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\begin{document}
\title{Engineering sub-Poisson light in a simple mirror and beam splitter system}
\author{Sun-Hyun Youn\footnote{E-mail: [email protected], fax: +82-62-530-3369}} \address{Department of Physics, Chonnam National University, Gwangju 500-757, Korea}
\begin{abstract}
Vacuum fluctuation, which is the intrinsic nature of an electric field
can be measured via homodyne detection. Moreover, electric field intensity fluctuation are also related to vacuum fluctuations.
Squeezed vacuum and sub-Poisson light can be obtained by controlling the vacuum fluctuation using noble nonlinear interaction. Based on the squeezed vacuum by inserting a mirror on the unused part of the beam splitter was proposed in 1994, we present the mode matching method for the vacuum and light fields. Light intensity fluctuations also can be reduced by inserting a mirror on the unused part of the beam splitter. To obtain sub-Poisson light as a function of the distance between the mirror and detector, a detector with a thinner active layer than the wavelength is required.
\pacs{03.67.-a,03.70.+k, 03.65.Yz}
\keywords{Quantum optics, Squeezed State, Vacuum fluctuation, Sub-Poisson, Beam splitter and Mirror}
\end{abstract}
\maketitle
\section{Introduction}
When a single photon is in a particular mode, according to the particle nature of light, photons will be sequentially found in that mode. The probability of finding a photon is proportional to the absolute square of the wave function related to the electromagnetic wave. Vacuum fluctuations are related to the spatial characteristics of the electromagnetic wave. The spontaneous decay caused by the vacuum can be suppressed in cavities \cite{Jhe1987}. Theoretical and experimental studies have beem conducted on methods to change the vacuum fluctuations near mirrors\cite{sun1995,sun1994,wadood}.
In this study, in contrast to previous studies on the vacuum noise characteristics of light using a homodyne detector, we calculate the intensity fluctuations when photons are directly measured using photon counter. The obtained results are similar to those obtained in previous studies, but herein we predict the results considering mode matching in the experiment.
In section II, the fluctuation of light that can be measured using a detector is calculated with a mirror placed on one side of the beam splitter. In section III, an experimental device is proposed for perfect mode matching, and in the last section, the practical limits of the vacuum fluctuation near the mirror are discussed.
\section{Vacuum fluctuation near a mirror.}
\begin{figure}
\caption{Vacuum mode relations in the beam splitter with a mirror. BS: Beam splitter, M: mirror}
\label{DoubleP}
\end{figure}
An electric field can be written as
\begin{eqnarray} \hat{E}_{L} = \hat{E}_{cl} + \hat{E}_{Q} ,
\label{eLo} \end{eqnarray}
where
\begin{eqnarray} \hat{ E} _{cl} &=& i \sqrt{\frac{\hbar \omega }{2 \epsilon_0 V }} ( \alpha e^{i(\omega t - k_0 z )} - \alpha^{*} e^{i(\omega t - k_0 z )} ) \vec{x}, \nonumber \\ \hat{ E} _{Q} &=& i \sum_{k} \sqrt{\frac{\hbar \omega_k }{2 \epsilon_0 V }} ( \hat{b}_{k} e^{- i (\omega_k t - k z) } -\hat{b}_{k}^{\dagger} e^{ i (\omega_k t - k z) } )\vec{x} .
\label{efcleq} \end{eqnarray}
Here, $k_0$ and $\omega$ are the wave number and angular frequency of the laser, respectively, $\hbar$ and $\epsilon_0$ have usual meanings, and $V$ is the normalization volume\cite{yariv}. Considering the laser mode in Fig. \ref{DoubleP}, the modes $a_1 ^{out} $ and $ a_2 ^{out} $ can be written as \begin{eqnarray} a_1 ^{out} &=& \sqrt{T} b + \sqrt{R} c , \nonumber \\ a_2 ^{out} &=& -\sqrt{R} b + \sqrt{T} c , \label{a1a2out} \end{eqnarray}
where the modes $c$ and $c^{out} $ can be written as
\begin{eqnarray} c &=& \sqrt{T_m} d - \sqrt{R_m} c^{out} , \nonumber \\
c^{out} &=& \sqrt{ R }a_1 + \sqrt{T} a_2 .\nonumber \label{cmode} \end{eqnarray}
Then the electric field in fluctuating vacuum modes at $a_1$ is
\begin{eqnarray} \hat{ E} _{vac,1} ^{(+)} &=& \sum_{k} i \sqrt{\frac{\hbar \omega_k }{4 \epsilon_0 V }} \{ \sqrt{T} \hat{b}_{k}^{\dagger} e^{i(\omega_k t - k Z_1)} + \mu \hat{a}_{1,k}^{\dagger} e^{i(\omega_k t + k z_1 )} \nonumber \\ & &- R \sqrt{R_m} \hat{a}_{1,k}^{\dagger} e^{i(\omega_k t - k z_1 )} - \sqrt{RT } \sqrt{R_m} \hat{a}_{2,k}^{\dagger} e^{i(\omega_k t - k z_1) } + \sqrt{ R T_m} \hat{d}_{k} ^{\dagger} e^{i(\omega_k t - k Z_M )} \}
\label{efield1} \end{eqnarray}
where $R_m$($T_m$) is the reflectance(transmittance) of the mirror and $R$($T$) is the reflectance (transmittance) of the beam splitter, $z_1$($Z_1$) is the distance from the mirror (laser) to the detector. $Z_M$ is related to the vacuum source behind the mirror and it can be any number.
We add the factor $\frac{1}{\sqrt{2}}$ for the normalization of the vacuum fluctuation. The vacuum mode ($\hat a_1 ^{\dagger} e^{i ( \omega t - k z_1 )}$) at the detector is the reflected vacuum mode ($\hat a_1 ^{\dagger} e^{i ( \omega t + k z_1 )}$) at the mirror. If two modes are perfectly matched the $\mu$ in Eq. \ref{efield1} is 1 and the two counterpropagating modes yield the standing wave mode\cite{sun1994,sun1995}.
If $\mu = 0 $, the fluctuation value from Eq. \ref{fluc} becomes $\frac{|\alpha|^2 T}{2}$, it is the square of the constant dc current $ \frac{T |\alpha |^2}{2} $. In other words, if we directly measure the fluctuation of the laser intensity, the fluctuation is dependent on the distance ($z_1$) between the mirror and the detector. Even in photo counting experiments, the photon number fluctuation is related to the vacuum fluctuation, therefor, the photon number fluctuation is also depend on the distance $z_1$.
If we used the photodetetion theory \cite{detection} with instantaneous response of the photodetector \cite{response},
\begin{eqnarray}
\hat{I}_1 = \{ \sqrt{T} \hat{E}_{cl}^{(+)} + \hat{E}_{vac,1}^{(+)} \} \times \{ \sqrt{T} \hat{E}_{cl}^{(-)} + \hat{E}_{vac,1}^{(-)}\},
\label{current} \end{eqnarray}
where we normalize the photocurrent. If the electric field of the local oscillator is considerably greater than the vacuum field, the terms containig $\alpha$ have physical significance. When the constant dc current $ \frac{T |\alpha |^2}{2} $ is neglected, Eq. \ref{current} yields
\begin{eqnarray}
&& \hat {I}_{1}^{o} (z_1, Z_1) = \frac{|\alpha|}{\sqrt{2}} [\sqrt{T} e^{i \phi} \{ (\mu e^{- i k (Z_1 + z_1 )} - e^{- i k (Z_1 - z_1 )} R \sqrt{R_m} ) \hat{a}_1 - e^{- i k (Z_1 - z_1 )} \hat{a}_2 \} \nonumber \\ &+& \sqrt{T} e^{- i \phi } \{ (\mu e^{ i k (Z_1 + z_1 )} - e^ { i k ( Z_1 - z_1 )} R \sqrt{R_m} ) \hat{a}_1 ^{\dagger} - e^{- k (Z_1 - z_1 )} \hat{a}_2 ^{\dagger} \} \nonumber \\ &+& e^{ i \phi} T \hat{b} +e^{- i \phi} T \hat{b}^{\dagger} + e^{i \phi} e^{i k (Z_M - Z_1 )} \sqrt{T R T_m } \hat{d} + e^{-i \phi} e^{- i k (Z_M - z_1 )} \sqrt{T R T_m } \hat{d}^{\dagger} ], \label{photocurrentA} \end{eqnarray}
We then evaluate the square of the photocurrent to determine the fluctuation. After squaring Eq. \ref{photocurrentA}, we find the photocurrent fluctuation as follows: \begin{eqnarray}
\langle (\hat {I}_1 ^{o})^2 \rangle =\frac{|\alpha|^2 T}{2} \{ 1+\mu^2 - 2 \mu R \sqrt{R_m} \cos( 2 k z_1 )\} \label{fluc} \end{eqnarray}
If $\mu = 0 $, the fluctuation value from Eq. \ref{fluc} becomes $\frac{|\alpha|^2 T}{2}$, which is the square of the constant dc current $ \frac{\sqrt{T} |\alpha |}{\sqrt{2}} $. In other words, if we directly measure the laser intensity fluctuation, the fluctuation is dependent on the distance ($z_1$) between the mirror and detector. Even in the photo counting experiment, the photon number fluctuation is related to the vacuum fluctuation; therefore, the photon number fluctuation is also dependent on the distance $z_1$.
If we consider practical limits such as finite linewidth and finite absorption length, Eq. \ref{fluc} will change as follows\cite{ref7, sun1995}.
\begin{eqnarray}
\langle (\hat {I}_1 ^{o})^2 \rangle_{P} &=&\frac{|\alpha|^2 T}{2} \{ 1+\mu^2 - 2 \mu R \sqrt{R_m} e^{- z_1 ^2 \Delta k^2} \nonumber \\ &\times& \frac{\kappa [ \cos(2 k_0 z_1 + \phi_0 ) - e^{-\kappa D } \cos(2 k_0 (z_1 + D) + \phi_0 )]}{\sqrt{4 k_0 ^2 + \kappa ^2}} \} \label{flucPr}, \end{eqnarray}
where $\Delta k$ is the line width of the local oscillator beam with Gaussian line width distribution functions. $\kappa$ is the absorption coefficient, $D$ is the detector active length, and $\phi_0 = \arctan \frac{2k}{\kappa} $. We assumed that the probability that a photon is converted into an electron hole pair at distance $\eta$ from the surface of the detector's active region is $\kappa e^{- \kappa \eta} $\cite{ref8}.
The two coefficients $\sqrt{R_m}$ and $\mu$ depend on the mode matching condition. Even when we used the total mirror, if the mode from the mirror is not perfectly matched with the mode from the laser, the effective reflectance $ \sqrt{R_m} $ can not be $1$. Furthermore, the mode $a_1$ to the mirror is reflected by the mirror and then meets at the detector. At the detector, if two counter-propagating modes are not exactly matched, the coefficient $\mu$ cannot be $1$. To evaluate this mode matching condition, we assume that the amplitude envelope of the electromagnetic wave in the transverse plane is given by a Gaussian function.
Considering the Gaussian modes \cite{saleh}
\begin{eqnarray} E(\rho, z ) = E_0 \frac{w_0}{w(z)} \exp [- \frac{\rho^2}{w(z)^2} ] \exp [ - i k z - i k \frac{\rho^2}{2 R(z)} + i \zeta(z)] \label{gfield} \end{eqnarray} , where $w_0$ is the radius of the beam waist and \begin{eqnarray} w(z) &=& w_0 \sqrt{1+ (\frac{z}{z_0})^2} \nonumber \\ R(z) &=& z (1+ (\frac{z_0}{z})^2 ) \nonumber \\ \zeta(z) &=& \tan^{-1} \frac{z}{z_0} \label{gfieldsub} \end{eqnarray} and $z_0$ is defined as follows: \begin{eqnarray} z_0 &=& \frac{\pi}{\lambda } w_0 ^2. \label{z0} \end{eqnarray}
First, we assume that the laser and vacuu modes have the same beam waist $w_0$ at the detector. Then the laser and vacuum modes are perfectly matched; thus, $\sqrt{R_m} = 1$. On the other hand, the vacuum $E_v (0) $ starting from the detector propagates to the mirror and reflects at the mirror. The returned vacuum $E_v (2 z_1 )$ is not the same
$E_v(0)$. The coefficient $\mu$ can be calculated as follow:
\begin{eqnarray}
\mu &=&\frac{|<E_v (0) E_v (2 z_1 )^{*} >| }{\sqrt{<E_v (0)^2 > <E_v (2 z_1 )^2>}}
\nonumber \\
&=& \frac{(1+ \frac{4 z_1 ^2 }{z_0 ^2 })^{\frac{1}{4}} }{(1+ 5 \frac{z_1 ^2 }{z_0 ^2 }+ 4 \frac{z_1 ^4 }{z_0 ^4 })^{\frac{1}{4}} } \label{muF} \end{eqnarray}
\begin{figure}
\caption{ Mode matching value $\mu$ as a function of $w_0$ and $z_1$. }
\label{modeMu}
\end{figure}
In Fig. \ref{modeMu}, $\mu$ is plotted as a function of $z_1$ and $w_0$, where $z_1$ is the distance between the mirror and detector
We assume that the detector and mirror are large enough that all the waves are detected and reflected. If the distance between the mirror and detector and the size of the beam waist are small enough, the coefficient $\mu$ remains near $1$.
If we consider the case where the vacuum field has waist at the mirror, the coefficient $\mu$ automatically becomes $1$ due to the symmetry, but the vacuum field $E_v (z_1)$ at the detector does not matche the laser field $E_L (0)$. We assumed that the laser field has beam waist $w_0$ at the detector, and the vacuum field has a beam waist $w_m$ at the mirror. Then the effective reflectance $\sqrt{R_m} $ becomes
\begin{eqnarray}
\sqrt{R_m} &=& \frac{|<E_v (z_1) E_L (0 )^{*} >| }{\sqrt{<E_v (z_1)^2 > <E_L (0 )^2>}} \nonumber \\ &=& \frac{\sqrt{2} \sqrt{\frac{w_m }{w_0}} (1+ \frac{ z_1 ^2 }{z_m ^2 })^{\frac{1}{4}} } {(\{(1+ \frac{w_m ^2}{w_0 ^2})^2 + \frac{z_1 ^2 }{z_0 ^2 }\}\{1 + \frac{z_1 ^2 }{z_m ^2 }\})^{\frac{1}{4}} }, \label{mirR} \end{eqnarray} where $z_m = \frac{\pi}{\lambda } w_m ^2$.
\begin{figure}
\caption{Mode matching value $\sqrt{R_m} $ as a function of $w_0$ and $z_1$, with $w_0$ equal to $100 \lambda$ }
\label{modeR}
\end{figure}
In Fig. \ref{modeR}, $\sqrt{R_m}$ is plotted as a function of $z_1$ and $w_m$, where $z_1$ is the distance between the mirror and detector. We set $w_0$ to $100 \lambda$. Additionally, we also assume that the detector and mirror are large enough that all the waves are detected and reflected. The coefficient $\sqrt{R_m}$ can be 1 only when the distance between the mirror and detector is small and the size of the beam waist is sufficiently small.
The mode matching condition is crucial for detecting the modulation effect of the vacuum fluctuation near the mirror, as denoted by Eq. \ref{flucPr}. With the usual setup,
we can not satisfy the conditions $\mu=1$ and $\sqrt{R_m} = 1$. In the next section, we suggest a noble experimental setup that satisfies two mode-matching conditions.
\section{Set up for mode matching}
For a laser that has a Gaussian transverse mode, we have to establish a vacuum mode that also has a Gaussian transverse mode. Fig. \ref{mode} displays the setup for perfect mode matching between the laser light mode and a vacuum mode.
The laser used in the experiment passes through lens $L_1$ and is divided into two by the beam splitter ($BS_1$).
The laser is a Gaussian beam and it proceeds according to the Gaussian approximation. The light passing through $BS_1$ and traveling to mirror $M_2$ reaches the partial mirror $B$ and yields a beam waist on the $L_3$ side surface of $B$. Similarly, the light reflecting from the mirror $M_1$ passes through the partial reflector $A$ and yields a beam waist on the $L_2$ side surface of $A$.
The light passing through $A$ and $B$ passes through the $L_2$ and $L_3$ of the same focal length, respectively, and yields another beam waist on the detector surface. The transmittance of light passing through $A$ from $M_1$ is almost 0, and the reflectance of light stemming from the $L_2$ side is almost 1. In this way, if the mode is perfectly matched using the light passing through $B$ and $A$, an experimental setup can be established wherein one side of the beam splitter $BS_2$ is a mirror ($A$).
Using this method, the degree of mode matching can be increased compared to that when the experiment is performed by simply placing a plane mirror on one side of the beam splitter. Additionally the experimental constraints caused by the mode matching can be overcome. The experimental setup in Fig. \ref{mode} enables the measurement of how the vacuum fluctuations of the light passing through the beam splitter change when a mirror is placed on one side of the beam splitter.
\begin{figure}
\caption{Mode matching setup }
\label{mode}
\end{figure}
\section{Conclusion and Discussion.}
The quantum nature of photons is highly dependent on their vacuum fluctuations. Vacuum fluctuations can be directly measured via homodyne detection. The fluctuation of one quadrature of the vacuum can be less than that of the usual vacuum, e.g., squeezed vacuum. Light intensity fluctuations are also dependent on vacuum fluctuations. Sub-Poisson light can be generated by controlling the vacuum fluctuations based on the nonlinear interaction of light and matter. In this study, we proposed the modulation of vacuum fluctuations by inserting a mirror on the unused part of the beam splitter in a homodyne measuring system. Furthermore, we calculated the effect of the line width of the laser and the thickness of the detector layer. The line width can be practically reduced to modulate vacuum fluctuations, but the decrease of the thickness of the detector to modulate vacuum fluctuations is challenging. We calculated the effect of mode matching between the vacuum and light fields and showed that the degree of mode matching obtained by adding a simple mirror in the unused beam splitter may not be sufficient to modulate the vacuum fluctuations. We present the perfect mode matching method for the vacuum and light fields. Then, the light intensity fluctuations can be reduced by inserting a beam splitter and a mirror. We still require a detector with an active layer thinner than the wavelength to obtain a sub-Poisson light as a function of the distance between the mirror and detector. We expect that our simple method of reducing vacuum fluctuations will play a great role in quantum information science.
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\bibitem{yariv} A. Yariv, {\it Quantum Electronics 3rd ed.}, John Wiley \& Sons. Inc, (1989)
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\bibitem{response} B. Yurke, Phys. Rev. A {\bf 32}, 311 (1985)
\bibitem{ref7} A. E. Siegman, {\it Laser} (Oxford University Press, Oxford, 1986 )
\bibitem{ref8} S. M. Sze, { \it Semiconductor Devices Physics and Technology} (AT\&T Bell Lab. Murray Hill, New Jersey, 1985)
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\end{document} | arXiv |
Movie genome: alleviating new item cold start in movie recommendation
Yashar Deldjoo ORCID: orcid.org/0000-0002-6767-358X1,
Maurizio Ferrari Dacrema1,
Mihai Gabriel Constantin2,
Hamid Eghbal-zadeh3,
Stefano Cereda1,
Markus Schedl ORCID: orcid.org/0000-0003-1706-34063,
Bogdan Ionescu2 &
Paolo Cremonesi1
User Modeling and User-Adapted Interaction volume 29, pages 291–343 (2019)Cite this article
As of today, most movie recommendation services base their recommendations on collaborative filtering (CF) and/or content-based filtering (CBF) models that use metadata (e.g., genre or cast). In most video-on-demand and streaming services, however, new movies and TV series are continuously added. CF models are unable to make predictions in such a scenario, since the newly added videos lack interactions—a problem technically known as new item cold start (CS). Currently, the most common approach to this problem is to switch to a purely CBF method, usually by exploiting textual metadata. This approach is known to have lower accuracy than CF because it ignores useful collaborative information and relies on human-generated textual metadata, which are expensive to collect and often prone to errors. User-generated content, such as tags, can also be rare or absent in CS situations. In this paper, we introduce a new movie recommender system that addresses the new item problem in the movie domain by (i) integrating state-of-the-art audio and visual descriptors, which can be automatically extracted from video content and constitute what we call the movie genome; (ii) exploiting an effective data fusion method named canonical correlation analysis, which was successfully tested in our previous works Deldjoo et al. (in: International Conference on Electronic Commerce and Web Technologies. Springer, Berlin, pp 34–45, 2016b; Proceedings of the Twelfth ACM Conference on Recommender Systems. ACM, 2018b), to better exploit complementary information between different modalities; (iii) proposing a two-step hybrid approach which trains a CF model on warm items (items with interactions) and leverages the learned model on the movie genome to recommend cold items (items without interactions). Experimental validation is carried out using a system-centric study on a large-scale, real-world movie recommendation dataset both in an absolute cold start and in a cold to warm transition; and a user-centric online experiment measuring different subjective aspects, such as satisfaction and diversity. Results show the benefits of this approach compared to existing approaches.
A dramatic rise in the generation of video content has occurred in recent years. According to Cisco, the largest networking company across the globe, by 2020 more than \(75\%\) of the world's mobile data traffic will be video, or even \(80\%\) when video and audio data are considered together (Cisco visual networking index 2016). This rise has been fueled by online social network users who upload/post a staggering amount of user-generated video on a daily basis. For instance, as of 2018, YouTubeFootnote 1 users upload over 400 h of video every minute. This translates to about 3 years of non-stop watching in order to consume all videos uploaded to YouTube in a single hour. Similarly, InstagramFootnote 2 users post nearly 70 million photos and videos each day (Xu et al. 2017).
In this context, video recommender systems play an important role in helping users of online streaming services, as well as of social networks, cope with this rapidly increasing volume of videos and provide them with personalized experiences. Nevertheless, the growing availability of digital videos has not been fully accompanied by comfort in their accessibility via video recommender systems. The causes of this problem are twofold: (i) the type of recommendation models in service today, which are heavily dependent on usage data (in particular, implicit or explicit preference feedback) and/or metadata (e.g., genre and cast associated with the videos) (cf. Sect. 1.1), and (ii) the nature of video data, which are information intensive when compared to other media types, such as music or images (cf. Sect. 1.2). In the following article, we analyze each of these dimensions. Throughout this paper, we will use a number of abbreviations, which, for convenience are summarized in Table 1.
Table 1 List of abbreviations used throughout the paper
New item cold-start recommendation in the movie domain
To date, collaborative filtering (CF) methods (Koren and Bell 2015) lie at the core of most real-word movie recommendation engines, due to their state-of-the-art accuracy (McFee et al. 2012; Yuan et al. 2016). In most video-streaming services, however, new movies and TV series are continuously added. CF models are not capable of providing meaningful recommendations when items in the catalogue contain few interactions, a problem commonly known as the cold start (CS) problem. The most severe case of CS is when new items are added that lack any interactions, technically known as the new item CS problem.Footnote 3 In such a situation, CF models are completely unable to make predictions. As such, these new items are not recommended, go unnoticed by a large part of the user community, and remain unrated, creating a vicious circle in which a set of items in the RS is left out of the vote/recommendation process (Bobadilla et al. 2012). Being able to provide high-quality recommendations for cold items has several advantages. Firstly, it will increase the novelty of the recommendations, which is a highly desirable property and inherent in the user-centric and business-centric goals of RS, i.e., the discovery of new content and the increase of revenues (Aggarwal 2016b; Liu et al. 2014). Secondly, providing good new movie recommendations will allow enough interactions/feedbacks to be collected in a brief amount of time enabling effective CF recommendation. Despite previous efforts, the new item CS problem remains far from being solved in the general case, and most existing approaches suffer from it (Bobadilla et al. 2012; Zhou et al. 2011; Zhang et al. 2011).
Currently, the most common approach to counteracting the new item CS problem is to switch to a pure CBF (de Gemmis et al. 2015; Lops et al. 2011) method by using additional attribute content for items, usually by resorting to metadata provided in textual form (Liu et al. 2011). This approach is known to have lower accuracy than CF because it ignores potentially useful collaborative information and typically relies on human-generated textual metadata, which are often noisy, expensive to collect, and sparse. More importantly, extra information for cold items is not always available on the web (especially in user-generated form), even if it is available in abundance for warm items (Zhang et al. 2015). In addition, given the unstructured or semi-structured nature of metadata, they often require complex natural language processing (NLP) techniques for pre-processing, e.g., syntactic and semantic analysis or topic modeling (Aggarwal 2016a).
Many approaches have been proposed to address the new item CS issue, mainly based on hybrid CF and CBF models (Lika et al. 2014; Cella et al. 2017; Sharma et al. 2015; Ferrari Dacrema et al. 2018). Most recent work relies on machine learning to combine content and collaborative data. We focus on feature weighting rather than on other types of hybrids (e.g., joint matrix factorization) because we aim to build a hybridization strategy that can be easily applied to a CBF model. For instance, the authors in Gantner et al. (2010) proposed a method to map item features into the item embeddings learned in a matrix factorization algorithm, while the authors in Schein et al. (2002) defined a probabilistic model trained via expectation minimization. Another example is Sharma et al. (2015), where the authors proposed a feature weighting model that learns feature weight by optimizing the ranking of the recommendations over the user interactions for warm items.
Addressing this issue, the main contribution of the present work is to improve the current state of the art by presenting a generalized, two-step machine learning approach to feature weighting and by testing its effectiveness on both editorial features and state-of-the-art multimedia (MM) descriptors. Hereafter, for simplicity, we refer to items without interactions as cold items and items containing interactions as warm items.
Video as an information-intensive multimodal media type
When we watch a movie, we can effortlessly register many details conveyed to us through different multimedia channels—in particular, the audio and visual channels. As a result, the perception of a film in the eyes of viewers is influenced by many factors related not only related to, e.g., the genre, cast, and plot, but also according to the overall film style (Bordwell et al. 1997). These factors affect the viewer's experience. For example, two movies may be from the same genre and director, but they can be different based on the movie style. Consider as an example Empire of the Sun and Schindler's List, both dramatic movies directed by Steven Spielberg and both describing historical events. However, they are completely different in style, with Schindler's List shot like a documentary in black and white, while Empire of the Sun is shot using bright colors and makes heavy use of special effects. Although these two movies are similar with respect to traditional metadata (e.g., director, genre, year of production), their different styles are likely to affect the viewers' feelings and opinions differently (Deldjoo et al. 2016d). In fact, the film story is first created by the author and the comprehension of the cinematographical language by the spectator reshapes the story (Fatemi and Mulhem 1999). The notion of story in a movie depends on semantic content (reflected better in metadata) reshaped through stylistic cinematography elements (reflected better in multimedia content). These discernible characteristics of movie content meet users' different information needs.
The extent to which content-based approaches are used, and even the way "content" is interpreted, varies between domains. While extracting descriptive item features from text, audio, image, and video content is a well-established research domain in the multimedia community (Lew et al. 2006), the recommender system community has long considered metadata, such as the title, genre, tags, actors, or plot of a movie, as the single source for content-based recommendation models, thereby disregarding the wealth of information encoded in the actual content signals. In order for MRS to make progress in recommending the right movies to the right user(s), they need to be able to interpret such multimodal signals as an ensemble and utilize item models that take into account the maximum possible amount of this information. We refer to such a holistic description of a movie, taking into account all available modalities, as its movie genome, since it can be considered the footprint of both content and style (Bronstein et al. 2010; Jalili et al. 2018).Footnote 4
In this paper, we specifically address the above-mentioned shortcomings of purely metadata-based MRS by proposing a practical solution for the new item CS challenge that exploits the movie genome. We set out to answer the following research questions:
RQ1 Can the exploitation of movie genome describing rich item information as a whole, provide better recommendation quality compared with traditional approaches that use editorial metadata such as genre and cast in CS scenarios?
RQ2 Which visual and audio information better captures users' movie preferences in CS scenarios?
RQ3 Can we effectively leverage past user behavior data on warm items (items with interactions) to enrich the overall item representation and improve our ability to recommend cold items when interactions are not available?
The remainder of this article is structured as follows. Section 2 positions our work in the context of the state of the art and highlights its novel contributions. Section 3 introduces the proposed general content-based recommendation framework. Sections 4 and 5 report on the experimental validation, namely the experimental setup and parameter tuning, offline experimentation, and a user study in a web survey, respectively. Section 6 concludes the article in the context of the research questions and discusses limitations and future perspectives.
One main contribution of this work is the introduction of a solution for the new item CS problem in the multimodal movie domain. In this section, we therefore review the existing, state-of-the-art approaches in content-based multimedia recommender systems (Sect. 2.1) and feature weighting for CS recommender systems (Sect. 2.2) and position our contribution (Sect. 2.3).
Content-based multimedia recommendation
A multimedia recommendation system is a system that recommends a particular media type, such as audio, image, video, and/or text, to the users (Deldjoo et al. 2018e, f). We therefore organize the state-of-the-art CB-MMRS based on the target media type, namely: (i) audio recommendation, (ii) image recommendation, and (iii) video recommendation. In the following subsections, we describe each of these systems.
Audio recommendation
The most common example of audio recommendation is music recommendation (Schedl et al. 2018; Vall et al. 2019). Over the past several years, a wealth of approaches, including CF, CBF, context-aware recommenders, and hybrid methods, have been proposed to address this task. An overview of popular approaches can be found in Schedl et al. (2015, 2018). Perhaps more than in other MM domains, CB recommenders have attracted substantial interest from researchers in the music domain, not least due to their superior performance in CS scenarios.
Recent work has proposed deep learning-based CB approaches. For instance, the authors in van den Oord et al. (2013) use a deep convolutional neural network (CNN) trained on audio features, more precisely, on the log-scaled Mel spectrograms extracted from 3-second-snippets of the audio, resulting in a latent factor representation for each song. The authors evaluate their approach for tag prediction and music recommendation using the Million Song Dataset (Bertin-Mahieux et al. 2011). In tenfold cross-validation experiments using 50-dimensional latent factors, they show that the CNN outperforms both metric learning to rank and a multilayer perceptron trained on bag-of-words representations of vector-quantized Mel frequency cepstral coefficients (MFCC) (Logan 2000a) in both tasks.
In contrast to such automatic feature learning approaches, some systems use human-made annotations of music. Perhaps, the most notable and well-known is the proprietary Music Genome Project (MGP),Footnote 5 which is used by music streaming major Pandora.Footnote 6 MGP captures various attributes of music and uses them in a CBF recommender system. These attributes are created by musical experts who manually annotate songs. Pandora uses up to 450 specific descriptors per song, such as "aggressive female vocalist", "prominent backup vocals", or "use of unusual harmonies".
In our approach, we follow a strategy in between these two extremes (i.e., fully automated feature learning by deep learning and pure manual expert annotations). The proposed movie genome uses well-established, state-of-the-art audio descriptors that are semantically more meaningful than deep learned features, but at the same time do not require a massive number of human annotators.
Image recommendation
Some interesting use-case scenarios of image recommendation can be mentioned in the fashion domain (e.g., recommending clothes) and the cultural heritage domain (e.g., recommending paintings in museums). For fashion, recommendation can be performed in two main manners: finding a piece of clothing that matches a given garment image shown to the system as a visual query (such as two pairs of jeans which are similar to each other considering their visual appearance) and finding the clothing, which complements the given query (such as recommending a pair of jeans that match a shirt). The authors in McAuley et al. (2015) propose a CB-MMRS which provides personalized fashion recommendations by considering the visual appearance of clothes. The main novelty, besides focusing on this novel fashion recommendation scenario, is examining the visual appearance of the items under investigation to overcome the CS problem.
The authors of Bartolini et al. (2013) propose a multimedia (image—video—document) recommender platform to address the cultural heritage domain: in particular, a recommender system to provide personalized visiting paths to tourists visiting the Paestum ruins, one of the major Greco-Roman cities in the South of Italy. The proposed system is able to uniformly combine heterogeneous multimedia data and to provide context-aware recommendation techniques. This paper provides interesting insights for building context-aware multimedia systems using content information, with explicit focus on contextualization. The authors exploit high-level metadata extracted in an automatic or semi-automatic manner from low-level (signal-level) features and compare it with user preferences. The main shortcoming of this research is the lack of an experimental study on a larger multimedia dataset.
Visual descriptors have also been used in restaurant recommendation systems by the authors of Chu and Tsai (2017), in which images collected from a restaurant-based social platform were first processed by an SVM-based image classification system that used both low-level and deep features and split the images into four classes, indoor, outdoor, food and drink images, based on the idea that these different categories of pictures may have different influences on restaurant recommendation. This content-based approach was used to successfully enhance the performance of matrix factorization, Bayesian personalized ranking matrix factorization and FM approaches.
In our approach, we follow a strategy that also recognizes the importance of low-level content (visual and audio) for movie recommendation and leverages it for new item CS movie recommendation.
Video recommendation
As one of the earliest approaches to the problem of video recommendations, the authors of Yang et al. (2007) and Mei et al. (2007, 2011) propose a video recommender system named "Video Reach". Given an online video and related information (query, title, tags and surrounding text), the system recommends relevant videos in terms of multimodal relevance and user feedback. Two types of user feedback are leveraged: browsing behavior and playback on different portions of the video (the latter is specific to Mei et al. 2011). These approaches are interesting from the perspective of using multimodal video content (audio, visual, and textual) and a fusion scheme based on user behavior. However, they have some limitations as well. Firstly, according to the properties required by the attention fusion function, the proposed Video Reach system filters out videos with low textual similarity to ensure that all videos are more or less relevant and then only calculates the visual similarity of the filtered videos; this may result in losing important information. Secondly, it uses only one type of visual feature, namely the basic color histogram. Thirdly, an empirical set of weights is chosen to serve as importance weights in a linear feature/modality fusion; for example, the textual keywords are given a much higher weight than the visual and aural keywords, without investigating the opposite arrangement. Although the authors show that this assumption is sufficient to make recommendation via adjusting weights, it is not clear what effect such an empirical assumption has.
In our approach, we introduce a video recommendation system that leverages all video properties (i.e., audio, visual, and textual) and an effective fusion method based on canonical correlation analysis (CCA) to exploit the complementary information between modalities in order to produce more powerful combined descriptors. More importantly, we propose an approach for new item recommendation that leverages the collaborative knowledge about warm items for the CBF of cold items, using the combined descriptors.
Feature weighting for cold-start recommender systems
Relying on CBF algorithms to address cold items has two main drawbacks: firstly, it is limited by the availability and quality of item features, and secondly, it is difficult to connect the content and collaborative information. One way to build a hybrid of content and collaborative information is via feature weighting. We focus on feature weighting rather than on other types of hybrids because we aim to build a hybridization strategy that can be easily applied to a CBF model. Feature weighting algorithms can be either embedded methods, which learn feature weights as part of the model training, or wrapper methods, which learn weights in a second phase on top of an already available model. Examples of embedded methods are user-specific feature-based similarity models (UFSM) (Elbadrawy and Karypis 2015) and factorized bilinear similarity models (FBSM) (Sharma et al. 2015). Among embedded methods, the main drawbacks are the complex training phase and a sensitivity to noise due to the strong coupling of features and interactions. UFSM learns a personalized linear combination of similarity functions, known as global similarity functions for cold-start top-N item recommendations. UFSM can be considered a special case of FM (Elbadrawy and Karypis 2015; Rendle 2012). FBSM was proposed as an evolution of UFSM that aims to discover relations among item features instead of building user-specific item similarities. The model builds an item-item similarity matrix which models how how well a feature of an item interacts with all the features of the second item.
Wrapper methods, meanwhile, rely instead on a two-step approach by learning feature weights on top of an already available model. An example of this is least-square feature weights (LFW) (Cella et al. 2017), which learns feature weights from a SLIM item-item similarity matrix using a simpler model than FBSM:
$$\begin{aligned} sim(i,j) = \mathbf {f}_i^T \mathbf {D} \mathbf {f}_j \end{aligned}$$
where f is the feature vector of an item and D is a diagonal matrix having as dimension the number of features. Another example of a wrapper method is HP3 (Bernardis et al. 2018), which builds a hybrid recommender on top of a graph-based collaborative model. A generalization of LFW has recently been published by the authors in Ferrari Dacrema et al. (2018). They demonstrate the effectiveness of wrapper methods in learning from a wider variety of collaborative models and present a comparative study of some state-of-the-art algorithms. Their paper further shows that wrapper methods with no latent factor component (i.e., matrix \(\mathbf {V}\), as in FBSM) tend to outperform others. In our approach, we therefore choose to adopt this simpler model, as it combines good recommendation quality with fast training time.
Similar strategies are available for matrix factorization models. Collective matrix factorization (Singh and Gordon 2008) allows the joint factorization of both collaborative and content data, which is applied in Saveski and Mantrach (2014) to propose local collective embedding, a joint matrix factorization that enforces the manifold structure exhibited by the collective embedding in the content data as well as allowing collaborative interactions to be mapped to topics. An example of a wrapper method is attribute to feature mapping (Gantner et al. 2010), an attribute-aware matrix factorization model which maps item features to its latent factors via a two-step approach. All previous approaches rely on the availability of some descriptors for each item, which in some cases can be an issue.
Other proposals to address the CS problem make use of other relations between users or items, i.e., social networks. For example, the authors in Zhang et al. (2010) use social tags to enrich the descriptions of items in a user-tag-object tripartite graph model; while the authors in Ma et al. (2011) instead use a social trust network to enrich the user profile. Another example is Victor et al. (2008), where authors analyze the impact of the connections on the quality of recommendations. While this group of techniques shows promising results, it is still limited by the fact that obtaining fine-grained and accurate features is a complex and time-consuming task. Moreover, those other existing relationships might not always be available or meaningful for the target domain. See Elahi et al. (2018) for a good and general introduction to recommendation complicating scenarios (e.g., the CS problem).
In this work, we adopt feature weighting techniques because they have shown promising results in recent years to the point of becoming the current state of the art.
Contributions of this work
The work at hand builds on foundations and results realized in our previous work, but considerably extends it. We therefore present in the following our novel contributions, and connect them to previous work.
In Deldjoo et al. (2015a, b, 2016a, d), Elahi et al. (2017) and Cremonesi et al. (2018), we proposed a CB-MRS that implements a movie filter according to average shot length (measure of camera motion), color variation, lighting key (measure of contrast), and motion (measure of object and camera motion). The proposed features were originally used in the field of multimedia retrieval for movie genre classification (Rasheed et al. 2005) and have a stylistic nature which is believed to be in accordance with applied media aesthetics (Zettl 2013) for conveying communication effects and simulating different feelings in the viewers. For this reason, these features were named mise-en-scène features.
Since full movies can be unavailable, costly or difficult to obtain, in Deldjoo et al. (2016d) it was studied whether movie trailers can be used to extract mise-en-scène visual features. The results indicated that they are indeed correlated with the corresponding features extracted from full-length movies and that feeding the features extracted from movie trailers and full movies into a similar CB-MRS results in a comparable quality of recommendations (both superior to the genre baseline). The main shortcoming of this work is that it used a small dataset for evaluation (containing only 167 movies and the corresponding trailers). Additionally, the number of visual features was limited (only five features, cf. Rasheed et al. 2005). Due to these restrictions, the generalizability of our findings in Deldjoo et al. (2016d) may be limited; also see Sect. 6.3 for a discussion of limitations.
In Deldjoo et al. (2016b, c, 2018d) we specifically addressed the under-researched problem of combining visual features extracted from movies with available semantic information embedded in metadata or collaborative data available in users' interaction patterns in order to improve offline recommendation quality. To this end, for multimodal fusion (i.e., fusing features from different modalities) in Deldjoo et al. (2016b), for the first time, we investigated adoption of an effective data fusion technique named canonical correlation analysis (CCA) to fuse visual and textual features extracted from movie trailers. A detailed discussion about CCA can be found in Sect. 3.2. Although a small number of visual features were used to represent the trailer content (similar to Deldjoo et al. 2016d), the results of offline recommendation using 14K trailers suggested the merits of the proposed fusion approach for the recommendation task. In Deldjoo et al. (2018d) we extended (Deldjoo et al. 2016b) and used both low-level visual features (color- and texture-based) using the MPEG-7 standard together with deep learning features in a hybrid CF and CBF approach. The aggregated and fused features were ultimately used as input for a collective sparse linear method (SLIM) (Ning and Karypis 2011) method, generating an enhancement for the CF method. While the results for each of these two features improved the genre and tag baselines, the best results were achieved with the CCA fusion approach. Although (Deldjoo et al. 2018d) significantly extended the previous works (Deldjoo et al. 2016b) both in terms of the content and the core recommendation model, it ignored the role of the audio modality in the entire item modeling.
Finally, in Deldjoo et al. (2016c), we used factorization machines (FM) (Rendle 2012) as the core recommendation technique. FM is a general predictor working with any real valued feature vector and has the power of capturing all interactions between variables using factorized parameters. FM was used specifically with the goal of encoding the interactions between mise-en-scène visual features and metadata features for the recommendation task. Please note that in the present work, we neither use FM nor SILM, specifically because one of the main contributions of the work at hand is to propose and simulate a novel technique for new item recommendation for which FM or SLIM are not applicable.
In a different research line, in Elahi et al. (2017), we designed an online movie recommender system which incorporates mise-en-scène visual features for the evaluation of recommendations by real users. We performed an offline performance assessment by implementing a pure CB-MRS with three different versions of the same algorithm, respectively based on (i) conventional movie attributes, (ii) mise-en-scène visual features, and (iii) a hybrid method that interleaves recommendations based on the previously noted features. As a second contribution, we designed an empirical study and collected data regarding the quality perceived by the users. Results from both studies showed that the introduction of mise-en-scène, together with traditional movie attributes, improves the quality of both offline and online recommendations. However, the main limitation of Elahi et al. (2017) is that we used basic late fusion by interleaving the recommendations to combine recommendations generated by different CBF systems.
In summary, although we achieved relevant progress, some limitations of our previous work remain unsolved: (i) solely visual and/or text modalities were considered, forgetting the rich audio information (e.g., conversations or music); (ii) better fusion techniques are required to fully exploit the complementary information from (several) modalities; (iii) visual content can be represented with richer descriptors; and (iv) the recommendation model used was either a CBF model based on KNN or a CBF+CF model based on SLIM, both of which are not capable to deal with new item CS scenarios.
In this paper, we enhance these previous achievements and go beyond the state of the art in the following directions:
We propose a multimodal movie recommendation system which exploits established multimedia aesthetic-visual features; block-level audio features; state-of-the-art deep visual features; and i-vectors audio features. Apart from the use of automated content descriptors, the system uses as input movie trailers instead of complete movies, which makes it more versatile, as trailers are more readily available than full movies. We show that the proposed CB-MRS outperforms the traditional use of metadata. To the best of our knowledge, this has not previously been achieved, existing systems being limited to the use of either visual and/or textual modalities (Deldjoo et al. 2016c, d, 2017a) or basic low-level descriptors (Yang et al. 2007; Mei et al. 2011);
We propose a practical solution to the CS new-item problem where user behavior data are unavailable, and therefore neither CF nor CBF using user-generated content are applicable. Our solution consists of a two-step approach named collaborative-filtering-enriched content-based filtering (CFeCBF) to leverage the collaborative knowledge about warm items and exploit it for CBF on cold items.
To achieve multimodal MRS, we adopt an early fusion approach using canonical correlation analysis (CCA), which was successfully tested in our previous works (Deldjoo et al. 2016b, 2018d) for combining heterogeneous features extraced from different modalities (audio, visual and textual). CCA is often used when two types of data (feature vectors in training) are assumed to correlate. We hypothesize that this is relevant in the movie domain and that combining audio, visual, and textual data enriches the recommendations.
We evaluate the quality of the proposed movie genome descriptors by two comprehensive wide and articulated empirical studies: (i) a system-centric experiment to measure the offline quality of recommendations in terms of accuracy-related metrics, i.e., mean average precision (MAP) and normalized discounted cumulative gain (NDCG); and beyond-accuracy metrics (Kaminskas and Bridge 2016), i.e., list diversity, distributional diversity, and coverage; (ii) a user-centric online experiment involving 101 users, computing different subjective metrics, including relevance, satisfaction, and diversity.
We publicly release the resources of this work to allow researchers to test their own recommendation models. The dataset was already released partly in Deldjoo et al. (2018c) while the code is now available on Github.Footnote 7
The proposed collaborative-filtering-enriched content-based filtering (CFeCBF) movie recommender system framework
Proposed recommendation framework
The main processing stages involved in our proposed CFeCBF-MRS are presented in Fig. 1. As previously mentioned, the only input information, apart from the collaborative one, is the movie trailers. First, we perform pre-processing that consists of decomposing the visual and audio channels into smaller and semantically more meaningful units. We use frame-level and block-level segmentation for the audio channel. For video, we use the frames captured at 1 fps. The next step consists of computing meaningful content descriptors (cf. Sect. 3.1), namely: (i) multimedia—audio and visual features; and (ii) metadata—movie genres. Features are aggregated temporally using different video-level aggregation techniques, such as statistical summarization, Gaussian mixture models (GMM), and vectors of locally aggregated descriptors (VLAD) (Jégou et al. 2010). Features are fused by using the early fusion method CCA (cf. Sect. 3.2). At this stage, each video is represented by a feature vector of fixed length, which is referred to as the item profile. A collaborative recommender is trained on all available user-item interactions in order to model the correlations encoded in users' interaction patterns, using the similarity of ratings as an indicator of similar preference. As the last step, the CFeCBF weighting scheme is trained on the given item profile and collaborative model to discover the hybrid feature weights. The learned feature weights are then applied to a CBF recommender able to provide recommendations for cold items. Each of these steps is detailed in the following sections.
Rich item descriptions to model the movie genome
Similar to biological DNA, which represents a living being, multimedia content information can be seen as the genome of video recommendation, i.e., the footprint of both content and style. In this section, we present the rich content descriptors integrated into the proposed movie recommendation system to boost its performance. These features were selected based on their effectiveness in representing multimedia content in various domains and comprise both audio and visual features (Deldjoo et al. 2018b, c).
The exploited audio features are inspired by the fields of speech processing and music information retrieval (MIR) and by their successful application in MIR-related tasks, including music retrieval, music classification, and music recommendation (Knees and Schedl 2016). We investigate two kinds of audio features: (i) block-level features (Seyerlehner et al. 2011) which consider chunks of the audio signal known as blocks and are therefore capable of exploiting temporal aspects of the signal; and (ii) i-vector features (Eghbal-Zadeh et al. 2015) which are extracted at the level of audio segments using audio frames. Both approaches eventually model the feature at the level of the entire audio piece; by aggregating the individual feature vectors across time.
Overview of the feature extraction process in the block-level features (BLF), according to Seyerlehner et al. (2010)
Obtaining a global feature representation from individual blocks in the block-level framework, according to Seyerlehner et al. (2010)
Block-level features We extract block-level features (BLF) from larger audio segments (several seconds long) as proposed in Seyerlehner et al. (2010). They can capture temporal aspects of an audio recording and have been shown to perform very well in audio and music retrieval and similarity tasks (Seyerlehner et al. 2011) and can be considered state of the art in this domain.
The BLF framework (Seyerlehner et al. 2010) defines six features. These capture spectral aspects (spectral pattern, delta spectral pattern, variance delta spectral pattern), harmonic aspects (correlation pattern), rhythmic aspects (logarithmic fluctuation pattern), and tonal aspects (spectral contrast pattern). The feature extraction process in the block-level framework is illustrated in Fig. 2. Based on the spectrogram, blocks of fixed length are extracted and processed one at a time. The block width defines how many temporally ordered feature vectors comprise a block. The hop size is used to account for possible information loss due to windowing. After having computed the feature vectors for each block, a global representation is created by aggregating the feature values along each dimension of the individual feature vectors via a summarization function, which is usually expressed as a percentile, as illustrated in Fig. 3. A more technical and algorithmic discussion can be found in Seyerlehner et al. (2010). The extraction process results in a 9948-dimensional feature vector per video.
I-vector features I-vector is a fixed-length and low-dimensional representation containing rich acoustic information, which is usually extracted from short segments (typically from 10 s to 5 min) of acoustic signals such as speech, music, and acoustic scene. The i-vector features are computed using frame-level features such as mel-frequency cepstral coefficients (MFCCs). In a movie recommendation system, we define total variability as the deviation of a video clip representation from the average representation of all video clips. I-vectors are latent variables that capture total variability to represent how much an audio excerpt is shifted from the average clip. The main idea is to first learn a universal background model (UBM) to capture the average distribution of all the clips in the acoustic feature space using a dataset containing a sufficient amount of data consisting of different movie clips. The UBM is usually a Gaussian Mixture Model (GMM) and serves as a reference to measure the amount of shift for each segment where the i-vector is the estimated shift.
Block diagram of i-vector FA pipeline for, both, supervised and unsupervised approaches
The block-diagram of the i-vector pipeline, from frame-level feature extraction to i-vector extraction and finally to recommendation, is shown in Fig. 4. The framework can be decomposed into several stages: (i) Frame-level feature extraction MFCCs have proven to be useful features for many audio and music processing tasks (Logan et al. 2000b; Ellis 2007; Eghbal-Zadeh et al. 2015). They provide a compact representation of the spectral envelope are also a musically meaningful representation (Eghbal-Zadeh et al. 2015), and are used to capture acoustic scenes (Eghbal-Zadeh et al. 2016). Even though it is possible to use other features (Suh et al. 2011), we avoid the challenges involved in feature engineering and instead focus on the timbral modeling technique. We used a 20-dimensional MFCCs feature; (ii) Computation of Baum–Welch statistics In this step, we collect sufficient statistics by adapting UBM to a specific segment. This is a process in which a sequence of MFCC feature is represented by the Baum–Welch (BW) statistics (0-th and 1-st order Baum–Welch statistics) (Lei et al. 2014; Kenny 2012) using a GMM as prior; (iii) I-vector extraction I-vector extraction refers to the extraction of total factors from BW statistics. This step reduces the dimensionality of the movie clip representations and improves the representation for a recommendation task; (iv) Recommendation Recommendation is effected by integrating the extracted i-vector features in a CBRS.
During the training phase, the UBM is trained on the items in the training dataset and is used as an external knowledge source for the test dataset. In the testing step, test i-vectors are extracted using the models from the training step and the MFCCs of the test set. In the supervised approach, these i-vectors are projected by LDA in the training step. For the i-vector extraction, we used 20-dimensional MFCCs. For the items in the training set (in each fold), we trained a UBM with either 256 or 512 Gaussian components and a different dimensionality of latent factors (40, 100, 200, 400). We performed a hyper-parameter search and reported the best results obtained over fivefold cross-validation for each evaluation metric.
Visual features
The visual features we selected for our experiments were previously used in other domains, including image aesthetics, media interestingness, object recognition, and affect classification. We selected two types of visual features: (i) aesthetic visual features, a set of features mostly associated with media aesthetics, and (ii) deep learning features extracted from the fc7 layer of the AlexNet deep neural network, initially developed for visual object recognition, but extended and used in numerous other domains. Several aggregation methods were also performed with these features, with the goal of obtaining video-level descriptors from the frame-level set of extracted features.
Aesthetic-visual features the three groups of features and their early fusion combinations were aggregated in a standard statistical aggregation scheme based on mean, median, variance, and median absolute deviation. In a work discussing the measurement of coral reef aesthetics, the authors in Haas et al. (2015) propose a set of features inspired by the aesthetic analysis of artwork (Li and Chen 2009) and photographic aesthetics (Datta et al. 2006; Ke et al. 2006). This collection of features is derived from related domains, such as photographic style, composition, and the human perception of images, and was grouped into three general features types: color-related, texture-related and object-related.
The color-related features have 8 main components. The first elements consist of the average channel values extracted from the HSL and HSV color spaces. A colorfulness measure was created by calculating the Earth Mover's Distance, Quadratic Distance and standard deviation between two distributions: the color frequency in each of the 64 divisions of the RGB spectrum and an equal reference distribution. The hue descriptors contained statistical calculations for pixel hues: number of hues present, number of significant hues for the image etc. The hue models are based on the distance between the current picture and a set of nine hue models considered appealing for humans inspired by the models presented in Matsuda (1995). The brightness descriptor calculates statistics regarding image brightness, including average brightness values and brightness/contrast across the image. Finally, average HSV and HSL values were calculated while taking into account the main focus region and rule of thirds compositional guideline (Obrador et al. 2010).
The texture-related features have 6 components. The edge component calculates statistics based on edge distribution and energy, while the texture component calculates statistics based on texture range and deviation. Also entropy measures were calculated on each channel of the RGB color space, generating a measure of randomness. A three-level Daubechies wavelet transform (Daubechies 1992) was calculated for each channel of the HSV space along with the values for the average wavelet. A final texture component was based on the low depth-of-field photographic composition rule, according to the method described by Datta et al. (2006).
The object-related features have 11 components. These components are mostly based on the largest segments in an image obtained through the method proposed in Datta et al. (2006), which is based on the k-means clustering algorithm. The area, centroids, values for the hue, saturation, and value channels, average brightness values, horizontal and vertical coordinates, mass variance and skewness for the largest, and therefore most salient, segments each constitutes a component of this feature type. Color spread and complementarity also represented a component, while the last component calculates hue, saturation, and brightness contrast between the resulting segments.
As previously mentioned, this set of features is highly correlated to the human observer, some components being heavily based on psychological or aesthetic aspects of visual communication. For example, the hue model component calculates the distance between the hue model of a certain image and models considered appealing to humans, inspired by the work of Matsuda (1995). Also, some general rules of photographic style were used, rules previously shown to have a high impact on human aesthetic perception, therefore generating more pleasant images and videos (Krages 2012). For example, the authors in Liu et al. (2010) modify images in order to achieve a better aesthetic score, one of the rules applied for this optimization being the rule of thirds.
We used these features in our experiments, both separated into the three main feature types (color, texture and object) and in an early fusion concatenated descriptor for each image in the video. Regarding the aggregation method, we used four standard statistical aggregation schemes based on mean, median, variance and median absolute deviation.
Deep-learning features Deep neural networks have become an important part of the computer vision community, gathering interest and gaining importance as their results started performing better than more traditional approaches in different domains. The ImageNet Large Scale Visual Recognition Competition (ILSVRC) gives the opportunity to test different object recognition algorithms on the same dataset, consisting of a subset of 1.2 million images and 1000 different classes taken from the ImageNetFootnote 8 database. The AlexNet (Krizhevsky et al. 2010) deep neural network was the winner of the competition in 2012, achieving a top-5 error rate of 15.3%—a significant improvement over the second—best entry—that year. The authors also ran experiments on the ILSVRC 2010 dataset, concluding that the top-1 and top-5 error rates of 37.5% and 17% were again improvements on previous state-of-the-art approaches. One of the novelties introduced by this network was the ReLU (Rectified Linear Units) nonlinearity output function, which was able to achieve faster training times than networks working with more standard functions like \(f(x) = tanh(x)\) or \(f(x) = (1 + e^{-x})^{-1}\), instead using \(f(x) = max(0, x)\).
AlexNet consists of 5 convolutional layers and 3 fully connected layers, ending with a final, 1000-dimensional softmax layer. The input of the network consists of a \(224 \times 224 \times 3\) image, therefore requiring the original image to be resized if the resolution is different. The five convolutional layers have the following structure: the first layer has 96 kernels of size \(11 \times 11 \times 3\); the second, 256 kernels of size \(5 \times 5 \times 48\); the third, 384 kernels of size \(3 \times 3 \times 256\); the fourth, 384 kernels of size \(3 \times 3 \times 192\); and the final, fifth convolutional layer, 256 kernels of size \(3 \times 3 \times 192\). The fully connected layers all have 4096 neurons, and the output of the final one is fed into a softmax layer that creates a distribution for the 1000 labeled classes. This generates a network with 60 million parameters and 650, 000 neurons; thus, in order to reduce overfitting on the original dataset, some data augmentation solutions were employed, including image translations, horizontal reflections, and the alteration of the intensity of the RGB channels and a dropout technique (Hinton et al. 2012).
Given the good performance of the fc7 layer in tasks related to human preference, we chose to extract the outputs of this layer for each frame of our videos, thus obtaining a 4096-dimensional descriptor for each image. We then obtain a video-level descriptor through two types of aggregation methods: standard statistical aggregation, where we calculate the mean, median, variance, and median absolute deviation, and VLAD (Jégou et al. 2010) aggregation followed by PCA for dimensional reduction, with three different sizes for the visual word codebook: \(k \in \{32, 64, 128\}\).
Metadata features
We also use two types of editorial metadata features to serve as baselines: movie genre and cast/crew features.
Genre features For every movie, genre features are used to serve as metadata baselines. Genre Features (18 categories): Action, Adventure, Animation, Children's, Comedy, Crime, Documentary, Drama, Fantasy, Film Noir, Horror, Musical, Mystery, Romance, Sci-Fi, Thriller, War, and Western. The final genre feature vector is a binary, 18-dimensional vector.Footnote 9
Cast/crew features For every movie, the corresponding cast and crew have been downloaded from TMDBFootnote 10 using the available API and movie ID mapping provided by Movielens20M. The feature vector contains 162K Boolean features. Each movie is associated, on average, with 25 features.
Multimodal fusion
Two main paradigms of fusion exist in the literature of multimedia processing (Snoek et al. 2005): (i) late fusion which generates separate candidate results created by different systems and fuses them into a final set of results; the main limitation of late fusion methods is that they do not consider the correlation among features and are computationally more expensive during training; (ii) early fusion which tries to map multiple feature spaces to a unified space, in which conventional similarity-based evaluation can be conducted.
Motivated by the above, in current work we exploit a multimodal early fusion method based on canonical correlation analysis (CCA) that was successfully tested in our previous works (Deldjoo et al. 2016b, 2018d). CCA is a technique for joint fusion and dimensionality reduction across two or more (heterogeneous) feature spaces, which is often used when two set of data are believed to have some underlying correlation. We hypothesize that this is relevant in movie domain and combining audio, visual and textual data enriches the recommendations and training. Additionally, since the focus of the recommendation model in our work is on a CF-enriched CBF model (see Sect. 3.3), we have realized that currently the proposed method functions better with a lower size of the feature vectors. As CCA reduces the dimensionality of the final descriptor, it is is leveraged greatly in the proposed recommendation framework. Finally, CCA can be pre-computed and used in an off-the-shelf manner making it a convenient descriptor in offline experiments (as opposed to late fusion methods Deldjoo et al. 2018b).
We review the concept of CCA here for our methodology. Let \(X\in \mathbb {R}^{p \times n}\) and \(Y\in \mathbb {R}^{q \times n}\) be two sets of features in which p and q are the dimensions of features extracted from the n items. Let \(S_{xx} = \mathrm {cov}(x) \in \mathbb {R}^{p \times p}\) and \(S_{yy} = \mathrm {cov}(y) \in \mathbb {R}^{q \times q}\) be the within-set and \(S_{xy} = \mathrm {cov}(x,y) \in R^{p \times q}\) be the between-set covariance matrix. Let us further define \(S \in \mathbb {R}^{(p+q) \times (p+q)}\) as the overall covariance matrix—a complete matrix which contains information about associations between pairs of features—represented as follows:
$$\begin{aligned} S= \begin{pmatrix} S_{xx} &{}\quad S_{xy} \\ S_{yx} &{}\quad S_{yy} \end{pmatrix} = \left( \begin{array}{l@{\quad }l} \mathrm {cov}(x) &{}\mathrm {cov}(x,y)\\ \mathrm {cov}(y,x) &{} \mathrm {cov}(y) \end{array}\right) \end{aligned}$$
The aim of CCA is to identify a pair of linear transformations, represented by \(X^{*}=W_{x}^T X\) and \(Y^{*}=W_{y}^T Y\), that maximizes the pairwise correlation across two feature sets given by
$$\begin{aligned} \arg \max _{W_{x},W_{y}} \mathrm {corr}(X^{*},Y^{*}) = \frac{\mathrm {cov}(X^{*},Y^{*})}{\mathrm {var}(X^{*}) \cdot \mathrm {var}(Y^{*})} \end{aligned}$$
where \(\mathrm {cov}(X^{*},Y^{*}) = W_{x}^T S_{xy} W_{y}\) and \(\mathrm {var}(X^{*}) = W_{x}^T S_{xx} W_{x}\) and \(\mathrm {var}(Y^{*}) = W_{y}^T S_{yy} W_{y}\).
In order to solve the above optimization problem, we use the maximization procedure described in Haghighat et al. (2016). The CCA model parameters \(W_{x}\) and \(W_{x}\) are learned on trained items (warm items) and leveraged both in the training and test phases. We investigate two ways to perform fusion: (i) via concatenation (abbreviated by 'ccat') and (ii) via summation (abbreviated by 'sum') of the transformed features.
The cold-start recommendation model
The core recommendation model in our system is a standard pure CBF system using Eq. (4) to compute similarities between different pair of videos:
$$\begin{aligned} sim(i,j) = \frac{\mathbf {f}_i \, D \, \mathbf {f}_j}{\left||\mathbf {f}_i\right||^2_F \left||\mathbf {f}_j\right||^2_F} \end{aligned}$$
where \({\mathbf {f}_i \in \mathbb {R}^{n_F}}\) is the feature vector for video i, \(\left||\right||^2_F\) is the Frobenius norm and \(n_F\) is the number of features. We are interested in finding the diagonal weight matrix \(D \in \mathbb {R}^ {n_F \times n_F}\), which represents the importance of each feature.
An underlying assumption is that a CF model will achieve much higher recommendation quality than CBF and will be better able to capture the user's point-of-view. We use a CF model to learn D, cast into the following optimization problem:
$$\begin{aligned} \begin{aligned}&\mathop {\hbox {argmin}}\limits _{\mathbf {D}}&\left||\mathbf {S^{(CF)}} - \mathbf {S^{(D)}}\right||^2_F + \alpha \left||\mathbf {D}\right||^2_F + \beta \left||\mathbf {D}\right|| \end{aligned} \end{aligned}$$
where \(\mathbf {S^{(CF)}}\) is the item-item collaborative similarity matrix from which we want to learn, \(\mathbf {S^{(D)}}\) is the item-item hybrid similarity metric presented via Eq. (4), \(\mathbf {D}\) is the feature weight matrix, \(\alpha \) and \(\beta \) are the weights of the regularization terms.
We call this model collaborative-filtering-enriched content-based filtering (CFeCBF). The optimal \(\mathbf {D}\) is learned via machine learning, applying stochastic gradient descent with Adam (Kingma and Ba 2014), which is well suited for sparse and noisy gradients. The code is available on Github.Footnote 11 CFeCBF is a wrapper method for feature weighting; therefore, it does not learn weights while building the model but rather relies on a previously trained model and then learns feature weights as a subsequent step. Since the model we rely on is collaborative, we can only learn weights associated with features that occur in warm items. This affects how well the algorithm can perform in scenarios where the available features are too sparse; in this case, the number of features appearing in s but not in warm items will tend to increase, reducing the number of parameters in the model.
It is important to point out that while it will be possible to learn a zero collaborative similarity for items having a common feature, it will not be possible to learn anything for items with no common features. Therefore, content-based similarity poses a hard constraint on the extent to which collaborative information can be learned. As content-based similarity is a function of the item features, the sparser this matrix is the less information will be learnable from a collaborative model. This could be a challenge when using Boolean features that tend to be sparse, but much less of one when using real-valued attributes like the multimedia descriptors, which result in dense feature vectors. A consequence of this is that the success of applying CFeCBF on a given dataset depends not only on how accurate the collaborative model is, but also on whether its similarity structure, resulting from the items having common features, is sufficiently compatible with that of the content-based model.Footnote 12 CFeCBF requires a two-step training procedure. In the first step, we aim to find the optimal hyper-parameters for a collaborative model by training it on warm items and selecting the optimal hyper-parameters via cross-validation. Since we want a single hyper-parameters set, not one for each fold, we chose those with the best average recommendation quality across all the training folds.
User interactions' item-wise split; A contains warm items while B contains s and refers to a subset of the users
Once the collaborative model is available, the second step is to learn weights by solving the minimization problem described in Eq. (5). As the purpose of this method is to learn \(\mathbf {D}\), or feature weights, the optimal hyper-parameters for the machine learning phase are chosen via a cold item split to improve the CBF on new items. Figure 5 shows how a cold item split is performed: split A represents the warm items, that is, items for which we have interactions and that we can use to train the collaborative model, and split B represents cold items that we use only for testing the weights. All reported results for pure CBF and CFeCBF are reported on split B.
Experimental study A: Offline experiment
In this experiment, we investigate offline recommendation in cold- and warm-start scenarios. The specific experimental setup is presented in the following section.
Table 2 Characteristics of the evaluation dataset used in the offline study: \(\left| \mathcal {U} \right| \) is the number of users, \(\left| \mathcal {I} \right| \) the number of items, \(\left| \mathcal {R} \right| \) the number of ratings
We evaluated the performance of the proposed MRS on the MovieLens-20M (ML-20M) dataset (Harper and Konstan 2016), which contains user-item interactions between users and an up-and-running movie recommender system. We employ fivefold cross-validation (CV) in our experiments by partitioning the items in our dataset into 5 non-overlapping subsets (item-wise splitting of the user-rating matrix). Different folds will have different cold items. Similar to Adomavicius and Zhang (2012), we built the test split by randomly selecting 3000 users, each having a minimum of 50 ratings in their rating profile, in order to speed up the experiments on the many feature sets. The items those users interacted with will be considered cold items; see split B in Fig. 5. The remaining items and interactions will be part of the training set. The reported results are referred to split B. Meanwhile, split A is used to perform parameter tuning. The characteristics of the data split are shown in Table 2. The significantly higher number of ratings per item in the training set (A) is due to the fact that it contains more users, and hence more interactions, than the test set (B).
Objective evaluation metrics
For assessing performance in the offline experiments, we compute the two categories of metrics, accuracy metrics (cf. Sect. 4.2.1) and beyond-accuracy metrics (cf. Sect. 4.2.2). The name and definition of the specific metrics computed is provided in the corresponding sections.
Accuracy metrics
Mean average precision (MAP) is a metric that computes the overall precision of a recommender system, based on precision at different recall levels (Li et al. 2010). It is computed as the arithmetic mean of the average precision (AP) over the entire set of users in the test set, where AP is defined as follows:
$$\begin{aligned} AP = \frac{1}{\min (M,N)} \sum _{k=1}^{N} {P@k \, \cdot \, rel(k)} \end{aligned}$$
where rel(k) is an indicator signaling if the \(k{\mathrm {th}}\) recommended item is relevant, i.e., \(rel(k)=1\), or not, i.e., \(rel(k)=0\); M is the number of relevant items; and N is the number of recommended items in the top N recommendation list. Note that AP implicitly incorporates recall, because it considers relevant items not in the recommendation list. Finally, given the AP equation, MAP will be defined as follows:
$$\begin{aligned} {\textit{MAP}} = \frac{1}{|U|}\sum _{u \in |U|} {\textit{AP}}_u \end{aligned}$$
Normalized discounted cumulative gain (NDCG) is a measure for the ranking quality of the recommendations. This metric was originally proposed to evaluate the effectiveness of information retrieval systems (Järvelin and Kekäläinen 2002). It is nowadays also frequently used for evaluating music recommender systems (Liu and Yang 2008; Park and Chu 2009; Weimer et al. 2008). Assuming that the recommendations for user u are sorted according to the predicted rating values in descending order, \(DCG_u\) is defined as follows:
$$\begin{aligned} {\textit{DCG}}_u = \sum _{i=1}^N \frac{r_{u,i}}{log_{2} (i+1)} \end{aligned}$$
where \(r_{u,i}\) is the true rating (as found in test set T) for the item ranked at position i for user u, and N is the length of the recommendation list. Since the rating distribution depends on users' behavior, the DCG values for different users are not directly comparable. Therefore, the cumulative gain for each user should be normalized. This is done by computing the ideal DCG for user u, denoted as \( {IDCG}_u\), which is the \(DCG_u\) value that provides the best possible ranking, obtained by ordering the items by true ratings in descending order. Normalized discounted cumulative gain for user u is then computed as follows:
$$\begin{aligned} {\textit{NDCG}}_u = \frac{DCG_u}{IDCG_u} \end{aligned}$$
Finally, the overall normalized discounted cumulative gain \( {NDCG}\) is computed by averaging \( {NDCG}_u\) over the entire set of users.
Beyond-accuracy metrics
The purpose of a recommender system is not only to recommend relevant items to the user based on their past behavior but also to facilitate exploration of the catalogue, helping to discover new items that the user might find interesting. Beyond-accuracy metrics try to assess if the recommender is able to diversify its recommendations for different users and leverage the whole catalogue or if it is focused on just a few highly popular items. In this study, we focus on the following measures:
Coverage of a recommender system is defined as the proportion of items which have been recommended to at least one user (Herlocker et al. 2004):
$$\begin{aligned} coverage = \frac{|\hat{I}|}{|I|} \end{aligned}$$
where |I| is the cardinality of the test item set and \(|\hat{I}|\) is the number of items in I which have been recommended at least once. Recommender systems with lower coverage are limited in the number of items they recommend.
Intra-list diversity Is another important beyond-accuracy measure. It gauges the extent to which recommended items are different from each other, where difference can relate to various aspects, e.g., genre, style or composition. Diversity can be defined in several ways. One of the most common is to compute the pairwise distance between all items in the recommendation set, either averaged (Ziegler et al. 2005) or summed (Smyth and McClave 2001). In the former case, the diversity of a recommendation list L is calculated as follows:
$$\begin{aligned} IntraL(L) = \frac{ \displaystyle \sum \nolimits _{i \in L} \sum _{j \in L {\setminus } i} dist_{i,j}}{|L| \cdot \left( |L|-1\right) } \end{aligned}$$
where \(dist_{i,j}\) is some distance function defined between items i and j. Common choices are inverse cosine similarity (Ribeiro et al. 2012), inverse Pearson correlation (Vargas and Castells 2011), or Hamming distance (Kelly and Bridge 2006). In our experiments we report a diversity computed using the genre of the movies and cosine similarity.
Inter-list diversity or inter-user diversity measures the uniqueness of different users recommendation lists (Zhou et al. 2010). Given two users i and j, and their recommendation list L, the inter-list distance can be calculated by:
$$\begin{aligned} InterL(L_i, L_j) = 1 - \frac{q(L_i, L_j)}{|L|} \end{aligned}$$
where \(q(L_i, L_j)\) is the number of common items in recommendation lists of length |L|. \(InterL(L_i, L_j)=0\) indicates identical lists and \(InterL(L_i, L_j)=1\), completely different ones. The mean distance is obtained by averaging \(InterL(L_i, L_j)\) over all pairs of users such that \(i \ne j\).
A model which tends to frequently recommend the same set of items will result in similar recommendation lists and low diversity, whereas a recommender better able to tailor its recommendations to each user will exhibit higher diversity (Zhou et al. 2010). In this respect, inter-list diversity and intra-list diversity are complementary. Consider a Top Popular recommender (i.e., one that recommends the most popular items). Its recommendations might have high intra-list diversity if they involve movies with different characteristics; therefore, a user will perceive them as diverse. However, all users will receive the same recommendations and both item coverage and inter-list diversity will be very low.
While an increase in diversity can indicate that the recommender is better able to offer personalized recommendations, it should be taken into account that the lowest diversity, and item coverage, will be obtained by always recommending the same items, whereas the highest will be obtained by a random recommender. This is another example of the accuracy-diversity trade-off.
In order to better understand how much the proposed techniques truly contribute towards more diverse and idiosyncratic recommendations across all users, in addition to the above beyond-accuracy metric, we also computed the metrics entropy, Gini coefficient, and Herfindahl (HHI) index (Adomavicius and Kwon 2012). These metrics provide different means for measuring distributional dispersion of recommended items across all users, and are therefore referred to as aggregate diversity. If recommendations are concentrated on a few popular items, the recommender will have low coverage and low diversity in terms of entropy and HHI but high Gini Index. If recommendations are more equally spread out across all candidate items, the recommender will exhibit high diversity and coverage but low Gini Index (Adomavicius and Kwon 2012). These metrics provide an overview of the recommender system from a system-wide point of view and are useful for assessing its behavior when deployed on a real, business-oriented system.
The distributional dispersion metrics are defined as follows:
$$\begin{aligned}&Entropy = - \sum _{i \in I} \frac{rec(i)}{rec_t} \cdot \ln \frac{rec(i)}{rec_t} \end{aligned}$$
$$\begin{aligned}&Gini-index = \sum _{i=1}^{|I|} \frac{2i - |I| - 1 }{|I|} \cdot \frac{rec(i)}{rec_t} \end{aligned}$$
$$\begin{aligned}&Herfindahl-index = 1- \frac{1}{rec_t^2}\sum _{i \in I}rec(i)^2 \end{aligned}$$
where rec(i) refers to the number of times item i has been recommended over all users, \(rec_t\) the total number of recommendations (i.e., cutoff value times the number of test users), I the cold items set, and |I| its cardinality. Note that while the Gini index and Herfindahl index have a value range between 0 and 1, Shannon entropy is not bounded by 1.
Collaborative filtering model
Following the results of Ferrari Dacrema et al. (2018) we chose as collaborative model RP3beta (Paudel et al. 2017) which demonstrated a very competitive recommendation quality at a very small computational cost, since it does not require ML. RP3beta is a graph-based algorithm which models a random walk between two sets of nodes, users and items. Each user is connected to the items he/she interacted with and each item is similarly connected to the users. The model consists of an item-item similarity matrix which represents the transition probability between the two items, computed directly via the graph adjacency matrix, easily obtainable from the URM. The similarity values are are elevated at a coefficient alpha and divided by each item's popularity elevated to a coefficient beta, the latter acting as a reranking phase which takes the popularity bias into account.
Hyper-parameter tuning
The proposed approach requires two types of parameter tuning. Firstly, it is necessary to train and tune the CF model. Since we want a single optimal hyper-parameter set we train the CF recommender on all the train folds separately and then select the hyper-parameters corresponding to the best average recommendation quality on all folds, measured with MAP. This constitutes a robust validation and testing methodology, and reduces the risk to overfit. Each fold will be associated with its own collaborative model since different folds will correspond to different cold items split. Secondly, the tuning of the hyper-parameters of the feature weighting machine learning is performed in a similar way, again optimizing MAP. We searched the optimal hyper-parameters via a Bayesian search (Antenucci et al. 2018) using the implementation of Scickit-optimize.Footnote 13 As for different aggregation methods designed for the audio and visual features, we chose the best performing ones with regards to the metric under study.
Overall computational time and complexity
In this section, we provide general information regarding runtimes and overall computational complexity of the subsystems in the proposed framework.
Regarding the extraction of the visual features, this process performs above the real-time frame rate of the movies (25 or 30 frames per second). We have performed feature extraction on a computer with Intel Xeon E5-1680 processor with 8 cores, 16 threads and a base frequency of 3.00 GHz, 192 GB RAM and an NVIDIA 1080TI GPU card with 3584 CUDA cores. While the extraction of AlexNet features was handled by the GPU, with an average speed of 62.8 processed frames per second, the extraction of the aesthetic visual features was done on the CPU, in parallel, using 7 of the 8 available cores and recording an average speed of 38.3 processed frames per second.
The feature weighting phase has a low computational complexity as it requires, for each epoch, to compute the gradient for each collaborative similarity value and compute the prediction error by using the item features. It is therefore linear in terms of both the number of descriptors and in terms of the number of similarities which in turn grows quadratically on the items. In terms of runtime, on an Intel Xeon E3-1246 3.50 GHz with 32 GB RAM, learning the weights on the descriptors of length 200 takes 15 min on a single core, including the time required to perform the validations needed by early stopping.
Table 3 Performance of various features: i-vector (Audio), BLF (Audio), Deep (Visual), and AVF (Visual), editorial-metadata, in terms of accuracy metrics NDCG and MAP. For fusion, we report the results for the CCA fusion variation (either ccat or sum) that lead to the best performance (cf. Sect. 3.2). The features (or feature combination) which outperform genre significantly are shown in bold (\(p<0.05\))
Performance analysis: accuracy metrics
The experiments performed in Study A can be divided into four different categories, as presented in Table 3: baseline experiments using the genre and cast/crew metadata features, both editorially created (cf. Sect. 3.1.3)Footnote 14; unimodal experiments using traditional and state-of-the-art (SoA) audio and visual features (cf. Sect. 3.1); content-based multimodal experiments, where the proposed canonical correlation analysis (CCA) is used as an early fusion method (cf. Sect. 3.2); and finally, collaborative-filtering enhanced multimodal experiments, where the systems from the previous multimodal experiments are enhanced through the use of collaborative filtering (cf. Sect. 3.3). In the latter two, multimodal, categories, we report and analyze the performance of all combinations from the proposed unimodal features and the genre baseline.Footnote 15
As a general observation, we see that the unimodal visual and audio features constantly outperform the baseline metadata systems. The best performance is obtained by Deep visual features, improving the genre baseline by 53.0% in terms of NDCG and by 42.8% in terms of MAP. Even the lowest performing unimodal feature, i.e., i-vector, still achieves a 14.4% increase for NDCG and a 7.1% increase for MAP over the baseline. We further observe that the Deep feature outperforms the traditional AVF feature in the visual category, while in the audio category, the reverse pattern occurs, i.e., the traditional BLF feature has a better performance than the i-vector audio feature for both metrics.
As presented in Sect. 3.2, our multimodal approaches use CCA as a fusion method. We compared the CCA approach with a simple concatenation method, as well as with a weighted late fusion Borda count method, as described in Deldjoo et al. (2018b). We chose CCA as our early fusion method because all results were better for the CCA approach. For example, in the case of the i-vec + genre multimodal combination, CCA achieved a 9.5% MAP increase and a 20.2% NDCG increase over the simple concatenation method in the pure CBF approach, while in the CFeCBF approach, the CCA fusion method achieved a 151.6% increase in terms of MAP and a 181.8% increase in terms of NDCG. These results confirm not only that CCA fusion produces good results on its own but also that it increases the power of collaborative filtering approaches by heavily reducing the size of the feature vector. Furthermore, the use of an early fusion method such as CCA allows us to easily create systems that outperform the late fusion method mentioned in Deldjoo et al. (2018b), in both accuracy metrics.
For the multimodal CBF approach, we observe that the CCA fusion of the best performing unimodal audio and visual features (i.e., Deep and BLF) leads to the best multimodal results. More precisely, Deep + BLF achieves a 22.8% improvement over the baseline (0.0102 vs. 0.0083) in terms of NDCG and a 26.1% increase in terms of MAP (0.0053 vs. 0.0042). Similarly, the combination i-vec + genre performed strongly, improving on the baseline by 21.6% for NDCG (0.0101 vs. 0.0083) and 9.5% for MAP (0.0046 vs. 0.0042). This result was surprising, since both individual features, genre and i-vec, had a weaker performance in the unimodal experiment. In fact, in all genre combinations, such as AVF + genre, BLF + genre, and i-vec + genre, we can see an improvement in performance. This suggests that the genre feature has an information-complementary nature with other modalities, which can be leveraged using the CCA fusion. However, the combination of Deep + genre is an exception, as one can observe a decrease in performance. This may be due to the correlation between the two.
The multimodal CFeCBF approach aims to enable the recommendation of cold items by leveraging collaborative knowledge of warm items. The proposed method was applied on CCA multimodal approaches, as presented in the CBF multimedia approach. Looking at the performance globally, one can observe that the CFeCBF multimodal approach improves the pure CBF multimodal systems in all 10 combinations along NDCG and in 8 combinations along MAP; the few non-improved feature combinations, i.e., AVF + BLF and Deep + BLF, already performed well in pure CBF experiments. For NDCG, the average growth factor is 67%, with the minimum equal to 7% for Deep + BLF and the maximum equal to 123% for AVF + Genre. For MAP, the average growth factor is 68%, with the minimum equal to − 7% for AVF + BLF and the maximum equal to 148% for AVF + Genre. When compared with the genre baseline, the proposed CFeCBF method improves the features, on average, by 79.75% for MAP and 72.6% for NDCG.
One final step was taken for the validation of these results, namely performing the significance tests as pairwise comparisons between the best performing systems and the best performing baseline genre. For both NDCG and MAP metrics, we performed statistical significance tests using the multiple comparison test provided by the statistical and machine learning toolbox in MATLABFootnote 16 (function multcompare()), in which we adopted Fisher's least significant difference to compensate for multiple tests when performing all pairwise comparisons. Detailed information about the test can be found in Sheskin (2003). The three best performing systems, i-vec + genre, AVF + genre, and AVF + Deep, show significant improvements over the baseline with \(p < 0.05\), where the improvement along NDCG is 124.1%, 131.33%, and 130.12%, respectively, and that along MAP equal to 85.7%, 130.12%, and 130.12%, respectively. These results indicate the effectiveness of the proposed approach in dealing with very different kinds of features and its ability to embed collaborative knowledge in a CBF recommender. In particular, the systems showing significant improvements have lower dimensionality for the descriptors than the others. This suggests that learning feature weights becomes harder as the number of dimensions increases. Applying dimensionality reduction techniques is therefore beneficial when dealing with very long descriptors.
Performance analysis: beyond-accuracy metrics
In this section, we report the results for beyond-accuracy metrics. The results are summarized in Table 4 (reports diversity metrics computed on the various recommendation lists: inter-list diversity and intra-list diversity) and Table 5 (reports all the aggregate diversity metrics, which are instead computed on the overall number of times each item was recommended to any user: Item coverage, Shannon entropy, Gini index, and Herfindahl Index).
Table 4 Performance of various features in terms of beyond-accuracy metrics for list diversity. For fusion, we report the results for the CCA fusion variation (either ccat or sum) that lead to the the best performance (cf. Sect. 3.2). Results in bold show the features (or feature combinations) that outperform genre significantly (\(p<0.05\)) along the respective metric
Table 5 Performance of various features in terms of beyond-accuracy metrics for aggregate diversity. Results in bold show the features (or feature combinations) that outperform genre significantly (\(p<0.05\)). For each feature combination, we only report the results for the CCA method that has the best performance (either ccat or sum)
From Table 4, we can observe that intra-list diversity (intraL) exhibits similar values across all cases. As previously mentioned, this diversity is computed with respect to the genre of movies, so a higher diversity would mean recommendations of heterogeneous genres, while a lower diversity would mean recommendations of the same genre. Following this definition, we expect that a recommender based only on genre as a feature will exhibit the lowest intraL diversity, which is in fact what we do observe. If we consider that as baseline value, we can see that all other features—metadata, unimodal or multimodal—achieve slightly higher diversity while not penalizing recommendation accuracy; this increase is significant in all cases. In terms of inter-list diversity (interL), results are more varied. We can see that multimodal recommenders, both pure CBF and hybrid CFeCBF, yield higher diversity in most cases, meaning that given any two users, the average number of items they have in common in their recommendation lists is going to be lower. The increased InterL diversity for CFeCBF is statistically significant in almost all cases. This suggests that multimodal recommenders will be less prone to concentrate their recommendations on a small subset of items.
From Table 5, we can see the results for aggregate diversity metrics. Note that while greater diversity will result in higher values for Item coverage, Shannon entropy, and Herfindahl Index, it will drive Gini index closer to zero. These metrics allow us to look at the recommender from the point of view of the whole system instead of that of the user, which is important when deploying recommenders as a part of a business model. We first focus on Item coverage, which tells us the portion of cold items the system was able to recommend. We can immediately see that the baseline recommenders using metadata have poor coverage: only half of the available items were recommended at least once. Most models based on multimodal features, instead, exhibit significantly higher coverage—up to more than 90%, meaning they are able to explore the catalogue much better without sacrificing recommendation quality. The other metrics measure the number of times each item has been recommended. Compared to the coverage, they provide the additional information about the number of occurrences. Within a certain coverage value, the distribution of items can be very different. For example, in the case of a Top Popular recommender in a warm item scenario, the final coverage will be higher than the length of the recommendation list because some users will already have already interacted with those items and therefore other, less popular, items will be recommended to them. Distribution diversity metrics allow us to determine the extent to which the recommender is trying to diversify its recommendations. As an example, consider the 4 cases having coverage between 94.5 and 96.5%, with an interval of just 2% of all items. These cases exhibit a Gini index varying between .65 and .78, meaning that there is a difference in the number of times those items were recommended. In particular, the increase in coverage was accompanied in this case by more unbalanced, and therefore less diverse, item occurrence.
We can see how there is a significant difference between Multimodal and Base recommenders in terms of Gini index, meaning that the multimodal recommenders, both pure and hybrid, have more balanced item distribution. The combination of very high item coverage and improved distributional diversity metrics suggest that the collaborative machine learning step does not add a popularity bias to the feature weights, on the contrary CFeCBF is less subject to it than the Base recommenders. Moreover, we see that Shannon Entropy increases, meaning that the recommender is getting less "predictable" in the recommendations it will provide. This confirms what was observed in terms of interL diversity. The Herfindahl index is known to have a small value range when applied to recommender systems, as we can see in our experiments where its value ranges from 0.96 to 0.99. Compared to the other indices, it is less sensitive to items being recommended only a few times, due to its quadratic nature, but more sensitive to items being recommended a high number of times. Its values confirm the increased diversity achievable by Multimodal recommenders in almost all cases for pure CBF and in all cases for hybrid CFeCBF.
Cold to warm item transition
While the core of our experimental study is aimed at cold start items, in a real case scenario we expect some interactions to become available over time as the users interact with the cold items. For practical use it is interesting to assess when it is appropriate to change the recommendation model from a content based, either pure or CFeCBF, to a collaborative model. To this end we design a brief study, aiming to assess at which interaction density an item transition from cold to warm, allowing the use of CF methods.
It is already well known that, depending on the dataset, even a few interactions may be sufficient to outperform CBF approaches (Pilászy and Tikk 2009).
Experimental protocol
To simulate a realistic cold to warm transition we add some interactions to the cold items. Those interactions are taken from the original test set of that fold. Since this study requires to create a new data split, with a denser train and a sparser test set, the results here reported are not comparable to the ones reported in the previous study.
We report two different experimental settings, one preserves the popularity distribution of the items, the other does not. The reader should notice that, being sampled in different, ways, the test set of the two experiments are different and the results are not directly comparable.
Random sampling In order to preserve the statistical distribution of the interactions and the impact of the item's popularity, the new train interactions for the cold items are randomly sampled, with no constraints applied. This will result in a mixture of popular items having a few interactions and unpopular items having none. This experiment allows to assess what happens in a realistic case in which some cold items will be popular and therefore collect interactions much faster, while others will not. This is motivated by the fact that CF algorithms, which CFeCBF is learning from, are sensitive to the popularity distribution and altering such distribution will result in biased CF models. The original test data is sampled so that 2% of its interactions become new train data and 98% constitute the new test data. To show the behaviour at different densities, the train data is further divided in a smaller set only containing 0.5% of the original test interactions.
Fixed number of interactions While the previous experiment models a real case scenario more accurately, it leaves open the question of how significant is the effect of the popularity bias on the results. To this end also build a different split which contains a fixed number of train interactions for the cold items. This creates an artificial popularity distribution which will change the behaviour of the CF model. The number of interactions we chose is 1 and 5. This will result in a perfectly balanced train set. In this case the test data is composed by the original test data minus 4 interactions for each item.
This new train data is therefore composed by the original train data plus the interactions sampled from the test set and is used to train all algorithms: CBF, CF and CFeCBF. The optimal parameters remain those selected in the previous phase when no interactions were available. In a real case scenario it would be impractical to run a new tuning of the model's parameters after each few interactions are added. It is instead more realistic for this tuning phase to be executed again only once a sufficient amount of new data is available.
Table 6 Results for the cold to warm transition scenario for accuracy metrics and Item Coverage. In evaluation scenario Cold the test items are cold. In Warm 0.5% the 0.5% of existing interactions have been added to the cold items, while in Warm 2.0% its the 2.0%
Result discussion
The results for the random split are reported in Table 6 for both accuracy metrics and Item Coverage.Footnote 17 As it is possible to see, in terms of accuracy metrics the recommendation quality of pure CBF remains constant as the transition progresses. CFeCBF, instead, changes its recommendation quality, in some cases improving over the cold item case, in others not. This is due to the evolving CF model it is learning from.
The most important thing to observe is that the pure collaborative algorithm, RP3beta, is immediately able to outperform all CBF and CFeCBF models in terms of accuracy metrics. It should be noted that Movielens, the dataset from which the interactions are taken, tends to exhibit high recommendation quality for collaborative algorithms which makes this cold to warm transition very fast. Consider that Warm 0.5% corresponds to an average of \(1\times 10^{-1}\) interactions per item and Warm 2.0% of \(4\times 10^{-1}\) interactions per item. Looking at the recommendation quality alone is however misleading. In terms of diversity it is possible to see that CF has a remarkably low item coverage. This means that the CF algorithm is still not able to explore the catalogue, being confined to a marginal \(6\%\) of the available items. The result can be explained by the significant popularity bias of the dataset, hence a few items account for a sizable quota of the interactions, while many others have much fewer. This behaviour means that the CF model is recommending only a few popular items, being unable to recommend the vast majority of them. CF fails completely to allow the user a broad exploration of the catalogue and offers very little personalization. Moreover, if the items are not seen by the users, it will be very difficult to collect the interactions needed for them to become warm items, the risk being to keep them in a cold state for very long. CFeCBF, on the other hand, has a very high Item Coverage, which allows a broader exploration of the catalogue, yielding to a higher probability cold items will be rated and a more effective CF model could be applied at a later stage.
If we look at the results for the fixed number of interactions experiment in Table 7, we can observe a different behaviour. The CBF and CFeCBF models maintain their almost stable recommendation quality while CF increases. However, as opposed to the previous case, we can see that the CF advantage grows less steeply with respect to CFeCBF even though the train data is much denser, 1 and 4 as opposed to \(4\times 10^{-1}\). Moreover, the CF Item Coverage is comparable or higher than CFeCBF. This allows to state that the behaviour of the CF algorithm in the random sampling experiment is strongly influenced by the significant popularity bias of the dataset.
To summarize, in terms of accuracy metrics CF algorithms are able to outperform CBF and CFeCBF when even just a few interactions are available, more so if the dataset has a strong popularity bias. However CBF and CFeCBF maintain a sizable advantage in terms of diversity metrics and Item Coverage. Depending on the specific use-case or application, and therefore the desired balance between accuracy and catalogue exploration, a different strategy may be adopted. If the main focus is on accuracy, then as soon as the item has an interaction it can be considered as warm. The reader should note that, while Movielens has a high popularity bias, other datasets with a less pronounced bias will exhibit a less steep CF quality improvement. If the focus is on improving catalogue exploration to reduce the popularity bias effect then the target number of interactions per item may be pushed further.
Table 7 Results for the cold to warm transition scenario for accuracy metrics and Item Coverage. In evaluation scenario Cold the test items are cold, in Warm 1 each test item has exactly 1 interaction in the train set, in Warm 5 each test items has 5 interactions in the train set
Experimental study B: insights from a preliminary user study about perceived quality
In this section, we describe an empirical study whose goal is not to recommend new movies, as in the experimental study A, but to understand to what extent the proposed movie genome is perceived as useful when deployed in a real MRS. The developed system uses a pure CBF recommender based on the KNN algorithm and measures the utility of the recommendation as perceived by the user in terms of accuracy, novelty, diversity, level of personalization, and overall satisfaction. In this study, we intentionally avoid the discussion of hybridization and focus instead only on six unimodal recommendation approaches, classifiable in 3 categories: (i) metadata: genre and tag, (ii) audio: i-vectors and BLF, and (iii) visual: Deep features and AVF. We use only the unimodal recommendation schemes presented in the experimental study A. The reason for this is to avoid overloading users with too many recommendation choices, and thus to be able to obtain more reliable responses from users collectively. Note that in this study, the tags feature is considered because, as stated, the study's focus is no longer on new movie recommendation (as in study A) and tags serve as a rich semantic baseline.
Our preliminary studies in a similar direction have been published in Elahi et al. (2017) which focused on a single visual modality (Elahi et al. 2017), and in Deldjoo et al. (2018b), which used a lower number of participants (74 vs. 101). In addition, compared to Deldjoo et al. (2018b), we performed better sanity checks and removed unreliable user input. Further information is provided in the following sections.
Perceived quality metrics
The goal of the current study is to measure how the user perceives the quality of the proposed recommender system. Perceived quality is as an indirect indicator of a recommenders potential for persuasion (Cremonesi et al. 2012). It is defined as the degree to which the users judge recommendations positively and appreciate the overall experience of the recommender system. We operationalize the notion of perceived quality in terms of five metrics (Ekstrand et al. 2014): Perceived accuracy (also called Relevance)—measures how much the recommendations match users' interests, preferences, and tastes; Satisfaction—measures global users' feelings about their experience with the recommender system; Understands me—relates to perceived personalization or the user's perception that the recommender understands their tastes and can effectively adapt to them; NoveltyFootnote 18—measures the extent to which users receive new (unknown) recommendations; Diversity—measures how much users perceive recommendations as different from each other, e.g., movies from different genres.
Evaluation protocol
To measure the user's perception of the recommendation lists according to the five quality metrics explained above, we adopt the questionnaire proposed in Knijnenburg et al. (2012). This instrument contains 22 questions to assess various aspects of the recommendation lists. For convenience, these questions are shown in Table 8. As suggested by the authors from Ekstrand et al. (2014), the questions are asked in a comparative mode instead of seeking absolute values.
Table 8 The list of questions (Ekstrand et al. 2014; Knijnenburg et al. 2012) used to measure the perceived quality of recommendations. Note that answers/scores given to questions marked with a \(+\) contribute positively to the final score, whereas scores to questions marked with a − are subtracted
Screenshots of the MISRec web application, designed for movie recommendation and empirical studies. The user needs to register, answer demographic and personality questionnaires, select his/her favorite genre, and rate some movies by looking at their trailers. Then, he/she is presented with 3 recommendation lists and a list of questions about perceived quality
We developed MISRec (Mise-en-Scène Movie Recommender), a web-based testing framework for the movie search and recommendation domain, which can easily be configured to facilitate the execution of controlled empirical studies. Some screenshots of the system are presented in Fig. 6. MISRec is powered by a pure CBF algorithm based on KNN and supports users with a wide range of functionalities common in online video-streaming services such as Netflix.Footnote 19 MISRec contains the same catalog of movies used in the first study (see Sect. 4). Users can browse the catalog of movies, retrieve detailed descriptions of each, rate them, and receive recommendations. MISRec also embeds an online questionnaire system that allows researchers to easily collect quantitative and qualitative information from the user. The first prototype of MISRec was used for conducting an empirical study on the contribution of stylistic visual features to movie recommendation, and the results were published in Elahi et al. (2017). A more recent development of MISRec powered by the proposed movie genome features was published in Deldjoo et al. (2018b). An extension of the system was also developed in Deldjoo et al. (2017b) to use the system in an interactive manner e.g., for kid movie recommendation using cover photos of the movies as the system activator.
Our main target audience is users aged between 19 and 54 who have some familiarity with the use of the web but have never used MISRec before the study (to control for the potentially confounding factor of biases or misconceptions derived from previous uses of the system). The total number of recruited subjects who also completed the task was 101 (73 male, 28 female, mean age 25.64 years, std. 6.61 years, min. 19 years, max. 54 years). Data collection were carried out mostly from master students at three universities: Politecnico Di Milano Italy, JKU Linz Austria and Politehnica di Bucharest, Romania attending the course of Recommender Systems or similarly related courses. They were trained to perform the study, were given written instructions on the evaluation procedure, and were regularly supervised by Ph.D. students and a PostDoc researcher during their activities. The interaction begins with a sign-up process, where each participant (user) is asked to specify his/her e-mail address, user name, and password (see Fig. 6 top-middle). For users who wish to remain anonymous, we provide the option to conceal their true email address. Afterwards, the user is asked to provide basic demographics (age, gender, education, nationality, and number of movies watched per month, consumption channels, some optional social media IDs, such as Facebook, Twitter, and Instagram). After the user has registered for the system and provided his/her basic demographic information, he/she is asked to fill out the Ten-Item Personality Inventory (TIPI) questionnaire (see Fig. 6 middle-left) so that the system can assess his/her Big Five personality traits (openness, conscientiousness, extroversion, agreeableness, and neuroticism) (McCrae and John 1992). Then, for preference elicitation (Chen and Pu 2004), the user is invited to browse the movie catalog from his/her favorite genre and to scroll through productions from different years in a user-friendly manner (see Fig. 6 center and middle-right). The user initially selects four movies as his/her favorites.
The user can watch the trailers for the selected movies and provide ratings for them using a 5-level Likert scale (1 = low interest in/appreciation for the movie to 5 = high interest in/appreciation for the movie). The user can also report a movie (if the trailer is not correctly displayed) and the movie will be skipped (see Fig. 6 bottom-left). After that, on the basis of these ratings and the content features described in Sect. 3.1, three categories of recommendation lists are presented to the user: (i) audio-based recommendation using BLF or i-vectors as features, (ii) visual-based recommendation using AVF or Deep as features, (iii) metadata-based recommendation using genre or tag as features. In each of the three recommendation categories, the recommendations are created using one of the two recommendation approaches (e.g., BLF or i-vectors for (i), and so on), chosen randomly. Since watching trailers is a time-consuming process, we decided to show only four recommendations in each of the three lists.
It is important to note that since we do not wish to overload the user with too much information, we avoid presenting him/her with six recommendation lists using all of the features. This would be the case in a within-subject design, where each subject uses all variants of the factorial designs simultaneously, i.e., six recommendation approaches in this case. Instead, we decided to use a between-subject design, where factorial designs are randomized for a given subject. Since our final goal is to have the user compare the three recommendation classes (i.e., audio vs. visual vs. metadata) at the same time, the way we implemented the between-subject design randomizes each of the two instances of each category for a given user. Therefore, each user compares one out of eight possible combinations: (BLF, AVF, genre), (i-vector, AVF, genre), (BLF, AVF, tag), (i-vector, AVF, tag) and so forth.Footnote 20 This gives us more flexibility in handling all this information and obtaining reliable responses. Finally, to avoid possible biases or learning effects, the positions of the recommendation lists are randomized for each user.
In this section, we present the user-perceived accuracy, satisfaction, personalization, diversity, and novelty. Before analyzing the survey responses, we cleaned the data by removing users who did not complete the questionnaire. We also removed users who were too fast in giving answers (less than 15% of the median time of all users) since we do not consider these users reliable. As the results of these filtering steps, 21 users are filtered out. Furthermore, users were asked to specify how many of the movies in each recommendation list they have seen. A list is included in the analysis only if the user has seen at least one movie from it. For example, if a user chooses a list as the recommendation most accurately matching his/her taste but has previously specified that he/she has not seen any movie from that list, we discard that list from his/her responses.
We compute a score for each recommender/feature with respect to the five performance measures. When recommendation lists are presented to the user, he/she has to choose one list out of the three as an answer to each question (cf. Table 8). Each selected list counts for a vote for the respective recommender that has created the list. Note that answers/scores given to questions marked with a \(+\) contribute positively to the final score, whereas scores to questions marked with a − contribute negatively. Finally, all votes given to each recommender are summed along each dimension (performance measure) and expressed as percentages, i.e., the relative frequency with which each recommender has been selected as the best one. The final results for the five dimensions are presented in Table 9 and discussed below.
Table 9 Results of the user study with respect to the five tested perceived quality criteria in a real movie recommender system
Perceived accuracy/relevance The following algorithms are perceived as the most accurate (relevant) by the subjects: tag, genre, and the SoA visual deep feature, with 26%, 25%, and 24% of the votes, respectively. User-generated tags are rich semantic descriptors and, as expected, the respective feature is evaluated the best by the subjects; however, the difference from genre and deep features remains very small (1–2%). Meanwhile, the lowest performance is obtained by the traditional audio and visual features BLF and AVF with 3% and 8% of the votes, respectively. I-vector aggregates 13% of the votes. These results are in agreement with our expectations in that, as a standalone feature, the proposed SoA feature, deep, and i-vector show the most promising results compared with traditional multimedia features; e.g., Deep achieves a result of 24% in comparison with 8% for AVF, which represents an improvement of about 300%.
Understands me and satisfaction The results of users' perceived personalization (captured by the questions in the "Understands Me" category) and the overall feeling of the experience with the recommender system (captured by the questions in the "Satisfaction" category) show superior performance for Deep and tag features, with 32% and 31% of the votes, while genre is ranked lower, with 24% of the votes. For user satisfaction, the best performance is perceived for tag, deep, and genre features, with 25%, 24%, and 24% of the user votes, respectively. The lowest performance is obtained by the traditional audio and visual features (between 7 and 10%). We can also note that the results along the above perceived quality metrics are highly correlated (Pearson's correlation coefficient is 0.9735). The only exception is audio, in which we can find a difference in two dimensions between the performance obtained by SoA i-vectors (compare 3.6% vs. 11%) and by traditional BLFs (compare 1.2% vs. 6%). The results of "Understands me" and "Satisfaction" are also highly correlated with perceived accuracy (Pearson's correlation coefficients are 0.9390 and 0.9897, respectively.). This can indicate that the users' perception of personalization and satisfaction is the same as accuracy and that users respond to the questions belonging to these categories in a similar way.
Diversity The results for the perceived diversity indicate that the best performance is achieved by genre (29%)—substantially higher than i-vector, Deep, and tag, with 19%, 18%, and 16% of the votes, respectively. On the other hand, both traditional visual and audio features, AVF and BLF, show the lowest perceived diversity, attracting only 13% and 6% of the votes, respectively. The results for diversification are slightly different than those gained in our original user study (Deldjoo et al. 2018b) and show that users perceived recommendation by genre the most diverse (while perceived highly relevant too). Perhaps this is because users do not mentally compute list diversification based on genre diversity but also consider other attributes (e.g., the appearance of the DVD cover) when they are asked to indicate the most diversified recommendation list. Another reason could be that one of the questions explicitly asks for diversity of mood, and the same genre can have movies with very different moods (e.g., in sci-fi).
Novelty Results for novelty are surprising in several ways. Firstly, it is the traditional visual features, AVF, which have the highest amount of perceived novelty, gaining as much as 31% of votes, followed by the SoA audio and visual features i-vector and deep with 21% and 19% of the votes, respectively. Meanwhile, the tag feature has attracted a very small amount, i.e., only 5%, of the scores for perceived novelty. Since tags are user-assigned, they have a high semantic content and capture something specific about the user perception of the movie. Therefore, similar tags may yield to recommendations not perceived as novel.
Globally, the results of our study on perceived recommendation quality indicate that perceived quality of recommendations is high for the SoA visual and audio features (Deep and i-vector) along most investigated performance measures. The exception is the user's perceived personalization ("Understands Me") for which i-vector performs poorly (but Deep visual performs best). For the remaining dimensions, these SoA features are ranked in the top 3 of all investigated features. Especially when it comes to novelty, SoA audio and visual features by far outperform metadata features. Overall, each feature has its merits, which again support our proposal for multimodal recommendation approaches.
Conclusions and future perspectives
In this work, we presented a framework for new movie recommendation by exploiting rich item descriptors and a novel recommendation model. We compared our system to some standard metadata-based methods that use genres and casts (editorial metadata). Specifically, the proposed system integrates multimedia aesthetic visual features and audio block-level features, as well as novel, state-of-the-art deep visual features and i-vector audio features, together with genre and cast features, all of which are referred to as the movie genome. For exploiting the complementary information of different modalities, we proposed CCA to fuse movie genome descriptors into shorter and stronger descriptors. Lastly, we presented a novel recommendation model that leverages a two-step approach named collaborative-filtering-enriched content-based filtering (CFeCBF). It exploits the collaborative knowledge of warm items (videos with interactions) to weight content information for cold items (videos without interactions) and improve the ability to recommend cold videos, for which interactions and user-generated content are rare or unavailable. The proposed system represents a practical solution for alleviating the CS problem, in particular, the extreme CS new item problem, where newly added items lack any interaction and/or user-generated content.
Discussion of the results
For evaluation, we conducted two empirical studies: (i) a system-centric study to measure the offline quality of recommendations in terms of accuracy (NDCG and MAP) and beyond accuracy (list diversity, distributional diversity, and item coverage) (cf. Sect. 4); (ii) a preliminary, user-centric online experiment to measure different subjective metrics, including relevance, satisfaction, and diversity (cf. Sect. 5). In both studies, we used a dataset of more than 4000 movie trailers, which makes our approach more versatile, because trailers are more readily available than full movies.
In the first study, visual and audio features generally outperform the metadata features with respect to the two tested accuracy measures, with an average growth factor of 32% along NDCG (min 14% and max 53%) and 23% along MAP (min 7% and max 42%). The real improvement, however, is in the final system performance, in which the proposed system outperforms the baseline by a substantial margin of 80% along NDCG and 73% along MAP and also outperforms the simpler multimodal recommender model using CCA in a pure CBF system by 67% for NDCG and 68% for MAP. These results are promising and indicate the capability of our recommendation model to improve the utility of new item recommendation by leveraging rich CF data for existing warm items and utilizing them as feature weights to improve the content information in pure CBF.
Moreover, in terms of beyond-accuracy measures, we can see that the genre-based recommender exhibits the lowest diversity, as could be expected. In addition, our results show that the multimodal recommender is able to provide substantially higher coverage and improved distributional diversity on all reported metrics. This means that a multimodal recommender is less prone to popularity bias; in particular, multimodal recommendations generated by our CFeCBF model show a significant improvement along (almost) all reported beyond-accuracy metrics, while not penalizing the accuracy and even improving it substantially.
When an item transition from cold to warm we can see that CF is able to outperform CFeCBF very soon in terms of accuracy metrics on a dataset with significant popularity bias, while CFeCBF still exhibit much better ability to leverage all the available items. The strength of the two algorithms may be combined allowing to exploit the superior recommendation quality of CF for warm items and the much greater coverage of CFeCBF to recommend cold items, whose low popularity renders the transition to warm slower.
In the user study, results show that the perceived recommendation for state-of-the-art visual (Deep) and audio (i-vector) features are meaningful. With the exception of the user's perceived personalization, in which i-vector performed poorly, these audio and visual features are ranked in the top 3 of all investigated features. In some cases, such as for the perceived novelty, the improvement of these features over metadata was significantly high. Overall, the results of the user study show that each feature has advantages and supports our proposal for multimodal recommendation approaches.
Answers to research questions
RQ1: Can the exploitation of movie genome describing rich item information as a whole, provide better recommendation quality compared with traditional approaches that use editorial metadata such as genre and cast in CS scenarios? As the experiments have shown, multimedia features can provide a good alternative to editorial metadata such as genre and cast in terms of both accuracy and beyond-accuracy measures. The use of multimedia features can allow to increase the recommendation quality in terms of accuracy while also improving the ability of the recommender to leverage the whole catalogue of items.
RQ2: Which visual and audio information better captures users' movie preferences in CS scenarios? The most important improvement for the accuracy metric was achieved by exploiting the state-of-the-art deep features for the visual modality but traditional block-level features for the audio modality.
RQ3: Could we leverage user interaction to enrich cold item information? We proved that it is possible to effectively leverage user interactions and enrich the item descriptors by learning a set of feature weights associated with the descriptors. This would result in improving the recommendation quality of cold items over current editorial baselines (genre and cast).
Recommendation model The proposed recommender model has a few limitations. Firstly, since it leverages item features, the quality and noisiness of item features have an impact on the ability to learn good feature weights. If an item has too few features, the resulting recommendations will exhibit limited diversity and the weights might embed some popularity bias. This is visible in Table 4 for AVF + Genre, which, while having good recommendation quality, exhibits lower InterL diversity with respect to the other cases. On the other hand, if the number of features is too high, the number of collaborative similarities might not be enough to ensure good weighs are learned.
Secondly, as the model leverages a collaborative model, this feature weighting scheme will not be applicable to any scenario. If the user-item interactions are too few, it is well known that the collaborative model will perform poorly in comparison to a pure CBF recommender. If this is the case, the learned weights will be approximating a poor collaborative model and therefore the resulting recommendations will not improve. Even so, however, it may still be possible to leverage a collaborative model on a smaller and denser portion of the dataset to learn only some of the weights. This is an aspect that can be studied more in detail.
Thirdly, in the case of Boolean features, CFeCBF is sensitive to items with very sparse features due to the fact that it can learn weights only for features available for cold items. Feature sparsity has the dual effect of both increasing the probability of new items having many new features, previously unobserved, and reducing the degree of freedom of the model.
Finally, in our previous study (Deldjoo et al. 2016d), we concluded concluded that trailers and movies share similar characteristics in the recommendation scenario. However, the dataset used in Deldjoo et al. (2016d) was rather small (167 full movies and corresponding trailers were used for comparison). Also, the number of visual features was limited (only five features, cf. Rasheed et al. 2005). Due to these restrictions, the generalizability of our findings in Deldjoo et al. (2016d) might be restricted. Nevertheless, we argue that using trailers instead of full-length movies serves as a good proxy and has several advantages: trailers are accessible, are sensibly shorter that the entire movie, and preserve the main idea of the movie since they are designed to trigger the viewers interest in watching the entire movie. Results in the paper at hand show that the performance recommendation system that exploits movie genome is better in comparison with editorial metadata (using genre or cast). We believe this can be seen as a breakthrough to demonstrate that they can effectively replace the full movies. Lastly, depending on the strength of the video descriptors with respect to the CF information, the items may transition from cold to warm after even a single interaction. In popularity biased datasets a premature switch from CFeCBF to CF may result in poor catalogue exploration and therefore limited overall recommender effectiveness. This effect can be minimized by adopting strategies to allow a gradual switch between the two allowing the less popular items more time to collect the interactions they need to become warm, while benefitting from the higher recommendation quality of a CF for warm items. The choice of an optimal point where to switch between CFeCBF and CF remains challenging.
User study limitations The reported user study results should be considered preliminary. In fact, given the relatively low number of participants, the results may not be statistically significant. Given the complexity of the questionnaire, which takes more than half an hour to complete, as well as due to the specificity of the movie dataset used, i.e., the movies tend to be classic ones not easily available to the younger generation, it is very difficult to find reliable users and motivate them to participate in the study, even when considering a paying platform such as crowdsourcing.
We believe our proposed movie recommendation framework can pave the way for a new paradigm in new product recommendation by exploiting CFeCBF models built on top of rich item descriptors extracted from content. Examples of such products include fashion (images), music (audio), and tourism (both images and audio) and generic videos. As a related future research line, we would like to understand in what ways affective metadata (metadata that describe the user's emotions) can be used for CBF of videos/movies, similar to the research (Tkalčič et al. 2010) carried out for images.
Regarding the carried out user study, currently it involves 101 subjects. This is while according to Knijnenburg and Willemsen (2015), approximately 73 subjects are necessary in every configuration to ensure statistical significance of results (i.e., about 600 subjects in total). This is an important limitation of our current work, which we plan to overcome in the future by hiring a larger number of reliable subjects. Furthermore, we plan to validate the generalization power of our new movie recommender model on video datasets of a different nature, such as full-length movies, movie clips and user-generated videos. An initial attempt at the former was published in our work (Deldjoo et al. 2018b) and at the latter in Deldjoo et al. (2018a), whose authors plan to release a publicly available dataset of movie clips. Part of these data is used in the MediaEval 2018 task "Recommending Movies Using Content".Footnote 21
Last but not least, a feature analysis will be conducted to better understand how movie genome features contribute to the success of the combined features as part of future work.
https://www.youtube.com/.
https://www.instagram.com/.
Note that videos without interactions can also be old videos that have been never been watched by a user.
Similar to biological DNA composed of long sequences of four letters A, T, C, G referred to as nucleotides.
http://www.pandora.com/about/mgp.
Pandora might also use automatically extracted content (and other) features in their system, but the MGP is arguable the approach for which Pandora is best known.
https://github.com/MaurizioFD/CFeCBF.
http://www.image-net.org/.
While presenting the results, we will use the genre metadata as the baseline for evaluation, as it is prevalent in the domain. Furthermore, we refrain from using user-generated metadata such as tag features in this work, since in a new item CS situation these features cannot exist.
https://www.themoviedb.org/.
As our experiments have showed, the best collaborative similarity will not necessarily yield the best weights.
https://scikit-optimize.github.io/.
Note that we could not use tags as a feature in Study A, since the tags available in this dataset are user-generated. For cold items, no interactions with users have occurred yet, so no tags could be provided. While it could be possible the users added tags without providing a rating, this does not solve the underlying problem as it presumes a kind of interaction. Therefore the available tags for each items will be related to its popularity, some items will acquire tags easily while others may have none for quite some time.
Note that we use the genre features as the main baseline due to their widespread usage and the fact that genre and cast had similar performance in almost all reported metrics.
https://www.mathworks.com/help/stats/multiple-comparisons.html.
For brevity we did not report all beyond accuracy metrics.
Note that we could not use Novelty as an evaluation criterion in study A because Novelty is defined in terms of item popularity, which is available for warm items but not for cold items.
https://www.netflix.com.
Note that we could not use tag as a feature in Study A since tags are user-generated content. In cold items, no interactions with users have occurred yet; therefore, no tag could have been provided as a feature. Tags could be obtained via cross-domain techniques, but those are a vast research area and outside the scope of this paper. Tags could also be obtained by manual/editorial tagging, but that would be time-consuming and expensive, and therefore not suitable for a high rate of new items, which is the scenario of main interest for this paper.
http://www.multimediaeval.org/mediaeval2018/content4recsys.
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Open access funding provided by Johannes Kepler University Linz. This work was supported by the Austrian Ministry for Transport, Innovation and Technology, and the Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH. The work of Mihai Gabriel Constantin and Bogdan Ionescu was partially supported by the Ministry of Innovation and Research, UEFISCDI, project SPIA-VA, agreement 2SOL/2017, Grant PN-III-P2-2.1-SOL-2016-02-0002.
Politecnico di Milano, Milan, Italy
Yashar Deldjoo, Maurizio Ferrari Dacrema, Stefano Cereda & Paolo Cremonesi
University Politehnica of Bucharest, Bucharest, Romania
Mihai Gabriel Constantin & Bogdan Ionescu
Department of Computational Perception, Johannes Kepler University Linz, Linz, Austria
Hamid Eghbal-zadeh & Markus Schedl
Yashar Deldjoo
Maurizio Ferrari Dacrema
Mihai Gabriel Constantin
Hamid Eghbal-zadeh
Stefano Cereda
Markus Schedl
Bogdan Ionescu
Paolo Cremonesi
Correspondence to Markus Schedl.
OpenAccess This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Deldjoo, Y., Dacrema, M.F., Constantin, M.G. et al. Movie genome: alleviating new item cold start in movie recommendation. User Model User-Adap Inter 29, 291–343 (2019). https://doi.org/10.1007/s11257-019-09221-y
Issue Date: 01 April 2019
DOI: https://doi.org/10.1007/s11257-019-09221-y
Movie recommender systems
Cold start
Warm start
Semi-cold start
Content-based
Audio descriptors
Visual descriptors
Hybrid recommender system
Feature weighting
Collaborative-enriched content-based filtering
Canonical correlations analysis | CommonCrawl |
Area and Perimeter of an Isosceles Triangle – Formulas and Examples
The perimeter of an isosceles triangle represents the total length around the triangle. On the other hand, the area represents the two-dimensional space occupied by the figure. We can find the perimeter of an isosceles triangle by adding the lengths of its three sides, and we can find its area by multiplying the product of its base and height by one-half.
In this article, we will learn how to calculate the perimeter and area of an isosceles triangle. We will explore their formulas and use them to solve some practice problems.
Learning about the perimeter and area of an isosceles triangle.
How to find the perimeter of an isosceles triangle?
How to find the area of an isosceles triangle?
Perimeter and area of an isosceles triangle – Examples with answers
Perimeter and area of an isosceles triangle – Practice problems
We can calculate the perimeter of an isosceles triangle by adding the lengths of its three sides. This means we can use the following formula:
$latex p=a+b+c$
where, $latex a,~b,~c$ are the lengths of the sides of the triangle.
However, since an isosceles triangle has two sides of equal length, we can simplify the formula for the perimeter as follows:
$latex p=b+2a$
where, b is the length of the base and a is the length of the congruent sides.
We can calculate the area of an isosceles triangle by multiplying one-half times the product of its base and height. That is, we multiply the lengths of its base and its height and divide by 2:
$latex \text{Area}= \frac{1}{2} \times \text{base} \times \text{height}$
$latex A=\frac{1}{2} \times b \times h$
where, b is the length of the base and h is the length of the height.
Calculate the area of an isosceles triangle if we only know its sides
To find the area of an isosceles triangle in terms of its sides only, we need to find an expression for the height of the triangle in terms of its sides and then plug it into the area formula.
The Height of an Isosceles Triangle can be calculated using the following formula:
$latex h=\sqrt{{{a}^2}-\frac{{{b}^2}}{4}}$
Substituting this expression for height into the area formula, we have:
$latex A=\frac{1}{2}(\sqrt{{{a}^2}-\frac{{{b}^2}}{4}}\times b)$
b is the length of the base of the isosceles triangle
h is the height of the triangle
a is the length of the congruent sides of the isosceles triangle
Area and perimeter of an isosceles triangle – Examples with answers
Find the perimeter of an isosceles triangle that has a base with a length of 11 inches and congruent sides of 8 inches.
We have the following lengths:
Base, $latex b=11$ in
Length, $latex a=8$ in
Using the perimeter formula with these values, we have:
$latex p=11+2(8)$
$latex p=11+16$
$latex p=27$
The perimeter of the triangle is equal to 27 in.
Find the area of an isosceles triangle that has a base with a length of 6 yards and a height of 7 yards.
We have the following information:
Height, $latex h=7$ yd
Base, $latex b=6$ yd
Using the area formula with these values, we have:
$latex A= \frac{1}{2} \times b \times h$
$latex A= \frac{1}{2} (6)(7)$
$latex A=21$
The area of the triangle is equal to 21 yd².
What is the perimeter of an isosceles triangle that has a base of 12 feet and congruent sides of 15 feet?
Base, $latex b=12$ ft
Sides, $latex a=15$ ft
Applying the formula for the perimeter with the given information, we have:
$latex p=12+2(15)$
The perimeter of the triangle is equal to 42 ft.
Find the area of an isosceles triangle that has a base with a length of 10 inches and a height of 11 inches.
Height, $latex h=11$ in
Substituting these values into the formula for the area, we have:
$latex A= \frac{1}{2} (10)(11)$
The area of the triangle is equal to 55 in².
What is the perimeter of an isosceles triangle that has congruent sides of 22 yards and a base of 15 yards?
We can observe the following lengths:
Base, $latex b=15$ yd
Sides, $latex a=22$ yd
Using the formula for the perimeter with these values, we have:
The perimeter of the triangle is equal to 59 yd.
What is the area of an isosceles triangle that has a height of 13 feet and a base of 15 feet?
Height, $latex h=13$ ft
Using the formula for the area with these values, we have:
$latex A=97.5$
The area of the triangle is equal to 97.5 ft².
What is the length of the base of an isosceles triangle that has a perimeter of 38 inches and congruent sides of 13 inches?
We have the following:
Perimeter, $latex p=38$ in
Sides, $latex a=13$ in
In this case, we know the perimeter of the triangle, and we want to find the length of the base, so we use the formula for the perimeter and solve for b:
$latex 38=b+2(13)$
$latex 38=b+26$
$latex b=12$
The length of the base is 12 in.
What is the area of an isosceles triangle that has a base with a length of 8 feet and congruent sides of 10 feet?
Base, $latex b=8$ ft
Congruent sides, $latex a=10$ ft
Since we only know the lengths of the sides of the triangle and not its height, we can use the second formula for the area, and we have:
$latex h=\frac{1}{2}(\sqrt{{{a}^2}-\frac{{{b}^2}}{4}}\times b)$
$latex h=\frac{1}{2}(\sqrt{{{10}^2}-\frac{{{8}^2}}{4}}\times 8)$
$latex h=\frac{1}{2}(\sqrt{100-\frac{64}{4}}\times 8)$
$latex h=\frac{1}{2}(\sqrt{100-16}\times 8)$
$latex h=\frac{1}{2}(\sqrt{84}\times 8)$
$latex h=\frac{1}{2}(9.17\times 8)$
$latex h=\frac{1}{2}(73.36)$
$latex h=36.68$
The area of the triangle is equal to 36.68 ft².
If an isosceles triangle has a base with a length of 25 inches and a perimeter of 55 inches, what is the length of one of the triangle's congruent sides?
We can use the formula for the perimeter and solve for a:
$latex 55=25+2a$
$latex 2a=30$
The length of one of the congruent sides of the triangle is equal to 15 in.
What is the area of an isosceles triangle that has a base with a length of 12 inches and congruent sides with a length of 14 inches?
Congruent sides, $latex a=14$ in
Using the second formula for the area, we have:
$latex h=\frac{1}{2}(\sqrt{{{14}^2}-\frac{{{12}^2}}{4}}\times 12)$
$latex h=\frac{1}{2}(\sqrt{196-\frac{144}{4}}\times 12)$
$latex h=\frac{1}{2}(\sqrt{196-36}\times 12)$
$latex h=\frac{1}{2}(\sqrt{160}\times 12)$
$latex h=\frac{1}{2}(12.65 \times 12)$
$latex h=\frac{1}{2}(151.8)$
$latex h=75.9$
The area of the triangle is equal to 75.9 in².
Area and perimeter of an isosceles triangle – Practice problems
Find the perimeter of an isosceles triangle with a base of 12 inches and congruent sides with a length of 18 inches.
$latex p=38~in$
What is the area of an isosceles triangle that has a base with a length of 8 feet and a height of 12 feet?
$latex A=36~{{ft}^2}$
$latex A=140~{{in}^2}$
What is the length of the base of an isosceles triangle that has a perimeter of 27 feet and congruent sides with a length of 8 feet?
$latex b=8~ft$
$latex b=10~ft$
What is the area of an isosceles triangle that has a base with a length of 9 yards and congruent sides with a length of 5 yards?
$latex A=8.87~{{yd}^2}$
$latex A=10.21~{{yd}^2}$
Interested in learning more about isosceles triangles? Take a look at these pages:
Altura del Triángulo Isósceles
Características del triángulo isósceles
Triángulo Rectángulo Isósceles | CommonCrawl |
Let $x_1, x_2, \ldots, x_n$ be real numbers which satisfy $|x_i| < 1$ for $i = 1, 2, \dots, n,$ and \[|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.\]What is the smallest possible value of $n$?
The left-hand side satisfies \[|x_1| + |x_2| + \dots + |x_n| < 1 + 1 + \dots + 1 = n,\]while the right-hand side satisfies \[19 + |x_1 + x_2 + \dots + x_n| \ge 19.\]Therefore, $n > 19,$ so $n \ge 20.$ It is possible that $n=20,$ since, for example, we can choose \[\begin{aligned} x_1 = x_2 = \dots = x_{10} &= \tfrac{19}{20}, \\ x_{11} =x_{12} = \dots =x_{20}& = -\tfrac{19}{20}, \end{aligned}\]which makes $|x_1| + |x_2| + \dots = |x_{20}| = 19$ and $|x_1 + x_2 + \dots + x_{20}| = 0.$ Therefore the answer is $\boxed{20}.$ | Math Dataset |
Alea iacta est
Isaac B. Manfred always dreamed about being a terribly rich man. Recently, he started to study dice games. He found several of them similar to a trademarked game called Yahtzee! The rules sometimes vary but basic principles are the same. To give you an idea, we will describe a simplified version of such rules.
The game consists of rounds. In each round, a player rolls five dice. After the first roll, it is possible to keep some of the dice and re-roll the rest of them. Any number of dice can be rerolled (including none or all of them). If the re-rolled dice still do not fit the player's intentions, it is possible to re-roll some of them again, for the third and final time. After at most two such re-rolls, the player must assign the result to one of possible combinations and the round is scored according to that combination.
At least one 1.
One point for each 1.
Twos
Two points for each 2.
Three points for each 3.
Four points for each 4.
Five points for each 5.
Six points for each 6.
(1 2 3 4 5) or (2 3 4 5 6).
Thirty points.
Three of the same value and
Sum of all dice values.
a pair of another value.
Four of the same value,
the fifth one different.
Five of a kind
All five of the same value.
Fifty points.
Figure 1: The list of combinations, conditions that must be satisfied to use them, and the number of points scored when the combination is used.
A small example: The player rolls 2, 3, 6, 6, 5. The two 6's are kept and the three remaining dice re-rolled, they give new values: 1, 1, 6. The player may now choose to score 20 points immediately for a Full House. Instead, he or she decides to re-roll the two 1's again, in hope there will be another 6. The dice give 4 and 5 and the player will score either 18 points for Sixes or 27 points for Chance.
