TITLE: Using linearization to calculate the thickness of a layer of paint on a spherical ball
QUESTION [1 upvotes]: The volume of a sphere with radius $r$ is given by the formula $V(r) = \frac{4 \pi}{3} r^3$.
a) If $a$ is a given fixed value for $r$, write the formula for the linearization of the volume function $V(r)$ at $a$.
b) Use this linearization to calculate the thickness $\Delta r$ (in $cm$) of a layer of paint on the surface of a spherical ball with radius $r=52cm$ if the total volume of paint used is $340cm^3$.
The first part is easy to calculate, but I don't know exactly how to get the second part?

REPLY [0 votes]: For second part by logarithmic differentiation of volume/radius
$$\dfrac{\Delta V}{V}=3\dfrac{\Delta r}{r}$$
$$\dfrac{340}{\dfrac{4\pi}{3} 52^3}=3\dfrac{\Delta r}{52}$$
from which we calculate the paint thickness $\Delta r \approx 0.01 $ cm.
The same would be the case if instead $a=r$ would represent the side of a cube.


TITLE: A textbook for numerical analysis (with solution manual)
QUESTION [3 upvotes]: This semester I'm studying numerical analysis in university. Our textbook is Numerical Analysis by David.R. Kincaid and E. Ward. Cheney. It is a very good and strong book in numerical analysis with a lot of good exercises but the problem is that it doesn't have any solution manual and I cannot check whether I solved problems true or false. As the main important thing in this course is the ability of solving problem, I'm looking for a good textbook in numerical analysis with strong exercises and also with solution manual.
furthermore, I know that Kincaid's book 6th edition has solution manual but our textbook is its 1991 edition which doesn't have any solution manual and its exercises are completely different with 6th edition. Also the 6th edition of this book as I know doesn't have as many questions as its 1991 edition has, so it is not a good book for me.
Any help would be appreciated.

REPLY [3 votes]: You can read "An introduction to Numerical Analysis" by David.F.Mayers. Also another would be "Numerical Methods" by Dahlquist and Bjorck which I think has solutions.


TITLE: Limit of a sequence. Help finding the epsilon
QUESTION [0 upvotes]: So I'm trying to find the limit of the sequence $n$/($n^2$ + $1$). So I need to prove that
$|$ $n$ / ($n^2$ + 1) | < ε .
I'm having trouble finding the n for when this is true. How do I simply the equation so that I can find n in terms of ε? Thanks.

REPLY [2 votes]: Note that $\dfrac{n}{n^2+1} \leq \dfrac{n}{n^2}=1/n.$ So choose $n$ such that $ n \geq 1/ \epsilon$


text:
\section{Properties of Hadamard Product}
Tags: Hadamard Product

\begin{theorem}
Let $\map {\MM_S} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $S$ over an [[Definition:Algebraic Structure|algebraic structure]] $\struct {S, \cdot}$.
Let $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$.
Let $\mathbf A \circ \mathbf B$ be defined as the [[Definition:Hadamard Product|Hadamard product]] of $\mathbf A$ and $\mathbf B$.
The operation $\circ$ of [[Definition:Hadamard Product|Hadamard product]] satisfies the following properties:
:$\circ$ is [[Definition:Closure (Abstract Algebra)|closed]] on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is [[Definition:Closure (Abstract Algebra)|closed]] on $\struct {S, \cdot}$
:$\circ$ is [[Definition:Associative|associative]] on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is [[Definition:Associative|associative]] on $\struct {S, \cdot}$
:$\circ$ is [[Definition:Commutative Operation|commutative]] on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is [[Definition:Commutative Operation|commutative]] on $\struct {S, \cdot}$.
\end{theorem}

\begin{proof}
=== [[Closure of Hadamard Product]] ===
{{:Closure of Hadamard Product}}
=== [[Associativity of Hadamard Product]] ===
{{:Associativity of Hadamard Product}}
=== [[Commutativity of Hadamard Product]] ===
{{:Commutativity of Hadamard Product}}
[[Category:Hadamard Product]]
9o8hi3biogmwh2i4ezufn49x1df57f7
\end{proof}

config: wiki


TITLE: Higgs self-coupling
QUESTION [0 upvotes]: The measurement of the Higgs particle mass at the LHC has made the determination of the value of the Higgs self-coupling possible: with $M_H \approx 125$ GeV, one gets $\lambda \approx 1/8$.
Does this particular value imply anything special for the Standard Model, except that now books can be filled with the values of $M_H$ and $\lambda$?

REPLY [2 votes]: Does this particular value imply anything special for the Standard Model, except that now books can be filled with the values of MH and λ?

The standard model is an ongoing model fitting  current at the time experimental data. It has been continuously updated since its inception when new data invalidate it. So, from  Cosmas Zachos' link a diagram like this

needs very high energy experiments, in order for two on mass Higgs to appear from a virtual one,  the mass of the Higgs being so large. Such experiments in addition to refining the value of $λ$ will also:

Higgs pair production provides a unique handle for measuring the strength of the Higgs self interaction and constraining the shape of the Higgs potential.

Physics history shows that when the energy of experiments is increased, new and unexpected phenomena can also appear, so the future will tell how useful $λ$ is.


text:
\section{One-Shot Optimization of Multi-Step Trees}
\label{sec:OST}
In this section, we describe our multi-step lookahead acquisition function, a differentiable, tree-based approximation to $v_k(x\given\data)$. We then propose a one-shot optimization technique for effectively optimizing the acquisition function and extracting a first-stage decision.
\subsection{Multi-Step Trees}
\label{subsec:OST:MST}

Solving the $k$-step problem requires recursive maximization and integration over continuous domains:
\begin{equation}
    v_k(x\given \data)
    =
    v_1(x \given \data)
    +
    \E_y\Big[\max_{x_2} \Big\{v_1(x_2 \given \data_1) +
    \E_{y_2}\big[ \max_{x_3} \big\{ v_1(x_3 \given \data_2) +

    \cdots  \Big].
    \label{eq:recursive_max_and_exp}
\end{equation}

Since under a \gp surrogate, these nested expectations are analytically intractable (except the last step for  \ei), we must resort to numerical integration.

If we use Monte Carlo integration, this essentially means building a discrete scenario tree (Figure \ref{fig:decision_tree}), where each branch in a node corresponds to a particular \emph{fantasized} outcome drawn from the model posterior, and then averaging across scenarios. Letting $m_t$, $t=1,\dots, k-1$ denote the number of \emph{fantasy samples} from the posterior in step $t$,

we have the approximation
\[
    \bar{v}_k(x\given \data)
    =
    v_1(x \given \data)
    +
    \frac{1}{m_1}\sum_{j_1=1}^{m_1} \Big[\max_{x_2} \Big\{v_1(x_2 \given \data_1^{j_1}) +
    \frac{1}{m_2}\sum_{j_2=1}^{m_2} \big[ \max_{x_3} \big\{ v_1(x_3 \given \data_2^{j_1 j_2}) +
    \cdots  \Big].
\label{eq:multistagesaa}
\]

where $\data^{j_1}_1 = \data \cup \{(x, y^{j_1}) \}$,
$\data^{j_1\dotsc j_t}_t = \data^{j_1\dotsc j_{t-1}}_{t-1} \cup \{(x_{t}^{j_1\dotsc j_{t-1}}, y_t^{j_1\dotsc j_{t}})\}$, with fantasy samples $y_t^{j_1\dotsc j_{t}} \sim p(y_t \given x_{t}^{j_1\dotsc j_t}, \data^{j_1\dotsc j_{t-1}}_{t-1})$.

As the distribution of the fantasy samples depends on the query locations $x, x_1, x_2, \ldots$, we cannot directly optimize $\bar{v}_k(x\given\data)$. To make $\bar{v}_k(x\given\data)$ amenable to optimization, we leverage the re-parameterization trick \cite{kingma2013reparam, wilson2018maxbo} to write $y = h_{\data}(x, z)$, where $h_{\data}$ is a deterministic function and $z$ is a random variable independent of both $x$ and $\mathcal{D}$. Specifically, for a \gp posterior, we have $h_{\data}(x, z) = \mu_{\data}(x) + L_{\data}(x) z$,

where $\mu_{\data}(x)$ is the posterior mean, $L_\data(x)$ is a root decomposition of the posterior covariance $\Sigma_\data(x)$ such that $L_\data(x)L_\data^T(x) = \Sigma_\data(x)$, and $z \sim \mathcal{N}(0, I)$.
Since a \gp conditioned on additional samples remains a \gp, we can perform a similar re-parameterization for every dataset $\data^{j_1\dotsc j_{t}}_t$ in the tree.
We refer to the $z$'s as \emph{base samples}.

