https://drive.google.com/file/d/0B2JyzqDGBuZ3QnlKa0s2VGY2ZDg/view?usp=drivesdk

Above is link for the image of Book "ComplexNumbers from A to Z " by Titu Andreescu ,there i have highlighted a Mathematical step which am unable to get ,its a product of two sets but seems tricky to me, i would appreciate your talent if you grab me out of this doubt.

Let $F\in M_k(\Gamma_0(N),\chi)$, not necessarily an eigenform nor cuspidal, but assume that $\Bbb Q(F)$ is a number field $K$. What can one say of $\Bbb Q(F|_kW_N)$, where $W_N$ is the Fricke involution ? Experiments seem to show that the Fourier coefficients of $F|_kW_N$ divided by $\sqrt{Q}$ for some positive or negative divisor of $N$ (probably linked to the conductor of $\chi$) also lie in $K$.

More generally same question for a general Atkin--Lehner involution $W_Q$, and also for $1/2$-integral weight (in which case even a fourth root may be necessary).

It may be possible to start with a corresponding result for newforms, but I have not seen how to complete the argument.

P.S. Since I tested mainly with real characters, $\sqrt{Q}$ may of course be the Gauss sum associated to $\chi$. But the questions remain.

**Edit :** According to the comments of Michael Renardy and Christian Remling I revise the question as follows:

Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such that $X $ has a closed orbit and is in the form $X=f(\bar{z})$ where $f$ is a holomorphic function on $\overline{U}=\{\bar{z}\mid z\in U\}$?

**Added after the answer by Prof. Duchon:** Is there an example of such vector field with an Isochronous band of closed orbits?

Recently, I read the following result:

Let $\Omega$ be a bounded Lipschitz domain, $u\in H^1(\Omega)$ and $\Delta u=0$. Then the following conditions are equivalent:

(a) $u\mid_{\partial \Omega}\in H^{1}(\partial \Omega)$

(b) $\frac{\partial u}{\partial n}\mid_{\partial \Omega}\in L^2(\partial \Omega)$

Moreover, these conditions imply $u\in H^{3/2}(\Omega)$.

I have found a paper on this result. I'm interested in finding a book or lecture notes over this topic, i.e., boundary value problems for general elliptic equations with $L^2$ boundary data.

A classical theorem of Thierry Aubin states that:

**Theorem (Aubin, T. 1979):** If the Ricci curvature of a compact Riemannian manifold is
non-negative and positive at a point, then the manifold carries a metric of positive
Ricci curvature.

In the study of structures on manifolds (such as Hermitian, Kaehlerian, symplectic,...) is the above theorem true? e.g.

**Question:** Does "the Ricci curvature of a compact *Hermitian* manifold is
non-negative and positive at a point", imply "the manifold carries a *Hermitian metric* of positive
Ricci curvature"?

Your suggestions will be appreciated.

I want to solve the following optimization problem

\begin{align} \inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] \end{align} where $X^\prime$ is an independent copy of $X$ and $a>0$ is some constant.

How would one approach such a problem? Is the solution easy to find?

At some point I thought that the optimal distribution is given by $X=\{-a,a\}$ equally likely. In which case, the solution is given by \begin{align} \inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] \le \frac{1}{2}\frac{1}{1+4a^2}+\frac{1}{2}. \end{align} However, I don't have any supporting arguments for this.

The following might be useful. Note that by Jensens' inequality

\begin{align} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] \ge \frac{1}{1+E[(X-X^\prime)^2]} =\frac{1}{1+2Var(X)}. \end{align}

Therefore,

\begin{align} \inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] \ge \frac{1}{1+2 \sup_{ X: |X| \le a \text{ a.s.}} Var(X)}=\frac{1}{1+2a^2}, \end{align}

where in the last optimization step we used \begin{align} Var(X) \le E[X^2] \le a^2, \end{align} which is achievale with $X=\{-a,a\}$ equally likely.

Let $\mathcal{C}$ be the set of compact convex centrally symmetric sets in $\mathbb{R}^d$, and let $\mathcal{E} \subset \mathcal{C}$ be the set of ellipsoids centered at the origin.