The main point of the game is that there are eleven combinations and eleven rounds. During the whole game, each combination must be used exactly once. It may happen that some result would not fit into any available combination. In such a case, the player must select some combination anyway, scoring zero points for that round and losing the possibility to use that combination later. These rules make the game very tricky, especially at the end, when the combinations have been almost exhausted.
Now, we get back to Isaac. He found a casino with an electronic version of this dice game. After carefully watching many games of other players, he was able to crack the random-number generator used in the machine. Therefore, he is able to predict the following rolls exactly. What an opportunity! However, it is still not easy to find the optimal strategy. If you write a program that would help him to become rich, he may share some of his money with you.
The input contains several scenarios (at most 12), each of them specified on a single line. The line contains three numbers separated by a space: $A$, $C$, and $X_0$. These numbers describe the random-number generator: $A$ is called a multiplier $(1\leq A\leq 2^{31})$, $C$ is an increment $(0\leq C\leq 2^{31})$, and $X_0$ is the initial seed $(0\leq X_0\leq 2^{31})$. The last scenario is followed by a line containing three zeros.
The generator is a linear congruential generator, which means that the next random number is calculated from the previous one using the following formula:
\[ X_{n+1} = (A\cdot X_ n + C) \bmod 2^{32} \]
The modulo operation specifies that only the lowest 32 bits of the result are used, the rest is discarded. Numbers $X_1, X_2, X_3, \ldots $ constitute a pseudo-random sequence, each of them determines the result of one individual roll of a dice. With congruential generators, the "randomness" of the numbers is in their higher bits only – therefore, to get a result of the $n$-th roll (starting with $n = 1$), we discard lower 16 bits of the number $X_ n$ and compute the remainder when the number in bits 16–31 is divided by six. This gives a number between 0 and 5, by adding one, we get a number shown on a dice:
\[ roll(n) = (\lfloor X_ n/2^{16}\rfloor \bmod 6) + 1 \]
For example, when $A = 69069, C = 5,$ and the $X_0 = 0$ is zero, we get the following sequence of "random" rolls: $1, 6, 6, 3, 2, 4, 3, 2, 3, 5, 1, 6, 6, 4, 5, 1, 3, 4, 1, \ldots $.
For each scenario, print one integer number: the maximal number of points that may be scored in a game determined by the given generator. The score is calculated after 11 rounds as the sum of scores in all combinations.
1664525 1013904223 177
1103515245 12345 67890
Problem ID: alea
CPU Time limit: 4 seconds
Authors: Josef Cibulka, Martin Kačer, and Jan Stoklasa
Source: CTU Open 2008 | CommonCrawl |
Lock Optimisation
In a simple combination lock, a sequence of several digits is used as a password, with one wheel per digit. Generally, it is not possible to unlock the lock without knowing the password, but due to the workings of this lock's mechanism, a "click" is heard whenever a wheel is rotated into the correct position.
For example, the following lock has 3 wheels, each with the digits from 0 to 9. Trying all possible combinations would require up to 1000 attempts to unlock the lock. But due to the clicking mechanism, it is possible to so in at most 30 tries (up to 10 tries for each wheel, where a "try" refers to a single rotation of a wheel).
Now, instead of 10 digits, you can design the lock to have $k$ digits on each wheel ($k \in \mathbb{N},\ k\geq 2$). For security reasons, we want the lock to have at least $N=10^{10}$ different possible combinations. The lock can have as many wheels as needed.
What is the optimal value of $k$ to minimize the number of tries (wheel rotations) needed to open the lock in the worst case? What about the average case?
Now, suppose that the lock can have different numbers of digits at different places. For example: The first wheel can hold 7 values (0 to 6), the second can hold 4 values and so on. Again, how can we minimize the number of tries in the average and worst cases?
Can you generalize for any $N$?
mathematics combinatorics optimization keys-and-locks
ghosts_in_the_code
RohcanaRohcana
$\begingroup$ Are the only three possible results from testing the lock "nothing," "click," and "open?" Or do we get some information about how many wheels are in the correct position? Also, is the lock guaranteed to be in a non-clicking position at the start or are the wheels randomly positioned initially? $\endgroup$ – 2012rcampion Aug 22 '15 at 23:03
$\begingroup$ Yes, "nothing", "click" and "open". You may have to turn the whole wheel to find a click. The lock is in a random position. $\endgroup$ – Rohcana Aug 22 '15 at 23:04
$\begingroup$ I remember finding a lock like that in my basement one day. And yes, I went through the 999 attempts way. I eventually found it around 600ish $\endgroup$ – warspyking Aug 22 '15 at 23:13
$\begingroup$ Does your 30 tries worst case take into account the cases when you start in a position that clicks? Also, shouldn't it be 9 tries per digit (since if 0-8 haven't clicked, 9 is guaranteed to be correct)? $\endgroup$ – 2012rcampion Aug 23 '15 at 1:25
$\begingroup$ @2012rcampion Remember that, even if we know 9 is the correct answer, we still have turn the wheel to make it 9, otherwise we won't be able to unlock the lock. In other words, our aim is to unlock the lock, not find the key. $\endgroup$ – Rohcana Aug 23 '15 at 1:28
Consider a lock $L$ consisting of $m$ wheels, where the $i$th wheel has $k_i$ digits (assume $k_i\geq 2$). I will use the notation $L=\{k_1,k_2,\ldots,k_m\}$.
The number of combinations $L$ can be set to, which I call the size of $L$ which I write as $|L|$, is simply the product of the $k_i$:
$$ |L|=\prod_{i=1}^m k_i $$
The worst-case number of turns to unlock the lock is:
$$ \mathsf{WC}(L)=\sum_{i=1}^mk_i $$
And the average-case is:
$$ \mathsf{AC}(L)=\sum_{i=1}^m\frac{k_i+1}2 $$
An important fact is that there are no 'cross terms' in any of the quantities associated with a lock: any subset of the wheels may be considered separately from the rest. (This also implies that the order of the $k_i$ is unimportant.)
I'll also define four types of locks:
A worst-case minimal lock for $N$ is a lock with at least $N$ combinations and the smallest possible number of worst-case turns. Formally, a lock $L$ is worst-case minimial if $|L|\geq N$ and $\forall L'\left(|L'|\geq N\implies\textsf{WC}(L')\geq\textsf{WC}(L)\right)$.
A worst-case optimal lock for $N$ is a minimal lock with the greatest size. Formally, a lock $L$ is worst-case optimal if $|L|\geq N$ and $\forall L'\left(|L'|\geq|L|\iff\textsf{WC}(L')\geq\textsf{WC}(L)\right)$. The set of worst-case optimal locks is the smallest set of locks which contains a worst-case minimal lock for all $N$.
Average-case minimal locks and average-case optimal locks are defined analogously to the first two.
Let us consider the replacement:
$$ \{k,\ldots\}\to\{2,\frac{k}{2},\ldots\} $$
(Assume that $k$ is even.) We can see that the size is unchanged; however, the worst-case number of turns decreases by this replacement when:
$$ \frac{k}{2}+2<k \\ k+4<2k \\ k>4 $$
We can perform a similar replacement for odd $k$:
$$ \{k,\ldots\}\to\{2,\frac{k+1}{2},\ldots\} $$
This replacement increases the size, and also improves the worst-case count when:
$$ \frac{k+1}{2}+2<k \\ k+5<2k \\ k>5 $$
These replacements tell us an important fact: no worst-case minimal lock has $k_i>5$.
Now consider the replacement:
$$ \{4,\ldots\}\to\{2,2,\ldots\} $$
Under this replacement both the size and worst-case count of the lock are unchanged. This means that it is sufficient to consider locks without $4$s for now.
Now we move onto replacements dealing with optimal locks. The replacement:
increases the size of a lock without changing it's worst-case count. Therefore no optimal lock will contain a $5$. Similarly, the replacement:
$$ \{2,2,2,\ldots\}\to\{3,3,\ldots\} $$
shows that no optimal locks have more than two $2$s.
This gives us enough information to determine that all worst-case optimal locks take one of the forms:
$$ \{3\ldots\} \\ \{2,3\ldots\} \\ \{2,2,3\ldots\} \\ $$
(Note that the last form is equivalent to $\{4,3\ldots\}$) $L$, the worst-case optimal lock for $N$, has $\lceil\log_3N\rceil$ wheels. The number of $2$ wheels can be determined by looking at the fractional part, $f=\log_3N-\lfloor\log_3N\rfloor$:
if $f\leq\log_32^2-1\approx.2619\ldots$ then $L$ has two $2$s
if $f\leq\log_32^1-0\approx .6309\ldots$ then $L$ has one $2$
all other wheels are $3$
Take $N=10^{10}$. We have $\log_310^{10}\approx 20.9590\ldots$, so the worst-case optimal lock has $21$ wheels. The fractional part $.9590\ldots$ is larger than both cutoffs, so all wheels have $3$ digits.
The actual number of combinations is $3^{21}=10\,460\,353\,203$ and the worst-case number of turns is $3\times 21=63$.
Average Case
The arguments for the average-case minimal locks work similarly. Again we consider the transformation for even $k$:
The average-case number of turns decreases under this replacement when:
$$ \frac{k/2+1}2+\frac{2+1}2<\frac{k+1}2 \\ k/2+1+3<k+1 \\ k>6 \\ $$
For odd $k$:
This replacement decreases the average-case count when:
$$ \frac{(k+1)/2+1}2+\frac{2+1}2< \frac{k+1}2 \\ (k+1)/2+2+3< k+1 \\ k> 7 \\ $$
Thus an average-case minimal lock will consist of no wheels greater than $7$.
As before, we have an additional replacement under which both size and average count are unchanged:
which shows that it is sufficient to consider locks without any $6$s.
The replacement:
increases the number of combinations while leaving the average-case count unchanged, implying that no average-case optimal lock contains a $7$. Similarly, the replacements
$$ \{2,2,\ldots\}\to\{5,\ldots\} \\ \{2,3,3,\ldots\}\to\{4,5,\ldots\} \\ \{2,4,\ldots\}\to\{3,3,\ldots\} \\ \{2,5,\ldots\}\to\{3,4,\ldots\} $$
show that an average-case optimal lock will have no $2$s, except in the special cases $\{2\}$ and $\{2,3\}$. Following a similar pattern of replacements:
$$ \{5,3,\ldots\}\to\{4,4,\ldots\} \\ \{5,4,4,\ldots\}\to\{3,3,3,3,\ldots\} \\ \{5,5,\ldots\}\to\{3,3,3,\ldots\} $$
we show that an average-case optimal lock will have no $5$s, except in the special cases $\{5\}$ and $\{5,4\}$. At this point, we have shown that all average-case optimal locks consist only of $3$s and $4$s (excepting the aforementioned special cases).
Finally, consider the replacement:
$$ \{3,3,3,3,3,\ldots\}\to\{4,4,4,4,\ldots\} $$
As before, this shows that an average-case optimal lock will have at most four $3$s.
All average-case optimal locks take one of the forms:
$$ \{2\} \\ \{5\} \\ \{2,3\} \\ \{6\} \\ \{4,5\} \\ \{4,\ldots\} \\ \{3,4,\ldots\} \\ \{3,3,4,\ldots\} \\ \{3,3,3,4,\ldots\} \\ \{3,3,3,3,4,\ldots\} \\ $$
Given a number of combinations $N\leq 5$, the smallest average-case optimal lock is simply $\{N\}$.
For $N=6$ there are two optimal locks: $\{2,3\}$ and $\{6\}$.
For $N>6$, the smallest average-case optimal lock has $\lceil\log_4N\rceil$ wheels. The number of $3$s can be found by looking at the fractional part $f$ of $\log_4N$:
$f\leq\log_43^4-3\approx.1699$: four $3$s
$f\leq\log_43^3-2\approx.3774$: three $3$s
$f\leq\log_43^2-1\approx.5849$: two $3$s
$f\leq\log_43^1-0\approx.7925$: one $3$
All other wheels have $4$ digits.
Except when $17\leq N\leq 20$, where $2.0437<\log_4N<2.1610$. The above procedure dictates that we should have three wheels but four $3$s; in this case the average-case optimal lock is actually $\{4,5\}$.
Take $N=10^{10}$. We have $\log_410^{10}\approx 16.6096\ldots$, so the average-case optimal lock has $17$ wheels. The fractional part, $.6096\ldots$, tells us that there is one $3$ and 16 $4$s.
The actual number of combinations is $3\times 4^{16}=12\,884\,901\,888$ and the average-case number of turns is $\frac{3+1}2+16\times\frac{4+1}2=42$.
2012rcampion2012rcampion
$\begingroup$ I wish I could upvote it a thousand times. This is thousand times better than my intended solution. This is the definition of beautiful. $\endgroup$ – Rohcana Aug 23 '15 at 6:21
$\begingroup$ @Anachor What was the intended solution? (Alternatively: is my answer "beautiful" but also wrong?) $\endgroup$ – 2012rcampion Aug 23 '15 at 6:25
$\begingroup$ The "all wheels equal" part was just equation (and some calculas like in Peter's answer). The general part was derived from the previous parts with a lot of casework, but nowhere as elegantly as these. There seems to be a slight inconsistency with my results, but I couldn't find any error in yours, so I will recheck mine and let you know. $\endgroup$ – Rohcana Aug 23 '15 at 6:30
$\begingroup$ Yes, it is perfectly correct. The inconsistency results because when faced with multiple optimal choices, my solution sticks to the smallest one. $\endgroup$ – Rohcana Aug 23 '15 at 9:46
$\begingroup$ @Anachor I see, my solution goes exactly the opposite way (i.e. an optimal lock is the largest minimal lock for a given $N$). $\endgroup$ – 2012rcampion Aug 23 '15 at 17:12
For each $k$-digit dial, the number of tries in the worst case is $k$, and the number of tries in the average case is $\frac{k+1}{2}$.
We get $k = 3$, giving 63 tries in the worst case (beating $k=2$ and $k=4$, both on 68) and 42 tries in the average case (beating $k=2$ on 51 and $k=4$ on 42.5). This is probably dependent on $N$: $3^{21} = 10460353203$ is just larger than $N$, but $2^{34} = 17179869184$ is much larger than $N$.
I haven't managed to do better than for 1.
Generalising the situation where all the dials are the same:
the number of dials is $\left\lceil{\frac {\ln N}{\ln k}}\right\rceil$; the worst case number of trials is $\left\lceil{\frac {\ln N}{\ln k}}\right\rceil k$, which is not fun to differentiate with respect to $k$. However, the minimum of $\frac{k}{\ln k}$ is $k = e$, so it's probably reasonable to expect the answer to be 3 for most values of $N$ and 2 for the rest.
I made a mistake with the average case, but 2012rcampion's answer goes further anyway so I won't correct it.
$\begingroup$ You are correct on $N=10^{10}$. If you experiment further, you should reveal something. $\endgroup$ – Rohcana Aug 22 '15 at 22:06
$\begingroup$ @Anachor, I think I will mainly reveal that I should have been in bed half an hour ago. $\endgroup$ – Peter Taylor Aug 22 '15 at 22:09
$\begingroup$ I'm getting a different result for the minimum of $(k+1)/\log k$: $$\partial_k(k+1)/\log k=0 \\ \frac{\log k-1-1/k}{\log^2 k}=0 \\ \log k=1+1/ k \\ k\approx 3.5911$$ $\endgroup$ – 2012rcampion Aug 23 '15 at 6:38
$\begingroup$ @2012rcampion, which proves my point about being too tired. I minimised $k+k/\log k$. $\endgroup$ – Peter Taylor Aug 23 '15 at 7:07
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# Introduction to the Geometry of the Triangle
Paul Yiu
Summer 2001
Department of Mathematics
Florida Atlantic University
## Chapter 5 Circles I 61
5.1 Isogonal conjugates 61
5.2 The circumcircle as the isogonal conjugate of the line at infinity 62
5.3 Simson lines 65
5.4 Equation of the nine-point circle 67
5.5 Equation of a general circle 68
5.6 Appendix: Miquel theory 69
Chapter 6 Circles II 73
6.1 Equation of the incircle 73
6.2 Intersection of incircle and nine-point circle 74
6.3 The excircles 78
6.4 The Brocard points 80
6.5 Appendix: The circle triad $(A(a), B(b), C(c)) \quad 83$
Chapter 7 Circles III 87
7.1 The distance formula 87
7.2 Circle equation 88
7.3 Radical circle of a triad of circles 90
7.4 The Lucas circles 93
7.5 Appendix: More triads of circles 94
Chapter 8 Some Basic Constructions 97
8.1 Barycentric product 97
8.2 Harmonic associates 100
8.3 Cevian quotient 102
8.4 Brocardians 103 Chapter 9 Circumconics 105
9.1 Circumconic as isogonal transform of lines 105
9.2 The infinite points of a circum-hyperbola 108
9.3 The perspector and center of a circumconic 109
9.4 Appendix: Ruler construction of tangent 112
Chapter 10 General Conics 113
10.1 Equation of conics 113
10.2 Inscribed conics 115
10.3 The adjoint of a matrix 116
10.4 Conics parametrized by quadratic equations 117
10.5 The matrix of a conic 118
10.6 The dual conic 119
10.7 The type, center and perspector of a conic 121
Chapter 11 Some Special Conics 125
11.1 Inscribed conic with prescribed foci 125
11.2 Inscribed parabola 127
11.3 Some special conics 129
11.4 Envelopes 133
Chapter 12 Some More Conics 137
12.1 Conics associated with parallel intercepts 137
12.2 Lines simultaneously bisecting perimeter and area 140
12.3 Parabolas with vertices as foci and sides as directrices 142
12.4 The Soddy hyperbolas 143
12.5 Appendix: Constructions with conics 144
## Chapter 1
## The Circumcircle and the Incircle
### Preliminaries
#### Coordinatization of points on a line
Let $B$ and $C$ be two fixed points on a line $\mathcal{L}$. Every point $X$ on $\mathcal{L}$ can be coordinatized in one of several ways:
(1) the ratio of division $t=\frac{B X}{X C}$,
(2) the absolute barycentric coordinates: an expression of $X$ as a convex combination of $B$ and $C$ :
$$
X=(1-t) B+t C,
$$
which expresses for an arbitrary point $P$ outside the line $\mathcal{L}$, the vector $\mathbf{P X}$ as a combination of the vectors $\mathbf{P B}$ and $\mathbf{P C}$.
(3) the homogeneous barycentric coordinates: the proportion $X C: B X$, which are masses at $B$ and $C$ so that the resulting system (of two particles) has balance point at $X$.
#### Centers of similitude of two circles
Consider two circles $O(R)$ and $I(r)$, whose centers $O$ and $I$ are at a distance $d$ apart. Animate a point $X$ on $O(R)$ and construct a ray through $I$ oppositely parallel to the ray $O X$ to intersect the circle $I(r)$ at a point $Y$. You will find that the line $X Y$ always intersects the line $O I$ at the same point $P$. This we call the internal center of similitude of the two circles. It divides the segment $O I$ in the ratio $O P: P I=R: r$. The absolute barycentric coordinates of $P$ with respect to $O I$ are
$$
P=\frac{R \cdot I+r \cdot O}{R+r} .
$$
If, on the other hand, we construct a ray through $I$ directly parallel to the ray $O X$ to intersect the circle $I(r)$ at $Y^{\prime}$, the line $X Y^{\prime}$ always intersects $O I$ at another point $Q$. This is the external center of similitude of the two circles. It divides the segment $O I$ in the ratio $O Q: Q I=R:-r$, and has absolute barycentric coordinates
$$
Q=\frac{R \cdot I-r \cdot O}{R-r} .
$$
#### Harmonic division
Two points $X$ and $Y$ are said to divide two other points $B$ and $C$ harmonically if
$$
\frac{B X}{X C}=-\frac{B Y}{Y C}
$$
They are harmonic conjugates of each other with respect to the segment $B C$.
## Exercises
1. If $X, Y$ divide $B, C$ harmonically, then $B, C$ divide $X, Y$ harmonically. 2. Given a point $X$ on the line $B C$, construct its harmonic associate with respect to the segment $B C$. Distinguish between two cases when $X$ divides $B C$ internally and externally. ${ }^{1}$
2. Given two fixed points $B$ and $C$, the locus of the points $P$ for which $|B P|:|C P|=k$ (constant) is a circle.
#### Menelaus and Ceva Theorems
Consider a triangle $A B C$ with points $X, Y, Z$ on the side lines $B C, C A$, $A B$ respectively.
## Menelaus Theorem
The points $X, Y, Z$ are collinear if and only if
$$
\frac{B X}{X C} \cdot \frac{C Y}{Y A} \cdot \frac{A Z}{Z B}=-1
$$
## Ceva Theorem
The lines $A X, B Y, C Z$ are concurrent if and only if
$$
\frac{B X}{X C} \cdot \frac{C Y}{Y A} \cdot \frac{A Z}{Z B}=+1
$$
## Ruler construction of harmonic conjugate
Let $X$ be a point on the line $B C$. To construct the harmonic conjugate of $X$ with respect to the segment $B C$, we proceed as follows.
(1) Take any point $A$ outside the line $B C$ and construct the lines $A B$ and $A C$.
${ }^{1}$ Make use of the notion of centers of similitude of two circles. (2) Mark an arbitrary point $P$ on the line $A X$ and construct the lines $B P$ and $C P$ to intersect respectively the lines $C A$ and $A B$ at $Y$ and $Z$.
(3) Construct the line $Y Z$ to intersect $B C$ at $X^{\prime}$.
Then $X$ and $X^{\prime}$ divide $B$ and $C$ harmonically.
#### The power of a point with respect to a circle
The power of a point $P$ with respect to a circle $\mathcal{C}=O(R)$ is the quantity $\mathcal{C}(P):=O P^{2}-R^{2}$. This is positive, zero, or negative according as $P$ is outside, on, or inside the circle $\mathcal{C}$. If it is positive, it is the square of the length of a tangent from $P$ to the circle.
## Theorem (Intersecting chords)
If a line $\mathcal{L}$ through $P$ intersects a circle $\mathcal{C}$ at two points $X$ and $Y$, the product $P X \cdot P Y$ (of signed lengths) is equal to the power of $P$ with respect to the circle.
### The circumcircle and the incircle of a triangle
For a generic triangle $A B C$, we shall denote the lengths of the sides $B C$, $C A, A B$ by $a, b, c$ respectively.
#### The circumcircle
The circumcircle of triangle $A B C$ is the unique circle passing through the three vertices $A, B, C$. Its center, the circumcenter $O$, is the intersection of the perpendicular bisectors of the three sides. The circumradius $R$ is given by the law of sines:
$$
2 R=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
$$
#### The incircle
The incircle is tangent to each of the three sides $B C, C A, A B$ (without extension). Its center, the incenter $I$, is the intersection of the bisectors of the three angles. The inradius $r$ is related to the area $\frac{1}{2} S$ by
$$
S=(a+b+c) r
$$
If the incircle is tangent to the sides $B C$ at $X, C A$ at $Y$, and $A B$ at $Z$, then
$$
A Y=A Z=\frac{b+c-a}{2}, \quad B Z=B X=\frac{c+a-b}{2}, \quad C X=C Y=\frac{a+b-c}{2} .
$$
These expressions are usually simplified by introducing the semiperimeter $s=\frac{1}{2}(a+b+c)$ :
$$
A Y=A Z=s-a, \quad B Z=B X=s-b, \quad C X=C Y=s-c .
$$
Also, $r=\frac{S}{2 s}$.
#### The centers of similitude of $(O)$ and $(I)$
Denote by $T$ and $T^{\prime}$ respectively the internal and external centers of similitude of the circumcircle and incircle of triangle $A B C$.
These are points dividing the segment $O I$ harmonically in the ratios
$$
O T: T I=R: r, \quad O T^{\prime}: T^{\prime} I=R:-r .
$$
## Exercises
1. Use the Ceva theorem to show that the lines $A X, B Y, C Z$ are concurrent. (The intersection is called the Gergonne point of the triangle).
2. Construct the three circles each passing through the Gergonne point and tangent to two sides of triangle $A B C$. The 6 points of tangency lie on a circle.
3. Given three points $A, B, C$ not on the same line, construct three circles, with centers at $A, B, C$, mutually tangent to each other externally.
4. Two circles are orthogonal to each other if their tangents at an intersection are perpendicular to each other. Given three points $A, B, C$ not on a line, construct three circles with these as centers and orthogonal to each other.
5. The centers $A$ and $B$ of two circles $A(a)$ and $B(b)$ are at a distance $d$ apart. The line $A B$ intersect the circles at $A^{\prime}$ and $B^{\prime}$ respectively, so that $A, B$ are between $A^{\prime}, B^{\prime}$.
(1) Construct the tangents from $A^{\prime}$ to the circle $B(b)$, and the circle tangent to these two lines and to $A(a)$ internally.
(2) Construct the tangents from $B^{\prime}$ to the circle $A(a)$, and the circle tangent to these two lines and to $B(b)$ internally.
(3) The two circles in (1) and (2) are congruent.
6. Given a point $Z$ on a line segment $A B$, construct a right-angled triangle $A B C$ whose incircle touches the hypotenuse $A B$ at $Z .{ }^{2}$
7. (Paper Folding) The figure below shows a rectangular sheet of paper containing a border of uniform width. The paper may be any size and shape, but the border must be of such a width that the area of the inner rectangle is exactly half that of the sheet. You have no ruler or compasses, or even a pencil. You must determine the inner rectangle purely by paper folding. ${ }^{3}$
8. Let $A B C$ be a triangle with incenter $I$.
(1a) Construct a tangent to the incircle at the point diametrically opposite to its point of contact with the side $B C$. Let this tangent intersect $C A$ at $Y_{1}$ and $A B$ at $Z_{1}$.
${ }^{2}$ P. Yiu, G. Leversha, and T. Seimiya, Problem 2415 and solution, Crux Math. 25 (1999) $110 ; 26$ (2000) $62-64$.
${ }^{3}$ Problem 2519, Journal of Recreational Mathematics, 30 (1999-2000) 151 - 152. (1b) Same in part (a), for the side $C A$, and let the tangent intersect $A B$ at $Z_{2}$ and $B C$ at $X_{2}$.
(1c) Same in part (a), for the side $A B$, and let the tangent intersect $B C$ at $X_{3}$ and $C A$ at $Y_{3}$.
(2) Note that $A Y_{3}=A Z_{2}$. Construct the circle tangent to $A C$ and $A B$ at $Y_{3}$ and $Z_{2}$. How does this circle intersect the circumcircle of triangle $A B C$ ?
9. The incircle of $\triangle A B C$ touches the sides $B C, C A, A B$ at $D, E, F$ respectively. $X$ is a point inside $\triangle A B C$ such that the incircle of $\triangle X B C$ touches $B C$ at $D$ also, and touches $C X$ and $X B$ at $Y$ and $Z$ respectively.
(1) The four points $E, F, Z, Y$ are concyclic. ${ }^{4}$
(2) What is the locus of the center of the circle EFZY? ${ }^{5}$
#### The Heron formula
The area of triangle $A B C$ is given by
$$
\frac{S}{2}=\sqrt{s(s-a)(s-b)(s-c)} .
$$
This formula can be easily derived from a computation of the inradius $r$ and the radius of one of the tritangent circles of the triangle. Consider the excircle $I_{a}\left(r_{a}\right)$ whose center is the intersection of the bisector of angle $A$ and the external bisectors of angles $B$ and $C$. If the incircle $I(r)$ and this excircle are tangent to the line $A C$ at $Y$ and $Y^{\prime}$ respectively, then
(1) from the similarity of triangles $A I Y$ and $A I_{a} Y^{\prime}$,
$$
\frac{r}{r_{a}}=\frac{s-a}{s}
$$
(2) from the similarity of triangles $C I Y$ and $I_{a} C Y^{\prime}$,
$$
r \cdot r_{a}=(s-b)(s-c) .
$$
${ }^{4}$ International Mathematical Olympiad 1996.
${ }^{5}$ IMO 1996.
It follows that
$$
r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}} .
$$
From this we obtain the famous Heron formula for the area of a triangle:
$$
\frac{S}{2}=r s=\sqrt{s(s-a)(s-b)(s-c)} .
$$
## Exercises
1. $R=\frac{a b c}{2 S}$.
2. $r_{a}=\frac{S}{b+c-a}$.
3. Suppose the incircle of triangle $A B C$ touches its sides $B C, C A, A B$ at the points $X, Y, Z$ respectively. Let $X^{\prime}, Y^{\prime}, Z^{\prime}$ be the antipodal points of $X, Y, Z$ on the incircle. Construct the rays $A X^{\prime}, B Y^{\prime}$, and $C Z^{\prime}$.
Explain the concurrency of these rays by considering also the points of contact of the excircles of the triangle with the sides.
4. Construct the tritangent circles of a triangle $A B C$.
(1) Join each excenter to the midpoint of the corresponding side of $A B C$. These three lines intersect at a point $P$. (This is called the Mittenpunkt of the triangle).
(2) Join each excenter to the point of tangency of the incircle with the corresponding side. These three lines are concurrent at another point $Q$.
(3) The lines $A P$ and $A Q$ are symmetric with respect to the bisector of angle $A$; so are the lines $B P, B Q$ and $C P, C Q$ (with respect to the bisectors of angles $B$ and $C$ ). 5. Construct the excircles of a triangle $A B C$.
(1) Let $D, E, F$ be the midpoints of the sides $B C, C A, A B$. Construct the incenter $S$ of triangle $D E F,{ }^{6}$ and the tangents from $S$ to each of the three excircles.
(2) The 6 points of tangency are on a circle, which is orthogonal to each of the excircles.
### Euler's formula and Steiner's porism
#### Euler's formula
The distance between the circumcenter and the incenter of a triangle is given by
$$
O I^{2}=R^{2}-2 R r
$$
Construct the circumcircle $O(R)$ of triangle $A B C$. Bisect angle $A$ and mark the intersection $M$ of the bisector with the circumcircle. Construct the circle $M(B)$ to intersect this bisector at a point $I$. This is the incenter since
$$
\angle I B C=\frac{1}{2} \angle I M C=\frac{1}{2} \angle A M C=\frac{1}{2} \angle A B C,
$$
and for the same reason $\angle I C B=\frac{1}{2} \angle A C B$. Note that
(1) $I M=M B=M C=2 R \sin \frac{A}{2}$,
(2) $I A=\frac{r}{\sin \frac{A}{2}}$, and
(3) by the theorem of intersecting chords, $R^{2}-O I^{2}=$ the power of $I$ with respect to the circumcircle $=I A \cdot I M=2 R r$.
${ }^{6}$ This is called the Spieker point of triangle $A B C$.
#### Steiner's porism ${ }^{7}$
Construct the circumcircle $(O)$ and the incircle $(I)$ of triangle $A B C$. Animate a point $A^{\prime}$ on the circumcircle, and construct the tangents from $A^{\prime}$ to the incircle $(I)$. Extend these tangents to intersect the circumcircle again at $B^{\prime}$ and $C^{\prime}$. The lines $B^{\prime} C^{\prime}$ is always tangent to the incircle. This is the famous theorem on Steiner porism: if two given circles are the circumcircle and incircle of one triangle, then they are the circumcircle and incircle of a continuous family of poristic triangles.
## Exercises
1. $r \leq \frac{1}{2} R$. When does equality hold?
2. Suppose $O I=d$. Show that there is a right-angled triangle whose sides are $d, r$ and $R-r$. Which one of these is the hypotenuse?
3. Given a point $I$ inside a circle $O(R)$, construct a circle $I(r)$ so that $O(R)$ and $I(r)$ are the circumcircle and incircle of a (family of poristic) triangle(s).
4. Given the circumcenter, incenter, and one vertex of a triangle, construct the triangle.
5. Construct an animation picture of a triangle whose circumcenter lies on the incircle. ${ }^{8}$
### Appendix: Mixtilinear incircles ${ }^{9}$
A mixtilinear incircle of triangle $A B C$ is one that is tangent to two sides of the triangle and to the circumcircle internally. Denote by $A^{\prime}$ the point of tangency of the mixtilinear incircle $K(\rho)$ in angle $A$ with the circumcircle. The center $K$ clearly lies on the bisector of angle $A$, and $A K: K I=\rho$ : $-(\rho-r)$. In terms of barycentric coordinates,
$$
K=\frac{1}{r}[-(\rho-r) A+\rho I]
$$
Also, since the circumcircle $O\left(A^{\prime}\right)$ and the mixtilinear incircle $K\left(A^{\prime}\right)$ touch each other at $A^{\prime}$, we have $O K: K A^{\prime}=R-\rho: \rho$, where $R$ is the circumradius.
${ }^{7}$ Also known as Poncelet's porism.
${ }^{8}$ Hint: $O I=r$.
${ }^{9}$ P.Yiu, Mixtilinear incircles, Amer. Math. Monthly 106 (1999) 952 - 955. From this,
$$
K=\frac{1}{R}\left[\rho O+(R-\rho) A^{\prime}\right]
$$
Comparing these two equations, we obtain, by rearranging terms,
$$
\frac{R I-r O}{R-r}=\frac{R(\rho-r) A+r(R-\rho) A^{\prime}}{\rho(R-r)} .
$$
We note some interesting consequences of this formula. First of all, it gives the intersection of the lines joining $A A^{\prime}$ and $O I$. Note that the point on the line $O I$ represented by the left hand side is $T^{\prime}$.
This leads to a simple construction of the mixtilinear incircle:
Given a triangle $A B C$, let $P$ be the external center of similitude of the circumcircle $(O)$ and incircle $(I)$. Extend $A P$ to intersect the circumcircle at $A^{\prime}$. The intersection of $A I$ and $A^{\prime} O$ is the center $K_{A}$ of the mixtilinear incircle in angle $A$.
The other two mixtilinear incircles can be constructed similarly.
## Exercises
1. Can any of the centers of similitude of $(O)$ and $(I)$ lie outside triangle $A B C$ ?
2. There are three circles each tangent internally to the circumcircle at a vertex, and externally to the incircle. It is known that the three lines joining the points of tangency of each circle with $(O)$ and $(I)$ pass through the internal center $T$ of similitude of $(O)$ and $(I)$. Construct these three circles. 10
${ }^{10}$ A.P. Hatzipolakis and P. Yiu, Triads of circles, preprint.
3. Let $T$ be the internal center of similitude of $(O)$ and $(I)$. Suppose $B T$ intersects $C A$ at $Y$ and $C T$ intersect $A B$ at $Z$. Construct perpendiculars from $Y$ and $Z$ to intersect $B C$ at $Y^{\prime}$ and $Z^{\prime}$ respectively. Calculate the length of $Y^{\prime} Z^{\prime} .{ }^{11}$
${ }^{11}$ A.P. Hatzipolakis and P. Yiu, Pedal triangles and their shadows, Forum Geom., 1 (2001) $81-90$.
## Chapter 2
## The Euler Line and the Nine-point Circle
### The Euler line
#### Homothety
The similarity transformation $\mathrm{h}(T, r)$ which carries a point $X$ to the point $X^{\prime}$ which divides $T X^{\prime}: T X=r: 1$ is called the homothety with center $T$ and ratio $r$.
#### The centroid
The three medians of a triangle intersect at the centroid, which divides each median in the ratio $2: 1$. If $D, E, F$ are the midpoints of the sides $B C, C A$, $A B$ of triangle $A B C$, the centroid $G$ divides the median $A D$ in the ratio $A G: G D=2: 1$. The medial triangle $D E F$ is the image of triangle $A B C$ under the homothety $\mathrm{h}\left(G,-\frac{1}{2}\right)$. The circumcircle of the medial triangle has radius $\frac{1}{2} R$. Its center is the point $N=\mathrm{h}\left(G,-\frac{1}{2}\right)(O)$. This divides the segement $O G$ in the ratio $O G: G N=2: 1$.
#### The orthocenter
The dilated triangle $A^{\prime} B^{\prime} C^{\prime}$ is the image of $A B C$ under the homothety $\mathrm{h}(G,-2) .{ }^{1}$ Since the altitudes of triangle $A B C$ are the perpendicular bisectors of the sides of triangle $A^{\prime} B^{\prime} C^{\prime}$, they intersect at the homothetic image of the circumcenter $O$. This point is called the orthocenter of triangle $A B C$, and is usually denoted by $H$. Note that
$$
O G: G H=1: 2 \text {. }
$$
The line containing $O, G, H$ is called the Euler line of triangle $A B C$. The Euler line is undefined for the equilateral triangle, since these points coincide.
## Exercises
1. A triangle is equilateral if and only if two of its circumcenter, centroid, and orthocenter coincide.
2. The circumcenter $N$ of the medial triangle is the midpoint of $O H$.
3. The Euler lines of triangles $H B C, H C A, H A B$ intersect at a point on the Euler line of triangle $A B C$. What is this intersection?
4. The Euler lines of triangles $I B C, I C A, I A B$ also intersect at a point on the Euler line of triangle $A B C .{ }^{2}$
5. (Gossard's Theorem) Suppose the Euler line of triangle $A B C$ intersects the side lines $B C, C A, A B$ at $X, Y, Z$ respectively. The Euler lines of the triangles $A Y Z, B Z X$ and $C X Y$ bound a triangle homothetic to $A B C$ with ratio -1 and with homothetic center on the Euler line of $A B C$.
6. What is the locus of the centroids of the poristic triangles with the same circumcircle and incircle of triangle $A B C$ ? How about the orthocenter?
${ }^{1}$ It is also called the anticomplementary triangle.
${ }^{2}$ Problem 1018, Crux Mathematicorum. 7. Let $A^{\prime} B^{\prime} C^{\prime}$ be a poristic triangle with the same circumcircle and incircle of triangle $A B C$, and let the sides of $B^{\prime} C^{\prime}, C^{\prime} A^{\prime}, A^{\prime} B^{\prime}$ touch the incircle at $X, Y, Z$.
(i) What is the locus of the centroid of $X Y Z$ ?
(ii) What is the locus of the orthocenter of $X Y Z$ ?
(iii) What can you say about the Euler line of the triangle $X Y Z$ ?
### The nine-point circle
#### The Euler triangle as a midway triangle
The image of $A B C$ under the homothety $\mathrm{h}\left(P, \frac{1}{2}\right)$ is called the midway triangle of $P$. The midway triangle of the orthocenter $H$ is called the Euler triangle. The circumcenter of the midway triangle of $P$ is the midpoint of $O P$. In particular, the circumcenter of the Euler triangle is the midpoint of $O H$, which is the same as $N$. The medial triangle and the Euler triangle have the same circumcircle.
#### The orthic triangle as a pedal triangle
The pedals of a point are the intersections of the sidelines with the corresponding perpendiculars through $P$. They form the pedal triangle of $P$. The pedal triangle of the orthocenter $H$ is called the orthic triangle of $A B C$.
The pedal $X$ of the orthocenter $H$ on the side $B C$ is also the pedal of $A$ on the same line, and can be regarded as the reflection of $A$ in the line $E F$. It follows that
$$
\angle E X F=\angle E A F=\angle E D F
$$
since $A E D F$ is a parallelogram. From this, the point $X$ lies on the circle $D E F$; similarly for the pedals $Y$ and $Z$ of $H$ on the other two sides $C A$ and $A B$.
#### The nine-point circle
From $\S 2.2 .1,2$ above, the medial triangle, the Euler triangle, and the orthic triangle have the same circumcircle. This is called the nine-point circle of triangle $A B C$. Its center $N$, the midpoint of $O H$, is called the nine-point center of triangle $A B C$.
## Exercises
1. On the Euler line,
$$
O G: G N: N H=2: 1: 3 .
$$
2. Let $P$ be a point on the circumcircle. What is the locus of the midpoint of $H P$ ? Can you give a proof? 3. Let $A B C$ be a triangle and $P$ a point. The perpendiculars at $P$ to $P A, P B, P C$ intersect $B C, C A, A B$ respectively at $A^{\prime}, B^{\prime}, C^{\prime}$.
(1) $A^{\prime}, B^{\prime}, C^{\prime}$ are collinear. ${ }^{3}$
(2) The nine-point circles of the (right-angled) triangles $P A A^{\prime}, P B B^{\prime}$, $P C C^{\prime}$ are concurrent at $P$ and another point $P^{\prime}$. Equivalently, their centers are collinear. ${ }^{4}$
4. If the midpoints of $A P, B P, C P$ are all on the nine-point circle, must $P$ be the orthocenter of triangle $A B C ?^{5}$
5. (Paper folding) Let $N$ be the nine-point center of triangle $A B C$.
(1) Fold the perpendicular to $A N$ at $N$ to intersect $C A$ at $Y$ and $A B$ at $Z$.
(2) Fold the reflection $A^{\prime}$ of $A$ in the line $Y Z$.
(3) Fold the reflections of $B$ in $A^{\prime} Z$ and $C$ in $A^{\prime} Y$.
What do you observe about these reflections?
#### Triangles with nine-point center on the circumcircle
We begin with a circle, center $O$ and a point $N$ on it, and construct a family of triangles with $(O)$ as circumcircle and $N$ as nine-point center.
(1) Construct the nine-point circle, which has center $N$, and passes through the midpoint $M$ of $O N$.
(2) Animate a point $D$ on the minor arc of the nine-point circle inside the circumcircle.
(3) Construct the chord $B C$ of the circumcircle with $D$ as midpoint. (This is simply the perpendicular to $O D$ at $D$ ).
(4) Let $X$ be the point on the nine-point circle antipodal to $D$. Complete the parallelogram $O D X A$ (by translating the vector $\mathbf{D O}$ to $X$ ).
The point $A$ lies on the circumcircle and the triangle $A B C$ has nine-point center $N$ on the circumcircle.
Here is an curious property of triangles constructed in this way: let $A^{\prime}, B^{\prime}, C^{\prime}$ be the reflections of $A, B, C$ in their own opposite sides. The
${ }^{3}$ B. Gibert, Hyacinthos 1158, 8/5/00.
${ }^{4}$ A.P. Hatzipolakis, Hyacinthos 3166, 6/27/01. The three midpoints of $A A^{\prime}, B B^{\prime}, C C^{\prime}$ are collinear. The three nine-point circles intersect at $P$ and its pedal on this line.
${ }^{5}$ Yes. See P. Yiu and J. Young, Problem 2437 and solution, Crux Math. 25 (1999) 173; 26 (2000) 192. reflection triangle $A^{\prime} B^{\prime} C^{\prime}$ degenerates, i.e., the three points $A^{\prime}, B^{\prime}, C^{\prime}$ are collinear. ${ }^{6}$
### Simson lines and reflections
#### Simson lines
Let $P$ on the circumcircle of triangle $A B C$.
(1) Construct its pedals on the side lines. These pedals are always collinear. The line containing them is called the Simson line $\mathrm{s}(P)$ of $P$.
(2) Let $P^{\prime}$ be the point on the cirucmcircle antipodal to $P$. Construct the Simson line $\left(P^{\prime}\right)$ and trace the intersection point $\mathrm{s}(P) \cap\left(P^{\prime}\right)$. Can you identify this locus?
(3) Let the Simson line $\mathrm{s}(P)$ intersect the side lines $B C, C A, A B$ at $X, Y$, $Z$ respectively. The circumcenters of the triangles $A Y Z, B Z X$, and $C X Y$ form a triangle homothetic to $A B C$ at $P$, with ratio $\frac{1}{2}$. These circumcenters therefore lie on a circle tangent to the circumcircle at $P$.
#### Line of reflections
Construct the reflections of the $P$ in the side lines. These reflections are always collinear, and the line containing them always passes through the orthocenter $H$, and is parallel to the Simson line $\mathbf{s}(P)$.
${ }^{6}$ O. Bottema, Hoofdstukken uit de Elementaire Meetkunde, Chapter 16.
#### Musselman's Theorem: Point with given line of reflec- tions
Let $\mathcal{L}$ be a line through the orthocenter $H$.
(1) Choose an arbitrary point $Q$ on the line $\mathcal{L}$ and reflect it in the side lines $B C, C A, A B$ to obtain the points $X, Y, Z$.
(2) Construct the circumcircles of $A Y Z, B Z X$ and $C X Y$. These circles have a common point $P$, which happens to lie on the circumcircle.
(3) Construct the reflections of $P$ in the side lines of triangle $A B C$.
#### Musselman's Theorem: Point with given line of reflec- tions (Alternative)
Animate a point $Q$ on the circumcircle, together with its antipode $Q^{\prime}$.
(1) The reflections $X, Y, Z$ of $Q$ on the side lines $B C, C A, A B$ are collinear; so are those $X^{\prime}, Y^{\prime}, Z^{\prime}$ of $Q^{\prime}$.
(2) The lines $X X^{\prime}, Y Y^{\prime}, Z Z^{\prime}$ intersect at a point $P$, which happens to be on the circumcircle.
(3) Construct the reflections of $P$ in the side lines of triangle $A B C$.
#### Blanc's Theorem
Animate a point $P$ on the circumcircle, together with its antipodal point $P^{\prime}$.
(1) Construct the line $P P^{\prime}$ to intersect the side lines $B C, C A, A B$ at $X, Y, Z$ respectively.
(2) Construct the circles with diameters $A X, B Y, C Z$. These three circles have two common points. One of these is on the circumcircle. Label this point $P^{*}$, and the other common point $Q$.
(3) What is the locus of $Q$ ?
(4) The line $P^{*} Q$ passes through the orthocenter $H$. As such, it is the line of reflection of a point on the circumcircle. What is this point?
(5) Construct the Simson lines of $P$ and $P^{\prime}$. They intersect at a point on the nine-point circle. What is this point?
## Exercises
1. Let $P$ be a given point, and $A^{\prime} B^{\prime} C^{\prime}$ the homothetic image of $A B C$ under $\mathrm{h}(P,-1)$ (so that $P$ is the common midpoint of $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ ). (1) The circles $A B^{\prime} C^{\prime}, B C^{\prime} A^{\prime}$ and $C A^{\prime} B^{\prime}$ intersect at a point $Q$ on the circumcircle;
(2) The circles $A B C^{\prime}, B C A^{\prime}$ and $C A B^{\prime}$ intersect at a point $Q^{\prime}$ such that $P$ is the midpoint of $Q Q^{\prime}$. ${ }^{7}$
### Appendix: Homothety
Two triangles are homothetic if the corresponding sides are parallel.
#### Three congruent circles with a common point and each tangent to two sides of a triangle ${ }^{8}$
Given a triangle $A B C$, to construct three congruent circles passing through a common point $P$, each tangent to two sides of the triangle.
Let $t$ be the common radius of these congruent circles. The centers of these circles, $I_{1}, I_{2}, I_{3}$, lie on the bisectors $I A, I B, I C$ respectively. Note that the lines $I_{2} I_{3}$ and $B C$ are parallel; so are the pairs $I_{3} I_{1}, C A$, and $I_{1} I_{2}, A B$. It follows that $\triangle I_{1} I_{2} I_{3}$ and $A B C$ are similar. Indeed, they are in homothetic from their common incenter $I$. The ratio of homothety can be determined in two ways, by considering their circumcircles and their incircles. Since the circumradii are $t$ and $R$, and the inradii are $r-t$ and $r$, we have $\frac{r-t}{r}=\frac{r}{R}$. From this, $t=\frac{R r}{R+r}$.
${ }^{7}$ Musselman, Amer. Math. Monthly, 47 (1940) $354-361$. If $P=(u: v: w)$, the intersection of the three circles in (1) is the point
$$
\left(\frac{1}{b^{2}(u+v-w) w-c^{2}(w+u-v) v}: \cdots: \cdots\right)
$$
on the circumcircle. This is the isogonal conjugate of the infinite point of the line
$$
\sum_{\text {cyclic }} \frac{u(v+w-u)}{a^{2}} x=0
$$
${ }^{8}$ Problem 2137, Crux Mathematicorum. How does this help constructing the circles? Note that the line joining the circumcenters $P$ and $O$ passes through the center of homothety $I$, and indeed,
$$
O I: I P=R: t=R: \frac{R r}{R+r}=R+r: r .
$$
Rewriting this as $O P: P I=R: r$, we see that $P$ is indeed the internal center of similitude of $(O)$ and $(I)$.
Now the construction is easy.
#### Squares inscribed in a triangle and the Lucas circles
Given a triangle $A B C$, to construct the inscribed square with a side along $B C$ we contract the square erected externally on the same side by a homothety at vertex $A$. The ratio of the homothety is $h_{a}: h_{a}+a$, where $h_{a}$ is the altitude on $B C$. Since $h_{a}=\frac{S}{a}$, we have
$$
\frac{h_{a}}{h_{a}+a}=\frac{S}{S+a^{2}} \text {. }
$$
The circumcircle is contracted into a circle of radius
$$
R_{a}=R \cdot \frac{S}{S+a^{2}}=\frac{a b c}{2 S} \cdot \frac{S}{S+a^{2}}=\frac{a b c}{2\left(S+a^{2}\right)},
$$
and this passes through the two vertices of the inscribed on the sides $A B$ and $A C$. Similarly, there are two other inscribed squares on the sides $C A$ and $A B$, and two corresponding circles, tangent to the circumcircle at $B$ and $C$ respectively. It is remarkable that these three circles are mutually tangent to each other. These are called the Lucas circles of the triangle. ${ }^{9}$
${ }^{9}$ See A.P. Hatzipolakis and P. Yiu, The Lucas circles, Amer. Math. Monthly, 108 (2001) 444 - 446. After the publication of this note, we recently learned that Eduoard Lucas (1842 - 1891) wrote about this triad of circles, considered by an anonymous author, as the three circles mutually tangent to each other and each tangent to the circumcircle at a vertex of $A B C$. The connection with the inscribed squares were found by Victor Thébault $(1883-1960)$.
#### More on reflections
(1) The reflections of a line $\mathcal{L}$ in the side lines of triangle $A B C$ are concurrent if and only if $\mathcal{L}$ passes through the orthocenter. In this case, the intersection is a point on the circumcircle. 10
(2) Construct parallel lines $\mathcal{L}_{a}, \mathcal{L}_{b}$, and $\mathcal{L}_{c}$ through the $D, E, F$ be the midpoints of the sides $B C, C A, A B$ of triangle $A B C$. Reflect the lines $B C$ in $\mathcal{L}_{a}, C A$ in $\mathcal{L}_{b}$, and $A B$ in $\mathcal{L}_{c}$. These three reflection lines intersect at a point on the nine-point circle. ${ }^{11}$
(3) Construct parallel lines $\mathcal{L}_{a}, \mathcal{L}_{b}$, and $\mathcal{L}_{c}$ through the pedals of the vertices $A, B, C$ on their opposite sides. Reflect these lines in the respective side lines of triangle $A B C$. The three reflection lines intersect at a point on the nine-point circle. ${ }^{12}$
${ }^{10}$ S.N. Collings, Reflections on a triangle, part 1, Math. Gazette, 57 (1973) 291 - 293; M.S. Longuet-Higgins, Reflections on a triangle, part 2, ibid., 293 - 296.
${ }^{11}$ This was first discovered in May, 1999 by a high school student, Adam Bliss, in Atlanta, Georgia. A proof can be found in F.M. van Lamoen, Morley related triangles on the nine-point circle, Amer. Math. Monthly, 107 (2000) 941 - 945. See also, B. Shawyer, A remarkable concurrence, Forum Geom., 1 (2001) 69 - 74.
${ }^{12}$ Ibid.
## Chapter 3
## Homogeneous Barycentric Coordinates
### Barycentric coordinates with reference to a triangle
#### Homogeneous barycentric coordinates
The notion of barycentric coordinates dates back to Möbius. In a given triangle $A B C$, every point $P$ is coordinatized by a triple of numbers ( $u$ : $v: w)$ in such a way that the system of masses $u$ at $A, v$ at $B$, and $w$ at $C$ will have its balance point at $P$. These masses can be taken in the proportions of the areas of triangle $P B C, P C A$ and $P A B$. Allowing the point $P$ to be outside the triangle, we use signed areas of oriented triangles. The homogeneous barycentric coordinates of $P$ with reference to $A B C$ is a triple of numbers $(x: y: z)$ such that
$$
x: y: z=\triangle P B C: \triangle P C A: \triangle P A B .
$$
## Examples
1. The centroid $G$ has homogeneous barycentric coordinates $(1: 1: 1)$. The areas of the triangles $G B C, G C A$, and $G A B$ are equal. ${ }^{1}$
2. The incenter $I$ has homogeneous barycentric coordinates $(a: b: c)$. If $r$ denotes the inradius, the areas of triangles $I B C, I C A$ and $I A B$ are respectively $\frac{1}{2} r a, \frac{1}{2} r b$, and $\frac{1}{2} r c .{ }^{2}$
${ }^{1}$ In Kimberling's Encyclopedia of Triangle Centers, [ETC], the centroid appears as $X_{2}$.
${ }^{2}$ In ETC, the incenter appears as $X_{1}$. 3. The circumcenter. If $R$ denotes the circumradius, the coordinates of the circumcenter $O$ are ${ }^{3}$
$$
\begin{aligned}
& \triangle O B C: \triangle O C A: \triangle O A B \\
= & \frac{1}{2} R^{2} \sin 2 A: \frac{1}{2} R^{2} \sin 2 B: \frac{1}{2} R^{2} \sin 2 C \\
= & \sin A \cos A: \sin B \cos B: \sin C \cos C \\
= & a \cdot \frac{b^{2}+c^{2}-a^{2}}{2 b c}: b \cdot \frac{c^{2}+a^{2}-b^{2}}{2 c a}: \frac{a^{2}+b^{2}-c^{2}}{2 a b} \\
= & a^{2}\left(b^{2}+c^{2}-a^{2}\right): b^{2}\left(c^{2}+a^{2}-b^{2}\right): c^{2}\left(a^{2}+b^{2}-c^{2}\right) .
\end{aligned}
$$
4. Points on the line $B C$ have coordinates of the form $(0: y: z)$. Likewise, points on $C A$ and $A B$ have coordinates of the forms $(x: 0: z)$ and $(x: y: 0)$ respectively.
## Exercise
1. Verify that the sum of the coordinates of the circumcenter given above is $4 S^{2}$ :
$$
a^{2}\left(b^{2}+c^{2}-a^{2}\right)+b^{2}\left(c^{2}+a^{2}-b^{2}\right)+c^{2}\left(a^{2}+b^{2}-c^{2}\right)=4 S^{2},
$$
where $S$ is twice the area of triangle $A B C$.
2. Find the coordinates of the excenters. ${ }^{4}$
${ }^{3}$ In ETC, the circumcenter appears as $X_{3}$.
${ }^{4} I_{a}=(-a: b: c), I_{b}=(a:-b: c), I_{c}=(a: b:-c)$.
#### Absolute barycentric coordinates
Let $P$ be a point with (homogeneous barycentric) coordinates $(x: y: z)$. If $x+y+z \neq 0$, we obtain the absolute barycentric coordinates by scaling the coefficients to have a unit sum:
$$
P=\frac{x \cdot A+y \cdot B+z \cdot C}{x+y+z} .
$$
If $P$ and $Q$ are given in absolute barycentric coordinates, the point $X$ which divides $P Q$ in the ratio $P X: X Q=p: q$ has absolute barycentric coordinates $\frac{q \cdot P+p \cdot Q}{p+q}$. It is, however, convenient to perform calculations avoiding denominators of fractions. We therefore adapt this formula in the following way: if $P=(u: v: w)$ and $Q=\left(u^{\prime}: v^{\prime}: w^{\prime}\right)$ are the homogeneous barycentric coordinates satisfying $u+v+w=u^{\prime}+v^{\prime}+w^{\prime}$, the point $X$ dividing $P Q$ in the ratio $P X: X Q=p: q$ has homogeneous barycentric coordinates
$$
\left(q u+p u^{\prime}: q v+p v^{\prime}: q w+p w^{\prime}\right)
$$
## Example: Internal center of similitudes of the circumcircle and the incircle
These points, $T$ and $T^{\prime}$, divide the segment $O I$ harmonically in the ratio of the circumradius $R=\frac{a b c}{2 S}$ and the inradius $\frac{S}{2 s}$. Note that $R: r=\frac{a b c}{2 S}: \frac{S}{2 s}=$ sabc $: S^{2}$.
Since
$$
O=\left(a^{2}\left(b^{2}+c^{2}-a^{2}\right): \cdots: \cdots\right)
$$
with coordinates sum $4 S^{2}$ and $I=(a: b: c)$ with coordinates sum $2 s$, we equalize their sums and work with
$$
\begin{aligned}
O & =\left(s a^{2}\left(b^{2}+c^{2}-a^{2}\right): \cdots: \cdots\right) \\
I & =\left(2 S^{2} a: 2 S^{2} b: 2 S^{2} c\right)
\end{aligned}
$$
The internal center of similitude $T$ divides $O I$ in the ratio $O T: T I=R: r$, the $a$-component of its homogeneous barycentric coordinates can be taken as
$$
S^{2} \cdot s a^{2}\left(b^{2}+c^{2}-a^{2}\right)+s a b c \cdot 2 S^{2} a .
$$
The simplification turns out to be easier than we would normally expect:
$$
\begin{aligned}
& S^{2} \cdot s a^{2}\left(b^{2}+c^{2}-a^{2}\right)+s a b c \cdot 2 S^{2} a \\
= & s S^{2} a^{2}\left(b^{2}+c^{2}-a^{2}+2 b c\right)
\end{aligned}
$$
$$
\begin{aligned}
& =s S^{2} a^{2}\left((b+c)^{2}-a^{2}\right) \\
& =s S^{2} a^{2}(b+c+a)(b+c-a) \\
& =2 s^{2} S^{2} \cdot a^{2}(b+c-a) .
\end{aligned}
$$
The other two components have similar expressions obtained by cyclically permuting $a, b, c$. It is clear that $2 s^{2} S^{2}$ is a factor common to the three components. Thus, the homogeneous barycentric coordinates of the internal center of similitude are ${ }^{5}$
$$
\left(a^{2}(b+c-a): b^{2}(c+a-b): c^{2}(a+b-c)\right) .
$$
## Exercises
1. The external center of similitude of $(O)$ and $(I)$ has homogeneous barycentric coordinates 6
$$
\left(a^{2}(a+b-c)(c+a-b): b^{2}(b+c-a)(a+b-c): c^{2}(c+a-b)(b+c-a)\right),
$$
which can be taken as
$$
\left(\frac{a^{2}}{b+c-a}: \frac{b^{2}}{c+a-b}: \frac{c^{2}}{a+b-c}\right) .
$$
2. The orthocenter $H$ lies on the Euler line and divides the segment $O G$ externally in the ratio $O H: H G=3:-2.7$ Show that its homogeneous barycentric coordinates can be written as
$$
H=(\tan A: \tan B: \tan C),
$$
or equivalently,
$$
H=\left(\frac{1}{b^{2}+c^{2}-a^{2}}: \frac{1}{c^{2}+a^{2}-b^{2}}: \frac{1}{a^{2}+b^{2}-c^{2}}\right) .
$$
3. Make use of the fact that the nine-point center $N$ divides the segment $O G$ in the ratio $O N: G N=3:-1$ to show that its barycentric coordinates can be written as 8
$$
N=(a \cos (B-C): b \cos (C-A): c \cos (A-B)) .
$$
${ }^{5}$ In ETC, the internal center of similitude of the circumcircle and the incircle appears as the point $X_{55}$.
${ }^{6}$ In ETC, the external center of similitude of the circumcircle and the incircle appears as the point $X_{56}$.
${ }^{7}$ In ETC, the orthocenter appears as the point $X_{4}$.
${ }^{8}$ In ETC, the nine-point center appears as the point $X_{5}$.
### Cevians and traces
Because of the fundamental importance of the Ceva theorem in triangle geometry, we shall follow traditions and call the three lines joining a point $P$ to the vertices of the reference triangle $A B C$ the cevians of $P$. The intersections $A_{P}, B_{P}, C_{P}$ of these cevians with the side lines are called the traces of $P$. The coordinates of the traces can be very easily written down:
$$
A_{P}=(0: y: z), \quad B_{P}=(x: 0: z), \quad C_{P}=(x: y: 0) .
$$
#### Ceva Theorem
Three points $X, Y, Z$ on $B C, C A, A B$ respectively are the traces of a point if and only if they have coordinates of the form
$$
\begin{aligned}
& X=0: y: z, \\
& Y=x: 0: z, \\
& Z=x: y: 0
\end{aligned}
$$
for some $x, y, z$.
#### Examples
## The Gergonne point
The points of tangency of the incircle with the side lines are
$$
\begin{aligned}
& X=0: s-c: s-b, \\
& Y=s-c: 0: s-a, \\
& Z=s-b: s-a: 0 .
\end{aligned}
$$
These can be reorganized as
$$
\begin{gathered}
X=0: \frac{1}{s-b}: \frac{1}{s-c}, \\
Y=\frac{1}{s-a}: 00: \frac{1}{s-c}, \\
Z=\frac{1}{s-a}: \frac{1}{s-b}: c
\end{gathered}
$$
It follows that $A X, B Y, C Z$ intersect at a point with coordinates
$$
\left(\frac{1}{s-a}: \frac{1}{s-b}: \frac{1}{s-c}\right) .
$$
This is called the Gergonne point $G_{e}$ of triangle $A B C .{ }^{9}$
## The Nagel point
The points of tangency of the excircles with the corresponding sides have coordinates
$$
\begin{aligned}
& X^{\prime}=(0: s-b: s-c), \\
& Y^{\prime}=(s-a: 0: s-c), \\
& Z^{\prime}=(s-a: s-b: 0) .
\end{aligned}
$$
These are the traces of the point with coordinates
$$
(s-a: s-b: s-c) .
$$
This is the Nagel point $N_{a}$ of triangle $A B C .{ }^{10}$
## Exercises
1. The Nagel point $N_{a}$ lies on the line joining the incenter to the centroid; it divides $I G$ in the ratio $I N_{a}: N_{a} G=3:-2$.
${ }^{9}$ The Gergonne point appears in ETC as the point $X_{7}$.
${ }^{10}$ The Nagel point appears in ETC as the point $X_{8}$.
### Isotomic conjugates
The Gergonne and Nagel points are examples of isotomic conjugates. Two points $P$ and $Q$ (not on any of the side lines of the reference triangle) are said to be isotomic conjugates if their respective traces are symmetric with respect to the midpoints of the corresponding sides. Thus,
$$
B A_{P}=A_{Q} C, \quad C B_{P}=B_{Q} A, \quad A C_{P}=C_{Q} B .
$$
We shall denote the isotomic conjugate of $P$ by $P^{\bullet}$. If $P=(x: y: z)$, then
$$
P^{\bullet}=\left(\frac{1}{x}: \frac{1}{y}: \frac{1}{z}\right)
$$
#### Equal-parallelian point
Given triangle $A B C$, we want to construct a point $P$ the three lines through which parallel to the sides cut out equal intercepts. Let $P=x A+y B+z C$ in absolute barycentric coordinates. The parallel to $B C$ cuts out an intercept of length $(1-x) a$. It follows that the three intercepts parallel to the sides are equal if and only if
$$
1-x: 1-y: 1-z=\frac{1}{a}: \frac{1}{b}: \frac{1}{c} .
$$
The right hand side clearly gives the homogeneous barycentric coordinates of $I^{\bullet}$, the isotomic conjugate of the incenter $I$. ${ }^{11}$ This is a point we can easily construct. Now, translating into absolute barycentric coordinates:
$$
I^{\bullet}=\frac{1}{2}[(1-x) A+(1-y) B+(1-z) C]=\frac{1}{2}(3 G-P) .
$$
we obtain $P=3 G-2 I^{\bullet}$, and can be easily constructed as the point dividing the segment $I^{\bullet} G$ externally in the ratio $I^{\bullet} P: P G=3:-2$. The point $P$ is called the equal-parallelian noint of trianole $A B C .{ }^{12}$
${ }^{11}$ The isotomic conjugate of the incenter appears in ETC as the point $X_{75}$.
${ }^{12}$ It appears in ETC as the point $X_{192}$.
## Exercises
1. Calculate the homogeneous barycentric coordinates of the equal-parallelian point and the length of the equal parallelians. ${ }^{13}$
2. Let $A^{\prime} B^{\prime} C^{\prime}$ be the midway triangle of a point $P$. The line $B^{\prime} C^{\prime}$ intersects $C A$ at
$$
\begin{array}{ll}
B_{a}=B^{\prime} C^{\prime} \cap C A, & C_{a}=B^{\prime} C^{\prime} \cap A B, \\
C_{b}=C^{\prime} A^{\prime} \cap A B, & A_{b}=C^{\prime} A^{\prime} \cap B C, \\
A_{c}=A^{\prime} B^{\prime} \cap B C, & B_{c}=A^{\prime} B^{\prime} \cap C A .
\end{array}
$$
Determine $P$ for which the three segments $B_{a} C_{a}, C_{b} A_{b}$ and $A_{c} B_{c}$ have equal lengths. ${ }^{14}$
#### Yff's analogue of the Brocard points
Consider a point $P=(x: y: z)$ satisfying $B A_{P}=C B_{P}=A C_{P}=w$. This means that
$$
\frac{z}{y+z} a=\frac{x}{z+x} b=\frac{y}{x+y} c=w .
$$
Elimination of $x, x, x$ leads to
$$
0=\left|\begin{array}{ccc}
& -w & a-w \\
b-w & & -w \\
-w & c-w &
\end{array}\right|=(a-w)(b-w)(c-w)-w^{3} .
$$
Indeed, $w$ is the unique positive root of the cubic polynomial
$$
(a-t)(b-t)(c-t)-t^{3} .
$$
This gives the point
$$
P=\left(\left(\frac{c-w}{b-w}\right)^{\frac{1}{3}}:\left(\frac{a-w}{c-w}\right)^{\frac{1}{3}}:\left(\frac{b-w}{a-w}\right)^{\frac{1}{3}}\right) .
$$
The isotomic conjugate
$$
P^{\bullet}=\left(\left(\frac{b-w}{c-w}\right)^{\frac{1}{3}}:\left(\frac{c-w}{a-w}\right)^{\frac{1}{3}}:\left(\frac{a-w}{b-w}\right)^{\frac{1}{3}}\right)
$$
${ }^{13}(c a+a b-b c: a b+b c-c a: b c+c a-a b)$. The common length of the equal parallelians is $\frac{2 a b c}{a b+b c+c a}$.
${ }^{14}$ A.P. Hatzipolakis, Hyacinthos, message 3190, 7/13/01. $P=(3 b c-c a-a b: 3 c a-$ $a b-b c: 3 a b-b c-c a)$. This point is not in the current edition of ETC. It is the reflection of the equal-parallelian point in $I^{\bullet}$. In this case, the common length of the segment is $\frac{2 a b c}{a b+b c+c a}$, as in the equal-parallelian case. satisfies
$$
C A_{P}=A B_{P}=B C_{P}=w .
$$
These points are usually called the Yff analogues of the Brocard points. ${ }^{15}$ They were briefly considered by A.L. Crelle. ${ }^{16}$
### Conway's formula
#### Notation
Let $S$ denote twice the area of triangle $A B C$. For a real number $\theta$, denote $S \cdot \cot \theta$ by $S_{\theta} \cdot$ In particular,
$$
S_{A}=\frac{b^{2}+c^{2}-a^{2}}{2}, \quad S_{B}=\frac{c^{2}+a^{2}-b^{2}}{2}, \quad S_{C}=\frac{a^{2}+b^{2}-c^{2}}{2} .
$$
For arbitrary $\theta$ and $\varphi$, we shall simply write $S_{\theta \varphi}$ for $S_{\theta} \cdot S_{\varphi}$.
We shall mainly make use of the following relations.
## Lemma
(1) $S_{B}+S_{C}=a^{2}, S_{C}+S_{A}=b^{2}, S_{A}+S_{B}=c^{2}$.
(2) $S_{A B}+S_{B C}+S_{C A}=S^{2}$.
Proof. (1) is clear. For (2), since $A+B+C=180^{\circ}, \cot (A+B+C)$ is infinite. Its denominator
$$
\cot A \cdot \cot B+\cot B \cdot \cot C+\cot C \cdot \cot A-1=0 .
$$
From this, $S_{A B}+S_{B C}+S_{C A}=S^{2}(\cot A \cdot \cot B+\cot B \cdot \cot C+\cot C \cdot \cot A)=$ $S^{2}$.
## Examples
(1) The orthocenter has coordinates
$$
\left(\frac{1}{S_{A}}: \frac{1}{S_{B}}: \frac{1}{S_{C}}\right)=\left(S_{B C}: S_{C A}: S_{A B}\right) .
$$
${ }^{15} \mathrm{P}$. Yff, An analogue of the Brocard points, Amer. Math. Monthly, 70 (1963) $495-$ 501.
${ }^{16}$ A.L. Crelle, 1815. Note that in the last expression, the coordinate sum is $S_{B C}+S_{C A}+S_{A B}=$ $S^{2}$.
(2) The circumcenter, on the other hand, is the point
$$
O=\left(a^{2} S_{A}: b^{2} S_{B}: c^{2} S_{C}\right)=\left(S_{A}\left(S_{B}+S_{C}\right): S_{B}\left(S_{C}+S_{A}\right): S_{C}\left(S_{A}+S_{B}\right)\right)
$$
Note that in this form, the coordinate sum is $2\left(S_{A B}+S_{B C}+S_{C A}\right)=2 S^{2}$.
## Exercises
1. Calculate the coordinates of the nine-point center in terms of $S_{A}, S_{B}$, $S_{C} \cdot 17$
2. Calculate the coordinates of the reflection of the orthocenter in the circumcenter, i.e., the point $L$ which divides the segment $H O$ in the ratio $H L: L O=2:-1$. This is called the de Longchamps point of triangle $A B C$.
#### Conway's formula
If the swing angles of a point $P$ on the side $B C$ are $\angle C B P=\theta$ and $\angle B C P=$ $\varphi$, the coordinates of $P$ are
$$
\left(-a^{2}: S_{C}+S_{\varphi}: S_{B}+S_{\theta}\right)
$$
The swing angles are chosen in the rangle $-\frac{\pi}{2} \leq \theta, \varphi \leq \frac{\pi}{2}$. The angle $\theta$ is positive or negative according as angles $\angle C B P$ and $\angle C B A$ have different or the same orientation.
${ }^{17} N=\left(S^{2}+S_{B C}: S^{2}+S_{C A}: S^{2}+S_{A B}\right)$.
${ }^{18} L=\left(S_{C A}+S_{A B}-S_{B C}: \cdots: \cdots\right)=\left(\frac{1}{S_{B}}+\frac{1}{S_{C}}-\frac{1}{S_{A}}: \cdots: \cdots\right)$. It appears in ETC as the point $X_{20}$.
#### Examples
## Squares erected on the sides of a triangle
Consider the square $B C X_{1} X_{2}$ erected externally on the side $B C$ of triangle $A B C$. The swing angles of $X_{1}$ with respect to the side $B C$ are
$$
\angle C B X_{1}=\frac{\pi}{4}, \quad \angle B C X_{1}=\frac{\pi}{2} .
$$
Since $\cot \frac{\pi}{4}=1$ and $\cot \frac{\pi}{2}=0$,
$$
X_{1}=\left(-a^{2}: S_{C}: S_{B}+S\right) .
$$
Similarly,
$$
X_{2}=\left(-a^{2}: S_{C}+S: S_{B}\right)
$$
## Exercises
1. Find the midpoint of $X_{1} X_{2}$.
2. Find the vertices of the inscribed squares with a side along $B C .{ }^{19}$.
### The Kiepert perspectors
#### The Fermat points
Consider the equilateral triangle $B C X$ erected externally on the side $B C$ of triangle $A B C$. The swing angles are $\angle C B X=\angle B C X=\frac{\pi}{3}$. Since
${ }^{19}$ Recall that this can be obtained from applying the homothety $\mathrm{h}\left(A, \frac{S}{S+a^{2}}\right)$ to the square $B C X_{1} X_{2}$ $\cot \frac{\pi}{3}=\frac{1}{\sqrt{3}}$
$$
X=\left(-a^{2}: S_{C}+\frac{S}{\sqrt{3}}: S_{B}+\frac{S}{\sqrt{3}}\right)
$$
which can be rearranged in the form
$$
X=\left(\frac{-a^{2}}{\left(S_{B}+\frac{S}{\sqrt{3}}\right)\left(S_{C}+\frac{S}{\sqrt{3}}\right)}: \frac{1}{S_{B}+\frac{S}{\sqrt{3}}}: \frac{1}{S_{C}+\frac{S}{\sqrt{3}}}\right) .
$$
Similarly, we write down the coordinates of the apexes $Y, Z$ of the equilateral triangles $C A Y$ and $A B Z$ erected externally on the other two sides. These are
$$
Y=\left(\frac{1}{S_{A}+\frac{S}{\sqrt{3}}}: * * * * *: \frac{1}{S_{C}+\frac{S}{\sqrt{3}}}\right)
$$
and
$$
Z=\left(\frac{1}{S_{A}+\frac{S}{\sqrt{3}}}: \frac{1}{S_{B}+\frac{S}{\sqrt{3}}}: * * * * *\right) .
$$
Here we simply write $* * * * *$ in places where the exact values of the coordinates are not important. This is a particular case of the following general situation.
#### Perspective triangles
Suppose $X, Y, Z$ are points whose coordinates can be written in the form
$$
\begin{aligned}
& X=* * * * *: \quad y \quad: \quad z, \\
& Y=x: * * * * *: \quad z,
\end{aligned}
$$
The lines $A X, B Y, C Z$ are concurrent at the point $P=(x: y: z)$.
Proof. The intersection of $A X$ and $B C$ is the trace of $X$ on the side $B C$. It is the point $(0: y: z)$. Similarly, the intersections $B Y \cap C A$ and $C Z \cap A B$ are the points $(x: 0: z)$ and $(x: y: 0)$. These three points are in turn the traces of $P=(x: y: z)$.
Q.E.D.
We say that triangle $X Y Z$ is perspective with $A B C$, and call the point $P$ the perspector of $X Y Z$.
We conclude therefore that the apexes of the equilateral triangles erected externally on the sides of a triangle $A B C$ form a triangle perspective with $A B C$ at the point
$$
F_{+}=\left(\frac{1}{\sqrt{3} S_{A}+S}: \frac{1}{\sqrt{3} S_{B}+S}: \frac{1}{\sqrt{3} S_{C}+S}\right) .
$$
This is called the (positive) Fermat point of triangle $A B C .{ }^{20}$
## Exercises
1. If the equilateral triangles are erected "internally" on the sides, the apexes again form a triangle with perspector
$$
F_{-}=\left(\frac{1}{\sqrt{3} S_{A}-S}: \frac{1}{\sqrt{3} S_{B}-S}: \frac{1}{\sqrt{3} S_{C}-S}\right)
$$
the negative Fermat point of triangle $A B C .{ }^{21}$
2. Given triangle $A B C$, extend the sides $A C$ to $B_{a}$ and $A B$ to $C_{a}$ such that $C B_{a}=B C_{a}=a$. Similarly define $C_{b}, A_{b}, A_{c}$, and $B_{c}$.
(a) Write down the coordinates of $B_{a}$ and $C_{a}$, and the coordinates of the intersection $A^{\prime}$ of $B B_{a}$ and $C C_{a}$.
(b) Similarly define $B^{\prime}$ and $C^{\prime}$, and show that $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$. Calculate the coordinates of the perspector. ${ }^{22}$
#### Isosceles triangles erected on the sides and Kiepert perspectors
More generally, consider an isosceles triangle $Y C A$ of base angle $\angle Y C A=$ $\angle Y A C=\theta$. The vertex $Y$ has coordinates
$$
\left(S_{C}+S_{\theta}:-b^{2}: S_{A}+S_{\theta}\right) .
$$
If similar isosceles triangles $X B C$ and $Z A B$ are erected on the other two sides (with the same orientation), the lines $A X, B Y$, and $C Z$ are concurrent at the point
$$
K(\theta)=\left(\frac{1}{S_{A}+S_{\theta}}: \frac{1}{S_{B}+S_{\theta}}: \frac{1}{S_{C}+S_{\theta}}\right) .
$$
We call $X Y Z$ the Kiepert triangle and $K(\theta)$ the Kiepert perspector of parameter $\theta$.
${ }^{20}$ The positive Fermat point is also known as the first isogonic center. It appears in ETC as the point $X_{13}$.
${ }^{21}$ The negative Fermat point is also known as the second isogonic center. It appears in ETC as the point $X_{14}$.
${ }^{22}$ The Spieker point.
#### The Napoleon points
The famous Napoleon theorem states that the centers of the equilateral triangles erected externally on the sides of a triangle form an equilateral triangle. These centers are the apexes of similar isosceles triangles of base angle $30^{\circ}$ erected externally on the sides. They give the Kiepert perspector
$$
\left(\frac{1}{S_{A}+\sqrt{3} S}: \frac{1}{S_{B}+\sqrt{3} S}: \frac{1}{S_{C}+\sqrt{3} S}\right) .
$$
This is called the (positive) Napoleon point of the triangle. ${ }^{23}$ Analogous results hold for equilateral triangles erected internally, leading to the negative Napoleon point ${ }^{24}$
$$
\left(\frac{1}{S_{A}-\sqrt{3} S}: \frac{1}{S_{R}-\sqrt{3} S}: \frac{1}{S_{C}-\sqrt{3} S}\right) .
$$
${ }^{23}$ The positive Napoleon point appears in ETC as the point $X_{17}$.
${ }^{24}$ The negative Napoleon point appears in ETC as the point $X_{18}$.
## Exercises
1. The centers of the three squares erected externally on the sides of triangle $A B C$ form a triangle perspective with $A B C$. The perspector is called the (positive) Vecten point. Why is this a Kiepert perspector? Identify its Kiepert parameter, and write down its coordinates? ${ }^{25}$
2. Let $A B C$ be a given triangle. Construct a small semicircle with $B$ as center and a diameter perpendicular to $B C$, intersecting the side $B C$. Animate a point $T$ on this semicircle, and hide the semicircle.
(a) Construct the ray $B T$ and let it intersect the perpendicular bisector of $B C$ at $X$.
(b) Reflect the ray $B T$ in the bisector of angle $B$, and construct the perpendicular bisector of $A B$ to intersect this reflection at $Z$.
(c) Reflect $A Z$ in the bisector of angle $A$, and reflect $C X$ in the bisector of angle $C$. Label the intersection of these two reflections $Y$.
(d) Construct the perspector $P$ of the triangle $X Y Z$.
(e) What is the locus of $P$ as $T$ traverses the semicircle?
3. Calculate the coordinates of the midpoint of the segment $F_{+} F_{-}{ }^{26}$
4. Inside triangle $A B C$, consider two congruent circles $I_{a b}\left(r_{1}\right)$ and $I_{a c}\left(r_{1}\right)$ tangent to each other (externally), both to the side $B C$, and to $C A$ and $A B$ respectively. Note that the centers $I_{a b}$ and $I_{a c}$, together with their pedals on $B C$, form a rectangle of sides $2: 1$. This rectangle can be constructed as the image under the homothety $\mathrm{h}\left(I, \frac{2 r}{a}\right)$ of a similar rectangle erected externally on the side $B C$.
${ }^{25}$ This is $K\left(\frac{\pi}{4}\right)$, the positive Vecten point. It appears in ETC as $X_{485}$.
${ }^{26}\left(\left(b^{2}-c^{2}\right)^{2}:\left(c^{2}-a^{2}\right)^{2}:\left(a^{2}-b^{2}\right)^{2}\right)$. This points appears in ETC as $X_{115}$. It lies on the nine-point circle. (a) Make use of these to construct the two circles.
(b) Calculate the homogeneous barycentric coordinates of the point of tangency of the two circles. ${ }^{27}$
(c) Similarly, there are two other pairs of congruent circles on the sides $C A$ and $A B$. The points of tangency of the three pairs have a perspector 28
$$
\left(\frac{1}{b c+S}: \frac{1}{c a+S}: \frac{1}{a b+S}\right)
$$
(d) Show that the pedals of the points of tangency on the respective side lines of $A B C$ are the traces of 29
$$
\left(\frac{1}{b c+S+S_{A}}: \frac{1}{c a+S+S_{B}}: \frac{1}{a b+S+S_{C}}\right) .
$$
#### Nagel's Theorem
Suppose $X, Y, Z$ are such that
$$
\begin{aligned}
& \angle C A Y=\angle B A Z=\theta, \\
& \angle A B Z=\angle C B X=\varphi, \\
& \angle B C X=\angle A C Y=\psi
\end{aligned}
$$
The lines $A X, B Y, C Z$ are concurrent at the point
$$
\left(\frac{1}{S_{A}+S_{\theta}}: \frac{1}{S_{B}+S_{\varphi}}: \frac{1}{S_{C}+S_{\psi}}\right) .
$$
${ }^{27}$ This divides $I D(D=$ midpoint of $B C)$ in the ratio $2 r: a$ and has coordinates $\left(a^{2}: a b+S: a c+S\right)$.
${ }^{28}$ This point is not in the current edition of ETC.
${ }^{29}$ This point is not in the current edition of ETC.
## Exercises
1. Let $X^{\prime}, Y^{\prime}, Z^{\prime}$ be respectively the pedals of $X$ on $B C, Y$ on $C A$, and $Z$ on $A B$. Show that $X^{\prime} Y^{\prime} Z^{\prime}$ is a cevian triangle. ${ }^{30}$
2. For $i=1,2$, let $X_{i} Y_{i} Z_{i}$ be the triangle formed with given angles $\theta_{i}, \varphi_{i}$ and $\psi_{i}$. Show that the intersections
$$
X=X_{1} X_{2} \cap B C, \quad Y=Y_{1} Y_{2} \cap C A, \quad Z=Z_{1} Z_{2} \cap A B
$$
form a cevian triangle. 31
${ }^{30}$ Floor van Lamoen.
${ }^{31}$ Floor van Lamoen. $X=\left(0: S_{\psi_{1}}-S_{\psi_{2}}: S_{\varphi_{1}}-S_{\varphi_{2}}\right)$.
## Chapter 4
## Straight Lines
### The equation of a line
#### Two-point form
The equation of the line joining two points with coordinates $\left(x_{1}: y_{1}: z_{1}\right)$ and $\left(x_{2}: y_{2}: z_{2}\right)$ is
$$
\left|\begin{array}{ccc}
x_{1} & y_{1} & z_{1} \\
x_{2} & y_{2} & z_{2} \\
x & y & z
\end{array}\right|=0
$$
or
$$
\left(y_{1} z_{2}-y_{2} z_{1}\right) x+\left(z_{1} x_{2}-z_{2} x_{1}\right) y+\left(x_{1} y_{2}-x_{2} y_{1}\right) z=0
$$
#### Examples
1. The equations of the side lines $B C, C A, A B$ are respectively $x=0$, $y=0, z=0$.
2. The perpendicular bisector of $B C$ is the line joining the circumcenter $O=\left(a^{2} S_{A}: b^{2} S_{B}: c^{2} S_{C}\right)$ to the midpoint of $B C$, which has coordinates $(0: 1: 1)$. By the two point form, it has equation
$$
\left(b^{2} S_{B}-c^{2} S_{C}\right) x-a^{2} S_{A} y+a^{2} S_{A} z=0,
$$
Since $b^{2} S_{B}-c^{2} S_{C}=\cdots=S_{A}\left(S_{B}-S_{C}\right)=-S_{A}\left(b^{2}-c^{2}\right)$, this equation can be rewritten as
$$
\left(b^{2}-c^{2}\right) x+a^{2}(y-z)=0 .
$$
3. The equation of the Euler line, as the line joining the centroid $(1: 1: 1)$ to the orthocenter $\left(S_{B C}: S_{C A}: S_{A B}\right)$ is
$$
\left(S_{A B}-S_{C A}\right) x+\left(S_{B C}-S_{A B}\right) y+\left(S_{C A}-S_{B C}\right) z=0,
$$
or
$$
\sum_{\text {cyclic }} S_{A}\left(S_{B}-S_{C}\right) x=0
$$
4. The equation of the $O I$-line joining the circumcenter $\left(a^{2} S_{A}: b^{2} S_{B}\right.$ : $\left.c^{2} S_{C}\right)$ to and the incenter $(a: b: c)$ is
$$
0=\sum_{\text {cyclic }}\left(b^{2} S_{B} c-c^{2} S_{C} b\right) x=\sum_{\text {cyclic }} b c\left(b S_{B}-c S_{C}\right) x .
$$
Since $b S_{B}-c S_{C}=\cdots=-2(b-c) s(s-a)$ (exercise), this equation can be rewritten as
$$
\sum_{\text {cyclic }} b c(b-c) s(s-a) x=0
$$
or
$$
\sum_{\text {cyclic }} \frac{(b-c)(s-a)}{a} x=0 .
$$
5. The line joining the two Fermat points
$$
\begin{aligned}
F_{ \pm} & =\left(\frac{1}{\sqrt{3} S_{A} \pm S}: \frac{1}{\sqrt{3} S_{B} \pm S}: \frac{1}{\sqrt{3} S_{C} \pm S}\right) \\
& =\left(\left(\sqrt{3} S_{B} \pm S\right)\left(\sqrt{3} S_{C} \pm S\right): \cdots: \cdots\right)
\end{aligned}
$$
has equation
$$
\begin{aligned}
0 & =\sum_{\text {cyclic }}\left(\frac{1}{\left(\sqrt{3} S_{B}+S\right)\left(\sqrt{3} S_{C}-S\right)}-\frac{1}{\left(\sqrt{3} S_{B}+S\right)\left(\sqrt{3} S_{C}-S\right)}\right) x \\
& =\sum_{\text {cyclic }}\left(\frac{\left(\sqrt{3} S_{B}-S\right)\left(\sqrt{3} S_{C}+S\right)-\left(\sqrt{3} S_{B}-S\right)\left(\sqrt{3} S_{C}+S\right)}{\left(3 S_{B B}-S^{2}\right)\left(3 S_{C C}-S^{2}\right)}\right) x \\
& =\sum_{\text {cyclic }}\left(\frac{2 \sqrt{3}\left(S_{B}-S_{C}\right) S}{\left(3 S_{B B}-S^{2}\right)\left(3 S_{C C}-S^{2}\right)}\right) x .
\end{aligned}
$$
Clearing denominators, we obtain
$$
\sum_{\text {cyclic }}\left(S_{B}-S_{C}\right)\left(3 S_{A A}-S^{2}\right) x=0 .
$$
#### Intercept form: tripole and tripolar
If the intersections of a line $\mathcal{L}$ with the side lines are
$$
X=(0: v:-w), \quad Y=(-u: 0: w), \quad Z=(u:-v: 0),
$$
the equation of the line $\mathcal{L}$ is
$$
\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=0
$$
We shall call the point $P=(u: v: w)$ the tripole of $\mathcal{L}$, and the line $\mathcal{L}$ the tripolar of $P$.
## Construction of tripole
Given a line $\mathcal{L}$ intersecting $B C, C A, A B$ at $X, Y, Z$ respectively, let
$$
A^{\prime}=B Y \cap C Z, \quad B^{\prime}=C Z \cap A X, \quad C^{\prime}=A X \cap B Y .
$$
The lines $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ intersect at the tripole $P$ of $\mathcal{L}$.
## Construction of tripolar
Given $P$ with traces $A_{P}, B_{P}$, and $C_{P}$ on the side lines, let
$$
X=B_{P} C_{P} \cap B C, \quad Y=C_{P} A_{P} \cap C A, \quad Z=A_{P} B_{P} \cap A B .
$$
These points $X, Y, Z$ lie on the tripolar of $P$.
## Exercises
1. Find the equation of the line joining the centroid to a given point $P=(u: v: w){ }^{1}$
2. Find the equations of the cevians of a point $P=(u: v: w)$.
3. Find the equations of the angle bisectors.
### Infinite points and parallel lines
#### The infinite point of a line
The infinite point of a line $\mathcal{L}$ has homogeneous coordinates given by the difference of the absolute barycentric coordinates of two distinct points on the line. As such, the coordinate sum of an infinite point is zero. We think of all infinite points constituting the line at infinity, $\mathcal{L}_{\infty}$, which has equation $x+y+z=0$.
## Examples
1. The infinite points of the side lines $B C, C A, A B$ are $(0:-1: 1)$, $(1: 0:-1),(-1: 1: 0)$ respectively.
2. The infinite point of the $A$-altitude has homogeneous coordinates
$$
\left(0: S_{C}: S_{B}\right)-a^{2}(1: 0: 0)=\left(-a^{2}: S_{C}: S_{B}\right) .
$$
3. More generally, the infinite point of the line $p x+q y+r z=0$ is
$$
(q-r: r-p: p-q) .
$$
4. The infinite point of the Euler line is the point
$$
3\left(S_{B C}: S_{C A}: S_{A B}\right)-S S(1: 1: 1) \sim\left(3 S_{B C}-S S: 3 S_{C A}-S S: 3 S_{A B}-S S\right) .
$$
5. The infinite point of the $O I$-line is
$$
\begin{gathered}
(c a(c-a)(s-b)-a b(a-b)(s-c): \cdots: \cdots) \\
\sim \quad\left(a\left(a^{2}(b+c)-2 a b c-(b+c)(b-c)^{2}\right): \cdots: \cdots\right) .
\end{gathered}
$$
${ }^{1}$ Equation: $(v-w) x+(w-u) y+(u-v) z=0$.
#### Parallel lines
Parallel lines have the same infinite point. The line through $P=(u: v: w)$ parallel to $\mathcal{L}: p x+q y+r z=0$ has equation
$$
\left|\begin{array}{ccc}
q-r & r-p & p-q \\
u & v & w \\
x & y & z
\end{array}\right|=0 .
$$
## Exercises
1. Find the equations of the lines through $P=(u: v: w)$ parallel to the side lines.
2. Let $D E F$ be the medial triangle of $A B C$, and $P$ a point with cevian triangle $X Y Z$ (with respect to $A B C$. Find $P$ such that the lines $D X, E Y, F Z$ are parallel to the internal bisectors of angles $A, B, C$ respectively. ${ }^{2}$
### Intersection of two lines
The intersection of the two lines
$$
\begin{aligned}
& p_{1} x+q_{1} y+r_{1} z=0 \\
& p_{2} x+q_{2} y+r_{2} z=0
\end{aligned}
$$
is the point
$$
\left(q_{1} r_{2}-q_{2} r_{1}: r_{1} p_{2}-r_{2} p_{1}: p_{1} q_{2}-p_{2} q_{1}\right) .
$$
The infinite point of a line $\mathcal{L}$ can be regarded as the intersection of $\mathcal{L}$ with the line at infinity $\mathcal{L}_{\infty}: x+y+z=0$.
## Theorem
Three lines $p_{i} x+q_{i} y+r_{i} z=0, i=1,2,3$, are concurrent if and only if
$$
\left|\begin{array}{lll}
p_{1} & q_{1} & r_{1} \\
p_{2} & q_{2} & r_{2} \\
p_{3} & q_{3} & r_{3}
\end{array}\right|=0
$$
${ }^{2}$ The Nagel point $P=(b+c-a: c+a-b: a+b-c)$. N.Dergiades, Hyacinthos, message $3677,8 / 31 / 01$.
#### Intersection of the Euler and Fermat lines
Recall that these lines have equations
$$
\sum_{\text {cyclic }} S_{A}\left(S_{B}-S_{C}\right) x=0
$$
and
$$
\sum_{\text {cyclic }}\left(S_{B}-S_{C}\right)\left(3 S_{A A}-S^{2}\right) x=0
$$
The $A$-coordinate of their intersection
$$
\begin{aligned}
= & S_{B}\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right)\left(3 S_{C C}-S^{2}\right) \\
& -S_{C}\left(S_{A}-S_{B}\right)\left(S_{C}-S_{A}\right)\left(3 S_{B B}-S^{2}\right) \\
= & \left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right)\left[S_{B}\left(3 S_{C C}-S^{2}\right)-S_{C}\left(3 S_{B B}-S^{2}\right)\right] \\
= & \left.\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right)\left[3 S_{B C}\left(S_{C}-S_{B}\right)-S^{2}\left(S_{B}-S_{C}\right)\right)\right] \\
= & -\left(S_{B}-S_{C}\right)\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right)\left(3 S_{B C}+S^{2}\right) .
\end{aligned}
$$
This intersection is the point
$$
\left(3 S_{B C}+S^{2}: 3 S_{C A}+S^{2}: 3 S_{A B}+S^{2}\right) .
$$
Since $\left(3 S_{B C}: 3 S_{C A}: 3 S_{A B}\right)$ and $\left(S^{2}: S^{2}: S^{2}\right)$ represent $H$ and $G$, with equal coordinate sums, this point is the midpoint of $G H .{ }^{3}$
## Remark
Lester has discovered that there is a circle passing the two Fermat points, the circumcenter, and the nine-point center. ${ }^{4}$ The circle with $G H$ as diameter,
${ }^{3}$ This point appears in ETC as $X_{381}$.
${ }^{4}$ J.A. Lester, Triangles, III: complex centre functions and Ceva's theorem, Aequationes Math., 53 (1997) 4-35. whose center is the intersection of the Fermat and Euler line as we have shown above, is orthogonal to the Lester circle. ${ }^{5}$ It is also interesting to note that the midpoint between the Fermat points is a point on the ninepoint circle. It has coordinates $\left(\left(b^{2}-c^{2}\right)^{2}:\left(c^{2}-a^{2}\right)^{2}:\left(a^{2}-b^{2}\right)^{2}\right)$.
#### Triangle bounded by the outer side lines of the squares erected externally
Consider the square $B C X_{1} X_{2}$ erected externally on $B C$. Since $X_{1}=\left(-a^{2}\right.$ : $S_{C}: S_{B}+S$ ), and the line $X_{1} X_{2}$, being parallel to $B C$, has infinite point $(0:-1: 1)$, this line has equation
$$
\left(S_{C}+S_{B}+S\right) x+a^{2} y+a^{2} z=0 .
$$
Since $S_{B}+S_{C}=a^{2}$, this can be rewritten as
$$
a^{2}(x+y+z)+S x=0 .
$$
Similarly, if $C A Y_{1} Y_{2}$ and $A B Z_{1} Z_{2}$ are squares erected externally on the other two sides, the lines $Y_{1} Y_{2}$ and $Z_{1} Z_{2}$ have equations
$$
b^{2}(x+y+z)+S y=0
$$
and
$$
c^{2}(x+y+z)+S z=0
$$
respectively. These two latter lines intersect at the point
$$
X=\left(-\left(b^{2}+c^{2}+S\right): b^{2}: c^{2}\right) .
$$
Similarly, the lines $Z_{1} Z_{2}$ and $X_{1} X_{2}$ intersect at
$$
Y=\left(a^{2}:-\left(c^{2}+a^{2}+S\right): c^{2}\right),
$$
${ }^{5}$ P. Yiu, Hyacinthos, message 1258, August 21, 2000. and the lines $X_{1} X_{2}$ and $Y_{1} Y_{2}$ intersect at
$$
Z=\left(a^{2}: b^{2}:-\left(a^{2}+b^{2}+S\right)\right)
$$
The triangle $X Y Z$ is perspective with $A B C$, at the point
$$
K=\left(a^{2}: b^{2}: c^{2}\right) .
$$
This is called the symmedian point of triangle $A B C .{ }^{6}$
## Exercises
1. The symmedian point lies on the line joining the Fermat points.
2. The line joining the two Kiepert perspectors $K( \pm \theta)$ has equation
$$
\sum_{\text {cyclic }}\left(S_{B}-S_{C}\right)\left(S_{A A}-S^{2} \cot ^{2} \theta\right) x=0 .
$$
Show that this line passes through a fixed point. ${ }^{7}$
3. Show that triangle $A^{\theta} B^{\theta} C^{\theta}$ has the same centroid as triangle $A B C$.
4. Construct the parallels to the side lines through the symmedian point. The 6 intersections on the side lines lie on a circle. The symmedian point is the unique point with this property. ${ }^{8}$
5. Let $D E F$ be the medial triangle of $A B C$. Find the equation of the line joining $D$ to the excenter $I_{a}=(-a: b: c)$. Similarly write down the equation of the lines joining to $E$ to $I_{b}$ and $F$ to $I_{c}$. Show that these three lines are concurrent by working out the coordinates of their common point. ${ }^{9}$
6. The perpendiculars from the excenters to the corresponding sides are concurrent. Find the coordinates of the intersection by noting how it is related to the circumcenter and the incenter. 10
${ }^{6}$ It is also known as the Grebe point, and appears in ETC as the point $X_{6}$.
${ }^{7}$ The symmedian point.
${ }^{8}$ This was first discovered by Lemoine in 1883.
${ }^{9}$ This is the Mittenpunkt $(a(s-a): \cdots: \cdots)$.
${ }^{10} \mathrm{Th}$ is is the reflection of $I$ in $O$. As such, it is the point $2 O-I$, and has coordinates
$$
\left(a\left(a^{3}+a^{2}(b+c)-a(b+c)^{2}-(b+c)(b-c)^{2}\right): \cdots: \cdots\right) .
$$
7. Let $D, E, F$ be the midpoints of the sides $B C, C A, A B$ of triangle $A B C$. For a point $P$ with traces $A_{P}, B_{P}, C_{P}$, let $X, Y, Z$ be the midpoints of $B_{P} C_{P}, C_{P} A_{P}, A_{P} B_{P}$ respectively. Find the equations of the lines $D X, E Y, F Z$, and show that they are concurrent. What are the coordinates of their intersection? ${ }^{11}$
8. Let $D, E, F$ be the midpoints of the sides of $B C, C A, A B$ of triangle $A B C$, and $X, Y, Z$ the midpoints of the altitudes from $A, B, C$ respeectively. Find the equations of the lines $D X, E Y, F Z$, and show that they are concurrent. What are the coordinates of their intersection? ${ }^{12}$
9. Given triangle $A B C$, extend the sides $A C$ to $B_{a}$ and $A B$ to $C_{a}$ such that $C B_{a}=B C_{a}=a$. Similarly define $C_{b}, A_{b}, A_{c}$, and $B_{c}$. The lines $B_{c} C_{b}, C_{b} A_{b}$, and $A_{c} B_{c}$ bound a triangle perspective with $A B C$. Calculate the coordinate of the perspector. ${ }^{13}$
### Pedal triangle
The pedals of a point $P=(u: v: w)$ are the intersections of the side lines with the corresponding perpendiculars through $P$. The $A$-altitude has infinite point $A_{H}-A=\left(0: S_{C}: S_{B}\right)-\left(S_{B}+S_{C}: 0: 0\right)=\left(-a^{2}: S_{C}: S_{B}\right)$. The perpendicular through $P$ to $B C$ is the line
$$
\left|\begin{array}{ccc}
-a^{2} & S_{C} & S_{B} \\
u & v & w \\
x & y & z
\end{array}\right|=0,
$$
or
$$
-\left(S_{B} v-S_{C} w\right) x+\left(S_{B} u+a^{2} w\right) y-\left(S_{C} u+a^{2} v\right) z=0
$$
${ }^{11}$ The intersection is the point dividing the segment $P G$ in the ratio $3: 1$.
${ }^{12}$ This intersection is the symmedian point $K=\left(a^{2}: b^{2}: c^{2}\right)$.
${ }^{13}\left(\frac{a(b+c)}{b+c-a}: \cdots: \cdots\right)$. This appears in ETC as $X_{65}$. This intersects $B C$ at the point
$$
A_{[P]}=\left(0: S_{C} u+a^{2} v: S_{B} u+a^{2} w\right) .
$$
Similarly the coordinates of the pedals on $C A$ and $A B$ can be written down. The triangle $A_{[P]} B_{[P]} C_{[P]}$ is called the pedal triangle of triangle $A B C$ :
$$
\left(\begin{array}{c}
A_{[P]} \\
B_{[P]} \\
C_{[P]}
\end{array}\right)=\left(\begin{array}{ccc}
0 & S_{C} u+a^{2} v & S_{B} u+a^{2} w \\
S_{C} v+b^{2} u & 0 & S_{A} v+b^{2} w \\
S_{B} w+c^{2} u & S_{A} w+c^{2} v & 0
\end{array}\right)
$$
#### Examples
1. The pedal triangle of the circumcenter is clearly the medial triangle.
2. The pedal triangle of the orthocenter is called the orthic triangle. Its vertices are clearly the traces of $H$, namely, the points $\left(0: S_{C}: S_{B}\right)$, $\left(S_{C}: 0: S_{A}\right)$, and $\left(S_{B}: S_{A}: 0\right)$.
3. Let $L$ be the reflection of the orthocenter $H$ in the circumcenter $O$. This is called the de Longchamps point. ${ }^{14}$ Show that the pedal triangle of $L$ is the cevian triangle of some point $P$. What are the coordinat of $n \rightarrow 15$
4. Let $L$ be the de Longchamps point again, with homogeneous barycentric coordinates
$$
\left(S_{C A}+S_{A B}-S_{B C}: S_{A B}+S_{B C}-S_{C A}: S_{B C}+S_{C A}-S_{A B}\right) .
$$
Find the equations of the perpendiculars to the side lines at the corresponding traces of $L$. Show that these are concurrent, and find the coordinates of the intersection.
${ }^{14}$ The de Longchamps point appears as $X_{20}$ in ETC.
${ }^{15} P=\left(S_{A}: S_{B}: S_{C}\right)$ is the isotomic conjugate of the orthocenter. It appears in ETC as the point $X_{69}$. The perpendicular to $B C$ at $A_{L}=\left(0: S_{A B}+S_{B C}-S_{C A}: S_{B C}+\right.$ $\left.S_{C A}-S_{A B}\right)$ is the line
$$
\left|\begin{array}{ccc}
-\left(S_{B}+S_{C}\right) & S_{C} & S_{B} \\
0 & S_{A B}+S_{B C}-S_{C A} & S_{B C}+S_{C A}-S_{A B} \\
x & y & z
\end{array}\right|=0 .
$$
This is
$$
S^{2}\left(S_{B}-S_{C}\right) x-a^{2}\left(S_{B C}+S_{C A}-S_{A B}\right) y+a^{2}\left(S_{B C}-S_{C A}+S_{A B}\right) z=0 .
$$
Similarly, we write down the equations of the perpendiculars at the other two traces. The three perpendiculars intersect at the point ${ }^{16}$
$$
\left(a^{2}\left(S_{C}^{2} S_{A}^{2}+S_{A}^{2} S_{B}^{2}-S_{B}^{2} S_{C}^{2}\right): \cdots: \cdots\right) .
$$
## Exercises
1. Let $D, E, F$ be the midpoints of the sides $B C, C A, A B$, and $A^{\prime}$, $B^{\prime}, C^{\prime}$ the pedals of $A, B, C$ on their opposite sides. Show that $X=E C^{\prime} \cap F B^{\prime}, Y=F A^{\prime} \cap D C^{\prime}$, and $Z=D B^{\prime} \cap E C^{\prime}$ are collinear. ${ }^{17}$
2. Let $X$ be the pedal of $A$ on the side $B C$ of triangle $A B C$. Complete the squares $A X X_{b} A_{b}$ and $A X X_{c} A_{c}$ with $X_{b}$ and $X_{c}$ on the line $B C .{ }^{18}$
(a) Calculate the coordinates of $A_{b}$ and $A_{c} \cdot{ }^{19}$
(b) Calculate the coordinates of $A^{\prime}=B A_{c} \cap C A_{b} .{ }^{20}$
(c) Similarly define $B^{\prime}$ and $C^{\prime}$. Triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$. What is the perspector? ${ }^{21}$
(d) Let $A^{\prime \prime}$ be the pedal of $A^{\prime}$ on the side $B C$. Similarly define $B^{\prime \prime}$ and $C^{\prime \prime}$. Show that $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime}$ is perspective with $A B C$ by calculating the coordinates of the perspector. ${ }^{22}$
${ }^{16}$ This point appears in ETC as $X_{1078}$. Conway calls this point the logarithm of the de Longchamps point.
${ }^{17}$ These are all on the Euler line. See G. Leversha, Problem 2358 and solution, Crux Mathematicorum, 24 (1998) 303; 25 (1999) 371 -372.
${ }^{18}$ A.P. Hatzipolakis, Hyacinthos, message $3370,8 / 7 / 01$.
${ }^{19} A_{b}=\left(a^{2}:-S: S\right)$ and $A_{c}=\left(a^{2}: S:-S\right)$.
${ }^{20} A^{\prime}=\left(a^{2}: S: S\right)$.
${ }^{21}$ The centroid.
${ }^{22}\left(\frac{1}{S_{A}+S}: \frac{1}{S_{B}+S}: \frac{1}{S_{C}+S}\right)$.
### Perpendicular lines
Given a line $\mathcal{L}: p x+q y+r z=0$, we determine the infinite point of lines perpendicular to it. ${ }^{23}$ The line $\mathcal{L}$ intersects the side lines $C A$ and $A B$ at the points $Y=(-r: 0: p)$ and $Z=(q:-p: 0)$. To find the perpendicular from $A$ to $\mathcal{L}$, we first find the equations of the perpendiculars from $Y$ to $A B$ and from $Z$ to $C A$. These are
$$
\left|\begin{array}{ccc}
S_{B} & S_{A} & -c^{2} \\
-r & 0 & p \\
x & y & z
\end{array}\right|=0 \text { and }\left|\begin{array}{ccc}
S_{C} & -b^{2} & S_{A} \\
q & -p & 0 \\
x & y & z
\end{array}\right|=0
$$
These are
$$
\begin{aligned}
& S_{A} p x+\left(c^{2} r-S_{B} p\right) y+S_{A} r z=0 \\
& S_{A} p x+S_{A} q y+\left(b^{2} q-S_{C} p\right) z=0 .
\end{aligned}
$$
These two perpendiculars intersect at the orthocenter of triangle $A Y Z$, which is the point
$$
\begin{aligned}
X^{\prime} & =\left(* * * * *: S_{A} p\left(S_{A} r-b^{2} q+S_{C} p\right): S_{A} p\left(S_{A} q+S_{B} p-c^{2} r\right)\right. \\
& \sim\left(* * * * *: S_{C}(p-q)-S_{A}(q-r): S_{A}(q-r)-S_{B}(r-p)\right) .
\end{aligned}
$$
The perpendicular from $A$ to $\mathcal{L}$ is the line $A X^{\prime}$, which has equation
$$
\left|\begin{array}{ccc}
1 & 0 & 0 \\
* * * & S_{C}(p-q)-S_{A}(q-r) & -S_{A}(q-r)+S_{B}(r-p) \\
x & y & z
\end{array}\right|=0
$$
or
$$
-\left(S_{A}(q-r)-S_{B}(r-p)\right) y+\left(S_{C}(p-q)-S_{A}(q-r)\right) z=0 .
$$
This has infinite point
$$
\left(S_{B}(r-p)-S_{C}(p-q): S_{C}(p-q)-S_{A}(q-r): S_{A}(q-r)-S_{B}(r-p)\right) .
$$
Note that the infinite point of $\mathcal{L}$ is $(q-r: r-p: p-q)$. We summarize this in the following theorem.
${ }^{23}$ I learned of this method from Floor van Lamoen.
## Theorem
If a line $\mathcal{L}$ has infinite point $(f: g: h)$, the lines perpendicular to $\mathcal{L}$ have infinite points
$$
\left(f^{\prime}: g^{\prime}: h^{\prime}\right)=\left(S_{B} g-S_{C} h: S_{C} h-S_{A} f: S_{A} f-S_{B} g\right) .
$$
Equivalently, two lines with infinite points $(f: g: h)$ and $\left(f^{\prime}: g^{\prime}: h^{\prime}\right)$ are perpendicular to each other if and only if
$$
S_{A} f f^{\prime}+S_{B} g g^{\prime}+S_{C} h h^{\prime}=0 .
$$
#### The tangential triangle
Consider the tangents to the circumcircle at the vertices. The radius $O A$ has infinite point
$$
\left(a^{2} S_{A}: b^{2} S_{B}: c^{2} S_{C}\right)-\left(2 S^{2}: 0: 0\right)=\left(-\left(b^{2} S_{B}+c^{2} S_{C}\right): b^{2} S_{B}: c^{2} S_{C}\right) .
$$
The infinite point of the tangent at $A$ is
$\left(b^{2} S_{B B}-c^{2} S_{C C}: c^{2} S_{C C}+S_{A}\left(b^{2} S_{B}+c^{2} S_{C}\right):-S_{A}\left(b^{2} S_{B}+c^{2} S_{C}\right)-b^{2} S_{B B}\right)$.
Consider the $B$-coordinate:
$c^{2} S_{C C}+S_{A}\left(b^{2} S_{B}+c^{2} S_{C}\right)=c^{2} S_{C}\left(S_{C}+S_{A}\right)+b^{2} S_{A B}=b^{2}\left(c^{2} S_{C}+S_{A B}\right)=b^{2} S^{2}$.
Similarly, the $C$-coordinate $=-c^{2} S^{2}$. It follows that this infinite point is $\left(-\left(b^{2}-c^{2}\right): b^{2}:-c^{2}\right)$, and the tangent at $A$ is the line
$$
\left|\begin{array}{ccc}
1 & 0 & 0 \\
-\left(b^{2}-c^{2}\right) & b^{2} & -c^{2} \\
x & y & z
\end{array}\right|=0,
$$
or simply $c^{2} y+b^{2} z=0$. The other two tangents are $c^{2} x+a^{2} z=0$, and $b^{2} x+a^{2} y=0$. These three tangents bound a triangle with vertices
$$
A^{\prime}=\left(0: b^{2}: c^{2}\right), \quad B^{\prime}=\left(a^{2}: 0: c^{2}\right), \quad C^{\prime}=\left(a^{2}: b^{2}: 0\right) .
$$
This is called the tangential triangle of $A B C$. It is perspective with $A B C$ at the point $\left(a^{2}: b^{2}: c^{2}\right)$, the symmedian point.
#### Line of ortho-intercepts ${ }^{24}$
Let $P=(u: v: w)$. We consider the line perpendicular to $A P$ at $P$. Since the line $A P$ has equation $w y-v z=0$ and infinite point $(-(v+w): v: w)$, the perpendicular has infinite point $\left(S_{B} v-S_{C} w: S_{C} w+S_{A}(v+w):-S_{A}(v+\right.$ $\left.w)-S_{B} v\right) \sim\left(S_{B} v-S_{C} w: S_{A} v+b^{2} w:-S_{A} w-c^{2} v\right)$. It is the line
$$
\left|\begin{array}{ccc}
u & v & w \\
S_{B} v-S_{C} w & S_{A} v+b^{2} w & -S_{A} w-c^{2} v \\
x & y & z
\end{array}\right|=0 .
$$
This perpendicular line intersects the side line $B C$ at the point
$$
\begin{aligned}
& \left(0: u\left(S_{A} v+b^{2} w\right)-v\left(S_{B} v-S_{C} w\right):-u\left(S_{A} w+c^{2} v\right)-w\left(S_{B} v-S_{C} w\right)\right) \\
\sim & \left(0:\left(S_{A} u-S_{B} v+S_{C} w\right) v+b^{2} w u:-\left(\left(S_{A} u+S_{b} v-S_{C} w\right) w+c^{2} u v\right)\right) .
\end{aligned}
$$
Similarly, the line perpendicular to $B P$ at $P$ intersects $C A$ at
$$
\left.\left(-\left(-S_{A} u+S_{B} v+S_{C} w\right) u+a^{2} v w\right): 0:\left(S_{A} u-S_{B} v+S_{C} w\right) w+c^{2} u v\right)
$$
and
$$
\left.\left.\left(\left(-S_{A} u+S_{B} v+S_{C} w\right) u+a^{2} v w\right):-\left(S_{A} u-S_{B} v+S_{C} w\right) v+b^{2} w u\right): 0\right) .
$$
These three points are collinear. The line containing them has equation
$$
\sum_{\text {cyclic }} \frac{x}{\left(-S_{A} u+S_{B} v+S_{C} w\right) u+a^{2} v w}=0 .
$$
## Exercises
1. If triangle $A B C$ is acute-angled, the symmedian point is the Gergonne point of the tangential triangle.
2. Given a line $\mathcal{L}$, construct the two points each having $\mathcal{L}$ as its line of ortho-intercepts. ${ }^{25}$
${ }^{24}$ B. Gibert, Hyacinthos, message 1158, August 5, 2000.
${ }^{25}$ One of these points lies on the circumcircle, and the other on the nine-point circle. 3. The tripole of the line of ortho-intercepts of the incenter is the point $\left(\frac{a}{s-a}: \frac{b}{s-b}: \frac{c}{s-c}\right) .{ }^{26}$
4. Calculate the coordinates of the tripole of the line of ortho-intercepts of the nine-point center. ${ }^{27}$
5. Consider a line $\mathcal{L}: p x+q y+r z=0$.
(1) Calculate the coordinates of the pedals of $A, B, C$ on the line $\mathcal{L}$. Label these points $X, Y, Z$.
(2) Find the equations of the perpendiculars from $X, Y, Z$ to the corresponding side lines.
(3) Show that these three perpendiculars are concurrent, and determine the coordinates of the common point.
This is called the orthopole of $\mathcal{L}$.
6. Animate a point $P$ on the circumcircle. Contruct the orthopole of the diameter $O P$. This orthopole lies on the nine-point circle.
7. Consider triangle $A B C$ with its incircle $I(r)$.
(a) Construct a circle $X_{b}\left(\rho_{b}\right)$ tangent to $B C$ at $B$, and also externally to the incircle.
(b) Show that the radius of the circle $\left(X_{b}\right)$ is $\rho_{b}=\frac{(-s b)^{2}}{4 r}$.
(c) Let $X_{c}\left(\rho_{c}\right)$ be the circle tangent to $B C$ at $C$, and also externally to the incircle. Calculate the coordinates of the pedal $A^{\prime}$ of the intersection $B X_{c} \cap C X_{b}$ on the line $B C .{ }^{28}$
${ }^{26}$ This is a point on the $O I$-line of triangle $A B C$. It appears in ETC as $X_{57}$. This point divides $O I$ in the ratio $O X_{57}: O I=2 R+r: 2 R-r$.
${ }^{27}\left(a^{2}\left(3 S^{2}-S_{A A}\right): \cdots: \cdots\right)$. This point is not in the current edition of ETC.
${ }^{28}\left(0:(s-c)^{2}:(s-b)^{2}\right)$. (d) Define $B^{\prime}$ and $C^{\prime}$. Show that $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$ and find the perspector. ${ }^{29}$
### Appendices
#### The excentral triangle
The vertices of the excentral triangle of $A B C$ are the excenters $I_{a}, I_{b}, I_{c}$.
(1) Identify the following elements of the excentral triangle in terms of the elements of triangle $A B C$.
Excentral triangle $I_{a} I_{b} I_{c} \quad$ Triangle $A B C$
Orthocenter
Orthic triangle
Nine-point circle
Euler line
Circumradius
Circumcenter
Centroid Triangle $A B C$
Circumcircle
$O I$-line
$2 R$
$I^{\prime}=$ Reflection of $I$ in $O$ centroid of $I^{\prime} I N_{a}^{30}$
(2) Let $Y$ be the intersection of the circumcircle $(O)$ with the line $I_{c} I_{a}$ (other than $B$ ). Note that $Y$ is the midpoint of $I_{c} I_{a}$. The line $Y O$ intersects $C A$ at its midpoint $E$ and the circumcircle again at its antipode $Y^{\prime}$. Since $E$ is the common midpoint of the segments $Q_{c} Q_{a}$ and $Q Q_{b}$,
(i) $Y E=\frac{1}{2}\left(r_{c}+r_{a}\right)$;
(ii) $E Y^{\prime}=\frac{1}{2}\left(r_{a}-r\right)$.
${ }^{29}\left(\frac{1}{(s-a)^{2}}: \frac{1}{(s-b)^{2}}: \frac{1}{(s-c)^{2}}\right)$. This point appears in ETC as $X_{279}$. See P. Yiu, Hyacinthos, message $3359,8 / 6 / 01$. Since $Y Y^{\prime}=2 R$, we obtain the relation
$$
r_{a}+r_{b}+r_{c}=4 R+r
$$
#### Centroid of pedal triangle
We determine the centroid of the pedal triangle of $P$ by first equalizing the coordinate sums of the pedals:
$$
\begin{aligned}
& A_{[P]}=\left(0: S_{C} u+a^{2} v: S_{B} u+a^{2} w\right) \sim\left(0: b^{2} c^{2}\left(S_{C} u+a^{2} v\right): b^{2} c^{2}\left(S_{B} u+a^{2} w\right)\right) \\
& B_{[P]}=\left(S_{C} v+b^{2} u: 0: S_{A} v+b^{2} w\right) \sim\left(c^{2} a^{2}\left(S_{C} v+b^{2} u\right): 0: c^{2} a^{2}\left(S_{A} v+b^{2} w\right)\right) \\
& C_{[P]}=\left(S_{B} w+c^{2} u: S_{A} w+c^{2} v: 0\right) \sim\left(a^{2} b^{2}\left(S_{B} w+c^{2} u\right): a^{2} b^{2}\left(S_{A} w+c^{2} v\right): 0\right) .
\end{aligned}
$$
The centroid is the point
$$
\left(2 a^{2} b^{2} c^{2} u+a^{2} c^{2} S_{C} v+a^{2} b^{2} S_{B} w: b^{2} c^{2} S_{C} u+2 a^{2} b^{2} c^{2} v+a^{2} b^{2} S_{A} w: b^{2} c^{2} S_{B} u+c^{a} a^{2} S_{A} v+2 a^{2} b^{2} c^{2} w\right) .
$$
This is the same point as $P$ if and only if
$$
\begin{aligned}
2 a^{2} b^{2} c^{2} u+a^{2} c^{2} S_{C} v+a^{2} b^{2} S_{B} w & =k u \\
b^{2} c^{2} S_{C} u+2 a^{2} b^{2} c^{2} v+a^{2} b^{2} S_{A} w & =k v \\
b^{2} c^{2} S_{B} u+c^{2} a^{2} S_{A} v+2 a^{2} b^{2} c^{2} w & =k w
\end{aligned}
$$
for some $k$. Adding these equations, we obtain
$$
3 a^{2} b^{2} c^{2}(u+v+w)=k(u+v+w)
$$
If $P=(u: v: w)$ is a finite point, we must have $k=3 a^{2} b^{2} c^{2}$. The system of equations becomes
$$
\begin{aligned}
-a^{2} b^{2} c^{2} u+a^{2} c^{2} S_{C} v+a^{2} b^{2} S_{B} w & =0 \\
b^{2} c^{2} S_{C} u-a^{2} b^{2} c^{2} v+a^{2} b^{2} S_{A} w & =0 \\
b^{2} c^{2} S_{B} u+c^{2} a^{2} S_{A} v-a^{2} b^{2} c^{2} w & =0
\end{aligned}
$$
Now it it easy to see that
$$
\begin{aligned}
b^{2} c^{2} u: c^{2} a^{2} v: a^{2} b^{2} w & =\left|\begin{array}{cc}
-b^{2} & S_{A} \\
S_{A} & -c^{2}
\end{array}\right|:-\left|\begin{array}{cc}
S_{C} & S_{A} \\
S_{B} & -c^{2}
\end{array}\right|:\left|\begin{array}{cc}
S_{C} & -b^{2} \\
S_{B} & S_{A}
\end{array}\right| \\
& =b^{2} c^{2}-S_{A A}: c^{2} S_{C}+S_{A B}: S_{C A}+b^{2} S_{B} \\
& =S^{2}: S^{2}: S^{2} \\
& =1: 1: 1 .
\end{aligned}
$$
It follows that $u: v: w=a^{2}: b^{2}: c^{2}$, and $P$ is the symmedian point.
## Theorem (Lemoine)
The symmedian point is the only point which is the centroid of its own pedal triangle.
#### Perspectors associated with inscribed squares
Consider the square $A_{b} A_{c} A_{c}^{\prime} A_{b}^{\prime}$ inscribed in triangle $A B C$, with $A_{b}, A_{c}$ on $B C$. These have coordinates
$$
\begin{array}{ll}
A_{b}=\left(0: S_{C}+S: S_{B}\right), & A_{c}=\left(0: S_{C}: S_{B}+S\right), \\
A_{b}^{\prime}=\left(a^{2}: S: 0\right), & A_{c}^{\prime}=\left(a^{2}: 0: S\right) .
\end{array}
$$
Similarly, there are inscribed squares $B_{c} B_{a} B_{a}^{\prime} B_{c}^{\prime}$ and $C_{a} C_{b} C_{b}^{\prime} C_{a}^{\prime}$ on the other two sides.
Here is a number of perspective triangles associated with these squares. ${ }^{31}$ In each case, we give the definition of $A_{n}$ only.
| $n$ | $A_{n}$ | Perspector of $A_{n} B_{n} C_{n}$ |
| :--- | :--- | :--- |
| 1 | $B B_{c} \cap C C_{b}$ | orthocenter |
| 2 | $B A_{c}^{\prime} \cap C A_{b}^{\prime}$ | circumcenter |
| 3 | $B C_{a}^{\prime} \cap C B_{a}^{\prime}$ | symmedian point |
| 4 | $B_{c}^{\prime \prime} B_{a}^{\prime \prime} \cap C_{a}^{\prime \prime} C_{b}^{\prime \prime}$ | symmedian point |
| 5 | $B_{c}^{\prime} B_{a}^{\prime} \cap C_{a}^{\prime} C_{b}^{\prime}$ | $X_{493}=\left(\frac{a^{2}}{S+b^{2}}: \cdots: \cdots\right)$ |
| 6 | $C_{b} A_{b} \cap A_{c} B_{c}$ | Kiepert perspector $K(\arctan 2)$ |
| 7 | $C_{a} A_{c} \cap A_{b} B_{a}$ | Kiepert perspector $K(\arctan 2)$ |
| 8 | $C_{a} A_{c}^{\prime} \cap B_{a} A_{b}^{\prime}$ | $\left(\frac{S_{A}+S}{S_{A}}: \cdots: \cdots\right)$ |
| 9 | $C_{a}^{\prime} A_{b}^{\prime} \cap B_{a}^{\prime} A_{c}^{\prime}$ | $X_{394}=\left(a^{2} S_{A A}: b^{2} S_{B B}: c^{2} S_{C C}\right)$ |
For $A_{4}, B C A_{c}^{\prime \prime} A_{b}^{\prime \prime}, C A B_{a}^{\prime \prime} B_{c}^{\prime \prime}$ and $A B C_{b}^{\prime \prime} C_{a}^{\prime \prime}$ are the squares constructed externally on the sides of triangle $A B C$.
${ }^{31}$ K.R. Dean, Hyacinthos, message 3247, July 18, 2001.
## Chapter 5
## Circles I
### Isogonal conjugates
Let $P$ be a point with homogeneous barycentric coordinates $(x: y: z)$.
(1) The reflection of the cevian $A P$ in the bisector of angle $A$ intersects the line $B C$ at the point $X^{\prime}=\left(0: \frac{b^{2}}{y}: \frac{c^{2}}{z}\right)$.
Proof. Let $X$ be the $A$-trace of $P$, with $\angle B A P=\theta$. This is the point $X=(0: y: z)=\left(0: S_{A}-S_{\theta}:-c^{2}\right)$ in Conway's notation. It follows that $S_{A}-S_{\theta}:-c^{2}=y: z$. If the reflection of $A X$ (in the bisector of angle $A)$ intersects $B C$ at $X^{\prime}$, we have $X^{\prime}=\left(0:-b^{2}: S_{A}-S_{\theta}\right)=\left(0:-b^{2} c^{2}\right.$ : $\left.c^{2}\left(S_{A}-S_{\theta}\right)\right)=\left(0: b^{2} z: c^{2} y\right)=\left(0: \frac{b^{2}}{y}: \frac{c^{2}}{z}\right)$.
(2) Similarly, the reflections of the cevians $B P$ and $C P$ in the respective angle bisectors intersect $C A$ at $Y^{\prime}=\left(\frac{a^{2}}{x}: 0: \frac{c^{2}}{z}\right)$ and $A B$ at $Z^{\prime}=\left(\frac{a^{2}}{x}: \frac{b^{2}}{y}\right.$ : $0)$.
(3) These points $X^{\prime}, Y^{\prime}, Z^{\prime}$ are the traces of
$$
P^{*}=\left(\frac{a^{2}}{x}: \frac{b^{2}}{y}: \frac{c^{2}}{z}\right)=\left(a^{2} y z: b^{2} z x: c^{2} x y\right) .
$$
The point $P^{*}$ is called the isogonal conjugate of $P$. Clearly, $P$ is the isogonal conjugate of $P^{*}$.