\tikzset{
  decision/.style = {minimum size=1.5em,shape=rectangle,draw,yshift=-0.5em},

  empty/.style = {draw=none}
}
\begin{figure}
    \centering
\begin{tikzpicture}[
      grow                    = right,
      sibling distance        = 2.1em,
      level distance          = 5.5em,
    ]
    \node [decision] {$\data$}
    child { node [empty] {$x$}
        child { node [empty] {$z_1^3$}
            child { node [empty] {$\cdots$ }
            }
        }
        child { node [empty] {$z_1^2$}
            child { node [empty] {$x^2_2$}
                child { node [empty] {$z_2^{2,3}$}
                    child { node [empty] {$x^{2,3}_3$}
                        child { node [empty] {$\dots$} }
                        child { node [empty] {$\cdots$} }
                        child { node [empty] {$\cdots$} }
                    }
                }
                child { node [empty] {$z_2^{2,2}$}
                    child { node [empty] {$x^{2,2}_3$}
                        child { node [empty] {$\cdots$} }
                        child { node [empty] {$\cdots$} }
                        child { node [empty] {$\cdots$} }
                    }
                }
                child { node [empty] {$z_2^{2,1}$}
                    child { node [empty] {$x^{2,1}_3$}
                        child { node [empty] {$\cdots$} }
                        child { node [empty] {$\cdots$} }
                        child { node [empty] {$\dots$} }
                    }
                }
            }
        }
        child { node [empty] {$z_1^1$}
            child { node [empty] {$\cdots$}
            }
        }
    };
  \end{tikzpicture}
      \caption{Illustration of the decision tree with three base samples in each stage.}
    \label{fig:decision_tree}
\end{figure}

\subsection{One-Shot Optimization}
\label{subsec:OST:OSO}
Despite re-parameterizing $\bar{v}_k(x\given\data)$ using base samples, it still involves nested maximization steps. Particularly when each optimization must be performed numerically using sequential approaches (as is the case when auto-differentiation and gradient-based methods are used), this becomes cumbersome. Observe that conditional on the base samples, $\bar{v}_k$ is a \emph{deterministic} function of the decision variables.

\begin{proposition}
\label{prop:oneshot}
Fix a set of base samples and consider $\bar{v}_k(x\given\data)$. Let $x_t^{j_1\dots j_{t-1}}$ be an instance of $x_t$ for each realization of $\data_{t-1}^{j_1\dots j_{t-1}}$ and let
\begin{multline}
    x^*, \mathbf{x}^*_2, \mathbf{x}^*_3, \dots, \mathbf{x}^*_k
    =
    \argmax_{x, \mathbf{x}_2, \mathbf{x}_3, \dots, \mathbf{x}_k} \Biggl\{
    v_1(x \given \data)
    +
   \frac{1}{m_1}\sum_{j_1=1}^{m_1} v_1(x_2^{j_1} \given \data_1^{j_1})
   + \cdots
   +  \\
    \frac{1}{\prod_{\ell=1}^{k-1} m_{\ell}}
    \sum_{j_1=1}^{m_1}
    \cdots
    \sum_{j_{k-1}=1}^{m_{k-1}}
    v_1(x_k^{j_1\cdots j_{k-1}} \given \data_{k-1}^{j_1\cdots j_{k-1}})  \Biggr\}, \label{eq:one-shot-objective}
\end{multline}
where we compactly represent $\mathbf{x}_2 = \{x^{j_1}_2 \}_{j_1=1\dots m_1}$,
$\mathbf{x}_3 = \{x^{j_1 j_2}_3\}_{j_1=1\dots m_1, j_2=1\dots m_2}$, and so on.

Then,

$x^* = \argmax_x \bar{v}_k(x \given \data)$.
\end{proposition}
Proposition~\ref{prop:oneshot} suggests that rather than solving a nested optimization problem, we can solve a joint optimization problem of higher dimension and subsequently extract the optimizer. We call this the \emph{one-shot multi-step} approach. A single-stage version of this was used in \cite{balandat2019botorch} for optimizing the Knowledge Gradient (\kg) acquisition function \citep{wu2016parallel}, which also has a nested maximization (of the posterior mean). Here, we generalize the idea to its full extent for efficient multi-step \bo.

\section{Fast, Differentiable, Multi-Step Fantasization}
\label{sec:fastdifffantasies}
Computing the one-shot objective~\eqref{eq:one-shot-objective}
requires us to repeatedly condition the model on the fantasy samples as we traverse the tree to deeper levels. Our ability to solve multi-step lookahead problems efficiently is made feasible by linear algebra insights and careful use of efficient batched computation on modern parallelizable hardware.

Typically, conditioning a \gp on additional data in a computationally efficient fashion is done by performing rank-1 updates to the Cholesky decomposition of the input covariance matrix.
In this paper, we develop a related approach, which we call \emph{multi-step fast fantasies}, in order to efficiently construct fantasy models for \gpytorch~\citep{gardner2018gpytorch} \gp models representing the full lookahead tree. A core ingredient of this approach is a novel linear algebra method for efficiently updating \gpytorch's \LOVE caches~\citep{pleiss2018love} for posterior inference in each step.

\subsection{Background: Lanczos Variance Estimates}
\label{subsec:FastFantasies:Background}

We start by providing a brief review of the main concepts for the Lanczos Variance Estimates (\LOVE) as introduced in~\cite{pleiss2018constant}. The \gp predictive covariance between two test points $x^*_i$ and $x^*_j$ is given by:
\[
  k_{f\mid\dset}(x^*_i, x^*_j) = k_{x^*_i x^*_j} - \bk_{\X \! x^*_i}^\top (K_{\X\!\X} + \Sigma)^{-1} \bk_{\X \! x^*_j},
\]
$X = (x_1, \dotsc, x_n)$ is the set of training points, $K_{XX}$ is the kernel matrix at~$X$, and $\Sigma$ is the noise covariance.\footnote{Typically, $\Sigma = \sigma^2 I$, but other formulations, including heteroskedastic noise models, are also compatible with fast fantasies described here.}

\LOVE achieves fast (co-)variances by decomposing $K_{\X\!\X} + \Sigma = RR^{\top}$ in $\bigo{r\nu(K_{\X\!\X})}$ time, where $R \in \mathbb{R}^{n\times r}$ and  $\nu(K_{\X\!\X})$ is the time complexity of a matrix vector multiplication $K_{\X\!\X}v$. This allows us to compute the second term of the predictive covariance as:
\[
  k_{f\mid\dset}(x^*_i, x^*_j) = k_{x^*_i x^*_j} - \bk_{\X \! x^*_i}^\top R^{-\top}R^{-1} \bk_{\X \! x^*_j},
\]
where $R^{-1}$ denotes a pseudoinverse if~$R$ is low-rank.\footnote{Additional approximations can be performed when using Spectral Kernel Interpolation (SKI), which result in constant time predictive covariances. For simplicity, we only detail the case of exact \gps here.}

The main operation to perform is decomposing $\tilde{K}_{\X\!\X} = RR^{\top}$, where $\tilde{K}_{\X\X} := K_{\X\!\X} + \Sigma \in \mathbb{R}^{n \times n}$.
Computing this decomposition can be done from scratch in $\bigo{nr^{2}}$ time. After forming~$R$, additional $\bigo{nr^{2}}$ time is required to perform a \QR decomposition of~$R$ so that a linear least squares systems can be solved efficiently (i.e., approximate $R^{-1}$). $R$ and its \QR decomposition are referred to as the \emph{\LOVE cache}.

\subsection{Fast Cache Updates}
\label{subsec:FastFantasies:FastCache}

If $R$ were a full Cholesky decomposition of $\tilde{K}_{\X\X}$, it could be updated in $\bigo{n^2}$ time using well-known procedures for rank 1 updates to Cholesky decompositions. This is advantageous, because computing the Cholesky decomposition requires $\bigo{n^3}$ time. However, for dense matrices, the \LOVE cache requires only $\bigo{n^2r}$ time to compute. Therefore, for it to be useful to multi-step lookahead, we must demonstrate that it can be updated in $o(n^2)$ time.

Updating the \LOVE caches is in particular complicated by the fact that $R$ is not necessarily triangular (or even square). Therefore, unlike with a Cholesky decomposition, updating $R$ itself in linear time is not sufficient, as recomputing a \QR decomposition of $R$ to update the pseudoinverse $R^{\dagger}$ would itself take quadratic time. In the Appendix \ref{appdx:sec:FastFantasies}, we demonstrate that the following proposition is true:
\begin{proposition}
Suppose $(K_{\X\!\X} + \Sigma)^{-1}$ has been decomposed using LOVE into $R^{-\top}R^{-1}$, with $R^{-1} \in \mathbb{R}^{n \times r}$. Suppose we wish to augment $\X$ with $q$ data points, thereby augmenting $K_{\X\!\X}$ with $q$ rows and columns, yielding $K_{\hat{\X}\!\hat{\X}}$. A rank $r+q$ decomposition $\hat{R}^{-1}$ of the inverse, $\hat{R}^{-\top}\hat{R}^{-1} \approx (K_{\hat{\X}\!\hat{\X}} + \Sigma)^{-1}$, can be computed from $R$ in $\bigo{nrq}$ time.
\end{proposition}

\begin{figure*}[t]
    \begin{minipage}[c]{0.45\columnwidth}
        \centering
        \includegraphics[width=\textwidth]{figures/CPU_FF_eval.pdf}\\[-1.5ex]
        \caption{\acro{CPU} times for constructing fantasy model and evaluating its posterior at a single point (variance negligible relative to the mean).

