text:
\section{Stochastic Thunderstorm Modeling}
\label{sect:stoch_mod}

The unsafe-to-fly regions to be avoided by the aircraft are hazardous convective weather regions, referred to as \textit{thunderstorm cells}. The actual position and size of these regions, as well as an extrapolation of their position at different forecast horizons, are provided in the form of \textit{nowcasts}. A nowcast file contains observations of the current state of the hazardous thunderstorm cells that are present in a particular portion of the airspace (here denoted by the 2D flight level $X \times Y$). These observations include, among others, the position of each thunderstorm cell center (in longitude, latitude coordinates), the extremities of each thunderstorm cell in all four directions (given in longitude, latitude), the size of each thunderstorm cell (expressed in number of ``pixels''), a mean radius, as well as the speed and direction of each thunderstorm cell. Figure \ref{fig:storm_cell} depicts  these thunderstorm cell characteristics. A nowcast file also provides a series of extrapolated predictions at discrete forecast horizons. However, these predictions are limited to the position of the thunderstorm cell centers. No other thunderstorm cell parameters are predicted. Every 10 minutes, a new nowcast file is published. Each nowcast file gives predictions for 6 forecast horizons from 10 to 60 minutes with a 10 minute spacing. Compared to the eventual observations of the thunderstorms, these forecasts feature significant prediction errors, mainly due to the nonlinear growth and decay of rainfall field size, which according to \cite{Browning1982} account for up to a quarter of the forecast error. This motivates the use of a stochastic model to represent the thunderstorm obstacles.

We consider a constant altitude flight, and thus, a two-dimensional domain. We aim to evaluate the probability that any discrete region of the flight level will be affected by hazardous thunderstorm at different forecast horizons $\tau=1, 2\dots,N$, corresponding to the forecast times 10, 20, $\dots$, 60 minutes and interpolated time instances in between. Figure \ref{fig:timeline} illustrates the nowcast structure and notation. Let $\Xi_{t_0+\tau|t_0}$ be the set of 2D \textst{obstacles} regions capturing the thunderstorm forecast for time $t_0+\tau$ in the nowcast published at time $t_0$. For the sake of brevity, we define $\square_{\tau} \coloneqq \square_{t_0+\tau|t_0}$ throughout the paper. Let $X \times Y$ denote the 2D airspace under consideration. At each time instance $t$, we assign to each discretized coordinate $(x,y) \in X \times Y$ a thunderstorm probability, that is, the probability that the position $(x,y)$ is in a thunderstorm cell. For this purpose we use the function $p_{\Xi_{\tau}}(x,y): X\times Y \mapsto [0,1]$. To compute $p_{\Xi_{\tau}}(x,y)$ for $\tau=1,\dots,N$, starting from a known and deterministic initial obstacle set $\Xi_{0}$, a stochastic model that describes the evolution of the thunderstorm is needed.

All information on thunderstorm position, size and movement is drawn from the available nowcasts. The analysis herein relies on a data sample of 720 nowcast files collected between December 15th and December 19th 2016, provided by the Spanish meteorological agency, \textit{Agencia Estatal de Meteorolog\'ia} (AEMET). AEMET's YRADAR application produces both a 2D and a 3D analysis. Due to the simplifying assumption of constant altitude, the following analysis will be constrained to the 2D dataset. Due to the short average lifespan of thunderstorm cells, a significant proportion of detected cells disappear after the 40 minute horizon, rendering the data for the 50 and 60 minute horizons sparse. We will therefore restrict the analysis to a forecast horizon up to 40 minutes. The most relevant nowcast parameters used for the stochastic characterization of the storm cells are presented in Table \ref{tab:parameters_nowcast}.
\begin{table}[b!]
\centering
\begin{scriptsize}
\begin{tabular}{p{2.95cm} p{4.5cm}}
	\hline
	\textbf{Nowcast parameter} & \textbf{Description}\\
	\hline
	NUM & ID of the cell\\
	NUPIX & Number of pixels of the cell\\
	LATCEN, LONCEN & Latitude/longitude of cell's center\\
	RADIOE & Effective radius of the cell\\
	LATNOR, LATSUR & Limits of the cell in latitude\\
	LONOES, LONEST & Limits of the cell in longitude\\
	LAT10, LON10 & Forecast for center after 10 min.\\
	LAT20, LON20 & Forecast for center after 20 min.\\
	\vdots & \vdots\\
	LAT60, LON60 & Forecast for center after 60 min.\\
	DIRN & Direction of displacement of the storm cell\\
	V(KMH) & Velocity of displacement of the storm cell\\
	\hline
\end{tabular}
\end{scriptsize}
\caption{Overview of relevant nowcast parameters.}
\label{tab:parameters_nowcast}
\end{table}

\begin{figure}
	\centering
	\includegraphics[width=0.8 \linewidth]{figures/storm_cell_round_named_pixels-eps-converted-to.pdf}
	\caption{Thunderstorm cell representation}
	\label{fig:storm_cell}
\end{figure}

We represent the thunderstorm cells using two states:
\begin{enumerate}
	\item the position of the cell center $(x_t^c,y_t^c)$ at time $t$,
	\item the position of the thunderstorm cell's lateral and longitudinal extremities $(x^{(E)}_t, x^{(W)}_t,y^{(N)}_t,y^{(S)}_t)$ at time $t$ .
\end{enumerate}
 While no information on the thunderstorm cell's shape is available, the four extremities enable us to formulate a rectangular bound of the thunderstorm cell. At time $t=t_0$ ($t_0$ being the time at which the nowcast is published), we have deterministic knowledge (assumed exact) of all six state variables. For times $t=t_0 + \tau$ $(\tau =1,\dots,N)$, the nowcast files provide a prediction of the thunderstorm cell centers but no information on the thunderstorm cell extremities. To account for the prediction error on the center position (increasingly significant with growing forecast horizon) and the lack of extremity predictions, we aim to formulate a stochastic dynamic model, which will allow us to predict the evolution of the thunderstorm cell state in a probabilistic fashion. By using this model in combination with a sampling-based method, we will be able to compute the probability of thunderstorm at each $(x_t,y_t) \in X\times Y$ for each time instance $t = t_0 + \tau$, that is, $p_{\Xi_\tau}(x,y)$.

\subsection{Modeling Stochastic Thunderstorm Cell Dynamics}
\label{subsec:StormDyn}

Let $\mathbf{c}_t = (x^c_t,y^c_t) \in \R^2$ denote the observed center of the thunderstorm cell at time $t$, where $c$ denotes the center. Let $\mathbf{c}_{\tau}$ denote the forecast made at time $t_0$ for the thunderstorm cell center $\mathbf{c}$ at time $t_0+\tau$. Recall that $\tau=1,2,\dots,N$ is the forecast horizon corresponding to the forecast times 10, 20, $\dots$, 40 minutes and interpolations in between.

\begin{figure}
	\centering
	\includegraphics[width=0.8\linewidth]{figures/timeline-01.png}
	\caption{Nowcast structure.}
	\label{fig:timeline}
\end{figure}

The forecast dynamics for the (observed) thunderstorm cell center position is given by $\mathbf{c}_{t_0+\tau} = \mathbf{c}_{\tau} + \eta_{\tau}$,
where $\eta_\tau = (\eta_{\tau,x}, \eta_{\tau,y})$ denotes the prediction error associated with the forecast horizon $\tau$ in x and y directions. The prediction error grows with the length of the forecast horizon (\cite{Sauer2014,Radhakrishna2012}). It may also be dependent on current meteorological conditions, such as strong wind in a particular direction. In the developed model, this error is accounted for by sampling it from a parameterized probability distribution and adding it to the nowcast prediction. The distribution type is determined by analyzing historical predictions and comparing them with the observations. Once an appropriate distribution type has been determined, the parameters of the distribution are determined and updated in real time using maximum likelihood estimation.

The forecast error at forecast horizon $\tau$ is given by $\eta_{\tau} = \mathbf{c}_{t_0+\tau} - \mathbf{c}_{\tau}$. Figure \ref{fig:scatterhist_center} shows combined scatter plots and histograms illustrating the deviation in longitude and latitude of the forecast center positions compared to the observed realizations in the training set for forecast horizons 10, 20, 30 and 40 minutes. Using Matlab's \texttt{allfitdist} function, all valid parametric probability distributions are fitted over the empirical data and compared for goodness-of-fit, as quantified by the Bayesian information criterion (BIC), a criterion for selecting a model among a finite set of models. The \textit{logistic} distribution offers the best fit for the available data (see \cite[Chapter 3]{HentzenThesis} for a detailed discussion, where logistic, tlocationscale, normal and generalized extreme value distribution fittings are presented, compared, and discussed). In particular, it is able to better account for the heavier tails compared to a normal distribution.

\begin{figure*}[t!]
\includegraphics[width=1\linewidth]{figures/hist_group.pdf}
\vspace*{-0.5cm}
\caption{Longitudinal and lateral deviation of forecast thunderstorm cell center positions to observed realizations over growing forecast horizon.}
\label{fig:scatterhist_center}
\end{figure*}

The probability density function (PDF) $f_{\mathcal{L}}$ of the \textit{logistic} distribution $\m{L}$ is given by
\begin{equation*}
f_{\mathcal{L}}(\eta;m,s) = \frac{1}{4s} \text{sech}^2\bigg(\frac{\eta-m}{2s}\bigg) = \frac{e^{-\frac{\eta-m}{s}}}{s\big(1+e^{-\frac{\eta-m}{s}}\big)^2},
\end{equation*}
where $\eta \in \mathbb{R}$ is the forecast error, $m$ is the mean of the distribution and $s\geq0$ is the \textit{scale} or \textit{dispersion} parameter. The standard deviation $\sigma$ is expressed by $\sigma = \frac{s \pi}{\sqrt{3}}$.  The distribution's parameters are determined using maximum likelihood estimation with a defined number of historical data points.
Based on the studied data samples, we conclude:
\begin{enumerate}
	\item The thunderstorm cell center position prediction error in x and y directions $\eta_{\tau,x}$ and $\eta_{\tau,y}$ at forecast horizon $\tau$ can be approximated by sampling from the logistic PDFs $\m{L}_{x}^c$ and $\m{L}_{y}^c$ associated to the forecast horizon $\tau$: $\eta_{\tau,x} \sim \m{L}_{x}^c(\tau)$, $\eta_{\tau,y} \sim \m{L}_{y}^c(\tau)$.
	\item The parameters of these distributions (mean and scale) at different forecast horizons are computed using maximum likelihood estimation.
\end{enumerate}

 As the available nowcasts offer no forecast on thunderstorm cell size, past observations have to be analyzed to identify and extrapolate a growth or decay trend.

In the following analysis, growth and decay are expressed in terms of the change in lateral and longitudinal size of a single thunderstorm cell. Longitudinal size (width $w$) and lateral size (height $h$) at time $t$ are defined using the nowcast extremity data:
\begin{align*}
\begin{split}
w_t &= x^{(E)}_t - x^{(W)}_t, \quad h_t = y^{(N)}_t - y^{(S)}_t.
\end{split}
\end{align*}
The dynamics of the thunderstorm cell size are given by
\begin{align*}
\begin{split}
w_{t+1} &= w_{t} + \Delta w, \quad h_{t+1} = h_{t} + \Delta h,\\
\end{split}
\end{align*}
where $\Delta w$ and $\Delta h$ denote the change in width and height, respectively. They are the stochastic parameters to be extrapolated from historical data.

Similar to the analysis of the thunderstorm cell center prediction error above, we determine an appropriate parameterized probability distribution to model $\Delta w$ and $\Delta h$. The BIC indicates that the logistic probability distribution offers a superior fit to the normal distribution (see \cite{HentzenThesis} for a detailed discussion). We observe that thunderstorm cells featuring a large surface area (expressed using the number of pixels on the radar images) grow or decay faster than smaller cells. We thus choose the parameters of the \textit{logistic} distribution as a function of the thunderstorm cell size in pixels ($s_{pix}$):
\begin{align*}
&\Delta w \sim \m{L}^{w}(s_{pix}) \quad \Delta h \sim \m{L}^{h}(s_{pix}).
\end{align*}
The scale parameter of the $\Delta h$ and $\Delta w$ distributions is assumed to grow logarithmically with thunderstorm cell size (see \cite{HentzenThesis}).

\label{para:stoch_model}
The derived stochastic model of the thunderstorm cell dynamics is given by two steps as follows.
\begin{enumerate}
	\item Sample step:
	\begin{align*}
	&\eta_{\tau,x} \sim \m{L}_{x}^c(\tau), \quad \eta_{\tau,y} \sim \m{L}_{y}^c(\tau)\\
	&\Delta w \sim \m{L}^{w}(s_{pix}), \quad \Delta h \sim \m{L}^{h}(s_{pix})
	\end{align*}
	\item Update step:
	\begin{equation*} \label{eq:cell_dyn}
	\begin{cases}
	\underbrace{\begin{bmatrix}
		x^c_{t_0+\tau}\\
		y^c_{t_0+\tau}\\
		\end{bmatrix}}_{\substack{\text{corrected centers} \\ \text{at forecast horizon $\tau$}}} =
	\underbrace{\begin{bmatrix}
		x^{c}_{\tau}\\
		y^{c}_{\tau}\\
		\end{bmatrix}}_{\substack{\text{uncorrected centers} \\ \text{at forecast horizon $\tau$}}} +
	\underbrace{\begin{bmatrix}
		\eta_{\tau,x}\\
		\eta_{\tau,y}\\
		\end{bmatrix}}_\text{sampled error}\\
	\qquad\\
	\underbrace{\begin{bmatrix}
		x_{t_0+\tau}^{(W)}\\
		x_{t_0+\tau}^{(E)}\\
		y_{t_0+\tau}^{(S)}\\
		y_{t_0+\tau}^{(N)}\\
		\end{bmatrix}}_{\substack{\text{corrected extremities} \\ \text{at forecast horizon $\tau$}}} =
	\begin{bmatrix}
		x^c_{t_0+\tau}\\
		x^c_{t_0+\tau}\\
		y^c_{t_0+\tau}\\
		y^c_{t_0+\tau}\\
		\end{bmatrix} +
	\underbrace{\begin{bmatrix}
		-x^c_{t_0} + x_{t_0}^{(W)}\\
		x_{t_0}^{(E)} -x^c_{t_0}\\
		-y^c_{t_0} + y_{t_0}^{(S)}\\
		y_{t_0}^{(N)} -y^c_{t_0}\\
		\end{bmatrix}}_{\substack{\text{width/height} \\ \text{at $t_0$}}}+\frac{1}{2}
	\underbrace{\begin{bmatrix}
		- \sum_{i=1}^{\tau} \Delta w(s_{pix})\\
		\sum_{i=1}^{\tau}\Delta w(s_{pix})\\
		-\sum_{i=1}^{\tau} \Delta h(s_{pix})\\
		\sum_{i=1}^{\tau} \Delta h(s_{pix})
		\end{bmatrix}}_{\substack{\text{sampled change} \\ \text{in width/height} \\ \text{over forecast horizon $\tau$}}}
	\end{cases}
	\end{equation*}
\end{enumerate}
.

