TITLE: $\liminf_{n\to\infty} g_n(x)\ge 0$ for all $x>0$ implies $\liminf_{n\to\infty} g_n(x_n)\ge 0$ for some $x_n\to 0$?
QUESTION [0 upvotes]: As the title suggests, the problem is: if $\liminf_{n\to\infty} g_n(x)\ge 0$ for each $x>0$, then we may choose a sequence $\{x_n\}$ such that $x_n\to 0$ and $\liminf_{n\to\infty} g_n(x_n)\ge 0$. Intuitively, this is right as long as $x_n$ tends to 0 sufficiently slow. But how to rigorously show this? Thanks!

REPLY [1 votes]: For all $x$ and all $\epsilon>0$, there are only finitely many $n$ with $g_n(x)<-\epsilon$.
For $k\in\mathbb N$, let $y_k=\frac1k$ (or virtually any sequence with $y_k\to 0)$ and let $m_k$ be the least integer such that $g_n(y_k)>-\frac1k$ for all $n>m_k$.
Now let $\nu_1=0$ and recursively $\nu_{n+1}=\min\{\nu_n,m_{n+1}\}+1$.
Then $\nu_k\to +\infty$ strictly monotonically.
Especially, for each $n$ there exists a maximal $k$ with $\nu_k\le n$. Let $x_n=y_k$ with this maximal $k$.
Then $x_n\to 0$ because $(\nu_k)$ exceeds every $n$ soner or later.
More proecisely, given $\epsilon>0$ we find $K$ such that $|y_k|<\epsilon$ for all $k>K$. Then for $n>\nu_{K+1}$ we have $x_n=y_k$ with $k>K$, hence $|x_n|<\epsilon$. In other words, $x_n\to 0$.
Since $x_n=y_k$ with $n\ge \nu_k>m_{k}$, we have $g_n(x_n)=g_n(y_k)>-\frac 1k$.
Thus if $\epsilon>0$ is given, we have $g_n(x_n)>\epsilon$ as soon as $n>\nu_{\lceil\frac1\epsilon\rceil}$. In other words, $\liminf_{n\to 0} g_n(x_n)\ge0$.


TITLE: Does there exist a relative compactification with flat boundary?
QUESTION [6 upvotes]: Let $S$ be the spectrum of a DVR, $X/S$ a flat finite type affine scheme. Fix a closed immersion $X\subseteq \mathbb{A}^n_S$ and consider the closure $\bar X$ of $X$ in $\mathbb{P}^n_S$. Then $\bar{X}$ is flat over $S$, as $X$ is dense in it. However, the boundary $Y = \bar X - X$ does not have to be flat over $S$ (see the examples below).

Does every point of the closed fiber of $X$ have an open neighborhood $X'$ which can be compactified to a projective $\bar X'/S$ with $Y' := \bar X' - X'$ flat over $S$?

Example. Take $X$ to be the generic point of $S$, $\bar X = S$ and $Y$ is the closed point of $S$. But as the closed fiber $X$ is empty, the condition is trivially satisfied.
Example. Let $X \subseteq{A}^2_S$ be given by the equation $x = \pi y^2$ where $\pi$ is a uniformizer, then $\bar X \subseteq \mathbb{P}^n$ is given by $xz = \pi y^2$ and $Y$ is given by $z = 0 = \pi y^2$, a union of the $z=0$ line in the closed fiber and the section $(1:0:0)$, so not flat. Of course, $X$ is isomorphic to $\mathbb{A}^1_S$, so we can find a compactification for which $Y$ is flat, namely $\bar X = \mathbb{P}^1_S$.

REPLY [6 votes]: I believe the answer is no.  All of the following will be relative to an algebraically closed field $k$.  You may assume $k$ has characteristic $0$, but I think that is unnecessary (use the work of Stefan Schröer).
Let $g',g''>2$ be positive, distinct integers.  Let $g$ be $g'+g''$.  Let $\overline{M}_g$ denote the Deligne-Mumford moduli stack of stable curves of (arithmetic) genus $g$.  Let $\overline{M}_g^o\subset \overline{M}_g$ be the maximal open substack that is a scheme.  In particular $\overline{M}_g^o$ intersects the boundary divisor $\Delta_{g',g''}$ since $g'\neq g''$; denote the intersection by $\Delta_{g',g''}^o$.  Denote the universal curve over $\overline{M}_g^o$ by $$\pi:\mathcal{C}\to \overline{M}_g^o.$$  Over $\Delta_{g',g''}^o$, the restriction of the universal curve is a union of two proper closed subschemes, $\mathcal{C}'$ and $\mathcal{C}''$, intersecting along divisors, $Z'\subset \mathcal{C}'$, respectively $Z''\subset \mathcal{C}''$, such that $$(\pi':\mathcal{C}'\to \Delta_{g',g''}^o,Z'), \ \text{resp.} \ (\pi'':\mathcal{C}''\to \Delta_{g',g''},Z''),$$ is a family of stable, $1$-pointed curves of (arithmetic) genus $g'$, resp. $g''$.
Let $\delta_{g',g''}$ denote the generic point of $\Delta_{g',g''}^o$, let $R$ be $\mathcal{O}_{\overline{M}_g^o,\delta_{g',g''}}$, the local ring of $\overline{M}_g^o$ at this codimension $1$ point.  Of course $R$ is a DVR.  Let $S$ be $\text{Spec}(R)$ with its natural morphism to $\overline{M}_g^o$.   Denote the closed point of $S$ by $0$, and denote the generic point by $\eta$. Let $$ \pi_S:\mathcal{C}_S\to S $$ be the base change of $\pi$ over $S$.  Denote the closed fiber by $C_0$, with its two irreducible components $C'_0$ and $C''_0$ as above.  Let $X$ be $\mathcal{C}_S \setminus C''_0$.
By way of contradiction, assume that there exists a Zariski open subset $X'\subset X$ that intersects $C'_0$ and that has a projective compactification $\overline{X}'_S$ over $S$ whose boundary $Y'$ is flat over $S$.  In particular, $Y'$ is finite over $S$.  Denote by $$u:\widetilde{X}\to \overline{X}'$$ the normalization of $\widetilde{X}$.  In particular, the generic fiber $\widetilde{X}_\eta$ is isomorphic to the generic fiber $\widetilde{C}_\eta$, i.e., $\widetilde{X}_\eta \cong \mathcal{C}_\eta$.  Moreover, because the closed fiber $\widetilde{X}_0$ is some compactification of an open subset of the genus $g'>0$ curve $C'_0$, in particular $\widetilde{X}_0$ contains no genus $0$ curves.  Moreover, $\widetilde{C}_S$ is regular.
Thus, by Abhyankar's lemma, etc., the isomorphism of generic fibers extends to an $S$-morphism, $$ v :\mathcal{C}_S \to \widetilde{X}, $$ that is automatically projective.  Moreover, this morphism contracts the irreducible component $C''_0$ of the closed fiber of $\mathcal{C}_S$.  In particular, for every invertible sheaf $\mathcal{L}$ on $\widetilde{X}$, $v^*\mathcal{L}$ restricts to the trivial invertible sheaf on $C''_0$.  In particular, since $\widetilde{X}$ is projective, there exists an invertible sheaf $\mathcal{L}$ that is ample, so that $v^*\mathcal{L}$ is big.
This is impossible.  By Franchetta's conjecture / Harer's theorem, the Picard group of $\mathcal{C}_\eta$ is generated by the relative dualizing sheaf, and the Picard group of $\mathcal{C}_S$ is generated by the relative dualizing sheaf together with the invertible sheaf $\mathcal{O}_{\mathcal{C}_S}(\underline{C'}_0)$.  The restrictions of these invertible sheaves to $C''_0$ are $\omega_{C''_0}(\underline{Z}'')$ and $\mathcal{O}_{C''_0}(\underline{Z}'')$.  But these are linearly independent in $\text{Pic}(C''_0)$, i.e., the universal curve over the generic point of $\Delta_{g',g''}$.  Since $v^*\mathcal{L}$ has trivial restriction to $C''_0$, $v^*\mathcal{L}$ is trivial.  In particular, the restriction of $v^*\mathcal{L}$ to the generic fiber $\widetilde{C}_\eta$ is trivial.  However, by construction, it is also big.  This is a contradiction, proving that $\overline{X}'$ does not exist.
Observe the following points.  First, $S$ is not Henselian.  Second, the residue field of $S$ is not algebraically closed.  Third, this argument only works if $\overline{X}'$ is a projective scheme; the argument fails if $\overline{X}'$ is only a proper algebraic space.  So, if you really need something like this to hold, there are still directions to explore.


TITLE: Is metallic hydrogen an example of exciton?
QUESTION [0 upvotes]: I know that in exciton an electron is excited and goes from valence band to conducting band leaving an electron hole which is positively charged, soon Jupiter came into my mind and then this question. Also what is the different between plasma and exciton?

REPLY [0 votes]: An exciton is a quasi-particle describing the excited electronic state od insulators or semiconductors.
Metallic Hydrogen, at temperatures and pressures it is a stable phase, is a conductor in its ground state.


TITLE: Sign of work done in an $L$-$R$ circuit
QUESTION [2 upvotes]: Suppose we have a L-R circuit with an inductor of inductance $L$ and resistance $R$ and emf $\epsilon$. Now initially when the switch is closed no current is present. Let the maximum current be $I_{max}$ then the Energy stored in the inductor after sufficient time  is $U=\frac{1}{2}L{I_{max}}^2$. It is also clear that the magnitude of work done by the battery on the inductor and the magnetic field are both equal to $U$.
But I don’t know which of the two does 'positive' and which does 'negative work'  Also is power supplied or delivered to/from an inductor when current through inductor increases?

REPLY [2 votes]: Firstly, it is necessary to note that LC circuit is not a conservative system - while the current in the inductor is brought to a steady level some energy is necessarily dissipated in the resistance. Moreover, with a steady current in the inductor the energy continues to be dissipated.
Textbooks would normally distinguish work done by the system and work done on the system, which have opposite signs. Thus, one usually discusses only one of these (which @Dale refers to as positive/negative sign conventions - his/her answer is certainly correct). This is rather general in energy discussions - e.g., in thermodynamics one similarly distinguishes the work done by/on the gas.
Thus, if we adopt the point of view of the battery - it spends energy, which goes into building the current in the inductor and producing Joule's heat in the resistance. Thus, the battery is doing positive work, while the resistor and the inductor is doing negative work on the battery.
Conversely, if we look from the point of view of the inductor, its energy increases, i.e., it has a work done on it, which means that the inductor itself is doing negative work.
To summarize: the sign of the work is a matter of prospective: we do not say that work is positive/negative, but rather that A does positive/negative work on B, which means that A transfers energy to B. When the work is called simply positive/negative, it means that there is an agree convention about who does the work.

REPLY [1 votes]: It is also clear that the magnitude of work done by the battery and the magnetic field are both equal to U.

Actually, this is incorrect. You are neglecting the energy that goes into the resistor.