I'm looking for a mapping $\pi:\mathcal{C}\to\mathcal{E}$ that satisfies the following properties:

*continuity*(with respect to the Hausdorff topology, say);*equivariance*under linear isomorphisms of $\mathbb{R}^d$, i.e., $$C\in \mathcal{C},\ L \in GL(d,\mathbb{R}) \ \Rightarrow \ \pi(L(C))=L(\pi(C));$$*mononicity*, i.e., $$C \subseteq D \ \Rightarrow \ \pi(C) \subseteq \pi(D).$$

The Löwner ellipsoid (meaning the unique ellipsoid of minimal volume containing the given compact set) is continuous and equivariant, but unfortunately it is not monotone: see this MO question and answer.

**Question:** Is there another kind of ellipsoid that satisfies the three properties? Or, if no such construction is known explicitly, can it proved abstractly (feel free to use axiom of choice) that such a map $\pi$ exists?

PS: One would expect $\pi$ to be a projection, but I don't need that property.

I want to calculate the following sum:

$$\sum_{\substack{d \leq x\\P^+(d)\le\sqrt{x}}} \mu(d) \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1,$$

where $P^+(d)$ is the largest prime divisor of $d\geq 2$, and $P^+(1)=0$.

I am currently studying the following inequality involving the square of the modulus of a specific Dirichlet polynomial:

$$\left( \sum_{1}^{N}\frac{1}{n} \right)^2 \ \ - \ \left| \sum_{1}^{N}\frac{(-1)^{n-1}}{n^{1/2+it}} \right| ^2 \ > \ \ 0$$ which, for arbitrary combinations of $N$ and $t$, is in general false. Just take the example $t=749.2$ plotting the above difference for $1<N<10000$ will show that it is negative up to about $N=3100$ (the exact value might be affected by the numerical accuracy of the particular Math SW tool). The inequality appears instead to hold true for $N$ greater than that. I then wondered whether there might exist simple functions $N(t)$ such that the above inequality is always satisfied. I played quite a lot with numerical simulations by first trying the very simple $N(t) = \lceil t\rceil^2 $ $\Rightarrow $ I was unable to find any violation. But of course, this is just an "experimental" approach. I wonder whether anybody may suggest specific references useful for the theoretical study, verification, or rebuttal, of the above inequality for particular $N(t)$ functions.

Many thanks.

Let $A \cong L^\infty[0,1]$ be a non-atomic maximal abelian *-subalgebra in $M \cong B(L^2[0,1])$ (or any von Neumann algebra $M$). Is the following true? For every $T \in M$ and $\epsilon>0$, there are non-zero projections $p,q \in A$ such that $\| qTp \| \le \epsilon \| T \|$.

Let $X$ be an algebraic variety, $at(E) \in Ext^1(E, E \otimes T)$ the Atiyah class of a complex $E \in D(Coh X)$ (see Markaryan, $\S$1.1-1.2).

Then $at(\Omega[1])$ gives a "shifted Lie algebra" on $T_X[-1]$, that is a morphism $T_X[-1] \otimes T_X[-1] \to T_X[-1]$ in the derived category $D(Coh X)$, satisfying certain properties.

How can one think about such an object? For example, can one explicitly (but, maybe up to quasi-isomorphism or something?) describe it for $X=\mathbb P^1$?

I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "rolling" behavior; that is, the bounces get shorter and shorter and it essentially sticks to the surface. The simulation basically can't progress beyond this point as a result. Below is the part of the trajectory that illustrates this behavior; the trajectory goes from chaotic to convergent along the surface of the torus:

On the left can be seen the discrete bounces, and on the right is where the particle sticks to the surface (until program termination). It seems counter-intuitive that a the trajectory could become trapped like this after demonstrating chaotic behavior for thousands of prior bounces. Closer examination reveals that the angles become more and more grazing with each bounce, and the distances between bounces shorter, until no progress can be made.

If I had infinite numerical precision to peek past this event horizon, would I eventually find that the particle breaks free again? Or is it possible that this is also a theoretical limit, and the motion of the particle has transitioned from finite segments to a continuous curve constrained to the surface? Can this happen in 3D billiards problems, or does chaos dictate that this behavior would reverse after some unknown amount of time?

A cardinal register machine is like a finite register machine:

* Finite internal state.

* Finite set of registers.

* The ability to

- halt

- zero a register

- increment: add a new element to a register

- test whether two registers have the same number of elements

*but with a twist:*

* The machine can run for a transfinite time. At limit steps,

- the internal state is set to the initial state

- each register keeps the elements added since the supremum of times it was zeroed.

Thus, each register stores a set of elements, with the ability to add a new element (elements are never repeated), remove all elements, and test whether two registers have the same (cardinal) number of elements. At every stage, an element is in a register iff it was added but not removed.

