text:
\section{Preliminaries}\label{sec:Preliminaries}

This section reviews all preliminary definitions and results which are required throughout the paper.  Section \ref{sec:Notation} establishes our notation for Weyl groups, root systems, and alcoves, using the conventions from \cite{Bourbaki4-6}.  In Section \ref{sec:NewtonMap}, we review the Newton and Kottwitz maps, which determine the two invariants that parameterize $\sigma$-conjugacy classes. Following \cite{GHKRadlvs}, we discuss the standard representative for a given $\sigma$-conjugacy class in Section \ref{sec:StandardReps-General}. We provide a brief review of affine Deligne--Lusztig varieties in Section \ref{sec:ADLVs}. We conclude the preliminary material in Section \ref{sec:galleriesChimneys} by recalling the necessary background on combinatorial galleries and chimneys from \cite{MST1, MNST}.

\subsection{Notation}\label{sec:Notation}

Let $G$ be a split connected reductive group over $\F_q$, and let $T$ be a split maximal torus of $G$. Let $F=\overline{\mathbb{F}}_q((t))$ be the discretely valued field with ring of integers $\cO = \overline{\mathbb{F}}_q[[t]]$.  The choice of $T$ corresponds to fixing an apartment $\cA$ in the Bruhat--Tits building for $G(F)$, which we identify with $\mathfrak{a} := X_*(T)\otimes_{\Z}\R \cong \R^n$ where $n$ is the rank of $G$.  Fix a Borel subgroup $B$ and let $P=MN$ be a standard parabolic subgroup, with $M$ the unique Levi subgroup containing $T$ and unipotent radical $N$.  In the special case $P=B$, we write $B=TU.$ The finite or spherical Weyl group of $T$ in $G$ equals $\sW = N_G(T)/T$.  The extended affine Weyl group is then $\eW = N_GT(F)/T(\cO) \cong X_*(T) \rtimes \sW.$ Given any $\lambda \in X_*(T)$, we write $t^\lambda$ to denote the image of $t$ under the homomorphism $\lambda: \mathbb{G}_m \rightarrow T$, and $t^\lambda$ acts on $\mathfrak{a}$ by translation by $\lambda$.  Any element $x \in \eW$ can  be uniquely written as $x = t^\lambda w$ for some $\lambda \in X_*(T)$ and spherical direction $w \in \sW$.

Denote by $\Phi$ the set of roots of $T$ in $G$, which we assume is irreducible throughout the paper; equivalently, the associated Dynkin diagram is assumed to be connected. Denote by $\Phi^+$ those roots which are positive with respect to the opposite Borel subgroup $B^-$.  Let $\Delta = \{\alpha_i\}_{i=1}^n$ be a basis of simple roots in $\Phi^+$.  The set of indices on the elements of $\Delta$ shall be denoted by $[n] = \{1,2, \dots, n\}$. Denote by $\rho$ the half-sum of the positive roots in $\Phi^+$, and let $\tilde{\alpha}$ denote the highest root.  The coroot associated to $\alpha \in \Phi$ with respect to the evaluation pairing $\langle \cdot, \cdot \rangle: X^*(T) \times X_*(T) \rightarrow \Z$ is then $\alpha^\vee = 2\alpha/ (\alpha, \alpha )$, where $(\cdot , \cdot )$ denotes the standard Euclidean dot product. Denote by $\Delta^\vee$ the corresponding simple coroots. The fundamental weights $\{\omega_i\}$ and the fundamental coweights $\{\varpi_i\}$ are dual bases to $\Delta^\vee$ and $\Delta$, respectively, with respect to $\langle \cdot, \cdot \rangle$.   The dominant Weyl chamber is defined as $\Cf = \{v \in \mathfrak{a} \mid \langle \alpha, v \rangle \geq 0 \ \text{for all}\ \alpha \in \Phi^+\}$. The shrunken dominant Weyl chamber $\Cfs$ is defined by the same inequalities, after replacing the $0$ by $1$.

Let $R^\vee = \bigoplus \Z\alpha_i^\vee \subset X_*(T)$ denote the coroot lattice, and let $Q^\vee = \bigoplus \Z \varpi_i \subset X_*(T)$ denote the coweight lattice.  Denote by $\check{\rho}$ the element of $Q^\vee$ defined as the half-sum of the positive coroots, or equivalently the sum of the fundamental coweights; note that $\check{\rho}$ may not be a scalar multiple of $\rho$. An element of $Q^\vee$ which lies in $\Cf$ is called a dominant coweight, and we remark that $\check{\rho}$ is dominant. The dominance ordering on $R^\vee$ defines $\lambda \geq \mu$ if and only if $\lambda-\mu$ is a non-negative $\Z$-linear combination of positive coroots. The partial ordering $\geq$ is then extended to $Q^\vee$ and $\mathfrak{a}$ by replacing $\Z$ with $\Q$ and $\R$, respectively. Denote by $\Lambda_G = X_*(T)/R^\vee$.

The finite Weyl group $\sW$ is a Coxeter group, generated by the set $S = \{ s_i\}_{i=1}^n$ of (simple) reflections across the hyperplanes orthogonal to the respective simple roots in $\Delta$.  Denote by $\ell(w)$ the length of an element $w \in \sW$ with respect to $S$, and denote by $w_0$ the longest element in $\sW$. We may also view $\sW$ as a finite reflection group acting on $\mathfrak{a} \cong \R^n$. Given any coweight $\mu \in Q^\vee$, we denote by $\mu^+$ the unique dominant coweight in the $\sW$-orbit of $\mu$ under this action.

Given any root $\alpha \in \Phi$ and any integer $k \in \Z$, define $H_{\alpha, k} = \{v \in \mathfrak{a} \mid \langle \alpha, v \rangle = k\}.$ In particular, note that $\check{\rho} \in H_{\alpha_i,1}$ for all $\alpha_i \in \Delta$. The complement in $\mathfrak{a}$ of the collection of all the affine hyperplanes $H_{\alpha,k}$ are called alcoves. The base alcove is defined to be the unique alcove in $\Cf$ whose closure contains the origin, denoted by $\fa_0 =\{v \in \Cf \mid \langle \alpha, v \rangle \leq 1 \ \text{for all}\ \alpha \in \Phi^+\}$.   The reflection across the affine hyperplane $H_{\alpha, k}$ defined by $s_{\alpha, k}(v) = v - (\langle \alpha, v \rangle - k)\alpha^\vee$ is an affine transformation of $\mathfrak{a}$, and the collection of all such reflections generates the (non-extended) affine Weyl group $\aW$. When $k=0$, we typically write $H_{\alpha, 0} = H_{\alpha}$ and $s_{\alpha, 0} = s_\alpha$. We remark that choosing $\Phi^+$ to be positive with respect to $B^-$ implies that the positive roots lie on the same side of the hyperplane $H_{\tilde{\alpha}}$ as $\Cf$.  Even when $k \neq 0$, we refer to $H_{\alpha,k}$ as an $\alpha$-hyperplane.

The action of the affine Weyl group $\aW$ on $\mathfrak{a}$ preserves the collection of hyperplanes $H_{\alpha, k}$, and so the element $x \in \aW$ naturally corresponds to the alcove $x\fa_0$, which we typically denote in bold by $\x$. The extended affine Weyl group $\eW$ acts simply transitively on the set of extended alcoves, which may be viewed as $|\Lambda_G|$ distinct copies of $\mathfrak{a}$, where each copy is endowed with the same configuration of alcoves on which the affine Weyl group $\aW$ acts simply transitively. The affine Weyl group can also be viewed as $\aW \cong R^\vee \rtimes \sW$, and so every element $x \in \aW$ can be uniquely expressed as $x = t^\lambda w$ for some $\lambda \in R^\vee$ and $w \in \sW$. Similarly, the extended affine Weyl group is isomorphic to $\eW \cong Q^\vee \rtimes \sW$.

The affine Weyl group $\aW$ is a Coxeter group with generating set $\tilde{S} = S \cup \{ s_{\tilde{\alpha},1}\},$ and the additional affine generator is  denoted by $s_0 :=s_{\tilde{\alpha},1}$.  We again denote the corresponding length function by $\ell: \aW \rightarrow \Z$. Although the extended affine Weyl group $\eW$ is no longer a Coxeter group, the length function $\ell$ extends to $\eW$ by defining those elements whose extended alcove projects onto the base alcove $\fa_0$ in $\mathfrak{a}$ to have length zero. Denote by $\Omega_G$ the subgroup of elements $\omega \in \eW$ such that $\ell(\omega) = 0$, and note that $|\Omega_G| = |\Lambda_G|$. An element $\omega \in \Omega_G$ acts on the base alcove $\fa_0$ by an automorphism, which induces a permutation action on the set $\tilde I = \{0,1,\dots, n\}$ indexing the elements of $\tilde S$. We then also have $\eW \cong \aW \rtimes \Omega_G$, so that for any $x \in \eW$, we may also write $x = y \omega$ for unique $y \in \aW$ and $\omega \in \Omega_G$.