#### Examples
1. The isogonal conjugate of the centroid $G$ is the symmedian point $K=$ $\left(a^{2}: b^{2}: c^{2}\right)$.
2. The incenter is its own isogonal conjugate; so are the excenters.
3. The isogonal conjugate of the orthocenter $H=\left(\frac{1}{S_{A}}: \frac{1}{S_{B}}: \frac{1}{S_{C}}\right)$ is $\left(a^{2} S_{A}: b^{2} S_{B}: c^{2} S_{C}\right)$, the circumcenter.
4. The isogonal conjugate of the Gergonne point $G_{e}=\left(\frac{1}{s-a}: \frac{1}{s-b}: \frac{1}{s-c}\right)$ is the point $\left(a^{2}(s-a): b^{2}(s-b): c^{2}(s-c)\right)$, the internal center of similitude of the circumcircle and the incircle.
5. The isogonal conjugate of the Nagel point is the external center of similitude of $(O)$ and $(I)$.
## Exercises
1. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the circumcenters of the triangles $O B C, O C A, O A B$. The triangle $A^{\prime} B^{\prime} C^{\prime}$ has perspector the isogonal conjugate of the ninepoint center. ${ }^{1}$
2. Let $P$ be a given point. Construct the circumcircles of the pedal triangles of $P$ and of $P^{*}$. What can you say about these circles and their centers?
3. The isodynamic points are the isogonal conjugates of the Fermat points. ${ }^{2}$ (a) Construct the positive isodynamic point $F_{+}^{*}$. This is a point on the line joining $O$ and $K$. How does this point divide the segment $O K$ ?
(b) Construct the pedal triangle of $F_{+}^{*}$. What can you say about this triangle?
4. Show that the isogonal conjugate of the Kiepert perspector $K(\theta)=$ $\left(\frac{1}{S_{A}+S_{\theta}}: \frac{1}{S_{B}+S_{\theta}}: \frac{1}{S_{C}+S_{\theta}}\right)$ is always on the line $O K$. How does this point divide the segment $O K$ ?
5. The perpendiculars from the vertices of $A B C$ to the corresponding sides of the pedal triangle of a point $P$ concur at the isogonal conjugate of $P$.
${ }^{1}$ This is also known as the Kosnita point, and appears in ETC as the point $X_{54}$.
${ }^{2}$ These appear in ETC as the points $X_{15}$ and $X_{16}$.
### The circumcircle as the isogonal conjugate of the line at infinity
Let $P$ be a point on the circumcircle.
(1) If $A X$ and $A P$ are symmetric with respect to the bisector of angle $A$, and $B Y, B P$ symmetric with respect to the bisector of angle $B$, then $A X$ and $B Y$ are parallel.
Proof. Suppose $\angle P A B=\theta$ and $\angle P B A=\varphi$. Note that $\theta+\varphi=C$. Since $\angle X A B=A+\theta$ and $\angle Y B A=B+\varphi$, we have $\angle X A B+\angle Y B A=180^{\circ}$ and $A X, B Y$ are parallel.
(2) Similarly, if $C Z$ and $C P$ are symmetric with respect to the bisector of angle $C$, then $C Z$ is parallel to $A X$ and $B Y$.
It follows that the isogonal conjugate of a point on the circumcircle is an infinite point, and conversely. We may therefore regard the circumcircle as the isogonal conjugate of the line at infinity. As such, the circumcircle has equation
$$
a^{2} y z+b^{2} z x+c^{2} x y=0
$$
## Exercises
1. Animate a point $P$ on the circumcircle.
(1) Construct the locus of isogonal conjugates of points on the line $O P$.
(2) Construct the isogonal conjugate $Q$ of the infinite point of the line $O P$.
The point lies on the locus in (1).
2. Animate a point $P$ on the circumcircle. Find the locus of the isotomic conjugate $P^{\bullet}{ }^{3}$
${ }^{3}$ The line $a^{2} x+b^{2} y+c^{2} z=0$. 3. Let $P$ and $Q$ be antipodal points on the circumcircle. The lines $P Q^{\bullet}$ and $Q P^{\bullet}$ joining each of these points to the isotomic conjugate of the other intersect orthogonally on the circumcircle.
4. Let $P$ and $Q$ be antipodal points on the circumcircle. What is the locus of the intersection of $P P^{\bullet}$ and $Q Q^{\bullet}$ ?
5. Let $P=(u: v: w)$. The lines $A P, B P, C P$ intersect the circumcircle again at the points
$$
\begin{aligned}
A^{(P)} & =\left(\frac{-a^{2} v w}{c^{2} v+b^{2} w}: v: w\right), \\
B^{(P)} & =\left(u: \frac{-b^{2} w u}{a^{2} w+c^{2} u}: w\right), \\
C^{(P)} & =\left(u: v: \frac{-c^{2} u v}{b^{2} u+a^{2} v}\right) .
\end{aligned}
$$
These form the vertices of the Circumcevian triangle of $P$.
(a) The circumcevian triangle of $P$ is always similar to the pedal triangle.
(b) The circumcevian triangle of the incenter is perspective with $A B C$. What is the perspector? 4
(c) The circumcevian triangle of $P$ is always perspective with the tangential triangle. What is the perspector? ${ }^{5}$
${ }^{4}$ The external center of similitude of the circumcircle and the incircle.
${ }^{5}\left(a^{2}\left(-\frac{a^{4}}{u^{2}}+\frac{b^{4}}{v^{2}}+\frac{c^{4}}{w^{2}}\right): \cdots: \cdots\right)$.
### Simson lines
Consider the pedals of a point $P=(u: v: w)$ :
$$
\begin{aligned}
& A_{[P]}=\left(0: S_{C} u+a^{2} v: S_{B} u+a^{2} w\right), \\
& B_{[P]}=\left(S_{C} v+b^{2} u: 0: S_{A} v+b^{2} w\right), \\
& C_{[P]}=\left(S_{B} w+c^{2} u: S_{A} w+c^{2} v: 0\right) .
\end{aligned}
$$
These pedals of $P$ are collinear if and only if $P$ lies on the circumcircle, since
$$
\begin{array}{rlcc}
& \left|\begin{array}{ccc}
0 & S_{C} u+a^{2} v & S_{B} u+a^{2} w \\
S_{C} v+b^{2} u & 0 & S_{A} v+b^{2} w \\
S_{B} w+c^{2} u & S_{A} w+c^{2} v & 0
\end{array}\right| \\
= & (u+v+w)\left|\begin{array}{ccc}
a^{2} & S_{C} u+a^{2} v & S_{B} u+a^{2} w \\
b^{2} & 0 & S_{A} v+b^{2} w \\
c^{2} & S_{A} w+c^{2} v & 0
\end{array}\right| \\
\vdots & \cdots \\
= & (u+v+w)\left(S_{A B}+S_{B C}+S_{C A}\right)\left(a^{2} v w+b^{2} w u+c^{2} u v\right) .
\end{array}
$$
If $P$ lies on the circumcircle, the line containing the pedals is called the Simson line $\mathrm{s}(P)$ of $P$. If we write the coordinates of $P$ in the form $\left(\frac{a^{2}}{f}: \frac{b^{2}}{g}: \frac{c^{2}}{h}\right)=\left(a^{2} g h: b^{2} h f: c^{2} f g\right)$ for an infinite point $(f: g: h)$, then
$$
\begin{aligned}
A_{[P]} & =\left(0: a^{2} S_{C} g h+a^{2} b^{2} h f: a^{2} S_{B} g h+a^{2} c^{2} f g\right) \\
& \sim\left(0: h\left(-S_{C}(h+f)+\left(S_{C}+S_{A}\right) f\right): g\left(-S_{B}(f+g)+\left(S_{A}+S_{B}\right) f\right)\right) \\
& \sim\left(0:-h\left(S_{C} h-S_{A} f\right): g\left(S_{A} f-S_{B} g\right)\right) .
\end{aligned}
$$
This becomes $A_{[P]}=\left(0:-h g^{\prime}: g h^{\prime}\right)$ if we write $\left(f^{\prime}: g^{\prime}: h^{\prime}\right)=\left(S_{B} g-S_{C} h:\right.$ $\left.S_{C} h-S_{A} f: S_{A} f-S_{B} g\right)$ for the infinite point of lines in the direction perpendicular to $(f: g: h)$. Similarly, $B_{[P]}=\left(h f^{\prime}: 0:-f h^{\prime}\right)$ and $C_{[P]}=$ $\left(-g f^{\prime}: f g^{\prime}: 0\right)$. The equation of the Simson line is
$$
\frac{f}{f^{\prime}} x+\frac{g}{g^{\prime}} y+\frac{h}{h^{\prime}} z=0 .
$$
It is easy to determine the infinite point of the Simson line:
$$
\begin{aligned}
B_{P]}-C_{[P]} & =c^{2}\left(S_{C} v+b^{2} u: 0: S_{A} v+b^{2} w\right)-b^{2}\left(S_{B} w+c^{2} u: S_{A} w+c^{2} v: 0\right) \\
& =\left(* * *:-b^{2}\left(S_{A} w+c^{2} v\right): c^{2}\left(S_{A} v+b^{2} w\right)\right) \\
& \vdots \\
& =\left(* * *: S_{C} h-S_{A} f: S_{A} f-S_{B} g\right) \\
& =\left(f^{\prime}: g^{\prime}: h^{\prime}\right) .
\end{aligned}
$$
The Simson line $\mathrm{s}(P)$ is therefore perpendicular to the line defining $P$. It passes through, as we have noted, the midpoint between $H$ and $P$, which lies on the nine-point circle.
#### Simson lines of antipodal points
Let $P$ and $Q$ be antipodal points on the circumcircle. They are isogonal conjugates of the infinite points of perpendicular lines.
Therefore, the Simson lines $\mathrm{s}(P)$ and $\mathrm{s}(Q)$ are perpendicular to each other. Since the midpoints of $H P$ and $H Q$ are antipodal on the nine-point circle, the two Simson lines intersect on the nine-point circle.
## Exercises
1. Animate a point $P$ on the circumcircle of triangle $A B C$ and trace its Simson line.
2. Let $H$ be the orthocenter of triangle $A B C$, and $P$ a point on the circumcircle. Show that the midpoint of $H P$ lies on the Simson line $\mathrm{s}(P)$ and on the nine-point circle of triangle $A B C$.
3. Let $\mathcal{L}$ be the line $\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=0$, intersecting the side lines $B C, C A$, $A B$ of triangle $A B C$ at $U, V, W$ respectively.
(a) Find the equation of the perpendiculars to $B C, C A, A B$ at $U$, $V, W$ respectively. ${ }^{6}$
(b) Find the coordinates of the vertices of the triangle bounded by these three perpendiculars. ${ }^{7}$
(c) Show that this triangle is perspective with $A B C$ at a point $P$ on the circumcircle. ${ }^{8}$
(d) Show that the Simson line of the point $P$ is parallel to $\mathcal{L}$.
### Equation of the nine-point circle
To find the equation of the nine-point circle, we make use of the fact that it is obtained from the circumcircle by applying the homothety $\mathrm{h}\left(G, \frac{1}{2}\right)$. If $P=(x: y: z)$ is a point on the nine-point circle, then the point
$Q=3 G-2 P=(x+y+z)(1: 1: 1)-2(x: y: z)=(y+z-x: z+x-y: x+y-z)$
is on the circumcircle. From the equation of the circumcircle, we obtain
$a^{2}(z+x-y)(x+y-z)+b^{2}(x+y-z)(y+z-x)+c^{2}(y+z-x)(z+x-y)=0$.
Simplifying this equation, we have
$$
0=\sum_{\text {cyclic }} a^{2}\left(x^{2}-y^{2}+2 y z-z^{2}\right)=\sum_{\text {cyclic }}\left(a^{2}-c^{2}-b^{2}\right) x^{2}+2 a^{2} y z,
$$
or
$$
\sum_{\text {cyclic }} S_{A} x^{2}-a^{2} y z=0 .
$$
${ }^{6}\left(S_{B} v+S_{C} w\right) x+a^{2} w y+a^{2} v z=0$, etc.
${ }^{7}\left(-S^{2} u^{2}+S_{A B} u v+S_{B C} v w+S_{C A} w u: b^{2}\left(c^{2} u v-S_{A} u w-S+B v w\right): c^{2}\left(b^{2} u w-S_{A} u v-\right.\right.$ $\left.S_{C} v w\right)$.
${ }^{8} P=\left(\frac{a^{2}}{-a^{2} v w+S_{B} u v+S_{C} u w}: \cdots: \cdots\right)$.
## Exercises
1. Verify that the midpoint between the Fermat points, namely, the point with coordinates
$$
\left(\left(b^{2}-c^{2}\right)^{2}:\left(c^{2}-a^{2}\right)^{2}:\left(a^{2}-b^{2}\right)^{2}\right),
$$
lies on the nine-point circle.
### Equation of a general circle
Every circle $\mathcal{C}$ is homothetic to the circumcircle by a homothety, say $\mathrm{h}(T, k)$, where $T=u A+v B+w C$ (in absolute barycentric coordinate) is a center of similitude of $\mathcal{C}$ and the circumcircle. This means that if $P(x: y: z)$ is a point on the circle $\mathcal{C}$, then
$\mathrm{h}(T, k)(P)=k P+(1-k) T \sim(x+t u(x+y+z): y+t v(x+y+z): z+t w(x+y+z))$, where $t=\frac{1-k}{k}$, lies on the circumcircle. In other words,
$$
\begin{aligned}
0= & \sum_{\text {cyclic }} a^{2}(t y+v(x+y+z))(t z+w(x+y+z)) \\
= & \sum_{\text {cyclic }} a^{2}\left(y z+t(w y+v z)(x+y+z)+t^{2} v w(x+y+z)^{2}\right) \\
= & \left(a^{2} y z+b^{2} z x+c^{2} x y\right)+t\left(\sum_{\text {cyclic }} a^{2}(w y+v z)\right)(x+y+z) \\
& +t^{2}\left(a^{2} v w+b^{2} w u+c^{2} u v\right)(x+y+z)^{2}
\end{aligned}
$$
Note that the last two terms factor as the product of $x+y+z$ and another linear form. It follows that every circle can be represented by an equation of the form
$$
a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)(p x+q y+r z)=0 .
$$
The line $p x+q y+r z=0$ is the radical axis of $\mathcal{C}$ and the circumcircle.
## Exercises
1. The radical axis of the circumcircle and the nine-point circle is the line
$$
S_{A} x+S_{B} y+S_{C} z=0 .
$$
2. The circle through the excenters has center at the reflection of $I$ in $O$, and radius $2 R$. Find its equation. ${ }^{9}$
${ }^{9} a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)(b c x+c a y+a b z)=0$.
### Appendix: Miquel Theory
#### Miquel Theorem
Let $X, Y, Z$ be points on the lines $B C, C A$, and $A B$ respectively. The three circles $A Y Z, B Z X$, and $C X Y$ pass through a common point.
#### Miquel associate
Suppose $X, Y, Z$ are the traces of $P=(u: v: w)$. We determine the equation of the circle $A Y Z$. ${ }^{10}$ Writing it in the form
$$
a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)(p x+q y+r z)=0
$$
we note that $p=0$ since it passes through $A=(1: 0: 0)$. Also, with $(x: y: z)=(u: 0: w)$, we obtain $r=-\frac{b^{2} u}{w+u}$. Similarly, with $(x: y: z)=$ $(u: v: 0)$, we obtain $q=-\frac{c^{2} u}{u+v}$. The equation of the circle
$$
\mathcal{C}_{A Y Z}: \quad \quad a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(\frac{c^{2} u}{u+v} y+\frac{b^{2} u}{w+u} z\right)=0 .
$$
Likewise, the equations of the other two circles are
$$
\mathcal{C}_{B Z X}: \quad a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(\frac{c^{2} v}{u+v} x+\frac{a^{2} v}{v+w} z\right)=0,
$$
and the one through $C, X$, and $Y$ has equation
$$
\mathcal{C}_{C X Y}: \quad a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(\frac{b^{2} w}{w+u} x+\frac{a^{2} w}{v+w} y\right)=0 .
$$
By Miquel's Theorem, the three circles intersect at a point $P^{\prime}$, which we call the Miquel associate of $P$. The coordinates of $P^{\prime}$ satisfy the equations
$$
\frac{c^{2} u}{u+v} y+\frac{b^{2} u}{w+u} z=\frac{c^{2} v}{u+v} x+\frac{a^{2} v}{v+w} z=\frac{b^{2} w}{w+u} x+\frac{a^{2} w}{v+w} y .
$$
${ }^{10}$ For the case when $X, Y, Z$ are the intercepts of a line, see J.P. Ehrmann, Steiner's theorems on the complete quadrilateral, Forum Geometricorum, forthcoming. Solving these equations, we have
$$
\begin{aligned}
P^{\prime}= & \left(\frac{a^{2}}{v+w}\left(-\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}\right),\right. \\
& : \frac{b^{2}}{w+u}\left(\frac{a^{2} v w}{v+w}-\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}\right) \\
& \left.: \frac{c^{2}}{u+v}\left(\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}-\frac{c^{2} u v}{u+v}\right)\right) .
\end{aligned}
$$
## Examples
| $P$ | Miquel associate $P^{\prime}$ |
| :---: | :---: |
| centroid | circumcenter |
| orthocenter | orthocenter |
| Gergonne point | incenter |
| incenter | $\left(\frac{a^{2}\left(a^{3}+a^{2}(b+c)-a\left(b^{2}+b c+c^{2}\right)-(b+c)\left(b^{2}+c^{2}\right)\right)}{b+c}: \cdots: \cdots\right)$ |
| Nagel Point | $\left(a\left(a^{3}+a^{2}(b+c)-a(b+c)^{2}-(b+c)(b-c)^{2}\right): \cdots: \cdots\right)$ |
#### Cevian circumcircle
The cevian circumcircle of $P$ is the circle through its traces. This has equation
$$
\left(a^{2} y z+b^{2} z x+c^{2} x y\right)-(x+y+z)(p x+q y+r z)=0,
$$
where
$$
v q+w r=\frac{a^{2} v w}{v+w}, \quad u p+w r=\frac{b^{2} w u}{w+u}, \quad u p+v q=\frac{c^{2} u v}{u+v} .
$$
Solving these equations, we have
$$
\begin{aligned}
p & =\frac{1}{2 u}\left(-\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}\right), \\
q & =\frac{1}{2 v}\left(\frac{a^{2} v w}{v+w}-\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}\right), \\
r & =\frac{1}{2 w}\left(\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}-\frac{c^{2} u v}{u+v}\right) .
\end{aligned}
$$
#### Cyclocevian conjugate
The cevian circumcircle intersects the line $B C$ at the points given by
$$
a^{2} y z-(y+z)(q y+r z)=0 .
$$
This can be rearranged as
$$
q y^{2}+\left(q+r-a^{2}\right) y z+r z^{2}=0 .
$$
The product of the two roots of $y: z$ is $\frac{r}{q}$. Since one of the roots $y: z=v: w$, the other root is $\frac{r w}{q v}$. The second intersection is therefore the point
$$
X^{\prime}=0: r w: q v=0: \frac{1}{q v}: \frac{1}{r w} .
$$
Similarly, the "second" intersections of the circle $X Y Z$ with the other two sides can be found. The cevians $A X^{\prime}, B Y^{\prime}$, and $C Z^{\prime}$ intersect at the point $\left(\frac{1}{p u}: \frac{1}{q v}: \frac{1}{r w}\right)$. We denote this by $\mathrm{c}(P)$ and call it the cyclocevian conjugate of $P$. Explicitly,
$$
\mathrm{c}(P)=\left(\frac{1}{-\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}}: \frac{1}{\frac{a^{2} v w}{v+w}-\frac{b^{2} w u}{w+u}+\frac{c^{2} u v}{u+v}}: \frac{1}{\frac{a^{2} v w}{v+w}+\frac{b^{2} w u}{w+u}-\frac{c^{2} u v}{u+v}}\right) .
$$
## Examples
1. The centroid and the orthocenter are cyclocevian conjugates, their common cevian circumcircle being the nine-point circle.
2. The cyclocevian conjugate of the incenter is the point
$$
\left(\frac{1}{a^{3}+a^{2}(b+c)-a\left(b^{2}+b c+c^{2}\right)-(b+c)\left(b^{2}+c^{2}\right)}: \cdots: \cdots\right) .
$$
## Theorem
Given a point $P$, let $P^{\prime}$ be its Miquel associate and $Q$ its cyclocevian conjugate, with Miquel associate $Q^{\prime}$.
(a) $P^{\prime}$ and $Q^{\prime}$ are isogonal conjugates.
(b) The lines $P Q$ and $P^{\prime} Q^{\prime}$ are parallel.
(c) The "second intersections" of the pairs of circles $A Y Z, A Y^{\prime} Z^{\prime} ; B Z X$, $B Z^{\prime} X^{\prime}$; and $C X Y, C X^{\prime} Y^{\prime}$ form a triangle $A^{\prime} B^{\prime} C^{\prime}$ perspective with $A B C$.
(e) The "Miquel perspector" in (c) is the intersection of the trilinear polars of $P$ and $Q$ with respect to triangle $A B C$.
## Exercises
1. For a real number $t$, we consider the triad of points
$$
X_{t}=(0: 1-t: t), \quad Y_{t}=(t: 0: 1-t), \quad Z_{t}=(1-t: t: 0)
$$
on the sides of the reference triangle.
(a) The circles $A Y_{t} Z_{t}, B Z_{t} X_{t}$ and $C X_{t} Y_{t}$ intersect at the point
$$
\begin{aligned}
M_{t}= & \left(a^{2}\left(-a^{2} t(1-t)+b^{2} t^{2}+c^{2}(1-t)^{2}\right)\right. \\
& : b^{2}\left(a^{2}(1-t)^{2}-b^{2} t(1-t)+c^{2} t^{2}\right) \\
& : c^{2}\left(a^{2} t^{2}+b^{2}(1-t)^{2}-c^{2} t(1-t)\right) .
\end{aligned}
$$
(b) Writing $M_{t}=(x: y: z)$, eliminate $t$ to obtain the following equation in $x, y, z$ :
$$
b^{2} c^{2} x^{2}+c^{2} a^{2} y^{2}+a^{2} b^{2} z^{2}-c^{4} x y-b^{4} z x-a^{4} y z=0 .
$$
(c) Show that the locus of $M_{t}$ is a circle.
(d) Verify that this circle contains the circumcenter, the symmedian point, and the two Brocard points
$$
\Omega_{\leftarrow}=\left(\frac{1}{b^{2}}: \frac{1}{c^{2}}: \frac{1}{a^{2}}\right),
$$
and
$$
\Omega_{\rightarrow}=\left(\frac{1}{c^{2}}: \frac{1}{a^{2}}: \frac{1}{b^{2}}\right)
$$
## Chapter 6
## Circles II
### Equation of the incircle
Write the equation of the incircle in the form
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)(p x+q y+r z)=0
$$
for some undetermined coefficients $p, q, r$. Since the incircle touches the side $B C$ at the point $(0: s-c: s-b), y: z=s-c: s-b$ is the only root of the quadratic equation $a^{2} y z+(y+z)(q y+r z)=0$. This means that
$$
q y^{2}+\left(q+r-a^{2}\right) y z+r z^{2}=k((s-b) y-(s-c) z)^{2}
$$
for some scalar $k$.
Comparison of coefficients gives $k=1$ and $q=(s-b)^{2}, r=(s-c)^{2}$. Similarly, by considering the tangency with the line $C A$, we obtain $p=$ $(s-a)^{2}$ and (consistently) $r=(s-c)^{2}$. It follows that the equation of the incircle is
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left((s-a)^{2} x+(s-b)^{2} y+(s-c)^{2} z\right)=0 .
$$
The radical axis with the circumcircle is the line
$$
(s-a)^{2} x+(s-b)^{2} y+(s-c)^{2} z=0 .
$$
#### The excircles
The same method gives the equations of the excircles:
$$
\begin{aligned}
& a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(s^{2} x+(s-c)^{2} y+(s-b)^{2} z\right)=0 \\
& a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left((s-c)^{2} x+s^{2} y+(s-a)^{2} z\right)=0 \\
& a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left((s-b)^{2} x+(s-a)^{2} y+s^{2} z\right)=0
\end{aligned}
$$
## Exercises
1. Show that the Nagel point of triangle $A B C$ lies on its incircle if and only if one of its sides is equal to $\frac{s}{2}$. Make use of this to design an animation picture showing a triangle with its Nagel point on the incircle.
2. (a) Show that the centroid of triangle $A B C$ lies on the incircle if and only if $5\left(a^{2}+b^{2}+c^{2}\right)=6(a b+b c+c a)$.
(b) Let $A B C$ be an equilateral triangle with center $O$, and $\mathcal{C}$ the circle, center $O$, radius half that of the incirle of $A B C$. Show that the distances from an arbitrary point $P$ on $\mathcal{C}$ to the sidelines of $A B C$ are the lengths of the sides of a triangle whose centroid is on the incircle.
### Intersection of the incircle and the nine-point circle
We consider how the incircle and the nine-point circle intersect. The intersections of the two circles can be found by solving their equations simultaneously:
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left((s-a)^{2} x+(s-b)^{2} y+(s-c)^{2} z\right)=0,
$$
$$
a^{2} y z+b^{2} z x+c^{2} x y-\frac{1}{2}(x+y+z)\left(S_{A} x+S_{B} y+S_{C} z\right)=0
$$
#### Radical axis of $(I)$ and $(N)$
Note that
$$
(s-a)^{2}-\frac{1}{2} S_{A}=\frac{1}{4}\left((b+c-a)^{2}-\left(b^{2}+c^{2}-a^{2}\right)\right)=\frac{1}{2}\left(a^{2}-a(b+c)+b c\right)=\frac{1}{2}(a-b)(a-c) .
$$
Subtracting the two equations we obtain the equation of the radical axis of the two circles:
$$
\mathcal{L}: \quad(a-b)(a-c) x+(b-a)(b-c) y+(c-a)(c-b) z=0 .
$$
We rewrite this as
$$
\frac{x}{b-c}+\frac{y}{c-a}+\frac{z}{a-b}=0 .
$$
There are two points with simple coordinates on this line:
$$
P=\left((b-c)^{2}:(c-a)^{2}:(a-b)^{2}\right),
$$
and
$$
Q=\left(a(b-c)^{2}: b(c-a)^{2}: c(a-b)^{2}\right) .
$$
Making use of these points we obtain a very simple parametrization of points on the radical axis $\mathcal{L}$, except $P$ :
$$
(x: y: z)=\left((a+t)(b-c)^{2}:(b+t)(c-a)^{2}:(c+t)(a-b)^{2}\right)
$$
for some $t$.
#### The line joining the incenter and the nine-point center
We find the intersection of the radical axis $\mathcal{L}$ and the line joining the centers $I$ and $N$. It is convenient to write the coordinates of the nine-point center in terms of $a, b, c$. Thus,
$$
N=\left(a^{2}\left(b^{2}+c^{2}\right)-\left(b^{2}-c^{2}\right)^{2}: b^{2}\left(c^{2}+a^{2}\right)-\left(c^{2}-a^{2}\right)^{2}: c^{2}\left(a^{2}+b^{2}\right)-\left(a^{2}-b^{2}\right)^{2}\right)
$$
with coordinate sum $8 S^{2}$.
${ }^{1}$ Start with $N=\left(S^{2}+S_{B C}: \cdots: \cdots\right)$ (with coordinate sum $\left.4 S^{2}\right)$ and rewrite $S^{2}+$ $S_{B C}=\cdots=\frac{1}{2}\left(a^{2}\left(b^{2}+c^{2}\right)-\left(b^{2}-c^{2}\right)^{2}\right)$. We seek a real number $k$ for which the point
$$
\begin{aligned}
& \left(a^{2}\left(b^{2}+c^{2}\right)-\left(b^{2}-c^{2}\right)^{2}+k a\right. \\
: & b^{2}\left(c^{2}+a^{2}\right)-\left(c^{2}-a^{2}\right)^{2}+k b \\
: & \left.c^{2}\left(a^{2}+b^{2}\right)-\left(a^{2}-b^{2}\right)^{2}+k c\right)
\end{aligned}
$$
on the line $I N$ also lies on the radical axis $\mathcal{L}$. With $k=-2 a b c$, we have
$$
\begin{aligned}
& a^{2}\left(b^{2}+c^{2}\right)-\left(b^{2}-c^{2}\right)^{2}-2 a^{2} b c \\
= & a^{2}(b-c)^{2}-\left(b^{2}-c^{2}\right)^{2} \\
= & (b-c)^{2}\left(a^{2}-(b+c)^{2}\right) \\
= & 4 s(a-s)(b-c)^{2},
\end{aligned}
$$
and two analogous expressions by cyclic permutations of $a, b, c$. These give the coordinates of a point on $\mathcal{L}$ with $t=-s$, and we conclude that the two lines intersect at the Feuerbach point
$$
F=\left((s-a)(b-c)^{2}:(s-b)(c-a)^{2}:(s-c)(a-b)^{2}\right) .
$$
We proceed to determine the ratio of division $I F: F N$. From the choice of $k$, we have
$$
F \sim 8 S^{2} \cdot N-2 a b c \cdot 2 s \cdot I=8 S^{2} \cdot N-4 s a b c \cdot I .
$$
This means that
$$
N F: F I=-4 s a b c: 8 S^{2}=-8 s R S: 8 S^{2}=-s R: S=R:-2 r=\frac{R}{2}:-r \text {. }
$$
The point $F$ is the external center of similitude of the nine-point circle and the incircle.
However, if a center of similitude of two circles lies on their radical axis, the circles must be tangent to each other (at that center). ${ }^{2}$
${ }^{2}$ Proof: Consider two circles of radii $p$ and $q$, centers at a distance $d$ apart. Suppose the intersection of the radical axis and the center line is at a distance $x$ from the center of the circle of radius $p$, then $x^{2}-p^{2}=(d-x)^{2}-q^{2}$. From this, $x=\frac{d^{2}+p^{2}-q^{2}}{2 d}$, and $d-x=\frac{d^{2}-p^{2}+q^{2}}{2 d}$. The division ratio is $x: d-x=d^{2}+p^{2}-q^{2}: d^{2}-p^{2}+q^{2}$. If this is equal to $p:-q$, then $p\left(d^{2}-p^{2}+q^{2}\right)+q\left(d^{2}+p^{2}-q^{2}\right)=0,(p+q)\left(d^{2}-(p-q)^{2}\right)=0$. From this $d=|p-q|$, and the circles are tangent internally.
## Feuerbach's Theorem
The nine-point circle and the incircle are tangent internally to each other at the point $F$, the common tangent being the line
$$
\frac{x}{b-c}+\frac{y}{c-a}+\frac{z}{a-b}=0 .
$$
The nine-point circle is tangent to each of the excircles externally. The points of tangency form a triangle perspective with $A B C$ at the point
$$
F^{\prime}=\left(\frac{(b+c)^{2}}{s-a}: \frac{(c+a)^{2}}{s-b}: \frac{(a+b)^{2}}{s-c}\right) .
$$
## Exercises
1. Show that $F$ and $F^{\prime}$ divide $I$ and $N$ harmonically.
2. Find the equations of the common tangent of the nine-point circle and the excircles. ${ }^{3}$
${ }^{3}$ Tangent to the $A$-excircle: $\frac{x}{b-c}+\frac{y}{c+a}-\frac{z}{a+b}=0$. 3. Apart from the common external tangent, the nine-point circle and the $A$-circle have another pair of common internal tangent, intersecting at their excenter of similitude $A^{\prime}$. Similarly define $B^{\prime}$ and $C^{\prime}$. The triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$. What is the perspector? ${ }^{4}$
4. Let $\ell$ be a diameter of the circumcircle of triangle $A B C$. Animate a point $P$ on $\ell$ and construct its pedal circle, the circle through the pedals of $P$ on the side lines. The pedal circle always passes through a fixed point on the nine-point circle.
What is this fixed point if the diameter passes through the incenter?
### The excircles
Consider the radical axes of the excircles with the circumcircle. These are the lines
$$
\begin{aligned}
& s^{2} x+(s-c)^{2} y+(s-b)^{2} z=0 \\
& (s-c)^{2} x+s^{2} y+(s-a)^{2} z=0 \\
& (s-b)^{2} x+(s-a)^{2} y+s^{2} z=0
\end{aligned}
$$
These three lines bound a triangle with vertices
The triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$ at the Clawson point ${ }^{5}$
$$
\left(\frac{a}{S_{A}}: \frac{b}{S_{B}}: \frac{c}{S_{C}}\right) .
$$
${ }^{4}$ The Feuerbach point.
${ }^{5}$ This point appears in ETC as the point $X_{19}$.
## Exercises
1. Let $A_{H}$ be the pedal of $A$ on the opposite side $B C$ of triangle $A B C$. Construct circle $B\left(A_{H}\right)$ to intersect $A B$ at $C_{b}$ and $C_{b}^{\prime}$ (so that $C_{b}^{\prime}$ in on the extension of $A B$ ), and circle $C\left(A_{H}\right)$ to intersect $A C$ at and $B_{c}$ and $B_{c}^{\prime}$ (so that $B_{c}^{\prime}$ in n o thn nutannion of $A C$ ).
(a) Let $A_{1}$ be the intersection of the lines $B_{c} C_{b}^{\prime}$ and $C_{b} B_{c}^{\prime}$. Similarly define $B_{1}$ and $C_{1}$. Show that $A_{1} B_{1} C_{1}$ is perspective with $A B C$ at the Clawson point. ${ }^{6}$
(b) Let $A_{2}=B B_{c} \cap C C_{b}, B_{2}=C C_{a} \cap A A_{c}$, and $C_{2}=A A_{b} \cap B B_{a}$. Show that $A_{2} B_{2} C_{2}$ is perspective with $A B C$. Calculate the coordinates of the perspector. 7
(c) Let $A_{3}=B B_{c}^{\prime} \cap C C_{b}^{\prime}, B_{3}=C C_{a}^{\prime} \cap A A_{c}^{\prime}$, and $C_{3}=A A_{b}^{\prime} \cap B B_{a}^{\prime}$. Show that $A_{3} B_{3} C_{3}$ is perspective with $A B C$. Calculate the coordinates of the perspector. 8
2. Consider the $B$ - and $C$-excircles of triangle $A B C$. Three of their common tangents are the side lines of triangle $A B C$. The fourth common tangent is the reflection of the line $B C$ in the line joining the excenters $I_{b}$ and $I_{c}$.
(a) Find the equation of this fourth common tangent, and write down the equations of the fourth common tangents of the other two pairs of excircles.
(b) Show that the triangle bounded by these 3 fourth common tangents is homothetic to the orthic triangle, and determine the homothetic center. ${ }^{9}$
${ }^{6}$ A.P.Hatzipolakis, Hyacinthos, message 1663, October 25, 2000.
${ }^{7} X_{278}=\left(\frac{1}{(s-a) S_{A}}: \cdots: \cdots\right)$
${ }^{8} X_{281}=\left(\frac{s-a}{S_{A}}: \cdots: \cdots\right)$
${ }^{9}$ The Clawson point. See R. Lyness and G.R. Veldkamp, Problem 682 and solution,
### The Brocard points
Consider the circle through the vertices $A$ and $B$ and tangent to the side $A C$ at the vertex $A$. Since the circle passes through $A$ and $B$, its equation is of the form
$$
a^{2} y z+b^{2} z x+c^{2} x y-r z(x+y+z)=0
$$
for some constant $r$. Since it is tangent to $A C$ at $A$, when we set $y=0$, the equation should reduce to $z^{2}=0$. This means that $r=b^{2}$ and the circle is
$$
\mathcal{C}_{A A B}: \quad a^{2} y z+b^{2} z x+c^{2} x y-b^{2} z(x+y+z)=0 .
$$
Similarly, we consider the analogous circles
$$
\mathcal{C}_{B B C}: \quad \quad a^{2} y z+b^{2} z x+c^{2} x y-c^{2} x(x+y+z)=0 .
$$
and
$$
\mathcal{C}_{C C A}: \quad \quad a^{2} y z+b^{2} z x+c^{2} x y-a^{2} y(x+y+z)=0 .
$$
These three circles intersect at the forward Brocard point
$$
\Omega_{\rightarrow}=\left(\frac{1}{c^{2}}: \frac{1}{a^{2}}: \frac{1}{b^{2}}\right) .
$$
This point has the property that
$$
\angle A B \Omega_{\rightarrow}=\angle B C \Omega_{\rightarrow}=\angle C A \Omega_{\rightarrow} .
$$
Crux Math. 9 (1983) $23-24$. In reverse orientations there are three circles $\mathcal{C}_{A B B}, \mathcal{C}_{B C C}$, and $\mathcal{C}_{C A A}$ intersecting at the backward Brocard point
$$
\Omega_{\leftarrow}=\left(\frac{1}{b^{2}}: \frac{1}{c^{2}}: \frac{1}{a^{2}}\right) .
$$
satisfying
$$
\angle B A \Omega_{\leftarrow}=\angle C B \Omega_{\leftarrow}=\angle C B \Omega_{\leftarrow} .
$$
Note from their coordinates that the two Brocard points are isogonal conjugates. This means that the 6 angles listed above are all equal. We denote the common value by $\omega$ and call this the Brocard angle of triangle $A B C$. By writing the coordinates of $\Omega_{\rightarrow}$ in Conway's notation, it is easy to see that
$$
\cot \omega=\frac{1}{2}\left(S_{A}+S_{B}+S_{C}\right) .
$$
The lines $B \Omega_{\rightarrow}$ and $C \Omega_{\leftarrow}$ intersect at $A_{-\omega}$. Similarly, we have $B_{-\omega}=$ $C \Omega_{\rightarrow} \cap A \Omega_{\leftarrow}$, and $C_{-\omega}=A \Omega_{\rightarrow} \cap B \Omega_{\leftarrow}$. Clearly the triangle $A_{-\omega} B_{-\omega} C_{-\omega}$ is perspective to $A B C$ at the point
$$
K(-\omega)=\left(\frac{1}{S_{A}-S_{\omega}}: \cdots: \cdots\right) \sim \cdots \sim\left(\frac{1}{a^{2}}: \cdots: \cdots\right),
$$
which is the isotomic conjugate of the symmedian point. ${ }^{10}$
${ }^{10}$ This is also known as the third Brocard point. It appears as the point $X_{76}$ in ETC.
## Exercises
1. The midpoint of the segment $\Omega_{\rightarrow} \Omega_{\leftarrow}$ is the Brocard midpoint ${ }^{11}$
$$
\left(a^{2}\left(b^{2}+c^{2}\right): b^{2}\left(c^{2}+a^{2}\right): c^{2}\left(a^{2}+b^{2}\right)\right) .
$$
Show that this is a point on the line $O K$.
2. The Brocard circle is the circle through the three points $A_{-\omega}, B_{-\omega}$, and $C_{-\omega}$. It has equation
$$
a^{2} y z+b^{2} z x+c^{2} x y-\frac{a^{2} b^{2} c^{2}}{a^{2}+b^{2}+c^{2}}(x+y+z)\left(\frac{x}{a^{2}}+\frac{y}{b^{2}}+\frac{z}{c^{2}}\right)=0 .
$$
Show that this circle also contains the two Brocard point $\Omega_{\rightarrow}$ and $\Omega_{\leftarrow}$, the circumcenter, and the symmedian point.
3. Let $X Y Z$ be the pedal triangle of $\Omega_{\rightarrow}$ and $X^{\prime} Y^{\prime} Z^{\prime}$ be that of $\Omega_{\leftarrow}$.
(a) Find the coordinates of these pedals.
(b) Show that $Y Z^{\prime}$ is parallel to $B C$.
${ }^{11}$ The Brocard midpoint appears in ETC as the point $X_{39}$. (c) The triangle bounded by the three lines $Y Z^{\prime}, Z X^{\prime}$ and $X Y^{\prime}$ is homothetic to triangle $A B C$. What is the homothetic center? ${ }^{12}$
(d) The triangles $Z X Y$ and $Y^{\prime} Z^{\prime} X^{\prime}$ are congruent.
### Appendix: The circle triad $(A(a), B(b), C(c))$
Consider the circle $A(a)$. This circle intersects the line $A B$ at the two points $(c+a:-a: 0),(c-a: a: 0)$, and $A C$ at $(a+b: 0:-a)$ and $(b-a: 0: a)$. It has equation
$\mathcal{C}_{a}: a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)\left(a^{2} x+\left(a^{2}-c^{2}\right) y+\left(a^{2}-b^{2}\right) z\right)=0$.
Similarly, the circles $B(b)$ and $C(c)$ have equations
$\mathcal{C}_{b}: a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)\left(\left(b^{2}-c^{2}\right) x+b^{2} y+\left(b^{2}-a^{2}\right) z\right)=0$,
and
$\mathcal{C}_{c}: a^{2} y z+b^{2} z x+c^{2} x y+(x+y+z)\left(\left(c^{2}-b^{2}\right) x+\left(c^{2}-a^{2}\right) y+c^{2} z\right)=0$.
These are called the de Longchamps circles of triangle $A B C$. The radical center $L$ of the circles is the point $(x: y: z)$ given by
$a^{2} x+\left(a^{2}-c^{2}\right) y+\left(a^{2}-b^{2}\right) z=\left(b^{2}-c^{2}\right) x+b^{2} y+\left(b^{2}-a^{2}\right) z=\left(c^{2}-b^{2}\right) x+\left(c^{2}-a^{2}\right) y+c^{2} z$.
Forming the pairwise sums of these expressions we obtain
$$
S_{A}(y+z)=S_{B}(z+x)=S_{C}(x+y) .
$$
From these,
$$
y+z: z+x: x+y=\frac{1}{S_{A}}: \frac{1}{S_{B}}: \frac{1}{S_{C}}=S_{B C}: S_{C A}: S_{A B},
$$
and
$$
x: y: z=S_{C A}+S_{A B}-S_{B C}: S_{A B}+S_{B C}-S_{C A}: S_{B C}+S_{C A}-S_{A B} .
$$
This is called the de Longchamps point of the triangle. ${ }^{13}$ It is the reflection of the orthocenter in the circumcenter, i.e., $L=2 \cdot O-H$.
${ }^{12}$ The symmedian point.
${ }^{13}$ The de Longchamps point appears as the point $X_{20}$ in ETC.
## Exercises
1. Show that the intersections of $\mathcal{C}_{b}$ and $\mathcal{C}_{c}$ are
(i) the reflection of $A$ in the midpoint of $B C$, and
(ii) the reflection $A^{\prime}$ in the perpendicular bisector of $B C$.
What are the coordinates of these points? ${ }^{14}$
2. The circle $\mathcal{C}_{a}$ intersects the circumcircle at $B^{\prime}$ and $C^{\prime}$.
3. The de Longchamps point $L$ is the orthocenter of the anticomplementary triangle, and triangle $A^{\prime} B^{\prime} C^{\prime}$ is the orthic triangle.
#### The Steiner point
The radical axis of the circumcircle and the circle $\mathcal{C}_{a}$ is the line
$$
a^{2} x+\left(a^{2}-c^{2}\right) y+\left(a^{2}-b^{2}\right) z=0 .
$$
This line intersects the side line $B C$ at point
$$
A^{\prime}=\left(0: \frac{1}{c^{2}-a^{2}}: \frac{1}{a^{2}-b^{2}}\right) .
$$
Similarly, the radical axis of $\mathcal{C}_{b}$ has $b$-intercept
$$
B^{\prime}=\left(\frac{1}{b^{2}-c^{2}}: 0: \frac{1}{a^{2}-b^{2}}\right)
$$
and that of $\mathcal{C}_{c}$ has $c$-intercept
$$
C^{\prime}=\left(\frac{1}{b^{2}-c^{2}}: \frac{1}{c^{2}-a^{2}}: 0\right) .
$$
These three points $A^{\prime}, B^{\prime}, C^{\prime}$ are the traces of the point with coordinates
$$
\left(\frac{1}{b^{2}-c^{2}}: \frac{1}{c^{2}-a^{2}}: \frac{1}{a^{2}-b^{2}}\right) \text {. }
$$
This is a point on the circumcircle, called the Steiner point. ${ }^{15}$
${ }^{14}(-1: 1: 1)$ and $A^{\prime}=\left(-a^{2}: b^{2}-c^{2}: c^{2}-b^{2}\right)$.
${ }^{15}$ This point appears as $X_{99}$ in ETC.
## Exercises
1. The antipode of the Steiner point on the circumcircle is called the Tarry point. Calculate its coordinates. ${ }^{16}$
2. Reflect the vertices $A, B, C$ in the centroid $G$ to form the points $A^{\prime}$, $B^{\prime}, C^{\prime}$ respectively. Use the five-point conic command to construct the conic through $A, B, C, A^{\prime}, B^{\prime}, C^{\prime \prime}$. This is the Steiner circumellipse. Apart from the vertices, it intersects the circumcircle at the Steiner point.
3. Use the five-point conic command to construct the conic through the vertices of triangle $A B C$, its centroid, and orthocenter. This is a rectangular hyperbola called the Kiepert hyperbola which intersect the circumcircle, apart from the vertices, at the Tarry point.
${ }^{16}\left(\frac{1}{a^{2}\left(b^{2}+c^{2}\right)-\left(b^{4}+c^{4}\right)}: \cdots: \cdots\right)$. The Tarry point appears the point $X_{98}$ in ETC.
## Chapter 7
## Circles III
### The distance formula
Let $P=u A+v B+w C$ and $Q=u^{\prime} A+v^{\prime} B+w^{\prime} C$ be given in absolute barycentric coordinates. The distance between them is given by
$$
P Q^{2}=S_{A}\left(u-u^{\prime}\right)^{2}+S_{B}\left(v-v^{\prime}\right)^{2}+S_{C}\left(w-w^{\prime}\right)^{2}
$$
Proof. Through $P$ and $Q$ draw lines parallel to $A B$ and $A C$ respectively, intersecting at $R$. The barycentric coordinates of $R$ can be determined in two ways. $R=P+h(B-C)=Q+k(A-C)$ for some $h$ and $k$. It follows that $R=u A+(v+h) B+(w-h) C=\left(u^{\prime}+k\right) A+v^{\prime} B+\left(w^{\prime}-k\right) C$, from which $h=-(v-v)^{\prime}$ and $k=u-u^{\prime}$. Applying the law of cosines to triangle $P Q R$, we have
$$
\begin{aligned}
P Q^{2}= & (h a)^{2}+(k b)^{2}-2(h a)(k b) \cos C \\
= & h^{2} a^{2}+k^{2} b^{2}-2 h k S_{C} \\
= & \left(S_{B}+S_{C}\right)\left(v-v^{\prime}\right)^{2}+\left(S_{C}+S_{A}\right)\left(u-u^{\prime}\right)^{2}+2\left(u-u^{\prime}\right)\left(v-v^{\prime}\right) S_{C} \\
= & S_{A}\left(u-u^{\prime}\right)^{2}+S_{B}\left(v-v^{\prime}\right)^{2} \\
& +S_{C}\left[\left(u-u^{\prime}\right)^{2}+2\left(u-u^{\prime}\right)\left(v-v^{\prime}\right)+\left(v-v^{\prime}\right)^{2}\right] .
\end{aligned}
$$
The result follows since
$$
\left(u-u^{\prime}\right)+\left(v-v^{\prime}\right)=(u+v)-\left(u^{\prime}+v^{\prime}\right)=(1-w)-\left(1-w^{\prime}\right)=-\left(w-w^{\prime}\right) .
$$
## The distance formula in homogeneous coordinates
If $P=(x: y: z)$ and $Q=(u: v: w)$, the distance between $P$ and $Q$ is given by
$$
|P Q|^{2}=\frac{1}{(u+v+w)^{2}(x+y+z)^{2}} \sum_{\text {cyclic }} S_{A}((v+w) x-u(y+z))^{2} .
$$
## Exercises
1. The distance from $P=(x: y: z)$ to the vertices of triangle $A B C$ are given by
$$
\begin{aligned}
A P^{2} & =\frac{c^{2} y^{2}+2 S_{A} y z+b^{2} z^{2}}{(x+y+z)^{2}}, \\
B P^{2} & =\frac{a^{2} z^{2}+2 S_{B} z x+c^{2} x^{2}}{(x+y+z)^{2}}, \\
C P^{2} & =\frac{b^{2} x^{2}+2 S_{C} x y+a^{2} y^{2}}{(x+y+z)^{2}} .
\end{aligned}
$$
2. The distance between $P=(x: y: z)$ and $Q=(u: v: w)$ can be written as
$$
|P Q|^{2}=\frac{1}{x+y+z} \cdot\left(\sum_{\text {cyclic }} \frac{c^{2} v^{2}+2 S_{A} v w+b^{2} w^{2}}{(u+v+w)^{2}} x\right)-\frac{a^{2} y z+b^{2} z x+c^{2} x y}{(x+y+z)^{2}} .
$$
3. Compute the distance between the incenter and the nine-point center $N=\left(S^{2}+S_{A}: S^{2}+S_{B}: S^{2}+S_{C}\right)$. Deduce Feuerbach's theorem by showing that this is $\frac{R}{2}-r$. Find the coordinates of the Feuerbach point $F$ as the point dividing $N I$ externally in the ratio $R:-2 r$.
### Circle equations
#### Equation of circle with center $(u: v: w)$ and radius $\rho$ :
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z) \sum_{\text {cyclic }}\left(\frac{c^{2} v^{2}+2 S_{A} v w+b^{2} w^{2}}{(u+v+w)^{2}}-\rho^{2}\right) x=0 .
$$
#### The power of a point with respect to a circle
Consider a circle $\mathcal{C}:=O(\rho)$ and a point $P$. By the theorem on intersecting chords, for any line through $P$ intersecting $\mathcal{C}$ at two points $X$ and $Y$, the product $|P X||P Y|$ of signed lengths is constant. We call this product the power of $P$ with respect to $\mathcal{C}$. By considering the diameter through $P$, we obtain $|O P|^{2}-\rho^{2}$ for the power of a point $P$ with respect to $O(\rho)$.
#### Proposition
Let $p, q, r$ be the powers of $A, B, C$ with respect to a circle $\mathcal{C}$.
(1) The equation of the circle is
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)(p x+q y+r z)=0 .
$$
(2) The center of the circle is the point
$$
\left(a^{2} S_{A}+S_{B}(r-p)-S_{C}(p-q): b^{2} S_{B}+S_{C}(p-q)-S_{A}(r-p): c^{2} S_{C}+S_{A}(q-r)-S_{B}(r-p) .\right.
$$
(3) The radius $\rho$ of the circle is given by
$$
\rho^{2}=\frac{a^{2} b^{2} c^{2}-2\left(a^{2} S_{A} p+b^{2} S_{B} q+c^{2} S_{C} r\right)+S_{A}(q-r)^{2}+S_{B}(r-p)^{2}+S_{C}(p-q)^{2}}{4 S^{2}} .
$$
## Exercises
1. Let $X, Y, Z$ be the pedals of $A, B, C$ on their opposite sides. The pedals of $X$ on $C A$ and $A B, Y$ on $A B, B C$, and $Z$ on $C A, B C$ are on a circle. Show that the equation of the circle is 1
$$
a^{2} y z+b^{2} z x+c^{2} x y-\frac{1}{4 R^{2}}(x+y+z)\left(S_{A A} x+S_{B B} y+S_{C C} z\right)=0 .
$$
${ }^{1}$ This is called the Taylor circle of triangle $A B C$. Its center is the point $X_{389}$ in ETC. This point is also the intersection of the three lines through the midpoint of each side of the orthic triangle perpendicular to the corresponding side of $A B C$. 2. Let $P=(u: v: w)$.
(a) Find the equations of the circles $A B Y$ and $A C Z$, and the coordinates of their second intersection $A^{\prime}$.
(b) Similarly define $B^{\prime}$ and $C^{\prime}$. Show that triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$. Identify the perspector. ${ }^{2}$
### Radical circle of a triad of circles
Consider three circles with equations
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(p_{i} x+q_{i} y+r_{i} z\right)=0, \quad i=1,2,3 .
$$
#### Radical center
The radical center $P$ is the point with equal powers with respect to the three circles. Its coordinates are given by the solutions of the system of equations.
$$
p_{1} x+q_{1} y+r_{1} z=p_{2} x+q_{2} y+r_{2} z=p_{3} x+q_{3} y+r_{3} z .
$$
Explicitly, if we write
$$
M=\left(\begin{array}{lll}
p_{1} & q_{1} & r_{1} \\
p_{2} & q_{2} & r_{2} \\
p_{3} & q_{3} & r_{3}
\end{array}\right)
$$
then, $P=(u: v: w)$ with $^{3}$
$$
u=\left(\begin{array}{lll}
1 & q_{1} & r_{1} \\
1 & q_{2} & r_{2} \\
1 & q_{3} & r_{3}
\end{array}\right), \quad v=\left(\begin{array}{lll}
p_{1} & 1 & r_{1} \\
p_{2} & 1 & r_{2} \\
p_{3} & 1 & r_{3}
\end{array}\right), \quad w=\left(\begin{array}{lll}
p_{1} & q_{1} & 1 \\
p_{2} & q_{2} & 1 \\
p_{3} & q_{3} & 1
\end{array}\right) .
$$
#### Radical circle
There is a circle orthogonal to each of the circles $\mathcal{C}_{i}, i=1,2,3$. The center is the radical center $P$ above, and its square radius is the negative of the common power of $P$ with respect to the circles, i.e.,
$$
\frac{a^{2} v w+b^{2} w u+c^{2} u v}{(u+v+w)^{2}}-\frac{\operatorname{det} M}{u+v+w} .
$$
${ }^{2}\left(\frac{a^{2}}{v+w}: \cdots: \cdots\right)$. See Tatiana Emelyanov, Hyacinthos, message 3309, 7/27/01.
${ }^{3}$ Proof: $p_{1} u+q_{1} v+r_{1} w=p_{2} u+q_{2} v+r_{2} w=p_{3} u+q_{3} v+r_{3} w=\operatorname{det} M$. This circle, which we call the radical circle of the given triad, has equation
$$
\sum_{\text {cyclic }}\left(c^{2} v+b^{2} w\right) x^{2}+2 S_{A} u y z-\operatorname{det}(M)(x+y+z)^{2}=0 .
$$
In standard form, it is
$$
a^{2} y z+b^{2} z x+c^{2} x y-\frac{1}{u+v+w} \cdot(x+y+z)\left(\sum_{\text {cyclic }}\left(c^{2} v+b^{2} w-\operatorname{det}(M)\right) x\right)=0 .
$$
The radical circle is real if and only if
$$
(u+v+w)\left(p_{i} u+q_{i} v+r_{i} w\right)-\left(a^{2} v w+b^{2} w u+c^{2} u v\right) \geq 0
$$
for any $i=1,2,3$.
#### The excircles
The radical center of the excircles is the point $P=(u: v: w)$ given by
$$
\begin{aligned}
u & =\left(\begin{array}{ccc}
1 & (s-c)^{2} & (s-b)^{2} \\
1 & s^{2} & (s-a)^{2} \\
1 & (s-a)^{2} & s^{2}
\end{array}\right)=\left(\begin{array}{ccc}
1 & (s-c)^{2} & (s-a)^{2} \\
0 & c(a+b) & -c(a-b) \\
0 & b(c-a) & b(c+a)
\end{array}\right) \\
& =b c(a+b)(c+a)+b c(a-b)(c-a)=2 a b c(b+c),
\end{aligned}
$$
and, likewise, $v=2 a b c(c+a)$ and $w=2 a b c(a+b)$. This is the point $(b+c: c+a: a+b)$, called the Spieker center. It is the incenter of the medial triangle.
Since, with $(u, v, w)=(b+c, c+a, a+b)$,
$$
(u+v+w)\left(s^{2} u+(s-c)^{2} v+(s-b)^{2} w\right)-\left(a^{2} v w+b^{2} w u+c^{2} u v\right)
$$
$$
\begin{aligned}
& =(a+b+c)\left(2 a b c+\sum a^{3}+\sum a^{2}(b+c)\right)-(a+b+c)\left(a b c+\sum a^{3}\right) \\
& =(a+b+c)\left(a b c+\sum a^{2}(b+c)\right),
\end{aligned}
$$
the square radius of the orthogonal circle is
$$
\frac{a b c+\sum a^{2}(b+c)}{a+b+c}=\cdots=\frac{1}{4}\left(r^{2}+s^{2}\right) .
$$
The equation of the radical circle can be written as
$$
\sum_{\text {cyclic }}(s-b)(s-c) x^{2}+a s y z=0
$$
#### The de Longchamps circle
The radical center $L$ of the circle triad $(A(a), B(b), C(c))$ is the point $(x$ : $y: z)$ given by
$$
a^{2} x+\left(a^{2}-c^{2}\right) y+\left(a^{2}-b^{2}\right) z=\left(b^{2}-c^{2}\right) x+b^{2} y+\left(b^{2}-a^{2}\right) z=\left(c^{2}-b^{2}\right) x+\left(c^{2}-a^{2}\right) y+c^{2} z .
$$
Forming the pairwise sums of these expressions we obtain
$$
S_{A}(y+z)=S_{B}(z+x)=S_{C}(x+y) .
$$
From these,
$$
y+z: z+x: x+y=\frac{1}{S_{A}}: \frac{1}{S_{B}}: \frac{1}{S_{C}}=S_{B C}: S_{C A}: S_{A B},
$$
and
$$
x: y: z=S_{C A}+S_{A B}-S_{B C}: S_{A B}+S_{B C}-S_{C A}: S_{B C}+S_{C A}-S_{A B} .
$$
This is called the de Longchamps point of the triangle. ${ }^{4}$ It is the reflection of the orthocenter in the circumcenter, i.e., $L=2 \cdot O-H$. The de Longchamps circle is the radical circle of the triad $A(a), B(b)$ and $C(c)$. It has equation
$$
a^{2} y z+b^{2} z x+c^{2} x y-(x+y+z)\left(a^{2} x+b^{2} y+c^{2} z\right)=0 .
$$
This circle is real if and only if triangle $A B C$ is obtuse - angled.
It is also orthogonal to the triad of circles $(D(A), E(B), F(C)) .{ }^{5}$
${ }^{4}$ The de Longchamps point appears as the point $X_{20}$ in ETC.
${ }^{5}$ G. de Longchamps, Sur un nouveau cercle remarquable du plan d'un triangle, Journal de Math. Spéciales, 1886, pp. 57 -60, 85 -87, 100 - 104, 126 - 134.
## Exercises
1. The radical center of the triad of circles $A\left(R_{a}\right), B\left(R_{b}\right)$, and $C\left(R_{c}\right)$ is the point
$$
2 S^{2} \cdot O-a^{2} R_{a}^{2}\left(A-A_{H}\right)-b^{2} R_{b}^{2}\left(B-B_{H}\right)-c^{2} R_{c}^{2}\left(C-C_{H}\right) .
$$
### The Lucas circles ${ }^{6}$
Consider the square $A_{b} A_{c} A_{c}^{\prime} A_{b}^{\prime}$ inscribed in triangle $A B C$, with $A_{b}, A_{c}$ on $B C$. Since this square can be obtained from the square erected externally on $B C$ via the homothety $\mathrm{h}\left(A, \frac{S}{a^{2}+S}\right)$, the equation of the circle $\mathcal{C}_{A}$ through $A, A_{b}^{\prime}$ and $A_{c}^{\prime}$ can be easily written down:
$\mathcal{C}_{A}: \quad a^{2} y z+b^{2} z x+c^{2} x y-\frac{a^{2}}{a^{2}+S} \cdot(x+y+z)\left(c^{2} y+b^{2} z\right)=0$.
Likewise if we construct inscribed squares $B_{c} B_{a} B_{a}^{\prime} B_{c}^{\prime}$ and $C_{a} C_{b} C_{b}^{\prime} C_{a}^{\prime}$ on the other two sides, the corresponding Lucas circles are
$\mathcal{C}_{B}: \quad a^{2} y z+b^{2} z x+c^{2} x y-\frac{b^{2}}{b^{2}+S} \cdot(x+y+z)\left(c^{2} x+a^{2} z\right)=0$,
and
$\mathcal{C}_{C}: \quad a^{2} y z+b^{2} z x+c^{2} x y-\frac{c^{2}}{c^{2}+S} \cdot(x+y+z)\left(b^{2} x+a^{2} y\right)=0$.
The coordinates of the radical center satisfy the equations
$$
\frac{a^{2}\left(c^{2} y+b^{2} z\right)}{a^{2}+S}=\frac{b^{2}\left(a^{2} z+c^{2} x\right)}{b^{2}+S}=\frac{c^{2}\left(b^{2} x+a^{2} y\right)}{c^{2}+S}
$$
Since this can be rewritten as
$$
\frac{y}{b^{2}}+\frac{z}{c^{2}}: \frac{z}{c^{2}}+\frac{x}{a^{2}}: \frac{x}{a^{2}}+\frac{y}{b^{2}}=a^{2}+S: b^{2}+S: c^{2}+S,
$$
it follows that
$$
\frac{x}{a^{2}}: \frac{y}{b^{2}}: \frac{z}{c^{2}}=b^{2}+c^{2}-a^{2}+S: c^{2}+a^{2}-b^{2}+S: a^{2}+b^{2}-c^{2}+S,
$$
${ }^{6}$ A.P. Hatzipolakis and P. Yiu, The Lucas circles, Amer. Math. Monthly, 108 (2001) $444-446$. and the radical center is the point
$$
\left(a^{2}\left(2 S_{A}+S\right): b^{2}\left(2 S_{B}+S\right): c^{2}\left(2 S_{C}+S\right)\right) .
$$
The three Lucas circles are mutually tangent to each other, the points of tangency being
$$
\begin{aligned}
& A^{\prime}=\left(a^{2} S_{A}: b^{2}\left(S_{B}+S\right): c^{2}\left(S_{C}+S\right)\right) \\
& B^{\prime}=\left(b^{2}\left(S_{A}+S\right): b^{2} S_{B}: c^{2}\left(S_{C}+S\right)\right) \\
& C^{\prime}=\left(a^{2}\left(S_{A}+S\right): b^{2}\left(S_{B}+S\right): c^{2} S_{C}\right)
\end{aligned}
$$
## Exercises
1. These point of tangency form a triangle perspective with $A B C$. Calculate the coordinates of the perspector. ${ }^{7}$
### Appendix: More triads of circles
1. (a) Construct the circle tangent to the circumcircle internally at $A$ and also to the side $B C$.
(b) Find the coordinates of the point of tangency with the side $B C$.
(c) Find the equation of the circle. ${ }^{8}$
(d) Similarly, construct the two other circles, each tangent internally to the circumcircle at a vertex and also to the opposite side.
(e) Find the coordinates of the radical center of the three circles. ${ }^{9}$
2. Construct the three circles each tangent to the circumcircle externally at a vertex and also to the opposite side. Identify the radical center, which is a point on the circumcircle. ${ }^{10}$
3. Let $X, Y, Z$ be the traces of a point $P$ on the side lines $B C, C A, A B$ of triangle $A B C$.
(a) Construct the three circles, each passing through a vertex of $A B C$ and tangent to opposite side at the trace of $P$.
${ }^{7}\left(a^{2}\left(S_{A}+S\right): b^{2}\left(S_{B}+S\right): c^{2}\left(S_{C}+S\right)\right.$. This point appears in ETC as $X_{371}$, and is called the Kenmotu point. It is the isogonal conjugate of the Vecten point $\left(\frac{1}{S_{A}+S}: \frac{1}{S_{B}+S}\right.$ : $\left.\frac{1}{S_{C}+S}\right)$.
${ }^{8} a^{2} y z+b^{2} z x+c^{2} x y-\frac{a^{2}}{(b+c)^{2}}(x+y+z)\left(c^{2} y+b^{2} z\right)=0$.
${ }^{9}\left(a^{2}\left(a^{2}+a(b+c)-b c\right): \cdots: \cdots\right)$.
${ }^{10} \frac{a^{2}}{b-c}: \frac{b^{2}}{c-a}: \frac{c^{2}}{a-b}$. (b) Find the equations of these three circles.
(c) The radical center of these three circles is a point independent of $P$. What is this point?
4. Find the equations of the three circles each through a vertex and the traces of the incenter and the Gergonne point on the opposite side. What is the radical center of the triad of circles? ${ }^{11}$
5. Let $P=(u: v: w)$. Find the equations of the three circles with the cevian segments $A A_{P}, B B_{P}, C C_{P}$ as diameters. What is the radical center of the triad ? ${ }^{12}$
6. Given a point $P$. The perpendicular from $P$ to $B C$ intersects $C A$ at $Y_{a}$ and $A B$ at $Z_{a}$. Similarly define $Z_{b}, X_{b}$, and $X_{c}, Y_{c}$. Show that the circles $A Y_{a} Z_{A}, B Z_{b} X_{b}$ and $C X_{c} Y_{c}$ intersect at a point on the circumcircle of $A B C .{ }^{13}$
## Exercises
1. Consider triangle $A B C$ with three circles $A\left(R_{a}\right), B\left(R_{b}\right)$, and $C\left(R_{c}\right)$. The circle $B\left(R_{b}\right)$ intersects $A B$ at $Z_{a+}=\left(R_{b}: c-R_{b}: 0\right)$ and $Z_{a-}=$ $\left(-R_{b}: c+R_{b}: 0\right)$. Similarly, $C\left(R_{c}\right)$ intersects $A C$ at $Y_{a+}=\left(R_{c}: 0\right.$ : $\left.b-R_{c}\right)$ and $Y_{a-}=\left(-R_{c}: 0: b+R_{c}\right) .{ }^{14}$
(a) Show that the centers of the circles $A Y_{a+} Z_{a+}$ and $A Y_{a-} Z_{a-}$ are symmetric with respect to the circumcenter $O$.
(b) Find the equations of the circles $A Y_{a+} Z_{a+}$ and $A Y_{a-} Z_{a-} \cdot{ }^{15}$
(c) Show that these two circles intersect at
$$
Q=\left(\frac{-a^{2}}{b R_{b}-c R_{c}}: \frac{b}{R_{b}}: \frac{-c}{R_{c}}\right)
$$
on the circumcircle.
${ }^{11}$ The external center of similitude of the circumcircle and incircle.
${ }^{12}$ Floor van Lamoen, Hyacinthos, message $214,1 / 24 / 00$.
${ }^{13}$ If $P=(u ; v: w)$, this intersection is $\left(\frac{a^{2}}{v S_{B}-w S_{C}}: \frac{b^{2}}{w S_{C}-u S_{A}}: \frac{c^{2}}{u S_{A}-v S_{B}}\right)$; it is the infinite point of the line perpendicular to HP. A.P. Hatzipolakis and P. Yiu, Hyacinthos, messages $1213,1214,1215,8 / 17 / 00$.
${ }^{14}$ A.P. Hatzipolakis, Hyacinthos, message 3408, 8/10/01.
${ }^{15} a^{2} y z+b^{2} z x+c^{2} x y-\epsilon(x+y+z)\left(c \cdot R_{b} y+b \cdot R_{c} z\right)=0$ for $\epsilon= \pm 1$. (d) Find the equations of the circles $A Y_{a+} Z_{a-}$ and $A Y_{a-} Z_{a+}$ and show that they intersect at
$$
Q^{\prime}=\left(\frac{-a^{2}}{b R_{b}+c R_{c}}: \frac{b}{R_{b}}: \frac{c}{R_{c}}\right)
$$
on the circumcircle. ${ }^{16}$
(e) Show that the line $Q Q^{\prime}$ passes through the points $\left(-a^{2}: b^{2}: c^{2}\right)$ and 17
$$
P=\left(a^{2}\left(-a^{2} R_{a}^{2}+b^{2} R_{b}^{2}+c^{2} R_{c}^{2}\right): \cdots: \cdots\right) .
$$
(f) If $W$ is the radical center of the three circles $A\left(R_{a}\right), B\left(R_{b}\right)$, and $C\left(R_{c}\right)$, then $P=(1-t) O+t \cdot W$ for
$$
t=\frac{2 a^{2} b^{2} c^{2}}{R_{a}^{2} a^{2} S_{A}+R_{b}^{2} b^{2} S_{B}+R_{c}^{2} c^{2} S_{C}} .
$$
(g) Find $P$ if $R_{a}=a, R_{b}=b$, and $R_{c}=c$. $^{18}$
(h) Find $P$ if $R_{a}=s-a, R_{b}=s-b$, and $R_{c}=s-c$. ${ }^{19}$
(i) If the three circles $A\left(R_{a}\right), B\left(R_{b}\right)$, and $C\left(R_{c}\right)$ intersect at $W=$ $(u: v: w)$, then
$$
P=\left(a^{2}\left(b^{2} c^{2} u^{2}-a^{2} S_{A} v w+b^{2} S_{B} w u+c^{2} S_{C} u v\right): \cdots: \cdots\right) .
$$
(j) Find $P$ if $W$ is the incenter. ${ }^{20}$
(k) If $W=(u: v: w)$ is on the circumcircle, then $P=Q=Q^{\prime}=W$.
${ }^{16} a^{2} y z+b^{2} z x+c^{2} x y-\epsilon(x+y+z)\left(c \cdot R_{b} y-b \cdot R_{c} z\right)=0$ for $\epsilon= \pm 1$.
${ }^{17} Q Q^{\prime}:\left(b^{2} R_{b}^{2}-c^{2} R_{c}^{2}\right) x+a^{2}\left(R_{b}^{2} y-R_{c}^{2} z\right)=0$.
${ }^{18}\left(a^{2}\left(b^{4}+c^{4}-a^{4}\right): b^{2}\left(c^{4}+a^{4}-b^{4}\right): c^{2}\left(a^{4}+b^{4}-c^{4}\right)\right)$. This point appears as $X_{22}$ in ETC.
${ }^{19}\left(\frac{a^{2}\left(a^{2}-2 a(b+c)+\left(b^{2}+c^{2}\right)\right)}{s-a}: \cdots: \cdots\right)$. This point does not appear in the current edition of ETC.
$$
{ }^{20}\left(\frac{a^{2}}{s-a}: \frac{b^{2}}{s-b}: \frac{c^{2}}{s-c}\right) .
$$
## Chapter 8
## Some Basic Constructions
### Barycentric product
Let $X_{1}, X_{2}$ be two points on the line $B C$, distinct from the vertices $B, C$, with homogeneous coordinates $\left(0: y_{1}: z_{1}\right)$ and $\left(0: y_{2}: z_{2}\right)$. For $i=1,2$, complete parallelograms $A K_{i} X_{i} H_{i}$ with $K_{i}$ on $A B$ and $H_{i}$ on $A C$. The coordinates of the points $H_{i}, K_{i}$ are
$$
\begin{array}{ll}
H_{1}=\left(y_{1}: 0: z_{1}\right), & K_{1}=\left(z_{1}: y_{1}: 0\right) \\
H_{2}=\left(y_{2}: 0: z_{2}\right), & K_{2}=\left(z_{2}: y_{2}: 0\right)
\end{array}
$$
From these,
$$
\begin{aligned}
& B H_{1} \cap C K_{2}=\left(y_{1} z_{2}: y_{1} y_{2}: z_{1} z_{2}\right), \\
& B H_{2} \cap C K_{1}=\left(y_{2} z_{1}: y_{1} y_{2}: z_{1} z_{2}\right) .
\end{aligned}
$$
Both of these points have $A$-trace $\left(0: y_{1} y_{2}: z_{1} z_{2}\right)$. This means that the line joining these intersections passes through $A$.
Given two points $P=(x: y: z)$ and $Q=(u: v: w)$, the above construction (applied to the traces on each side line) gives the traces of the point with coordinates $(x u: y v: z w)$. We shall call this point the barycentric product of $P$ and $Q$, and denote it by $P \cdot Q$.
In particular, the barycentric square of a point $P=(u: v: w)$, with coordinates $\left(u^{2}: v^{2}: w^{2}\right)$ can be constructed as follows:
(1) Complete a parallelogram $A B_{a} A_{P} C_{a}$ with $B_{a}$ on $C A$ and $C_{a}$ on $A B$.
(2) Construct $B B_{a} \cap C C_{a}$, and join it to $A$ to intersect $B C$ at $X$.
(3) Repeat the same constructions using the traces on $C A$ and $A B$ respectively to obtain $Y$ on $C A$ and $Z$ on $A B$.
Then, $X, Y, Z$ are the traces of the barycentric square of $P$.
#### Examples
(1) The Clawson point $\left(\frac{a}{S_{A}}: \frac{b}{S_{B}}: \frac{c}{S_{C}}\right)$ can be constructed as the barycentric product of the incenter and the orthocenter.
(2) The symmedian point can be constructed as the barycentric square of the incenter.
(3) If $P=(u+v+w)$ is an infinite point, its barycentric square can also be constructed as the barycentric product of $P$ and its inferior $(v+w$ : $w+u: u+v)$
$$
\begin{aligned}
P^{2} & =\left(u^{2}: v^{2}: w^{2}\right) \\
& =(-u(v+w):-v(w+u):-w(u+v)) \\
& =(u: v: w) \cdot(v+w: w+u: u+v) .
\end{aligned}
$$
#### Barycentric square root
Let $P=(u: v: w)$ be a point in the interior of triangle $A B C$, the barycentric square root $\sqrt{P}$ is the point $Q$ in the interior such that $Q^{2}=P$. This can be constructed as follows.
(1) Construct the circle with $B C$ as diameter.
(2) Construct the perpendicular to $B C$ at the trace $A_{P}$ to intersect the circle at $X .{ }^{1}$ Bisect angle $B X C$ to intersect $B C$ at $X^{\prime}$.
(3) Similarly obtain $Y^{\prime}$ on $C A$ and $Z^{\prime}$ on $A B$.
The points $X^{\prime}, Y^{\prime}, Z^{\prime}$ are the traces of the barycentric square root of $P$.
${ }^{1}$ It does not matter which of the two intersections is chosen.
## The square root of the orthocenter
Let $A B C$ be an acute angled triangle so that the orthocenter $H$ is an interior point. Let $X$ be the $A$-trace of $\sqrt{H}$. The circle through the pedals $B_{[H]}$, $C_{[H]}$ and $X$ is tangent to the side $B C$.
#### Exercises
1. Construct a point whose distances from the side lines are proportional to the radii of the excircles. ${ }^{2}$
2. Find the equation of the circle through $B$ and $C$, tangent (internally) to incircle. Show that the point of tangency has coordinates
$$
\left(\frac{a^{2}}{s-a}: \frac{(s-c)^{2}}{s-b}: \frac{(s-b)^{2}}{s-c}\right) .
$$
Construct this circle by making use of the barycentric "third power" of the Gergonne point.
3. Construct the square of an infinite point.
${ }^{2}$ This has coordindates $\left(\frac{a}{s-a}: \cdots: \cdots\right)$ and can be construced as the barycentric product of the incenter and the Gergonne point. 4. A circle is tangent to the side $B C$ of triangle $A B C$ at the $A$-trace of a point $P=(u: v: w)$ and internally to the circumcircle at $A^{\prime}$. Show that the line $A A^{\prime}$ passes through the point (au:bv:vw).
Make use of this to construct the three circles each tangent internally to the circumcircle and to the side lines at the traces of $P$.
5. Two circles each passing through the incenter $I$ are tangent to $B C$ at $B$ and $C$ respectively. A circle $\left(J_{a}\right)$ is tangent externally to each of these, and to $B C$ at $X$. Similarly define $Y$ and $Z$. Show that $X Y Z$ is perspective with $A B C$, and find the perspector. ${ }^{3}$
6. Let $P_{1}=\left(f_{1}: g_{1}: h_{1}\right)$ and $P_{2}=\left(f_{2}: g_{2}: h_{2}\right)$ be two given points. Denote by $X_{i}, Y_{i}, Z_{i}$ the traces of these points on the sides of the reference triangle $A B C$.
(a) Find the coordinates of the intersections $X_{+}=B Y_{1} \cap C Z_{2}$ and $X_{-}=B Y_{2} \cap C Z_{1} . \quad 4$
(b) Find the equation of the line $X_{+} X_{-} \cdot{ }^{5}$
(c) Similarly define points $Y_{+}, Y_{-}, Z_{+}$and $Z_{-}$. Show that the three lines $X_{+} X_{-}, Y_{+} Y_{-}$, and $Z_{+} Z_{-}$intersect at the point
$$
\left(f_{1} f_{2}\left(g_{1} h_{2}+h_{1} g_{2}\right): g_{1} g_{2}\left(h_{1} f_{2}+f_{1} h_{2}\right): h_{1} h_{2}\left(f_{1} g_{2}+g_{1} f_{2}\right)\right) .
$$
### Harmonic associates
The harmonic associates of a point $P=(u: v: w)$ are the points
$$
A^{P}=(-u: v: w), \quad B^{P}=(u:-v: w), \quad C^{P}=(u: v:-w) .
$$
The point $A^{P}$ is the harmonic conjugate of $P$ with respect to the cevian segment $A A_{P}$, i.e.,
$$
A P: P A_{P}=-A A^{P}:-A^{P} A_{P} ;
$$
similarly for $B^{P}$ and $C^{P}$. The triangle $A^{P} C^{P} C^{P}$ is called the precevian triangle of $P$. This terminology is justified by the fact that $A B C$ is the cevian triangle $P$ in $A^{P} B^{P} C^{P}$. It is also convenient to regard $P, A^{P}, B^{P}, C^{P}$ as a
${ }^{3}$ The barycentric square root of $\left(\frac{a}{s-a}: \frac{b}{s-b}: \frac{c}{s-c}\right)$. See Hyacinthos, message 3394, $8 / 9 / 01$.
${ }^{4} X_{+}=f_{1} f_{2}: f_{1} g_{2}: h_{1} f_{2} ; X_{-}=f_{1} f_{2}: g_{1} f_{2}: f_{1} h_{2}$
${ }^{5}\left(f_{1}^{2} g_{2} h_{2}-f_{2}^{2} g_{1} h_{1}\right) x-f_{1} f_{2}\left(f_{1} h_{2}-h_{1} f_{2}\right) y+f_{1} f_{2}\left(g_{1} f_{2}-f_{1} g_{2}\right) z=0$. harmonic quadruple in the sense that any three of the points constitute the harmonic associates of the remaining point.
## Examples
(1) The harmonic associates of the centroid, can be constructed as the intersection of the parallels to the side lines through their opposite vertices. They form the superior triangle of $A B C$.
(2) The harmonic associates of the incenter are the excenters.
(3) If $P$ is an interior point with square root $Q$. The harmonic associates of $Q$ can also be regarded as square roots of the same point.
#### Superior and inferior triangles
The precevian triangle of the centroid is called the superior triangle of $A B C$. If $P=(u: v: w)$, the point $(-u+v+w: u-v+w: u+v-w)$, which divides $P G$ in the ratio $3:-2$, has coordinates $(u: v: w)$ relative to the superior triangle, and is called the superior of $P$.
Along with the superior triangle, we also consider the cevian triangle of $G$ as the inferior triangle. The point $(v+w: w+u: u+v)$, which divides $P G$ in the ratio $3:-1$, has coordinates $(u: v: w)$ relative to the inferior triangle, and is called the inferior of $P$.
## Exercises
1. If $P$ is the centroid of its precevian triangle, show that $P$ is the centroid of triangle $A B C$. 2. The incenter and the excenters form the only harmonic quadruple which is also orthocentric, i.e., each one of them is the orthocenter of the triangle formed by the remaining three points.
### Cevian quotient
## Theorem
For any two points $P$ and $Q$ not on the side lines of $A B C$, the cevian triangle of $P$ and precevian triangle $Q$ are perspective. If $P=(u: v: w)$ and $Q=(x: y: z)$, the perspector is the point
$$
P / Q=\left(x\left(-\frac{x}{u}+\frac{y}{v}+\frac{z}{w}\right): y\left(\frac{x}{u}-\frac{y}{v}+\frac{z}{w}\right): z\left(\frac{x}{u}+\frac{y}{v}-\frac{z}{w}\right)\right) .
$$
## Proposition
$P /(P / Q)=Q$.
Proof. Direct verification.
This means that if $P / Q=Q^{\prime}$, then $P / Q^{\prime}=Q$.
## Exercises
1. Show that $P /(P \cdot P)=P \cdot(G / P)$.
2. Identify the following cevian quotients.
| $P$ | $Q$ | $P / Q$ |
| :--- | :--- | :--- |
| incenter | centroid | |
| incenter | symmedian point | |
| incenter | Feuerbach point | |
| centroid | circumcenter | |
| centroid | symmedian point | |
| centroid | Feuerbach point | |
| orthocenter | symmedian point | |
| orthocenter | $(a(b-c): \cdots: \cdots)$ | |
| Gergonne point | incenter | |
3. Let $P=(u: v: w)$ and $Q=\left(u^{\prime}: v^{\prime}: w^{\prime}\right)$ be two given points. If
$$
X=B_{P} C_{P} \cap A A_{Q}, \quad Y=C_{P} A_{P} \cap B B_{Q}, \quad Z=A_{P} B_{P} \cap C C_{Q},
$$
show that $A_{P} X, B_{P} Y$ and $C_{P} Z$ are concurrent. Calculate the coordinates of the intersection. ${ }^{6}$
### The Brocardians
The Brocardians of a point $P=(u: v: w)$ are the points
$$
P_{\rightarrow}=\left(\frac{1}{w}: \frac{1}{u}: \frac{1}{v}\right) \quad \text { and } \quad P_{\rightarrow}=\left(\frac{1}{v}: \frac{1}{w}: \frac{1}{u}\right)
$$
## Construction of Brocardian points
${ }^{6}\left(u u^{\prime}\left(v w^{\prime}+w v^{\prime}\right): \cdots: \cdots\right)$; see J.H. Tummers, Points remarquables, associ'es à un triangle, Nieuw Archief voor Wiskunde IV 4 (1956) 132 - 139. O. Bottema, Une construction par rapport à un triangle, ibid., IV 5 (1957) 68 - 70, has subsequently shown that this is the pole of the line $P Q$ with respect to the circumconic through $P$ and $Q$.
## Examples
(1) The Brocard points $\Omega_{\rightarrow}$ and $\Omega_{\leftarrow}$ are the Brocardians of the symmedian point $K$.
(2) The Brocardians of the incenter are called the Jerabek points:
$$
I_{\rightarrow}=\left(\frac{1}{c}: \frac{1}{a}: \frac{1}{b}\right) \quad \text { and } \quad I_{\leftarrow}=\left(\frac{1}{b}: \frac{1}{c}: \frac{1}{a}\right) .
$$
The oriented parallels through $I_{\rightarrow}$ to $B C, C A, A B$ intersect the sides $C A$, $B C, A B$ at $Y, Z, X$ such that $I_{\rightarrow} Y=I_{\rightarrow} Z=I_{\rightarrow} X$. Likewise, the parallels through $I_{\leftarrow}$ to $B C, C A, A B$ intersect the sides $A B, B C, C A$ at $Z, X, Y$ such that $I_{\leftarrow} Z=I_{\leftarrow} X=I_{\leftarrow} Y$. These 6 segments have length $\ell$ satisfying $\frac{1}{\ell}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, one half of the length of the equal parallelians drawn through $\left(-\frac{1}{a}+\frac{1}{b}+\frac{1}{c}: \cdots: \cdots\right)$.
(3) If oriented parallels are drawn through the forward Broadian point of the (positive) Fermat point $F_{+}$, and intersect the sides $C A, A B, B C$ at $X, Y, Z$ respectively, then the triangle $X Y Z$ is equilateral. ${ }^{7}$
${ }^{7}$ S. Bier, Equilateral triangles formed by oriented parallelians, Forum Geometricorum, $1(2001) 25-32$
## Chapter 9
## Circumconics
### Circumconics as isogonal transforms of lines
A circumconic is one that passes through the vertices of the reference triangle. As such it is represented by an equation of the form
$\mathcal{C}:$
$$
p y z+q z x+r x y=0
$$
and can be regarded as the isogonal transform of the line
$\mathcal{L}:$
$$
\frac{p}{a^{2}} x+\frac{q}{b^{2}} y+\frac{r}{c^{2}} z=0
$$
The circumcircle is the isogonal transform of the line at infinity. Therefore, a circumconic is an ellipse, a parabola, or a hyperbola according as its isogonal transform intersects the circumcircle at 0,1 , or 2 real points.
Apart from the three vertices, the circumconic intersects the circumcircle at the isogonal conjugate of the infinite point of the line $\mathcal{L}$ :
$$
\left(\frac{1}{b^{2} r-c^{2} q}: \frac{1}{c^{2} p-a^{2} r}: \frac{1}{a^{2} q-b^{2} p}\right) .
$$
We call this the fourth intersection of the circumconic with the circumcircle.
## Examples
(1) The Lemoine axis is the tripolar of the Lemoine (symmedian) point, the line with equation
$$
\frac{x}{a^{2}}+\frac{y}{b^{2}}+\frac{z}{c^{2}}=0 .
$$
Its isogonal transform is the Steiner circum-ellipse
$$
y z+z x+x y=0 .
$$
The fourth intersection with the circumcircle at the Steiner point 1
$$
\left(\frac{1}{b^{2}-c^{2}}: \frac{1}{c^{2}-a^{2}}: \frac{1}{a^{2}-b^{2}}\right)
$$
(1) The Euler line $\sum_{\text {cyclic }}\left(b^{2}-c^{2}\right) S_{A} x=0$ transforms into the Jerabek hyperbola
$$
\sum_{\text {cyclic }} a^{2}\left(b^{2}-c^{2}\right) S_{A} y z=0
$$
Since the Euler infinity point $=\left(S S-3 S_{B C}: S S-3 S_{C A}: S S-3 S_{A B}\right)=$ $\left(S_{C A}+S_{A B}-2 S_{B C}: \cdots: \cdots\right)$, the fourth intersection with the circumcircle is the point ${ }^{2}$
$$
\left(\frac{a^{2}}{S_{C A}+S_{A B}-2 S_{B C}}: \cdots: \cdots\right) .
$$
${ }^{1}$ The Steiner point appears as $X_{99}$ in ETC.
${ }^{2}$ This is the point $X_{74}$ in ETC. (2) The Brocard axis $O K$ has equation
$$
b^{2} c^{2}\left(b^{2}-c^{2}\right) x+c^{2} a^{2}\left(c^{2}-a^{2}\right) y+a^{2} b^{2}\left(a^{2}-b^{2}\right) z=0 .
$$
Its isogonal transform is the Kiepert hyperbola
$$
\left(b^{2}-c^{2}\right) y z+\left(c^{2}-a^{2}\right) z x+\left(a^{2}-b^{2}\right) x y=0 .
$$
The fourth intersection with the circumcircle is the Tarry point 3
$$
\left(\frac{1}{S_{B C}-S_{A A}}: \frac{1}{S_{C A}-S_{B B}}: \frac{1}{S_{A B}-S_{C C}}\right)
$$
This is antipodal to the Steiner point, since the Eule line and the Lemoine axis are perpendicular to each other. ${ }^{4}$
(4) Recall that the tangent to the nine-point circle at the Feuerbach point $F=\left((b-c)^{2}(b+c-a):(c-a)^{2}(c+a-b):(a-b)^{2}(a+b-c)\right)$ is the line
$$
\frac{x}{b-c}+\frac{y}{c-a}+\frac{z}{a-b}=0 .
$$
Applying the homothety $\mathrm{h}(G,-2)$, we obtain the line
$$
(b-c)^{2} x+(c-a)^{2} y+(a-b)^{2} z=0
$$
tangent to the point $\left(\frac{a}{b-c}: \frac{b}{c-a}: \frac{c}{a-b}\right)$ at the circumcircle. 5
The isogonal transform of this line is the parabola
$$
a^{2}(b-c)^{2} y z+b^{2}(c-a)^{2} z x+c^{2}(a-b)^{2} x y=0 .
$$
## Exercises
1. Let $P$ be a point. The first trisection point of the cevian $A P$ is the point $A^{\prime}$ dividing $A A_{P}$ in the ratio $1: 2$, i.e., $A A^{\prime}: A^{\prime} A_{P}=1: 2$. Find the locus of $P$ for which the first trisection points of the three cevians are collinear. For each such $P$, the line containing the first trisection points always passes through the centroid.
2. Show that the Tarry point as a Kiepert perspector is $K\left(-\left(\frac{\pi}{2}-\omega\right)\right)$.
${ }^{3}$ The Tarry point appears as the point $X_{98}$ in ETC.
${ }^{4}$ The Lemoine axis is the radical axis of the circumcircle and the nine-point; it is perpendicular to the Euler line joining the centers of the two circles.
${ }^{5}$ This point appears as $X_{100}$ in ETC. 3. Show that the circumconic $p y z+q z x+r x y=0$ is a parabola if and only if
$$
p^{2}+q^{2}+r^{2}-2 q r-2 r p-2 p q=0 .
$$
4. Animate a point $P$ on the circumcircle of triangle $A B C$ and draw the line $O P$.
(a) Construct the point $Q$ on the circumcircle which is the isogonal conjugate of the infinite point of $O P$.
(b) Construct the tangent at $Q$.
(c) Choose a point $X$ on the tangent line at $Q$, and construct the isogonal conjugate $X^{*}$ of $X$.
(d) Find the locus of $X^{*}$.
### The infinite points of a circum-hyperbola
Consider a line $\mathcal{L}$ intersecting the circumcircle at two points $P$ and $Q$. The isogonal transform of $\mathcal{L}$ is a circum-hyperbola $\mathcal{C}$. The directions of the asymptotes of the hyperbola are given by its two infinite points, which are the isogonal conjugates of $P$ and $Q$. The angle between them is one half of that of the arc $P Q$.
These asymptotes are perpendicular to each other if and only if $P$ and $Q$ are antipodal. In other words, the circum-hyperbola is rectangular, if and only if its isogonal transform is a diameter of the circumcircle. This is also equivalent to saying that the circum-hyperbola is rectangular if and only if it contains the orthocenter of triangle $A B C$.
## Theorem
Let $P$ and $Q$ be antipodal points on the circumcircle. The asymptotes of the rectangular circum-hyperbola which is the isogonal transform of $P Q$ are the Simson lines of $P$ and $Q$. It follows that the center of the circum-hyperbola is the intersection of these Simson lines, and is a point on the nine-point circle.
## Exercises
1. Let $P=(u: v: w)$ be a point other than the orthocenter and the vertices of triangle $A B C$. The rectangular circum-hyperbola through $P$ has equation
$$
\sum_{\text {cyclic }} u\left(S_{B} v-S_{C} w\right) y z=0
$$
### The perspector and center of a circumconic
The tangents at the vertices of the circumconic
$$
p y z+q z x+r x y=0
$$
are the lines
$$
r y+q z=0, \quad r x+p z=0, \quad q x+p y=0 .
$$
These bound the triangle with vertices
$$
(-p: q: r), \quad(p:-q: r), \quad(p: q:-r)
$$
This is perspective with $A B C$ at the point $P=(p: q: r)$, which we shall call the perspector of the circumconic.
We shall show in a later section that the center of the circumconic is the cevian quotient
$$
Q=G / P=(u(v+w-u): v(w+u-v): w(u+v-w)) .
$$
Here we consider some interesting examples based on the fact that $P=G / Q$ if $Q=G / P$. This means that the circumconics with centers $P$ and $Q$ have perspectors at the other point. The two circumconics intersect at
$$
\left(\frac{u}{v-w}: \frac{v}{w-u}: \frac{w}{u-v}\right)
$$
#### Examples
## Circumconic with center $K$
Since the circumcircle (with center $O$ ) has perspector at the symmedian point $K$, the circumconic with center $K$ has $O$ as perspector. This intersects the circumcircle at the point ${ }^{6}$
$$
\left(\frac{a^{2}}{b^{2}-c^{2}}: \frac{b^{2}}{c^{2}-a^{2}}: \frac{c^{2}}{a^{2}-b^{2}}\right) .
$$
This point can be $\mathrm{c}$ the Euler infinity p
## Circumconic with incenter as perspector
The circumconic with incenter as perspector has equation
$$
a y z+b z x+c x y=0 .
$$
This has center $G / I=(a(b+c-a): b(c+a-b): c(a+b-c))$, the Mittenpunkt. The circumconic with the incenter as center has equation
$$
a(s-a) y z+b(s-b) z x+c(s-c) x y=0 .
$$
The two intersect at the point ${ }^{7}$
$$
\left(\frac{a}{b-c}: \frac{b}{c-a}: \frac{c}{a-b}\right)
$$
which is a point on the circumcircle.
${ }^{6}$ This point appears as $X_{110}$ in ETC.
${ }^{7}$ This point appears as $X_{100}$ in ETC.
## Exercises
1. Let $P$ be the Spieker center, with coordinates $(b+c: c+a: a+b)$.
(a) Show that the circumconic with perspector $P$ is an ellipse.
(b) Find the center $Q$ of the conic. ${ }^{8}$
(c) Show that the circumconic with center $P$ (and perspector $Q$ ) is also an ellipse.
(d) Find the intersection of the two conics. ${ }^{9}$
2. If $P$ is the midpoint of the Brocard points $\Omega_{\rightarrow}$ and $\Omega_{\leftarrow}$, what is the point $Q=G / P$ ? What is the common point of the two circumconics with centers and perspectors at $P$ and $Q$ ? ${ }^{10}$
3. Let $P$ and $Q$ be the center and perspector of the Kiepert hyperbola. Why is the circumconic with center $Q$ and perspector $P$ a parabola? What is the intersection of the two conics? ${ }^{11}$
4. Animate a point $P$ on the circumcircle and construct the circumconic with $P$ as center. What can you say about the type of the conic as $P$ varies on the circumcircle?
${ }^{8} Q=(a(b+c): b(c+a): c(a+b))$. This point appears in ETC as $X_{37}$.
$9\left(\frac{b-c}{b+c}: \frac{c-a}{c+a}: \frac{a-b}{a+b}\right)$. This point does not appear in the current edition of ETC.
${ }^{10} Q=$ symmedian point of medial triangle; common point $=\left(\frac{b^{2}-c^{2}}{b^{2}+c^{2}}: \cdots: \cdots\right)$. This point does not appear in the current edition of ETC.
${ }^{11}\left(\frac{b^{2}-c^{2}}{b^{2}+c^{2}-2 a^{2}}: \cdots: \cdots\right)$. This point does not appear in the current edition of ETC. 5. Animate a point $P$ on the circumcircle and construct the circumconic with $P$ as perspector. What can you say about the type of the conic as $P$ varies on the circumcircle?
### Appendix: Ruler construction of tangent at $A$
(1) $P=A C \cap B D$;
(2) $Q=A D \cap C E$;
(3) $R=P Q \cap B E$.
Then $A R$ is the tangent at $A$.
## Chapter 10
## General Conics
### Equation of conics
#### Carnot's Theorem
Suppose a conic $\mathcal{C}$ intersect the side lines $B C$ at $X, X^{\prime}, C A$ at $Y, Y^{\prime}$, and $A B$ at $Z, Z^{\prime}$, then
$$
\frac{B X}{X C} \cdot \frac{B X^{\prime}}{X^{\prime} C} \cdot \frac{C Y}{Y A} \cdot \frac{C Y^{\prime}}{Y^{\prime} A} \cdot \frac{A Z}{Z B} \cdot \frac{A Z^{\prime}}{Z^{\prime} B}=1
$$
Proof. Write the equation of the conic as
$$
f x^{2}+g y^{2}+h z^{2}+2 p y z+2 q z x+2 r x y=0 .
$$
The intersections with the line $B C$ are the two points $\left(0: y_{1}: z_{1}\right)$ and $\left(0: y_{2}: z_{2}\right)$ satisfying
$$
g y^{2}+h z^{2}+2 p y z=0
$$
From this,
$$
\frac{B X}{X C} \cdot \frac{B X^{\prime}}{X^{\prime} C}=\frac{z_{1} z_{2}}{y_{1} y_{2}}=\frac{g}{h} .
$$
Similarly, for the other two pairs of intersections, we have
$$
\frac{C Y}{Y A} \cdot \frac{C Y^{\prime}}{Y^{\prime} A}=\frac{h}{f}, \quad \frac{A Z}{Z B} \cdot \frac{A Z^{\prime}}{Z^{\prime} B}=\frac{f}{g} .
$$
The product of these division ratios is clearly 1 .
The converse of Carnot's theorem is also true: if $X, X^{\prime}, Y, Y^{\prime}, Z, Z^{\prime}$ are points on the side lines such that
$$
\frac{B X}{X C} \cdot \frac{B X^{\prime}}{X^{\prime} C} \cdot \frac{C Y}{Y A} \cdot \frac{C Y^{\prime}}{Y^{\prime} A} \cdot \frac{A Z}{Z B} \cdot \frac{A Z^{\prime}}{Z^{\prime} B}=1,
$$
then the 6 points are on a conic.
## Corollary
If $X, Y, Z$ are the traces of a point $P$, then $X^{\prime}, Y^{\prime}, Z^{\prime}$ are the traces of another point $Q$.
#### Conic through the traces of $P$ and $Q$
Let $P=(u: v: w)$ and $Q=\left(u^{\prime}: v^{\prime}: w^{\prime}\right)$. By Carnot's theorem, there is a conic through the 6 points. The equation of the conic is
$$
\sum_{\text {cyclic }} \frac{x^{2}}{u u^{\prime}}-\left(\frac{1}{v w^{\prime}}+\frac{1}{v^{\prime} w}\right) y z=0 .
$$
## Exercises
1. Show that the points of tangency of the $A$-excircle with $A B, A C$, the $B$-excircle with $B C, A B$, and the $C$-excircle with $C A, C B$ lie on a conic. Find the equation of the conic. ${ }^{1}$
2. Let $P=(u: v: w)$ be a point not on the side lines of triangle $A B C$.
(a) Find the equation of the conic through the traces of $P$ and the midpoints of the three sides. ${ }^{2}$
(b) Show that this conic passes through the midpoints of $A P, B P$ and $C P$.
(c) For which points is the conic an ellipse, a hyperbola?
$$
\begin{aligned}
& \sum_{\mathrm{cyclic}} x^{2}+\frac{s^{2}+(s-a)^{2}}{s(s-a)} y z=0 . \\
& { }^{2} \sum_{\mathrm{cyclic}}-v w x^{2}+u(v+w) y z=0 .
\end{aligned}
$$
3. Given two points $P=(u: v: w)$ and a line $\mathcal{L}: \frac{x}{u^{\prime}}+\frac{y}{v^{\prime}}+\frac{z}{w^{\prime}}=0$, find the locus of the pole of $\mathcal{L}$ with respect to the circumconics through $P$. 3
### Inscribed conics
An inscribed conic is one tangent to the three side lines of triangle $A B C$. By Carnot's theorem, the points of tangency must either be the traces of a point $P$ (Ceva Theorem) or the intercepts of a line (Menelaus Theorem). Indeed, if the conic is non-degenerate, the former is always the case. If the conic is tangent to $B C$ at $(0: q: r)$ and to $C A$ at $(p: 0: r)$, then its equation must be
$$
\frac{x^{2}}{p^{2}}+\frac{y^{2}}{q^{2}}+\frac{z^{2}}{r^{2}}-\frac{2 y z}{q r}-\frac{2 z x}{r p}-\epsilon \frac{2 x y}{p q}=0
$$
for $\epsilon= \pm 1$. If $\epsilon=-1$, then the equation becomes
$$
\left(-\frac{x}{p}+\frac{y}{q}+\frac{z}{r}\right)^{2}=0
$$
and the conic is degenerate. The inscribed conic therefore has equation
$$
\frac{x^{2}}{p^{2}}+\frac{y^{2}}{q^{2}}+\frac{z^{2}}{r^{2}}-\frac{2 y z}{q r}-\frac{2 z x}{r p}-\frac{2 x y}{p q}=0
$$
and touches $B C$ at $(0: q: r)$. The points of tangency form a triangle perspective with $A B C$ at $(p: q: r)$, which we call the perspector of the inscribed conic.
${ }^{3}$ The conic through the traces of $P$ and $Q=\left(u^{\prime}: v^{\prime}: w\right)$; Jean-Pierre Ehrmann, Hyacinthos, message 1326, 9/1/00.
#### The Steiner in-ellipse
The Steiner in-ellipse is the inscribed conic with perspector $G$. It has equation
$$
x^{2}+y^{2}+z^{2}-2 y z-2 z x-2 x y=0 .
$$
## Exercises
1. The locus of the squares of infinite points is the Steiner in-ellipse
$$
x^{2}+y^{2}+z^{2}-2 y z-2 z x-2 x y=0 .
$$
2. Let $\mathcal{C}$ be the inscribed conic
$$
\sum_{\text {cyclic }} \frac{x^{2}}{p^{2}}-\frac{2 y z}{q r}=0,
$$
tangent to the side lines at $X=(0: q: r), Y=(p: 0: r)$, and $Z=$ $(0: p: q)$ respectively. Consider an arbitrary point $Q=(u: v: w)$.
(a) Find the coordinates of the second intersection $A^{\prime}$ of $\mathcal{C}$ with $X Q$. 4
(b) Similarly define $B^{\prime}$ and $C^{\prime}$. Show that triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$, and find the perspector. ${ }^{5}$
### The adjoint of a matrix
The adjoint of a matrix (not necessarily symmetric)
$$
M=\left(\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right)
$$
is the transpose of the matrix formed by the cofactors of $M$ :
$$
\begin{aligned}
& M^{\#}=\left(\begin{array}{ccc}
a_{22} a_{33}-a_{23} a_{32} & -a_{12} a_{33}+a_{13} a_{32} & a_{12} a_{23}-a_{22} a_{13} \\
-a_{21} a_{33}+a_{23} a_{31} & a_{11} a_{33}-a_{13} a_{31} & -a_{11} a_{23}+a_{21} a_{13} \\
a_{21} a_{32}-a_{31} a_{22} & -a_{11} a_{32}+a_{31} a_{12} & a_{11} a_{22}-a_{12} a_{21}
\end{array}\right) \\
& { }^{4}\left(\frac{4 u^{2}}{p}: q\left(\frac{u}{p}+\frac{v}{q}-\frac{w}{r}\right)^{2}: r\left(\frac{u}{p}-\frac{v}{q}+\frac{w}{r}\right)^{2}\right) . \\
& { }^{5}\left(\frac{p}{\left(-\frac{u}{p}+\frac{v}{q}+\frac{w}{r}\right)^{2}}: \cdots: \cdots\right) .
\end{aligned}
$$
## Proposition
(1) $M M^{\#}=M^{\#} M=\operatorname{det}(M) I$.
(2) $M^{\# \#}=(\operatorname{det} M) M$.
## Proposition
Let $(i, j, k)$ be a permutation of the indices $1,2,3$.
(1) If the rows of a matrix $M$ are the coordinates of three points, the line joining $P_{i}$ and $P_{k}$ has coordinates given by the $k$-th column of $M^{\#}$.
(2) If the columns of a matrix $M$ are the coordinates of three lines, the intersection of $L_{i}$ and $L_{j}$ is given by the $k$-row of $M^{\#}$.
### Conics parametrized by quadratic functions
Suppose
$$
x: y: z=a_{0}+a_{1} t+a_{2} t^{2}: b_{0}+b_{1} t+b_{2} t^{2}: c_{0}+c_{1} t+c_{2} t^{2}
$$
Elimination of $t$ gives
$$
\left(p_{1} x+q_{1} y+r_{1} z\right)^{2}-\left(p_{0} x+q_{0} y+r_{0} z\right)\left(p_{2} x+q_{2} y+r_{2} z\right)=0,
$$
where the coefficients are given by the entries of the adjoint of the matrix
$$
M=\left(\begin{array}{lll}
a_{0} & a_{1} & a_{2} \\
b_{0} & b_{1} & b_{2} \\
c_{0} & c_{1} & c_{2}
\end{array}\right),
$$
namely,
$$
M^{\#}=\left(\begin{array}{lll}
p_{0} & q_{0} & r_{0} \\
p_{1} & q_{1} & r_{1} \\
p_{2} & q_{2} & r_{2}
\end{array}\right) .
$$
This conic is nondegenerate provided $\operatorname{det}(M) \neq 0$.
#### Locus of Kiepert perspectors
Recall that the apexes of similar isosceles triangles of base angles $\theta$ constructed on the sides of triangle $A B C$ form a triangle $A^{\theta} B^{\theta} C^{\theta}$ with perspector
$$
K(\theta)=\left(\frac{1}{S_{A}+S_{\theta}}: \frac{1}{S_{B}+S_{\theta}}: \frac{1}{S_{C}+S_{\theta}}\right) .
$$
Writing $t=S_{\theta}$, and clearing denominators, we may take
$$
(x: y: z)=\left(S_{B C}+a^{2} t+t^{2}: S_{C A}+b^{2} t+t^{2}: S_{A B}+c^{2} t+t^{2}\right) .
$$
With
$$
M=\left(\begin{array}{ccc}
S_{B C} & a^{2} & 1 \\
S_{C A} & b^{2} & 1 \\
S_{A B} & c^{2} & 1
\end{array}\right)
$$
we have
$$
M^{\#}=\left(\begin{array}{ccc}
b^{2}-c^{2} & c^{2}-a^{2} & a^{2}-b^{2} \\
-S_{A}\left(b^{2}-c^{2}\right) & -S_{B}\left(c^{2}-a^{2}\right) & -S_{C}\left(a^{2}-b^{2}\right) \\
S_{A A}\left(b^{2}-c^{2}\right) & S_{B B}\left(c^{2}-a^{2}\right) & S_{C C}\left(a^{2}-b^{2}\right)
\end{array}\right)
$$
Writing $u=\left(b^{2}-c^{2}\right) x, v=\left(c^{2}-a^{2}\right) y$, and $w=\left(a^{2}-b^{2}\right) z$, we have
$$
\left(S_{A} u+S_{B} v+S_{C} w\right)^{2}-(u+v+w)\left(S_{A A} u+S_{B B} v+S_{C C} w\right)=0,
$$
which simplifies into
$$
0=\sum_{\text {cyclic }}\left(2 S_{B C}-S_{B B}-S_{C C}\right) v w=-\sum_{\text {cyclic }}\left(b^{2}-c^{2}\right)^{2} v w .
$$
In terms of $x, y, z$, we have, after deleting a common factor $-\left(a^{2}-b^{2}\right)\left(b^{2}-\right.$ $\left.c^{2}\right)\left(c^{2}-a^{2}\right)$,
$$
\sum_{\text {cyclic }}\left(b^{2}-c^{2}\right) y z=0 .
$$
This is the circum-hyperbola which is the isogonal transform of the line
$$
\sum_{\text {cyclic }} b^{2} c^{2}\left(b^{2}-c^{2}\right) x=0 .
$$
### The matrix of a conic
#### Line coordinates
In working with conics, we shall find it convenient to use matrix notations. We shall identify the homogeneous coordinates of a point $P=(x: y: z)$ with the row matrix $\left(\begin{array}{lll}x & y & z\end{array}\right)$, and denote it by the same $P$. A line $\mathcal{L}$ with equation $p x+q y+r z=0$ is represented by the column matrix
$$
L=\left\{\begin{array}{c}
p \\
q \\
r
\end{array}\right\}
$$
(so that $P L=0$ ). We shall call $L$ the line coordinates of $\mathcal{L}$.
#### The matrix of a conic
A conic given by a quadratic equation
$$
f x^{2}+g y^{2}+h z^{2}+2 p y z+2 q z x+2 r x y=0
$$
can be represented by in matrix form $P M P^{t}=0$, with
$$
M=\left(\begin{array}{lll}
f & r & q \\
r & g & p \\
q & p & h
\end{array}\right)
$$
We shall denote the conic by $\mathcal{C}(M)$.
#### Tangent at a point
Let $P$ be a point on the conic $\mathcal{C}$. The tangent at $P$ is the line $M P^{t}$.
### The dual conic
#### Pole and polar
The polar of a point $P$ (with respect to the conic $\mathcal{C}(M)$ ) is the line $M P^{t}$, and the pole of a line $L$ is the point $L^{t} M^{\#}$.
Conversely, if $L$ intersects a conic $\mathcal{C}$ at two points $P$ and $Q$, the pole of $L$ with respect to $\mathcal{C}$ is the intersection of the tangents at $P$ and $Q$.
## Exercises
1. A conic is self-polar if each vertex is the pole of its opposite side. Show that the matrix of a self-polar conic is a diagonal matrix.
2. If $P$ lies on the polar of $Q$, then $Q$ lies on the polar of $P$.
#### Condition for a line to be tangent to a conic
A line $L: p x+q y+r z=0$ is tangent to the conic $\mathcal{C}(M)$ if and only if $L^{t} M^{\#} L=0$. If this condition is satisfied, the point of tangency is $L^{t} M^{\#}$.
#### The dual conic
Let $M$ be the symmetric matrix
$$
\left(\begin{array}{lll}
f & r & q \\
r & g & p \\
q & p & h
\end{array}\right)
$$
The dual conic of $\mathcal{C}=\mathcal{C}(M)$ is the conic represented by the adjoint matrix
$$
M^{\#}=\left(\begin{array}{ccc}
g h-p^{2} & p q-r h & r p-g q \\
p q-h r & h f-q^{2} & q r-f p \\
r p-g q & q r-f p & f g-r^{2}
\end{array}\right)
$$
Therefore, a line $L: p x+q y+r z=0$ is tangent to $\mathcal{C}(M)$ if and only if the point $L^{t}=(p: q: r)$ is on the dual conic $\mathcal{C}\left(M^{\#}\right)$.
#### The dual conic of a circumconic
The dual conic of the circumconic $p y z+q z x+r x y=0$ (with perspector $P=(p: q: r))$ is the inscribed conic
$$
\sum_{\text {cyclic }}-p^{2} x^{2}+2 q r y z=0
$$
with perspector $P^{\bullet}=\left(\frac{1}{p}: \frac{1}{q}: \frac{1}{r}\right)$. The center is the point $(q+r: r+p: p+q)$.
## Exercises
1. The polar of $(u: v: w)$ with respect to the circumconic $p y z+q z x+$ $r x y=0$ is the line
$$
p(w y+v z)+q(u z+w x)+r(v x+u y)=0 .
$$
2. Find the equation of the dual conic of the incircle. Deduce Feuerbach's theorem by showing that the radical axis of the nine-point circle and the incircle, namely, the line
$$
\frac{x}{b-c}+\frac{y}{c-a}+\frac{z}{a-b}=0
$$
is tangent to the incircle. ${ }^{6}$
3. Show that the common tangent to the incircle and the nine-point circle is also tangent to the Steiner in-ellipse. Find the coordinates of the point of tangency. 7
4. Let $P=(u: v: w)$ and $Q=\left(u^{\prime}: v^{\prime}: w^{\prime}\right)$ be two given points. If
$$
X=B_{P} C_{P} \cap A A_{Q}, \quad Y=C_{P} A_{P} \cap B B_{Q}, \quad Z=A_{P} B_{P} \cap C C_{Q},
$$
show that $A_{P} X, B_{P} Y$ and $C_{P} Z$ are concurrent at the pole of $P Q$ with respect to the circumconic through $P$ and $Q .{ }^{8}$
5. The tangents at the vertices to the circumcircle of triangle $A B C$ intersect the side lines $B C, C A, A B$ at $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. The second tagents from $A^{\prime}, B^{\prime}, C^{\prime}$ to the circumcircle have points of tangency $X$, $Y, Z$ respectively. Show that $X Y Z$ is perspective with $A B C$ and find the perspector. ${ }^{9}$
${ }^{6} \sum_{\text {cyclic }}(s-a) y z=0$.
${ }^{7}\left((b-c)^{2}:(c-a)^{2}:(a-b)^{2}\right)$. This point appears as $X_{1086}$ in ETC.
${ }^{8} \mathrm{O}$. Bottema, Une construction par rapport à un triangle, Nieuw Archief voor Wiskunde, IV 5 (1957) 68-70.
${ }^{9}\left(a^{2}\left(b^{4}+c^{4}-a^{4}\right): \cdots: \cdots\right)$. This is a point on the Euler line. It appears as $X_{22}$ in ETC. See D.J. Smeenk and C.J. Bradley, Problem 2096 and solution, Crux Mathematicorum, 21 (1995) 344; $22(1996) 374-375$.
### The type, center and perspector of a conic
#### The type of a conic
The conic $\mathcal{C}(M)$ is an ellipse, a parabola, or a hyperbola according as the characteristic $G M^{\#} G$ is positive, zero, or negative.
Proof. Setting $z=-(x+y)$, we reduce the equation of the conic into
$$
(h+f-2 q) x^{2}+2(h-p-q+r) x y+(g+h-2 p) y^{2}=0 .
$$
This has discriminant
$$
\begin{aligned}
& (h-p-q+r)^{2}-(g+h-2 p)(h+f-2 q) \\
= & h^{2}-(g+h)(h+f)-2 h(p+q-r) \\
& +2(h+f) p+2(g+h) q+(p+q-r)^{2}+4 p q \\
= & -(f g+g h+h f)+2(f p+g q+h r)+\left(p^{2}+q^{2}+r^{2}-2 p q-2 q r-2 r p\right)
\end{aligned}
$$
which is the negative of the sum of the entries of $M^{\#}$. From this the result follows.
#### The center of a conic
The center of a conic is the pole of the line at infinity. As such, the center of $\mathcal{C}(M)$ has coordinates $G M^{\#}$, formed by the column sums of $M^{\#}$ :
$$
(p(q+r-p)-(q g+r h)+g h: q(r+p-q)-(r h+p f)+h f: r(p+q-r)-(p f+q g)+f g)
$$
#### The perspector of a conic
## Theorem (Conway)
Let $\mathcal{C}=\mathcal{C}(M)$ be a nondegenerate, non-self-polar conic. The triangle formed by the polars of the vertices is perspective with $A B C$, and has perspector $(p: q: r)$.
Proof. Since the polars are represented by the columns of $M^{\#}$, their intersections are represented by the rows of $M^{\# \#}=(\operatorname{det} M) M$. The result follows since $\operatorname{det} M \neq 0$.
The point $(p: q: r)$ is called the perspector of the conic $\mathcal{C}(M)$.
## Proposition
The center of the inscribed conic with perspector $P$ is the inferior of $P^{\bullet}$.
Proof. The inscribed conic with perspector $P$ has equation
$$
\sum_{\text {cyclic }} \frac{x^{2}}{p^{2}}-\frac{2 y z}{q r}=0 .
$$
## Exercises
1. Let $(f: g: h)$ be an infinite point. What type of conic does the equation
$$
\frac{a^{2} x^{2}}{f}+\frac{b^{2} y^{2}}{g}+\frac{c^{2} z^{2}}{h}=0
$$
represent? ${ }^{10}$
2. Find the perspector of the conic through the traces of $P$ and $Q$.
3. Find the perspector of the conic through the 6 points of tangency of the excircles with the side lines. ${ }^{11}$
4. A circumconic is an ellipse, a parabola or a hyperbola according as the perspector is inside, on, or outside the Steiner in-ellipse.
5. Let $\mathcal{C}$ be a conic tangent to the side lines $A B$ and $A C$ at $B$ and $C$ respectively.
(a) Show that the equation of $\mathcal{C}$ is of the form $x^{2}-k y z=0$ for some $k$.
(b) Show that the center of the conic lies on the $A$-median.
(c) Construct the parabola in this family as a five-point conic. ${ }^{12}$
${ }^{10}$ Parabola.
${ }^{11}\left(\frac{a^{2}+(b+c)^{2}}{b+c-a}: \cdots: \cdots\right)$. This points appears in ETC as $X_{388}$.
${ }^{12}$ The parabola has equation $x^{2}-4 y z=0$. (d) Design an animation of the conic as its center traverses the $A$ median. 13
6. Prove that the locus of the centers of circumconics through $P$ is the conic through the traces of $P$ and the midpoints of the sides. ${ }^{14}$
${ }^{13}$ If the center is $(t: 1: 1)$, then the conic contains $(t:-2: t)$.
${ }^{14}$ Floor van Lamoen and Paul Yiu, Conics loci associated with conics, Forum Geometricorum, forthcoming.
## Chapter 11
## Some Special Conics
### Inscribed conic with prescribed foci
#### Theorem
The foci of an inscribed central conic are isogonal conjugates.
Proof. Let $F_{1}$ and $F_{2}$ be the foci of a conic, and $T_{1}, T_{2}$ the points of tangency from a point $P$. Then $\angle F_{1} P T_{1}=\angle F_{2} P T_{2}$. Indeed, if $Q_{1}, Q_{2}$ are the pedals of $F_{1}, F_{2}$ on the tangents, the product of the distances $F_{1} Q_{1}$ and $F_{2} Q_{2}$ to the tangents is constant, being the square of the semi-minor axis.
Given a pair of isogonal conjugates, there is an inscribed conic with foci at the two points. The center of the conic is the midpoint of the segment.
#### The Brocard ellipse
$$
\sum_{\text {cyclic }} b^{4} c^{4} x^{2}-2 a^{4} b^{2} c^{2} y z=0
$$
The Brocard ellipse is the inscribed ellipse with the Brocard points
$$
\begin{aligned}
& \Omega_{\rightarrow}=\left(a^{2} b^{2}: b^{2} c^{2}: c^{2} a^{2}\right), \\
& \Omega_{\leftarrow}=\left(c^{2} a^{2}: a^{2} b^{2}: b^{2} c^{2}\right) .
\end{aligned}
$$
Its center is the Brocard midpoint
$$
\left(a^{2}\left(b^{2}+c^{2}\right): b^{2}\left(c^{2}+a^{2}\right): c^{2}\left(a^{2}+b^{2}\right)\right),
$$
which is the inferior of $\left(b^{2} c^{2}: c^{2} a^{2}: a^{2} b^{2}\right)$, the isotomic conjugate of the symmedian point. It follows that the perspector is the symmedian point.
## Exercises
1. Show that the equation of the Brocard ellipse is as given above.
2. The minor auxiliary circle is tangent to the nine-point circle. ${ }^{1}$ What is the point of tangency? ${ }^{2}$
#### The de Longchamps ellipse ${ }^{3}$
$$
\sum_{\text {cyclic }} b^{2} c^{2}(b+c-a) x^{2}-2 a^{3} b c y z=0,
$$
The de Longchamps ellipse is the conic through the traces of the incenter $I$, and has center at $I$.
## Exercises
1. Given that the equation of the conic is show that it is always an ellipse.
2. By Carnot's theorem, the "second" intersections of the ellipse with the side lines are the traces of a point $P$. What is this point? ${ }^{4}$
3. The minor axis is the ellipse is along the line $O I$. What are the lengths of the semi-major and semi-minor axes of the ellipse? ${ }^{5}$
${ }^{1} \mathrm{~V}$. Thébault, Problem 3857, American Mathematical Monthly, APH,205.
${ }^{2}$ Jean-Pierre Ehrmann, Hyacinthos, message 209, 1/22/00.
${ }^{3}$ E. Catalan, Note sur l'ellipse de Longchamps, Journal Math. Spéciales, IV 2 (1893) $28-30$.
$$
\begin{aligned}
& 4\left(\frac{a}{s-a}: \frac{b}{s-b}: \frac{c}{s-c}\right) . \\
& 5 \frac{R}{2} \text { and } r
\end{aligned}
$$
#### The Lemoine ellipse
Construct the inscribed conic with foci $G$ and $K$.
Find the coordinates of the center and the perspector.
The points of tangency with the side lines are the traces of the $G$ symmedians of triangles $G B C, G C A$, and $G A B$.
#### The inscribed conic with center $N$
This has foci $O$ and $H$. The perspector is the isotomic conjugate of the circumcenter. It is the envelope of the perpendicular bisectors of the segments joining $H$ to a point on the circumcircle. The major auxiliary circle is the nine-point circle.
## Exercises
1. Show that the equation of the Lemoine ellipse is
$$
\sum_{\text {cyclic }} m_{a}^{4} x^{2}-2 m_{b}^{2} m_{c}^{2} y z=0
$$
where $m_{a}, m_{b}, m_{c}$ are the lengths of the medians of triangle $A B C$.
### Inscribed parabola
Consider the inscribed parabola tangent to a given line, which we regard as the tripolar of a point $P=(u: v: w)$. Thus, $\ell: \frac{x}{u}+\frac{y}{v}+\frac{z}{w}=0$. The dual conic is the circumconic passes through the centroid $(1: 1: 1)$ and $P^{\bullet}=\left(\frac{1}{u}: \frac{1}{v}: \frac{1}{w}\right)$. It is the circumconic
$\mathcal{C} \#$
$$
\frac{v-w}{x}+\frac{w-u}{y}+\frac{u-v}{z}=0 .
$$
The inscribed parabola, being the dual of $\mathcal{C}^{\#}$, is
$$
\sum_{\text {cyclic }}-(v-w)^{2} x^{2}+2(w-u)(u-v) y z=0
$$
The perspector is the isotomic conjugate of that of its dual. This is the point
$$
\left(\frac{1}{v-w}: \frac{1}{w-u}: \frac{1}{u-v}\right)
$$
on the Steiner circum-ellipse.
The center of the parabola is the infinite point $(v-w: w-u: u-v)$. This gives the direction of the axis of the parabola. It can also be regarded the infinite focus of the parabola. The other focus is the isogonal conjugate
$$
\frac{a^{2}}{v-w}: \frac{b^{2}}{w-u}: \frac{c^{2}}{u-v}
$$
on the circumcircle.
The axis is the line through this point parallel to $u x+v y+w z=0$. The intersection of the axis with the parabola is the vertex
$$
\left(\frac{\left(S_{B}(w-u)-S_{C}(u-v)\right)^{2}}{v-w}: \cdots: \cdots\right) .
$$
The directrix, being the polar of the focus, is the line
$$
S_{A}(v-w) x+S_{B}(w-u) y+S_{C}(u-v) z=0 .
$$
This passes through the orthocenter, and is perpendicular to the line
$$
u x+v y+w z=0 .
$$
It is in fact the line of reflections of the focus. The tangent at the vertex is the Simson line of the focus.
Where does the parabola touch the given line?
$$
\left(u^{2}(v-w): v^{2}(w-u): w^{2}(u-v)\right),
$$
the barycentric product of $P$ and the infinite point of its tripolar, the given tangent, or equivalently the barycentric product of the infinite point of the tangent and its tripole.
## Exercises
1. Animate a point $P$ on the Steiner circum-ellipse and construct the inscribed parabola with perspector $P$.
### Some special conics
#### The Steiner circum-ellipse $x y+y z+z x=0$
Construct the Steiner circum-ellipse which has center at the centroid $G$.
The fourth intersection with the circumcircle is the Steiner point, which has coordinates
$$
\left(\frac{1}{b^{2}-c^{2}}: \frac{1}{c^{2}-a^{2}}: \frac{1}{a^{2}-b^{2}}\right)
$$
Construct this point as the isotomic conjugate of an infinite point.
The axes of the ellipse are the bisectors of the angle $K G S .{ }^{6}$ Construct these axes, and the vertices of the ellipse.
Construct the foci of the ellipse. ${ }^{7}$
These foci are called the Bickart points. Each of them has the property that three cevian segments are equal in length. ${ }^{8}$
#### The Steiner in-ellipse $\sum_{\text {cyclic }} x^{2}-2 y z=0$
## Exercises
1. Let $\mathcal{C}$ be a circumconic through the centroid $G$. The tangents at $A$, $B, C$ intersect the sidelines $B C, C A, A B$ at $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. Show that the line $A^{\prime} B^{\prime} C^{\prime}$ is tangent to the Steiner in-ellipse at the center of $\mathcal{C} .9$
${ }^{6}$ J.H. Conway, Hyacinthos, message 1237, 8/18/00.
${ }^{7}$ The principal axis of the Steiner circum-ellipse containing the foci is the least square line for the three vertices of the triangle. See F. Gremmen, Hyacinthos, message 260, $2 / 1 / 00$.
${ }^{8}$ O. Bottema, On some remarkable points of a triangle, Nieuw Archief voor Wiskunde, 19 (1971) 46 - 57; J.R. Pounder, Equal cevians, Crux Mathematicorum, 6 (1980) 98 - 104; postscript, ibid. $239-240$.
${ }^{9}$ J.H. Tummers, Problem 32, Wiskundige Opgaven met de Oplossingen, 20-1 (1955) $31-32$.