What is the complexity of the halting problem? Initially, all registers are 0 (alternatively, finite).

A computationally equivalent description is that each register stores an ordinal number, is incremented as an ordinal, with liminf behavior at limit stages, but with equality (as tested by the machine) being cardinal equality.

This question is in Q/A format as I was able to solve it before posting, but feel free to contribute additional answers. For example, my answer does not have

* The restriction to countable time (or other bounded time computations).

* The minimum number of registers.

* The spectrum of complexities of $Σ_1(\mathrm{Card})$ without the large cardinal assumption

It seems that the following claim is true, but I did not manage to prove it neither to find a reference.

**Claim** Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its Hessian is Lipschitz continuous:
$$
\|\nabla^2 f(x)-\nabla^2 f(y)\|\le M\|x-y\|\qquad\forall x,y\in\mathbb R^p,
$$
where the matrix norm is the largest singular value while the vector norm is the usual Euclidean norm. Then, for every $x\in\mathbb R^p$ we have
$$
\|\boldsymbol\Delta [\nabla f(x)]\|^2=\sum_{i=1}^p \bigg(
\sum_{j=1}^p \frac{\partial^3 f}{\partial x_i\partial x_j^2} (x)\bigg)^2 \le pM^2.
$$

I can prove such an inequality with $p^2M^2$ instead of $pM^2$, but do not see how one can remove one power of $p$.

I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\sigma(s,x_s)\,dW_s}_{=:z_t}+y_t
\end{equation}
where $y_t=\sup_{0\le s\le t}(z_s)^{-}$ is the *regulator term* which ensures that the positivity constraint is respected and the stochastic integral is a Young integral.

Let us suppose everything real valued.

The following hypothesis are considered: $\xi_0\in\Bbb R_+^d:=\{(x_1,\dots,x_d)\in\Bbb R^d\;:\;x_i>0\;\;\forall i=1,\cdots,d\}$ is fixed, \begin{align*} &b:[0,L]\times\Bbb R\to\Bbb R\;\;\;\;,\\ &\sigma:[0,L]\times\Bbb R\to\Bbb R \end{align*} are measurable and bounded functions which satisfy \begin{equation}\label{hyp1}%hyp1 |b(t,x)-b(t,y)|\le K_0|x-y|\;\;\forall x,y\in\Bbb R,\;\forall t\in[0,L] \end{equation} \begin{equation}\label{hyp2}%hyp2 |\sigma(t,x)-\sigma(t,y)|\le K_0|x-y|\;\;\forall x,y\in\Bbb R,\;\forall t\in[0,L] \end{equation} \begin{equation}\label{hyp3}%hyp3 |\sigma(t,x)-\sigma(s,x)|\le K_0|t-s|^{\nu}\;\;\forall x\in\Bbb R,\;\forall s,t\in[0,L] \end{equation} where $\nu\in]\frac12,1]$ and $K_0>0$.

Hence, both $b$ and $\sigma$ are Lipschitz in space, moreover $\sigma$ is $\nu$-Holder continous in time.

Next we take a fractional Brownian motion $W$ of Hurst parameter $H$ that is, $W\in\mathcal C^{H-\varepsilon}([0,L],\Bbb R)$ for every $\varepsilon>0$, with $1/2<H\le\nu$.

We will consider an arbitrary fixed trajectory of this FBM (i.e. $\omega$ is fixed and we mean $W=W(\omega)$).

Then it is proved that for every fixed $\lambda\in]\frac12,H[$, the equation at the beginning admits a solution $x$ such that, a.s. $x(\omega)\in\mathcal C^{\lambda}([0,L],\Bbb R)$.

We summarize here the relations between the parameters considered above, in order to be clear: $$ \frac12<\lambda<H\le\nu\le1\;\;. $$

I would like to prove uniqueness for this SDE.

In the following we will consider $x^{(1)},x^{(2)}$ two $\lambda$-solutions of the SDE on $[0,L]$, writing $z^{(i)},y^{(i)}$ with the obvious meaning: $x^{(i)}=z^{(i)}+y^{(i)}\;\;i=1,2$.

We will take $0\le T\le L$, which will be chosen conveniently later.

I will skip the details (if someone is interested, tell me this!) in order not to annoy you, going directly to the core of the problem: setting $$ H_T:=\|x^{(1)}-x^{(2)}\|_{\infty,[0,T]} $$ I was able to prove that $$ H_T\le2K_0H_TT+\eta_TT^{\lambda+H-\varepsilon} $$ where $$ \eta_T:=K_0\|W\|_{H-\varepsilon,[0,T]}\left(\|x^{(1)}\|_{\infty,[0,T]}+\|x^{(2)}\|_{\infty,[0,T]}+2T^{1-\lambda}\right) $$ which is a bounded non-negative increasing function of $T$.