Following \cite{Ram}, we view the elements in $\Omega_G$ as deck transformations, which move between the sheets of extended alcoves in the product $\Omega_G \times \mathfrak{a}$. For any $\omega \in \Omega_G$, the extended alcove $\omega \times \fa_0$ then projects onto the extended base alcove $0 \times \fa_0 \in \Omega_G \times \mathfrak{a}$. Given $\omega \in \Omega_G$, we denote the corresponding extended base alcove by $\fa_\omega = \omega \times \fa_0$. By abuse of notation, we often denote the extended base alcoves by $\fa_0 = 0\times \fa_0$ and $\fa = \fa_\omega$ for nonzero $\omega \in \Omega_G$. Similarly, the extended alcove in $\Omega_G \times \mathfrak{a}$ corresponding to the element $x \in \eW$ is typically denoted in bold by $\x=x\fa_0$. We occasionally refer to an extended alcove simply as an alcove, since the precise meaning should always be clear from context. Further, we present some figures which overlay extended alcoves from different sheets together in a single copy of $\mathfrak{a}$.

Since $\eW \cong Q^\vee \rtimes W \cong \aW \rtimes \Omega_G$ and $\aW \cong R^\vee \rtimes W$, then an element $\omega \in \Omega_G$ also determines a nonzero coset in $Q^\vee /R^\vee \cong \Omega_G$. As such, we shall typically index distinct sheets of extended alcoves by the elements of $Q^\vee/R^\vee$.   In particular, the image in $Q^\vee/R^\vee$ of the translation part of $x \in \eW$  uniquely identifies the sheet containing the extended alcove $\x$. By abuse of notation, we also frequently write $\lambda \in \omega + R^\vee$ for $\omega \in \Omega_G$ to denote the image of a given $\lambda \in Q^\vee$ in the quotient $Q^\vee/R^\vee$.

Given a Levi decomposition for a standard (spherical) parabolic subgroup $P=MN$, the set of simple roots for $G$ decomposes as $\Delta = \Delta_M \sqcup \Delta_N,$ where $\Delta_M$ is the set of simple roots for the reductive group $M$, and $\Delta_N$ is the set of simple roots which occur in the Lie algebra of $N$. Note that although $\Phi$ is assumed to be irreducible, the set $\Phi_M$ of roots of $T$ in the Levi subgroup $M$ may not be.  Define $\mathfrak{a}_P = \{ v \in \mathfrak{a} \mid \langle \alpha_i, v \rangle = 0,\ \text{for all}\ \alpha_i \in \Delta_M\};$ equivalently, $\mathfrak{a}_P$ is the intersection of all hyperplanes $H_{\alpha_i}$ corresponding to those simple roots in $\Delta_M$.  The choice of standard parabolic then determines an open chamber in $\mathfrak{a}_P$ defined as
\begin{equation}\label{eq:aP+}
     \mathfrak{a}_P^+ = \{ v \in \mathfrak{a}_P \mid \langle \alpha, v \rangle>0\ \text{for all}\ \alpha \in \Delta_N \}.
\end{equation}
The dominant Weyl chamber is a disjoint union of these open chambers $\Cf = \coprod \mathfrak{a}_P^+,$ where $P$ ranges over all standard parabolic subgroups $P$.  Using the identification $\mathfrak{a}_P \cong \Lambda_M \otimes_{\Z}\R$, denote by $\Lambda_M^+$ the subset of $\Lambda_M$ whose image under this isomorphism lies in $\mathfrak{a}_P^+$.

The Iwahori subgroup $I$ of $G(F)$ is the stabilizer of the base alcove $\fa_0$. Equivalently, $I$ is the preimage of the opposite Borel under the projection $G(\cO) \rightarrow G(\overline{\mathbb{F}}_q)$ defined by $t \mapsto 0$. Note that the base alcove $\fa_0$ coincides with the basepoint of the affine flag variety $G(F)/I$. Given any standard parabolic subgroup $P=MN$, we define an associated subgroup of $G(F)$ by $I_P = (I\cap M(F))N(F)$.

The Frobenius automorphism $a \mapsto a^q$ on $\overline{\mathbb{F}}_q$ can be extended to $\sigma: F \rightarrow F$ by defining $\sigma$ to raise each coefficient to the $q^{\text{th}}$ power. Two elements $g,h \in G(F)$ are said to be $\sigma$-conjugate if $g = xh\sigma(x)^{-1}$ for some $x \in G(F)$.  The collection of all $\sigma$-conjugates of an element $g \in G(F)$ will typically be denoted by $[g]$.  Given any subgroup $H$ of $G(F)$ and any $g,h \in G(F)$, we shall denote the usual conjugation action by $H^g = gHg^{-1}$ and $h^g = ghg^{-1}$. Given any $x,y \in \eW$, we also denote by $\x^y$ the extended alcove corresponding to the element $x^y$.

\subsection{The Newton and Kottwitz maps}\label{sec:NewtonMap}

Given any $b \in G(F)$, we denote by $[b]$ the $\sigma$-conjugacy class
\[ [b] = \{ g^{-1}b\sigma(g) \mid g \in G(F) \}.\]
The set $B(G)$ of $\sigma$-conjugacy classes in $G(F)$ is characterized by a pair of invariants: the Newton point and the Kottwitz point. In this section we briefly review the key properties of these two $\sigma$-conjugacy invariants, originally defined in \cite{KotIsoI, KotIsoII}.

Denote by $\kappa_G: G(F) \rightarrow \Lambda_G$ the natural surjection defined in \cite{KotIsoII}, subsequently called the \emph{Kottwitz homomorphism}. By \cite[Lem.~7.2.1]{GHKRadlvs}, the map $\kappa_G$ induces a bijection $\Lambda_G \longleftrightarrow \Omega_G$.  Given any $b \in G(F)$, we may thus view the \emph{Kottwitz point} $\kappa_G(b)$ as specifying an extended base alcove $\fa_\omega$ for some $\omega \in \Omega_G$. Equivalently, $\kappa_G(b)$ determines a coset in $Q^\vee /R^\vee \cong \Omega_G$.

By \cite[Cor.~7.2.2]{GHKRadlvs}, the restriction of the map $G(F) \rightarrow B(G)$ to $N_GT(F)$ factors through $\eW$, and the induced map $\eW \twoheadrightarrow B(G)$ is surjective.  We thus define the Newton map only on the extended affine Weyl group, where it has an especially elementary formula; see, for example, \cite[Sec.~4.2]{GoertzSurvey}. For $x = t^\lambda w \in \eW$ where $\lambda \in Q^\vee$ and the order of $w \in \sW$ equals $m$, the \emph{Newton point} for $x$ (viewed as an element of $G$) is given by
\begin{equation}\label{eq:avgFormula}
    \nu_G(x) = \left( \frac{1}{m} \sum\limits_{i=1}^m w^i(\lambda)\right)^+.
\end{equation}
In particular, a Newton point $\nu$ is an element of $\mathfrak{a}_{\Q}^+ := \left(X_*(T)\otimes_{\Z}\Q\right)^+$, which is a subset of the dominant Weyl chamber. As with an element of the integral coweight lattice, we say that $\nu \in \mathfrak{a}_{\Q}^+$ is \emph{regular} if and only if the stabilizer of $\nu$ in $\sW$ is trivial.

Choosing any representative $\dot{x} \in [b]$ such that $\dot{x} \in \eW$, and any representative $x \in G(F)$ of $\dot{x} \in N_GT(F)/T(\cO)$, the map
\begin{align}\label{eq:NewtonKottMap}
    B(G) & \rightarrow \mathfrak{a}^+_{\Q} \times \Lambda_G \nonumber \\
    [b] & \mapsto (\nu_G(x), \kappa_G(b))
\end{align}
is injective \cite[Sec.~4.13]{KotIsoII}. By abuse of notation, we often write $\nu_G(b)$ to denote the image of $[b]\in B(G)$ under the Newton map, without passing to a representative in $\eW$ or referencing its $\sigma$-conjugacy class.  When the context is clear and no further reference to the ambient group is required, we may also abbreviate $\nu_b :=\nu_G(b)$.

\begin{definition}\label{defn:integralNP}
Given any $[b] \in B(G)$, denote its Newton point by $\nu_b \in \mathfrak{a}_{\Q}^+$ and its Kottwitz point by $\omega_b \in \Lambda_G$. By definition, $\nu_b \in \bigoplus \Q\varpi_i$, where the coefficients on the fundamental coweights are all non-negative. If $\nu_b \in \bigoplus \Z \varpi_i$ and $\nu_b$ has the same image as $\omega_b$ in $Q^\vee/R^\vee$, then we say that $b$ has an \emph{integral} Newton point; equivalently, $\nu_b$ is an \emph{integral} Newton point.  Otherwise, we say that $b$ has a \emph{non-integral} Newton point; equivalently, $\nu_b$ is \emph{non-integral}.
\end{definition}

Although our definition of integrality formally requires the additional input of the Kottwitz point, note that if $Q^\vee = R^\vee$, then $\nu_b$ is integral if and only if $\nu_b$ is an element of the integral weight lattice.  Example \ref{ex:C2integral} below illustrates how the same Newton point (representing two different $\sigma$-conjugacy classes) can be both integral and non-integral, if more generally $Q^\vee \neq R^\vee$.  The primary motivation for splitting the cases as we have in Definition \ref{defn:integralNP} arises later in Proposition \ref{prop:stdReps}, in which our formulas for standard representatives fall precisely into these two distinct cases.

\begin{figure}[ht]
\centering
\begin{overpic}[width=0.6\textwidth]{C2StdReps}
\put(23,20.5){\footnotesize{\color{black}$\fa_0$}}
\put(13,28){\footnotesize{\color{red}$\alpha_2^\vee$}}
\put(25,4){\footnotesize{\color{red}$\alpha_1^\vee$}}
\put(23,29){\footnotesize{\color{DarkGreen}$\varpi_2$}}
\put(32,15){\footnotesize{\color{DarkGreen}$\varpi_1$}}
\end{overpic}
\caption{Newton points and their standard representatives in type $\tilde{C}_2$.}
\label{fig:C2StdReps}
\end{figure}

\begin{example}\label{ex:C2integral}
In Figure~\ref{fig:C2StdReps}, the coroot lattice $R^\vee = \Z^2$ for type $\tilde{C}_2$ is shown in black, and the lattice $\varpi_2 + R^\vee$ is shown in blue.
Consider $b = t^{3\varpi_2} \in \eW$ whose (extended) alcove $\bb$ is shaded teal.   The element $b$ has Newton point $\nu_b = \left( \frac{3}{2}, \frac{3}{2} \right) = 3\varpi_2$ by \eqref{eq:avgFormula}.
The translation part $3\varpi_2$ of $b$ is an element of the blue lattice, and so the element $b$ has Kottwitz point $\omega_b = \varpi_2$.  Since $2\varpi_2 \in R^\vee$, the Newton point $\nu_b$ maps to $\varpi_2 \in Q^\vee/R^\vee$, and so $\nu_b=3\varpi_2$ is integral by Definition \ref{defn:integralNP}.

