TITLE: Let $s(n)$ be the number of divisors of $n\in\mathbb N$ that are squares. Show that $\frac{1}{n}\sum_{i=1}^n s(i) < \frac{(\pi^2)}{6}$.
QUESTION [1 upvotes]: I believe I need to use the rule of double counting but I am struggling to find an answer. Thanks!

REPLY [0 votes]: Simply use double counting: $$\sum_{k=1}^n{s(k)}=\sum_{1 \leq d \leq n}{\sum_{a^2|d}{1}} = \sum_{a \geq 1}{\sum_{1 \leq d \leq n,a^2|d}{1}}=\sum_{1 \leq a \leq \sqrt{n}}{\sum_{k \geq 1,ka^2 \leq n}{1}} \leq \sum_{1 \leq a \leq \sqrt{n}}{\frac{n}{a^2}}\leq n\sum_{i=1}^n{i^{-2}} < n\zeta(2)=n\frac{\pi^2}{6}.$$

REPLY [0 votes]: HINT: Try proving that
$$\sum_{i=1}^n s(i) = \sum_{i=1}^\infty \bigg\lfloor\frac{n}{i^2}\bigg\rfloor$$


TITLE: How can an interval be an open ball?
QUESTION [2 upvotes]: I have been asked this question, but I do not understand it.
"For what values of $\alpha$ and $\beta$ are the sets $(\alpha, \beta)$, $[\alpha, \beta)$, $(\alpha, \beta]$ and $[\alpha, \beta]$ open balls in the metric space $[a, b]$?"
We have our interval $[a,b]$ which can be interpreted as a line along the x-axis. How can, for example, $(\alpha, \beta)$ which is another interval along the x-axis be considered a ball if it only exists on the x-axis and doesn't extend to the y-axis?

REPLY [0 votes]: Balls aren't always circles or spheres. Consider $(x-\epsilon,x+\epsilon)$. It is a ball centered at $x$ and radius $\epsilon$. It is an open ball by definition. (You can view it as the intersection of a circle (or sphere) of radius $\epsilon$ centered at $x$ in $\mathbb{R}^2$ (or $\mathbb{R}^3)$,and its itersection with $\mathbb{R}$)


text:
\section{Introduction}
\label{intro}

Photonic crystals are to electromagnetic waves what crystalline solids are to an electron, and therefore it is not surprising that periodic electromagnetic media exhibit analogs to various phenomena from condensed matter physics. One particularly intriguing example of such a quantum-wave analogy is the \emph{“Quantum Hall Effect for light”} in media with broken time-reversal symmetry. While nowadays whole sub communities are working on topological phenomena in classical waves, Raghu's and Haldane's seminal idea \cite{Raghu_Haldane:quantum_Hall_effect_photonic_crystals:2008} was initially met with a lot of skepticism. That is until researchers at MIT confirmed Haldane's prediction in experiment \cite{Wang_et_al:unidirectional_backscattering_photonic_crystal:2009,Ozawa_et_al:review_topological_photonics:2018}.

Just like in case of the well-known Integer Quantum Hall Effect \cite{von_Klitzing_Dorda_Pepper:quantum_hall_effect:1980,von_Klitzing:quantum_Hall_effect:2004,Thouless_Kohmoto_Nightingale_Den_Nijs:quantized_hall_conductance:1982}, Haldane proposed to explain the existence and robustness of unidirectional, back-scattering-free edge modes that are at the heart of this phenomenon by means of a \emph{bulk-edge correspondence} \cite{Hatsugai:Chern_number_edge_states:1993,Hatsugai:edge_states_Riemann_surface:1993}:

\begin{conjecture}[Raghu and Haldane's Photonic Bulk-Edge Correspondence \cite{Raghu_Haldane:quantum_Hall_effect_photonic_crystals:2008}]\label{intro:conjecture:photonic_bulk_edge_correspondence}
	In a two-dimensional photonic crystals with boundary the difference of the number of left- and right-moving boundary modes in bulk band gaps is a topologically protected quantity; it equals the bulk Chern number associated to the positive frequency bands below the bulk band gap.
\end{conjecture}

Chern numbers are topological invariants that are associated to families of frequency bands, the \emph{relevant bands}, that are separated from the others by a \emph{spectral gap} (the Gap Condition~\ref{periodic:assumption:gap_condition} makes this precise); they cannot change under continuous, gap-preserving transformations, which explains their robustness under rather strong perturbations.