\subsection{Obstacle Clustering}
\label{subsec:StochNoFly}
The dynamic thunderstorm cell model provides a sequence of predictions of thunderstorm cell positions and sizes for $t=t_0 + \tau$ with $\tau = 1, \dots, N$. However, the individual thunderstorm cells alone are not suitable obstacles  for scenarios with a high density of convective weather. The thunderstorm cells are often grouped closely together in the airspace affected by convective weather. If every thunderstorm cell were to be considered as an individual obstacle, the resulting safety-optimal trajectory would likely contain a high number of turn advisories, as individual obstacles would have to be avoided every few minutes. Although such a trajectory might be feasible in theory, it is not an option in a realistic ATM setting. Furthermore, individual thunderstorm cells are likely to split into sub-cells or merge with other neighboring or overlapping cells. It is also likely that new thunderstorm cells appear in regions with high levels of convective weather. Nowcasts  do not contain information on these processes. Considering the thunderstorm cells as individual obstacles would thus result in potentially unsafe trajectories. For the these reasons, we propose a more robust approach in which the individual thunderstorm cells predicted for forecast horizons $\tau=1,\dots, N$ by the dynamic thunderstorm cell model at time $t_0$, are clustered into larger \textst{obstacles} unsafe-to-fly regions. This clustering is updated at each forecast horizon $\tau$ to account for the movement of the individual thunderstorm cells. The clustering results in a sequence of \textst{obstacles} unsafe-to-fly regions that represent areas of the airspace that feature a high density of thunderstorm cells and a high probability of thunderstorm occurrence. We use K-means clustering for this purpose.

At time $t_0 + \tau$, we forecast the individual thunderstorms $(\mathbf{o}^1_\tau, \mathbf{o}^2_\tau, \dots, \mathbf{o}^n_\tau)$, where each element is a $d$-dimensional real vector. The $n$ forecast thunderstorms are partitioned into $K \leq n$ clusters $\{C^1_\tau, C^2_\tau, \dots, C^K_\tau\}$ with centroids (that is, mean of points within the cluster) $\{\bar{\mathbf{o}}^1_\tau, \bar{\mathbf{o}}^2_\tau,\dots, \bar{\mathbf{o}}^K_\tau\}$ such that the variance of the distance of the observations within a cluster to the cluster's centroid is minimized:
\begin{equation*}
\arg \min_{C} \sum_{k=1}^K \sum_{\mathbf{o^i_\tau} \in C^k_\tau } \lVert \mathbf{o}^i_\tau-\bar{\mathbf{o}}^k_\tau \rVert^2_2.
\end{equation*}
Above,  $\lVert \cdot \rVert_2$ is the Euclidean $2$-norm. For our application, we consider $\mathbf{o}_\tau =(x^c_{\tau}, y^c_{\tau}, \xi_{\tau})$, where $x^c$ and $y^c$ are the $x$ and $y$ coordinates of the thunderstorm center and $\xi$ is the heading of the thunderstorm, as provided in the nowcast. This choice allows to group the thunderstorms based on their distance to each other and based on the directions in which they are heading.

Choosing the number of clusters $K$ is a hard algorithmic problem. A possible metric for the quality of the resulting clusters is the sum of squared errors (SSE), that is, the sum of the squared distance between each member of the cluster and its centroid. The SSE decreases with growing $K$, with $\text{SSE} = 0$ for $K=n$. Applying the so-called \textit{elbow method}, we choose $K$ at the \textit{elbow point}, that is, where the SSE starts decreasing at a significantly slower rate. The elbow method is heuristic and is not guaranteed to give satisfactory results in all cases. We discuss an alternative, empirical method to determine $K$ based on a parameter study with historical data in \cite{HentzenThesis}.

Let $\Xi_{\tau} = \{{C^k_{\tau}}\}$ for $k=1,\dots,K$ and $\tau = 1,\dots,N$ be the new obstacle set containing the K elliptical \textst{obstacles} unsafe-to-fly regions forecast for time $t_0 + \tau$ by the nowcast published at time $t_0$. We now require a stochastic representation of the obstacle set, expressed by $p_{\Xi_\tau}(x,y)$. Recall that this function returns the probability that the coordinate $(x,y)$ is within any obstacle cluster $C^k_{\tau}$ ($k \in {1,2,\dots,K}$)  in $\Xi_\tau$ at time $t_0 + \tau$.

\subsection{Stochastic Representation of Unsafe-to-fly Region Clusters}
\label{subsec:CovFunc}

The computation of $p_{\Xi_\tau}(x,y)$ requires a prediction of the obstacle clusters' position at forecast horizons $\tau$. This prediction in turn calls for a geometric description of the clusters. For this purpose, we choose minimum-volume ellipses (MVE) that enclose the extremities of the thunderstorm cells present within one cluster $C^k_{\tau}$. Ellipses are preferred over polygons for the geometric obstacle characterization due to their efficient mathematical description. Figure \ref{fig:cluster_ex} shows an example of MVE enclosing for two clusters.

\begin{figure}[t!]
	\centering
	\includegraphics[width= 1\columnwidth]{figures/mve_2.pdf}

	\caption{Minimum volume ellipse enclosing of clusters in nowcast data of December 19th 2016, 10:30 UTC}
	\label{fig:cluster_ex}
\end{figure}

Each ellipse $\varepsilon_\tau^k$ is  parameterized by its center $\mathbf{m}_\tau^k \in \R^2$ and its positive definite eccentricity matrix $M_\tau^k$. The position given by coordinate $\mathbf{x}=(x,y)$  is within an elliptic \textst{obstacle} unsafe-to-fly region at time $t_0 + \tau$ if the following condition is fulfilled for any $k$:
\begin{equation*}
\mathbf{x} \in \varepsilon_\tau^k(\mathbf{m}_\tau^k,M_\tau^k) \iff (\mathbf{x}-\mathbf{m}_\tau^k)^T M_\tau^k (\mathbf{x}-\mathbf{m}_\tau^k) \leq 1 .
\end{equation*}
The objective is to compute $p_{\Xi_\tau}$ for the defined obstacle set at forecast horizons $\tau=1,\dots,N$, that is, the probability
\begin{equation*}
p_{\Xi_\tau}=P\{\mathbf{x} \in \varepsilon_\tau^k(m_\tau^k,M_\tau^k) \} \quad \text{for some} \quad k \in \{1,\dots,K\}.
\end{equation*}

\label{subsec:MVE_sampling}
\begin{figure}[t!]
	\centering
	\includegraphics[width= 1\columnwidth]{figures/mve_sampling_2.pdf}

	\caption{50 MVE samples for forecast horizon 10 min.}
	\label{fig:MVE_sampling}
\end{figure}
To achieve this objective we introduce a sampling-based approach, here referred to as the \textit{Minimum-Volume-Ellipse (MVE)} sampling. Every time a new nowcast file is published (at time instance $t_0$), clustering is applied for $t=t_0$ (the current observations of the thunderstorms) and for $t=t_0+\tau$ with $\tau = 1, \dots, N$ (the thunderstorm's forecast at forecast horizons $\tau$). Then, for each of the $K$ clusters present at time $t + \tau$, we predict the position and size of all thunderstorms assigned to that cluster at forecast horizon $\tau$ using the stochastic thunderstorm cell model in subsection \ref{para:stoch_model}. We then enclose the predicted thunderstorms with K MVEs. This procedure is repeated $n_s$ times at each forecast horizon $\tau$, resulting in $n_s$ different MVEs per obstacle cluster. We can then apply Monte Carlo methods to compute the probability $p_{C_\tau^k}(x,y)$ of coordinate $(x,y)\in  X \times Y$ being within the obstacle cluster $C_\tau^k$ at time $t_0 + \tau$.

Figure \ref{fig:MVE_sampling} shows an example for such MVE realizations for a forecast horizon index $\tau$ equivalent to 10 minutes. The black ellipses bound the clustered thunderstorms as provided by the deterministic (uncorrected) data in the nowcast. For each cluster, $n_s = 50$ realizations of the thunderstorm development up to the forecast horizon of 10 minutes are sampled and bounded by an MVE (depicted in grey). This allows to account for the uncertain evolution of the thunderstorms over the 10 minute horizon.

Once the \textit{MVE Sampling} algorithm has been applied to all $K$ obstacle clusters within a nowcast for all forecast horizons $\tau=1,2,\dots,N$, the $K$ individual functions $p_{C^k_\tau}$ for $k=1,\dots,K$ at forecast horizon $\tau$ need to be  merged into the function $p_{\Xi_\tau}$ that returns the probability of $(x,y)$ being in any of the $K$ clusters. We can now approximate the  probability of a coordinate $(x,y)$ being in any of the obstacle clusters as
\begin{equation}
\label{eq:prob_event}
p_{\Xi_\tau}(x,y) \approx \underbrace{1-\underbrace{\prod_{k=1}^K (1-p_{C^k_\tau}(x,y))}_{\text{probability of no thunderstorm at $(x,y)$}}}_{\text{probability of at least one thunderstorm at $(x,y)$}}.
\end{equation}
 In the approximation above, we assume that the event of a coordinate $(x,y)$ falling into a certain cluster is independent of the event of the same coordinate falling into another cluster. While the event that a storm cell is present in a certain region and travels in a certain direction is certainly not independent from the presence of other storm cells in the same region and their movements, we minimize this dependence within the clusters using K-means. Indeed, K-means maximizes the between-cluster variance of all the storm observation vectors, that is, $\mathbf{o} =(x^c, y^c, \xi_{\tau})$. Furthermore, the above inequality is safely conservative. In particular,  denoting stochastic events of $(x,y)$ falling into clusters $C^k_\tau$ and $C^l_\tau$, for $k, l \in \{ 1, \dots, K\}$, with $A$ and $B$ respectively, we have $P(A \cap B) = P(A | B) P(B)$. Since in this setting it is more plausible that $P(A|B) \geq P(A)$ (knowing a cell belongs to one storm cluster increases its probability of belonging to the other storm cluster at the same time), we conclude that $P(A \cap B) \geq P(A) P(B)$. Hence, $1-P(A \cap B) \leq 1-P(A) P(B)$ and $p_{\Xi_\tau} \geq p_{\Xi_{\tau, \text{real}}}$, where $p_{\Xi_{\tau, \text{real}}}$ is the true (unknown) probability. Therefore, our approximation yields an upper bound on the true probability of coordinate $(x,y)$ being in any of the obstacle clusters. This will lead to a more conservative solutions but will not degrade safety.

Figure \ref{fig:covfct20} shows an example of the probability function $p_{\Xi_\tau}(x,y)$ that results for the nowcast provided at 00:00 UTC on December 19th 2016, at a forecast horizon of 30 minutes. The pink, dark boxes indicate thunderstorm cells that were predicted in the original nowcast and indeed observed 30 minutes after the nowcast was issued. The green, light boxes indicate thunderstorm cells that were observed 30 minutes after the nowcast was issued, but not predicted in the nowcast. The figure illustrates the predictive performance of the developed algorithm and the robustness of the clustering approach to newly appearing thunderstorms that are not predicted in the nowcast.
\begin{figure}[t!]
	\centering
		\includegraphics[width=1\linewidth]{figures/covering_fct_2.pdf}
\caption{Probability of storm $p_{\Xi_{30}}$ for nowcast published on Dec. 19th 2016 at 00:00 UTC.}
	\label{fig:covfct20}
\end{figure}


TITLE: Global applications of eigenvarieties
QUESTION [28 upvotes]: A Sunday question for specialists of eigenverieties:
In their important paper "the eigencurve", Coleman and Mazur globalized the earlier construction of Coleman's families, constructing a beautiful eponymous rigid analytic space
that parametrizes all systems of Hecke eigenvalues of finite-slope overconvergent modular forms. Since them many generalizations of this construction have been performed (the "eigenvarieties") which each time appear as globalizations of local constructions generalizing Coleman's families. This process of globalization has even been axiomatized by Buzzard ("the eigenvariety machine").
Yet, I wonder:

What are the benefits of working with a global object (which is
  considerably more difficult to construct and to deal with) rather than just the
  local objects with which it is constructed (the families of Coleman and their
  generalizations) ?

Of course, having a global, canonical, object is much more satisfying on esthetic grounds.
As a mathematician formed after Grothendieck's revolution, this reason alone would be for me
a sufficient one to consent the effort of constructing global eigenvarieties.
But my question is meant to be understood a little bit more specifically:

What are the applications, or expected applications (to our knowledge of the arithmetic of automorphic forms, Galois representations, L-functions, etc.) of the global existence and geometry of eigenverities that are not already consequences of the existence and geometry
  of their local pieces?

Of course, there are already an enormous amount, still growing fast, of arithmetic informations obtained from the local pieces of eigenvarieties. But what for the global structure? Let me mention the only one I know: the global existence of the eigencurve (say)
is necessary to be able to attached to any overconvergent finite slope modular form
a Galois representation. With  Coleman's families alone, we could construct those representations only for these forms having a weight sufficiently close p-adically to a non-negative integer (for example the one with negative weights). Yet  I find this application
not very convincing, as why do we care about overconvergent form with weight far away from integers except for their being the "flesh" of the eigencurve?
So what other applications do you have in mind?
(edited for one typo)

REPLY [7 votes]: This is something I've told people privately for a while, and now have enough ingredients written up (jointly with Liang Xiao) to claim on the web somewhere: the validity of the parity conjecture is constant in $p$-adic analytic families.  (Fine print: the family must be symplectic self-dual, and be equipped with a sort of Panchishkin triangulation, but that's all.)  Now that the triangulation of the entire eigencurve has been constructed, and the conjecture is known in weight two by any number of authors, it follows that the parity conjecture holds for any finite-slope form lying on an irreducible component of the eigencurve that admits a classical weight two point.  Thus, the parity conjecture for all finite-slope forms reduces to the claim that Buzzard's observation noted above in $p=2,N=1$ holds generally (apply Coleman's classicality theorem to the low-slope, weight two points near the boundary), which is a question of the global geometry of the eigencurve.


TITLE: Show that $f(a)=\frac{ ((a+1)^k+a^k+1 )^2}{2} -a^{2k}- (a+1)^{2k}-1$ is negative for $k>2$.
QUESTION [1 upvotes]: I would like to show the for
\begin{align}
f(a)=\frac{ ((a+1)^k+a^k+1 )^2}{2} -a^{2k}- (a+1)^{2k}-1
\end{align}
for $a\ge 0$ we have that
\begin{align}
f(a) \ge 0, \text{ for } k< 2\\
f(a) =0, \text{ for } k=2\\
f(a) \le 0 , \text{ for } k> 2
\end{align}
I am not quite sure fot inequality to use?
For example using $(c+d)^k \le 2^k(c+d)$ is not enough.

