TITLE: deduce that there exists a unique polynomial q(x) of degree at most n such that$ q(c_i)=a_i$ for $0 \leq i \leq n$.
QUESTION [1 upvotes]: Let $V=P_n(F)$ (the vector space of polynomials with coeffficients in R of degree at most n), and let $c_0,c_1,...,c_n$ be distinct scalars in F. For any scalars $a_1,...,a_n$, deduce that there exists a unique polynomial q(x) of degree at most n such that$ q(c_i)=a_i$ for $0 \leq i \leq n$. In fact $q(x)=\sum_{i=1}^n a_i p_i(x)$.

The hint says for uniqueness, prove that ${p_0(x),p_1(x),...,p_n(x)}$ is a basis for $V=P_n(F)$. My question is how to prove the existence? And after that we need to use the hint right?

REPLY [2 votes]: The same problem gave you the recipe of how to construct the desired polynomial, just put
$$q(x) = \sum_{j=0}^n a_jp_j(x)$$
and we check that this satisfy $q(c_i) = a_i$. But this is easy, since
$$q(c_i) = \sum_{j=0}^n a_jp_j(c_i) = \sum_{j=0}^n a_j \delta_{ij} = a_i.$$


TITLE: What is the difference between a transparent material and an opaque one at the atomic level?
QUESTION [13 upvotes]: What is the difference between a transparent material and an opaque one at the atomic level? I know that something is transparent to light because the energy of the photons that correspond to visible light does not match the transition energy of the atoms that make up that body. What happens when something is opaque? And why the angle of incidence is always equal to the angle of reflection? I know about the Fermat's principle and all the mathematical arguments, but I was looking for a "physical picture" of the phenomenon.

REPLY [4 votes]: First, you are asking about the material being opaque or transparent to visible light, so we are talking about visible wavelength photons (400-700nm). And you are asking about QM level explanation. You are right to ask the question, because most explanations on this site are more classical, and in this case, you might be curious about what happens at the levels of atoms and photons.
You are asking why the angle of reflection is always equal to the angle of incidence. This is a common misconception.

This is only true for specular reflection (mirror-like), where the relative phase and angle of photons are kept. Most of the photons, in this case, will have an angle relatively (to each other) same.
For diffuse reflection the angle of reflection is random. This way the relative angle of the photons is not kept, and you do not see a mirror image.

Now you are asking why some materials are opaque. First of all, there are materials that are partially transparent, like some liquids, some gases, but let's disregard that for now, and just take the simple two cases of opaque and transparent materials. Opaque materials are opaque because:

opaque materials reflect (or absorb) most of the visible wavelength photons at their surface, and do not let visible light propagate through them (from the opposite direction into your eyes). For example, a normal mirror is opaque too. There are some mirrors (interrogation room), that is opaque from one side, and transparent from the other side (it is basically a trick because from one side they let more than 50% of the photons through, and from the other side less then 50%).
in the case of metals, opaqueness comes from reflection (elastic scattering), and for some other materials, it comes from absorption and re-emission (at the surface) of visible light. That is why metals do not have their own color, they reflect all visible light. Other non-metallic materials absorb and re-emit visible light, that is how we see their colors.

Now transparent materials are transparent because:

they do not reflect most visible light, but let most visible light refract, that is travel inside them. Some transparent media (like air) and glass, let light travel so, that the photons keep their relative phases and energies (and relative angles) inside the media so that you see a mirror image on the other side of the media. That is why glass (and air) lets light keep a mirror image inside them.
In the case of transparent media, it is elastic scattering. As the photons travel through the glass, air, water, they elastically scatter off the molecules and atoms of the material and keep their relative phases, energies, and angles. It is a common misconception that inside the air, glass or water, photons get absorbed and re-emitted. That would not keep their energy levels, and relative phases, and angles. Only elastic scattering keeps a mirror image.