Our ultimate goal is to furnish a proof to Haldane's conjecture, and our earlier publications \cite{DeNittis_Lein:adiabatic_periodic_Maxwell_PsiDO:2013,DeNittis_Lein:sapt_photonic_crystals:2013,DeNittis_Lein:ray_optics_photonic_crystals:2014,DeNittis_Lein:symmetries_Maxwell:2014,DeNittis_Lein:Schroedinger_formalism_classical_waves:2017,DeNittis_Lein:symmetries_electromagnetism:2017} all systematically work towards this goal.

This paper addresses one aspect of this endeavor: Maxwell's equations~\eqref{setup:eqn:Maxwell_equations} for the \emph{electromagnetic} field are \emph{first} order in time and space. When bi-anisotropic coupling between electric and magnetic fields is absent, the dynamical Maxwell equations~\eqref{setup:eqn:Maxwell_equations:dynamics} are block-\emph{off}diagonal in $(\mathbf{E},\mathbf{H})$, and it is possible and indeed, sometimes preferable to work with \emph{second}-order wave equations for the \emph{electric} or \emph{magnetic} fields instead (equations~\eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:wave_equation_magnetic}, respectively). That makes the second-order formalism appealing for numerical schemes, because it is inherently more efficient to work with $\C^3$-valued vector fields rather than $\C^6$-valued vector fields. Moreover, for certain geometries (\cf \eg \cite[Section~2.4]{DeNittis_Lein:symmetries_Maxwell:2014}) the second-order formalism allows us to describe all of the physics via two \emph{scalar} equations as opposed to $\C^6$-valued vector fields. In view of this, it is not surprising that in many applications the Chern numbers that enter Haldane's photonic bulk-boundary conjecture are computed on the basis of the electric or magnetic field alone (see \eg \cite{Joannopoulos_Johnson_Winn_Meade:photonic_crystals:2008,Wang_et_al:edge_modes_photonic_crystal:2008,Wang_et_al:unidirectional_backscattering_photonic_crystal:2009}); we will denote those electric and magnetic Chern numbers with $\mathrm{Ch}^E$ and $\mathrm{Ch}^H$, respectively.

On the other hand, analogies to quantum mechanics are most readily apparent in the \emph{first}-order formalism. Indeed, Haldane's own work uses the \emph{electromagnetic} field to compute Chern numbers $\mathrm{Ch}^{EH}$. And we have argued in \cite[Section~5.2.1]{DeNittis_Lein:symmetries_electromagnetism:2017} that conceptually speaking, this is the right quantity to start with in a first-principles approach. In the same place, we also raised the question that we will answer in the affirmative here:

\begin{theorem}[Electric, magnetic and electromagnetic bulk Chern numbers agree]\label{intro:thm:Chern_numbers_agree}~\\
	Suppose the material weights which describe the medium satisfy Assumption~\ref{periodic:assumption:periodic_material_weights}, \ie the medium is periodic, lossless, not bi-anisotropic and has positive index.
	Then the electric, magnetic and electromagnetic Chern numbers associated to any family of frequency bands satisfying the Gap Condition~\ref{periodic:assumption:gap_condition} agree,

	\begin{align*}
		\mathrm{Ch}^E = \mathrm{Ch}^H = \mathrm{Ch}^{EH}
		.
	\end{align*}

\end{theorem}

While the validity of this result is often tacitly assumed in the literature, to the best of our knowledge there is no publication that actually tries to \emph{derive} this from first principles. Indeed, we rephrase the problem in the — for physicists — more abstract language of vector bundles and then the proof is straightforward. In contrast, a direct verification of this Theorem by computing the electric/magnetic/electromagnetic Chern number from the electric/magnetic/electromagnetic Berry curvature seems unfeasible — even in the simplest case of a single, non-degenerate band.