REPLY [1 votes]: If my computations are correct you have
$$-2f(a)=((a+1)^k-(a^{k/2}+1)^2)((a+1)^k-(a^{k/2}-1)^2) $$
hence this is
$$((a+1)^{k/2}-a^{k/2}-1)((a+1)^{k/2}+a^{k/2}+1)((a+1)^{k/2}-a^{k/2}+1)((a+1)^{k/2}+a^{k/2}-1)$$
Hence you have only to look at the sign of these $4$ expressions, and this seems not difficult (I have not done these last computations)
Added:
Note that $a^{2k}f(1/a)=f(a)$, so wlog we may suppose (if needed) that $a\geq 1$. Put $G(x)=(a+1)^x -a^x-1$. The derivative of $G$ is $G^{\prime}(x)=\log(a+1)(a+1)^x-(\log a) a^x=a^{x}(\log(a+1)(1+1/a)^x-(\log a))$, this is $\geq 0$ for $x\geq 0$. As $G(1)=0$, this prove the inequality for $0\leq x\leq 1$. For $x<0$, we have $(a+1)^x<1$, hence the inequality.


TITLE: Compute $\lim_{x\to0^+} \sqrt{x\left[\cos^2\left(\frac1{x^3}\right)-3\right]}$ using Squeeze theorem
QUESTION [1 upvotes]: This is for a first year calc course, and we are using the squeeze theorem to determine the limits of composite functions containing trig functions. I'm familiar with the examples posted in other questions on this website, but haven't seen anything addressing the squeeze theorem as applied to trig functions squared.
My question is how to use the squeeze theorem on the example below:

$$\lim_{x\to0^+} \sqrt{x\left[\cos^2\left(\frac1{x^3}\right)-3\right]}$$

Normally, I would set the upper and lower bounds of this as being $-1 ≤ f(x) ≤ 1$ because that is the range of the cosine function,
but I wasn't sure how the $\cos^2$ would affect this.
Any help would be greatly appreciated!

REPLY [0 votes]: Let's define your function as $f(x) = \sqrt{x\cos^2(x^{-3}) - 3x}$.
One way to think about it is to consider $f$ as a composition of several functions, and just "build up" the necessary inequality by bounding one of the inner functions, and transforming the inequality by each composing function, being sure to flip the signs whenever multiplying by a negative/composing by any decreasing function.
Specifically, for $x \le 0$:
$$
    -\infty \le x^{-3} \le \infty  \\
    -1 \le \cos(x^{-3}) \le 1  \\
    0 \le \cos^2(x^{-3}) \le 1  \\
    0 \ge x\cos^2(x^{-3}) \ge x  \\
    -3x \ge x\cos^2(x^{-3}) - 3x \ge -2x  \\
    \sqrt{-3x} \ge \sqrt{x\cos^2(x^{-3}) - 3x} \ge \sqrt{-2x}
$$
So you can use the squeeze theorem between the functions $\sqrt{-3x}$ and $\sqrt{-2x}$ to show that the limit from the left is $0$.
Regarding your question about the range of $\cos^2(x)$... without worrying about cosine for just a moment, how would the function $g(x) = x^2$ affect the real interval $[-1, 1]$?
In the same vein as above - think of $\cos^2$ as a composition of two functions, $\cos^2 = h\circ g$, where $g : x \mapsto x^2$, and where $h : x \mapsto \cos(x)$.

The image of the real number line $\mathbb{R}$ under $h$ is: $\;h(\mathbb{R}) = [-1, 1]$.
The image of $[-1, 1]$ under $g$ (think of the range along the $y$-axis of a parabola from $(-1, 1)$ down to $(0, 0)$ and back up to $(1, 1)$) is: $\;g([-1, 1]) = g(h(\mathbb{R})) = [0, 1]$.
Thus, the image of $\mathbb{R}$ under $h \circ g$ is: $\;(h \circ g)(\mathbb{R}) = [0, 1]$.


TITLE: Bayes' formula conditional probabilities
QUESTION [1 upvotes]: I am having troubles finding out $P(A)$ and $P(B)$ in the following problem:

The specificity of a test is $99.6\%$ and the sensitivity is
$99.7\%$. $1\%$ is expected to have had an infection
at some point. We are
interested in the probability of having been infected
. Define the events $A = \text{"Have had an infection”}$ and $B = \text{”test is positive”}$.
What are the odds for $A$ given the evidence $B$?

I know that I need to calculate $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ but how do I find $P(A)$ and $P(B)$ from the information given?

REPLY [1 votes]: In the information it is given that '$1\%$ is expected to have had it at some point', which in this question translates to $\mathbb{P}(A) = 0.01$.
A trick in using Bayes' formula is to rewrite the denominator in terms of conditional probabilities. That is, we can write:
\begin{align*}
\mathbb{P}(B) = \mathbb{P}(B | A^c)\mathbb{P}(A^c) + \mathbb{P}(B | A)\mathbb{P}(A) =
(1-\mathbb{P}(B^c | A^c))\mathbb{P}(A^c) + \mathbb{P}(B | A)\mathbb{P}(A)
\end{align*}
and all the probabilities in the latter expression are given in the question.


text:
\section{Matching Time $\sim$ Driver and Rider Flow}\label{app:explain}

Equation (\ref{eq:waiting}) can be better explained graphically in Figure \ref{fig:match_time}. Line 1 and 2 indicates the accumulative drivers (slope $f_{1}$) and riders (slope $f_{2}$) arrival curves, respectively. Line 3 represents the accumulative matching (with slope $m$ calculated from Cobb-Douglas matching function). The area of the shadow areas indicate the total delay of drivers (or riders) till time period $T$, from which, the average delay per driver ($t_1$) and per rider ($t_2$) can be calculated. By varying $f_1$ and $f_2$, we are able to calculate $t_1$ and $t_2$ numerically, which will be used for estimating parameters in (\ref{eq:waiting_time_flow}).
\begin{figure}[htbp]
\begin{center}
    \includegraphics[width=0.5\textwidth]{./Figures/Graph.pdf}
    \vspace{-1cm}
\caption{Graphical Representation of Matching Time}
\label{fig:match_time}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
    \includegraphics[width=0.7\textwidth]{./Figures/regression.pdf}
\caption{Regression Summary Output of (\ref{eq:waiting_time_flow})}
\label{fig:regression_output}
\end{center}
\end{figure}

config: arxiv


TITLE: Inequality prove $ e^x \ge 1+x $
QUESTION [1 upvotes]: I want to prove that $ e^x \ge 1+x $ for all $ x \in R $ , using Mean Value Theorem it can be proved for $ x \gt 0 $, and equality holds for $ x = 0$, however I can't solve it for $ x \lt 0 $

REPLY [1 votes]: First we take note that $e^0=1+0$ satisfies our inequality. Then as Dr. Sonnhard pointed out let us consider the function $f(x)=e^x-x-1$. The derivative of this function would be:
$$f'(x)=e^x-1$$
If we consider $x \leq 0$, this implies  $e^x \leq 1$. Which means $e^x-1 \leq 0$, thus $f(x)$ is decreasing for any negative $x$.
So now we are sure (because $f(0)=0$ and $f(x)$ is decreasing as $x$ increases) that $f(x<0)>0$. So now we have.
For $x<0$,  $e^x-x-1>0$
With some algebra:
For $x<0$, $e^x > x+1$


TITLE: How to calculate percentage overlap between labels when given the number of labels, number of labeling cycles and number or runs?
QUESTION [0 upvotes]: I need some help with a problem that i think it involves combinatorics and integration.
Given: a number k of labels, a number n of objects, a number c of labeling cycles;
Each of the n objects receives by random assignment with replacement (pardon my R-like language) one of the labels. This is repeats c times (number of labeling cycles). At the end each object will have a combination of c labels out of k available labels. In the process some of the objects will end up having identical label combinations. Let's call these events barcode collisions.
My question is how to derive a formula to calculate barcode collision rate given k labels, n objects and c labeling cycles?
N.B. This applies to a real problem involving labeling biological cells for RNA sequencing. The known variables in this techniques are the number of labels, the number of labeling cycles and the number of cells. HOWEVER here the number of cells is not the same as n in the problem description. This number of cells includes barcode collisions therefore the real number of cells that were analysed is actually n + (collision rate)/2.

REPLY [1 votes]: After all cycles, an object has one among $\binom{k}{c}$ label configurations.
There are $\binom{k}{c}^2$ ways that 2 objects can be configurated. Among these there are $\binom{k}{c}$ cases where both object has the same labels. So the probability that 2 objects constitude a barcode collision $$\binom{k}{c}^2/\binom{k}{c}=\binom{k}{c}$$


TITLE: Finding a number of twin primes less than a certain number
QUESTION [1 upvotes]: I was doing some problems on number theory, and I came across the following question: "How many twin primes less than 100 exist?" I was wondering if anyone could tell me what method would be used to solve this problem and others too.
Thanks!

REPLY [1 votes]: You can count the twin primes less than $n$ by hand.
You can find the number of twin primes less than $p_k!!$ (the product of primes not exceeding the $k$th prime) using the sieve S as modified in S' in this question, but there may not be any twin primes beyond some very large $n$. In any event the size of the numbers makes computation (of the prime pairs, if not their cardinality) difficult for even modest $k.$
For small $n$ the sieve S above is useful. For the original question, by S there are 3 pairs (n,n+2) co-prime to $2,3,5$ on an interval of length 30. Then we have 12 co-prime pairs on (1,120) and so on $[1,100]$  we cannot have more than 12 pairs of twin primes. We can quickly compute co-prime pairs and  we see that 10 pairs are in fact twin primes, 8 less than 100.
Version II of Chen's 1973 theorem says that for any positive even integer $h$ there are infinitely many primes $p$ such that $p+h$ is either a prime or semiprime. Since $h$ can be 2 there are infinitely many such pairs which are potentially twin primes. Up to boundaries which would not be much use for the OP, this may be the best we can currently say regarding existence of infinitely many twin primes. This is just a gloss on 2 above.

Halberstam and Richert's Sieve Methods is a really good resource for questions in this area.


TITLE: How to I solve this limit to -3
QUESTION [2 upvotes]: I need help solving this limit $\lim \limits_{x \to -3} \frac{4x+12}{3x^3-27x}$.
I know that I am suppose to factor the function and then plug in -3 to calculate the result.
$\lim \limits_{x \to -3} \frac{4(x+3)}{3x^3-27x}$. But don't know how to factor $3x^3-27x$.
How do I factor $3x^3-27x$? I'm I on the right path?
Thanks!

REPLY [4 votes]: $$
\begin{split}
\mathop {\lim }\limits_{x \to  - 3} \frac{{4x + 12}}
{{3{x^3} - 27x}} &= \mathop {\lim }\limits_{x \to  - 3} \frac{{4\left( {x + 3} \right)}}
{{3x\left( {{x^2} - 9} \right)}}\\
&= \mathop {\lim }\limits_{x \to  - 3} \frac{{4\left( {x + 3} \right)}}
{{3x\left( {x - 3} \right)\left( {x + 3} \right)}} \\
&= \mathop {\lim }\limits_{x \to  - 3} \frac{4}
{{3x\left( {x - 3} \right)}} \\
&= \frac{4}
{{3\cdot\left( { - 3} \right)\cdot\left( { - 3 - 3} \right)}} = \frac{2}
{{27}}
\end{split}
$$

REPLY [0 votes]: As your limit is of type $[\frac 00]$, you may use l'Hospitale rule, it's the most simple way to find this limit. You should just differentiate numerator and denominator and the limit will be the same. Hope you can continue by yourself from this.


TITLE: Graph of summation of absolute functions
QUESTION [0 upvotes]: Let $f$ be defined on $[-2,2]$ as
  $$f(x)=\begin{cases}-1&-2\le x\le0\\x-1&0\le x\le2\end{cases}$$
  If
  $g(x)=f(|x|)+|f(x)|$, then find g(x).

I have drawn $f(x)$:

How do I draw $g(x)$ from here?

REPLY [2 votes]: I think that a picture will be the clearest possible answer, but if you need more explanations just ask!


TITLE: What is the function associated with the following power series: $\sum_{n=2}^{\infty}x^n\;\;?$
QUESTION [2 upvotes]: What is the function associated with the following power series:
$$\sum_{n=2}^{\infty}x^n$$
If the sum started at 0 (if $n=0$) the function would be $\frac{1}{1-x}$, but with $n=2$, I am a bit confused.
Please pardon my poor english.

REPLY [3 votes]: $$ \sum_{n=2}^{\infty}x^n = x^2+x^3+x^4+...= x^2( 1+x+x^2+...)= \frac {x^2}{1-x}$$
Which converges for $|x|<1.$


TITLE: How to solve geometry question on internal tangency
QUESTION [0 upvotes]: Let $\Gamma_1$ be a circle with centre at the Point $O$ and radius $R$. Two other circles $\Gamma_2$ and $\Gamma_3$ with centres $O_2$ and $O_3$ respectively are internally tangent to $\Gamma_1$ and meet each other at Points $A$ and $B$. Find the sum of the radii of $\Gamma_2$ and $\Gamma_3$, given that angle $OAB=90^{\circ}$.

REPLY [0 votes]: The sum of radii of $\Gamma_2$ and $\Gamma_3$ is that of $\Gamma_1$. Let $C_2$ and $C_3$ be the points of intersection of $OA$ and the circles $O_2$ and $O_3$ respectively. We have that $C_2B$ and $C_3B$ are the diameters of circles $O_2$ and $O_3$ respectively, and $O_2$ and $O_3$ are their respective mid-points. So $O_2O_3$ is parallel to $C_2C_3$ and thus perpendicular to $AB$. Moreover $O_2O_3$ bisects $AB$. Consider the triangles $BO_2O_3$ and $OO_3O_2$. They share the common base $O_2O_3$ and by the above have the same heights. Moreover, by the internal tangency condition, $OO_3+R_3=OO_3+O_3B=OO_2+O_2B=OO_2+R_2$. So $O_2B-O_3B=OO_3-OO_2$. This forces $O_2B=OO_3$ and $O_3B=OO_2$, and $R_2+R_3=O_2B+O_3B=OO_3+O_3B=R$.


TITLE: Is the anti-derivative another way of saying the derivative?
QUESTION [4 upvotes]: This question is being posted so that I can understand a definition and concept.

The definition - F(x) is an anti-derivative of f(x) if F'(x) = f(x)

Therefore, $$F(x) = x^2$$ is an anti-derivative of $$f(x) = 2x$$ since $$F'(x) = 2x = f(x)$$

REPLY [0 votes]: Okay, this is a decent question as the terminology can get confusing so lets be very clear and precise about this: $$x^2+k$$ is the 'indefinite integral' of $$2x$$ since $$x^2+k$$ represents a "family" of quadratic curves all differing by a constant $k$. If we knew the value of $k$ we would then know the 'definite' integral.
$2x$ has many anti-derivatives (infinitely many); for example $x^2$ is an anti-derivative of $2x$ when $k=0$ and $x^2+4$ is another anti-derivative of $2x$ for the case when $k=4$ and so on.
Therefore an anti-derivative is just a particular case of the indefinite integral where you know the constant $k$; and hence an anti-derivative means exactly the same thing as a definite integral. Just two different words with the same meaning.
So the answer to the question in the title: "Is the anti-derivative another way of saying the derivative?" is no. But the rest you mentioned is fine. Just remember that indefinite integrals need a constant.