        }
        \label{fig:ParallelHardware:FastFantasies:WallTimes2}
    \end{minipage}
    \hfill
    \begin{minipage}[c]{0.528\columnwidth}
        \begin{algorithm}[H]
        \small
        \SetKwFunction{step}{\textsc{Step}}
        \SetKwFunction{sample}{\textsc{sample}}
        \SetKwFunction{stageval}{\textsc{value}}
        \SetKwFunction{correlate}{\textsc{correlate}}
        \SetKwFunction{fantasize}{\textsc{fantasize}}

        \stageval{$\mathcal{M}_t$, $\bm X_t$, $\mathcal{D}_{t-1}$}:\\
        \Indp
        $y^*_{t-1} = \max_{(x, y)\in \mathcal{D}_{t-1}} y$\\
        \Return $\mathbb{E}_{y\sim \mathcal{M}_t(\bm X_t)}\bigl[(y - y^*_{t-1})^+ \bigr]$
        \BlankLine
        \Indm

        \step{$\alpha_t$, $\mathcal{M}_t$, $\bm X_{t:k}$, $\bm Z_{t:k}$, $\mathcal{D}_{t-1}$}:\\
        \Indp
        $\alpha_{t+1}$ = $\alpha_t$ + \stageval{$\mathcal{M}_t$, $\bm X_t$, $\mathcal{D}_{t-1}$}\\
        \If{$t = k-1$}{
            \Return $\alpha_{t+1}$
        }
            $Y_t$ = \correlate{$\mathcal{M}_t(\bm X_t)$, $\bm Z_t$}\\
            $\mathcal{M}_{t+1}$ = \fantasize{$\mathcal{M}_t$, $\bm X_t$, $\bm Y_t$}\\
            $\mathcal{D}_{t} = \mathcal{D}_{t-1} \cup \{(\bm X_t, \bm Y_t)\}$\\
            \Return \step{$\alpha_{t+1}$, $\mathcal{M}_{t+1}$, $\bm X_{t+1:k}$, $\bm Z_{t+1:k}$, $\mathcal{D}_t$}\\

        \caption{Multi-Step Tree Evaluation}
        \label{algo:OST:FullAlgo}
        \end{algorithm}

    \caption{
    The recursive procedure for evaluating multi-step trees by repeatedly sampling from the posterior
     (\texttt{\textsc{correlate}}), conditioning (\texttt{\textsc{fantasize}}),
    and evaluating stage-values (\texttt{\textsc{value}}). }
    \end{minipage}

\end{figure*}

\subsection{Multi-Step Fantasies and Scalability}
Our other core insight is that the different levels of the lookahead tree can be represented by the batch dimensions of batched \gp models; this allows us to exclusively use batched linear algebra tensor operations that heavily exploit parallelization and hardware acceleration for our computations.

This optimized implementation is crucial in order to scale to non-trivial multi-step problems.

Algorithm~\ref{algo:OST:FullAlgo}
shows our recursive implementation of~\eqref{eq:one-shot-objective}.\footnote{Here $\mathcal{M}_t(\bm x_t)$ denotes the posterior of the model $\mathcal{M}_t$ evaluate at $\bm x_t$, and $\bm X_{t:k} := \{\bm x_i\}_{i=t}^k$ and $\bm Z_{t:k} := \{\bm z_i\}_{i=t}^k$ are collections of decision variables and base samples for lookahead steps~$t$ through~$k$, respectively. \mbox{\textsc{correlate}($\mathcal{M}_t(\bm X_t)$, $\bm Z_t$)} generates fantasy samples by correlating the base samples $\bm Z_t$ via the model posterior $\mathcal{M}_t(\bm X_t)$, and \mbox{\textsc{fantasize}($\mathcal{M}_t$, $\bm X_t$, $\bm Y_t$)} produces a new fantasy model (with an additional batch dimension) by conditioning on the fantasized observations.

To compute $k$-step one-shot lookahead conditional on base samples $\bm Z_{0:k}$ at decision variables $\bm X_{0:k}$, we simply need to call \textsc{Step}($0$, $\mathcal{M}$, $\bm X_{0:k}$, $\bm Z_{0:k}$, $\mathcal{D}$).}

Using reparameterization, we retain the dependence of the value functions in all stages on $x, \mathbf{x}_2, \dots, \mathbf{x}_k$, and can auto-differentiate through Algorithm~\ref{algo:OST:FullAlgo}.

Figure~\ref{fig:ParallelHardware:FastFantasies:WallTimes2} compares the overall wall time (on a logarithmic scale) for constructing fantasy models and performing posterior inference, for both standard and fast fantasy implementations. Fast fantasies achieve substantial speedups with growing model size and number of fantasies (up to 16x for $n=1024$, $m=128$). In Appendix~\ref{appdx:sec:FastFantasies}, we show that speedups on a \acro{GPU} are even higher.

\section{Special Instances of Multi-Step Trees}
\label{sec:pseudo}

The general one-shot optimization problem is of dimension $d + d \sum_{t=1}^{k}\prod_{i=1}^t m_i$,

which grows exponentially in the lookahead horizon~$k$. Therefore, in practice we are limited to relatively small horizons $k$ (nevertheless, we are able to show experimental results for up to $k=4$, which has never been considered in the literature). In this section, we describe two alternative approaches that have dimension only linear in~$k$ and have the potential to be even more scalable.

\textbf{Multi-Step Path.} Suppose only a single path is allowed in each subtree rooted at each fantasy sample $y^{j_1}, j_1=1,\dots, m_1$, that is, let $m_t=1$ for $t\geq 2$, then the number of variables is linear in $k$ and $m_1$. An even more extreme case is further setting $m_1=1$, that is, we assume there is only one possible future path.
When Gauss-Hermite (\gh) quadrature rules are used (one sample is exactly the mean of the Gaussian), this approach has a strong connection with \emph{certainty equivalent control} \citep{bertsekas2017dynamic}, an approximation technique for solving optimal control problems.
It also relates to some of the notable batch construction heuristics such as Kriging Believer \citep{ginsbourger2010kriging}, or \acro{GP-BUCB} \citep{desautels2014parallelizing}, where one fantasizes (or ``hallucinates'' as is called in \cite{desautels2014parallelizing}) the posterior mean and continue simulating future steps as if it were the actual observed value.
We will see that this degenerate tree can work surprisingly well in practice.

\textbf{Non-Adaptive Approximation.}

If we approximate the adaptive decisions after the current step by non-adaptive decisions,

the \emph{one-shot} optimization would be
\begin{gather}
   \SwapAboveDisplaySkip
    \max_{x, X^{(1)}, \dots, X^{(m_1)} }
    v_1(x\given \data)
    +
    \frac{1}{m_1} \sum_{i=1}^{m_1} V^{k-1}_1(X^{(i)} \given \data^{(i)}_1),  \label{eq:one-shot-ensbo}
\end{gather}
where we replaced the adaptive value function $v_{k-1}$ by the one-step batch value function $V^{k-1}_1$ with batch size $k-1$, i.e., $|X| = k-1$.
The dimension of \eqref{eq:one-shot-ensbo} is $d + m_1 (k-1) d$.
Since non-adaptive expected utility is no greater than the adaptive expected utility, \eqref{eq:one-shot-ensbo} is a lower bound of the adaptive expected utility.
Such non-adaptive approximation is actually a proven idea for \emph{efficient nonmyopic search} (\ens) \citep{jiang2017efficient, jiang2018efficient}, a problem setting closely related to \bo.
We refer to~\eqref{eq:one-shot-ensbo} as \eno, for \emph{efficient nonmyopic optimization}.
See Appendix \ref{app:lbub} for further discussions of these two special instances.

config: arxiv


text:
\begin{document}
\pagestyle{fancy}
\thispagestyle{empty}

\lhead{\textsf{A Filippov Framework for an Extended Budyko Model\rhead \today}}

\vspace{2mm}

\hrule

\vspace{3mm}
\noindent {\Large \textbf{\textsf{A Filippov Framework for an Extended Budyko Model}}}

\vspace{3mm}

\noindent \textbf{A.M. Barry, E. Widiasih, and R. McGehee}

\vspace{3mm}

\noindent\today

\vspace{3mm}

\hrule

\vspace{1cm}

\begin{abstract}

In latitude-dependent energy balance models, ice-free and ice-covered conditions form physical boundaries of the system.  With carbon dioxide treated as a bifurcation parameter, the resulting bifurcation diagram is nonsmooth with curves of equilibria and boundaries forming corners at points of intersection.  Over long time scales, atmospheric carbon dioxide varies dynamically and the nonsmooth diagram becomes a set of quasi-equilibria.  In this article, we extend an energy balance model to include slowly varying carbon dynamics in a way that allows for a nonsmooth dynamical systems treatment.  In our analysis within this framework, we prove existence and forward uniqueness of solutions and show that the physical region is forward invariant with sliding motion on the physical boundaries.
\end{abstract}

\section{Introduction}

There is significant evidence that large glaciation events took place during the Proterozoic era (2500-540 million years ago).  In particular, this evidence points to the existence of glacial formations at low latitudes, see the review articles \cite{hoffmanschrag2002terranova, pierrehumbert2011climate} and the references therein. One theory on the exodus from such an extreme climate was put forward by Joseph Kirschvink \cite{kirschvink1992protero}, who advocated that there was accumulation of greenhouse gases in the atmosphere, e.g. CO$_2$.  His theory purports that during a large glaciation chemical weathering processes would be shut down, thus eliminating a CO$_2$ sink.  Moreover, volcanic activity would continue during the glaciated state.  The combination of these effects would \textit{slowly} lead to enough build-up of atmospheric carbon dioxide to warm the planet and start the melting of the glaciers. Once a melt began, a deglaciation would follow \textit{rapidly} due to ice-albedo feedback.

There has been a wealth of modelling work on ``snowball'' events ranging from computationally intensive global circulation models (GCMs) to low dimensional conceptual climate models (CCMs).  In 1969, Mikhail Budyko and William Sellers independently proposed energy balance models (EBMs); these were CCMs capturing the evolution of the temperature profile of an idealized Earth \cite{budyko2010effect, sellersglobal}. Many others, for example \cite{ caldeira1992susceptibility, abbot2011, north1975theory}, have followed in the footsteps of Budyko and Sellers and used similar conceptual models capable of exhibiting snowball events. The low dimensionality of CCMs allows for a dynamical systems analysis, and hence a deeper investigation into some of the key feedbacks such as greenhouse gas and the ice-albedo effect.