#### The Kiepert hyperbola $\sum_{\text {cyclic }}\left(b^{2}-c^{2}\right) y z=0$
The asymptotes are the Simson lines of the intersections of the Brocard axis $O K$ with the circumcircle. ${ }^{10}$ These intersect at the center which is on the nine-point circle. An easy way to construct the center as the intersection of the nine-point circle with the pedal circle of the centroid, nearer to the orthocenter. ${ }^{11}$
## Exercises
1. Find the fourth intersection of the Kiepert hyperbola with the circumcircle, and show that it is antipodal to the Steiner point. ${ }^{12}$
2. Show that the Kiepert hyperbola is the locus of points whose tripolars are perpendicular to the Euler line. ${ }^{13}$
3. Let $A^{\prime} B^{\prime} C^{\prime}$ be the orthic triangle. The Brocard axes (the line joining the circumcenter and the symmedian point) of the triangles $A B^{\prime} C^{\prime}$, $A^{\prime} B C^{\prime}$, and $A^{\prime} B^{\prime} C$ intersect at the Kiepert center. ${ }^{14}$
#### The superior Kiepert hyperbola $\sum_{\text {cyclic }}\left(b^{2}-c^{2}\right) x^{2}=0$
Consider the locus of points $P$ for which the three points $P, P^{\bullet}$ (isotomic conjugate) and $P^{*}$ (isogonal conjugate) are collinear. If $P=(x: y: z)$, then we require
$$
\begin{aligned}
0 & =\left|\begin{array}{ccc}
x & y & z \\
y z & z x & x y \\
a^{2} y z & b^{2} z x & c^{2} x y
\end{array}\right| \\
& =a^{2} x y z\left(y^{2}-z^{2}\right)+b^{2} z x y\left(z^{2}-x^{2}\right)+c^{2} x y z\left(x^{2}-y^{2}\right) \\
& =-x y z\left(\left(b^{2}-c^{2}\right) x^{2}+\left(c^{2}-a^{2}\right) y^{2}+\left(a^{2}-b^{2}\right) z^{2}\right) .
\end{aligned}
$$
${ }^{10}$ These asymptotes are also parallel to the axes of the Steiner ellipses. See, J.H. Conway, Hyacinthos, message 1237, 8/18/00.
${ }^{11}$ The other intersection is the center of the Jerabek hyperbola. This is based on the following theorem: Let $P$ be a point on a rectangular circum-hyperbola $\mathcal{C}$. The pedal circle of $P$ intersects the nine-point circle at the centers of $\mathcal{C}$ and of (the rectangular circumhyperbola which is) the isogonal conjugate of the line $O P$. See A.P. Hatzipolakis and P. Yiu, Hyacinthos, messages 1243 and 1249, 8/19/00.
${ }^{12}$ The Tarry point.
${ }^{13}$ O. Bottema and M.C. van Hoorn, Problem 664, Nieuw Archief voor Wiskunde, IV 1 (1983) 79. See also R.H. Eddy and R. Fritsch, On a problem of Bottema and van Hoorn, ibid., IV 13 (1995) 165 - 172.
${ }^{14}$ Floor van Lamoen, Hyacinthos, message 1251, 8/19/00. Excluding points on the side lines, the locus of $P$ is the conic
$$
\left(b^{2}-c^{2}\right) x^{2}+\left(c^{2}-a^{2}\right) y^{2}+\left(a^{2}-b^{2}\right) z^{2}=0 .
$$
We note some interesting properties of this conic:
- It passes through the centroid and the vertices of the superior triangle, namely, the four points $( \pm 1: \pm 1: \pm 1)$.
- It passes through the four incenters, namely, the four points $( \pm a: \pm b$ : $\pm c$ ). Since these four points form an orthocentric quadruple, the conic is a rectangular hyperbola.
- Since the matrix representing the conic is diagonal, the center of the conic has coordinates $\left(\frac{1}{b^{2}-c^{2}}: \frac{1}{c^{2}-a^{2}}: \frac{1}{a^{2}-b^{2}}\right)$, which is the Steiner point.
## Exercises
1. All conics passing through the four incenters are tangent to four fixed straight lines. What are these lines? ${ }^{15}$
2. Let $P$ be a given point other than the incenters. Show that the center of the conic through $P$ and the four incenters is the fourth intersection of the circumcircle and the circumconic with perspector $P \cdot P$ (barycentric square of $P$ ). ${ }^{16}$
3. Let $X$ be the pedal of $A$ on the side $B C$ of triangle $A B C$. For a real number $t$, let $A_{t}$ be the point on the altitude through $A$ such that $X A_{t}=t \cdot X A$. Complete the squares $A_{t} X X_{b} A_{b}$ and $A_{t} X X_{c} A_{c}$ with $X_{b}$ and $X_{c}$ on the line $B C \cdot{ }^{17}$ Let $A_{t}^{\prime}=B A_{c} \cap C A_{b}$, and $A_{t}^{\prime \prime}$ be the pedal of $A_{t}^{\prime}$ on the side $B C$. Similarly define $B_{t}^{\prime \prime}$ and $C_{t}^{\prime \prime}$. Show that as $t$ varies, triangle $A_{t}^{\prime \prime} B_{t}^{\prime \prime} C_{t}^{\prime \prime}$ is perspective with $A B C$, and the perspector traverses the Kiepert hyperbola. ${ }^{18}$
#### The Feuerbach hyperbola
$$
\sum_{\text {cyclic }} a(b-c)(s-a) y z=0
$$
${ }^{15}$ The conic $\mathcal{C}$ is self-polar. Its dual conic passes through the four incenters. This means that the conic $\mathcal{C}$ are tangent to the 4 lines $\pm a x+ \pm b y+ \pm c z=0$.
${ }^{16}$ Floor van Lamoen, Hyacinthos, message 1401, 9/11/00.
${ }^{17}$ A.P. Hatzipolakis, Hyacinthos, message 3370, 8/7/01.
${ }^{18}$ A.P. Hatzipolakis, Hyacinthos, message 3370, 8/7/01. This is the isogonal transform of the $O I$-line. The rectangular hyperbola through the incenter. Its center is the Feuerbach point.
#### The Jerabek hyperbola
The Jerabek hyperbola
$$
\sum_{\text {cyclic }} \frac{a^{2}\left(b^{2}-c^{2}\right) S_{A}}{x}=0
$$
is the isogonal transform of the Euler line. Its center is the point
$$
\left(\left(b^{2}-c^{2}\right)^{2} S_{A}:\left(c^{2}-a^{2}\right)^{2} S_{B}:\left(a^{2}-b^{2}\right)^{2} S_{C}\right)
$$
on the nine-point circle. ${ }^{19}$
## Exercises
1. Find the coordinates of the fourth intersection of the Feuerbach hyperbola with the circumcircle. ${ }^{20}$
2. Animate a point $P$ on the Feuerbach hyperbola, and construct its pedal circle. This pedal circle always passes through the Feuerbach point.
3. Three particles are moving at equal speeds along the perpendiculars from $I$ to the side lines. They form a triangle perspective with $A B C$. The locus of the perspector is the Feuerbach hyperbola.
4. The Feuerbach hyperbola is the locus of point $P$ for which the cevian quotient $I / P$ lies on the $O I$-line. ${ }^{21}$
5. Find the fourth intersection of the Jerabek hyperbola with the circumcircle. ${ }^{22}$
6. Let $\ell$ be a line through $O$. The tangent at $H$ to the rectangular hyperbola which is the isogonal conjugate of $\ell$ intersects $\ell$ at a point on the Jerabek hyperbola. ${ }^{23}$
${ }^{19}$ The Jerabek center appears as $X_{125}$ in ETC.
${ }^{20}\left(\frac{a}{a^{2}(b+c)-2 a b c-(b+c)(b-c)^{2}}: \cdots: \cdots\right)$. This point appears as $X_{104}$ in ETC.
${ }^{21}$ P. Yiu, Hyacinthos, message 1013, 6/13/00.
${ }^{22}\left(\frac{a^{2}}{2 a^{4}-a^{2}\left(b^{2}+c^{2}\right)-\left(b^{2}-c^{2}\right)^{2}}: \cdots: \cdots\right)$. This point appears as $X_{74}$ in ETC.
${ }^{23}$ B. Gibert, Hyacinthos, message 4247, 10/30/01.
### Envelopes
The envelope of the parametrized family of lines
$$
\left(a_{0}+a_{1} t+a_{2} t^{2}\right) x+\left(b_{0}+b_{1} t+b_{2} t^{2}\right) y+\left(c_{0}+c_{1} t+c_{2} t^{2}\right) z=0
$$
is the $\operatorname{conic}^{24}$
$$
\left(a_{1} x+b_{1} y+c_{1} z\right)^{2}-4\left(a_{0} x+b_{0} y+c_{0} z\right)\left(a_{2} x+b_{2} y+c_{2} z\right)=0,
$$
provided that the determinant
$$
\left|\begin{array}{lll}
a_{1} & a_{1} & a_{2} \\
b_{0} & b_{1} & b_{2} \\
c_{0} & c_{1} & c_{2}
\end{array}\right| \neq 0
$$
Proof. This is the dual conic of the conic parametrized by
$$
x: y: z=a_{0}+a_{1} t+a_{2} t^{2}: b_{0}+b_{1} t+b_{2} t^{2}: c_{0}+c_{1} t+c_{2} t^{2} .
$$
#### The Artzt parabolas
Consider similar isosceles triangles $A^{\theta} B C, A B^{\theta} C$ and $A B C^{\theta}$ constructed on the sides of triangle $A B C$. The equation of the line $B^{\theta} C^{\theta}$ is
$$
\left(S^{2}-2 S_{A} t-t^{2}\right) x+\left(S^{2}+2\left(S_{A}+S_{B}\right) t+t^{2}\right) y+\left(S^{2}+2\left(S_{C}+S_{A}\right) t+t^{2}\right) z=0,
$$
where $t=S_{\theta}=S \cdot \cot \theta$. As $\theta$ varies, this envelopes the conic
$$
\left(-S_{A} x+c^{2} y+b^{2} z\right)^{2}-S^{2}(x+y+z)(-x+y+z)=0
$$
#### Envelope of area-bisecting lines
Let $Y$ be a point on the line $A C$. There is a unique point $Z$ on $A B$ such that the signed area of $A Z Y$ is half of triangle $A B C$. We call $Y Z$ an areabisecting line. If $Y=(1-t: 0: t)$, then $Z=\left(1-\frac{1}{2 t}: \frac{1}{2 t}: 0\right)=(2 t-1: 1: 0$. The line $Y Z$ has equation
$$
0=\left|\begin{array}{ccc}
1-t & 0 & t \\
2 t-1 & 1 & 0 \\
x & y & z
\end{array}\right|=-t x+\left(-t+2 t^{2}\right) y+(1-t) z .
$$
${ }^{24}$ This can be rewritten as $\sum\left(4 a_{0} a_{2}-a_{1}^{2}\right) x^{2}+2\left(2\left(b_{0} c_{2}+b_{2} c_{0}\right)-b_{1} c_{1}\right) y z=0$. This envelopes the conic
$$
(x+y+z)^{2}-8 y z=0 .
$$
This conic has representing matrix
$$
M=\left(\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & -3 \\
1 & -3 & 1
\end{array}\right)
$$
with adjoint matrix
$$
M^{\#}=-4\left(\begin{array}{ccc}
2 & 1 & 1 \\
1 & 0 & -1 \\
1 & -1 & 0
\end{array}\right) .
$$
This is a hyperbola with center at the vertex $A$.
To construct this as a 5-point conic, we need only find 3 points on the hyperbola. Here are three obvious points: the centroid $G,(1:-1: 0)$ and $(1: 0:-1)$. Unfortunately the latter two are infinite point: they give the lines $A B$ and $A C$ as asymptotes of the hyperbola. This means that the axes of the hyperbola are the bisectors of angle $A$. Thus images of $G$ in these axes give three more points on the hyperbola. To find a fifth point, we set $x=0$ and obtain $(y+z)^{2}-8 y z=0, \ldots, y-3 z: z= \pm 2 \sqrt{2}: 1$,
$$
y: z=3 \pm 2 \sqrt{2}: 1=(\sqrt{2} \pm 1)^{2}: 1=\sqrt{2} \pm 1: \sqrt{2} \mp 1 .
$$
#### Envelope of perimeter-bisecting lines
Let $Y$ be a point on the line $A C$. There is a unique point $Z$ on $A B$ such that the (signed) lengths of the segments $A Y$ and $A Z$ add up to the semiperimeter of triangle $A B C$. We call $Y Z$ a perimeter-bisecting line. If $A Y=t$, then $A Z=s-t$. The coordinates of the points are $Y=(b-t: 0: t)$ and $Z=(c-s+t: s-t: 0)$. The line $Y Z$ has equation
$$
\left(t^{2}-s t\right) x+\left(t^{2}-(s-c) t\right) y+\left(t^{2}-(s+b) t+b s\right) z=0 .
$$
These lines envelopes the conic
$$
(s x+(s-c) y+(s+b) z)^{2}-4 b s z(x+y+z)=0
$$
with representing matrix
$$
\left(\begin{array}{ccc}
s^{2} & s(s-c) & s(s-b) \\
s(s-c) & (s-c)^{2} & (s-b)(s-c) \\
s(s-b) & (s-b)(s-c) & (s-b)^{2}
\end{array}\right)
$$
with adjoint matrix
$$
M^{\#}=-8 b c s\left(\begin{array}{ccc}
2(s-a) & s-b & s-c \\
s-b & 0 & -s \\
s-c & -s & 0
\end{array}\right) .
$$
This conic is a parabola tangent to the lines $C A$ and $A B$ at the points $(-(s-b): 0: s)$ and $(-(s-c): s: 0) .{ }^{25}$
#### The tripolars of points on the Euler line
A typical point on the Euler line
$$
\sum_{\text {cyclic }} S_{A}\left(S_{B}-S_{C}\right) x=0
$$
has coordinates $\left(S_{B C}+t: S_{C A}+t: S_{A B}+t\right)$, with tripolar
$$
\sum_{\text {cyclic }} \frac{1}{S_{B C}+t} x=0
$$
or
$$
0=\sum_{\text {cyclic }}(v+t)(w+t) x=\sum_{\text {cyclic }}\left(S_{B C}+a^{2} S_{A} t+t^{2}\right) x .
$$
The envelope is the conic
$$
\left(a^{2} S_{A} x+b^{2} S_{B} y+c^{2} S_{C} z\right)^{2}-4 S_{A B C}(x+y+z)\left(S_{A} x+S_{B} y+S_{C} z\right)=0 .
$$
This can be rewritten as
$$
\sum_{\text {cyclic }} S_{A A}\left(S_{B}-S_{C}\right)^{2} x^{2}-2 S_{B C}\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right) y z=0 .
$$
This can be rewritten as
$$
\sum_{\text {cyclic }} S_{A A}\left(S_{B}-S_{C}\right)^{2} x^{2}-2 S_{B C}\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right) y z=0 .
$$
It is represented by the matrix
$$
M=\left(\begin{array}{ccc}
S_{A A}\left(S_{B}-S_{C}\right)^{2} & -S_{A B}\left(S_{B}-S_{C}\right)\left(S_{C}-S_{A}\right) & -S_{C A}\left(S_{A}-S_{B}\right)\left(S_{B}-S_{C}\right) \\
-S_{A B}\left(S_{B}-S_{C}\right)\left(S_{C}-S_{A}\right) & S_{B B}\left(S_{C}-S_{A}\right) & -S_{B C}\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right) \\
S_{C A}\left(S_{A}-S_{B}\right)\left(S_{B}-S_{C}\right) & -S_{B C}\left(S_{C}-S_{A}\right)\left(S_{A}-S_{B}\right) & S_{C C}\left(S_{A}-S_{B}\right)
\end{array}\right) .
$$
${ }^{25}$ These are the points of tangency of the $A$-excircle with the side lines. This is clearly an inscribed conic, tangent to the side lines at the points $\left(0: S_{C}\left(S_{A}-S_{B}\right): S_{B}\left(S_{C}-S_{A}\right)\right),\left(S_{C}\left(S_{A}-S_{B}\right): 0: S_{A}\left(S_{B}-S_{C}\right)\right)$, and $\left(S_{B}\left(S_{C}-S_{A}\right): S_{A}\left(S_{B}-S_{C}\right): 0\right)$. The perspector is the point ${ }^{26}$
$$
\left(\frac{1}{S_{A}\left(S_{B}-S_{C}\right)}: \frac{1}{S_{B}\left(S_{C}-S_{A}\right)}: \frac{1}{S_{C}\left(S_{A}-S_{B}\right)}\right) .
$$
The isotomic conjugate of this perspector being an infinite point, the conic is a parabola. 27
## Exercises
1. Animate a point $P$ on the circumcircle, and construct a circle $\mathcal{C}(P)$, center $P$, and radius half of the inradius. Find the envelope of the radical axis of $\mathcal{C}(P)$ and the incircle.
2. Animate a point $P$ on the circumcircle. Construct the isotomic conjugate of its isogonal conjugate, i.e., the point $Q=\left(P^{*}\right)^{\bullet}$. What is the envelope of the line joining $P Q ?{ }^{28}$
${ }^{26}$ This point appears as $X_{648}$ in ETC.
${ }^{27}$ The focus is the point $X_{112}$ in ETC:
$$
\left(\frac{a^{2}}{S_{A}\left(S_{B}-S_{C}\right)}: \frac{b^{2}}{S_{B}\left(S_{C}-S_{A}\right)}: \frac{c^{2}}{S_{C}\left(S_{A}-S_{B}\right)}\right) .
$$
Its directrix is the line of reflection of the focus, i.e.,
$$
\sum_{\text {cyclic }} S_{A A}\left(S_{B}-S_{C}\right) x=0 .
$$
${ }^{28}$ The Steiner point.
## Chapter 12
## Some More Conics
### Conics associated with parallel intercepts
#### Lemoine's thorem
Let $P=(u: v: w)$ be a given point. Construct parallels through $P$ to the side lines, intersecting the side lines at the points
$$
\begin{aligned}
& Y_{a}=(u: 0: v+w), \quad Z_{a}=(u: v+w: 0) ; \\
& Z_{b}=(w+u: v: 0), \quad X_{b}=(0: v: w+u) \text {; } \\
& X_{c}=(0: u+v: w), \quad Y_{c}=(u+v: 0: w) \text {. }
\end{aligned}
$$
These 6 points lie on a conic $\mathcal{C}_{P}$, with equation
$$
\sum_{\text {cyclic }} v w(v+w) x^{2}-u(v w+(w+u)(u+v)) y z=0 .
$$
This equation can be rewritten as
$$
\begin{aligned}
& -(u+v+w)^{2}(u y z+v z x+w x y) \\
& +(x+y+z)(v w(v+w) x+w u(w+u) y+u v(u+v) z)=0 .
\end{aligned}
$$
From this we obtain
## Theorem (Lemoine)
The conic through the 6 parallel intercepts of $P$ is a circle if and only if $P$ is the symmedian point.
## Exercises
1. Show that the conic $\mathcal{C}_{P}$ through the 6 parallel intercepts through $P$ is an ellipse, a parabola, or a hyperbola according as $P$ is inside, on, or outside the Steiner in-ellipse, and that its center is the midpoint of the $P$ and the cevian quotient $G / P .{ }^{1}$
2. Show that the Lemoine circle is concentric with the Brocard circle. ${ }^{2}$
#### A conic inscribed in the hexagon $W(P)$
While $\mathcal{C}_{P}$ is a conic circumscribing the hexagon $W(P)=Y_{a} Y_{c} Z_{b} Z_{a} X_{c} X_{b}$, there is another conic inscribed in the same hexagon. The sides of the hexagon have equations
$$
\begin{aligned}
Y_{a} Y_{c}: & y=0 ; & Y_{c} Z_{b}: & -v w x+w(w+u) y+v(u+v) z=0 ; \\
Z_{b} Z_{a}: & z=0 ; & Z_{a} X_{c}: & w(v+w) x-w u y+u(u+v) z=0 ; \\
X_{c} X_{b}: & x=0 ; & X_{b} Y_{a}: & v(v+w) x+u(w+u) y-u v z=0 .
\end{aligned}
$$
These correspond to the following points on the dual conic: the vertices and
$$
\left(-1: \frac{w+u}{v}: \frac{u+v}{w}\right), \quad\left(\frac{v+w}{u}:-1: \frac{u+v}{w}\right), \quad\left(\frac{v+w}{u}: \frac{w+u}{v}:-1\right) .
$$
It is easy to note that these six points lie on the circumconic
$$
\frac{v+w}{x}+\frac{w+u}{y}+\frac{u+v}{z}=0 .
$$
It follows that the 6 lines are tangent to the incribed conic
$$
\sum_{\text {cyclic }}(v+w)^{2} x^{2}-2(w+u)(u+v) y z=0,
$$
with center $(2 u+v+w: u+2 v+w: u+v+2 w)$ and perspector
$$
\left(\frac{1}{v+w}: \frac{1}{w+u}: \frac{1}{u+v}\right) \text {. }
$$
${ }^{1}$ The center has coordinates $(u(2 v w+u(v+w-u)): v(2 w u+v(w+u-v)): w(2 u v+$ $w(u+v-w))$.
${ }^{2}$ The center of the Lemoine circle is the midpoint between $K$ and $G / K=O$.
## Exercises
1. Find the coordinates of the points of tangency of this inscribed conic with the $Y_{c} Z_{b}, Z_{a} X_{c}$ and $X_{b} Y_{a}$, and show that they form a triangle perspective with $A B C$ at ${ }^{3}$
$$
\left(\frac{u^{2}}{v+w}: \frac{v^{2}}{w+u}: \frac{w^{2}}{u+v}\right) .
$$
#### Centers of inscribed rectangles
Let $P=(x: y: z)$ be a given point. Construct the inscribed rectangle whose top edge is the parallel to $B C$ through $P$. The vertices of the rectangle on the sides $A C$ and $A B$ are the points $(x: y+z: 0)$ and $(x: 0: y+z)$.
The center of the rectangle is the point
$$
A^{\prime}=\left(a^{2} x: a^{2}(x+y+z)-S_{B} x: a^{2}(x+y+z)-S_{C} x\right) .
$$
Similarly, consider the two other rectangles with top edges through $P$ parallel to $C A$ and $A B$ respectively, with centers $B^{\prime}$ and $C^{\prime}$. The triangle $A^{\prime} B^{\prime} C^{\prime}$ is perspective with $A B C$ if and only if
$$
=\begin{aligned}
& \left(a^{2}(x+y+z)-S_{B} x\right)\left(b^{2}(x+y+z)-S_{C} y\right)\left(c^{2}(x+y+z)-S_{A} z\right) \\
= & \left(a^{2}(x+y+z)-S_{C} x\right)\left(b^{2}(x+y+z)-S_{A} y\right)\left(c^{2}(x+y+z)-S_{B} z\right) .
\end{aligned}
$$
The first terms of these expressions cancel one another, so do the last terms. Further cancelling a common factor $x+y+z$, we obtain the quadratic equation
$$
\begin{aligned}
& \sum a^{2} S_{A}\left(S_{B}-S_{C}\right) y z+(x+y+z) \sum_{\text {cyclic }} b^{2} c^{2}\left(S_{B}-S_{C}\right) x=0 . \\
& { }^{3}\left(v+w: \frac{v^{2}}{w+u}: \frac{w^{2}}{u+v}\right),\left(\frac{u^{2}}{v+w}: w+u: \frac{w^{2}}{u+v}\right), \text { and }\left(\frac{u^{2}}{v+w}: \frac{v^{2}}{w+u}: u+v\right) .
\end{aligned}
$$
This means that the locus of $P$ for which the centers of the inscribed rectangles form a perspective triangle is a hyperbola in the pencil generated by the Jerabek hyperbola
$$
\sum a^{2} S_{A}\left(S_{B}-S_{C}\right) y z=0
$$
and the Brocard axis $O K$
$$
\sum_{\text {cyclic }} b^{2} c^{2}\left(S_{B}-S_{C}\right) x=0
$$
Since the Jerabek hyperbola is the isogonal transform of the Euler line, it contains the point $H^{*}=O$ and $G^{*}=K$. The conic therefore passes through $O$ and $K$. It also contains the de Longchamps point $L=\left(-S_{B C}+S_{C A}+\right.$ $\left.S_{A B}: \cdots: \cdots\right)$ and the point $\left(S_{B}+S_{C}-S_{A}: S_{C}+S_{A}-S_{B}: S_{A}+S_{B}-S_{C}\right) .{ }^{4}$
$$
\begin{array}{ll}
P & \text { Perspector } \\
\hline \text { circumcenter } & \left(\frac{1}{2 S^{2}-S_{B C}}: \frac{1}{2 S^{2}-S_{C A}}: \frac{1}{2 S^{2}-S_{A B}}\right) \\
\text { symmedian point } & \left(3 a^{2}+b^{2}+c^{2}: a^{2}+3 b^{2}+c^{2}: a^{2}+b^{2}+3 c^{2}\right) \\
\text { de Longchamps point } & \left(S_{B C}\left(S^{2}+2 S_{A A}\right): \cdots: \cdots\right) \\
\left(3 a^{2}-b^{2}-c^{2}: \cdots: \cdots\right) & \left(\frac{1}{S^{2}+S_{A A}+S_{B C}}: \cdots: \cdots\right)
\end{array}
$$
## Exercises
1. Show that the three inscribed rectangles are similar if and only if $P$ is the point
$$
\left(\frac{a^{2}}{t+a^{2}}: \frac{b^{2}}{t+b^{2}}: \frac{c^{2}}{t+c^{2}}\right),
$$
where $t$ is the unique positive root of the cubic equation
$$
2 t^{3}+\left(a^{2}+b^{2}+c^{2}\right) t^{2}-a^{2} b^{2} c^{2}=0
$$
### Lines simultaneously bisecting perimeter and area
Recall from $\S 11.3$ that the $A$-area-bisecting lines envelope the conic whose dual is represented by the matrix
$$
M_{1}=\left(\begin{array}{ccc}
2 & 1 & 1 \\
1 & 0 & -1 \\
1 & -1 & 0
\end{array}\right) .
$$
${ }^{4}$ None of these perspectors appears in the current edition of ETC. On the other hand, the $A$-perimeter-bisecting lines envelope another conic whose dual is represented by
$$
M_{2}=\left(\begin{array}{ccc}
2(s-a) & s-b & s-c \\
s-b & 0 & -s \\
s-c & -s & 0
\end{array}\right)
$$
To find a line simultaneously bisecting the area and perimeter, we seek an intersection of of the two dual conics represented by $M_{1}$ and $M_{2}$. In the pencil of conics generated by these two, namely, the conics represented by matrices of the form $t M_{1}+M_{2}$, there is at least one member which degenerates into a union of two lines. The intersections of the conics are the same as those of these lines with any one of them. Now, for any real parameter $t$,
$$
\begin{aligned}
\operatorname{det}\left(t M_{1}+M_{2}\right) & =\left|\begin{array}{ccc}
2(t+s-a) & t+s-b & t+s-c \\
t+s-b & 0 & -(t+s) \\
t+s-c & -(t+s) & 0
\end{array}\right| \\
& =-2(t+s)(t+s-b)(t+s-c)-2(t+s)^{2}(t+s-a) \\
& =-2(t+s)[(t+s-b)(t+s-c)+(t+s)(t+s-a)] \\
& =-2(t+s)\left[2(t+s)^{2}-2 s(t+s)+b c\right]
\end{aligned}
$$
By choosing $t=-s$, we obtain
$$
-s M_{1}+M_{2}=\left|\begin{array}{ccc}
-2 a & -b & -c \\
-b & 0 & 0 \\
-c & 0 & 0
\end{array}\right|
$$
which represents the degenerate conic
$$
2 a x^{2}+2 b x y+2 c x y=2 x(a x+b y+c z)=0 .
$$
In other words, the intersections of the two dual conics are the same as those
$$
x^{2}+x y+x z-y z=0
$$
(represented by $M_{1}$ ) and the lines $x=0$ and $z x+b y+c z=0$.
(i) With $x=0$ we obtain $y=0$ and $z=0$, and hence the points $(0: 0: 1)$ and $(0: 1: 0)$ respectively on the dual conic. These correspond to the line This means that such a line must pass through the incenter $I$, and as an area-bisecting line,
$$
2 b t^{2}-(a+b+c) t+c=0
$$
and
$$
t=\frac{(a+b+c) \pm \sqrt{(a+b+c)^{2}-8 b c}}{4 b}=\frac{s \pm \sqrt{s^{2}-2 b c}}{2 b} .
$$
The division point on $A C$ are
$$
(1-t: 0: t)=\left(2 b-s \mp \sqrt{s^{2}-2 b c}: 0: s \pm \sqrt{s^{2}-2 b c}\right) .
$$
### Parabolas with vertices of a triangle as foci and sides as directrices
Given triangle $A B C$, consider the three parabolas each with one vertex as focus and the opposite side as directrix, and call these the $a-, b-$, and $c$-parabolas respectively. The vertices are clearly the midpoints of the altitudes. No two of these parabolas intersect. Each pair of them, however, has a unique common tangent, which is the perpendicular bisector of a side of the triangle. The three common tangents therefore intersect at the circumcenter.
The points of tangency of the perpendicular bisector $B C$ with the $b-$ and $c$-parabolas are inverse with respect to the circumcircle, for they are at distances $\frac{b R}{c}$ and $\frac{c R}{b}$ from the circumcenter $O$. These points of tangency can be easily constructed as follows. Let $H$ be the orthocenter of triangle $A B C, H_{a}$ its reflection in the side $B C$. It is well known that $H_{a}$ lies on the circumcircle. The intersections of $B H_{a}$ and $C_{a}$ with the perpendicular bisector of $B C$ are the points of tangency with the $b$ - and $c$-parabolas respectively.
## Exercises
1. Find the equation of the $a$-parabola. ${ }^{5}$
### The Soddy hyperbolas
#### Equations of the hyperbolas
Given triangle $A B C$, consider the hyperbola passing through $A$, and with foci at $B$ and $C$. We shall call this the $a$-Soddy hyperbola of the triangle, since this and related hyperbolas lead to the construction of the famous Soddy circle. The reflections of $A$ in the side $B C$ and its perpendicular bisector are clearly points on the same hyperbola, so is the symmetric of $A$ with respect to the midpoint of $B C$. The vertices of the hyperbola on the transverse axis $B C$ are the points $(0: s-b: s-c)$, and $(0: s-c: s-b)$, the points of tangency of the side $B C$ with the incircle and the $A$-excircle.
Likewise, we speak of the $B$ - and $C$-Soddy hyperbolas of the same triangle, and locate obvious points on these hyperbolas.
#### Soddy circles
Given triangle $A B C$, there are three circles centered at the vertices and mutually tangent to each other externally. These are the circles $A(s-a)$, $B(s-b)$, and $C(s-c)$. The inner Soddy circle of triangle $A B C$ is the circle tangent externally to each of these three circles. The center of the inner Soddy circle clearly is an intersection of the three Soddy hyperbolas.
${ }^{5}-S^{2} x^{2}+a^{2}\left(c^{2} y^{2}+2 S_{A} y z+b^{2} z^{2}\right)=0$.
## Exercises
1. Show that the equation of $A$-Soddy hyperbola is
$$
\begin{aligned}
F_{a}= & (c+a-b)(a+b-c)\left(y^{2}+z^{2}\right) \\
& -2\left(a^{2}+(b-c)^{2}\right) y z+4(b-c) c x y-4 b(b-c) z x=0 .
\end{aligned}
$$
### Appendix: Constructions with conics
Given 5 points $A, B, C, D, E$, no three of which are collinear, and no four concyclic, the conic $\mathcal{C}$. Through these 5 points is either an ellipse, a parabola, or a hyperbola.
#### The tangent at a point on $\mathcal{C}$
(1) $P:=A C \cap B D$;
(2) $Q:=A D \cap C E$;
(3) $R:=P Q \cap B E$.
$A R$ is the tangent at $A$.
12.5.2 The second intersection of $\mathcal{C}$ and a line $\ell$ through $A$
(1) $P:=A C \cap B E$;
(2) $Q:=\ell \cap B D$; (3) $R:=P Q \cap C D$;
(4) $A^{\prime}:=\ell \cap E R$.
$A^{\prime}$ is the second intersection of $\mathcal{C}$ and $\ell$.
#### The center of $\mathcal{C}$
(1) $B^{\prime}:=$ the second intersection of $\mathcal{C}$ with the parallel through $B$ to $A C$;
(2) $\ell_{b}:=$ the line joining the midpoints of $B B^{\prime}$ and $A C$;
(3) $C^{\prime}:=$ the second intersection of $\mathcal{C}$ with the parallel through $C$ to $A B$
(4) $\ell_{c}:=$ the line joining the midpoints of $C C^{\prime}$ and $A B$;
(5) $O:=\ell_{b} \cap \ell_{c}$ is the center of the conic $\mathcal{C}$.
#### Principal axes of $\mathcal{C}$
(1) $K(O):=$ any circle through the center $O$ of the conic $\mathcal{C}$.
(2) Let $M$ be the midpoint of $A B$. Construct (i) $O M$ and (ii) the parallel through $O$ to $A B$ each to intersect the circle at a point. Join these two points to form a line $\ell$.
(3) Repeat (2) for another chord $A C$, to form a line $\ell^{\prime}$.
(4) $P:=\ell \cap \ell^{\prime}$.
(5) Let $K P$ intersect the circle $K(O)$ at $X$ and $Y$.
Then the lines $O X$ and $O Y$ are the principal axes of the conic $\mathcal{C}$.
#### Vertices of $\mathcal{C}$
(1) Construct the tangent at $A$ to intersect to the axes $O X$ and $O Y$ at $P$ and $Q$ respectively.
(2) Construct the perpendicular feet $P^{\prime}$ and $Q^{\prime}$ of $A$ on the axes $O X$ and $O Y$.
(3) Construct a tangent $O T$ to the circle with diameter $P P^{\prime}$. The intersections of the line $O X$ with the circle $O(T)$ are the vertices on this axis.
(4) Repeat (3) for the circle with diameter $Q Q^{\prime}$.
#### Intersection of $\mathcal{C}$ with a line $\mathcal{L}$
Let $F$ be a focus, $\ell$ a directrix, and $e=$ the eccentricity.
(1) Let $H=\mathcal{L} \cap \ell$.
(2) Take an arbitrary point $P$ with pedal $Q$ on the directrix.
(3) Construct a circle, center $P$, radius $e \cdot P Q$. (4) Through $P$ construct the parallel to $\mathcal{L}$, intersecting the directrix at $O$.
(5) Through $O$ construct the parallel to $F H$, intersecting the circle above in $X$ and $Y$.
(6) The parallels through $F$ to $P X$ and $P Y$ intersect the given line $\mathcal{L}$ at two points on the conic.
| Textbooks |
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
For normal vector space, see normal (geometry).
Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
Hausdorff)
T6 (perfectly normal
Hausdorff)
• History
Definitions
A topological space X is a normal space if, given any disjoint closed sets E and F, there are neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.
A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.
A completely normal space, or hereditarily normal space, is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology.
A T5 space, or completely T4 space, is a completely normal T1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.
A perfectly normal space is a topological space $X$ in which every two disjoint closed sets $E$ and $F$ can be precisely separated by a function, in the sense that there is a continuous function $f$ from $X$ to the interval $[0,1]$ such that $f^{-1}(0)=E$ and $f^{-1}(1)=F$.[1] This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of $E\subseteq f^{-1}(0)$ and $F\subseteq f^{-1}(1)$, but not precisely separated in general. It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is the zero set of a continuous function. The equivalence between these three characterizations is called Vedenissoff's theorem.[2][3] Every perfectly normal space is completely normal, because perfect normality is a hereditary property.[4][5]
A T6 space, or perfectly T4 space, is a perfectly normal Hausdorff space.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.
Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.
A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.
Examples of normal spaces
Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
• All compact Hausdorff spaces are normal;
• In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
• All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.
Also, all fully normal spaces are normal (even if not regular). Sierpiński space is an example of a normal space that is not regular.
Examples of non-normal spaces
An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.
A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.
Properties
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[6]
The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X.
Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be entirely within the unit interval [0,1]. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.
More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R that extends f in the sense that F(x) = f(x) for all x in A.
The map $\emptyset \rightarrow X$ has the lifting property with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.[7]
If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U. This shows the relationship of normal spaces to paracompactness.
In fact, any space that satisfies any one of these three conditions must be normal.
A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. In fact, since there exist spaces which are Dowker, a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T2 axiom are preserved under arbitrary products.[8]
Relationships to other separation axioms
If a normal space is R0, then it is in fact completely regular. Thus, anything from "normal R0" to "normal completely regular" is the same as what we usually call normal regular. Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff. These are what we usually call normal Hausdorff spaces.
A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.
Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpiński space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
See also
• Collectionwise normal space – Property of topological spaces stronger than normality
• Monotonically normal space – Property of topological spaces stronger than normality
Citations
1. Willard, Exercise 15C
2. Engelking, Theorem 1.5.19. This is stated under the assumption of a T1 space, but the proof does not make use of that assumption.
3. "Why are these two definitions of a perfectly normal space equivalent?".
4. Engelking, Theorem 2.1.6, p. 68
5. Munkres 2000, p. 213
6. Willard 1970, pp. 100–101.
7. "separation axioms in nLab". ncatlab.org. Retrieved 2021-10-12.
8. Willard 1970, Section 17.
References
• Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989. ISBN 3-88538-006-4
• Kemoto, Nobuyuki (2004). "Higher Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.). Encyclopedia of General Topology. Amsterdam: Elsevier Science. ISBN 978-0-444-50355-8.
• Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 978-0-13-181629-9.
• Sorgenfrey, R.H. (1947). "On the topological product of paracompact spaces". Bull. Amer. Math. Soc. 53 (6): 631–632. doi:10.1090/S0002-9904-1947-08858-3.
• Stone, A. H. (1948). "Paracompactness and product spaces". Bull. Amer. Math. Soc. 54 (10): 977–982. doi:10.1090/S0002-9904-1948-09118-2.
• Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 978-0-486-43479-7.
| Wikipedia |
\begin{document}
\title{H\"older continuity of normal cycles and of support measures of convex bodies}
\begin{abstract} We provide an estimate of the distance (in the dual flat seminorm) of the normal cycles of convex bodies with given Hausdorff distance. We also give an estimate (in the bounded Lipschitz metric) of the support measures of convex bodies. \\[1mm] {\em 2010 Mathematics Subject Classification:} primary 52A20, secondary 52A22\\[1mm] {\em Keywords:} Normal cycle, flat metric, weak convergence, valuation, support measure \end{abstract}
\section{Introduction}\label{sec1}
In 1986, Martina Z\"ahle \cite{Zae86} presented a current representation of Federer's curvature measures. For $k=0,\dots,n-1$, she defined a differential form $\varphi_k$ of degree $n-1$ on the Euclidean space ${\mathbb R}^{2n}$ such that, for each compact set $K$ of positive reach in ${\mathbb R}^n$ and for each Borel set $\beta$ in ${\mathbb R}^n$, the evaluation of ${\bf 1}_\beta \varphi_k$ at the normal cycle of $K$ yields the $k$th curvature measure of $K$, evaluated at $\beta$. The curvature measures had previously been introduced by Federer \cite{Fed59} in a different way. The approach using currents has later been investigated, applied and considerably extended in work of Z\"ahle \cite{Zae87, Zae90}, Rataj and Z\"ahle \cite{RZ01, RZ03, RZ05}, Fu \cite{Fu94}, Pokorn\'{y} and Rataj \cite{PR12}, and others. The normal cycle has thus become an important tool for the treatment of curvature properties of very general classes of sets, and it has found many applications in integral geometry. In this note, we restrict ourselves to convex bodies. The normal cycle $T_K$ of a convex body $K$ in ${\mathbb R}^n$ (we recall the definition in Section \ref{sec2}) has a useful continuity property. If $K_i$, $i\in\ensuremath{\mathbb{N}}$, and $K$ are convex bodies in ${\mathbb R}^n$ and $K_i\to K$ in the Hausdorff metric, as $i\to\infty$, then $T_{K_i}\to T_K$ in the dual flat seminorm for currents. This was stated without proof in \cite[p.~251]{Zae90} and was proved in \cite[Thm.~3.1]{RZ01}; see also \cite[Thm.~3.1]{Fu10}. The continuity property is used, for example, in \cite{HS13} in the course of the proof for a classification theorem for local tensor valuations on the space of convex bodies.
The purpose of this note is to obtain a quantitative improvement of the preceding continuity result, in the form of a H\"older estimate. To formulate it, we denote by ${\mathcal K}^n$ the space of convex bodies (nonempty compact convex subsets) in Euclidean space ${\mathbb R}^n$, as usual equipped with the Hausdorff metric $d_H$. We write $B^n$ and ${\mathbb S}^{n-1}$ for the unit ball and the unit sphere, respectively, in ${\mathbb R}^n$, and ${\mathcal H}^k$ for the $k$-dimensional Hausdorff measure. For $\rho>0$, $K_\rho:=K+\rho B^n$ is the parallel body of the convex body $K$ at distance $\rho$. Below, we identify ${\mathbb R}^n\times{\mathbb R}^n$ with ${\mathbb R}^{2n}$ and denote by $\mathcal{E}^{n-1}({\mathbb R}^{2n})=\mathcal{E}({\mathbb R}^{2n},\bigwedge^{n-1}{\mathbb R}^{2n})$ the vector space of all differential forms of degree $n-1$ on ${\mathbb R}^{2n}$ with real coefficients and of class $C^\infty$.
\begin{theorem}\label{T1} Let $K,L\in{\mathcal K}^n$, and let $M\subset {\mathbb R}^{2n}$ be a compact convex set containing $K_1\times{\mathbb S}^{n-1}$ and $L_1\times{\mathbb S}^{n-1}$. Then, for each $\varphi\in \mathcal{E}^{n-1}({\mathbb R}^{2n})$,
$$ |T_K(\varphi)-T_L(\varphi)|\le C(M,\varphi)\,d_H(K,L)^{\frac{1}{2n+1}},$$ w\textsc{}here $C(M,\varphi)$ is a constant which depends (for given dimension) on $M$ and on the Lipschitz constant and the sup-norm of $\varphi$ on $M$. \end{theorem}
According to the definition of the dual flat seminorm, this result can be interpreted as local H\"older continuity of the normal cycles of convex bodies with respect to the Hausdorff metric and the dual flat seminorm. A similar, but essentially different quantitative result is obtained in \cite[Thm. 2]{CM06}. It refers to more general sets and is, therefore, less explicit. On the other hand, its restriction to convex bodies does not yield the present result, since at least one of the sets in \cite{CM06} has to be bounded by a submanifold of class $C^2$.
As mentioned, normal cycles are useful for introducing curvature measures for quite general classes of sets. In the theory of convex bodies, they have been used to introduce a generalization of curvature measures, the support measures. For ${\mathbb R}^n$, these are Borel measures on the product of ${\mathbb R}^n$ and the unit sphere ${\mathbb S}^{n-1}$, with the property that their marginal measures are the curvature measures on one hand and the surface area measures on the other hand. On the space of convex bodies with the Hausdorff metric, the support measures are weakly continuous. We improve this statement by showing that the support measures are locally H\"older continuous with respect to the bounded Lipschitz metric $d_{bL}$ (we recall its definition in Section 4). Let $\Lambda_i(K,\cdot)$ denote the $i$th support measure of $K\in{\mathcal K}^n$, normalized as explained in Section 4.
\begin{theorem}\label{T2} Let $K,L\in{\mathcal K}^n$ be convex bodies, and let $R$ be the radius of a ball containing $K_2$ and $L_2$. Then $$ d_{bL}(\Lambda_i(K,\cdot), \Lambda_i(L,\cdot)) \le C(R)\, d_H(K,L)^{1/2}$$ for $i\in\{0,\dots,n-1\}$, where $C(R)$ is a constant which (for given dimension) depends only on $R$. \end{theorem}
An estimate of this type, though with a smaller exponent of the Hausdorff distance, could be derived directly from Theorem \ref{T1}. We shall obtain the stronger result of Theorem \ref{T2} by adapting an approach due to Chazal, Cohen--Steiner and M\'{e}rigot \cite{CCM10}.
A special case of Theorem \ref{T2} concerns the area measure $S_{n-1}(K,\cdot)$. If $\omega\subset{\mathbb S}^{n-1}$ is a Borel set, then $S_{n-1}(K,\omega)=2\Lambda_{n-1}(K,{\mathbb R}_n\times\omega)$. From Theorem \ref{T2} it follows under the same assumptions on $K$ and $L$ that \begin{equation}\label{4.14} d_{bL}(S_{n-1}(K,\cdot),S_{n-1}(L,\cdot))\le C'(R)\,d_H(K,L)^{1/2}. \end{equation} We want to present some motivation for proving such an inequality.
The area measure is the subject of a famous existence and uniqueness theorem due to Minkowski (see, e.g., \cite[Sec.~7.1]{Sch93}). The uniqueness assertion has been improved by some stability results. One of these (going back to Diskant; see \cite[Thm.~7.2.2]{Sch93}) says that for convex bodies $K,L\in {\mathcal K}^n$ one has \begin{equation}\label{4.15}
d_H(K,L') \le \gamma \,\|S_{n-1}(K,\cdot)-S_{n-1}(L,\cdot)\|_{\rm TV}^{1/n} \end{equation}
for a suitable translate $L'$ of $L$, where $\|\cdot\|_{\rm TV}$ denotes the total variation norm. Here $\gamma>0$ is a constant depending only on the dimension and on a-priori bounds for the inradius and circumradius of $K$ and $L$.
The stability result (\ref{4.15}) has the blemish that its assumption is too strong: the left side can be small even if the right side is large. For example, if $K$ is a unit cube and $L$ is a rotated image of $K$, arbitrarily close to $K$ but not a translate of it, then $\|S_{n-1}(K,\cdot)-S_{n-1}(L,\cdot)\|_{\rm TV}\ge 1$. It seems, therefore, more meaningful to replace the right-hand side in (\ref{4.15}) by an expression involving a metric for measures that metrizes the weak convergence. For the L\'{e}vy--Prokhorov metric, such a stability result was proved in \cite{HS02}. It was deduced from a corresponding stability result for the bounded Lipschitz metric (which is implicit in the proof, though it was not stated explicitly), namely \begin{equation}\label{4.16} d_H(K,L') \le \gamma\,d_{bL}(S_{n-1}(K,\cdot),S_{n-1}(L,\cdot))^{1/n} \end{equation} for a suitable translate $L'$ of $L$, with a constant $\gamma$ as above. It appears that the H\"older continuity (\ref{4.14}) is, in principle, a more elementary fact than its reverse, the stability estimate (\ref{4.16}), and should therefore have preceded it.
\section{Notation and preliminaries}\label{sec2}
We have to use several definitions and results from geometric measure theory, therefore we choose most of our notation as in Federer's \cite{Fed69} book, in order to facilitate the comparison. For example, we denote the scalar product in ${\mathbb R}^n$ by $\bullet$ and the induced norm by $|\cdot|$. The same notation is used also for other Euclidean spaces which will come up in the following. We identify ${\mathbb R}^n$ and its dual space via the given scalar product.
Next we recall some notation and basic facts from multilinear algebra. Let $V$ be finite-dimensional real vector spaces. Then $\bigwedge_m V$, for $m\in{\mathbb N}_0$, denotes the vector space of {\em $m$-vectors} of $V$, and $\bigwedge^m V$ is the vector space of all $m$-linear alternating maps from $V^m$ to ${\mathbb R}$, whose elements are called {\em $m$-covectors}. The map $\bigwedge^m V \to\text{Hom}(\bigwedge_mV,{\mathbb R})$, which assigns to $f\in \bigwedge^m V$ the homomorphism $v_1\wedge \ldots\wedge v_m\mapsto f(v_1,\ldots,v_m)$, allows us to identify $\bigwedge^m V$ and $\text{Hom}(\bigwedge_mV,{\mathbb R})$. By this identification, the dual pairing of elements $a\in\bigwedge_m V$ and $\varphi\in \bigwedge^m (V,{\mathbb R})$ can be defined by $\langle a,\varphi\rangle :=\varphi(a)$. If $V'$ is another finite-dimensional vector space and $f:V\to V'$ is a linear map, then a linear map $\bigwedge_mf:\bigwedge_mV\to\bigwedge_mV'$ is determined by $$ (\mbox{$\bigwedge$}_mf)(v_1\wedge\ldots\wedge v_m)=f(v_1)\wedge\ldots\wedge f(v_m), $$ for all $v_1,\ldots,v_m\in V$.
Given an inner product space $(V,\bullet)$ with norm $|\cdot|$ we obtain an inner product on $\bigwedge_mV$. For $\xi,\eta\in \bigwedge_mV$ with $\xi=v_1\wedge\ldots\wedge v_m$ and $\eta=w_1\wedge\ldots\wedge w_m$, where $v_i,w_j\in V$, we define $$ \xi\bullet \eta=\det\left(\langle v_i,w_j\rangle_{i,j=1}^m\right). $$
This is independent of the particular representation of $\xi,\eta$. For general $\xi,\eta\in \bigwedge_mV$ the inner product is defined by linear extension, and then we put $|\xi|:=\sqrt{\xi\bullet \xi}$ for $\xi\in \bigwedge_mV$. If $(b_1,\ldots,b_n)$ is an orthonormal basis of $V$, then the $m$-vectors $b_{i_1}\wedge\ldots\wedge b_{i_m}$ with $1\le i_1<\ldots<i_m\le n$ form an orthonormal basis of $\bigwedge_mV$. Moreover, if $\xi\in \bigwedge_pV$ or $\eta\in \bigwedge_qV$ is simple, then \begin{equation}\label{eq2}
|\xi\wedge \eta|\le |\xi|\, |\eta|. \end{equation} Let $(b_1,\ldots,b_n)$ be an orthonormal basis of $V$, and let $(b_1^*,\ldots,b_n^*)$ be the dual basis in $V^*= \bigwedge^1 V$. We endow $\bigwedge^m V$ with the inner product for which the vectors $b_{i_1}^*\wedge\ldots\wedge b_{i_m}^*$, for $1\le i_1<\ldots<i_m\le n$, are an orthonormal basis. Then \begin{equation}\label{eq3}
|\langle \xi,\Phi\rangle|\le |\xi|\,|\Phi| \end{equation} for $\xi\in\bigwedge_mV$ and $\Phi\in \bigwedge^mV$.
The preceding facts are essentially taken from \cite[Section 1.7]{Fed69}.
Let $V$ be an $n$-dimensional inner product space. Then {\em comass} and {\em mass} are defined as in \cite[Section 1.8]{Fed69}. In particular, for $\Phi\in\bigwedge^m V$ the comass $\|\Phi\|$ of $\Phi$ satisfies $\|\Phi\|=|\Phi|$ if $\Phi$ is simple. Moreover, for $\xi\in\bigwedge_m V$ the mass $\|\xi\|$ of $\xi$ satisfies $\|\xi\|=|\xi|$ if $\xi$ is simple.
Now we turn to convex bodies. For notions from the theory of convex bodies which are not explained here, we refer to \cite{Sch93}. Let $K\in{\mathcal K}^n$. The metric projection, which is denoted by $p(K,\cdot)$, maps ${\mathbb R}^n$ to $ K$, and $u(K,x):=|x-p(K,x)|^{-1}(x-p(K,x))$ is defined for $x\in{\mathbb R}^n\setminus K$. Let $\partial K$ denote the topological boundary of $K$. The map $F:\partial K_1\to{\mathbb R}^n\times {\mathbb S}^{n-1}$ given by $F(x):= (p(K,x),u(K,x))$ is bi-Lipschitz, and the image is the {\em normal bundle} $\Nor K$ of $K$, which is an $(n-1)$ rectifiable subset of ${\mathbb R}^{2n}$. Hence, for $\mathcal{H}^{n-1}$-almost all $(x,u)\in\Nor K$, the set of $(\mathcal{H}^{n-1}\fed \Nor K,n-1)$ approximate tangent vectors at $(x,u)$ is an $(n-1)$-dimensional linear subspace of $\ensuremath{\mathbb{R}}^{2n}$, which is denoted by $\textrm{Tan}^{n-1}(\mathcal{H}^{n-1}\fed \Nor K,(x,u))$. Let $\Pi_1:{\mathbb R}^n\times{\mathbb R}^n\to{\mathbb R}^n$, $(x,u)\mapsto x$, and $\Pi_2:{\mathbb R}^n\times{\mathbb R}^n\to{\mathbb R}^n$, $(x,u)\mapsto u$, be projection maps and $\Omega_n$ the volume form for which $\Omega_n(e_1,\ldots,e_n)=1$ for the standard basis $(e_1,\ldots,e_n)$ of ${\mathbb R}^n$. The following statements hold for $\mathcal{H}^{n-1}$-almost all $(x,u)\in\Nor K$. First, we can choose an orthonormal basis $(a_1(x,u),\ldots,a_{n-1}(x,u))$ of $\textrm{Tan}^{n-1}(\mathcal{H}^{n-1}\fed \Nor K,(x,u))$ such that the $(n-1)$-vector $a_K(x,u):=a_1(x,u)\wedge\ldots\wedge a_{n-1}(x,u)$ satisfies \begin{equation}\label{orientations} \left\langle \mbox{$\bigwedge$}_{n-1}(\Pi_1+\varrho\,\Pi_2) a_K(x,u)\wedge u,\Omega_n\right\rangle>0 \end{equation} for all $\varrho>0$ and thus determines an orientation of $\textrm{Tan}^{n-1}(\mathcal{H}^{n-1}\fed \Nor(K),(x,u))$. Then the {\em normal cycle} associated with the convex body $K$ is the $(n-1)$-dimensional current in ${\mathbb R}^{2n}$ which is defined by $$ T_K:=\left(\mathcal{H}^{n-1}\fed\Nor K\right)\wedge a_K. $$ More generally, we can define $$ T_K(\varphi)=\int_{\Nor K}\langle a_K(x,u),\varphi(x,u)\rangle\, \mathcal{H}^{n-1}(d(x,u)) $$ for all $\mathcal{H}^{n-1}\fed \Nor K$-integrable functions $\varphi:{\mathbb R}^n\times{\mathbb R}^n\to \bigwedge^{n-1}{\mathbb R}^{2n}$. Here we use that $T_K$ is a rectifiable current, which has compact support, and thus $T_K$ can be defined for a larger class of functions than just for smooth differential forms.
Sometimes it is convenient to work with the orthonormal basis of the approximate tangent space of $\Nor K$ at $(x,u)$ that is given by $$ a_i(x,u):=\left( \alpha_i(x,u)\,b_i(x,u),\sqrt{1-\alpha_i(x,u)^2}\,b_i(x,u)\right), $$ where $(b_1(x,u),\ldots,b_{n-1}(x,u))$ is a suitably chosen orthonormal basis of $u^\perp$ (the orthogonal complement of the linear subspace spanned by $u$) such that $(b_1,\ldots,b_{n-1},u)$ has the same orientation as the standard basis $(e_1,\ldots,e_n)$ of ${\mathbb R}^n$, and $\alpha_i(x,u)\in [0,1]$ for $i=1,\ldots,n-1$. Note that the dependence of $a_i,b_i,\alpha_i$ on $K$ is not made explicit by our notation. The data $b_i, \alpha_i$, $i=1,\ldots,n-1$, are essentially uniquely determined (cf.~\cite[Proposition 3 and Lemma 2]{RZ05}). Moreover, we can assume that $b_i(x+\varepsilon u,u)=b_i(x,u)$, independent of $\varepsilon>0$, where $(x,u)\in\Nor K$ and $(x+\varepsilon u,u)\in\Nor K_\varepsilon$ with $K_\varepsilon:=K+\varepsilon B^n$. However, in general $\alpha_i(x+\varepsilon u,u)$ is not independent of $\varepsilon$ and will be positive for $\varepsilon>0$. See \cite{Zae86, RZ01, RZ05, Hug95, Hug98} for a geometric description of the numbers $\alpha_i(x,u)$ in terms of generalized curvatures and for arguments establishing the facts stated here.
\section{Proof of Theorem \ref{T1}}\label{sec3}
In order to obtain an upper bound for $|T_K-T_L|$, we first establish an upper bound for $|T_{A_\varepsilon}-T_A|$, for $A\in\{K,L\}$ and $\varepsilon\in [0,1]$, which is done in Lemma \ref{Lemma4.3}. Then we derive an upper bound for $|T_{K_\varepsilon}-T_{L_\varepsilon}|$ under the assumption that the Hausdorff distance of $K$ and $L$ is sufficiently small. This bound is provided in Lemma \ref{Lemma4.6}, which in turn is based on four preparatory lemmas.
\begin{lemma}\label{Lemma4.3} Let $K\in{\mathcal K}^n$ and $\varepsilon\in [0,1]$. Let $\varphi\in \mathcal{E}^{n-1}({\mathbb R}^{2n})$. Then $$
\left|T_{K_{\varepsilon}}(\varphi)-T_K(\varphi)\right|\le C(K,\varphi)\, \varepsilon, $$ where $C(K,\varphi)$ is a real constant, which depends on the maximum and the Lipschitz constant of $\varphi$ on $K_1\times {\mathbb S}^{n-1}$ and on $\mathcal{H}^{n-1}(\partial K_1)$. \end{lemma}
\begin{proof} We consider the bi-Lipschitz map $$ F_\varepsilon:\Nor K\to\Nor K_\varepsilon,\qquad (x,u)\mapsto (x+\varepsilon u,u). $$ The extension of $F_\varepsilon$ to all $(x,u)\in{\mathbb R}^{2n}$ by $F_\varepsilon(x,u):= (x+\varepsilon u,u)$ is differentiable for all $(x,u)\in{\mathbb R}^{2n}$. By \cite[Theorem 3.2.22 (1)]{Fed69}, for $\mathcal{H}^{n-1}$-almost all $(x,u)\in\Nor K$ the approximate Jacobian of $F_\varepsilon$ satisfies \begin{equation}\label{eq2.1}
\textrm{ap}\,J_{n-1}F_\varepsilon (x,u)=\left\|\mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u) a_K(x,u)\right\|>0, \end{equation} and the simple orienting $(n-1)$-vectors $a_K(x,u)$ and $a_{K_\varepsilon}(x+\varepsilon u,u)$ are related by \begin{equation}\label{eq2.2}
a_{K_\varepsilon}(x+\varepsilon u,u)=\frac{\bigwedge_{n-1}\textrm{ap}\,DF_\varepsilon (x,u) a_K(x,u)}{\left\|\bigwedge_{n-1}\textrm{ap}\,DF_\varepsilon (x,u) a_K(x,u)\right\|}. \end{equation} It follows from \eqref{orientations} that the orientations coincide. Thus, first using the coarea theorem \cite[Theorem 3.2.22]{Fed69} and then \eqref{eq2.1} and \eqref{eq2.2}, we get \begin{align*} T_{K_\varepsilon}(\varphi)&=\int_{\Nor K_\varepsilon}\langle a_{K_\varepsilon},\varphi\rangle\, {\rm d}\mathcal{H}^{n-1}\\ &=\int_{\Nor K}\langle a_{K_\varepsilon}\circ F_\varepsilon (x,u),\varphi\circ F_\varepsilon (x,u)\rangle\, \textrm{ap}\,J_{n-1}F_\varepsilon (x,u)\, \mathcal{H}^{n-1}({\rm d} (x,u))\\ &=\int_{\Nor K}\left\langle \mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u) a_K(x,u),\varphi\circ F_\varepsilon (x,u)\right\rangle \mathcal{H}^{n-1}({\rm d} (x,u)). \end{align*} By the triangle inequality, we obtain \begin{eqnarray*}
\left|T_{K_\varepsilon}(\varphi)-T_K(\varphi)\right| &\le& \int_{\Nor K} \Big\{\left|\left\langle\left( \mbox{$\bigwedge$}_{n-1} \textrm{ap}\,DF_\varepsilon (x,u)-\mbox{$\bigwedge$}_{n-1}\textrm{id}\right) a_K(x,u),\varphi\circ F_\varepsilon (x,u)\right\rangle\right|\\
& & +\left|\left\langle a_K(x,u),\varphi(x+\varepsilon u,u)-\varphi(x,u)\right\rangle\right|\Big\}\,
\mathcal{H}^{n-1}({\rm d} (x,u)). \end{eqnarray*} We have \begin{eqnarray*}
& & \left|\left\langle\left( \mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u)-\mbox{$\bigwedge$}_{n-1}\textrm{id}\right) a_K(x,u),\varphi\circ F_\varepsilon (x,u)\right\rangle\right|\\
& & \le |\varphi(x+\varepsilon u,u)|\,
\left|\left( \mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u)-\mbox{$\bigwedge$}_{n-1}\textrm{id}\right) a_K(x,u)\right|, \end{eqnarray*}
where we used \eqref{eq3}. Now $a_K(x,u)$ is of the form $\mbox{$\bigwedge$}_{i=1}^{n-1}(v_i,w_i)$ with suitable $(v_i,w_i)\in {\mathbb R}^{2n}$ and $|v_i|^2+|w_i|^2=1$. Moreover, we have $DF_\varepsilon(x,u)(v,w)=(v+\varepsilon w,w)$, for all $(v,w)\in {\mathbb R}^{2n}$. Writing $z_i^0:=v_i$, $z_i^1:=w_i$, we have \begin{align*}
& \left| \left( \mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u)-\mbox{$\bigwedge$}_{n-1}\textrm{id}\right) a_K(x,u)\right|
=\left|\bigwedge_{i=1}^{n-1}(v_i+\varepsilon w_i,w_i)-\bigwedge_{i=1}^{n-1}(v_i,w_i)\right| \\
&= \left| \sum_{\alpha_i\in\{0,1\}} \varepsilon^{\sum\alpha_i} \bigwedge_{i=1}^{n-1}(z_i^{\alpha_i},w_i) - \bigwedge_{i=1}^{n-1}(z_i^0,w_i)\right|
\le \varepsilon \sum_{\alpha_i\in\{0,1\}, \,\sum \alpha_i\ge 1} \, \left|\bigwedge_{i=1}^{n-1}(z_i^{\alpha_i},w_i) \right| \le c(n)\varepsilon, \end{align*}
where we used \eqref{eq2} and the fact that $|(v_i,w_i)|=1$ and $|(w_i,w_i)|\le 2$. We deduce that $$
|\varphi(x+\varepsilon u,u)|\, \left|\left( \mbox{$\bigwedge$}_{n-1}\textrm{ap}\,DF_\varepsilon (x,u)-\mbox{$\bigwedge$}_{n-1}\textrm{id}\right) a_K(x,u)\right| \le C_1(K,\varphi)\varepsilon. $$ Furthermore, again by \eqref{eq3} we get $$
\left|\left\langle a_K(x,u),\varphi(x+\varepsilon u,u)-\varphi(x,u)\right\rangle\right|
\le |\varphi(x+\varepsilon u,u)-\varphi(x,u)|\le C_2(K,\varphi)\, \varepsilon . $$ Thus we conclude that $$
\left|T_{K_\varepsilon}(\varphi)-T_K(\varphi)\right|\le C_3(K,\varphi)\,\varepsilon\, \mathcal{H}^{n-1}(\Nor K). $$ Since $F:\partial K_1\to\Nor K$, $z\mapsto (p(K,z),z-p(K,z))$, is Lipschitz with Lipschitz constant bounded from above by 3, the assertion follows. \end{proof}
A convex body $K\in{\mathcal K}^n$ is said to be $\varepsilon$-smooth (for some $\varepsilon>0$), if $K=K'+\varepsilon B^n$ for some $K'\in {\mathcal K}^n$. For a nonempty set $A\subset{\mathbb R}^n$, we define the distance from $A$ to $x\in{\mathbb R}^n$ by $d(A,x):= \inf\{|a-x|:a\in A\}$. The signed distance is defined by $d^*(A,x):=d(A,x)-d({\mathbb R}^n\setminus A,x)$, $x\in{\mathbb R}^n$, if $A,{\mathbb R}^n \setminus A\neq \emptyset$. If $K$ is $\varepsilon$-smooth, then $\partial K$ has positive reach. More precisely, if $x\in {\mathbb R}^n$ satisfies $d(\partial K,x)<\varepsilon$, then there is a unique point $p(\partial K,x)\in\partial K$ such that $d(\partial K,x)=|p(\partial K,x)-x|$.
\begin{lemma}\label{Lemma4.4} Let $\varepsilon\in(0,1)$ and $\delta\in(0,\varepsilon/2)$. Let $K,L\in{\mathcal K}^n$ be $\varepsilon$-smooth and assume that $d_H(K,L)\le\delta$. Then $$ p:\partial K\to \partial L,\qquad x\mapsto p(\partial L,x), $$
is well-defined, bijective, bi-Lipschitz with $\textrm{\rm Lip}(p)\le \varepsilon/(\varepsilon-\delta)$, and $|p(x)-x|\le \delta$ for all $x\in\partial K$. \end{lemma}
\begin{proof} Since $d_H(K,L)\le \delta$, we have $K\subset L+\delta B^n$, $L\subset K+\delta B^n$, and a separation argument yields that \begin{equation}\label{eqneighbour} \{x\in L:d(\partial L,x)\ge \delta\}\subset K. \end{equation} This shows that $\partial K\subset \{z\in{\mathbb R}^n:d(\partial L,z)\le \delta\}$
and therefore the map $p$ is well-defined on $\partial K$ and $|p(x)-x|\le \delta$ for all $x\in \partial K$. By \cite[Theorem 4.8 (8)]{Fed59} it follows that $\textrm{Lip}(p)\le \varepsilon/(\varepsilon-\delta)$. Since $L$ is $\varepsilon$-smooth, for $y\in \partial L$ there is a unique exterior unit normal of $L$ at $y$, which we denote by $u=u(L,y)=:u_L(y)$. Put $y_0:=y-\varepsilon u$ and note that $y_0+(\varepsilon-\delta)B^n\subset K\cap L$ by \eqref{eqneighbour}. Then $x\in\partial K$ is uniquely determined by the condition $\{x\}=\left(y_0+[0,\infty)u\right)\cap \partial K$ and satisfies $p(x)=y$. This shows that $p$ is surjective.
Now let $x_1,x_2\in\partial K$ satisfy $p(x_1)=p(x_2)=:p_0\in\partial L$. Since there is a ball $B$ of radius $\varepsilon$ with $p_0\in B\subset L$, the points $x_1,x_2\in\partial K$ are on the line through $p_0$ and the center of $B$. By \eqref{eqneighbour}, they cannot be on different sides of $p_0$, hence $x_1=x_2$. This shows that the map $p$ is also injective. If $d^*(\partial K,\cdot):{\mathbb R}^n\to\partial K$ denotes the signed distance function of $\partial K$, then $q:\partial L\to\partial K$, $z\mapsto z-d^*(\partial K,z)u_L(z)$, is the inverse of $p$. Since the signed distance function is Lipschitz, Lemma \ref{Lespaet} shows that $q$ is Lipschitz as well. \end{proof}
The following lemma provides a simple argument for the fact that the spherical image map of an $\varepsilon$-smooth convex body is Lipschitz with Lipschitz constant at most $1/\varepsilon$. A less explicit assertion is contained in \cite[Hilfssatz 1]{Lei86}.
\begin{lemma}\label{Lespaet} Let $K\in {\mathcal K}^n$ be $\varepsilon$-smooth, $\varepsilon>0$. Then the spherical image map $u_K$ is Lipschitz with Lipschitz constant $1/\varepsilon$. \end{lemma}
\begin{proof} Let $x,y\in \partial K$, and define $u:=u_K(x)$, $v:=u_K(y)$. Then $$ x-\varepsilon u+\varepsilon v\in x-\varepsilon u+ \varepsilon B^n \subset K, $$ and hence $( x-\varepsilon u+\varepsilon v-y)\bullet v \leq 0$. This yields \begin{equation}\label{Gluno} \varepsilon ( v-u)\bullet v\leq (y-x)\bullet v. \end{equation} By symmetry, we also have $\varepsilon ( u-v)\bullet u\leq (x-y)\bullet u$, and therefore \begin{equation}\label{Gldue} \varepsilon ( v-u)\bullet(-u)\leq ( y-x)\bullet (-u). \end{equation} Addition of (\ref{Gluno}) and (\ref{Gldue}) yields $$
\varepsilon\,|v-u|^{2}\leq ( y-x)\bullet(v-u)\leq |y-x|\,|v-u|, $$ which implies the assertion. \end{proof}
\begin{lemma}\label{Lemma4.5} Let $\varepsilon\in(0,1)$ and $\delta\in(0,\varepsilon/2)$. Let $K,L\in{\mathcal K}^n$ be $\varepsilon$-smooth and assume that $d_H(K,L)\le\delta$. Put $p(x):=p(\partial L,x)$ for $x\in\partial K$. Then $$ G:\Nor K\to \Nor L,\qquad (x,u)\mapsto (p(x),u_L(p(x))), $$
is bijective, bi-Lipschitz with $\textrm{\rm Lip}(G)\le 2/(\varepsilon-\delta)\le 4/\varepsilon$, and $|G(x,u)-(x,u)|\le \delta+2\sqrt{\delta/\varepsilon}$ for all $(x,u)\in\Nor K$. \end{lemma}
\begin{proof} It follows from Lemma \ref{Lemma4.4} that $G$ is bijective. Then, for $(x,u),(y,v)\in\Nor K$ we get \begin{align*}
|G(x,u)-G(y,v)|&\le |p(x)-p(y)|+|u_L(p(x))-u_L(p(y))|\\
&\le \frac{\varepsilon}{\varepsilon-\delta}|x-y|+\frac{1}{\varepsilon}\frac{\varepsilon}{\varepsilon-\delta}|x-y|\le \frac{\varepsilon+1}{\varepsilon-\delta}
|x-y|\\
&\le \frac{2}{\varepsilon-\delta}|(x,u)-(y,v)|, \end{align*} where we have used again Lemma \ref{Lemma4.4} and Lemma \ref{Lespaet}. Let $x\in\partial K$ and $z:=p(x)\in\partial L$. We want to bound $ u_L(z)\bullet u_K(x)$ from below. If $x\notin L$, then $\textrm{conv}(\{x\}\cup(z-\varepsilon u_L(z)+(\varepsilon-\delta)B^n))\subset K$, and therefore $$
u_L(z)\bullet u_K(x) \ge \frac{\varepsilon-\delta}{\varepsilon+\delta}\ge 1-\frac{2\delta}{\varepsilon}. $$ If $x\in L$, then in a similar way we obtain $$
u_L(z)\bullet u_K(x) \ge \frac{\varepsilon-\delta}{\varepsilon}\ge 1-\frac{\delta}{\varepsilon}, $$ hence \begin{equation}\label{angle}
u_L(z)\bullet u_K(x) \ge 1-\frac{2\delta}{\varepsilon} \end{equation} holds for all $x\in\partial K$. Thus $$
|u_L(z)-u_K(x)|\le 2\sqrt{\delta/\varepsilon}, $$ which finally implies that, for all $(x,u)\in \Nor K$, $$
|G(x,u)-(x,u)|\le |p(x)-x|+|u_L(p(x))-u_K(x)|\le \delta+2\sqrt{\delta/\varepsilon}. $$ Since $G^{-1}:\Nor L\to\Nor K$ is given by $G^{-1}(z,u)=(q(z),u_K(q(z)))$, it follows that also $G^{-1}$ is Lipschitz. \end{proof}
Next we show that, under the assumptions of the subsequent lemma, $\mbox{$\bigwedge$}_{n-1}DG(x,u)$ is an orientation preserving map from the approximate tangent space of $\Nor K$ to the approximate tangent space of $\Nor L$. It seems that a corresponding fact is not provided in the proofs of related assertions in the literature.
\begin{lemma}\label{orient} Let $\varepsilon\in(0,1)$ and $\delta\in(0,\varepsilon/(4n))$. Let $K,L\in{\mathcal K}^n$ be $\varepsilon$-smooth and assume that $d_H(K,L)\le\delta$. Then, for $\mathcal{H}^{n-1}$-almost all $(x,u)\in \Nor K$, the $(n-1)$-vector $\mbox{$\bigwedge$}_{n-1}DG(x,u) a_K(x,u)\in \textrm{\rm Tan}^{n-1}(\mathcal{H}^{n-1}\fed \Nor L,G(x,u))$ has the same orientation as $a_L(G(x,u))$. \end{lemma}
\begin{proof}
Let $x\in\partial K$, $u:=u_K(x)$, and $\bar{x}:=p(x)$, hence $d(\partial L,x)=|x-\bar x|$. The orientation of $\textrm{Tan}^{n-1}(\partial K,x)$ is determined by an arbitrary orthonormal basis $b_1(x),\ldots,b_{n-1}(x)$ of $u^\perp$ with $\Omega_n(b_1(x),\ldots,b_{n-1}(x),u)=1$. Similarly, any orthonormal basis $\bar b_1(\bar x),\ldots,\bar b_{n-1}(\bar x),\bar u$ with $\bar u:=u_L(p(x))$ determines the orientation of $\textrm{Tan}^{n-1}(\partial L,p(x))$. Since $G$ is bi-Lipschitz, we can assume that $(x,u)\in\Nor K$ is such that all differentials exist that are encountered in the proof. Moreover, we can also assume that $\mbox{$\bigwedge$}_{n-1}DG(x,u) a_K(x,u)$ spans $ \textrm{\rm Tan}^{n-1}(\mathcal{H}^{n-1}\fed \Nor L,G(x,u))$, where we write again $G$ for a Lipschitz extension of the given map $G$ to ${\mathbb R}^{2n}$. In the following, we put $b_i:=b_i(x)$ and $\bar b_i:=\bar b_i(\bar x)$ for $i=1,\ldots,n-1$.
The differentials of the maps $\Nor K\to\partial K$, $(x,u)\mapsto x$, and $\partial L\to\Nor L$, $z\mapsto (z,u_L(z))$, are orientation preserving, which follows for instance from the discussion at the end of Section 2. Hence, it remains to be shown that the differential of $p:\partial K\to\partial L$, $x\mapsto p(x)$, is orientation preserving, that is, $$ \Delta:=\Omega_n(Dp(x)(b_1),\ldots,Dp(x)(b_{n-1}),\bar u)>0. $$
First, we assume that $x\neq \bar x$, that is, $x\notin \partial L$. Since $Dp(x)(\bar u)=o$, we get $$ Dp(x)(b_i)=\sum_{j=1}^{n-1} b_i\bullet \bar b_j \, Dp(x)(\bar b_j), $$ and thus $$ \Delta=\det(B)\,\Omega_n(Dp(x)(\bar b_1),\ldots,Dp(x)(\bar b_{n-1}),\bar u), $$ where $B=(B_{ij})$ with $B_{ij}:= b_i\bullet \bar b_j $ for $i,j\in\{1,\ldots,n-1\}$. We choose $\bar b_1,\ldots,\bar b_{n-1}$ as principal directions of curvature of $\partial L$ at $\bar x=p(x)$. Then $Dp(x)(\bar b_i)=\tau_i\, \bar b_i$ with $$
\tau_i:=1-d(\partial L,x)k_i\left(\partial L,\bar x,\frac{x-\bar x}{|x-\bar x|}\right)>0, $$ for $i=1,\ldots,n-1$. Here we use that $L$ is $\varepsilon$-smooth, hence $\partial L$ has positive reach, $d(\partial L,x)<\varepsilon$ and $$
\left|k_i\left(\partial L,\bar x,\frac{x-\bar x}{|x-\bar x|}\right)\right|\le 1/\varepsilon. $$ Hence it follows that $\Delta>0$ if we can show that $\det(B)>0$. Let $\tilde B=(\tilde B_{ij})$ be defined by $\tilde B_{ij}:=b_{ij}$, $\tilde B_{in}:= b_i \bullet \bar u $, $\tilde B_{nj}:= u\bullet \bar b_j $, and $\tilde B_{nn}:= u\bullet \bar u $, for $i,j\in\{1,\ldots,n-1\}$. Then \begin{align*} 1&=\Omega_n(b_1,\ldots,b_{n-1},u)\, \Omega_n(\bar b_1,\ldots,\bar b_{n-1},\bar u)=\det(\tilde B)\\
&\le u\bullet \bar u\, \det(B)+\sum_{i=1}^{n-1}| b_i\bullet \bar u |\cdot 1 \le u \bullet \bar u \det(B)+\sqrt{n-1}\sqrt{1- (u\bullet \bar u) ^2}. \end{align*} From \eqref{angle} and our assumptions, we get $u\bullet \bar u\ge 1-(2\delta)/\varepsilon\ge 1-1/(2n)$, and therefore $$ \sqrt{1- (u\bullet \bar u) ^2}\le \sqrt{1/n}. $$ Thus $$ 1< u\bullet \bar u \,\det(B) +1, $$ which implies that $\det(B)>0$.
Finally, we have to consider the case where $x\in\partial L$. For $\mathcal{H}^{n-1}$-almost all $x\in \partial K\cap \partial L$, we have $\textrm{\rm Tan}^{n-1}(\mathcal{H}^{n-1}\fed (\partial K\cap\partial L),x)=u^\perp$ and $Dp(x)=\textrm{id}_{u^\perp}$, since $p(z)=z$ for all $z\in \partial K\cap\partial L$. Hence, $\Delta=\Omega_n(b_1,\ldots,b_{n-1},\bar u)=u\bullet \bar u>0$. \end{proof}
\begin{lemma}\label{Lemma4.6} Let $\varepsilon\in(0,1)$ and $\delta\in(0,\varepsilon/(4n))$. Let $K,L\in{\mathcal K}^n$ be $\varepsilon$-smooth and assume that $d_H(K,L)\le\delta$. Let $M\subset{\mathbb R}^{2n}$ be a compact convex set containing $K_{1-\varepsilon}\times {\mathbb S}^{n-1}$ and $L_{1-\varepsilon}\times {\mathbb S}^{n-1}$ in its interior. Then $$
|T_K(\varphi)-T_L(\varphi)|\le C(M,\varphi)\,\left(\frac{4}{\varepsilon}\right)^{n-1}\,\left(\delta+2\sqrt{\delta/\varepsilon}\right) $$ for $\varphi\in\mathcal{E}^{n-1}({\mathbb R}^{2n})$, where $C(M,\varphi)$ is a constant which depends on the sup-norm and the Lipschitz constant of $\varphi$ on $M$, and on $\mathcal{H}^{n-1}(\partial K_1)$. \end{lemma}
\begin{proof} Let $G$ be as in Lemma \ref{Lemma4.5} (or a Lipschitz extension to the whole space with the same Lipschitz constant). Then \cite[Theorem 4.1.30]{Fed69} implies that $$ T_L=G_\sharp T_K, $$ since $\mbox{$\bigwedge$}_{n-1}DG$ preserves the orientation of the orienting $(n-1)$-vectors, by Lemma \ref{orient}. (In \cite{RZ01} a corresponding fact is stated without further comment.) Recall the definitions of the dual flat metric $\mathbf{F}_{M}$ from \cite[4.1.12]{Fed69} and of the mass $\mathbf{M}$ (of a current) from \cite[p.~358]{Fed69}. Using \cite[4.1.14]{Fed69}, $\partial T_K=0$, the fact that $T_K$ has compact support contained in the interior of $M$ and Lemma \ref{Lemma4.5}, we get \begin{align*}
\mathbf{F}_{M}(T_L-T_K)&=\mathbf{F}_{M}(G_\sharp T_K-T_K)\le \mathbf{M}\,( T_K)\cdot \|G-\textrm{id}\|_{\Nor K,\infty}\cdot \left(\frac{4}{\varepsilon}\right)^{n-1}\\ &\le \mathcal{H}^{n-1}(\partial K_1)\, \left( \frac{4}{\varepsilon}\right)^{n-1}\, \left(\delta+2\sqrt{\delta/\varepsilon}\right), \end{align*}
where $\|G-\textrm{id}\|_{\Nor K,\infty}:=\sup\{|G(x,u)-(x,u)|:(x,u)\in\Nor K\}$. The assertion now follows from the
definition of $\mathbf{F}_{M}$, since $\|d\varphi\|$ can be bounded in terms of the sup-norm and the Lipschitz constant of $\varphi$ on $M$. \end{proof}
Now we are in a position to complete the proof of Theorem \ref{T1}.
\begin{proof}[Proof of Theorem $\ref{T1}$] Let $\varphi\in\mathcal{E}^{n-1}({\mathbb R}^{2n})$. Let $\delta:=d_H(K,L)>0$ and $\varepsilon:=\delta^{\frac{1}{2n+1}}$. Assume that $\delta<(4n)^{-\frac{2n+1}{2n}}$. Then $\delta<\varepsilon/(4n)$. Lemma \ref{Lemma4.3} implies that \begin{align*}
\left|T_K(\varphi)-T_{K_{\varepsilon}}(\varphi)\right|&\le C(M,\varphi)\, \varepsilon,\\
\left|T_L(\varphi)-T_{L_{\varepsilon}}(\varphi)\right|&\le C(M,\varphi)\, \varepsilon. \end{align*} Since $K_\varepsilon$ and $L_\varepsilon$ are $\varepsilon$-smooth, $d_H(K_\varepsilon,L_\varepsilon)=\delta$, $(K_{\varepsilon})_{1-\varepsilon}=K$ and $(L_{\varepsilon})_{1-\varepsilon}=L$, Lemma \ref{Lemma4.6} shows that $$
|T_{K_{\varepsilon}}(\varphi)-T_{L_{\varepsilon}}(\varphi)|\le C(M,\varphi)\,\left(\frac{4}{\varepsilon}\right)^{n-1}\,\left(\delta+2\sqrt{\delta/\varepsilon}\right). $$ The triangle inequality then yields $$
|T_K(\varphi)-T_L(\varphi)|\le C_4(M,\varphi)\,\left(\varepsilon+\frac{\delta}{\varepsilon^{n-1}}+\frac{1}{\varepsilon^{n-1}}\sqrt{\frac{\delta}{\varepsilon}}\right) \le C_5(M,\varphi)\, \delta^{\frac{1}{2n+1}}. $$ If $\delta\ge (4n)^{-\frac{2n+1}{2n}}$, we simply adjust the constant. \end{proof}
\section{Proof of Theorem \ref{T2}}\label{sec4}
In the theory of convex bodies, Federer's curvature measures are supplemented by the area measures, which are finite Borel measures on the unit sphere (and were, in fact, introduced more than 20 years earlier). The two series of measures are generalized by the support measures. We briefly recall their definition (see \cite[Chap.~4]{Sch93}).
For $K\in{\mathcal K}^n$, we have already used the metric projection $p(K,\cdot)$ and the vector function $u(K,x):= (x-p(K,x))/d(K,x)$, where $d(K,x):= |x-p(K,x)|$ denotes the distance of the point $x$ from $K$. We also write $p(K,\cdot)=:p_K$, $u(K,\cdot)=:u_K$ and $d(K,\cdot)=:d_K$. For $\rho>0$, we set $K^\rho:=K_\rho\setminus K$, where $K_\rho=K+\rho B^n$ is the already defined parallel body of $K$ at distance $\rho$. The product space ${\mathbb R}^n\times {\mathbb S}^{n-1}$ is denoted by $\Sigma$. For given $K$, the map $f_\rho:K^\rho\to\Sigma$ is defined by $f_\rho(x):= (p_K(x),u_K(x))$ for $x\in K^\rho$, and $\mu_{K,\rho}=\mu_\rho(K,\cdot)$ is the image measure of ${\mathcal H}^n\fed K^\rho$ under $f_\rho$. This is a finite Borel measure on $\Sigma$, concentrated on ${\rm Nor}\,K$. The {\em support measures} $\Lambda_0(K,\cdot),\dots,\Lambda_{n-1}(K,\cdot)$ of $K$ can be defined by \begin{equation}\label{4.9} \mu_{K,\rho} =\sum_{i=0}^{n-1} \rho^{n-i}\kappa_{n-i}\Lambda_i(K,\cdot). \end{equation} Thus, the normalization is different from \cite[(4.2.4)]{Sch93}; the connection is given by $n\kappa_{n-i} \Lambda_i(K,\cdot)=\binom{n}{i}\Theta_i(K,\cdot)$. The support measures have the property of weak continuity: if a sequence $(K_j)_{j\in{\mathbb N}}$ of convex bodies converges to a convex body $K$ in the Hausdorff metric, then the sequence $(\Lambda_i(K_j,\cdot))_{j\in{\mathbb N}}$ converges weakly to $\Lambda_i(K,\cdot)$. The topology of weak convergence can be metrized by the bounded Lipschitz metric $d_{bL}$ or the L\'{e}vy--Prokhorov metric $d_{LP}$ (see, e.g., Dudley \cite[Sec.~11.3]{Dud02}). Therefore, the question arises whether the weak continuity of the support measures can be improved to H\"older continuity with respect to one of these metrics.
For bounded real functions $f$ on $\Sigma$ we define
$$ \|f\|_L:=\sup_{x\not= y}\frac{|f(x)-f(y)|}{|x-y|},\qquad \|f\|_\infty:=\sup_x |f(x)|.$$ For finite Borel measures $\mu,\nu$ on $\Sigma$, their {\em bounded Lipschitz distance} is defined by
$$ d_{bL}(\mu,\nu):= \sup\left\{ \left|\int_\Sigma f\,{\rm d}\mu- \int_\Sigma f\,{\rm d}\nu\right|:f:\Sigma\to{\mathbb R},\; \|f\|_L\le 1,\;\|f\|_\infty\le 1\right\}.$$
The following lemma is modeled after Proposition 4.1 of Chazal, Cohen--Steiner and M\'{e}rigot \cite{CCM10}. Under the restriction to convex bodies, it extends the latter to the measures $\mu_{K,\rho}$.
\begin{lemma}\label{L4.1} If $K,L\in{\mathcal K}^n$ are convex bodies and $\rho>0$, then $$
d_{bL}(\mu_{K,\rho},\mu_{L,\rho}) \le \int_{K^\rho\cap L^\rho} |p_K-p_L|\,{\rm d}{\mathcal H}^n+ \int_{K^\rho\cap L^\rho} |u_K-u_L|\,{\rm d}{\mathcal H}^n +{\mathcal H}^n(K^\rho\triangle L^\rho), $$ where $\triangle$ denotes the symmetric difference. \end{lemma}
\begin{proof}
Let $f:\Sigma\to{\mathbb R}$ be a function with $\|f\|_L\le 1$ and $\|f\|_\infty\le 1$. Using the transformation formula for integrals and the properties of $f$, we obtain \begin{eqnarray*}
& & \left|\int_\Sigma f\,{\rm d}\mu_{K,\rho} -\int_\Sigma f\,{\rm d}\mu_{L,\rho} \right|\\
& & = \left|\int_{K^\rho} f\circ(p_K,u_K)\,{\rm d}{\mathcal H}^n -\int_{L^\rho} f\circ(p_L,u_L)\,{\rm d}{\mathcal H}^n\right|\\
& & \le \int_{K^\rho\cap L^\rho} \left|f\circ(p_K,u_K)-f\circ(p_L,u_L)\right|\,{\rm d}{\mathcal H}^n\\
& & \hspace{4mm}+\int_{K^\rho\setminus L^\rho} \left|f\circ(p_K,u_K)\right|\,{\rm d}{\mathcal H}^n +\int_{L^\rho\setminus K^\rho} \left|f\circ(p_L,u_L)\right|\,{\rm d}{\mathcal H}^n \\
& & \le \int_{K^\rho\cap L^\rho} |(p_K,u_K)-(p_L,u_L)|\,{\rm d}{\mathcal H}^n + \int_{K^\rho\setminus L^\rho} 1\,{\rm d}{\mathcal H}^n +\int_{L^\rho\setminus K^\rho} 1\,{\rm d}{\mathcal H}^n \\
& & \le \int_{K^\rho\cap L^\rho} (|p_K-p_L|+|u_K-u_L|)\,{\rm d}{\mathcal H}^n +{\mathcal H}^n(K^\rho\triangle L^\rho), \end{eqnarray*} from which the assertion follows. \end{proof}
\noindent{\em Proof of Theorem} \ref{T2}. We assume that $K,L\in{\mathcal K}^n$ and $d_H(K,L)=:\delta<1$. Let $R$ be the radius of a ball containing $K_2$ and $L_2$. For $0<\rho\le 1$ we use Lemma \ref{L4.1} (where for convex bodies, the estimation of the first and the third term on the right-hand side is easier than for the case of general compact sets considered in \cite{CCM10}). First, from Lemma 1.8.9 in \cite{Sch93} we get \begin{equation}\label{4.10}
\int_{K^\rho\cap L^\rho} |p_K-p_L|\,{\rm d}{\mathcal H}^n\le\sqrt{5D}{\mathcal H}^n(K^\rho\cap L^\rho)\sqrt{\delta} \le C_1(R)\sqrt{\delta}, \end{equation} where $D={\rm diam}(K_\rho\cup L_\rho)$ and the constant $C_1(R)$ depends only on $R$.
About the distance function $d_K$, it is well known that
$$ \sup_{x\in{\mathbb R}^n}|d_K(x)-d_L(x)|=d_H(K,L)=\delta$$ and that $$ \nabla d_K=u_K \qquad\mbox{on }{\mathbb R}^n\setminus K.$$ Therefore, it follows immediately from Theorem 3.5 of Chazal, Cohen--Steiner and M\'{e}rigot \cite{CCM10} (applied to $E={\rm int}(K^\rho\cap L^\rho)$) that \begin{equation}\label{4.11}
\int_{K^\rho\cap L^\rho} |u_K-u_L|\,{\rm d}{\mathcal H}^n\le C_2(R)\sqrt{\delta}. \end{equation}
For the estimation of ${\mathcal H}^n(K^\rho\triangle L^\rho)$, let $x\in K^\rho\setminus L^\rho$; then $x\in K_\rho\setminus K$ and $x\notin L_\rho\setminus L$. If $x\in L$, then $d(K,x)\le\delta$, hence $x\in K_\delta\setminus K$. If $x\notin L$, then $x\notin L_\rho$ but $x\in K_\rho$, $K_\rho\subset(L_\delta)_\rho=L_{\rho+\delta}$, and hence $x\in L_{\rho+\delta}\setminus L_\rho$. It follows that $$ K^\rho \setminus L^\rho \subset (K_\delta\setminus K) \cup (L_{\rho+\delta}\setminus L_\rho)$$ and hence \begin{eqnarray*} {\mathcal H}^n(K^\rho\setminus L^\rho) &\le & {\mathcal H}^n(K_\delta)-{\mathcal H}^n(K) + {\mathcal H}^n(L_{\rho+\delta}) -{\mathcal H}^n(L_\rho)\\ &\le &C_3(R)\delta\le C_3(R)\sqrt{\delta}. \end{eqnarray*} Here $K$ and $L$ can be interchanged, and together with (\ref{4.10}), (\ref{4.11}) and Lemma \ref{L4.1} this gives \begin{equation}\label{4.12} d_{bL}(\mu_{K,\rho},\mu_{L,\rho}) \le C_4(R)\sqrt{\delta}. \end{equation}
To deduce an estimate for the support measures, we apply the usual procedure (e.g., \cite{Sch93}, p. 202) and choose in (\ref{4.9}) for $\rho$ each of the $n$ fixed values $\rho_j=j/n$, $j=1,\dots,n$, and solve the resulting system of linear equations (which has a non-zero Vandermonde determinant), to obtain representations $$ \Lambda_i(K,\cdot) = \sum_{j=1}^n a_{i,j}\mu_{K,\rho_j},\qquad i=0,\dots,n-1,$$ with constants $a_{ij}$ depending only on $i,j$. Using the definition of the bounded Lipschitz metric, we deduce that \begin{equation}\label{4.13}
d_{bL}(\Lambda_i(K,\cdot),\Lambda_i(L,\cdot)) \le \sum_{j=1}^n|a_{ij}|d_{bL}(\mu_{K,\rho_j},\mu_{L,\rho_j}) \le C(R)\sqrt{\delta}. \end{equation} This completes the proof of Theorem \ref{T2}. \qed
\noindent Authors' addresses:\\[2mm] Daniel Hug\\ Karlsruhe Institute of Technology, Department of Mathematics\\ D-76128 Karlsruhe, Germany\\ E-mail: [email protected]\\[3mm] Rolf Schneider\\ Mathematisches Institut, Albert-Ludwigs-Universit{\"a}t\\ D-79104 Freiburg i. Br., Germany\\ E-mail: [email protected]
\end{document} | arXiv |
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".[1]
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer n, the ring of real square matrices of order n is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.
An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (unital) ring that is also a vector space.
Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra.
Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.
Definition and motivation
Motivating examples
Algebra vector space bilinear operator associativity commutativity
complex numbers $\mathbb {R} ^{2}$ product of complex numbers
$\left(a+ib\right)\cdot \left(c+id\right)$
Yes Yes
cross product of 3D vectors $\mathbb {R} ^{3}$ cross product
${\vec {a}}\times {\vec {b}}$
No No (anticommutative)
quaternions $\mathbb {R} ^{4}$ Hamilton product
$(a+{\vec {v}})(b+{\vec {w}})$
Yes No
polynomials $\mathbb {R} [X]$ polynomial multiplication Yes Yes
square matrices $\mathbb {R} ^{n\times n}$ matrix multiplication Yes No
Definition
Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (that is, if x and y are any two elements of A, then x · y is an element of A that is called the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, z in A , and all elements (often called scalars) a and b in K:
• Right distributivity: (x + y) · z = x · z + y · z
• Left distributivity: z · (x + y) = z · x + z · y
• Compatibility with scalars: (ax) · (by) = (ab) (x · y).
These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative, although some authors use the term algebra to refer to an associative algebra.
When a binary operation on a vector space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.
Basic concepts
Algebra homomorphisms
Main article: Algebra homomorphism
Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x, y in A. The space of all K-algebra homomorphisms between A and B is frequently written as
$\mathbf {Hom} _{K{\text{-alg}}}(A,B).$
A K-algebra isomorphism is a bijective K-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.
Subalgebras and ideals
Main article: Substructure (mathematics)
A subalgebra of an algebra over a field K is a linear subspace that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a K-algebra A is a subalgebra if for every x, y in L and c in K, we have that x · y, x + y, and cx are all in L.
In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.
A left ideal of a K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.
1. x + y is in L (L is closed under addition),
2. cx is in L (L is closed under scalar multiplication),
3. z · x is in L (L is closed under left multiplication by arbitrary elements).
If (3) were replaced with x · z is in L, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. The term ideal on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to L being a linear subspace of A. It follows from condition (3) that every left or right ideal is a subalgebra.
This definition is different from the definition of an ideal of a ring, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).
Extension of scalars
Main article: Extension of scalars
If we have a field extension F/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product $V_{F}:=V\otimes _{K}F$. So if A is an algebra over K, then $A_{F}$ is an algebra over F.
Kinds of algebras and examples
Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.
Unital algebra
An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.
Zero algebra
An algebra is called zero algebra if uv = 0 for all u, v in the algebra,[2] not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.
One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero. That is, if λ, μ ∈ K and u, v ∈ V, then (λ + u) (μ + v) = λμ + (λv + μu). If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E1, ..., En] by the ideal generated by the EiEj for every pair (i, j).
An example of unital zero algebra is the algebra of dual numbers, the unital zero R-algebra built from a one dimensional real vector space.
These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or modules. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x1, ..., xn] over a field. The construction of the unital zero algebra over a free R-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.
Associative algebra
Main article: Associative algebra
Examples of associative algebras include
• the algebra of all n-by-n matrices over a field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication.
• group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication.
• the commutative algebra K[x] of all polynomials over K (see polynomial ring).
• algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.
• Incidence algebras are built on certain partially ordered sets.
• algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied in functional analysis.
Non-associative algebra
Main article: Non-associative algebra
A non-associative algebra[3] (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map $A\times A\rightarrow A$. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".
Examples detailed in the main article include:
• Euclidean space R3 with multiplication given by the vector cross product
• Octonions
• Lie algebras
• Jordan algebras
• Alternative algebras
• Flexible algebras
• Power-associative algebras
Algebras and rings
The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism
$\eta \colon K\to Z(A),$
where Z(A) is the center of A. Since η is a ring homomorphism, then one must have either that A is the zero ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication
$K\times A\to A$
given by
$(k,a)\mapsto \eta (k)a.$
Given two such associative unital K-algebras A and B, a unital K-algebra homomorphism f: A → B is a ring homomorphism that commutes with the scalar multiplication defined by η, which one may write as
$f(ka)=kf(a)$
for all $k\in K$ and $a\in A$. In other words, the following diagram commutes:
${\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}$
Structure coefficients
Main article: Structure constants
For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisfies the algebra laws.
Thus, given the field K, any finite-dimensional algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n3 structure coefficients ci,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:
$\mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}$
where e1,...,en form a basis of A.
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
In mathematical physics, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as
eiej = ci,jkek.
If you apply this to vectors written in index notation, then this becomes
(xy)k = ci,jkxiyj.
If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
Classification of low-dimensional unital associative algebras over the complex numbers
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.[4]
There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element,
$\textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.$
It remains to specify
$\textstyle aa=1$ for the first algebra,
$\textstyle aa=0$ for the second algebra.
There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify
$\textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0$ for the first algebra,
$\textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0$ for the second algebra,
$\textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0$ for the third algebra,
$\textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b$ for the fourth algebra,
$\textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0$ for the fifth algebra.
The fourth of these algebras is non-commutative, and the others are commutative.
Generalization: algebra over a ring
In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an R-module (instead of a K-vector space).
Associative algebras over rings
Main article: Associative algebra
A ring A is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to $\mathbb {H} \times \mathbb {H} $, the direct product of two quaternion algebras. The center of that ring is $\mathbb {R} \times \mathbb {R} $, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional $\mathbb {R} $-algebra.
In commutative algebra, if A is a commutative ring, then any unital ring homomorphism $R\to A$ defines an R-module structure on A, and this is what is known as the R-algebra structure.[5] So a ring comes with a natural $\mathbb {Z} $-module structure, since one can take the unique homomorphism $\mathbb {Z} \to A$.[6] On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See Field with one element for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.
See also
• Algebra over an operad
• Alternative algebra
• Clifford algebra
• Differential algebra
• Free algebra
• Geometric algebra
• Max-plus algebra
• Mutation (algebra)
• Operator algebra
• Zariski's lemma
Notes
1. See also Hazewinkel, Gubareni & Kirichenko 2004, p. 3 Proposition 1.1.1
2. Prolla, João B. (2011) [1977]. "Lemma 4.10". Approximation of Vector Valued Functions. Elsevier. p. 65. ISBN 978-0-08-087136-3.
3. Schafer, Richard D. (1996). An Introduction to Nonassociative Algebras. ISBN 0-486-68813-5.
4. Study, E. (1890), "Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen", Monatshefte für Mathematik, 1 (1): 283–354, doi:10.1007/BF01692479, S2CID 121426669
5. Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press. ISBN 978-0-521-36764-6.
6. Kunz, Ernst (1985). Introduction to Commutative algebra and algebraic geometry. Birkhauser. ISBN 0-8176-3065-1.
References
• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. Vol. 1. Springer. ISBN 1-4020-2690-0.
| Wikipedia |
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Are sun-synchronous orbits always North to South?
So someone told me that weather satellites orbit from north to south generally because that helps them to obtain a sun-synchronous orbit (his explanation for why this was true was too complicated for me to understand, something to do with inclinations, something retrograde, and the earth's relation to the sun, and more). At least, that's how I heard their comment.
But does this make any sense given that once a satellite reaches the South Pole it will just begin ascending in a south to north pattern? Was that person right and I just don't understand something about how satellites are described? Any explanation here is much appreciated. Thanks!
artificial-satellite orbit low-earth-orbit sun-synchronous
Wayne Conrad
appleLoverappleLover
$\begingroup$ Welcome to Space! "Explain X to me" questions should contain some evidence of research for several reasons. One is that it helps those who would consider answering have a better idea at what level to write and which specific issues to address, so that their answer doesn't have to be a book chapter in length. You might check Wikipedia, or just search this site by typing "sun-synchronous" in the search bar. Once you find something specific you'd like explained, you can edit your question and describe it. $\endgroup$
You are correct in your understanding,
Once it reaches the vicinity of the South Pole, specifically the most southern point in its orbit it will start going north.
The reason they said north to south is to differentiate it from the orbits going west to east (which by the way stay west to east).
This is also very neatly demonstrated by the satellite's ground track: The satellite alternates between going south and north.
(Source: tornado.sfsu.edu)
The north-south motion (or south-north) itself doesn't produce a sun-synchronous orbit. It's actually the deviation from straight north-south, coupled with the primary's (the planet or other body the satellite is orbiting) oblateness, that allows sun-synchronous orbits.
The wikipedia article on nodal precession is a good general source on this topic.
If a planet is located far from any significant gravitating object (such as the sun) and isn't rotating, gravity makes it assume a spherical shape. Orbits around a spherically symmetric object are very simple: they don't change shape, they don't change orientation, etc. But no planets are exactly spherical, and a planet's deviation from sphericity can make for interesting evolution of orbits around it.
If a planet is rotating, centrifugal force makes the equator bulge. That situation can be viewed as a somewhat smaller spherical mass, with the bulge being "extra mass" (enough to make the planet's actual mass) centered around the rotating planet's equator. If an object is orbiting with an inclination of, say, 45°, when over the northern hemisphere, that extra mass gently pulls the satellite toward the south. Eventually it gets to the equator, where it "crosses a node". Nodes are the places where the orbit intersects the equatorial plane. The ascending node is where the satellite crosses from the southern to the northern hemisphere, and the descending node is the opposite. Because of that southerly pull while over the northern hemisphere, the satellite reaches the descending node a bit earlier, farther to the west, than it would have were the planet purely spherical. The plane of its orbit has rotated westward!
While over the southern hemisphere, the pull of the bulge is northward, making it arrive at the ascending node even earlier, so the orbit plane has rotated even farther, in the same direction. The bulge (and the torque on the satellite arising from it) causes precession of the orbit plane.
The equation that relates the rate of that precession to the planet's and orbit's characteristics is $$\omega_p = -\frac{3}{2} J_2\frac{R_p^2}{p^2}\omega \cos i$$ where $\omega_p$ is the precession rate in radians per second, $J_2$ is a parameter describing the gravity field's deviation from spherical resulting from the bulge, $R_p$ is the planet's average radius, $p$ is the orbit's "semi-latus rectum", a parameter related to the orbit's size and eccentricity, $\omega$ is the orbiting object's average angular velocity around the primary ($2\pi$ radians divided by the orbit period), and $i$ is the orbit's inclination. The Wikipedia article gives a slightly different version of this equation, but they are equivalent.
For a qualitative description of precession you don't need to pay attention to most of that equation. If the orbit's eccentricity and size remains fixed, then everything to the left of $\cos i$ is constant. With a positive $i$ ("prograde orbit"), the ascending node migrates westward (the "negative" direction), as in the qualitative example above.
But as Earth orbits around the sun, the direction from the center of Earth to the sun migrates eastward in a reference frame fixed to the stars (an "inertial" frame). To establish a sun-synchronous orbit, the inclination has to make $\cos i$ negative, reversing that westward precession direction. To make $\cos i$ negative, $i$ must be larger than 90° ($\frac{\pi}{2}$ radians), or retrograde—but only slightly.
If you plug all the parameter values into the equation and assume an object in low circular Earth orbit (circular LEO), $i$ winds up being somewhere around 97-98°, depending on the precise orbit altitude. This is only 7-8° away from straight north-south, so it is generally referred to as a polar orbit. But that 7-8° of retrograde component, the deviation from exactly polar, is critical for being sun-synchronous. Indeed, if the orbit is exactly polar, $i$ is 90° so $\cos i$ is zero, and no precession occurs.
For orbits at higher altitudes $\omega$ is smaller, so to maintain $\omega_p$ at the sun-synchronous value $\cos i$ must have a larger negative value. This means its orbit inclination must be farther from exactly polar.
asdfex
Tom SpilkerTom Spilker
$\begingroup$ your equations are better looking than my equations ;-) space.stackexchange.com/a/34558/12102 $\endgroup$
$\begingroup$ great answer, but doesn't answer the question asked $\endgroup$
$\begingroup$ @JCRM Read the first paragraph for context. $\endgroup$
– Tom Spilker
$\begingroup$ I did. That sentence could equally apply two two classes of orbits, the north-south orbits and the south-north orbits. $\endgroup$
$\begingroup$ This is an excellent description of sun-synchronous orbits both in the qualitative and quantitative sense. @JCRM is right that it doesn't answer the question since technically the question is just asking what is meant by a "north-south obrit." The correct answer is alluded to in your first sentence by essentially implying north-south and south-north are equivalent. $\endgroup$
In addition to Hans' answer, these are the terms used in the spaceflight community to describe the orbits you're referring to.
Satellites that orbit "north to south" are called polar orbits. These are all orbits that place the satellite over the poles. These orbits have an inclination (angle between the orbit's plane and the equator) of about 90º.
A sun-synchronous orbit belongs to the group of polar orbits. Its inclination is slightly more than 90º (depending on the orbit's altitude). This ensures that the orbit stays in the same position relative to the sun. This means the satellite will pass over a given spot on Earth at the same time of day every day, which is valuable for some applications.
HobbesHobbes
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\begin{definition}[Definition:Decimal Expansion/Size Less than 1]
A number $x$ such that $\size x < 0$ has a units digit which is zero.
Such a number may be expressed either with or without the zero, for example:
:$0 \cdotp 568$
or:
:$\cdotp 568$
While both are commonplace, the form with the zero is less prone to the mistake where decimal point is missed when reading it.
\end{definition} | ProofWiki |
Aims and Scopes
Publishing Ethic Statement
Journal of Data and Information Science, 2019, 4(4): 1-12
doi: 10.2478/jdis-2019-0017
Infrastructure of Scientometrics:The Big and Network Picture
Jinshan Wu †
School of Systems Science, Beijing Normal University, Beijing 100875, China
Jinshan Wu. Infrastructure of Scientometrics:The Big and Network Picture. Journal of Data and Information Science[J], 2019, 4(4): 1-12 doi:10.2478/jdis-2019-0017
2019 Editorial office of Journal of Data and Information Science
†Jinshan Wu (E-mail: [email protected]).
Jinshan Wu, Professor of School of Systems Science at Beijing Normal University, China. He received his Ph.D. degree from University of British Columbia, Canada. Prof. Wu is a physicist who works in non-equilibrium quantum statistical physics, game theory and experiments, network science and scientometrics. He sees connections beyond boundaries.
Received: Accepted: Online:
A network is a set of nodes connected via edges, with possibly directions and weights on the edges. Sometimes, in a multi-layer network, the nodes can also be heterogeneous. In this perspective, based on previous studies, we argue that networks can be regarded as the infrastructure of scientometrics in the sense that networks can be used to represent scientometric data. Then the task of answering various scientometric questions related to this data becomes an algorithmic problem in the corresponding network.
Key words: Network science ; Scientometrics
Scientometric analyses are often used in evaluating the scientific performance of journals, universities or institutes, countries, and sometimes even individual papers and authors. We refer to them as research/publication units at various levels. These research units are often ranked according to one or more indicators. In generating such rankings, often simple counting and statistics are used, for example, counting how many citations each paper received and from there to count how many citations other research units received in total or on average. Of course, a final step in such counting and statistics often involves a weighted average to incorporate various counting schemes into a single number. Due to this prevailing paradigm of scientometric studies, scientometrics often appears to be a scientific discipline mainly providing tools and results for evaluating science with simple statistical analysis, such as calculating mean, median, mode, distribution, and sometimes also hypojournal testing as its main tools.
In this perspective, we will argue that scientometrics is a scientific discipline that collects and makes use of data related to scientific activities to help the development of the sciences, and for that purpose, we need a framework to represent relational data and to answer the questions that are potentially helpful to the development of other sciences. We claim that this framework is a network.
Of course, we notice that many other scientometric studies have already looked into regularities or laws of various scientific activities aiming to help the development of other sciences and also scientometrics as a science, rather than directly working towards evaluation. See, for example, Waltman (2016) and Mingers & Leydesdorff (2015), Rousseau et al. (2018) and references therein.
Moreover, there exist already quite some studies using network analysis as a tool to answer scientometric questions. For example, Otte and Rousseau point out that many network analysis techniques are applicable to scientometric studies (Otte & Rousseau, 2002, Rousseau et al., 2018). Waltman and van Eck use clustering algorithms to classify papers into clusters of topics in the citation network of papers (Waltman & van Eck, 2012). Chen applies pathfinding methods of network analysis to get a better presentation of the paths over the co-citing or co-cited network of papers or authors (Chen, 2006). West et al. apply PageRank algorithm (Brin & Page, 1998) for networks of web pages as a ranking method in the citation network of journals, authors, or papers (West et al., 2010). In the case of citation networks, the PageRank algorithm assumes that papers cited by more influential papers are themselves more influential. PageRank is derived from the Pinski-Narin approach introduced in the fields of bibliometrics, much earlier (Pinski & Narin, 1976). Amjad et al. extend the PageRank algorithm to multi-layer networks, including both authors and papers (Amjad et al., 2015). In this extension, the PageRank algorithm further assumes that papers cited by more influential papers and written by more influential authors are themselves more influential. Shen et al. (Shen et al., 2016) maps the interrelation among subfields of physics using the citation network of the subfields and a PageRank-like algorithm developed from the Leontief input-output analysis in economics (Leontief, 1941). Shen et al. apply network embedding and clustering method to cluster journals (Shen et al., 2019). We refer the reader to the review by Zeng et al. (Zeng et al., 2017) for more network-related studies in scientometrics.
However, networks can offer much more than just being a technique applicable to scientometric studies. We will show in the following sections how networks can be used as a language to express relational data and hence answer questions in scientometrics. They can provide at the same time a framework, a language, and a platform to develop algorithms to solve interesting - and maybe even essential - questions within a scientometric framework.
2 Core ideas of network science and how they fit into a scientometric framework
Before explaining why networks can serve as the language or infrastructure of scientometrics, let us briefly recall the core ideas behind networks. First, a network is an abstraction of entities and their relations. A set of entities with pairwise (or one-to-one) relations can be represented as a network. Here pairwise (one-to-one) means that the relationship is between two entities, or more accurately from one entity to another. Many relations that might seem to be multi-party can often be seen as a collection of several pairwise relations. For example, the famous three-body question in physics, where three stars are moving under the gravity interaction among them, is, in fact, a collection of three pairwise gravity interactions. When a speaker is talking to a room full of audience, it may seem that the speaker is interacting with all of them together, and there are even interactions among the audience. However, it is also possible to see this as a collection of one-to-one interactions between the speaker and each member of the audience, plus possibly interactions between each pair in the audience. Of course, there might be some inherently multi-party interactions, which current networks cannot describe. Often when using networks, people simplify the interaction even further, for example,by discarding the strength of the interaction and keeping only its existence. Conventionally, the mathematical notation $A_{j}^{j}$ = 1 means there is an interaction between entity i and entity j (or directed from i to j) and $A_{j}^{j}$ = 0 otherwise. Thus, this matrix A = ($A_{j}^{j}$)N×N describes the existence of an interaction between each pair of N entities. Often such an overly simplified description serves as a good platform for many investigations involving these entities. For example, the citation network sets $A_{j}^{j}$ = 1 when paper i is cited by paper j while we clearly know that not all citations should be counted equally. On one hand, this is indeed a limit, and thus, one might want to consider extensions such as assigning a weight to $A_{j}^{j}$ by taking into account the number of citations from j to i and where the citations are in j to a certain degree. On the other hand, however, many analyses can already be done over this limited network. Besides weighted networks, a further extension will be to consider heterogeneous nodes and heterogeneous edges. For example, one might want to present authors and papers all together in a single network. In this case, the relation among papers are citations, the relation among authors are mentor-mentee or other academic or social relations, and the relation between authors and papers are "writing" (authorship). We can see that there are three kinds of relations and two kinds of nodes.
$ A_{a_{2}}^{a_{1}}=\sum_{P=1}^{N}W_{p}^{a_{1}}W_{p}^{a_{2}}=WW^T$ (1)
where a1, a2 are two authors and p is a paper. This is one example of "calculating" an induced relation from a more fundamental network. In network analysis, given the fundamental network, which ideally captures all necessary relations, researchers always try to answer other questions via definitions of certain more advanced structures and related algorithms, especially those making use of direct and indirect connections via various orders of the fundamental network just like the WWT in Eq. (1). Given a simple network A, we call $A_{j}^{j}$ the direct (length-1) connections,(A2)ij the length-2 indirect connections, and so on.
Researchers from network science already defined quite some structural quantities over networks, for example, degree, PageRank score, and community. Using again a citation network as an example, we see that the counting of the number of publications of a research unit is the zeroth order quantity of the network, taking no connections in the citation network into consideration at all. A counting of the number of received citations is the first order quantity of the network, considering only the direct connections in the citation network. A PageRank score, say of papers, on the other hand, measures the influence of the papers to other papers along the citation path with all possible lengths,
$P=\frac{αe}{1-(1-α) \mathscr{F}}=αe+x(1-α)e\mathscr{F}+α(1-α)^2e\mathscr{F}^2+···$ (2)
$\mathscr{F}_{j}^{i}=\frac { A_{j}^{j}}{\sum_{k} A_{k}^{i}} $ (3),
and ae is the zeroth order influence score of each paper. Sometimes e is taken to be e = [1,1,…]; $α(1-α) e\mathscr{F}$ is the length-1 path influence (the papers directly citing the targeted paper), $α(1-α)^2e\mathscr{F}^2$ is the length-2 path influence and so on. There are other structural quantities and algorithms that take various orders of indirect connections together with direct connections into account, for example the General Input-Output analysis (Shen et al., 2016) and the K-core (Alvarez-Hamelin et al., 2008).
We hope that the above examples sufficiently illustrate why networks can serve as a language or infrastructure of scientometrics. Networks describe relations among entities, and often network analysis starts from some fundamental network and makes use of both direct and indirect connections in the fundamental network via proper mathematics/algorithms to answer more advanced questions. We refer to this collection of connected entities as a system. From the detailed connections,we observe the micro-level structures of the system, and from the more advanced network analysis, which takes both direct and indirect connections into consideration, we can see the macro-level structures of the system. Therefore, the network framework and network analysis serve as a bridge between micro and macro-level structures.
Equipped with these two core ideas of network science, which, from now on, we refer to respectively as "relation" and "propagation", we now try to use networks to represent, at least many if not all, scientometric data, research questions, and analysis.
3 The three-layer fundamental network of scientometrics
As we mentioned earlier, scientometrics studies all kinds of scientific activities to help the development of the sciences. What are the major actors - to become nodes in the fundamental network of scientometrics - of those scientific activities? Since the core scientific activities are researchers performing and publishing their researches, researchers, research questions (also methods, instruments, and materials), and papers should be considered as the major actors. We call them respectively, the authors, concepts, and papers, and represent them as a three-layer network in Fig. 1. What are the relations between them? Within each layer, among authors, there are academic or social relations, such as the mentor-mentee relation; among the papers, of course, there are citations; among the concepts, there are logic relations from disciplinary knowledge. Between the layers, authors "write" papers; papers "work on" or "use" certain concepts.
2096-157X-4-4-1/img_2.png
Figure 1. A three-layer network of scientometric relational data: Authors, papers and concepts on the one hand; and inventors, patents, and technology concepts on the other.
Figure 1.2096-157X-4-4-1/img_2.png
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A three-layer network of scientometric relational data: Authors, papers and concepts on the one hand; and inventors, patents, and technology concepts on the other.
Let us denote the network as a matrix W, where $W_{a_{2}}^{a_{1}}$ means author a1 supervises author a2; $W_{ p _{2}}^{p_{1}}$ means paper p1 is cited by paper p2; $W_{ c _{2}}^{c_{1}}$ means concept c1 provides logically the basis of concept c2, while $W_{p}^{a}$ means author a write paper p; $W_{c}^{p}$ means paper p works on concept c.
Then other common relations studied in scientometrics can be defined according to this fundamental network. For example, co-authorship becomes an induced relation from the fundamental network W, $A_{α_{2}}^{α_{1}}=\sum_{p}W_{p}^{α_{1}}W_{p}^{α_{2}}=\sum_{p}W_{p}^{α_{1}}(W^T)_{α_{2}}^{P}$. Co-cited relation between papers becomes $C_{P_{2}}^{P_{1}}=\sum_{p}W_{p}^{p_{1}}W_{p}^{p_{2}}=\sum_{p}W_{p}^{p_{1}}(W^T)_{p_{2}}^{P}$,while co-citing relation becomes $B_{P_{2}}^{P_{1}}=\sum_{p}W_{p_{1}}^{p}W_{{p_{2}}}^{p}=\sum_{p}(W^{T})_{p}^{p_{1}}W_{p}^{{p_{2}}}$. Co-occurrence (or co-studying) of concepts becomes $CO_{c_{2}}^{c_{1}}=\sum_{p}W_{c_{1}}^{p}W_{{c_{2}}}^{p}=\sum_{p}(W^T)_{p}^{c_{1}}W_{c_{2}}^{p}$. Author a's expertise, which are the concepts that the author has been working on can be find from $E_{c}^{α}=\sum_{p}W_{p}^{α}W_{c}^{p}$.
Other properties of the entities can also be added to the fundamental network. For example, research groups are sets of authors; countries, and also journals are sets of papers; and disciplines are clusters of concepts. All those sets can be either pre-defined or generated from the fundamental network via clustering algorithms. For example, a research group is in principle a set of authors who often publish the same papers and similarly disciplines or fields are sets of concepts, often studied by the same papers. In network science, clustering algorithms are designed to find communities of nodes, where there are more intra-community links than inter-community links. Therefore, it is quite possible that those communities can be found via clustering algorithms on the network, instead of defining them ahead of time using certain heuristic rules or conventions.
Besides unifying various relations, the fundamental network can also provide new insights concerning some scientometric questions. For example, we might be able to measure the creativity of a paper by considering the concepts on which the paper works. If a paper p proposes a new concept c or a new connection such as a theorem between existing concepts ci and cj, denoted as a new concept cij. Then, we should see in the fundamental network that $W_{c}^{p}=1$,$W_{c_{ij}}^{p}=1$,$W_{c}^{q}=0$ and $W_{c_{ij}}^{q}=0$ for all q≠p. Therefore, we may define a creativity metric based on those quantities, for instance by comparing $W_{c}^{p}$ and $\sum_{ q≠p }W_{c}^{p}$.
Consider next the task of recommending the most relevant papers to researchers as another example. For such a system, we first need to rank all papers according to their quality in a certain way, and secondly, we also need to measure the relevance of all fields (concepts) for each given author. We can then choose to recommend the top papers in each most relevant field to the researcher. For the first task, i.e., the quality measure we can, roughly speaking, extend the PageRank-like algorithm to the three-layer network, implying that papers working on more influential concepts are themselves more influential, and the concepts worked on by more influential papers are themselves more influential. The second task, i.e., relevance measure, might be done simply by covering all the fields that author a has been working on, i.e., the cs such that $ E_{c}^{α}≠0$, or ranking the cs according to the value of $ E_{c}^{α}$. A more complex but potentially valuable algorithm would be to propagate E_{c}^{α}$ on the concept layer, meaning that not only the fields cs that have been directly worked on by author a but also the fields that are logically close to these should be taken into consideration. For the task of relevance measure, we can even try to solve it by turning it into a task of more general similarity measure between papers. Once we have such a similarity measure between papers, we can choose to recommend those papers which are most similar to author a's papers and also of high quality. Working on the same (or closely related) concepts, citing the same (or closely related) papers and being cited by the same (or closely related) papers, are all attributes that make papers more similar. Therefore, for such a similarity measure, it is natural to consider a propagation algorithm over the whole three-layer fundamental network. From this example, although we have not implemented and explained all the details, we can already see that the task of recommending papers becomes an algorithmic problem in the fundamental network and we can also see how the two key ideas of networks - "relation" and "propagation" - can potentially help in designing a better recommendation system of papers to researchers. This example shows how the fundamental network can be used to represent scientometric data, rephrase scientometric questions, and to establish concepts and algorithms to answer the questions with the data.
The fundamental network can be expanded further if we need to look into other scientific activities by including new entities and relations into the network. For example, a similar three-layer network can be established for patents: inventors, patents, and technology concepts. We can even link the two three-layer networks by connecting the papers and patents via their citations. Furthermore, if we have data on how each patent and paper are used in producing products, we can even add another layer, the product layer, to the fundamental network. Papers and patents are connected to products via the "used-in" relation while the products themselves are connected by the "made-from" relation. In this way, we might be able to measure the contribution of papers and/or patents in final products. Moreover, if we have a citation network of textbooks and corresponding citations between textbooks and papers (patents), we can add another layer of textbooks so that we can even measure the contribution of papers and patents to the accumulated knowledge of humanity.
We hope that the above examples illustrated the value of the fundamental network of scientometrics. However, in reality, it is quite challenging to establish such a network. Besides the citation network, we will need to disambiguate authors and find their academic relations, build a concept network for human knowledge, and annotate each paper with their concepts. With the accumulated data, like the open data from academictree1(1http://www.academictree.org) and dblp2(2 https://dblp.uni-trier.de/), and with the fast development of natural language processing, we are getting closer to the dream of really establishing such a network. One possible proxy of the fundamental network can be the network of researchers, papers, and reactants/reactions in chemistry. In the concept layer of this proxy, chemical reactants care connected via chemical reactions, and it is also much easier to link papers to specific chemical reactants and reactions than linking papers to more general concepts. In fact, there are already commercial products such as Reaxys3(3https://www.reaxys.com/)and SciFinder4(4 https://scifinder.cas.org)that realize at least part of these ideas. Creating an open data version is, in principle, plausible. It will be interesting to create a three-layer fundamental network for the discipline of scientometrics itself.
4 The closed-system and open-system approach
Often in our investigations, we try to collect and represent all necessary data, and then we search for concepts and algorithms to answer the research questions. We want to make sure that all entities relevant to the research questions are covered by the framework and data. This is called a closed-system approach. Using the network language, this means that our network is complete such that all entities and their relations relevant to the research questions become nodes and edges in the network. We do not need any other information beyond the network, i.e., the system, to answer the research questions.
On the other hand, there might be situations where some of the relevant entities or some relations are not covered by the framework and data, but we still need to answer those research questions. In that case we call this an open-system approach. Sometimes, we have to take the open-system approach due to the limitation of available data. Alternatively, we might do so on purpose. When we work with open systems, often, we need to take some exogenous information/quantities of the missing entities into account via, for example, propagation algorithms. Rephrased in another way, for a presumed closed system, if for some reason we think that the closed system is missing some very relevant entities and we cannot easily get full data about those entities, it is then a good idea to treat the presumed closed system as an open system and make use of some available data about the missing entities. We will illustrate this via examples.
Second, we want to show that an open-system approach, which makes use of partial data when the full data of the missing entities are not available, often
helps us to go beyond the presumed closed system. Let us continue with the above example of the citation network. Now we want to measure the reliability or trustworthiness of papers. Taking the closed-system approach implies that the citation network has captured all the relevant entities and relations regarding the reliability of papers. However, this assumption very likely does not hold. There might be some correlation between citation counts and reliability, but these notions are not the same, and the citation network only represents citations among papers but not reliability. Therefore, the citation network is not enough for the task of finding or determining a reliability measure. One way out of this would be to define a new network that captures the essence of reliability, about which we do not really know much. An open-system approach will be making use of the citation network together with some exogenous information. Let us assume that we have a small set of papers whose reliability scores have been evaluated by human experts. Then if we assume that papers citing more reliable papers are themselves more reliable or papers cited by more reliable papers are themselves more reliable, we can propagate the exogenous reliability scores of a selected set of papers to all papers via the citation network. This is exactly the idea of the TrustRank algorithm (Gyöngyi et al., 2004) in computer science, which has been used to rank most and least trustworthy web pages to either recommend the highly trustworthy papers or filter out the least trustworthy ones in search engines.
Besides measuring reliability, we might be able to extend the idea of propagation and open systems to measure the contribution of research papers towards human knowledge. For example, we can still use the citation network of papers as the open system, and we take the number of citations from textbooks to papers as the exogenous contribution scores. We then propagate this exogenous contribution scores to all papers via the citation network. Of course, if we also have the full data on the citations between textbooks, we turn back again to the closed-system approach, and it will even be better. However, the open-system approach only requires citation counts from textbooks to papers and the citation network of papers, but not the full citation network of textbooks and papers.
What is the relation between networks on the one hand, and open and closed systems on the other? With networks, it is explicit that the algorithms that we are using are closed-system ones or open-system ones depending on whether or not all the relevant entities and relations are covered by the network. Networks also inspire us to make use of partial data when some relevant entities and some or all of their data are missing, but partial data is available. Furthermore, networks also provide a natural platform to develop algorithms to take partial data into account via, for example, propagation.