But i dont know which of the two does 'positive' and which does 'negative work' Also is power supplied or delivered to/from an inductor when current through inductor increases

This depends on the sign convention. There are two basic sign conventions that you can use: the passive and the active convention. Here is an image of the passive sign convention, note that the current is going into the positive terminal

And here is an image of the active sign convention, note that the current is going into the negative terminal

(both images from https://en.wikipedia.org/wiki/Passive_sign_convention )
If you are using the passive sign convention then a positive power is energy that is being supplied to the component. If you are using the active sign convention then a positive power is energy that is being delivered from the component.
So for example in the circuits below note that in (b) the current is going into the positive terminal of the inductor, which is the passive sign convention. So if $P=iv$ is positive for circuit (b) then that represents energy being supplied to the inductor. In contrast note that in (c) the current is going into the negative terminal of the inductor, which is the active sign convention. So if $P=iv$ is positive for circuit (c) then that represents energy being delivered from the inductor.


TITLE: Is sum of differences bounded for $C^0[a,b]$ functions?
QUESTION [1 upvotes]: Let $a < b$ and suppose that $f \in C^{0}([a,b]).$ Is it true that
$$ \limsup_{h \rightarrow 0} \sum_{i}| f(t_{i+1})-f(t_i)|^2 = 0 $$
where $a < \{ t_i\} < b$ and $h = t_{i+1} - t_i$?
This is certainly true for $C^1([a,b])$ functions since they are Lipschitz but I cannot prove anything for functions in $C^0([a,b])$.

REPLY [1 votes]: Do you consider one partition of $[a,b]$ with shrinking width or the strong quadratic variation, i.e.
$$
\limsup_{ h \downarrow 0} \, \sup \{ \sum_{i=1}^n |f(t_{i+1})-f(t_i)|^2 :  a \le t_1 < \ldots t_n \le b, \, \Delta t_i \le h \}?
$$
One famous example is the Browian motion which has $\mathbb{P}$-a.s. $\infty$ strong quadratic variation - that's was proven by Levy in  1948. However, the quadratic variation in the sense of stochastic analysis is $\mathbb{P}$-a.s. equal to $t$ and for any partition with $\Delta t_i = o(1/\log(n))$ [result of Dudley] one has
$$
\sum_{i=1}^n |f(t_{i+1})-f(t_i)|^2 \rightarrow t
$$
$\mathbb{P}$-almost sure.
We can also construct an explicit example as follows: Take $f(x) = x^{\alpha} \sin(1/x)$ for $\alpha \in (0,1/2)$. Then for $x_k = ( \pi k + \pi + 2 \pi n )^{-1}$ with $k=1,\ldots, n-1$ we have $0 \le x_{k}-x_{k+1} \le 1/n$ and
\begin{align}
  \sum_{k=1}^{n-1} |f(x_k) - f(x_{k+1})|^2 &= \sum_{k=1}^{n-1} |x_k^\alpha + x_{k+1}^\alpha|^2 \\
& \ge  \sum_{k=1}^n x_k^{2\alpha} \\
& \ge \int_1^n \frac{1}{(\pi(x+1) + 2 \pi n)^{2 \alpha}} \mathrm{d} x \\
& \gg_{\alpha} n^{-2\alpha+1}.
\end{align}
This diverges if $\alpha \in (0,1/2)$.


text:
\begin{document}
\keywords{affine flag variety, $k$-Schur function, nilCoxeter algebra, Pieri rule, strong Schur function}
\maketitle
\begin{abstract}
We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono.
We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.
\end{abstract}

\section{Introduction}

Affine Schubert calculus is a subject that ties combinatorics, algebraic geometry and representation theory together. Its modern development is motivated by the relation between $k$-Schur functions and the (co)homology of the affine Grassmannian of $SL(n)$. $k$-Schur functions were introduced by Lapointe, Lascoux, Morse \cite{LLM03} in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory.
\\

 Peterson \cite{Pet97} identified the equivariant homology of the affine Grassmannian with a subalgebra of the affine nilHecke algebra $\mathbb{A}$, now called the Peterson algebra. Lam \cite{Lam08} proved that $k$-Schur functions can be identified with the Schubert classes of the homology of the affine Grassmannian of $SL(n)$. The nilHecke ring acts as divided difference operators on the equivariant cohomology of Kac-Moody partial flag varieties. By using the correspondence, one can investigate problems about the (co)homology of the affine Grassmannian of $SL(n)$ by translating them into the theory of $k$-Schur functions and understanding the combinatorics of $k$-Schur functions. There are affine analogues of the classical theory of Pieri rules, tableaux, and Stanley symmetric functions \cite{Lam06,LM05,LM07}. \\

Lam, Lapointe, Morse and Shimozono \cite{LLMS10} introduced the strong Schur functions, indexed by elements
in the affine symmetric group $W_\af$. These strong Schur functions generalize the $k$-Schur functions combinatorially. They conjectured a series of properties of strong Schur functions such as the symmetry of the strong Schur functions. Berg, Saliola and Serrano \cite{BSS11,BSS12} studied the Pieri operators acting on the affine nilCoxeter ring $\mathbb{A}_0$ to establish some of conjectures in \cite{LLMS10}. \\

In this paper, we prove the affine Pieri rule for the cohomology of the affine flag variety conjectured in \cite{LLMS10}. We introduce the cap operators acting on the affine nilCoxeter ring $\mathbb{A}_0$ by investigating the work of Kostant and Kumar \cite{KK86} and show that the cap operators for Pieri elements are the same as the Pieri operators defined in \cite{BSS12} by using the strong strips. The affine Pieri rule gives us geometric interpretation of the skew strong Schur functions as an affine Grassmannian part of the cap product of the Schubert classes in (co)homology of the affine flag variety. We now describe these two operators.

\subsection{Pieri operators}
For $u,v\in W_\af$, a \emph{marked strong cover} $C=\left(u\overset{a}\longrightarrow v \right)$ consists of $u,v\in W_\af$ and an integer $a$ such that $u=vt_{ij},v\lessdot u, a= u(j)=v(i)$ where $i\leq 0 <j$ and $\lessdot$ is the Bruhat order. We use the notation inside$(C)=u$ and outside$(C)=v$.\\
Lam, Lapointe, Morse and Shimozono \cite{LLMS10} introduced the notion of \emph{strong strips} to define $k$-Schur functions and strong Schur functions. A \emph{strong strip} $S$ of length $i$ from $u$ to $v$, denoted by $u\longrightarrow_{(i)} v$, is a path
\[u\overset{a_1}\longrightarrow u_1 \overset{a_2}\longrightarrow\cdots \overset{a_i}\longrightarrow u_i=v\]
where $a_1>a_2>\ldots>a_i$. We let inside$(S)=u$, outside$(S)=v$.\\

Berg, Saliola and Serrano \cite{BSS12} studied the \emph{Pieri operators} $D'_i$ acting on the affine nilCoxeter ring $\mathbb{A}_0$ defined by
\[ D'_i(A_w)= \sum_{w\longrightarrow_{(i)}u} A_u\]
where the sum is over all strong strips from $w$ to $u$ of length $i$. They proved that the operators $D'_i$ commute for all $i$ so that the symmetry of strong Schur functions follows. They also showed a series of properties of $D'_i$ that uniquely determine the operator $D'_i$.

\subsection{Cap operators}
Let us consider the affine type $A$ root datum and corresponding Kac-Moody flag variety $\widetilde{Fl}$ (See \cite{KK86} for details). Let $\{\xi^w | w\in W_\af\}$ denote the Schubert basis for the equivariant cohomology $H^*_T(\widetilde{Fl})$ of $\widetilde{Fl}$. One of main problems in affine Schubert calculus is to find a combinatorial formula for the structure constants $p_{u,v}^w$ where $ \xi^u \xi^v=\sum_w p_{u,v}^w\xi^w$. In \cite{KK86}, Kumar and Kostant identified the torus-equivariant cohomology $H^*_T(\widetilde{Fl})$ of the affine flag variety and the \emph{dual} of the nilHecke ring. Using this connection, one can compute the structure constants $p_{u,v}^w$ by calculating the coproduct structure constants of $\mathbb{A}$. More precisely, we have
\[\Delta(A_w)=\sum_{u,v\in W_\af} p_{u,v}^w A_u\otimes A_v. \]
For $u\in W_\af$, a \emph{cap operator} $D_u$ on $\mathbb{A}_0$ is defined by
$$D_u(A_w)=\sum_{\substack{v\in W_\af \\ \ell(v)=\ell(w)-\ell(u)}} p_{u,v}^w A_v.$$
Geometrically, the cap operator is the cap product on the ordinary homology and cohomology of the affine flag variety. More precisely, the cap operator $D_u$ can be considered as an element in $H^*(\widetilde{Fl})$ and $A_w$ can be considered in $H_*(\widetilde{Fl})$ so that the cap product $H^*(\widetilde{Fl})\times H_*(\widetilde{Fl})\rightarrow H_*(\widetilde{Fl})$ can be described by $(D_u,A_w)=D_u(A_w)$. Note that the cap operator only keep track of ordinary cohomology since $p_{u,v}^w$ is constant when $\ell(v)=\ell(w)-\ell(u)$. \\
 Let $\rho_i$ be the Pieri element $s_{i-1}\ldots s_1 s_0$ in $W_\af$ where indices are taken modulo $n$. We study the cap operators $D_{\rho_i}$ for $\rho_i$ and show that $D_{\rho_i}$ satisfy the properties of the Pieri operators $D'_i$ that Berg, Saliola and Serrano proved in \cite{BSS12}. This establishes the following main theorems equivalent to the affine Pieri rule conjectured in \cite{LLMS10}.

\begin{thm}\label{affinePieri}
For $w,u \in W_\af$ with $i=\ell(w)-\ell(u)\in \mathbb{N}$, $p^w_{\rho_i,u}$ counts the number of strong strips from $w$ to $u$.
\end{thm}
As a corollary, one can compute $p^w_{v,u}$ for $v=\rho_m^{-1}$ or $v=s_{i+m-1}s_{i+m-2}\cdots s_{i+1}s_i$ for any $i$ by applying automorphisms on Dynkin diagram of affine type $A$, namely, the map $j\mapsto -j$ and $j\mapsto j+i$ for $j\in \mathbb{Z}/n\mathbb{Z}$.\\
\smallskip

The paper is structured as follows. In section 2, we recall some notions about the affine symmetric groups, $k$-Schur functions, strong Schur functions
and root systems. In section 3, we define the affine nilHecke ring $\mathbb{A}$ and study its properties. In section 4, we recall some statements concerning the affine flag variety and its equivariant cohomology as well as the relationship between the equivariant cohomology of affine flag variety and the coproduct structure of the affine nilHecke ring. In section 5, we define the cap operator. We show that these cap operators for Pieri elements agree with the Pieri operators defined by Berg, Saliola and Serrano. We also prove the affine Pieri rule for the ordinary cohomology of the affine flag variety. In section 6, we apply the affine Pieri rule to show that the structure constants $p^w_{u,v}$ for the cohomology of the affine flag variety can be described in terms of strong Schur functions when $u$ is $0$-Grassmannian. This also gives a geometric interpretation of the skew strong Schur functions.
\section*{Acknowledgments}
I would like to thank Thomas Lam for helpful discussions and for introducing affine Schubert calculus to me. I also wish to thank KIAS for valuable support.
\section{Preliminaries}
\subsection{Affine symmetric group}