From the point of view of dynamical systems, many of these early works share a similar theme by focusing on a bifurcation analysis with respect to the \textit{radiative forcing} parameter, one that depends on changes in atmospheric CO$_2$ and other greenhouse gas levels.  The reader may find figures similar to those displayed in Figure \ref{icf} in earlier works \cite{abbot2011, hoffmanschrag2002terranova, pollard2005snowball}.  These figures illustrate the glaciation state of an Earth with symmetric ice caps in terms of atmospheric CO$_2$ i.e. ice line latitude at $90^o$ means Earth is ice free while $0^o$ means Earth is fully glaciated. In both figures, the effect of the radiative forcing due to CO$_2$ and other greenhouse gases is treated statically as a parameter in a simple bifurcation analysis.  Mathematically, the bifurcation analysis already extends beyond the realm of smooth dynamical systems. On one hand, the bifurcation diagrams are obtained conventionally, with the bold curves indicating the stable branches and the dotted curves showing the unstable ones.   On the other hand, the extreme states (ice-free and ice-covered) are not true equilibria of the system, but treated as such in the literature as evidenced by the labeling of "ice-free branch" and "ice-covered branch". Indeed, the extreme states of the ice line are special because they serve as physical boundaries for the dynamics: the ice line cannot extend beyond the pole nor, because of the north-south symmetry assumed in the models, can it move downward of the equator. Therefore, mathematical models of snowball earth must reflect this physical imposition and be treated with a nonsmooth systems perspective.

Since glacial extent varies over time and dynamic processes such as chemical weathering affect the level of atmospheric CO$_2$, the bifurcation diagrams in Figure \ref{icf}  can be viewed as phase planes with dynamic variables consisting of the glacier extent (ice line) and  radiative forcing due to greenhouse gases. In particular, if a global glaciation did occur, and Kirschvink's argument about the accumulation of atmospheric  CO$_2$ held, then we should expect orbits of this dynamical system to traverse the extreme ice latitudes, i.e. the equator and the pole.  Following Kirschvink's idea further, we treat greenhouse gas effects on the energy balance by incorporating a slow CO$_2$ variable and treat the ice latitude as a relatively fast variable. The goal of the present article is to analyze such an interplay using a nonsmooth systems framework, thus providing mathematical support for Kirschvink's hypothesis.  In this work, we treat the bifurcation diagram (similar to those in Figure \ref{icf}) as a set of quasi-steady states of a slow-fast system and utilize the theory of Filippov.

 \begin{figure}[ht]
  \centering
   \subfloat[Figure 10 from Abbot, Voigt, and Koll \cite{abbot2011} ]  {\includegraphics[width=0.4\textwidth]{fig1-abbot-2011-Aeta.pdf} {\label{fig11}} }  \quad \quad
   \subfloat [Figure 6 from Hoffman and Schrag \cite{hoffmanschrag2002terranova}]  {\includegraphics[width=0.4\textwidth]{fig1-terranova.pdf} {\label{fig12}}} \quad
  \caption{Bifurcation diagrams from energy balance models illustrating hysteresis in the climate system.  In each figure, solid lines correspond to stable steady states while dashed lines correspond to unstable steady states.  The positive horizontal axis can be thought of as increasing atmospheric carbon dioxide, and the vertical axis is the latitude of the ice line.  Stability of snowball and ice-free states is \textit{inferred}; these are physical boundaries and not true equilibria of the models.}
\label{icf}
 \end{figure}

We present the topics as follows.  In the next section, we motivate the model of interest.  In Section \ref{sect:fil}, we propose an extension of the model that is amenable to a nonsmooth dynamical systems treatment.  In particular, we use Filippov's theory for differential equations with discontinuous right-hand sides to show that an ice line model based on a latitude-dependent EBM coupled with a simple equation for greenhouse gas evolution can be embedded in the plane to form a system that has unique forward-time solutions.  We end Section \ref{sect:fil} with an analysis of the system dynamics and conclude the paper with a discussion in Section \ref{sect:discussion}.

\section{The Equations of Motion}
\subsection{The Budyko Energy Balance Model and the Ice Line Equation}
The Budyko EBM describes the evolution of the annual temperature profile $T=T(y,t)$, where $t$ denotes time and $y$ denotes the sine of the latitude. The governing equation may be written:

\begin{equation} \label{bud}
R \frac{\partial }{\partial t}T(y,t) =  Qs(y)(1-\alpha(\eta,y))-(A+BT(y,t))+C \left( \int_0^1 T(y,t)dy-T(y,t) \right).
\end{equation}

The main idea is that the change in temperature is proportional to the imbalance in the energy received by the planet.  The amount of short wave radiation entering the atmosphere is given by $Qs(y)(1-\alpha(\eta,y))$, where Q is the total solar radiation (treated as constant), $s(y)$ is the distribution of the solar radiation, and $\alpha(\eta,y)$ is the albedo at latitude $y$ given that the ice line is at latitude $\eta$. The outgoing longwave radiation is the term $A+BT(y,t)$; it turns out that the highly complex nature of greenhouse gas effects on the Earth's atmosphere can be better approximated by a linear function of surface temperature than by the Stephan-Boltzmann law for blackbody radiation ($\sigma T^4$), see the discussion in \cite{graves1993new}. The parameter $A$ is particularly interesting here, because it is related to greenhouse gas effects on the climate system, which we will describe further in Section \ref{sec-ghg}.  The transport term $C ( \int_0^1 T(y,t)dy - T )$ redistributes heat by a relaxation process to the global average temperature.  Assuming symmetry of the hemispheres, one may take $y \in [0,1]$ and $T(y,t)$ as a symmetric temperature profile over the interval $[-1,1]$, hence, an even function of $y$.  A more detailed discussion of this model can be found in \cite{tung2007topics}. For interested readers, Table \ref{tab:ParValues} lists the polynomial expressions of some key functions in this model using standard parameter values.

The evolution of the ice-water boundary, or what is often called the \textit{ice line}, $\eta$, affects the albedo function $\alpha(\eta,y)$.  In \cite{tung2007topics} and \cite{north1975theory}, a critical ice line annual average temperature, $T_c$, is specified.  Above this temperature ice melts, causing the ice line to retreat toward the pole, and below it ice forms, allowing glaciers to advance toward the equator. One way to model this is described in \cite{widiasih2013dynamics}, where the augmented ice line equation governing $\eta$ is written as

\begin{equation} \label{etadot}
\frac{d \eta}{dt} = \rho ( T(\eta)-T_c ).
\end{equation}

The exact value of $\rho$ is unspecified in our analysis, except that it is assumed to be large in comparison to the timescale governed by volcanism and weathering processes. The interested reader will find a detailed discussion of the value of $\rho$ in \cite{mcgwid2014simplification}.
Because \eqref{bud} is an integro-differential equation, the phase space must contain the function space of the temperature profile. Therefore, the dynamics of this coupled system take place in some infinite-dimensional space.  The work by Widiasih in \cite{widiasih2013dynamics} treated the model in a discrete time framework and showed the existence of a one-dimensional attracting manifold.

McGehee and Widiasih \cite{mcgwid2014simplification} imposed equatorial symmetry by using even Legendre polynomials with a discontinuity at the iceline $y=\eta$, and showed the existence of a one-dimensional attracting invariant manifold, similar to that shown in \cite{widiasih2013dynamics}. The invariant manifold is parametrized by the ice line, thereby reducing the dynamics to a single equation:

\begin{equation} \label{etadot-reduced}
\frac{d\eta}{dt}= h_0(\eta; A)
\end{equation}
\noindent with
\begin{equation}\label{h}
h_0(\eta;A):= \rho\left(\frac{Q}{B+C} \left(s(\eta)(1-\alpha(\eta,\eta))+\frac{C}{B}(1-\overline{\alpha}(\eta))\right)-\frac{A}{B}-T_c \right)
\end{equation}
\noindent where $\overline{\alpha}(\eta)=\int_0^1 s(y) \alpha(\eta,y)dy$. The parameter $A$ appearing in \eqref{etadot-reduced} is indeed the greenhouse gas parameter in \cite{mcgwid2014simplification}, and is playing a similar role as that in the bifurcation diagram shown in Figure \ref{fig11}.

In what follows, we apply the invariant manifold result of McGehee and Widiasih \cite{mcgwid2014simplification} by coupling the ice line equation \eqref{etadot-reduced} to an equation for atmospheric greenhouse gas evolution through the (former) parameter $A$. From here on, we highlight the explicit dependence of the equation for $\eta$ on $A$ by writing
\begin{equation*}
\frac{d\eta}{dt}=h(A,\eta)
\end{equation*}
where $h(A,\eta)=h_0(\eta;A)$ and $h(A,\eta)$ is thought of as a real valued function over the plane $\mathbb R \times \mathbb R$.

\begin{table}[htb]
\centering
\begin{tabular}{|ccc||c|}
\hline
\textbf{Parameters} & Value & Units &  \textbf{Functions} \\ \hline \hline
&&& \\
$Q$  &  321  &  $\text{W}\text{m}^{-2}$ & $s(y) = 1 - \frac{0.482}{2} (3 y^2 - 1)$\\
$s_1$ & 1 & dimensionless & \\
$s_2$ &  -0.482  &  dimensionless & $h(A,\eta)=\rho\left(112.88+56.91\eta-24.31\eta^2-11.05\eta^3-\frac{A}{1.5}\right)$   \\
$B$ & 1.5  &  $\text{W}\text{m}^{-2}\text{K}^{-1}$ &    \\
$C$ & 2.5B & $\text{W}\text{m}^{-2}\text{K}^{-1}$ &  $g(A,\eta)=\delta(\eta-\eta_c)$  \\
$\alpha_1$ & 0.32 & dimensionless &  \\
$\alpha_2$ & 0.62 & dimensionless &
$\alpha(\eta,y)=\begin{cases}
&\alpha_1 \text { when } y< \eta \\
& \frac{\alpha_1+\alpha_2}{2} \text{ when } y=\eta\\
&\alpha_2 \text{ when } y>\eta
  \end{cases}$ \\
$T_c$ & $-10$ & ${}^\circ\text{C}$  & \\
&&& \\
\hline
\end{tabular}
\caption{Parameter values as in \cite{abbot2011}  and functions as in \cite{mcgwid2014simplification}.}   \label{tab:ParValues}
\end{table}

\subsection{Incorporating Greenhouse Gases} \label{sec-ghg}

We now make an argument for a very simple form of an equation for greenhouse gas evolution. In the Budyko and Sellers models \cite{  budyko2010effect,sellersglobal}, the parameter $A$ plays the important role of reradiation constant.  The Earth absorbs shortwave radiation from the sun, and some of this is reradiated in the form of longwave radiation.  The current value of $A$ is measured using satellite data to be approximately $202$ $W/m^2$ \cite{ graves1993new, tung2007topics}.

