Recent MathOverflow Questions

prove that if lim x approaches infinity f(x)=a that lim x approaches infinity f'(x)=0 [on hold]

Math Overflow Recent Questions - Wed, 05/29/2019 - 14:33

can somebody please answer this question for me I am having a lot of trouble proving this. I need proof

Existence of nonparabolic ends

Math Overflow Recent Questions - Wed, 05/29/2019 - 13:47

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the space of constant functions. Well, by the existence of nonparabolic end $E$, i can build a harmonic $g$ function in $M- \Omega$( where $\Omega$ is subdomain compact of $M$ such that $E$ is a nonparabolic end of $M$). The function $g$ is limited and has finite Dirichlet integral. But i don't know as to use the fact which has only one nonparabolic end of $M$.

Norms of elements in a quadratic order - can you do it elementarily?

Math Overflow Recent Questions - Wed, 05/29/2019 - 13:20

Let $\mathcal O$ be an order in an imaginary quadratic field $K$.

  1. Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is not a square?

  2. Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is squarefree and not equal to $1$?

  3. Is there an elementary solution for 1. and 2.?

Note that if there was a simple proof of the first statement, then we could perhaps simplify the proof of the integrality of the $j$-invariant at $CM$ points by avoiding reduction to the case of the maximal order.

Are there some relations between F-polynomials and theta functions?

Math Overflow Recent Questions - Wed, 05/29/2019 - 13:14

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to cluster variables. Are there some relations between F-polynomials and theta functions? Thank you very much.

Linearization of a PDE

Math Overflow Recent Questions - Wed, 05/29/2019 - 13:03

I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us consider the NLS equation $$ 2i\phi_t+\Delta\phi+\vert\phi\vert^{2\sigma}\phi=0, \quad 0<\sigma<\tfrac{2}{n-2}.$$ This equation has very interesting "localized" solutions of the form: $$\phi(t,x)=u(x)e^{it/2},$$ where $u(t,x)$ solves $\Delta u-u+\vert u\vert^{2\sigma}u=0$. Besides, the latter equation has an even more interesting real, positive and radial $H^1(\mathbb{R}^n)$ solution called "Ground state" and denoted by $R(x)$.

Now let me try to explain my question. Consider the perturbed Initial Valued Problem (IVP): $$2i\phi_t^\varepsilon+\Delta \phi^\varepsilon+\vert \phi^\varepsilon\vert^{2\sigma}\phi^\varepsilon=\varepsilon F(\vert \phi^\varepsilon\vert)\phi^\varepsilon, \quad \phi^\varepsilon(t=0,x)=R(x)+\varepsilon S(x)$$ We will seek solutions of the previous equation of the form $$\phi^\varepsilon(t,x)=(R(x)+\varepsilon w_1+\varepsilon^2 w_2+...)e^{it/2}.$$ According to Weinstein if you reeplace this function into the perturbed equation and linearize you will get the following IVP for the linearized perturbation $w$: $$ 2iw_t+\Delta w-w+(\sigma+1)R^{2\sigma}w+\sigma R^{2\sigma}\overline{w}=F(R)R, \quad w(0,x)=0.$$ Now my problem is: I do not really understand how to obtain this linearization, can someone explain a little bit how to do it? Or recommend some references to learn about it. I tried replacing $\phi^\varepsilon$ (truncated after $\varepsilon w$) and then I took the derivative with respect to $\varepsilon$ and evaluate at $\varepsilon=0$, but I cannot recover the equation claimed by Weinstein, so I think that I'm not undersitanding how to linearize.

Note2: The parameter $n$ denotes the dimension.

Is there a two-dimensional Higman's Lemma?

Math Overflow Recent Questions - Wed, 05/29/2019 - 13:02

A string over a finite alphabet $A$ can be thought of as a function $f:\{1,2,...,m\} \rightarrow A$ for some natural number $m$.

A 2-Dim string over $A$ is a function $f$ where $f:\{1,2,\ldots,m\}\times\{1,2,\ldots,n\} \rightarrow A$ for some natural numbers $m$ and $n$.

If $f$ and $f'$ are 2-Dims (over $A$), with dimensions $(m,n)$ and $(m'n')$ respectively, we say $f\leq f'$ if there is are increasing functions $D:\{1,2,\ldots,m\}\rightarrow\{1,2,\ldots, m'\}$ and $E:\{1,2,\ldots,n\}\rightarrow\{1,2,\ldots, n'\}$ such that whenever $1\leq i\leq m$ and $1\leq j\leq n$, $f(i,j)=f'(D(i),E(j))$.