Now you would think that glass only refracts. But in reality, when you look at a glass window, it will both refract and reflect, and in both cases usually will give you a perfect mirror image. Even if you stand on both sides of the glass, you might get a mirror image of what is on the other side, and what is behind you (reflection). When you look at the glass window you can sometimes see a mirror image of yourself too. This is because glass reflects and refracts.
Now you could ask, then what is the real difference between reflection and refraction in the case of glass, when it does the same thing at the same time. Both reflection and refraction is elastic scattering. This is the only way the photons keep their energies, relative phases and angles. The only difference between refraction and reflection in the case of glass is the angle. For reflection, the angle is the same as the angle of incidence (and the photons move backwards in the original medium, air). For refraction, the angle changes (but the relative angle of the photons is kept), as the photons pass through the glass.
The question then comes down at the QM level to why some materials refract and why some reflect (most visible light). Here is a good explanation:
Why certain objects reflect while some refract?


TITLE: Find the series for $\csc^2(x)$
QUESTION [1 upvotes]: In a problem I'm being asked to find the series representation for $$\csc^2(x)=\sum_{-\infty}^{\infty} \frac{1}{(x-n \pi)^{2}}$$
I'm supposed to solve this problem by using complex analysis. However, I have no idea where to start.
How do I approach this problem?
I tried using the residue theorem, however, I didn't get very far.

REPLY [0 votes]: The log-derivative of $\sin x=x\prod_{n\ge1}(1-x^2/(n\pi)^2)$ is$$\cot x=\frac1x+\sum_{n\ge1}\frac{2x}{x^2-(n\pi)^2}=\sum_{n\in\Bbb Z}\frac{1}{x-n\pi}$$by partial fractions. Now apply $-\frac{d}{dx}$.


text:
\section{The Heisenberg double associated to positive self-adjoint Hopf algebras}
\label{sec:algebraFock}

\subsection{The Heisenberg double}
\label{ssec:algebraFock}

Given a pair of Hopf algebras $H,H'$ and a pairing between them satisfying certain requirements one can endow the space $H\otimes H'$ with an algebra structure and form an algebra called the \emph{Heisenberg double}. The description of the general notion can be found in \cite{STSMHeisdouble} and \cite{kapranovheisenberg}. In this article we are concerned with the specific case of this construction in the case of positive selfadjoint Hopf algebras introduced by Zelevinsky in \cite{Zelbook}. Let us recall the definitions:
\begin{Definition}
A positive selfadjoint Hopf (PSH) algebra is a graded connected Hopf algebra over $\ZZ$ with an inner product and a distinguished finite orthogonal $\ZZ$ basis in each grade s.t. multiplication and comultiplication are adjoint and take elements with positive coefficients to elements with positive coefficients.
\end{Definition}

\begin{Definition}
A graded Hopf algebra over $\ZZ$ $A=\bigoplus_{n\geq 0}A_n$ is called \emph{connected} if the unit morphism and the counit morphism restricted to $A_0$ give an isomorphism of $A_0$ with $\ZZ$
\end{Definition}

For a positive self-adjoint Hopf algebra $A$ let us consider the dual pair of Hopf algebras $(A,A)$ and outline the construction of the Heisenberg double in this case. Simultaneously we will show that this algebra has a natural action on $A$ (considered as a $\ZZ$ module). We will refer to this action as \emph{Fock space action} by the analogy with the case of infinite dimensional Heisenberg algebra, which is a special case of this construction as noted in \autoref{ssec:generators}. Applying the analog of this construction in the categorical setting will allow us to construct the natural categorical Fock space action for any SSH category in \autoref{sec:heisenbergaction}.

Consider a PSH algebra $A$ and denote the adjoint multiplication and comultiplication maps by
\begin{align*}
m: A\otimes A &\rightarrow A &\Delta:A&\rightarrow A\otimes A
\end{align*}
Note that $A$ is commutative and cocommutative, a result proven in \cite{Zelbook}.