5 Working towards a unified framework of scientometrics
We hope that the above discussion has already shown that the three-layer network of scientometrics can be a fundamental network and lead to a unified framework of scientometrics. Firstly, the fundamental network is capable of representing most,if not all, relevant entities and their relations regarding various scientific activities of scientometrics in a broad sense, which is to discover regularities for all kinds of scientific activities to help the development of science. Secondly, with the fundamental network already in place, scientometric questions can be rephrased as questions waiting to be solved via concepts and algorithms over the network. Finally, with the two key ideas, i.e., relation and propagation, the fundamental network also provides a good platform to develop concepts and algorithms to solve these scientometric questions. Furthermore, with the concept layer and the links between the papers and the concepts of this fundamental network, we can now dive into a contents-based scientometric analysis rather than only making use of metadata of publications.
However, as we mentioned earlier, building up such a unified framework for scientometrics is a very demanding task. Mapping a concept network of human knowledge is challenging. Connecting each paper to the concept network is also not easy. Even author name disambiguation and getting correct academic relations are not simple tasks. To illustrate the power and beauty of this unified framework of scientometrics, we may start from a small scale example or a proxy of it. That is the ultimate purpose of this article: Let us implement a small scale example and work together towards a unified framework of a general scientometric theory (some may refer to it as informetrics, or as a science of science) that goes beyond the evaluation of various research units.
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Alvarez-Hamelin J.I., Dall'Asta L., Barrat A., & Vespignani A. (2008). K-core decomposition of Internet graphs: Hierarchies, selfsimilarity and measurement biases. Networks and Heterogeneous Media, 3(2), 371-393. doi:10.3934/nhm.2008.3.371. Annual European Conference on Complex Systems, Dresden, GERMANY, OCT 01-06, 2007.
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The paper studies the incumbent-entrant problem in a fully dynamic setting. We find that under an open-loop information structure the incumbent anticipates entry by overinvesting, whereas in the Markov perfect equilibrium the incumbent slightly underinvests in the period before the entry. The entry cost level where entry accommodation passes into entry deterrence is lower in the Markov perfect equilibrium. Further we find that the incumbent's capital stock level needed to deter entry is hump shaped as a function of the entry time, whereas the corresponding entry cost, where the entrant is indifferent between entry and non-entry, is U-shaped.
Otte ,E.,& Rousseau ,R. (2002). Social network analysis: A powerful strategy, also for the information sciences. Journal of Information Science, 28(6), 441-453. doi:10.1177/016555102762202123.
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Cross-sector community partnerships are a potentially powerful strategy to address population health problems, including health disparities. US immigrants - commonly employed in low-wage jobs that pose high risks to their health - experience such disparities because of hazardous exposures in the workplace. Hazardous exposures contribute to chronic health problems and complicate disease management. Moreover, prevention strategies such as worksite wellness programs are not effective for low-wage immigrant groups. The purpose of this article was to describe an innovative application of social network analysis to characterize interagency connections and knowledge needed to design and deliver a comprehensive community-based chronic disease prevention program for immigrant workers.
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We present the application of limited one-time sampling irregularity map (LOTS-IM): a fully automatic unsupervised approach to extract brain tissue irregularities in magnetic resonance images (MRI), for quantitatively assessing white matter hyperintensities (WMH) of presumed vascular origin, and multiple sclerosis (MS) lesions and their progression. LOTS-IM generates an irregularity map (IM) that represents all voxels as irregularity values with respect to the ones considered &"normal&". Unlike probability values, IM represents both regular and irregular regions in the brain based on the original MRI's texture information. We evaluated and compared the use of IM for WMH and MS lesions segmentation on T2-FLAIR MRI with the state-of-the-art unsupervised lesions' segmentation method, Lesion Growth Algorithm from the public toolbox Lesion Segmentation Toolbox (LST-LGA), with several well established conventional supervised machine learning schemes and with state-of-the-art supervised deep learning methods for WMH segmentation. In our experiments, LOTS-IM outperformed unsupervised method LST-LGA on WMH segmentation, both in performance and processing speed, thanks to the limited one-time sampling scheme and its implementation on GPU. Our method also outperformed supervised conventional machine learning algorithms (i.e., support vector machine (SVM) and random forest (RF)) and deep learning algorithms (i.e., deep Boltzmann machine (DBM) and convolutional encoder network (CEN)), while yielding comparable results to the convolutional neural network schemes that rank top of the algorithms developed up to date for this purpose (i.e., UResNet and UNet). LOTS-IM also performed well on MS lesions segmentation, performing similar to LST-LGA. On the other hand, the high sensitivity of IM on depicting signal change deems suitable for assessing MS progression, although care must be taken with signal changes not reflective of a true pathology.
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Shen Z., Yang L., Pei J., Li M., Wu C., Bao J., Wei T., Di Z., Rousseau R., & Wu J. (2016). Interrelations among scientific fields and their relative influences revealed by an input-output analysis. Journal of Informetrics, 10(1), 82-97. doi:10.1016/j.joi.2015.11.002.
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There has been an unprecedented increase in critical care research recently and there is a need for an organized and systematic approach to surgical critical care research planning. The purpose of this paper was to establish a surgical critical care research agenda via a systematic review of the literature and needs assessment.
Waltman , L., & van Eck,N.J.(2012). A new methodology for constructing a publication-level classification system of science. Journal of the American Society for Information Science and Technology, 63(12), 2378-2392. doi:10.1002/asi.22748.
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The purpose of our study was to better understand why parents/caregivers might not practice safe sleep behaviours. In autumn 2016, we conducted 'pulse' interviews with 124 parents/caregivers of children under the age of one year at a variety of local community events, festivals and meetings in cities with high infant mortality rates around the Midwestern US state of Ohio. Through an inductive approach, pulse interviews were analysed using thematic coding and an iterative process which followed for further clarification of themes (Qualitative Research in Psychology, 2006, 3, 77; BMC Medical Research Methodology, 2013, 13, 117). The six major themes of underlying reasons why parents/caregivers might not practice safe sleep behaviours that were identified in our coding process included the following: (a) culture and family tradition, (b) knowledge about safe sleep practices, (c) resource access, (d) stressed out parents, (f) lack of support and (g) fear for safety of baby. Using the descriptive findings from the pulse interviews, qualitative themes and key informant validation feedback, we developed four diverse fictional characters or personas of parents/caregivers who are most likely to practice unsafe sleep behaviours. These personas are characteristic scenarios which imitate parent and caregiver experiences with unsafe sleep behaviours. The personas are currently being used to influence development of health promotion and education programs personalised for parents/caregivers of infants less than one year to encourage safe sleep practices.
... and ae is the zeroth order influence score of each paper. Sometimes e is taken to be e = [1,1,…]; $α(1-α) e\mathscr{F}$ is the length-1 path influence (the papers directly citing the targeted paper), $α(1-α)^2e\mathscr{F}^2$ is the length-2 path influence and so on. There are other structural quantities and algorithms that take various orders of indirect connections together with direct connections into account, for example the General Input-Output analysis (Shen et al., 2016) and the K-core (Alvarez-Hamelin et al., 2008). ...
... Moreover, there exist already quite some studies using network analysis as a tool to answer scientometric questions. For example, Otte and Rousseau point out that many network analysis techniques are applicable to scientometric studies (Otte & Rousseau, 2002, Rousseau et al., 2018). Waltman and van Eck use clustering algorithms to classify papers into clusters of topics in the citation network of papers (Waltman & van Eck, 2012). Chen applies pathfinding methods of network analysis to get a better presentation of the paths over the co-citing or co-cited network of papers or authors (Chen, 2006). West et al. apply PageRank algorithm (Brin & Page, 1998) for networks of web pages as a ranking method in the citation network of journals, authors, or papers (West et al., 2010). In the case of citation networks, the PageRank algorithm assumes that papers cited by more influential papers are themselves more influential. PageRank is derived from the Pinski-Narin approach introduced in the fields of bibliometrics, much earlier (Pinski & Narin, 1976). Amjad et al. extend the PageRank algorithm to multi-layer networks, including both authors and papers (Amjad et al., 2015). In this extension, the PageRank algorithm further assumes that papers cited by more influential papers and written by more influential authors are themselves more influential. Shen et al. (Shen et al., 2016) maps the interrelation among subfields of physics using the citation network of the subfields and a PageRank-like algorithm developed from the Leontief input-output analysis in economics (Leontief, 1941). Shen et al. apply network embedding and clustering method to cluster journals (Shen et al., 2019). We refer the reader to the review by Zeng et al. (Zeng et al., 2017) for more network-related studies in scientometrics. ...
... Second, we want to show that an open-system approach, which makes use of partial data when the full data of the missing entities are not available, often
helps us to go beyond the presumed closed system. Let us continue with the above example of the citation network. Now we want to measure the reliability or trustworthiness of papers. Taking the closed-system approach implies that the citation network has captured all the relevant entities and relations regarding the reliability of papers. However, this assumption very likely does not hold. There might be some correlation between citation counts and reliability, but these notions are not the same, and the citation network only represents citations among papers but not reliability. Therefore, the citation network is not enough for the task of finding or determining a reliability measure. One way out of this would be to define a new network that captures the essence of reliability, about which we do not really know much. An open-system approach will be making use of the citation network together with some exogenous information. Let us assume that we have a small set of papers whose reliability scores have been evaluated by human experts. Then if we assume that papers citing more reliable papers are themselves more reliable or papers cited by more reliable papers are themselves more reliable, we can propagate the exogenous reliability scores of a selected set of papers to all papers via the citation network. This is exactly the idea of the TrustRank algorithm (Gyöngyi et al., 2004) in computer science, which has been used to rank most and least trustworthy web pages to either recommend the highly trustworthy papers or filter out the least trustworthy ones in search engines. ...
... Of course, we notice that many other scientometric studies have already looked into regularities or laws of various scientific activities aiming to help the development of other sciences and also scientometrics as a science, rather than directly working towards evaluation. See, for example, Waltman (2016) and Mingers & Leydesdorff (2015), Rousseau et al. (2018) and references therein. ...
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In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((7,4)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,4),NW);
label("$Q$",(7,4),NE);
label("$R$",(12,0),E);
[/asy]
Since $PQ$ is parallel to $SR,$ the height of $\triangle PQS$ (considering $PQ$ as the base) and the height of $\triangle SRQ$ (considering $SR$ as the base) are the same (that is, the vertical distance between $PQ$ and $SR$).
Since $SR$ is twice the length of $PQ$ and the heights are the same, the area of $\triangle SRQ$ is twice the area of $\triangle PQS.$ In other words, the area of $\triangle PQS$ is $\frac{1}{3}$ of the total area of the trapezoid, or $\frac{1}{3}\times 12 = \boxed{4}.$ | Math Dataset |
E-dense semigroup
In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse x, meaning that xax = x.[1] The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that axa=a).
The above definition of an E-inversive semigroup S is equivalent with any of the following:[1]
• for every element a ∈ S there exists another element b ∈ S such that ab is an idempotent.
• for every element a ∈ S there exists another element c ∈ S such that ca is an idempotent.
This explains the name of the notion as the set of idempotents of a semigroup S is typically denoted by E(S).[1]
The concept of E-inversive semigroup was introduced by Gabriel Thierrin in 1955.[2][3][4] Some authors use E-dense to refer only to E-inversive semigroups in which the idempotents commute.[5]
More generally, a subsemigroup T of S is said dense in S if, for all x ∈ S, there exists y ∈ S such that both xy ∈ T and yx ∈ T.
A semigroup with zero is said to be an E*-dense semigroup if every element other than the zero has at least one non-zero weak inverse. Semigroups in this class have also been called 0-inversive semigroups.[6]
Examples
• Any regular semigroup is E-dense (but not vice versa).[1]
• Any eventually regular semigroup is E-dense.[1]
• Any periodic semigroup (and in particular, any finite semigroup) is E-dense.[1]
See also
• Dense set
• E-semigroup
References
1. John Fountain (2002). "An introduction to covers for semigrops". In Gracinda M. S. Gomes (ed.). Semigroups, Algorithms, Automata and Languages. World Scientific. pp. 167–168. ISBN 978-981-277-688-4. preprint
2. Mitsch, H. (2009). "Subdirect products of E–inversive semigroups". Journal of the Australian Mathematical Society. 48: 66. doi:10.1017/S1446788700035199.
3. Manoj Siripitukdet and Supavinee Sattayaporn Semilattice Congruences on E-inversive Semigroups Archived 2014-09-03 at the Wayback Machine, NU Science Journal 2007; 4(S1): 40 - 44
4. G. Thierrin (1955), 'Demigroupes inverses et rectangularies', Bull. Cl. Sci. Acad. Roy. Belgique 41, 83-92.
5. Weipoltshammer, B. (2002). "Certain congruences on E-inversive E-semigroups". Semigroup Forum. 65 (2): 233. doi:10.1007/s002330010131.
6. Fountain, J.; Hayes, A. (2014). "E ∗-dense E-semigroups". Semigroup Forum. 89: 105. doi:10.1007/s00233-013-9562-z. preprint
Further reading
• Mitsch, H. "Introduction to E-inversive semigroups." Semigroups : proceedings of the international conference ; Braga, Portugal, June 18–23, 1999. World Scientific, Singapore. 2000. ISBN 9810243928
| Wikipedia |
\begin{document}
\thanks{MSC (2000): 16W35, \;16W40}\thanks{The research was supported by CEx05-D11-11/04.10.05.}
\begin{abstract} The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given. \end{abstract} \maketitle
\section*{Introduction} In this paper we introduce a notion of double coset for semisimple finite dimensional Hopf algebras, similar to the one for groups. This is achieved considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. The equivalence relation that we define generalizes the equivalence relation introduced in \cite{NR}. Using Frobenius-Perron theory for nonnegative Hopf algebras the results from \cite{NR} are generalized and proved in a simpler manner.
The paper is organized as follows. In the first Section we recall some basic results for finite dimensional semisimple Hopf algebras that we need in the other sections.
Section \ref{dcsf} introduce the equivalence relation on the set of irreducible characters of the dual Hopf algebra and it proves the coset decomposition. Using this coset decomposition in the next section we prove a result concerning the restriction of a module to a normal Hopf subalgebra. A formula for the induction from a normal Hopf subalgebra is also described. In the situation of a unique double coset a formula equivalent to the Mackey decomposition formula for groups is described.
Section \ref{dr} considers one of the above equivalence relations but for the dual Hopf algebra. In the situation of normal Hopf subalgebras this relation can be written in terms of the restriction of the characters to normal Hopf subalgebras. Some results similar to those in group theory are proved.
The next sections studies the restriction functor from a semisimple Hopf algebra to a normal Hopf subalgebra. We define a notion of conjugate module similar to the one for modules over normal subgroups of a group. Some results from group theory hold in this more general setting. In particular we show that the induced module restricted back to the original normal Hopf subalgebra has as irreducible constituents the constituents of all the conjugate modules.
Algebras and coalgebras are defined over the algebraically closed ground field $k=\mathbb{C}$. For a vector space $V$ over $k$ by $|V|$ is denoted the dimension $\mathrm{dim}_kV$. The comultiplication, counit and antipode of a Hopf algebra are denoted by $\Delta$, $\epsilon$ and $S$, respectively. We use Sweedler's notation $\Delta(x)=\sum x_1\otimes x_2$ for all $x\in H$. All the other notations are those used in \cite{Montg}. \section{Preliminaries}\label{prel} Throughout this paper $H$ denotes a finite dimensional semisimple Hopf algebra over $K=\mathbb{C}$. It follows that $H$ is also cosemisimple \cite{Lard}. If $K$ is a Hopf subalgebra of $H$ then $K$ is also semisimple and cosemisimple Hopf algebra \cite{Montg}.
Denote by $\mathrm{Irr}(H)$ the set of irreducible characters of $H$ and $C(H)$ the character ring of $H$. Then $C(H)$ is a semisimple subalgebra of $H^*$ \cite{Z} and $C(H)=\mathrm{Cocom}(H^*)$, the space of cocommutative elements of $H^*$. By duality, the character ring of $H^*$ is a semisimple subalgebra of $H$ and under this identification it follows that $C(H^*)=\mathrm{Cocom}(H)$.
If $M$ is an $H$-module with character $\chi$ then $M^*$ is also an $H$-module with character $\chi^*=\chi \circ S$. \\This induces an involution $``\;^*\;":C(H)\rightarrow C(H)$ on $C(H)$.
For a finite dimensional semisimple Hopf algebra $H$ use the notation $\Lambda_H \in H$ for the integral of $H$ with $\epsilon(\Lambda_H)=|H|$ and $t_H \in H^*$ for the integral of $H^*$ with $t_H(1)=|H|$. It follows from \cite{Montg} that the regular character of $H$ is given by the formula \begin{equation*}\label{f1}t_{ _H}=\sum_{\chi \in \mathrm{Irr}(H)}\chi(1)\chi\end{equation*} The dual formula is \begin{equation*}\label{f2}\Lambda_{ _H}=\sum_{d \in \mathrm{Irr}(H^*)}\epsilon(d)d\end{equation*}
If $W \in H^*$-mod then $W$ becomes a right $H$-comodule via $\rho :W \rightarrow W \otimes H$ given by $\rho(w)=\sum w_{0}\otimes w_1$ if and only if $fw=\sum f(w_1)w_0$ for all $w \in W$ and $f \in H^*$.
Let $W$ be a left $H^*$-module. Then $W$ is a right $H$-comodule and one can associate to it a subcoalgebra of $H$ denoted by $C_{_W}$ \cite{Lar}. If $W$ is simple and $q=|W|$ then $|C_{_W}|=q^2$ and it is a matrix coalgebra. It has a basis $\{x_{ij}\}_{1\leq i,j \leq q}$ such that $\Delta(x_{ij})=\sum_{l=0}^qx_{il}\otimes x_{lj}$ for all $1\leq i,j \leq q$. Moreover $W \cong k<x_{1i}|\;1\leq i \leq q>$ as right $H$-comodule where $\rho(x_{1i})=\Delta(x_{1i})=\sum_{l=0}^qx_{1l}\otimes x_{li}$ for all $1\leq i \leq q$. The character of $W$ as left $H^*$-module is $d \in C(H^*)\subset H$ and it is given by $d=\sum_{i=1}^qx_{ii}$. Then $\epsilon(d)=q$ and the simple subcoalgebra $C_{ _W}$ is sometimes denoted by $C_d$.
If $M$ and $N$ are right $H$-comodules the the tensor product $M \otimes N$ is also a right $H$-comodule. The associated coalgebra of $M \otimes N$ is is $CD$ where $C$ and $D$ are the associated subcoalgebras of $M$ and $N$ respectively (see \cite{NR'}).
\section{Double coset formula for cosemisimple Hopf algebras}\label{dcsf} In this section let $H$ be a semisimple finite dimensional Hopf algebra as before and $K$ and $L$ be two Hopf subalgebras. Then $H$ can be decomposed as sum of $K-L$ bimodules which are free both as left $K$-modules and right $L$-modules and are analogues of double cosets in group theory. To the end of this section we give an application in the situation of a unique double coset.
There is a bilinear form $m : C(H^*)\otimes C(H^*) \rightarrow k$ defined as follows (see \cite{NR}). If $M$ and $N$ are two $H$-comodules with characters $c$ and $d$ then $m(c,\;d)$ is defined as $\mathrm{dim}_k\mathrm{Hom}^H(M,\;N)$. The following properties of $m$ (see \cite{NR}) will be used later: $$m(x, \;yz)=m(y^*,\;zx^*)=m(z^*,\;x^*y)\;\;\text{and}\;\;$$$$m(x,y)=m(y,\;x)=m(y^*,\;x^*)$$ for all $x,y,z \in C(H^*)$.
Let $H$ be a finite dimensional cosemisimple Hopf algebra and $K$, $L$ be two Hopf subalgebras of $H$. We define an equivalence relation $r_{ _{K,\;L}}^H$ on the set of simple coalgebras of $H$ as following: $C \sim D$ if $ C \subset \mathrm{KDL}$.
Since the set of simple subcoalgebras is in bijection with $\mathrm{Irr}(H^*)$ the above relation in terms of $H^*$-characters becomes the following: $c \sim d$ if $m(c\;,\Lambda_{ _K}d\Lambda_{ _L}) > 0$
where $\Lambda_{ _K}$ and $\Lambda_{ _L}$ are the integrals of $K$ and $L$ with $\epsilon(\Lambda_{ _K})=|K|$ and $\epsilon(\Lambda_{ _L})=|L|$ and $c,\;d \in \mathrm{Irr}(H^*)$.
It is easy to see that $\sim$ is an equivalence relation. Clearly $c \sim c$ for any $c \in \mathrm{Irr}(H^*) $ since both $\Lambda_{ _K}$ and $\Lambda_{ _L}$ contain the trivial character.
Using the above properties of the bilinear form $m$, one has that if $c \sim d$ then $m(d,\; \Lambda_{ _K}c\Lambda_{ _L})=m(\Lambda_{ _K}^*,\;c\Lambda_{ _L}d^*)=m(c^*,\;\Lambda_{ _L}d^*\Lambda_{ _K})=m(c,\;\Lambda_{ _K}^*d\Lambda_{ _L}^*)=m(c,\;\Lambda_{ _K}d\Lambda_{ _L})$ since $\Lambda_{ _K}^*=\Lambda_{ _K}$ and $\Lambda_{ _L}^*=\Lambda_{ _L}$. Thus $d \sim c$.
The transitivity can be easier seen that holds in terms of simple subcoalgebras. Suppose that $c\sim d$ and $d \sim e$ and $c,\;d,$ and $e$ are three irreducible characters associated to the simple subcoalgebras $C,\;D$ and $E$ respectively. Then $C\subset KDL$ and $D \subset KEL$. The last relation implies that $KDL \subset K^2EL^2=KEL$. Thus $C \subset KEL$ and $c \sim e$.
If $\mathcal{C}_1,\mathcal{C}_2,\cdots \mathcal{C}_l$ are the equivalence classes of $r_{ _{K,\;L}}^H$ on $\mathrm{Irr}(H^*)$ then let \begin{equation}\label{pregnv}a_i=\sum\limits_{d \in \mathcal{C}_i}\epsilon(d)d\end{equation} for $0 \leq i \leq l$.
For any character $d \in C(H^*)$ let $L_d$ and $R_d$ be the left and right multiplication with $d$ on $C(H^*)$.
\begin{prop} \label{eigenvv}With the above notations it follows that $a_i$ are eigenvectors of the operator $T=L_{ _{\Lambda_{ _K}}}\circ R_{ _{\Lambda_{ _L}}}$ on $C(H^*)$ corresponding to the eigenvalue $|K||L|$. \end{prop}
\begin{proof} Definition of $r_{ _{K,\;L}}^H$ implies that $d \sim d'$ if and only if $m(d',\;T(d)) >0$. Since $T(\Lambda_{ _H})=|H||K|\Lambda_{ _H}$ and $\Lambda_{ _H}=\sum_{i=1}^la_i$ it follows that $T(a_i)=|K||L|a_i$ for all $1 \leq i \leq \l$. \end{proof} \begin{rem}Let $C_1$ and $C_2$ be two subcoalgebras of $H$ and $K=\sum_{n\geq 0}C_1^n$ and $L=\sum_{n \geq 0}C_2^n$ be the two Hopf subalgebras of $H$ generated by them \cite{NR}. The above equivalence relation can be written in terms of characters as follows: $c \sim d$ if $m(c,\;c_1^ndc_2^m) > 0$ for some natural numbers $m,n\geq 0$. \end{rem}
In the sequel, we use the Frobenius-Perron theorem for matrices with nonnegative entries (see \cite{F}). If $A$ is such a matrix then $A$ has a positive eigenvalue $\lambda$ which has the biggest absolute value among all the other eigenvalues of $A$. The eigenspace corresponding to $\lambda$ has a unique vector with all entries positive. $\lambda$ is called the principal value of $A$ and the corresponding positive vector is called the principal vector of $A$. Also the eigenspace of $A$ corresponding to $\lambda$ is called the principal eigenspace of $A$.
The following result is also needed:
\begin{prop}(\cite{F}, Proposition 5.)\label{transpose} Let $A$ be a matrix with nonnegative entries such that $A$ and $A^t$ have the same principal eigenvalue and the same principal vector. Then after a permutation of the rows and the same permutation of the columns $A$ can be decomposed in diagonal blocks $A=\{A_1,\; A_2,\;\cdots,\;A_s\}$ with each block an indecomposable matrix. \end{prop}
Recall from \cite{F} that a matrix $A\in\mathrm{M}_n(k)$ is called decomposable if the set $I=\{1,2, \cdots , n\}$ can be written as a disjoint union $I=J_1\bigcup J_2$ such that $a_{uv}=0$ whenever $u \in J_1$ and $v \in J_2$. Otherwise the matrix $A$ is called indecomposable.
\begin{thm}\label{mainMack} Let $H$ be a finite dimensional semisimple Hopf algebra over the algebraically closed field $k$ and $K$, $L$ be two Hopf subalgebras of $H$. Consider the linear operator $T=L_{ _{\Lambda_{ _K}}}\circ R_{ _{\Lambda_{ _L}}}$ on the character ring $C(H^*)$ and $[T]$ the matrix associated to $T$ with respect to the standard basis of $C(H^*)$ given by the irreducible characters of $H^*$. \begin{enumerate}
\item The principal eigenvalue of $[T]$ is $|K||L|$.
\item The eigenspace corresponding to the eigenvalue $|K||L|$ has \\ ${(a_i)}_{1 \leq i \leq l}$ as $k$- basis were $a_i$ are defined in \ref{pregnv}. \end{enumerate} \end{thm}
\begin{proof}
1) Let $\lambda$ be the biggest eigenvalue of $T$ and $v$ the principal eigenvector corresponding to $\lambda$. Then $\Lambda_{ _K}v\Lambda_{ _L}=\lambda v$. Applying $\epsilon$ on both sides of this relation it follows that
$|K||L|v=\lambda v$. Since $v$ has positive entries it follows that
$\lambda=|K||L|$.
2) It is easy to see that the transpose of the matrix $[T]$ is also $[T]$. To check that let $x_1,\;\cdots,\;x_s$ be the basis of $C(H^*)$ given by the irreducible characters of $H^*$ and suppose that $T(x_i)=\sum_{j=1}^st_{ij}x_j$. Thus $t_{ij}=m(x_j,\;\Lambda_{ _K}x_i\Lambda_{ _L})$ and $t_{ji}=m(x_i,\;\Lambda_{ _K}x_j\Lambda_{ _L})=m(\Lambda_{ _K}^*,\;x_j\Lambda_{ _L}x_i^*)= m(x_j^ *,\;\Lambda_{ _L}x_i^*\Lambda_{ _K})=m(x_j, \Lambda_{ _K}^*x_i\Lambda_{ _L}^*)=t_{ij}$ since $\Lambda_{ _K}^*=S(\Lambda_{ _K})=\Lambda_{ _K}$ and also $\Lambda_{ _L}^*=\Lambda_{ _L}$. Proposition \ref{transpose} implies that after a permutation of the rows and the same permutation of the columns the matrix $[T]$ decomposes in diagonal blocks $A=\{A_1,\; A_2,\;\cdots,\;A_s\}$ with each block an indecomposable matrix. This decomposition of $[T]$ in diagonal blocks gives a partition $\mathrm{Irr}(H^*)=\cup_{i=1}^s\mathcal{A}_i$ on the set of irreducible characters of $H^*$, where each $\mathcal{A}_i$ corresponds to the rows (or columns) indexing the block $A_i$.
The eigenspace of $[T]$ corresponding to the eigenvalue $\lambda$ is the sum of the eigenspaces of the diagonal blocks $A_1, A_2,\cdots A_l$ corresponding to the same value. Since each $A_i$ is an indecomposable matrix it follows that the eigenspace of $A_i$ corresponding to $\lambda$ is one dimensional (see \cite{F}). If $b_j=\sum_{d \in \mathcal{A}_j}\epsilon(d)d$ then as in the proof of Proposition \ref{eigenvv} it can be seen that $b_j$ is eigenvector of $T$ corresponding to the eigenvalue $\lambda=|K||L|$. Thus $b_j$ is the unique eigenvector of $A_j$ corresponding to the eigenvalue $|K||L|$. Therefore each $a_i$ is a linear combination of these vectors. But if $d \in \mathcal{A}_i$ and $d'\in \mathcal{A}_j$ with $i \neq j$ then $m(d', \; T(d))=0$ and the definition of $r_{ _{K,\;L}}^H$ implies that $d \nsim d'$. This means that $a_i$ is a scalar multiple of some $b_j$ and this defines a bijective correspondence between the diagonal blocks and the equivalence classes of the relation $r_{ _{K,\;L}}^H$. Thus the eigenspace corresponding to the principal eigenvalue $|K||L|$ has a $k$- basis given by $a_i$ with $0 \leq i \leq l$.
\end{proof}
\begin{cor}\label{bim} Let $H$ be a finite dimensional semisimple Hopf algebra and $K,\;L$ be two Hopf subalgebras of $H$. Then $H$ can be decomposed as \begin{equation*} \label{decomp} H=\bigoplus_{i=1}^lB_i \end{equation*} where each $B_i$ is a $(K,L)$- bimodule free as both left $K$-module and right $L$-module. \end{cor} \begin{proof}Consider the above equivalence relation $r_{ _{K,\;L}}^H$, relative to the Hopf subalgebras $K$ and $L$. For each equivalence class $\mathcal{C}_i$ let $B_i=\bigoplus_{C \in \mathcal{C}_i}C$. Then $KB_iL=B_i$ from the definition of the equivalence relation. It follows that $B_i=KB_iL \in {}_{K}\mathcal{M}_L^H$ which implies that $B_i$ is free as left $K$-module and right $L$-module \cite{NZ}. \end{proof} The bimodules $B_i$ from the above corollary will be called a double coset for $H$ with respect to $K$ and $L$.
\begin{cor}With the above notations, if $d \in \mathcal{C}_i$ then
\begin{equation}\label{formula}\frac{\Lambda_{ _K}}{|K|}d\frac{\Lambda_{ _L}}{|L|}=\epsilon(d)\frac{a_i}{\epsilon(a_i)}\end{equation} \end{cor} \begin{proof}
One has that $\Lambda_{ _K}d\Lambda_{ _L}$ is an eigenvector of $T=L_{ _{\Lambda_{ _K}}}\circ R_{ _{\Lambda_{ _L}}}$ with the maximal eigenvalue $|K||L|$. From Theorem \ref{mainMack} it follows that $\Lambda_{ _K}d\Lambda_{ _L}$ is a linear combination of the elements $a_j$. But $\Lambda_{ _K}d\Lambda_{ _L}$ cannot contain any $a_j$ with $j \neq i$ because all the irreducible characters entering in the decomposition of the product are in $\mathcal{C}_i$. Thus $\Lambda_{ _K}d\Lambda_{ _L}$ is a scalar multiple of $a_i$ and formula \ref{formula} follows. \end{proof}
\begin{rem}\label{oneside} Setting $C_1=k$ in Theorem \ref{mainMack} we obtain Theorem 7 \cite{NR}. The above equivalence relation becomes $c \sim d$ if and only if $m(c,\;dc_2^m) > 0$ for some natural number $m \geq 0$. The equivalence class corresponding to the simple coalgebra $k1$ consists of the simple subcoalgebras of the powers $C_2^m$ for $m \geq 0$. Without loss of generality we may assume that this equivalence class is $\mathcal{C}_1$. It follows that \begin{equation*} \frac{d}{\epsilon(d)}\frac{a_1}{\epsilon(a_1)}=\frac{a_i}{\epsilon(a_i)} \end{equation*}for any irreducible character $d\in \mathcal{C}_i$. \end{rem}
Let $K$ be a Hopf subalgebra of $H$ and $s=|H|/|K|$. Then $H$ is free as left $K$-module \cite{NZ}. If $\{a_i\}_{i=1,\;s}$ is a basis of $H$ as left $K$-module then $H=Ka_1\oplus Ka_2\cdots\oplus Ka_s$ as left $K$ modules. Consider the operator $L_{ _{\Lambda_ {_K}}}$ given by left multiplication on $H$. The eigenspace corresponding to the eigenvalue $|K|$ has a basis given by $\Lambda_{ _K}a_i$, thus it has dimension $s$. If we restrict the operator $L_{ _{\Lambda_ {_K}}}$ on $C(H^*)\subset H$ then Theorem \ref{mainMack} implies that the number of equivalence classes of $r_{ _{K,\;k}}^H$ is equal to the dimension of the eigenspace of $L_{ _{\Lambda_ {_K}}}$ corresponding to the eigenvalue $|K|$. Thus the number of the equivalence classes of $r_{ _{K,\;k}}^H$ is always less or equal then the index of $K$ in $H$.
\begin{example}Let $H=kQ_8\#^{\alpha}kC_2$ the 16-dimensional Hopf algebra described in \cite{Kash}. Then $G(H^*)=<x> \times <y>\cong \mathbb{Z}_2\times \mathbb{Z}_2$ and $\mathrm{Irr}(H^*)$ is given by the four one dimensional characters $1,x,y,xy$ and $3$ two dimensional characters denoted by $d_1,d_2,d_3$. The algebra structure of $C(H^*)$ is given by: $$x.d_1=d_3=d_1.x,\;\;x.d_2=d_2=d_2.x,\;\;x.d_3=d_1=d_3.x$$ $$y.d_i=d_i=d_i.y \;\;\text{for all}\;i=1,3$$ $$d_1^2=d_3^2=x+xy+d_2,\;\;d_2^2=1+x+y+xy,\;d_1d_2=d_2d_1=d_1+d_3$$ $$d_1d_3=d_3d_1=1+y+d_2$$ Consider $K=k<x>$ as Hopf subalgebra of $H$ and the equivalence relation $r_{ _{k,\;K}}^H$ on $\mathrm{Irr}(H^*)$. The equivalence classes are given by $\{1,x\}$, $\{y,xy\}$, $\{d_2\}$ and $\{d_1,d_3\}$ and the number of them is strictly less than the index of $K$ and $H$. If $C_2 \subset H$ is the coalgebra associated to $d_2$ then the third equivalence class determine in the decomposition \ref{decomp} the free $K$-module $C_2K=C_2$ whose rank is less then the dimension of $C_2$. \end{example} \section{More on coset decomposition}\label{mcd} Let $H$ be a semisimple Hopf algebra and $A$ be a Hopf subalgebra. Define $H//A=H/HA^+$ and let $\pi : H \rightarrow H//A$ be the natural module projection. Since $HA^+$ is a coideal of $H$ it follows that $H//A$ is a coalgebra and $\pi$ is also a coalgebra map.
Let $k$ be the trivial $A$-module via the counit $\epsilon$. It can be checked that $H//A\cong H\otimes_{ _A}k$ as $H$-modules via the map $\hat{h}\rightarrow h\otimes_{ _A}1$. Thus $\mathrm{dim}_kH//A=\mathrm{rank}_{ _A}H$.
If $L$ and $K$ are Hopf subalgebras of $H$ define $LK//K:=LK/LK^+$. $LK$ is a right free $K$-module since $LK \in \mathcal{M}_K^H$. A similar argument to the one above shows that $LK//K\cong LK\otimes_{ _K}k$ as left $L$-modules where $k$ is the trivial $K$-module. Thus $\mathrm{dim}_kLK//K=\mathrm{rank}_{ _K}LK $. It can be checked that $LK^+$ is a coideal in $LK$ thus $LK//K$ has a natural coalgebra structure.
\begin{thm} Let $H$ be a semisimple Hopf algebra and $K,\;L$ be two Hopf subalgebras of $H$. Then $L//L\cap K\cong LK//K$ as coalgebras and left $L$-modules. \end{thm}
\begin{proof} Define the map $\phi: L \rightarrow LK//K$ by $\phi(l)=\hat{l}$. Then $\phi$ is the composition of $L \hookrightarrow LK \rightarrow LK//K$ and is a coalgebra map as well as a morphism of left $L$-modules. Moreover $\phi$ is surjective since $\widehat{lk}=\epsilon(k)\hat{l}$ for all $l \in L$ and $k \in K$. Clearly $L(L\cap K)^+ \subset \mathrm{ker}(\phi)$ and thus $\phi$ induces a surjective map $\phi: L//L\cap K\rightarrow LK//K$.
Next it will be shown that
\begin{equation}\label{dims}\frac{|L|}{|L\cap K|}=\frac{|LK|}{|K|}\end{equation} which implies that $\phi$ is bijective since both spaces have the same dimension. Consider on $\mathrm{Irr}(H^*)$ the equivalence relation introduced above and corresponding to the linear operator $L_{ _{\Lambda_{ _L}}}\circ R_{ _{\Lambda_{ _K}}}$. Assume without loss of generality that $\mathcal{C}_1$ is the equivalence class of the character $1$ and put $d=1$ the trivial character, in the formula \ref{formula}. Thus $\frac{\Lambda_{ _L}}{|L|}\frac{\Lambda_{ _K}}{|K|}=\frac{a_1}{\epsilon(a_1)}$. But from the definition of $\sim$ it follows that $a_1$ is formed by the characters of the coalgebra $LK$. On the other hand $\Lambda_{ _L}=\sum_{d \in Irr(L^*)}\epsilon(d)d$ and $\Lambda_{ _K}=\sum_{d \in Irr(K^*)}\epsilon(d)d$
(see \cite{Lar}). Equality \ref{dims} follows counting the multiplicity of the irreducible character $1$ in $\Lambda_{ _K}\Lambda_{ _L}$. Using \cite{NR} we know that $m(1\;, dd')>0$ if and only if $d'=d^*$ in which case $m(1\;, dd')=1$. Then $m(1, \;\frac{\Lambda_{ _L}}{|L|}\frac{\Lambda_{ _K}}{|K|})=\frac{1}{|L||K|}\sum_{d \in \mathrm{Irr}(L\cap K)}\epsilon(d)^2=\frac{|L\cap K|}{|L||K|}$ and $m(1, \;\frac{a_1}{\epsilon(a_1)})=\frac{1}{\epsilon(a_1)}=\frac{1}{|LK|}$. \end{proof}
\begin{cor} If $K$ and $L$ are Hopf subalgebras of $H$ then $\mathrm{rank}_{K}LK=\mathrm{rank}_{L\cap K}L$. \end{cor} \begin{prop}Let $H$ be a finite dimensional cosemisimple Hopf algebra and $K,\;L$ be two Hopf subalgebras of $H$ such that $KL=LK$. If $M$ is a $K$-module then \begin{equation*} M\uparrow_K^{LK}\downarrow_L \cong (M \downarrow_{L\cap K})\uparrow^L \end{equation*} as left $L$-modules. \end{prop} \begin{proof}
For any $K$-module $M$ one has $$M\uparrow^{LK}\downarrow_L=LK\otimes_KM$$ while $$(M\downarrow_{L\cap K})\uparrow^L=L\otimes_{L\cap K}M$$ The previous Corollary implies that $\mathrm{rank}_{_ K}LK=\mathrm{rank}_{_ {L\cap K}}L$ thus both modules above have the same dimension.
Define the map $\phi: L\otimes_{L\cap K}M \rightarrow LK\otimes_KM$ by $\phi(l\otimes_{L\cap K} m)=l\otimes_K m$ which is the composition of $L\otimes_{L\cap K}M \hookrightarrow LK\otimes_{L\cap K}M \rightarrow LK\otimes_KM$. Clearly $\phi$ is a surjective homomorphism of $L$-modules. Equality of dimensions implies that $\phi$ is an isomorphism.
\end{proof} If $LK=H$ then the previous theorem is the generalization of Mackey's theorem decomposition for groups in the situation of a unique double coset. corollary
\section{A dual relation}\label{dr}
Let $K$ be a normal Hopf subalgebra of $H$ and $L=H//K$. Then the natural projection $\pi:H\rightarrow L$ is a surjective Hopf map and then $\pi^* :L^* \rightarrow H^*$ is an injective Hopf map. We identify $L^*$ with its image $\pi^*(L^*)$ in $H^*$. This is a normal Hopf subalgebra of $H^*$. In this section we will study the equivalence relation $r_{ _{L^*,\;k}}^{H^*}$ on $\mathrm{Irr}(H^{**})=\mathrm{Irr}(H)$.
The following result was proved in \cite{Bker}.
\begin{prop} \label{resn} Let $K$ be a normal Hopf subalgebra of a finite dimensional semisimple Hopf algebra $H$ and $L=H//K$. If $t_L \in L^*$ is the integral on $L$ with $t_L(1)=|L|$ then $\epsilon_{ _K}\uparrow_K^H=t_L$ and $t_L\downarrow_K^H=\frac{|H|}{|K|}\epsilon_{ _K}$ \end{prop}
\begin{prop}Let $K$ be a normal Hopf subalgebra of a semisimple Hopf algebra $H$ and $L=H//K$. Consider the equivalence relation $r_{ _{L^*,\;k}}^{H^*}$ on $\mathrm{Irr}(H)$. Then $\chi \sim \mu$ if and only if their restriction to $K$ have a common constituent. \end{prop} \begin{proof} The equivalence relation $r_{ _{L^*,\;k}}^{H^*}$ on $\mathrm{Irr}(H)$ becomes the following: $\chi \sim \mu$ if and only if $m_{ _H}(\chi,\;t_{ _L}\mu)>0$. On the other hand, applying the previous Proposition it follows that: \begin{eqnarray*} m_{ _H}(\chi,\;t_{ _L}\mu)& \!=\! & m_{ _H}(t_{ _L}^*,\;\mu\chi^*)=m_{ _H}(t_{ _L},\;\mu\chi^*) \\ & = & m_{ _H}(\epsilon \uparrow_{ _{K}}^H,\;\mu\chi^*)=m_{ _K}(\epsilon_{ _{K}},\;(\mu\chi^*)\downarrow_{ _K}) \\ & = & m_{ _K}(\epsilon_{ _{K}},\;\mu\downarrow_{ _K}\chi^*\downarrow_{ _K})=m_{ _K}(\chi\downarrow_{ _K},\:\mu\downarrow_{ _K}) \end{eqnarray*} Thus $\chi \sim \mu$ if and only if their restriction to $K$ have a common constituent. \end{proof}
\begin{thm}\label{restrres}Let $K$ be a normal Hopf subalgebra of a semisimple Hopf algebra $H$ and $L=H//K$. Consider the equivalence relation $r_{ _{L^*,\;k}}^{H^*}$ on $\mathrm{Irr}(H)$. Then $\chi \sim \mu$ if and only if $\frac{1}{\chi(1)}\chi\downarrow_K=\frac{1}{\mu(1)}\mu\downarrow_K$ \end{thm} \begin{proof} Let $\mathcal{C}_1,\mathcal{C}_2,\cdots \mathcal{C}_l$ be the equivalence classes of $\sim$ on $\mathrm{Irr}(H)$ and let \begin{equation}\label{pregnvddual}a_i=\sum\limits_{\chi \in \mathcal{C}_i}\chi(1)\chi\end{equation} for $0 \leq i \leq l$. If $\mathcal{C}_1$ is the equivalence class of the trivial character $\epsilon$ then the definition of $r_{ _{L^*,\;k}}^{H^*}$ implies that $a_1=t_{ _L}$. Formula from Remark \ref{oneside} becomes
\begin{equation*}
\frac{\chi}{\chi(1)}\frac{t_{ _L}}{|L|}=\frac{a_i}{a_i(1)} \end{equation*}
for any irreducible character $\chi \in \mathcal{C}_i$.
Restriction to $K$ of the above relation combined with Proposition \ref{resn} gives: \begin{equation}\label{scform}\frac{\chi\downarrow_{ _K}}{\chi(1)}=\frac{a_i\downarrow_{ _K}}{a_i(1)}\end{equation} Thus $\chi \sim \mu$ if and only if $\frac{\chi\downarrow_{ _K}}{\chi(1)}=\frac{\mu\downarrow_{ _K}}{\mu(1)}$.
\subsection{Formulae for restriction and induction}\label{indrestr} The previous theorem implies that the restriction of two irreducible $H$-characters to $K$ either have the same common constituents or they have no common constituents. Let $t_H$ be the integral on $H$ with $t_H(1)=1$. One has that $|H |t_{ _H}=\sum_{i=1}^la_i$ as $|H|t_{ _H}$ is the regular character of $H$. Since $H$ is free as $K$-module \cite{NZ} it follows that the restriction of $|H|t_{ _H}$ to $K$ is the regular character of $K$ multiplied by $|H|/|K|$. Thus $(|H|t_{ _H})\downarrow_{ _K}=|H|/|K|(|K|t_{ _K})$. But $|K|t_{ _K}=\sum_{\alpha \in \mathrm{Irr}(K)}\alpha(1)\alpha$ and Theorem \ref{restrres} implies that the set of the irreducible characters of $K$ can be partitioned in disjoint subsets $\mathcal{A}_i$ with $1\leq i \leq l$ such that \begin{equation*} \label{restrform}
a_i\downarrow_{ _K}=\frac{|H|}{|K|}\sum_{\alpha\in \mathcal{A}_i}\alpha(1)\alpha \end{equation*} Then if $\chi \in \mathcal{C}_i$ formula \ref{scform} implies that \begin{equation*}
\chi\downarrow_{ _K}=\frac{\chi(1)}{a_i(1)}\frac{|H|}{|K|}\sum_{\alpha\in \mathcal{A}_i}\alpha(1)\alpha \end{equation*}
Let $|\mathcal{A}_i|=\sum_{\alpha \in \mathcal{A}_i}\alpha^2(1)$. Evaluating at $1$ the above equality one gets $a_i(1)=\frac{|H|}{|K|}|\mathcal{A}_i|$. By Frobenius reciprocity the above restriction formula implies that if $\alpha \in \mathcal{A}_i$ then
\begin{equation}\label{indform}\alpha\uparrow_{ _K}^H=\frac{\alpha(1)}{a_i(1)}\frac{|H|}{|K|}\sum_{\chi \in \mathcal{C}_i}\chi(1)\chi=\frac{\alpha(1)}{a_i(1)}\frac{|H|}{|K|}a_i\end{equation} \end{proof}
\section{Restriction of modules to normal Hopf subalgebras}\label{restr}
Let $G$ a finite group and $H$ a normal subgroup of $G$. If $M$ is an irreducible $H$-module then \begin{equation*} M\uparrow_H^G\downarrow_H^G={\oplus_{i=1}^s} \;^{g_i}N \end{equation*} where $^gN$ is a conjugate module of $M$ and $\{g_i\}_{i=1,s} $ is a set of representatives for the left cosets of $H$ in $G$. For $g \in G$ the $H$-module $\;^gN$ has the same underlying vector space as $N$ and the multiplication with $h \in H$ is given by $h.n=(g^{-1}hg)n$ for all $n \in N$. It is easy to see that $\;^gN \cong \;^{g'}N$ if $gN=g'N$.
Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be an irreducible $K$-module. In this section we will define the notion of conjugate module to $M$ similar to group situation. If $d\in \mathrm{Irr}(H^*)$ we define a conjugate module $\;^dM$. The left cosets of $K$ in $H$ correspond to the equivalence classes of $r_{ _{K,\;k}}^H$. We will show that if $d,\;d'$ are two irreducible characters in the same equivalence class of $r_{ _{K,\;k}}^H$ then the modules $\;^dM$ and $\;^{d'}M$ have the same constituents. We will show that the irreducible constituents of $M\uparrow_K^H\downarrow_K^H$ and $\oplus_{d \in \mathrm{Irr}(H^*)}\;^dM$ are the same.
\begin{rem} Since $K$ is a normal Hopf subalgebra it follows that $\Lambda_{ _K}$ is a central element of $H$ (see \cite{Mas'}) and by their definition $r_{ _{K,\;k}}^H=r_{ _{k,\;K}}^H$. Thus the left cosets are the same with the right cosets in this situation.
\end{rem} \subsection{}\label{conjdef} Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be a $K$-module. If $W$ is an $H^*$-module then $W\otimes M$ becomes a $K$-module with
\begin{equation}\label{def}
k(w\otimes m)=w_0\otimes(S(w_1)kw_2)m \end{equation}
In order to check that $W\otimes M$ is a $K$-module one has that \begin{eqnarray*} k.(k'.(w\otimes m))& \!=\! &k(w_0\otimes(S(w_1)k'w_2)m)\\ & = & w_0\otimes (S(w_1)kw_2)(S(w_3)k'w_4)\\ & = & w_0\otimes(S(w_1)kk'w_2)m\\ & = & (kk').(w\otimes m) \end{eqnarray*} for all $k,k' \in K$, $w \in W$ and $m \in M$.
It can be checked that if $W \cong W'$ as $H^*$-modules then $W\otimes M \cong W'\otimes M$. Thus for any irreducible character $d \in \mathrm{Irr}(H^*)$ associated to a simple $H$-comodule $W$ one can define the $K$-module $\;^dM \cong W\otimes M$. \begin{prop}Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be an irreducible $K$-module with character $\alpha \in C(K)$. Suppose that $W$ is a simple $H^*$-module with character $d \in \mathrm{Irr}(H^*)$. Then the character $\alpha_{ _d}$ of the $K$-module $\;^d M$ is given by the following formula: \begin{equation*} \alpha_{ _d}(x)=\alpha(Sd_1xd_2) \end{equation*} for all $x \in K$. \end{prop} \begin{proof}
Indeed one may suppose that $W\cong k<x_{1i}\;|\;1\leq i \leq q>$ where
$C_{ _d}=k<x_{ij} \;|\; 1\leq i,j\leq q>$ is the coalgebra associated to $W$ and $q=\epsilon(d)=|W|$. Then formula \ref{def} becomes $k(x_{1i}\otimes w)=\sum_{j,\;l=1}^qx_{1j}\otimes (S(x_{jl})kx_{lj})m$. Since $d=\sum_{i=1}^qx_{ii}$ one gets the the formula for the character $\alpha_{ _d}$. \end{proof}
For any $d \in \mathrm{Irr}(H^*)$ define the linear operator $c_{ _d}:C(K)\rightarrow C(K)$ which on the basis given by the irreducible characters is given by $c_{ _d}(\alpha)=\alpha_{ _d}$ for all $\alpha \in \mathrm{Irr}(K)$. \begin{rem} From the above formula it can be directly checked that $^{dd'}\alpha=\;^{d}(\;^{d'}\alpha)$ for all $d, d' \in \mathrm{Irr}(H^*)$ and $\alpha \in C(K)$. This shows that $C(K)$ is a left $C(H^*)$-module. Also one can verify that $\;^d(\alpha^*)=(\;^d\alpha)^*$. \end{rem} \begin{prop} Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be an irreducible $K$-module with character $\alpha \in C(K)$. If $d,\;d' \in \mathrm{Irr}(H^*)$ lie in the same coset of $r^H_{ _{k,\;K}}$ then $\;^dM$ and $\;^{d'}M$ have the same irreducible constituents. Moreover $\frac{\alpha_d}{\epsilon(d)}=\frac{\alpha_{d'}}{\epsilon(d')}$ \end{prop} \begin{proof}
Consider the equivalence relation $r_{ _{k,\;K}}^H$ from section \ref{dcsf} and $H=\oplus_{i=1}^sB_i$ the decomposition from Corollary \ref{bim}. Let $\mathcal{B}_1,\cdots,\mathcal{B}_s$ be the equivalence classes and $b_i$ defined as in \ref{pregnv}. Then formula \ref{oneside} becomes \begin{equation}
\frac{d}{\epsilon(d)}\frac{\Lambda{ _K}}{|K|}=\frac{b_i}{\epsilon(b_i)} \end{equation}
where $\Lambda{ _K}$ is the integral in $K$ with $\epsilon(\Lambda{ _K})=|K|$ and $d \in \mathcal{B}_i$. Thus
\begin{eqnarray*} \alpha_{ _{b_i}}(x)\! & = & \! \alpha(S(b_i)_1x(b_i)_2)=
\\ & = & \frac{\epsilon(b_i)}{\epsilon(d)|K|}\alpha(S({d\Lambda_{ _K})}_1)x(d{\Lambda_{ _K}})_2)=
\\ & = & \frac{\epsilon(b_i)}{\epsilon(d)|K|}\alpha(S(({\Lambda_{ _K}})_1)S(d_1)xd_2({\Lambda_{ _K}})_2)= \\ & =& \frac{\epsilon(b_i)}{\epsilon(d)}\alpha(S(d_1)xd_2)= \\ & =& \frac{\epsilon(b_i)}{\epsilon(d)}\alpha_{ _d}(x) \end{eqnarray*} for all $d \in \mathcal{B}_i$.
This implies that $d\sim d'$ then $\frac{\alpha_{ _d}}{\epsilon(d)}=\frac{\alpha_{ _{d'}}}{\epsilon(d')}$. \end{proof}
Let $N$ be an $H$-module and $W$ an $H^*$-module. Then $W\otimes N$ becomes an $H$-module such that
\begin{equation}\label{def'}
h(w\otimes m)=w_0\otimes(S(w_1)hw_2)m \end{equation}
It can be checked that $W\otimes N\cong N^{|W|}$ as $H$-modules. Indeed the map $$\phi: W\otimes N \rightarrow \; _{ _{\epsilon}}W\otimes N, \;\;\;w\otimes n \mapsto w_0 \otimes w_1n$$ is an isomorphism of $H$-modules where $\;_{ _{\epsilon}}W$ is considered left $H$-module with the trivial action. Its inverse is given by $w\otimes m \mapsto w_0 \otimes S(w_1)m$. To check that $\phi$ is an $H$-module map one has that \begin{eqnarray*} \phi(h.(w\otimes n))\! & = & \! \phi(w_0 \otimes (S(w_1)hw_2)m) \\ & = & w_0 \otimes w_1 (S(w_2)hw_3)m \\ & = & w_0 \otimes hw_1m \\ & = & h.( w_0 \otimes w_1m) \\ & = &h\phi(w\otimes m) \end{eqnarray*} for all $w \in W$, $m \in M$ and $h \in H$. \begin{prop} Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be an irreducible $K$-module with character $\alpha \in C(K)$. If $d\in \mathrm{Irr}(H^*)$ then \begin{equation*} \frac{1}{\epsilon(d)}\alpha_{ _d}\uparrow_{ K}^H=\alpha\uparrow_{ _K}^H \end{equation*} \end{prop} \begin{proof} Using the notations from subsection \ref{indrestr} let $\mathcal{A}_i$ be the subset of $\mathrm{Irr}(K)$ which contains $\alpha$. It is enough to show that the constituents of $\alpha_{ _d}$ are contained in this set and then the induction formula \ref{indform} from the same subsection can be applied. For this, suppose $N$ is an irreducible $H$-module and \begin{equation}N\downarrow_{ _K}=\oplus_{i=1}^sN_i \end{equation}
where $N_i$ are irreducible $K$-modules. The above result implies that $W\otimes N \cong N^{|W|}$ as $H$-modules. Therefore $(W\otimes N)\downarrow_{ _K}={N\downarrow_{ _K}}^{|W|}$ as $K$-modules. But $(W\otimes N)\downarrow_{ _K}=\oplus_{i=1}^s(W \otimes N_i)$ where each $W \otimes N_i$ is a $K$-module by \ref{conjdef}. Thus \begin{equation}
\oplus_{i=1}^sN_i^{|W|}=\oplus_{i=1}^s(W \otimes N_i) \end{equation} This shows that if $N_i$ is a constituent of $N\downarrow_{ _K}$ then $W\otimes N_i$ has all the irreducible $K$- constituents among those of $N\downarrow_{ _K}$. Formula \ref{indform} applied for each irreducible constituent of $\alpha_{ _d}$ gives that \begin{equation} \frac{1}{\epsilon(d)}\alpha_{ _d}\uparrow_{ K}^H=\alpha\uparrow_{ _K}^H \end{equation} for all $\alpha \in \mathrm{Irr}(K)$ and $d \in \mathrm{Irr}(H^*)$. \end{proof} \begin{prop} Let $K$ be a normal Hopf subalgebra of $H$ and $M$ be an irreducible $K$-module. Then $M\uparrow_K^H\downarrow_K^H$ and $\oplus_{d \in \mathrm{Irr}(H^*)}\;^dM$ have the same irreducible constituents. \end{prop} \begin{proof} Consider the equivalence relation $r_{ _{k,\;K}}^H$ from Section \ref{dcsf} and let $\mathcal{B}_1,\cdots,\mathcal{B}_s$ be its equivalence classes. Pick an irreducible character $d_i \in \mathcal{B}_i$ in each equivalence class of $r_{ _{k,\;K}}^H$ and let $C_i$ be its associated simple coalgebra. Then Corollary \ref{bim} implies that $H=\oplus_{i=1}^sC_iK$. It follows that the induced module $M\uparrow_{ _K}^H$ is given by $$M\uparrow_{ _K}^H=H\otimes_{ _K}M=\oplus_{i=1}^sC_iK\otimes_{ _K}M$$ Each $C_iK\otimes_{ _K}M$ is a $K$-module by left multiplication with elements of $K$ since $$k.(ck'\otimes_Km)=c_1(Sc_2kc_3)k'\otimes_Km=c_1\otimes_K(Sc_2kc_3)k'm$$ for all $k,k' \in K$, $c \in C_i$ and $m \in M$. Thus $M\uparrow_{ _K}^H$ restricted to $K$ is the sum of the $K$-modules $C_iK\otimes_{ _K}M$. On the other hand the composition of the canonical maps $C_i\otimes K \hookrightarrow C_iK\otimes M \rightarrow C_iK\otimes_{_ K}M$ is a surjective morphism of $K$-modules which implies that $C_iK\otimes_{_ K}M$ is a homeomorphic image of $\epsilon(d_i)$ copies of $^{d_i}M$. Therefore the irreducible constituents of $M\uparrow_K^H\downarrow_K^H$ are among those of $\oplus_{d \in \mathrm{Irr}(H^*)}\;^dM$. In the proof of the previous Proposition we showed the other inclusion. Thus $M\uparrow_K^H\downarrow_K^H$ and $\oplus_{d \in \mathrm{Irr}(H^*)}\;^dM$ have the same irreducible constituents. \end{proof}
\end{document}
Dear Sebastian, I have sent the paper to a referee. However I gave a look to it myself and have the following comments:
1) I do not see why the Hopf algebra $U(D, \;\lambda)$ is $Z^theta$-graded, page 4, subsection 1.1, since the linking relations are not homogeneous (I do not remember saying this in our paper).
2) Cor. 3.11, page 12, was more-or-less known to me, I believe that Beattie included such a statement in her paper [4] (a more general statement should exists, say for pointed Hopf algebras fin dim and abelian group). Indeed, this really goes back to the very first paper by Drinfeld-- where he introduced the double just to deal with this kind of examples.
3) I have the impression that the results in Section 4 can be presented in a more general and clear way as follows:
i) Suppose that $A = R \# H $ is a bosonization of a braided Hopf algebra $R \in {\mathcal{M}^H}_HYD$ by a Hopf algebra $H$, all fin-dim. Consider left integrals $\Lambda$ in $H$, $x$ in R. Then $\Lambda x$ is a left integral in A (this should be well-known, integrals in braided Hopf algebras are considered in several papers, e.g. in Fischman, Montgomery, Schneider, Trans AMS and also in Turaev et al, JPAA).
ii) Suppose further that $R = k \oplus R(1) \oplus ... \oplus R(n)$ is a GRADED braided Hopf algebra, connected, $R(n)$ the top non-zero homogeneous component. Then $\mathrm{dim} R(n) = 1$ and this is the space of left and right integrals; this is easy, we have observed this in Braided Hopf algebras over non-abelian groups. (With M. Graña). Bol. Acad. Nacional de Ciencias (Córdoba) 63, 45-78 (1999).
iii) Now $R(n)$ determines a unique group-like $g$ and a unique character chi, since it has dim 1 (whether H is a group algebra or not...). I guess that $\chi$ or a related one is the distinguished group-like.
iv) Now dualize, $A^* = R^* \# H^*$ and repeat the argument.
In this way you should probably obtain a sufficient condition for ribbon, and perhaps a necessary one. This would include all fin-dim pointed Hopf algebras... at least theoretically.
Best,
\end{document}
\end{document} | arXiv |
\begin{definition}[Definition:Type Space]
Let $\MM$ be an $\LL$-structure, and let $A$ be a subset of the universe of $\MM$.
Let $\map {S_n^\MM} A$ be the set of complete $n$-types over $A$.
The '''space of $n$-types over $A$''' is the topological space formed by the set $\map {S_n^\MM} A$ together with the topology arising from the basis which consists of the sets:
:$\sqbrk \phi := \set {p \in \map {S_n^\MM} A:\phi \in p}$
for each $\LL_A$-formula $\phi$ with $n$ free variables.
Note that each $\sqbrk \phi$ is also closed in this topology, since $\sqbrk \phi$ is the complement of $\sqbrk {\neg \phi}$ in $\map {S_n^\MM} A$.
{{LinkWanted|Definition of the notation $\sqbrk \phi$}}
\end{definition} | ProofWiki |
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The Archimedes Code: a dialogue between science, practice, design theory and systems engineering
Yoram Reich
Journal: Design Science / Volume 9 / 2023
Published online by Cambridge University Press: 27 January 2023, e2
Archimedes, the founder of statics and hydrostatics, in his mathematics and physics studies, created methods related to his inventions of new machines, for example, the method of mechanical theorems based on his lever invention. He also used the principles of decomposition and replication underlying his heat ray invention, and these two principles permeate his work. Analysis of Archimedes' work shows how he was perhaps the first to use methodically a strategy for solving diverse complex problems. In this article, we use the term Archimedes Code to encompass the way Archimedes approached problems including those two principles. Archimedes was perhaps the first design theorist and the first to think systematically about how to address design challenges. Furthermore, his work demonstrates the fundamental role of engineering practice in advancing science. The new insights regarding the Archimedes Code and its value in design practice may inspire both design researchers and practitioners.
A decomposition for Lévy processes inspected at Poisson moments
Onno Boxma, Michel Mandjes
Journal: Journal of Applied Probability , First View
Published online by Cambridge University Press: 15 November 2022, pp. 1-13
We consider a Lévy process Y(t) that is not continuously observed, but rather inspected at Poisson( $\omega$ ) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$ . The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$ , denoted by $Y_{\beta,\omega}$ . Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$ . Concretely, $\overline{Y}(T_\beta)$ can be written as the sum of two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$ . The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér–Lundberg insurance risk model.
What factors drive gender differences in the body mass index? Evidence from Turkish adults
Ebru Caglayan-Akay, Merve Ertok-Onurlu, Fulden Komuryakan
Journal: Journal of Biosocial Science , First View
Published online by Cambridge University Press: 05 May 2022, pp. 1-26
In recent years, studies show that obesity has become an important health condition, especially among adults. The first aim of this study is to examine socio-demographic and behavioural factors on body mass index distribution of male and female adults over 20 years old in Turkey. The second aim is to determine the body mass index disparity by gender and the socio-demographic and behavioural factors that might wider or narrow it. This study adopts unconditional quantile regression and decomposition methods, and the data set covers the Turkish Health Surveys for 2014, 2016, and 2019. The findings document that high level of body mass index are associated with being married, aging, and physical inactivity. Interestingly, employment status has different contributions on the body mass index of males and females. The results also claim a body mass index gap among males and females as a result of differences in some potential socio-demographic and behavioural factors, and the gap gets higher at the upper and lower quantiles of BMI distribution. This study may provide a clear understanding for policymakers on how to design efficacious obesity policies considering the differences in the effect of socio-demographic and behavioural factors on the distribution of body mass index across females and males. The results suggest that the Ministry of Health should specifically target different groups for males and females and should reduce the differences in socio-demographic and behavioural determinants between females and males to prevent and reduce obesity prevalence in Turkey.
4 - Measuring Market Efficiency
Carlo Altomonte, Università Commerciale Luigi Bocconi, Milan, Filippo di Mauro
Book: The Economics of Firm Productivity
Published online: 14 April 2022
Print publication: 21 April 2022, pp 54-63
We present in this chapter the empirical techniques used to measure market efficiency and its implications, starting with the so-called OP gap and its dynamic version. We will then discuss the Foster productivity decomposition method and the Hsieh and Klenow techniques to measure allocative inefficiency. Market inefficiency is also related to concentration and market power. When competition in a market is reduced, aggregate productivity growth may decrease, reducing consumer welfare.
Design without representation
Sotirios D. Kotsopoulos
Journal: AI EDAM / Volume 36 / 2022
Published online by Cambridge University Press: 09 February 2022, e12
Shapes are perceived unanalyzed, without rigid representation of their parts. They do not comply with standard symbolic knowledge representation criteria; they are treated and judged by appearance. Resolving the relationship of parts to parts and parts to wholes has a constructive role in perception and design. This paper presents a computational account of part–whole figuration in design. To this end, shape rules are used to show how a shape is seen, and shape decompositions having structures of topologies and Boolean algebras reveal alternative structures for parts. Four examples of shape computation are presented. Topologies demonstrate the relationships of wholes, parts, and subparts, in the computations enabling the comparison and relativization of structures, and lattice diagrams are used to present their order. Retrospectively, the topologies help to recall the generative history and establish computational continuity. When the parts are modified to recognize emergent squares locally, other emergent shapes are highlighted globally as the topology is re-adjusted. Two types of emergence are identified: local and global. Seeing the local parts modifies how we analyze the global whole, and thus, a local observation yields a global order.
2 - Mixed-Integer Programming
from Part I - Background
Christos T. Maravelias, Princeton University, New Jersey
Book: Chemical Production Scheduling
Print publication: 06 May 2021, pp 32-64
This chapter provides an overview of mixed-integer programming (MIP) modeling and solution methods. In Section 2.1, we present some preliminary concepts on optimization and mixed-integer programming. In Section 2.2, we discuss how binary variables can be used to model features commonly found in optimization problems. In Section 2.3, we present some basic MIP problems and models. Finally, in Section 2.4, we overview the basic approaches to solving MIP models and present some concepts regarding formulation tightness and decomposition methods. Finally, we discuss software tools for modeling and solving MIP models in Section 2.5.
12 - Solution Methods: Sequential Environments
from Part IV - Special Topics
Print publication: 06 May 2021, pp 289-317
The goal of the present chapter, as well as Chapter 13, is to illustrate how problem features can be exploited to develop more efficient models and/or specialized algorithms. We start, in the present chapter, with solution methods for problems in sequential environments. Specifically, we discuss four methods: (1) a decomposition approach, in Section 12.1; (2) preprocessing algorithms and tightening constraints, in Section 12.2; (3) a reformulation and tightening constraints based on time windows, in Section 12.3; and (4) a two-step algorithm, combining the advantages of discrete and continuous time models, in Section 12.4. While all presented methods can be applied to a wide range of problems, we present them for a subset of problems for the sake of brevity. Also, all methods can be applied to problems under different processing features, but to keep the presentation simple, we discuss problems with no shared utilities and no storage constraints.
13 - Climatic Impacts on Salt Marsh Vegetation
from Part III - Marsh Response to Stress
By Katrina L. Poppe, John M. Rybczyk
Edited by Duncan M. FitzGerald, Boston University, Zoe J. Hughes, Boston University
Book: Salt Marshes
The salt marsh response to a changing climate may be more complex than that of either terrestrial or marine ecosystems because salt marshes exist at the interface of land and sea and both bring changes to the marsh. Climate change may exacerbate anthropogenic-related stresses that salt marsh plants are already experiencing, limiting their resilience (Keddy 2011). In this chapter we discuss major climate change impacts likely to affect salt marshes including temperature, sea level rise (SLR), salinity, CO2, freshwater flow, sediment, and nutrients, and consider how salt marsh plants respond to these impacts and potential interactions of these impacts. Specifically, we explore changes in plant productivity and decomposition rates, aboveground and belowground biomass, and stem density as they are central to understanding marsh responses on a larger scale, with implications for species composition, elevation change, nutrient cycling, carbon sequestration, food webs, and ultimately marsh survival. Although this chapter is focused on salt marshes, examples from tidal fresh and brackish marshes are also included to a limited extent where relevant.
Synchronized Lévy queues
Operations research and management science
Offer Kella, Onno Boxma
Journal: Journal of Applied Probability / Volume 57 / Issue 4 / December 2020
We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.
Decomposition of degenerate Gromov–Witten invariants
Projective and enumerative geometry
Families, fibrations
Dan Abramovich, Qile Chen, Mark Gross, Bernd Siebert
Journal: Compositio Mathematica / Volume 156 / Issue 10 / October 2020
We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.
Thermal degradation kinetics of sepiolite
Yüksel Sarıkaya, Müşerref Önal, Abdullah Devrim Pekdemir
Journal: Clay Minerals / Volume 55 / Issue 1 / March 2020
Published online by Cambridge University Press: 13 March 2020, pp. 96-100
The kinetic parameters of the thermal degradation of sepiolite were evaluated with a new method based on thermal analysis data. Thermogravimetric/differential thermal analysis curves were recorded for the natural and preheated sepiolite samples in the temperature range 25–800°C for 4 h. The temperature-dependent height of the exothermic heat flow peak for the thermal decomposition of sepiolite located at ~850°C on the differential thermal analysis curve was taken as a kinetic variable for the thermal degradation. A thermal change coefficient was defined depending on this variable because this coefficient fit to the Arrhenius equation was assumed as a rate constant for the thermal degradation. The Arrhenius plot showed that the degradation occurs in three steps. Two of these are due to stepwise dehydration and the third originated from dehydroxylation of sepiolite. Three activation energies were obtained that increase with the increasing temperature interval of the steps.
Products of Involutions of an Infinite-dimensional Vector Space
Basic linear algebra
Clément de Seguins Pazzis
Journal: Canadian Journal of Mathematics / Volume 73 / Issue 1 / February 2021
Print publication: February 2021
We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.
Socio-economic inequality in unhealthy snacks consumption among adolescent students in Iran: a concentration index decomposition analysis
Vahid Yazdi-Feyzabadi, Arash Rashidian, Mostafa Amini Rarani
Journal: Public Health Nutrition / Volume 22 / Issue 12 / August 2019
Published online by Cambridge University Press: 14 June 2019, pp. 2179-2188
The present study aimed to assess and decompose the socio-economic inequality in unhealthy snacks consumption among adolescent students in Kerman, Iran.
The data were obtained from a cross-sectional study. Principal component analysis was done to measure the socio-economic status (SES) of the adolescents' families and the normalized concentration index (NCI) was used to measure the inequality in unhealthy snacks consumption among adolescent students of different SES. The contributions of environmental and individual explanatory variables to inequality were assessed by decomposing the concentration index.
Forty secondary schools of Kerman Province in Iran in 2015.
Eighth-grade adolescent students (n 1320).
The data of 1242 adolescent students were completed for the current study. Unhealthy snacks consumption was unequally distributed among adolescent students and was concentrated mainly among the high-SES adolescents (NCI = 0·179; 95 % CI 0·056, 0·119). The decomposition showed that higher SES (62 %) and receiving pocket money allowance (31 %), as environmental variables, had the highest positive contributions to the measured inequality in unhealthy snacks consumption. Taste and sensory perception (7 %) as well as cost sensitivity (5 %), as individual variables, followed them in terms of their contribution importance.
It is highly suggested that both environmental and individual factors should be addressed at different settings including schools, families and suppliers of unhealthy snacks. These findings can help future health promotion strategies in Iran to tackle the observed inequality in unhealthy snacks consumption.
The decline in China's fertility level: a decomposition analysis
Quanbao Jiang, Shucai Yang, Shuzhuo Li, Marcus W. Feldman
Journal: Journal of Biosocial Science / Volume 51 / Issue 6 / November 2019
Many factors have contributed to the decline in China's fertility level. Using China's population census data from 1990, 2000 and 2010, the present study investigates the factors causing the decline in China's fertility rate by decomposing changes in two fertility indices: the total fertility rate (TFR) and the net reproduction rate (NRR). The change in the TFR is decomposed into the change in the marital fertility rate (MFR) and the change in the proportion of married women (PMW). Four factors contribute to the change in the NRR. The following are the main findings. A drop in the MFR caused a decrease in the TFR and the NRR between 1989 and 2000. However, the change in MFR increased TFR and NRR between 2000 and 2010. Marriage postponement caused a decline in the fertility level between 1989 and 2000 as well as between 2000 and 2010. The effect of the MFR and marriage postponement varied with age and region and also between urban and rural areas.
Quantifying Uncertainty from Mass-Peak Overlaps in Atom Probe Microscopy
Andrew J. London
Journal: Microscopy and Microanalysis / Volume 25 / Issue 2 / April 2019
Print publication: April 2019
There are many sources of random and systematic error in composition quantification by atom probe microscopy, often, however, only statistical error is reported. Significantly larger errors can occur from the misidentification of ions and overlaps or interferences of peaks in the mass spectrum. These overlaps can be solved using maximum likelihood estimation (MLE), improving the accuracy of the result, but with an unknown effect on the precision. An analytical expression for the uncertainty of the MLE solution is presented and it is demonstrated to be much more accurate than the existing methods. In one example, the commonly used error estimate was five times too small.
Literature results containing overlaps most likely underestimate composition uncertainty because of the complexity of correctly dealing with stochastic effects and error propagation. The uncertainty depends on the amount of overlapped intensity, for example being ten times worse for the CO/Fe overlap than the Cr/Fe overlap. Using the methods described here, accurate estimation of error, and the minimization of this could be achieved, providing a key milestone in quantitative atom probe. Accurate estimation of the composition uncertainty in the presence of overlaps is crucial for planning experiments and scientific interpretation of the measurements.
The Decline of British Manufacturing, 1973–2012: The Role of Total Factor Productivity
Richard Harris, John Moffat
Journal: National Institute Economic Review / Volume 247 / February 2019
Published online by Cambridge University Press: 01 January 2020, pp. R19-R31
This paper uses plant-level estimates of total factor productivity covering 40 years to examine what role, if any, productivity has played in the decline of output share and employment in British manufacturing. The results show that TFP growth in British manufacturing was negative between 1973 and 1982, marginally positive between 1982 and 1994 and strongly positive between 1994 and 2012. Poor TFP performance therefore does not appear to be the main cause of the decline of UK manufacturing. Productivity growth decompositions show that, in the latter period, the largest contributions to TFP growth come from foreign-owned plants, industries that are heavily involved in trade, and industries with high levels of intangible assets.
Effects of soil temperature and tidal condition on variation in carbon dioxide flux from soil sediment in a subtropical mangrove forest
Mitsutoshi Tomotsune, Shinpei Yoshitake, Yasuo Iimura, Morimaru Kida, Nobuhide Fujitake, Hiroshi Koizumi, Toshiyuki Ohtsuka
Journal: Journal of Tropical Ecology / Volume 34 / Issue 4 / July 2018
Published online by Cambridge University Press: 26 July 2018, pp. 268-275
The variation in CO2 flux from the forest floor is important in understanding the role of mangrove forests as a carbon sink. To clarify the effects of soil temperature and tidal conditions on variation in CO2 flux, sediment–atmosphere CO2 fluxes were measured between June 2012 and May 2013. We used the closed chamber method for two plots, with a 0.5 m difference in elevation (B, high elevation; R-B, low elevation), in a mangrove forest in south-western Japan. CO2 fluxes were highest in the warm season and showed a weak positive correlation with soil temperature in both forests. Estimated monthly CO2 flux showed moderate seasonal variation in accordance with the exposure duration of the soil surface under tidal fluctuation. Additionally, measured CO2 flux and soil temperature were slightly higher in the R-B plot than the B plot, although estimated annual CO2 flux was higher in the B plot than the R-B plot due to different exposure durations. These results suggest that variation in the exposure duration of the forest floor, which changes seasonally and microgeographically, is important in evaluating the annual CO2 flux at a local scale and understanding the role of mangrove ecosystems as regulators of atmospheric CO2.
Fourier transform infrared spectroscopy study of acid birnessites before and after Pb2+ adsorption
Wei Zhao, Fan Liu, Xionghan Feng, Wenfeng Tan, Guohong Qiu, Xiuhua Chen
Journal: Clay Minerals / Volume 47 / Issue 2 / June 2012
To provide fundamental knowledge for studying the relative content of vacant sites and exploring the mechanism of interaction between Pb2+ and birnessite, Fourier transform infrared spectroscopy (FTIR) of birnessites with different Mn average oxidation states (AOS) before and after Pb2+ adsorption were investigated. The number of absorption bands of FTIR spectra was determined by using the second derivatives of the original spectra. The band at 899–920 cm–1 was assigned to the bending vibration of -OH located at vacancies. The bands at 1059–1070, 1115–1124 and 1165–1171 cm–1 could be attributed to the vibrations of Mn(III)-OH in MnO6 layers, and the intensities of these bands increased with decreasing Mn AOS. The bands at 990 and 1023–1027 cm–1 were ascribed to the vibrations of Mn(III)-OH in the interlayers. Mn(III) in MnO6 layers partially migrated to interlayers during Pb2+ adsorption, which led to an increased intensity of the band at 990 cm–1. The band at 564–567cm–1 was assigned to the vibration of Mn-O located at vacancies. This band could split by coupling of vibrations due to Pb2+ and/or Mn2+ adsorbed at vacant sites. The large distance between the band at 610–626 cm–1 and that at 638–659 cm–1 might reflect small Mn(III) ions located in Mn(III)-rich rows.
Composition variation of illite-vermiculitesmectite mixed-layer minerals in a bentonite bed from Charente (France)
A. Meunier, B. Lanson, B. Velde
Journal: Clay Minerals / Volume 39 / Issue 3 / September 2004
Mineralogical and chemical variations were studied in the upper half of a 1 m thick discontinuous bentonite bed interlaminated in the Lower Cenomanian sedimentary formations of the northern Aquitaine basin (France). X-ray diffraction patterns obtained from the <2 mm fraction in the Ca and K-saturated states were decomposed and compared to those calculated from decomposition parameters. They revealed the presence of two highly expandable illite-expandable (I-Exp) mixedlayer minerals (MLMs). The relative proportions of the two MLMs evolve steadily with depth leading to the decrease of the cation exchange capacity and of the (Na + Ca) content towards the centre of the bentonite bed. However, the system is essentially isochemical and Mg, Al, Si, K and Fe are roughly constant in the bulk samples. It is thought that the mineralogical zonation results from the initial stages of the smectite formation in an ash layer.
In the Ca-saturated state, the expandable component of these MLMs was for the most part homogeneous with the presence of 2 sheets of ethylene glycol molecules in the interlayer. However, the heterogeneous hydration behaviour of these expandable layers was enhanced by the K-saturation test. From this test, the presence of three layer types with contrasting layer charge was evidenced from their contrasting swelling abilities. The C12-alkylammonium saturation test applied to samples in which the octahedral charge had previously been neutralized (Hofmann-Klemen treatment) showed that the tetrahedral charge is located on specific layers. These layers are responsible for the heterogeneous hydration behaviour. Low-charge smectite layers are mostly octahedrally substituted, whereas for intermediate- and high-charge layers this montmorillonitic charge is complemented by additional tetrahedral substitutions (0.30 and 0.35–0.40 charge per O10(OH)2, respectively).
Mösbauer spectroscopic study of the decomposition mechanism of ankerite in CO2 atmosphere
A. E. Milodowski, B. A. Goodman, D. J. Morgan
Journal: Mineralogical Magazine / Volume 53 / Issue 372 / September 1989
Mössbauer spectroscopy has been used to identify the iron-containing products that are formed during the thermal decomposition of ankerite in a CO2 atmosphere. The decomposition takes place in three stages and evidence is produced to show that the first stage involves decomposition of ankerite to yield a periclase-wustite solid solution, (Mg,Fe)O, along with calcite and CO2, the periclase-wustite then reacting with CO2 to produce magnesioferrite (MgFe2O4) and CO. In the second stage the magnesioferrite and calcite react to produce periclase and dicalcium ferrite. The third stage does not involve reaction of Fe-containing phases and corresponds to the decomposition of calcite to CaO. | CommonCrawl |
\begin{document}
\title{Quantum and quantum-inspired optimization \\ for solving the minimum bin packing problem}
\author{A.A~Bozhedarov} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\author{A.S.~Boev} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\author{S.R.~Usmanov} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\author{G.V.~Salahov} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\author{E.O.~Kiktenko} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\author{A.K.~Fedorov} \affiliation{Russian Quantum Center, Skolkovo, Moscow 143025, Russia}
\begin{abstract} Quantum computing devices are believed to be powerful in solving hard computational tasks, in particular, combinatorial optimization problems. In the present work, we consider a particular type of the minimum bin packing problem, which can be used for solving the problem of filling spent nuclear fuel in deep-repository canisters that is relevant for atomic energy industry. We first redefine the aforementioned problem it in terms of quadratic unconstrained binary optimization. Such a representation is natively compatible with existing quantum annealing devices as well as quantum-inspired algorithms. We then present the results of the numerical comparison of quantum and quantum-inspired methods. Results of our study indicate on the possibility to solve industry-relevant problems of atomic energy industry using quantum and quantum-inspired optimization. \end{abstract}
\maketitle
\section{Introduction}
Optimization is a primary tool with numerous applications across various industries~\cite{Paschos2014}. Specific attention is traditionally paid to combinatorial optimization problems, which are especially difficult in the view of the so-called curse of dimensionality --- a dramatic increase of the complexity with increasing problem size. One of the notable classes of combinatorial optimization problems is quadratic unconstrained binary optimization (QUBO)~\cite{Lucas2014,Fedorov2022}, which appears in various applications. Quantum computing devices, both universal and specialized, are considered to be useful in solving such computational problems~\cite{Farhi2000,Das2008,Lidar2018,Fedorov2022,Farhi2014}. An idea behind, generally speaking, is to encode a cost function in a quantum Hamiltonian~\cite{Lucas2014}, so that its low-energy state corresponds to the minimum of the cost function. Several architectures of quantum computing devices, which are of interests for solving optimization problems, have been developed~\cite{Fedorov2022}. Specifically, quantum annealing devices based on superconducting qubits, which are able to solve problems of a non-trivial size~\cite{Amin2021}, have been used to tackle various industry-relevant tasks, including quantum chemistry calculations~\cite{Leib2019,Chermoshentsev2021}, (lattice) protein folding~\cite{Aspuru-Guzik2012-2,Fingerhuth2018}, genome assembly~\cite{Fedorov2021,Sarkar2021}, solving polynomial~\cite{Sota2019} and linear systems of equations~\cite{Sota2019}, financial optimization~\cite{Orus2019,Orus2020,Grant2021,Alexeev2022,Orus2019-2,Rosenberg2016-2,Rosenberg2016,Rounds2017,Vesely2022}, traffic optimization~\cite{Neukart2017,Inoue2021,Hussain2020}, scheduling~\cite{Venturelli2016,Ikeda2019,Sadhu2020,Botter2020,Domino2021,Domino2021-2}, railway conflict management~\cite{Domino2021,Domino2021-2}, and many others (for a review, see Ref.~\cite{Fedorov2022}). An alternative approach is to use programmable quantum simulators based on atomic arrays~\cite{Browaeys2020-2}, where the most recent advances include a demonstration of a superlinear quantum speedup in finding exact solutions for the hardest maximum independent set graphs~\cite{Lukin2022}. One may also note that gate-based running variational optimization algorithms, mainly quantum approximate optimization algorithm~\cite{Farhi2014}, also offer interesting possibilities for combinatorial optimization~\cite{Babbush2021,Bharti2021}. Although such devices in principle are able to demonstrate quantum computational advantage in near future, still various limitations make it challenging to use them for solving problems of industry relevant sizes.
The problem of a clear comparison between quantum and classical algorithms, which can be used to highlight the quantum origin of the speed up, is also nontrivial~\cite{Troyer2014}. As a result of such a comparison, a new class of algorithms and techniques, know as {\it quantum-inspired}, has been developed~\cite{Tiunov2019,Lloyd2019}. As soon as these algorithms are compatible with currently existing (classical) hardware, analyzing their limiting capabilities and advantages over classical approaches are required towards their use in practice. Recently, solving the wavelength assignment problem in telecommunication using quantum-inspired algorithm SimCIM~\cite{Tiunov2019} has been demonstrated~\cite{Boev2022}. For a wide range of benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization, namely D-Wave Hybrid Solver Service, Toshiba Simulated Bifurcation Machine, Fujitsu Digital Annealer, and simulated annealing, see Ref.~\cite{Oshiyama2022}.
A specific class of a combinatorial optimization problem that appear across many industry application is the minimum bin packing problem, where items of different sizes must be allocated into a finite number of bins (containers), each of a fixed given capacity, in a way that minimizes the number of bins~\cite{Paschos2014}; this problem is known to be NP-hard. A particular application of this problem is optimization of spent nuclear fuel (SNF) filling in canisters for the deep repository. According to existing standards, the deposing should be realized by using special (deep-repository) canisters, so that the maximum heat output per canister does not exceed the limiting value. The tasks of the optimization of the SNF using canister filling (CF) is then clearly linked to the aforementioned minimum bin packing problem~\cite{Zerovnik2009}. The use of combinatorial methods to optimize the filling of SNF in metal canisters for the final deep repository, according to the maximal allowed thermal power per canister and the limit in the number of spent-fuel assemblies per canister, has been demonstrated~\cite{Zerovnik2009}. In this context, quantum and quantum-inspired tools are now considered as a way to solve this problem for larger sizes; in particular, optimization of fuel arrangements in nuclear power plants using quantum tools has been considered~\cite{Whyte2021}.
In this work, we present a method for solving the SNF management problem using quantum and quantum-inspired annealing. We first formulate the problem in the QUBO form, which allows solving this problem using various annealing tools, including quantum annealing. We then benchmark its solution using quantum annealing device from D-Wave\footnote{The results of the present paper are based on the data that have been collected during the availability of the device.}, quantum-inspired algorithm SimCIM~\cite{Tiunov2019}, and quantum-inspired Simulated Bifurcation Machine (SBM)\footnote{The results of the present paper are based on the data that have been collected during the availability of the algorithm.}~\cite{Goto2019}. Our results indicate the possibility to solve such an industry-relevant problem using quantum and quantum-inspired annealing.
Our work is organized as follows. In Sec.~\ref{sec:CF}, we formulate the CF optimization problem in the QUBO form, which makes it suitable for solving this using quantum and quantum-inspired annealing. Sec.~\ref{sec:bench}, describes the numerical analysis setup; there we also benchmark a solution of the CF problem using available the quantum annealer and quantum-inspired annealing algorithms. We summarize our results and conclude in Sec.~\ref{sec:conclusion}.
\section{Canister filling optimization problem}\label{sec:CF}
The SNF is a subject of the deposition for further safe keeping. Existing industrial standards require that the deposing should be realized using special (deep-repository) canisters, so that the total heat output per canister does not exceed the limiting value $P_{\max}$. At the same time, there is a minimum number of spent fuel elements that can be stored in one canister $N_{\min}$. This is a subject of the SNF management problem, which is important for the optimal use of existing canisters without violating standards. The SNF management problem can be formulated as a combinatorial optimization problem as follows.
Let $n$ be the total number of spent fuel elements, $m$ is the total number of available canisters, and $p_i$ is the heat output of the $i$-th fuel element. Let us introduce additional variables for indication of fuel element location and canister usage as following: \begin{align}
x_{ij} &=
\begin{cases}
1, & \text{if $i$-th fuel element is in $j$-th canister}, \\
0, & \text{otherwise};
\end{cases} \\
y_{j} &=
\begin{cases}
1, & \text{if $j$-th canister is being used}, \\
0, & \text{otherwise}.
\end{cases} \end{align}
Then, one may formulate optimization problem in the following way: \begin{align}
&M=\sum_{j=1}^{m} y_{j} \rightarrow \min, \label{cnst:0}\\
& \quad \text{such that} \nonumber \\
&\sum_{i=1}^{n} p_{i} x_{i j} \leq P_{\max } \quad \forall j \in\{1, \ldots, m\}, \label{cnst:1}\\
&\sum_{j=1}^{m} x_{i j}=1 \quad \forall i \in\{1, \ldots, n\}, \label{cnst:2} \\
&\sum_{i=1}^{n} x_{i j} \geq N_{\min } y_{j} \quad \forall j \in\{1, \ldots, m\}, \label{cnst:3}\\
&x_{i j} \leq y_{j} \quad \forall i \in\{1, \ldots, n\}, \forall j \in\{1, \ldots, m\}, \label{cnst:4} \end{align} where condition~(\ref{cnst:1}) restricts the maximum heat output per one canister, condition~(\ref{cnst:2}) implies that every fuel element placed only in one canister, condition~(\ref{cnst:3}) stands for minimal filling of every used canister, and condition~(\ref{cnst:4}) binds the variables $x_{i j}$ and $y_j$ so that the placement of the fuel elements matches the vector of the used canisters.
The main step in solving an optimization problem using quantum and quantum-inspired annealing is to map the problem of interest to the energy Hamiltonian, so the quantum device could find the ground state that corresponds to the optimum value of the objective function. The natural way of mathematical description of a quantum annealer is the Ising spin Hamiltonian that can be transformed into QUBO problem in a straightforward way. We are to formulate mapping of the CF problem into QUBO form. In general, a QUBO problem may be formulated using matrix notation as following: \begin{equation}
z^T Q z \to \min, \end{equation} where $z$ is the vector of binary decision variables and $Q$ is a square symmetric matrix of constants.
It is necessary to include optimization constraints by adding penalty terms to the objective function. Let us represent optimization constraints~(\ref{cnst:1})--(\ref{cnst:4}) in the QUBO form. Constraint~\eqref{cnst:1} can be represented as \begin{equation} \label{qubo:1}
\mathcal{H}_1 = \sum_{j=1}^{m}\left(\sum_{i=1}^{n} p_{i} x_{ij}+\sum_{l=0}^{s-1} 2^{l} a_{l j}-P_{\max}\right)^{2}, \end{equation} where $s=\lceil\log _{2} P_{\max }\rceil$ and $a_{lj}$ denotes auxiliary binary variables, which are required to represent~\eqref{cnst:1} in the form of equality: $\sum_{l=0}^{s} 2^{l} a_{l j}$ is certain non-negative integer that corresponds to a difference between $P_{\max}$ and $\sum_{i=1}^{n} p_{i} x_{ij}$. Constraints~\eqref{cnst:2} and \eqref{cnst:3} take the following form: \begin{equation} \label{qubo:2}
\mathcal{H}_2 = \sum_{i=1}^{n}\left(\sum_{j=1}^{m} x_{i j}-1\right)^{2}, \end{equation} and \begin{equation} \label{qubo:3}
\mathcal{H}_3 = \sum_{j=1}^{m}\left(\sum_{i=1}^{n} x_{ij}-\sum_{l=0}^{k-1} 2^{l} b_{l j}-N_{\min}y_j\right)^{2}, \end{equation} correspondingly. Here $k=\lceil\log_{2} N_{\min}\rceil$ and $b_{l j}$ are another auxiliary binary variables used to represent non-negative difference $\sum_{l=0}^{k} 2^{l} b_{l j}$ between $\sum_{i=1}^{n} x_{ij}$ and $N_{\min}y_j$. The final constraint~\eqref{cnst:4} takes the form \begin{equation} \label{qubo:4}
\mathcal{H}_4 = \sum_{i=1}^n\sum_{j=1}^m \left(x_{i j}-x_{i j} y_{j}\right) \end{equation}
The problem Hamiltonian then consist of two main components: \begin{equation}
\mathcal{H} = A \sum_{j=1}^{m}{y_j} + B \sum_{r=1}^4 \mathcal{H}_r \end{equation} where $A$ is a positive constant and $B$ stands for a positive penalty value. Parameters $A$ and $B$ should be set manually, using the following criteria. Penalty value should be high enough to keep final solution from violating constraints. At the same time, too large penalty value may overwhelm the objective function so it becomes hard to distinguish solutions of different quality. Therefore, the solution of the optimization problem requires finding optimal values of the variables $x_{ij}, y_{j}, a_{lj}$ and $b_{lj}$. More details about the total number of binary variables may be found in Subsec.~\ref{app:dataset}.
One may transform QUBO problem into Ising Hamiltonian using following approach: \begin{equation}
z_i = \frac{\sigma^{Z}_i + 1}{2} \in\{0,1\}, \end{equation} where $\sigma_i^{Z} = \pm 1$ and vector $z$ contains all optimized variables $x_{ij}, y_{j}, a_{lj}$, and $b_{lj}$.
\section{Benchmarking procedure}\label{sec:bench}
In order to evaluate the feasibility of the proposed scheme of solving the SNF problem in the QUBO form, we conduct a comparison of existing quantum annealing device and quantum-inspired annealing simulator as instruments to solving CF combinatorial problem.
\subsection{Generating of synthetic dataset}\label{app:dataset}
We first prepare a synthetic dataset of 80 problem instances with number of fuel elements ranging from 3 to 10 and the maximum number of canisters equal to 3. For each problem size, 10 different cell configurations with various heat output are prepared (plus single trivial case with 2 elements). The optimal allocation for all instances is known in advance and requires at most 2 canisters.
The general idea of dataset is to create problem with minimal possible QUBO sizes for guaranteed best solution achievement by annealers. Using notation from Eqs.(\ref{qubo:1})--(\ref{qubo:4}), the total number of logical transformation variables may be represented as follows: \begin{equation}
D = m \left(1 + n + s + k \right), \end{equation} where $m$ is the available number of canisters, $n$ is the number of fuel elements, $s=\lceil\log _{2} P_{\max }\rceil$ and $k=\lceil\log_{2} N_{\min}\rceil$ is the number of auxiliary variables used in equalities~(\ref{qubo:1}) and~(\ref{qubo:3}), correspondingly. As a result, the smallest-size problem with $m=2, n=2, s=2, k=0$ requires 10 logical variables, we mark it as a trivial case (see Fig.~\ref{fig:comparison}).
We use only fixed configurations, where the optimal number of canisters is 2 and the feasible solution without constraint violation can have 3 canisters, in other words, we restrict problem samples to cases, where $m=3$ and $M=2$. The maximum capacities of canisters are equal with limit $2^{s=4}-1$, where $k$ is also fixed and equals zero, since we do not use minimum elements constraint when $N_{\min}=1$. This compression allows us to compute model problems on present quantum hardware and evaluate the dependence of problem size and performance. Tasks with the same elements quantity are also have identical QUBO size for avoiding additional deviation in data; see Table~\ref{tab:dataset}. We use public access to D-Wave 5000 Advantage system with Pegasus topology processor~\cite{Boothby2020} to run our quantum annealing algorithm.
\begin{table}[ht]
\begin{tabular}{|c|c|c|c|c|} \hline \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}
Elements\\ quantity \end{tabular}} &
\multirow{3}{*}{QUBO size} &
\multicolumn{3}{c|}{Physical qubits} \\ \cline{3-5}
&
&
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}
Number \\ of qubits\end{tabular}} &
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}
Heuristic, \\ mean\end{tabular}} &
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}
Heuristic, \\ std \end{tabular}} \\
& & & & \\ \hline 2 & 10 & 20 & 18.4 & 2.3 \\ \hline 3 & 21 & 80 & 64.0 & 3.5 \\ \hline 4 & 24 & 90 & 85.0 & 1.6 \\ \hline 5 & 27 & 102 & 103.8 & 8.6 \\ \hline 6 & 33 & 157 & 153.6 & 5.5 \\ \hline 7 & 36 & 170 & 185.6 & 7.6 \\ \hline 8 & 39 & 185 & 226.6 & 15.6 \\ \hline 9 & 42 & 246 & 246.0 & 10.6 \\ \hline 10 & 45 & 262 & 280.0 & 5.3 \\ \hline \end{tabular} \caption{Synthetic dataset scheme. Heuristic embeddings on physical qubits of D-Wave device were found via D-Wave Leap SDK for 5 different random samples.} \label{tab:dataset} \end{table}
\subsection{Benchmarking}
\begin{figure*}
\caption{Comparison of the performance of quantum and quantum-inspired methods for bin packing problem based on synthetic data:
we compare TTS (mean and standard deviation) for quantum device D-Wave and two quantum-inspired optimization algorithms, SimCIM and SBM.}
\label{fig:comparison}
\end{figure*}
Each problem instance has been further transformed into the QUBO matrix and run through both quantum and quantum-inspired instruments, specifically, the D-Wave quantum annealer and two quantum-inspired algorithms (SimCIM~\cite{Tiunov2019} and SBM~\cite{Goto2019}). SimCIM algorithm~\cite{Tiunov2019} is based on the method of efficient simulation of Coherent Ising Machine~\cite{Yamamoto2017} using classical computer. As it has been shown, SimCIM outperforms Coherent Ising Machine in terms of samples quality and speed of computation, and that is why it has been chosen as a benchmarking tool for comparative analysis.
Simulated Bifurcation algorithm (SBM)~\cite{Goto2019} is a heuristic algorithm for combinatorial optimization. Its workflow is inspired by quantum bifurcation machine~\cite{Goto2016} that is based on nonlinear oscillators and implements quantum adiabatic algorithm for solving optimization problems.
We run D-Wave experiments in a pure quantum mode using the Advantage chip featuring 5000 qubits with 15-way connectivity. In order to embed QUBO problem into physical qubit layout we utilized clique embedding supported by D-Wave Leap SDK. SimCIM was run on Xeon E3-1230v5 4x3,4GHz, 16 GB DDR4, GeForce GTX 1080. Comparison results are shown in Fig.~\ref{fig:comparison}.
\subsection{Analysis}
As a figure of merit in our benchmarking procedure, we use time-to-solution (TTS). The TTS means a time that is needed to a heuristic algorithm to find the solution (ground state energy) with 99\% success probability. It is given by \begin{equation}
{\rm TTS} = t_a R_{99}, \end{equation} where $t_a$ is the annealing time (default value of $t_a$ for D-Wave is $20 \mu s$), and $R_{99}$ stands for the number of repetition that is needed for the desired success probability~\cite{Katzgraber2019,Botter2020}. It can be calculated as follows: \begin{equation}\label{r99}
R_{99} = \frac{\log(1-0.99)}{\log(1-\theta)}, \end{equation} where $\theta$ is the estimated success probability of each run.
All tasks were grouped by fuel elements quantity for demonstrating results (see Fig.~\ref{fig:comparison}). We note that the D-Wave annealer shows good result in problem solving in the small-size cases (2 possible canisters and 2 elements), while optimal solution was not found for 6 and more elements problem. The standard deviation of TTS is significantly increase with elements quantity for all methods. This is caused by an exponential growth of the space of possible solutions, leading to a decrease in the probability of finding the optimal solution. As a result, the annealing process often terminates at a suboptimal point instead of the ground state. This is especially true for complex problems that require a large number of variables to be taken into consideration.
While the main obstacle of quantum-inspired optimization methods is complexity and the size of the space of possible solutions, for quantum annealing a very important parameter is the gap between the ground state and the first excited state of the Hamiltonian. The smaller the gap, the slower the adiabatic evolution of the quantum system should proceed in order to stay in the ground state. However, a long evolution time increases the influence of quantum decoherence and can lead to incorrect solutions.
\section{Conclusion}\label{sec:conclusion}
Quantum computing is a promising technique for solving combinatorial optimization problems. In our work, we have demonstrated the potential of quantum and quantum-inspired tools to solve computational problems of the minimum bin packing problem, which is formulated as the problem of atomic energy industry, As a target problem, we have chosen the optimization of spent nuclear fuel filling in canisters for the deep repository. The CF problem has been formulated in a QUBO matrix form (see Eqs.~(\ref{qubo:1})--(\ref{qubo:4})) and was solved using existing quantum annealer and quantum-inspired annealing algorithms. We note that the current development level of quantum computing devices does not allow to solve large-scale practical problems, however, it is possible to scale a size of the problem for the next generation of quantum computers. Moreover, such research helps to identify practical-oriented tasks that may be solved more efficiently by quantum computing.
\begin{table}[t]
\begin{tabular}{|c|c|c|} \hline Annealing time, $\mu s$ & Success probability & TTS, $\mu s$ \\ \hline 20 & 0.277 & 284 \\ \hline 40 & 0.303 & 511 \\ \hline 80 & 0.343 & 878 \\ \hline \end{tabular} \caption{Dependence of success probability and TTS for trivial case (2 fuel elements) for D-Wave device.} \label{tab:annealing-time} \end{table}
\section*{Appendix}
In our benchmarking procedure, we use various accessible parameters (particularly, annealing time, embedding type and chain strength) of the quantum hardware on the final solution quality. For the simplest task, we have observed that increasing annealing time gives a better success probability (see Table~\ref{tab:annealing-time}), but in the same time TTS is getting worse, so annealing time was set to 20 $\mu$s (the. default value). The number of annealing runs is set to $10^4$ (maximum possible value). Comparing different embeddings (see Table~\ref{tab:dataset}), we analyze deviation of physical qubits number on five different random samples and decided that for better experiment performance using stable stable clique embedding is more preferable. According to Ref.~\cite{Botter2020} we choose custom optimized chain strength instead of other variants such as maximum absolute value in QUBO. Thereby we have used mostly the standard configuration of the D-Wave processor during our experiments, so we do not have any specific requirements on the weights/couplers in the model. We use only pure quantum regime to obtain solution for each task.
\end{document} | arXiv |
How does the momentum operator act on state kets?
Active 11 days ago
I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that $$\langle x|\alpha\rangle=f(x)$$ for some known function $f$. ($|\alpha\rangle$ is a Gaussian wave packet.) Sakurai says that this is given by:
$$\langle p\rangle = \int\limits_{-\infty}^{+\infty}\langle\alpha|x\rangle\left(-i\hbar\frac{\partial}{\partial x}\right)\langle x|\alpha\rangle dx.$$
I am wondering how we get to this expression. I know that we can express
$$|\alpha\rangle =\int dx|x\rangle\langle x|\alpha\rangle$$
$$\langle\alpha|=\int dx\langle\alpha|x\rangle\langle x|,$$
so my thinking is that we have:
$$\langle\alpha|\hat{p}|\alpha\rangle =\iint dx dx'\langle\alpha|x\rangle\langle x|\hat{p}|x'\rangle \langle x'|\alpha\rangle, $$
and if we can 'commute' $|x\rangle$ and $\hat{p}$ this would become: $$\iint dxdx'\langle\alpha |x\rangle\hat{p}\langle x|x'\rangle \langle x'|\alpha\rangle,$$ which is the desired result as $$\langle x|x'\rangle=\delta(x-x').$$ Is this approach valid?
I think my question boils down to: Does the operator $\hat{p}$ act on the basis kets $|x\rangle$ or on their coefficients? In the latter case, if we had some state $|\psi\rangle = |x_0\rangle$ for some position $x_0$, then would we say that for this state $$\langle p\rangle =\langle x_0|\left(-i\hbar\frac{\partial}{\partial x}\right)|x_0\rangle = 0$$ as the single coefficient is $1$ and the derivative of $1$ is $0$?
quantum-mechanics operators momentum wavefunction hilbert-space
Qmechanic♦
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Ruvi LecamwasamRuvi Lecamwasam
Looking at the question
I think my question boils down to: does $\hat p$ act on the basis kets $|x\rangle $ or on their coefficients?
one can safely identify that you're confused about something but it's harder to figure out what the question really is. So let me repeat some basic things here – I am confident that you must be confused about one of them, despite their basic character.
The symbol $\hat p$ is an operator. It means an object that acts on any ket vector and gives you another (or the same) ket vector. The map must be linear and so on. So $\hat p$ surely acts on vectors, not on "coefficients".
On the other hand, when it acts on a basis vector such as $|x\rangle$, the result may be expressed as a linear combination of the basis vectors in the same basis, $$\hat p |x\rangle = \int dx' f_x(x') |x'\rangle$$ with some coefficients $f_x(x')$. Every ket vector, and $\hat p | x \rangle$ is a ket vector, may be expressed using a basis in a way.
So while you may identify $|x\rangle$ with the "wave function" equal to $\psi(x'') = \delta (x''-x)$ where $x$ is a fixed value of the position, the acted-upon ket vector $\hat p|x\rangle$ is given in terms of the function $f_x(x')$ which encodes the coefficients in front of $|x'\rangle$. This function (storing the coefficients) is fully given by the meaning of the operator $\hat p$ and by the value of $x$ and it replaces the delta-function encoding $|x\rangle$ itself, so in this sense, operators also act on coefficients. One must just know the basic rules how they act on a basis etc. and then he knows everything!
Another thing you may be confused by is even more elementary, what a derivative is. A derivative is not an operator acting on the Hilbert space. A derivative is an operation that takes a function of a real variable and maps it to another function of the real variable $$ \frac{\partial}{\partial x} : f(x)\mapsto f'(x) = \lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)-f(x)}{\epsilon} $$ You must be confused about this definition of a derivative, otherwise you wouldn't write the meaningless derivatives in the last sentence. Something should generally depend on the variable with respect to which we are differentiating, and then we differentiate it as a function using the general definition above.
The kernel (or "matrix elements") of $\hat p$ is $$\langle x | \hat p | x'\rangle = -i\hbar \delta'(x-x')=f_{x'}(x)$$ which is the derivative of the delta-function. It's a delta-function whose argument is the difference of the two values $x,x'$ that specify the bra vector and the ket vector between which $\hat p$ was sandwiched. The delta-function is equal to the inner product of the bra vector $\langle x |$ and the vector $\hat p |x\rangle$ which results from the action of $\hat p$ on $|x\rangle$.
The kernel is enough to calculate anything involving $\hat p$ and bra and ket vectors in the $x$-basis. For example, you may multiply my equation for the kernel above by $|x\rangle$ from the left and integrate over $x$. Then one gets (after noticing that $1$ was constructed on LHS via the completeness relation) $$\hat p |x'\rangle = \int dx (-i\hbar) \delta(x-x') |x\rangle = -i\hbar\frac{\partial}{\partial x}|x\rangle_{x= x'} $$ Sorry if there's a sign error anywhere. It makes sense to differentiate with respect to $x$ because the object actually is a function of $x$. If a general wave function is rewritten as a combination of such $|x'\rangle$ vectors from the LHS of the equation above, via an integral and with the coefficients called $\psi(x')$, the equation above becomes the usual $$\hat p:\psi(x')\to -i\hbar \psi'(x')$$ in terms of the coefficients. This doesn't mean that a linear operator is the same thing as a derivative of functions. It just says that in the $x$-basis, when acting on a general combination of these basis vectors, the coefficients transform in this derivative-like way. But that's a special property of this particular operator. Other operators, like $\hat x$, act differently.
In my opinion, manipulations involving $\hat p$ and position bras and kets are most easily done by considering the action of $\hat p$ on the position bras, which is simply $$ \boxed{ \vphantom{\begin{array}{}make\\the box\\taller\end{array}} \quad\,\,\, \langle x|\hat p=-i\hbar\frac{\text d}{\text dx}\langle x|. \quad\,\,\,} \tag 1 $$
You can get this easily by seeing that for any state $|\psi\rangle$ with position-representation wavefunction $\psi(x)=\langle x|\psi\rangle$, the action of the momentum operator on the state gives a derivative on the wavefunction. That is, $$\langle x|\hat p|\psi\rangle =-i\hbar\frac{\text d}{\text dx}\langle x|\psi\rangle.$$ Since this equation holds for all states $|\psi\rangle\in\mathcal H$, you can "cancel $|\psi\rangle$ out". (More technically, since the action of the bras $\langle x|\hat p$ and $-i\hbar\frac{\text d}{\text dx}\langle x|$ is the same for all vectors, they must be equal as linear functionals.)
Emilio PisantyEmilio Pisanty
There are many good answers already. This answer is basically an expanded version of Emilio Pisanty's answer.
Let us start by recalling the standard convention to write the position wavefunction $$ \tag{1} \psi(x)~=~\langle x | \psi \rangle$$ as an overlap with a position bra state $\langle x |$.
The CCR $$\tag{2} [\hat{x},\hat{p}]~=~i\hbar{\bf 1}$$ is the first principle of canonical quantization.
The position Schrödinger representation $$\tag{3}\hat{x}~=~x , \qquad \hat{p}~=~\frac{\hbar}{i}\frac{\partial}{\partial x},$$ is the most common representation of the CCR (2), although it is far from being unique, cf. e.g. this Phys.SE post. However, see also the Stone-von Neumann theorem.
It's important to realize that it's implicitly understood that the operators (3) act on bras (as opposed to kets). (However, see eq. (6) below.) Hence it is more proper to write (3) as $$\tag{4}\langle x |\hat{x}~=~x\langle x | , \qquad \langle x |\hat{p} ~=~\frac{\hbar}{i}\frac{\partial \langle x |}{\partial x} ~=~\lim_{\varepsilon\to 0}\frac{\hbar}{i}\frac{ \langle x+\varepsilon |- \langle x |}{\varepsilon}.$$
Note that the position Schrödinger representation (4) on bras realizes the CCR (2) $$\tag{5}\langle x |[\hat{x},\hat{p}]~=~ \lim_{\varepsilon\to 0}\frac{\hbar}{i}\left\{ x\frac{ \langle x+\varepsilon |- \langle x |}{\varepsilon} -\frac{ ( x+\varepsilon)\langle x+\varepsilon |- x\langle x |}{\varepsilon}\right\}~=~i\hbar \langle x | ~, $$ with the correct sign, as it should.
Note that the position Schrödinger representation (4) on bras and the convention (1) implies the standard formulas
$$\tag{6}\hat{x}\psi(x)~=~x\psi(x) , \qquad \hat{p}\psi(x) ~=~\frac{\hbar}{i}\frac{\partial\psi(x)}{\partial x} .$$
Note in particular that if one insists to act on kets (as opposed to bras), then the position Schrödinger representation comes with the opposite sign: $$\tag{7}\hat{x}|x\rangle ~=~|x\rangle x , \qquad \hat{p}|x\rangle ~=~i\hbar\frac{\partial |x\rangle}{\partial x} ~=~\lim_{\varepsilon\to 0}i\hbar \frac{ |x+\varepsilon \rangle- | x\rangle }{\varepsilon}.$$
Qmechanic♦Qmechanic
$\begingroup$ I don't understand your point 4: aren't the operators usually introduced to operate on kets? For a given quantum system, we have a Hilbert space containing its (ket) states, basis of that state (e.g. $|x\rangle$, the position basis), and linear operators such as $\hat{x}$ or $\hat{p}$ that act on those kets, i.e. that change the state of the system. I don't think that it is "more proper" to have them act on bras rather than kets. $\endgroup$ – Frank Feb 20 '18 at 16:30
$\begingroup$ Acting on kets is covered in point 7. $\endgroup$ – Qmechanic♦ Feb 20 '18 at 17:33
States are vectors, $|\alpha\rangle$ and the basis $|x\rangle$ are vectors.
The notation $\langle x|\alpha\rangle$ is equivalent to $\alpha(x)$, this is the coordinate of the state $|\alpha\rangle$ on the basis $|x\rangle$, $\alpha(x)$ is the probability amplitude or wavefunction .
The operator $\hat p$, applying to a state or a function dependent of $x$, has the representation $-i \frac{\partial}{\partial x}$ (we choose here the unit $\hbar = 1$ for simplicity).
So, for instance, we have : $\langle x|\hat p|x'\rangle= -i \langle x|\frac{\partial}{\partial x'}|x'\rangle = = -i\frac{\partial}{\partial x'}\langle x|x'\rangle =+i(\frac{\partial}{\partial x'}\delta)(x-x')\tag{1}$
You have :
$$\langle\alpha|\hat{p}|\alpha\rangle=\iint dx dx'\langle\alpha|x\rangle\langle x|\hat{p}|x'\rangle\langle x'|\alpha\rangle$$
$$=\iint dx dx'~~\alpha^*(x)~~i(\frac{\partial}{\partial x'}\delta)(x-x') ~~\alpha(x')$$
$$=\iint dx dx'~~\alpha^*(x)~~-i\delta(x-x') ~~\frac{\partial}{\partial x'}\alpha(x')\tag{2}$$
$$=\int dx~~ \alpha^*(x)~~(-i\frac{\partial}{\partial x})~~\alpha(x)$$ In $(2)$, we have used an integration by parts, supposing that the wavefunction is decreasing sufficiently quickly at the boudary.
The equation $\hat p|x\rangle = (-i\frac{\partial}{\partial x})|x\rangle$, is correct, but not useful, because we have no expression for $\frac{\partial}{\partial x}|x\rangle$. A more useful equation is about translation operations, and is : $e^{-i\hat p.a}|x\rangle = |x+a\rangle$ or $\langle x |e^{i\hat p.a}=\langle x+a| $
Finally, looking at the state $|\psi\rangle =|x_0\rangle$, the associated wavefunction is $\langle x|x_0\rangle = \delta(x-x_0)$, so the mean value of the momentum in this state is :
$$ \langle\psi|\hat p|\psi\rangle\tag{3}=\int dx \delta(x-x_0) (-i\frac{\partial}{\partial x})\delta(x-x_0) = -i \delta'(0)=0$$
This can be understood easily, because if you fix the position ($x=x_0$), the uncertainty of the momentum is infinite, so all momenta are authorized with the same propability, so the momenta mean is zero.
TrimokTrimok
Consider the commutation relation $[\hat x,\hat p_x]=i\hbar$. Its matrix element between states $\langle x|$ and $|x'\rangle$, \begin{equation} \langle x|\hat x \,\hat p_x - \hat p_x\hat x|x'\rangle =i\hbar \langle x|x'\rangle , \end{equation} gives \begin{equation} (x-x')\langle x|\hat p_x |x'\rangle =i\hbar \delta (x-x'), \end{equation} so that \begin{equation} \langle x|\hat p_x |x'\rangle =i\hbar \frac{\delta (x-x')}{x-x'}. \end{equation} Substituting this into $\langle x|\hat p_x|\psi \rangle $, where $|\psi \rangle $ is an arbitrary ket state with the wavefunction $\psi (x)\equiv \langle x|\psi \rangle $, we have \begin{eqnarray} \langle x|\hat p_x|\psi \rangle &=&\int \langle x|\hat p_x|x'\rangle \langle x'|\psi \rangle dx'\\ &=&i\hbar \int \frac{\delta (x-x')}{x-x'}\psi (x') dx'. \end{eqnarray} The only contribution to the integral comes from $x'$ very close to $x$, so we can expand the wavefunction in Taylor's series to first order, $\psi (x')\simeq \psi (x)+\psi '(x)(x'-x)$. This gives \begin{eqnarray} \langle x|\hat p_x|\psi \rangle =i\hbar \int \frac{\delta (x-x')}{x-x'}\psi (x) dx'-i\hbar \int \delta (x-x')\psi '(x)dx'=-i\hbar \frac{\partial \psi(x)}{\partial x}, \end{eqnarray} where the first integral vanishes because $\delta (x-x')$ is an even function and $1/(x-x')$ is odd.
Gleb GribakinGleb Gribakin
$\begingroup$ Isn't it true that $\frac{d}{da}\delta(a) = \frac{1}{a} \delta(a)$? Wouldn't using this be a more direct way of arrive at the answer? $\endgroup$ – Greg.Paul Dec 8 '17 at 6:14
$\begingroup$ I do not think that this is true. As always with relations that involve the delta function, you have to include it under the integral multiplied by some arbitrary function, to check. $\endgroup$ – Gleb Gribakin Mar 26 '18 at 12:29
Note that $\hat{p}=-i\hbar\frac{\partial}{\partial x}$.
Put this in given formula:
$$\langle p\rangle = \int\limits_{-\infty}^{+\infty}\langle\alpha|x\rangle\left(-i\hbar\frac{\partial}{\partial x}\right)\langle x|\alpha\rangle dx,$$
which gives:
$$\langle p\rangle = \int\limits_{-\infty}^{+\infty}\langle\alpha|x\rangle\hat{p}\langle x|\alpha\rangle dx.$$
Now it's a well-known result in quantum mechanics (completeness relation) that:
$$\int\limits_{-\infty}^{+\infty}|x\rangle \langle x|dx=1,$$
So when we put this in the expression for $\langle p\rangle$ we get:
$$\langle p\rangle =\langle\alpha|\hat p|\rangle\alpha,$$
which we had to prove.
You can also start with
place the completeness relation
$$\int\limits_{-\infty}^{+\infty}|x\rangle \langle x|dx=1$$
in it and replace $\hat{p}$ by $-i\hbar\frac{\partial}{\partial x}$.
descheleschilderdescheleschilder
I would say that one part of the question is still unanswered. That would be, how the operator $\hat{p}$ acts on a state that is not a trivial linear combination of the position eigenstates.
Let's say we are trying to calculate $$\hat{\frac{\partial}{\partial x}}|n\rangle=\hat{\frac{\partial}{\partial x}} \int dx' \langle x'|n\rangle |x'\rangle, $$ where $|n\rangle$ is some state that can be represented as a linear combination of states $|x\rangle$. To grasp the intuition we can check the case of $$\hat{x}|n\rangle=\hat{x} \int dx' \langle x'|n\rangle |x'\rangle, $$ where the answer is very simple, as the $|x\rangle$ is eigenstate of the $\hat{x}$ operator with eigenvalue $x$ $$\hat{x}|n\rangle=\int dx' \langle x'|n\rangle \hat{x}|x'\rangle = \int dx' \langle x'|n\rangle x'|x'\rangle. $$ Applying the same reasoning to the initial case, we get $$\hat{\frac{\partial}{\partial x}} \int dx' \langle x'|n\rangle |x'\rangle=\int dx' \langle x'|n\rangle \hat{\frac{\partial}{\partial x}}|x'\rangle=\int dx' \langle x'|n\rangle \lim_{h\rightarrow 0}\frac{1}{h}\Big(|x'+h\rangle - |x'\rangle\Big), $$ where I simply used the definition of the derivative on a vector. Then we simply separate the integral on two parts, and redefine the integration variables so that we can extract the state $$=\lim_{h\rightarrow 0}\frac{1}{h}\Big(\int dx' \langle x'|n\rangle|x'+h\rangle - \int dx' \langle x'|n\rangle|x'\rangle\Big), $$ $$=\lim_{h\rightarrow 0}\frac{1}{h}\Big(\int dx'' \langle x''-h|n\rangle|x''\rangle - \int dx' \langle x'|n\rangle|x'\rangle\Big), $$ $$=\lim_{h\rightarrow 0}\frac{1}{h}\int dx' \Big(\langle x'-h|n\rangle - \langle x'|n\rangle\Big)|x'\rangle, $$ but this is the derivative of the coefficients $$=-\int dx' \Big(\frac{\partial\langle x|n\rangle}{\partial x}\Big)\Bigg|_{x = x'}|x'\rangle, $$ or in a more familiar form with the wave function defined as $\langle x|n\rangle = \psi_{n}(x)$ $$\hat{\frac{\partial}{\partial x}}|n\rangle=\hat{\frac{\partial}{\partial x}} \int dx' \psi_{n}(x') |x'\rangle = -\int dx' \psi'_{n}(x') |x'\rangle. $$ Therefore acting with the spatial derivative on a state gives you the derivative of a wave function, or in other words, the derivative of the coefficient that gives you the mapping from a state you are differentiating to the position basis.
edited Jan 9 at 11:32
Miha SMiha S
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In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem.
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• Integral (improper)
• Riemann integral
• Lebesgue integration
• Contour integration
• Integral of inverse functions
Integration by
• Parts
• Discs
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• Euler's formula
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• Risch algorithm
Series
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Convergence tests
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Vector
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Theorems
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Definitions
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Advanced
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Specialized
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Miscellaneous
• Precalculus
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• Integration Bee
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• Nonstandard analysis
Green's first identity
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X ) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ Rd, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Using the product rule above, but letting X = ∇φ, integrate ∇⋅(ψ∇φ) over U. Then[1]
$\int _{U}\left(\psi \,\Delta \varphi +\nabla \psi \cdot \nabla \varphi \right)\,dV=\oint _{\partial U}\psi \left(\nabla \varphi \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \,\nabla \varphi \cdot d\mathbf {S} $
where ∆ ≡ ∇2 is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal to the surface element dS and dS = ndS is the oriented surface element.
This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with ψ and the gradient of φ replacing u and v.
Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting F = ψΓ,
$\int _{U}\left(\psi \,\nabla \cdot \mathbf {\Gamma } +\mathbf {\Gamma } \cdot \nabla \psi \right)\,dV=\oint _{\partial U}\psi \left(\mathbf {\Gamma } \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \mathbf {\Gamma } \cdot d\mathbf {S} ~.$
Green's second identity
If φ and ψ are both twice continuously differentiable on U ⊂ R3, and ε is once continuously differentiable, one may choose F = ψε ∇φ − φε ∇ψ to obtain
$\int _{U}\left[\psi \,\nabla \cdot \left(\varepsilon \,\nabla \varphi \right)-\varphi \,\nabla \cdot \left(\varepsilon \,\nabla \psi \right)\right]\,dV=\oint _{\partial U}\varepsilon \left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.$
For the special case of ε = 1 all across U ⊂ R3, then,
$\int _{U}\left(\psi \,\nabla ^{2}\varphi -\varphi \,\nabla ^{2}\psi \right)\,dV=\oint _{\partial U}\left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.$
In the equation above, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface normal n of the surface element dS,
${\partial \varphi \over \partial \mathbf {n} }=\nabla \varphi \cdot \mathbf {n} =\nabla _{\mathbf {n} }\varphi .$
Explicitly incorporating this definition in the Green's second identity with ε = 1 results in
$\int _{U}\left(\psi \,\nabla ^{2}\varphi -\varphi \,\nabla ^{2}\psi \right)\,dV=\oint _{\partial U}\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot d\mathbf {S} .$
In particular, this demonstrates that the Laplacian is a self-adjoint operator in the L2 inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.
Green's third identity
Green's third identity derives from the second identity by choosing φ = G, where the Green's function G is taken to be a fundamental solution of the Laplace operator, ∆. This means that:
$\Delta G(\mathbf {x} ,{\boldsymbol {\eta }})=\delta (\mathbf {x} -{\boldsymbol {\eta }})~.$
For example, in R3, a solution has the form
$G(\mathbf {x} ,{\boldsymbol {\eta }})={\frac {-1}{4\pi \|\mathbf {x} -{\boldsymbol {\eta }}\|}}~.$
Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
$\int _{U}\left[G(\mathbf {y} ,{\boldsymbol {\eta }})\,\Delta \psi (\mathbf {y} )\right]\,dV_{\mathbf {y} }-\psi ({\boldsymbol {\eta }})=\oint _{\partial U}\left[G(\mathbf {y} ,{\boldsymbol {\eta }}){\partial \psi \over \partial \mathbf {n} }(\mathbf {y} )-\psi (\mathbf {y} ){\partial G(\mathbf {y} ,{\boldsymbol {\eta }}) \over \partial \mathbf {n} }\right]\,dS_{\mathbf {y} }.$
A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation. Then ∇2ψ = 0 and the identity simplifies to
$\psi ({\boldsymbol {\eta }})=\oint _{\partial U}\left[\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}-G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \psi }{\partial \mathbf {n} }}(\mathbf {y} )\right]\,dS_{\mathbf {y} }.$
The second term in the integral above can be eliminated if G is chosen to be the Green's function that vanishes on the boundary of U (Dirichlet boundary condition),
$\psi ({\boldsymbol {\eta }})=\oint _{\partial U}\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}\,dS_{\mathbf {y} }~.$
This form is used to construct solutions to Dirichlet boundary condition problems. Solutions for Neumann boundary condition problems may also be simplified, though the Divergence theorem applied to the differential equation defining Green's functions shows that the Green's function cannot integrate to zero on the boundary, and hence cannot vanish on the boundary. See Green's functions for the Laplacian or [2] for a detailed argument, with an alternative.
It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens principle, and leads to Kirchhoff's diffraction formula and other approximations.
On manifolds
Green's identities hold on a Riemannian manifold. In this setting, the first two are
${\begin{aligned}\int _{M}u\,\Delta v\,dV+\int _{M}\langle \nabla u,\nabla v\rangle \,dV&=\int _{\partial M}uNv\,d{\widetilde {V}}\\\int _{M}\left(u\,\Delta v-v\,\Delta u\right)\,dV&=\int _{\partial M}(uNv-vNu)\,d{\widetilde {V}}\end{aligned}}$
where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric, $d{\widetilde {V}}$ is the induced volume form on the boundary of M, N is the outward oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian.
Green's vector identity
Green's second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form
$p_{m}\,\Delta q_{m}-q_{m}\,\Delta p_{m}=\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\,\nabla p_{m}\right),$
where pm and qm are two arbitrary twice continuously differentiable scalar fields. This identity is of great importance in physics because continuity equations can thus be established for scalar fields such as mass or energy.[3]
In vector diffraction theory, two versions of Green's second identity are introduced.