Is there a proof or disproof of the version of Higman's lemma that states that for every infinite sequence $f_1,f_2,\ldots$ of 2-Dims over $A$, there is $1\leq u<v$ such that $f_u\leq f_v$?

Kastanas' game and completely Ramsey sets

Math Overflow Recent Questions - Wed, 05/29/2019 - 11:49

recently I was reading the article ''On the Ramsey property for sets of reals'' of Ilias Kastanas (https://www.jstor.org/stable/2273667?seq=1#metadata_info_tab_contents), in this article the author characterizes the sets completely Ramsey by a game created by Kastanas, the proof seemed very good. But I think it can be formalized better, for example in the proof of that Player 1 has a winning strategy in the Kastanas game then there is a homogeneous set in $\psi$, to build that set the author use a tree, but I think that the proof is more intuitive than formal, I would like to know if someone knows any more formal proof, or more bibliography about it.

Thank you

Estimate related to the Möbius function

Math Overflow Recent Questions - Wed, 05/29/2019 - 11:43

I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less o equal than $x$ that are square free and with exactly $k$ primes on its decomposition. That is the cardinal of the following set $$ \mathcal{J}_T(x,k) = \{ n \in \mathbb{N} : n \le x, \Omega(n)=k , n \mbox{ is square free } \}.$$ Other way to describe this cardinal, using the Möbius function, is $$ |\mathcal{J}_T(x,k) | = \sum_{\Omega(n) = k, n \le x } |\mu(n)|.$$

I am looking for the asymptotic behaviour on $x$, but this will depend also on $k$ in some way. The bound given using $$ \sum_{\Omega(n) = k, n \le x } |\mu(n)| \le \sum_{n \le x } |\mu(n)| \le \frac{6x}{\pi^2} + O(\sqrt{x}) \ll x, $$
is not good enought for my purposes.

Thanks in advanced, any reference or idea is helpful

Hermitian sectional curvature

Math Overflow Recent Questions - Wed, 05/29/2019 - 11:22

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.

Consider the complexified tangent space $TM \otimes \mathbb{C}$ and the complex-linear extension of $R$, which we still denote $R$. By definition, $N$ has nonpositive Hermitian sectional curvature if $R(X, Y, \bar{X}, \bar{Y}) \leqslant 0$ for all $X, Y \in TM \otimes \mathbb{C}$.

Obviously, nonpositive Hermitian sectional curvature is stronger than nonpositive sectional curvature.

QUESTION. Is nonpositive Hermitian curvature strictly stronger than nonpositive curvature?

In other words, are there examples of Riemannian manifolds with nonpositive sectional curvature, but not nonpositive Hermitian sectional curvature?

I expect the answer easily yes, in fact it is claimed in e.g. [1] or [8], but I couldn't find an example in the relevant literature, e.g. [1][2][3][4][5][6][7][8][9].

NB: Yau-Zheng [8] showed that the answer is no for manifolds with negative $\delta$-pinched sectional curvature with $\delta \geqslant 1/4$. According to [9, Theorem 9.26], the answer is no for Kähler surfaces.

$$$$

FOLLOW UP QUESTIONS

Following (almost) the terminology of Siu [6], a Riemannian manifold with nonpositive Hermitian sectional curvature has strongly nonpositive curvature. He also introduces very strongly nonpositive curvature: Consider the curvature operator $$ \begin{aligned} Q \colon \otimes^2 TM \times \otimes^2 TM \to \mathbb{R} \end{aligned} $$ such that $Q$ is defined for decomposable tensors by $Q(X\otimes Y, Z \otimes W) = R(X , Y, Z , W)$. $N$ has very strongly nonpositive curvature if $Q(\sigma, \sigma) \leqslant 0$ for all $\sigma \in \otimes^2 TM$. In other words, the curvature operator is negative semidefinite. In this case, the complex-linear extension of $Q$ is still negative semidefinite, which clearly implies that $M$ has strongly nonpositive curvature.

Question 2. Is there an example showing that very strongly nonpositive curvature is strictly stronger than strongly nonpositive curvature?