However, throughout the span of millions of years $A$ is not constant, as the amount of heat reradiated to space depends crucially on greenhouse gases, especially carbon dioxide.  In fact, it was posited in \cite{caldeira1992susceptibility} that $A$ behaves like a function of the logarithm of atmospheric carbon dioxide, measured in parts per million.  Intuitively, adding carbon dioxide to the atmosphere decreases its emissivity, allowing less energy to escape into space. Therefore, outgoing longwave radiation should vary inversely with CO$_2$.

Since the land masses were concentrated in middle and low latitudes prior to the global glaciation period, Kirschvink postulated that the stage was set for an ice-covered Earth.  Then \lq On a snowball Earth, volcanoes would continue to pump CO$_2$ into the atmosphere (and ocean), but the sinks for CO2 – silicate weathering and photosynthesis would be largely eliminated \rq   \cite{ hoffmanschrag2002terranova, kirschvink1992protero}. In  more recent work,  Hogg \cite{hogg} put forth an elementary model  for the evolution of greenhouse gases consistent with Kirschvink's theory.  In short, he argued that the main sources for atmospheric CO$_2$ were due to an averaged volcanism rate and ocean outgassing.  The main carbon dioxide sink was said to be due to the weathering of silicate rocks. For our purposes, it will be enough to work with volcanism and weathering.  Let $V$ denote the rate of volcanism and $W$ the weathering rate, where the latter is assumed to depend on the location of the iceline, or more specifically the amount of available land or rock to be weathered.  Putting this together with the assumption that the reradiation variable $A$ varies inversely with CO$_2$ gives

\begin{align}
\frac{dA}{dt}&=-(V-W\eta)\nonumber\\
&:=\delta(\eta-\eta_c). \label{Adoteq}
\end{align}

where $\delta>0$ and $\eta_c=\frac{V}{W}$, the ratio of volcanism to weathering. Let $g(A,\eta):=\delta (\eta -\eta_c)$.

\begin{remark} In what follows, we often replace the right hand side of equation \eqref{Adoteq} with the more general function $g(A,\eta)$.  While our analysis in Sections \ref{sect:dynamics} and \ref{sect:jorm} uses the linear function of ice line alone, it is conceivable that greenhouse gas feedbacks depend directly on their atmospheric concentration, see eg. \cite{caldeira1992susceptibility, pierrehumbert2011climate}. We leave this avenue for future exploration and extension of the current work. \end{remark}

Next, allowing $A$ to vary in the ice line equation \eqref{h}, we now have a system of equations for $A$ and $\eta$:
\begin{equation} \label{Aeta}
\begin{cases}
&\dot{A}=g(A,\eta)\\
&\dot{\eta}=h(A, \eta).\\
\end{cases}
\end{equation}

To understand the timescale of $A$, we refer to the elucidation of the snowball scenario by Hoffman and Schrag in which they argue that weathering took place at a much slower rate than the ice-albedo feedback (see Fig. 7 on page 137 \cite{hoffmanschrag2002terranova} ).

The idea of packaging the essence of the long term carbon cycle into one simple equation is not novel, and is consistent with earlier findings \cite{  edmond1996fluvial, hogg, kump2000chemical}.  The novelty comes from connecting such an equation to an energy balance model and analyzing the dynamics of the coupled system.  Indeed, the parameter $A_{ex}$ on page 644 \cite{kump2000chemical} is the variable $\eta$ in equation \eqref{Aeta} and it represents \textit{the effective area of exposure of fresh minerals}. The parameter $\eta_c$ can therefore be thought of as a \textit{critical} area. The coupling of the ice line $\eta$ and the greenhouse gas variable $A$ follows naturally.

In its current form, the model does not restrict dynamics to the physical region $0\leq \eta\leq 1$.  There are orbits that exit this interval on either end, and we must therefore create reasonable assumptions for projection of this motion onto the boundaries.  Furthermore, $\eta$ should evolve along the physical boundaries, as suggested by  the bifurcation diagrams in Figure \ref{icf}. This will be guaranteed by first carefully extending the vector field to the whole plane, and then utilizing Filippov's theory for the resulting nonsmooth system.

\section{A Filippov System for a Glaciated Planet\label{sect:fil}}

Before we dive into the analysis of the coupled $(A, \eta)$ system,  we shall first build an intuition for the type of system of our focus and prove a general result for such a system.  We then apply this result to an extended version of \eqref{Aeta} and study its dynamics.  We find that this framework both ensures that trajectories with initial conditions in the physical region remain in this region for all time and produces the expected sliding motion on the boundaries.

Various frameworks for analyzing non-smooth systems have been developed, eg. by Caratheodory, Rosenthal, Viktorovskii, Utkin \cite{filippov1964, cortes2008}. Here, we choose to use the framework developed by A.F. Filippov \cite{filippov1964}.

\subsection{Casting the Model in a Filippov framework}

As mentioned previously, the pole and the equator define physical constraints of the ice line;  it must stay in the unit interval $[0,1]$. However, the model as it stands does not take this into account.  A natural choice is to assume that $\dot{\eta}=0$ when $\eta$ reaches a physical boundary.  In addition to this, one needs to account for the stability/instability of these states, thus allowing orbits to exit the boundary when it becomes unstable.  More specifically, we should take

\begin{equation}\label{AetaFillipov}
\begin{cases}
\dot{\eta}&= \begin{cases}
 0 & \{ \eta=0 \text{ and } h(\eta,A)<0 \}\text{ \text{or} }\{ \eta=1 \text{ and } h(A, \eta)>0\}\\
      h(A, \eta) & \text{otherwise}\end{cases} \\
\dot{A}&=g(A,\eta)
\end{cases}
\end{equation}

The first equation forces $\eta$ to stop at the physical boundary exactly when it is about to cross that boundary.  When this happens, $\eta$ should remain constant while $A$ continues to evolve.  As $A$ evolves, so does the value of $h$ and this should result in the $\dot{\eta}$ equation ``turning back on'' as soon as $h$ becomes positive on the lower boundary, $\eta=0$, or negative on the upper boundary, $\eta=1$. This is analogy with Kirschvink's hypothesis, as we expect orbits that enter the snowball state to recover when enough carbon dioxide has built up in the atmosphere. However, it is not clear that classical results from dynamical systems apply because the system is nonsmooth as well as undefined beyond the physical boundary. Fundamentally, the existence and uniqueness of solutions needs to be verified.  Our analysis takes the approach of \textit{extending} the vector field to the nonphysical region in a way that makes the system amenable to the theory of Filippov.  This mathematical extension is used to obtain exactly the expected dynamics of the system \eqref{AetaFillipov} outlined above, including boundary motion.  There are many possible modeling choices for the boundary, however, as will be discussed in Remark \ref{remark}, our proposed extension ensures the consistency of the model dynamics with the existing snowball theory demonstrated by the literature e.g. in \cite{abbot2011, caldeira1992susceptibility}. The type of extension we adopt is highlighted in Section \ref{sect:ex-un} below.  Much of the proof relies on the theory developed by Fillipov \cite{filippov1964, filippov1988}.

\subsection{Existence and uniqueness of solutions \label{sect:ex-un}}

The following general result will motivate and apply to an extended version of the system \eqref{Aeta}, as well as guarantee the desired motion described in \eqref{AetaFillipov}.

\begin{theorem} \label{filippov-bud}
Let $G(x,y), H(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}$ be $C^1$ functions.  Define the vector field on $\mathbb{R} \times \mathbb{R}$ as follows:
\begin{align} \label{gen-pws}
&\dot{x} = G(x,y)\\
&\dot{y} = \begin{cases}
-|H| \quad &\text{when } y > 1\\
\frac{H-|H|}{2}  \quad &\text{when } y=1 \\
H \quad &\text{when } 0< y < 1\\
\frac{H+|H|}{2} \quad &\text{when } y = 0\\\label{gen-pws1}
|H| \quad &\text{when } y < 0. \\
\end{cases}\\\nonumber
\end{align}

Then, an initial value problem with time derivative \eqref{gen-pws}-\eqref{gen-pws1} has a unique forward-time Filippov solution. Furthermore, the strip $\mathbb{R} \times [0,1]$ is forward invariant.
\end{theorem}

\begin{proof}

First, to simplify notation, let $F(x,y)$ denote the vector-valued function $[G(x,y),H(x,y)]$. To show the existence of a Filippov solution, one must show that $F$ is locally essentially bounded or equivalently, satisfies what Filippov called \textit{Condition B} on p. 207 of \cite{filippov1964}. This is straight forward to check since $F$ is Lipschitz continuous everywhere except at the boundaries $y=0,1$.