On the other hand, consider $\mu = (2,1) = 2\alpha_1^\vee + 3\alpha_2^\vee$ and define $b' = t^{\mu}s_1 \in \eW$, which corresponds to a (non-extended) alcove $\bb'$ shaded dark blue, Again by \eqref{eq:avgFormula}, we have $\nu_{b'} = 3\varpi_2$. In particular, $b$ and $b'$ have the same Newton points.  However, since $\mu \in R^\vee$, then $\omega_{b'} = 0$, which is not equal to the image of $\nu_{b'}$ in $Q^\vee/R^\vee$.  Therefore, $\nu_{b'}=3\varpi_2$ is non-integral in this case. Figure \ref{fig:C2StdReps} is discussed in more detail later in Example \ref{ex:C2stdreps}.
\end{example}

We continue by reviewing the geometric method for associating a standard parabolic subgroup to a Newton point; see \cite[Sec.~7]{GHKRadlvs} for equivalent algebraic interpretations. Recall that the dominant Weyl chamber decomposes as $\Cf = \coprod \mathfrak{a}^+_P$, where $P$ ranges over the standard parabolic subgroups.

\begin{definition}\label{defn:assocParabolic} Given any $[b] \in B(G)$, denote the associated Newton point by $\nu = \nu_G(b)$.
The \emph{parabolic subgroup associated to $\nu$} is defined to be the unique standard parabolic subgroup $P_\nu$ such that $\nu \in \mathfrak{a}^+_P$. Equivalently, $P_{\nu} = MN$ is defined by the criterion
    \begin{equation}\label{eq:PnuRoots}
        \alpha_i \in \Delta_M  \iff  \nu \in H_{\alpha_i}.
    \end{equation}
\end{definition}

We now mention several important special cases which arise elsewhere in the literature; compare \cite[Ex.~11.3.6]{GHKRadlvs}, for example.

\begin{example}\label{ex:Pnu}
Let $[b] \in B(G)$, and denote by $\nu = \nu_G(b)$ the associated Newton point.
 \begin{enumerate}
     \item $P_{\nu} = B$ if and only if $\nu$ is regular.
 \item $P_\nu = G$ if and only if $\nu$ in the $\omega$-sheet projects onto the origin in the 0-sheet.
 \end{enumerate}
\end{example}

\subsection{Standard representatives for Newton points}\label{sec:StandardReps-General}

Denote by $B(G)_P$ the subset of $B(G)$ consisting of all elements $[b]$ such that $P_{\nu_b}$ equals $P$, and note that $B(G) = \coprod B(G)_P$, where $P$ runs over the set of all standard parabolic subgroups. In this section, we review the fact that the elements of $B(G)_P$ can be represented by certain convenient elements in the corresponding Levi subgroup $M(F)$.

An element $b \in G(F)$ is called \emph{basic} if its image under the Newton map $\nu_b$ factors through the center $Z(G)$ of $G$.  Fix a standard parabolic subgroup $P$ with Levi decomposition $P = MN$.  For any basic element $[b_m] \in B(M)$, the image under the Newton map with respect to $M$ satisfies $\nu_M(b_m) \in \mathfrak{a}_P$, although it may not additionally satisfy $\nu_M(b_m) \in \mathfrak{a}_P^+$.  Denote by $B(M)_{\bas}^+$ the subset of basic elements $[b_m] \in B(M)$ such that $\nu_M(b_m) \in \mathfrak{a}_P^+$.  Then by \cite[Sec.~5]{KotIsoII}, there is a bijection $B(M)_{\bas}^+ \longleftrightarrow B(G)_P.$

By \cite[Lemma 7.2.1]{GHKRadlvs}, the elements in $\Omega_M \subset M(F)$ are $M$-basic, and furthermore, the canonical map $\Omega_M \rightarrow B(M)_{\bas}$ is a bijection. Denote by $\Omega_M^+$ the subset of $\Omega_M$ which is in bijection with $B(M)_{\bas}^+$ under this map, so that
\begin{equation}\label{eq:StdRepMaps}
    \Omega_M^+ \longleftrightarrow B(M)_{\bas}^+ \longleftrightarrow B(G)_P.
\end{equation}
The following is a slight reformulation of \cite[Def.~7.2.3]{GHKRadlvs}.

\begin{definition}\label{defn:StdRep} Given any $[b] \in B(G)_P \subset B(G)$ associated to the Newton point $\nu = \nu_G(b)$, the element $b_{\nu} \in \Omega_M^+$ obtained from the bijections in \eqref{eq:StdRepMaps} is called the \emph{standard representative of $[b]$}.
\end{definition}

The next lemma simply rephrases the criteria characterizing the standard representative from the bijections in \eqref{eq:StdRepMaps} as a more convenient checklist, although we include a proof for the sake of completeness.

\begin{lemma}\label{lem:stdRepList}
Let $[b] \in B(G)_P$, and denote by $(\nu_b, \omega_b)$ the associated Newton and Kottwitz points. The element $b_\nu \in \eW$ is the standard representative of $[b]$ if and only if the following three conditions hold:
\begin{enumerate}
    \item $\kappa_G(b_\nu)=\omega_b$,
    \item $b_\nu \in \Omega_M$, and
    \item $\nu_M(b_\nu)=\nu_b$.
\end{enumerate}
In particular, these conditions imply that $b_\nu \in [b]$.
\end{lemma}

\begin{proof}
If $b_\nu$ is the standard representative of $[b]$, then in particular $b_\nu \in [b]$.  By \eqref{eq:NewtonKottMap}, we thus have $\nu_G(b_\nu) = \nu_b$ and $\kappa_G(b_\nu) = \omega_b$.  In particular, condition (1) holds.
Condition (2) is necessary by \eqref{eq:StdRepMaps}. In addition, $\nu_M(b_\nu) \in \mathfrak{a}_P^+$ is already $G$-dominant, and so $\nu_b = \nu_G(b_\nu) = \nu_M(b_\nu)$, which means that (3) holds as well.

Conversely, assume that the three above conditions hold.  Condition (2) guarantees that $b_\nu \in \Omega_M$ is an $M$-basic element. Condition (3) then implies that $\nu_M(b_\nu)=\nu_b$ is already $G$-dominant, and so $\nu_M(b_\nu) = \nu_G(b_\nu) = \nu_b$.  Together with condition (1), we thus have by \eqref{eq:NewtonKottMap} that $b_\nu \in [b]$.  Therefore, $[b_\nu] \in B(M)_{\bas}^+$ represents $[b]\in B(G)_P$, as required for $b_\nu$ to be the standard representative.
\end{proof}

We now record the standard representatives for $\sigma$-conjugacy classes having one of the two associated parabolic subgroups studied in Example \ref{ex:Pnu}; compare \cite[Ex.~11.3.6]{GHKRadlvs}.

\begin{example}\label{ex:stdReps} Let $[b] \in B(G)_P$.  Denote by $\nu = \nu_G(b)$ the Newton point, and let $b_\nu$ denote the standard representative for $[b]$.
\begin{enumerate}
\item If $P = B$, then $M=T\cong (\mathbb{G}_m)^n$ and $b_\nu$ is the translation $t^\nu=(t^{\varpi_1})^{\nu_1}\cdots (t^{\varpi_n})^{\nu_n} \in \Omega_M$, where $\nu = \sum \nu_i \varpi_i \mapsto (\nu_1, \dots, \nu_n)$ under the isomorphism $X_*(T) \cong \Z^n$.
\item If $P = G$, then $b_\nu = \id$ if $\nu=0$; otherwise $\nu=\nu_G(\omega)$ where $\omega = t^\eta v \in \Omega_G$ is such that $\eta \in Q^\vee$ projects onto a vertex of the base alcove $\fa_0$, and $b_\nu = \omega$.
\end{enumerate}
\end{example}

More generally, given $[b] \in B(G)_P$ with Newton point $\nu$, we shall typically express the standard representative as $b_\nu = t^\eta v \in \eW$, where $\eta \in Q^\vee$ and $v \in \sW_M$.   As discussed in \cite[Sec.~7]{GHKRadlvs}, the coweight $\eta$ is both $M$-dominant and $M$-minuscule, meaning that $\langle \alpha_i, \eta \rangle \in \{0,1\}$ for all $\alpha_i \in \Delta_M$.  The colored alcoves in Figure \ref{fig:C2StdReps} depict the standard representatives in type $\tilde{C}_2$, which are discussed in more detail in Example \ref{ex:C2stdreps} after we have explicit formulas.