Positive integers $n\geq 2$ and $k=n-1$ will be fixed throughout the paper. Let $W_\af$ denote the affine symmetric group with simple generators $s_0,s_1,\ldots,s_{n-1}$ satisfying the relations
\begin{align*}
s_i^2&=1\\
s_is_{i+1}s_i&=s_{i+1}s_is_{i+1}\\
s_is_j&=s_js_i&& \text{if } i-j \neq 1,-1.
\end{align*}
where indices are taken modulo $n$. An element of the affine symmetric group may be written as a word in the generators $s_i$. A \emph{reduced word} of the element is a word of minimal length. The \emph{length} of $w$, denoted $\ell(w)$, is the number of generators in any reduced word of $w$. \\
The \emph{Bruhat order}, also called \emph{strong order}, on affine symmetric group elements is a partial order where $u<w$ if there is a reduced word for $u$ that is a subword of a reduced word for $w$. If $u<w$ and $\ell(u)=\ell(w)-1$, we write $u\lessdot w$. It is well-known that $u\lessdot w$ if and only if there exists a reflection $t\in \{ws_iw^{-1}|w\in W_\af, 0 \leq i<n \}$ such that $w=ut$ and $\ell(u)=\ell(w)-1$. For type $A$, the set $\{ws_iw^{-1}|w\in W_\af\}$ consists of transpositions $t_{ab}$. See \cite{BB05} for instance.\\
The subgroup of $W_\af$ generated by $\{s_1,\cdots,s_{n-1}\}$ is naturally isomorphic to the symmetric group $W$. The $0$-Grassmannian elements are minimal length coset representatives of $W_\af/ W$.

\subsection{Symmetric functions}
Let $\Lambda$ denote the ring of symmetric functions. For a partition $\lambda$, we let $m_\lambda,h_\lambda,e_\lambda,s_\lambda$ denote the monomial, homogeneous, elementary and Schur symmetric functions, respectively, indexed by $\lambda$. Each of these families forms a basis of $\Lambda$.\\
Let $\Lambda_{(k)}$ denote the subalgebra generated by $h_1,h_2,\ldots,h_k$. The elements $h_\lambda$ with $\lambda_1\leq k$ form a basis of $\Lambda_{(k)}$. Let $\Lambda^{(k)}=\Lambda/I_k$ denote the quotient of $\Lambda$ by the ideal $I_k$ generated by $m_\lambda$ with $\lambda_1>k$. The image of elements $m_\lambda$ with $\lambda_1\leq k$ form a basis of $\Lambda^{(k)}$. Lam \cite[Theorem 7.1]{Lam08} showed that $\Lambda_{(k)}$(resp. $\Lambda^{(k)}$) are isomorphic to homology(resp. cohomology) of affine Grassmannian as Hopf-algebras.\\

\subsection{$k$-Schur and Strong Schur functions}
There are many conjecturally equivalent definitions of $k$-Schur functions in \cite{LLMSSZ}. We follow the definition of $k$-Schur functions in \cite{LLMS10} using strong strips. Note that the marking of strong covers and strong strips defined in this extended abstract follows the notation of \cite{BSS11}. This differs from the definition of strong covers and strips in \cite{LLMS10} by reversing direction.\\

A \emph{marked strong cover} $C=\left(u\overset{a}\longrightarrow v \right)$ consists of $u,v\in W_\af$ and an integer $a$ such that $u=vt_{ij},v\lessdot u, a= u(j)=v(i)$ where $i\leq 0<j$. We use the notation inside$(C)=u$ and outside$(C)=v$.\\
A \emph{strong strip} S of length $i$ from $u$ to $v$, denoted by $u\longrightarrow_{(i)} v$, is a path
\[u\overset{a_1}\longrightarrow u_1 \overset{a_2}\longrightarrow\cdots \overset{a_i}\longrightarrow u_i=v\]
where $a_1>a_2>\ldots>a_i$. We let inside$(S)=u$, outside$(S)=v$, and size$(S)=i$.\\
A \emph{strong tableau} is a sequence $T=(S_1,S_2,\ldots)$ of strong strips $S_i$ such that outside$(S_j)=$ inside$(S_{j+1})$ for all $j \in \mathbb{Z}_{>0}$ and size$(S_i)=0$ for all sufficiently large $i$. We define inside$(T)=$ inside$(S_1)$ and outside$(T)=$ outside$(S_i)$ for $i$ large. The \emph{weight} $wt(T)$ of $T$ is the sequence
\[wt(T)=(\text{size}(S_1),\text{size}(S_2),\ldots).\]
We say that $T$ has \emph{shape} $u/v$ where $u=$ inside$(T)$ and $v=$ outside$(T)$ so that we have $u>v$.
\begin{definition}
For fixed  $u,v \in W_\af$, define the Strong Schur function
\[\emph{Strong}_{u/v}(x)=\sum_T x^{\text{wt}(T)}\]
where T runs over the strong tableaux of shape $u/v$.

\end{definition}

If $u$ is $0$-Grassmannian and $v$ is the identity element, $\textrm{Strong}_u(x)$ is a $k$-Schur function $s_{{\bf c}(u)}^{(k)}$ where ${\bf c}$ is a bijection between the set of 0-Grassmannian elements and the set of $k$-bounded partitions. For details, see \cite{LLMS10} for instance. \\

\subsection{Root systems}

We shall assume basic familiarity with Weyl groups, root systems, and weights. See \cite{Hum90} for details. \\
Let $A=(a_{ij})_{i,j\in I_\af}$ denote an affine Cartan matrix, where $I_{\af}=I\cup\{0\}$, so that $(a_{ij})_{i,j\in I}$ is corresponding finite Cartan matrix. For type $\widetilde{A}_{n-1}$(corresponding to $W_\af$), we have $I_{\text{af}}=\mathbb{Z}/n\mathbb{Z}$ and
\[a_{ij}= \left\{\begin{array}{rl} 2 & \mbox{if } i=j   \\ -1 & \mbox{if }  |i-j|=1 \\ 0  &\mbox{otherwise}.\end{array} \right.\]
Let $R$ be the root system for $W$. Let $R^+, R^-$ denote the set of positive roots, negative roots respectively. Let $\{ \alpha_i\mid i\in I\}$ denote the simple roots and $\{\alpha_i^\vee\mid i\in I\}$ denote the simple coroots. Let $\theta$ denote the highest root $\alpha_1+\alpha_2+\ldots + \alpha_{n-1}$ of $R^+$. \\
Let $R_{\text{af}}$ and $R_{\af}^+$ denote the affine root system and positive affine roots. The positive simple affine roots (resp. coroots) are  $\{\alpha_i\mid i\in I_{\af}\}$ (resp. $\{\alpha_i^\vee\mid i\in I_{\af}\}$). The null root $\delta$ is given by $\delta=\alpha_0+\theta=\alpha_0+\cdots+ \alpha_{n-1}$. Similarily, the canonical central element $c$ is given by $\alpha_0^\vee+\cdots+ \alpha_{n-1}^\vee$. A root $\alpha$ is real if it is in $W_\af$-orbit of the simple affine roots, and imaginary otherwise. The imaginary roots are exactly $\{k\delta \mid k\in \mathbb{Z}\backslash \{0\}\}$. Every real affine root is of the form $\alpha+k\delta$, where $\alpha\in R$ and $k\in \mathbb{Z}$. The root $\alpha+k\delta$ is positive if $k>0$, or if $k=0$ and $\alpha\in R^+$. Let $R_{\text{re}}^+$ denote the set of positive roots in $R_\af$.\\
Let $Q=\oplus_{i\in I} \mathbb{Z}\cdot \alpha_i$ denote the root lattice and let $Q^\vee=\oplus _{i \in I} \mathbb{Z} \cdot \alpha_i^\vee$ denote the coroot lattice. Let $P$ and $P^\vee$ be the weight lattice and coweight lattice respectively. We have inclusions $Q \subset P, Q^\vee \subset P^\vee$ and a map $Q_{\af}=\oplus_{i \in I_\af} \mathbb{Z} \cdot \alpha_i \rightarrow P$ given by sending $\delta$ to $0$. Let $\langle\cdot,\cdot\rangle$ denote the pairing between $P$ and $P^\vee$ satisfying $\langle \alpha_i^\vee, \alpha_j\rangle=a_{ij}$.\\

Let $P_{\af}$ be the affine weight lattice and $\{\Lambda_i | i\in I_{\af} \}$ be the set of affine fundamental weights. Then
\[P_{\af}= \mathbb{Z}\delta \oplus \bigoplus_{i\in I_{\af}} \mathbb{Z}\Lambda_i. \]

The affine Weyl group $W_\af$ acts on the affine weight lattice $P_\af$ and the affine coweight lattice by

\begin{align*}
s_i\cdot \lambda&=\lambda- \langle\alpha_i^\vee, \lambda\rangle \alpha_i\\
s_i \cdot \mu&=\mu - \langle\mu,\alpha_i\rangle\alpha_i^\vee.
\end{align*}
This is called the \emph{level zero action}. An element $w\in W_\af$ can be uniquely written of the form $ut_\mu$ where $u\in W$ and $\mu \in Q^\vee$. Then the level zero action on $P\subset P_\af$ is given by
$$ut_\mu\cdot \lambda = u \cdot \lambda.$$
For a real root $\alpha$, we let $s_\alpha$ denote the corresponding reflection, defined by $s_\alpha=w s_i w^{-1}$ if $\alpha=w\cdot\alpha_i$. The reflection $s_\alpha$ acts on weights by $s_{\alpha}\lambda=\lambda- \langle\alpha,\lambda\rangle\alpha$. For a strong cover $v\lessdot w$, let $\alpha_{v,w}$ denote the unique positive root satisfying the equation $v^{-1}w=s_{a_{v,w}}$.