One variant invokes the divergence of a cross product [4][5][6] and states a relationship in terms of the curl-curl of the field
$\mathbf {P} \cdot \left(\nabla \times \nabla \times \mathbf {Q} \right)-\mathbf {Q} \cdot \left(\nabla \times \nabla \times \mathbf {P} \right)=\nabla \cdot \left(\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)-\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)\right).$
This equation can be written in terms of the Laplacians,
$\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} +\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]=\nabla \cdot \left(\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right).$
However, the terms
$\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right],$
could not be readily written in terms of a divergence.
The other approach introduces bi-vectors, this formulation requires a dyadic Green function.[7][8] The derivation presented here avoids these problems.[9]
Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e.,
$\mathbf {P} =\sum _{m}p_{m}{\hat {\mathbf {e} }}_{m},\qquad \mathbf {Q} =\sum _{m}q_{m}{\hat {\mathbf {e} }}_{m}.$
Summing up the equation for each component, we obtain
$\sum _{m}\left[p_{m}\Delta q_{m}-q_{m}\Delta p_{m}\right]=\sum _{m}\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right).$
The LHS according to the definition of the dot product may be written in vector form as
$\sum _{m}\left[p_{m}\,\Delta q_{m}-q_{m}\,\Delta p_{m}\right]=\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} .$
The RHS is a bit more awkward to express in terms of vector operators. Due to the distributivity of the divergence operator over addition, the sum of the divergence is equal to the divergence of the sum, i.e.,
$\sum _{m}\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)=\nabla \cdot \left(\sum _{m}p_{m}\nabla q_{m}-\sum _{m}q_{m}\nabla p_{m}\right).$
Recall the vector identity for the gradient of a dot product,
$\nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)+\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right),$
which, written out in vector components is given by
$\nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\nabla \sum _{m}p_{m}q_{m}=\sum _{m}p_{m}\nabla q_{m}+\sum _{m}q_{m}\nabla p_{m}.$
This result is similar to what we wish to evince in vector terms 'except' for the minus sign. Since the differential operators in each term act either over one vector (say $p_{m}$’s) or the other ($q_{m}$’s), the contribution to each term must be
$\sum _{m}p_{m}\nabla q_{m}=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right),$
$\sum _{m}q_{m}\nabla p_{m}=\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right).$
These results can be rigorously proven to be correct through evaluation of the vector components. Therefore, the RHS can be written in vector form as
$\sum _{m}p_{m}\nabla q_{m}-\sum _{m}q_{m}\nabla p_{m}=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right).$
Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained,
Theorem for vector fields:
$\color {OliveGreen}\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].$
The curl of a cross product can be written as
$\nabla \times \left(\mathbf {P} \times \mathbf {Q} \right)=\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right);$
Green's vector identity can then be rewritten as
$\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)-\nabla \times \left(\mathbf {P} \times \mathbf {Q} \right)+\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].$
Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity:
$\color {OliveGreen}\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)+\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].$
With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors
$\Delta \left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} +2\nabla \cdot \left[\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P} \right].$
As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation,
$\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]-\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]=\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)\right].$
This result can be verified by expanding the divergence of a scalar times a vector on the RHS.
See also
• Green's function
• Kirchhoff integral theorem
• Lagrange's identity (boundary value problem)
References
1. Strauss, Walter. Partial Differential Equations: An Introduction. Wiley.
2. Jackson, John David (1998-08-14). Classical Electrodynamics. John Wiley & Sons. p. 39.
3. Guasti, M Fernández (2004-03-17). "Complementary fields conservation equation derived from the scalar wave equation". Journal of Physics A: Mathematical and General. IOP Publishing. 37 (13): 4107–4121. Bibcode:2004JPhA...37.4107F. doi:10.1088/0305-4470/37/13/013. ISSN 0305-4470.
4. Love, Augustus E. H. (1901). "I. The integration of the equations of propagation of electric waves". Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character. The Royal Society. 197 (287–299): 1–45. doi:10.1098/rsta.1901.0013. ISSN 0264-3952.
5. Stratton, J. A.; Chu, L. J. (1939-07-01). "Diffraction Theory of Electromagnetic Waves". Physical Review. American Physical Society (APS). 56 (1): 99–107. Bibcode:1939PhRv...56...99S. doi:10.1103/physrev.56.99. ISSN 0031-899X.
6. Bruce, Neil C (2010-07-22). "Double scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes". Journal of Optics. IOP Publishing. 12 (8): 085701. Bibcode:2010JOpt...12h5701B. doi:10.1088/2040-8978/12/8/085701. ISSN 2040-8978. S2CID 120636008.
7. Franz, W (1950-09-01). "On the Theory of Diffraction". Proceedings of the Physical Society. Section A. IOP Publishing. 63 (9): 925–939. Bibcode:1950PPSA...63..925F. doi:10.1088/0370-1298/63/9/301. ISSN 0370-1298.
8. Chen-To Tai (1972). "Kirchhoff theory: Scalar, vector, or dyadic?". IEEE Transactions on Antennas and Propagation. Institute of Electrical and Electronics Engineers (IEEE). 20 (1): 114–115. Bibcode:1972ITAP...20..114T. doi:10.1109/tap.1972.1140146. ISSN 0096-1973.
9. Fernández-Guasti, M. (2012). "Green's Second Identity for Vector Fields". ISRN Mathematical Physics. Hindawi Limited. 2012: 1–7. doi:10.5402/2012/973968. ISSN 2090-4681.
External links
• "Green formulas", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Green's Identities at Wolfram MathWorld
| Wikipedia |
Ten Great Ideas about Chance
This book grew out of a course given by the authors, a statistician (Diaconis) and a philosopher (Skyrms) to a mixed audience of students. The idea is that the reader should be familiar with a basic course on probability theory. If that has been a while and it needs some refreshing, then an appendix summarizes the main definitions and formulas in a brief tutorial. The true subject of the course are the foundations and the philosophical aspects of probability and statistics and how they evolved in the course of time. These are summarised in the form of ten "great ideas" developed in as many chapters that have formed the subject as we know it today. They are introduced in an approximately chronological order, so that the book is also a history book that shows the genesis of the ideas from Cardano to the 21st century.
A mathematical introduction to probability starts with a probability space. That is a triple $(Ω,\mathcal{A},P)$ with $Ω$ a non-empty set (sample space), $\mathcal{A}$ a σ-algebra (event space) and $P$ a probability measure defined on $\mathcal{A}$. This is more or less how A.N. Kolmogorov has defined probability, but the idea of chance has evolved in several stages before this definition emerged.
The real origin of probability is commonly believed to be in the correspondence between Pascal and Fermat (1654). Chance was measured for the first time by counting and defining a frequency of an event occurring in many repetitions of the same experiment like throwing dice. Huygens and later Jacob Bernoulli wrote the first books on probability (1713) and even defined conditional probability. In fact, they also considered another problem where throwing the dice was part of a game where winning or loosing was the issue. The gambler had to estimate his chance of winning and adapt his stakes as the game went on. In that setting probability was expected to predict the future. What will be the future outcome of a dice throw and eventually of the game, or of the economical evolution, or of the political elections, if we stretch it to the present day. Your actions today will depend on your expectations. You buy on the financial market when expecting A and you sell when expecting B. It depends on your judgement and how much you value the outcome A by the money you hope to win if A occurs. That is judgemental probability and this can be considered a measure of probability on condition that it is coherent. If not, you get a Dutch book and you can be drained of all your money. Clearly this is a dynamical system, thus as new evidence becomes available, your judgement should be adapted. But people are not always rational or do not act accordingly. Their acts depend also on psychology. The Allais paradox shows that people value certainty more. Even if it is objectively preferable to choose A involving risk, rather than B which involves certainty, people will choose for certainty. They also prefer a choice formulated in terms of gains over an equivalent formulation in terms of losses. All this makes financial markets difficult to manage on a pure rational basis.
In Bernoulli's view, just counting the frequency of an event happening defines its probability, but what he did was basically the inverse. His law of large numbers is saying that for a given probability of an event, this will show up eventually as the frequency of that event when the experiment is repeated a large number of times. In fact, this is a bit of a shaky definition, not built on a sound ground. Sufficiently often repetition of identical experiments basically means an infinite number of times and that is practically impossible. It was John Venn who made such a statement mathematically more rigorous but still a vivid debate was started in Victorian England whether frequency and probability should be identified or whatever relation existed between both. Von Mises tried to answer Hilbert and attempted to give an axiomatic definition but the discrepancy between reality and ideal randomness remained.
The definition of Kolmogorov (1933) was a release. He defined a random variable, and probability was not a frequency but it could take any value in an interval $[0,1]$ and he used the newly developed notion of measure. But it still needed infinity and the controversy about the inference from frequency to chance continued. This controversy is related to inductive reasoning. Is what we observe in a large number of experiment guaranteed to be consistently observed if the number of experiments is increased to infinity? Because the sun has risen every morning till today, are we sure that will happen again tomorrow? Can we deduce behaviour from observation? Especially David Hume was sceptical about inductive reasoning. Bayes applied conditional probability to use the evidence that becomes available with every observation to reconsider the probability that it will happen the next time. So one starts from a prior hypothetical distribution, a subjective belief so to speak, and its probability is updated permanently. This a posteriori updated model can then express a degree of belief (probability) to what will be the next observation. Laplace in France came to a similar conclusion. Bruno di Finetti's theorem proves the existence of such an a priory distribution that perfectly predicts the outcome of an exchangeable sequence (i.e. independent of reordering).
With the inference problem more or less settled, it remains to find a computer algorithm to generate a truly random sequence. Per Martin-Löf gave an appropriate answer (1966) using the theory of computability developed in the 1930's by Church and Turing. Consider a set of countably computable binary sequences and select a nested sequence of subsets whose probability behaves like $2^{-n}$, and with probability 1, you will end up with a set of sequences that are perfectly random (i.e. that have probability 0).
Statistical mechanics and quantum theory placed probability at the center of physical phenomena. The second law of thermodynamics explained on a statistical basis required a notion of ergodicity: the time average equals the ensemble average like Boltzmann believed in his model of gas dynamics. A full proof that it is indeed ergodic is still missing today. Poincaré introduced chaos in such systems and the sensitivity on the initial conditions will eventually remove the prior probability on the initial conditions, which leads to a Boltzmann-Liouville uniform distribution. Quantum physics leaves us with many puzzling mysteries, even for the designers and the specialists, but still it is possible to place it in a framework of classical probability. The EPR (Einstein, Podolsky, Rosen) paper of 1935 was based on probability and so stated the nonlocality (the spooky action at a distance) that was confirmed in subsequent experiments. We also need a form of quantum ergodicity, but it can be shown that that follows from classical ergodicity.
In a last chapter, inductive reasoning is reconsidered revising what has been explained in the previous chapters with the visions of Hume, Kant, and Popper and the sceptics Bayes, and Laplace. But there are always assumptions. Should we be sceptic about these assumptions too. But then one can be sceptic about anything making absolute scepticism impossible.
In summary, in this book the authors focus on the philosophical aspects of probability and statistics. The ten great ideas are captured in the titles of the chapters: measurement, judgement, psychology, frequency, mathematics, inverse inference, unification, algorithmic randomness, physical chance, and induction. But these are of course just keywords and much more is covered in each chapter. Their account is partially historical because they describe the origin and the controversies that arose about some concepts. Besides technical graphics, pictures of some of the main historical contributors are used as illustrations but the historical component is not so thorough that we get biographies of the mathematicians involved (not even short ones). Besides the summary on probability at the end of the book, several of the chapters have appendices in which some topic is further elaborated. There are a few short footnotes but most of the notes (mostly mentioning a reference) are collected per chapter at the end of the book. An annotated list of references per chapter is added separately. Often the vision of some historical person is formulated in his own words (or their English translations) as easily recognizable quotes. The typesetting is done very carefully (although I could spot a few typos) and they are just long enough to catch the main idea without boring, as if they were written to cover one lecture each.
Although the text is discussing formulas that are used in everyday practical applications, the reader should be warned that it is mainly about their foundations and philosophical aspects. Thus a reader who is just a practitioner may have difficulties if not interested in the philosophy. Philosophers not very familiar with the underlying mathematics may have a hard time too. But if the reading is tough, the tough go on reading, and when you worked your way through it, it is a very enriching journey. Your vision will be broadened assimilating all these issues and solutions as well as open problems from the early history of probability, game theory, financial markets, politics, thermodynamics, quantum theory and much much more.
Adhemar Bultheel
The authors start from the early days of probability applied in gambling and work their way through ten great ideas that have shaped the current day scenery of probability and statistics. The discussion is fundamental and covers the different philosophical interpretations of concepts such as frequency versus mathematical probability, Bayesian probability, subjective probability, entropy, quantum physics, and inductive reasoning. For the reader who needs to refresh notions from elementary probability a short tutorial is provided.
persi diaconis
Brian Skyrms
978-0-6911-7416-7 (hbk)
£ 22.95 (hbk)
https://press.princeton.edu/titles/11082.html
60 Probability theory and stochastic processes
Submitted by Adhemar Bultheel | 8 / Jan / 2018 | CommonCrawl |
Reversible superdense ordering of lithium between two graphene sheets
Matthias Kühne1 na1 nAff6,
Felix Börrnert2 na1,
Sven Fecher1,
Mahdi Ghorbani-Asl3,
Johannes Biskupek2,
Dominik Samuelis1 nAff7,
Arkady V. Krasheninnikov3,4,5,
Ute Kaiser2 &
Jurgen H. Smet1
Nature volume 564, pages234–239(2018)Cite this article
Materials for energy and catalysis
Surfaces, interfaces and thin films
Two-dimensional materials
Many carbon allotropes can act as host materials for reversible lithium uptake1,2, thereby laying the foundations for existing and future electrochemical energy storage. However, insight into how lithium is arranged within these hosts is difficult to obtain from a working system. For example, the use of in situ transmission electron microscopy3,4,5 to probe light elements (especially lithium)6,7 is severely hampered by their low scattering cross-section for impinging electrons and their susceptibility to knock-on damage8. Here we study the reversible intercalation of lithium into bilayer graphene by in situ low-voltage transmission electron microscopy, using both spherical and chromatic aberration correction9 to enhance contrast and resolution to the required levels. The microscopy is supported by electron energy-loss spectroscopy and density functional theory calculations. On their remote insertion from an electrochemical cell covering one end of the long but narrow bilayer, we observe lithium atoms to assume multi-layered close-packed order between the two carbon sheets. The lithium storage capacity associated with this superdense phase far exceeds that expected from formation of LiC6, which is the densest configuration known under normal conditions for lithium intercalation within bulk graphitic carbon10. Our findings thus point to the possible existence of distinct storage arrangements of ions in two-dimensional layered materials as compared to their bulk parent compounds.
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Fig. 1: Device layout and working principle.
Fig. 2: In situ TEM measurements.
Fig. 3: Atomistic models of Li crystals between AB-stacked graphene sheets obtained from DFT calculations.
Fig. 4: Li crystal growth between two graphene sheets.
The data that support the findings of this study are available from the corresponding authors on request.
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We acknowledge financial support from the Baden-Württemberg Stiftung gGmbH (project CT 5) and from the European Union Graphene Flagship. We are grateful to FEI/ThermoFisher Scientific for providing drawings and specifications of the NanoEx-i/v holder. F.B., J.B. and U.K. acknowledge funding from the German Research Foundation (DFG) and the Ministry of Science, Research and the Arts (MWK) of the federal state of Baden-Württemberg, Germany, in the frame of the SALVE project. A.V.K. thanks the Academy of Finland for support under project number 286279 and the DFG under project KR 4866/1-1. The theoretical study of Li diffusion (A.V.K.) was supported by the Russian Science Foundation (project identifier, 17-72-20223). We thank K. v. Klitzing for discussions and support and J. Popovic for useful comments on the manuscript. We acknowledge CSC Finland and PRACE (HLRS, Stuttgart, Germany) for generous grants of CPU time.
Nature thanks I. Honma and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Matthias Kühne
Present address: Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
Dominik Samuelis
Present address: Heraeus Battery Technology, Hanau, Germany
These authors contributed equally: M. Kühne, F. Börrnert
Max Planck Institute for Solid State Research, Stuttgart, Germany
Matthias Kühne, Sven Fecher, Dominik Samuelis & Jurgen H. Smet
Materialwissenschaftliche Elektronenmikroskopie, Universität Ulm, Ulm, Germany
Felix Börrnert, Johannes Biskupek & Ute Kaiser
Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany
Mahdi Ghorbani-Asl & Arkady V. Krasheninnikov
Department of Applied Physics, Aalto University, Aalto, Finland
Arkady V. Krasheninnikov
National University of Science and Technology MISiS, Moscow, Russia
Felix Börrnert
Sven Fecher
Mahdi Ghorbani-Asl
Johannes Biskupek
Ute Kaiser
Jurgen H. Smet
J.H.S. and U.K. composed the project. M.K. and S.F. fabricated samples and performed electrochemical measurements. F.B. performed the TEM and EELS experiments. M.G. and A.V.K. did the DFT calculations. J.B. helped with TEM and EELS experiments and did TEM imaging simulations. U.K. supervised the TEM work. D.S. contributed to the electrochemical design of the experiment. M.K. and J.H.S. wrote the manuscript and all authors contributed to it.
Correspondence to Ute Kaiser or Jurgen H. Smet.
The authors declare no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Atomic resolution SALVE TEM images of lithium.
a, Original TEM image (acquired between Fig. 2b, c). b, Same as a, but with graphene signals filtered out. c, Magnified view of area shown boxed in b. d, Slightly Gauss-filtered version of c. e, Same as d, but with red and blue circles indicating single Li atoms and symmetric positions without Li atoms, respectively. The latter show that the contrasts are not delocalization artefacts. f, Line profiles of the imaging contrast, centred on two neighbouring, individual Li atoms (left panel) and on the negative atomic contrast of a missing C atom in one graphene sheet, artificially inserted by the filtering procedure (right panel). The red arrows and red dashed lines indicate the signal intensity of the respective atomic species. g, Fourier-filtered version (bilayer graphene lattice removed) of a TEM image acquired during lithiation at t = 266 s (bottom image), between Fig. 2b, c. The top row shows two zoom-ins centred on locations where the line profiles in f have been extracted. h, Temporal evolution of the C vacancy in one of the two graphene sheets from which the line profile in g has been extracted at t = 266 s. Each row corresponds to data from a single time: left, magnified sections of our TEM images as acquired; right, the same sections after the removal of the graphene lattice by Fourier filtering. Scale bars, 1 nm.
Extended Data Fig. 2 80 kV SALVE TEM image simulations.
a, b, Atomistic model of a hexagonal close-packed Li wedge (red), with five additional single Li atom rows at its left edge as well as with single-atom vacancies, between two graphene sheets (cyan). Shown are a three-dimensional representation (a, top), a side view (a, bottom), and a top view of its thin front without C lattices (b). c–e, 80 kV dose-limited (applied dose 2 × 106 e− nm−2) TEM image simulations. c, Embedded in graphene. d, Fourier-filtered image with graphene removed. e, Unfiltered image simulated without graphene. In d and e, the yellow arrows mark single Li atoms and the white arrows point to a vacancy in the Li lattice. f, g, Image simulation of wedges of two different close-packed Li systems (cubic and hexagonal shown at left and right, respectively). f, Atomistic model used for the simulations, showing side and top views. The thickness gradually increases by one layer from left (one layer) to right (six layers). g, Image simulations for two different values of defocus, seen along [111] for cubic close-packed Li and along [0001] for hexagonal close-packed Li. The von Hann-filtered Fourier transforms on the right (diffractograms) are calculated for 4 layers at +6 nm overfocus. For the simulation the corresponding experimentally measured electron dose was applied.
Extended Data Fig. 3 Fourier filtering.
Filtering mask for removing the graphene lattice and the moiré effects. a, Portion of the signal cut out by the applied mask. b, Remaining Fourier transform. The streaking (indicated by yellow arrows) results from the edges of the real image and was not Fourier-filtered to avoid masking too-large portions of Fourier space.
Extended Data Fig. 4 Thickness determination of crystalline Li.
a, Main panel, Fourier-filtered TEM image. Dashed lines demarcate major edges enclosing regions of the Li phase with approximately constant thickness. Two-coloured lines demarcate grain boundaries. Fourier transforms from two selected regions of different thickness (yellow boxes) are shown on the right. Signals from the Li phase in the Fourier transforms for both regions lying within one grain are identical. b–d, Magnified views of selected areas of the TEM image (white boxes) in a. e, f, Relative thickness determination from contrast quantification. e, Fourier-filtered TEM image (graphene lattice removed), identical to the main panel in a. Dashed lines demarcate major edges enclosing regions of the Li phase with approximately constant thickness. Arrowheads point outwards from thicker areas. f, Area-normalized intensity histograms acquired from the shaded regions in e that are labelled i, ii and iii. The full-widths at half-maximum (FWHMs) are indicated by white double-headed arrows.
Extended Data Fig. 5 Electron energy-loss spectroscopy.
a, Low-loss EEL spectra of pristine (blue) and lithiated (yellow) bilayer graphene, with the close-packed Li phase present in the latter case. b, Calculated ELNES (electron energy loss near-edge structure; Methods) of the Li-K edge for bilayer graphene containing 1, 2 and 3 Li layers compared to the experimental ELNES. The spectrometer broadening is taken into account by convoluting the result with a Gaussian function. Two different broadenings have been considered (left and right panels). The spectrometer broadening is given as the FWHM of the respective Gaussian function.
Extended Data Fig. 6 Atomic configurations and energetics of a single layer of Li atoms between two graphene sheets.
a, b, Structural evolution of a finite single-layer cluster of Li atoms between two graphene sheets with different stacking: AA-stacking (a) and AB-stacking (b). Left and right structures are before and after relaxation (that is, energy minimization); top and bottom are top and side views, respectively. The relaxation gives rise to the formation of a system with a geometry close to the C6LiC6 configuration. Note that the configurations correspond to a local, not global, energy minimum. c, d, The periodic C6LiC6 configuration with Li atoms arranged in a commensurate \(\left(\sqrt{3}\times \sqrt{3}\right){\rm{R}}3{0}^{\circ }\) superstructure between graphene sheets for AA stacking (c) and AB stacking (d). dLi-Li refers to the separation of Li atoms. Double-headed arrows indicate the spacing between graphene sheets. e, f, Electron density difference between the combined system and its isolated parts. Red colour corresponds to a decrease in the electron density, blue to an increase. Charge transfer between Li and graphene (with an average value of 0.85 electrons per Li atom) is clearly observable.
Extended Data Fig. 7 Atomic configurations and energetics of Li bilayers between two graphene sheets.
a, The geometric arrangement of the atoms of a finite bilayer cluster of Li atoms encapsulated between two graphene sheets after energy minimization for two different rotation angles θ between the Li and C lattices. The original configuration of the cluster was the perfect h.c.p. lattice. The structure is largely preserved during the relaxation. The distance between the Li atoms is denoted as dLi-Li. A very weak dependence on the angle between graphene and h.c.p. lattice is found, as shown in the table at right. b, Atomic configurations and energetics of the infinite commensurate Li bilayer h.c.p./graphene structure and the dependence of the energy on the orientation angle θ between surfaces. Very weak dependence on the angle between graphene and the h.c.p. Li lattice is found. A small amount of strain introduced into the system to make the graphene and Li lattices commensurate affects the results by no more than 0.01 eV, as evident from the tables presenting Ef for different θ.
Extended Data Fig. 8 Atomic configurations and energetics of Li multilayers between two graphene sheets.
a, b, Atomic configurations and energetics of the infinite commensurate Li trilayer with h.c.p. structure between two graphene sheets for AA stacking (a) and AB stacking (b). c, Main panel, charge transfer from Li to graphene as a function of the number of close-packed Li layers NLi between two graphene sheets. The corresponding atomic configurations are shown above the plot. The blue values are obtained as ΔqLi = q0 − qLi, where qLi is the charge of Li after intercalation and q0 is the charge of the isolated Li atom. Since the charge transfer is relevant for outermost Li layers only, we also plot the full charge transfer renormalized by the number of Li atoms in these outermost Li layers without considering the other Li layers (purple). d, e, Energy difference between different possible close-packed configurations (stacking orders), calculated for three layers of Li between two graphene sheets (d) and four layers of Li between two graphene sheets (e). To illustrate the stacking order, below each top view we include a side view of atoms within the red rectangle. The energy differences with respect to each f.c.c. case are stated below the side views.
Extended Data Fig. 9 Registry of graphene layers.
a, b, Successive SALVE TEM images at different defocus values before lithiation (a) and after delithiation (b). The red lines are guides for the eye. We do not observe a change in registry of the two graphene sheets when comparing TEM images of the graphene lattice before lithiation and after delithiation. The registry can be checked at sites where one of the two graphene sheets features a moderately big hole. The stacking is AB without any sign of change throughout the experiment.
Extended Data Fig. 10 Lateral diffusion of a Li atom inside the close-packed Li system confined between two graphene sheets.
a, b, Schematic atomistic configuration (top and side view). The red sphere represents the extra interstitial Li atom. c, Schematic of the diffusion process. d, The actual initial/final positions of atoms. The yellow and blue transparent triangles in c and d mark the initial and final configurations of the interstitial atom with regard to the nearest Li atoms in the undistorted (c) and the optimized (d) structures. The red and blue arrows connect the initial and final positions of the diffusing atoms.
TEM image sequence TEM image sequence showing the propagation front of a Li crystal forming inside bilayer graphene during lithiation
Kühne, M., Börrnert, F., Fecher, S. et al. Reversible superdense ordering of lithium between two graphene sheets. Nature 564, 234–239 (2018). https://doi.org/10.1038/s41586-018-0754-2
Issue Date: 13 December 2018
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What is the best approach to balance this equation? Why I cannot do it efficiently?
What is the best approach to balance this equation?
$$\ce{CO2 + H2O <--> C2H2 + O2}$$
I did it at first this way and I was not successful: \begin{align} \ce{4CO2 + H2O &<--> C2H2 + 3O2}\\ \ce{2CO2 + 6H2O &<--> 6C2H2 + 5O2}\\ \ce{12CO2 + 6H2O &<--> 6C2H2 + 5O2}\\ \end{align}
Should I start from right side or left? They say start from the most complicated molecule? If it is like that, then what is the most complicated one and why?
equilibrium stoichiometry
Martin - マーチン♦
$\begingroup$ @airhuff Please do not add the reaction tag any more. We are in the process of blacklisting it. See meta for more information. $\endgroup$ – Martin - マーチン♦ Mar 29 '17 at 7:15
$\begingroup$ Related: chemistry.stackexchange.com/questions/68924/… (Personally, I'm biased toward my own answer to this question, but I think all of them address how to balance a reaction in general without guesswork). $\endgroup$ – Tyberius Mar 29 '17 at 15:32
This may be a bit dumb method, but sometimes I do use it as it is pretty straightforward and doesn't require any guessing. I setup a few equations and try solving over integers. $$\ce{aCO2 + bH2O <--> cC2H2 + dO2}$$
Balancing $\ce{C}$ atoms both sides, $$a = 2c\\$$ Balancing $\ce{H}$ atoms both sides, $$2b=2c\\$$ Balancing $\ce{O}$ atoms both sides,
$$2a+b=2d$$
Solving gives this. Not hard to see that $a=4$ does the job.
HiiiiHiiii
The key for this type of equation is to balance the diatomic molecule last, and then just add a 1/2 of the diatomic molecule to finish the balance. Then, double all the coefficients and you're done.
balance everything but the diatomic molecule: $$\ce{2CO2 + H2O <=> C2H2 + O2}$$
Then add the right fraction of the diatomic molecule: $$\ce{2CO2 + H2O <=> C2H2 + 2.5O2}$$
Notice that 2.5 oxygen molecules is 5 oxygen atoms.
Double all the coefficients to get whole numbers $$\ce{4CO2 + 2H2O <=> 2C2H2 + 5O2}$$
Mr RoyMr Roy
There is a standard equation for balancing such reactions: $$\ce{C_xH_y + [x +(y/4)]O2 -> x CO2 + y/2 H2O}$$
Hopefully, now you should be able to apply it to your reaction. It's just the reverse reaction that you're dealing with.
SupernovaSupernova
$\begingroup$ Please don't use shorthand or text speak. $\endgroup$ – Martin - マーチン♦ Mar 29 '17 at 7:19
$\begingroup$ This answer would be more helpful if it demonstrated how the stoichiometric coefficients are found in the first place. $\endgroup$ – Nicolau Saker Neto Mar 29 '17 at 7:52
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\(L_{p}\)-Dual geominimal surface area and general \(L_{p}\)-centroid bodies
Jinsheng Guo1 &
Yibin Feng1
In this article, we consider the Shephard type problems and obtain the affirmative and negative parts of the version of \(L_{p}\)-dual geominimal surface area for general \(L_{p}\)-centroid bodies. Combining with the \(L_{p}\)-dual geominimal surface area we also give a negative form of the Shephard type problems for \(L_{p}\)-centroid bodies.
Introduction and main results
Let \(\mathcal{K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space \(\mathbb{R}^{n}\). For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in \(\mathbb{R}^{n}\), we write \(\mathcal{K}^{n}_{o}\) and \(\mathcal{K}^{n}_{c}\), respectively. \(S^{n}_{o}\) and \(S^{n}_{c}\), respectively, denote the set of star bodies (about the origin) and the set of origin-symmetric star bodies in \(\mathbb{R}^{n}\). Let \(S^{n-1}\) denote the unit sphere in \(\mathbb{R}^{n}\), and let \(V(K)\) denote the n-dimensional volume of a body K. For the standard unit ball B in \(\mathbb{R}^{n}\), we use \(\omega_{n} = V(B)\) to denote its volume.
The notion of geominimal surface area was discovered by Petty (see [1]). For \(K\in\mathcal{K}^{n}\), the geominimal surface area, \(G(K)\), of K is defined by
$$\omega_{n}^{\frac{1}{n}}G(K)= \inf\bigl\{ nV_{1}(K,Q)V \bigl(Q^{\ast}\bigr)^{\frac{1}{n}}: Q\in\mathcal{K}^{n}\bigr\} . $$
Here \(Q^{\ast}\) denotes the polar of body Q and \(V_{1}(M, N)\) denotes the mixed volume of \(M, N\in\mathcal{K}^{n}\) (see [2]).
The geominimal surface area serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative geometry, and Minkowskian geometry. Hence it receives a lot of attention (see, e.g., [3, 4]). Lutwak in [5] showed that there were natural extensions of geominimal surface areas in the Brunn-Minkowski-Firey theory. It motivates extensions of some known inequalities for geominimal surface areas to \(L_{p}\)-geominimal surface areas. The inequalities for \(L_{p}\)-geominimal surface areas are stronger than their classical counterparts (see [6–10]).
Based on \(L_{p}\)-mixed volume, Lutwak [5] introduced the notion of \(L_{p}\)-geominimal surface area. For \(K\in\mathcal{K}^{n}_{o}\), \(p\geq1\), the \(L_{p}\)-geominimal surface area, \(G_{p}(K)\), of K is defined by
$$\omega_{n}^{\frac{p}{n}}G_{p}(K)= \inf\bigl\{ nV_{p}(K,Q)V\bigl(Q^{\ast}\bigr)^{\frac{p}{n}}: Q\in \mathcal{K}^{n}_{o}\bigr\} . $$
Here \(V_{p}(M, N)\) denotes the \(L_{p}\)-mixed volume of \(M, N\in\mathcal{K}^{n}_{o}\) (see [5, 11]). Obviously, if \(p=1\), \(G_{p}(K)\) is just the geominimal surface area \(G(K)\).
Recently, Wang and Qi [12] introduced a concept of \(L_{p}\)-dual geominimal surface area, which is a dual concept for \(L_{p}\)-geominimal surface area and belongs to the dual \(L_{p}\)-Brunn-Minkowski theory for star bodies also developed by Lutwak (see [13, 14]). For \(K\in S_{o}^{n}\), and \(p\geq1\), the \(L_{p}\)-dual geominimal surface area, \(\widetilde{G}_{-p}(K)\), of K is defined by
$$ \omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)= \inf\bigl\{ n \widetilde{V}_{-p}(K,Q)V\bigl(Q^{\ast}\bigr)^{-\frac{p}{n}}: Q\in \mathcal{K}_{c}^{n}\bigr\} . $$
Here, \(\widetilde{V}_{-p}(M,N)\) denotes the \(L_{p}\)-dual mixed volume of \(M, N\in S_{o}^{n}\) (see [5]).
Centroid bodies are a classical notion from geometry which have attracted increased attention in recent years (see [13, 15–22]). In particular, Lutwak and Zhang [18] introduced the notion of \(L_{p}\)-centroid bodies. For each compact star-shaped (about the origin) K in \(\mathbb{R}^{n}\) and real number \(p\geq1\), the \(L_{p}\)-centroid body, \(\Gamma_{p} K\), of K is an origin-symmetric convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}|u\cdot x|^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}|u\cdot v|^{p} \rho_{K}^{n+p}(v)\,dS(v) \end{aligned}$$
for all \(u\in S^{n-1}\), where
$$ c_{n, p}=\omega_{n+p}/\omega_{2}\omega_{n} \omega_{p-1}, \quad\mbox{and}\quad \omega _{n}= \pi^{\frac{n}{2}}/\Gamma\biggl(1+\frac{n}{2}\biggr). $$
More recently, Feng et al. [23] defined a new notion of general \(L_{p}\)-centriod bodies, which generalized the concept of \(L_{p}\)-centroid bodies. For \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), the general \(L_{p}\)-centroid body, \(\Gamma_{p}^{\tau}K\), of K is a convex body whose support function is defined by
$$\begin{aligned} h^{p}_{\Gamma_{p}^{\tau}K}(u)&=\frac{1}{c_{n, p}(\tau)V(K)}\int _{K}\varphi _{\tau}(u\cdot x)^{p} \,dx \\ & =\frac{1}{c_{n, p}(\tau)(n+p)V(K)}\int_{S^{n-1}}\varphi_{\tau}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv, \end{aligned}$$
$$c_{n, p}(\tau)=\frac{1}{2}c_{n, p}\bigl[(1+ \tau)^{p}+(1-\tau)^{p}\bigr], $$
and \(\varphi_{\tau}: \mathbb{R}\rightarrow[0, \infty)\) is a function defined by \(\varphi_{\tau}(t)=|t|+\tau t\). We note that general \(L_{p}\)-centroid bodies are an essential part of the rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [20, 24–32]).
The normalization is chosen such that \(\Gamma_{p}^{\tau}B=B\) for every \(\tau\in[-1, 1]\), and \(\Gamma_{p}^{0} K=\Gamma_{p} K\). Let \(\varphi_{+}(u\cdot x)=\max\{u\cdot x, 0\}\) (\(\tau=1\)) in (1.4), then a special case of the definition of \(\Gamma_{p}^{\tau}K\) is \(\Gamma_{p}^{+} K\), i.e.,
$$\begin{aligned} h^{p}_{\Gamma_{p}^{+} K}(u)&=\frac{1}{c_{n, p}V(K)}\int _{K}\varphi_{+}(u\cdot x)^{p} \,dx \\ &=\frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p}\rho_{K}^{n+p}(v)\,dv. \end{aligned}$$
Besides, we also define
$$ \Gamma_{p}^{-}K=\Gamma_{p}^{+}(-K). $$
From the definition of \(\Gamma^{\pm}_{p} K\) and (1.4), we see that if \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
$$ \Gamma_{p}^{\tau}K=f_{1}(\tau)\cdot \Gamma_{p}^{+} K+_{p}f_{2}(\tau)\cdot \Gamma_{p}^{-} K, $$
where '\(+_{p}\)' denotes the Firey \(L_{p}\)-combination of convex bodies, and
$$ f_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}, \qquad f_{2}(\tau )= \frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}. $$
If \(\tau=\pm1\) in (1.7) and using (1.8), then
$$\Gamma_{p}^{+1} K=\Gamma_{p}^{+} K, \qquad \Gamma_{p}^{-1} K=\Gamma_{p}^{-} K. $$
In [16] Grinberg and Zhang discussed an investigation of Shephard type problems for \(L_{p}\)-centriod bodies. Namely, let K and L be two origin-symmetric star bodies such that
$$\Gamma_{p} K\subset\Gamma_{p} L. $$
They proved that if the space \((\mathbb{R}^{n}, \|\cdot\|_{L})\) embeds in \(L_{p}\), then we necessarily have
$$V(K)\leq V(L). $$
On the other hand, if \((\mathbb{R}^{n}, \|\cdot\|_{K})\) does not embed in \(L_{p}\), then there is a body L so that \(\Gamma_{p} K\subset\Gamma_{p} L\), but \(V(K)\leq V(L)\).
In this article, we first investigate the Shephard type problems for general \(L_{p}\)-centroid bodies and give the affirmative and negative parts of the version of \(L_{p}\)-dual geominimal surface area.
For \(K \in\mathcal{K}_{o}^{n}\), \(L\in\mathcal{K}_{c}^{n}\), and \(p\geq1\), if \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), then
$$ \widetilde{G}_{-p}(K)\leq\widetilde{G}_{-p}(L), $$
with equality if and only if \(K=L\).
For \(L \in S_{o}^{n}\), \(p\geq1\) and \(\tau\in (-1, 1)\), if L is not origin-symmetric, then there exists \(K\in S_{o}^{n}\), such that
$$\Gamma_{p}^{+} K\subset\Gamma_{p}^{\tau}L,\qquad \Gamma_{p}^{-} K\subset\Gamma _{p}^{-\tau} L. $$
$$\widetilde{G}_{-p}(K)>\widetilde{G}_{-p}(L). $$
Further, taking together the \(L_{p}\)-dual geominimal surface area with \(L_{p}\)-centroid bodies we establish the following Shephard type problem.
For \(L \in S_{o}^{n}\) and \(1\leq p< n\), if L is not origin-symmetric star body, then there exists \(K\in S_{o}^{n}\), such that
$$\Gamma_{p}K\subset\Gamma_{p} L. $$
The proofs of Theorems 1.1-1.3 will be given in Section 3.
Support functions, radial functions, and polars of convex bodies
The support function, \(h_{K} = h(K,\cdot):\mathbb{R}^{n}\rightarrow(-\infty ,\infty)\), of \(K\in\mathcal{K}^{n}\) is defined by (see [33, 34])
$$ h(K,x)=\max\{x \cdot y: y\in K\},\quad x\in\mathbb{R}^{n}, $$
where \(x\cdot y\) denotes the standard inner product of x and y.
If K is a compact star-shaped (about the origin) set in \(\mathbb {R}^{n}\), then its radial function, \(\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\setminus\{0\}\rightarrow[0,\infty )\), is defined by (see [33, 34])
$$ \rho(K,u)=\max\{\lambda\geq0: \lambda\cdot u\in K\},\quad u\in S^{n-1}. $$
If \(\rho_{K}\) is continuous and positive, then K will be called a star body. Two star bodies K, L are said to be dilates (of one another) if \(\rho_{K}(u)\diagup\rho_{L}(u)\) is independent of \(u\in S^{n-1}\).
If \(K\in\mathcal{K}_{o}^{n}\), the polar body, \(K^{\ast}\), of K is defined by (see [33, 34])
$$ K^{\ast}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y\leq1,y\in K\bigr\} . $$
For \(K, L\in\mathcal{K}_{o}^{n}\), \(p\geq1\), and \(\lambda, \mu\geq0\) (not both zero), the Firey \(L_{p}\)-combination, \(\lambda\cdot K +_{p}\mu\cdot L\), of K and L is defined by (see [35])
$$ h(\lambda\cdot K +_{p}\mu\cdot L, \cdot)^{p}=\lambda h(K, \cdot)^{p}+\mu h(L, \cdot)^{p}, $$
where ' ⋅ ' in \(\lambda\cdot K\) denotes the Firey scalar multiplication. Obviously, the \(L_{p}\)-Firey and the usual scalar multiplications are related by \(\lambda\cdot K=\lambda^{\frac{1}{p}}K\).
For \({K, L}\in S_{o}^{n}\), \(p\geq1\), and \({\lambda, \mu} \geq0\) (not both zero), the \(L_{p}\)-harmonic radial combination, \(\lambda\star K +_{-p} \mu\star L\in S_{o}^{n}\), of K and L is defined by (see [5])
$$ \rho(\lambda\star K +_{-p}\mu\star L, \cdot)^{-p} = \lambda \rho(K, \cdot )^{-p} +\mu \rho(L, \cdot)^{-p}, $$
where \(\lambda\star K\) denotes the \(L_{p}\)-harmonic radial scalar multiplication. Here, we have \(\lambda\star K=\lambda^{-\frac{1}{p}}K\).
\(L_{p}\)-Dual mixed volume
Using \(L_{p}\)-harmonic radial combination, Lutwak [5] introduced the notion of \(L_{p}\)-dual mixed volume. For \({K, L}\in S_{o}^{n}\), \(p \geq1\), and \(\varepsilon> 0\), the \(L_{p}\)-dual mixed volume, \(\widetilde{V}_{-p}(K, L)\), of K and L is defined by
$$\frac{n}{-p}\widetilde{V}_{-p}(K, L)=\lim_{\varepsilon\rightarrow 0^{+}} \frac{V(K+_{-p}\varepsilon\star L)-V(K)}{\varepsilon}. $$
The definition above and de l'Hospital's rule yield the following integral representation of \(L_{p}\)-dual mixed volume (see [5]):
$$ \widetilde{V}_{-p}(K, L)=\frac{1}{n}\int_{S^{n-1}} \rho_{K}^{n+p}(u)\rho _{L}^{-p}(u)\,du, $$
where the integration is with respect to spherical Lebesgue measure on \(S^{n-1}\).
From (2.6), it follows immediately that, for each \(K\in S_{o}^{n}\) and \(p\geq1\),
$$ \widetilde{V}_{-p}(K, K)=V(K)=\frac{1}{n}\int _{S^{n-1}}\rho _{K}^{n}(u)\,du. $$
Minkowski's inequality for a \(L_{p}\)-dual mixed volume can be stated as follows (see [5]).
Theorem 2.A
If \({K, L}\in S_{o}^{n}\), \(p \geq1\), then
$$ \widetilde{V}_{-p}(K, L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac{p}{n}}, $$
with equality if and only if K and L are dilates.
General \(L_{p}\)-harmonic Blaschke bodies
For \(K\in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), the general \(L_{p}\)-harmonic Blaschke body, \(\widehat{{\nabla}}_{p}^{\tau}K\), of K is defined by (see [36])
$$ \frac{\rho(\widehat{\nabla}_{p}^{\tau}K, \cdot)^{n+p}}{V(\widehat{\nabla }_{p}^{\tau}K)}=f_{1}(\tau)\frac{\rho(K, \cdot)^{n+p}}{V(K)}+f_{2}( \tau)\frac {\rho(-K, \cdot)^{n+p}}{V(-K)}. $$
Operators of this type and related maps compatible with linear transformations appear essentially in the theory of valuations in connection with isoperimetric and analytic inequalities (see [37–43]).
Theorem 2.B
If \(K \in S_{o}^{n}\), \(p\geq1\), and \(\tau\in(-1, 1)\), then
$$ \widetilde{G}_{-p}\bigl(\widehat{\nabla}_{p}^{\tau}K\bigr)\geq\widetilde {G}_{-p}(K), $$
with equality if and only if K is origin-symmetric.
Proofs of main results
In this section, we complete the proofs of Theorems 1.1-1.3. The proof of Theorem 1.1 requires the following lemma.
If \(K, L \in S_{o}^{n}\) and \(p\geq1\), if \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), then for any \(Q\in S_{c}^{n}\)
$$ \frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(K)}. $$
Since \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), it easily follows that for any \(u\in S^{n-1}\)
$$h^{p}_{\Gamma_{p}^{+} K}(u)+h^{p}_{\Gamma_{p}^{-} K}(u)=h^{p}_{\Gamma_{p}^{+} L}(u)+h^{p}_{\Gamma_{p}^{-} L}(u). $$
Together (1.5) with (1.6), we get
$$\int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p} \biggl[ \frac{\rho _{K}^{n+p}(v)}{V(K)}+\frac{\rho_{-K}^{n+p}(v)}{V(-K)} -\frac{\rho_{L}^{n+p}(v)}{V(L)}-\frac{\rho_{-L}^{n+p}(v)}{V(-L)} \biggr]\,dv=0. $$
$$\mu(v)=\frac{\rho_{K}^{n+p}(v)}{V(K)}+\frac{\rho_{-K}^{n+p}(v)}{V(-K)} -\frac{\rho_{L}^{n+p}(v)}{V(L)}- \frac{\rho_{-L}^{n+p}(v)}{V(-L)}, $$
then have
$$ \int_{S^{n-1}}\varphi_{+}(u\cdot v)^{p} \mu(v)\,dv=0. $$
Notice that \(\rho_{-K}(v)=\rho_{K}(-v)\) for all \(v\in S^{n-1}\), thus we know that \(\mu(v)\) is a finite even Borel measure. Together with (3.2), then \(\mu(v)=0\), i.e.,
$$\frac{\rho_{K}^{n+p}(v)}{V(K)}+\frac{\rho_{K}^{n+p}(-v)}{V(-K)}= \frac{\rho_{L}^{n+p}(v)}{V(L)}+\frac{\rho_{L}^{n+p}(-v)}{V(L)}. $$
For any \(Q\in S_{c}^{n}\), then use \(\rho_{Q}(v)=\rho_{-Q}(v)=\rho_{Q}(-v)\) to get
$$\frac{\rho_{K}^{n+p}(v)\rho_{Q}^{-p}(v)}{V(K)}+\frac{\rho_{K}^{n+p} (-v)\rho_{Q}^{-p}(-v)}{V(K)}= \frac{\rho_{L}^{n+p}(v)\rho_{Q}^{-p}(v)}{V(L)}+\frac{\rho_{L}^{n+p} (-v)\rho_{Q}^{-p}(-v)}{V(L)}. $$
From (2.6), this yields for any \(Q\in S_{c}^{n}\)
$$\frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(L)}. $$
Proof of Theorem 1.1
Together with definition (1.1), we know
$$ \frac{\omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)}{V(K)}= \inf \biggl\{ n\frac{\widetilde{V}_{-p}(K,Q)}{V(K)}V\bigl(Q^{\ast}\bigr)^{-\frac{p}{n}}: Q\in\mathcal{K}_{c}^{n} \biggr\} . $$
Since \(\Gamma_{p}^{+} K=\Gamma_{p}^{+} L\) and \(\Gamma_{p}^{-} K=\Gamma_{p}^{-} L\), from (3.1), we get, for any \(Q\in\mathcal{K}_{c}^{n}\),
$$ \frac{\widetilde{V}_{-p}(K, Q)}{V(K)}=\frac{\widetilde{V}_{-p}(L, Q)}{V(L)}. $$
Hence, from (3.3) and (3.4), we can get
$$\frac{\widetilde{G}_{-p}(K)}{V(K)}=\frac{\widetilde{G}_{-p}(L)}{V(L)}, $$
i.e.,
$$ \frac{\widetilde{G}_{-p}(K)}{\widetilde{G}_{-p}(L)}=\frac {V(K)}{V(L)}. $$
Taking \(Q=L\) in (3.4) and associating this with (2.8), since \(L\in\mathcal{K}_{c}^{n}\), we obtain
$$V(K)=\widetilde{V}_{-p}(K, L)\geq V(K)^{\frac{n+p}{n}}V(L)^{-\frac{p}{n}}, $$
$$ V(K)\leq V(L). $$
Combining (3.5) with (3.6), we get (1.9).
According to the equality condition of (3.6), we see that equality holds in (1.9) if and only if \(K=L\). □
If \(K \in S_{o}^{n}\), \(p\geq1\), \(\tau \in(-1, 1)\), then
$$ \Gamma_{p}^{+}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}^{\tau}K $$
$$ \Gamma_{p}^{-}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}^{-\tau} K. $$
Since L is not origin-symmetric and \(\tau \in(-1, 1)\), it follows from Theorem 2.B that \(\widetilde {G}_{-p}(\widehat{\nabla}_{p}^{\tau}L)> \widetilde{G}_{-p}(L)\). Choose \(\varepsilon>0\), such that \(K=(1-\varepsilon)\widehat{\nabla}_{p}^{\tau}L\) satisfies
$$\widetilde{G}_{-p}(K)=\widetilde{G}_{-p}\bigl((1- \varepsilon)\widehat{\nabla }_{p}^{\tau}L\bigr)> \widetilde{G}_{-p}(L). $$
By (3.7) and (3.8), we, respectively, have
$$\Gamma_{p}^{+} K=\Gamma_{p}^{+}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p}^{+} \bigl( \widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p}^{\tau}L \subset\Gamma_{p}^{\tau}L $$
$$\Gamma_{p}^{-} K=\Gamma_{p}^{-}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p}^{-} \bigl(\widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p}^{-\tau} L \subset\Gamma_{p}^{-\tau} L. $$
If \(K \in S_{o}^{n}\), \(p\geq1\), and \(\tau\in[-1, 1]\), then
$$ \Gamma_{p}\bigl(\widehat{{\nabla}}^{\tau}_{p}K \bigr)=\Gamma_{p}K. $$
Since L is not origin-symmetric, Theorem 2.B has \(\widetilde{G}_{-p}(\widehat{\nabla}_{p}^{\tau}L)> \widetilde {G}_{-p}(L)\) for \(\tau\in(-1, 1)\). Take \(\varepsilon>0\), and let \(K=(1-\varepsilon)\widehat{\nabla}_{p}^{\tau}L\) such that
It follows from (3.9) that
$$\Gamma_{p} K=\Gamma_{p}\bigl[(1-\varepsilon)\widehat{ \nabla}_{p}^{\tau}L\bigr]=(1-\varepsilon)\Gamma_{p} \bigl(\widehat{\nabla}_{p}^{\tau}L\bigr)=(1-\varepsilon ) \Gamma_{p} L \subset\Gamma_{p} L. $$
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This work was partly supported by the National Natural Science Foundation of China (Grant No. 11561020 and No. 11371224) and the Young Foundation of Hexi University (Grant No. QN2014-12). The referees of this paper proposed many very valuable comments and suggestions to improve the accuracy and readability of the original manuscript. We would like to express our most sincere thanks to the anonymous referees.
School of Mathematics and Statistics, Hexi University, Zhangye, 734000, China
Jinsheng Guo & Yibin Feng
Jinsheng Guo
Yibin Feng
Correspondence to Yibin Feng.
All authors completed the writing of this paper and read and approved the final manuscript.
Guo, J., Feng, Y. \(L_{p}\)-Dual geominimal surface area and general \(L_{p}\)-centroid bodies. J Inequal Appl 2015, 358 (2015). https://doi.org/10.1186/s13660-015-0888-9
\(L_{p}\)-dual geominimal surface area
general \(L_{p}\)-centroid bodies
\(L_{p}\)-centroid bodies
Shephard type problems
Recent Advances in Inequalities and Applications | CommonCrawl |
Detection of high frequency of MAD20 allelic variants of Plasmodium falciparum merozoite surface protein 1 gene from Adama and its surroundings, Oromia, Ethiopia
Temesgen File ORCID: orcid.org/0000-0003-2126-56171,
Tsegaye Chekol2,
Gezahegn Solomon2,
Hunduma Dinka1 &
Lemu Golassa2
Malaria Journal volume 20, Article number: 385 (2021) Cite this article
One of the major challenges in developing an effective vaccine against asexual stages of Plasmodium falciparum is genetic polymorphism within parasite population. Understanding the genetic polymorphism like block 2 region of merozoite surface protein-1 (msp-1) gene of P. falciparum enlighten mechanisms underlining disease pathology, identification of the parasite clone profile from the isolates, transmission intensity and potential deficiencies of the ongoing malaria control and elimination efforts in the locality. Detailed understanding of local genetic polymorphism is an input to pave the way for better management, control and elimination of malaria. The aim of this study was to detect the most frequent allelic variant of the msp-1 gene of P. falciparum clinical isolates from selected health facilities in Adama town and its surroundings, Oromia, Ethiopia.
One hundred thirty-nine clinical isolates were successfully amplified for msp-1 gene using specific primers. Nested PCR amplification was conducted targeting K1, MAD20, and R033 alleles followed by gel electrophoresis for fragment analysis. Based on the detection of a PCR fragment, infections were classified as monoclonal or multiple infections.
19 different size polymorphism of msp-1 gene were identified in the study, with 67(48%) MAD20, 18 (13%) K-1 and 18 (13%) RO33 allelic family. Whereas, the multiple infections were 21(15%), 8 (5.8%), 4(2.9%), 3(2.2%) for MAD20 + K-1, MAD20 + RO33, K-1 + RO33, and MAD20 + K-1, RO33, respectively. The overall Multiplicity of infection (MOI) was 1.3 and the expected heterozygosity (He) was 0.39 indicating slightly low falciparum malaria transmission.
The status of msp-1 allele size polymorphism, MOI and He observed in the study revealed the presence of slightly low genetic diversity of P. falciparum clinical isolates. However, highly frequent MAD20 allelic variant was detected from clinical isolates in the study area. Moreover, the driving force that led to high predominance of MAD20 allelic variant revealed in such malaria declining region demands further research.
Despite an enormous effort to control and eventually eliminate malaria, studies reveal that it is still a major public health problem, especially in sub- Saharan Africa (SSA) where more than 90% of the disease burden prevails [1, 2]. About 68% of Ethiopian population inhabits in 75% of the countries land mass that is malarious, where Plasmodium falciparum and Plasmodium vivax accounts for 70% and 30%, respectively [3]. Studies revealed that, multiple factors greatly affected malaria control and elimination efforts. From which the frequent emergence and spread of genetic diversity of P. falciparum is prominent. High genetic diversity is not only an indicator of its evolutionary success [4] but also, the intensity of transmission [5] that pose potential challenges in malaria control programmes [6].
Molecular characterization of P. falciparum enables us to investigate the genetic diversity of infection with consideration of various factors, such as disease phenotype, age and host immunity [7]. Genetic diversity of P. falciparum is usually determined through genotyping of the polymorphic regions block 2 of msp-1 [6, 8, 9]. MSP1 is involved in erythrocyte invasion and is one of the major P. falciparum blood-stage malaria vaccine targets [10,11,12]. MSP1 is a 190 KDa surface protein encoded by the msp1 gene located on chromosome 9 and contains 17 blocks of sequences flanked by conserved regions [9, 13, 14]. The precise functional role of msp1 during invasion has not been fully evaluated, and its macromolecular characterization is incomplete [15].
msp-1 markers are useful to investigate genetic diversity, multiplicity of infection (MOI) and parasite carriage. Polymorphism in msp1 and msp2 have been frequently reported from different parts of the world. Of the 17 blocks of msp1, block 2 is the most polymorphic region characterized into three allelic families (K1, MDA20 and R033). Based on the variation in length and sequence diversity, this region is a commonly targeted part in determining genetic diversity and MOI in clinical isolates of P. falciparum.
Even though genetic diversity of P. falciparum has been extensively studied in different parts of the world, limited data are available from Ethiopia. The aim of this study was, to assess genetic diversity of block 2 region of msp-1 gene of P. falciparum clinical isolates from three districts in central Ethiopia.
Health facility based cross-sectional study conducted at Adama, Modjo, Wonji, Awash Malkasa and Olanciti towns from September 2019 to August 2020. These sample collection sites include Adama city administration, Adama district which includes Wonji and Awash Malkasa, Modjo town capital of Lume district, Olanciti the capital of Boset district. The location of these study sites is as shown in (Fig. 1). Patient data and sample collection, was performed from purposively selected health facilities at each site depending on their patient caseload, physical location and the availability of qualified and experienced medical laboratory technologist previously participated in similar research work. Adama is the major town next to the main capital in central Ethiopia. It is found at about 99 km southeast of Addis Ababa. The location of the other sites are Modjo at 16 km northwest, Wonji at 8 km south, Awash Malkasa at 15 km southeast and Olanciti at about 23 km northeast of Adama.
Map of the study area, showing sample collection sites (Adama, Modjo, Wonji, Awash Malkasa, and Olanciti)
The study sites are located in the Rift Valley areas having rain fall pattern that is heavy from mid-June to mid-September followed by major malaria transmission season and shorter rainy period in March accounting for minor malaria transmission [16]. The catchment population of the study site estimated to reach 800,000 inhabitants.
Adama and its surrounding area is a well-known malaria endemic region in central Ethiopia. The major factors that account for such malaria endemicity are; its physical location in Rift Valley area of Ethiopia, latitudinal location which varies from1436 to 1850 m above sea level, rainfall patterns, average annual temperature that varies from 16–32 °C that is favourable for the breeding of Anopheles arabiensis (the predominant malaria vector in the region), and various micro ecological factors that favour mosquito breeding [17].
Clinical data, blood sample collection and processing
One hundred thirty-nine microscopically-confirmed P. falciparum infected patients were included in the study. The age of study participants were ranged from 1 to 66. The inclusion criteria for the study were uncomplicated malaria patient with the history of fever onset since 24 h of the clinical examination. Blood samples were collected through finger prick for dry blood spot (DBS) preparation was held from September 2019 to August 2020 by trained medical laboratory technologists from the catchment area of those selected health facilities in the study area. After consent of the patients or guardians, the spotted blood on Whatman TM 3MM filter paper was allowed to air dry in dust free area. The DBS were placed in a zip-lock bag with silica gel temporarily stored at 0–4 °C to prevent DNA degradation. For longer time storage of DBS was kept in deep freeze (− 20 °C) at Adama regional research laboratory.
Microscopy and parasite count
Microscopic examination was conducted for both thick and thin blood smear following the national malaria microscopy protocol. All slides examined by two independent laboratory technologists to determine species identification for P. falciparum and its parasite density. In case of discordance, a third laboratory technologist read the slides. In addition, parasite density was estimated by counting and recording the asexual stage of the parasite per 200 white blood cells (WBC) in thick film. Moreover, when the sexual form (gametocytes) have been seen, the slides were excluded from the count. The parasite density of the asexual stage was estimated by counting the number of WBC by field examined assuming that 8000 WBC were present in 1 µl of blood. Thus, parasite density per microlitre (μl) of blood was calculated by using the following formula [7, 18, 19]:
$${\text{Parasite density}}/{\upmu} {\text{l}}=\frac{\begin{array}{c} \\ {\text{Number of asexual parasite per}}200{\text{WBC}} \\ \times\, {\text{absolute or assumed WBC}}/{\upmu} {\text{l}} \end{array}}{200}$$
For comparison with ranked order variables, parasitaemia were categorized in to five levels: L1 (< 50 parasite/µl blood), L2 (50–499 parasite /µl blood), L3 (500- 4999 parasite /µl blood), L4 (5000–49,999 parasite/µl blood), and L5 (≥ 50,000 parasite/µl blood) [20].
Extraction of the parasite DNA
Extraction of the parasite genomic DNA and genotyping the polymorphic region of msp-1 were conducted at malaria research laboratory, Akililu Lemma Pathobiology Institute, Addis Ababa University (AAU). Genomic DNA of P. falciparum extracted from approximately 200 µl of frozen blood samples spotted on Whatman 3 filter paper for nested PCR amplification. 0.5% Tween® 20 (Sigma-Aldrich, USA) was used to lyse RBC; tracked by treatment with 6% chelex ® 100 (Sigma-Aldrich, USA) and heat treatment in water bath at 96 °C following the optimized standard operating procedure (SOP) to free the parasites DNA [9].
PCR amplification for genotyping of msp-1 gene and gel electrophoresis
Nested PCR amplification targeting the unique sequence of 18 srRNA gene was held by using specific primer pairs for molecular detection of P. falciparum from the isolates [21]. In the present study, the polymorphic region of the confirmed P. falciparum msp-1 gene (block 2) was used as a genetic marker for the genotyping of parasite populations. The primers and PCR conditions used during this study were slightly modified from the previously described work [9, 22] (Additional file 1). Briefly, all reactions carried out in a final volume of 20 µl. In the first round (N1) reaction containing 0.5 µl of each primer, 5 × FIREPol®.Master Mix (MM), 11 µl of nuclease free water aliquot to 16 µl to which 4 µl of DNA template was added. In nested (N2) reaction, 2 µl of the amplicon product was used. The PCR amplification profile for both N1 and N2 reactions includes; initial denaturation at 95 °C at 3 min, denaturation at 94 °C for 1 min, annealing 58 °C for 1 min, elongation 72 °C for 2 min and final elongation at 5 min [9, 22]. The PCR reaction mixture incubated in a thermal cycler (VWR) Schmidt, Germany. To monitor the quality the protocol allele specific positive control 3D7 and DNA free negative control were included in each reaction. Separation of the PCR product was performed on 2% agarose gel electrophoresis stained with ethidium bromide [9]. Stained agarose gels visualized under Benchtop 2UV trans-illuminator (UVP) USA and photographed to estimate band size in relation to 50 bp DNA ladder (Invitrogen, by thermal Fisher-scientific) (Additional file 2). Infections considered as monoclonal when a single PCR fragment was detected on each locus and polyclonal when more than one fragment identified on a locus. Polymorphism in each allele family was analysed by assuming that one band represented one amplified DNA fragment derived from a single copy of P. falciparum msp-1. Multiplicity of infection (MOI) was defined as the average number of detected P. falciparum genotypes per infected patient. Allele for each family were considered the same when the fragment size is less than 20 bp [23].
The data were analysed, after entering and processing it by using Statistical Package for Social Sciences (SPSS version 20). To examine endemicity or potential importation of the msp-1 allelic variants, confirmed malaria patients with P. falciparum were categorized in those having travel history to other places where malaria is endemic and those not having travel history in the preceding 30 days. MOI for P. falciparum was calculated as a total number of parasite genotypes for the same gene and the number of PCR positive isolates. Descriptive analysis performed to compare the distribution of different allele families in relation to patient data.
To test the correlation of two variables, Pearson correlation test was used. Pearson Chi square test was also conducted for statistical comparison of categorical variables. P < 0.05 was used to test the level of statistical significance to accept or reject the hypothesis.
The expected heterogeneity \((He)\) was calculated by the formula;
$$He=\left(\frac{n}{n-1}\right)\left(1-\sum {p}^{2}\right),$$
where "n" stands for the number of the isolates analyzed and "p" represents the frequency of each different allele at a locus.
Ethical approval of the study was obtained from Institutional Ethical Review Board of ASTU, certificate reference number RECSoANS/BIO/01/2019 and approval of Oromia Regional State Health Bureau. In addition, written informed consent obtained from parents or guardian prior to recruitment.
Socio-demographic and parasitological data
One hundred thirty-nine samples from P. falciparum patients were successfully analysed for msp-1 allelic diversity. A total of 68.3% of the study participant were males. The age of the study participants ranged from 1 to 66. Mean ± SD (27.0 ± 13.6*) years. Asexual parasite density ranged from 64 – 104,320 parasites/µl with a geometric mean of 5,654 parasites/µl. Of all study participants 83 (60%) were from urban inhabitants, and only 15 (11%) were having recent travel history to other malarious area. Of all the study subjects, by occupation 74% P. falciparum malaria cases were detected from students, daily labourers and farmers (Table 1).
Table 1 Socio-demographic characteristics and parasitological data of the study population at Adama and its surroundings (n = 139)
Geometric mean of the parasite density across different age groups
Analysis of the geometric mean of P. falciparum parasite density across patients of different age groups has shown that school aged children (5–14 years) carry disproportionate burden of the infection (Fig. 2). However, the correlation between parasite density with patient's age is not statistically significant (Pearson's correlation = 0.12, P = 0.6).
Relationship between geometric mean of the parasite density of P. falciparum patients with age groups in Adama and its surroundings (n = 139)
Allele typing and diversity profile across different age groups
From all the age groups, 74% of the isolates had monoclonal infections (Fig. 3A). The prevalence of multiple infections slightly increases with age group (Fig. 3B). However, no significant correlation exists between parasite density and multiple infections (Pearson's correlation = − 0.07, X2 = 0.6) and age of the patient with parasite density (Pearson's correlation = 0.12, X2 = 0.6) (Fig. 2). Similarly, there was no significant variation in msp-1 allelic families with age (X2 = 0.5), sex (X2 = 0.56), residence (X2 = 0.2), travel history (X2 = 0.9), educational level (X2 = 0.8) and occupation (X2 = 0.5) (Table 1).
The frequency of monoclonal (A) and polyclonal (B) allele typing of msp-1 gene across different age groups of malaria patients due to P. falciparum in Adama and its surroundings (n = 139)
Allelic Polymorphism of block 2 region of msp-1 gene and their level of severity, spatial and seasonal features
From the total of 139 successfully genotyped samples by nested PCR; the frequency of msp-1 allelic families detected in monoclonal isolates were 48%, 13%, 13% for MAD20, K1 & RO33, respectively, and the remaining 24% were diclonal (MAD20 + K-1, MAD20 + RO33, K-1 + RO33) and 2% triclonal (MAD20 + K1 + RO33) infections. From all P. falciparum msp-1 gene amplified by nested PCR for block 2 region, 19 different alleles were identified of which 8 alleles were MAD20 (160–280 bp), 6 alleles were K-1(100–270 bp), and 5 alleles of RO33 type (100–200 bp). The overall MOI was 1.3, with the expected heterozygosity of 0.39 (Table 2).
Table 2 Genetic diversity and genotype multiplicity of P. falciparum clinical isolates from Symptomatic uncomplicated malaria patients in Adama and its surroundings (n = 139)
Of the total multiclonal infections 29 (80%) were detected during the major malaria season (September to December) and the rest were from the isolates of minor malaria season in the region. No statistically significant variation in the seasonal distribution of polyclonal infection (X2 = 0.8) in the study area. Moreover, 33 (92%) patients with polyclonal infection were having no travel history to other malaria endemic places. Thus, there was no statistically significant variation in the distribution of allelic variants in relation to patient's travel history in the study area (X2 = 0.9) (Table 3).
Table 3 The relationship between polyclonal infections, its seasonality and travel history of malaria patient due to P. falciparum in Adama and its surroundings (n = 139)
In this study, of all P. falciparum isolates; 83 (60%) were from the urban locality, and the rest were from rural area (Table 4). Allelic variants of msp-1 did not show significant variation between urban and rural areas; and seasonal variations were not statistically significant (X2 = 0.23) and (X2 = 0.57), respectively.
Table 4 Rural, urban and seasonal variations in the distribution of P. falciparum msp-1 block 2 region allelic variants in Adama and its surroundings (n = 139)
Analysis of the spatial feature of msp-1 allelic variants and MOI from the study sites has shown that 65 (47%), 18 (13%), 17 (12%), 18 (13%), 21(15%) isolates were from Adama, Modjo, Wonji, Malkasa, and Olanciti sites, respectively. The spatial variation of the distribution of msp-1 allelic variant across sample collection sites was significantly related (P = 0.000) (Fig. 3) showing heterogeneity in their distribution (Fig. 4).
Distribution P. falciparum msp-1 gene allelic families isolated in Adama and its surroundings (n = 139)
In Ethiopia, even though considerable efforts have been made at national and local levels to control and eventually eliminate malaria, limited molecular data exists on genetic polymorphism of P. falciparum, the most predominant and virulent malaria parasite in the region. The present study aimed to assess the genetic polymorphism of P. falciparum clinical isolates based on block 2 region msp-1 genotypes and multiplicity of infection. This is the first study that widely investigated the status of P. falciparum genetic diversity from three districts of the study areas in central Ethiopia. Moreover, the study examined the spatial and seasonality of such polymorphism in relation to parasite density and other patient characteristics.
The study revealed that; geometric mean of parasite density was disproportionately high in school age children (SAC) and relatively stable afterwards (Fig. 2). In addition, there was no statistically significant correlation existed between parasite density and age of the patients (Pearson's correlation = 0.12, P = 0.6). Even though a number of factors may contribute to the fluctuation of parasitaemia level overtime in symptomatic patients, the geometric mean of microscopically detectable parasitaemia levels could be used to explain the finding of this study [24]. The major factor that mainly contributed for higher parasitaemia level in SAC is delayed acquisition of protective immunity during this immunological transition age making this age group more vulnerable to malaria infection than adults [25].
In the present study, multiple infections slightly increased with age group (Fig. 2B), although the variation was not statistically significant (X2 = 0.5). This finding is in congruent with the report from Burkina Faso [26] and Tanzania [27], where they explained that episodes of infection in children is commonly for very short duration and the duration of episodes of infection increases with age contributing to the multiple infections. Other reports suggested that multiple infections vary with parasite density, immunity status, the overall prevalence of infection in the population and transmission intensity as reviewed by [28,29,30]. Other studies have shown an inverse association. Therefore, the relationship between malaria patient age, level of parasitaemia, number of clones of infection, transmission intensity and status of immunity to malaria parasite needs further investigation.
In the present study, there was no significant correlation existed between multiple clone infections of P. falciparum with seasonal variation of malaria incidence and travel history of patients (Table 3). In favour of this finding, report from southwestern Ethiopia [31], has shown the absence of correlation or negative correlation between the proportion of multi-clonal infections and parasite prevalence. On the other hand, reports from Indonesia [32], and Papua New Guinea [33], show the presence of positive correlation between the rate of polyclonal infections and annual parasite incidence. The predominance of polyclonarity (92%) in those patients having no travel history depicts real features of malaria epidemiology with respect to the genetic marker of msp-1 gene in the study area.
In this study, 26% of the isolates having multiple genotype infections. The overall MOI of 1.3 and the expected heterozygosity of 0.39 (Table 2). This finding differs from north western Ethiopia and southwestern Ethiopia reported by Mohammed et al. [23] and Abamecha et al. [34] with 75% and 80% frequency of multi-clonal infections, and 1.8 MOI with He (0.79), 2.0 MOI and He (0.43), respectively. This shows that malaria transmission in the study under report exhibits slightly low genetic diversity, compared with northwestern and southwestern Ethiopia. This could be due the locational advantage of central Ethiopia to better health services, differences in local epidemiology, demographic and environmental conditions that might have resulted in observed reduced genetic diversity pattern in Adama and its surroundings. In the present study, from 139 samples 19 different length polymorphism of msp-1 allelic variant was revealed; 8 MAD20 (160–330 bp), 6 K-1 (100–270) bp, and 5 RO33 (100–220 bp). This shows the level of size polymorphism of msp-1 alleles in the study area. However, the number of alleles identified may have been under estimated due to a number of limitations like sensitivity of PCR technique used, inability to differentiate minor fragments, the possible existence of similar size fragments and the same size fragment having different amino acid motifs [34, 35].
Size polymorphism of msp-1 allelic variant identified in the present study is slightly higher than the report from Chewaka district of southwestern Ethiopia [34] and Humera of north-western Ethiopia [6]. This was less diverse than Kolla Shele district of south western part of Ethiopia [23], but more or less similar to reports from Equatorial Guinea [22], and Bobo-Dioulasso in Burkina Faso [36]. The major factor that may account for such variation could be the scope of study sites covered and local malaria transmission patterns might have contributed. Gel- analysis of the present study revealed that 103 out of 139 msp-1 amplicon (74%) were monoclonal infections, whereas the remaining 36 (26%) was poly-allelic type, with 15% for (MAD20 + K-1), 5.7% for (MAD20 + RO33), 2.8% for (K-1 + RO33), and 2.1% were MAD20 + K-1 + RO33 type. The proportion of monoclonal infection was 48% MAD20, 13% K-1 and 13% RO33 (Table 2). This finding differ from the report from southwestern Ethiopia [23, 34], where they reported that K-1 was the most prevalent allelic family. Similarly, report from Cameroon, Gambia, Nigeria and Gabon has shown that MAD20 allelic variant was the least predominant [37, 38]. On the other hand, in agreement with the present study report from northwestern part of Ethiopia [6], Sudan by [7] and Equatorial Guinea [22] of the three msp-1 gene allelic families MAD20 was the predominant allelic type. Although the deriving forces for such variation needs further investigation; the difference in micro-ecological factors and the local transmission intensity [39, 40], could play a significant role. Moreover, evolutionary process like genetic drift resulting uneven reproduction of the parasite lineages, types and rate of mutations, inbreeding and the contribution of allelic variants in reproductive success are some of the factors that might have contributed for such variation [41]. In addition, in the present study when the spatial feature of the distribution of msp-1 gene allelic variant in urban and rural areas (Table 4) was examined, no statistically significant (P = 0.2) variation was revealed. This finding could be taken as an evidence to show similar malaria epidemiology and the possible crossbreeding of the parasite populations between urban and rural settings in the study area, demanding similar intervention endeavours. Similarly in the present study, no statistically significant variation of multi-clonal infection of msp-1 gene with parasite density (P = 0.6), and seasonality of transmission (P = 0.8). This could be due to the characteristic feature of low transmission settings in such malaria endemic regions [42, 43]. On the other hand, study sites based distribution of allelic variants has shown a highly significant variation (P = 0.000), (Fig. 4). This could be due to the difference in local micro-ecology of the areas, intensity of local transmission pattern, and differences in the age of the study population [36, 44] and the relative potential differences and challenges on the ongoing malaria control and elimination endeavours in those sites.
This study is the first attempt to analyse the most polymorphic gene (msp-1) of P. falciparum population in the study area. However, further characterization of this gene needs to be designed by increasing the sample size, use of the most powerful techniques, such as microsatellite DNA sequencing and capillary electrophoresis that would provide strong molecular evidence for malaria parasite genetic profile.
The study revealed that slightly low genetic diversity of P. falciparum clinical isolates found in the study area. Moreover, high frequency of MAD20 allelic variant form was detected. The driving force for such selective advantage for this allele under declining malaria prevalence in our study area demand further investigation. Thus, this information will serve as a baseline molecular evidence for further research on areas having similar malaria epidemiology for targeted interventions to make the control and elimination efforts more effective (Additional files 1, 2).
All relevant data is included in manuscript, and the datasets analyzed in the study are available from the corresponding author on reasonable request. Additional data uploaded with main document.
Bp:
Base pair
msp-1:
Merozoite surface protein-1
MOI:
Multiplicity of infection
PCR:
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We thank data collectors from all health facilities in the study area. We are grateful to all study participants, parents and guardian for the children who participated in the study. We strongly acknowledge Oromia Health Bureau for their consent and support in data collection from the selected health facilities in the study area. We also thank Mr. Ahimed Tola (Geomatics Engineer) for his kind assistance in sketching map of study areas by using Arc-GIS Desktop version 10.4.
No funding was obtained for this study.
Department of Applied Biology, Adama Science and Technology University, P.O.Box 1888, Adama, Ethiopia
Temesgen File & Hunduma Dinka
Aklilu Lemma Institute of Pathobiology, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia
Tsegaye Chekol, Gezahegn Solomon & Lemu Golassa
Temesgen File
Tsegaye Chekol
Gezahegn Solomon
Hunduma Dinka
Lemu Golassa
TF designed, conducted and analysed the study, drafted and wrote the manuscript. TC assisted the molecular laboratory work. GS organized the molecular laboratory work. HD designed the study and finally reviewed and approved the manuscript. LG conceived the idea, designed the study, supervised the molecular laboratory work, and reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Temesgen File.
The research and ethical committee of Adama Science and Technology University reviewed and approved the study protocol, as verified through certificate reference number RECSoANS/BIO/01/2019. Oromia Health Bureau also approved the study protocol.
The authors declare that they have no competing of interest.
Primer design.
msp-1 allelic fragment size using 50 bp ladder identified by gel electrophoresis.
File, T., Chekol, T., Solomon, G. et al. Detection of high frequency of MAD20 allelic variants of Plasmodium falciparum merozoite surface protein 1 gene from Adama and its surroundings, Oromia, Ethiopia. Malar J 20, 385 (2021). https://doi.org/10.1186/s12936-021-03914-9
Genetic polymorphism
Msp-1
P. falciparum | CommonCrawl |
\begin{document}
\title{ Comments on: Asymptotic Bound for Heat-Bath Algorithmic Cooling }
\author{Nayeli Azucena Rodr\'iÂguez-Briones} \email{[email protected]} \affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada} \affiliation{Department of Physics \& Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada}
\author{Jun Li}
\affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China}
\author{Xinhua Peng} \affiliation{Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China} \affiliation{Synergetic Innovation Center of Quantum Information \& Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China}
\author{Tal Mor} \affiliation{Computer Science Department, Technion, Haifa 320008, Israel}
\author{Yossi Weinstein} \affiliation{Computer Science Department, Technion, Haifa 320008, Israel}
\author{Raymond Laflamme} \email{[email protected]} \affiliation{Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada} \affiliation{Department of Physics \& Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada} \affiliation{Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada} \affiliation{Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada}
\date{\today}
\pacs{03.67.Pp} \maketitle
In a recent paper \cite{Raesi:2014kx}, Raeisi and Mosca gave a limit for cooling with Heat-Bath Algorithmic Cooling (HBAC). Here we show how to exceed that limit by having correlation in the qubits-bath interaction.
Schulman, Mor and Weinstein (SMW)~\cite{schulman2005physical,schulman2007physical} were the first to (wrongly) claim that the Partner Pairing Algorithm (the PPA, which consists of iterations of SORT + free-relaxation) will achieve the optimal physical cooling of HBAC. SMW provided bounds on the cooling levels that PPA could obtain, but the bounds were not tight, hence it was not possible to observe that PPA is not optimal. SMW assumed that the system-bath interaction proceeds through a swap of qubits, but it turns out that more general form of couplings to the bath exist.
For two qubits (one target qubit, which is going to be cooled, and one reset qubit), starting in the maximally mixed state, the PPA gives a steady state with the qubits at the temperature of the bath and no polarization gain (above the one from the bath) is observed. The first refresh step polarizes the reset qubit, then the first PPA entropy compression transfers that polarization to the target qubit. The reset qubit is then in a fully mixed state and can be repolarised by a thermal contact with the heat bath. It turns out that, after these steps, the diagonal terms are already ordered with non-increasing probability (SORT), such that PPA does not increase the polarization anymore. We thus end up with two qubits in a thermal state, the same as the bath.
In a recent paper\cite{Li:2014ys}, Jun Li and collaborators studied the efficiency of Liouville space polarization transfer in the presence of a bath, and showed cases/experiments where utilizing relaxation effects does offer and enhancement. In looking at the maximum polarization (or purity), they (re)-discovered that it is possible to enhance the polarization of one of two qubits beyond the bath polarization in presence of relaxation and cross relaxation of the quantum system. This effect was discovered by Overhauser in 1953 \cite{Overhauser:1953fk} and has been observed many times experimentally. It is possible to enhance the polarization of one spin (qubit) at the expense of a second spin (qubit) when their coupling to the bath leads to cross relaxation \cite{Solomon:1955uq}. In the limit of low polarization, the expectation of the Z Pauli operator $S_Z^1$ ($\langle S^1_z \rangle$), obeys the equation (see \cite{Solomon:1955uq}):
\begin{equation} \frac{d \langle S_z^1\rangle}{dt}= -\rho_1 (\langle S_z^1\rangle - \langle S_z^1\rangle_0) -\sigma (\langle S_z^2\rangle - \langle S_z^2\rangle_0), \label{szevol} \end{equation} where $\langle S_z^i\rangle_0$ is the expectation of $S_z^i$ at equilibrium when the other spin is not driven (not rotated), $\rho_1$ is the relaxation parameter for the first spin, and $\sigma$ is the cross relaxation parameter.
It is possible to drive (rotate) the second spin so that on the relevant timescale (related to $\rho_2$ and $\sigma$) the expectation of $\langle S_z^2\rangle\approx 0$. Then the steady state of eq.\eqref{szevol} implies that
\begin{equation} \langle S_z^1\rangle = \langle S_z^1\rangle_0 + \frac{\sigma}{\rho_1} \langle S_z^2\rangle_0. \end{equation} Note that $\langle S_z^1\rangle $ corresponds to the polarization of the first qubit, this gives an enhancement compared to what the PPA gives, as long as $\sigma /{\rho_1}$ is positive (which happens in nature).
One way to understand the process from an algorithmic point of view is to realize that the cross relaxation effectively provides a state relaxation/equilibration (``state reset'') between $|11\rangle $ and $|00\rangle$, without touching the other states, analogous to the qubit reset. This form of reset accompanied by a rotation of the second qubit can however boost the polarization of the first qubit beyond what would be obtained by a qubit reset from the bath as in the PPA.
Thus the PPA, at least for two qubits, gives only a lower bound on maximum polarization achievable for HBAC. It is possible to generalize this idea to enhance the polarization of three qubits beyond the PPA, and the details will be provided in a forthcoming paper.
In conclusion, we have presented a Heat Bath Algorithmic Cooling technique that can have a better polarization enhancement than the one obtained by the PPA. As mentioned in \cite{Briones:2014vn}, the polarization achieved using the PPA should be interpreted as a lower bound on the maximum amount of polarization that can be achieved. Its importance is due to the simplicity of the PPA when the initial state is totally mixed or in an equilibrium thermal state. From that, it is possible to get analytical results that describe both the steady state and its polarization from which we can determine a variety of properties, e.g. how far it is from polarization of one and explicitly show how much resources are needed \cite{Briones:2014vn}. It will be interesting to see if we can generalize the Overhauser effect and what advantages it can give as we increase the number of qubits.
\end{document} | arXiv |
Komornik–Loreti constant
In the mathematical theory of non-standard positional numeral systems, the Komornik–Loreti constant is a mathematical constant that represents the smallest base q for which the number 1 has a unique representation, called its q-development. The constant is named after Vilmos Komornik and Paola Loreti, who defined it in 1998.[1]
Definition
Given a real number q > 1, the series
$x=\sum _{n=0}^{\infty }a_{n}q^{-n}$
is called the q-expansion, or $\beta $-expansion, of the positive real number x if, for all $n\geq 0$, $0\leq a_{n}\leq \lfloor q\rfloor $, where $\lfloor q\rfloor $ is the floor function and $a_{n}$ need not be an integer. Any real number $x$ such that $0\leq x\leq q\lfloor q\rfloor /(q-1)$ has such an expansion, as can be found using the greedy algorithm.
The special case of $x=1$, $a_{0}=0$, and $a_{n}=0$ or 1 is sometimes called a $q$-development. $a_{n}=1$ gives the only 2-development. However, for almost all $1<q<2$, there are an infinite number of different $q$-developments. Even more surprisingly though, there exist exceptional $q\in (1,2)$ for which there exists only a single $q$-development. Furthermore, there is a smallest number $1<q<2$ known as the Komornik–Loreti constant for which there exists a unique $q$-development.[2]
Value
The Komornik–Loreti constant is the value $q$ such that
$1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}$
where $t_{k}$ is the Thue–Morse sequence, i.e., $t_{k}$ is the parity of the number of 1's in the binary representation of $k$. It has approximate value
$q=1.787231650\ldots .\,$[3]
The constant $q$ is also the unique positive real root of
$\prod _{k=0}^{\infty }\left(1-{\frac {1}{q^{2^{k}}}}\right)=\left(1-{\frac {1}{q}}\right)^{-1}-2.$
This constant is transcendental.[4]
See also
• Euler-Mascheroni constant
• Fibonacci word
• Golay–Rudin–Shapiro sequence
• Prouhet–Thue–Morse constant
References
1. Komornik, Vilmos; Loreti, Paola (1998), "Unique developments in non-integer bases", American Mathematical Monthly, 105 (7): 636–639, doi:10.2307/2589246, JSTOR 2589246, MR 1633077
2. Weissman, Eric W. "q-expansion" From Wolfram MathWorld. Retrieved on 2009-10-18.
3. Weissman, Eric W. "Komornik–Loreti Constant." From Wolfram MathWorld. Retrieved on 2010-12-27.
4. Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449, doi:10.2307/2695302, JSTOR 2695302, MR 1763399
| Wikipedia |
Looking for a set implementation with small memory footprint
I am looking for implementation of the set data type. That is, we have to
maintain a dynamic subset $S$ (of size $n$) from the universe $U = \{0, 1, 2, 3, \dots , u – 1\}$ of size $u$ with
operations insert(x) (add an element x to $S$) and find(x) (checks whether element x is a member of $S$).
I don't care about other operations. For orientation, in applications I'm working with we have $u \approx 10^{10}$.
I know of implementations that provide both operations in time $O(1)$, so I worry mostly about the size of data structure. I expect billions of entries but want to avoid swapping as much as possible.
I am willing to sacrifice runtime if necessary. Amortised runtime of $O(\log n)$ is what I can admit; expected runtimes or runtimes in $\omega(\log n)$ are not admissable.
One idea I have is that if $S$ can be represented as a union of ranges [xmin, xmax], then we will be able to save on storage size with the price of some performance decrease. Also, some other data patterns are possible, like [0, 2, 4, 6].
Could you please point me to data structures which can do something like that?
data-structures efficiency space-complexity sets dictionaries
HEKTOHEKTO
$\begingroup$ let us continue this discussion in chat $\endgroup$ – Raphael♦ Jan 29 '14 at 20:22
$\begingroup$ How does the number $n$ of elements enter the picture? I.e., what happens if an element is inserted and there are already $n$? $\endgroup$ – vonbrand Jan 30 '14 at 9:57
$\begingroup$ @vonbrand - the n is size of the set S. It can increase with every insert, or it can remain the same if element x is already in the set. $\endgroup$ – HEKTO Jan 30 '14 at 14:43
$\begingroup$ Can you accept a small probability of false positives? If so, a bloom filter might be ideal: en.wikipedia.org/wiki/Bloom_filter $\endgroup$ – Joe Jan 30 '14 at 21:56
$\begingroup$ @AlekseyYakovlev, the false positive rate of a bloom filter has nothing to do with the universe size (only with the number of hash functions $k$, the size of the data structure $m$, and the number of items $n$), but if $n$ really is close to $u$ (say $u = n\cdot c$ for a small constant $c$), you will be hard pressed to do better than a simple bit vector I think, with only $cn$ total bits of space. $\endgroup$ – Joe Jan 31 '14 at 22:59
Joe's answer is extremely good, and gives you all the important keywords.
You should be aware that succinct data structure research is still in an early stage, and many of the results are largely theoretical. Many of the proposed data structures are quite complex to implement, but most of the complexity is due to the fact that you need to maintain asymptotic complexity both over the universe size and the number of elements stored. If either one of these is relatively constant, then a lot of the complexity goes away.
If the collection is semi-static (that is, inserts are rare, or at least low-volume), then it's certainly worth considering an easy-to-implement static data structure (Sadakane's sdarray is a fine choice) in conjunction with an update cache. Basically, you record updates in a traditional data structure (e.g. B-tree, trie, hash table), and periodically bulk-update the "main" data structure. This is a very popular technique in information retrieval, since inverted indexes have many advantages for searching but are hard to update in-place. If this is the case, please let me know in a comment and I'll amend this answer to give you some pointers.
If inserts are more frequent, then I suggest succinct hashing. The basic idea is straightforward enough to explain here, so I will do so.
So the basic information theoretic result is that if you're storing $n$ elements from a universe of $u$ items, and there is no other information (e.g. no correlation between the elements) then you need $\log {u \choose n} + O(1)$ bits to store it. (All logarithms are base-2 unless otherwise specified.) You need this many bits. There is no way around it.
Now some terminology:
If you have a data structure which can store the data and support your operations in $\log {u \choose n} + O(1)$ bits of space, we call this an implicit data structure.
If you have a data structure which can store the data and support your operations in $\log {u \choose n} + O(\log {u \choose n}) = (1 + O(1)) \log {u \choose n}$ bits of space, we call this a compact data structure. Note that in practice this means that the relative overhead (relative to the theoretical minimum) is within a constant. It could be 5% overhead, or 10% overhead, or 10 times overhead.
If you have a data structure which can store the data and support your operations in $\log {u \choose n} + o(\log {u \choose n}) = (1 + o(1)) \log {u \choose n}$ bits of space, we call this a succinct data structure.
The difference between succinct and compact is the difference between little-oh and big-oh. Ignoring the absolute-value thing for a moment...
$g(n) = O(f(n))$ means that there exists a constant $c$ and a number $n_0$ such that for all $n > n_0$, $g(n) < c \cdot f(n)$.
$g(n) = o(f(n))$ means that for all constants $c$ there exists a number $n_0$ such that for all $n > n_0$, $g(n) < c \cdot f(n)$.
Informally, big-oh and little-oh are both "within a constant factor", but with big-oh the constant is chosen for you (by the algorithm designer, the CPU manufacturer, the laws of physics or whatever), but with little-oh you pick the constant yourself and it can be as small as you like. To put it another way, with succinct data structures, the relative overhead gets arbitrarily small as the size of the problem increases.
Of course, the size of the problem may have to get huge to realise the relative overhead that you want, but you can't have everything.
OK, with that under our belts, let's put some numbers on the problem. Let's supposed that keys are $n$-bit integers (so the universe size is $2^n$), and we want to store $2^m$ of these integers. Let's suppose that we can magically arrange an idealised hash table with full occupancy and no wastage, so that we need exactly $2^m$ hash slots.
A lookup operation would hash the $n$-bit key, mask off $m$ bits to find the hash slots, and then check to see if the value in the table matched the key. So far, so good.
Such a hash table uses $n 2^m$ bits. Can we do better than this?
Suppose that the hash function $h$ is invertible. Then we don't have to store the whole key in each hash slot. The location of the hash slot gives you $m$ bits of the hash value, so if you only stored the $n-m$ remaining bits, you can reconstruct the key from those two pieces of information (the hash slot location and the value stored there). So you would only need $(n - m) 2^m$ bits of storage.
If $2^m$ is small compared with $2^n$, Stirling's approximation and a little arithmetic (proof is an exercise!) reveals that:
$$(n - m) 2^m = \log {2^n \choose 2^m} + o\left(\log {2^n \choose 2^m}\right)$$
So this data structure is succinct.
However, there are two catches.
The first catch is constructing "good" invertible hash functions. Fortunately, this is much easier than it looks; cryptographers make invertible functions all the time, only they call them "cyphers". You could, for example, base a hash function on a Feistel network, which is a straightforward way to construct an invertible hash functions from non-invertible hash functions.
The second catch is that real hash tables are not ideal, thanks to the Birthday paradox. So you'd want to use a more sophisticated type of hash table which gets you closer to full occupancy with no spilling. Cuckoo hashing is perfect for this, as it lets you get arbitrarily close to ideal in theory, and quite close in practice.
Cuckoo hashing does require multiple hash functions, and it requires that values in hash slots be tagged with which hash function was used. So if you use four hash functions, for example, you need to store an additional two bits in each hash slot. This is still succinct as $m$ grows, so it's not a problem in practice, and still beats storing whole keys.
Oh, you might also want to look at van Emde Boas trees.
MORE THOUGHTS
If $n$ is somewhere around $\frac{u}{2}$, then $\log {u \choose n }$ is approximately $u$, so (once again) assuming that there is no further correlation between the values, you basically can't do any better than a bit vector. You will note that the hashing solution above does effectively degenerate to that case (you end up storing one bit per hash slot), but it's cheaper just to use the key as the address rather than using a hash function.
If $n$ is very close to $u$, all of the succinct data structures literature advises you to to invert the sense of the dictionary. Store the values that don't occur in the set. However, now you effectively have to support the delete operation, and to maintain succinct behaviour you also need to be able to shrink the data structure as more elements get "added". Expanding a hash table is a well-understood operation, but contracting it is not.
PseudonymPseudonym
$\begingroup$ Hi, as for the second paragraph of your answer - I expect that every call to insert will be accompanied by a call to find with the same argument. So, if the find returns true, then we just skip the insert. So, frequency of find calls is more then frequency of insert calls, also when n becomes close to u, then insert calls become very rare. $\endgroup$ – HEKTO Feb 5 '14 at 2:59
$\begingroup$ But you expect $u$ to get close to $n$ eventually? $\endgroup$ – Pseudonym Feb 5 '14 at 3:08
$\begingroup$ In the real world n is growing until it reaches u, however we can't predict will it happen or not. The data structure should work well for any n <= u $\endgroup$ – HEKTO Feb 5 '14 at 4:48
$\begingroup$ Right. Then it's fair to say that we don't know of a single data structure which is succinct (in the above sense) and which achieves this over the whole range of $\frac{n}{u}$. I think you're going to want a sparse data structure when $n < u$, then switch to a dense one (e.g. a bit vector) when $n$ is around $\frac{u}{2}$, then a sparse data structure with an inverted sense when $n$ is close to $u$. $\endgroup$ – Pseudonym Feb 5 '14 at 5:27
It sounds like you want a succinct data structure for the dynamic membership problem.
Recall that a succinct data structure is one for which the space requirement is "close" to the information-theoretic lower bound, but unlike a compressed data structure, still allows for efficient queries.
The membership problem is exactly what you describe in your question:
maintain a subset $S$ (of size $n$) from the universe $U = \{0, 1, 2, 3, \dots , u – 1\}$ of size $u$ with operations:
find(x) (checks whether element x is a member of $S$).
insert(x) (add an element x to $S$)
delete(x) (remove an element x from $S$)
If only the find operation is supported, then this is the static membership problem. If either insert or delete are supported, but not both, it is called semi-dynamic, and if all three operations are supported, then it is called the dynamic membership problem.
Techincally, I think you only asked for a data structure for the semi-dynamic membership problem, but I don't know of any data structures which take advantage of this constraint and also meet your other requirements. However, I do have the following reference:
In Theorem 5.1 of the article Membership in Constant Time and Almost-Minimum Space, Brodnik and Munro give the following result:
There exists a data structure requiring $O(B)$ bits which supports searches in constant time and insertions and deletions in constant expected amortized time.
where $B = \lceil \log {u \choose n} \rceil$ is the information theoretic minimum number of bits required.
The basic idea is that they recursively split the universe into ranges of carefully chosen sizes, so this even sounds like the techniques might be along the lines that you are thinking of.
However, if you're looking for something you can actually implement, I don't know if this is going to be your best bet. I only skimmed the paper, and trying to explain the details is way beyond the scope of this answer. They parameterize their solution, using different strategies depending on the relative sizes of $u$ and $n$. And the dynamic version of the data structure is only sketched in the paper.
$\begingroup$ The Brodnik & Munro paper abstract doesn't say anything about inserts. But their result is what we can expect, right? If n = u/2, then the needed space is maximal. $\endgroup$ – HEKTO Jan 31 '14 at 16:15
$\begingroup$ @AlekseyYakovlev They don't really mention the dynamic case in the abstract, but the theorem that deals with the dynamic case is quoted in my answer ( from section 5). $\endgroup$ – Joe Jan 31 '14 at 22:48
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Semiconductors/Diodes from an Engineering Perspective
< Semiconductors
A P-N junction is useful in electrical engineering as a Diode. Diodes have various purposes, including: - Preventing current from flowing in one direction but not the other - Regulating voltage (when used in reverse bias) - Voltage-dependent capacitors (when used in reverse bias)
1 Diode construction and properties
1.1 Biasing diodes
2 Circuit models of diodes
2.1 Ideal diode model
2.2 Voltage drop model
2.3 Empirical model
2.4 VRC model
3 Diode analysis
4 Voltage regulator
5 Voltage dependent capacitors
Diode construction and propertiesEdit
As noted in the previous chapter, diodes are constructed at the physical interface of a P-doped semiconductor and an N-doped semiconductor. When these two regions meet, holes and electrons diffuse across the junction due to the differing concentration at the junction:
Holes and electrons drift across the junction between the P and N doped semiconductors.
This diffusion does not continue indefinitely. Instead, as the electrons diffuse to the P-side and the holes diffuse in the N-side, a charge imbalance arises, creating a voltage which eventually prevents any further diffusion from occuring. The voltage is an inherent property of the diode and depends on the amount that the P-side is doped relative to the N-side. From this arises what is known as the Built-in potential, $V_{bi}$.
Biasing diodesEdit
- When a diode is subject to an external electric field, it is said to be biased. - If the field is in the opposite direction to the built-in voltage, it will counteract the diffusion, making the depletion region thinner. This is known as forward biasing.
- In addition to counteracting diffusion, forward biasing also injects charge carriers (holes and electrons) into the P and N regions of the diode. This encourages further diffusion from P to N, leading to current flow.
- However, if the field is in the same direction as the built-in voltage, it will encourage the diffusion, making the depletion region thicker. This is known as reverse biasing.
- In this mode, charge carriers are injected into the opposite regions of the diode (holes in N and electrons in P), where they instantly recombine. This prevents any current flowing, up to a certain point.
Circuit models of diodesEdit
When analysing diodes, a number of models can be used, including:
Ideal diode modelEdit
In the ideal diode model, there are two states: on and off. In the on-state, the diode acts as a short circuit, allowing any amount of current to flow through and forcing the potential between its ends to be 0. In the off-state, the diode acts as an open circuit, so that no current can flow through the diode.
Voltage drop modelEdit
In the voltage drop model, there are similarly two states: on and off. The off state is the same as above; the circuit prevents any current from flowing. In the on-state however, the diode acts as a voltage source that maintains a constant voltage between its two terminals equal to its built-in voltage.
Empirical modelEdit
In the empirical model, we treat the diode as a circuit component governed by $I = I_s (e^{\frac{V_f}{V_T}}-1)$, giving us an exponential curve. This model typically requires numerical iteration to evaluate.
VRC modelEdit
In the VRC model, we model the diode as a voltage source, capacitor and resistor. This is typically used when the diode is being used in reverse-bias.
Diode analysisEdit
Voltage regulatorEdit
Voltage dependent capacitorsEdit
Retrieved from "https://en.wikibooks.org/w/index.php?title=Semiconductors/Diodes_from_an_Engineering_Perspective&oldid=3772445" | CommonCrawl |
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Equivalence principle doubt
There is something about Einstein Equivalence Principle that I don't quite get. This is my reasoning:
Equivalence principle $\rightarrow$ locally, acceleration is equivalent to a gravitational field
Forces (which each observer, inertial or not inertial, agree about) causes particles to have a proper acceleration (they don't follow geodesics)
These particles which have a proper acceleration, from their point of view, feel a gravitational field (point one).
Now... is this gravitational field real or fictitious?
If it is real $\rightarrow$ it must depend on mass distribution around the object. So the force which caused the acceleration must be linked to the mass distribution somehow (since these two actions balance themselves so that the object is in equilibrium, from his point of view).
If it is fictitious $\rightarrow$ then it has nothing to do with "real" gravity, which depends on mass and is manifested as the curvature of spacetime. In this case, the Equivalence Principle seems to me just a coincidence which has nothing in common with the geometrized view of curved spacetime.
If my assumptions are correct... which of the two options is true?
general-relativity gravity spacetime curvature equivalence-principle
MarianD
Federico TosoFederico Toso
You've somewhat misstated the equivalence principle. It says that the effects of a gravitational field cannot be distinguished from the effects of having an accelerating frame of references. That's different than saying they are equivalent, and it's enough of a difference to break your logic chain at your point #3.
Another route to seeing this is that equivalence principle says that the $m$ in $F=ma$ (inertial mass) is the same as the $m$ in Newton's gravitational law (gravitational mass). It didn't have to be that way in theory, but it is. That does not preclude, however, the existence of other forces.
A bit more, following the comments, by way of explanation. I prefer to think of the equivalence principle as the mathematical statement that $m_{\mathrm{inertial}} = m_{\mathrm{grav}}$, and the (probably more popularly stated) description about indistinguishability between gravitational force and an accelerating frame as a consequence of this equivalence between the two conceptual types of mass. One implies the other though, so I think there's no unique argument for starting with one or the other, other than historical convention.
Note the last statement, which I'll elaborate here: The equivalence between gravitational and inertial mass implies the indistinguishability between a gravitational force and an accelerating frame.
If you imagine that you're in a box with no information about what's happening outside of your box, you now try to construct an experiment to determine if your box is in an inertial frame. No problem with this. Hold a pencil out, let it "drop" and see if it heads toward a wall of the box. If it does, then you're in a non-inertial frame.
Assume now that it does move toward a wall (meaning that it accelerates since it started at rest in your hand) and try to figure out if you're in a "stationary" box subject to gravity or in an "accelerating" box. Now you're stuck. If the box is not subject to gravity and the frame (e.g. the box) is accelerating due to some other force, then the pencil will move toward the side with acceleration equal to that of the box. If the box is subject to gravity but "stationary" you have $m_{\mathrm{inertial}} a = m_{\mathrm{grav}} g$, where $g$ gives the local strength of the gravitational field (and can be signed to account for direction of the field). But since the two masses are equal, this just gives $a=g$, which is non-informative since you don't have an independent measure of $g$. In either case (or, by extension, any case that includes some elements of both), all you can tell is that the pencil accelerated according to $a$.
The rest of your question about whether the force is "real" or "fictitious" seems to be trying to apply Newtonian reasoning to a relativistic question, and also seems to be based on your subtle misstatement of the principle. A frame exists in a (possibly small) neighborhood of a point but not at a single point. To say that the frame is accelerating is to say that all of the points in the frame are (up to some order) moving rigidly "together" with a single acceleration. That's distinct from looking at individual particles (possibly described in a frame) where each particle has a different relative acceleration. Your question on this part seems to be either comparing different frames (if you build a frame around each particle separately) or confounding the motion of the frame with the motion of the various particles in the frame (if you build one frame around your collection of particles). That's different than what the principle describes.
BrickBrick
$\begingroup$ this makes me wonder why it is called "Equivalence" Principle at all... so in this case, the gravitational field they feel is fictitious... does that imply my last deduction ("if it is fictitious --> etc)? $\endgroup$ – Federico Toso Sep 11 '19 at 14:52
$\begingroup$ The "equivalence" is between inertial and gravitational mass. $\endgroup$ – Brick Sep 11 '19 at 14:53
$\begingroup$ Your use of the word "fictitious" here is probably not wrong, but it's not especially relativistic in my view. There's no invariant way to describe that - if anything it's an expression of coordinate-dependent concepts that relativity would like to supersede with coordinate-invariant, geometric concepts. $\endgroup$ – Brick Sep 11 '19 at 14:56
$\begingroup$ Ok... but even in this light, I don't see an explanation of the "coincident similarity" between local acceleration effect (due to forces) and local gravitational effects (due to mass-energy). The geometrization of curved spacetime does not attempt to explain this coincidence. Do you agree? $\endgroup$ – Federico Toso Sep 11 '19 at 15:19
$\begingroup$ Updated answer with more detail. Writing quickly, so hopefully I got a good balance of precision and detail. $\endgroup$ – Brick Sep 11 '19 at 16:15
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\begin{document}
\begin{abstract}
We consider solutions to the Cauchy problem for an internal-wave model derived by Camassa-Choi \cite{MR1389977}. This model is a natural generalization of the Benjamin-Ono and Intermediate Long Wave equations in the case of weak transverse effects. We prove the existence and long-time dynamics of global solutions from small, smooth, spatially localized initial data on \(\mathbb R^2\).
\end{abstract}
\maketitle
\section{Introduction}
In this article we consider real-valued solutions \(u\colon\mathbb R_t\times\mathbb R^2_{(x,y)}\rightarrow\mathbb R\) to the Cauchy problem for an internal-wave model derived by Camassa-Choi \cite{MR1389977},
\begin{equation} \label{eqn:cc-fd} \left( u_t + \mathcal T_h^{-1}u_{xx} + h^{-1}u_x - u u_x \right)_x + u_{yy} = 0, \end{equation}
where \(h>0\) is the depth and the operator \(\mathcal T_h^{-1}\) has symbol \(i\coth(h\xi)\). In the limit \(h \rightarrow \infty\) we obtain the infinite depth equation,
\begin{equation} \label{eqn:cc} \left( u_t + \mathcal H^{-1}u_{xx} - u u_x \right)_x + u_{yy} = 0, \end{equation}
where the inverse of the Hilbert transform \(\mathcal H^{-1} = - \mathcal H\) has symbol \(i\sgn\xi\). These are natural \(2\)-dimensional versions of the Intermediate Long Wave (ILW) and Benjamin-Ono equations in the case of weak transverse effects. Our goal is to investigate the long-time dynamics of solutions with sufficiently small, smooth and spatially localized initial data.
The infinite depth equation \eqref{eqn:cc} is a special case of the dispersion-generalized (or fractional) Kadomtsev-Petviashvili II (KP-II) equation,
\begin{equation}\label{eqn:gKPII}
\left( u_t - |D_x|^\alpha u_x + uu_x\right)_x + u_{yy} = 0. \end{equation}
The original KP-II equation corresponds to the case \(\alpha = 2\) and is completely integrable in the sense that it possesses both a Lax pair and an infinite number of formally conserved quantities (see for example the survey article \cite{Klein2015}). To authors' knowledge, a Lax pair is not known for \eqref{eqn:cc-fd} or \eqref{eqn:cc} although both of their \(1d\) counterparts, the ILW and Benjamin-Ono, are integrable in this sense.
Both the finite and infinite depth equations are Hamiltonian with formally conserved energies,
\begin{gather} E_h[u] = \int \left( u\mathcal T_h^{-1}\partial_xu + h^{-1}u^2 + (\partial_x^{-1}\partial_yu)^2 - \frac13 u^3\right)\,dxdy,\\ E_\infty[u] = \int \left(u\mathcal H^{-1}\partial_xu + (\partial_x^{-1}\partial_yu)^2 - \frac13u^3\right)\,dxdy, \end{gather}
respectively. Both equations also conserve the \(L^2\)-norm,
\begin{equation} M[u] = \int u^2\,dxdy. \end{equation}
The infinite depth equation is invariant with respect to the scaling
\begin{equation}\label{Scaling} u(t,x,y)\mapsto \lambda u(\lambda^2t,\lambda x,\lambda^{\frac{3}{2}}y),\qquad \lambda>0. \end{equation}
Taking \(\lambda = h\), this scaling also maps solutions to the finite depth equation with depth \(h\) to solutions with depth \(1\). Both the finite and infinite depth equations are invariant with respect to Galilean shifts of the form,
\begin{equation}\label{Galilean} u(t,x,y)\mapsto u(t,x+cy - c^2t,y - 2ct),\qquad c\in\mathbb R. \end{equation}
The natural spaces in which to consider local well-posedness for the infinite depth equation are the homogeneous anisotropic Sobolev spaces \(\dot H^{s_1,s_2} = \dot H^{s_1}_x\dot H^{s_2}_y\) with norm
\[
\|u\|_{\dot H^{s_1,s_2}} = \||D_x|^{s_1}|D_y|^{s_2}u\|_{L^2_{x,y}}, \]
for which the scaling-critical, Galilean-invariant space is given by \((s_1,s_2) = (\frac14,0)\).
Small data global well-posedness and scattering were proved for the KP-II at the scaling-critical regularity \((s_1,s_2) = (-\frac12,0)\) by Hadac-Herr-Koch \cite{MR2526409,MR2629889}. Local well-posedness results are also available in higher dimensions \cite{2016arXiv160806730K} and for the dispersion generalized equation \eqref{eqn:gKPII} on \(\mathbb R_{x,y}^2\) provided \(\alpha>\frac43\) \cite{MR2434299}. While preparing this paper we also learned of a recent result of Linares-Pilod-Saut~\cite{2017arXiv170509744L} who prove several local well-posedness and ill-posedness results for \eqref{eqn:cc-fd}, \eqref{eqn:cc} and other similar generalizations of the KP-II.
We define the linear operator
\[ \mathcal L_h = \partial_t + \mathcal T_h^{-1}\partial_x^2 + h^{-1}\partial_x + \partial_x^{-1}\partial_y^2, \]
with the corresponding modification when \(h = \infty\). Here \(\partial_x^{-1}\) is interpreted as the Fourier multiplier \(\mathrm{p.v. }(i\xi)^{-1}\), which for \(f\in L^1\) gives us
\[ \partial_x^{-1}f = \frac12\int_{-\infty}^xf(y)\,dy - \frac12\int_x^\infty f(y)\,dy. \]
For \(\xi\neq0\), the dispersion relation associated to \eqref{eqn:cc-fd} is given by
\begin{equation}\label{OMEGAh} \omega_h(\mathbf k) = \xi^2\coth(h \xi) - h^{-1}\xi - \xi^{-1}\eta^2, \end{equation}
where \(\mathbf k = (\xi,\eta)\), and in the limit \(h = \infty\) we obtain
\begin{equation}\label{OMEGAinf}
\omega_\infty(\mathbf k) = \xi|\xi| - \xi^{-1}\eta^2. \end{equation}
We take \(S_h(t)\) to be the associated linear propagator, defined using the Fourier transform\footnote{We use the isometric normalization of the Fourier transform, \(\hat f(\mathbf k) = \frac1{2\pi}\int f(x,y)e^{-i(x,y)\cdot\mathbf k}\,d\mathbf k\).} as
\begin{equation}\label{Propagator}
S_h(t)f = \frac{1}{2\pi}\lim\limits_{\epsilon\downarrow0}\int_{|\xi|>\epsilon} e^{it\omega_h(\mathbf k)}\hat f(\mathbf k)e^{i(x,y)\cdot\mathbf k}\,d\mathbf k, \end{equation}
with the corresponding modification when \(h = \infty\). We note that the linear propagator \eqref{Propagator} extends to a well-defined unitary map \(S_h(t)\colon L^2(\mathbb R^2)\rightarrow L^2(\mathbb R^2)\) without the need for additional moment assumptions on \(f\).
Linear solutions satisfy the dispersive estimates,
\begin{equation}\label{est:Dispersive}
\|h^{\frac12}\langle hD_x\rangle^{-\frac12} S_h(t)u_0\|_{L^\infty}\lesssim |t|^{-1}\|u_0\|_{L^1}, \end{equation}
which may be readily seen from the fact that the kernel \(K_h\) of the linear propagator \(S_h\) is given by
\[ K_h(t,x,y) = \frac1{\sqrt{2ht}}\ k\left(h^{-2}t,h^{-1}(x + \frac1{4t}y^2)\right), \]
where the oscillatory integral
\[
k(t,x) = \frac1{\sqrt{2\pi}}\lim\limits_{\epsilon\downarrow0}\int_{|\xi|>\epsilon} |\xi|^{\frac12}e^{it(\xi^2\coth\xi - \xi) + ix\xi}\,d\xi. \]
For a more detailed proof see \cite[Lemmas~4.7,~4.8]{2017arXiv170509744L}.
Due to the \(O(|t|^{-1})\) decay, bilinear interactions are long-range so it is natural to seek a normal form transformation that upgrades the quadratic nonlinearity to a cubic one. Resonant nonlinear interactions correspond to solutions of the system
\[ \begin{cases} \omega_h(\mathbf k_1) + \omega_h(\mathbf k_2) + \omega_h(\mathbf k_3) = 0
\\ \mathbf k_1+\mathbf k_2+\mathbf k_3 = 0, \end{cases} \]
and some elementary algebraic manipulations show that this cannot be satisfied for non-zero \(\xi_j\). As a consequence the Camassa-Choi nonlinearity is non-resonant and formally we may construct a normal form leading to enhanced lifespan solutions.
Given the non-resonance of the bilinear interactions, one might expect the methods used for large values of \(\alpha\) to apply to the Camassa-Choi. However, in the corresponding \(1\)-dimensional cases of the ILW and Benjamin-Ono it is known that Picard iteration methods fail \cite{MR1885293,MR2172940} due to strong high-low bilinear interactions. While the additional dispersion in \(2d\) should allow for improved results over the corresponding \(1d\) case, one may apply the methods of \cite{MR1885293,2016arXiv160806730K} to show that Picard iteration still fails in the infinite depth case \eqref{eqn:cc} in the anisotropic Sobolev space \(\dot H^{\frac14,0}\) and almost all of the Besov-type refinements considered in \cite{2016arXiv160806730K}. For completeness we provide a brief proof of this ill-posedness result in Appendix~\ref{app:IP}.
Instead we will assume additional spatial localization on the initial data and establish global existence using a similar approach to work of the first author with Ifrim and Tataru on the KP-I equation \cite{2014arXiv1409.4487H}. Here we will apply the \emph{method of testing by wave packets}, originally developed in work of Ifrim-Tataru on the \(1d\) cubic NLS \cite{MR3382579} and \(2d\) gravity water waves \cite{2014arXiv1404.7583I}, and subsequently applied in several other contexts \cite{MR3462131,2014arXiv1406.5471I,2014arXiv1409.4487H}. The key difficulty we encounter when adapting this method to the setting of the Camassa-Choi is the presence of the non-local operator \(\mathcal T_h^{-1}\). Indeed, a testament to the robust nature of this approach is that it may be applied to obtain global solutions even in this context. We note that a related approach to establishing global well-posedness is via the space-time resonances method, simultaneously developed by Germain-Masmoudi-Shatah \cite{MR2542891,MR2482120} and Guftason-Nakanishi-Tsai \cite{MR2360438} as a significant upgrade to the method of normal forms, originally applied in the context of dispersive PDE by Shatah \cite{MR803256}.
We expect that linear solutions initially localized in space near \((x,y) = 0\) and at frequency \(\mathbf k = (\xi,\eta)\) will travel along rays of the Hamiltonian flow
\[ \Gamma_{\mathbf v} = \{(x,y) = t\mathbf v\}, \]
where the group velocity
\[ \mathbf v = - \nabla\omega_h(\mathbf k). \]
In order to measure this localization we define the operators \(L_{x,h}\), \(L_y\) with corresponding symbols \(x + t\partial_\xi\omega_h(\mathbf k)\), \(y + t\partial_\eta\omega_h(\mathbf k)\) respectively. A simple computation yields the explicit expressions,
\[ L_{x,h} = x - 2t\mathcal T_h^{-1}\partial_x - th\partial_x^2(1 + \mathcal T_h^{-2}) - th^{-1} + t\partial_x^{-2}\partial_y^2,\qquad L_y = y - 2t\partial_x^{-1}\partial_y, \]
and in the limit \(h = \infty\),
\[ L_{x,\infty} = x - 2t\mathcal H^{-1}\partial_x + t\partial_x^{-2}\partial_y^2. \]
Further, by construction the operators \(L_{x,h},L_y\) commute with \(\mathcal L_h\).
As the equations possesses a Galilean invariance, in the spirit of \cite{MR3382579,2014arXiv1409.4487H} we will consider well-posedness in Galilean invariant spaces. The vector field \(\partial_xL_y\) is the generator of the Galilean symmetry. However, the operators \(\partial_y\), \(L_{x,h}\) do not commute with the Galilean group so we will instead measure \(x\)-localization with the Galilean-invariant operator
\[ J_h = L_{x,h}\partial_x + L_y\partial_y. \]
We then define the time-dependent space \(X_h\) of distributions on \(\mathbb R^2\) with finite norm,
\[
\|u\|_{X_h}^2 = h^{-\frac12}\|u\|_{L^2}^2 + h^{\frac{15}2}\|\partial_x^4u\|_{L^2}^2 + h^{-\frac92}\|L_y^2\partial_x u\|_{L^2}^2 + h^{-\frac12}\|J_hu\|_{L^2}^2. \]
We note that this space is uniform with respect to \(h\) and that \(S_h(-t)X_h = X_h(0)\), where the initial data space
\[
\|u_0\|_{X_h(0)} = h^{-\frac12}\|u_0\|_{L^2}^2 + h^{\frac{15}2}\|\partial_x^4u_0\|_{L^2}^2 + h^{-\frac92}\|y^2\partial_xu_0\|_{L^2}^2 + h^{-\frac12}\|(x\partial_x + y\partial_y)u_0\|_{L^2}^2. \]
Our main result in the finite depth case is the following:
\begin{thm}\label{thm:Main}
There exists \(0<\epsilon\ll1\) so that for all \(h>0\) and \(u(0)\in X_h(0)\) satisfying the estimate
\begin{equation}\label{est:Init}
\|u(0)\|_{X_h(0)} \leq\epsilon, \end{equation}
there exists a unique global solution \(u\in C(\mathbb R;X_h)\) to \eqref{eqn:cc-fd} satisfying the energy estimate
\begin{equation}\label{Fdn}
\|u\|_{X_h(t)}\lesssim \epsilon \langle h^{-2}t\rangle^{C\epsilon} \end{equation}
and the pointwise decay estimate
\begin{equation}\label{est:PTWISEDECAY}
h^{2}\|u_x\|_{L^\infty}\lesssim \epsilon |h^{-2}t|^{-\frac12}\langle h^{-2}t\rangle^{-\frac12}. \end{equation}
Further, this solution scatters in the sense that there exists some \(W\in L^2\) so that \(\|W\|_{L^2} = \|u_0\|_{L^2}\) and
\begin{equation}\label{Fds}
h^{-\frac14}\|S_h(-t)u - W\|_{L^2} \lesssim \epsilon^2 (h^{-2}t)^{-\frac1{96} + C\epsilon},\qquad t\rightarrow+\infty. \end{equation}
\end{thm}
\begin{rem} As in~\cite{MR3382579,2014arXiv1409.4487H,MR3462131} we may interpolate between the estimates \eqref{Fdn} and \eqref{Fds} to show that \(W\in (X_h(0),h^{\frac14}L^2)_{C\epsilon,2}\) for some \(C>0\) where \((X_h(0),h^{\frac14}L^2)_{C\epsilon,2}\) is the usual real interpolation space. \end{rem}
In the infinite depth case we define the modification
\[
\|u\|_{X_\infty}^2 = \|u\|_{L^2}^2 +\|\partial_x^4u\|_{L^2}^2 + \|L_y^2\partial_x u\|_{L^2}^2 + \|J_\infty u\|_{L^2}^2, \]
for which the initial data space is given by
\[
\|u_0\|_{X_\infty(0)} = \|u_0\|_{L^2}^2 + \|\partial_x^4u_0\|_{L^2}^2 + \|y^2\partial_xu_0\|_{L^2}^2 + \|(x\partial_x + y\partial_y)u_0\|_{L^2}^2. \]
Our main result in the infinite depth case is then:
\begin{thm}\label{thm:MainInf}
There exists \(0<\epsilon\ll1\) so that for all \(u(0)\in X_\infty(0)\) satisfying the estimate
\begin{equation}
\|u(0)\|_{X_\infty(0)} \leq\epsilon, \end{equation}
there exists a unique global solution \(u\in C(\mathbb R;X_\infty)\) to \eqref{eqn:cc-fd} satisfying the energy estimate
\begin{equation}
\|u\|_{X_\infty(t)}\lesssim \epsilon\langle t\rangle^{C\epsilon} \end{equation}
and the pointwise decay estimate
\begin{equation}
\|u_x\|_{L^\infty}\lesssim \epsilon |t|^{-\frac12}\langle t\rangle^{-\frac12}. \end{equation}
Further, this solution scatters in the sense that there exists some \(W\in L^2\) so that
\begin{equation}
\|S_\infty(-t)u - W\|_{L^2} \lesssim \epsilon t^{-\frac1{48} + C\epsilon},\qquad t\rightarrow+\infty. \end{equation}
\end{thm}
\begin{rem} The spaces \(X_h\) and \(X_\infty\) are essentially identical to the spaces considered in \cite{2014arXiv1409.4487H}. However, due to the reduced dispersion of \eqref{eqn:cc-fd} and \eqref{eqn:cc} at high frequency, we require an additional \(x\)-derivative to obtain sufficient decay. \end{rem}
For simplicity we now restrict our attention to the case \(h = 1\) and drop the dependence on \(h\) throughout our notation. We remark the remaining finite depth cases may be obtained by scaling. Namely, we have \[ u^{(1)}(t,x,y) = hu^{(h)}(h^2t,hx,h^{\frac32}y). \] The proof in the infinite depth case (Theorem~\ref{thm:MainInf}) is essentially identical and is thus omitted. However, for completeness we will outline a few of the required modifications when they deviate sufficiently from the finite depth proof.
The remainder of the paper is structured as follows: in Section~\ref{sect:Prelim} we give some brief technical preliminaries; in Section~\ref{sect:AP} we prove local well-posedness and a priori energy estimates for solutions to \eqref{eqn:cc-fd}; in Section~\ref{sect:KS} we prove pointwise bounds in the spirit of Klainerman-Sobolev estimates; in Section~\ref{sect:TWP} we complete the proof of Theorem~\ref{thm:Main} using the method of testing by wave packets. In Appendix~\ref{app:IP} we prove an ill-posedness result for \eqref{eqn:cc}.
\section{Preliminaries}\label{sect:Prelim}
In this section we briefly state several technical preliminaries required for the proof.
\subsection{The resonance function}
We define the resonance function
\[ \Omega(\mathbf k_1,\mathbf k_2,\mathbf k_3) = \omega(\mathbf k_1) + \omega(\mathbf k_2) + \omega(\mathbf k_3). \]
If we restrict the resonance function to the hyperplane \(\{\mathbf k_1 + \mathbf k_2 + \mathbf k_3 = 0\}\) we may compute a simplified expression for
\( \Omega(\mathbf k_1,\mathbf k_2) = \Omega(\mathbf k_1,\mathbf k_2,-(\mathbf k_1 + \mathbf k_2)), \)
\begin{equation}\label{ResonanceFunction}
\Omega(\mathbf k_1,\mathbf k_2) = - \xi_1\xi_2(\xi_1 + \xi_2)\left(\frac{(\xi_1 + \xi_2)^2\coth(\xi_1 + \xi_2) - \xi_1^2\coth\xi_1 - \xi_2^2\coth\xi_2}{\xi_1\xi_2(\xi_1 + \xi_2)} + \frac{\left|\dfrac{\eta_1}{\xi_1} - \dfrac{\eta_2}{\xi_2}\right|^2}{(\xi_1 + \xi_2)^2}\right). \end{equation}
By considering the asymptotic behavior of \(\coth\xi\) we obtain the lower bound
\[
\frac{(\xi_1 + \xi_2)^2\coth(\xi_1 + \xi_2) - \xi_1^2\coth\xi_1 - \xi_2^2\coth\xi_2}{\xi_1\xi_2(\xi_1 + \xi_2)} \gtrsim \frac1{1 + \max\{|\xi_1|,|\xi_2|,|\xi_1 + \xi_2|\}}, \]
and hence we obtain lower bounds for the resonance function \(\Omega_h\) in the low-high (\(|\xi_1|\ll|\xi_2|\)) and high-high (\(|\xi_1 + \xi_2|\ll|\xi_2|\)) asymptotic regions,
\begin{equation}\label{FDResonanceLB}
|\Omega(\mathbf k_1,\mathbf k_2)|\gtrsim \begin{cases}\dfrac{|\xi_1||\xi_2|^2}{\langle \xi_2\rangle}\left(1 + \dfrac{\langle\xi_2\rangle}{|\xi_1|^2}\left|\dfrac{\eta_1 + \eta_2}{\xi_1 + \xi_2} - \dfrac{\eta_2}{\xi_2}\right|^2\right),&\quad |\xi_1|\ll|\xi_2|,
\\\dfrac{|\xi_1 + \xi_2||\xi_2|^2}{\langle \xi_2\rangle}\left(1 + \dfrac{\langle\xi_2\rangle }{|\xi_1 + \xi_2|^2}\left|\dfrac{\eta_1}{\xi_1} - \dfrac{\eta_2}{\xi_2}\right|^2\right),&\quad |\xi_1 + \xi_2|\ll|\xi_2|.\end{cases} \end{equation}
The resonance function in the infinite depth case is given by the slightly more straightforward expression,
\begin{equation}\label{ResonanceInf}
\Omega_\infty(\mathbf k_1,\mathbf k_2) = \xi_1\xi_2\xi_3\left(\frac{2}{\max\{|\xi_1|,|\xi_2|,|\xi_3|\}} + \frac{\left|\dfrac{\eta_1}{\xi_1} - \dfrac{\eta_2}{\xi_2}\right|^2}{\xi_3^2}\right). \end{equation}
\subsection{Littlewood-Paley decomposition} We take \(0<\delta\ll1\) to be a fixed universal constant that determines the resolution of our frequency decomposition. The size of \(\delta\) will be determined in the ODE estimates of Section~\ref{sect:TWP} but will otherwise be irrelevant. For \(N\in 2^{\delta \mathbb Z}\) we take \(P_N\) to project to \(x\)-frequencies \(2^{-\delta}N<|\xi|<2^\delta N\) so that
\[ 1 = \sum\limits_{N\in 2^{\delta\mathbb Z}}P_N. \]
We write \(u_N = P_Nu\), and for \(A>0\) we take \(u_{<A} = \sum_{N<A}u_N\) and \(u_{\geq A} = \sum_{N\geq A}u_N\), where the sums are understood to be over \(N\in 2^{\delta\mathbb Z}\). We observe that for any \(A>0\) we have
\[
\|u\|_X^2 \sim \|u_{<A}\|_X^2 + \sum\limits_{N\geq A}\|u_N\|_X^2. \]
We may further decompose \(u_N = u_N^+ + u_N^-\) where \(u_N^+\) is the projection to positive wavenumbers in \(x\)-frequency. For real-valued \(u\) we have \(u_N = 2\mathop{\rm Re}\nolimits(u_N^+)\) and hence
\[
\|u_N\|_X \sim \|u_N^+\|_X \sim \|u_N^-\|_X. \]
\subsection{Symbol classes and elliptic operators}
Given \(d\geq1\) we define the symbol class \(\mathscr S\) to consist of functions \(p\in C^\infty(\mathbb R^d\times \mathbb R^d\backslash\{0\})\) so that, writing \(x = (x_1,\dots,x_d)\) and \(\xi = (\xi_1,\dots,\xi_d)\), we have
\[
|D_x^\beta D_\xi^\alpha p(x,\xi)|\lesssim_{|\alpha|,|\beta|} |\xi|^{-|\alpha|}. \]
Given a symbol \(p\in\mathscr S\) we may define a pseudo-differential operator
\[ p(x,D)u = \frac1{(2\pi)^{\frac d2}}\int_{\mathbb R^d} p(x,\xi)\hat f(\xi) e^{ix\cdot\xi}\,d\xi, \]
and then have the estimate (see for example \cite{MR1766415})
\begin{equation}
\|p(x,D)u\|_{L^2}\lesssim \|u\|_{L^2}. \end{equation}
We also recall that if \(r(x,\xi) = p(x,\xi)q(x,\xi)\) for \(p,q\in \mathscr S\) then we have the estimate
\begin{equation}\label{ProductRule}
\|r(x,D)f\|_{L^2} \lesssim \|p(x,D)q(x,D)f\|_{L^2} + \|f\|_{H^{-1}}. \end{equation}
Further, if \(p\in\mathscr S\) satisfies the estimate (again see \cite{MR1766415})
\[
|p(x,\xi)^{-1}|\lesssim 1, \]
then we say \(p\) is \emph{elliptic} and for \(f\in L^2(\mathbb R^d)\) and any \(s\in \mathbb R\) we have the estimate
\begin{equation}\label{OPSElliptic}
\|f\|_{L^2} \lesssim_s \|p(x,D)f\|_{L^2} + \|f\|_{H^s}. \end{equation}
\subsection{Multilinear Fourier multipliers}
If \(d = 2n\) and \(m(\mathbf k_1,\dots,\mathbf k_n)\in\mathscr S\) is independent of the spatial variables, we may define a multilinear Fourier multiplier \(M\) with symbol \(m\) by
\[ M[u_1,\dots,u_n] = \frac1{(2\pi)^n}\int_{\mathbb R^{2n}} m(\mathbf k_1,\dots,\mathbf k_n) \hat u_1(\mathbf k_1)\dots\hat u_n(\mathbf k_n) e^{i(x,y)\cdot(\mathbf k_1 + \dots + \mathbf k_n)}\,d\mathbf k_1\dots d\mathbf k_n. \]
We then recall the Coifman-Meyer Theorem (see for example~\cite{MuscaluPipherTaoThiele} and references therein)
\begin{equation}\label{CM}
\|M[u_1,\dots,u_n]\|_{L^p}\lesssim \|u_1\|_{L^{p_1}}\dots\|u_n\|_{L^{p_n}}, \end{equation}
provided \(\frac 1p = \frac1{p_1} + \dots + \frac1{p_n}\), \(1\leq p <\infty\), \(1<p_j\leq\infty\).