TITLE: Using visible light to get (weak) diffraction information about crystals?
QUESTION [1 upvotes]: If a crystal has diameter roughly 0.25mm containing unit cells (say cubes) of about 1 angstrom on a side, there are about 2.5 million unit cells in the diameter of the crystal. Bragg's law (n$\lambda = 2d\sin\theta)$ applies, and tells us that we would like to use wavelength on the order of 1 angstrom to probe the crystal.
But if $\lambda = 5000$ angstroms (a laser) it seems we could still get refraction of parallel waves separated by about 2500 angstroms, or about $0.25$mm/$2500$ angstrom $= 1000$ reinforcing refractions in the above crystal, considering only the one dimension. Hence my question--
Is it possible that while we cannot resolve structures smaller than a few angstroms without using x-rays, we might get information (albeit attenuated) about identical repeating structures (crystals) using longer wavelengths?
Edit:
The images below may help clarify. Above is radial average intensity calculated in python for the laser diffraction of zircon image (below, with central beam edited out). One one hand I wouldn't try to correlate the calculation with XRD calculations, on the other hand there seems to be some non-random information in the image. If nothing else, this may clarify my misconceptions.

Edit 2
Am adding the
 of ZrO2.
unedited photo of ZrO2.

REPLY [2 votes]: The following isn't really an answer, rather a rebuttal of sorts.

Using visible light to get (weak) diffraction information about
crystals

would of course a very interesting thing to do, which is why as a chemist it piqued my attention. But I fear that in accordance with common perception it isn't really possible and that which is on display in the question is something else.
I first ran a quick test with a monocrystalline Calcite crystal, about the size of my thumb. I shone a $532\mathrm{nm}$ ($8000\mathrm{mW}$) green pointer laser through it and obtained two weakened central beams and a completely random 'halo' of scattered light. No diffraction pattern whatsoever.
I then finely ground up some kitchen salt (chemically $\text{NaCl}$) in a mortar and pestle and loaded it into a glass capillary tube.
But trying to shine a laser beam through that is a waste of time: like most bulk transparent materials (glass, sucrose crystals, salt crystals etc) their ground up versions simply aren't transparent to VIS.
They would be to X-rays but that would yield circular diffraction patterns. Only monocrystallin materials would show the kind of diffraction pattern the OP shows in his question.
So I don't know what's causing the interesting pattern but I don't think it is diffraction related. But I'd like to know more about the OP's experimental set up to understand what might be producing the observed effect.


TITLE: When is "Inertia" Negligible?
QUESTION [10 upvotes]: I have started to read Strogatz's Nonlinear Dynamics and Chaos and I have come across an interesting bit. He states certain damped oscillators may be modeled as having no inertia term, I.E.
$$m \ddot{x} + b \dot{x} = F(x)$$
But if $m \ddot{x}\approx 0$ (he calls this the "Inertia Term")Then it is equally valid to write
$$b \dot{x} = F(x)$$
To me, this implies that the velocity is allowed to vary depending on the spatial coordinate. However I just said a second ago that either the acceleration or the mass is negligible. So shouldn't velocity be constant?
I think what is actually going on is while a temporal change in velocity is not allowed, a spatial one still may exist. Anyways, this is still a little strange to me so does anyone know of good resources on such subjects or times when this approximation is valid?
The equation appears on page 29, concerning the impossibility of oscillations for first order ordinary differential equations.
UPDATE:
After doing a bit of reflection and dimensional analysis, I have something else to add. $m$ has dimensions of [mass], while $b$ has dimensions of $\frac{[mass]}{[time]}$. Does this mean that a viable situation when this approximation holds is for some small time scales (how small and corresponding to what, I do not know). For if some time corresponding to $b$ is small, then $b$ is large, and maybe $m \ddot{x}$ can be neglected? Can someone verify this?

REPLY [2 votes]: Here's a "mathsy" explaination, but with a physical basis:
I don't know the book to which you refer but, in general, one can find which terms of an equation are dominant under given conditions through dimensional analysis.  Firstly, let us imagine that there are characteristic ("distinguishing") scales that describe the amplitude, period etc. of your system so that we may write $x = \chi\hat{x}$, $t = \tau\hat{t}$ and $F=\mathcal{F}\hat{F}$, where a "hat" denotes a dimensionless variable.  Then we rewrite your equation as
\begin{align}
  m \ddot{x} + b\dot{x} & = F(x), \\
  \frac{m\chi}{\tau^2}\frac{d^2\hat{x}}{d\hat{t}^2} + \frac{b\chi}{\tau}\frac{d\hat{x}}{d\hat{t}} & = \mathcal{F}\hat{F}(\hat{x}), \\
  \frac{m}{b\tau}\frac{d^2\hat{x}}{d\hat{t}^2} + \frac{d\hat{x}}{d\hat{t}} & = \frac{\mathcal{F}\tau}{b\chi}\hat{F}(\hat{x}). \\
\end{align}
Then chosing the distinguishing scale for $F$ to be $\mathcal{F} = b\chi/\tau$ we have
$$ \frac{m}{b\tau}\frac{d^2\hat{x}}{d\hat{t}^2} + \frac{d\hat{x}}{d\hat{t}} = \hat{F}(\hat{x}), $$
which is now a dimensionless equation, i.e. any pre-multiplying group (such as $m/b\tau$) has no dimensions, nor do the derivatives.  If the distinguishing scales for $x$ and $t$ are carefully chosen, the derivatives (without pre-multiplying groups) are said to be of "order 1" (i.e. $d\hat{x}/d\hat{t} = O(1)$ etc.)  It may now be apparent that if the group $m/b\tau \ll 1$ then the second-order derivative is of lesser importance than the other terms and can be "dropped" from the leading-order balance, leading to
$$ \frac{d\hat{x}}{d\hat{t}} = \hat{F}(x), $$
which is equivalent to your second equation.  What this is telling you physically is not that the inertial term "vanishes" (i.e. we are not setting $m\ddot{x} = 0$), but rather that its numerical value is so small that it plays only a minor role in determining the dynamics of the system.  In this case, the major dynamics are determined by "forcing" and "damping", or in other words the damping term dominates the inertial term in the physical balance.  Other conditions will give rise to other leading-order balances which is where the "inner" and "outer" solutions mentioned by another correspondent come in.


TITLE: Paths along graded posets
QUESTION [1 upvotes]: I'm wondering whether or not there are any resources available which deal with the number of distinct paths along a graded poset.  I am relatively new to this area so I don't know if this is an interesting question to ask in the first place.  But I've found some fairly interesting sequences from counting these paths on various posets, so I was curious as to whether this has been studied.
Edit: Here is an example: https://oeis.org/A002846.

REPLY [1 votes]: The topic comes up in Richard Stanley's Enumerative Combinatorics Volume 1, chapter 3. Edition 2 is freely available: http://math.mit.edu/~rstan/ec/ec1.pdf
Regarding examples, there are many. Consider the poset of an $(n+1)$-by-$(k+1)$ grid ordered by $(x,y) \leq (x',y') \iff x \leq x' \text{ and } y \leq y'$ graded by $\rho(x,y) = x + y$. The number of paths from $(0,0)$ to $(n,k)$ is $\binom{n+k}{k}$.
For an $(n+1)$-by-$(n+1)$ grid without points below the diagonal, using the same order and grading as in the previous example, the number of paths from $(0,0)$ to $(n,n)$ is the $n$-th Catalan number.
The number of paths from bottom to top in the Boolean lattice is $n!$.
For general facts, there are some nice theorems. For example, a distributive lattice and the poset of antichains in such a lattice are closely related. The number of maximal chains (ie paths) in a distributive lattice is the same as the number of linear extensions of the poset of antichains.
For a graded poset with unique top and bottom element, let $f(v)$ be the number of maximal chains from the bottom to vertex $v$. If $w_1,\ldots,w_k$ are predecessors of $v$, then $f(v) = f(w_1) + \cdots + f(w_k)$. Thus, even when there isn't a nice formula, the number of maximal chains in such a graded poset can be counted recursively.


text:
\begin{document}
\selectlanguage{english}
\bibliographystyle{amsplain}
{\vglue-12mm
\footnotesize
\leftline{\href{http://www.univie.ac.at/EMIS/proceedings/6ICDGA/IV/}
{\textbf{DIFFERENTIAL GEOMETRY AND APPLICATIONS}}}
\leftline{Proc. Conf., Aug.~28 -- Sept.~1, 1995, Brno, Czech Republic}
\leftline{Masaryk University, Brno 1996, 403--415}
\vskip32mm}
\begin{abstract}A variational equation of the third order in three-dimensional
space is proposed which describes autoparallel
curves of some connection
\end{abstract}
\maketitle
We shall focus on three-dimensional \hbox{(pseudo-)} \!Euclidean
space and consider the problem of finding a third-order variational-type
equation which can be put down in the form of the autoparallel transport equation
for some non-linear connection. It is common to introduce the latter
in the following form:
\begin{equation}
\dddot x^{\,\rho}=f^{\rho}(x^{\beta}, \dot x^{\beta}, \ddot x^{\beta})\,.
\label{1}
\end{equation}
On the other side, an Euler-Poisson
third-order equation is always of the affine type,
\begin{equation}
{\mathcal{A}}_{\rho\mu}\dddot x^{\,\mu}+k_{\rho}(x^{\beta}, \dot x^{\beta}, \ddot x^{\beta})=0\,,
\label{2}
\end{equation}
with a skew-symmetric matrix $\pmb{{\mathcal{A}}}=({\mathcal{A}}_{\rho\beta})$, and, consequently,
in the case when the number of equations equals {\it three\/}, can not be solved
with respect to the derivatives of the third order. What one can undertake in this
situation is at most to look for such a variational equation, which
describes the geodesic curve only up to reparametrization.

\section{General setting.}
One algorithm for building up an attached connection to a third order differential
equation of a certain class was presented in \cite{EvtushikKenerovo}, and we shall follow it here.
Although only \hbox{(pseudo-)} \!Euclidean space will be considered, to give the Reader a sense
of general setting, some constructions will be described as developed over
an $n$-dimensional manifold $M$. A differential equation of the third order
will be understood to be a cross-section of the third-order velocity
manifold
$T''M=J_{\sssize 3}({\mathbb{R}}_{\sssize 0};\, M)=\{\x;\u,\du,\ddu\}$,
fibred over the second order one,
$T'M=J_{\sssize 2}({\mathbb{R}}_{\sssize 0};\, M)=\{\x;\u,\du\}$.
These fibred manifolds are associated, as fibre bundles, to the principle
fibre bundles of the third-order and second-order frames,
$H''=\tilde J_{\sssize 3}({\mathbb{R}}^{n}_{\sssize 0};\, M)=\{x^{\rho};r^{\rho}_{\beta},r^{\rho}_{\beta\gamma},r^{\rho}_{\beta\gamma\delta}\}$,
and
$H'=\tilde J_{\sssize 2}({\mathbb{R}}^{n}_{\sssize 0};\, M)=\{x^{\rho};r^{\rho}_{\beta},r^{\rho}_{\beta\gamma}\}$,
where the tilde means that only invertible jets count, and also we shall denote
the inverse to the matrix $(r^{\rho}_{\beta})$ by $(\r^{\gamma}_{\delta})$.
The cotangent space to the manifold $H'$ is spanned by the following set
of differential forms (with coefficients from above the manifold $H''$)
\begin{align*}
\omega^{\rho}&\Doteq\r^{\rho}_{\mu}dx^{\mu}\,,\\
\omega^{\rho}_{\beta}&\Doteq\r^{\rho}_{\mu}dr^{\mu}_{\beta}
-\r^{\rho}_{\mu}r^{\mu}_{\beta\nu}\r^{\nu}_{\lambda}dx^{\lambda}\,,
\\
\begin{split}
\omega^{\rho}_{\beta\gamma}&\Doteq\r^{\rho}_{\mu}dr^{\mu}_{\beta\gamma}
-\r^{\rho}_{\mu}r^{\mu}_{\beta\nu}\r^{\nu}_{\lambda}dr^{\lambda}_{\gamma}
-\r^{\rho}_{\mu}r^{\mu}_{\gamma\nu}\r^{\nu}_{\lambda}dr^{\lambda}_{\beta}\,,
\\
&\qquad+\r^{\rho}_{\mu}r^{\mu}_{\beta\nu}\r^{\nu}_{\lambda}r^{\lambda}_{\gamma\iota}\r^{\iota}_{\sigma}dx^{\sigma}
+\r^{\rho}_{\mu}r^{\mu}_{\gamma\nu}\r^{\nu}_{\lambda}r^{\lambda}_{\beta\iota}\r^{\iota}_{\sigma}dx^{\sigma}
-\r^{\rho}_{\mu}r^{\mu}_{\nu\beta\gamma}\r^{\nu}_{\lambda}dx^{\lambda}\,.
\end{split}
\end{align*}
To span the cotangent space to the manifold $H''$ one more string of forms drops in
(we present their definition through a recurcive relation, which appears
more simple for any order as well),
$$
r^{\rho}_{\delta}\omega^{\delta}_{\iota\beta\gamma}=dr^{\rho}_{\iota\beta\gamma}
-r^{\rho}_{\mu\iota}\omega^{\mu}_{\beta\gamma}-r^{\rho}_{\mu\beta}\omega^{\mu}_{\iota\gamma}
-r^{\rho}_{\mu\gamma}\omega^{\mu}_{\beta\iota}-r^{\rho}_{\mu\iota\beta}\omega^{\mu}_{\gamma}
-r^{\rho}_{\mu\gamma\beta}\omega^{\mu}_{\iota}-r^{\rho}_{\mu\iota\gamma}\omega^{\mu}_{\beta}
-r^{\rho}_{\mu\iota\beta\gamma}\omega^{\mu}\,.
$$
These differential forms constitute a global object, intrinsically defined
in \cite{KobayashiCanonicalforms}.