\section{NilHecke algebra}
\subsection{Definition}
Let $S=\text{Sym}(P)$ be the polynomial ring having a variable for each free generator of $P$ and let $F=\text{Frac}(S)$ be the fraction field. \\

Define the $F$-vector space $F_{W_{\af}}=\bigoplus_{w\in W_{\af}} F w$ with basis $W_{\af}$, with product given by
\[(pv)(qw)=\left( p(v\cdot q)\right) (vw) \quad \mbox{for } p,q \in F \mbox{ and } v,w \in W_{\af}.\]
For any real root $\alpha \in R_{\text{re}}$ define the element $A_\alpha\in F_{W_{\af}}$ by
\[A_\alpha= \alpha^{-1} (1-s_\alpha).\]
We write
\[ A_i = A_{\alpha_i} \quad \mbox{for } i\in I.\]
For $\alpha=w \cdot \alpha_i \in R_{\text{re}}^+$, we have
\[\begin{array}{rcl}
wA_i w^{-1} & = & A_\alpha  \\ s_\alpha A_\alpha & = &  A_\alpha \\ A_\alpha s_\alpha&=& - A_\alpha \\ A_\alpha^2&=&0. \end{array}\]
The $A_i$ satisfy the braid relations as the $s_i$ in $W_{\af}$, i.e., $A_iA_{i+1}A_i=A_{i+1}A_iA_{i+1}$. Therefore it makes sense to define
\[ \begin{array} {rlll} A_w&=&A_{i_1}\cdots A_{i_l} & \mbox{where} \\ w&=&s_{i_1}\cdots s_{i_l} & \mbox{is a reduced decomposition.} \end{array} \]
One can check that
\[A_vA_w=\left\{ \begin{array}{ll} A_{vw} & \mbox{if } \ell(vw)=\ell(v)+\ell(w) \\ 0 & \mbox{otherwise.}\end{array} \right. \]
The \emph{nilCoxeter algebra} $\mathbb{A}_0$ is the subring of $F_{W_{\af}}$ generated by $A_i$ over $\mathbb{Z}$ for $i\in I_{\af}$. The set $\{A_w|w\in W_\af\}$ forms a basis of $\mathbb{A}_0$ over $\mathbb{Z}$. The \emph{nilHecke algebra} $\mathbb{A}$ is the subring of $F_{W_{\af}}$ generated by $S$ and $\mathbb{A}_0$.

We state the following result \cite[Proposition 4.30]{KK86}. Let $w\in W_\af$ and $\lambda\in P$. Then
\begin{align} A_w\lambda= (w\cdot \lambda) A_w + \sum_{ws_\alpha:\ell(ws_\alpha)=\ell(w)-1} \langle\alpha^\vee, \lambda\rangle A_{ws_\alpha} \end{align}
where $\alpha$ is always taken to be a positive root in $R_\af^+$. The coefficients $\langle\alpha^\vee,\lambda\rangle$ are known as \emph{Chevalley coefficients}.

\begin{rmk}
The affine nilHecke algebra $\mathbb{A}$ defined above is slightly different from those defined in \cite{KK86}. In \cite{KK86}, Kostant and Kumar set $S=\text{Sym}(P_{\text{af}})$ instead of $\text{Sym}(P)$. We have a projection $P_{\text{af}} \rightarrow P$ and the map induces $\text{Sym}(P_{\text{af}})\rightarrow \text{Sym}(P)$ which is compatible with Theorem \ref{cohomology}. The image of the null root $\delta=\alpha_0+\theta$ in $P$ is 0. Geometric interpretation of $P$ and $P_\af$ will be discussed in section \ref{affineflagvariety}.
\end{rmk}

\subsection{Cyclically decreasing elements and Peterson algebra}
A word $s_{i_1}s_{i_2}\cdots s_{i_l}$ with indices in $\mathbb{Z}/n\mathbb{Z}$ is called \emph{cyclically decreasing} if each letter occurs at most once and whenever $s_i$ and $s_{i+1}$ both occurs in the word, $s_{i+1}$ precedes $s_i$. For $J\varsubsetneq\mathbb{Z}/n\mathbb{Z}$ there is the unique cyclically decreasing element $w_J$ with letters $\{s_j|j\in J\}$. For $i \in \{ 0,1,\ldots,n-1\}$, let
\[\h_i=\sum\limits_{\substack{ J\subset I_{\af} \\ |J|=i }} A_{w_J} \in \mathbb{A}_0 \]
where $\h_0=1$ and $\h_i=0$ for $i< 0$ by convention. Lam \cite{Lam06} showed that the elements $\{\h_i\}_{i<n}$ commute and freely generate a subalgebra $\mathbb{B}$ of $\mathbb{A}_0$ called \emph{affine Fomin-Stanley algebra}. It is well-known that $\mathbb{B}$ is isomorphic to $\Lambda_{(k)}$ via the map sending $\h_i$ to $h_i$. Therefore, the set $\{\h_\lambda=\h_{\lambda_1}\ldots \h_{\lambda_l} | \lambda_1\leq k\}$ forms a basis of $\mathbb{B}$.
\\

\subsection{Coproduct structure of $\mathbb{A}$} \label{coproduct}
There is a coproduct structure on the nilHecke ring $\mathbb{A}$. Kostant and Kumar showed that the structure constants of the coproduct in $\mathbb{A}$ are the same as the structure constants of the equivariant cohomology ring of $\widetilde{Fl}$. Details and proofs can be found in \cite{KK86,LLMSSZ,Pet97}.\\
Let $\Delta:F_{W_\af}\rightarrow F_{W_\af}\otimes_F F_{W_\af}$ be the left $F$-linear map defined by
\[\Delta(w)=w \otimes w \quad \mbox{for all } w\in W_\af\]

\begin{thm} \label{delta}
The map $\Delta:F_{W_\af}\rightarrow F_{W_\af}\otimes_F F_{W_\af}$ induces the unique left $S$-module homomorphism $\Delta: \mathbb{A} \rightarrow \mathbb{A}\otimes_S \mathbb{A}$ such that

\[\begin{array}{rll}
\Delta(A_i)&=A_i\otimes 1 + s_i \otimes A_i & \\
 &= 1\otimes A_i + A_i \otimes s_i & \text{ for all } i \in I \\
\Delta(ab)&=\Delta(a)\Delta(b) & \text{ for all } a,b\in \mathbb{A}. \\
\end{array}
\]
\end{thm}
For $a,b \in F_{W_\af}$ and any expression of $\Delta(a)$ and $\Delta(b)$ of the form
\begin{align*} \Delta(a)&=\sum_{a} a_{(1)}\otimes a_{(2)}\\ \Delta(b)&=\sum_{b} b_{(1)}\otimes b_{(2)}. \end{align*}
The product of $\Delta(a)$ and $\Delta(b)$ can be computed by the naive componentwise product
\begin{align}\label{naive}
\Delta(ab)=\Delta(a)\Delta(b)= \sum_{(a),(b)} a_{(1)}b_{(1)}\otimes a_{(2)}b_{(2)}.
\end{align}

For $w,u_1,u_2\in W_\af$, the \emph{equivariant Schubert structure constants} $p^w_{u_1,u_2}\in S$ are defined as the coefficients in the expansion $\Delta(A_w)=\sum p^w_{u_1,u_2} A_{u_1} \otimes A_{u_2}$. The followings are properties of $p^w_{u_1,u_2}$.

\begin{thm} \ \begin{enumerate}
\item $p^w_{u_1,u_2}=0$ unless $w\geq u_1$ and $w\geq u_2$.\\
\item $p^w_{u_1,u_2}$ is homogeneous of degree $\ell(u_1)+\ell(u_2)-\ell(w)$. \\
\item \cite{Gra01,Kum02} $(-1)^{\ell(u_1)+\ell(u_2)-\ell(w)}p^w_{u_1,u_2}\in \mathbb{Z}_{\geq 0} [\alpha_i | i\in I_\af].$
\end{enumerate} \label{essc}
\end{thm}
The last property in Theorem \ref{essc} is called \emph{Graham positivity}.
One can use Theorem \ref{delta} to compute $p^w_{u_1,u_2}$ explicitly.
\begin{thm}\label{p}
\cite{KK86}
Let $w=s_{i_1}s_{i_2}\ldots s_{i_l}$ and $l=\ell(w)$. Then
$$p^w_{u_1,u_2} = \sum\limits_{\substack{ A_{s_{i_{j_1}}} A_{s_{i_{j_2}}}\ldots A_{s_{i_{j_k}}} = A_{u_1} \\
 j_1<j_2<\ldots<j_k }}\prod_{p=1}^l f_{j_1,j_2,\ldots,j_k}(A_{s_{i_{p}}})\Bigg|_{A_{u_2}}$$
where $f_{j_1,j_2,\ldots,j_k}(A_{s_{i_{p}}})= s_{i_{p}}$ if $p \in \{j_1,j_2,\ldots,j_k\}$ and $=A_{s_{i_{p}}}$ otherwise, and $\sum c_w A_w |_{A_u}=c_u$.

\end{thm}

\section{Affine flag varieties} \label{affineflagvariety}
In this section, we define the Kac-Moody flag variety $\widetilde{Fl}$ and establish the relationship between the equivariant cohomology of $\widetilde{Fl}$ and the coproduct structure on $\mathbb{A}$. There are two definitions of Kac-Moody flag variety $\widetilde{Fl}$ in \cite{LLMSSZ}, but we only recall $\widetilde{Fl}$ as the Kac-Moody flag ind-variety in \cite{KK86,Kum02}. \\
Let $G_\af$ denote the Kac-Moody group of affine type associated with $SL(n)$ and let $B_\af$ denote its Borel subgroup. The Kac-Moody flag ind-variety $\fl=G_\af/B_\af$ is paved by cells $B_\af\dot{w}B_\af/B_\af\cong\mathbb{C}^{\ell(w)}$ whose closure $X_w$ is called the Schubert variety. Schubert variety defines a Schubert class $[X^w]_{T_\af} \in H^*_{T_\af}(\widetilde{Fl})$ where $T_\af$ is the maximal torus in $B_\af$. For our setting, we consider $H^*_{T}(\widetilde{Fl})$ where $T$ is a maximal torus in $SL(n)$. Kumar and Kostant \cite{KK86} identified the equivariant cohomology ring of $\widetilde{Fl}$ with the \emph{dual} of nilHecke ring $\mathbb{A}$. By restricting the group action from $T_\af$ to $T$, this identification shows that the structure constant of $H^*_T(\widetilde{Fl})$ is the same as $p_{u,v}^w$ defined in Section \ref{coproduct}.

\begin{thm} \label{cohomology}
\cite{KK86} The $T$-equivariant cohomology of $\widetilde{Fl}$ has a basis $\{[X^w]_T \in H^*_T(\widetilde{Fl})\}$ over $S\cong H^*_T(pt)$. Moreover, the structure constants of $H^*_T(\widetilde{Fl})$ are $p^w_{u_1,u_2}$, i.e.,
$$ [X^{u_1}]_T [X^{u_2}]_T=\sum_w p^w_{u_1,u_2}[X^w]_T. $$
\end{thm}
\begin{rmk}\label{ordinary}

The ordinary cohomology of $\widetilde{Fl}$ has a basis $\{[X^w] \in H^*(\widetilde{Fl})\}$ over $\mathbb{Z}$ and corresponding structure constants are just $\phi(p^w_{u_1,u_2})$ where $\phi:S\rightarrow\mathbb{Z}$ is the evaluation at $0$.
\end{rmk}

\section{Pieri operators and Affine Pieri rule for type $A$}

Let $\phi: S\rightarrow \mathbb{Z}$ denote the map sending a polynomial to its constant term, called the \emph{evaluation map} at $0$. It extends to the map $\phi: \mathbb{A}\rightarrow\mathbb{A}_0$ given by $\phi(\sum_w a_wA_w)=\phi(a_w)A_w$. \emph{Pieri elements} $\rho_i$ are defined by $s_{i-1}s_{i-2}\ldots s_{1}s_0$ for $i>0$.
For $w\in W_{\text{af}}$, let us define \emph{the cap operator} $D_{w}$ on $\mathbb{A}$ (and $\mathbb{A}_0$) by

\[D_w(A_v):= \phi\left(\sum_{u\in W_\af} p^v_{w,u} A_u\right) =\sum_{\substack{u\in W_\af\\\ell(u)=\ell(v)-\ell(w) }} p^v_{w,u} A_u.  \]
 If $w=\rho_i$, we call $D_{\rho_i}$ the \emph{Pieri operator} and denote by $D_i$.

