Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{g_0}$.

Suppose $g_{\epsilon}$ is close to $g_0$ in the $C^0$ sense. Is it true that $H^k_{g_0}$ is "close" to $H^k_{g_{\epsilon}}$ in some sense?

**Edit:**

On a second thought, Perhaps we need to assume $g_{\epsilon}$ is close to $g_0$ in the $C^1$ sense. The reason is that $\delta_g=d^*_g=\pm \star_g d \star_g$, so we differentiate the metric's components (which are encoded in the Hodge dual operator).

In other words, I am interested in the behavior of the space of harmonic $k$-forms under a small perturbation of the metric. Are there any results of this kind?

**Here is one way of quantifying what does it mean for $H^k_g$ to be close $H^k_{g_{\epsilon}}$:** (I guess there are other ways, this seemed natural to me).

Take the sup norm on $\Omega^k(M)$ (where the pointwise norm is the one induced by $g_0$), and let $S$ be the unit sphere of $\Omega^k(M)$ w.r.t this norm. Define $S_0=H^k_{g_0} \cap S, S_{\epsilon}=H^k_{g_{\epsilon}} \cap S.$ Set

$$ d(H^k_{g_{\epsilon}},H^k_{g_{0}}):=d_H(S_{\epsilon},S_0)$$ where $d_H$ is the Hausdorff distance of $H^k_{g_{\epsilon}},H^k_{g_{0}}$ inside $\Omega^k(M)$.

Is it true that if $g_{\epsilon}$ is close to $g_0$ then $H^k_{g_{\epsilon}}$ is close to $H^k_{g_{0}}$ in this sense?

(I guess that I am asking whether the map $g \to H^k_{g}$ is "continuous" in some sense).

I am not particularly fussed on this specific proximity measure, so results using other notions of distance are also welcomed.

Let $E$ be a set. In a lot of proof it is useful to take another set $S$ such thar there is a bijection $E\rightarrow S$ and $E\cap S= \emptyset$. But how to create a set like that ?

I tried to take $\omega \not\in E$ and define $S= E\times \{\omega\}$ but this can not work because for example if $E=\{ 1,(1,2)\}$ then as $\omega$ can be anything, it could happen $\omega = 2$ and then $E\cap S = \{(1,2\}$. A guess would be that $\mathfrak{P}(E)$ is huge compared to $E$ thus I could find $X\subset \mathfrak{P}(E)$ such that $E\cap X= \emptyset$ and $X \approx E$.

For example the the Euclidian metric and Alcubierre metrics.

As the image presented below, the reddish point set is **totally** separated from the blueish one and the greenish one, while the blueish point set is **quite** mixed with the greenish one.

A number of point sets on the plane. Each point set takes up a simply connected domain concave or convex. The points' coordinates is known and the points are not necessarily on the grid which might be implied by the image. The separation or mixture degree of two sets is all that matters.

How to value the extent of separation or mixing of point sets? Any suggestion or reference will be greatly appreciated. Statistical method would be preferred if optional.

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of Serre generalizes this theorem to the situation that you can take $GL_2$ of the coordinate ring of a smooth projective curve minus a point and it will be equal to the fundamental group of a graph group.

I wonder whether there are higher dimensional versions of this theorem or not, in the sense that you can both consider $GL_n$ instead of $GL_2$ and higher dimensional varieties instead of curves. I would really appreciate it if you could point me to some references about these kinds of theorems.

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often see assertions of things like "the prime number theorem is equivalent to the fact that the Riemann zeta function has no zeros whose real part is at least $1$" and "the Riemann hypothesis is equivalent to the best possible error term in the prime number theorem".

However, whenever I have attempted to learn justifications for these slogans, I am immediately confronted with huge masses of complicated and unmotivated formulas. I can verify these calculations line-by-line, but I don't seem to learn anything from them. I always feel that there must be a bigger picture in the back of the mind of the writers, but I don't see it.

**Question**: Can anyone recommend a motivated account of these topics that is accessible to people whose backgrounds are not in analysis? Since they are fundamentally analytic facts, I expect that this will involve learning some analysis. But at a fundamental level I prefer to think either algebraically or geometrically, so I have difficulty attaining enlightenment from complicated formulas/estimates without having the meaning of the various terms explicitly spelled out.