Rather then proceed with the cross-section $f:\:T'M\to T''M$, one could wish to
develop some calculus on the corresponding principle bundles.
By the commutative diagram
\begin{equation}
\CD
H''\times{\mathbb{V}}''@>\rho''>>T''M\\
@A\Phi AA@AAfA\\
H'\times{\mathbb{V}}'@>>\rho'>T'M
\endCD
\label{3}
\end{equation}
the mapping $\Phi$ has to be both an equivariant one and a cross-section.
The typical fibre ${\mathbb{V}}''=J_{\sssize 3}({\mathbb{R}}_{\sssize 0};\, {\mathbb{R}}^{n}_{\sssize 0})
=\{U^{\rho},\dot U^{\rho},\ddot U^{\rho}\}$ undergoes such a left action
of the group
$GL''(n)=\tilde J_{\sssize 3}({\mathbb{R}}^{n}_{\sssize 0};\, {\mathbb{R}}^{n}_{\sssize 0})
=\{s^{\rho}_{\beta},s^{\rho}_{\beta\gamma},s^{\rho}_{\beta\gamma\delta}\}$,
that the quotient map $\rho''(r.s,s^{\sssize-1}.U)=\rho''(r,U)$ is described
explicitly by
\begin{align*}
u^{\rho}&=r^{\rho}_{\mu}U^{\mu}\,,\\
\dot u^{\rho}&=r^{\rho}_{\mu}\dot U^{\mu}+r^{\rho}_{\mu\nu}U^{\mu}U^{\nu}\,,\\
\ddot u^{\rho}&=r^{\rho}_{\mu}\ddot U^{\mu}+3r^{\rho}_{\mu\nu}\dot U^{\mu}U^{\nu}
+r^{\rho}_{\mu\nu\lambda}U^{\mu}U^{\nu}U^{\lambda}\,.
\end{align*}
A tangent vector to the product manifold $H''\times{\mathbb{V}}''$,
$$
\pmb{{\frak a}}=a^{\rho}\p{x^{\rho}}+a^{\rho}_{\mu}\p{r^{\rho}_{\mu}}
+a^{\rho}_{\mu\nu}\p{r^{\rho}_{\mu\nu}}
+a^{\rho}_{\mu\nu\lambda}\p{r^{\rho}_{\mu\nu\lambda}}
+\upsilon^{\rho}\p{U^{\rho}}+\dot\upsilon^{\rho}\p{\dot U^{\rho}}
+\ddot\upsilon^{\rho}\p{\ddot U^{\rho}}\,,
$$
is vertical with respect to the projection $\rho''$ if and only if
\begin{gather*}
a^{\rho}=0\,,
\\
r^{\rho}_{\mu}\upsilon^{\mu}+a^{\rho}_{\mu}U^{\mu}=0\,,\\
r^{\rho}_{\mu}\dot\upsilon^{\mu}+a^{\rho}_{\mu}\dot U^{\mu}
+a^{\rho}_{\mu\nu}U^{\mu}U^{\nu}-2r^{\rho}_{\mu\nu}\r^{\mu}_{\lambda}a^{\lambda}_{\iota}U^{\nu}U^{\iota}=0\,,\\
\begin{split}
r^{\rho}_{\mu}\ddot\upsilon^{\mu}+a^{\rho}_{\mu}\ddot U^{\mu}+3a^{\rho}_{\mu\nu}\dot U^{\mu}U^{\nu}
+a^{\rho}_{\mu\nu\lambda}U^{\mu}U^{\nu}U^{\lambda}
\qquad\qquad\qquad\qquad\\
-3r^{\rho}_{\mu\nu}\r^{\mu}_{\lambda}U^{\nu}(a^{\lambda}_{\iota}\dot U^{\iota}
	+a^{\lambda}_{\iota\sigma}U^{\iota}U^{\sigma}
	-2r^{\lambda}_{\iota\sigma}\r^{\iota}_{\beta}a^{\beta}_{\delta}U^{\sigma}U^{\delta})
\qquad&\\
\qquad\qquad
-3r^{\rho}_{\mu\nu}\r^{\nu}_{\lambda}a^{\lambda}_{\iota}\dot U^{\mu}U^{\iota}
-3r^{\rho}_{\mu\nu\lambda}\r^{\lambda}_{\iota}a^{\iota}_{\sigma}U^{\mu}U^{\nu}U^{\sigma}
&=0\,.
\end{split}
\end{gather*}

If the map $\Phi$ is equivariant, then its Lie derivative with respect to
an arbitrary pair of vertical vector fields $\pmb{{\frak a}}^{\boldsymbol\prime\boldsymbol\prime}$
and $\pmb{{\frak a}}^{\boldsymbol\prime}$ on the manifolds $H''\times{\mathbb{V}}''$ and $H'\times{\mathbb{V}}'$
is zero (i.e. vector fields $\pmb{{\frak a}}^{\boldsymbol\prime\boldsymbol\prime}$
and $\pmb{{\frak a}}^{\boldsymbol\prime}$ are $\Phi$-related):
$$
T\Phi\,\circ\,\pmb{{\frak a}}^{\boldsymbol\prime\boldsymbol\prime}\,=\,\pmb{{\frak a}}^{\boldsymbol\prime}\,\circ\,\Phi\,.
$$
The map $\eta^{\sssize-1}\Phi:\:H''\times{\mathbb{V}}'\to H''\times{\mathbb{V}}''$,
induced by the projection $\eta:\:H''\to H'$,
$$
\eta^{\sssize-1}\Phi(r,U)=\Phi(\eta r,U)\,,
$$
is not fibred over the identity in $H''$. Nevertheless, there can always be found
an element $(\delta^{\rho}_{\mu},0,s^{\rho}_{\mu\nu\lambda})\in GL''(n)$
such that
\begin{equation}
r^{\rho}_{\sigma}s^{\sigma}_{\mu\nu\lambda}+r^{\rho}_{\mu\nu\lambda}
=\Phi^{\rho}_{\mu\nu\lambda}(x^{\beta};r^{\beta}_{\iota},r^{\beta}_{\iota\gamma},U^{\beta},\dot U^{\beta})\,.
\label{4}
\end{equation}
We define the fibred morphism $F$ over the identity in $H''$ as a family of
cross-sections $F(r)$ of the fibration $\pi:\:{\mathbb{V}}''\to{\mathbb{V}}'$ by means of
$$
F^{\rho}(x^{\beta};r^{\beta}_{\mu},r^{\beta}_{\mu\nu},r^{\beta}_{\mu\nu\lambda},U^{\beta},\dot U^{\beta})
=\ddot\Phi^{\rho}+s^{\rho}_{\iota\sigma\gamma}U^{\iota}U^{\sigma}U^{\gamma}\,,
$$
with $s^{\rho}_{\iota\sigma\gamma}$ defined from~(\ref{4}).
Then, by virtue of $\Phi$ equivariant, for every vertical vector field
$\pmb{{\frak a}}$ on the manifold $H''\times{\mathbb{V}}''$ we have
\begin{equation}
TF\,\circ\,(\,T(id)\times T\pi)\,\circ\,\pmb{{\frak a}}\,
=\,
\pmb{{\frak a}}
\,\circ\,F\,\circ\,(id\times\pi)\,,
\label{5}
\end{equation}
as can be seen from the next diagram by an appropriate `diagram chasing',
$$
\CD
TH''\times T{\mathbb{V}}''@<TF<<TH''\times T{\mathbb{V}}'\\
@| @|\\
TH''\times T{\mathbb{V}}''@>T(id)\times T\pi>>TH''\times T{\mathbb{V}}'\\
@A\pmb{{\frak a}}AA @AA(T(id)\times T\pi)\,\circ\,\pmb{{\frak a}}A\\
H''\times {\mathbb{V}}''@>id\times \pi>>H''\times{\mathbb{V}}'\\
@| @|\\
H''\times {\mathbb{V}}''@<F<<H''\times{\mathbb{V}}'
\endCD
$$

The kernel of $T\rho''$ annuls the following one-forms:
\begin{equation}
\begin{alignedat}{2}
&\omega^{\rho}\,,\\
&\triangle U^{\rho}&&\Doteq dU^{\rho}+U^{\mu}\omega^{\rho}_{\mu}\,,\\
&\triangle \dot U^{\rho}&&\Doteq d\dot U^{\rho}+\dot U^{\mu}\omega^{\rho}_{\mu}
+U^{\mu}U^{\nu}\omega^{\rho}_{\mu\nu}\,,\\
&\triangle \ddot U^{\rho}&&\Doteq d\ddot U^{\rho}+\ddot U^{\mu}\omega^{\rho}_{\mu}
+3U^{\mu}\dot U^{\nu}\omega^{\rho}_{\mu\nu}+U^{\mu}U^{\nu}U^{\lambda}\omega^{\rho}_{\mu\nu\lambda}\,.
\end{alignedat}
\label{6}
\end{equation}
By calculating the Lie derivative of the differential forms~(\ref{6}) along a vertical
vector field it turns out that the exterior differential system, generated by~(\ref{6}) is invariant under the action of the group $GL''(n)$ upon the manifold
$H''\times{\mathbb{V}}''$.

{\it If the functions $F^{\rho}$ satisfy~(\ref{5}), then the differential
forms
$$
\triangle F^{\rho}\Doteq dF^{\rho}+\ddot U^{\mu}\omega^{\rho}_{\mu}
+3U^{\mu}\dot U^{\nu}\omega^{\rho}_{\mu\nu}+U^{\mu}U^{\nu}U^{\lambda}\omega^{\rho}_{\mu\nu\lambda}
$$
expand into the differential forms~(\ref{6}) alone:}
\begin{equation}
\triangle F^{\rho}
=F^{\rho}{}^{\sssize2}_{\mu}\triangle\dot U^{\mu}
+F^{\rho}{}^{\sssize1}_{\mu}\triangle U^{\mu}
+F^{\rho}{}^{\sssize0}_{\mu}\omega^{\mu}\,.
\label{7}
\end{equation}

The concept of second order connection involves the quotient manifold
$\Gamma H'=TH'/GL'(n)$ with respect to the standard action of the group $GL'(n)$:
$$
a^{\rho}\p{x^{\rho}}+a^{\rho}_{\mu}\p{r^{\rho}_{\mu}}
+a^{\rho}_{\mu\nu}\p{r^{\rho}_{\mu\nu}}\mapsto
a^{\rho}\p{x^{\rho}}+s^{\nu}_{\mu}a^{\rho}_{\nu}\p{r^{\rho}_{\mu}}
+(s^{\lambda}_{\mu\nu}a^{\rho}_{\lambda}
+s^{\lambda}_{\mu}s^{\iota}_{\nu}a^{\rho}_{\lambda\iota})\p{r^{\rho}_{\mu\nu}}\,.
$$
The manifold $\Gamma H'$ projects onto the manifold $T''M$ by means of the following
mapping of $TH'$, compatible with this action:
$$
(a^{\rho},a^{\rho}_{\mu},a^{\rho}_{\mu\nu})\mapsto
(a^{\rho},a^{\rho}_{\mu}\r^{\mu}_{\nu}a^{\nu},
a^{\rho}_{\lambda\iota}\r^{\lambda}_{\mu}\r^{\iota}_{\nu}a^{\mu}a^{\nu}
\!\!-\!a^{\rho}_{\delta}\r^{\delta}_{\sigma}r^{\sigma}_{\lambda\iota}\r^{\lambda}_{\mu}\r^{\iota}_{\nu}a^{\mu}a^{\nu}
\!\!+\!a^{\rho}_{\iota}\r^{\iota}_{\lambda}a^{\lambda}_{\mu}\r^{\mu}_{\nu}a^{\nu})\,.
$$
\begin{definition}[\cite{EvtushikTretyakov_translation}]
A second order connection is given by a map $\gamma:\:T'M\to\Gamma H'$,
which is identity in $TM$.
\end{definition}
By means of the commutative diagram
$$
\CD
\Gamma H'@>>>T''M\\
@A\gamma AA@AA\widehat\gamma A\\
T'M@=T'M
\endCD
$$
every connection $\gamma$ defines a morphism of manifolds $\widehat\gamma$.
The very similar way to~(\ref{7}) this map $\gamma$
may be described through the structure equations,
\begin{alignat*}{2}
&\triangle\Gamma^{\rho}_{\beta}
&&=\Gamma^{\rho}_{\beta}{}^{\sssize2}_{\mu}\triangle\dot U^{\mu}
+\Gamma^{\rho}_{\beta}{}^{\sssize1}_{\mu}\triangle U^{\mu}
+\Gamma^{\rho}_{\beta}{}^{\sssize0}_{\mu}\omega^{\mu}\,,\\
&\triangle\Gamma^{\rho}_{\beta\gamma}
&&=\Gamma^{\rho}_{\beta\gamma}{}^{\sssize2}_{\mu}\triangle\dot U^{\mu}
+\Gamma^{\rho}_{\beta\gamma}{}^{\sssize1}_{\mu}\triangle U^{\mu}
+\Gamma^{\rho}_{\beta\gamma}{}^{\sssize0}_{\mu}\omega^{\mu}\,,
\end{alignat*}
where the differential forms $\triangle\Gamma^{\rho}_{\beta}$, $\triangle\Gamma^{\rho}_{\beta\gamma}$
are build up from the differentials of the functions ($\Gamma^{\rho}_{\beta},\, \Gamma^{\rho}_{\beta\gamma}$),
which represent the map $\gamma$, as follows:
\begin{alignat*}{2}
&\triangle\Gamma^{\rho}_{\beta}&&\Doteq d\Gamma^{\rho}_{\beta}
+\Gamma^{\mu}_{\beta}\omega^{\rho}_{\mu}
-\Gamma^{\rho}_{\mu}\omega^{\mu}_{\beta}
+U^{\mu}\omega^{\rho}_{\beta\mu}\,,\\
&\triangle\Gamma^{\rho}_{\beta\gamma}&&\Doteq d\Gamma^{\rho}_{\beta\gamma}
+\Gamma^{\mu}_{\beta\gamma}\omega^{\rho}_{\mu}
-\Gamma^{\rho}_{\beta\mu}\omega^{\mu}_{\gamma}
-\Gamma^{\rho}_{\mu\gamma}\omega^{\mu}_{\beta}
+\Gamma^{\mu}_{\beta}\omega^{\rho}_{\mu\gamma}
+\Gamma^{\mu}_{\gamma}\omega^{\rho}_{\mu\beta}
-\Gamma^{\rho}_{\mu}\omega^{\mu}_{\beta\gamma}
+U^{\mu}\omega^{\rho}_{\beta\gamma\mu}\,.
\end{alignat*}

Connection $\gamma$ is called {\it stable\/} if it projects onto the identity
in $T'M$. In this case the morphism $\widehat\gamma$ is a cross-section and
thus defines a third-order differential equation of type~(\ref{1}).

To discuss a weaker condition of a quasi-stable connection, we recall that
the group $GL'(1)$ acts on the right upon the space $T'M$ by parameter transformations.
The generators are:
\begin{align*}
\pmb{{\frak p}}_{\sssize\bold1}&=u^{\rho}\p{u^{\rho}}+2\dot u^{\rho}\p{\dot u^{\rho}}\,,\\
\pmb{{\frak p}}_{\sssize\bold2}&=u^{\rho}\p{\dot u^{\rho}}\,.
\end{align*}
The quotient space with respect to this action is the manifold $C'M$ of contact
elements, locally arranged as ${\mathbb{R}}\times T'{\mathbb{R}}^{n-1}$. Connection $\gamma$
is said to be {\it quasi-stable\/} if it projects onto the identity in $C'M$.

In case of quasi-stable connection it is possible to introduce\cite{EvtushikKenerovo,EvtushikTretyakov_translation} the
notion of parallel transport in such a way, that the autoparallel curves of this
connection will be described in the typical fibre ${\mathbb{V}}''$ of the fibre bundle $T''M$
by means of the equation
\begin{equation}
\ddot U^{\rho}=\Gamma^{\rho}_{\mu}\dot U^{\mu}+\Gamma^{\rho}_{\mu\nu}U^{\mu}U^{\nu}
+\lambda^{\sssize(2)}\dot U^{\rho}+\lambda^{\sssize(1)}U^{\rho}\,.
\label{9}
\end{equation}
If the quasi-stable connection $\gamma$ is stable, the
functions $\lambda^{\sssize(1)}$ and $\lambda^{\sssize(2)}$ both vanish.