\subsection{Sobolev estimates}
We recall the Sobolev estimate,
\begin{equation}
\|f\|_{L^\infty} \lesssim \|f\|_{L^2}^{\frac14}\|f_x\|_{L^2}^{\frac12}\|f_{yy}\|_{L^2}^{\frac14},\label{est:Sobolev} \end{equation}
and the H\"older space modification,
\begin{equation}
\|f\|_{\dot C^{0,\alpha}} \lesssim \left(\|f\|_{L^2}^{\frac14 - \alpha}\|f_x\|_{L^2}^\alpha + \|f\|_{L^2}^{\frac14 - \frac\alpha 2}\|f_{yy}\|_{L^2}^{\frac\alpha2}\right)\|f_x\|_{L^2}^{\frac12}\|f_{yy}\|_{L^2}^{\frac14},\qquad 0<\alpha\leq\frac14.\label{est:Holder} \end{equation}
\section{Local well-posedness and energy estimates}\label{sect:AP}
In this section we prove a priori estimates for the solution to \eqref{eqn:cc-fd} in the case \(h = 1\). As a consequence of the usual energy method we obtain local well-posedness in the spaces \(Z^k\), where the norm
\begin{equation}
\|u\|_{Z^k}^2 = \|u\|_{L^2}^2 + \|\partial_x^ku\|_{L^2}^2 + \|L_y^2\partial_x u\|_{L^2}^2, \end{equation}
and we note that \(X\subset Z^4\). Our local well-posedness result is the following:
\begin{thm}\label{thm:LWP}
Let \(h = 1\) and \(k\geq3\). Then for all \(u_0\in Z^k(0)\), the equation \eqref{eqn:cc-fd} is locally well-posed in \(Z^k\) and the solution exists at least as long as \(\left|\int_0^t \|u_x(s)\|_{L^\infty}\,ds\right| <\infty\). \end{thm}
\begin{rem} Our definition of local well-posedness in Theorem~\ref{thm:LWP} includes:
\begin{itemize} \item \emph{Existence.} There exists a solution \(S(-t)u\in C([-T,T];Z^k(0))\). \item \emph{Uniqueness.} The solution \(S(-t)u\) is unique in the space \(Z^k(0)\). \item \emph{Continuity.} The solution map \(u_0 \mapsto S(-t)u(t)\) is continuous in the \(Z^k(0)\) topology. \item \emph{Persistence of regularity.} If \(u_0\in Z^{k'}(0)\) for \(k'\geq k\) then \(u\in Z^{k'}\). \end{itemize}
\end{rem}
\begin{rem} The result of Theorem~\ref{thm:LWP} is certainly not optimal in terms of regularity but will suffice for the purposes of establishing the existence of global solutions. Indeed, an elementary application of the usual Littlewood-Paley trichotomy allows us to obtain local well-posedness in \(Z^{\frac 72+}\). We also mention several other local well-posedness results in other topologies are proved in~\cite{2017arXiv170509744L}. \end{rem}
\begin{rem}
As usual, it suffices to consider times \(t\geq0\) as the equation is invariant under the transformation
\( u(t,x,y)\mapsto u(-t,-x,y). \)
\end{rem}
The key ingredient for local well-posedness will be a priori estimates for the solution in the space \(Z^k\). We will supplement these a priori bounds with a further estimate when the initial data \(u_0\in X\) satisfies the smallness condition \eqref{est:Init} and obtain energy estimates for the solution depending upon the size of
\begin{equation}\label{ControlQuantity}
\mathcal M = \frac1\epsilon\sup\limits_{t\in[0,T]} |t|^{\frac12}\langle t\rangle^{\frac12}\|u_x\|_{L^\infty} \end{equation}
Our main a priori bound is the following:
\begin{prop}\label{prop:NRG}
Let \(h=1\) and \(u\) be a smooth solution to \eqref{eqn:cc-fd} on the time interval \([0,T]\). Then we have the a priori estimate,
\begin{equation}\label{est:Zk}
\|u(t)\|_{Z^k} \leq \|u_0\|_{Z^k(0)} \exp\left(C\int_0^t \|u_x(s)\|_{L^\infty}\,ds\right). \end{equation}
Further, if the initial data \(u_0\in X(0)\) satisfies \eqref{est:Init} and \(0<\epsilon\ll \mathcal M^{-1}\) is sufficiently small, we have the improved estimate
\begin{equation}\label{est:NRG}
\|u(t)\|_X\lesssim \epsilon \langle t\rangle^{C\mathcal M\epsilon}. \end{equation}
\end{prop} \begin{proof}
In order to justify the various computations we note that by standard approximation arguments it suffices to assume that \(u\) is a Schwartz function and that for some \(0<\nu\ll1\) we have \(P_{< \nu}u = 0\).
\emph{Estimates for \(\partial_x^ku\).}
Differentiating the equation \(k\) times we obtain
\[ \mathcal L\partial_x^ku = u\partial_x^{k+1}u + \frac12\sum\limits_{j=1}^k\binom{k+1}{j}\partial_x^ju\cdot\partial_x^{k+1-j}u. \]
Integrating by parts for the first term and using the elementary interpolation estimate
\[
\|\partial_x^ju\|_{L^{\frac{2(k-1)}{j-1}}}^{k-1}\lesssim \|u_x\|_{L^\infty}^{k-j}\|\partial_x^ku\|_{L^2}^{j - 1},\qquad 1\leq j\leq k, \]
for the second term we obtain the bound
\begin{equation}\label{est:D4u}
\frac{d}{dt}\|\partial_x^ku\|_{L^2}^2\lesssim \|u_x\|_{L^\infty}\|\partial_x^ku\|_{L^2}^2. \end{equation}
\emph{Estimates for \(L_y^2\partial_xu\).} Again we start by calculating
\[ \mathcal LL_y^2\partial_xu = uL_y^2\partial_x^2u + (L_y\partial_xu)^2. \]
For the first term we may simply integrate by parts. For the second term it suffices to show that
\begin{equation}\label{L4Bound}
\|L_yu_x\|_{L^4}^2\lesssim \|u_x\|_{L^\infty}\|L_y^2u_x\|_{L^2}, \end{equation}
from which we obtain the estimate
\begin{equation}\label{est:Lyu}
\frac{d}{dt}\|L_y^2\partial_xu\|_{L^2}^2\lesssim \|u_x\|_{L^\infty}\|L_y^2\partial_xu\|_{L^2}^2. \end{equation}
To prove the estimate \eqref{L4Bound} we first make a change of variables \(f(t,x,y) = u(t,x - \frac1{4t}y^2,y)\) so that
\[
\|f_x\|_{L^\infty} = \|u_x\|_{L^\infty},\qquad\|f_y\|_{L^4} = \|L_y\partial_xu\|_{L^4},\qquad \|\partial_x^{-1}f_{yy}\|_{L^2} = \|L_y^2\partial_xu\|_{L^2}. \]
We first observe that by symmetry we may decompose by frequency as
\[
\|f_y\|_{L^4}^4 = \sum\limits_{N_1\sim N_2}\int f_{N_1,y} f_{N_2,y} (f_{\leq N_2,y})^2\,dxdy. \]
We then integrate by parts to obtain
\begin{align*}
\|f_y\|_{L^4}^4 &= - \sum\limits_{N_1\sim N_2}\int f_{N_1} f_{N_2,yy} (f_{\leq N_2,y})^2\,dxdy - 2\sum\limits_{N_1\sim N_2}\int f_{N_1} f_{N_2,y} f_{\leq N_2,y} f_{\leq N_2,yy}\,dxdy\\
&\lesssim \|f_x\|_{L^\infty}\left(\|C_1[\partial_x^{-1}f_{yy},f_y,f_y]\|_{L^1} + \|C_2[f_y,f_y,\partial_x^{-1}f_{yy}]\|_{L^1}\right), \end{align*}
where the trilinear Fourier multipliers
\[ C_1[f,g,h] = \sum\limits_{N_1\sim N_2}\partial_x^{-1}P_{N_1}(\partial_xf_{N_2} g_{\leq N_2}h_{\leq N_2}),\quad C_2[f,g,h] = \sum\limits_{N_1\sim N_2}\partial_x^{-1}P_{N_1}(f_{N_2}g_{\leq N_2}\partial_xh_{\leq N_2}), \]
may be bounded using the Coifman-Meyer Theorem \eqref{CM} to obtain
\[
\|C_1[\partial_x^{-1}f_{yy},f_y,f_y]\|_{L^1} + \|C_2[f_y,f_y,\partial_x^{-1}f_{yy}]\|_{L^1}\lesssim \|f_y\|_{L^4}^2\|\partial_x^{-1}f_{yy}\|_{L^2}. \]
\emph{Proof of \eqref{est:Zk}.} The estimate \eqref{est:Zk} then follows from the conservation of mass and estimates \eqref{est:D4u}, \eqref{est:Lyu} and Gronwall's inequality.
\emph{Estimates for \(Ju\): Short times.} For short times (\(0<t<1\)) we take \(w = Ju + tuu_x\) and calculate
\[ \mathcal Lw = (uw)_x - t\partial_x[\mathcal T^{-1}\partial_x,u]u_x - t\partial_x[(1 + \mathcal T^{-2})\partial_x^2,u]u_x. \]
We observe that \(\mathcal T^{-1}\partial_x\) is a smooth Fourier multiplier with principle symbol homogeneous of order \(1\) and that \((1 + \mathcal T^{-2})\partial_x^2\) is a smooth Fourier multiplier with Schwartz symbol. Standard commutator estimates (see for example~\cite{MR1766415}) then yield the bounds
\begin{align*}
\|\partial_x[\mathcal T^{-1}\partial_x,u]u_x\|_{L^2} &\lesssim \|u_x\|_{L^\infty}\left(\|u_{xx}\|_{L^2} + \|u\|_{L^2}\right),\\
\|\partial_x[(1 + \mathcal T^{-2})\partial_x^2,u]u_x\|_{L^2} &\lesssim \|u_x\|_{L^\infty}\|u\|_{L^2}. \end{align*}
Integrating by parts in the first term we then obtain the estimate
\begin{equation}\label{est:JShort}
\frac{d}{dt}\|w\|_{L^2}^2\lesssim \|u_x\|_{L^\infty}\|w\|_{L^2}\left(\|w\|_{L^2} + \|u_{xx}\|_{L^2} + \|u\|_{L^2}\right). \end{equation}
\emph{Estimates for \(Ju\): Long times.}
For times \(t\geq1\) we write
\[ J = \mathbf S - \frac12 L_y\partial_y - 2t\mathcal L, \]
where the operator
\[ \mathbf S = 2t\partial_t + x\partial_x + \frac32 y\partial_y - t\partial_x^3(1 + \mathcal T^{-2}) + t\partial_x, \]
satisfies
\[ [\mathcal L,\mathbf S] = 2\mathcal L. \]
We note that in the limit \(h = \infty\) we obtain the operator \(\mathbf S_\infty = 2t\partial_t + x\partial_x + \frac32y\partial_y\), which is the generator of the scaling symmetry \eqref{Scaling}.
As a consequence, we define
\[ w = \mathbf Su - \frac12 L_y\partial_y u + \frac12u = Ju + \frac12u - 2tuu_x, \]
and calculate
\[ \mathcal Lw = (uw)_x - \frac1{4t}\left(u_xL_y^2\partial_xu - (L_y\partial_xu)^2\right) - t\partial_x[\partial_x^2(1 + \mathcal T^{-2}),u]u_x \]
In order to obtain long time bounds for the commutator term we will take advantage of the non-resonance and use a normal form transformation to upgrade it to a cubic term. We start by using the Fourier transform to write
\[ \mathcal F\left[\partial_x[\partial_x^2(1 + \mathcal T^{-2}),u]v_x\right] = \int_{\mathbf k_1 + \mathbf k_2 = \mathbf k}k(\xi_1,\xi_2)\hat u(\mathbf k_1)\hat v(\mathbf k_2)\,d\mathbf k, \]
where the symbol
\[ k(\xi_1,\xi_2) = (\xi_1 + \xi_2)\xi_2\left((\xi_1 + \xi_2)^2\cosech^2(\xi_1 + \xi_2) - \xi_2^2\cosech^2(\xi_2) \right). \]
Next we symmetrize to obtain
\begin{align*} k_{\mathrm{sym}}(\xi_1,\xi_2) &= \frac12k(\xi_1,\xi_2) + \frac12k(\xi_2,\xi_1)\\ &= \frac12(\xi_1 + \xi_2)^4\cosech^2(\xi_1 + \xi_2) - \frac12(\xi_1 + \xi_2)\xi_1^3\cosech^2\xi_1 - \frac12(\xi_1 + \xi_2)\xi_2^3\cosech^2\xi_2. \end{align*}
We then construct a symmetric bilinear Fourier multiplier \(B[u,v]\) with symbol
\[ b(\mathbf k_1,\mathbf k_2) = \frac{k_{\mathrm{sym}}(\xi_1,\xi_2)}{\Omega(\mathbf k_1,\mathbf k_2)}, \]
where the resonance function \(\Omega\) is defined as in \eqref{ResonanceFunction}. The symbol \(b\) may be readily seen to be rapidly decaying at high frequencies. However, due to the commutator structure of \(k\) it also has additional smallness at low frequencies that will allow us to obtain bounds in terms of pointwise norms of \(u_x\) rather than \(u\). We remark that in the infinite depth case, we replace \(1 + \mathcal T^{-2}\) by \(1 + \mathcal H^{-2}\) and hence this term vanishes, i.e. \(k\equiv0\).
By construction we have
\[ \mathcal L B[u,u] = \partial_x[\partial_x^2(1 + \mathcal T^{-2}),u]u_x + 2B[u,uu_x], \]
so taking \(q = w + tB[u,u]\) we obtain
\[ \mathcal Lq = (uq)_x - \frac1{4t}\left(u_xL_y^2\partial_xu - (L_y\partial_xu)^2\right) + (1 - tu_x)B[u,u] + 2t \left(B[u,uu_x] - uB[u,u_x]\right). \]
We claim that
\begin{align}
\|B[u,u]\|_{L^2} &\lesssim \|u_x\|_{L^\infty}\|u\|_{L^2},\label{GoodNFBounds1}\\
\|B[u,uu_x] - uB[u,u_x]\|_{L^2} &\lesssim \|u_x\|_{L^\infty}^2\|u\|_{L^2},\label{GoodNFBounds2} \end{align}
so applying these bounds along with the \(L^4\) estimate \eqref{L4Bound} and integrating by parts in the first term we obtain the estimate
\[
\frac d{dt}\|q\|_{L^2}^2\lesssim \|u_x\|_{L^\infty}\|q\|_{L^2}^2 + \|u_x\|_{L^\infty}(1 + t\|u_x\|_{L^\infty})\|q\|_{L^2}\left(\|u\|_{L^2} + \|L_y^2\partial_xu\|_{L^2}\right), \]
which suffices to complete the proof of \eqref{est:NRG}.
It remains to prove the estimates \eqref{GoodNFBounds1}, \eqref{GoodNFBounds2}. By considering the asymptotic behavior of \(\coth\xi\) we obtain the following asymptotic behavior in the low-high and high-high regimes,
\[
|k_{\mathrm{sym}}(\xi_1,\xi_2)|\lesssim \begin{cases}|\xi_1||\xi_2|^3\langle \xi_2\rangle^{-2},&\quad |\xi_1|\ll|\xi_2|,
\\|\xi_1 + \xi_2|^2|\xi_2|^2\langle \xi_2\rangle^{-2},&\quad |\xi_1 + \xi_2|\ll|\xi_2|.\end{cases} \]
Combining these bounds with the estimate \eqref{FDResonanceLB} for the resonance function we obtain a (crude) bound for the symbol \(b\) of the bilinear operator \(B\),
\[
|b(\mathbf k_1,\mathbf k_2)|\lesssim \begin{cases}|\xi_2|\langle\xi_2\rangle^{-1},&\quad |\xi_1|\ll|\xi_2|,
\\|\xi_1 + \xi_2|\langle\xi_2\rangle^{-1},&\quad |\xi_1 + \xi_2|\ll|\xi_2|.\end{cases} \]
Next we decompose the operator \(B\) using the Littlewood-Paley trichotomy as
\[ B[u,v] = Q_1[u,v_x] + Q_1[v,u_x] + \partial_xQ_2[u,v], \]
where we define the bilinear forms
\[ Q_1[u,v] = \sum\limits_N B[u_{\ll N},\partial_x^{-1}v_N],\qquad Q_2[u,v] =\sum\limits_{N_1\sim N_2} \partial_x^{-1}B[u_{N_1},v_{N_2}]. \]
From the above estimates for \(b\) we see that the corresponding symbols \(q_1,q_2\) are bounded and applying similar estimates for the derivatives we obtain \(q_1,q_2\in\mathscr S\). We may then apply the Coifman-Meyer Theorem \eqref{CM} to obtain the estimates
\[
\|Q_1[u,u_x]\|_{L^2}\lesssim \|u_x\|_{L^\infty}\|u\|_{L^2},\qquad \|\partial_xQ_2[u,u]\|_{L^2}\lesssim \|u_x\|_{L^\infty}\|u\|_{L^2}, \]
which suffice to complete the proof of \eqref{GoodNFBounds1}.
The proof of the estimate \eqref{GoodNFBounds2} is similar, taking advantage of the commutator structure. We first write that the difference
\[ B[u,uu_x] - uB[u,u_x] = C[u,u,u_x], \]
where the operator \(C\) has symbol,
\[ C(\mathbf k_1,\mathbf k_2,\mathbf k_3) = b(\mathbf k_1,\mathbf k_2 + \mathbf k_3) - b(\mathbf k_1,\mathbf k_3). \]
Next we decompose \(C\) according to frequency balance of the last two terms,
\[ C[u,u,u_x] = R_1[u,u_x,u_x] + R_2[u,u_x,u_x], \]
where we define the trilinear forms,
\[ R_1[u,v,w] = \sum\limits_NC[u,\partial_x^{-1}v_N,w_{\leq N}],\qquad R_2[u,v,w] = \sum\limits_NC[u,\partial_x^{-1}v_N,w_{> N}]. \]
For the first term we may use the above estimates for the symbol \(b\) to see that \(R_1\) has symbol \(r_1\in\mathscr S \). We then apply the Coifman-Meyer Theorem \eqref{CM} to obtain the estimate
\[
\|R_1[u,u_x,u_x]\|_{L^2}\lesssim \|u_x\|_{L^\infty}^2\|u\|_{L^2}. \]
For the second term we instead use the commutator structure of \(C\), writing
\[ C(\mathbf k_1,\mathbf k_2,\mathbf k_3) = \int_0^1 \nabla_{\mathbf k_2}b(\mathbf k_1,h\mathbf k_2 + \mathbf k_3)\cdot\mathbf k_2\,dh, \]
and using similar computations to above in the region \(|\xi_2|\ll|\xi_3|\) we have the estimate
\[
|C(\mathbf k_1,\mathbf k_2,\mathbf k_3)|\lesssim |\xi_2|. \]
Applying similar estimates for the derivatives we may show that \(r_2\in\mathscr S\) and once again we apply the Coifman-Meyer Theorem \eqref{CM} to obtain the estimate
\[
\|R_2[u,u_x,u_x]\|_{L^2}\lesssim \|u_x\|_{L^\infty}^2\|u\|_{L^2}, \]
which completes the proof of \eqref{GoodNFBounds2}.
\end{proof}
The proof of Theorem~\ref{thm:LWP} now follows from a standard application of the energy method using the a priori estimate \eqref{prop:NRG} and the following Sobolev estimate:
\begin{lem}\label{lem:ShortTimes}
For times \(t>0\) we have the estimate
\begin{equation}\label{ShortTime}
\|u\|_{L^\infty} + \|u_x\|_{L^\infty}\lesssim t^{-\frac12}\|u\|_{Z^3} \end{equation}
\end{lem} \begin{proof}
We start by decomposing \(u\) with respect to \(x\)-frequency as
\[ u = u_{<1} + \sum\limits_{N\geq1}u_N. \]
Writing
\[ f(t,x,y) = u_{<1}(t,x - \frac{1}{4t}y^2,y), \]
we may apply the Sobolev estimate \eqref{est:Sobolev} to obtain
\[
\|u_{<1}\|_{L^\infty}\lesssim t^{-\frac12}\|u\|_{Z^0},\qquad \|\partial_xu_{<1}\|_{L^\infty}\lesssim t^{-\frac12}\|u\|_{Z^0}. \]
Replacing \(u_{<1}\) by \(u_N\) for \(N\in 2^{\delta\mathbb Z}\) we obtain a similar bound,
\[
\|u_N\|_{L^\infty}\lesssim t^{-\frac12}N^{-\frac32}\|u_N\|_{Z^3},\qquad \|\partial_xu_N\|_{L^\infty}\lesssim t^{-\frac12}N^{-\frac12}\|u_N\|_{Z^3}. \]
Summing over \(N\geq1\) we obtain the estimate \eqref{ShortTime}.
\end{proof}
\section{Pointwise bounds}\label{sect:KS}
In this section we prove that the energy estimates for solutions proved in Section~\ref{sect:AP} lead to corresponding pointwise bounds. In particular, we have the following result:
\begin{prop}\label{prop:BasicPointwise}
For \(t>0\) we have the estimate
\begin{equation}\label{Starter410}
\|u_x\|_{L^\infty}\lesssim |t|^{-\frac12}\langle t\rangle^{-\frac12}\|u\|_X. \end{equation}
\end{prop}
\begin{rem} By combining the a priori estimates of Proposition~\ref{prop:NRG} with Proposition~\ref{prop:BasicPointwise} and a standard bootstrap argument, we obtain the local well-posedness of \eqref{eqn:cc-fd} on \emph{almost} global timescales \(T \approx e^{C \epsilon^{-1}}\). \end{rem}
For times \(0<t< 1\), Proposition~\ref{prop:BasicPointwise} is corollary of the estimate \eqref{ShortTime}. As a consequence, it suffices to consider times \(t\geq 1\). Here we will prove a slightly more involved result that we will subsequently use to upgrade the almost global existence to global existence via a bootstrap argument.
We recall that linear solutions initially localized near the origin in space will propagate along the rays \(\Gamma_{\mathbf v}\) of the Hamiltonian flow. In particular, if the solution is localized at \(x\)-frequency \(N\in 2^{\delta \mathbb Z}\) then at time \(t\) it should be localized in the spatial region \(\{z\approx t m(N)\}\), where we define the spatial variable
\begin{equation}\label{zVar} z = -\left(x + \frac1{4t}y^2\right), \end{equation}
and the non-negative symbol
\begin{equation}\label{LittleM} m(\xi) = 2\xi\coth\xi - \xi^2\cosech^2\xi - 1. \end{equation}
With this heuristic in mind we decompose
\[ u = u^\mr{hyp} + u^\mr{ell}, \]
where the part of the hyperbolic piece localized at frequency \(N\in 2^{\delta\mathbb Z}\) is localized in space so that
\[ t^{-1}z\in B_N^\mr{hyp} := \{v>0: v \sim m(N)\}. \]
We note that
\[ m(\xi)\sim \frac{\xi^2}{\langle\xi\rangle}, \]
so due to the uncertainty principle such a localization is only possible when \(N\geq t^{-\frac13}\). As a consequence we include the low frequencies \(N<t^{-\frac13}\) (for which we may obtain improved decay) in the elliptic piece.
To make this construction rigorous, for each \(N\geq t^{-\frac13}\) we take a smooth bump function \(\chi_N^\mr{hyp} \in C^\infty_0(\mathbb R_+)\), identically \(1\) on the set \(B_N^\mr{hyp}\) and localized up to rapidly decaying tails at frequencies \(|\xi|\lesssim \frac1{m(N)}\). We then define
\[ u^\mr{hyp} = \sum\limits_{N>t^{-\frac13},\pm}u_N^{\mr{hyp},\pm},\qquad u^\mr{ell} = u - u^\mr{hyp}, \]
where, for each \(N\geq t^{-\frac13}\),
\[ u_N^{\mr{hyp},\pm}(t,x,y) = \chi_N^\mr{hyp}(t^{-1}z) u_N^\pm(t,x,y),\qquad u_N^{\mr{ell}}(t,x,y) = \left(1 - \chi_N^\mr{hyp}(t^{-1}z)\right)u_N(t,x,y). \]
\begin{figure}
\caption{A phase space illustration of the hyperbolic region \(B_N^\mr{hyp}\) for a fixed time \(t\geq 1\).}
\label{fig:PhaseSpace1}
\end{figure}
If the solution \(u\) behaves like a linear wave we expect that most of its energy will be concentrated in the hyperbolic piece \(u^\mr{hyp}\) and hence the elliptic piece \(u^\mr{ell}\) will be decay faster than the expected \(O(t^{-1})\) linear decay. As a consequence, we obtain the following pointwise bounds, similar to \cite[Proposition 3.1]{2014arXiv1409.4487H}:
\begin{lem}\label{lem:FDR2} For \(t\geq 1\) and a.e. \((x,y)\in\mathbb R^2\) we have the estimates
\begin{gather}
|u^\mr{hyp}| \lesssim t^{-1}v^{-\frac38}\langle v\rangle^{-\frac78}\|u\|_X,\quad |u_x^\mr{hyp}|\lesssim t^{-1}v^{\frac18}\langle v\rangle^{-\frac38}\|u\|_X,\label{est:FDR2-HYPERBOLIC-2}\\
|u^\mr{ell}| \lesssim t^{-\frac34}\langle t^{\frac23}v\rangle^{-\frac34}\left(1 + \log\langle t^{\frac23}v\rangle\right)\|u\|_X,\quad |u_x^\mr{ell}|\lesssim t^{-\frac{13}{12}}\langle t^{\frac23}v\rangle^{-\frac14}\langle t^{\frac12}v\rangle^{\frac14}\langle t^{-\frac12}v\rangle^{-\frac5{12}}\|u\|_X\label{est:FDR2-ELLIPTIC-2}. \end{gather}
\end{lem}
\begin{rem}
The unusual scaling of the estimate \eqref{est:FDR2-ELLIPTIC-2} is a consequence of the fact that due to the weaker dispersion we must use additional derivatives to control the high \(x\)-frequencies rather than just the vector fields. This breaks the natural scaling of the other estimates.
\end{rem}
\begin{rem}
In the infinite depth case we have \(m(\xi) = 2|\xi|\) and hence the high frequency threshold is \(N\geq t^{-\frac12}\) rather than \(N\geq t^{-\frac13}\). The slightly different form of the function \(m\) leads to minor adjustments to the numerology of Lemma~\ref{lem:FDR2}:
\begin{lem} If \(h = \infty\) then for \(t\geq 1\) and a.e. \((x,y)\in\mathbb R^2\) we have the estimates
\begin{gather}
|u^\mr{hyp}| \lesssim t^{-1}v^{-\frac14}\langle v\rangle^{-1}\|u\|_X,\quad |u_x^\mr{hyp}|\lesssim t^{-1}v^{\frac34}\langle v\rangle^{-1}\|u\|_X,\\
|u^\mr{ell}| \lesssim t^{-\frac78}\langle t^{\frac12}v\rangle^{-\frac34}\left(1 + \log\langle t^{\frac12}v\rangle\right)\|u\|_X,\quad |u_x^\mr{ell}|\lesssim t^{-\frac98}\langle t^{-\frac12}v\rangle^{-\frac5{12}}\|u\|_X. \end{gather}
\end{lem} \end{rem}
In order to prove Lemma~\ref{lem:FDR2} we require some auxiliary estimates so we delay the proof until Section~\ref{sect:ProofPointwise}.
\subsection{The solution to the eikonal equation}
In this section we construct the solution \(\phi\) to the eikonal equation
\begin{equation}\label{eikonal} \ell(\phi_t,\phi_x,\phi_y) = 0, \end{equation}
where \(\ell\) is the symbol of the linear operator \(\mathcal L\), given by
\[ \ell(\tau,\xi,\eta) = \tau - \xi^2\coth\xi + \xi + \xi^{-1}\eta^2. \]
If we make the ansatz that for \(z>0\),
\[ \phi(t,x,y) = t\Phi(t^{-1}z), \]
then we obtain an ODE for \(\Phi = \Phi(v)\),
\begin{equation}\label{EikonalODE} \Phi - v\Phi' + (\Phi')^2\coth(\Phi') - \Phi' = 0. \end{equation}
Differentiating we obtain
\[ (m(\Phi') - v)\Phi'' = 0, \]
and hence
\[ \Phi'(v) = m^{-1}(v), \]
where the positive inverse \(m^{-1}(v)>0\) may be defined using the Inverse Function Theorem for \(v>0\). As a consequence we have the following lemma:
\begin{lem} For \(z>0\) the solution to the eikonal equation \eqref{eikonal} is given by
\begin{equation}\label{PHASE} \phi(t,x,y) = t\Phi(t^{-1}z), \end{equation}
where
\[ \Phi(v) = (m^{-1}(v))^2\coth(m^{-1}(v)) - (1 + v)m^{-1}(v). \]
\end{lem}
We finish this section by noting that we have the estimates
\[ 0< m(\xi)\sim \frac{\xi^2}{\langle \xi\rangle}, \] and \[ 0<m^{-1}(v)\sim \langle v\rangle^{\frac12}v^{\frac12}. \]
In particular, we may show that the solution to the eikonal equation satisfies the estimate
\[
|\Phi(v)|\sim \langle v\rangle^{\frac12}v^{\frac32}. \]
\begin{rem} In the infinite depth case we have \(\Phi(v) = -\frac14 v^2\) and hence the solution to the eikonal equation is simply
\[ \phi = -\frac{z^2}{4t}. \]
\end{rem}
\begin{figure}
\caption{The graph of the function \(v = m(\xi)\).}
\end{figure}
\subsection{Elliptic estimates}
In this section we establish weighted \(L^2\)-estimates for the frequency localized pieces \(u_N\). As we expect to obtain improved decay at very low frequencies \(N< t^{-\frac13}\) regardless, we restrict out attention to high frequencies \(N\geq t^{-\frac13}\).
In order to control the localization of solutions we define an operator adapted to the hyperbolic/elliptic decomposition of \(u\) by
\begin{equation}\label{Lz} L_z = z - tm(D_x), \end{equation}
so that the symbol of \(L_zP_N\) is elliptic away from the set \(B_N^\mr{hyp}\). We note that we may write
\[ L_z\partial_x = - \left(J + \frac{1}{4t}L_y^2\partial_x + \frac12\right), \]
and hence for \(t\geq 1\) we have the estimate
\begin{equation}\label{est:LzBound}
\|L_z\partial_xu_N\|_{L^2}\lesssim \|u_N\|_X. \end{equation}
In order to obtain more detailed estimates for the hyperbolic piece we observe that for a given \(z>0\) there exist two solutions to the equation \(m(\xi) = t^{-1}z\). As a consequence we construct operators adapted to each of these roots,
\[ L_z^\pm = m^{-1}(t^{-1}z) \pm i\partial_x, \]
where \(L_z^-\) (respectively \(L_z^+\)) will be elliptic if \(u_N\) is localized to positive (respectively negative) wavenumbers. A useful observation, and indeed our main motivation for introducing these operators is that if \(\phi\) is the solution to the eikonal equation \eqref{eikonal} defined as in \eqref{PHASE} then we have
\[ \partial_x(e^{-i\phi}f) = -ie^{-i\phi}L_z^+f. \]
Our main elliptic estimates are then the following:
\begin{lem}\label{lem:Elliptic}
For \(t\geq1\) and \(N\geq t^{-\frac13}\) we have the estimates
\begin{align}
\frac N{\langle N\rangle}\|L_z^+ u_N^{\mr{hyp},+}\|_{L^2} &\lesssim \frac1{tN}\|u_N\|_X ,\label{est:FDR2-Elliptic1}\\
\frac{N^2}{\langle N\rangle}\|u_N^\mr{ell}\|_{L^2} + \|v u_N^\mr{ell}\|_{L^2} &\lesssim \frac1{tN}\|u_N\|_X,\label{est:FDR2-Elliptic2} \end{align}
where \(v = t^{-1}z\).
\end{lem}
\begin{rem} In the infinite depth case the operator
\[
L_z = z - 2t|D_x|. \]
As a consequence, the natural analogues,
\( L_z^\pm = \dfrac z{2t} \pm i\partial_x \),
satisfy
\[ L_z^\pm P_\pm = \frac1{2t}L_z P_\pm, \]
so we do not expect a gain of regularity for the hyperbolic piece as in \eqref{est:FDR2-Elliptic1} (see also~\cite{MR3382579}). For the elliptic piece we instead have the estimate
\[
N\|u_N^\mr{ell}\|_{L^2} + \|v u_N^\mr{ell}\|_{L^2} \lesssim \frac1{tN}\|u_N\|_X. \]
\end{rem}
\begin{proof}[Proof of Lemma~\ref{lem:Elliptic}]
As these are effectively one-dimensional estimates, we ignore the dependence upon \(y\) and treat \(t\geq1\) as a fixed parameter.
\emph{Proof of \eqref{est:FDR2-Elliptic1}.}
For high, positive wavenumbers \(\xi\approx N\geq1\) we may write the symbol \(m^{-1}(t^{-1}z) - \xi\) of \(L_z^+\) in terms of the symbol \(z - tm(\xi)\) of \(L_z\) as
\[ m^{-1}(t^{-1}z) - \xi = t^{-1}(z - tm(\xi))p(t^{-1}z,\xi), \]
where the smooth function
\[ p(v,\xi) = \frac{m^{-1}(v) - \xi}{v - m(\xi)}, \]
is elliptic in the region \(v\sim m(\xi)\) for \(\xi\geq1\). We recall that \(u_{N,+}^\mr{hyp}\) is localized in space in the set \(B_N^\mr{hyp}\) and in frequency at positive wavenumbers \(\xi\approx N\) up to rapidly decaying tails at scale \(tN\). In particular we may harmlessly localize \(p(v,\xi)\) using cutoffs in space and frequency and then apply the product rule \eqref{ProductRule} and the elliptic estimate \eqref{OPSElliptic} to obtain
\[
\|L_{z,+}u_N^\mr{hyp}\|_{L^2}\lesssim \frac1t\left(\|L_zu_N^\mr{hyp}\|_{L^2} + \|u_N^\mr{hyp}\|_{L^2}\right) \lesssim \frac1{tN}\|u_N\|_X. \]
For low, positive wavenumbers \(\xi\approx N\) so that \(t^{-\frac13}\leq N < 1\) we instead write the product of the symbols of \(L_z^-L_z^+\) as
\[ (m^{-1}(t^{-1}z) + \xi)(m^{-1}(t^{-1}z) - \xi) = t^{-1}(z - tm(\xi))q(t^{-1}z,\xi), \]
where
\[ q(v,\xi) = \frac{(m^{-1}(v))^2 - \xi^2}{v - m(\xi)}\in\mathscr S \]
A similar estimate to above yields the bound
\[
\|L_z^-L_z^+u_{N}^{\mr{hyp},+}\|_{L^2}\lesssim \frac1{tN}\left(\|L_z\partial_xu_N\|_{L^2} + \|u_N\|_{L^2}\right). \]
For sufficiently smooth \(w\) we then calculate
\[
\|wf\|_{L^2}^2 + \|f_x\|_{L^2}^2 = \|L_z^-f\|_{L^2}^2 + 2\mathop{\rm Im}\nolimits\int wf\cdot\bar f_x. \]
Applying this with \(w = m^{-1}(t^{-1}z)\) and \(f = L_z^+u_{N}^{\mr{hyp},+}\), which is localized at positive wavenumbers \(\xi \sim N\) and in space in the region \(B_N^\mr{hyp}\) up to rapidly decaying tails, we obtain the estimate
\begin{align*}
\|f\|_{L^2}^2 &\lesssim \frac1{N^2}\left(\|wf\|_{L^2}^2 + \|f_x\|_{L^2}^2\right) +\frac1{t^2N^4}\|u_N\|_{L^2}^2\\
&\lesssim \frac1{N^2} \|L_z^-f\|_{L^2}^2 + \frac1{N^2}\mathop{\rm Im}\nolimits\int wf\cdot\bar f_x + \frac1{t^2N^4}\|u_N\|_{L^2}^2\\
&\lesssim \frac1{t^2N^4}\left(\|L_z\partial_xu_N\|_{L^2}^2 + \|u_N\|_{L^2}^2\right) \end{align*}
Combining these estimates we obtain \eqref{est:FDR2-Elliptic1}.
\emph{Proof of \eqref{est:FDR2-Elliptic2}.}
We define the Fourier multiplier
\[ a(\xi) = \sqrt{m(\xi)}, \]
and take \(A = a(D_x)\) so that
\[ L_z = z - tA^2. \]
Integrating by parts for real-valued \(f\) we obtain the identity
\begin{equation}\label{est:FDR2-IBP}
\|v f_x\|_{L^2}^2 - 2\int v|Af_x|^2\,dx + \|A^2f_x\|_{L^2}^2 = t^{-2}\|L_z\partial_xf\|_{L^2}^2, \end{equation}
where have used that the symbol of the operator \(A[A,v]\) is skew-adjoint.
We then smoothly decompose the elliptic part of \(u_N\)
\[
u_N^\mr{ell} = \chi_{\{|z|\ll tm(N)\}}u_N + \chi_{\{z\approx - tm(N)\}}u_N + \chi_{\{|z|\gg tm(N)\}}u_N, \]
where the smooth cutoffs are assumed to have compact support and be localized in frequency near zero at the scale of uncertainty.
For the first and last piece we apply the estimate \eqref{est:FDR2-IBP} with \(f = \chi_{\{|v|\ll m(N)\}}u_N\), \(\chi_{\{|z|\gg tm(N)\}}u_N\) respectively to obtain
\[
\|v f_x\|_{L^2} + m(N)\|f_x\|_{L^2} \lesssim \frac 1t\|u_N\|_X + \sqrt{m(N)}\|\sqrt vf_x\|_{L^2}, \]
so from the spatial localization of \(f\) we obtain,
\[
\|v f_x\|_{L^2} + m(N)\|f_x\|_{L^2}\lesssim \frac 1t\|u_N^\mr{ell}\|_X. \]
For the remaining piece we use that for \(f = \chi_{\{z\approx - tm(N)\}}u_N\) the function \(Af_x\) is localized in the spatial region \(v <0 \) up to rapidly decaying tails to obtain a similar estimate,
\[
\|v f_x\|_{L^2} + m(N)\|f_x\|_{L^2}\lesssim \frac 1t\|u_N^\mr{ell}\|_X. \]
Combining these bounds with the fact that \(f\) is localized at frequencies \(\sim N\) up to rapidly decaying tails, we obtain the estimate \eqref{est:FDR2-Elliptic2}.
\end{proof}
\begin{rem}
We obeserve that we may combine the estimate \eqref{est:FDR2-Elliptic2} with the elementary low frequency estimate
\[
\|\partial_xu_{\leq t^{-\frac13}}\|_{L^2}\lesssim t^{-\frac13}\|u\|_{L^2}, \]
to obtain the estimate for the elliptic piece
\begin{equation}\label{est:EllipticGain}
\|\partial_xu^{\mr{ell}}\|_{L^2}\lesssim t^{-\frac13}\|u\|_X. \end{equation}
Further, in the infinite depth case we have the corresponding estimate,
\begin{equation}
\|\partial_xu^{\mr{ell}}\|_{L^2}\lesssim t^{-\frac12}\|u\|_X. \end{equation}
\end{rem}
\subsection{Proof of Lemma~\ref{lem:FDR2}}\label{sect:ProofPointwise}
We now apply the elliptic estimates of Lemma~\ref{lem:Elliptic} to prove Lemma~\ref{lem:FDR2}. As the estimates are linear in \(u\) we will assume that \(\|u\|_X = 1\). Throughout this section we use the notation \(v = t^{-1}z\).
\emph{Low frequencies.} We first consider the low frequency part \(u_{\leq t^{-\frac13}}\). We recall that at low frequency \(m(\xi)\approx \xi^2\) and hence the operator \(t^{-1}L_z = v - m(D_x) \approx v\) whenever \(v\gg t^{-\frac23}\) whereas \(t^{-1}L_z\approx \partial_x^2\) whenever \(v\ll t^{-\frac23}\).
If \(|v|< t^{-\frac23}\) we take \(f(t,x,y) = u_{< t^{-\frac13}}(t,x - \frac{1}{4t}y^2,y)\). We then apply Bernstein's inequality to obtain the bounds
\begin{gather*}
\|f\|_{L^2}\lesssim \|u_{<t^{-\frac13}}\|_{L^2}\lesssim 1,\qquad \|f_x\|_{L^2}\lesssim \|\partial_xu_{<t^{-\frac13}}\|_{L^2}\lesssim t^{-\frac13},\\ \|f_{yy}\|_{L^2}\lesssim t^{-2}\|L_y^2\partial_x^2u_{<t^{-\frac13}}\|_{L^2}\lesssim t^{-\frac73}. \end{gather*}
Applying the Sobolev estimate as in Lemma~\ref{lem:ShortTimes} we obtain the estimate,
\[
\|u_{<t^{-\frac13}}\|_{L^\infty}\lesssim t^{-\frac34}. \]
Essentially identical estimates to \(f_x\) yield a similar bound,
\[
\|\partial_xu_{< t^{-\frac13}}\|_{L^\infty}\lesssim t^{-\frac{13}{12}}. \]
If instead \(|v|\geq t^{-\frac23}\) we dyadically decompose in space, taking \(\chi_M\) to localize to the spatial region \(\{|v|\sim M\}\) for each \(M>t^{-\frac23}\). We then decompose \(\chi_Mu_{\leq t^{-\frac13}}\) at the scale of the uncertainty principle as
\[ \chi_Mu_{< t^{-\frac13}} = \chi_Mu_{< (tM)^{-1}} + \sum\limits_{(tM)^{-1}\leq N< t^{-\frac13}}\chi_Mu_N. \]
For any \(N<t^{-\frac13}\) we have the estimate
\[
\|v\partial_xu_N\|_{L^2}\lesssim t^{-1}\left(\|L_z\partial_xu\|_{L^2} + tNm(N)\|u\|_{L^2}\right) \lesssim t^{-1}, \]
where we have used that \(Nm(N)\approx N^3\lesssim t^{-1} \) whenever \(N<t^{-\frac13}\). As a consequence we obtain the bounds,
\[
\|\partial_x(\chi_Mu_{< (tM)^{-1}})\|_{L^2}\lesssim (tM)^{-1},\qquad \|\partial_x(\chi_Mu_N)\|_{L^2}\lesssim (tM)^{-1}. \]
We then apply the Sobolev estimate \eqref{est:Sobolev} as before to
\[ f(t,x,y) = \chi_M(t^{-1}x)u_{< (tM)^{-1}}(t,x - \frac{1}{4t}y^2,y),\qquad f(t,x,y) = \chi_M(t^{-1}x)u_N(t,x - \frac{1}{4t}y^2,y), \]
respectively to obtain the bounds,
\[
\|\chi_M u_{< (tM)^{-1}}\|_{L^\infty} + \|\chi_M u_N\|_{L^\infty}\lesssim t^{-\frac12}(tM)^{-\frac34}. \]
Replacing \(u\) by \(u_x\) we obtain similar bounds,
\[
\|\chi_M \partial_xu_{< (tM)^{-1}}\|_{L^\infty}\lesssim t^{-\frac12}(tM)^{-\frac74},\qquad \|\chi_M \partial_xu_N\|_{L^\infty} \lesssim t^{-\frac12}(tM)^{-\frac34}N. \]
Summing over \(N\) we obtain the estimates,
\[
|\chi_Mu_{<t^{-\frac13}}|\lesssim t^{-\frac34}(t^{\frac23}M)^{-\frac34}\log(t^{\frac23}M),\qquad |\chi_M\partial_xu_{<t^{-\frac13}}|\lesssim t^{-\frac{13}{12}}(t^{\frac23}M)^{-\frac34}, \]
where the logarithmic loss arises due to summation over frequencies \((tM)^{-1}\leq N\leq t^{-\frac13}\) (see also the corresponding bound in \cite[Proposition~3.1]{2014arXiv1409.4487H}).
Combining these bounds with the estimate in the region for \(|v|<t^{-\frac23}\) we obtain the low frequency estimates
\begin{equation}
|u_{< t^{-\frac13}}|\lesssim t^{-\frac34}\langle t^{\frac23} v\rangle^{-\frac34}\left(1 + \log\langle t^{\frac23} v\rangle\right),\qquad |\partial_xu_{< t^{-\frac13}}|\lesssim t^{-\frac{13}{12}}\langle t^{\frac23} v\rangle^{-\frac34}.\label{est:FDR2-LF-2} \end{equation}
\emph{Elliptic piece.} For the high frequency part of the elliptic piece we proceed similarly to the low frequency piece. For \(N\geq t^{-\frac13}\) we apply the Sobolev estimate \eqref{est:Sobolev} and the elliptic estimate \eqref{est:FDR2-Elliptic2} to \(f(t,x,y) = u_N^\mr{ell}(t,x - \frac{1}{4t}y^2,y)\) on dyadic spatial intervals (as for the low frequency piece) to obtain the pointwise bound,
\[
|u^\mr{ell}_N|\lesssim t^{-\frac54}\min\{N^{-\frac32}\langle N\rangle^{\frac34},|v|^{-\frac34}\}. \]
If \(|v|< t^{-\frac23}\) then we sum in \(N\) to obtain
\[
|u^\mr{ell}_{\geq t^{-\frac13}}|\lesssim \sum\limits_{N\geq t^{-\frac13}}t^{-\frac54}N^{-\frac32}\langle N\rangle^{\frac34}\lesssim t^{-\frac34}. \]
If \(|v|\geq t^{-\frac23}\) then we decompose the sum as
\[
|u^\mr{ell}_{\geq t^{-\frac13}}|\lesssim \sum\limits_{t^{-\frac13}\leq N\leq v^{\frac12}\langle v\rangle^{\frac12}} t^{-\frac54}|v|^{-\frac34} + \sum\limits_{N\geq v^{\frac12}\langle v\rangle^{\frac12}}t^{-\frac54}N^{-\frac32}\langle N\rangle^{\frac34} \lesssim t^{-\frac34}(t^{\frac23}|v|)^{-\frac34}\left(1 + \log (t^{\frac13}v^{\frac12}\langle v\rangle^{\frac12})\right). \]
Combining these, we obtain the bound,
\[
|u^\mr{ell}_{\geq t^{-\frac13}}|\lesssim t^{-\frac34}\langle t^{\frac23}v\rangle^{-\frac34}\left(1 + \log\langle t^{\frac23}v\rangle\right). \]
As \(u_N^\mr{ell}\) is localized at frequencies \(|\xi|\sim N\) up to rapidly decaying tails, we obtain a similar estimate for the derivative,
\[
|\partial_xu^\mr{ell}_N|\lesssim t^{-\frac54}N\min\{N^{-\frac32}\langle N\rangle^{\frac34},|v|^{-\frac34}\}, \]
Summing over \(t^{-\frac13}\leq N<1\) we then obtain the bound,
\[
|\partial_xu^\mr{ell}_{t^{-\frac13}\leq \cdot <1}|\lesssim t^{-\frac{13}{12}}\langle t^{\frac23}v\rangle^{-\frac14}. \]
For \(N>1\) we may use the fact that \(\|\partial_x^4u_N\|_{L^2}\lesssim 1\) in lieu of the elliptic estimate \eqref{est:FDR2-Elliptic2} to obtain the slight modification
\[
|\partial_xu^\mr{ell}_N|\lesssim t^{-\frac12}N\min\{(tN)^{-\frac34},(t|v|)^{-\frac34},N^{-\frac94}\}, \]
from which we obtain the bound
\[
|\partial_xu^\mr{ell}_{>1}|\lesssim t^{-\frac98}\langle t^{-\frac12}v\rangle^{-\frac5{12}}, \]
completing the proof of \eqref{est:FDR2-ELLIPTIC-2}.
\emph{Hyperbolic piece.} We define the phase function \(\phi\) as in \eqref{PHASE} and observe that if we apply the Sobolev estimate \eqref{est:Sobolev} to \(f(t,x,y) = e^{-i\phi}u_N^{\mr{hyp},+}(t,x - \frac{1}{4t}y^2,y)\) we obtain the estimate
\[
|u_N^{\mr{hyp},+}|\lesssim t^{-\frac12}\|u_N\|_{L^2}^{\frac14}\|L_z^+u_N^{\mr{hyp},+}\|_{L^2}^{\frac12}\|(L_y\partial_x)^2u_N\|_{L^2}^{\frac14}. \]
We then use the elliptic estimate \eqref{est:FDR2-Elliptic1} and that \(u_N^{\mr{hyp},+}\) is localized in the spatial region \(v\approx m(N)\) and at frequencies \(\sim N\) up to rapidly decaying tails to obtain
\[
|u_N^{\mr{hyp},+}|\lesssim t^{-1}N^{-\frac34}\langle N\rangle^{-\frac12},\qquad |\partial_xu_N^{\mr{hyp},+}|\lesssim t^{-1}N^{\frac14}\langle N\rangle^{-\frac12} \]
If \(t^{-\frac13}\leq N<1\) then on the support of \(u_N^{\mr{hyp},+}\) we have \(v\approx N^2\) so using that the supports of the \(u_N^{\mr{hyp},+}\) are essentially disjoint we may sum to obtain
\[
|u_{t^{-\frac13}\leq \cdot<1}^{\mr{hyp},+}|\lesssim t^{-1}v^{-\frac38},\qquad |\partial_xu_{t^{-\frac13}\leq \cdot<1}^{\mr{hyp},+}|\lesssim t^{-1}v^{\frac18}. \]
Similarly, if \(N\geq1\) we have \(v\approx N\) on the support of \(u_N^{\mr{hyp},+}\) and hence
\[
|u_{\geq 1}^{\mr{hyp},+}|\lesssim t^{-1}v^{-\frac54},\qquad |\partial_xu_{\geq 1}^{\mr{hyp},+}|\lesssim t^{-1}v^{-\frac14}. \]
By combining these estimates we obtain the bound \eqref{est:FDR2-HYPERBOLIC-2}, which completes the proof of Lemma~\ref{lem:FDR2}.\qed
\section{Testing by wave packets}\label{sect:TWP}
We now turn to the problem of proving the existence of global solutions to \eqref{eqn:cc-fd} using a bootstrap argument. We assume that for some \(T>1\) there exists a solution \(S(-t)u\in C([0,T];X(0))\) to \eqref{eqn:cc-fd} satisfying the bootstrap assumption,
\begin{equation}\label{BS}
\sup\limits_{t\in[0,T]}\|u_x\|_{L^\infty}\leq \mathcal M\epsilon t^{-\frac12}\langle t\rangle^{-\frac12}. \end{equation}
Applying the a priori bound \eqref{est:NRG} we obtain the estimate
\begin{equation}\label{AP}
\|u\|_X\lesssim \epsilon \langle t\rangle^{C\mathcal M\epsilon}, \end{equation}
where the constants are independent of \(\mathcal M\).
From the pointwise bounds proved in Lemma~\ref{lem:FDR2}, we see that the worst decay occurs in the region \(\{z\approx t\}\). As a consequence we define the time-dependent set
\[ \Sigma_t = \{v\in\mathbb R: t^{-\frac1{12}}< v < t^{\frac1{12}}\}, \]
so that the worst behavior will occur whenever \(t^{-1}z\in \Sigma_t\). If we take \(\chi_{\Sigma_t^c}\) to be a smooth bump function supported in the complement \(\Sigma_t^c = \mathbb R^2\backslash\Sigma_t\) we may then apply the estimates of Lemma~\ref{lem:FDR2} to obtain
\begin{equation}\label{ImprovedDecay}
\|u_x\ \chi_{\Sigma_t^c}(t^{-1}z)\|_{L^\infty}\lesssim t^{-\frac{97}{96}}\|u\|_X. \end{equation}
The additional \(t\)-decay in the estimate \eqref{ImprovedDecay} leads to an improvement of \eqref{BS} for sufficiently large times. As a consequence it remains to consider improved pointwise bounds for \(u_x\) in the region \(\Sigma_t\).
\begin{figure}
\caption{The region \(\{\frac zt\in \Sigma_t\}\).}
\end{figure}
\subsection{Construction of the wave packets} Given a time \(t\geq 1\) and a velocity \(\mathbf v = (v_x,v_y)\in\mathbb R^2\) such that
\[ v = - \left(v_x + \frac 14 v_y^2\right) \in \Sigma_t, \]
we construct a wave packet adapted to the associated ray \(\Gamma_{\mathbf v} = \{(x,y) = t\mathbf v\}\) of the Hamiltonian flow by
\[ \Psi_{\mathbf v}(t,x,y) = \partial_x\left(\frac1{i\partial_x\phi}e^{i\phi}\chi\left(\lambda_z(z - tv),\lambda_y(y - tv_y)\right)\right), \]
where \(\chi\in C^\infty_0(\mathbb R^2)\) is a smooth, non-negative, real-valued, compactly supported function, localized near \(0\) in space and frequency at scale \(\lesssim 1\), the phase \(\phi\) is defined as in \eqref{PHASE} and the scales
\[ \lambda_z = t^{-\frac12}v^{-\frac14}\langle v\rangle^{\frac14},\qquad \lambda_y = t^{-\frac12}v^{\frac14}\langle v\rangle^{\frac14}. \]
For simplicity we normalize \(\int_{\mathbb R^2} \chi(z,y)\,dzdy = 1\). We also note that by a slight abuse of notation we consider \(v\) to be a fixed parameter (independent of \(t,z\)) in this section.
\begin{rem}
If our initial data is localized near the origin in space and at frequency \(\mathbf k_0 = (\xi_0,\eta_0)\) then the corresponding linear solution will be spatially localized on the ray \(\Gamma_{\mathbf v} = \{z = t\mathbf v\}\) of the Hamiltonian flow, where the group velocity
\[ \mathbf v = - \nabla\omega(\mathbf k_0) = (- m(\xi_0) - \xi_0^{-2}\eta_0^2,2\xi_0^{-1}\eta_0). \]
We note that the frequency may then be written in terms of the velocity as
\[ \mathbf k_0 = \left(m^{-1}(v),\frac12v_y m^{-1}(v)\right). \]
If a linear solution is localized near the ray \(\Gamma_{\mathbf v}\) in space at scale \(\lambda_z^{-1}\) in \(z\) and \(\lambda_y^{-1}\) in \(y\) then from the uncertainty principle, the Fourier transform may be localized at most such that
\[
\left|\xi - \xi_0\right|\lesssim\lambda_z,\qquad \left|(\eta - \eta_0) - \frac{\eta_0}{\xi_0}(\xi - \xi_0)\right|\lesssim \lambda_y. \]
In order for a function to be coherent on timescales \(\approx T\) we require that \(\omega(\mathbf k)\) may be well-approximated by its linearization, with errors of size \(\ll T^{-1}\). Computing the Taylor expansion of the dispersion relation \(\omega(\mathbf k)\) at frequency \(\mathbf k = \mathbf k_0\) we obtain
\[ \omega(\mathbf k) = \omega(\mathbf k_0) - \mathbf v\cdot(\mathbf k - \mathbf k_0) + \frac12(\mathbf k - \mathbf k_0)\cdot\nabla^2\omega(\mathbf k_0) (\mathbf k - \mathbf k_0) + \dots. \]
With the above localization we calculate
\[
\left|(\mathbf k - \mathbf k_0)\cdot\nabla^2\omega(\mathbf k_0) (\mathbf k - \mathbf k_0)\right|\lesssim m'(\xi_0) \lambda_z^2 + \xi_0^{-1} \lambda_y^2. \]
Thus we require,
\[ T\lambda_z^2\lesssim \frac1{m'(\xi_0)}\sim \frac{\langle v\rangle^{\frac12}}{v^{\frac12}},\qquad T\lambda_y^2\lesssim \xi_0\sim v^{\frac12}\langle v\rangle^{\frac12}, \]
which motivates the choice of scales. \end{rem}
\begin{rem}
In the infinite depth case \(h = \infty\) essentially identical reasoning yields the scales
\[ \lambda_z = t^{-\frac12},\qquad \lambda_y = t^{-\frac12}v^{\frac12}. \]
\end{rem}
\begin{figure}
\caption{An illustration of the localization of the wave packet \(\Psi_{\mathbf v}\) in \((z,\xi)\)-phase space.}
\end{figure}
In order to clarify these heuristics we first make the following definition:
\begin{defn}
Given a fixed time \(t\geq1\), a velocity \(\mathbf v\in \mathbb R^2\) so that \(v\in \Sigma_t\) and a (possibly \(t,v\)-dependent) constant \(\Upsilon>0\), we say \(f\in WP(t,\mathbf v,\Upsilon)\) if \(f\in C^\infty_0(\mathbb R^2)\) is supported in the set \(\{\frac zt\in \tilde\Sigma_t\}\), where \(\tilde \Sigma_t\) is a slight dilation of \(\Sigma_t\), and for all \(\alpha,\beta,\mu,\nu\geq 0\) we have the estimate,
\begin{equation}
\left|(z - tv)^\mu (y - tv_y)^\nu(\partial_x - i\partial_x\phi)^\alpha (\partial_xL_y)^\beta f\right|\lesssim_{\alpha,\beta,\mu,\nu}\lambda_z^{\alpha - \mu}\lambda_y^{\beta - \nu}\Upsilon. \end{equation}
\end{defn}
\noindent We note that if \(f\in WP(t,\mathbf v,\Upsilon)\) then it is localized in space near the ray \(\Gamma_{\mathbf v}\) and in frequency near the corresponding frequency \(\mathbf k = \left(m^{-1}(v),\frac12v_y m^{-1}(v)\right)\) at the scale of uncertainty. Using this definition we may clarify the structure of the wave packet \(\Psi_{\mathbf v}\):
\begin{lem}\label{lem:Microlocal}
For all times \(t\gg1\) sufficiently large and all \(\mathbf v\in \mathbb R^2\) be chosen such that \(v\in\Sigma_t\), the associated wave packet \(\Psi_{\mathbf v}\in WP(t,\mathbf v,1)\) and \(\mathcal L\Psi_{\mathbf v}\in WP(t,\mathbf v,t^{-1})\).
Further, writing \(\chi = \chi(\lambda_z(z - tv),\lambda_y(y - tv_y))\), we have the decomposition
\begin{equation}\label{LinearForm} \begin{aligned} e^{-i\phi}\mathcal L\Psi_{\mathbf v} &= - \partial_x\left(\frac1{2t}(z - tv)\chi\right) - \partial_xL_y\left(\frac1{4t^2}(y - tv_y)\chi\right) + \partial_x\left(\frac12im'(\partial_x\phi)\partial_x\chi\right)\\ &\quad + (\partial_xL_y)^2\left(\frac1{i4t^2\partial_x\phi}\chi\right) + \mb{err}, \end{aligned} \end{equation}
where the error term \(\mb{err}\in WP(t,\mathbf v,t^{-\frac32}v^{-\frac34}\langle v\rangle^{-\frac14})\).
\end{lem} \begin{proof}
We may write the wave packet in the form
\begin{equation}\label{LeadingOrderDecomp} e^{-i\phi}\Psi_{\mathbf v} = \chi\left(\lambda_z(z - tv),\lambda_y(y - tv_y)\right) + \frac{\lambda_z}{i\partial_x\phi}\chi_z\left(\lambda_z(z - tv),\lambda_y(y - tv_y)\right). \end{equation}
We recall that \(\partial_x\phi = m^{-1}(t^{-1}z)\) and a simple application of the Inverse Function Theorem yields the estimates
\begin{gather*}
m^{-1}(v)\sim v^{\frac12}\langle v\rangle^{\frac12},\qquad |\partial_vm^{-1}(v)|\lesssim v^{-\frac12}\langle v\rangle^{\frac12},\\
|\partial_v^\alpha m^{-1}(v)|\lesssim_k v^{-(\frac12 + \alpha)}\langle v\rangle^{-k},\quad \alpha\geq 2. \end{gather*}
We then recall that if \((x,y)\in \operatorname{supp}\Psi_{\mathbf v}\) then \(|z - tv|\lesssim \lambda_z^{-1} \lesssim t^{\frac12}v^{-\frac14}\langle v\rangle^{\frac14}\) so provided \(t\gg1\) we have
\[ m^{-1}(t^{-1}z)\sim m^{-1}(v) \]
As a consequence we may differentiate to obtain
\[
\left|\partial_x^\alpha\left(\frac{\lambda_z}{i\partial_x\phi}\right)\right|\lesssim_\alpha t^{-(\frac12+\alpha)} v^{-(\frac34 + \alpha)}\langle v\rangle^{-\frac14}\ll \lambda_z^\alpha, \]
whenever \(\alpha\geq 0\), \(t\gg1\), \((x,y)\in \operatorname{supp}\Psi_{\mathbf v}\) and \(v\in \Sigma_t\). Differentiating \eqref{LeadingOrderDecomp} with respect to \(\partial_x\), \(\partial_xL_y\) and using the fact that \(\chi\) is compactly supported near \(0\), we obtain \(\Psi_{\mathbf v}\in WP(t,\mathbf v,1)\).
In order to calculate \(\mathcal L\Psi_{\mathbf v}\), we first define the Fourier multiplier
\[ M(\xi) = \xi^2\coth\xi - \xi, \]
so that \(M(D_x) = i(\partial_x^2\mathcal T^{-1} - \partial_x)\) and \(m = \partial_\xi M\). Using the Taylor expansion of the symbol about the point \(\xi = \partial_x\phi\) we obtain
\begin{equation}\label{NonlocalExpansion} \begin{aligned} M(D_x)\Psi_{\mathbf v} &= M(\partial_x\phi)\Psi_{\mathbf v} + m(\partial_x\phi)\left(D_x - \partial_x\phi\right)\Psi_{\mathbf v} + \frac12m'(\partial_x\phi)\left(D_x - \partial_x\phi\right)^2\Psi_{\mathbf v}\\ &\quad + \frac i{2t}\Psi_{\mathbf v} + \mb{err}_0, \end{aligned} \end{equation}
where we have used that \(\partial_x\phi = m^{-1}(t^{-1}z)\) so \(m'(\partial_x\phi)\partial_x^2\phi = -\frac1t\), and the error term \(\mb{err}_0\) may be written using the Fourier transform as
\begin{align*} \mb{err}_0 &= \frac1{2\pi}\int \left(\frac12\int_0^1m''\left(\partial_x\phi + \tau(\xi - \partial_x\phi)\right)(1 - \tau)^2(\xi - \partial_x\phi)^3\,d\tau\right)\hat\Psi_{\mathbf v}(t,\xi,\eta)e^{i(\xi x + \eta y)}\,d\xi d\eta. \end{align*}
Using the facts that \(\Psi_{\mathbf v}\in WP(t,\mathbf v,1)\) and \(|m''(\xi)|\lesssim_k \langle \xi\rangle^{-k}\) it is quickly observed that the error term satisfies \(\mb{err}_0\in WP(t,\mathbf v,t^{-\frac32}v^{-\frac34}\langle v\rangle^{-k})\).
As a consequence it remains to consider the approximate linear operator,
\[ \tilde{\mathcal L} = \partial_t - iM(\partial_x\phi) - im(\partial_x\phi)\left(D_x - \partial_x\phi\right) - \frac12im'(\partial_x\phi)\left(D_x^2 - \partial_x\phi\right)^2 + \frac1{2t} + \partial_x^{-1}\partial_y^2, \]
which satisfies
\( \mathcal L\Psi_{\mathbf v} = \tilde{\mathcal L}\Psi_{\mathbf v} + \mb{err}_0. \)
Next we change variables \((t,x,y)\mapsto (t,z,y)\) so that with \(\phi = \phi(t,z) = t\Phi(t^{-1}z)\) we may write the approximate linear operator \(\tilde{\mathcal L}\) as
\begin{align*} \mathcal{\tilde L} = \left(\partial_t - i\partial_t\phi\right) + \frac zt\left(\partial_z - i\partial_z\phi\right) + \frac12m'(-\partial_z\phi)\left(\partial_z - i\partial_z\phi\right)^2 + \frac1t - \partial_z^{-1}\partial_y^2 + \frac yt\partial_y. \end{align*}
As a consequence, for a function \(f = f(t,z,y)\) we obtain
\begin{align*} e^{-i\phi}\tilde{\mathcal L}\partial_z\left(\frac{e^{i\phi}}{i\partial_z\phi}f\right) &= \left(\partial_t + \frac zt\partial_z + \frac12m'(-\partial_z\phi)\partial_z^2 + \frac1t + \frac yt\partial_y\right)\left(f + \frac1{i\partial_z\phi}\partial_zf\right) - \partial_y^2\left(\frac1{i\partial_z\phi}f\right). \end{align*}
Taking \(f(t,z,y) = \chi\left(\lambda_z(z - tv),\lambda_y(y - tv_y)\right)\) we calculate
\[ \partial_tf = - \frac1{2t}(z - tv)\partial_z\chi - \frac1{2t}(y - tv_y)\partial_y\chi - v\partial_z\chi - v_y\partial_y\chi, \]
and similarly for \(\partial_zf\). Plugging these into the above expression we obtain
\begin{align*} e^{-i\phi}\tilde{\mathcal L}\partial_z\left(\frac{e^{i\phi}}{i\partial_z\phi}\chi\right) &= \partial_z\left(\frac1{2t}(z - tv)\chi\right) + \partial_y\left(\frac1{2t}(y - tv_y)\chi\right) + \partial_z\left(\frac12m'(-\partial_z\phi)\partial_z\chi\right)\\ &\quad - \partial_y^2\left(\frac1{i\partial_z\phi}\chi\right) + \mb{err}_1, \end{align*}
where the error term is given by
\begin{align*} \mb{err}_1 &= \partial_z\left(\frac1{2t}(z - tv)\left(\frac1{i\partial_z\phi}\partial_z\chi\right)\right) + \partial_y\left(\frac1{2t}(y - tv_y)\left(\frac1{i\partial_z\phi}\partial_z\chi\right)\right) \\ &\quad + \partial_z\left(\frac12m'(-\partial_z\phi)\partial_z\left(\frac1{i\partial_z\phi}\partial_z\chi\right)\right) + \frac12m''(- \partial_z\phi)\partial_z^2\phi \partial_z\left(\chi + \frac1{i\partial_z\phi}\partial_z\chi\right)\\ &\quad + i\frac zt\frac{\partial_z^2\phi}{(\partial_z\phi)^2}\partial_z\chi, \end{align*}
and we have used the fact that \(\partial_z\partial_t\phi = - \frac zt\partial_z^2\phi\).
Returning to the original variables, \((t,z,y)\mapsto (t,x,y)\), we obtain
\begin{align*} e^{-i\phi}\tilde{\mathcal L}\Psi_{\mathbf v} &= - \partial_x\left(\frac1{2t}(z - tv)\chi\right) - \partial_xL_y\left(\frac1{4t^2}(y - tv_y)\chi\right)\\ &\quad + \partial_x\left(\frac12im'(\partial_x\phi)\partial_x\chi\right) + (\partial_xL_y)^2\left(\frac1{i4t^2\partial_x\phi}\chi\right) + \mb{err}_1, \end{align*}
where we may use the bounds
\begin{gather*}
\partial_x\phi \sim v^{\frac12}\langle v\rangle^{\frac12},\qquad |\partial_x^2\phi|\lesssim t^{-1}v^{-\frac12}\langle v\rangle^{\frac12}\\
|\partial_x^\alpha \phi|\lesssim_k t^{-(1 + \alpha)}v^{\frac12 - \alpha}\langle v\rangle^{-k},\quad \alpha\geq2, \end{gather*}
to show that whenever \(v\in \Sigma_t\) and \(t\gg1\), the error term \(\mb{err}_1\in WP(t,\mathbf v,t^{-\frac32}v^{-\frac34}\langle v\rangle^{-\frac14})\).
Finally, we observe that we may expand the leading order terms in this expression to obtain
\[ \tilde{\mathcal L}\Psi_{\mathbf v}\in WP(t,\mathbf v,t^{-1}), \]
which completes the proof.
\end{proof}
\subsection{Testing by wave packets}
We recall that for a given velocity \(\mathbf v\) the wave packet has similar spatial localization to the hyperbolic part \(u^{\mr{hyp},+}_N\) of \(u\) localized at positive \(x\)-frequency \(N\sim m^{-1}(v)\). From the pointwise estimates of Lemma~\ref{lem:FDR2} we expect that in the region \((x,y)\approx t\mathbf v\) the leading order part of \(u(t,x,y)\) is given by \(u_N^{\mr{hyp},\pm}(t,x,y)\). As a consequence, we should be able to recover the leading order behavior of \(u\) by testing it against \(\Psi_{\mathbf v}\).
This heuristic motivates the definition of the function
\[ \gamma(t,\mathbf v) = \int u_x\bar\Psi_{\mathbf v}\,dxdy. \]
Due to the normalization that \(\int \chi\,dxdy = 1\) we then expect that
\[ u(t,t\mathbf v)\approx 2t^{-1}\langle v\rangle^{\frac12}\mathop{\rm Re}\nolimits\left(e^{i\phi}\gamma(t,\mathbf v)\right), \]
where we note that \(t^{-1}\langle v\rangle^{\frac12} = \lambda_z\lambda_y\). To make this heuristic precise we prove the following lemma:
\begin{lem}\label{lem:GammaBounds}
For \(t\gg1\) we have the estimate
\begin{equation}\label{est:BasicGammaBound}
\|\langle v\rangle^{\frac12}\gamma\chi_{\Sigma_t}\|_{L^\infty_{\mathbf v}}\lesssim t\|u_x\|_{L^\infty_{x,y}}. \end{equation}
as well as the estimate for the difference,
\begin{equation}
\|(u_x(t,t\mathbf v) - 2t^{-1}\langle v\rangle^{\frac12}\mathop{\rm Re}\nolimits(e^{i\phi}\gamma(t,\mathbf v)))\chi_{\Sigma_t}\|_{L^\infty_{\mathbf v}} \lesssim t^{-\frac{13}{12}}\|u\|_X.\label{est:DifferenceLInf} \end{equation}
\end{lem} \begin{proof}
The pointwise estimate \eqref{est:BasicGammaBound} follows from the fact that for \(v\in \Sigma_t\) we have
\[
\|\Psi_{\mathbf v}\|_{L^1_{x,y}}\lesssim t\langle v\rangle^{-\frac12}. \]
For the pointwise difference \eqref{est:DifferenceLInf} we first define
\[ u_{x,v}^{\mr{hyp},+} = \sum\limits_{m(N)\sim v}\partial_xu_N^{\mr{hyp},+}, \]
and use the pointwise bound \eqref{est:FDR2-ELLIPTIC-2} for the elliptic piece \(u^\mr{ell}\) as well as the spatial localization of \(u_N^\mr{hyp} = 2\mathop{\rm Re}\nolimits u_N^{\mr{hyp},+}\) to obtain
\[
\left|u_x(t,t\mathbf v) - 2\mathop{\rm Re}\nolimits u_{x,v}^{\mr{hyp},+}(t,t\mathbf v)\right|\lesssim t^{-\frac{13}{12}}\|u\|_X. \]
Next we use the pointwise bound \eqref{est:FDR2-ELLIPTIC-2} for the elliptic piece to obtain
\[
|t^{-1}\langle v\rangle^{\frac12}\langle u_x^\mr{ell},\psi_{\mathbf v}\rangle|\lesssim t^{-\frac{13}{12}}. \]
We may then use the spatial localization of \(\Psi_{\mathbf v}\) to obtain
\[
|t^{-1}\langle v\rangle^{\frac12}\langle u^\mr{hyp}_x - u_{x,v}^\mr{hyp},\Psi_{\mathbf v}\rangle| \lesssim_k t^{-k}\|u\|_X, \]
and similarly, recalling that \(u_N^{\mr{hyp},-} = \chi_N^\mr{hyp} u_N^-\), we have
\[
|\langle u_{x,v}^{\mr{hyp},-} ,\Psi_{\mathbf v}\rangle |\lesssim \sum\limits_{m(N)\sim v}\langle \partial_xu_N^-,\chi_N^\mr{hyp} \Psi_{\mathbf v}\rangle|\lesssim t^{-k}\|u\|_X, \]
where the rapid decay follows from the fact that \(\chi_N^\mr{hyp}\Psi_{\mathbf v}\in WP(t,\mathbf v,1)\) is localized at \emph{positive} wavenumbers \(\sim m^{-1}(v)\) up to rapidly decaying tails at scale \(\lambda_z\lesssim t^{-\frac{23}{48}}\). Combining these bounds we obtain
\[
t^{-1}\langle v \rangle^{\frac12}\left|\gamma(t,\mathbf v) - \langle u_{x,v}^{\mr{hyp},+},\Psi_{\mathbf v}\rangle\right| \lesssim t^{-\frac{13}{12}}\|u\|_X. \]
Thus it remains to consider the difference
\[
\mathfrak D = \left|e^{-i\phi}u_{x,v}^{\mr{hyp},+}(t,t\mathbf v) - t^{-1}\langle v\rangle^{\frac12}\gamma(t,\mathbf v)\right| \]
Next we define
\[ w(t,z,y) = e^{-i\phi}u_{x,v}^{\mr{hyp},+}(t,x,y), \]
and write the difference as
\begin{align*} \mathfrak D &= w(t,tv,tv_y) - t^{-1}\langle v\rangle^{\frac12}\int w(t,z,y)\chi(\lambda_z(z - tv),\lambda_y(y - tv_y))\,dzdy\\ &= t^{-1}\langle v\rangle^{\frac12}\int \left( w(t,tv,tv_y) - w(t,z,y) \right) \chi(\lambda_z(z - tv),\lambda_y(y - tv_y))\,dzdy \end{align*}
Applying the elliptic estimate \eqref{est:FDR2-Elliptic1} with the frequency localization of \(u_{x,v}^{\mr{hyp},+}\) we obtain the estimates
\begin{gather*}
\|w_z\|_{L^2}\lesssim \|L_+u_{v,x}^{\mr{hyp},+}\|_{L^2}\lesssim t^{-1}v^{-\frac12}\langle v\rangle^{\frac12}\|u\|_X,\\
\|w_{yy}\|_{L^2}\lesssim t^{-2}\|(L_y\partial_x)^2u_{v,x}^{\mr{hyp},+}\|_{L^2}\lesssim t^{-2}v\langle v\rangle\|u\|_X, \end{gather*}
so we may apply the Sobolev estimate \eqref{est:Holder} with \(\alpha = \frac14\) to \(w\) to obtain
\[
|w(t,tv,tv_y) - w(t,z,y)|\lesssim t^{-\frac54}v^{-\frac18}\langle v\rangle^{\frac78}\left(|z - tv|^{\frac14} + |y - tv|^{\frac14}\right)\|u\|_X. \]
As a consequence we obtain the estimate
\[
|\mathfrak D|\lesssim t^{-\frac98}v^{-\frac3{16}}\langle v\rangle^{\frac{15}{16}}\|u\|_X, \]
which completes the proof of \eqref{est:DifferenceLInf}.
\end{proof}
\subsection{The ODE for \(\gamma\)}
From Lemma~\ref{lem:GammaBounds} we see that \(\gamma\) may be used to estimate the size of \(u_x\) up to errors that decay in time. In order to obtain bounds for \(\gamma\) we will treat \(\mathbf v\) as a fixed parameter and consider the ODE satisfied by \(\gamma\)
\begin{equation}\label{ResonantODE} \dot\gamma(t,\mathbf v) = \langle (uu_x)_x,\Psi_{\mathbf v}\rangle + \langle u_x,\mathcal L\Psi_{\mathbf v}\rangle. \end{equation}
For the first of these terms we may use that there are no parallel resonances to show that at least one of the \(u\) terms must be elliptic and have improved decay. For the second term we use the expression \eqref{LinearForm} to see that to leading order \(\mathcal L\Psi_{\mathbf v}\) has a divergence-type structure so we may integrate by parts to obtain improved decay. As a consequence we obtain the following lemma:
\begin{lem}
If \(u\) is a solution to \eqref{eqn:cc-fd} then for \(t\gg1\) we have the estimate
\begin{equation}
\|\dot\gamma\chi_{\Sigma_t}\|_{L^\infty_{\mathbf v}} \lesssim t^{-\frac{13}{12}} \|u\|_X\left(1 + \|u\|_X\right).\label{ODEestPwse} \end{equation}
\end{lem} \begin{proof}
We start by considering the nonlinear term appearing in \eqref{ResonantODE}. Integrating by parts we obtain
\[ \frac12\langle (u^2)_{xx},\Psi_{\mathbf v}\rangle = - \frac12\langle ((u^\mr{hyp})^2)_x,\partial_x\Psi_{\mathbf v}\rangle - \langle (u^\mr{hyp} u^\mr{ell})_x,\partial_x\Psi_{\mathbf v}\rangle - \frac12\langle ((u^\mr{ell})^2)_x,\partial_x\Psi_{\mathbf v}\rangle. \]
For the second and third terms we may apply the pointwise bounds of Lemma~\ref{lem:FDR2} to obtain
\[
|\langle (u^\mr{hyp} u^\mr{ell})_x,\partial_x\Psi_{\mathbf v}\rangle| + |\langle ((u^\mr{ell})^2)_x,\partial_x\Psi_{\mathbf v}\rangle| \lesssim t^{-\frac{13}{12}}\|u\|_{X}^2. \]
For the remaining term we first use the spatial localization of \(\Psi_{\mathbf v}\) to replace \(u^\mr{hyp}\) by \(u^\mr{hyp}_v\), where we recall that
\[ u^\mr{hyp}_v = \sum\limits_{\substack{m(N)\sim v}} u_N^\mr{hyp}. \]
Recalling the definition of \(u_N^\mr{hyp} = \chi_N^\mr{hyp} u_N\) we may write
\[ \langle (u^\mr{hyp}_v)^2,\Psi_{\mathbf v}\rangle = \sum\limits_{m(N_1),m(N_2)\sim v}\langle u_{N_1}u_{N_2},\chi_{N_1}^\mr{hyp}\chi_{N_2}^\mr{hyp}\Psi_{\mathbf v}\rangle. \]
We observe that for sufficiently large \(t\gg1\) (independent of \(v\)) the function \(\Theta = \chi_{N_1}^\mr{hyp}\chi_{N_2}^\mr{hyp}\Psi_{\mathbf v}\in WP(t,\mathbf v,1)\) and hence is localized at frequency \(m^{-1}(v)\) up to rapidly decaying tails at scale \(\lambda_z\lesssim t^{- \frac{23}{48}}\). In particular, for \(t\gg1\) sufficiently large (independently of \(t,v\)) we have
\[
|P_{< \frac12m^{-1}(v)}\Theta |\lesssim_k t^{-k},\qquad |P_{\geq \frac32 m^{-1}(v)}\Theta|\lesssim_k t^{-k}. \]
However, the product \(u_{N_1}u_{N_2}\) has compact Fourier support in neighborhoods of size \(O(\delta m^{-1}(v))\) about the frequencies \(0, \pm 2m^{-1}(v)\). In particular, by choosing \(0<\delta\ll1\) sufficiently small (independently of \(v\)) we may ensure that
\[ P_{\frac12 m^{-1}(v)\leq \cdot < \frac32 m^{-1}(v)}(u_{N_1}u_{N_2}) = 0, \]
and hence
\[
|\langle (u^\mr{hyp}_v)^2,\Psi_{\mathbf v}\rangle|\lesssim t^{-k}\|u\|_X^2. \]
To complete the estimate we consider the linear term. We first recall from Lemma~\ref{lem:Microlocal} that \(\mathcal L\Psi_{\mathbf v}\in WP(t,\mathbf v,t^{-1})\) and hence satisfies \(\|\langle v\rangle^{\frac12}\mathcal L\Psi_{\mathbf v}\|_{L^1_{x,y}}\lesssim 1\). Estimating as in Lemma~\ref{lem:GammaBounds} we then obtain
\[
|\langle u_x^\mr{hyp} - u_{x,v}^{\mr{hyp},+},\mathcal L\Psi_{\mathbf v}\rangle|\lesssim_k t^{-k}\|u\|_X. \]
Next we recall the expression \eqref{LinearForm} for the operator \(e^{-i\phi}\mathcal L\Psi_{\mathbf v}\). In particular, we may take \(w = e^{-\phi}u_{x,v}^{\mr{hyp},+}(t,x,y)\) as before and integrate by parts to obtain
\begin{align*} \langle u_{x,v}^{\mr{hyp},+},\mathcal L\Psi_{\mathbf v}\rangle &= \langle w_z,\frac1{2t}(z - tv)\chi\rangle + \langle w_y,\frac1{4t^2}(y - tv_y)\chi\rangle\\ &\quad - \langle w_z,\frac12im'(\partial_x\phi) \partial_x\chi\rangle + \langle w_{yy},\frac1{4it^2\partial_x\phi}\chi\rangle + \langle w,\mb{err}\rangle, \end{align*}
where the error term \(\mb{err}\in WP(t,\mathbf v,t^{-\frac32}v^{-\frac34}\langle v\rangle^{-\frac14})\). Applying the \(L^2\)-estimates for \(w\) as in Lemma~\ref{lem:GammaBounds} and the pointwise estimate \eqref{est:FDR2-HYPERBOLIC-2} for the final term, we obtain the estimate \eqref{ODEestPwse}.
\end{proof}
\subsection{Proof of global existence}~
We now complete the proof of Theorem~\ref{thm:Main}. We choose \(\mathcal T_0\geq 1\) and by taking \(\mathcal M\gg1\) sufficiently large and \(0<\epsilon\ll1\) sufficiently small we may find a solution \(S(-t)u\in C([0,T];X(0))\) to \eqref{eqn:cc-fd} for some \(T\geq \mathcal T_0\). Next we assume that the bootstrap assumption \eqref{BS} holds on the interval \([0,T]\) from which we obtain the energy estimate \eqref{AP}.
Next we use the estimate \eqref{est:BasicGammaBound} to bound \(\gamma\) at time \(\mathcal T_0\) in terms of \(\|u_x\|_{L^\infty}\) and the Sobolev estimate \eqref{Starter410} to obtain
\[
|\gamma(\mathcal T_0,\mathbf v)|\lesssim \epsilon \mathcal T_0^{C\mathcal M\epsilon}. \]
We may then solve the ODE satisfied by \(\gamma\) on the time interval \([\mathcal T_0,T]\) using the estimate \eqref{ODEestPwse} to obtain
\[
\|\langle v\rangle^{\frac12}\gamma \chi_{\Sigma_t}\|_{L^\infty_{\mathbf v}}\lesssim \epsilon \mathcal T_0^{C\mathcal M\epsilon} + \int_{\mathcal T_0}^t \|\langle v\rangle^{\frac12}\dot\gamma \chi_{\Sigma_t}\|_{L^\infty_{\mathbf v}}\lesssim \epsilon \mathcal T_0^{C\mathcal M\epsilon} + \epsilon \mathcal T_0^{-\frac1{12} + 2C\mathcal M \epsilon}, \]
provided \(0<\epsilon\ll1\) is sufficiently small. We may then apply the estimate \eqref{est:DifferenceLInf} for the difference between \(u_x\) and \(2t^{-1}\langle v\rangle^{\frac12}\mathop{\rm Re}\nolimits(e^{i\phi}\gamma)\) to obtain
\[
\|u_x\chi_{\Sigma_t}\|_{L^\infty}\lesssim \epsilon t^{-1}\left(\mathcal T_0^{C\mathcal M\epsilon} + \mathcal T_0^{-\frac1{12} + 2C\mathcal M \epsilon}\right). \]
By choosing \(\mathcal M\gg1\) sufficiently large and \(0<\epsilon\ll1\) sufficiently small we may combine this with the estimate \eqref{ImprovedDecay} for \(u_x\) in the region \(\Sigma_t^c\) to obtain the bound
\[
\|u_x\|_{L^\infty}\leq \frac12\mathcal M\epsilon t^{-\frac12}\langle t\rangle^{-\frac12}, \]
which closes the bootstrap. The solution \(u\) then exists globally and satisfies the energy estimate \eqref{Fdn} and the pointwise estimate \eqref{est:PTWISEDECAY}.
\subsection{Proof of scattering}~
It remains to prove that our solutions scatters in \(L^2\). As in~\cite{2014arXiv1409.4487H} we do not have scattering in the sense that \(\mathcal L u\in L^1_tL^2_{x,y}\) but we are able to construct a normal form correction to remove the worst bilinear interactions and show that \(S(-t)u(t)\) converges in \(L^2\) as \(t\rightarrow\infty\). We note that for translation invariant initial data the worst nonlinear interactions are the high-low interactions (see Appendix~\ref{app:IP}). However, the spatial localization ensures that these interactions can only occur on very short timescales, thus attenuating their effect. From the pointwise and elliptic estimates of Section~\ref{sect:KS} we see that the worst nonlinear interactions for spatially localized initial data are the high-high (hyperbolic) interactions for which we may construct a well-defined normal form.
We first define the leading order part of \(w\) by
\[ w = P_{t^{-\frac16}<\cdot\leq t^{\frac1{12}}}u, \]
and then have the following lemma:
\begin{lem}
For \(t\gg1\) we have the estimate
\begin{equation}\label{Reduction}
\|uu_x - 2\mathop{\rm Re}\nolimits(w^+w_x^+)\|_{L^2}\lesssim t^{-\frac{97}{96}}\|u\|_X^2. \end{equation}
\end{lem} \begin{proof}
We start by using the estimate \eqref{ImprovedDecay} to reduce the estimate to the region \(\Sigma_t\),
\[
\|uu_x\chi_{\Sigma_t^c}\|_{L^2}\lesssim \|u_x \chi_{\Sigma_t^c}\|_{L^\infty}\|u\|_{L^2}\lesssim t^{-\frac{97}{96}}\|u\|_X^2. \]
Next we use the pointwise estimates of Lemma~\ref{lem:FDR2} to reduce to the hyperbolic parts,
\[
\|(uu_x - u^\mr{hyp} u^\mr{hyp}_x) \chi_{\Sigma_t}\|_{L^2}\lesssim\|u_x^\mr{ell}\|_{L^\infty} \|u\|_{L^2} + \|u_\mr{ell}\chi_{\Sigma_t}\|_{L^\infty}\|u_x^\mr{hyp}\|_{L^2}\lesssim t^{-\frac{97}{96}}\|u\|_X^2. \]
We observe that
\[
u^\mr{hyp} u_x^\mr{hyp} = 2\mathop{\rm Re}\nolimits(u^{\mr{hyp},+}u_x^{\mr{hyp},+}) + \partial_x|u^{\mr{hyp},+}|^2, \]
and that
\[
\partial_x|u^{\mr{hyp},+}|^2 = 2\mathop{\rm Im}\nolimits (\bar u^{\mr{hyp},+}L_z^+u^{\mr{hyp},+}), \]
so applying the elliptic estimate \eqref{est:FDR2-Elliptic1} we obtain
\[
\|(u^\mr{hyp} u_x^\mr{hyp} - 2\mathop{\rm Re}\nolimits(u^{\mr{hyp},+}u_x^{\mr{hyp},+}))\chi_{\Sigma_t}\|_{L^2} \lesssim \|u^\mr{hyp} \chi_{\Sigma_t}\|_{L^\infty}\|L_z^+u^{\mr{hyp},+}\chi_\mr{hyp}^+\|_{L^2}\lesssim t^{-\frac{97}{96}}\|u\|_X^2. \]
Taking \(w^{\mr{hyp},+} = \sum\limits_N \chi_N^\mr{hyp} w_N^+\) as above we see that
\[ w^{\mr{hyp},+} \chi_{\Sigma_t} = u^{\mr{hyp},+} \chi_{\Sigma_t}, \]
and hence
\[
\|uu_x - 2\mathop{\rm Re}\nolimits(w^{\mr{hyp},+} w_x^{\mr{hyp},+})\|_{L^2} \lesssim t^{-\frac{97}{96}}\|u\|_X^2. \]
Finally we may once again apply the pointwise estimates of Lemma~\ref{lem:FDR2} to obtain the bound
\[
\|w^+w_x^+ - w^{\mr{hyp},+}w_x^{\mr{hyp},+}\|_{L^2}\lesssim t^{-\frac{97}{96}}\|u\|_X^2, \]
which completes the proof.
\end{proof}
We now construct a normal form for the nonlinear term \(2\mathop{\rm Re}\nolimits(w^+w_x^+)\). Here we essentially proceed as in Proposition~\ref{prop:NRG} and define a symmetric bilinear form \(B[u,v]\) with symbol
\[ b(\mathbf k_1,\mathbf k_2) = \frac{\xi_1 + \xi_2}{2\Omega(\mathbf k_1,\mathbf k_2)}, \]
where \(\Omega\) is the resonance function defined as in \eqref{ResonanceFunction}. By construction we have
\[ \mathcal L B[f,g] = \frac12(fg)_x + B[f,\mathcal Lg] + B[\mathcal L f,g]. \]
Further, we have the following lemma:
\begin{lem}
We have the estimates
\begin{align}
\|B[w^+,w^+]\|_{L^2} &\lesssim t^{-\frac23}\|u\|_X^2,\label{AMC}\\
\|B[w_+,\mathcal L w_+]\|_{L^2} &\lesssim t^{-\frac32}\|u\|_X^2(1 + \|u\|_X).\label{BMC} \end{align}
\end{lem} \begin{proof}
We note that here we need only consider high frequency outputs as \(\xi_1,\xi_2>0\) have the same sign. From the estimate \eqref{FDResonanceLB} we see that for \(0<\xi_1\lesssim \xi_2\), the symbol \(b\) satisfies the bounds
\[ b(\mathbf k_1,\mathbf k_2)\lesssim \frac{\langle \xi_2\rangle}{\xi_1\xi_2}, \]
and hence we may decompose
\[ B[u^+,v^+] = Q[\partial_x^{-1}u^+,v^+] + Q[v^+,\partial_x^{-1}u^+], \]
where \(Q\) is given by
\[ Q[u^+,v^+] = \sum\limits_{N}B[u_{<N}^+,v_N^+]. \]
We may then verify that the corresponding symbol \(q\in \mathscr S\) and applying the Coifman-Meyer Theorem \eqref{CM} with the frequency localization of \(w^+\) we obtain the estimates
\begin{align*}
\|B[w^+,w^+]\|_{L^2}&\lesssim \|w_+\|_{L^\infty}\|\partial_x^{-1}w_+\|_{L^2},\\
\|B[w_+,\mathcal L w_+]\|_{L^2} &\lesssim \|w_+\|_{L^\infty}\|\partial_x^{-1}\mathcal Lw^+\|_{L^2}. \end{align*}
For the estimate \eqref{AMC} we may use the pointwise estimates of Lemma~\ref{lem:FDR2} and the frequency localization to obtain
\[
\|w^+\|_{L^\infty}\lesssim t^{-\frac34}\|u\|_X,\qquad \|\partial_x^{-1}w^+\|_{L^2}\lesssim t^{\frac16}\|u\|_X. \]
For the estimate \eqref{BMC} we instead compute
\[ \partial_x^{-1}\mathcal L w^+ = \frac12 P_{t^{-\frac16}<\cdot\leq t^{\frac1{12}}}^+(u^2) + \partial_x^{-1}[\partial_t,P_{t^{-\frac16}<\cdot\leq t^{\frac1{12}}}^+]u. \]
Using the frequency localization we then obtain
\[
\|\partial_x^{-1}\mathcal L w^+\|_{L^2}\lesssim \|u\|_{L^\infty}\|u\|_{L^2} + t^{-\frac56}\|u\|_{L^2}\lesssim t^{-\frac34}\|u\|_X^2 + t^{-\frac56}\|u\|_X, \]
which completes the proof.
\end{proof}
To complete the proof of scattering we apply the estimates \eqref{Reduction}, \eqref{BMC} with the energy estimate \eqref{Fdn} to obtain the bound
\[
\|\mathcal L(u - 2\mathop{\rm Re}\nolimits B[w^+,w^+])\|_{L^2}\lesssim t^{-\frac{97}{96} + C\epsilon}\epsilon^2. \]
In particular, provided \(0<\epsilon\ll1\) is sufficiently small, we can see that given the integrability in time of the nonlinear interactions we can construct a Cauchy sequence for $S(-t)(u - 2\mathop{\rm Re}\nolimits B[w^+,w^+])$ in $L^2$ converging to a \(W\in L^2\) so that for \(t\gg1\),
\[
\|S(-t) (u - 2\mathop{\rm Re}\nolimits B[w^+,w^+]) - W\|_{L^2}\lesssim t^{-\frac1{96} + C\epsilon}\epsilon^2. \]
Applying the estimate \eqref{AMC} we have
\[
\|B[w^+,w^+]\|_{L^2}\lesssim t^{-\frac34 + C\epsilon}\epsilon^2, \]
and hence \(u\) satisfies the estimate \eqref{Fds}. Finally we note that \(\|W\|_{L^2} = \|u_0\|_{L^2}\) by conservation of mass.
\begin{appendix}
\section{Ill-posedness in Besov-type spaces}\label{app:IP}
In this section we show that the infinite depth equation \eqref{eqn:cc} is ill-posed in (almost all) the natural Galilean-invariant, scale-invariant Besov-type refinements of \(\dot H^{\frac14,0}\) considered in \cite{2016arXiv160806730K}.
To define these spaces we make an almost orthogonal decomposition
\[ u = \sum_{N\in2^\mathbb Z}\sum_{k\in\mathbb Z}u_{N,k}, \]
where each \(u_{N,k}\) has Fourier-support in the trapezium
\[
\mathcal Q_{N,k} = \left\{(\xi,\eta)\in\mathbb R^2: \frac12 N<|\xi|<2N,\ \left|\frac\eta\xi - kN^{\frac12}\right| < \frac34 N^{\frac12}\right\}. \]
We then define the space \(\ell^q\ell^p L^2\) with norm
\[
\|u\|_{\ell^q\ell^p L^2}^q = \sum\limits_{N\in 2^\mathbb Z} N^{\frac14 q}\left(\sum\limits_{k\in \mathbb Z}\|u_{N,k}\|_{L^2}^p\right)^{\frac qp}. \]
It is straightforward to verify that these spaces are indeed both scale-invariant and Galilean invariant by recalling that the Galilean shift \eqref{Galilean} corresponds to the map
\[ \hat u(t,\xi,\eta)\mapsto e^{-ic^2t\xi}e^{2ict\eta}\hat u(t,\xi,\eta - c\xi). \]
Further, it is clear that when \(p = q = 2\) we have \(\ell^2\ell^2L^2 = \dot H^{\frac14,0}\). We remark that analogously to \cite[Theorem~1.4]{2016arXiv160806730K} we may show that \(\ell^q\ell^pL^2\) embeds continuously into the space of distributions whenever \(1\leq q\leq\infty\) and \(1\leq p<\frac43\) and that it contains the Schwartz functions for all \(p > 1\).
Using similar ideas to \cite{2016arXiv160806730K,MR1885293}, we then obtain the following ill-posedness result:
\begin{thm}\label{thm:IP}
Let \(1\leq q\leq\infty\) and \(1<p\leq\infty\). Then there does not exist a continuously embedded space \(X_T\subset C([-T,T]:\ell^q\ell^pL^2)\) so that for all \(\phi\in \ell^q\ell^pL^2\),
\begin{gather}
\|S_\infty(t)\phi\|_{X_T}\lesssim \|\phi\|_{\ell^q\ell^pL^2},\\
\left\|\int_0^tS_\infty(t - t')[u(t')\ \partial_xu(t')]\,dt'\right\|_{X_T}\lesssim \|u\|_{X_T}^2, \end{gather}
where \(S_\infty(t)\) is the infinite depth linear propagator, defined as in \eqref{Propagator}.
In particular, for the infinite depth equation \eqref{eqn:cc}, the solution map \(u_0\mapsto u(t)\) (considered as a map on \(\ell^q\ell^pL^2\)) fails to be twice differentiable at \(u_0 = 0\). \end{thm} \begin{proof}
We proceed by contradiction. Suppose that such a space \(X_T\) does exist, then for any \(\phi\in \ell^q\ell^pL^2\) and \(t\in[-T,T]\) we have the estimate
\begin{equation}\label{est:BoundToFail}
\left\|\int_0^tS_\infty(t - t')\partial_x[(S_\infty(t')\phi)^2]\,dt'\right\|_{\ell^q\ell^pL^2}\lesssim \|\phi\|_{\ell^q\ell^pL^2}^2. \end{equation}
Our goal is to show that this estimate must fail for a suitable choice of \(\phi\). We note that as we will only work with \(O(1)\) choices of \(x\)-frequency, our argument is independent of the choice of \(q\).
We first choose low and high frequency parameters \(0<\delta\ll 1\ll N\), where for convenience we assume that both are dyadic integers. We then define the high and low frequency sets by
\begin{align*}
E_\mr{high} &:= \left\{\mathbf k\in\mathbb R^2:- \frac14\delta<|\xi| - N<\frac14\delta,\ \left|\frac\eta\xi\right|< N^{\frac12}\right\},\\
E_\mr{low} &:= \left\{\mathbf k\in\mathbb R^2:-\frac14\delta<|\xi| - \delta<\frac14\delta,\ \left|\frac\eta\xi\right|< N^{\frac12}\right\}. \end{align*}
We observe that
\[
|E_\mr{high}|\sim\delta N^{\frac32},\qquad |E_\mr{low}|\sim \delta^2N^{\frac12} \]
Further, an elementary algebraic calculation gives us that,
\begin{align*}
E_\mr{high} + E_\mr{low} &\subset \left\{\mathbf k\in\mathbb R^2:- \frac12\delta<|\xi| - N \mp \delta<\frac12\delta,\ \left|\frac\eta\xi\right|< 10 N^{\frac12}\right\},\\
E_\mr{high} + E_\mr{low}&\supset \left\{\mathbf k\in\mathbb R^2:- \frac12\delta<|\xi| - N \mp \delta<\frac12\delta,\ \left|\frac\eta\xi\right|< \frac1{10}N^{\frac12}\right\}, \end{align*}
as well as
\[ (E_\mr{high} + E_\mr{low})\cap (E_\mr{low} + E_\mr{low}) = \emptyset,\qquad (E_\mr{high} + E_\mr{low})\cap(E_\mr{high} + E_\mr{high}) = \emptyset. \]
We define functions associated to the sets \(E_\mr{high},E_\mr{low}\) by
\[ \hat\phi_\mr{high} = \delta^{-\frac12}N^{-1}\mb1_{E_\mr{high}},\qquad \hat\phi_\mr{low} = \delta^{\frac1{2p}-\frac32}N^{-\frac1{2p}}\mb1_{E_\mr{low}}, \]
and take
\[ \phi = \phi_\mr{high} + \phi_\mr{low}. \]
As the high frequency set \(E_\mr{high}\subset\mathcal Q_{N,0}\) we have the estimate
\begin{equation}\label{est:HiBound}
\|\phi_\mr{high}\|_{\ell^q\ell^pL^2}\sim 1. \end{equation}
As the low frequency set
\(
E_\mr{low}\subset \bigcup_{|k|\leq \delta^{-\frac12}N^{\frac12}}\mathcal Q_{\delta,k} \)
we obtain the low frequency bound
\begin{equation}\label{est:LoBound}
\|\phi_\mr{low}\|_{\ell^q\ell^pL^2}\sim 1. \end{equation}
Combining \eqref{est:HiBound} and \eqref{est:LoBound} with the fact that \(E_\mr{high}\cap E_\mr{low} = \emptyset\) we obtain
\begin{equation}
\|\phi\|_{\ell^q\ell^pL^2}\sim 1. \end{equation}
Further, from the bounds on \(|E_\mr{high} + E_\mr{low}|\) and \(|E_\mr{low}|\) we obtain the convolution estimate
\[
\|\hat\phi_\mr{high}*\hat\phi_\mr{low}\|_{L^2} \gtrsim \delta^{\frac12 + \frac1{2p}}N^{\frac14 - \frac1{2p}}. \]
We now consider the left hand side of \eqref{est:BoundToFail}. Using the support properties of the sums of \(E_\mr{high} + E_\mr{high}\), \(E_\mr{low} + E_\mr{low}\) and \(E_\mr{high} + E_\mr{low}\) we obtain the lower bound
\[
\left\|\int_0^tS_\infty(t - s)\partial_x[(S_\infty(s)\phi)^2]\,ds\right\|_{\ell^q\ell^pL^2} \gtrsim N^{\frac54}\left\|\int_0^t\int_{\mathbf k = \mathbf k_1 + \mathbf k_2}e^{is\Omega(\mathbf k_1,\mathbf k_2)}\hat\phi_\mr{high}(\mathbf k_1)\hat\phi_\mr{low}(\mathbf k_2)\,d\mathbf k_1\,ds\right\|_{L^2}, \]
where the resonance function \(\Omega = \Omega_\infty\) is defined as in \eqref{ResonanceInf}. If \(\mathbf k_1\in E_\mr{high}\), \(\mathbf k_2\in E_\mr{low}\) and \(\mathbf k = \mathbf k_1 + \mathbf k_2\) then provided \(N\gg 1\) we obtain the estimate
\[
|\Omega(\mathbf k_1,\mathbf k_2)|\lesssim N\delta. \]
If we choose \(\delta = N^{-(1 + \epsilon)}\) then for \(|t|\sim 1\) we obtain
\[
\left|\int_0^t e^{it'\Omega(\mathbf k_1,\mathbf k_2)}\,dt'\right| = \left|\frac{e^{it\Omega(\mathbf k_1,\mathbf k_2)} - 1}{i\Omega(\mathbf k_1,\mathbf k_2)}\right| \gtrsim 1 + O(N^{-\epsilon}). \]
As \(\hat\phi_\mr{high},\hat\phi_\mr{low}\geq0\) then for \(N\gg1\) we obtain
\[
\left\|\int_0^t\int_{\mathbf k = \mathbf k_1 + \mathbf k_2}e^{is\Omega(\mathbf k_1,\mathbf k_2)}\hat\phi_\mr{high}(\mathbf k_1)\hat\phi_\mr{low}(\mathbf k_2)\,d\mathbf k_1\,ds\right\|_{L^2}\gtrsim \|\phi_\mr{high}*\phi_\mr{low}\|_{L^2}, \]
and hence
\[
\left\|\int_0^tS_\infty(t - s)\partial_x[(S_\infty(s)\phi)^2]\,ds\right\|_{\ell^q\ell^pL^2}\gtrsim N^{1 - \frac1p}N^{- \epsilon(\frac12 + \frac1{2p})}. \]
If \(p>1\) then by choosing \(0<\epsilon\ll1\) sufficiently small we may take \(N\rightarrow+\infty\) to obtain a contradiction.
\end{proof}
\end{appendix}
\end{document} | arXiv |
Tag: classics
This Riddler puzzle is a simple twist on a classic.
You have a camel and 3,000 bananas. You would like to sell your bananas at the bazaar 1,000 miles away. Your loyal camel can carry at most 1,000 bananas at a time. However, it has an insatiable appetite and quite the nose for bananas — if you have bananas with you, it will demand one banana per mile traveled. In the absence of bananas on his back, it will happily walk as far as needed to get more bananas, loyal beast that it is. What should you do to get the largest number of bananas to the bazaar? What is that number?
This classic puzzle has appeared in various forms. Camels carrying bananas, monkeys carrying bananas, and even a boy carrying watermelons! In these instances, it costs bananas (or watermelons) for all movement. The twist in the present incarnation of the puzzle is that movement is free if no bananas are being carried.
Suppose we have $1000q+r$ bananas, where $0 \le r < 1000$. For example, with $6548$ bananas, $q=6$ and $r=548$. The key is to realize that moving those bananas will cost at least $q+1$ bananas per mile since the camel can carry at most $1000$ bananas at once, and so must make at least $q+1$ trips. We want to minimize the number of trips required because that will also serve to minimize the number of bananas lost in transit.
The optimal way to transport the bananas is to split the journey into segments. Using the example above, transportation initially costs us $7$ bananas per mile, until we drop to $6000$ bananas. Then, it will cost us $6$ bananas per mile until we drop to $5000$ bananas, and so on. Let's illustrate how this works for version asked about in the problem: $3000$ bananas that must be carried $1000$ miles.
our 3000 bananas can be moved at a cost of 3 bananas per mile. Moving them 333.33 miles, we are left with 2000 bananas and 666.66 miles to go.
our remaining 2000 bananas can be moved at a cost of 2 bananas per mile. We move them 500 miles, so we are left with 1000 bananas and 166.66 miles to go.
our remaining 1000 bananas can be moved at a cost of 1 banana per mile. We move them the remaining 166.66 miles, which leaves us with 833.33 bananas.
This is the most efficient way to move the bananas. Using this procedure, we also observe that the maximum distance we can travel before losing all our bananas is $333.33+500+1000=1833.33$ miles.
The above recipe can be extended to the case with $n$ bananas that must be carried a distance $d$. For each $(n,d)$, the plot below shows the maximum number of bananas can be carried to the destination.
The contours are piecewise-linear and approximate logarithmic curves. To understand why this is the case, we can consider $n = 1000q+r$ bananas and ask how far they can be carried. The total distance is:
d_\text{max} = \tfrac{r}{q+1} + \tfrac{1000}{q} + \tfrac{1000}{q-1} + \dots + \tfrac{1000}{2} + \tfrac{1000}{1}
\]The sums of reciprocals $H_m = 1+\frac{1}{2}+\dots+\frac{1}{m}$ are called harmonic numbers. A useful bound involving harmonic numbers is that
\log(m+1) \le H_m \le 1+\log(m)
\]So we can bound:
\log(q+1) \le H_q \le \frac{d_\text{max}}{1000} \le H_{q+1} \le 1+\log(q+1)
\]this explains the logarithmic shape of the top contour. We can also ask about efficiency by dividing the number of bananas carried to the destination by the total number of bananas at the start. This results in the following picture:
If you're interested in how I produced these images, I used IJulia and you can view the notebook here.
Infinite bananas
One might ask: what happens in the case where we have infinitely many bananas? how efficient can we be then? Based on the plot above, it appears that efficiency is eventually constant. To find the limit, let's assume we have $n = 1000 k$ bananas. Taking successive steps using our algorithm above until we reach the desired distance $d$, we must:
go $\tfrac{1000}{k}$ miles (at $k$ bananas/mile); $1000(k-1)$ bananas remaining.
go $\tfrac{1000}{k-1}$ miles (at $k-1$ bananas/mile); $1000(k-2)$ bananas remaining.
… and so on, until we have gone $d$ miles.
If it takes us $j$ steps to reach our goal, this means that $j-1$ steps wasn't enough, but that the $j^\text{th}$ step possibly put us over. An inequality representing this fact is:
H_k-H_{k-j}-\tfrac{1}{k-j}=\sum_{i=1}^{j-1} \tfrac{1}{k-i+1} < \tfrac{d}{1000} \le \sum_{i=1}^{j} \tfrac{1}{k-i+1} = H_k-H_{k-j} \]where the $H_m$ are the harmonic numbers we discussed earlier. Another useful property of harmonic series is that they grow logarithmically. More precisely,
\lim_{k\to\infty} \left( H_k-\log(k)\right) = \gamma \approx 0.5772
\]where $\gamma$ is known as the Euler-Mascheroni constant. Using properties of limits, this fact implies that for any fixed $j$, we have:
\lim_{k\to\infty} \left[ \left(H_{k-j}-H_{k}\right)-\log\left(\tfrac{k-j}{k}\right) \right] = 0
\]Rearranging our bound on $d$, we can write:
-\tfrac{d}{1000} \le H_{k-j}-H_k \le -\tfrac{d}{1000}+\tfrac{1}{k-j}
\]In the limit, we therefore have:
\lim_{k\to\infty} \left( H_{k-j}-H_k \right) \,=\, \lim_{k\to\infty}\log\left(\tfrac{k-j}{k}\right) \,=\, -\tfrac{d}{1000}
\]Notice that we will have $1000(k-j)$ bananas remaining when we arrive at our destination, and we started with $1000k$. Therefore, $\frac{k-j}{k}$ is precisely the efficiency we are trying to find; the fraction of bananas remaining once we reach our goal. As $k\to\infty$ (infinite banana limit), we obtain
(\text{infinite-banana efficiency}) = \lim_{k\to\infty}\tfrac{k-j}{k} = e^{-d/1000}
\] So, for example, if we must travel a distance $d=1000$, we can expect to keep roughly $1/e \approx 36.79\%$ of our bananas. It turns out that for the case of $3000$ bananas, our efficiency is only about $833.33/3000 \approx 27.78\%$, so there would be plenty of room for improvement if we had more bananas!
Author LaurentPosted on September 3, 2016 October 29, 2018 Categories The RiddlerTags classics, logic, Riddler5 Comments on How many bananas can the camel carry?
Today's Riddler problem is another classic. The current incarnation of the puzzle is about error-prone mathematicians, while the classic version is about blue-eyed islanders.
A university has 10 mathematicians, each one so proud that, if she learns that she made a mistake in a paper, no matter how long ago the mistake was made, she resigns the next Friday. To avoid resignations, when one of them detects a mistake in the work of another, she tells everyone else but doesn't inform the mistake-maker. All of them have made mistakes, so each one thinks only she is perfect. One Wednesday, a super-mathematician, whom all respect and believe, comes to visit. She looks at all the papers and says: "Someone here has made a mistake."
What happens then? Why?
Here is the solution:
On the tenth Friday, all mathematicians will resign. To understand why this happens, we'll use a recursive approach similar to what we did for the puzzle of the pirate booty.
For simplicity, we'll say a mathematician is flawed if they made a mistake, and perfect otherwise. Let's look at the world through the eyes of Alice, one of the ten mathematicians. Alice doesn't know whether she is flawed or not. She does, however, know how many of her colleagues are flawed.
According to the problem statement, all the mathematicians are flawed. So Alice sees that all nine of her colleagues are flawed. Let's solve a simpler version of the problem first. What if none of Alice's colleagues were flawed? In this case, the news that "someone here has made a mistake" can only mean one thing: that "someone" must be Alice! So Alice will resign on Friday.
What if exactly one of Alice's colleagues is flawed? Two possibilities:
If Alice is perfect, then the flawed mathematician will see that everyone else is perfect and resign on Friday, just as Alice would have done in the previous case when she saw her colleagues were all perfect.
If Alice is flawed, then nothing happens on Friday. Alice then deduces that she must be flawed and resigns on the next Friday.
What if exactly two of Alice's colleagues are flawed? Two possibilities:
If Alice is perfect, then each flawed mathematician sees one other flawed mathematician. As in the previous case, we expect them to both resign on the second Friday.
If Alice is flawed, then nothing happens on the second Friday. In that case, Alice will know of her flaw and resign on the third Friday.
The pattern is clear: if Alice has $k$ flawed colleagues but Alice is perfect, then the flawed mathematicians all resign on the $k^\text{th}$ Friday. If Alice is flawed, then nothing happens on the $k^\text{th}$ Friday, and Alice resigns on the $(k+1)^\text{st}$ Friday (along with all the other flawed mathematicians). In particular, if all ten mathematicians are flawed as in the problem statement, they all resign on the tenth Friday.
But but but…
It seems that we have a paradox — we aren't telling the mathematicians anything they don't already know! Each mathematician can clearly see that they have flawed colleagues, so why should explicitly announcing this fact make any difference? The answer lies in the distinction between mutual knowledge and common knowledge.
If everybody knows some proposition $P$ to be true, then $P$ is mutual knowledge. However, if everybody knows $P$, and everybody knows that everybody knows $P$, and everybody knows that everybody knows that everybody knows $P$, and so on, then $P$ is common knowledge. Although the fact that there is at least one flawed mathematician is mutual knowledge, it is not common knowledge. When the super-mathematician makes her announcement that there is at least one flawed mathematician, it becomes common knowledge, and this makes all the difference.
Rather than giving a detailed solution, I will defer to some other existing solutions online that are already well-explained. No need to reinvent the wheel! Here are some pointers.
The solution to the blue eyes puzzle featured on the xkcd website.
The wikipedia article on common knowledge, which uses the blue eyes puzzle as an illustrative example.
A post on the math stackexchange site that gives a detailed an formal proof of result.
Author LaurentPosted on June 11, 2016 September 3, 2016 Categories The RiddlerTags classics, dynamic programming, induction, recursion, Riddler2 Comments on The blue-eyed islanders
Today's puzzle was posed on the Riddler blog, but it's actually a classic among problem-solving enthusiasts, and is commonly known as the pirate game. Here is the formulation used in the Riddler:
Ten Perfectly Rational Pirate Logicians (PRPLs) find 10 (indivisible) gold pieces and wish to distribute the booty among themselves.
The pirates each have a unique rank, from the captain on down. The captain puts forth the first plan to divide up the gold, whereupon the pirates (including the captain) vote. If at least half the pirates vote for the plan, it is enacted, and the gold is distributed accordingly. If the plan gets fewer than half the votes, however, the captain is killed, the second-in-command is promoted, and the process starts over. (They're mutinous, these PRPLs.)
Pirates always vote by the following rules, with the earliest rule taking precedence in a conflict:
Self-preservation: A pirate values his life above all else.
Greed: A pirate seeks as much gold as possible.
Bloodthirst: Failing a threat to his life or bounty, a pirate always votes to kill.
Under this system, how do the PRPLs divide up their gold?
Extra credit: Solve the generalized problem where there are P pirates and G gold pieces.
Here is the solution to the main problem:
This problem is a perfect example of when to use recursion. If there are $P$ pirates, when each pirate contemplates how to vote, they are comparing what would happen if they voted yes (and agreed to the current proposal) to what would happen if they voted no (and the current pirate died, reducing to the $P-1$ case).
Let's start with two pirates and build up from there. For each case, here is a list of each pirates' share of the gold. We ordered the pirates from lowest rank to highest rank (captain).
&\quad\text{Gold distribution} &\text{Votes}&\text{ needed}\\ \hline
v_2 &: \begin{bmatrix} 0 & 10 \end{bmatrix} && 1\\
v_3 &: \begin{bmatrix} 1 & 0 & 9 \end{bmatrix} && 2\\
v_4 &: \begin{bmatrix} 0 & 1 & 0 & 9 \end{bmatrix} && 2\\
v_5 &: \begin{bmatrix} 1 & 0 & 1 & 0 & 8 \end{bmatrix} && 3\\
v_6 &: \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 8 \end{bmatrix} && 3\\
v_7 &: \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 7 \end{bmatrix} && 4\\
v_8 &: \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 7 \end{bmatrix} && 4\\
v_9 &: \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 6 \end{bmatrix} && 5 \\
v_{10} &: \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 6 \end{bmatrix} && 5
Let's examine this table one row at a time. If there are two pirates ($v_2$), a majority is decided by a single vote. Since the captain gets to vote, they can safely propose to keep all the gold.
If there are three pirates ($v_3$), a majority is decided by two votes. So the captain needs to secure one extra vote. The first pirate (lowest rank) won't win anything if the captain dies, because we would reduce to the $v_2$ case. The captain must give this pirate at least one gold piece to buy their vote and avoid death, but can keep the remaining $9$ pieces.
Similarly, if there are four pirates ($v_4$), a majority is decided by two votes. This time, the first pirate's vote costs two gold pieces because they stand to win one gold piece if we reduce to $v_3$. However, the second pirate's vote now costs one gold piece because they win nothing in the $v_3$ case. A similar argument applies as we continue to add pirates.
The final solution is that if we number the pirates $1,2,\dots,10$ in order from lowest rank to highest rank, then pirates $2,4,6,8$ each get one gold piece and the captain gets $6$ gold pieces.
And here is a solution to the general case:
As we saw in the previous solution, it makes sense to recurse on $P$, the number of pirates. If $G$ is the number of gold pieces, the pattern we observed in the simpler case still hold here, and we can continue it until the captain runs out of gold and is no longer able to bribe enough of his fellow pirates. So the table looks like:
v_2 &: \begin{bmatrix} 0 & G \end{bmatrix} && 1 \\
v_3 &: \begin{bmatrix} 1 & 0 & G-1 \end{bmatrix} && 2 \\
v_4 &: \begin{bmatrix} 0 & 1 & 0 & G-1 \end{bmatrix} && 2 \\
v_5 &: \begin{bmatrix} 1 & 0 & 1 & 0 & G-2 \end{bmatrix} && 3 \\
v_6 &: \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & G-2 \end{bmatrix} && 3 \\
&\vdots &&\vdots\\
v_{2G+1} &: \begin{bmatrix} 1 & 0 & \dots & 1 & 0 & 0 \end{bmatrix} && G+1\\
v_{2G+2} &: \begin{bmatrix} 0 & 1 & \dots & 0 & 1 & 0 & 0 \end{bmatrix} && G+1
In the last two rows of the table, there are $G$ 1's. In the case with $2G+2$ pirates, a majority is decided by $G+1$ votes, so the captain wins nothing; they must forfeit all the gold in order to buy the votes necessary to survive. From this point forward, it may seem that if we add any additional pirates, they will be killed because there isn't enough gold to buy enough votes. However, this isn't quite the case. Here is what happens if we keep adding pirates:
v_{2G+2} &: \begin{bmatrix} 0 & 1 & \dots & 0 & 1 & 0 & 0 \end{bmatrix} && G+1 \\
{\color{red}v_{\color{red} 2\color{red}G\color{red}+\color{red}3}} &: \begin{bmatrix} ? & ? & \dots & ? & ? & ? & ? & {\color{red}0} \end{bmatrix} && \color{red}G\color{red}+\color{red}2 \\
v_{2G+4} &: \begin{bmatrix} 1 & 0 & \dots & 1 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} && G+2 \\
{\color{red}v_{\color{red} 2\color{red}G\color{red}+\color{red}5}} &: \begin{bmatrix} ? & ? & \dots & ? & ? & ? & ? & ? & ? & {\color{red}0} \end{bmatrix} && \color{red}G\color{red}+\color{red}3 \\
{\color{red}v_{\color{red} 2\color{red}G\color{red}+\color{red}6}} &: \begin{bmatrix} ? & ? & \dots & ? & ? & ? & ? & ? & ? & {\color{red}0} & {\color{red}0} \end{bmatrix} && \color{red}G\color{red}+\color{red}3 \\
{\color{red}v_{\color{red} 2\color{red}G\color{red}+\color{red}7}} &: \begin{bmatrix} ? & ? & \dots & ? & ? & ? & ? & ? & ? & {\color{red}0} & {\color{red}0} & {\color{red}0} \end{bmatrix}\hspace{-2cm} && \color{red}G\color{red}+\color{red}4 \\
v_{2G+8} &: \begin{bmatrix} 0 & 1 & \dots & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\hspace{-2cm} && G+4
Once we get to $P = 2G+3$ pirates, the captain will always be killed (deaths are indicated in red) since there isn't enough gold to buy the required votes. In the $2G+4$ case, however, the second in command is doomed to die if he votes to kill the captain, so he will vote yes even if offered nothing. This gives the captain a free vote, which is enough to secure the required $G+2$ votes. The captain must make an offer that compares favorably to the $2G+2$ case since the intermediate case $2G+3$ always leads to $2G+2$.
In the $2G+5$ case, none of the pirates have an incentive to keep the captain alive (since everybody will be safe with $P=2G+4$). The same happens with $2G+6$ and $2G+7$. Once we arrive at $2G+8$, three of the pirates are doomed if the captain dies, so they will vote yes even if offered nothing. Once again, this allows the captain to barely survive. This pattern continues, each time doubling the interval length between safe captains.
The solution to the general problem. Number the pirates $1,2,\dots,P$ in order from lowest rank to highest rank. If we have $G$ gold pieces to distribute, then we consider two cases:
If $P \le 2G+2$, then:
If $P$ is odd, pirates $1,3,5,\dots,P-2$ each get one gold piece, and the captain gets the remaining $G-\frac{P-1}{2}$ pieces.
If $P$ is even, pirates $2,4,6,\dots,P-2$ each get one gold piece, and the captain gets the remaining $G-\frac{P-2}{2}$ pieces.
If $P > 2G+2$, then the highest ranked captains will die until there are $P = 2G+2^k$ pirates remaining with $k$ as large as possible. Then:
If $k$ is even, pirates $1,3,5,\dots,2G-1$ each get one gold piece and nobody else gets anything.
If $k$ is odd, pirates $2,4,6,\dots,2G$ each get one gold piece and nobody else gets anything.
In the case where $P \ge 2G+2$, the solution is not unique in the sense that once we arrive at $P=2G+2^k$, there are other admissible ways of dividing up the gold besides the one shown above.
If this sort of problem interests you, I recommend taking a crack at the riddle of the blue-eyed islanders, or the unfaithful husbands. More information here as well.
Author LaurentPosted on June 4, 2016 September 3, 2016 Categories The RiddlerTags classics, dynamic programming, induction, recursion, Riddler17 Comments on The puzzle of the pirate booty | CommonCrawl |
\begin{document}
\title[Prime exceptional divisors] {Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections} \author{Eyal Markman}
\address{Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003} \email{[email protected]}
\subjclass[2010]{Primary 14D05, 14J60; Secondary 14J28, 53C26, 14C20}
\begin{abstract} Let $X$ be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of $X$ is a lattice with respect to the Beauville-Bogomolov pairing. A divisor $E$ on $X$ is called a {\em prime exceptional divisor}, if $E$ is reduced and irreducible and of negative Beauville-Bogomolov degree.