Finally, there is a notion of (very) strongly negative curvature for Kähler manifolds, but it's not simply something like $Q(\sigma, \sigma) < 0$ for all nonzero $\sigma$. Indeed, still denoting $Q$ its complex-linear extension, we have $Q(\sigma, \bar{\sigma}) = 0$ for any $\sigma$ of type $(2,0)$ or $(0,2)$, e.g. $X \otimes Y$ with $X, Y \in T^{1,0} M$. $N$ has very strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$, and $N$ has strongly negative curvature if $Q(\sigma, \bar{\sigma}) < 0$ for all nonzero tensors $\sigma \in T^{1,0} M \otimes T^{0,1} M$ of length $\leqslant 2$, e.g. $\sigma = X \otimes \bar{Y} + Z \otimes {\bar{W}}$.

It is clear that $$\text{very strongly negative} ~\Rightarrow~ \text{strongly negative} ~\Rightarrow~ \text{negative sectional curvature}$$

Question 3. Are there examples proving that the converse implications are false?

Again, according to [9, Theorem 9.26], the answer is no for Kähler surfaces.

Remark: Of course, there are similar notions of (very) strong nonnegative / positive curvature and one could ask the same questions.

$$$$

[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo. Fundamental groups of compact Kähler manifolds. 1996.

[2] Eells and Lemaire. Two reports on harmonic maps. 1995

[3] Jost and Yau. Harmonic mappings and Kähler manifolds. 1983.

[4] Mostow and Siu. A compact Kähler surface of negative curvature not coveredby the ball. 1980.

[5] Ohnita and Udagawa. Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds. 1990.

[6] Siu. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. 1980.

[7] Xin. Geometry of harmonic maps. 1996

[8] Yau and Zheng. Negatively $\frac14$-pinched Riemannian metric on a compact Kähler manifold.

[9] F. Zheng, Complex differential geometry, 2000.

Reference request: When is the variance in the central limit theorem for Markov chains positive?

Math Overflow Recent Questions - Wed, 05/29/2019 - 10:11

I'm looking for a reference which gives sufficient conditions for the variance to be positive in the central limit theorem for Markov chains (cf https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem). In other words, using the notation on the wikipedia page, I'm looking for a sufficient condition for $\sqrt{n}(\hat\mu_n-\mu)$ to legitimately be a random variable in the limit.

In my situation, the Markov chain is simply a random walk on a finite, aperiodic, strongly-connected graph, so a result relating to this simpler situation would be good enough if nothing more general exists. Also, in my situation, actually attempting to estimate $\sigma^2$ via the formula is pretty much out of the question.

I beleive that I can argue that the variance will be positive if there are two directed cycles $v_0\to v_2\to\dots\to v_n=v_0$ and $u_0\to u_2\to\dots\to u_m=u_0$ with ${1\over n}\sum_{t=1}^n g(v_t)\neq{1\over m}\sum_{t=1}^m g(u_t)$. Intuitively, this is because the chain will always have a chance to move around either of these cycles and these cycles contribute different values to the sum.

I'm not looking for anyone to write down a proof since, if necessary, I can likely prove it myself. However, I would really like to avoid having to spend the necessary pages to do so in a paper I'm writing. It seems very likely that such a result must have appeared in either a paper or book somewhere, but I haven't had any luck in hunting down a reference so far.

Category of separated presheaves (over any site)

Math Overflow Recent Questions - Wed, 05/29/2019 - 09:57

It is well-known that the category of discrete fibrations over a category $\mathbb{C}$ is equivalent to the category of presheaves on $\mathbb{C}$.

More generally I think it is true, and probably well-known, though I can't find a reference, that the following two categories are equivalent:

  1. the category with objects: discrete fibrations $\mathbb{E} \to \mathbb{B}$ and morphisms: pairs of functors ($\mathbb{E}\to \mathbb{E}'$, $\mathbb{B} \to \mathbb{B}'$) making the square commute
  2. the category with objects: pairs ($\mathbb{B}$, $P_\mathbb{B}: \mathbb{B}^\mathrm{op} \to \mathbf{Set}$) consisting of a category and a presheaf on it, and morphisms: pairs $(f : \mathbb{B} \to \mathbb{B}', \alpha : P_\mathbb{B} \Rightarrow P_{\mathbb{B}'} \circ f^\mathrm{op})$ consisting of a functor and a natural transformation.