To show uniqueness of the Filippov solution, we must show the \textit{one-sided Lipschitz condition}, ie. that for almost all $z_1$ and $z_2$ in some small neighborhood in $\mathbb R^2$, the inequality
\begin{equation} \label{esslip}
 \left( F\left( z_1 \right)-F \left( z_2 \right)\right)^T \left( z_1-z_2 \right)  \le K \| z_1-z_2 \|^2
\end{equation}
holds for some constant $K$ (see inequality (51) p. 218 of \cite{filippov1964}). Since $F$ is Lipschitz away from the discontinuity boundary, one easily checks that this condition is satisfied in this region.

We now consider initial value problems with initial conditions in the discontinuity boundary. There are two such discontinuity boundaries: the lines $y=0$ and $y=1$.   First, let $z$ be a point on the line $y=1$. Let $\delta$ be small enough that $B_{\delta}(z)$ is a neighborhood contained within the set $\left\lbrace (x,y):y>0 \right\rbrace$. Suppose that $z_1=(x_1,y_1)$ satisfies $0<y_1<1$ and $z_2=(x_2,y_2)$ satisfies $y_2>1$.  We consider $\left( F(z_2)-F(z_1) \right)^T (z_2-z_1)$ for the following two cases:
\begin{enumerate}
\item If $H(z_1) \ge 0$, then $\left(F(z_2)-F(z_1)\right)^T(z_2-z_1)=\left(0, -|H(z_2)|-H(z_1) \right)^T(x_2-x_1, y_2-y_1)<0$, since $-|H(z_2)|-H(z_1) < 0$ and $y_2-y_1>0$. Then, inequality \eqref{esslip} is satisfied with $K=1$.

\item If $H(z_1)<0$, then $F(z_2)-F(z_1)=\left(0, -|H(z_2)|+|H(z_1)| \right)$. The second entry may be estimated further:
$-|H(z_2)|+|H(z_1)| \le |H(z_1)-H(z_2)| \le k_H \|z_1-z_2 \|$  where $k_H$ is the Lipschitz constant of the function $H$.
Therefore, the inequality \eqref{esslip} is satisfied with $K=k_H$.
\end{enumerate}

A similar argument holds for the other discontinuity boundary, and therefore, system \eqref{gen-pws}-\eqref{gen-pws1} satisfies the one-sided Lipschitz continuity condition \eqref{esslip}, guaranteeing forward-time uniqueness of Filippov solutions.

Filippov theory further guarantees that a sliding solution exists at the discontinuity boundary.  With the normal vector taken in the direction of increasing $y$, one checks that system \eqref{gen-pws}-\eqref{gen-pws1} satisfies the conditions of Lemma 3 in \cite{filippov1964}. This gives the sliding solution on the upper boundary $y=1$. If $z(t)$ denotes this solution, then, with $\alpha=1/2$,
\begin{align*}
\frac{dz}{dt}&=[G(x,1), \alpha (-|H|)+(1-\alpha) H] \\
&=[G(x,1), 0].
\end{align*}

Furthermore,  above the upper discontinuity boundary the vector field points downward, while below the lower boundary, the vector fields points up. Therefore, since any solution can only cross into or slide along the boundary, the strip is forward invariant.
\end{proof}

Armed with this result, we return to system \eqref{Aeta}.  Recall that $g$ and $h$ are both $C^1$ functions over the plane. Henceforth, we consider the initial value problem in the state space $(A,\eta) \in \mathbb{R} \times \mathbb{R}$ endowed with the following piecewise Lipschitz vector field:

\begin{align}\label{ig-pws}
&\dot{A} = g(A,\eta)\\
&\dot{\eta} = \begin{cases}
-|h| \quad &\text{when } \eta > 1\\
\frac{h-|h|}{2}  \quad &\text{when }
\eta=1 \\
h \quad &\text{when } 0< \eta < 1\\\label{ig-pws1}
\frac{h+|h|}{2} \quad &\text{when } \eta= 0\\
|h| \quad &\text{when } \eta < 0. \\
\end{cases}\\\nonumber
\end{align}
having an initial value $(A(0),\eta(0)) \in \mathbb{R} \times \mathbb{R}$.

An application  of Theorem \ref{filippov-bud} shows that a unique forward-time Filippov solution to the initial value problem \eqref{ig-pws}-\eqref{ig-pws1} exists.  Furthermore, the forward invariance of the strip $\mathbb R \times [0,1]$ ensures the physically relevant aspect of the ice line $\eta$.  We have arrived at the following conclusion:

\begin{corollary}
There exists a unique Filippov solution to \eqref{ig-pws}-\eqref{ig-pws1}. Furthermore,  the strip $\mathbb R \times [0,1]$ is forward time invariant.
\end{corollary}

\begin{remark}\label{remark}
There are many possible ways of extending system \eqref{Aeta} beyond the strip $\mathbb{R} \times [0,1]$. The choice of extension as done in system \eqref{ig-pws}-\eqref{ig-pws1} assures the following points:
\begin{enumerate}
\item The attracting nature of the strip $\mathbb{R} \times [0,1]$.
\item The switching from sliding to crossing is entirely determined by zeroes of $h(A, \eta)$. This point is especially important in the modeling of the physical system, since $h$ signals the ice line's advance or retreat and $h$ should  be ``off" for some positive amount of time at the extreme ice line locations (pole or equator) and should ``turn back on" when the system has reduced or increased enough greenhouse gases.
\item The time derivative of the sliding mode is entirely determined by the governing function of the greenhouse gas effect,  $g(A,\eta)$. Again, this aspect is especially important in the modeling of the physical system, since at the extreme ice line location (pole or equator), the ice-albedo feedback shuts off and the greenhouse gas effect is the only driver of the system.
\end{enumerate}
\end{remark}

\subsection{Dynamics of the model \label{sect:dynamics}}

We now focus our analysis on the system \eqref{ig-pws}-\eqref{ig-pws1}.  We show that its sliding dynamics guarantee exit from the snowball scenario, and use the Filippov framework to simulate stable small ice cap states and large amplitude periodic orbits.

The $\eta$-nullcline, given by $h(A,\eta)=0$, is cubic in $\eta$ and linear in $A$ (see Table \ref{tab:ParValues}).  It has a single fold in the region $0 \leq \eta \leq 1$ at $\eta =\eta_f \approx 0.77$, as can be seen in Figure \ref{critman}. The $A$-nullcline is the horizontal line $\eta=\eta_c$. If $0<\eta_c<1$, the system has a single fixed point. The eigenvalues associated with this fixed point are

\begin{equation*}
\lambda_{\pm}=\frac{1}{2}\left(\frac{\partial h}{\partial \eta}\pm \sqrt{\left(\frac{\partial h}{\partial \eta}\right)^2-\frac{4\delta}{B}}\right).
\end{equation*}

 From this and Figure \ref{critman}, we see that if the fixed point lies above the fold and in the physical region, i.e. $\eta_f<\eta_c<1$, then it is stable, and it is unstable when $0<\eta_c<\eta_f$.  This is reminiscent of the ``slope-stability theorem'' from the climate literature \cite{cahalan1979stability}. In that work, the authors related changes in slopes of equilibrium curves to changes in stability of equilibria for a globally averaged energy balance model.  They found that a small ice cap or ice-free solution could be stable, a large ice cap was unstable, and a snowball state was again stable.

\begin{remark} In the case that $\eta_c=0,1$, one of the physical boundaries is entirely composed of equilibria, and stability is determined by the direction of the vector field near it.  Physically, this degenerate scenario comes about when there is an absence of volcanism in the ice-covered state or a perfect balance between weathering and volcanism in the ice-free state. We do not discuss these cases further.\end{remark}

\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{{phase.png}}
\caption{The physical region of the phase space and possible fixed points of the system given by the $\eta$-nullcline, $h=0$.  The location of the equilibrium is determined by the critical effective area of exposed land $0<\eta_c<1$. Solid black portions of the curve represent stable equilibria while the dashed lines denote unstable equilibria.  Solid black portions of the boundary are attractive sliding regions and dashed boundaries are crossing regions.}\label{critman}
\end{figure}

\subsubsection{Sliding and escape from Snowball Earth}

Recall that $\eta_c$ was defined to be a ratio of volcanism to weathering.  We now prove that the system always recovers from the ice-covered regime provided this parameter is positive, i.e. there is some contribution from volcanic outgassing to atmospheric greenhouse gas content.

\begin{theorem}\label{exit} Suppose $\eta_c>0$.  Then any orbit of the system \eqref{ig-pws}-\eqref{ig-pws1} that enters the boundary $\eta=0$ must exit in finite time.\end{theorem}

\begin{proof}
Let $A_*$ be the $A$-coordinate of the intersection of the $\eta$-nullcline with the lower ice boundary $\eta=0$.
Suppose an orbit enters the boundary $\eta=0$.  At such an occurrence $A\geq A_*$, since this is where $\dot{\eta}=h(A,\eta)\leq 0$.  For $A\geq A_*$, the confinement vector field in the non-physical region $\eta<0$ is given by $\dot{\eta}=|h(A,\eta)|\geq 0$.  $A$ is decreasing on either side of $\eta=0$.  The Filippov convention induces sliding according to

\begin{align*}
\dot{\eta}&=\frac{h(A,0)+|h(A,0)|}{2}=0\\
\dot{A}&=-\delta \eta_c.
\end{align*}

At the intersection point $A=A_*$, the vector field is tangent to the boundary and sliding terminates. For $A<A_*$ the boundary is a crossing region, and the vector fields on either side of it are identical with $\dot{\eta}>0$.  Moreover, there is a single orbit of the smooth system that passes through the tangency point. Due to forward-time uniqueness from Theorem \ref{filippov-bud}, the sliding solution must follow this orbit and reenter the physical region.

