disprove this conjecture?

**Why I expect the conjecture to be true:**
Honestly, I have only very crude intuitive arguments for this. If $a\in A\subset\mathbb R^3$ is nonzero, then $a$ can be written as a gradient in $\Sigma$. This means that there cannot be a closed loop $\gamma$ in $\Sigma$ "going in the direction of $a$", meaning that the fundamental group class of that loop in $Y$ (considered as an element in $\mathbb R^3$) is non-orthogonal to $a$. But the existence of such a loop is obstructed only by $\Sigma^*$, which means that there must be some sort of two-dimensional plane inside $\Sigma^*$ which is "orthogonal" to $a$. But then, given any direction $b\in\mathbb R^3$ orthogonal to $a$, we can find a closed loop in that plane (thus in $\Sigma^*$) "going in the direction of $b$". This shows that $b\notin A$.

**What I have already done:**
I have already looked at the Mayer Vietoris sequence, but it does not seem to yield enough information. But it helps me to draw conclusions in case I already know the conjecture to be true. Indeed, denoting by $k$ the number of connected components of $\partial\Sigma$, we then know that
$$\begin{align*}
1 &\ge\dim A\cap B = \dim\ker(H^1(Y)\to H^1(\Sigma)\oplus H^1(\Sigma^*)) \\
&= \dim \operatorname{im} (H^0(\partial \Sigma)\to H^1(Y)) \\
&= k - \dim\ker(H^0(\partial\Sigma)\to H^1(Y)) \\
&= k - \dim\operatorname{im}(H^0(\Sigma)\oplus H^0(\Sigma^*)\to H^0(\partial\Sigma) = k-1,
\end{align*}
$$
showing that $\partial\Sigma$ can have at most $2$ connected components.
I have no independet proof of this result, so my second question would be if this statement is true.

I find the following mind-boggling.

Suppose that runner $R_1$ runs distance $[0,d_1]$ with average speed $v_1$. Runner $R_2$ runs $[0,d_2]$ with $d_2>d_1$ and with average speed $v_2 > v_1$. I would have thought that by some application of the intermediate value theorem we can find a subinterval $I\subseteq [0,d_2]$ having length $d_1$ such that $R_2$ had average speed at least $v_1$ on $I$. This is not necessarily so!

**Question.** What is the smallest value of $C\in\mathbb{R}$ with $C>1$ and the following property?

Whenever $d_2>d_1$, and $R_2$ runs $[0,d_2]$ with average speed $Cv_1$, then there is a subinterval $I\subseteq [0,d_2]$ having length $d_1$ such that $R_2$ had average speed at least $v_1$ on $I$.

Does there exist an explicit example of a

- Ricci-flat, non-flat metric on a closed manifold?
- Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and metrics on surfaces)?

Can we obtain the following result for ... $f(x)={x-\lfloor x \rfloor}$ ... ? Here ${\lfloor x \rfloor}$ is floor function with $a\in \mathbb{R}$ and $u \in \mathbb{R}$ . Thank you for your kind comment.

$$ \frac{{\frac {d} {du}}\left[\int_1^u f(x) \cdot x^{-a-1} dx\right]} {{\frac {d} {du}}\left[\int_1^u f(x) \cdot x^{a-2}dx\right]} =u^{1-2a} $$

$\newcommand{\Q}{\Bbb Q} \newcommand{\Z}{\Bbb Z}$

My question is the following:

What is known about number fields $K$ fulfilling the condition $C_{g,K}$ "there is a smooth projective curve of genus $g$ over $K$, having everywhere good reduction" for some $g \geq 1$ ?

By the work of Fontaine and Abrashkin, it is known that for every $g>0$, the field $K = \Q$ does not satisfy $C_{g,\Q}$. Notice that the condition $C_{g,K}$ (for some $g>0$) implies, by the functorial properties of the Jacobian, the property $A_K$ "there is a non-zero abelian variety over $K$ having good reduction everywhere". (Using the theory of Néron models, it should be equivalent to state that there is no non-trivial abelian scheme over the ring of integers $O_K$ (see here, where it is also explained that a CM abelian variety has potentially "good reduction everywhere")).