Suppose $\mathbb{B}$ is a site (i.e. has a topology), then the full subcategory of separated presheaves on $\mathbb{B}$ is a reflective subcategory of the category of all presheaves.

My question is: Consider the variation of (2) above where the objects are ($\mathbb{B}, P_\mathbb{B}$) where $\mathbb{B}$ is a site. Call this $\mathbf{C}$.
Is it true that the full subcategory of $\mathbf{C}$ whose objects are ($\mathbb{B}, P_\mathbb{B}$) with $P_\mathbb{B}$ separated is a reflective subcategory of $\mathbf{C}$?

Orientable with respect to complex cobordism?

Math Overflow Recent Questions - Wed, 05/29/2019 - 09:10

I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and that an orientation with respect to real K-theory is a spin structure. I think this is a very beautiful picture and I am wondering if orientations with respect to other theories like elliptic cohomology, G-equivariant cohomology, quaternionic K-theory, or spin cobordism correspond to interesting and well-studied differential-geometric structures.

Complex manifolds should be oriented with respect to any complex-oriented cohomology theory. Indeed, if $E$ is a complex-oriented cohomology theory then all complex vector bundles carry $E$-orientations. In particular, if $X$ is a complex manifold then its tangent bundle $TX$ has a complex structure making $X$ an $E$-oriented manifold.

Given that complex cobordism is the universal complex-oriented cohomology theory, I would guess that an orientation with respect to complex cobordism is a complex structure. I have been unable to find any literature on this and I am unsure how to approach the problem rigorously on my own. Maybe someone knows?

Rationally connected Kahler manifolds are projective

Math Overflow Recent Questions - Wed, 05/29/2019 - 08:56

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:

https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf

She writes in this remark the following:

Remark 0.5 A compact Kahler manifold $X$ which is rationally connected satisfies $H^2(X, {\cal O}_X) = 0$, hence is projective.

I understand that a Kahler manifold with $H^2(X, {\cal O}_X) = 0$ is projective. However, I don't understand why a Kahler manifold that is rationally connected has $H^2(X, {\cal O}_X) = 0$. Indeed, the definition for rational connectedness that Voisin is using is the following:

Definition 0.3 A compact Kahler manifold $X$ is rationally connected if for any two points $x, y\in X$, there exists a (maybe singular) rational curve $C\subset X$ with the property that $x\in C$, $y\in C$.

So my question is the following: How to prove that $H^2(X, {\cal O}_X)$ for a compact Kahler manifold $X$ that satisfies Definition 0.3? Is this easy/hard/well-known?

PS. As Donu Arapura correctly says below the vanishing of $H^2(X, {\cal O}_X)$ for rationally connected projective manifolds is a classical fact. However I want a proof of such a vanishing for Kahler manifolds (to show that they are protective). So I want to know if this vanishing is a well known fact or a couple of pages are needed to prove it?

Knight's tour problem

Math Overflow Recent Questions - Wed, 05/29/2019 - 08:45

It is known that on an infinite board, if all squares of the form $(ki,kj)$ are removed, $k$ even, $i,j\in\mathbf{Z}$, then there is no knight's tour due to unbalanced black and white squares.

My questions are the following:

  1. If $k$ is odd, does there exist a knight's tour? For example, $k=3$.

  2. If finitely many black squares are to be removed, can there possibly exist a knight's tour?

Any results on these problems? Thanks.

For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?

Math Overflow Recent Questions - Wed, 05/29/2019 - 04:17

Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the martingale. For example Theorem 3.5 of this paper of Pinelis leads to the following variant of the Azuma-Hoeffding inequality.

$$P(\|f_n\| \ge \epsilon \text{ for some }n\le N) \le \exp\left (-\frac{\epsilon^2}{2NL^2} \right).\tag{1}$$

In the paper such results are called tail inequalities for martingales. What I'm interested in could also be called a tail inequality, except I am interested in is the behaviour after $N$ rather than before. Of course there's no reason to believe since $f_1,f_2,\ldots$ should remain bounded, but if we instead focus on the normalised values $\frac{f_n}{n}$ we can get some bounds. For example taking $\epsilon = N$ the Pinelis theorem implies

$$P\bigg (\frac{1}{N}\|f_N\| \ge 1\bigg) \le \exp\left (-\frac{N}{2L^2} \right).$$

The crudest thing we can do is take a union bound to get

$$P\bigg (\frac{1}{n}\|f_n\| \ge 1\text{ for all }n\ge N\bigg) \le \sum_{n=N}^\infty\exp\left (-\frac{n}{2L^2} \right).$$

The sum can be bounded by the integral

$$\sum_{n=N}^\infty\exp\left (-\frac{n}{2L^2} \right) \le \int_{N-1}^\infty\exp\left (-\frac{x}{2L^2} \right) = 2L \exp\left (-\frac{N-1}{2L^2} \right)$$

which goes to zero as $N \to \infty$.