For each $x\in A$ we define operators $m_x,\Delta_x:A\rightarrow A$ by the formulas
\begin{equation}
\label{def:mxdeltax}
\begin{aligned}
m_{x} &=m\circ i_x & \Delta_x &=j_x\circ\Delta
\end{aligned}
\end{equation}
where
\begin{align*}
i_x(y)&=x\otimes y  & j_x(y\otimes z)&=z<x,y>
\end{align*}
\begin{Remark}
Note that $m_x, \Delta_x$ are adjoint for any $x \in A$.
\end{Remark}
We use these operators to define a morphism of $\ZZ$ groups
\begin{align*}
\varphi:A\otimes A &\rightarrow\End_\ZZ(A) &x\otimes y &\mapsto m_x\Delta_y
\end{align*}
\begin{Proposition}
\label{prop:heisdouble}
$\varphi$ is injective and its image is a subalgebra of $End_{\ZZ}(A)$
\end{Proposition}
\begin{Notation}
Since $\varphi$ is injective, it induces an algebra structure on $A\otimes A$. We denote the algebra $A\otimes A$, with the algebra structure given by $\varphi$, by $\Heis(A)$. The action of $\Heis(A)$ on $A$ defined by $\varphi$ is called the \emph{Fock space action}.
\end{Notation}
\begin{proof}
To prove that the image of $\varphi$ is a subalgebra note the we have the following relations $\forall x,y \in A$:
\begin{align}
\label{ref:heisenbergrelations}
m_xm_y &= m_{m(xy)}=m_{m(yx)}=m_ym_x\\
\Delta_x\Delta_y &= \Delta_{yx}=\Delta_{xy}=\Delta_y\Delta_x\\
\label{eqn:deltam}
\Delta_xm_y &= m\Delta^2_{\Delta(x)}i_y
\end{align}

where if $\Delta(x)=x_{(1)}\otimes x_{(2)}$ (in Sweedler notation) then
\begin{equation}
\Delta^2_{\Delta(x)}i_y(z)=\Delta^2_{\Delta(x)}(y\otimes z):=\Delta_{x_{(1)}}y\otimes\Delta_{x_{(2)}}z
\end{equation}

The first two relations hold since multiplication and comultiplication in $A$ are associative and commutative and the relation \eqref{eqn:deltam} holds since (as shown in \cite{Zelbook}) $\forall z,u \in A$:
\begin{multline*}
<\Delta_xm_yz,u>=<\Delta_xm(y\otimes z),u>=\\
<m(y \otimes z), m(x \otimes u)>=<y\otimes z, \Delta m(x \otimes u)>=\\
<y \otimes z,m(\Delta x \otimes \Delta u)>=<y \otimes z,m((x_{(1)}\otimes x_{(2)})\otimes\Delta u)>=\\
<\Delta_{x_{(1)}}y \otimes\Delta_{x_{(2)}}z,\Delta u>=<m(\Delta_{x_{(1)}}y \otimes\Delta_{x_{(2)}}z),u>
\end{multline*}
(we used the fact that $m_{x_{(1)}\otimes x_{(2)}}$ is adjoint to $\Delta_{x_{(1)}\otimes x_{(2)}}$ on $A^{\otimes 2}$)\\
So explicitly, the third relation gives us that
\[
\Delta_xm_y=m_{\Delta_{x_{(2)}}(y)}\Delta_{x_{(1)}}\in\varphi(A\otimes A)
\]

To prove that $\varphi$ is injective we use the fact that $A$ is graded and each grade has an orthogonal basis. Let $\sum_i x_iy_i \in Ker \varphi$ and assume without loss of generality that $y_i$ are the elements of the orthogonal basis of $A$. Let $y_{i_0}$ be an element of minimal degree so that $x_{i_0} \neq 0$ and let $r=\deg y_{i_0}$. It follows from the definition of $\Delta_y$ that for every $z \in A_n, y \in A_m$ $\Delta_yz=0$ when $n<m$, and $\Delta_yz=<y,z>$ for $n=m$. Hence by applying the operator $\sum_i m_{x_i}\Delta_{y_i}$ to $y_{i_0}$ we get $x_{i_0}=0$, a contradiction.
\end{proof}
To construct the categorification of the Fock space action in \autoref{ssec:heisenbergcategorification} we will use the following key
\begin{Observation}
\label{heisenbergrelation}
The relation $\Delta_xm_y = m\Delta^2_{\Delta(x)}i_y$ for any $x,y$ is equivalent to the relation $\Delta_xm = m\Delta^2_{\Delta(x)}$ for any $x$.
\end{Observation}
We will call this relation the \emph{Heisenberg relation} throughout this article. The reformulation above a first step towards the construction of the categorification of Heisenberg algebra from the SSH structure.