\begin{remark}
	The assumption that the medium is not bi-anisotropic is crucial. For otherwise, the electric and magnetic Chern numbers are not well-defined, and the question we set out to answer in this paper does not make sense. Indeed, when the Maxwell equations contain a non-zero bi-anisotropic coupling term, then electric and magnetic fields do \emph{not} decouple in the second-order formalism. Even block-diagonalizing the first-order equations in the $(\mathbf{E},\mathbf{H})$ splitting is usually problematic as that necessarily mixes positive and negative frequency states; for a more in-depth discussion we refer to \cite[Section~5.2.2]{DeNittis_Lein:symmetries_electromagnetism:2017}.
\end{remark}

Strictly speaking, we only provide a proof of this statement for a three-dimensional photonic crystal whereas the Quantum Hall Effect for light occurs in quasi-two-dimensional media. However, none of the arguments depend on the dimensionality and can be readily adapted to lower- or higher-dimensional photonic crystals. Moreover, it should be possible to go from periodic to include randomness, but we shall not do this here.

Our proof will consist of two steps: first, we will show the equivalence of the first- and second-order equations. While this is \emph{in principle} completely standard and is spelled out in the literature (including \cite{Wilcox:scattering_theory_classical_physics:1966,Reed_Simon:scattering_theory_wave_equations:1977,Figotin_Klein:localization_classical_waves_II:1997}), existing results do not cover the case we are interested in, media with broken time-reversal symmetry. Here, our only original contribution is to start with the correct, physically meaningful \emph{first}-order Maxwell equations that we have derived in  \cite[Section~2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}.

The second step is to use the language of vector bundles, and interpret the maps $\imath^{E,H}(k) : \varphi_n^{E,H}(k) \mapsto \bigl ( \varphi_n^E(k) , \varphi_n^H(k) \bigr )$, defined in Section~\ref{periodic:frequency_bands:equivalence_spectra} below, which reconstruct electromagnetic Bloch functions from the electric or magnetic field component alone. These maps can be interpreted as vector bundle isomorphisms between an electric, magnetic and electromagnetic Bloch bundle. Up to isomorphism these are characterized by Chern numbers, and since these three bundles are isomorphic, their Chern numbers necessarily agree.

config: arxiv


TITLE: Analyticity of Logarithmic Integrals
QUESTION [0 upvotes]: Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such integrals and their properties.

REPLY [1 votes]: The rough approach is to differentiate $g$ formally, and then look at whether the integral has a reason to converge. If it does, you should have the derivative of that order. If it does not , then you shouldn't expect to have that derivative.
So, you shouldn't  expect even one derivative here. Indeed, taking $f = \chi_{[0,1/2]}$ you will see that $g'(1/2)=\infty$. Just calculate $f$ explicitly: say, for $y>1/2$ it is
$$
g(y) =  y\ln (2y)+(1/2-y)\ln(2y-1)-(1/2)\ln(2) -\frac12
$$
where the second term has unbounded derivative at $1/2$.


TITLE: Determining equilibrium solutions for non-linear first order ODE
QUESTION [1 upvotes]: I have the first order ODE:
$$
\frac{dy}{dx}=y^2-r\sqrt{x}, ~~ x\geqslant 0
$$
How do I determine the possible equilibrium solutions? The answer is $y=0$ but I am unsure as to how one arrives at that.

REPLY [0 votes]: At each instant in time, a solution is growing if $y>\sqrt{r}\sqrt[4]x$ or $y<-\sqrt{r}\sqrt[4]x$ and falling in the interval in the middle. From these qualitative observations alone one sees that negative solutions tend towards the curve $y=-\sqrt{r}\sqrt[4]x$, while for large positive initial values you get divergence to $+\infty$. Somewhere in the smaller positive initial values is a point that divides these two possibilities.
In no case other than $r=0$ do you get stationary solutions $y(x)=y_*$. In the case $r=0$ the condition $y'=0$ is only satisfied for $y_*=0$.
WolframAlpha can be used for phase portraits, left for some nonzero $r$, right for $r$ nearly zero (the function needs x to be present in a non-trivial way)
streamplot[{1,y^2-2 sqrt(x)}, {x,0,10}, {y,-5,5}]
streamplot[{1,y^2-0.001*sqrt(x)}, {x,0,10}, {y,-5,5}]