Not every equation~(\ref{1}) can be rearranged in the form~(\ref{9}).
The crucial idea consists in applying a somewhat technical trick of reparametrization.
If the map $f$ in~(\ref{1}) or~(\ref{3}) defines in the consistent manner
some equation on the manifold $C'M$, and if we think of $f$ as of a vector field $\pmb{{\frak f}}$
on the manifold $T'M$ by the inclusion $T''M\hookrightarrow TT'M$, then the Lie brackets
$
[\pmb{{\frak p}}_{\sssize1},\pmb{{\frak f}}]$ and $[\pmb{{\frak p}}_{\sssize2},\pmb{{\frak f}}]
$
differ from a multiply of $\pmb{{\frak f}}$ by some vertical field with respect to the projection
\begin{equation}
\wp:\:T'M\to C'M\,.
\label{10}
\end{equation}
In fact, a stronger condition holds:
$$
\left\{
\begin{aligned}
(T\wp)\,[\pmb{{\frak p}}_{\sssize\bold1},\pmb{{\frak f}}]&=(T\wp)\,\pmb{{\frak f}}\\
(T\wp)\,[\pmb{{\frak p}}_{\sssize\bold2},\pmb{{\frak f}}]&=0
\end{aligned}
\right.
$$
In terms of the representation~(\ref{7}) the above condition amounts
to the following two equations with Lagrange multiplies $\mu$ and $\lambda$,
\begin{align}
3F^{\rho}-F^{\rho}{}^{\sssize1}_{\iota}U^{\iota}
-2F^{\rho}{}^{\sssize2}_{\iota}\dot U^{\iota}&=3\mu U^{\rho}
\label{11}
\\
3\dot U^{\rho}-F^{\rho}{}^{\sssize2}_{\iota}U^{\iota}&=3\lambda U^{\rho}\,.
\label{12}
\end{align}
The multipliers $\mu$ and $\lambda$ are functions on the manifold $H'\times{\mathbb{V}}'$,
and in order them to represent some well-defined functions on the manifold
$T'M$, they both have to satisfy the condition of $GL'(n)$-invariance of the type
$
\langle\pmb{{\frak a}},d\mu\rangle=0
$
for any $\rho'$-vertical vector $\pmb{{\frak a}}$, which amounts to the following system of partial
differential equations:
\begin{equation}
\begin{gathered}
\pp{\mu}{\dot U^{\rho}}U^{\beta}U^{\gamma}=r^{\nu}_{\rho}\pp{\mu}{r^{\nu}_{\beta\gamma}}\\
\pp{\mu}{U^{\rho}}U^{\beta}-\pp{\mu}{\dot U^{\rho}}\dot U^{\beta}
=2r^{\delta}_{\rho\nu}\pp{\mu}{r^{\delta}_{\beta\nu}}+r^{\delta}_{\rho}\pp{\mu}{r^{\delta}_{\beta}}\,.
\end{gathered}
\label{13}
\end{equation}
\begin{definition}[\cite{EvtushikTretyakov_translation}]
The equation~(\ref{1}) is {\it reducible\/} if (\ref{11}, \ref{12}) holds for the
representation~(\ref{7}). It will be called {\it strictly reducible\/}
if both $\mu=0$ and $\lambda=0$.
\end{definition}

Consider now a (second order nonlinear) connection,
the coefficients  $\Gamma^{\rho}_{\beta}$, $\Gamma^{\rho}_{\beta\gamma}$
of which are constructed from the coefficients of the first-order prolongation
of the differential system~(\ref{7}),
$$
dF^{\rho}{}^{\sssize2}_{\beta}+F^{\mu}{}^{\sssize2}_{\beta}\omega^{\rho}_{\mu}
-F^{\rho}{}^{\sssize2}_{\mu}\omega^{\mu}_{\beta}+3U^{\mu}\omega^{\rho}_{\beta\mu}
=F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize0}_{\mu}\omega^{\mu}+F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize1}_{\mu}
\triangle U^{\mu}+F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize2}_{\mu}\triangle \dot U^{\mu}\,,
$$
according to the following prescription:
\begin{equation}
\begin{alignedat}{2}
&\Gamma ^{\rho}_{\beta}&&={\tsize\frac13}F^{\rho}{}^{\sssize2}_{\beta}\,;\\
&\Gamma ^{\rho}{}_{\beta\gamma}&&={\tsize\frac12}(\Pi^{\rho}_{\beta\gamma}+\Pi^{\rho}_{\gamma\beta})\,,
\quad\text{where} \\
&\Pi^{\rho}_{\beta\gamma}&&={\tsize\frac13}F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize0}_{\gamma}
+{\tsize\frac19}(F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize2}_{\mu}F^{\mu}{}^{\sssize1}_{\gamma}
+F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize1}_{\mu}F^{\mu}{}^{\sssize2}_{\gamma})
+{\tsize\frac2{27}}F^{\rho}{}^{\sssize2}_{\beta}{}^{\sssize2}_{\mu}F^{\mu}{}^{\sssize2}_{\nu}F^{\nu}{}^{\sssize2}_{\gamma}\,.
\end{alignedat}
\label{14}
\end{equation}

Let us agree to call the connection, constructed accordingly to the formulae~(\ref{14}), as one, {\it attached\/} to the differential equation~(\ref{1})
\begin{proc}[\cite{EvtushikKenerovo}]
The connection, attached to a reducible differential equation, is quasi-stable.
The equation~(\ref{9}) of the autoparallel curves  of the connection,
attached to a reducible differential equation,
coincides with the initial equation~(\ref{1}).
If~(\ref{1}) is strictly reducible, then the attached connection is stable.
\end{proc}
The functions $\lambda^{\sssize(1)}$ and $\lambda^{\sssize(2)}$ in~(\ref{9})
are expressed through the functions $\mu$ from~(\ref{11}) and
$\lambda$ from~(\ref{12}) in terms
of the coefficients of the differential $d\lambda$,
$$
d\lambda=\lambda^{\sssize0}_{\mu}\omega^{\mu}+\lambda^{\sssize1}_{\mu}\triangle U^{\mu}
+\lambda^{\sssize2}_{\mu}\triangle\dot U^{\mu}\,,
$$
according to the formulae below:
\begin{gather*}
\lambda^{\sssize(1)}=\lambda^{\sssize0}_{\nu}U^{\nu}+\lambda^{\sssize1}_{\nu}\dot U^{\nu}
\lambda^{\sssize2}_{\nu}\ddot U^{\nu}+\mu(1-\lambda^{\sssize2}_{\nu}U^{\nu})
-\lambda(\lambda^{\sssize1}_{\nu}U^{\nu}
+{\tsize\frac23}\lambda^{\sssize2}_{\nu}F^{\nu}{}^{\sssize2}_{\iota}U^{\iota})-2\lambda^{2}\,,
\\\lambda^{\sssize(2)}=2\lambda\,.
\end{gather*}

In view of~(\ref{13}),
$$
\lambda^{\sssize1}_{\rho}=\pp{\lambda}{U^{\rho}}\,,
\qquad\lambda^{\sssize2}_{\rho}=\pp{\lambda}{\dot U^{\rho}}\,.
$$
\section{Euclidean space. Variational equation.}
As declared, we look for a third order differential equation in \hbox{(pseudo-)} \!Euclidean
space ${\bold E}^{\sssize 3}$, which would be derivable from a Lagrangian.
The dimension of the space is {\it three}. As mentioned at the very beginning
of the present contribution, we cannot expect such equation to exist in the form,
solved with respect to the highest (i.e of the third order) derivatives.
So we shall first settle down on the manifold
$$
{\mathbb{R}}\times T'{\bold E}^{\sssize 2}
$$
and afterwards go all the way back to the manifold $T'{\bold E}^{\sssize 3}$ along the
projection of~(\ref{10}), which in the canonical coordinates is so expressed:
\begin{align*}
\text{\ssm t}\circ\wp&=x^{\sssize0}\\
\text{\ssm x}^{a}\circ\wp&=x^{a}\\
\text{\ssm v}^{a}\circ\wp&=\frac{u^{a}}{u^{\sssize0}} \\
\text{\ssm v}'{}^{a}\circ\wp&=\frac{\dot u^{a}}{u_{\sssize0}{}^{2}}
	-\frac{\dot u_{\sssize0}}{u_{\sssize0}{}^{3}}u^{a}\,.
\end{align*}

Let us concentrate on a system of two third-order ordinary differential equations
\begin{equation}
{\text{\ssm E}}_{a}=0\,.
\label{15}
\end{equation}
We introduce a vector valued differential one-form
\begin{equation}
\boldsymbol\epsilon={\text{\ssm E}}_{a}\,{{d}}{\text{\ssm x}}^{a}\otimes{{d}}\text{\ssm t}\,,
\label{16}
\end{equation}
where the expressions ${\text{\ssm E}}_{a}$ are called the Euler-Poisson
expressions.\footnote{This is an alternative way to interpret the notion of
the Euler morphism, the latter having been considered by {\smc Kol\'a\v r}
in \cite{KolarNoveMesto}.}
Applying the general criterion of \cite{TulczyjewResolution} for an arbitrary system
of differential equations to be a system of Euler-Poisson equations, it was
established in \cite{Theses} that the vector expression ${\text{\ssb E}}=\{{\text{\ssm E}}_{a}\}$
in~(\ref{15}) must have the shape
\begin{equation}
{\text{\ssb E}}={\mathbb{A}}{\,\boldkey.\,}{\text{\ssb v}}^{\boldsymbol\prime\boldsymbol\prime}{\,+\,}({\text{\ssb v}}^{\boldsymbol\prime}{\!\boldkey.\,}{\boldsymbol\partial}_{\pmb{\text{\sssm v}}})\,
{\mathbb{A}}{\,\boldkey.\,}{\text{\ssb v}}^{\boldsymbol\prime}{\,+\,}{\mathbb{B}}{\,\boldkey.\,}{\text{\ssb v}}^{\boldsymbol\prime}{\,+\,}{\text{\ssb c}}\,,
\label{17}
\end{equation}
where  the skew-symmetric matrix ${\mathbb{A}}$, the matrix ${\mathbb{B}}$, and the column vector
${\text{\ssb c}}$ depend on the variables $\text{\ssm t}$, ${\text{\ssb x}}$, ${\text{\ssb v}}={d{\text{\ssb x}}}/d\text{\ssm t}$,
and satisfy the following system of partial differential equations in $\text{\ssm t}$, ${\text{\ssm x}}^a$,
and ${\text{\ssm v}}^a$ \cite{Theses,SovPhys30_6}
\begin{equation}
\begin{gathered}
       \partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{[a}{}{\text{\ssm A}}_{bc]}=0 \\[1\jot]
       2\,{\text{\ssm B}}_{[ab]}-3\,{\bold D}_{\sssize{\bold 1}}{\,}{\text{\ssm A}}_{ab}=0\\[1\jot]
       2\,\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{[a}{}{\text{\ssm B}}_{b]c}
-4\,\partial_{_{_{_{{\text{\sssm x}}}}}}{\!}_{[a}{}{\text{\ssm A}}_{b]c}
+{\partial_{_{_{_{{\text{\sssm x}}}}}}{\!}_{c}}{\,}{\text{\ssm A}}_{ab}
+2\,{\bold D}_{\sssize{\bold 1}}{\,}{\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{c}}{\,}{\text{\ssm A}}_{ab}=0 \\[1\jot]
       {\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{(a}}{}{\text{\ssm c}}_{b)}
-{\bold D}_{\sssize{\bold 1}}{\,}{\text{\ssm B}}_{(ab)}=0\\[1\jot]
       2\,{\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{c}}{\,}\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{[a}{}{\text{\ssm c}}_{b]}
-4\,\partial_{_{_{_{{\text{\sssm x}}}}}}{\!}_{[a}{}{\text{\ssm B}}_{b]c}
+{{\bold D}_{\sssize{\bold 1}}}^{2}{\,}{\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{c}}{\,}{\text{\ssm A}}_{ab}
+6\,{\bold D}_{\sssize{\bold 1}}{\,\,}\partial_{_{_{_{{\text{\sssm x}}}}}}{\!}_{[a}{}{\text{\ssm A}}_{bc]}=0 \\[1\jot]
       4\,\partial_{_{_{_{{\text{\sssm x}}}}}}{\!}_{[a}{}{\text{\ssm c}}_{b]}
-2\,{\bold D}_{\sssize{\bold 1}}{\,\,}\partial_{_{_{_{{\text{\sssm v}}}}}}{\!}_{[a}{}{\text{\ssm c}}_{b]}
-{{\bold D}_{\sssize{\bold 1}}}^{3}{\,}{\text{\ssm A}}_{ab}=0\,.
\end{gathered}
\label{18}
\end{equation}
In~(\ref{18}) ${\bold {}D_{\sssize1}}$ and farther below ${\bold {}D_{\sssize2}}$ denote the generators of the Cartan distribution,
\begin{align*}
{\bold {}D_{\sssize2}}={\text{\ssb v}}^{\boldsymbol\prime}{\!\boldkey.\,}{\boldsymbol\partial}_{\pmb{\text{\sssm v}}}
{\,+\,}&{\bold {}D_{\sssize1}}\,,\\
&{\bold {}D_{\sssize1}}=\partial_{\text{\sssm t}}{\,+\,}{\text{\ssb v}}{\,\boldkey.\,}{\boldsymbol\partial}_{\pmb{\text{\sssm x}}}\,.
\end{align*}

Let $\boldsymbol\theta_{\sssize{\bold 2}}$, $\boldsymbol\theta_{\sssize{\bold 3}}$ denote the canonical contact
forms
\begin{align*}
\boldsymbol\theta_{\sssize{\bold 3}}
=\frac\partial{\partial {{\text{\ssm v}}'}^{a}}\otimes({{d}}{\text{\ssm v}'}^{a}
-{\text{\ssm v}''}^{a}{{d}}\text{\ssm t}){\,+\,}&\boldsymbol\theta_{\sssize{\bold 2}}\,,  \\
&\boldsymbol\theta_{\sssize{\bold 2}}
=\frac\partial{\partial {\text{\ssm v}}^{a}}\otimes({{d}}\text{\ssm v}^{a}
-{\text{\ssm v}'}^{a}{{d}}\text{\ssm t})
{\,+\,}\frac\partial{\partial {\text{\ssm x}}^{a}}\otimes({{d}}\text{\ssm x}^{a}
-\text{\ssm v}^{a}{{d}}\text{\ssm t}) \,.
\end{align*}

Along with the differential form $\boldsymbol\epsilon$
we introduce another one, $\pmb{\underline{\epsilon}}$,
\begin{align*}
\pmb{\underline{\epsilon}}
=\text{\ssm A}_{ab}\,{{d}}\text{\ssm x}^{a}\otimes{{d}}{\text{\ssm v}'}^{b}
{\,+\,}&\text{\ssm k}_{a}\,{{d}}\text{\ssm x}^{a}\otimes{{d}}\text{\ssm t}\,,\\
&\text{\ssb k}
=({\text{\ssb v}}^{\boldsymbol\prime}{\!\boldkey.\,}{\boldsymbol\partial}_{\pmb{\text{\sssm v}}})\,
{\mathbb{A}}{\,\boldkey.\,}{\text{\ssb v}}^{\boldsymbol\prime}{\,+\,}{\mathbb{B}}{\,\boldkey.\,}{\text{\ssb v}}^{\boldsymbol\prime}{\,+\,}{\text{\ssb c}}\,.
\end{align*}
Exterior differential systems, generated by the forms $\boldsymbol\epsilon$ and
$\pmb{\underline{\epsilon}}$, are equivalent:
$$
\pmb{\underline{\epsilon}}-\boldsymbol\epsilon
=(\text{\ssm A}_{ab}\,{{d}}\text{\ssm x}^{a}\otimes{{d}}{\text{\ssm v}'}^{b})
\barwedge\boldsymbol\theta_{\sssize{\bold 3}}\,.
$$