**My background**: I am about 10 years post PhD, and my research is in topology and algebraic geometry. I have a good working knowledge of algebraic number theory (up to class field theory, though I have to admit that I have never carefully studied the proofs of the results in class field theory). I love complex analysis, though I fear that the way I think about that subject is more soft and geometric than analytic (and thus is not really the point of view of people working in analytic number theory).

After this question : Dominated convergence 2.0?

I want to know, what about the case when $h\in L^1([0,1])$.

The completed question :

Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$ and $\forall x \in [0,1], g(x)\in \mathbb R$.

Assume that:

$\forall n\in\mathbb N, f_n''<h$, where $h \in L^1([0,1])$.

Is it true that $\lim \int_0^1 f_n=\int_0^1 g$ ?

Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that $$ \sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty, $$ and suppose that $\mathcal Lg(x):=\int_0^\infty (g(x-y)-g(x))\ell(x,y)\,dy$ generates a decreasing Lévy-type process $s\mapsto L^{x}_s$, where $x\in\mathbb R$ denotes the starting point. Define the first exit time from $(0,\infty)$ as $\tau_0(x):=\inf\{s>0: L_s^{x}\le 0 \}$. Questions:

(i) Is this enough to prove that $\mathbf P[L_{\tau_0(x)}^{x} =0]=0$ ?

(ii) If (i) is not true, does it become true if we additionally assume that $L^{x}_s$ allows a density for every $s,x>0$ ?

iii) If (i) is not true, is (i) true for the Lévy case $\ell(x,y)=\ell(y)$ ? (I know $\mathbf P[L_{\tau_0(x)}^{x} =0]=0$ is true if $\ell(y) dy=\ell(dy)$ is an infinite measure)

Let $n$ be a nonnegative integer, and let $B$ be the $n \times n$-matrix (over the rational numbers) whose $\left(i, j\right)$-th entry is $\dbinom{n+1}{2j-i}$ for all $i, j \in \left\{ 1, 2, \ldots, n \right\}$.

For example, if $n = 5$, then \begin{equation} B = \left(\begin{array}{rrrrr} 6 & 20 & 6 & 0 & 0 \\ 1 & 15 & 15 & 1 & 0 \\ 0 & 6 & 20 & 6 & 0 \\ 0 & 1 & 15 & 15 & 1 \\ 0 & 0 & 6 & 20 & 6 \end{array}\right) . \end{equation}

**Question 1.** Prove that the eigenvalues of $B$ are $2^1, 2^2, \ldots, 2^n$. (I know how to do this -- I'll write up the answer soon -- but there might be other approaches too.)

**Question 2.** Find a left eigenvector for each of these eigenvalues. What I know is that the row vector $v$ whose $i$-th entry is $\left(-1\right)^{i-1} \dbinom{n-1}{i-1}$ (for $i \in \left\{1,2,\ldots,n\right\}$) is a left eigenvector for eigenvalue $2^1$ (that is, $v B = 2 v$). But the other left eigenvectors are a mystery to me.

**Question 3.** Find a right eigenvector for each of these eigenvalues. For example, it appears to me that the column vector $w$ whose $i$-th entry is $\left(-1\right)^{i-1} / \dbinom{n-1}{i-1}$ (for $i \in \left\{1,2,\ldots,n\right\}$) is a right eigenvector for eigenvalue $2^1$ (that is, $B w = 2 w$). This (if correct) boils down to the identity
\begin{equation}
\sum_{k=1}^n \left(-1\right)^{k-1} \left(k-1\right)! \left(n-k\right)! \dbinom{n+1}{2k-i} = 2 \left(-1\right)^{i-1} \left(i-1\right)! \left(n-i\right)!
\end{equation}
for all $i \in \left\{1,2,\ldots,n\right\}$.
Note that the entries of $w$ are the reciprocals to the corresponding entries of $v$ ! Needless to say, this pattern doesn't persist, but maybe there are subtler patterns.

I am going to put up an answer to Question 1 soon, as a stepping stone for the proof of https://math.stackexchange.com/questions/2886392 , but this shouldn't keep you from adding your ideas or answers.

It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric description of the projection to the sphere?

This manifold is actually a complex algebraic variety (namely a plane with one point blown up), so the question should make some sense. The best answer would probably be a meromorphic function, but I could not find one.

Let $(V,||.||)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $||f(x*y)||\ge ||f(x)+f(y)||,\forall x,y\in G$, is a group homomorphism i.e. $f(x*y)=f(x)+f(y),\forall x,y\in G$. Then is it true that the norm on $V$ comes from an inner product ?

Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \wedge\ \ \lambda^q_{3,4}> 0 \}$$

of all linear operators of rank 4 and exact two positive $\lambda_{3,4}$ and two negative $\lambda_{1,2}$ eigenvalues. The real dimension of $\mathcal{F}$ is $8d-16$.

Now we look at the spinor-space $S_q$ which is isomorphic to $\mathbb{C}^4$ and the linear operators $L(\mathcal{H},S_q)$. For some $A \in L(\mathcal{H},S_q)$ with $\text{rank}(A)=4$, $A^*A\in\mathcal{F}$ and an open subset $B_\epsilon(A)$ for $\epsilon > 0$ we define the map

$$R : B_\epsilon(A) \to \mathcal{F},$$ $$B \mapsto -B^*B$$

This map should be well-definied if $\epsilon$ is small enough such that $\forall C\in B_\epsilon(A): \ \text{rank}(C)=4$.

If we now look at the differential at the point $A$

$$DR_A: L(\mathcal{H},S_q) \to T_A\mathcal{F}$$.

Is this differential injectiv?

Let us consider the quantum scattering problem on the line with the Hamiltonian $$H=-\frac{d^2}{dx^2}+ V(x),$$ where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise.

It is easy to see that $H$ has no discrete spectrum (e.g. no bound states).

What explicitly are the Moeller operators $\Omega^{\pm}$?

A reference would be helpful.

Let $G$ be a group, and assume that there exist $a, b, c \in G$ such that $abc$, $acb$, $bac$, $bca$, $cab$ and $cba$ are precisely 5 distinct elements (i.e. that precisely two of the products are equal).

**Question 1:** Does it follow that there exist $d, e, f \in G$ such that
$def$, $dfe$, $edf$, $efd$, $fde$ and $fed$ are precisely 4 distinct elements?
And if not -- does the non-existence of such $d, e, f \in G$ at least imply
that $G$ is infinite?

*Remark:* When one replaces 5 by 6, the answer is *no*. -- The smallest group
which can be taken as an example here has order 54. It is
$$
G_{54,8} :=
\langle (1,4,7)(2,5,8)(3,6,9), (3,4,5)(6,8,7), (3,6)(4,7)(5,8) \rangle.
$$

**Question 2:** Let $G$ be as above, and assume further that there are no
$d, e, f \in G$ such that $def$, $dfe$, $edf$, $efd$, $fde$ and $fed$ are
pairwise distinct. If $G$ is finite, does it follow that the order of $G$
is a multiple of 5?

*Remark:* The groups of order up to 625 which fulfill the conditions have orders
20, 40, 60, 80, 100, 120, 125, 140, 160, 180, 200, 220, 240, 250, 260, 280,
300, 320, 340, 360, 375, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560,
580, 600, 620 and 625, respectively.

*Side note:* A related
earlier question of mine
remains unsolved so far.

**Added on Aug 21, 2018:** Given a group $G$, put
$$
{\rm P}_3(G) :=
\left\{ |\{abc, acb, bac, bca, cab, cba\}| \ \big| \ a,b,c \in G \right\}.
$$
Then clearly we have ${\rm P}_3(G) = \{1\}$ if and only if $G$ is abelian.
Computational investigations further suggest that ${\rm P}_3(G)$ is always one
of $\{1\}$, $\{1,2\}$, $\{1,2,3\}$, $\{1,2,3,4\}$, $\{1,2,3,4,5\}$,
$\{1,2,3,4,5,6\}$, $\{1,2,3,4,6\}$, $\{1,2,3,6\}$ and $\{1,2,6\}$ --
where $\{1,2,3,4\}$, $\{1,2,3,4,5,6\}$ and $\{1,2,3,4,6\}$ are all very
common, while $\{1,2,3,6\}$ and $\{1,2,3,4,5\}$ impose more-or-less
severe restrictions on the structure of the group.

Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is the algebraic closure of $K$.

This Galois action, call it $\phi$, gives rise to outer Galois actions on any quotient of $\pi_1(V_{\bar{K}})$ by a characteristic subgroup. In particular we get actions on its quotients by members of the derived series of $\pi_1(V_{\bar{K}})$ as well as its lower central series.