Note that $p^v_{w,u}\in \mathbb{Z}_{\geq 0}$ when $\ell(v)=\ell(w)+\ell(u)$ by Theorem \ref{essc}. Finding a combinatorial formula for such $p^v_{w,u}$ is one of the important problems in Schubert calculus and it is not completely known even for finite flag varieties. In this section, we will prove that $D_{i}$ is the same as the Pieri operator $D'_i$ defined by Berg, Saliola, Serrano in \cite{BSS12}. This identification gives a combinatorial description of the Pieri rule for the ordinary cohomology of $\widetilde{Fl}$ and therefore proves the main Theorem \ref{affinePieri}.\\
Let us state properties of $D'_i$ described in \cite{BSS12}.

\begin{thm} \cite[Theorem 4.8]{BSS12}\label{bss1}
Suppose $w \in W_{\text{af}}$ and $v \in W$. Then
\[D'_i (A_w A_v)=D'_i (A_w) A_v. \]

\end{thm}

\begin{lemma}\label{bss2} \cite[Lemma 4.5]{BSS12} For $r<n$ and $i\geq 1$,
\[D'_i(\h_r)=\h_{r-i}.\]
\end{lemma}

\begin{thm} \cite[Proposition 4.3]{BSS12} \label{bss3} For $p < n$,$w\in W_\af$ and $i\geq 1$, we have
\[ D'_i(\h_p A_w) = \sum_{j= 0}^i D'_j(\h_{p}) D'_{i-j}(A_w) =\sum_{j= 0}^i \h_{p-j} D'_{i-j}(A_w).\]
\end{thm}

For $w\in W_\af$, there is a unique decompostion $w=w_\lambda w(0)$  where $w_\lambda$ is $0$-Grassmannian and $w(0)\in W$. Note that $\lambda={\bf c}(w_\lambda)$ where ${\bf c}$ is the bijection between $0$-Grassmannian elements and $k$-bounded partitions mentioned in section 2. In this setup, the set $\{ \h_\lambda A_{w(0)} : w= w_\lambda w(0) \in W_{\text{af}} \}$ forms a basis of $\mathbb{A}$ (See \cite{BSS12}). Therefore, above theorems uniquely determine $D'_i$. In this section, we prove the same theorems for $D_{i}$. This implies that $D_i=D'_i$ and Theorem \ref{affinePieri} follows.

\begin{thm}\label{0}
Suppose $w \in W_{\text{af}}$ and $v \in W$. Then
\[D_{i} (A_w A_v)=D_{i}(A_w) A_v \]

\end{thm}

{\it Proof.} Let $F(A_w;j_1,j_2,\ldots,j_k) = \prod_{p=1}^l f_{j_1,j_2,\ldots,j_k}(A_{s_{i_{p}}})$ for $w=s_{i_1}s_{i_2}\ldots s_{i_l}$ and $1 \leq j_1<j_2<\ldots<j_k\leq l$.
It is enough to show that
$$ F(A_wA_v;j_1,j_2,\ldots,j_k) =F(A_w;j_1,j_2,\ldots,j_k) A_v $$
for all $1\leq j_1<j_2<\ldots <j_k \leq l$ by Theorem \ref{p}. Note that any reducd word of $v$ does not contain $s_0$ since $v\in W$, and $\rho_i$ has the unique reduced word ending with $s_0$, namely $s_i s_{i-1} \ldots s_1 s_0$. Let $w=s_{i_1}s_{i_2}\ldots s_{i_{\ell(w)}}$ and $v=s_{a_1}s_{a_2}\ldots s_{a_{\ell(v)}}$. Then $a_b\neq 0$ for $1\leq b\leq \ell(v)$. Therefore, we have

\begin{align*}
 F(A_wA_v;j_1,j_2,\ldots,j_k) &= \prod_{p=1}^{\ell(w)} f_{j_1,j_2,\ldots,j_k}(A_{s_{i_{p}}} A_v) \\
&= \prod_{p=1}^{\ell(w)} f_{j_1,j_2,\ldots,j_k}(A_{s_{i_{p}}}) A_v \\
&=F(A_w;j_1,j_2,\ldots,j_k) A_v
\end{align*}
and the theorem follows. \qed\\

\begin{thm}\label{dh}
For $r<n$ and all $i\geq 1$,
\[D_i(\h_r)=\h_{r-i}.\]
\end{thm}

{\it Proof.} This theorem follows as a corollary of \cite[Lemma 7.7]{Lam08}.
\begin{lemma}\cite[Lemma 7.7]{Lam08} For $r<n$,
$$\phi(\Delta(\h_r))=\sum_{0\leq j\leq r} \h_{r-j} \otimes \h_{j}.$$
\end{lemma}
Since $\rho_i$ appears in $\h_i$, terms appearing in $\phi(\Delta(\h_r))$ with second entry $\rho_i$ are exactly $\h_{r-i}\otimes \rho_i$.
\qed \\

In order to prove analogue of Theorem \ref{bss3} for $D_i$, we need the following lemmas.

\begin{lem}\label{hp} \cite[Lemma 7.2]{Lam08}
Let $b\in \mathbb{B}$ and $s\in S$. Then
$$\phi(b \cdot s )= \phi(s)b=b \cdot\phi(s).$$

\end{lem}

\begin{lem} \label{translation}
Let $\psi: W_\af \rightarrow W_\af$ be the group automorphism defined by $s_i\mapsto s_{i-1}$. The automorphism induces the ring automorphism $\psi: \mathbb{A}_0 \rightarrow \mathbb{A}_0$. For $b\in \mathbb{B}$, we have
$$\psi(b)=b.$$
\end{lem}
{\it Proof.} Since $\mathbb{B}$ is generated by $\h_i$ for $i<n$ and the equality $\phi(\h_i)=\h_i$ is obvious from the definition of $\h_i$, we are done.
\qed \\

Now we are ready to prove the following property of $D_i$.
\begin{thm}\label{dual} For $p < n$,$w\in W_\af$ and $i\geq 1$, we have
\begin{displaymath}D_{i}(\h_p A_w)= \sum_{j=0}^i D_{j} (\h_p) D_{i-j}(A_w).\end{displaymath}
\end{thm}

{\it Proof.} By the definition of the Pieri operator and Remark \ref{ordinary}, we have
\begin{align*}
D_{\rho_i}(\h_p A_w) &= \sum_{|S|=p} D_{\rho_i}(A_{w_s}A_w) \\
&=\sum_{|S|=p}\sum\limits_{\substack{u}}\phi\left(\text{coefficient of } A_u\otimes A_{\rho_i} \text{ in } \Delta(A_{w_s}A_w)\right)A_u \\
&=\sum_{|S|=p}\sum_{\substack{u}}\phi\left(\text{coefficient of } A_u\otimes A_{\rho_i} \text{ in } \Delta(A_{w_s})\Delta(A_w)\right)A_u.\\
\end{align*}
Note that the last equality follows from Theorem \ref{delta}. By (\ref{naive}), we can compute $ \Delta(A_{w_s})$ and $\Delta(A_w)$ seperately and multiply them together. Let $\widetilde{\rho}_{ij}$ denote the element $\rho_i \rho_{i-j}^{-1}=s_{i-1}s_{i-2}\ldots s_{i-j}$ for $i\geq j$. Then $\psi^{i-j}(\widetilde{\rho}_{ij})=\rho_j$. Therefore, we have
\begin{align*}
D_{\rho_i}(\h_p A_w)&=\phi\Big(\sum_{|S|=p}\sum_{j=0}^i\sum_{\substack{u_1,u_2 }} \left(\text{coefficient of } A_{u_1}\otimes A_{\widetilde{\rho}_{ij}} \text{ in }\Delta(A_{w_s})\right)A_{u_1}  \\
&\cdot \left(\text{coefficient of } A_{u_2}\otimes A_{\rho_{i-j}} \text{ in } \Delta(A_{w})\right) A_{u_2}\Big) \\
&=\phi\Big(\sum_{j=0}^i\sum_{\substack{u_1,u_2 }}\sum_{|S|=p} p_{u_1,\widetilde{\rho}_{ij}}^{w_s}A_{u_1}\cdot p_{u_2,\rho_{i-j}}^{w} A_{u_2}\Big) \\
&=\phi\Big(\sum_{j=0}^i\sum_{\substack{u_1,u_2 }}\sum_{|S|=p} \phi\left(p_{u_1,\widetilde{\rho}_{ij}}^{w_s}\right)A_{u_1}\cdot p_{u_2,\rho_j}^{w} A_{u_2}\Big) \\
&=\sum_{j=0}^i\sum_{\substack{u_2 }}\phi\Big(D_{\widetilde{\rho}_{ij}}(\h_{p})\cdot p_{u_2,\rho_j}^{w}\Big) A_{u_2} \\
&=\sum_{j=0}^i\sum_{\substack{u_2 }}\phi\Big(D_j(\h_{p})\cdot p_{u_2,\rho_j}^{w}\Big) A_{u_2} \mbox{ (By Lemma \ref{translation})}\\
&=\sum_{j=0}^i\sum_{\substack{u_2 }}\phi\Big(\h_{p-j}\cdot p_{u_2,\rho_j}^{w}\Big) A_{u_2} \mbox{ (By Theorem \ref{dh})}\\
&=\sum_{j=0}^i\sum_{\substack{u_2}}\h_{p-j}\cdot \phi\left(p_{u_2,\rho_j}^{w}\right) A_{u_2} \mbox{ (By Lemma \ref{hp} with $b=\h_{p-j}$)}\\
&=\sum_{j=0}^i \h_{p-j} D_{\rho_{i-j}}(A_w).\qed
\end{align*}
\\
{\it Proof of Theorem \ref{affinePieri}.} Theorem \ref{0}, \ref{dh}, \ref{dual} uniquely determine the Pieri operator $D_i$ and we have $D_i=D'_i$ by Theorem \ref{bss1}, \ref{bss2}, \ref{bss3}. Therefore, Theorem \ref{affinePieri} follows.\qed\\
\begin{rmk}
Theorem \ref{dual} generalizes \cite[Theorem 3.8]{BSS11}. In fact, the author believes that there is a gap in the first few lines in their proof of \cite[Theorem 3.8]{BSS11}, and Theorem \ref{dual} resolves the problem as well.
\end{rmk}