Now it is time to put in the concept of {\it symmetry}. Let
\begin{equation}
\pmb{{\frak x}}=\tau\frac\partial{\partial\text{\ssm t}}{\,+\,}{{\frak x}^{a}}\frac\partial{\partial\text{\ssm x}^{a}}
\label{19}
\end{equation}
denote the generator of some local group of transformations of the manifold ${\mathbb{R}}\times{{\bold E}}^{\sssize2}$,
its successive prolongations to the space
$J_{s}({\mathbb{R}};{{\bold E}}^{\sssize2})\approx{\mathbb{R}}\times T^{s}{{\bold E}}^{\sssize2}$
denoted by $\pmb{{\frak x}_{s}}$:
$$\pmb{{\frak x}_{\sssize 2}}={\frak v}^{a}\frac\partial{\partial{\text{\ssm v}'}^{a}}
{\,+\,}\pmb{{\frak x}_{\sssize 1}}\,.
$$

The demand that the exterior differential system, generated by the vector
valued differential form $\pmb{\underline{\epsilon}}$, be invariant under the
infinitesimal transformation $\pmb{{\frak x}}$ incarnates into the following
equation\footnote{The notion of vector bundle valued exterior differential systems
invariance was introduced in \cite{Methods}, see also \cite{SymVectorForms}.}
\begin{equation}
{\bold L}(\pmb{{\frak x}_{\sssize 2}})(\pmb{\underline{\epsilon}})=\boldsymbol\Xi{\,\boldkey.\,}
\pmb{\underline{\epsilon}}{\,+\,}\boldsymbol\beta\barwedge\boldsymbol\theta_{\sssize{\bold 2}}\,,
\label{20}
\end{equation}
where the elements of the matrix
$\boldsymbol\Xi$
and the coefficients of the semi-basic $T^{\ast}{{\bold E}}^{\sssize2}$-valued one-form
$\boldsymbol\beta$
depend upon the variables $\text{\ssm t}$, $\text{\ssb x}$, $\text{\ssb v}$, and ${\text{\ssb v}}^{\boldsymbol\prime}$.
Both $\boldsymbol\Xi$ and $\boldsymbol\beta$ play the role of Lagrange multipliers.
Splitting of equation~(\ref{20})
with respect to independent differentials
${{d}}\text{\ssm t}$, ${{d}}\text{\ssm x}^{a}$, ${{d}}\text{\ssm v}^{a}$, and
${{d}}{\text{\ssm v}'}^{a}$, results in the following system of partial differential
equations
\begin{equation}
\begin{aligned}
{\bold L}(\pmb{{\frak x}_{\sssize 1}})\text{\ssm A}_{ab}&={\Xi_{a}}^{c}\,\text{\ssm A}_{cb}
-\text{\ssm A}_{ac}\frac\partial{\partial{\text{\ssm v}'}^{b}}{\frak v}^{c}\\[1\jot]
{\bold L}(\pmb{{\frak x}_{\sssize 2}})\text{\ssm k}_{a}&={\Xi_{a}}^{b}\,\text{\ssm k}_{b}
-\text{\ssm A}_{ab}{\bold {}D_{\sssize2}}{\frak v}^{b}
-\text{\ssm k}_{a}{\bold {}D_{\sssize1}}\tau\,.
\end{aligned}
\label{21}
\end{equation}

\subsection{Variational problem in parametric form.}
Consider for a moment an $r^{\text th}$-order variational problem in parametric form, set by a Lagrangian
$$
\ell(\zeta, x^{\rho}, u^{\rho}, \dots,{\overset{\sssize r-1}u}{}^{\rho}){{d}}\zeta
$$
on the space $J_{r}({\mathbb{R}};M)$. As long as we limit ourselves only
to the case of autonomous Euler-Poisson equations,
\begin{equation}
{\mathcal{E}}_{\rho}=0\,,\;\text{see~(\ref{2})},
\label{22}
\end{equation}
the differential form
\begin{equation}
\boldsymbol\varepsilon={\mathcal{E}}_{\rho}\,{{d}}x^{\rho}\otimes{{d}}\zeta
\label{23}
\end{equation}
may {\it globally\/} be deprived of the factor ${{d}}\zeta$, constituting
thus a {\it globally\/} defined $T^{\ast}M$-valued density
\begin{equation}
\boldkey e={\mathcal{E}}_{\rho}{{d}}x^{\rho}\,.
\label{24}
\end{equation}

Now the projection $\wp:T^{r}M\to C^{r}M$ can be employed to
generate an autonomous variational problem set over $T^{r}M$ from every one
variational problem over $C^{r}M$.

\begin{proc}
In terms of a local chart, if in~(\ref{16}) the local semi-basic differential
form $\boldsymbol\epsilon$ corresponds to the Lagrangian
$$
L{{d}}\text{\ssm t}\,,
$$
then the vector valued density
\begin{equation}
\boldkey e=-u^{a}(\text{\ssm E}_{a}\circ\wp){{d}}x^{\sssize0}
{\,+\,}u^{\sssize0}(\text{\ssm E}_{a}\circ\wp){{d}}x^{a}
\label{25}
\end{equation}
corresponds to the Lagrangian
$$
\ell(\zeta, x^{\rho}, u^{\rho}, \dots,{\overset{\sssize r-1}u}{}^{\rho})\,{{d}}\zeta
={\mathcal{L}}(x^{\rho}, u^{\rho}, \dots,{\overset{\sssize r-1}u}{}^{\rho})\,{{d}}\zeta
$$
with the Lagrange function
\begin{equation}
{\mathcal{L}}=u^{\sssize0}L{\,\circ\,}\wp\,.
\label{26}
\end{equation}
\end{proc}

Let us return to the third-order case. The relations between quantities,
allocated on the space of contact elements $C'M\overset{\text{def}}{=}C^{2}M$ and the corresponding
quantities on the second-order velocity space $T'M\overset{\text{def}}{=}T^{2}M$,
expressed by (\ref{22}, \ref{24}, and~\ref{25}), say, that in~(\ref{2}) we have
$$
\boldkey k\;=\;(\,\bold{\dot{\boldkey u}}\,\boldkey.\,\boldsymbol\partial_{\boldkey u}\,)\;\pmb{{\mathcal{A}}}\,\boldkey.\,\bold{\dot{\boldkey u}}
\;+\;\pmb{{\mathcal{B}}}\,\boldkey.\,\bold{\dot{\boldkey u}}\;+\;\boldkey c
$$
with
\begin{equation}
{\mathcal{A}}_{ab}=(u^{\sssize0})^{-2}\cdot(\text{\ssm A}_{ab}\circ\wp),
\quad{\mathcal{B}}_{ab}=(u^{\sssize0})^{-1}(\text{\ssm B}_{ab}\circ\wp),\quad c_{a}=u^{\sssize0}(\text{\ssm c}_{a}\circ\wp)\,,
\label{27}
\end{equation}
and that the Weierstrass constraint holds:
$$
\pmb{{\mathcal{A}}}\,\boldkey.\,\boldkey u\equiv0,\quad\boldkey k\,\boldkey.\,\boldkey u\equiv0\,.
$$
\subsection{Circles and hyperbolae.}
Let in~(\ref{19}) generator $\pmb{{\frak x}}$ correspond to the \hbox{(pseudo-)} \!ortho\-gonal
transformations of a three-dimensional \hbox{(pseudo-)} \!Euclidean plain. Solving~(\ref{18}) together with~(\ref{21}),
we establish the expressions~(\ref{17}) for this case (see~\cite{CMPh1998} for more details):
\begin{equation}
{\text{\ssb E}}
=-\dfrac{\ast\text{\ssb v}^{\boldsymbol\prime\boldsymbol\prime}}
{(1+\text{\ssb v}\boldsymbol\cdot\text{\ssb v})^{3/2}}
\;+\;3\,\dfrac{\ast\text{\ssb v}^{\boldsymbol\prime}}
{(1+\text{\ssb v}\boldsymbol\cdot\text{\ssb v})^{5/2}}
\boldkey(\text{\ssb v}^{\boldsymbol\prime}\!\boldsymbol\cdot\text{\ssb v}\boldkey)
\;+\;\dfrac m{(1+\text{\ssb v}\boldsymbol\cdot\text{\ssb v})^{3/2}}
\bigl[(1+\text{\ssb v}\boldsymbol\cdot\text{\ssb v})\text{\ssb v}^{\boldsymbol\prime}
\,-\,\boldkey(\text{\ssb v}^{\boldsymbol\prime}\!\boldsymbol\cdot\text{\ssb v}\boldkey)\text{\ssb v}\bigr]\,.
\label{28}
\end{equation}
(The dual to some vector $\text{\ssb w}$ is known to be defined with the help of the
skew-symmetric Levi-Civita symbol $\text{\ssm e}_{ab}$ by means of
$(\ast\text{\ssb w})_{a}=\text{\ssm e}_{ba}\text{\ssm w}^{b}$.)
To convert~(\ref{28}) into a ``homogeneous'' three-dimensional form one applies~(\ref{27}) and obtains the final Euler-Poisson equations, which are
naturally connected to the second prolongation of the transformation group
$[{\bold E}(3,i),{{\bold E}}^{\sssize3}]$:
\begin{equation}
\boxed{
\pmb{{\mathcal{E}}}
=\dfrac{\bold{\ddot{\boldkey u}}\times\boldkey u}{\|\boldkey u\|^{3}}
\;-\;3\,\dfrac{\bold{\dot{\boldkey u}}\times\boldkey u}{\|\boldkey u\|^{5}}
\boldkey(\bold{\dot{\boldkey u}}\boldsymbol\cdot\boldkey u\boldkey)
\;+\;m\,\dfrac {\bold{\dot{\boldkey u}}\,\boldkey(\boldkey u\boldsymbol\cdot\boldkey u\boldkey)
\,-\,\boldkey u\,\boldkey(\bold{\dot{\boldkey u}}\boldsymbol\cdot\boldkey u\boldkey)}
{\|\boldkey u\|^{3}}=0
}
\label{29}
\end{equation}
Furthermore, we can indicate a general formula for the family of the Lagrange functions
which produce the expression~(\ref{29}):
$$
{\mathcal{L}}_{(\rho)}
\;=\;\dfrac{u^{\rho}[\bold{\dot{\boldkey u}},\boldkey u,\boldkey e_{(\rho)}]}
{\|\boldkey u\|\,\|\boldkey u\times\boldkey e_{(\rho)}\|^{2}}
\;-\;m\,\|\boldkey u\|
\;+\;\bold{\dot{\boldkey u}}\,\boldkey.\,\boldsymbol\partial_{\boldkey u}\;\phi
\;+\;\boldkey a\,\boldkey.\,\boldkey u\,,
$$
where an arbitrary row vector $\boldkey a$ is constant and a function $\phi$ depending on the variable
$\boldkey u$ is subject to the constraint $\boldkey u\,\boldkey.\,\boldsymbol\partial_{\boldkey u}\,\phi\;=\;0$.
(Recall the notation $[\;,\;,\;]$ for the parallelepipedal product of three vectors.)
The vector $\boldkey e_{\rho}$ denotes the $\rho$-th component of the
\hbox{(pseudo-)} \!Euclidean frame.
{\it Each ${\mathcal{L}}_{(\rho)}$ fits in.}

Although {\it there does not exist an invariant (even in extended sense)
Lagrange  function}, the equations~(\ref{29}) {\it are invariant\/} with respect to the group under consideration.
Namely, let
\begin{equation}
\pmb{{\frak x}''}\;=\;\boldkey[\boldsymbol\varpi, \boldkey x,
\boldsymbol\partial_{\boldkey x}\boldkey]
\;+\;\boldkey[\boldsymbol\varpi, \boldkey u, \boldsymbol\partial_{\boldkey u}\boldkey]
\;+\;\boldkey[\boldsymbol\varpi, \bold{\dot{\boldkey u}}, \boldsymbol\partial_{\bold{\dot{\boldkey u}}}\boldkey]
\;+\;\boldkey[\boldsymbol\varpi, \bold{\ddot{\boldkey u}}, \boldsymbol\partial_{\bold{\ddot{\boldkey u}}}\boldkey]
\label{30}
\end{equation}
stand for the third-order prolongation of the infinitesimal \hbox{(pseudo-)} \!Euclidean
transformations to the manifold $T''M$ with $\boldsymbol\varpi$ for the
group parameter. Then
$$
{\bold L}(\pmb{{\frak x}''})(\pmb{{\mathcal{E}}})\;=\;\boldsymbol\varpi\times\pmb{{\mathcal{E}}}\,.
$$
\begin{rem}
Assuming $m=0$ in~(\ref{29}), we recover geodesic circles as
integral curves, and in the case the index $i$ in ${\bold E}(3,i)$ equals $2$
this amounts to uniformly accelerated motion in three-dimensional special relativity.
\end{rem}
\section{Euclidean space. Connection.}
In a \hbox{(pseudo-)} \!orthonormal frame of reference, the corresponding third-order
frame takes on the shape
\begin{equation}
r^{\rho}_{\beta}=\delta^{\rho}_{\beta},\quad r^{\rho}_{\beta\gamma}=0,\quad
r^{\rho}_{\beta\gamma\nu}=0\,,
\label{31}
\end{equation}
so the structure forms $\omega^{\rho}_{\beta}$, $\omega^{\rho}_{\beta\gamma}$,
and $\omega^{\rho}_{\beta\gamma\nu}$ vanish and we can identify
$\u$, $\du$, and $\ddu$ with the ``invariant coordinates'' $U$, $\dot U$, and $\ddot U$ respectively.