\end{proof}

\begin{remark} A similar argument shows that any orbit that enters an ice-free state must exit in finite time.  The required condition is that there is \textit{not} a perfect balance between volcanism and weathering, i.e. $\eta_c\neq 1$.\end{remark}

\subsubsection{Numerical simulations\label{po}}

The Filippov framework also allows for numerical simulation of the model.  Let $(A_c,\eta_c)$ be the location of the single fixed point.  We find that small ice cap states corresponding to fixed points with $\eta_f<\eta_c<1$ are possible attractors of the system, as are periodic orbits born from the Hopf bifurcation at $\eta_c=\eta_f$.

The fold of the $\eta$-nullcline is a \textit{canard point}, and the mechanism that produces the large amplitude periodic orbit in Figure \ref{dynamics} is a \textit{canard explosion} \cite{ benoit1981chasse, dumortier1996canard, krupa2001extending}. While mathematically interesting, canard orbits are unlikely observables for planar systems because they occur within an exponentially small parameter range.  However, they can be robust in higher dimensions which makes more complex oscillatory behavior possible, see e.g.  \cite{benoit1983systemes,desroches2012mixed,szmolyan2001canards,wechselberger2005existence}.  With this in mind, we remark that an interesting avenue for future research involves studying an extended version of the system \eqref{ig-pws}-\eqref{ig-pws1} with an additional variable ($w$ in \cite{mcgwid2014simplification}) and foregoing their invariant manifold reduction.

 \begin{figure}[ht]
  \centering
   \subfloat[Small ice cap equilibrium]  {\includegraphics[width=0.4\textwidth]{sm_ice.png}}  \quad \quad
   \subfloat [Periodic orbit]  {\includegraphics[width=0.4\textwidth]{relax_osc.png}} \quad
  \caption{Attractors of the system when (a) $\eta_c=0.85$ and (b) $\eta_c=0.6$. The $+$ symbol marks the intial condition and the horizontal long-dashed line is the $A$-nullcline. In (a), the orbit reaches the ice-free state and slides until it reaches the intersection of the folded curve $h(A,\eta)=0$ with this boundary.  It then enters the physical region and approaches the small ice cap equilibrium.  In (b), the fixed point is unstable and the orbit oscillates between the ice-free and ice-covered boundaries.  Simulations were performed using Mathematica 9.}
\label{dynamics}
 \end{figure}

\subsection{Application to the Jormungand Model\label{sect:jorm}}

The previous occurrence of a complete snowball event is still a highly contested topic in geology.  In the model \eqref{ig-pws}-\eqref{ig-pws1}, the only possible stable fixed point is a small ice cap. In this section, we present a variant of the system, as introduced by Abbot, Voigt, and Koll \cite{abbot2011}, and employ the Filippov framework as above.  By modifying the albedo function $\alpha(\eta,y)$ so that it differentiates between the albedo of bare ice and snow-covered ice, the new system has an additional stable state that corresponds to a large ice cap. In \cite{abbot2011}, this was called the \textit{Jormungand} state because it allows for a snake-like band of open ocean at the equator.  Moreover, there are again attracting periodic orbits.  These can be seen in Figure \ref{jorm}.

In this version of the model, the albedo is defined as follows:

$$\al_J(\eta,y)=\left\{\begin{array}{ll}\al_w,& y<\eta\\
\frac{1}{2}(\al_w+\al_2(\eta)), & y=\eta\\ \al_2(y), & y>\eta, \end{array}\right.\eqno(14)$$

where $\al_2(y)=\frac{1}{2}(\al_s+\al_i)+\frac{1}{2}(\al_s-\al_i)\tanh M(y-0.35) $.

Here $\al_w$ is the albedo of open water, $\al_i$ is the albedo of  {\em bare} sea ice, and $\al_s$ is the albedo of {\em snow-covered} ice. The model assumes sea ice aquires a snow cover only for latitudes above $y=0.35$.  We modify $h(A,\eta)$ by replacing $\alpha$ with $\alpha_J$:

\begin{equation}\label{hj}
h_J(A,\eta)=\frac{Q}{B+C} \left(s(\eta)(1-\alpha_J(\eta,\eta))+\frac{C}{B}(1-\overline{\alpha_J}(\eta))\right)-\frac{A}{B}-T_c
\end{equation}
where we note that although $\alpha(\eta,y)$ has a discontinuity when $y=\eta$, both $\alpha(\eta,\eta)$ and $\overline{\alpha_J}(\eta)=\int_0^1 \alpha(\eta,y)s(y)dy$ are smooth functions of $\eta$.

For this system, we can immediately apply Theorems \ref{filippov-bud} and \ref{exit} to obtain the following result.

\begin{corollary} The physical region is forward invariant with respect to the system \eqref{ig-pws}-\eqref{ig-pws1} where $h$ is replaced by $h_J$. Solutions exist and are unique in forward time, and any trajectory that enters an ice-covered state must exit in finite time provided $\eta_c\neq 0$.\end{corollary}

 \begin{figure}[ht]
  \centering
   \subfloat[]  {\includegraphics[width=0.4\textwidth]{jorm1.png}}  \quad \quad
   \subfloat []  {\includegraphics[width=0.4\textwidth]{jorm2.png}} \quad
  \caption{Periodic orbits of the Jormungand system when (a) $\eta_c=0.8$ and (b) $\eta_c=0.15$. The folded curve is the $\eta$-nullcline $h_J(A,\eta)=0$ and dashing is as in Figures \ref{critman} and \ref{dynamics}.}
\label{jorm}
 \end{figure}

\begin{table}[htb]
\centering
\begin{tabular}{|ccc||c|}
\hline
\textbf{Parameters} & Value & Units \\ \hline \hline
&& \\
$T_c$ & 0 & ${}^\circ\text{C}$  \\
$M$ & 25 & dimensionless  \\
$\alpha_w$ & $0.35$ & dimensionless   \\
$\alpha_i$ & $0.45$ & dimensionless   \\
$\alpha_s$ & $0.8$ & dimensionless   \\
&& \\
\hline
\end{tabular}
\caption{Parameter values as in Table \ref{tab:ParValues} unless specified above.  Additional values taken from \cite{abbot2011}.}   \label{tab:ParJorm}
\end{table}

\section{Discussion\label{sect:discussion}}

In this article we have extended a class of energy balance models to include a greenhouse gas component.  The resulting system was nonsmooth due to physical constraints and did not immediately fit into an existing analytical framework.  However, we defined an extended system that allowed us to utilize the theory of Filippov and obtain the expected motion, in agreement with physical arguments from climate literature. More specifically, we proved existence and uniqueness of forward-time solutions and showed that the system was confined to the physical region with sliding on the ice-covered and ice-free boundaries.  We showed that the extended system always escapes from the ice-covered scenario, thus supporting Kirschvink's hypothesis about carbon dioxide accumulation in a snowball scenario due to a shut down of chemical weathering processes \cite{kirschvink1992protero}.  Using our extended system, we found that small ice-cap and large amplitude ice-cover oscillations were possible attractors of the system. We then applied our results to the case of Jormungand world and showed that it is possible to obtain further oscillatory dynamics between no ice cover, large ice-cover, and full ice-cover states.

General mathematical questions brought about by this work have to do with building a general framework for such models so that there is no need to appeal to artificial tools.  A first step toward dealing with physical boundaries might be a general theory for semiflows generated by smooth vector fields on manifolds with boundary.  In addition, the stable portions of the physical boundaries in our model should be thought of as equilibria of the fast subsystem, i.e. as part of the \textit{critical manifold} from geometric singular perturbation theory \cite{jones1995geometric}.  In fact, they are much more \textit{normally hyperbolic} than the curve $h=0$; they are reachable in finite time!  However, the current theory cannot be directly applied to the full discontinuous system and we remark that developing an analogous Fenichel theory \cite{fenichel1979geometric} for singularly perturbed discontinuous systems with sliding is an interesting avenue for future research.

Another interesting future direction is to study the explicit effect of the temperature (here we have considered it only through the equation for $\eta$ based on the reduction by McGehee and Widiasih \cite{mcgwid2014simplification}) on the dynamics of the system.  In a nonsmooth system, the addition of a stable dimension may destroy an existing stable periodic orbit \cite{sieber2010small}.  Moreover, an additional dimension could result in more interesting glacial dynamics, such as mixed mode oscillations.

Finally, it is natural to ask how the shape of the $\eta$-nullcline changes with physical parameters and how this affects system dynamics, e.g. could the lower fold in Figure \eqref{jorm} move to the left of the upper fold?  We refer the reader to \cite{widwal2014dynamics} for a number of examples.  \\

\textbf{Acknowledgements:} This research was supported in part by the Mathematics and Climate Research Network and NSF grants DMS-0940366, DMS-0940363.  AB was also supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.  We thank the members of the MCRN Paleoclimate and Nonsmooth Systems seminar groups for many useful discussions, especially Mary Lou Zeeman and Emma Cutler.  We also thank Andrew Roberts for his suggestions relating to the formulation of the problem.