\section{Applications}
Let $W^0$ denote the set of $0$-Grassmannian elements in $W_\af$. In this section, we prove that Theorem \ref{affinePieri} implies the following conjecture, as stated in \cite{LLMS10}.

\begin{thm}\cite[Conjecture 4.18]{LLMS10}\label{strong}
Let $w,u\in W_\af$ be two affine permutations. Then
\begin{enumerate}
\item $\Strong_{w/u}(x)\in \Lambda$.
\item $\Strong_{w/u}(x)\in \Lambda_{(n)}$.
\item $\Strong_{w/u}(x)=\sum_{v \in W^0} p^w_{u,v} \Strong_v(x)$.
\end{enumerate}
\end{thm}
Note that Berg, Saliola and Serrano \cite{BSS12} proved $(1),(2)$ of Theorem \ref{strong} and $(3)$ when $w,v$ are in $W^0$. To be more precise, they showed that for $w,v,u \in W^0$, $p^w_{u,v}$ is the coefficient of $\tilde F_w$ in $\tilde F_u \tilde F_v$, where $\tilde F_w$ is the \emph{affine Stanley symmetric function} labeled by $w$. Lam, Lapointe, Morse and Shimozono \cite{LLMS10} introduced \emph{weak Schur functions} defined by
$$ \Weak_{w/u}(x)= \sum_\lambda \langle \h_\lambda A_u , A_w \rangle_{\mathbb{A}_0} m_\lambda(x)$$
for $w,u \in W_\af$ where the summation is over all $k$-bounded partitions and $\langle \cdot, \cdot \rangle_{\mathbb{A}_0}$ is the inner product on $\mathbb{A}_0$ satisfying $\langle A_{w_1}, A_{w_2} \rangle_{\mathbb{A}_0} = \delta_{w_1, w_2}$. If $w \in W^0$ and $v$ is the identity element, $\Weak_{w/u}(x)$ is the affine Stanley symmetric function $\tilde F_w$. \\
Let $Gr$ denote the affine Grassmannian associated with $SL(n,\mathbb{C})$. Since $Gr \cong G_\af/\mathcal{P}$ for the affine Kac-Moody group $G_\af$ and its maximal parabolic subgroup $\mathcal{P}$, we have the Schubert bases
$$\{\zeta^w\in H^*(Gr,\mathbb{Z}) \mid w \in W^0 \}$$
$$\{\zeta_w\in H_*(Gr,\mathbb{Z}) \mid w \in W^0 \}.$$
Lam \cite{Lam08} showed that $\zeta_w$ and $\zeta^w$ can be represented by $k$-Schur functions $\Strong_w(x)$ and affine Stanley symmetric functions $\Weak_w(x)$ via the isomorphisms $H_*(Gr)\cong \Lambda_{(n)}$ and $H^*(Gr)\cong \Lambda^{(n)}$. In fact, \emph{affine insertion} \cite{LLMS10} offers a duality between weak and strong orders combinatorially.\\
Define the \emph{affine Cauchy kernel} $\Omega_n(x,y)$ by
\begin{align*}
\Omega_n(x,y)&= \prod_i \left( 1+ y_i h_1(x) + y_i^2 h_2(x) + \cdots + y_i^{n-1} h_{n-1}(x)\right)\\
&=\sum_{\lambda : \lambda_1<n} h_\lambda(x) m_\lambda(y)\\
&=\sum_{\lambda} m_\lambda(x) \tilde{h}_\lambda(y)
\end{align*}
as an element in $\Lambda_{(n)}(x)\widehat{\otimes}\Lambda^{(n)}(y)\subset\Lambda(x)\widehat{\otimes}\Lambda^{(n)}(y)$ where $\tilde{h}_\lambda(y)$ is the image of $h_\lambda(y)$ in $\Lambda^{(n)}$. The second equality follows from the well-known equality $\sum_\lambda h_\lambda(x)m_\lambda(y)=\sum_\lambda m_\lambda(x)h_\lambda(y)$ after taking the projection map $\Lambda(x)\widehat\otimes\Lambda(y)\rightarrow\Lambda(x)\widehat\otimes\Lambda^{(n)}(y)$.\\
The duality between strong and weak orders produces the following affine Cauchy identity \cite[Corollary 4.6]{LLMS10}.

\begin{cor} (Affine Cauchy Identity) \label{affinecauchy}
The following identity holds in the formal power series ring $\mathbb{Z}[[x_1,x_2,\ldots,y_1,y_2,\ldots]]$:
$$\Omega_n(x,y)=\sum_{w\in W^0} \Strong_w(x)\Weak_w(y).$$
\end{cor}
Now we are ready to prove Theorem \ref{strong}, following \cite[Proposition 4.19]{LLMS10}. \\
{\it Proof of Theorem \ref{strong}.}
Let $\{\xi^w\in H^*(\widetilde{Fl}) \mid w \in W \}$ $(\{\xi_w\in H_*(\widetilde{Fl}) \mid w \in W \})$ denote the Schubert bases for the cohomology (resp. homology) of the affine flag variety. In this setting, Theorem \ref{affinePieri} can be described as follows:
\begin{thm}\label{affinePieri2}
Let $w\in W_\af$ and $1\leq m$. Then in $H^*(\widetilde{Fl})$ we have
$$ \xi^{\rho_m} \xi^w = \sum_S \xi^{\outside(S)},$$
where the sum runs over strong strips $S$ of size $m$ such that $\inside(S)=w$.
\end{thm}
Theorem \ref{affinePieri2} completely determines the action of $H^*(Gr)$ on $H^*(\widetilde{Fl})$, obtained from the inclusion $H^*(Gr) \subset H^*(\widetilde{Fl})$. \\
We may consider the affine Cauchy kernel $\Omega_n$ as an element of the completion $\Lambda \widehat{\otimes} H^*(Gr)$ via the isomorphism $\Lambda^{(n)}\cong H^*(Gr)$. Let $\langle \cdot, \cdot \rangle_{\widetilde{Fl}}$ denote the inner product on $H^*(\widetilde{Fl})$ defined by $\langle \xi^w, \xi^u \rangle=\delta_{wu}$. By the definition of $\Omega_n$ and Theorem \ref{affinePieri2}, we have
$$ \Strong_{w/u}(x)= \langle \Omega_n \cdot \xi^u, \xi^w \rangle_{\widetilde{Fl}}$$
where $\Omega_n\cdot \xi^u \in \Lambda\widehat{\otimes} H^*(Gr)$. Note that we have also used the definition of the strong Schur functions and the equality $\Weak_{\rho_i}(y)=\tilde{h}_i(y)$ (See \cite{LLMS10}). By Corollary \ref{affinecauchy}, we may also write
$$ \Omega_n=\sum_{v\in W^0} \Strong_v(x)\otimes \xi^v$$
so that
\begin{align*}
 \Strong_{w/u}(x)&= \sum_{v \in W^0} \Strong_v(x) \langle \xi^v\xi^u,\xi^w\rangle_{\widetilde{Fl}}\\
&=\sum_{v \in W^0} p^w_{u,v} \Strong_v(x).
\end{align*}
Since $\Strong_v(x)$ are $k$-Schur functions in $\Lambda_{(n)}$, we have $\Strong_{w/u}\in \Lambda_{(n)}$. \qed\\
\begin{rmk}
The skew strong Schur function $\Strong_{w/u}(x)$ can be geometrically interpreted as the image of $D_u(\xi_w)$ under the projection $H_*(\widetilde{Fl})\rightarrow H_*(Gr)$ where $D_u$ is the cap product acting on $H_*(\widetilde{Fl})$. Indeed, by the definition of $D_u$ and Remark \ref{ordinary} we have $D_u(\xi_w)=\sum_v p^w_{uv}\xi_v$. Since the projection $H_*(\widetilde{Fl})\rightarrow H_*(Gr)$ maps $\xi_w$ to $\Strong_w(x)$, it sends $D_u(\xi_w)$ to $Strong_{w/u}(x)$ by Theorem \ref{strong}(3). Therefore, $\Strong_{w/u}(x)$ captures the $0$-Grassmannian part of $D_u(\xi_w)$. Note that when $w$ is not $0$-Grassmannian, we have $\Strong_w(x)=0$ (See \cite{LLMS10}).
\end{rmk}
Note that Lam, Lapointe, Morse and Shimozono \cite{LLMS10} defined the marked strong cover $wt_{ij} \rightarrow w$ where $i \leq l < j$ for a fixed constant $l$ instead of $i \leq 0 <j$ in our setup. The affine Pieri rule conjectured in \cite{LLMS10} is the following.
\begin{thm}\cite[Conjecture 4.15]{LLMS10}\label{affinePieri3}
Let $w\in W_\af$ and $1\leq m$. Then in $H^*(\widetilde{Fl})$ we have
$$ \xi^{\psi^{-l}(\rho_m)} \xi^w = \sum_S \xi^{\outside(S)},$$
where the sum runs over strong strips $S$ of size $m$ with respect to $l$ such that $\inside(S)=w$.
\end{thm}
We can obtain Theorem \ref{affinePieri3} by applying $\psi^l$ to Theorem \ref{affinePieri2}. Recall that $\omega$ is the automorphism of $W_\af$ sending $s_i$ to $s_{n-i}$. By applying $\omega$ to Theorem \ref{affinePieri2}, we get the dual affine Pieri rule.
\begin{thm}
Let $w\in W_\af$ and $1\leq m$. Then in $H^*(\widetilde{Fl})$ we have
$$ \xi^{\omega(\rho_m)} \xi^w = \sum_S \xi^{\omega(\outside(S))},$$
where the sum runs over strong strips $S$ of size $m$ such that $\inside(S)=\omega(w)$.
\end{thm}

Since the image of $\xi^{\omega(\rho_m)}$ under the projection $H^*(\widetilde{Fl})\rightarrow H^*(Gr)\cong \Lambda^{(n)}$ is  $\Weak_{\omega(\rho_m)}=\tilde{e}_m$, the image of $e_m\in \Lambda^{(n)}$, so we recover the dual Pieri rule for the affine Grassmannian in \cite{LLMS10}.

config: arxiv


TITLE: If $F(x) = \int_0^x e^{\sin(t)} dt$ , so $F'(0)$ is equal to:
QUESTION [2 upvotes]: If $F(x) = \int_0^x e^{\sin(t)} dt$ , so $F'(0)$ is equal to:

(A) $-2$  $\quad$  (B) $-1$  $\quad$  (C)  $0$  $\quad$  (D)  $\mathbf1$  $\quad$  (E)  $2$
My attempt:
$$\frac{d}{dt}F(x) = \frac{d}{dt}\int_0^x e^{\sin(t)} dt\Rightarrow F'(x) = \int_0^x \frac{d}{dt}e^{\sin(t)} dt$$
$$F'(x) = e^{\sin(t)}|_{0}^{x} = e^{x} - 1 \implies F'(0) = e^0-1 = 0$$
(wrong answer)
Where am I missing?