\subsection{Affine Deligne--Lusztig varieties}\label{sec:ADLVs}

This section provides a brief review of the definition of affine Deligne--Lusztig varieties in the affine flag variety, together with the key results from \cite{GHKRadlvs} upon which our work builds.

Given any $x \in \eW$ and $b \in G(F)$, the \emph{affine Deligne--Lusztig variety} associated to this pair of elements is defined as
\[ X_x(b) = \{ g \in G(F)/I \mid g^{-1}b\sigma(g) \in IxI \}. \]
In contrast to their classical counterparts, affine Deligne--Lusztig varieties may be empty, and it remains an open problem to fully characterize the precise conditions which determine nonemptiness.

In \cite{GHKRadlvs}, both the nonemptiness problem and dimension calculations are rephrased in terms of intersections of $I$ and conjugates of $I_P$-orbits. Below we provide a slight reformulation of the main results from \cite[Sec.~11]{GHKRadlvs}, which we shall reinterpret later in Section \ref{sec:alcoveWalks}. We include a short proof for the sake of completeness, since the statements do not appear there in precisely this form.

\begin{thm}[\cite{GHKRadlvs}]\label{thm:GHKRThm}
Let $[b] \in B(G)_P$, and denote the corresponding Newton point by $\nu$ and the standard representative by $b_\nu$. For any $x \in \eW$, we have
\begin{enumerate}
\item $X_x(b) \neq \emptyset$ if and only if there exists $y \in \eW$ such that $Ix\fa_0 \cap I_P^y b_\nu^y \fa_0 \neq \emptyset$.
\item If $X_x(b) \neq \emptyset$, then
\begin{equation*}
\dim X_x(b) = \sup\limits_{y \in \eW} \left(\dim\left( Ix\fa_0 \cap I_P^y b_\nu^y \fa_0 \right)\right)-\langle 2\rho, \nu\rangle.
\end{equation*}
\end{enumerate}
\end{thm}

\begin{proof}
 Since $[b] \in B(G)_P$, we have by \cite[Eq.~(11.3.1)]{GHKRadlvs} that
\begin{equation*}
    \dim X_x(b_\nu) = \sup\limits_{z \in \eW} \left(\dim \left( X_x(b_\nu)\cap I_P z\fa_0 \right) \right).
\end{equation*}
Next, \cite[Thm.~11.3.1]{GHKRadlvs} says that the dimensions on the righthand side equal
\begin{equation*}
\dim(X_x(b_\nu) \cap I_Pz\fa_0) = \dim\left( Ix\fa_0 \cap I_P^{z^{-1}} b_\nu^{z^{-1}} \fa_0 \right)-\langle \rho, \nu_G(b_\nu)+\nu_M(b_\nu)\rangle.
\end{equation*}
By property (3) of Lemma \ref{lem:stdRepList}, since $b_\nu$ is the standard representative for $[b]$, then in fact $\nu_M(b_\nu) = \nu_G(b_\nu)=\nu$. Since $b_\nu \in [b]$, then $X_x(b) \cong X_x(b_\nu)$, in which case $\dim X_x(b) = \dim X_x(b_\nu)$.  Finally, note that the value $\langle 2\rho, \nu \rangle$ is independent of $z \in \eW$. The result now follows by replacing $z=y^{-1}$.
\end{proof}

\subsection{Galleries and chimneys}\label{sec:galleriesChimneys}

We direct the reader to~\cite{MST1} and to our more recent work~\cite{MNST} with Naqvi for all background concerning combinatorial galleries and chimneys. In this section, we briefly recall only those concepts which play a key role this paper.

Every choice of a sub-root system, and hence choice of a standard spherical parabolic subgroup $P$, determines a $P$-\emph{chimney} in the $0$-sheet as defined in \cite[Def.~3.1]{MNST}. While formally a collection of half-apartments in the $0$-sheet, chimneys are represented by \emph{$P$-sectors}, which are regions of $\cA$ lying between pairs of adjacent hyperplanes for all positive roots in the defining sub-root system associated to $P$, and which go off to infinity in the other root directions. An important special case occurs when the sub-root system is the empty set and $P=B$, in which case chimneys are maximal simplices in the spherical building at infinity, represented by parallel sectors in $\cA$; equivalently, by Weyl chambers. See \cite[Rmk.~3.2]{MNST} for more background on the geometric interpretation.

\begin{figure}[ht]
\centering
\begin{overpic}[width=0.5\textwidth]{chimneyOrientations}
\put(59,65){\color{blue}{$\gamma$}}

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\end{overpic}
\caption{For $i = 1$ and $2$, the coroot $\alpha_i^\vee$ and the orientation induced by the $P_1$-chimney on $\alpha_i$-hyperplanes are shown in red and green, respectively.  The orientation induced on $(\alpha_1^\vee + \alpha_2^\vee)$-hyperplanes is shown in black.}
\label{fig:chimneyOrientations}
\end{figure}

Recall that the $\omega$-sheets overlay each other such that the fundamental alcoves $\fa_\omega$ and the Weyl chambers coincide; see Figure \ref{fig:chimneyOrientations} for an illustration for $G=PGL_3$, where an extended alcove is pictured as a triangle together with a choice of a colored vertex. In particular, base alcoves $\fa_\omega$ and $\fa_{\omega'}$ for $\omega\neq\omega'$ both correspond to the same grey triangle, but are based at lattice elements of two different colors. We say that an extended alcove (or panel) \emph{sits above} or \emph{below} another extended alcove (or panel) if they project to the same alcove (or panel) in the 0-sheet. In Figure~\ref{fig:chimneyOrientations}, such alcoves (or panels) then correspond to the same simplex with a different base vertex. As for alcoves and panels, we may speak of walls, half-apartments, chimneys, and sectors as \emph{sitting above} or \emph{below} one another.

The affine Weyl group $\aW$ admits a natural action on the set of chimneys and on the set of sectors, which is induced by the action of $\aW$ on the set of half-apartments; see \cite[Def.~3.7]{MNST} for a formal definition. We extend this action to a left-action of $\eW$ as follows.  The image of a $P$-chimney under $\omega\in\Omega_G$ is the collection of half-apartments in the $\omega$-sheet which projects to the given $P$-chimney in the $0$-sheet; i.e.\ the $\omega$-image of the $P$-chimney is the chimney in the $\omega$-sheet sitting above it. For any given $y=z\omega \in\eW \cong \aW \rtimes \Omega_G$, the \emph{$(P,y)$-chimney} is defined to be the $y$-image of the $P$-chimney under this action; equivalently, the $z$-image inside the $\omega$-sheet of the chimney sitting above the $P$-chimney.

\begin{figure}[ht]
\centering
\begin{overpic}[width=0.5\textwidth]{chimneys}
\put(30,53){\color{red}$\alpha_2^\vee$}
\put(55,53){\color{red}$\alpha_1^\vee$}
\put(43,56){\footnotesize{\color{black}$\fa_0$}}
\put(36,53){\footnotesize{\color{black}$\s_1$}}
\put(50,53){\footnotesize{\color{black}$\s_2$}}
\put(35,46){\footnotesize{\color{black}$\s_1\s_2$}}
\put(49,46){\footnotesize{\color{black}$\s_2 \s_1$}}
\put(43,41){\footnotesize{\color{black}$\w_0$}}
\put(70,33){\footnotesize{\color{black}$(P_1,\id)$}}
\put(35,20){\footnotesize{\color{red}$(P_1,s_1)$}}
\put(11,64){\footnotesize{\color{Purple}$(P_1,s_2)$}}
\put(33,80){\footnotesize{\color{DarkBlue}$(P_1,s_1s_2)$}}
\put(11,35){\footnotesize{\color{DarkGreen}$(P_1,s_2s_1)$}}
\put(70,67){\footnotesize{\color{DarkOrange}$(P_1,w_0)$}}
\end{overpic}
\caption{A sector representing the $(P_1,u)$-chimney, for each $u \in \sW$.}
\label{fig:chimneys}
\end{figure}

We illustrate the action of $\sW$ on a chimney in Figure \ref{fig:chimneys}. For each $u \in \sW$ we have shaded a sector representing the $(P_1,u)$-chimney. More generally, for any $y = t^\mu u \in \eW$ with $\mu \in Q^\vee$ and $u \in \sW$,  a sector representing the $(P_1,y)$-chimney can be obtained by translating by $\mu$ any sector representing the $(P_1,u)$-chimney.  Thus, every $(P_1,y)$-chimney is represented by a sector which is a translate to the $\omega$-sheet of one of the shaded regions in Figure~\ref{fig:chimneys} for some $\omega \in \Omega_G$.

Every chimney in a given $\omega$-sheet induces an orientation on that sheet as defined in \cite[Def.~3.23]{MNST}.  Formally, an orientation is a map that assigns either $+$ or $-$ to every pair of an extended alcove and a panel contained in it.
Fix a chimney in the $\omega$-sheet, and denote by $\phi$ the orientation it induces. We extend $\phi$ to all other sheets of $\Omega_G \times \mathfrak{a}$ by assigning to a pair of an alcove in the $\omega'$-sheet and a panel contained in it the same value as for the pair sitting above or below it in the $\omega$-sheet. Figure~\ref{fig:chimneyOrientations} shows the orientation of pairs of extended alcoves and panels for all overlaying sheets together in one picture.
The \emph{opposite standard orientation} is the orientation on $\Omega_G \times \mathfrak{a}$ corresponding to $P=B$. For the rest of this section, we fix an orientation $\phi$ on all sheets of $\Omega_G \times \mathfrak{a}$, which is induced by a particular $(P,y)$-chimney.