Thus a closely related question is:

Do we expect the existence of (quadratic?) number fields $K \neq \Q$ such that the assertion $A_K$ does not hold (so that in particular, there is no smooth projective curve of genus $>0$ over $K$ with everywhere good reduction)?

It is mentioned here that there is no elliptic curve over $\Q(\sqrt 2)$ with everywhere good reduction. The same happens with $\Q(i)$, see this answer. Examples of abelian surfaces with everywhere good reduction (i.e. the opposite of what I'm looking for) are mentioned here.

By some analogy discussed here, it may be useful to note that $\Q(i)$ and $\Q(\sqrt 2)$ have no non-trivial unramified extension (see here).

It was asked here whether *every* number field $K \neq \Q$ satisfies $C_{g,K}$ for some $g>0$, but the answer is only very partial.
Furtherfore, it is explained here that for every $g \geq 0$, there is *some* number field $K$ such that $C_{g,K}$ holds — this is different from my question, where I want $K$ to be fixed at the beginning.

Let $G=(V,E)$ be an infinite, simple, undirected graph, such that for all $v\in V$ we have $\text{deg}(v) \geq \aleph_0$. Given an integer $k\geq 1$, is there always $E^{(k)}\subseteq E$ such that $(V, E^{(k)})$ is $k$-regular, that is, every vertex has exactly $k$ neighbors?

Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic structure.

Call two primes 2-power-twins if their difference is (can you guess?) a power of 2. For example, 11 and 19 are 2-power-twins.

Is there a 2-power-twinless prime?

I would imagine that this is doable the following way.

If I take a prime of the form $3k+1$, then I know that adding an odd power of 2 or subtracting an even power of 2 cannot give a prime.

If I take a prime of the form $5k+1$, then I know that adding a power of 2 that is $2\bmod 4$ cannot give a prime.

Do such observations give enough conditions to conclude the existence of a 2-power-twinless prime from Dirichlet's theorem?

In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$

Joyal's isofibration theorem says precisely

An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model structure if and only if it has the right lifting property with respect to the vertex inclusion $e: \Delta^0=E^0 \hookrightarrow E^1$

An immmediate consequence of this theorem is that the fibrant objects in the Joyal model structure are precisely the quasi-categories.

In their paper Mapping Spaces in Quasi-Categories, Dugger and Spivak gave an alternative direct combinatorial argument of one of the lemmas in Joyal's paper, Lemma A.4.

The statement of this lemma is

For $n>0$, the corner product of $b^n:\partial \Delta^n \hookrightarrow \Delta^n$ with $e$, $$e\times^\lrcorner b^n:\Delta^0 \times \Delta^n\coprod_{\Delta^0 \times \partial \Delta^n}E^1 \times \partial \Delta^n\hookrightarrow E^1 \times \Delta^n$$ is special anodyne.

The definition of special anodyne here is not important, because this is the question: When reading through the proof of this statement, they only directly prove that the inclusion of stage $m$ of the filtration into stage $m+1$, written as $Y_m \hookrightarrow Y_{m+1}$, is *inner anodyne* for $0 < m < n,$ leaving the proof in the 'special' outer cases to the reader by just outlining the argument.

However, upon re-analyzing their argument myself, I found no reason why the maps $Y_0 \hookrightarrow Y_1$ and $Y_n \hookrightarrow Y_{n+1}$ are not also inner anodyne. Those two cases appear to be precisely identical as far as the structure of the proof goes.

It's now known by a completely different proof of Danny Stevenson (see Example 5.8) that in fact, the maps $e\times^\lrcorner b^n$ are all *inner* anodyne (not merely special anodyne) for $n>0$.