The above construction can be easily generalized to give the following general statement:
\begin{Proposition}
\label{prop:heisenbergaction}
Giving an action of $\Heis(A)$ on a space $V$ is the same as giving a morphism of spaces $A\otimes A\xrightarrow{a}\End(V)$ which satisfies
\begin{enumerate}
\item The restrictions of $a$ to $A\otimes 1$ and $1\otimes A$ are morphisms of algebras.
\item  Denote \begin{gather*}
\Delta^2_a:A\otimes A\rightarrow\End(A\otimes V) \\
\Delta^2_a(x\otimes y)(z\otimes v):=\Delta_x(z)\otimes (a(1\otimes y)(v))\\
m_a:A\otimes V\rightarrow V \\
m_a(z\otimes v):=a(z\otimes 1)(v)
\end{gather*}
then for any $x\in A$ we have $a(1\otimes x)\circ m_a=m_a\circ \Delta^2_a(\Delta(x))$
\end{enumerate}
\end{Proposition}

\subsection{The infinite-dimensional Heisenberg algebra}
\label{ssec:generators}
The classical one-variable Heisenberg algebra is the $\ZZ$-algebra with two generators $p,q$ and one defining relation $[p,q]=1$. The infinite version of this algebra is usually defined in terms of infinite number of generators and relations. There are several different versions used in different settings, some of which are described below. All of these are (sometimes non-isomorphic) $\ZZ$ forms of the same complex algebra.

We use the notion of of Heisenberg double for a positive self-adjoint Hopf algebra which we explored in the previous section to give an alternative description of the infinite-dimensional Heisenberg algebra. This description doesn't use the language of generators and relations, and thus lends itself more naturally to categorification.

Denote by $\Lambda$ the algebra of symmetric polynomials in a countable number of variables over $\ZZ$. $\Lambda$ has the structure of a PSH algebra. The $\ZZ$ basis is given by the Schur polynomials; this also defines an inner product. The multiplication map $m$ is given by the multiplication of polynomials.

\begin{Definition}
We define the Heisenberg algebra of infinite rank to be $\Heis(\Lambda)$, i.e. the the Heisenberg double corresponding to the pair $(\Lambda,\Lambda)$.
\end{Definition}

Let us describe how some of the commonly used definitions of Heisenberg algebra arise from the above definition:
\begin{enumerate}
\item A $\ZZ$-algebra with generators $p_n,q_n,n\in\NN$ and relations\begin{itemize}
\item
$[p_m,p_n]=[q_m,q_n]=0$
\item
$[p_m,q_n]=\delta_{mn}1$
\end{itemize}
This corresponds to taking $p_n$ to be the elementary symmetric function of degree $n$ ($\sum_{i_1<\dots<i_n} x_{i_1}\cdots x_{i_n}$) in the left $\Lambda$ and the $q_n$ to be the primitive symmetric function of degree $n$ (described in \cite{Zelbook}) in the right $\Lambda$.
\item A $\ZZ$-algebra with generators $c_k,k\in\ZZ$ and relations $[c_k,c_l]=k\delta_{k+l,0}$. This corresponds to taking $c_k$ with positive $k$ to be the primitive symmetric function
of degree $k$ in the right $\Lambda$ and $c_k$ with negative $k$ to be the primitive symmetric
function of degree $k$ in the left $\Lambda$, and taking $c_0=1$.
\item (The algebra which Khovanov categorifies in \cite{Khov})  Generators $a_n,b_n$, $n\in\NN$ with relations\begin{itemize}
\item
$a_0=b_0=1$
\item
$[a_m,a_n]=[b_m,b_n]=0$
\item
$[a_m,b_n]=b_{n-1}a_{m-1}$
\end{itemize}
This corresponds to taking $a_n$ to be the elementary symmetric function
of degree $n$ in the left $\Lambda$ and the $b_n$ to be the whole symmetric
function of degree $n$ ($\sum_{i_1\leq\dots\leq i_n} x_{i_1}\cdots x_{i_n}$) in the right $\Lambda$.
\end{enumerate}

config: arxiv