In order to construct a connection, consistent with the equation~(\ref{29}),
we first supplement the two independent expressions, entering in~(\ref{29}), with an arbitrary additional one. Without loss of
generality we can search for the latter in the form
\begin{equation}
\ddu\,\boldkey.\,\u\;=\;\|\u\|^{2}\cdot\Psi(\u,\du)\,.
\label{32}
\end{equation}
The system of equations~(\ref{29} and~\ref{32}) can now be solved with
respect to the third-order derivatives to produce:
\begin{equation}
\ddu\;=\;3\,\dfrac{\du\bd \u}{\|\u\|^{2}}\,\du\;-\;3\,\dfrac{(\du\bd \u)^{2}}{\|\u\|^{4}}\u
\;-\;m\,\u\times\du\;+\;\Psi\cdot\u\,.
\label{33}
\end{equation}
Now we proceed further to define more precisely the arbitrary function $\Psi$.
With (\ref{11} and~\ref{12}) we calculate $\mu$ and $\lambda$ for the equation~(\ref{33}):
\begin{align}
\mu\;&=\;\dfrac1{3}\,\left(\,2\Psi\,-\,2\du\pp{\Psi}{\du}\,-\,\u\pp{\Psi}{\u}\,\right)\,,
\label{34}
\\
\lambda\;&=\;\dfrac{\u\bd\du}{\|\u\|^{2}}\,-\,\dfrac1{3}\,\u\pp{\Psi}{\du}\,.
\label{35}
\end{align}
In the reference frame~(\ref{31}) by virtue of~(\ref{13}) we conclude that
$\mu$ and $\lambda$ are constant. Then the compatibility conditions for the system
of partial differential equations~(\ref{34}) and~(\ref{35}) show, that $\lambda$
must be equal to zero,
\begin{equation}
\lambda=0\,.
\label{36}
\end{equation}
Set
$$
\Psi\;=\;\dfrac3{\|\u\|^{2}}\,\left(\,\psi\;+\;\dfrac1{2}\|\du\|^{2}\right)\,.
$$
For the function $\psi$ we now get
\begin{equation}
\left(\u\bd\p{\du}\right)\,\psi\;=\;0\,.
\label{37}
\end{equation}
Introducing the intermediate variable $\z=\u\times\du$ we see by~(\ref{37})
that the function $\psi$ depends on $\u$ and $\du$ via the variable $\z$ only.
Now we express the equation~(\ref{34}) in terms of $\z$ to get
\begin{equation}
3\,\z\bd\pp{\psi}{\z}\;=\;4\psi\;-\;\|\u\|^{2}\mu\,.
\label{38}
\end{equation}
Again, the compatibility conditions for~(\ref{38}) turn $\mu$ to zero,
\begin{equation}
\mu=0\,.
\label{39}
\end{equation}

To settle the matter definitely, we call upon the demand of \hbox{(pseudo-)} \!Euclidean
symmetry (with the generator~(\ref{30})) for the equation~(\ref{33}), which gives
\begin{equation}
\z\times\p{\z}\;\psi\;=\;0\,.
\label{40}
\end{equation}
Altogether (\ref{38}, \ref{39}, and~\ref{40}) produce for the determination of the function $\psi$
the equation
$$
\pp{\psi}{\z}\;=\;=\dfrac4{3}\,\dfrac{\z}{\|\z\|^{2}}\,\psi
$$
with the solution $\psi=\|\z\|^{4/3}$. Thus the function $\Psi$ has been found,
$$
\Psi\;=\;\dfrac3{2}\,\dfrac{\|\du\|^{2}}{\|\u\|^{2}}\;+\;3A\,\dfrac{\|\du\times\u\|^{4/3}}{\|\u\|^{2}},\qquad
\text{$A$ is a real number}.
$$

Let us introduce the following obvious definition:
\begin{definition}
For any smooth transformation group $G$ acting upon a manifold $M$ let a curve
$\sigma:\:I\to M, \;I\subset{\mathbb{R}}$ be called an {\it autogeodesic path\/} if
\begin{enumerate}
\item the image $\sigma(I)$ is an extremal submanifold of some parameter-independent
variation problem;
\item the curve $\sigma$  is autoparallel with respect to some (nonlinear, higher-order)
connection on $M$;
\item the corresponding autoparallel transport equation is $G$-invariant.
\end{enumerate}
\end{definition}

In view of the preceding considerations we now are capable of calculating the coefficients
$(\Gamma^{\rho}_{\beta},\Gamma^{\rho}_{\beta\gamma})$ of the connection, given by~(\ref{14}).
Rather then make this, it appears more economic to accomplish only with the presentation
of the explicit expression for the corresponding autogeodesic path
equation.
\begin{proc}
\noindent\begin{enumerate}
\item
The third-order
autogeodesic paths of the three-dimensional \hbox{(pseudo-)} \!Eucli\-dean
space are the solutions of the next differential equation:
{
\begin{equation}
\boxed{
\ddu=3\dfrac{\du\bd \u}{\|\u\|^{2}}\du-3\left[\dfrac{(\du\bd \u)^{2}}{\|\u\|^{4}}
-\dfrac{\|\du\|^{2}}{2\|\u\|^{2}}-A\dfrac{\|\du\times\u\|^{4/3}}{\|\u\|^{2}}\right]\u
-m\,\u\times\du
}
\label{41}
\end{equation}
}
\item The corresponding connection is stable.
\end{enumerate}
\end{proc}
\begin{proof}
It is necessary to calculate $\Gamma^{\rho}_{\beta}$ and $\Gamma^{\rho}_{\beta\gamma}$ from
(\ref{14}) and to show that the right-hand side of~(\ref{9}) coincides with
the right-hand side of~(\ref{41}). The second statement follows from~(\ref{36} and~\ref{39}). \end{proof}

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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }

\providecommand{\MRhref}[2]{
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}


TITLE: Rectangles in rectangles and $(b^2-a^2)^2\le (ax-by)^2+(bx-ay)^2$
QUESTION [6 upvotes]: When does an $a\times b$ rectangle fit inside an $x\times y$ rectangle? I have an algebraic condition which I can diagram geometrically, and I'd like a good geometric argument.
Assume $0<a<b$, $0<x<y$. Then the one rectangle fits in the other iff $a\le x$ and either:
$$b\le y$$
or:
$$(b^2-a^2)^2 \le (ax-by)^2+(ay-bx)^2$$
which I found algebraically, with help from Mathematica. One possible diagram is

and then the last condition is equivalent to
$$
|\boldsymbol{\alpha \times \beta}|^2 \le
|\boldsymbol{\alpha \times \gamma}|^2 +
|\boldsymbol{\beta \times \gamma}|^2
$$
This can also be interpreted in terms of the lengths of these three vectors and the sines of the angles between them.
Is there a nice geometric argument from this diagram to fitting the one rectangle inside the other?

REPLY [7 votes]: The sense of this inequality is that it turns out into equality when one of rectangles is inscribed in another as on the picture.
We get $ax-by=a(m+n)-b(p+q)=am-bp=a^2\cos \varphi-b^2\cos \varphi=(a^2-b^2)\cos \varphi$, analogously $ay-bx=a(p+q)-b(m+n)=aq-bn=(a^2-b^2)\sin \varphi$, and your relation $(a^2-b^2)^2=(ax-by)^2+(ay-bx)^2$ reads as $\cos^2\varphi+\sin^2 \varphi=1$.


TITLE: Dual subgradient method - can we solve approximation of dual?
QUESTION [0 upvotes]: Consider the problem to minimize $f(x)$ under the constraints $x \leq b$ and $x \in X$.
I Lagrange relax the constraint $x \leq b$ getting
$L(x,u) = f(x) + u^t(x-b)$.
When using the subgradient method, we set $u^0$ to some arbitrary value and treat it as fixed in each iteration, find $q(u^k) = \inf_{x\in X}L(x,u^k)$ and then take a step in a subgradient direction.
Suppose calculating $q(u^k) = \inf_{x\in X}L(x,u^k)$ is still hard in some sense. Can we solve an approximative problem instead? Will it still converge to a dual optimum? Under what conditions? Is there some way to bound the error?
Answers or references both appreciated.

REPLY [0 votes]: In general, you need to be able to solve the subproblem to $\epsilon$ optimality in order for approximate subgradients to converge. In the specific case the poster (me) was interested in, it would not have been possible. For convergence, you need to be able to guarantee the iterations brings you geometrically closer to the optimal solution set. There is no way to verify this is the case here since I wanted to approximately solve an NP-problem in each iteration.


text:
\subsection{Substructures of a Network}\label{sec2}
A \emph{subnetwork} $G$ of a network is a subset of vertices and edges satisfying the following conditions:
\begin{enumerate}
    \item If $u \in G$, $x \leftarrow u$ and $x \rightarrow v$, then $v \in G$.
    \item If $u \in G$ and $u \to v$, then $G$ contains $v$ and the edge between $u$ and $v$.
    \item Let $u,v \in G$. The path from $u$ to $v$ does not include the root.
\end{enumerate}
The topology of a subnetwork can thus be characterized as follows: It is either a network itself or a \emph{leaf subnetwork} (a set containing only leaves) or a combination of both. Figure \ref{fig:stepsize} illustrates a network with both types of subnetworks.

\begin{figure}[h]
\centering
\begin{tikzpicture}

\def\w{0.8}
\def\u{0.6}

\node[circle,fill=black!20] (1) at (1*\w,0*\u) {1};
\node[circle,fill=black!20] (2) at (-1*\w,-2*\u) {2};
\node[circle,fill=black!20] (3) at (4*\w,-2*\u) {3};
\node[circle,fill=black!20] (4) at (-2*\w,-4*\u) {4};
\node[circle,fill=black!20] (5) at (0*\w,-4*\u) {5};
\node[circle,fill=black!20] (6) at (2*\w,-4*\u) {6};
\node[circle,fill=black!20] (7) at (4*\w,-4*\u) {7};
\node[circle,fill=black!20] (8) at (6*\w,-4*\u) {8};

\draw[-,thick] (1) to (2);
\draw[-,thick] (1) to (3);
\draw[-,thick] (2) to (4);
\draw[-,thick] (2) to (5);
\draw[-,thick] (3) to (6);
\draw[-,thick] (3) to (7);
\draw[-,thick] (3) to (8);

\draw[black,thick,dashed] (-1*\w,-3.3*\u) ellipse (1.7*\w cm and 1.9*\u cm);
\draw[black,thick,dashed] (4*\w,-4*\u) ellipse (2.5*\w cm and 1.1*\u cm);

\end{tikzpicture}
\caption{A network with the two types of subnetworks}
\label{fig:stepsize}
\end{figure}

Throughout the paper, we assume that every leaf is included in a subnetwork. The purpose of each subnetwork is to isolate a substructure of the network, so we assume that the subnetworks are pairwise disjoint. We denote the subnetworks by $G_1$, $G_2$, ..., $G_c$ and denote the vertex that immediately precedes $G_i$ by $g_i$. We further denote the leaves in $G_i$ by $\ell_{i,1}$, $\ell_{i,2}$, ...,$\ell_{i,t_i}$. We last denote the root of the largest tree in $G_i$ with the leaf $\ell_{i,j}$ by $r_{i,j}$. For example, in Figure \ref{fig:stepsize}, we have the following:

\begin{itemize}
\item $G_1 = \{2,4,5\}$, $G_2 = \{6,7,8\}$
\item $g_1 = 1$, $g_2 = 3$
\item $\ell_{1,1} = 4$, $\ell_{1,2} = 5$, $\ell_{2,1} = 6$, $\ell_{2,2} = 7$, $\ell_{2,3} = 8$
\item $r_{1,1} = 2$, $r_{1,2} = 2$, $r_{2,1} = 6$, $r_{2,2} = 7$, $r_{2,3} = 8$
\end{itemize}

As each subnetwork is a forest of trees, we may interpret an iteration of $\vec{x}^{(n)}_{g_i}$ through the subnetwork $G_i$ as a weighted average of the iterations through the corresponding trees. We therefore define the following operators:
\begin{align}
P_{G_i}^{\vec{\omega}} &=\sum_{j=1}^{t_i}w(g_i,\ell_{i,j})P_{\ell_{i,j}}^{\vec{\omega}}...P_{r_{i,j}}^{\vec{\omega}},
\label{eq15}\\
P^{\vec{\omega}}_{G_i,r} &=P^{\vec{\omega}}_{G_i} P^{\vec{\omega}}_{g_i}...P^{\vec{\omega}}_r, \label{eq16}\\
P^{\vec{\omega}}&=\sum_{i=1}^c w(r,g_i)P_{G_i,r}^{\vec{\omega}}, \label{Pomega}
\end{align}
where $P^{\vec{\omega}}_v...P^{\vec{\omega}}_u$ with $u \preceq v$ is the composition of those operators $P^{\vec{\omega}}_x$ where $u \preceq x \preceq v$ in the appropriate order designated by the path from $u$ to $v$. We define analogous operators in $Q$.

We will show in Section \ref{inconsistent} that the substructures in a network generally admit relatively large relaxation parameters for convergence. We assume that the relaxation parameters satisfy certain admissibility conditions.

\begin{definition} \label{D:admissible}
We say that the relaxation parameters $\omega_{v}$ are \emph{admissible} provided that:
\begin{enumerate}
    \item If $v\not\in G_i$ for every $i$, then $\omega_v\in (0,2)$.
    \item For each $i \in \{1,2,...,c\}$, there exists a constant $\alpha_i < 1$ such that
    \[
    \lVert P_{G_i}^{\vec{\omega}}\vec{x}\rVert\le \alpha_i \lVert \vec{x}\rVert
    \]
    for all $\vec{x
    }\in\mathrm{span}\{\vec{a}_u : u\in G_i\}$.
\end{enumerate}
\end{definition}

\begin{lemma}\label{contractions}
If the relaxation parameters are admissible, then $P_{G_i}^{\vec{\omega}}$, $P_{G_i,r}^{\vec{\omega}}$ and $P^{\vec{\omega}}$ are contractions.
\end{lemma}

\begin{proof}
Suppose that $\vec{x} \in \{\vec{a}_u\,:\,u \in G_i\}^\perp$, the subspace orthogonal to the vectors in the set $\{\vec{a}_u\,:\, u \in G_i\}$. Then $P^{\vec{\omega}}_u\vec{x} = \vec{x}$ for every $u \in G_i$, and we have
\[
P^{\vec{\omega}}_{G_i}\vec{x} = \sum_{j=1}^{t_i}w(g_i,\ell_{i,j})P^{\vec{\omega}}_{\ell_{i,j}}\cdots P^{\vec{\omega}}_{r_{i,j}}\vec{x}= \vec{x}.
\]
Since $\text{span}\{\vec{a}_u\,:\,u \in G_i\}$ is an invariant subspace for $P^{\vec{\omega}}_{G_i}$, the operator $P^{\vec{\omega}}_{G_i}$ is a contraction. Then, from Lemma \ref{contraction}, it follows that $P^{\vec{\omega}}_{G_i,r}$ and, subsequently, $P^{\vec{\omega}}$ are contractions.
\end{proof}


TITLE: ML-inequality: Why does this hold $|e^{-3y+3ix}| = e^{-3y}$ during numerator estimation of $f(z) = \frac{e^{3iz}}{z^2 + 1}$
QUESTION [1 upvotes]: Given the following related to an ML-inequality for $R > 1$:

Estimation of the numerator from the function $f(z)$ is supposed to develop as follows:

I'm wondering why and how exactly the power $3ix$ cancels out (why does it vanish/equal to zero) in the equality $|e^{-3y+3ix}| = e^{-3y}$ ?

REPLY [2 votes]: Let $z=\alpha+\beta i$ be a complex number. Then
\begin{align}
|e^z| &= |e^{\alpha +\beta i}|\\
&=|e^{\alpha}e^{\beta i}|\\
&=|e^\alpha| |e^{\beta i}|\\
&=e^\alpha|\cos\beta +i\sin\beta|\\
&=e^\alpha\sqrt{\cos^2\beta +\sin^2\beta}\\
&=e^\alpha\cdot \sqrt{1}\\
&=e^\alpha.
\end{align}