I contacted Dan Dugger and David Spivak by E-mail to ask them if in fact they had indeed proven the stronger statement:

For $n>0$, the corner product of $b^n:\partial \Delta^n \hookrightarrow \Delta^n$ with $e$, $$e\times^\lrcorner b^n:\Delta^0 \times \Delta^n\coprod_{\Delta^0 \times \partial \Delta^n}E^1 \times \partial \Delta^n\hookrightarrow E^1 \times \Delta^n$$ is *inner* anodyne,

but neither of them could remember the argument very well. When I suggested to David that perhaps they weakened their claim so that it would agree with the statement known at the time due to Joyal, he said that he had fuzzy memories of maybe doing something like this, but he wasn't at all sure.

He suggested I e-mail another mathematician closely acquainted with this combinatorial argument, but she also had forgotten the details of this rather technical combinatorial lemma.

So I ask, did they indeed prove the stronger claim? I have a use for this, since it actually exhibits $e\times^\lrcorner b^n$ with an even stronger property than that proved by Danny Stevenson, namely that it is a *relative cell complex* for the inner horns.

Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle.

This article Orbifolds as Stacks? by Eugene Lerman calls (in page $11$) this particular principal bundle to be the unit principal $\mathcal{G}$ bundle. So, for a Lie groupoid $\mathcal{G}$, this $\mathcal{G}$ bundle is a special element in $B\mathcal{G}$.

Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. For $B\mathcal{G}$, I have a special element $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$. For $B\mathcal{H}$, I have a special element $t:\mathcal{H}_1\rightarrow \mathcal{H}_0$.

Is it the case that any morphism of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$ should take this special element $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ to the special element $t:\mathcal{H}_1\rightarrow \mathcal{H}_0$? I could not see why this is true from definition of map of stacks but I feel this should be the case. Any comments are welcome.

Let $\mu$ be a probability measure on $I=[0,1]$, absolutely continuous with respect to Lebesgue measure. Denote by $T$ the "doubling angle map" on $I$, where $T(x)=2x \text{ mod }1$. Is it true, in general, that $\mu\left(T^{-n}\left[0,\frac12\right)\right)$ converges to $\frac12$?

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort of "integrability" condition. Integrability of what exactly? In the case of a distribution of a collection of ordinary (1-)vector fields, of course, this is just the Frobenius theorem. What is the interpretation of this "integrability" for multivector fields? Tagging PDEs because even if I know nothing, I assume they are involved in the answer of this question.

I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space, i.e., the closure of two disjoint open $F_{\sigma}$-sets are disjoint? I know that $\beta\mathbb{R}\setminus\mathbb{R}$ is a $\mathrm{F}$-space, but not if the whole space has this property.

Thank you for your help in advance :)

Consider an urn containing red, blue and green balls (the situation is the same illustrated in this post).

Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (of one ball at a time, with replacement) in correspondence of which we get at least one red ball, and at least one blue ball (event $E_X$), and $Y$ the non-negative, integer-valued random variable defined as the number of trials in correspondence of which we get at least one red ball, and at least one blue ball, and at least one green ball (event $E_Y$).

How to evalute $P(Y-X\geq 1)$?

My attempt:

I tried this: $P(Y-X\geq 1)=1-P(Y-X<1)=1-P(Y<X+1)$, but then I met this problem: How can it be $Y<X+1$? The event $E_Y$ can occur only if the event $E_X$ has already occurred, and this should have happened *at least one trial before* the occurrence of the event $E_Y$. Does this mean that $P(Y-X\geq 1)=1$? Also this is not convincing, because it can happen that the event $E_X$ occurred, but the event $E_Y$ not (yet).

Thanks for your help!

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:

Let two parallel lines $L_1 \parallel L_2$, let $A, B, C, D$ be four points in the plane. Let $L$ be a line on the plane such that $L \parallel AB$. Let points $E=L \cap L_1, F=L \cap L_2$. Let circle (center $E$, radius $ED$) meets the circle (center $F$, radius $FC$) at two points $H$, $G$.

**My question:** I am looking for a proof that the locus of $G$, $H$ is a conic when we moved $L$ on the plane

See also:

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller than 1. Here $\otimes$ is tensor product. These matrices $\bf J$, $\hat{\bf{G}}$, $\hat{\bf{H}}$ are defined below. I am thinking about if there is a matrix norm $\|\|$ such that $\|\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}\| < 1$, because spectral radius is smaller than any matrix norm.