Now from this I got $$ \frac{H_T}{\eta_TT^{\lambda+H-\varepsilon}}\le\frac1{1-2K_0T} $$ and thus, passing to the $\limsup_{T\to0+}$ we get \begin{align*} 1 &=\limsup_{T\to0+}\frac1{1-2K_0T}\\ &\ge\limsup_{T\to0+}\frac{H_T}{\eta_TT^{\lambda+H-\varepsilon}}\\ &\ge\limsup_{T\to0+}\frac{|x_T^{(1)}-x_T^{(2)}|}{\eta_TT^{\lambda+H-\varepsilon}} \end{align*} and setting $f_t:=x_t^{(1)}-x_t^{(2)}$ we have that $f_0=0$ and $f\in\mathcal C^{\lambda}[0,L]$, thus the last term can be rewritten as $$ \limsup_{T\to0+}\frac{|f_T-f_0|}{|T-0|}\frac1{\eta_T|T|^{\alpha}}=(*). $$ where $\alpha:=\lambda+H-\varepsilon-1>0$.

Now we have two cases: if $\limsup_{T\to0+}\frac{|f_T-f_0|}{|T-0|}>0$ we'd get $(*)=+\infty$ thus we would have reached a contradiction, which would allow to get uniqueness.

The other case is $\limsup_{T\to0+}\frac{|f_T-f_0|}{|T-0|}=0$, from which we would have $\exists\lim_{T\to0+}\frac{|f_T-f_0|}{|T-0|}=0$, and thus, $f$ would be differentiable at 0 on the right. But here I'm stuck.

I am stuck on this problem. I understand the fundamentals of induction proofs, but I am unfamiliar with induction on two variables. Here's the prompt:

Prove that for every positive integer k, the following is true: For every real number r > 0, there are finitely many solutions to (1/n1) + (1/n2) +...+ (1/nk) = r. In other words, there exists some number m (that depends on k and r) such that there are at most m ways of choosing a positive integer n1, and a (possibly different) positive integer n2, etc., that satisfy the equation.

I have no clue where to start, especially how to reference m in terms of k and r. All I know is we cannot induct on r because it is not a natural number.

Suppose $T \in V_1 \otimes \cdots \otimes V_k$ is a tensor, where each $V_i$ is a finite dimensional complex vector space. A $1$-flattening (or a flattening) is a realization of $T$ as a matrix in the space of matrices in $k$ essentially different ways as follows: \begin{equation} V^*_{i} \rightarrow V_{1} \otimes \cdots \otimes V_{k-1}. \end{equation} Is it true that the rank of a flattening of $T$ is always a lower bound for the border rank of $T$? If not in general, is it true when $T$ is a symmetric tensor?

On a compact manifold $X$, every $0$-th order pseudodifferential operator extends to a bounded operator on $L^2(X)$.

My question is: is there an example of a non-compact manifold $X$ and a $0$-th order pseudodifferential operator on $X$ that is not bounded on $L^2(X)$?

Are there an infinite number of primes $p$ such that $2^p = 2 \mod p^2$?

The only examples I could find are $p = 1093$ and $p = 3511$? Though I would love a proof, any heuristic for why there should be finite $p$ would also be greatly appreciated.

The motivation is trying show that for large enough $N$, all primes $p > N$, $\binom{2^p+1}{p+2} \neq 0 \mod p$.

In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, then the action of $ h $ projects onto the isotypic component of the trivial subrepresentation of $ V $.

If $ A $ is just a $k$-algebra, then we have no notion of isotypic component of the trivial representation. However, there is one interesting exception. Suppose that $ V, W $ are two $H$-modules, then $Hom_k(V,W) $ is an $ A^{op} \otimes A $ module and it contains the space of homomorphisms $ Hom_A(V, W) $. So it is natural to look for some element $ T \in A^{op} \otimes A $ which when acting on $ Hom_k(V,W) $ projects onto $ Hom_A(V, W) $.

There are a number of conditions that such an element $ T $ should satisfy, including $$ T (a \otimes 1) = (1 \otimes a) T \ \text{ for all } \, a \in A. $$

Is there a name for such an element $ T$? Are there any existence/uniqueness results regarding such elements?