A \emph{(combinatorial) gallery $\gamma$} is a sequence of (extended) alcoves $c_i$ and panels $p_i$, contained in $c_i$ and $c_{i-1}$. We say that $\gamma$ \emph{is contained in the $\omega$-sheet} if one (and hence all) of its alcoves is in the $\omega$-sheet.  The \emph{type} of $\gamma$ is the sequence in $\tilde S$ given by the types $s_{j_i}\in \tilde S$ of the panels $p_i$ in $\gamma$. More specifically, a minimal gallery $\gamma:\fa\rsa \x$ contained in the $\omega$-sheet has first alcove $\fa = \fa_\omega$ and end alcove $\x=x\fa_0$ for some $x = z\omega \in \eW \cong \aW \rtimes \Omega_G$. Writing the reduced expression $z=s_{i_1}s_{i_2} \cdots s_{i_k}\in \aW$ corresponding to the types in the 0-sheet of the panels sitting below those of $\gamma$, we have $x=s_{i_1}s_{i_2} \cdots s_{i_k}\omega\in \eW$. The type of $\gamma$ is then denoted by $\vec x = (s_{j_1}, s_{j_2}, \dots, s_{j_k})$, where $j_l = \omega(i_l)$ for all $1 \leq l \leq k$. There is thus a natural action of  $\eW \cong \aW \rtimes \Omega_G$ on the set of all combinatorial galleries. The subgroup $\aW$ acts in a type-preserving way on each individual sheet, while $\Omega_G$ permutes the sheets and acts on the type by the induced permutation action on the set of indices $\tilde I$.

A gallery is \emph{folded at $p_i$} if $c_i=c_{i-1}$, and we say that this fold is \emph{positive (with respect to the $(P,y)$-chimney)} if $c_i=c_{i-1}$ and the orientation $\phi$ assigns $+$  to the pair $(c_i, p_i)$. More generally, a gallery $\gamma$ is \emph{positively folded (with respect to the $(P,y)$-chimney)} if all of its folds are positive with respect to $\phi$. If $c_i \neq c_{i-1}$, the gallery has a \emph{crossing at $p_i$}.  A crossing is \emph{positive (with respect to the $(P,y)$-chimney)} if the orientation $\phi$ assigns $+$ to the pair $(c_i, p_i)$. For further details, see Section 2.2 in \cite{MNST}.
Positively folded galleries may be constructed using PRS-folding sequences, which the authors introduced in \cite[Def.~3.16]{MST1}. A \emph{PRS-folding sequence} is a sequence of galleries where the next entry in the sequence is obtained from the previous gallery by reflecting a shorter and shorter tail at a particular panel.

We further illustrate these concepts for the group $G=PGL_3$.
In Figure~\ref{fig:chimneyOrientations}, the shaded region is a sector representing the $P_1$-chimney, and we show using $+$ and $-$ signs the orientation induced by this chimney.  For each hyperplane, the $+$ signs are on the side of the hyperplane facing away from the ``point at infinity" corresponding to the chimney.  We also include in this figure an example of a gallery $\gamma$ which is positively folded with respect to the $P_1$-chimney.  Roughly speaking, this means that the blue gallery $\gamma$ is always ``folded away'' from the chimney.  Viewed as a gallery contained in the 0-sheet, the type of $\gamma$ is $(s_2,s_0,s_1,s_0,s_2,s_0,s_1,s_2,s_0,s_1,s_2,s_0,s_1,s_0,s_2,s_0)$. If we view $\gamma$ as a gallery contained in the $\varpi_1$-sheet based at the red vertex of $\fa_{\varpi_1}$, then the corresponding rotation of $\fa_0$ induces a 3-cycle on the set of indices $\tilde I = \{0,1,2\}$, and the type of $\gamma$ is instead $(s_1,s_2,s_0,s_2,s_1,s_2,s_0,s_1,s_2,s_0,s_1,s_2,s_0,s_2,s_1,s_2)$.


TITLE: How many triples satisfy $ab + bc + ca = 2 + abc $
QUESTION [1 upvotes]: $a^2 + b^2 + c^2 - \frac{a^3 + b^3 + c^3 - 3abc}{a+b+c} = 2 + abc$
  How many triples $(a,b,c)$ satisfies the statement? Here $a,b,c > 1$.

It is easy to simplify the statement to
$$ab + bc + ca = 2 + abc.$$
But now how to proceed I don't know. I think this is somehow related to stars and bars theorem but I don't know how to convert this to that problem. Any hint will be helpful.
Source this is a problem from BdMO 2015 Dhaka regional.

REPLY [1 votes]: I'm also a Bangladeshi, reading in class IX in DRMC. Probably the problem was:

How many solution triads $(a,b,c)$ are there for the equation $a^2 + b^2 + c^2 - \frac{a^3 + b^3 + c^3 - 3abc}{a+b+c} = 2 + abc?$ Where $(a,b,c)$ are positive integers and $a,b,c>1$.

After trying a lot, I've got a simple solution. Suppose that, $2≤a≤b≤c$.
Note that,
$$ab+bc+ca-2<3bc$$
i.e.
    $$abc<3bc$$      So,  $a=2$
Set the value of $a$. Therefore,
$$2b+2c = 2+bc$$
Now,     $2b+2c=2+bc>bc$
Hence,      $(b-2)(c-2)<4$
Solve the equation. Finally, $(a,b,c)=(2,3,4)$ is the only solution.


text:
\begin{definition}[Definition:Initial Point of Vector]
Let $\mathbf u$ be a [[Definition:Vector Quantity|vector quantity]] [[Definition:Arrow Representation of Vector Quantity|represented by an arrow]] $\vec {AB}$ embedded in [[Definition:The Plane|the plane]]:
:[[File:Vector.png|300px]]
The point $A$ is the '''initial point''' of $\mathbf u$.
\end{definition}

config: wiki


TITLE: Alternating Current/Electromototive Force equaling 0?
QUESTION [0 upvotes]: All alternating current diagrams are drawn as sinusoidal waves that intercept with x axis. Given our usual sine wave that equals to 0 at 180° it would mean the alternating current would equal to 0. If we are generating alternating current by creating electromototive force with Faraday's law of induction, a change in magnetic flux is needed. If we use a wire plate spinning at 'w' constant angular speed and send magnetic field lines through it we get our emf and alternating current. However, I can't understand how alternating current can equal to 0 since if the wire plate is contantly spinning there would always be a 'rate of change' in magnetic flux.

REPLY [1 votes]: Is a "wire plate" different from a wire loop in some way?
If you envision a circular wire loop of area $A$ spinning with a diameter of the loop along the $\hat{z}$ direction as the axis of rotation, with a constant magnetic field $\vec{B}$ in the $\hat{y}$ direction, and we take the normal vector to the loop of wire $\vec{n}$, where $\vec{n} = cos(\omega t)\ \hat{x} + sin(\omega t)\ \hat{y}$, then the magnetic flux $\Phi$ is given by
$$\Phi = \vec{B} \cdot A\ \vec{n}$$
$$\Phi = A\ B\ \hat{y} \cdot (cos(\omega t)\ \hat{x} + sin(\omega t)\ \hat{y})$$
$$\Phi = A\ B\ sin(\omega t)$$
Then using Faraday's law, the emf $\epsilon$ is
$$\epsilon = - A\ B\ \omega\ cos(\omega t)$$
Which is indeed zero twice per rotation.
Intuitively, you need only think about when the flux is changing the fastest or slowest.  The flux is changing the fastest when the flux is zero (the loop is facing perpendicular to the field).  The flux is changing slowest when the loop is facing the field or facing away, and in fact at those two moments the flux momentarily stops changing; the emf drops to zero.  It's similar to the way that when you throw a ball straight up, the ball's position momentarily stops changing right at the top of the trajectory.
It might help to consider that when the loop is facing directly away from the magnetic field (the point when the flux is greatest), the flux goes form "almost highest flux" to "highest flux" to "almost highest flux".  Since the flux goes up, then down, it must instantaneously stop changing.  That's the point when the rate of change of the flux is zero.


TITLE: The intersection of any set with the empty set is empty
QUESTION [1 upvotes]: I know this may be routine and simple, but I'm not great at proofs and was wondering if someone could look at my solution and offer corrections or tips?
If A and B are sets, prove that
$ A  \cap \emptyset = \emptyset. $
My proof:
Let $x \in A \cap \emptyset $. Then x $\in A$ and $x \in \emptyset$. Thus x $\in$ $\emptyset$.
Let x $\in \emptyset$. Because $\emptyset \subset A $, x $\in$ A. Thus x $\in A \cap \emptyset$

REPLY [2 votes]: It certainly works. Alternately, if you have proved that $\emptyset$ is a subset of every set and that $A\cap B\subseteq B$ regardless of $A,B,$ then you're basically done. There are other ways to proceed, as well, of course.


TITLE: A question regarding uniform integrability
QUESTION [1 upvotes]: We know that if $(h_n)$ is a sequence of nonnegative integrable functions on a measurable subset $E$ of $\mathbb R$ with finite measure such that $h_n\to 0$ pointwise a.e., then $\lim\limits_{n\to \infty}\int\limits_E h_n dm=0$ iff $(h_n)$ is uniformly integrable.
I want to have an example of non-uniformly integrable $(h_n)$, where $h_n$'s are not nonnegative and $h_n\to 0$, $\lim\limits_{n\to \infty}\int\limits_E h_n dm=0$. This question is from Royden's Real Analysis. Any hint will be appreciated.