REPLY [3 votes]: When you have $g(x) = \int_0^xf(t)dt$ then $g'(x) = \frac{\delta g}{\delta x} = (\int_0^xf(t)dt)^{'}=  f(x)$
So in your function:
$$F'(x) = e^{sin(x)}$$ and for x = 0:
$$F'(0) = e^{sin(0)} = e^0 = 1$$
That's completely different from $g(x)=\int f(x)dx$ where $g'(x) = f(x)$. That's your mistake in your approach of the problem.

REPLY [0 votes]: We have $(\int_0^xf)'=f(x)$ by the Fundamental Theorem of Calculus, but you applied the rule for integration
$$\int_a^bf=F(b) -F(a) $$
where $F'=f$.


TITLE: Prove that $(2n+1)(x+y+z)+6$ divides $x^3+y^3+z^3$
QUESTION [2 upvotes]: Let $n,x,y,z$ be positive integers such that $$(x-y)^2+(y-z)^2+(z-x)^2 = (2n+1)xyz.$$ Prove that $(2n+1)(x+y+z)+6$ divides $x^3+y^3+z^3$.

The original equation doesn't have $x^3,y^3,z^3$, so how can we show that $(2n+1)(x+y+z)+6$ divides $x^3+y^3+z^3$? Expanding the given equation seems to make the question more complicated, and also how do we get the $6$?

REPLY [1 votes]: This is linked to a well-known identity (a standard exercise in high-school):
\begin{align}x^3+y^3+z^3-3xyz&=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)\\&=\tfrac12(x+y+z)\bigl((x-y)^2+(y-z)^2+(z-x)^2\bigr)\end{align}


TITLE: Spectrometer vs. Spectrophotometer
QUESTION [5 upvotes]: I have been researching about the difference of a spectrometer and a spectrophotometer. They both sound the same. What is the difference?

REPLY [1 votes]: I thought this article looked credible and gave a thorough explanation:
Part of what makes this confusing for a lot of people is that all spectrophotometers incorporate a spectrometer. It’s also true that other analytical instruments use spectrometers. The spectrometer is the part of the spectrophotometer that is most responsible for measuring things. The spectrophotometer is a complete system that includes a light source along with a means to collect the light that has interacted with the things being tested, as well as a spectrometer for measurements.
Source: https://www.excedr.com/blog/spectrometer-vs-spectrophotometer/
I think the article explains both a spectrometer and a spectrophotometer in detail which may give more insight than simply the comparison.
The article addresses why it can be a little confusing (as well as a detailed answer), but in short, a spectrophotometer has a spectrometer in it, but the spectrophotometer is an entire system.
I believe this image/vector belows shows an entire spectrophotometer, where as the gauge at the end I believe is only a spectrometer.

Furthermore, I thought providing a concrete example in industry may be useful.  In laser safety, lenses are tested with a spectrophotometer.  The spectrophotometer will test the amount of light being absorbed against the amount of light being let through at each wavelength in question.  The amount being absorbed is a measurement known as Optical Density.  See the diagram below for an brief overview on how a spectrophotometer works.  Please note, the spectrophotometer depicted below is showing a spectrophotometer for visible light; different spectrophotometers test for different wavelength spans (UV and IR).


TITLE: Need help in showing that $\lim_{Δx\to 0}\frac{a^{Δx}-1}{Δx}=\ln(a)$
QUESTION [0 upvotes]: Defining $$e^x=\lim_{n\to0}(1+nx)^{\frac{1}{n}}$$ and $$e^y=x ;y=lnx $$ I want to show that $e$ is the unique number such that $\frac{de^x}{dx}=e^x$. I eventually obtain the expression $$\frac{da^x}{dx}=C(a)\cdot a^x$$ where $$C(a)=\lim_{Δx\to 0}\frac{a^{Δx}-1}{Δx}$$
Beyond this point, I am unable to show that $C(a)$ is infact $ln(a)$. One method that seems to work is to ignore "$\lim_{Δx\to0}$" , make $a^{Δx}$ the subject and then restore the limit operator to obtain $$a=\lim_{Δx\to0}(1+C(a)\cdotΔx)^{\frac{1}{Δx}}\Rightarrow a=e^{C(a)}\Rightarrow C(a)=\ln(a)$$but simply ignoring the limit operator feels extremely wrong. Could someone give me a hint on how to proceed?

REPLY [2 votes]: Let $\eta=a^{\delta x}-1$.
$$\lim_{\delta x\to0}\frac{a^{\delta x}-1}{\delta x}=\lim_{\delta x\to0}\frac{\eta}{\log_a(\eta+1)}=\lim_{\delta x\to0}\frac{1}{\frac{1}{\eta}\log_a(\eta+1)}=\cdots$$
You can go from here. I'll edit in a full answer if you wish.
Hints: consider what $\eta$ tends to as $\delta x\to0$, and remember your log properties!


TITLE: Intuitive Understanding of Order of Hyperreals
QUESTION [0 upvotes]: I'm trying to understand the ultrapower construction of the hyperreal numbers as described in Wikipedia.
The motivation is given as the surprising ability to find a total order of sequences of real numbers. Apparently if we select a free ultrafilter $U$ of the natural numbers, then it can be used to determine which indices "matter" in the set and then (from the article):

We write $(a_0, a_1, a_2, ...) ≤ (b_0, b_1, b_2, ...)$ if and only if the set of natural numbers $\{ n : a_n ≤ b_n \}$ is in $U$.

This is where I don't follow. Are we still working component-wise and then "$\land$-ing" the resultant sequence together as mentioned previously in the article?
The "intuitive explanation" which follows briefly mentions a parallel to Cantor's construction of the reals ... did I miss a step where this became a Cauchy sequence by implication?
In short: why does choosing a special set $U$ of indices of sequences to "matter" create a total order relation?

REPLY [2 votes]: $U$ isn't a set of indices, it's a set of sets of indices. The idea is that two sequences are equal (or rather, name the same element) if they agree "most of the time" in the sense of $U$. Similarly, one sequence $A$ (names an element of the ultrapower which) is less than or equal to (the element named by) another sequence $B$ iff "most of the time"each entry of $A$ is $\le$ the corresponding entry of $B$. And so forth.
For example, consider the following sequences:

$A_1=(1,0,0,0,0,...)$
$A_2=(0,1,0,0,0,...)$
$A_n$ has a $0$ in every place except the $n$th place, where it has a $1$.

I claim that these sequences are all equal (or rather, name the same element) in the ultrapower! For every $i,j$, the set of $n$ such that the $n$th elemenet of $A_i$ is $=$ the $n$th element of $A_j$ is cofinite, and every cofinite set is in $U$ since $U$ is a nonprincipal ultrafilter, so we always have $[A_i]_U\le [A_j]_U$ (this is a bit different from what you wrote - I'm being careful to distinguish between the sequence $A_i$ and the equivalence class $[A_i]_U$ which it belongs to; remember that elements of the ultrapower are equivalence classes of sequences, not individual sequences).

How does this generate a total order?
Well, suppose I have two sequences $A=(a_n)_{n\in\mathbb{N}}$ and $B=(b_n)_{n\in\mathbb{N}}$. Let $$X=\{n: a_n\le b_n\},\quad Y=\{n: a_n>b_n\}.$$ Clearly $X\sqcup Y=\mathbb{N}$, so since $U$ is an ultrafilter exactly one of $X$ and $Y$ is in $U$. If $X\in U$, then $[A]_U\le [B]_U$; if $Y\in U$, then $[A]_U>[B]_U$.


TITLE: Limits of sums of functions existence
QUESTION [1 upvotes]: Why is it that given two functions $ f(x)$ and $g(x)$, and some $a \in{\mathbb{R}}$, $$ \nexists \lim_{x \to a} f(x)  \ \land \  \exists\lim_{x \to a} g(x) \  \Rightarrow   \nexists \ \lim_{x \to a} \  [f(x)+g(x)] \  $$ yet $$ \nexists \ \lim_{x \to a} f(x) \   \ \land \ \nexists \lim_{x \to a} g(x) \ \not \Rightarrow \nexists\lim_{x \to a} \  [f(x)+g(x)] \   $$ How would you prove the first statment?

REPLY [0 votes]: If $\exists \lim_{x \to a}(f(x)+g(x))$ and $\exists \lim_{x \to a}g(x)$
then $\lim_{x \to a}f(x)=\lim_{x \to a}(f(x)+g(x)-g(x))=\lim_{x \to a}(f(x)+g(x))-\lim_{x \to a}g(x)$ which exists so we have a contradiction.
We used the fact that if two limits of two function exist then the difference of the limits is equal to the limit of the difference.
For the second we have a counterexample:
Take $$f(x)=\sin^2{\frac{1}{x}}$$ $$g(x)=\cos^2{\frac{1}{x}}$$
We have that $\lim_{x \to 0}f(x)+g(x)=1$ but $\lim_{x \to 0}f(x)$ and $\lim_{x \to 0}g(x)$ do not exist.


TITLE: Probability of intersecting paths on loop under random starting condition
QUESTION [2 upvotes]: Jim leaves his house at noon and walks at a constant rate of 4 miles per hour along a 4 mile loop (returning to his house at 1 PM). At a randomly chosen time, uniform between noon and 1 PM, Sally chooses a random location (uniform along Jim’s route) and begins running at a constant rate of 7 miles per hour along the route in the same direction that Jim is walking until she completes one 4-mile circuit of the route.
The probability that Sally runs past Jim while he is walking is given by $m/n$, where $m$ and $n$ are relatively prime integers.
Find $m + n$.

I find this problem interesting because of the condition that Sally completes just a single loop and cannot continue, even if it would allow her to catch Jim.  This makes finding the appropriate starting conditions a bit tricky.  Moreover, it is surprising to me that the answer will be the ratio of two primes.  This is certainly not obvious from the problem statement.

REPLY [1 votes]: At the time and place where Sally starts, Jim is $D$ miles ahead and will continue walking for $T$ hours.
Sally will run for $\frac47$ hour. Since she gains $3$ miles per hour on Jim, if $D > \frac{12}{7}$ she will not pass him before she stops running.
If $D > 3T$ then Sally will not pass Jim before he stops walking.
But if $D < 3T$ and $D < \frac{12}{7}$, Sally will pass Jim.
(I'm ignoring the cases $D = 3T$ and $D = \frac{12}{7}$ because they have zero probability and I don't care to quibble about the definition of "passing" in those cases.)
The joint distribution of $D$ and $T$ is uniform over the rectangle
$0 \leq D < 4$ and $0 \leq T \leq 1.$
Figure out the fraction of this region that lies under both of the lines
$D < 3T$ and $D < \frac{12}{7}$.