$\bf {J}$ is a general real-valued square matrix. We are NOT allowed to assume $\bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $\bf I$ is an identity matrix of the same size as $\bf J$.

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below:

Let $\hat{\mathbf{G}}=\begin{pmatrix} {\bf G} & & & & \\ & \ddots & & & \\ & & {\bf G} & & \\ & & & \ddots & \\ & & & & {\bf G} \\ \end{pmatrix}$ be a $p\hat{N}\times p\hat{N}$ diagonal block matrix repeating $p$ times of $\bf G$, which is an $\hat{N} \times \hat{N}$ matrix defined as the following:

where $h$ is some positive coefficient. The eigenvalues of ${\bf G}$ has analytical form $\frac{h}{{{\rm{2}}(\cos (\frac{{k\pi }}{{\widehat N + 1}}) - 1)}}$ where $k = 1,...,\hat{N}$.

Let $\hat{\mathbf{W}}=\begin{pmatrix} {\bf W} & & & & \\ & \ddots & & & \\ & & {\bf W} & & \\ & & & \ddots & \\ & & & & {\bf W} \\ \end{pmatrix}$ be a $p\hat{N}\times p\hat{N}$ diagonal weight matrix repeating $p$ times of $\bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$.

I have asked this question three weeks ago here

https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930 but received no relevant answers.

Let $n\geq 2$ and consider the improper integral $$I:=\int_{\mathbb{R}^{n}}F(x)dx$$ where $F$ is a continuous function.

If $I$ exists then

$$I=\lim_{R\rightarrow +\infty}\int_{B_{R}}F(x)dx,$$ where $B_{R}$ is a ball with radius $R$. So if this limit does not exist we know that the integral does not exist. Does the existence of this limit imply the existence of the integral ?

Motivation:

I am interested in the existence of the integral $$\int_{\mathbb{R}^{3}}\frac{e^{\dot{\imath}|x-y|^2}}{1+|y|}dy.$$

Using spherical coordinates (I do not even know if we are allowed to change variables here. Are we ? ) $$\int_{\mathbb{R}^{3}}\frac{e^{\dot{\imath}|x-y|^2}}{1+|y|}dy= \int_{\mathbb{S}^{2}}\int_{0}^{\infty} \frac{e^{\dot{\imath}|\rho\omega-x|^2}\rho^2}{1+\rho}d\rho d\omega\\ =e^{i|x|^{2}}\int_{\mathbb{S}^{2}}\int_{0}^{\infty} \frac{e^{\dot{\imath} (\rho^2-2x\cdot \omega\,\rho)}\rho^2}{1+\rho}d\rho d\omega.$$

Observations:

1-The inner integral does not exist for any $x$ and $\omega$.

2-We can not change order of integration

3-The limit

$$\lim_{R\rightarrow \infty}\int_{\mathbb{S}^{2}}\int_{0}^{R} \frac{e^{\dot{\imath} (\rho^2-2x\cdot \omega\,\rho)}\rho^2}{1+\rho}d\rho d\omega$$ exists. Simply apply the very nice formula [Grafakos, classical Fourier analysis-Appendix D]: $$\int_{\mathbb{S}^{n-1}} F(x.\omega)d\omega=c \int_{-1}^{1}(\sqrt{1-s^2})^{n-3} F(s|x|)ds.$$ then benefit from the oscillation in both variables $\rho$ and $\omega$ and integrate by parts in both variables.

Any ideas how to handle this ?

Thank you so much

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that $$ X(t) = \int_0^t H(s)dB(s).$$ Let's define step process $$M(t)=2 \sum_{j=1}^{\lfloor t \rfloor} 1_{\{ B(j)-B(j-1) > 0 \}} - \lfloor t \rfloor.$$ It's just a simple martingale, generated by fair coin tosses, just written in another way. One can see that $B(t)$ is the only source of uncertainty. What is $H(t)$ for $M(t)$ then? Or am I missing something and M.R.T. is not applicable here?