REPLY [1 votes]: Consider $E=(-3,3)$, and for $n=1,2, \ldots$ define the functions
\begin{align}h_n(x) :=
  \begin{cases}
-2n - n^2x, \; \text{ if } x \in \left[-\frac{2}{n}, \,-\frac{1}{n} \right] , \\
                                n^2x , \;  \text{ if } x \in \left[-\frac{1}{n} , \, \,\frac{1}{n}\right] ,  \\
                                   2n - n^2x , \;  \text{ if } x \in \left[\frac{1}{n} , \,\frac{2}{n}\right], \\
    0, \; \text{ if } x \in \left(-3, -\frac{2}{n}\right) \cup \left(\frac{2}{n}, 3 \right). \end{cases}
\end{align}
Notice $\lim\limits_{n\to \infty} h_n(x)=0$ and $\lim\limits_{n\to \infty}\int\limits_{E} h_n dm=0$. Let $\delta>0$ be given. By the Archimedean Property, we may find a positive integer $N$ such that $N \delta > 4$. So we have that $m\left( \left[-\frac{2}{N}, \frac{2}{N} \right]\right)< \delta$ but notice $\int_{-\frac{2}{N}}^{\frac{2}{N}} \left| h_n \right| dm=2$ whenever $n \geq N$*. Hence the sequence of functions $\{h_n\}_{n=1}^\infty$ is not uniformly integrable.
*Note: Technically it suffices to observe that $\int_{-\frac{2}{N}}^{\frac{2}{N}} \left| h_N \right| dm=2$ since
$\neg \left(\forall \varepsilon\left(\varepsilon>0 \implies \exists \delta
\left( \delta>0 \land \forall n \forall A \left(m(A)<\delta \implies \int_A |\,h_n| dm < \varepsilon \right)\right) \right) \right)$
$\iff \exists \varepsilon \left( \varepsilon>0 \land \forall \delta \left( \delta>0 \implies \exists N \exists A \left(m(A)<\delta \land \int_A |\,h_N| dm \geq \varepsilon\right)\right) \right)$.


TITLE: Counting points and lines in a plane
QUESTION [4 upvotes]: Let $P_1$ be a set of 4 points in the Euclidean plane.  Formally, $P_1$ determines a set $L_1$ of 6 lines, which then determine only 3 points not already in $P_1.$  Let $P_2$ be the set of 7 points thus far determined.  Formally, $P_2$ determines only 3 lines not already in $L_1.$  Let $L_2$ be the set of 9 lines thus far determined.  Continuing in this manner determines sets $P_n$ and $L_n.$  What can be said about the cardinalities $|P_n|$ and $|L_n|$?

REPLY [7 votes]: The question is discussed in Cooper, Joshua; Walters, Mark; Iterated point-line configurations grow doubly-exponentially, Discrete Comput. Geom. 43 (2010), no. 3, 554–562, MR2587837 (2011f:51016). I quote from the review by Tamas Szonyi:
Consider a quadrangle as the starting configuration (stage one). In each stage the process considers the lines joining pairs of points that were constructed in the previous stages and adds the (affine) intersections of these lines. The starting quadrangle has to be in such a general position that no lines in any stage are parallel.  The main results of the paper give doubly exponential upper and lower bounds on the number of points at each stage.
A preprint is also available on arXiv. See also https://oeis.org/A140468 and https://oeis.org/A243707. https://arxiv.org/abs/1406.5157 is entertaining.


TITLE: Negating a statement
QUESTION [1 upvotes]: State in words the negation of the following sentence: For every martian M, if M is green, then M is tall and ticklish.
I got the right answer to this, give or take a few words, but this is a question of form more than anything. After converting this statement to symbols and negating everything, I come up with: $\exists M(P \wedge (\neg \text{Tall } \vee \neg\text{ Ticklish})$ and so in word format that would be:

There exists a martian such that it is
  green and not tall or not ticklish.

However the really correct answer is:

There is a martian M such that M is
  green but M is not tall or M is not
  ticklish.

The difference between these two is a 'but' and an 'and'. Does this mean anything mathematically? Is my version correct?

REPLY [4 votes]: Your answer is "really correct". It is perhaps more usual in English to use "but" rather than "and" in a sentence as the one in your example. Formally (i.e., mathematically), there is no difference.


text:
\subsection{Lower bound on the likelihood of bias}

We now prove Proposition~\ref{prop:lower-bias},
on the limits on the extent to which
Theorem~\ref{lem:main-bias} can be improved.
The argument is analogous to that
of~\cite{DBLP:journals/cc/Ben-EliezerHL12} for the case of
$\mathbb{F}_2$.

\begin{proof}[Proof of Proposition~\ref{prop:lower-bias}]
We follow the same division of variables $[\sampledim]$ into parts
$L$ and $R$ with $|L|=\lfloor\frac{n}{d}\rfloor$ and
$|R|=\sampledim'=\lceil \sampledim(1-1/d)\rceil$ and $d'=d-1$ that
we used for the upper bound on the bias.
Define $\mathcal{L}$ to be the set of all polynomials in $\Pdm$ whose
monomials are from the set
$\mathcal{M}'\subseteq \monomial{p}{d}{\sampledim}$ (defined earlier)
that have degree 1 on $L$ and degree at most $d-1$ on $R$.
By Corollary~\ref{prop:mprime-bound}, there is some constant
$\gamma>0$ such that for sufficiently large $\sampledim$,
$|\mathcal{M}'|\ge \gamma\cdot |\monomial{p}{d}{\sampledim}|$ and
hence $|\mathcal{L}|\ge p^{\gamma |\monomial{p}{d}{\sampledim}|}$.
Therefore, we have
$|\mathcal{L}|/|\Pdm|\ge p^{-(1-\gamma)|\monomial{p}{d}{\sampledim}|}$.

Now consider the expected bias of polynomials in $\mathcal{L}$:
We can write $f$ chosen uniformly from $\mathcal{L}$ uniquely as
$$f(x)=\sum_{i\in L} x_i\cdot g_i(x_R)$$
where the $g_i$ are independently chosen polynomials over monomials
$\monomial{p}{d-1}{\sampledim'}$ on $R$.

For $j\in \mathbb{F}^*_p$,
\[\begin{split}
\E_{f\in_R \mathcal{L}}\ \biasj(f)=&
\E_{f\in_R \mathcal{L}}\ \E_{x\in_R \mathbb{F}_p^\sampledim}\  \omega^{j\cdot f(x)}\\
=&\E_{f\in_R \mathcal{L}}\
\E_{x_L\in_R \mathbb{F}_p^L}\ \E_{x_R\in_R \mathbb{F}_p^R }\
\omega^{j\cdot f(x)}\\
=&\E_{x_L\in_R \mathbb{F}_p^L}
\E_{x_R\in_R \mathbb{F}_p^R}\E_{f\in_R \mathcal{L}}\ \omega^{j\cdot f(x)}.
\end{split}
\]
Now with probability $p^{-|L|}$, all the $x_i$ for $i\in L$ are 0
and every $f\in \mathcal{L}$ evaluates to 0, so $\E_{f\in_R \mathcal{L}}\ \omega^{j\cdot f(0^L,x_R)}=1$.
With the remaining probability, $x_L\ne 0^L$ and hence there is some
$i \in L$ and $b_i\ne 0$ such that $x_i=b_i$.
For $f$ chosen at random from $\mathcal{L}$, for each fixed value of
$x_L=b_L$ with $b_i\ne 0$, we have
$$f(b_L,x_R)= b_i g_{i0} + f'(x_R)$$ where
$g_{i0}$ is the constant term of the polynomial $g_i$ and is chosen
independently of $f'$. Since $g_{i0}$ is uniformly chosen from
$\mathbb{F}_p$ for random $f$ in $\mathcal{L}$
and since $b_i\ne 0$,
$b_i g_{i0}$ is also uniformly chosen from $\mathbb{F}_p$.
Further,
since $g_{i0}$ is independent of $f'$, for every fixed $x_R$,
the value $\E_{f\in_R \mathcal{L}} \omega^{j\cdot f(b_L,x_R)}=0$.
Therefore,
$\E_{f\in_R \mathcal{L}} \biasj(f)=p^{-|L|}$.
Now since $|\biasj(f)|\le 1$,
we obtain
$$\Pr_{f\in_R \mathcal{L}}[\;|\biasj(f)| \ge p^{-|L|}/2\;]\ge p^{-|L|}/2.$$
Therefore
$$\Pr_{f\in_R \Pdm} [|\biasj(f)|\ge p^{-|L|}/2] \ge \frac{|\mathcal{L}|}{|\Pdm|}\cdot p^{-|L|}/2\ge p^{-c'|\monomial{p}{d}{\sampledim}|}$$
for some $c'<1$ since $|L|\ll
|\monomial{p}{d}{\sampledim}|$.
Since $|L|=\lfloor n/d \rfloor\ge 2$, we obtain $p^{-|L|}/2>p^{-c''n/d}$
for some constant $c''>0$ as required.
\end{proof}

config: arxiv


TITLE: Is there theoretical limit on the minimal achivable focal length?
QUESTION [2 upvotes]: There seems to be a limit, around 0.33, on the fastest possible aperture of a photographic lens made of glass. Is there a theoretical limit on the minimal focal length of a photographic lens (rectangular projection, not a fisheye)? Are the physical reasons that would prevent a, say, 1 mm focal length lens for a full frame sensor?
I also asked this question on Photography SE, but didn't get quite the answer I was looking for.