TITLE: Problem integrating in problem using the Poincaré Lemma
QUESTION [2 upvotes]: a) It is easy to show that $d\beta=0$.
b)
$\begin{align}\hat{\mathbb{X}}_t &= \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \hat{\Phi}_t^{-1} \\
&= \left(\frac{\partial}{\partial t}\hat{\Phi}_t\right) \left(\frac{x}{t},y,z\right) \\
&=\left(\frac{x}{t},0,0\right)
\end{align}$
Now $\Phi_t^* x =tx$, $\Phi_t^* y =y$, $\Phi_t^* z =z$ , so,
$$ \hat{\Phi}_t^* \beta = \hat{\Phi}_t^* \left(\frac{(t^2x^2-y^2)dy \wedge dz + 2txyt dz \wedge dx}{(t^2x^2+y^2)^2}\right)
\rightarrow \frac{-1}{y^2} dy \wedge dz
$$
So chose $\alpha_0$ to be $(t^2x^2+y^2)^2=\frac{1}{y}dz$
c) First want to compute that $i_{\mathbb{X}_t}\beta$.
$$\begin{align} i_{\mathbb{X}_t} \beta &= i_{\mathbb{X}_t} \left(\frac{(x^2-y^2)}{\rho^4} dy \wedge dz\right) + i_{\mathbb{X}_t}\left(\frac{2xy }{\rho^4} dz \wedge dx\right) \\
&= \left(\frac{(x^2-y^2)}{\rho^4}\right)\left[dy(\mathbb{X}_t)dz - dz(\mathbb{X}_t)dy\right]+\frac{2xy}{\rho^4}[dz(\mathbb{X}_t)dx-dx(\mathbb{X}_t)dz] \\
&= \left(\frac{2xy}{\rho^4}\right)[\frac{-x}{t}dz]
\end{align}$$
Then I have that,
$$\begin{align} \Phi_t^*(i_{\mathbb{X}_t}\beta) &= \frac{-2y}{t}\left(\frac{t^2x^2}{(t^2x^2+y^2)^2}\right)dz \\
&= -2yt \left(\frac{x^2}{(t^2x^2+y^2)^2}\right)dz
\end{align}$$
But this isnt very integrable. Not sure where I have gone wrong or how I should proceed with this question.

REPLY [2 votes]: You just made a silly mistake in the last line of computing $i_{\mathbb{X}_t}\beta$ and grabbed the wrong coefficient.
You should have $2xy$ in the numerator of the first factor instead of $x^2-y^2$. Then what you get after the pullback is easily integrated.


TITLE: Inverse of $f(x)=\sin(x)+x$
QUESTION [5 upvotes]: What is the inverse of
$$f(x)=\sin(x)+x.$$
I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.
What about
$$f(x)=\sin(a \cdot x)+x$$
where $a$ is a known real constant.
Thank you for taking the time to read this question!
Sorry if this has been asked before...

REPLY [0 votes]: That's a kind of Kepler's equation. Already Liouville proved that the inverse isn't an elementary function.
$$\sin(ax)+x=y$$
$$-\frac{1}{2}i\left(e^{aix}-e^{-aix}\right)+x=+y$$
$$-\frac{1}{2}i\left(e^{aix}\right)^2+\frac{1}{2}i+e^{aix}x-e^{aix}y=0$$
$$e^{aix}=-ix\pm\sqrt{-x^2+2yx-y^2+1}+iy$$
The inverse cannot be represented in closed form in terms of Lambert W either, and not by generalized Lambert W of Mezö et. al. But possibly by other generalized Lambert W - see the references below.
[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553
[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)
[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018
[Stoutemyer 2022] Stoutemyer, D. R.: Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses. 2022


TITLE: Compute the area of specific shapes
QUESTION [0 upvotes]: I'm trying to calculate the dashed area in the following pictures, and I can't solve them. I tried to guess the areas, subtract some shapes from others, but I'm confused if I calculated them wrong or correct.
Problem 1 http://i.stack.imgur.com/ldS9T.jpg
Problem 2 http://i.stack.imgur.com/2GAgM.jpg

REPLY [0 votes]: Problem 1. Let $x$ be the base of the small triangle & the side of trapezium then using simple geometry, we have $$x=a-a\left(\frac{\frac{\alpha\sqrt{3}}{2}}{\frac{a\sqrt{3}}{2}}\right)\implies \color{blue}{x=a-\alpha}$$ Now we have $$\text{area of trapezuim}=\frac{1}{2}(\text{sum of lengths of parallel sides})\left(\text{normal distance between parallel sides }\right)$$$$=\frac{1}{2}(a+x)\left(\frac{\alpha\sqrt{3}}{2}\right)$$  $$=\frac{1}{2}(a+a-\alpha)\left(\frac{\alpha\sqrt{3}}{2}\right)$$$$\color{blue}{=\frac{\alpha\sqrt{3}}{2}\left(a-\frac{\alpha}{2}\right)}$$
Problem 2.
Let $x$ be the base of the small triangle & the side of trapezium then using simple geometry, we have $$x=a-a\left(\frac{\frac{\alpha a}{2}}{\frac{a\sqrt{3}}{2}}\right)\implies \color{blue}{x=a-\frac{\alpha a}{\sqrt{3}}}$$ Now we have $$\text{area of trapezuim}=\frac{1}{2}(\text{sum of lengths of parallel sides})\left(\text{normal distance between parallel sides }\right)$$$$=\frac{1}{2}(a+x)\left(\frac{\alpha a}{2}\right)$$$$=\frac{1}{2}\left(a+a-\frac{\alpha a}{\sqrt{3}}\right)\left(\frac{\alpha a}{2}\right)$$  $$\color{blue}{=\frac{\alpha a^2}{4}\left(2-\frac{\alpha}{\sqrt{3}}\right)}$$


TITLE: Under what conditions, the following equation may not hold: $A = B X$, where $A \in M_{m,n}$ and $B \in M_{m,p}$ are given, and $X \in M_{p,n}$?
QUESTION [0 upvotes]: I am sorry for asking a basic question.
Under what (non-trivial) conditions the following equation will not hold?
\begin{align}
A = BX,
\end{align}
where $A \in M_{m,n}$ and $B \in M_{m,p}$ are given, and $X \in M_{p,n}$ is a variable?
The straight forward solution is
$X = B^{\dagger} A$, where $()^{\dagger}$ is pseudo inverse. But I wonder, will the solution always exist?

Numerically (in MATLAB), I see the solution exist for all my tests (see two them)...
A = rand(2,28) +1j*randn(2,28);
B = rand(2,4) +1j*randn(2,4);
X = pinv(B)*A;

% check whether the result is nearly zero. If yes, then good!
sum(sum((A - B*X)))

ans =

   4.6352e-15 - 1.6209e-15i

A = rand(1,4) +1j*randn(1,4);
B = rand(1,2) +1j*randn(1,2);
X = pinv(B)*A;

% check whether the result is nearly zero. If yes, then good!
sum(sum((A - B*X)))

ans =

  -9.5340e-15 + 2.1858e-15i

REPLY [1 votes]: Your solution does not work ! Consider the case $B = 0$. Then your solution gives $X = 0$.
You need the columns of $A$ to be in the image space of $B$. (Question : why ?).
You can also easily check that this condition is sufficient for a solution to exist.


TITLE: Proving $e^{-z}=\frac{1}{e^z}$
QUESTION [1 upvotes]: Prove $$e^{-z}=\frac{1}{e^z}$$.

Taking $z=z_1+z_2i$ then:
$e^{-z}=e^{-(z_1+z_2i)}=e^{-z_1-z_2i}=e^{-z_1}.e^{-z_2i}$
I know the first part is easy to see $e^{-z_1}=\frac{1}{e^{z_1}}$ since $z_1$ is a real number.
However I am not seeing how to proceed with $e^{-z_2i}$.
Question:
Since I have the imaginary unit. How should I treat $e^{-z_2i}$?
Thanks in advance!|

REPLY [1 votes]: $$\begin{array}{}
e^{z_1} e^{z_2}&=\sum_{k_1=0}^\infty\frac{z_1^{k_1}}{k_1!}\sum_{k_2=0}^\infty\frac{z_2^{k_2}}{k_2!}=\sum_{k_1=0}^\infty\sum_{k_2=0}^\infty\frac{z_1^{k_1}z_2^{k_2}}{k_1!k_2!}=\sum_{k=0}^\infty\sum_{k_1=0}^k\frac{z_1^{k_1}z_2^{k-k_1}}{k_1!(k-k_1)!}\\
&=\sum_{k=0}^\infty\frac1{k!}\sum_{k_1=0}^k\frac{k!}{k_1!(k-k_1)!}z_1^{k_1}z_2^{k-k_1}=\sum_{k=0}^\infty\frac{(z_1+z_2)^k}{k!}=e^{z_1+z_2}.\\
\end{array}
$$
Subsitute now $z_1=z$, $z_2=-z$.


TITLE: With regards to Zermelo-Fraenkel Set Theory, is the $0$ and $\emptyset$ in $0 =\{ \}= \emptyset$ equal?
QUESTION [2 upvotes]: I have been studying philosophy for some time now, although my knowledge with respect to mathematics is amateur at best. I recently starting learning set theory on my own, but have ran into an issue (at-least i think it's an issue). From what i understand (from one book and several youtube videos), $0 = \{\} = \emptyset$ in Zermelo-Fraenkel Set Theory. If that's the case, why are there two ways of representing the same thing? Instead of having both '$0$' and '$\emptyset$' representing one thing, why can't we have only one of those symbols represent that same exact thing? My apologies in advance for my lack of clarity; as I'm sure you noticed my mathematical jargon is not that good. Thanks for all of your help!

REPLY [0 votes]: Being incredibly pedantic, neither $\emptyset$, nor $0$, nor $\{\}$ are symbols in the ZF set theory. ZF set theory is axiomatized using the symbol $\in$ alone.
So, in a certain sense, everything that is not $\in$ is extra stuff that is added to ZF to make it easier to work with.
Set-builder notation $\{ x : x \in A \land \varphi(x)\}$ can be defined formally as the set of all values $x$ satisfying $\varphi$ and in $A$. We know that every set named using set builder notation exists by the axiom schema of comprehension and we know that the set named using set builder notation is unique by the axiom of extensionality.
Using this, we can define $\{\}$ as $\{x : x \in E \land \bot \}$, where $E$ is an arbitrary set. We know there's at least one set because our domain can't be empty. Alternatively, we can make $E$ the set promised to us by the axiom of infinity.
We can define $\emptyset$ the same way that we defined $\{\}$.
$0$ is a bit different. There are two commonly used set-based constructions for the natural numbers.
The Von Neumann naturals are $0 = \emptyset$, $1 = \{0\}$, $2 = \{0, 1\}$, $3=\{0,1,2\}$...
The Zermelo naturals are $0 = \emptyset$, $1 = \{0\}$, $2 = \{1\}$, $3=\{2\}$ ...
In both of these explicit constructions $0$ is the empty set, but there are trivial variations of them where $0$ is mapped to something else.
These constructions can be viewed as definitions of the natural numbers or as ways of interpreting natural numbers.