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TITLE: Proper way to write a mathematical theorem
QUESTION [1 upvotes]: Consider the Bernoulli Inequality, for instance. Basically it says the following:

For all $x\in\mathbb{R}$ such that $x\geq -1$ and all $n\in\mathbb{Z}^+$, we have that $(1+x)^n \geq 1+nx$.

I'm concerned about the equivalent ways to rephrase this result, and if they are correctly written. I'm gonna put here two of them.

1) Let $x\in\mathbb{R}$ be such that $x\geq -1$ and let $n\in\mathbb{Z}^+$. Then $(1+x)^n \geq 1+nx$.
2) If $x\in\mathbb{R}$ is such that $x\geq -1$ and $n\in\mathbb{Z}^+$, then $(1+x)^n \geq 1+nx$.

In the item 1), including a period and writing Then after it is grammatically correct? I see a lot of professors and books writing like this, but this Then is a conclusion from the hypothesis given. I always feels this is wrong in some way, but I'm never sure. The item 2) follows the format if P, then Q. In this case it's very clear, we don't use a period. It's precisely because of this that I feel strange about the first item, but there I didn't use any if, so maybe this is ok. I just want to be sure about the proper way to write this things.
Thank you.

REPLY [2 votes]: I'm actually a strong proponent of versions like (1) - at least, some of the time.
In a complicated theorem, both the hypothesis and conclusion might be a fair mouthful. As a reader, I'm happy to have them more cleanly separated by a period. Rules like "don't begin a sentence with 'then'" are (to my mind) only valuable insofar as they lead to clear writing, or prevent confusion - I care much more about whether I can follow the statement of a theorem than I do about whether it matches with Strunk & White.

For an example of such a complicated theorem: consider something of the form

If $x$ is such that, for all $x<y<2x$, if $y$ is prime then $gcd(x, y+1)>1$, then $x$ is odd.

(Ignore the fact that this "theorem" is bonkers.) The fact that the hypothesis itself contains a conditional makes this really awkward to express as a single sentence; I'd prefer it broken in two. And there are definitely theorems this complicated, or more complicated, so while this concern might not arise frequently, it will arise a noticeable amount of the time.


TITLE: How to describe this (paradoxical) motion by the hidden geometric series?
QUESTION [0 upvotes]: High-schooler here trying to work through an interesting problem:

A boy races an ant, giving the latter a head start. The motion of the
two bodies can be decomposed according to the following steps:
Step #1: The boy runs to the ant's starting point while the ant crawls
forward.
Step #2: The boy advances to where the ant was at the end of Step #1
while the ant goes yet further.
Step #3: The boy advances to where the ant was at the end of Step #2
while the ant goes yet further.
Step #4: The boy advances to where the ant was at the end of Step #3
while the ant goes yet further.
And so on and so forth.
Given that the boy is running at 8 m/s, the ant is crawling at 0.08
m/s and the latter has an 80-metre head start, calculate the time
taken for the boy to overtake the ant.

One may deduce that this lends itself to a geometric series. Since the boy does overtake the ant, the series must be convergent. How do we go about calculating the common ratio r and solving the rest of the problem?

REPLY [0 votes]: The relative velocity (in m/s)  between the boy and the ant is ...
$$v_r = 8 - 0.08 = 8\times \frac{99}{100} $$
So the time for the boy to catch up to the ant is ...
$$T = \frac{80}{v_r}  =\frac{1000}{99}=10.\dot 1\dot0s$$
We can use geometric series to show the the sequence of times converges to the same value.
$$t_1 = \frac{80}8 = 10s$$
at which time the ant has now moved on by $0.8$m
so if $t_2$ represents the time between steps #1 and #2...
$$t_2 = \frac{0.8}8 = 0.1s$$
the common ratio for this sequence is
$$r = \frac{t_2}{t_1}= 0.01$$
The total time can now be calculated using the formula for geometric series...
$$T  = \sum_{n=1}^{\infty}t_n= t_1\times \frac 1{1-r}= \frac{10s}{0.99}=  10.\dot 1\dot0s$$


TITLE: For $\sum_{n \geq 0}a_n z^n=f(z)=e^z /\cos{\frac{\pi}{2}z}$, what is $a_{100}$ with error $0.0001$ at most?
QUESTION [5 upvotes]: For $\displaystyle\sum\limits_{n \geq 0}a_n  z^n$ the Maclaurin series of $f(z)=\frac{e^z}{\cos{\frac{\pi}{2}z}}$, what is an approximation of $a_{100}$ with error $0.0001$ at most?
This is a past qualifying exam question, so no computer. I am looking for a solution that could be given in a timed exam setting, by a student with one semester of graduate Complex Analysis, familiar with first half of Stein's book, with Churchill's book, and somewhat with first half of Ahlfor's book.
I write $c(z)\,=\,\cos{\frac{\pi}{2}z}\,=\,\displaystyle\sum\limits_{\;n\geq0\\\text{(even)}}c_n  z^n,$ and also $\theta=\frac{2}{\pi}<1.$ We have $c_n\,=\,\frac{(-1)^{n/2}}{n!\theta^n}.$
$$f(z)\,=\,\frac{1\,+\,z\,+\,\frac{z^2}{2!}\,+\,\frac{z^3}{3!}\,+\,\frac{z^4}{4!}\,+\,\cdots}{1\;\quad-\quad\frac{z^2}{2!\theta^2}\quad+\quad\frac{z^4}{4!\theta^4}\,-+\,\cdots}\qquad\qquad(z\in\mathbb{D}).$$
Using the usual method, I found $a_0,a_1,a_2,a_3,a_4$ by hand, so we have $\,f(z)\,=\,$
$$\displaystyle\sum\limits_{n \geq 0}a_n  z^n\,=\,1+z+\left(\frac{1}{2}+\frac{1}{2\theta^2}\right)z^2+\left(\frac{1}{6}+\frac{1}{2\theta^2}\right)z^3+\left(\frac{1}{24}+\frac{1}{4\theta^2}+\frac{5}{24\theta^4}\right)z^4+\cdots.$$
I tried to discern a pattern by keeping the expressions more abstract, i.e. keeping the $a_n$ and $c_n$. Doing so we can write e.g.,
$$a_2\,=\,\frac{1}{2!}\,-\,c_2\,=\,\frac{1}{2!}\,-\,\displaystyle\sum\limits^{2}_{\,\ell\,=\,2\\\text{(even)}}c_\ell  a_{n-\ell},$$
$$a_7\,=\,\frac{1}{7!}\,-\,(c_2 a_5+c_4 a_3+c_6)\,=\,\frac{1}{7!}\,-\,\displaystyle\sum\limits^{7}_{\,\ell\,=\,2\\\text{(even)}}c_\ell  a_{n-\ell},$$
$$a_{100}\,=\,\frac{1}{100!}\,-\,(c_2 a_{98}+c_4 a_{96}+\cdots+c_{98}a_2+c_{100}).$$
I am not sure what to do next. I did consider Cauchy's Integral Inequalities, but could not see how to get a useful estimate from them. If $0<r<1,$
$$a_{100} = \frac{f^{(100)}(0)}{100!\,} = \frac{1}{2\pi i } \oint_{|z| = r} \frac{e^z}{c(z)\,z^{101}} dz,$$
$$|a_{100}| \leq \frac{1}{r^{100}}\max_{|z| = r}\left|\frac{e^z}{\cos(\frac{\pi}{2}z)} \right|,$$
but for either $r$ very near zero, or very near one, or say for $r=$ one-half, the estimate seems to not be useful.

REPLY [5 votes]: I think the idea is to subtract the principal parts $P$ at the poles at $z = \pm 1$, after which the resulting function $g=f-P$ is analytic in the disk $|z| < 3$, so that Cauchy inequalities should give you that the coefficient $b_{100}$ of $g$ is very small, certainly $|b_{100}| < 0.0001$. This means that within this small error, $a_{100}$ is just the corresponding coefficient of the sum of the principal parts $P$, which can be easily evaluated by geometric series.


TITLE: Are $c_0$ and $c$ duals of some spaces?
QUESTION [4 upvotes]: The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true.  That is, not all Banach spaces are isomorphic to the dual space of some normed vector space.  For instance $L^1$ is not isomorphic to any dual space.
My question is, are the sequence spaces $c_0$ and $c$ isomorphic to the duals of any spaces?

REPLY [2 votes]: $c_0$ is not the dual of some normed vector space.
sketch of proof:
This can be proven using the
Krein-Milman theorem.
If $c_0$ was the dual of a normed vector space, then its unit ball would be weakly-* compact.
However, it can be shown that the unit ball of $c_0$ has no extremal points,
therefore by Krein-Milman it is not weakly-* compact.
edit: for $c$ the situation is more complicated than I initially thought.
As uniquesolution pointed out in the comments, its unit ball has many extreme points,
and it is not clear to me if Krein-Milman can be used to show that the unit ball of $c$ cannot be weakly-* compact.


TITLE: evaluation of limit $\lim \frac 1n \cdot \frac {x^3}{1+x^2}$
QUESTION [1 upvotes]: $$\lim_{n \rightarrow \infty } \frac 1n \cdot \frac {x^3}{1+x^2}$$
I think that for any fixed x the limit is $0$ due to increasing n. But I don't feel confident. Could you help?

REPLY [1 votes]: sure, that's true. if you fix any constant $c$ then you have $\lim_{n \rightarrow \infty} \frac{c}{n} = 0$ because for any $\epsilon > 0$ for all $n$'s larger than $\frac{|c|}{\epsilon}$ you have $|\frac{c}{n}| < |\frac{c}{\frac{c}{\epsilon}}| = \epsilon$