I am wondering if there are any more sophisticated approaches to get better bounds?

One idea I had was, instead of forcing each $\frac{1}{n}\|f_n\| <1$ we force $\frac{1}{N}\|f_N\|_2 <1/2$. This forces the next $\frac{1}{n}\|f_n\| <1$ for all $n \le \left(\frac{2L}{2L-1/2}\right)N$. Then for $n_1 = \left(\frac{2L}{2L-1/2}\right)N$ we force $\frac{1}{n_1}\|f_{n_1}\|_2 <1/2$ and so on. Proceeding like this we get a union bound over $N,n_1,n_2,\ldots$ leading to a series of the form

$$\sum_{n=1}^\infty \exp \left( \frac{N}{8L^2} \left(\frac{2L}{2L-1/2}\right)^{n-1}\right).$$

and the integral

$$\int_{0}^\infty \exp \left( a b^{x-1}\right) = \int_{-1}^\infty \exp \left( a b^{x}\right)$$

for the obvious constants. Under the substitution $u =ab^{x}$ this becomes the exponential integral function

$$\frac{1}{\log b}\int_{a/b}^\infty \frac{e^{-t}}{t}dt = \frac{\text{Ei}_1(a/b)}{\log b}.$$

Using some special function inequalities I can bound the above by

$$\frac{e^{-a/b}}{(a/b)\log b}$$

which simplifies to something of the form

$$\frac{C \exp \left( \frac{N}{8L^2} \frac{2L-1/2}{2L}\right) }{N}$$.

We have acquired a $N$ in the denominator, and maybe a smaller coefficient $C$ than before. Unfortunately this makes no difference asymptotically because the coefficient inside the exponential is smaller than before.

You can also replace $1/2$ with any $\delta \in (0,1)$, perform the calculations, and try to minimise the result with respect to $\delta$. There is a closed form solution for such a $\delta$ but it is the solution of a cubic equation so doesn't offer much insight.

One could also try different $\delta_i$-values between each $n_i$ and $n_{i+1}$ but I cannot see how to bound the resulting series with an integral.

Has this problem been considered before? Could anyone provide a reference?

Quadratic diophantine equations and geometry of numbers

Math Overflow Recent Questions - Tue, 05/28/2019 - 07:18

Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system

$$ w^2 - ax^2 -by^2 + abz^2 = 1 $$ $$ \lvert w^2 + ax^2 +by^2 + abz^2 \rvert \ll \infty $$ $$ \lvert \bar{w}^2 + a\bar{x}^2 -b\bar{y}^2 - ab\bar{z}^2 \rvert \le C $$ for some constant $C$. Here $\overline{\alpha + \beta\sqrt{5}} = \alpha - \beta\sqrt{5}$ and ``$\ll \infty$'' means that ideally I would like to enumerate solutions in increasing order of this value. (Restriction of scalars turns this problem into a system of two quadratic equations and two inequalities in eight variables in $\mathbb{Z}$; if someone wants to see it, I can write it out including potential mistakes).

  1. What is the best (or even any practical) way to produce these?

I am aware that there is a lot of classical mathematics associated to this question but I don't quite manage to put it together. Perhaps a subquestion is:

  1. Can one enumerate the squares $s$ in $\mathbb{Z}[\varphi]$ with $\lvert \bar{s} \rvert \le C$ in increasing order of $\lvert s \rvert$?