REPLY [2 votes]: Just some thoughts...From the Wikipedia article Lens, we have the Lensmaker's Equation for a single lens:

where f is the focal length, n is the material index of refraction, d is the thickness and R1 and R2 are the radii of curvature. Let's for simplicity consider a plano-convex lens, with R2 infinite. The equation then reduces to f = R1/(n-1). So smaller and smaller f can be achieved with smaller and smaller radius of curvature, which implies smaller and smaller aperture since you can't have a big lens with a small radius. There doesn't appear to be any theoretical limit here.
Now if we are talking about a very short focal length photographic lens, it would have to be positioned very close to the sensor in order to achieve focus, which I would think would lead to handling and mounting issues.


TITLE: transitive relation question
QUESTION [2 upvotes]: I need to show if the following relation is transitive :
$$
R\subseteq \mathcal{P}(\mathbb{N})\times\mathcal{P}(\mathbb{N}) \space \text{with} \space XRY \space :\Leftrightarrow \exists x \in\mathbb{N}:\space x\in X \space \land \space x\in Y
$$
The answer says that it is not transitive because there is:
$$ \{1,2\}R\{2,3\}\space \text{and} \space \{2,3\}R\{3,4\} \space \text{but}\space \space \text{not} \space
 \{1,2\} \cap \{3,4\}=\emptyset $$
what does intersection have to do with it?

REPLY [1 votes]: When you say you have a relation $R$ on a set "A" you're basically saying that $R \subseteq A \times A$ that follows some specific conditions:
$$R = \{(a_1,a_2) \in A \times A: \text{such that $a_1$ and $a_2$ satisfy a condition}\}$$
In your case you have $R \subseteq \mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N})$ and as you might be aware $\mathcal{P}(\mathbb{N})$ is the set of all subsets of natural numbers so every element $a \in \mathcal{P}(\mathbb{N})$ is necessarily a set of natural numbers.
Thus, in this case our relation is set subset of $\mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N})$ given by
$$R = \{(A,B) \in \mathcal{P}(\mathbb{N}) \times \mathcal{P}(\mathbb{N}):  \text{such that the sets $A$ and $B$ satisfy a condition}\}$$
In this case the condition for $(A,B) \in R$ (in other words $ARB$) is that theres is at least a natural number $x$ that is simultaneously in $A$ and in $B$ therefore in order to $(A,B) \in R$ we must have $$ A \cap B \neq \emptyset$$
You want to prove that $\{1,2\}R\{3,4\}$ but this only happens if and only if {1,2} and {3,4} have elements in common and they certainly don't: $$\{1,2\} \cap \{3,4\} = \emptyset$$
Therefore $R$ is not transitive because you found a counter-example in $\mathcal{P}(\mathbb{N})$.


TITLE: Mastermind question with $1$ repeated color
QUESTION [1 upvotes]: I came across a combinatorics problem related to the game Mastermind. Here's the question:

Suppose your first guess uses $3$ different colors (one color is repeated) and
  its score is one black and $3$ whites. How many different secret codes are
  possible?

You score a black peg if you get both, the color and position right. You score a white peg if you get only the color right, but in the wrong position. There are $6$ possible colors to choose from, and $4$ positions to place your chosen colors in. Colors can be repeated, such as in this question.
So in the question above, we have $3$ colors and $4$ pegs. Now if one of these $4$ pegs is in the correct position and the other 3 are correct pegs but wrongly arranged, it's just a matter of rearranging them. So in how many ways can you do that?

REPLY [1 votes]: At least one of the duplicate colour pegs must be in its correct position as otherwise the other two pegs would have to take their places and there couldn't be a black peg.
Once we fix one of the duplicate pegs, the other peg can take either of the two remaining positions, and this already determines the rest. Henc ethere are four possible secret codes (two choices of correct peg times  two rearrangements of the rest).
Concretely, an inital guess of $aabc$ allows only $abca$, $acab$, $baca$, $caab$.


TITLE: solving for exponent of quadratic?
QUESTION [0 upvotes]: I have the following problem:
$$(2^x - 4) \cdot (2^x - 2) = 0$$
I really have no idea how to introduce logarithms to this to solve it...
It expands to:
$$2^{2x} - 2^{x + 1} - 2^{x+2}  + 8 = 0$$
How do I turn that into logarithms?

REPLY [2 votes]: Hint: Instead of expanding, use the fact that $$ab = 0 \iff a = 0 \text{ or } b = 0$$ That is, consider separately
$$2^x - 4 = 0$$
and $$2^x - 2 = 0$$


TITLE: L is a regular language. Is $L′ = \{w \in L: |w| \bmod2=1\}$ a regular language?
QUESTION [0 upvotes]: Let $L$ be a regular language.
$r$ is a regular expression that $L(r)=L$.
I'm trying to prove that $L′ = \{w \in L: |w| \bmod2=1\}$  is a regular language too.
To prove that language is a regular language I should prove 3 things:

There exists $r_1$ and $r_2$
such that $r = ((r_1) \cup (r_2))$
There exists $r_1$ and $r_2$
such that $r=((r_1)∘(r_2))$
There exists $r_1$ and $r_2$
such that $r = (r_1)^∗$

I'm having a problem to prove those 3 things.
Thank you for the help.

REPLY [2 votes]: Note that the language $L_1  =\{w: |w| \equiv 1 \pmod 2\}$ is regular. Now let $D$ be a finite state machine that recognizes $L$, and let $D_1$ be a macine that recognizes $L_1$. You can then construct a "product machine" $D' = D\otimes D_1$ (see below), which recognizes $L' = L\cap L_1$. This proves that the intersection of two regular languages is a regular language.
Given two finite state machines $A, B$, their product $A\otimes B$ works the following way (what follows is basically a formalisation of "$A$ and $B$ runs in parallel"): Its states are ordered pairs $(a, b)$ of states $a$ from $A$ and $b$ from $B$. If $a_i, a_a$ are the initial and accepting states of $A$, and analoguously $b_i, b_a$ for $B$, then $(a_i, b_i)$ is the initial state and $(a_a, b_a)$ is the accepting state of $A\otimes B$.
If the mahine is in state $(a_1, b_1)$ and recieves input $x$, then it transitions into state $(a_2, b_2)$, where $a_2$ is the state that machine $A$ transitions into if it is in state $a_1$ and recieves input $x$, and similarily for $b_2$.
If $A$ and $B$ are both machines that accepts some regular language, say $L_A$ and $L_B$, then $A\otimes B$ accepts their intersection $L_A\cap L_B$. To see this, let's take a word $w$ and look at two cases:

$w\in L_A\cap L_B$. In this case, the characters in the word $w$, if given to machine $A$, will take it to the state $a_a$. It will also take machine $B$ to the state $b_a$. By definition of $A\otimes B$, it will take the product machine to state $(a_a, b_a)$, so it is accepted.
$w \notin L_A \cap L_B$. We may WLOG assume that $w \notin  L_A$. This means that if we give the word $w$ to machine $A$, it will not end up in the accepting state $a_a$. This also means that if we give $w$ to the product machine, it cannot end up in the accepting state $(a_a, b_a)$. Thus the product machine does not accept $w$.


TITLE: Show matrix transformation, finding kernel & range of it
QUESTION [0 upvotes]: Let $a=(a1,a2,a3)$ be a fixed vector in $\Bbb R^3$. Define the cross product $a \times v$ of $a$ and another vector $v=(v_1,v_2,v_3) \in \Bbb R^3$ as $$a \times v = \det\begin{bmatrix}
        e_1 & e_2 & e_3 \\
        a_1 & a_2 & a_3 \\
        v_1 & v_2 & v_3 \\
        \end{bmatrix}$$
Define a function $T: \Bbb R^3 \to \Bbb R^3$ by $T(v) = a \times v$ for $v \in \Bbb R^3$.
a) Show that $T$ is a matrix transformation and calculate its representing matrix $M$
b) Find $\operatorname{ker}(T)$ and interpret its answer geometrically
c) find $\operatorname{range}(T)$ and interpret its answer geometrically

im really lost on this question. how do you find matrix M? i guess in order to find KerT you have to find matrix M first, and then find null space of it. and rangeT.. i dont know about that either..