First member of these series is the commutator subgroup whose quotient is $H_1(V_{\bar{K}})$. The corresponding outer Galois action on this abelan quotient is in fact a linear representation $\phi_1$ of $G_K$ on $H_1(V_{\bar{K}})$. Associated, thus, with the outer action $\phi$ on $\pi_1(V_{\bar{K}})$ is a sequence of outer Galois actions on bigger and bigger quotients by successive members of the given series, $\phi_i: i = 1, 2, ...$, the first of which is linear.

Suppose the linear representation $\phi_1$ proves to be automorphic (after dualizing to cohomology and taking an appropriate $l$-adic realization, say.) How can we "lift" its automorphy up the series to actions $\phi_i$, all the way perhaps to $\phi$? To begin with, what kind of objects would correspond to the automorphic representation associated to $\phi_1$ in such a lift?

What light does Deligne's work on $G_{\mathbb{Q}}$-action on nilpotent completion of $\pi_1(\mathbb{P}^1_{\mathbb{Q}} - \{0,1,\infty\})$ throw on this question?

It is a known fact that a 2nd countable compact Hausdorff space is metrizable. What if we weaken the 2nd countable to separable only - is the space still metrizable?

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: *why* Langlands? Why should arithmetic of polynomial equations over the rationals (and their generalizations) have anything to do with modular forms (and their generalizations), let alone have such a precise relationship as conjectured by arithmetic Langlands correspondence in its modern motivic form?

Even after decades of its formulation and spectacularly successful resolution in some cases - even more so for its geometric and $p$-adic analogues - my understanding of the literature is that the essential mystery remains in that beyond the equality of L-functions we don't have much of an inkling, at least in print, of a more direct relationship between the two sides, e.g., an action of (entities on) one side on the other, a direct map from one to the other, or both sides springing from a common source.

Of course, my reading of the literature may be wrong. It is certainly very incomplete. The proofs of special cases may well have provided the practitioners with insights into why these two sides seemingly so far apart should be so intimately related. If so, have they shared them publicly in a manner that I can access them? If not (perhaps because the insights are too speculative or far from formalizable to put into writing), this seemed to me to be a good - informal, widely shared and relatively anonymous - forum where to learn some of them.

Finally, to bring it back to where I had begun: is the recent work of Venkatesh and co-workers (e.g., https://arxiv.org/abs/1609.06370) a step in the direction of showing a more direct relationship? And related: among the myriad (if not quite thirteen) ways of looking at a modular form, are some more promising than others towards such an understanding in your opinion?

I want to share to everyone how to calculate primes and really want to ask who can test it and if you find anything I missed or mistakes to tell me?

p 301 =1993=11×33+196×10 p301=1993=11×33+196×10

p 51 =233=3×11+2×100 p51=233=3×11+2×100

19122013=31876×1000+1×3+1×10 19122013=31876×1000+1×3+1×10

311=11×1+3×100 311=11×1+3×100

31=11×1+2×10 31=11×1+2×10

29=3×3+2×10 29=3×3+2×10

17=7×1+1×10 17=7×1+1×10

13=3×1+1×10 13=3×1+1×10

11=1×10+1×1 11=1×10+1×1

19=3×3+1×10 19=3×3+1×10

23=2×10+1×3 23=2×10+1×3

p 314 =2083=43×43+10×10 p314=2083=43×43+10×10

How to help me with tests?

If we have two primes, x
x
and y
y
and both of them are two different sides of two geometric squares, x 2

x2
is the area on first square and y 2

y2
is the area on the second square. We can use math formulae to provide a new square with area z 2

z2
and that will be a new prime. We use

x 2 +y 2 +xy+xy, x2+y2+xy+xy,

we also can use

2(x+y)+x 2 +y 2

2(x+y)+x2+y2

the last prime that we can use before a zillion digits are 1946. After 1991 they reach a zillion digits and are close to infinity. Thank you all for the attention.

I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741).

Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ a deformation. Consider the Stokes equations written in the deformed configuration

\begin{align} - 2\mu \operatorname{div}(D(u)) + \nabla p &= f, \quad \text{in } \Phi(\Omega) \\ \operatorname{div}(u) &= 0, \quad \text{in } \Phi(\Omega) \\ \end{align}

where $u$ is the velocity of the fluid, $p$ the pressure, $\mu>0$ is the constant viscosity, $f$ is an external force and $D$ is the operator defined by

$$ D(u) = \frac{1}{2} (\nabla u + \nabla u^T). $$

How can the Stokes equations be written in the domain $\Omega$ using a change of variable ?

I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch.

Bonus points if the sketch and/or the colimit diagram is finite.