Context: Let $k = \mathbb{Q}(\varphi)$ and let $A$ be the quaternion algebra $(\frac{a,b}{k})$ with norm $\nu$. With the above values the algebra $A$ is a skew field but tensoring with $\mathbb{R}$ in the two possible ways (taking $\sqrt{5}$ to $\pm\sqrt{5}$) gives an isomorphism with $M_2(\mathbb{R})$ which we equip with the map $$ \left\lVert\left(\begin{array}{cc}\alpha&\beta\\\gamma&\delta\end{array}\right)\right\rVert = \frac{1}{2}\left(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\right). $$ The above system then asks for solutions $\lambda$ in the maximal order of $A$ for $\nu(\lambda) = 1$, $\lVert\lambda\rVert_{\sqrt{5} \mapsto \sqrt{5}} \ll \infty$ and $\lVert \lambda \rVert_{\sqrt{5} \mapsto -\sqrt{5}} \le C$.

How to calculate the volume of a section of a convex body?

Math Overflow Recent Questions - Tue, 05/28/2019 - 05:46

The following is essentially a partial case for my previous question.

Let $B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $\mathbb{R}^m$, say $l^p$-norm, $p\in (1,\infty)$. Let $v_1,...,v_n\in \mathbb{R}^m$ be linearly independent.

How to calculate the $n$-dimensional volume of $B\cap span\{v_1,...,v_n\}$?

I need to express this volume through the coordinates of $v_1,...,v_n$, or perhaps through some distances between certain combinations of them. I know that there is extensive literature on related matters, but I hope that this specific question has a specific answer..

Construction of elliptic equation with Neumann boundary condition from a minimization problem

Math Overflow Recent Questions - Mon, 05/27/2019 - 14:14

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : H^1(B) \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.

Does each integer $n>1$ have the form $2^k3^l+p_m-p_{m-1}+\ldots+(-1)^{m-1}p_1$?

Math Overflow Recent Questions - Sun, 05/26/2019 - 16:42

For each positive integer $n$, let $s_n$ be the alternating sum of the first $n$ primes given by $$s_n:=p_n-p_{n-1}+\ldots+(-1)^{n-1}p_1,$$ where $p_k$ denotes the $k$th prime. All the numbers $s_1,s_2,\ldots$ are pairwise distinct. The main term of $s_n$ as $n\to\infty$ is not known. It seems that $\lim_{n\to\infty}s_n/p_n=1/2$.

Here I ask a novel question involving $s_n$ on the basis of my computation.

Question. Is it true that $$\{2^k3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,3,\ldots\}=\{2,3,\ldots\}?$$

I conjecture that the question has a positive answer. Let $r(n)$ be the number of ways to write $n$ as $2^k3^l+s_m$ with $k,l\in\{0,1,\ldots\}$ and $m\in\{1,2,\ldots\}$. The sequence $r(1),r(2),\ldots$ is available from http://oeis.org/A308411. For example, $r(2)=1$ with $2=2^03^0+p_2-p_1$. On May 25, 2019 I verified $r(n)>0$ for all $n=2,\ldots,10^6$. On May 26, 2019 Prof. Qing-Hu Hou extended the verification to $2\times 10^7$ on my request. Based on Hou's program I have verified that $r(n)>0$ for all $n=2,\ldots,10^9$.

PS: By the way, I also note that the set $$\{6^k+3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,\ldots\}$$ contains $3,4,\ldots,10^9$, and conjecture that this set coincides with $\{3,4,\ldots\}$ (cf. http://oeis.org/A308403).

An application of Girsanov's Theorem

Math Overflow Recent Questions - Sun, 05/26/2019 - 15:39

Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, where $\mu$ is the Wiener measure. Now, clearly the most general case that we have is that $\mathbb{E}[\rho(-\delta a)]\leq1$ all the time where $\rho$ is the Wick exponential and $\delta$ is the adjoint of the weak (sobolev) derivative. I've seen that for any bounded measurable $f$, $\mathbb{E}[f\circ A]=\mathbb{E}[FL]$, where $L$ is the Radon-Nikodym derivative of $A\mu =\mu(A^{-1}(.)$ with respect to $\mu$. Then $\mathbb{E}[fL] \geq \mathbb{E}[f \circ A L \circ A \rho(-\delta a)]$, where the euqality only hold if $\mathbb{E}[\rho(-\delta a)]=1$ almost surely. How does this inequality exactly follows from Girsanov's theorem ? In the case when $\mathbb{E}[\rho(-\delta a)]=1$ it is directly comes from the definition of Girsanov's theorem but I could not deduce the inequality when this is not the case. It appears in page 9 of this paper : https://arxiv.org/pdf/0903.3891.pdf

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