TITLE: System of equations for vector valued functions problems
QUESTION [1 upvotes]: I am having trouble about determining the system to solve when using finite element method for vector fields.
Suppose we have the following problem:
$$-\Delta\mathbf{u}=\mathbf{f}\;\text{in}\;\Omega,$$
with some Dirichlet boundary conditions. The weak form would be:
$$\int_{\Omega}\nabla\mathbf{u}:\nabla\mathbf{v}=\int_{\Omega}\mathbf{fv},$$
where $\nabla\mathbf{u}$ and $\nabla\mathbf{v}$ are matrices.
If we discretize the functions like this:
$$\mathbf{u}^h=\sum_{j=1}^{N}\mathbf{u}_j\mathbf{\phi}_j,$$ where $N$ are the nodes of the discretization. Here is my first doubt, should be $\mathbf{\phi}_j$ scalar or vector valued function? In the first case I arrive to the following system of equation:
$$\begin{pmatrix}A & 0\\
0 & A\end{pmatrix} \begin{pmatrix}U^1\\U^2
\end{pmatrix}= \begin{pmatrix}f_1\phi_i\\
f_2\phi_i\end{pmatrix},$$where $A = \int\nabla\phi_i\nabla\phi_j$ and $f_1\phi_i,\;i,j=1...N$, is it correct?.
However, if $\mathbf{\phi}_j$ is a vector valued function then the discretization $u^h=\sum_{j=1}^{N}\mathbf{u}_j\cdot\mathbf{\phi}_j$ is not a vector but a scalar so I do not know what to do. Moreover, I do not know if it makes sense that $\mathbf{\phi}_j$ be a vector valued function.
Thanks!

REPLY [0 votes]: Finally, I found how to discretize the problem for vector valued functions. The following annotations were inspired by the great book Finite Element Methods for Flow Problems writen by Jean Donea and Antonio Huerta.
Let's define the canonical basis:$$\mathbf{e_1}=(1,0)^T,\;\mathbf{e_2}=(0,1)^T.$$
Then, the solution can be discretized as:$$\mathbf{u^h}=\sum_{i=1}^{2}u_i^h(\mathbf{x})\mathbf{e_i}=\sum_{i=1}^{2}\sum_{j=1}^{N}u_{ij}\phi_j(\mathbf{x})\mathbf{e_i},$$
where $N$ is the number of nodes in the discretization and $u_{ij}$ is the approximation of the solution for the node $j$ in dimension $i$ (not spatial dimension but the dimension of the vector valued function output).
So, the system to solve for the problem $$\int_{\Omega}\nabla\mathbf{u}:\nabla\mathbf{v}=\int_{\Omega}\mathbf{fv}\Rightarrow a(\mathbf{u},\mathbf{v})=l(\mathbf{v})$$is:
$$\sum_{i=1}^{2}\sum_{j=1}^{N}u_{ij}\;a(\phi_j\mathbf{e_i},\phi_s\mathbf{e_c}) = l(\phi_s\mathbf{e_c}),$$
and there are $N\times 2$ equations, that is, and equation for each $e$ and $s$ where $s=1,...,N$ and $c=1,2$.
I would like to say that @MSEoris was right about saying that the problem itself has no physical sense however I asked the question having in mind a Stokes problem and in order to summarize it I only wrote the laplacian part.


TITLE: Linear independence question variables
QUESTION [0 upvotes]: are $x^3 - x^2, x - x^2$, $1$ linearly independent and if not is there something similar to it that is? I think they are but I`m not exactly sure what to do

REPLY [0 votes]: Are the vectors $\begin{pmatrix}0\\0\\-1\\1\end{pmatrix}$, $\begin{pmatrix}0\\1\\-2\\0\end{pmatrix}$, $\begin{pmatrix}1\\0\\0\\0\end{pmatrix}$ linear independent? (Obviously.)