REPLY [1 votes]: $T:\Bbb R^3 \to \Bbb R^3$ is given by $T(\mathbf v) = \mathbf a \times \mathbf v = \det\begin{bmatrix}
        \mathbf e_1 & \mathbf e_2 & \mathbf e_3 \\
        a_1 & a_2 & a_3 \\
        v_1 & v_2 & v_3 \\
        \end{bmatrix}$ where $\{\mathbf e_1, \mathbf e_2, \mathbf e_3\}$ are the standard basis vectors for $\Bbb R^3$.  For instance, $\mathbf e_3 = (0,0,1)$.
Because $T$ is a linear transformation, if we can figure out how it transforms any basis then we can figure out how it transforms any vector -- and more importantly for this question, what its standard matrix is.
So let's see how $T$ transforms the standard basis (being the easiest one to check, usually).
$$\begin{align}T(\mathbf e_1) = \mathbf a \times \mathbf e_1 &= \det\begin{bmatrix} \mathbf e_1 & \mathbf e_2 & \mathbf e_3 \\ a_1 & a_2 & a_3 \\ 1 & 0 & 0 \end{bmatrix} \\ &= \det\begin{bmatrix}a_2 & a_3 \\ 0 & 0\end{bmatrix}\mathbf e_1 - \det\begin{bmatrix}a_1 & a_3 \\ 1 & 0\end{bmatrix}\mathbf e_2 + \det\begin{bmatrix}a_1 & a_2 \\ 1 & 0\end{bmatrix}\mathbf e_3 \\ &= a_3\mathbf e_2 -a_2\mathbf e_3\end{align}$$
Likewise, you can confirm for yourself that $$T(\mathbf e_2) = -a_3\mathbf e_1+a_1\mathbf e_3 \\ T(\mathbf e_3) = a_2\mathbf e_1 -a_1\mathbf e_2$$
Then we just need to find the matrix $M$ such that $M\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix} = \begin{bmatrix} 0 \\ a_3 \\ -a_2\end{bmatrix}$ and the similar expressions for the other basis vectors.
But $M\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}$ is just the first column of $M$ (Can you see why?).  Thus $$M=\begin{bmatrix} M\begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix} & M\begin{bmatrix} 0 \\ 1 \\ 0\end{bmatrix} & M\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}\end{bmatrix} = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0\end{bmatrix}$$

Here's how I'd find the kernel and range of $T$.
From the properties of the determinant we can see that $\mathbf a\times \mathbf v=\mathbf 0$ iff $\mathbf a \| \mathbf v$.  Thus $\ker(T) = \Bbb R^3$ if $\mathbf a=\mathbf 0$ or $\ker(T) = \operatorname{span}(\mathbf a)$ if $\mathbf a\ne \mathbf 0$.
Then we can use the rank-nullity theorem to get an idea of the range of $T$.  If $\mathbf a=\mathbf 0$ then we can immediately see that $\operatorname{range}(T) = \{\mathbf 0\}$.  If $\mathbf a\ne \mathbf 0$, then the range is $2$-dimensional.
Let's assume $\mathbf a\ne \mathbf 0$ and consider the dot product of $\mathbf a\times \mathbf v$ with $\mathbf a$.
$$\mathbf a \cdot (\mathbf a\times \mathbf v) = (a_1\mathbf e_1 + a_2\mathbf e_2 + a_3 \mathbf e_3)\cdot \left(\det\begin{bmatrix}a_2 & a_3 \\ v_2 & v_3\end{bmatrix}\mathbf e_1 - \det\begin{bmatrix}a_1 & a_3 \\ v_1 & v_3\end{bmatrix}\mathbf e_2 + \det\begin{bmatrix}a_1 & a_2 \\ v_1 & v_2\end{bmatrix}\mathbf e_3\right) \\ = \det\begin{bmatrix} a_1 & a_2 & a_3 \\ a_1 & a_2 & a_3 \\ v_1 & v_2 & v_3\end{bmatrix} = 0$$
Thus $T(\mathbf v)\ \bot\ \mathbf a$.  But we know that $\operatorname{span}(\mathbf a)^\bot$ is $2$-dimensional, thus this must be the range of $T$.
Geometrically speaking this means that $\ker(T)$ is the line parallel to $\mathbf a$ and $\operatorname{range}(T)$ is the plane orthogonal to that line.


TITLE: What does it imply when two lines cross each other in a level curve
QUESTION [0 upvotes]: I'm having trouble to articulate how the actual graph looks like when z = 0, meaning in the middle of the level curve where intersection occurs.
What shape would the graph look like and what curvature is it supposed to have?

REPLY [0 votes]: This hyperbolic paraboloid has has a pair of horizontal lines crossing at the central saddle point. That's a level set, but not a level curve.
A level set is the intersection of a surface with a plane.  If the surface is smooth that will be a one dimensional object in the plane. Sometimes it will be a single curve in the usual sense - then you would call it a level curve. But it might be an intersection of lines or curves, or come in several pieces. Then calling it a level curve is a mild abuse of language, but will not confuse anyone.

https://commons.wikimedia.org/wiki/File:Hyperbolic-paraboloid.svg


text:
\section{Union of Interiors and Boundary Equals Whole Space}
Tags: Boundaries, Set Interiors

\begin{theorem}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$S = \Int A \cup \partial A \cup \Int {A'}$
where:
:$A' = S \setminus A$ denotes the [[Definition:Relative Complement|complement of $A$ relative to $S$]]
:$\Int A$ denotes the [[Definition:Interior (Topology)|interior]] of $A$
:$\partial A$ denotes the [[Definition:Boundary (Topology)|boundary]] of $A$.
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \Int A \cup \partial A \cup \Int {A'}
      | r = \Int A \cup \Int {A'} \cup \partial A
      | c = [[Union is Associative]], [[Union is Commutative]]
}}
{{eqn | r = \Int A \cup \Int {A'} \cup \paren {\map \cl A \cap \map \cl {A'} }
      | c = [[Boundary is Intersection of Closure with Closure of Complement]]
}}
{{eqn | r = \paren {\Int A \cup \Int {A'} \cup \map \cl A} \cap \paren {\Int A \cup \Int {A'} \cup \map \cl {A'} }
      | c = [[Intersection Distributes over Union]]
}}
{{eqn | r = \paren {\Int A \cup \paren {\map \cl A}' \cup \map \cl A} \cap \paren {\Int A \cup \Int {A'} \cup \map \cl {A'} }
      | c = [[Complement of Interior equals Closure of Complement]]
}}
{{eqn | r = \paren {\Int A \cup \paren {\paren {\map \cl A}' \cup \map \cl A} } \cap \paren {\Int A \cup \Int {A'} \cup \map \cl {A'} }
      | c = [[Union is Associative]]
}}
{{eqn | r = \paren {\Int A \cup S} \cap \paren {\Int A \cup \Int {A'} \cup \map \cl {A'} }
      | c = [[Union with Relative Complement]]
}}
{{eqn | r = \paren {\Int A \cup S} \cap \paren {\Int A \cup \Int {A'} \cup \paren {\Int A}'}
      | c = [[Complement of Interior equals Closure of Complement]]
}}
{{eqn | r = \paren {\Int A \cup S} \cap \paren {\Int A \cup \paren {\Int A' \cup \Int {A'} } }
      | c = [[Union is Associative]], [[Union is Commutative]]
}}
{{eqn | r = \paren {\Int A \cup S} \cap \paren {S \cup \Int {A'} }
      | c = [[Union with Relative Complement]]
}}
{{eqn | r = S \cap \paren {S \cup \Int {A'} }
      | c = [[Union with Superset is Superset]]
}}
{{eqn | r = S \cap S
      | c = [[Union with Superset is Superset]]
}}
{{eqn | r = S
      | c = [[Intersection is Idempotent]]
}}
{{end-eqn}}
{{qed}}
\end{proof}


TITLE: The cube length of a vector
QUESTION [1 upvotes]: The form
$$
{\bf r} \cdot {\bf r} = \left|{\bf r}\right|^2 = \sum_{i,j} r_i r_j {\bf e}_i \cdot {\bf e}_j = \sum_{i,j} r_i r_j \delta_{ij} = \sum_i r_i r_i
$$
is defined via a bilinear form that takes two vectors and outputs a real number with certain conditions. This quantity is usually referred to as the square of a vector.
I have encountered here forms such as ${\bf r}^3, {\bf r}^4, ...$. Now ${\bf r} \cdot {\bf r} \equiv {\bf r}^2$ and is $SO(3)$ invariant. How does one show ${\bf r}^3$ is not $SO(3)$ invariant and consequently only even powers are invariant under $SO(3)$.
The question is tantamount to: How is ${\bf r}^j$ ($j>2$) defined?

REPLY [1 votes]: You want $\mathbf{r}^{j+k}=\mathbf{r}^j\otimes\mathbf{r}^k$ for a suitable definition of $\otimes$, right? The definition$$\mathbf{r}^{2k}=(\mathbf{r}\cdot\mathbf{r})^k,\,\mathbf{r}^{2k+1}=\mathbf{r}^{2k}r$$works (if $\otimes$ multiplies scalars by scalars and vectors in the usual ways, and takes the dot product of two vectors). This is the standard definition of $\mathbf{r}^n$. However, since $\mathbf{r}^{2k}=r^{2k}$ with $r:=|\mathbf{r}|$, $\mathbf{r}^{2k}$ is often simply written as $r^{2k}$.


TITLE: A subset of $\mathbb{C}^{n+1}$
QUESTION [0 upvotes]: I have the following topology question. Let $U$ be an open subset of $\mathbb{C}^n$. We consider the subset $U_1$ of $\mathbb C^{n+1} \colon U_1=\{a(1,z):a \in \mathbb{C}−\{0\},z \in U\}$. Could someone give me an elementary proof that $U_1$ is open or a counterexample if this not the case.

REPLY [1 votes]: A point $(z,w)\in \mathbb C\times \mathbb C^n$ belongs to $U_1$ if and only if $z\ne 0$ and $w/z\in U$. Therefore, $U_1$ is the preimage of $U$ under the map $(z,w)\mapsto w/z$ which is continuous on $(\mathbb C\setminus\{0\})\times \mathbb C$.


TITLE: What function does $f(x)=\sum_{k=0}^\infty k^3(x-2)^k$ represent?
QUESTION [0 upvotes]: $$f(x)=\sum_{k=0}^\infty k^3(x-2)^k$$
I am supposed to find the $f(x)$ that this Taylor polynomial represents. How do I do this? I've tried using standard polynomials and I've tried differentiating those for $(1-x)^{-1}$ but I haven't been able to get a proper function out of it.

REPLY [1 votes]: Try differentiating,
$$f(x)=\sum_{k=0}^{\infty} (x-2)^k$$
$$=\frac{1}{1-(x-2)}$$
If it converges.
Then multiply both sides by $x-2$ and repeat.