Let $E$ be a prime exceptional divisor on $X$. We first observe that associated to $E$ is a monodromy involution of the integral cohomology $H^*(X,{\mathbb Z})$, which acts on the second cohomology lattice as the reflection by the cohomology class $[E]$ of $E$ (Theorem \ref{cor-introduction}).
We then specialize to the case that $X$ is deformation equivalent to the Hilbert scheme of length $n$ zero-dimensional subschemes of a $K3$ surface, $n\geq 2$. We determine the set of classes of exceptional divisors on $X$ (Theorem \ref{conj-exceptional-line-bundles}). This leads to a determination of the closure of the movable cone of $X$.
\end{abstract}
\maketitle
\centerline{\sf In memoriam Professor Masaki Maruyama}
\tableofcontents
\section{Introduction} An {\em irreducible holomorphic symplectic manifold} is a simply connected compact K\"{a}hler manifold $X$, such that $H^0(X,\Omega^2_X)$ is generated by an everywhere non-degenerate holomorphic two-form \cite{beauville,huybrects-basic-results}. The dimension of $X$ is even, say $2n$. The second integral cohomology of $X$ is a lattice with respect to the Beauville-Bogomolov pairing \cite{beauville}. A divisor $E$ on $X$ is called a {\em prime exceptional divisor} if $E$ is reduced and irreducible and of negative Beauville-Bogomolov degree \cite{boucksom}.
\subsection{A prime exceptional divisor is monodromy-reflective} When $\dim(X)=2$, then $X$ is a K\"{a}hler $K3$ surface. Let $E$ be a prime divisor of negative degree on $X$. Then $E$ is necessarily a smooth rational curve. Its degree is thus $-2$. $E$ may be contracted, resulting in a surface $Y$ with an ordinary double point (\cite{BHPV}, Ch. III). A class $x\in H^2(X,{\mathbb Z})$ is {\em primitive} if it is not a multiple of another integral class by an integer larger than $1$. Let $c\in H^2(X,{\mathbb Z})$ be a primitive class of negative degree. Then $c$ has degree $-2$ if and only if the reflection $R_c:H^2(X,{\mathbb Q})\rightarrow H^2(X,{\mathbb Q})$, given by \begin{equation} \label{eq-reflection-R-c} R_c(x) \ \ = \ \ x - \frac{2(x,c)}{(c,c)}c, \end{equation} has integral values, since the lattice $H^2(X,{\mathbb Z})$ is even and unimodular.
Druel recently established the birational-contractibility of a prime exceptional divisor $E$ on a projective irreducible holomorphic symplectic manifold $X$ of arbitrary dimension $2n$. There exists a sequence of flops of $X$, resulting in a projective irreducible holomorphic symplectic manifold $X'$, and a projective birational morphism $\pi:X'\rightarrow Y$ onto a normal projective variety $Y$, such that the exceptional divisor $E'\subset X'$ of $\pi$ is the strict transform of $E$ (see \cite{druel} and Proposition \ref{prop-druel} below). The result relies on the work of several authors, in particular, on Boucksom's work on the divisorial Zariski decomposition and on recent results in the minimal model program \cite{boucksom,BCHM}.
Let $E$ be a prime exceptional divisor on a projective irreducible holomorphic symplectic manifold $X$. Let $c$ be the class of $E$ in $H^2(X,{\mathbb Z})$ and consider the reflection $R_c:H^2(X,{\mathbb Q})\rightarrow H^2(X,{\mathbb Q})$, given by (\ref{eq-reflection-R-c}).
Building on Druel's result, we prove the following statement.
\begin{thm} \label{cor-introduction} The reflection $R_c$ is a monodromy operator. In particular, $R_c$ is an integral isometry. Furthermore, $c$ is either a primitive class or two times a primitive class. \end{thm}
See Corollary \ref{cor-1} for a more detailed statement and a proof. The reflection $R_c$ arises as a monodromy operator as follows. Let $Def(Y)$ be the Kuranishi deformation space of $Y$ and $\bar{\psi}:{\mathcal Y}\rightarrow Def(Y)$ the semi-universal family.
Then $Def(Y)$ is smooth, as is the fiber $Y_t$ of $\bar{\psi}$, over a generic point $t\in Def(Y)$, and the smooth fiber $Y_t$ is deformation equivalent to $X$ \cite{namikawa}. Let $U\subset Def(Y)$ be the complement of the discriminant locus. $R_c$ is exhibited as a monodromy operator of the local system $R^2_{\bar{\psi}_*}{\mathbb Z}$ over $U$.
\subsection{Prime-exceptional divisors in the $K3^{[n]}$-type case} Theorem \ref{cor-introduction} imposes a rather strong numerical condition on a class $c$ to be the class of a prime exceptional divisor.
We get, for example, the following Theorem. Assume that $X$ is deformation equivalent to the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of a $K3$ surface $S$. We will abbreviate this statement by saying that $X$ is of {\em $K3^{[n]}$-type.} Assume that $n \geq 2$ and $X$ is projective.
\begin{thm}\label{thm-2} Let $e$ be a primitive class in $H^2(X,{\mathbb Z})$, with negative Beauville-Bogomolov degree $(e,e)<0$, such that some integer multiple of $e$ is the class of an irreducible divisor $E$. Then $(e,e)= -2$ or $(e,e)=2-2n$. If $(e,e)=2-2n$, then the class $(e,\bullet)$ in $H^2(X,{\mathbb Z})^*$ is divisible by $n-1$. \end{thm}
The Theorem is related to Theorem 22 in the beautiful paper \cite{hassett-tschinkel}. In particular, the case of fourfolds is settled in that paper. The hypothesis that the divisor $E$ is irreducible is necessary. There exist examples of pairs $(X,e)$, with $X$ of $K3^{[n]}$-type, $e\in H^2(X,{\mathbb Z})$, such that $2e$ is effective, the reflection by $e$ is an integral reflection of $H^2(X,{\mathbb Z})$, but $2-2n<(e,e)<-2$
(\cite{markman-constraints}, Example 4.8).
\begin{defi} \label{def-Mon-2} An isometry $g$ of $H^2(X,{\mathbb Z})$ is called a {\em monodromy operator}, if there exists a family ${\mathcal X} \rightarrow T$ (which may depend on $g$) of irreducible holomorphic symplectic manifolds, having $X$ as a fiber over a point $t_0\in T$, and such that $g$ belongs to the image of $\pi_1(T,t_0)$ under the monodromy representation. The {\em monodromy group} $Mon^2(X)$ of $X$ is the subgroup of $O[H^2(X,{\mathbb Z})]$ generated by all the monodromy operators. \end{defi}
\begin{defi} \label{def-monodromy-reflective} \begin{enumerate}
\item A class $e\in H^2(X,{\mathbb Z})$ is said to be {\em monodromy-reflective}, if $e$ is primitive, and the reflection $R_e(x):=x-\frac{2(x,e)}{(e,e)}e$, with respect to the class $e$, belongs to $Mon^2(X)$. \item A line bundle $L$ is said to be {\em monodromy-reflective}, if the class $c_1(L)$ is. \end{enumerate} \end{defi}
Theorem \ref{thm-2} is an immediate consequence of Theorem \ref{cor-introduction} and the following characterization of monodromy-reflective line bundles on $X$ of $K3^{[n]}$-type.
\begin{prop} \label{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon} Let $e\in H^2(X,{\mathbb Z})$ be a primitive class of negative degree $(e,e)$. Then the reflection $R_e$ belongs to $Mon^2(X)$, if and only if $e$ has one of the following two properties. \begin{enumerate} \item $(e,e)=-2$, or \item $(e,e)=2-2n$, and $n-1$ divides the class $(e,\bullet)\in H^2(X,{\mathbb Z})^*.$ \end{enumerate} \end{prop}
The proposition is proven in section \ref{sec-degrees-2-and-2-2n}. A class $e\in H^{1,1}(X,{\mathbb Z})$ is said to be {\em ${\mathbb Q}$-effective}, if some non-zero integer multiple of $e$ is the class of an effective divisor. Examples of monodromy-reflective line bundles, which are not ${\mathbb Q}$-effective, are exhibited in section \ref{sec-non-effective}.
\subsection{A classification of monodromy-reflective line bundles} \hspace{1ex}\\ When $X$ is a $K3$ surface, a monodromy-reflective line bundle $L$ has degree $-2$,
and precisely one of $L$ or $L^{-1}$ is isomorphic to $\StructureSheaf{X}(E)$, where $E$ is an effective divisor (see \cite{BHPV}, chapter VIII, Proposition 3.6). If ${\rm Pic}(X)$ is cyclic
then $E$ is necessarily a smooth rational curve. We may deform to this case upon deforming the pair $(X,L)$ to a nearby deformation equivalent pair.
Monodromy-reflective line bundles of degree $2-2n$, over $X$ of $K3^{[n]}$-type, need not be ${\mathbb Q}$-effective if $n> 1$. Whether the line bundle is or is not ${\mathbb Q}$-effective depends on a monodromy-invariant defined in Proposition \ref{prop-definition-of-rs} below. The definition depends on the following Theorem.
The topological $K$-group $K(S)$ of a $K3$ surface $S$, endowed with the {\em Mukai pairing} $(v,w):=-\chi(v^\vee\otimes w)$, is called the {\em Mukai lattice}. $K(S)$ is a rank $24$ even unimodular lattice isometric to the orthogonal direct sum $\widetilde{\Lambda}:=E_8(-1)^{\oplus 2}\oplus U^{\oplus 4}$, where $E_8(-1)$ is the negative definite $E_8$ lattice and $U$ is the rank $2$ lattice with Gram matrix {\scriptsize $\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array} \right)$}.
Let $\Lambda:=E_8(-1)^{\oplus 2}\oplus U^{\oplus 3}\oplus{\mathbb Z}\delta$, with $(\delta,\delta)=2-2n$. Then $H^2(X,{\mathbb Z})$ is isometric to $\Lambda$, for any $X$ of $K3^{[n]}$-type, $n>1$ \cite{beauville}. Let $O(\Lambda,\widetilde{\Lambda})$ be the set of primitive isometric embeddings $\iota:\Lambda\hookrightarrow\widetilde{\Lambda}$. $O(\widetilde{\Lambda})$ acts on $O(\Lambda,\widetilde{\Lambda})$ by compositions. If $n-1$ is a prime power, then $O(\Lambda,\widetilde{\Lambda})$ consists of a single $O(\widetilde{\Lambda})$-orbit. The Euler number $\eta:=\eta(n-1)$ is the number of distinct primes $p_1, \dots, p_\eta$ in the prime factorization $n-1=p_1^{e_1}\cdots p_\eta^{e_\eta}$, with positive integers $e_i$. For $n>2$, there are $2^{\eta-1}$ distinct $O(\widetilde{\Lambda})$-orbits in $O(\Lambda,\widetilde{\Lambda})$ (see \cite{oguiso} or \cite{markman-constraints}, Lemma 4.3).
\begin{thm} \label{thm-a-natural-orbit-of-embeddings-of-H-2-in-Mukai-lattice} (\cite{markman-constraints}, Theorem 1.10). An irreducible holomorphic symplectic manifold $X$ of $K3^{[n]}$-type, $n\geq 2$, comes with a natural choice of an $O(\widetilde{\Lambda})$-orbit of primitive isometric embeddings of $H^2(X,{\mathbb Z})$ in $\widetilde{\Lambda}$.
This orbit is monodromy-invariant, i.e., $\iota:H^2(X,{\mathbb Z})\hookrightarrow \widetilde{\Lambda}$ belongs to this orbit, if and only if $\iota\circ g$ does, for all $g\in Mon^2(X)$. \end{thm}
Let $S$ be a $K3$ surface, $H$ an ample line bundle on $S$, and $v\in K(S)$ a primitive class satisfying $(v,v)=2n-2$, $n\geq 2$. Assume that $X:=M_H(v)$ is a smooth and compact moduli space of $H$-stable sheaves of class $v$. Then $X$ is of $K3^{[n]}$-type and the orbit in the Theorem is that of Mukai's isometry $\iota:H^2(M_H(v),{\mathbb Z})\rightarrow v^\perp$, where $v^\perp\subset K(S)$ is the sub-lattice orthogonal to $v$ (see \cite{markman-constraints}, Theorem 1.14 or Theorem \ref{thm-item-orbit-of-inverse-of-Mukai-isom-is-natural} below). The monodromy-invariance, of the $O(K(S))$-orbit of Mukai's isometry, uniquely characterizes the orbit in the above Theorem, for every $X$ of $K3^{[n]}$-type.
Let $X$ be of $K3^{[n]}$-type, $n>1$. Let $I''_n(X)\subset H^2(X,{\mathbb Z})$ be the subset of monodromy-reflective classes of degree $2-2n$, and $e$ a class in $I''_n(X)$. Choose a primitive isometric embedding $\iota:H^2(X,{\mathbb Z})\hookrightarrow \widetilde{\Lambda}$, in the natural orbit of Theorem \ref{thm-a-natural-orbit-of-embeddings-of-H-2-in-Mukai-lattice}. Choose a generator $v$ of the rank $1$ sub-lattice of $\widetilde{\Lambda}$ orthogonal to the image of $\iota$. Then $(v,v)=2n-2$. Indeed, $(v,v)>0$, since the signature of $\widetilde{\Lambda}$ is $(4,20)$, and $(v,v)$ is equal to the order of $({\mathbb Z} v)^*/{\mathbb Z} v$, which is equal to the order of $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$, which is $2n-2$. Let $\rho$ be the positive integer, such that $(e+v)/\rho$ is an integral and primitive class in $\widetilde{\Lambda}$. Define the integer $\sigma$ similarly using $e-v$. Let ${\rm div}(e,\bullet)$ be the integer in $\{(e,e)/2,(e,e)\}$, such that the class $(e,\bullet)/{\rm div}(e,\bullet)$ is an integral and primitive class in $H^2(X,{\mathbb Z})^*$. Given a rational number $m$, let ${\mathcal F}(m)$ be the set of unordered pairs $\{r,s\}$ of positive integers, such that $rs=m$ and $\gcd(r,s)=1$. If $m$ is not a positive integer, then ${\mathcal F}(m)$ is empty. Set \[ \Sigma''_n \ \ := \ \ {\mathcal F}(n-1) \ \cup \ {\mathcal F}([n-1]/2) \ \cup \ {\mathcal F}([n-1]/4). \] Note that $\Sigma''_n$ is a singleton if and only if $n=2$ or $n-1$ is an odd prime power.
\begin{prop} \label{prop-definition-of-rs} If ${\rm div}(e,\bullet)=n-1$ and $n$ is even, set $\{r,s\}:=\{\rho,\sigma\}$. Otherwise, set $\{r,s\}:=\{\frac{\rho}{2},\frac{\sigma}{2}\}$. Then $\{r,s\}$ is a pair of relatively prime integers in $\Sigma''_n$, and the function \[ rs \ : \ I''_n(X) \ \ \ \longrightarrow \ \ \ \Sigma''_n, \] sending the class $e$ to the unordered pair $\{r,s\}$, is monodromy-invariant. The function $rs$ is surjective, if $n\equiv 1$ modulo $8$, and its image is ${\mathcal F}(n-1)\cup{\mathcal F}([n-1]/2)$ otherwise. \end{prop}
The proposition is proven in Lemmas \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} and \ref{lem-non-unimodular-rank-two-lattice}. A more conceptual definition of the monodromy-invariant $rs$ is provided in the statements of these Lemmas. The proof relies on the classification of the isometry classes of all possible pairs $(\widetilde{L},e)$, where $\widetilde{L}$ is the saturation in $\widetilde{\Lambda}$ of the rank $2$ sub-lattice ${\rm span}\{e,v\}$. The classification is summarized in the table following Lemma \ref{lemma-isometry-orbits-in-rank-2}. We finally arrive at the classification of monodromy-reflective line bundles.
\begin{prop} \label{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants} Let $X$ be of $K3^{[n]}$-type and $L$ a monodromy-reflective line bundle. Set $e:=c_1(L)$. \begin{enumerate} \item \label{prop-item-Mon-orbit-in-degree-2-minus-2n} If $(e,e)=2-2n$, then the $Mon^2(X)$-orbit of the class $e$ is determined by ${\rm div}(e,\bullet)$ and the value $rs(e)$. \item \label{prop-item-Mon-orbit-in-degree-minus-2} If $(e,e)=-2$, then the $Mon^2(X)$-orbit of the class $e$ is determined by ${\rm div}(e,\bullet)$. \end{enumerate} \end{prop}
Part \ref{prop-item-Mon-orbit-in-degree-2-minus-2n} is proven in Lemmas \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} and \ref{lem-non-unimodular-rank-two-lattice}. Part \ref{prop-item-Mon-orbit-in-degree-minus-2} is Lemma 8.9 in \cite{markman-monodromy-I}.
\subsection{A numerical characterization of exceptional \WithoutTorelli{classes} \WithTorelli{classes}}
\hspace{1ex}\\ Let $(X_1,L_1)$ and $(X_2,L_2)$ be two pairs as in Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants}. Set $e_i:=c_1(L_i)$. If $(e_i,e_i)=2-2n$, ${\rm div}(e_1,\bullet)={\rm div}(e_2,\bullet)$, and $rs(e_1)=rs(e_2)$, then
\WithoutTorelli{the pairs $(X_1,e_1)$ and $(X_2,e_2)$ are deformation equivalent (allowing $e_i$ to loose it Hodge type along the deformation), by Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants} and Lemma \ref{lem-monodromy-invariants-and-deformation-equivalence}. Furthermore, a deformation relating the two pairs and preserving the Hodge type exists as well, assuming a version of the Torelli Theorem holds (see section \ref{sec-deformation-equivalence-and-torelli}). Finally,}
\WithTorelli{the pairs $(X_1,L_1)$ and $(X_2,L_2^\epsilon)$ are deformation equivalent (requiring $e_i$ to preserve its Hodge type along the deformation), for $\epsilon=1$ or $\epsilon=-1$, by Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants} and Lemma \ref{lem-monodromy-invariants-and-deformation-equivalence} (the proof of the latter depends on the Torelli Theorem \cite{verbitsky}). Furthermore,}
if $X_1$ is projective and $L_1^{\otimes k}\cong\StructureSheaf{X_1}(E_1)$, for some $k>0$ and a prime exceptional divisor $E_1$, then a generic small deformation $(X,L)$ of $(X_2,L_2)$ consists of $L$ satisfying $L^{\otimes d}\cong\StructureSheaf{X}(E)$, for a prime exceptional divisor $E$ and for $d=k$ or $d=-k$
(Proposition \ref{prop-main-question-on-deformation-equivalence}). This leads us to the
\WithoutTorelli{conjectural}
numerical characterization of exceptional line bundles described in this section.
Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, $n\geq 2$. Let $L$ be a monodromy-reflective line bundle on $X$, $e:=c_1(L)$, and $R_e$ the reflection by $e$. $R_e$ preserves the Hodge structure, and so acts on $H^{1,1}(X)\cong H^1(X,T_X)$. The Kuranishi deformation space $Def(X)$ is an open neighborhood of $0$ in $H^1(X,T_X)$, which may be chosen to be $R_e$ invariant. Hence, $R_e$ acts on $Def(X)$. The Local Kuranishi deformation space $Def(X,L)$, of the pair $(X,L)$, is the smooth divisor $D_e\subset Def(X)$ of fixed points of $R_e$.
\begin{defi} \label{def-numerically-prime-exceptional} Let $h\in H^2(X,{\mathbb R})$ be a K\"{a}hler class. A line bundle $L\in {\rm Pic}(X)$ is called {\em numerically exceptional}, if its first Chern class $e:=c_1(L)$ is a primitive class in $H^2(X,{\mathbb Z})$, satisfying $(h,e)>0$ and the following properties. The Beauville-Bogomolov degree is either $(e,e)=-2$, or $(e,e)=2-2n$ and $n:=\dim_{\mathbb C}(X)/2>2$. In the latter case one of the following properties holds: \begin{enumerate} \item ${\rm div}(e,\bullet)=2n-2$ and $rs(e)=\{1,n-1\}$. \item ${\rm div}(e,\bullet)=2n-2$ and $rs(e)=\{2,(n-1)/2\}$. We must have $n\equiv 3$ (modulo $4$) for the pair $rs(e)$ to be relatively prime. \item ${\rm div}(e,\bullet)=n-1$, $n$ is even, and $rs(e)=\{1,n-1\}$. \item ${\rm div}(e,\bullet)=n-1$, $n$ is odd, and $rs(e)=\{1,(n-1)/2\}$. \end{enumerate} A cohomology class $e\in H^{1,1}(X,{\mathbb Z})$ is {\em numerically exceptional}, if $e=c_1(L)$, for a numerically exceptional line bundle $L$. \end{defi}
\begin{defi} \label{def-stably-prime-exceptional} \begin{enumerate} \item A line bundle $L\in {\rm Pic}(X)$ is called {\em stably-prime-exceptional}, if there exists a closed complex analytic subset $Z\subset D_e$, of codimension $\geq 1$, such that the linear system $\linsys{L_t}$ consists of a prime-exceptional divisor $E_t$, for all $t\in [D_e\setminus Z]$. \item $L$ is said to be {\em stably-${\mathbb Q}$-effective}, if there exists a non-zero integer $k$,
such that the linear system $\linsys{L_t^k}$ is non-empty, for all $t\in D_e$. \end{enumerate} \end{defi}
If $E$ is a prime exceptional divisor on a projective irreducible holomorphic symplectic manifold $X$, then $\StructureSheaf{X}(E)$ is stably-prime-exceptional, by Proposition \ref{prop-generic-prime-exceptional}.
Let $L$ be a line bundle on an irreducible holomorphic symplectic manifold $X$ of $K3^{[n]}$-type with a primitive first Chern class. Recall that a necessary condition for the linear system $\linsys{L^k}$ to consist of an exceptional divisor $E$, is that $L$ is monodromy-reflective (Definition \ref{def-monodromy-reflective}), by Theorem \ref{cor-introduction}. Assume that $L$ is monodromy-reflective. Set $e:=c_1(L)$.
Let $D_e\subset Def(X)$ be the divisor fixed by the reflection $R_e$.
\WithoutTorelli{\begin{conj}} \WithTorelli{\begin{thm}} \label{conj-exceptional-line-bundles} \begin{enumerate} \item \label{conj-item-effective} Assume that $L$ is numerically exceptional. Then $L^k$ is stably-prime-exceptional, where $k$ is determined as follows.
If the degree of $L$ is $2-2n$, then \[ k = \left\{ \begin{array}{ccl} 2, & \mbox{if} & {\rm div}(e,\bullet)=2n-2 \ \mbox{and} \ rs(e)=\{1,n-1\}, \\ 1, & \mbox{if} & {\rm div}(e,\bullet)=2n-2 \ \mbox{and} \ rs(e)=\{2,(n-1)/2\}, \\ 1, & \mbox{if} & {\rm div}(e,\bullet)=n-1. \end{array} \right. \] If the degree of $L$ is $-2$, then \[ k = \left\{ \begin{array}{ccl} 2, & \mbox{if} & {\rm div}(e,\bullet)=2 \ \mbox{and} \ n=2, \\ 1, & \mbox{if} & {\rm div}(e,\bullet)=2 \ \mbox{and} \ n>2, \\ 1, & \mbox{if} & {\rm div}(e,\bullet)=1. \end{array} \right. \]
\item \label{conj-item-vanishing} If $L$ is not numerically exceptional, then $L$ is not stably-${\mathbb Q}$-effective. I.e., for every non-zero integer $k$, there exists a dense open subset $U^k$ of $D_e$, such that $H^0(X_t,L_t^k)$ vanishes, for all $t\in U^k$. \end{enumerate} \WithoutTorelli{\end{conj}} \WithTorelli{\end{thm}}
See Remark \ref{rem-Euler-characteristic} for the Euler characteristic $\chi(L^k)$. Note that in part \ref{conj-item-effective} above $L^k$ is effective as well, for the specified integer $k$, by the semi-continuity theorem.
\WithoutTorelli{ \begin{thm} \label{thm-main-conjecture-follows-from-torelli} Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type and $L$ a monodromy-reflective line bundle on $X$. Assume an affirmative answer to the Torelli Question \ref{thm-torelli} [or the weaker Question \ref{question-connectedness} for the pair $(X,L)$].
Then Conjecture \ref{conj-exceptional-line-bundles} holds for $(X,L)$. \end{thm} Verbitsky recently posted a proof of an affirmative answer to Question \ref{thm-torelli} \cite{verbitsky}. Theorem \ref{thm-main-conjecture-follows-from-torelli} is proven in section \ref{sec-numerical-characterization-via-torelli}.}
\WithTorelli{Theorem \ref{conj-exceptional-line-bundles} is proven in section \ref{sec-numerical-characterization-via-torelli}.}
The proof relies both on the Torelli Theorem \cite{verbitsky} and the examples worked out in sections \ref{sec-examples} and \ref{sec-non-effective}. We exhibit an example of a pair $(X,L)$, for each possible value of the monodromy invariants $(e,e)$, ${\rm div}(e,\bullet)$, and $rs(e)$, and verify \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles} for $(X,L)$.
All values of the monodromy invariants are realized by examples where $X$ is a smooth and projective moduli space of sheaves on a $K3$ surface. See the table in section \ref{sec-numerical-characterization-via-torelli} for a reference to an example, for each value of the monodromy-invariants.
The vanishing in part \ref{conj-item-vanishing} of \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles} is verified in the examples as follows. In all the examples of monodromy-reflective but non-numerically-exceptional line bundles considered in section \ref{sec-non-effective}, $X$ admits a birational involution $\iota: X\rightarrow X$, inducing the reflection $R_{e}$.
The following simple observation is proven in section \ref{sec-non-effective}. \begin{observation} \label{observation-not-Q-effective} If $L$ is a monodromy reflective line bundle on $X$, and there exists a bimeromorphic involution $\iota:X\rightarrow X$ inducing the reflection $R_{e}$, $e=c_1(L)$, then the line bundle $L$ is not ${\mathbb Q}$-effective. \end{observation}
\subsection{Cones} Let $X$ be a projective irreducible holomorphic symplectic manifold. Set $N^1(X):=H^{1,1}(X,{\mathbb Z})\otimes_{\mathbb Z}{\mathbb R}$ and let ${\mathcal C}_X^{1,1}$ be the connected component of the cone $\{\lambda\in N^1(X) \ : \ (\lambda,\lambda)>0\}$, which contains the ample cone. Denote by $\overline{{\mathcal C}}_X^{1,1}$ its closure. A divisor $D$ on $X$ is called {\em movable}, if the base locus of the linear system $\linsys{D}$ has codimension $\geq 2$ in $X$. Denote by $\MV_X$ the convex cone in $N^1(X)$ generated by classes of movable divisors. Let $\overline{\MV}_X$ be its closure in $N^1(X)$. Then $\overline{\MV}_X$ is equal to the sub-cone of $\overline{{\mathcal C}}_X^{1,1}$, consisting of classes $\lambda$, such that $(\lambda,[E])\geq 0$, for every prime exceptional divisor $E$ \cite[Lemma 6.22]{boucksom,markman-torelli}.
The closure of the movable cone can be described also in terms of the set of stably-prime-exceptional divisors. $\overline{\MV}_X$ is the sub-cone of $\overline{{\mathcal C}}_X^{1,1}$, consisting of classes $\lambda$, such that $(\lambda,e)\geq 0$, for every stably-prime-exceptional class $e$ \cite[Theorem 6.17 and Lemma 6.22]{markman-torelli}. Hence, Theorem \ref{conj-exceptional-line-bundles} above determines the closure of the movable cone. Furthermore, a stably-prime-exceptional class $e$ is prime exceptional, if and only if the hyperplane orthogonal to $e$ intersects $\overline{\MV}_X$ along a face of codimension one of the latter \cite[Lemma 6.20]{markman-torelli}. In this sense Theorem \ref{conj-exceptional-line-bundles} determines the set of classes of prime exceptional divisors.
\subsection{The structure of the paper} The paper is organized as follows. In section \ref{sec-easy-examples} we provide a sequence of easy examples of monodromy-reflective line bundles on moduli spaces of sheaves on $K3$ surfaces. We calculate their invariants, and determine whether or not they are effective, illustrating
\WithTorelli{Theorem}
\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles}.
In section \ref{sec-monodromy-reflection} we prove Theorem \ref{cor-introduction} stating that associated to a prime exceptional divisor $E$ is a monodromy involution of the integral cohomology $H^*(X,{\mathbb Z})$, which acts on the second cohomology lattice as the reflection by the cohomology class $[E]$ of $E$ (Corollary \ref{cor-1}). In section \ref{sec-degrees-2-and-2-2n} we specialize to the $K3^{[n]}$-type case, $n\geq 2$, and prove Theorem \ref{thm-2} about the possible degrees of prime exceptional divisors.
Let $(X_i,E_i)$, $i=1,2$, be two pairs, each consisting of an irreducible holomorphic symplectic manifold $X_i$, and a prime exceptional divisor $E_i$. Let $e_i\in H^{2}(X,{\mathbb Z})$ be the class of $E_i$. In section \ref{sec-deformation-equivalence} we define two notions of deformation equivalence:
(1) Deformation equivalence of the two pairs $(X_i,E_i)$, $i=1,2$.
(2) Deformation equivalence of the two pairs $(X_i,e_i)$, $i=1,2$.
\noindent We relate these two notions via Torelli.
In section \ref{sec-Mukai} we return to the case where $X$ is of $K3^{[n]}$-type, $n\geq 2$. We associate, to each monodromy-reflective class $e\in H^2(X,{\mathbb Z})$ of degree $2-2n$, an isometry class of a pair $(\widetilde{L},\tilde{e})$, consisting of
a rank $2$ integral lattice $\widetilde{L}$ of signature $(1,1)$ and a primitive class $\tilde{e}\in \widetilde{L}$, with $(\tilde{e},\tilde{e})=2-2n$. The isometry class of the pair $(\widetilde{L},\tilde{e})$ is a monodromy-invariant, denoted by $f(X,e)$.
In section \ref{sec-invariant-rs} we calculate the monodromy-invariant $f(X,e)$ explicitly as the function $rs$ in Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants}. We then prove Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants}.
In section \ref{sec-numerical-characterization-via-torelli} we prove \WithTorelli{Theorem \ref{conj-exceptional-line-bundles}, which provides a numerical characterization of exceptional classes.} \WithoutTorelli{Theorem \ref{thm-main-conjecture-follows-from-torelli} stating that the numerical characterization of exceptional classes, suggested in Conjecture \ref{conj-exceptional-line-bundles}, follows from a version of Torelli.}
Sections \ref{sec-conditions-for-existence-ofslope-stable-vector-bundles}, \ref{sec-examples}, and \ref{sec-non-effective} are devoted to examples of monodromy-reflective line bundles over moduli spaces of stable sheaves. In section \ref{sec-conditions-for-existence-ofslope-stable-vector-bundles} we study the exceptional locus of Jun Li's morphism from certain moduli spaces, of Gieseker-Maruyama $H$-stable sheaves on a $K3$-surface, to the Uhlenbeck-Yau compactifications of the moduli spaces of $H$-slope stable vector-bundles.
In section \ref{sec-examples} we exhibit an example of a prime exceptional divisor, for each value of the invariants of a monodromy-reflective line bundle $L$, for which $L$ is stated to be stably-${\mathbb Q}$-effective in \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles}.
In section \ref{sec-non-effective} we exhibit an example of a monodromy-reflective line bundle $L$, which is not ${\mathbb Q}$-effective, for each value of the invariants for which $L$ is stated not to be ${\mathbb Q}$-effective in \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles}.
\hide{ In section \ref{sec-zariski-decomposition} we consider pairs $(X,L)$, with $X$ of $K3^{[n]}$-type and $L$ a line bundle of negative Beauville-Bogomolov degree, which is {\em not} monodromy-reflective. We review first the Zariski decomposition of effective divisors, due to Boucksom \cite{boucksom}. Theorem \ref{thm-2}, and the existence of a divisorial Zariski decomposition, imply that $L$ is not ${\mathbb Q}$-effective, for a generic such pair $(X,L)$ (Lemma \ref{lemma-generic-vanishing}).
}
\section{Easy examples of monodromy-reflective line bundles} \label{sec-easy-examples} In section \ref{sec-Mukai-notation} we review basic facts about moduli spaces of coherent sheaves on $K3$ surfaces. In section \ref{sec-sequence-of-examples} we briefly describe a sequence of examples of pairs $(X,e)$, with $X$ of $K3^{[n]}$-type, $e$ a monodromy-reflective class of degree $2-2n$ with ${\rm div}(e,\bullet)=2n-2$, for each $n\geq 2$, and for each value of the invariant $rs$. For details, references, as well as for examples of degree $-2$, or with ${\rm div}(e,\bullet)=n-1$, see sections \ref{sec-examples} and \ref{sec-non-effective}.
\subsection{The Mukai isomorphism} \label{sec-Mukai-notation} The group $K(S)$, endowed with the {\em Mukai pairing} \[ (v,w) \ \ := \ \ -\chi(v^\vee\otimes w), \] is called the {\em Mukai lattice}. Let us recall Mukai's notation for elements of $K(S)$. Identify the group $K(S)$ with $H^*(S,{\mathbb Z})$, via the isomorphism sending a class $F$ to its {\em Mukai vector} $ch(F)\sqrt{td_S}$. Using the grading of $H^*(S,{\mathbb Z})$, the Mukai vector is \begin{equation} \label{eq-Mukai-vector} ({\rm rank}(F),c_1(F),\chi(F)-{\rm rank}(F)), \end{equation} where the rank is considered in $H^0$ and $\chi(F)-{\rm rank}(F)$ in $H^4$ via multiplication by the orientation class of $S$. The homomorphism $ch(\bullet)\sqrt{td_S}:K(S)\rightarrow H^*(S,{\mathbb Z})$ is an isometry with respect to the Mukai pairing on $K(S)$ and the pairing \[ \left((r',c',s'),(r'',c'',s'')\right) \ \ = \ \ \int_{S}c'\cup c'' -r'\cup s''-s'\cup r'' \] on $H^*(S,{\mathbb Z})$ (by the Hirzebruch-Riemann-Roch Theorem). For example, $(1,0,1-n)$ is the Mukai vector in $H^*(S,{\mathbb Z})$, of the ideal sheaf of a length $n$ subscheme. Mukai defines a weight $2$ Hodge structure on the Mukai lattice $H^*(S,{\mathbb Z})$, and hence on $K(S)$, by extending that of $H^2(S,{\mathbb Z})$, so that the direct summands $H^0(S,{\mathbb Z})$ and $H^4(S,{\mathbb Z})$ are of type $(1,1)$.
Let $v\in K(S)$ be a primitive class with $c_1(v)$ of Hodge-type $(1,1)$. There is a system of hyperplanes in the ample cone of $S$, called $v$-walls, that is countable but locally finite \cite{huybrechts-lehn-book}, Ch. 4C. An ample class is called {\em $v$-generic}, if it does not belong to any $v$-wall. Choose a $v$-generic ample class $H$. Let $M_H(v)$ be the moduli space of $H$-stable sheaves on the $K3$ surface $S$ with class $v$. When non-empty, the moduli space $M_H(v)$ is a smooth projective irreducible holomorphic symplectic variety of $K3^{[n]}$ type, with $n=\frac{(v,v)+2}{2}$. This result is due to several people, including Huybrechts, Mukai, O'Grady, and Yoshioka. It can be found in its final form in \cite{yoshioka-abelian-surface}.
Over $S\times M_H(v)$ there exists a universal sheaf ${\mathcal F}$, possibly twisted with respect to a non-trivial Brauer class pulled-back from $M_H(v)$. Associated to ${\mathcal F}$ is a class $[{\mathcal F}]$ in $K(S\times M_H(v))$ (\cite{markman-integral-generators}, Definition 26). Let $\pi_i$ be the projection from $S\times M_H(v)$ onto the $i$-th factor. Assume that $(v,v)>0$. The second integral cohomology $H^2(M_H(v),{\mathbb Z})$, its Hodge structure, and its Beauville-Bogomolov pairing, are all described by Mukai's Hodge-isometry \begin{equation} \label{eq-Mukai-isomorphism} \theta \ : \ v^\perp \ \ \ \longrightarrow \ \ \ H^2(M_H(v),{\mathbb Z}), \end{equation} given by $\theta(x):=c_1\left(\pi_{2_!}\{\pi_1^!(x^\vee)\otimes [{\mathcal F}]\}\right)$ (see \cite{yoshioka-abelian-surface}).
Let $\widetilde{\Lambda}$ be the unimodular lattice $E_8(-1)^{\oplus 2}\oplus U^{\oplus 4}$, where $U$ is the rank two unimodular hyperbolic lattice. $\widetilde{\Lambda}$ is isometric to the Mukai lattice of a $K3$ surface. Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, $n\geq 2$. Recall that $X$ comes with a natural choice of an $O(\widetilde{\Lambda})$-orbit of primitive isometric embeddings of $H^2(X,{\mathbb Z})$ in $\widetilde{\Lambda}$, by Theorem \ref{thm-a-natural-orbit-of-embeddings-of-H-2-in-Mukai-lattice}.
\begin{thm} \label{thm-item-orbit-of-inverse-of-Mukai-isom-is-natural} (\cite{markman-constraints}, Theorem 1.14). When $X$ is isomorphic to the moduli space $M_H(v)$, of $H$-stable sheaves on a $K3$ surface of class $v\in K(S)$, then the above mentioned $O(\widetilde{\Lambda})$-orbit is that of the composition \begin{equation} \label{eq-iota-for-a-moduli-space} H^2(M_H(v),{\mathbb Z}) \LongRightArrowOf{\theta^{-1}}v^\perp \subset K(S) \cong \widetilde{\Lambda}, \end{equation} where $\theta^{-1}$ is the inverse of the Mukai isometry given in (\ref{eq-Mukai-isomorphism}). \end{thm}
The combination of Theorems \ref{thm-a-natural-orbit-of-embeddings-of-H-2-in-Mukai-lattice} and \ref{thm-item-orbit-of-inverse-of-Mukai-isom-is-natural} is an example of the following meta-principle guiding our study of the monodromy of holomorphic symplectic varieties of $K3^{[n]}$-type.
{\em Any topological construction, which can be performed uniformly and naturally for all smooth and compact moduli spaces of sheaves on any $K3$ surface $S$, and which is invariant under symmetries induced by equivalences of derived categories of $K3$-surfaces, is monodromy-invariant. }
\subsection{A representative sequence of examples} \label{sec-sequence-of-examples} Let $S$ be a projective $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Fix integers $r$ and $s$ satisfying $s\geq r\geq 1$, and $\gcd(r,s)=1$. Let $X$ be the moduli space $M_H(r,0,-s)$.
Then $X$ is a projective irreducible holomorphic symplectic manifold of $K3^{[n]}$-type with $n=1+\nolinebreak rs$ \cite{yoshioka-abelian-surface}. Set $e:=\theta(r,0,s)$. The weight two integral Hodge structure $H^2(M_H(r,0,-s),{\mathbb Z})$ is Hodge-isometric
to the orthogonal direct sum $H^2(S,{\mathbb Z})\oplus {\mathbb Z} e$, and the class $e$ is monodromy-reflective of Hodge-type $(1,1)$, $(e,e)=2-2n$, ${\rm div}(e,\bullet)=2n-2$, and $rs(e)=\{r,s\}$.
When $r=1$, then $X=S^{[n]}$ is the Hilbert scheme. Let $E\subset S^{[n]}$ be the big diagonal. Then $E$ is a prime divisor, which is the exceptional locus of the Hilbert-Chow morphism $\pi:S^{[n]}\rightarrow S^{(n)}$ onto the $n$-th symmetric product of $S$. The equality $e=\frac{1}{2}[E]$ was proven in \cite{beauville}.
When $r=2$, let $E\subset M_H(2,0,-s)$ be the locus of $H$-stable sheaves which are not locally free. Then $E$ is a prime divisor, which is the exceptional locus of Jun Li's morphism from $M_H(2,0,-s)$ onto the Uhlenbeck-Yau compactification of the moduli space of $H$-slope-stable vector bundles of that class. The equality $e=[E]$ holds, by Lemma \ref{lemma-class-of-exceptional-locus}.
When $r\geq 3$, let $Exc\subset M_H(r,0,-s)$ be the locus of $H$-stable sheaves, which are not locally free or not $H$-slope-stable. Then $Exc$ is a closed algebraic subset of $M_H(r,0,-s)$ of codimension $\geq 2$, by Lemma \ref{lem-codimension-of-Exc}. Jun Li's morphism is thus not a divisorial contraction. Set $U:=X\setminus Exc$.
Let $\iota:U\rightarrow U$ be the regular involution, which sends a locally free $H$-slope-stable sheaf $F$ to the dual sheaf $F^*$. The restriction homomorphism from $H^2(X,{\mathbb Z})$ to $H^2(U,{\mathbb Z})$ is an isomorphism, and the induced involution $\iota^*$ of $H^2(X,{\mathbb Z})$ is the reflection by the class $e$, by Proposition \ref{prop-vanishing-in-divisibility-2n-2}. The class $e$ is thus not ${\mathbb Q}$-effective, by Observation \ref{observation-not-Q-effective}.
\section{The monodromy reflection of a prime exceptional divisor} \label{sec-monodromy-reflection} Let $X$ be a projective irreducible holomorphic symplectic manifold and $E$ a reduced and irreducible divisor with negative Beauville-Bogomolov degree. The following result is due to S. Bouksom and S. Druel.
\begin{prop} \label{prop-druel} (\cite{druel}, Proposition 1.4) There exists a sequence of flops of $X$, resulting in a smooth birational model $X'$ of $X$, such that the strict transform $E'$ of $E$ in $X'$ is contractible via a projective birational morphism $\pi:X' \rightarrow Y$ to a normal projective variety $Y$. The exceptional locus of $\pi$ is equal to the support of $E'$. \end{prop}
The divisor $E$ is assumed to be {\em exceptional}, rather than to have negative Beauville-Bogomolov degree, in the statement of Proposition 1.4 in \cite{druel}. The technical term exceptional is in the sense of \cite{boucksom}, Definition 3.10, and is a precise measure of rigidity. Boucksom characterized exceptional divisors on irreducible holomorphic symplectic varieties by the following property, which we will use as a definition (\cite{boucksom}, Theorem 4.5).
\begin{defi} \label{def-rational-exceptional} A rational divisor $E\in Div(X)\otimes_{\mathbb Z}{\mathbb Q}$ is {\em exceptional}, if $E=\sum_{i=1}^k n_i E_i$, with positive rational coefficients $n_i$, prime divisors $E_i$, and a negative definite Gram-matrix $([E_i],[E_j])$. \end{defi}
In particular, a prime divisor is exceptional, if and only if it has negative Beauville-Bogomolov degree. Hence, we may replace in the above Proposition the hypothesis that $E$ is exceptional by the hypothesis that $E$ has negative Beauville-Bogomolov degree.
\begin{defi} \label{def-exceptional} A primitive class $e\in H^2(X,{\mathbb Z})$ is {\em (prime) exceptional} if some positive multiple of $e$ is the class of a (prime) exceptional divisor. A line bundle $L\in Pic(X)$ is {\em (prime) exceptional}, if $c_1(L)$ is. \end{defi}
Let $Def(X')$ and $Def(Y)$ be the Kuranishi deformation spaces of $X'$ and $Y$. Denote by $\psi:{\mathcal X}\rightarrow Def(X')$ the semi-universal deformation of $X'$, by $0\in Def(X')$ the point with fiber $X'$, and let $X_t$ be the fiber over $t\in Def(X')$. Let $\bar{\psi}:{\mathcal Y}\rightarrow Def(Y)$ be the semi-universal deformation of $Y$, $\bar{0}\in Def(Y)$ its special point with fiber $Y$, and $Y_t$ the fiber over $t\in Def(Y)$.
The variety $Y$ necessarily has rational Gorenstein singularities, by \cite{beuville-symplectic-singularities}, Proposition 1.3. The morphism $\pi:X'\rightarrow Y$ deforms as a morphism $\nu$ of the semi-universal families, which fits in a commutative diagram \begin{equation} \label{diagram-f-general} \begin{array}{ccc} {\mathcal X} & \RightArrowOf{\nu} & {\mathcal Y} \\ \psi \ \downarrow \hspace{1ex} & & \bar{\psi} \ \downarrow \ \hspace{1ex} \\ Def(X') & \RightArrowOf{f} & Def(Y), \end{array} \end{equation} by \cite{kollar-mori}, Proposition 11.4. The following is a fundamental theorem of Namikawa:
\begin{thm} \label{thm-namikawa} (\cite{namikawa}, Theorem 2.2) The Kuranishi deformation spaces $Def(X')$ and $Def(Y)$ are both smooth of the same dimension. They can be replaced by open neighborhoods of $0\in Def(X')$ and $\bar{0}\in Def(Y)$, and denoted again by $Def(X')$ and $Def(Y)$, in such a way that there exists a natural proper surjective map $f:Def(X')\rightarrow Def(Y)$ with finite fibers. Moreover, for a generic point $t\in Def(X')$, $Y_{f(t)}$ is isomorphic to $X_t$. \end{thm}
The morphism $f:Def(X')\rightarrow Def(Y)$ is in fact a branched Galois covering, by \cite{markman-galois}, Lemma 1.2. The Galois group $G$ is a product of Weyl groups of finite type, by \cite{markman-galois}, Theorem 1.4 (see also \cite{namikawa-galois}). Furthermore, $G$ acts on $H^*(X',{\mathbb Z})$ via monodromy operators preserving the Hodge structure. When the exceptional locus of $\pi:X'\rightarrow Y$ contains a single irreducible component of co-dimension one in $X'$, then the Galois group $G$ is ${\mathbb Z}/2{\mathbb Z}$.
Let $\Sigma\subset Y$ be the singular locus. The {\em dissident locus} $\Sigma_0\subset \Sigma$ is the locus along which the singularities of $Y$ fail to be of $ADE$ type. $\Sigma_0$ is a closed subvariety of $\Sigma$.
\begin{prop} \label{prop-dissident-locus} (\cite{namikawa}, Propositions 1.6, \cite{wierzba}) $Y$ has only canonical singularities. The dissident locus $\Sigma_0$ has codimension at least $4$ in $Y$. The complement $\Sigma\setminus\Sigma_0$ is either empty, or the disjoint union of codimension $2$ smooth and symplectic subvarieties of $Y\setminus \Sigma_0$. \end{prop}
Keep the notation of Proposition \ref{prop-druel}. Let $[E]$ be the class of $E$ in $H^2(X,{\mathbb Z})$ and $R$ the reflection of $H^2(X,{\mathbb Q})$ given by \[ R(x) \ \ := \ \ x-\left(\frac{2(x,[E])}{([E],[E])}\right)[E]. \] Consider the natural isomorphism $H^2(X,{\mathbb Z})^*\cong H_2(X,{\mathbb Z})$, given by the Universal Coefficients Theorem and the fact that $X$ is simply connected. Denote by \begin{equation} \label{eq-E-vee} [E]^\vee \ \ \ \in \ \ \ H_2(X,{\mathbb Q}) \end{equation} the class corresponding to $\frac{-2([E],\bullet)}{([E],[E])}$, where both pairings in the fraction are the Beauville-Bogomolov pairing.
We identify $H^2(X,{\mathbb Z})$ and $H^2(X',{\mathbb Z})$ via the graph of the birational map. This graph induces a Hodge isometry
and the isometry maps the class $[E]\in H^2(X,{\mathbb Z})$ to the class $[E']\in H^2(X',{\mathbb Z})$, by \cite{ogrady-weight-two}, Proposition 1.6.2. We get an identification of the dual groups $H_2(X,{\mathbb Z})$ and $H_2(X',{\mathbb Z})$. The following Corollary was proven, in case $E$ is an irreducible component of a contractible divisor, in \cite{markman-galois} Lemmas 4.10 and 4.23. We are now able to extend it to the more general case of a prime exceptional divisor $E$. The following is a Corollary of Proposition \ref{prop-druel}, Proposition \ref{prop-dissident-locus}, and \cite{markman-galois}, Lemmas 4.10 and 4.23.
\begin{cor}\label{cor-1} \begin{enumerate} \item \label{item-integrality} The class $[E]^\vee\in H_2(X,{\mathbb Z})$ corresponds to the class in $H_2(X',{\mathbb Z})$ of the generic fiber of the contraction $E'\rightarrow Y$ in Proposition \ref{prop-druel}. The generic fiber is either a smooth rational curve, or the union of two homologous smooth rational curves meeting at one point. In particular, $[E]^\vee$ is an integral class in $H_2(X,{\mathbb Z})$ and $R$ is an integral isometry. \item \label{item-monodromy-operator} The reflection $R$ is a monodromy operator in $Mon^2(X)$ as well as $Mon^2(X')$. $R$ preserves the Hodge structure. The action of $R$ on $H^{1,1}(X')\cong H^1(X',TX')$ induces an involution of $Def(X')$, which generates the Galois group of the double cover of the Kuranishi deformation spaces $Def(X')\rightarrow Def(Y)$. \item \label{item-divisibility-at-most-two} Either $[E]$ is a primitive class of $H^2(X,{\mathbb Z})$, or $[E]$ is twice a primitive class. Similarly, $[E]^\vee$ is either a primitive class or twice a primitive class. Finally, at least one of $[E]$ or $[E]^\vee$ is a primitive class \end{enumerate} \end{cor}
\begin{proof}
\ref{item-integrality}) The singular locus of $Y$ contains a unique irreducible component $\Sigma$ of codimension $2$, and $Y$ has singularities of type $A_1$ or $A_2$ along the Zariski dense open subset $\Sigma\setminus\Sigma_0$, by Proposition \ref{prop-dissident-locus} (see also the classification of singularities in section 1.8 of \cite{namikawa}). When $X=X'$, the class $[E']^\vee$ is the class of the fiber of the composite morphism $E'\hookrightarrow X'\rightarrow Y$, by Lemmas 4.10 and 4.23 in \cite{markman-galois}.
\ref{item-monodromy-operator}) $R$ is a monodromy operator in $Mon^2(X')$, by Lemmas 4.10 and 4.23 in \cite{markman-galois}. Now the isometry $Z_*:H^2(X,{\mathbb Z})\rightarrow H^2(X',{\mathbb Z})$, induced by the graph $Z$ of the birational map, is a parallel transport operator. This follows from the proof of Theorem 2.5 in \cite{huybrechts-kahler-cone}. In this proof Huybrechts constructs a correspondence $\Gamma:=Z+\sum_{i}Y_i$ in $X\times X'$ with the following properties. $\Gamma_*:H^2(X,{\mathbb Z})\rightarrow H^2(X',{\mathbb Z})$ is a parallel transport operator, $Z$ is the closure of the graph of the birational map as above, and the image of $Y_i$ in each factor $X$ and $X'$ has codimension $\geq 2$. It follows that the two isometries $Z_*$ and $\Gamma_*$ coincide.
\ref{item-divisibility-at-most-two}) The statements about the divisibility of $[E]$ and $[E]^\vee$ follow from the equality $\int_{[E]^\vee}[E]=-2$. \end{proof}
We denote by \[ e\in H^2(X,{\mathbb Z}) \] the primitive class, such that either $[E]=e$ or $[E]=2e$. Let $e^\vee$ be the primitive class in $H_2(X,{\mathbb Z})$, such that $[E]^\vee=e^\vee$ or $[E]^\vee=2e^\vee$. The {\em divisibility factor} ${\rm div}(e,\bullet)$, of the class $(e,\bullet)\in H^2(X,{\mathbb Z})^*$, is the positive number satisfying the equality $(e,\bullet)={\rm div}(e,\bullet)\cdot e^\vee$.
\begin{lem} \label{lemma-divisibility} We have \[ -{\rm div}(e,\bullet) \ \ = \ \ \left\{ \begin{array}{ccc} (e,e)/2 & \mbox{if} & [E]=e \ \mbox{and} \ [E]^\vee=e^\vee, \\ (e,e) & \mbox{if} & [E]=2e \ \mbox{and} \ [E]^\vee=e^\vee, \\ (e,e) & \mbox{if} & [E]=e \ \mbox{and} \ [E]^\vee=2e^\vee. \end{array} \right. \] \end{lem}
\begin{proof}
Let $[E]=k_1e$ and $[E]^\vee=k_2e^\vee$. Then \[ -(e,\bullet)= \frac{-1}{k_1}([E],\bullet)=\frac{([E],[E])}{2k_1}[E]^\vee= \frac{k_1(e,e)}{2}[E]^\vee=\frac{k_1k_2(e,e)}{2}e^\vee.
\] \end{proof}
\begin{rem} \label{rem-an-exceptional-linear-system-is-a-singleton} Let $L$ be the line bundle with $c_1(L)=e$. Then $\dim H^0(X,L^n)$ is either $0$ or $1$, for all $n\in{\mathbb Z}$, by \cite{boucksom}, Proposition 3.13. Hence, there exists at most one non-zero integer $n$, such that the linear system $\linsys{L^n}$ contains a prime divisor. In particular, for a given pair $(X,e)$, at most one of the equalities $([E],[E]^\vee)=(e,2e^\vee)$ or $([E],[E]^\vee)=(2e,e^\vee)$ can hold, for some prime divisor $E$ with $[E]\in{\rm span}_{\mathbb Z}\{e\}$. The same holds for an exceptional divisor, where the coefficients $n_i$ in Definition \ref{def-rational-exceptional} are integral and with $\gcd\{n_i \ : \ 1\leq i \leq k\}=1$. \end{rem}
\section{Holomorphic symplectic manifolds of $K3^{[n]}$-type} \label{sec-degrees-2-and-2-2n} We prove Proposition \ref{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon} in this section. This completes the proof of Theorem \ref{thm-2}. The lattice $H^2(X,{\mathbb Z})$ has signature $(3,20)$ \cite{beauville}. A $3$-dimensional subspace of $H^2(X,{\mathbb R})$ is said to be {\em positive-definite}, if the Beauville-Bogomolov pairing restricts to it as a positive definite pairing. The unit $2$-sphere, in any positive-definite $3$-dimensional subspace, is a deformation retract of the positive cone ${\mathcal C}_+\subset H^2(X,{\mathbb R})$, given by $ {\mathcal C}_+ := \{\lambda\in H^2(X,{\mathbb R}) \ : \ (\lambda,\lambda)>0\}. $ Hence, $H^2({\mathcal C}_+,{\mathbb Z})$ is isomorphic to ${\mathbb Z}$ and is a natural representation of the isometry group $OH^2(X,{\mathbb R})$. We denote by $O_+H^2(X,{\mathbb Z})$ the index two subgroup of $OH^2(X,{\mathbb Z})$, which acts trivially on $H^2({\mathcal C}_+,{\mathbb Z})$.
Let $X$ be of $K3^{[n]}$-type, $n\geq 2$. Embed the lattice $H^2(X,{\mathbb Z})$ in its dual lattice $H^2(X,{\mathbb Z})^*$, via the Beauville-Bogomolov form.
\begin{thm} \label{thm-monodromy-constraints} (\cite{markman-constraints}, Theorem 1.2 and Lemma 4.2). $Mon^2(X)$ is equal to the subgroup of $O_+H^2(X,{\mathbb Z})$, which acts via multiplication by $1$ or $-1$ on the quotient group $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$. \end{thm}
The quotient $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$ is a cyclic group of order $2n-2$. Indeed, we may deform $X$ to the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of a $K3$ surface $S$. $H^2(S^{[n]},{\mathbb Z})$ is Hodge-isometric to the orthogonal direct sum \begin{equation} \label{eq-orthogonal-direct-sum} H^2(S,{\mathbb Z})\oplus {\mathbb Z} \delta, \end{equation} where $\delta$ is half the class of the big diagonal, and $(\delta,\delta)=2-2n$ \cite{beauville}. Let $\pi:S^{[n]}\rightarrow S^{(n)}$ be the Hilbert-Chow morphism onto the symmetric product of $S$. The isometric embedding $H^2(S,{\mathbb Z})\hookrightarrow H^2(S^{[n]},{\mathbb Z})$ is given by the composition $ H^2(S,{\mathbb Z})\cong H^2(S^{(n)},{\mathbb Z})\LongRightArrowOf{\pi^*} H^2(S^{[n]},{\mathbb Z}). $
{\bf Proof\footnote{I thank V. Gritsenko for reference \cite[Corollary 3.4]{GHS-K3}, which drastically shortens the original proof.} of Proposition \ref{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon}:} The lattice $H^2(X,{\mathbb Z})$ is isometric to the orthogonal direct sum (\ref{eq-orthogonal-direct-sum}).
Let $e$ be a class in $H^2(X,{\mathbb Z})$ of negative Beauville-Bogomolov degree, and let $R_e(x):=x-2\frac{(x,e)}{(e,e)}e$ be the reflection by $e$. Then $R_e$ is an integral isometry of $H^2(X,{\mathbb Z})$, which acts by $1$ or $-1$ on the quotient $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$, if and only if $e$ has one of the two properties in the statement of Proposition \ref{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon}, by \cite[Corollary 3.4]{GHS-K3}. The proposition now follows from Theorem \ref{thm-monodromy-constraints}.
\hide{ \underline{Step 1:} Let $R(x):=x-2\frac{(x,e)}{(e,e)}e$ be the reflection by $e$. Assume that $R$ belongs to $Mon^2(X)$.
We may deform the pair $(X,e)$, loosing the Hodge type of $e$, so that $X$ is the Hilbert scheme $S^{[n]}$ of length $n$ subschemes of a $K3$ surface $S$. $H^2(S^{[n]},{\mathbb Z})$ is Hodge-isometric to the orthogonal direct sum \begin{equation} \label{eq-orthogonal-direct-sum} H^2(S,{\mathbb Z})\oplus {\mathbb Z} \delta, \end{equation} where $\delta$ is half the class of the big diagonal, and $(\delta,\delta)=2-2n$ \cite{beauville}. Let $\pi:S^{[n]}\rightarrow S^{(n)}$ be the Hilbert-Chow morphism onto the symmetric product of $S$. The isometric embedding $H^2(S,{\mathbb Z})\hookrightarrow H^2(S^{[n]},{\mathbb Z})$ is given by the composition $ H^2(S,{\mathbb Z})\cong H^2(S^{(n)},{\mathbb Z})\LongRightArrowOf{\pi^*} H^2(S^{[n]},{\mathbb Z}). $
Write \[ e \ \ = \ \ xa+y\delta, \] where the primitive class $a$ belongs to $H^2(S,{\mathbb Z})$ and $x$, $y$ are integers satisfying: \begin{equation} \label{*} \gcd(x,y) \ \ \ = \ \ \ 1. \end{equation}
The condition that the reflection $R$ is integral implies that \begin{equation} \label{**} \frac{2x}{(e,e)} \ \ \ \mbox{is an integer.} \end{equation} (Apply $R$ to a class $b$ in $H^2(S,{\mathbb Z})$ with $(a,b)=1$). It also implies that
\begin{equation} \label{***} \frac{4y(n-1)}{(e,e)} \ \ \ \mbox{is an integer.} \end{equation} (Apply $R$ to $\delta$).
The condition that $R$ is a monodromy operator implies that $R$ maps to either $1$ or $-1$ in the automorphism group of the quotient group $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$, by Theorem \ref{thm-monodromy-constraints}. We state in (\ref{****}) below a more explicit formulation of this condition. Write $R(\delta)=x'a'+y'\delta$, with $x'$, $y'$ integers, and the class $a'$ is in the $K3$ lattice. We get \[ y' \ \ \ := \ \ \ 1-\left[\frac{4y^2(1-n)}{(e,e)}\right]. \] The above condition on $R$ is then equivalent to the following statement. \begin{equation} \label{****} y' \ \mbox{is congruent to} \ 1 \ \mbox{or} \ -1 \ \mbox{modulo} \ (2n-2). \end{equation}
We need to prove that $(e,e)=-2$, or that $(e,e)=2-2n$ and $n-1$ divides ${\rm div}(e,\bullet)$.
This follows from equations (\ref{*}), (\ref{**}), (\ref{***}), and (\ref{****}) via the following elementary argument. Note first that (\ref{*}), (\ref{**}), and (\ref{***}) imply that $(e,e)$ divides $\gcd(2x,4n-4)$. On the other hand, $\gcd(2x,2n-2)$ divides $(e,e)=x^2(a,a)+(2-2n)y^2$. Write $n-1=t2^k$, with $t$ an odd integer. We get that \begin{eqnarray*} (e,e) &=& -\gcd(2x,2n-2) \ \mbox{or} \\ (e,e)&=& -2\gcd(2x,2n-2) \ \mbox{and} \ 2^{(k+1)} \ \mbox{divides} \ x. \end{eqnarray*} The rest follows from (\ref{****}) as follows.
If $y'$ is congruent to $1$ modulo $(2n-2)$, then $\frac{4y^2(n-1)}{(e,e)}$ is congruent to $0$ modulo $(2n-2)$. Hence $(e,e)$ divides $2y^2$. We conclude that $(e,e) \ \mbox{divides} \ \gcd(2x,2y^2)$. Thus $(e,e)=-2.$
Assume next that $y'$ is congruent to $-1$ modulo $(2n-2)$. Then $(2n-2)$ divides $1+y'=\frac{2x^2(a,a)}{(e,e)}$. So $(2n-2)$ divides $x^2(a,a)$. But $(e,e)=x^2(a,a)+y^2(2-2n)$. So \begin{equation} \label{6*} (2n-2) \ \mbox{divides} \ (e,e). \end{equation} If $(e,e)= -\gcd(2x,2n-2)$, then $(e,e)=2-2n$.
It remains to exclude the case where $y'$ is congruent to $-1$ modulo $(2n-2)$, $(e,e)= -2\gcd(2x,2n-2)$ and $2^{(k+1)}$ divides $x$. Now $(2n-2)$ divides $(e,e)$, by (\ref{6*}), and $(e,e)$ divides $2x$, by (\ref{**}). Thus $t$ divides $x$. So $\gcd(2x,2n-2)=t2^{(k+1)}=2n-2$. So $(e,e)=-2\gcd(2x,2n-2)=-(4n-4)$. We get the equality $\frac{4y^2(n-1)}{(e,e)}= -y^2.$ Condition (\ref{****}) implies that $\frac{4y^2(n-1)}{(e,e)}$ is congruent to $2$ modulo $(2n-2)$. But then $2$ divides $\gcd(x,y)$ contradicting (\ref{*}).
\underline{Step 2:} Assume next that $(e,e)=-2$, or that $(e,e)=2-2n$ and $n-1$ divides ${\rm div}(e,\bullet)$. If $(e,e)=-2$, then $R$ belongs to $Mon^2(X)$ by \cite{markman-monodromy-I}, Theorem 1.6. If $(e,e)=2-2n$, $n>2$, then $R$ belongs to $O_+H^2(X,{\mathbb Z})$, by the assumption that $n-1$ divides ${\rm div}(e,\bullet)$.
We may assume that $X=S^{[n]}$, for a $K3$ surface $S$. Then $H^2(X,{\mathbb Z})$ is the orthogonal direct sum $H^2(S,{\mathbb Z})\oplus{\mathbb Z} \delta$, with $(\delta,\delta)=2-2n$, as in (\ref{eq-orthogonal-direct-sum}). Write $e=x+t\delta$, $x\in H^2(S,{\mathbb Z})$ and $t\in{\mathbb Z}$. Then $x=(n-1)\xi$, for some class $\xi\in H^2(X,{\mathbb Z})$, by the assumption that $n-1$ divides ${\rm div}(e,\bullet)$. Set $w:=\frac{\delta}{2n-2}$. Then $w+H^2(X,{\mathbb Z})$ generates the residual group $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$, and the latter is isomorphic to ${\mathbb Z}/(2n-2){\mathbb Z}$. The equality $(e,e)=2-2n$ yields \begin{equation} \label{eq-t-square-1-equal-n-1-xi-xi-over-2} (t^2-1)=\frac{(n-1)(\xi,\xi)}{2}. \end{equation}
We claim that $R$ acts on $H^2(X,{\mathbb Z})^*/H^2(X,{\mathbb Z})$ via multiplication by $-1$. Once the claim is proven, then $R$ belongs to $Mon^2(X)$, by Theorem \ref{thm-monodromy-constraints}.
Compute \[ R(w)=w-\frac{2(w,e)}{(e,e)}e= w-\frac{t}{n-1}[(n-1)\xi+t\delta]\equiv (1-2t^2)w, \] where the last equivalence is modulo $H^2(X,{\mathbb Z})$. Equation (\ref{eq-t-square-1-equal-n-1-xi-xi-over-2}) yields $2t^2-2\equiv 0$ modulo $2n-2$. Hence, $1-2t^2\equiv -1$ modulo $2n-2$, and the claim is proven.
}
$\Box$
\begin{rem} \label{rem-Euler-characteristic} Let $X$ be of $K3^{[n]}$-type, $L$ a line bundle on $X$, and set $\alpha:=c_1(L)$. Then the sheaf-cohomology Euler characteristic of $L$ is given by the binomial coefficient $ \chi(L) = \Choose{[(\alpha,\alpha)/2]+n+1}{n}, $ by \cite{huybrechts-norway}, section 3.4, Example 7. We get the following equalities.\\ $ \chi(L) = \left\{ \begin{array}{ccl} 1 & \mbox{if} & (\alpha,\alpha)=-2 \\ 0 & \mbox{if} & (\alpha,\alpha)=2-2n \ \mbox{and} \ n\geq 3, \end{array}\right. $ \\ $ \begin{array}{ccccccc} \chi(L^2) & = & 0 & \mbox{if} & (\alpha,\alpha)=-2 & \mbox{and} & n\geq 3, \\ \chi(L^2) & < & 0 & \mbox{if} & (\alpha,\alpha)=2-2n & \mbox{and} & n\geq 2. \end{array} $
\end{rem}
\section{Deformation equivalence} \label{sec-deformation-equivalence} This section is influenced by an early draft of a paper of Brendan Hassett and Yuri Tschinkel, which was graciously communicated to the author \cite{hassett-tschinkel-monodromy}. \WithTorelli{The results rely heavily on Verbitsky's Torelli Theorem (Theorem \ref{thm-torelli}).}
\subsection{The prime-exceptional property of pairs $(X,L)$ is open} Let $X$ be a projective irreducible holomorphic symplectic manifold and $E$ a prime exceptional divisor. Set $c:=[E]\in H^2(X,{\mathbb Z})$. Given a point $t\in Def(X)$, let $c_t\in H^2(X_t,{\mathbb Z})$ be the class associated to $c$ via the parallel transport isomorphism\footnote{The local system $R^2\psi_*{\mathbb Z}$ over $Def(X)$ is trivial, since we may choose $Def(X)$ to be simply connected.} $H^2(X,{\mathbb Z})\rightarrow H^2(X_t,{\mathbb Z})$. Denote by $R_c$ both the reflection of $H^2(X,{\mathbb Z})$ with respect to $c$, as well as the involution of $Def(X)$. Let $D_c\subset Def(X)$ be the fixed locus of $R_c$. $D_c$ is a smooth divisor in $Def(X)$, which is characterized also as the subset \begin{equation} \label{eq-wall} D_c \ \ := \ \ \{t\in Def(X) \ : \ c_t \ \mbox{is of Hodge type} \ (1,1)\}. \end{equation}
\begin{lem} \label{lem-effectivity-along-a-wall} There exists an open subset $D_c^0\subset D_c$, containing $0$, such that for every $t\in D_c^0$ the class $c_t$ is Poincare dual to a prime exceptional divisor $E_t$. \end{lem}
\begin{proof} Let $X'$, $E'$, and $Y$ be as in Proposition \ref{prop-druel}. Denote the image of $E'\rightarrow Y$ by $B$. The generic fiber of $E'\rightarrow B$ is either a smooth rational curve $C$, whose normal bundle satisfies \[ N_{C/X'} \ \ \cong \ \ \omega_C\oplus \left(\oplus_{i=1}^{2n-2}\StructureSheaf{C}\right), \] or the union of two such curves meeting non-tangentially at one point, by Proposition \ref{prop-dissident-locus}. Druel shows that the exceptional locus of the birational map $X \rightarrow X'$ does not dominate $B$ (see the proof of \cite{druel}, Theorem 1.3). We conclude that a Zariski dense open subset of the original divisor $E$ in $X$ is covered by such rational curves. The proposition now follows from results of Ziv Ran about the deformations of such pairs $(X,C)$ (\cite{ziv-ran}, Theorem 1, with further comments in \cite{kawamata}). Our argument is inspired by \cite{hassett-tschinkel-conj}, Theorems 4.1 and 4.3. Note first that the class of the curve $C$ remains of type $(n-1,n-1)$ over $D_c$, by part \ref{item-integrality} of Corollary \ref{cor-1}. Let $\psi:{\mathcal X}\rightarrow D_c$ be the semi-universal family, ${\mathcal H}\rightarrow D_c$ the irreducible component of the relative Douady space containing the point $t_0$ representing the pair $(X,C)$, and ${\mathcal C}\subset {\mathcal H}\times_{D_c}{\mathcal X}$ the universal subscheme. We get the diagram \[ \begin{array}{ccccc} & & {\mathcal C} & \LongRightArrowOf{f} & {\mathcal X} \\ & & \alpha \ \downarrow \ \hspace{1ex} & & \hspace{1ex} \ \downarrow \ \psi \\ t_0 & \in & {\mathcal H} & \LongRightArrowOf{\beta} & D_c. \end{array} \] Let $\pi:H^1(C,N_{C/X})\rightarrow H^{2n}(\Omega_X^{2n-2})$ be the semi-regularity map. Then $\pi$ is an isomorphism of these one-dimensional vector spaces (Observation (a) before Corollary 4 in \cite{ziv-ran}). Theorem 1 of \cite{ziv-ran} implies that the morphism $\beta$ is surjective, of relative dimension $2n-2$, and it is smooth at the point $t_0$. It follows that ${\mathcal C}$ has relative dimension $2n-1$ over $D_c$, and ${\mathcal C}$ is smooth along the rational curve $C$ over $t_0$. Furthermore, the fiber $(\beta\circ \alpha)^{-1}(0)$ contains a unique irreducible component $\widetilde{E}$, which dominates $E$, and $f:\widetilde{E}\rightarrow E$ has degree $1$, by part \ref{item-integrality} of Corollary \nolinebreak\ref{cor-1}.
We claim that the differential $df:T{\mathcal C}\rightarrow f^*T{\mathcal X}$ is injective along $C$. $T{\mathcal C}$ comes with a natural filtration $T_{\alpha}\subset T_{\beta\circ\alpha}\subset T{\mathcal C}$. $(f^*T{\mathcal X}\restricted{)}{C}$ comes with the filtration $TC\subset (TX\restricted{)}{C}\subset (T{\mathcal X}\restricted{)}{C}.$ The homomorphism $df$ is compatible with the filtarations and induces the identity on the first and third graded summands $TC$ and $T_0(D_c)$. It suffices to prove injectivity on the middle graded summand. The above condition on $N_{C/X}$ implies, furthermore, that the evaluation homomorphism $H^0(N_{C/X})\otimes \StructureSheaf{C}\rightarrow N_{C/X}$ is injective. It follows that the differential $df$ is injective along $C$. Consequently, $f({\mathcal C})$ determines a divisor ${\mathcal E}$ in ${\mathcal X}$, possibly after eliminating embedded components of $f({\mathcal C})$, which are disjoint from the curve $C$. Furthermore, ${\mathcal E}$ intersects the fiber $X$ of $\psi$ along a divisor $E_0$ containing $E$ and $E_0$ is reduced along $E$.
It remains to prove that $E_0$ is irreducible. Now the fiber $X_t$ has a cyclic Picard group, for a generic $t \in D_c$. Hence, the generic fiber $E_t$ of ${\mathcal E}$ is of class $kc_t$, for some positive integer $k$. Thus $E_0$ is of class $kc$. But the linear system $\linsys{kE}$ consists of a single divisor $kE$, by \cite{boucksom}, Proposition 3.13. We get that $k=1$, since $E_0$ is reduced along $E$. \end{proof}
Let $\pi:{\mathcal X}\rightarrow T$ be a smooth family of irreducible holomorphic symplectic manifolds over a connected complex manifold $T$. Assume that there exists a section $e$ of $R^2_{\pi_*}{\mathbb Z}$, everywhere of Hodge type $(1,1)$. Given a point $t\in T$, denote by $L_t$ the line bundle on the fiber $X_t$ with class $e_t$.
\begin{prop} \label{prop-generic-prime-exceptional} Assume given a point $0\in T$, such that the fiber $X_0$ is projective and the linear system $\linsys{L_0^k}$, of the $k$-th tensor power, consists of a prime exceptional divisor $E_0\subset X_0$, for some positive integer $k$. Then $k=1$ or $2$. Let $Z\subset T$ be the subset of points $t\in T$, such that $h^0(X_t,L_t^k)>1$, or there exists a non-prime divisor, which is a member of the linear system $\linsys{L_t^k}$. Then $Z$ is a proper and closed analytic subset of $T$. Furthermore, there exists an irreducible divisor ${\mathcal E}$ in ${\mathcal X}\setminus \pi^{-1}(Z)$, such that ${\mathcal E}$ intersects the fiber $\pi^{-1}(t)$ along a prime exceptional divisor $E_t$ of class $ke_t$, for every $t$ in the complement $T\setminus Z$.
\end{prop}
\begin{proof} The integer $k$ is $1$ or $2$ by Corollary \ref{cor-1}. The dimension $h^0(X_t,L_t^k)$ is an upper-semi-continuous function on $T$, and so the locus where it is positive is a closed analytic subset of $T$. On the other hand, $h^0(X_t,L_t^k)$ is positive over an open subset, by Lemma \ref{lem-effectivity-along-a-wall}. Hence, it is positive everywhere and $L_t^k$ is effective for all $t$.
Let $Z_1\subset T$ be the closed analytic subset, where $h^0(X_t,L_t^k)>1$. We know that $h(X_0,L_0^k)=1$, by \cite{boucksom}, Proposition 3.13. Hence, we may assume, possibly after replacing $T$ by $T\setminus Z_1$, that $h^0(X_t,L_t^k)=1$, for all $t\in T$.
We prove next that the section $e$ lifts to a line bundle ${\mathcal L}\cong \StructureSheaf{{\mathcal X}}({\mathcal E})$, for a divisor ${\mathcal E}\subset{\mathcal X}$, which does not contain any fiber of $\pi$. The following is part of the edge exact sequence of the spectral sequence of the composite functor $\Gamma\circ\pi_*$ of push-forward and taking global sections. \[ H^1(T,\StructureSheaf{T}^*)\rightarrow H^1({\mathcal X},\StructureSheaf{{\mathcal X}}^*) \rightarrow H^0(T,R^1_{\pi_*}\StructureSheaf{{\mathcal X}}^*) \rightarrow H^2(T,\StructureSheaf{T}^*). \] Let $V$ be an open subset of $T$ satisfying $H^i(V,\StructureSheaf{V}^*)=0$, for $i=1,2$. Then the restriction of $e$ to $V$ lifts to a line bundle ${\mathcal L}_V$ over $\pi^{-1}(V)$. Now $\pi_*{\mathcal L}_V$ is a line bundle over $V$, which must be trivial, by the vanishing of $H^1(V,\StructureSheaf{V}^*)$. Hence, $H^0(\pi^{-1}(V),{\mathcal L}_V)\cong H^0(V,\StructureSheaf{V})$, and there exists a unique divisor ${\mathcal E}_V\subset \pi^{-1}(V)$, in the linear system $\linsys{{\mathcal L}_V}$, which does not contain any fiber of $\pi$. If $V_1$ and $V_2$ are two such open subsets of $T$, then the divisors ${\mathcal E}_{V_i}$ constructed above agree along $\pi^{-1}(V_1\cap V_2)$, since they are canonical over any subset $V$ of $V_1\cap V_2$, over which $H^i(V,\StructureSheaf{V}^*)=0$, for $i=1,2$. Hence, we get a global divisor ${\mathcal E}\subset {\mathcal X}$. Set ${\mathcal L}:=\StructureSheaf{{\mathcal X}}({\mathcal E})$.
We prove next that ${\mathcal E}$ is irreducible. Let $p:{\mathcal E}\rightarrow T$ be the restriction of $\pi$. Then $p$ is a proper morphism, which is also flat by \cite{matsumura} application 2 page 150.
All fibers of $p$ are connected, since $T$ is smooth, and in particular normal, and the fiber over $0$ is connected. The morphism $p$ is smooth along the smooth locus of $E_0$, and ${\mathcal E}$ is a local complete intersection in the smooth complex manifold ${\mathcal X}$. Hence, there exists precisely one irreducible component of ${\mathcal E}$ which contains $E_0$. Assume that there exists another irreducible component ${\mathcal E}'$. Then ${\mathcal E}'$ maps to a proper closed subset of $T$, which does not contain $0$. But $T$ is irreducible, and ${\mathcal E}'$ intersects each fiber of $\pi$ along a subset, which is either empty or of codimension at least one. Hence, the codimension of ${\mathcal E}'$ in ${\mathcal X}$ is larger than one. This contradicts the fact that ${\mathcal E}$ is a divisor. We conclude that ${\mathcal E}$ is irreducible.
The subset $Z\subset T$, consisting of points $t\in T$, where $E_t$ is reducible or non-reduced, is a closed analytic subset of $T$, which does not contain $0$.
\hide{ The last statement above means that the morphism $p$ is {\em geometrically-irreducible} and {\em geometrically-reduced}. We show the former property, as the proofs are similar.
We need to show that ${\mathcal E}$ remains irreducible after any base change by a finite morphism $f:\widetilde{T}\rightarrow T$ from a reduced, irreducible, and normal analytic space $\widetilde{T}$. Set $\widetilde{{\mathcal E}}:={\mathcal E}\times_T\widetilde{T}$ and let $\tilde{p}:\widetilde{{\mathcal E}}\rightarrow \widetilde{T}$ be the natural morphism.
We may assume that $\widetilde{T}$ is non-singular at some point over $0\in T$. This is clear if $\dim(T)=1$. If $\dim(T)>1$, let $\beta:\hat{T}\rightarrow T$ be the blow-up of the point $0$ in $T$ and denote by $D\subset \hat{T}$ the exceptional divisor. The geometric irreducibility of $p$ is equivalent to that of the base change via $\beta$. Now the singular locus of any normal finite covering $f:\widetilde{T}\rightarrow \hat{T}$ has codimension $\geq 2$. Hence, $\widetilde{T}$ is smooth at some point of $D$.
Assume that $\widetilde{T}$ is non-singular at the point $\tilde{0}$ over $0\in T$. Let $\tilde{\pi}:\widetilde{{\mathcal X}}\rightarrow \widetilde{T}$ be the base change of $\pi$. Then $\widetilde{{\mathcal X}}$ is smooth along the fiber $X_{\tilde{0}}$ over $\tilde{0}$. The morphism $\tilde{p}:\widetilde{{\mathcal E}}\rightarrow\widetilde{T}$ is again flat and proper. The argument used above again yields that $\widetilde{{\mathcal E}}$ has a unique irreducible component containing the fiber $E_{\tilde{0}}$ over $\tilde{0}$. Again any other component maps to a closed analytic subset of $\widetilde{T}$.
The rest of the proof is by contradiction. Assume that there exists an irreducible component of $\widetilde{{\mathcal E}}$, which does not contain $E_{\tilde{0}}$. Then the morphism $\widetilde{{\mathcal E}}\rightarrow {\mathcal E}$ must map it to a subset of ${\mathcal E}$ of lower dimension, as the image does not dominate $\widetilde{T}$. But the morphism $\widetilde{{\mathcal E}}\rightarrow {\mathcal E}$ is finite. A contradiction.
}
\hide{ Let $\nu:\widetilde{{\mathcal E}}\rightarrow {\mathcal E}$ be the normalization and set $\tilde{p}:=p\circ\nu:\widetilde{{\mathcal E}}\rightarrow T$. The normalization $\nu$ is an isomorphism in a neighborhood of the smooth locus of $E_0$. Hence, the fiber of $\tilde{p}$ over $0\in T$ is irreducible. Now $\tilde{p}$ factors as the composition $g\circ f$, where $f:\widetilde{{\mathcal E}}\rightarrow\widetilde{T}$ is proper with connected fibers and $g:\widetilde{T}\rightarrow T$ is finite, by the Stein Factorization Theorem. The morphism $\tilde{p}$ is smooth in a neighborhood of the smooth locus of $E_0$. Hence, $g$ is smooth at the unique point over $0\in T$. $T$ is smooth, and so $g$ is an isomorphism. Thus $\tilde{p}$ has connected fibers.
} \end{proof}
Proposition \ref{prop-generic-prime-exceptional} shows that the property that $L$ is prime exceptional is {\em open} in any smooth and connected base $T$ of a deformation of a pair $(X,L)$, as long as it holds for at least one projective pair. One limiting case is that of a pair $(X,L)$, where $L$ is exceptional, in the sense of Definition \ref{def-rational-exceptional}, but no longer prime.
However, the exceptional property is unfortunately not closed, as the following example shows.
\begin{example} \label{ex-being-exceptional-is-not-a-closed-property} Let $Y$ be the intersection of a quadric and a cubic in ${\mathbb P}^{4}$, which are tangent at one point $y_0$, such that $Y$ has an ordinary double point at $y_0$. Let $H$ be the hyperplane line bundle on $Y$. $Y$ is a singular $K3$ surface of degree $6$. Let $\pi:X\rightarrow Y$ be the blow-up of $Y$ at $y_0$ and $E\subset X$ its exceptional divisor. $X$ is a smooth $K3$ surface. Set $L_0:=\pi^*H\otimes \StructureSheaf{X}(2E)$. Then $L_0$ has degree $-2$, but $L_0$ is not exceptional.
Set $c:=c_1(L_0)$, let $D_c\subset Def(X)$ be the divisor defined in equation (\ref{eq-wall}), and denote by $L_t$ the line bundle on $X_t$ with class $c_t$, $t\in D_c$. Then $L_t$ has degree $-2$, and thus precisely one of $L_t$ or $L_t^{-1}$ is effective (\cite{BHPV}, Ch. VIII Prop. 3.6). The semi-continuity theorem implies that $L_t$ is effective, since $L_0^{-1}$ isn't and $D_c$ is connected. For a generic $t\in D_c$, the pair $(X_t,L_t)$ consists of a K\"{a}hler $K3$ surface, whose Picard group is generated by $L_t$. Hence, the linear system $\linsys{L_t}$ consists of a single smooth rational curve $E_t$. The analogue of Lemma \ref{lem-effectivity-along-a-wall} is known for such a pair $(X_t,L_t)$, even if $X_t$ is not projective. Hence, Proposition \ref{prop-generic-prime-exceptional} applies as well. Let $D_c^0\subset D_c$ be the subset of pairs $(X_t,L_t)$, such that $L_t\cong\StructureSheaf{X_t}(E_t)$, for a smooth connected rational curve $E_t$. We get that $D_c^0$ is non-empty and the complement $Z:=D_c\setminus D_c^0$ is a closed analytic subset containing $0\in D_c$. Consequently, the property of $L_t$ being exceptional is not closed.
\end{example}
\subsection{Deformation equivalence and Torelli} \label{sec-deformation-equivalence-and-torelli} We introduce and relate three notions of deformation equivalence of pairs.
\begin{defi} \label{def-deformation-equivalent-pairs-with-effective-divisors} Let $(X_i,E_i)$, $i=1,2$, be two pairs of an irreducible holomorphic symplectic manifold $X_i$, and an effective divisor $E_i\in Div(X_i)$. The two pairs are said to be {\em deformation equivalent}, if there exists a smooth proper family $\pi:{\mathcal X}\rightarrow T$ of irreducible holomorphic symplectic manifolds, over a connected analytic space $T$ with finitely many irreducible components, a holomorphic line bundle ${\mathcal L}$ over ${\mathcal X}$, a nowhere-vanishing section $s$ of $\pi_*{\mathcal L}$, points $t_i\in T$, and isomorphisms $f_i:{\mathcal X}_{t_i}\rightarrow X_i$, such that $f_i((s_{t_i})_0)=E_i$, $i=1,2$. Above $(s_{t_i})_0$ denotes the zero divisor of $s_{t_i}$ in $X_{t_i}$. \end{defi}
The relation is clearly symmetric and reflexive. It is also transitive, since we allow $T$ to be reducible.
\begin{defi} \label{def-deformation-equivalent-pairs-with-line-bundle} Let $(X_i,L_i)$, $i=1,2$, be two pairs of an irreducible holomorphic symplectic manifold $X_i$, and a line bundle $L_i$. The two pairs are said to be {\em deformation equivalent}, if there exists a smooth proper family $\pi:{\mathcal X}\rightarrow T$ of irreducible holomorphic symplectic manifolds, over a connected analytic space $T$ with finitely many irreducible components, and a section $e$ of $R^2\pi_*{\mathbb Z}$, which is everywhere of Hodge-type $(1,1)$, points $t_i\in T$, and isomorphisms $f_i:{\mathcal X}_{t_i}\rightarrow X_i$, such that $(f_i)_*(e_{t_i})=c_1(L_i)$. \end{defi}
\begin{defi} \label{def-deformation-equivalent-pairs-with-cohomology-class} Let $(X_i,e_i)$, $i=1,2$, be two pairs of an irreducible holomorphic symplectic manifold $X_i$, and a class $e_i\in H^2(X_i,{\mathbb Z})$. The two pairs are said to be {\em deformation equivalent}, if there exists a smooth proper family $\pi:{\mathcal X}\rightarrow T$ of irreducible holomorphic symplectic manifolds, over a connected analytic space $T$ with finitely many irreducible components, a section $e$ of $R^2\pi_*{\mathbb Z}$, points $t_i\in T$, and isomorphisms $f_i:{\mathcal X}_{t_i}\rightarrow X_i$, such that $(f_i)_*(e_{t_i})=e_i$. \end{defi}
The three Definitions, \ref{def-deformation-equivalent-pairs-with-line-bundle}, \ref{def-deformation-equivalent-pairs-with-effective-divisors}, and \ref{def-deformation-equivalent-pairs-with-cohomology-class}, fit in a hierarchy. If $L_i=\StructureSheaf{X_i}(E_i)$, and $e_i:=c_1(L_i)$, then \begin{eqnarray} \nonumber
(X_1,E_1)\equiv (X_2,E_2) & \Rightarrow & (X_1,L_1)\equiv (X_2,L_2), \\ \label{eq-equivalence-of-line-bundles-implies-that-of-classes} (X_1,L_1)\equiv (X_2,L_2) & \Rightarrow & (X_1,e_1)\equiv (X_2,e_2). \end{eqnarray}
Assume that the divisors $E_i$, $i=1,2$, are prime exceptional, and $X_1$ is projective. Then both implications above are equivalences, \WithoutTorelli{assuming an affirmative answer to a version of the Torelli Question,} by Proposition \ref{equivalence-of-deformation-equivalences-relations}. For a qualified ``converse'' to the second implication (\ref{eq-equivalence-of-line-bundles-implies-that-of-classes}), without assuming that $L_1$ and $L_2$ are effective, see Lemma \ref{lem-monodromy-invariants-and-deformation-equivalence}.
\subsubsection{Period maps} A {\em marking}, for an irreducible holomorphic symplectic manifold $X$, is a choice of an isometry $\eta: H^2(X,{\mathbb Z})\rightarrow \Lambda$ with a fixed lattice $\Lambda$. The {\em period}, of the marked manifold $(X,\eta)$, is the line $\eta[H^{2,0}(X)]$ considered as a point in the projective space ${\mathbb P}[\Lambda\otimes{\mathbb C}]$. The period lies in the period domain \begin{equation} \label{eq-period-domain} \Omega \ := \ \{ \ell \ : \ (\ell,\ell)=0 \ \ \ \mbox{and} \ \ \ (\ell,\bar{\ell}) > 0 \}. \end{equation} $\Omega$ is an open subset, in the classical topology, of the quadric in ${\mathbb P}[\Lambda\otimes{\mathbb C}]$ of isotropic lines \cite{beauville}.
There is a (non-Hausdorff) moduli space ${\mathfrak M}_\Lambda$ of marked irreducible holomorphic symplectic manifolds, with a second integral cohomology lattice isometric to $\Lambda$ \cite{huybrects-basic-results}. The period map \begin{eqnarray} \label{eq-period-map} P \ : \ {\mathfrak M}_\Lambda & \longrightarrow & \Omega, \\ \nonumber (X,\eta) & \mapsto & \eta[H^{2,0}(X)] \end{eqnarray} is a local isomorphism, by the Local Torelli Theorem \cite{beauville}. The Surjectivity Theorem states that the restriction of the period map to every connected component of ${\mathfrak M}_\Lambda$ is surjective \cite{huybrects-basic-results}.
Two points $x$ and $y$ of a topological space are {\em inseparable}, if every pair of open subsets $U$, $V$, with $x\in U$ and $y\in V$, have a non-empty intersection.
\WithoutTorelli{
Assume given a bimeromorphic map $f:X_1\rightarrow X_2$ and a marking $\eta_1$ for $X_1$. Let $f^* : H^2(X_2,{\mathbb Z}) \rightarrow H^2(X_1,{\mathbb Z})$ be the isometry induced by the bimeromorphic map $f$ (see the proof of Corollary \ref{cor-1}). Set $\eta_2=\eta_1\circ f^*$. Then $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are inseparable points of ${\mathfrak M}_\Lambda$ (\cite{huybrechts-kahler-cone}, Theorem 2.5). Conversely, Verbitsky recently posted a proof of an affirmative answer to the following version of the Torelli Question (\cite{verbitsky}, Theorem 4.24).
\begin{question} \label{thm-torelli} Let ${\mathfrak M}^0_\Lambda$ be a connected component of ${\mathfrak M}_\Lambda$. Let $(X_1,\eta_1)$ and $(X_2,\eta_2)$ be two pairs in ${\mathfrak M}^0_\Lambda$ such that $P(X_1,\eta_1)=P(X_2,\eta_2)$. Are $(X_1,\eta_1)$ and $(X_2,\eta_2)$ inseparable points of ${\mathfrak M}^0_\Lambda$?
\end{question}
} \WithTorelli{ Assume given a bimeromorphic map $f:X_1\rightarrow X_2$ and a marking $\eta_1$ for $X_1$. Let $f^* : H^2(X_2,{\mathbb Z}) \rightarrow H^2(X_1,{\mathbb Z})$ be the isometry induced by the bimeromorphic map $f$ (see the proof of Corollary \ref{cor-1}). Set $\eta_2=\eta_1\circ f^*$. Then $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are inseparable points of ${\mathfrak M}_\Lambda$ (\cite{huybrechts-kahler-cone}, Theorem 2.5). Conversely, Verbitsky recently proved the following version of the Torelli Theorem.
\begin{thm} \label{thm-torelli} (\cite[Theorem 4.24]{verbitsky}, \cite{huybrechts-bourbaki}). Let ${\mathfrak M}^0_\Lambda$ be a connected component of ${\mathfrak M}_\Lambda$. If $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are two pairs in ${\mathfrak M}^0_\Lambda$ and $P(X_1,\eta_1)=P(X_2,\eta_2)$, then $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are inseparable points of ${\mathfrak M}^0_\Lambda$.
\end{thm}
} A homomorphism $h:H^*(X_1,{\mathbb Z})\rightarrow H^*(X_2,{\mathbb Z})$ is a {\em parallel transport operator} if there exists a smooth and proper family $f:{\mathcal X}\rightarrow B$, of irreducible holomorphic symplectic manifolds over an analytic base $B$, points $b_1$, $b_2$ in $B$, isomorphisms $X_i\cong {\mathcal X}_{b_i}$, and a continuous path $\gamma$ from $b_1$ to $b_2$, such that parallel transport in the local system $R^*f_*{\mathbb Z}$ along $\gamma$ induces $h$. The following is a fundamental result of Huybrechts. \begin{thm} \label{thm-inseparable-are-birational} (\cite{huybrects-basic-results}, Theorem 4.3). Let $(X_1,\eta_1)$ and $(X_2,\eta_2)$ be two inseparable points of ${\mathfrak M}_\Lambda$, with $\dim(X_i)=2n$. Then there exists an effective cycle $\Gamma:=Z+\sum Y_j$ in $X_1\times X_2$, of pure dimension $2n$, with the following properties. \begin{enumerate} \item $Z$ is the graph of a bimeromorphic map from $X_1$ to $X_2$. \item The correspondence $[\Gamma]_*:H^*(X_1,{\mathbb Z})\rightarrow H^*(X_2,{\mathbb Z})$ is a parallel transport operator. Furthermore, the composition \[ \eta_2^{-1}\circ\eta_1:H^2(X_1,{\mathbb Z})\rightarrow H^2(X_2,{\mathbb Z}) \] is equal to the restriction of $[\Gamma]_*$. \item The image of the projection of each $Y_j$ into each $X_i$, $i=1,2$, has positive codimension in $X_i$. \end{enumerate}
\end{thm}
Assume given two deformation equivalent pairs $(X_i,e_i)$, $i=1,2$, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}. Then there exist isometries $\eta_i:H^2(X_i,{\mathbb Z})\rightarrow \Lambda$, having the following two properties: \begin{enumerate} \item $\eta_1(e_1)=\eta_2(e_2)$. \item The marked pairs $(X_i,\eta_i)$ belong to the same connected component ${\mathfrak M}^0_\Lambda$. \end{enumerate} Let $\lambda$ be the common value $\eta_i(e_i)$, $i=1,2$. If both classes $e_i$ belong to $H^{1,1}(X_i,{\mathbb Z})$, then the periods $P(X_i,\eta_i)$ belong to the hyperplane $\lambda^\perp\subset {\mathbb P}[\Lambda\otimes{\mathbb C}]$ orthogonal to $\lambda$. The intersection $\lambda^\perp\cap\Omega$ is connected.
Fix a primitive non-zero class $\lambda\in \Lambda$ with $(\lambda,\lambda)<0$. Let \[ {\mathfrak M}^0_{\Lambda,\lambda} \ \ \ \subset \ \ \ {\mathfrak M}^0_\Lambda \] be the subset parametrizing marked pairs $(X,\eta)$, such that $\eta^{-1}(\lambda)$ is of Hodge type $(1,1)$, and $(\kappa,\eta^{-1}(\lambda))>0$, for some K\"{a}hler class $\kappa$ on $X$.
\begin{claim} ${\mathfrak M}^0_{\Lambda,\lambda}$ is an open subset of $P^{-1}(\lambda^\perp\cap\Omega)$.
\end{claim}
\begin{proof} Let ${\mathfrak M}^0_+$ be the subset of ${\mathfrak M}^0_\Lambda$, consisting of marked pairs $(X,\eta)$, such that $(\kappa,\eta^{-1}(\lambda))>0$, for some K\"{a}hler class $\kappa$ on $X$. It suffices to prove that ${\mathfrak M}^0_+$ is an open subset of ${\mathfrak M}^0_\Lambda$. Let $(X_0,\eta_0)$ be a point of ${\mathfrak M}^0_+$ and $\kappa_0$ a K\"{a}hler class on $X_0$ satisfying $(\kappa_0,\eta_0^{-1}(\lambda))>0$. Let $Def(X_0)$ be the Kuranishi deformation space and $\psi:{\mathcal X}\rightarrow Def(X_0)$ the semi-universal family with fiber $X_0$ over $0\in Def(X_0)$. There exists an open subset $U$ of $Def(X_0)$, and a differentiable section $\kappa$ of the real vector bundle $(R^2_{\psi_*}{\mathbb R}\restricted{)}{U}$, over $U$, such that $\kappa_t$ is a K\"{a}hler class of $X_t$, for all $t\in U$, by the proof of the Stability of K\"{a}hler manifolds (\cite{voisin-book-vol1}, Theorem 9.3.3). We may identify $U$ with an open subset of ${\mathfrak M}^0_\Lambda$ containing $(X_0,\eta_0)$, by the Local Torelli Theorem. We get the continuous function $(\kappa_t,\eta_t^{-1}(\lambda))$ over $U$, which is positive at $(X_0,\eta_0)$. Hence, there is an open subset $W\subset U$, containing $(X_0,\eta_0)$, such that $(\kappa_t,\eta_t^{-1}(\lambda))>0$, for all $t\in W$. \end{proof}
The Local Torelli Theorem implies that the period map restricts to a local isomorphism \[ P_\lambda \ : \ {\mathfrak M}^0_{\Lambda,\lambda} \ \ \ \longrightarrow \ \ \ \lambda^\perp\cap \Omega. \] ${\mathfrak M}^0_{\Lambda,\lambda}$ is thus a non-Hausdorff smooth complex manifold of dimension $b_2(X)-3$.
\subsubsection{${\mathfrak M}^0_{\Lambda,\lambda}$ is path-connected \WithoutTorelli{if Torelli holds}} \hspace{1ex}\\ Given a point $t\in \Omega$, set $ \Lambda_t^{1,1} := \{x\in \Lambda \ : \ (x,t)=0\}. $ \WithTorelli{ The following statement is a Corollary of Theorem \ref{thm-torelli}. } \WithTorelli{\begin{cor}} \WithoutTorelli{\begin{lem}} \label{cor-torelli-for-M-Lambda-lambda} Let $t\in \lambda^\perp\cap\Omega$ be a point, such that $\Lambda_t^{1,1}={\rm span}_{\mathbb Z}\{\lambda\}$. Then the fiber $P_\lambda^{-1}(t)$ consists of the isomorphism class of a single marked \WithTorelli{pair.} \WithoutTorelli{pair, assuming an affirmative answer to Question \ref{thm-torelli}.} \WithTorelli{\end{cor}} \WithoutTorelli{\end{lem}}
\begin{proof} Let $(X,\eta)$ be a marked pair in $P_\lambda^{-1}(t)$. Set $\tilde{\lambda}:=\eta^{-1}(\lambda)$. Then $H^{1,1}(X,{\mathbb Z})$ is spanned by $\tilde{\lambda}$, and there exists a K\"{a}hler class $\kappa_0$, such that $(\kappa_0,\tilde{\lambda})>0$. Let us first describe the three possibilities for the K\"{a}hler cone ${\mathcal K}_X$ and the {\em birational K\"{a}hler cone} $\BK_X$ of $X$. Recall that $\BK_X$ is the union of the subsets $f^*({\mathcal K}_Y)\subset H^{1,1}(X,{\mathbb R})$, as $f$ varies over all bimeromorphic maps $f:X\rightarrow Y$ from X to another irreducible holomorphic symplectic manifold $Y$. Denote by $\tilde{\lambda}^\vee$ the primitive class in $H^2(X,{\mathbb Z})^*$, which is a positive multiple of $(\tilde{\lambda},\bullet)$. Let ${\mathcal C}_X$ be the connected component of the cone $\{\kappa\in H^{1,1}(X,{\mathbb R}) \ : \ (\kappa,\kappa)>0\}$, which contains the K\"{a}hler cone.
\underline{Case 1:} If $d\lambda^\vee$ is not represented by a rational curve, for any positive integer $d$, then $ \BK_X={\mathcal K}_X={\mathcal C}_X, $ by \cite{huybrechts-kahler-cone}, Corollary 3.3.
\underline{Case 2:} Assume that $\tilde{\lambda}$ is ${\mathbb Q}$-effective. Then $d\tilde{\lambda}$ is represented by a prime exceptional divisor $E\subset X$, for some positive integer $d$, which is uniruled, by \cite{boucksom}, Proposition 4.7. Then \[ \BK_X={\mathcal K}_X=\{\kappa\in {\mathcal C}_X \ : \ (\kappa,\tilde{\lambda})>0\}, \] by \cite{boucksom}, Theorem 4.3.
\underline{Case 3:} Assume that $d\tilde{\lambda}$ is not effective, for any positive integer $d$, but $d\tilde{\lambda}^\vee$ is represented by a rational curve, for some positive integer $d$. Then \begin{eqnarray*} {\mathcal K}_X & = & \{\kappa\in {\mathcal C}_X \ : \ (\kappa,\tilde{\lambda})>0\}, \\ \BK_X & = & {\mathcal K}_X\cup {\mathcal K}'_X, \ \ \mbox{where} \\ {\mathcal K}'_X & = & \{\kappa\in {\mathcal C}_X \ : \ (\kappa,\tilde{\lambda})<0\}, \end{eqnarray*} by \cite{boucksom}, Theorem 4.3.
Let $(X_1,\eta_1)$ and $(X_2,\eta_2)$ be two marked pairs in $P_\lambda^{-1}(t)$. Then $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are inseparable points, by \WithTorelli{Theorem} \WithoutTorelli{the assumed affirmative answer to Question} \ref{thm-torelli}. Hence, there exists a cycle $\Gamma:=Z+\sum Y_j$ in $X_1\times X_2$, satisfying the properties listed in Theorem \ref{thm-inseparable-are-birational}. Denote by $g:X_1\rightarrow X_2$ the bimeromorphic map with graph $Z$, and let $f:H^2(X_1,{\mathbb Z})\rightarrow H^2(X_2,{\mathbb Z})$ be the parallel-transport operator $[\Gamma]_*$, so that $f=\eta_2^{-1}\circ\eta_1$, by Theorem \ref{thm-inseparable-are-birational}. Set $\tilde{\lambda}_i:=\eta_i^{-1}(\lambda)$. Let $\kappa_i$ be a K\"{a}hler class on $X_i$, such that $(\tilde{\lambda}_i,\kappa_i)>0$.
In cases 1 and 3, $X_i$ does not contain any effective divisor, $i=1,2$. In particular, the image of each $Y_j$ has codimension $\geq 2$ in each $X_i$, and $f=g_*$. We have $ (\tilde{\lambda}_1,g^*(\kappa_2))=(\eta_1^{-1}(\lambda),g^*(\kappa_2))= (g_*\eta_1^{-1}(\lambda),\kappa_2)=(\eta_2^{-1}(\lambda),\kappa_2)= (\tilde{\lambda}_2,\kappa_2), $ since $g^*$ is an isometry. We conclude the inequality \begin{equation} \label{eq-lambda-tilde-1-paired-with-pull-back-of-kappa-2-is-positive} (\tilde{\lambda}_1,g^*(\kappa_2)) \ \ > \ \ 0. \end{equation}
If $g^*(\kappa_2)$ is not a K\"{a}hler class, then the birational K\"{a}hler cone $\BK_{X_1}$ consists of at least two connected components. Thus we must be in case 3. So $\kappa_1\in {\mathcal K}_{X_1}$ and $g^*(\kappa_2)$ belongs to ${\mathcal K}'_{X_1}$. Hence, $(g^*(\kappa_2),\tilde{\lambda}_1)<0$, by the characterization of ${\mathcal K}'_{X_1}$. This contradicts the inequality (\ref{eq-lambda-tilde-1-paired-with-pull-back-of-kappa-2-is-positive}).
We conclude that $g^*(\kappa_2)$ is a K\"{a}hler class. Thus $g$ is an isomorphism, by \cite{huybrechts-kahler-cone}, Proposition 2.1, and $(X_1,\eta_1)$ and $(X_2,\eta_2)$ are isomorphic marked pairs.
It remains to treat case 2. In that case $\BK_{X_i}={\mathcal K}_{X_i}$, and so $g$ is an isomorphism. Hence, $g_*(\tilde{\lambda}_1)=\tilde{\lambda}_2$, since the classes $\tilde{\lambda}_i$ are effective. On the other hand, $f(\tilde{\lambda}_1)=\eta_2^{-1}\eta_1(\eta_1^{-1}(\lambda))=\eta_2^{-1}(\lambda)= \tilde{\lambda}_2$. Hence, $f(\tilde{\lambda}_1)=g_*(\tilde{\lambda}_1)$. The subspace $\tilde{\lambda}_1^\perp$, orthogonal to $\tilde{\lambda}_1$, is necessarily in the kernel of the correspondence $[\sum Y_j]_*:H^2(X_1,{\mathbb Z})\rightarrow H^2(X_2,{\mathbb Z})$. Hence, $f(\alpha)=g_*(\alpha)$, for all $\alpha\in \tilde{\lambda}_1^\perp$. We conclude that $f=g_*$, and the two pairs $(X_1,\eta_1)$ and $(X_2,\eta_2)$
are isomorphic.
\end{proof}
\WithoutTorelli{ \begin{question} \label{question-connectedness} Let $(X_0,\eta_0)\in {\mathfrak M}^0_\Lambda$ be a marked pair, $L$ a monodromy reflective line bundle on $X$,
and set $\lambda:=\eta_0(c_1(L))$. Is ${\mathfrak M}^0_{\Lambda,\lambda}$ a path-connected subset of ${\mathfrak M}^0_\Lambda$? \end{question}
}
\WithTorelli{\begin{cor}} \WithoutTorelli{\begin{prop}} \label{cor-connectedness} ${\mathfrak M}^0_{\Lambda,\lambda}$ is a path-connected subset of \WithTorelli{${\mathfrak M}^0_\Lambda$.} \WithoutTorelli{${\mathfrak M}^0_\Lambda$, assuming an affirmative answer to Question \ref{thm-torelli}.} \WithTorelli{\end{cor}} \WithoutTorelli{\end{prop}}
\begin{proof} Let $(X,\eta)$ be a marked pair in ${\mathfrak M}^0_{\Lambda,\lambda}$. Then there exists a continuous path from $(X,\eta)$ to some $(X_0,\eta_0)$, where $H^{1,1}(X_0,{\mathbb Z})$ is spanned by $\eta_0^{-1}(\lambda)$, by the Local Torelli Theorem. Hence, it suffices to construct a continuous path between any two pairs $(X_0,\eta_0)$ and $(X_1,\eta_1)$ in ${\mathfrak M}^0_{\Lambda,\lambda}$, such that $H^{1,1}(X_i,{\mathbb Z})$ is cyclic, for $i=0,1$. Set $t_i:=P(X_i,\eta_i)$, $i=0,1$. Let $I$ be the closed interval $[0,1]$. Let $\gamma:I\rightarrow \lambda^\perp\cap\Omega$ be a continuous path from $t_0$ to $t_1$. Let $I_1\subset I$ be the subset of points $t$, such that $\Lambda_{\gamma(t)}^{1,1}$ is cyclic. We may choose $\gamma$ so that $I_1$ is a dense subset of $I$.
For each $t\in I_1$, there exists a unique isomorphism class of a marked pair $(X_t,\eta_t)$ in ${\mathfrak M}^0_{\Lambda,\lambda}$ with period $\gamma(t)$, by \WithTorelli{Corollary} \WithoutTorelli{Lemma} \ref{cor-torelli-for-M-Lambda-lambda}. Choose an open path-connected subset $U_t\subset {\mathfrak M}^0_{\Lambda,\lambda}$, containing $(X_t,\eta_t)$, such that $P_\lambda$ restricts to $U_t$ as an open embedding. This is possible, by the Local Torelli Theorem. We get the open covering $\gamma(I)\subset \cup_{t\in I_1} P_\lambda(U_{t})$. Choose a finite sub-covering $\cup_{j=0}^N P_\lambda(U_{t_j})$ of $\gamma(I)$, with $0=t_0<t_1< \ \cdots \ < t_N=1$. Choose an increasing subsequence $\tau_j:=t_{i_j}$, $0\leq j\leq k$, such that $\tau_0=t_0$, $\tau_k=t_N$, and $P_\lambda(U_{\tau_j})\cap P_\lambda(U_{\tau_{j+1}})$ is non-empty.
Choose points $s_{j,j+1}$ in $P_\lambda(U_{\tau_j})\cap P_\lambda(U_{\tau_{j+1}})$, such that $\Lambda^{1,1}_{s_{j,j+1}}$ is cyclic, and let $\tilde{s}_{j,j+1}$ be the unique point of ${\mathfrak M}^0_{\Lambda,\lambda}$ over $s_{j,j+1}$. Then $\tilde{s}_{j,j+1}$ belongs to $U_{\tau_j}\cap U_{\tau_{j+1}}$. Choose continuous paths $\alpha_j$ in $U_{\tau_j}$ from $\tilde{s}_{j-1,j}$ to $(X_{\tau_j},\eta_{\tau_j})$, if $j>0$, and $\beta_j$ in $U_{\tau_j}$ from $(X_{\tau_j},\eta_{\tau_j})$ to $\tilde{s}_{j,j+1}$, if $j<k$. Then the concatenated path $\beta_0\alpha_1\beta_1 \cdots \alpha_{k-1}\beta_{k-1}\alpha_k$ is a continuous path from the isomorphism class of $(X_0,\eta_0)$ to that of $(X_1,\eta_1)$. \end{proof}
\subsubsection{Deformation equivalent monodromy-reflective line bundles are simultaneously stably-${\mathbb Q}$-effective or not stably-${\mathbb Q}$-effective}
\begin{prop} \label{equivalence-of-deformation-equivalences-relations} Let $(X_1,E_1)$, $(X_2,E_2)$ be two pairs, of irreducible holomorphic symplectic manifolds $X_i$ and prime exceptional divisors $E_i\subset X_i$. Assume that $X_1$ is projective and $(X_1,[E_1])$ is deformation equivalent to $(X_2,[E_2])$, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}. \WithoutTorelli{ Assume that for some, and hence any, marking $\eta_1$ of $X_1$, Question \ref{question-connectedness} has an affirmative answer for the connected component ${\mathfrak M}^0_\Lambda$ of $(X_1,\eta_1)$ and for $\lambda=\eta_1([E_1])$. } Then $(X_1,E_1)$ and $(X_2,E_2)$ are deformation equivalent in the sense of Definition \ref{def-deformation-equivalent-pairs-with-effective-divisors}. \end{prop}
\begin{proof} We assume, for simplicity of notation, that the class $[E_i]$ is primitive. The generalization of the proof to the case $[E_i]=2e_i$ is straightforward. As noted above, we can choose a marking $\eta_2$ of $X_2$, such that $(X_2,\eta_2)$ belongs to ${\mathfrak M}^0_\Lambda$ and $\eta_2([E_2])=\lambda$. Any K\"{a}hler class $\kappa$ on $X_2$ satisfies $(\kappa,[E_2])>0$, since $E_2$ is effective \cite{huybrects-basic-results}. Hence, $(X_2,\eta_2)$ belongs to ${\mathfrak M}^0_{\Lambda,\lambda}$.
Choose a continuous path $\gamma:[0,1]\rightarrow {\mathfrak M}^0_{\Lambda,\lambda}$ from $(X_1,\eta_1)$ to $(X_2,\eta_2)$. Further choose a sufficiently fine partition of the unit interval \[ 0=t_0<t_1 < \ \cdots \ < t_N=1 \] and open connected subsets $U_i\subset {\mathfrak M}^0_{\Lambda,\lambda}$, $1\leq i \leq N$, such that $\gamma([t_{i-1},t_i])$ is contained in $U_i$, and the restriction of $P$ to $U_i$ is an open embedding $P_i:U_i\hookrightarrow \lambda^\perp\cap\Omega$. This is possible by the Local Torelli Theorem.
\begin{claim} For each $1\leq i \leq N-1$, there exists a marked pair $(Y_i,\varphi_i)$ in $U_i\cap U_{i+1}$, such that $Y_i$ is projective, and $\varphi_i^{-1}(\lambda)$ is the class of a prime exceptional divisor on $Y_i$. \end{claim}
\begin{proof} Following is an iterative process of constructing the pairs $(Y_i,\varphi_i)$. Set $(Y_0,\varphi_0)=(X_1,\eta_1)$. Assume that $i=1$, or that $1<i\leq N-1$ and $(Y_j,\varphi_j)$ exists for all $1\leq j<i$.
The pair $(Y_{i-1},\varphi_{i-1})$ belongs to $U_i$. Proposition \ref{prop-generic-prime-exceptional} implies that there exists a closed analytic subvariety $Z_i\subset U_i$, not containing $(Y_{i-1},\varphi_{i-1})$, such that for every $(X,\eta)$ in $U_i\setminus Z_i$, $\eta^{-1}(\lambda)$ is the class of a prime exceptional divisor $E\subset X$. The locus of projective marked pairs is dense in $U_i\cap U_{i+1}$, by \cite{huybrechts-norway}, Proposition 21. Hence, there exists a projective pair $(Y_i,\varphi_i)$ in $[U_i\setminus Z_i]\cap U_{i+1}$. \end{proof}
Set $(Y_N,\eta_N):=(X_2,\eta_2)$. Let $D_i\subset Y_i$ be the prime exceptional divisor with $[D_i]=\eta_i^{-1}(\lambda)$. It remains to prove that $(Y_{i-1},D_{i-1})$ is deformation equivalent to $(Y_i,D_i)$, for $1\leq i \leq N$. Both pairs $(Y_{i-1},\varphi_{i-1})$ and $(Y_i,\varphi_i)$ belong to $U_i\setminus Z_i$, by construction, for $i<N$, and by the characterization of $Z_N$ in Proposition \ref{prop-generic-prime-exceptional}, for $i=N$. Proposition \ref{prop-generic-prime-exceptional} exhibits a divisor ${\mathcal E}_i$ in the restriction of ${\mathcal X}$ to $U_i\setminus Z_i$, whose fiber over the pair $(Y_i,\varphi_i)$ is $D_i$, and whose fiber over the pair $(Y_{i-1},\varphi_{i-1})$ is $D_{i-1}$. This completes the proof of Proposition \ref{equivalence-of-deformation-equivalences-relations}. \end{proof}
The following variant of Proposition \ref{equivalence-of-deformation-equivalences-relations} will be used in the derivation of \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles} from Torelli.
\begin{prop} \label{prop-main-question-on-deformation-equivalence} Let $X$ and $Y$ be two irreducible holomorphic symplectic manifolds, with $X$ projective, $E\subset X$ a prime exceptional divisor, and $L$ a line bundle on $Y$. Set $c:=c_1(L)$. Assume that $(X,[E])$ and $(Y,c)$ are deformation equivalent in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}. Assume further that there exists a K\"{a}hler class $\kappa$ on $Y$, such that $(\kappa,c)>0$. \WithoutTorelli{ Finally assume an affirmative answer to Question \ref{question-connectedness} with a marking $\eta$ for $X$ and with $\lambda=\eta([E])$. } Then $L$ is stably-prime-exceptional (in the sense of \WithTorelli{Theorem}\WithoutTorelli{Conjecture} \ref{conj-exceptional-line-bundles}.)
\end{prop}
The above Proposition was proven in the course of proving Proposition \ref{equivalence-of-deformation-equivalences-relations}.
\begin{cor} \label{cor-if-L-1-is-prime-exceptional-L-2-not-Q-effective-then-not-def-equiv} Let $(X_1,L_1)$ and $(X_2,L_2)$ be two pairs, each of an irreducible holomorphic symplectic manifold $X_i$, and a monodromy-reflective line bundle $L_i$. Set $e_i:=c_1(L_i)$. Assume that $X_1$ is projective, $ke_1$ is the class of a prime exceptional divisor $E_1$, for some non-zero integer $k$, and $H^0(X_2,L_2^d)$ vanishes, for all non-zero integers $d$. \WithoutTorelli{ Finally assume an affirmative answer to Question \ref{question-connectedness} with a marking $\eta$ for $X_1$ and with $\lambda=\eta(e_1)$. } Then the pairs $(X_1,e_1)$ and $(X_2,e_2)$ are not deformation equivalent, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}. \end{cor}
\begin{proof} If $(X_1,e_1)$ and $(X_2,e_2)$ were deformation equivalent, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}, then $H^0(X_2,L_2^d)$ would not vanish for $d=k$ or $d=-k$, by Proposition \ref{prop-main-question-on-deformation-equivalence} and the semi-continuity theorem. \end{proof}
\subsection{Deformation equivalence and monodromy-invariants} \label{sec-deformation-equivalence-and-monodromy-invariants} Let $Mon^2(X)$ be the monodromy group, introduced in Definition \ref{def-Mon-2}. Let $I(X)\subset H^2(X,{\mathbb Z})$ be a $Mon^2(X)$-invariant subset and let $\Sigma$ be a set.
\begin{defi} \label{def-faithful} A function $f:I(X)\rightarrow \Sigma$ is a {\em monodromy-invariant}, if $f(e)=f(g(e))$, for all $g\in Mon^2(X)$.
The function $f$ is said to be a {\em faithful} monodromy-invariant, if the function $\bar{f}:I(X)/Mon^2(X)\rightarrow \Sigma$, induced by $f$, is injective. \end{defi}
Given an irreducible holomorphic symplectic manifold $X'$, deformation equivalent to $X$, denote by $I(X')\subset H^2(X',{\mathbb Z})$ the set of all classes $e'$, such that $(X',e')$ is deformation equivalent to $(X,e)$, for some $e\in I(X)$, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}.
Assume that $f:I(X)\rightarrow \Sigma$ is a monodromy-invariant function. Then $f$ admits a natural extension to a function $f:I(X')\rightarrow \Sigma$, for every irreducible holomorphic symplectic manifold $X'$ deformation equivalent to $X$. The extension is uniquely determined by the following condition. {\em Given any smooth and proper family $\pi:{\mathcal X}\rightarrow T$, of irreducible holomorphic symplectic manifolds deformation equivalent to $X$, and any flat section $e$ of the local system $R^2\pi_*{\mathbb Z}$, the function $f(e)$ is locally constant, in the classical topology of the analytic space $T$.} We denote this extension by $f$ as well. The following statement relates monodromy invariants to deformation equivalence.
\begin{lem} \label{lem-monodromy-invariants-and-deformation-equivalence} Let $f:I(X)\rightarrow \Sigma$ be a faithful monodromy-invariant function. Assume given two pairs $(X_i,e_i)$, $i=1,2$, with $X_i$ deformation equivalent to $X$ and $e_i\in I(X_i)$. \begin{enumerate} \item \label{lemma-item-monodromy-invariants-and-deformation-equivalence-of-classes} $f(e_1)=f(e_2)$ if and only if $(X_1,e_1)$ and $(X_2,e_2)$ are deformation equivalent, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}. \item \label{lemma-item-monodromy-invariants-and-deformation-equivalence-of-lb} Assume that $f(e_1)=f(e_2)$, $e_i=c_1(L_i)$, for holomorphic line bundles $L_i$ on $X_i$, and there exist K\"{a}hler classes $\kappa_i$ on $X_i$, satisfying $(\kappa_i,e_i)>0$, for $i=1,2$. \WithoutTorelli{Assume, furthermore, an affirmative answer to Question \ref{question-connectedness} with a marking $\eta$ for $X_1$ and with $\lambda=\eta(e_1)$.} Then $(X_1,L_1)$ is deformation equivalent to $(X_2,L_2)$, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-line-bundle}. \end{enumerate} \end{lem}
\begin{proof} Part \ref{lemma-item-monodromy-invariants-and-deformation-equivalence-of-classes} is evident. Part \ref{lemma-item-monodromy-invariants-and-deformation-equivalence-of-lb} follows from part \ref{lemma-item-monodromy-invariants-and-deformation-equivalence-of-classes} and \WithTorelli{Corollary \ref{cor-connectedness}.} \WithoutTorelli{the assumed connectedness of ${\mathfrak M}^0_{\Lambda,\lambda}$.} \end{proof}
\section{Monodromy-invariants from Mukai's isomorphism} \label{sec-Mukai} Let $S$ be a $K3$ surface and $M$ a smooth and projective moduli space of stable coherent sheaves on $S$. In section \ref{sec-Mukai-notation} we recalled Mukai's embedding $\theta^{-1}:H^2(M,{\mathbb Z})\rightarrow K(S)$, of the second cohomology of $M$, as a sub-lattice of the Mukai lattice. In section \ref{sec-a-rank-two-sub-lattice-of-the-Mukai-lattice} we use this embedding to define a monodromy invariant of a class in $H^2(M,{\mathbb Z})$. The values of this monodromy invariant, for monodromy-reflective classes, are calculated in sections \ref{sec-isometry-orbits} and \ref{sec-invariant-rs}.
\subsection{A rank two sub-lattice of the Mukai lattice} \label{sec-a-rank-two-sub-lattice-of-the-Mukai-lattice} Let $\widetilde{\Lambda}$ be the unimodular lattice $E_8(-1)^{\oplus 2}\oplus U^{\oplus 4}$, where $U$ is the rank two unimodular hyperbolic lattice. $\widetilde{\Lambda}$ is isometric to the Mukai lattice of a $K3$ surface. Let $X$ be an irreducible holomorphic symplectic manifold of $K3^{[n]}$-type, $n\geq 2$. Choose an embedding $\iota:H^2(X,{\mathbb Z})\hookrightarrow \widetilde{\Lambda}$ in the canonical $O(\widetilde{\Lambda})$-orbit of $X$ provided by Theorem \ref{thm-a-natural-orbit-of-embeddings-of-H-2-in-Mukai-lattice}. Let $v$ be a generator of the rank $1$ sub-lattice of $\widetilde{\Lambda}$ orthogonal to the image of $\iota$. Then $(v,v)=2n-2$. Let $e$ be a primitive class in $H^2(X,{\mathbb Z})$ satisfying $(e,e)=2-2n$. We get the sub-lattice \[ L \ \ := \ \ {\rm span}_{\mathbb Z}\{e,v\} \ \ \subset \ \ \widetilde{\Lambda}, \] where we denote by $e$ also the element $\iota(e)$. Let \begin{equation} \label{eq-saturation-of-L} \widetilde{L} \end{equation} be the saturation of $L$ in $\widetilde{\Lambda}$. Note that the pair $(\widetilde{L},e)$ determines the lattice $L$ via the equality $L={\mathbb Z} e+[e^\perp\cap \widetilde{L}]$.
\begin{defi} \label{def-isometric-pairs} Two pairs $(L_i,e_i)$, $i=1,2$, each consisting of a lattice $L_i$ and a class $e_i\in L_i$, are said to be {\em isometric}, if there exists an isometry $g:L_1\rightarrow L_2$, such that $g(e_1)=e_2$. \end{defi}
\begin{rem} Let $L_0$ be a lattice. The set of isometry classes of pairs $(L_1,e_1)$, with $L_1$ isometric to $L_0$, is in natural bijection with the orbit set $L_0/O(L_0)$. The bijection sends the isometry class of $(L_1,e_1)$ to the orbit $O(L_0)g(e_1)$, where $g:L_1\rightarrow L_0$ is some isometry. The orbit $O(L_0)g(e_1)$ is independent of the choice of $g$. \end{rem}
Let $U$ be the rank $2$ even unimodular hyperbolic lattice. Let $U(2)$ be the rank $2$ lattice with Gram-matrix $\left(\begin{array}{cc} 0 & -2 \\ -2 & 0 \end{array} \right)$. Let $H_{ev}$ be the rank $2$ lattice with Gram-matrix $\left(\begin{array}{cc} 2 & 0 \\ 0 & -2 \end{array} \right)$. Let $I''_n(X)\subset H^2(X,{\mathbb Z})$ be the subset of primitive classes of degree $2-2n$, such that ${\rm div}(e,\bullet)=n-1$ or ${\rm div}(e,\bullet)=2n-2$. Let $I_n(\widetilde{L})\subset \widetilde{L}$ be the subset of primitive classes of degree $2-2n$. Let $\rho$ be the largest positive integer, such that $(e+v)/\rho$ is an integral class. Define the integer $\sigma$ similarly using $(e-v)$.
\begin{prop} \label{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant} \begin{enumerate} \item \label{prop-item-three-possible-rank-2-lattices} The isometry class of the lattice $\widetilde{L}$ is determined as follows. \[ \widetilde{L} \ \ \cong \ \ \left\{ \begin{array}{ccl} U & \mbox{if} & {\rm div}(e,\bullet)=2n-2, \\ H_{ev} & \mbox{if} & {\rm div}(e,\bullet)=n-1 \ \mbox{and} \ n \ \mbox{is even}, \\ U(2) & \mbox{if} & {\rm div}(e,\bullet)=n-1 \ \mbox{and} \ n \ \mbox{is odd}, \ n\not\equiv 1\ (\mbox{mod} \ 8). \\ U(2) & \mbox{if} & {\rm div}(e,\bullet)=n-1, n\equiv 1\ (\mbox{mod} \ 8) \ \mbox{and} \ \rho\sigma=2n-2. \\ H_{ev} & \mbox{if} & {\rm div}(e,\bullet)=n-1, n\equiv 1\ (\mbox{mod} \ 8) \ \mbox{and} \ \rho\sigma=n-1. \end{array} \right. \] \item Consider the function \[ f : I''_n(X) \ \ \longrightarrow \ \ I_n(U)/O(U) \ \cup \ I_n(U(2))/O(U(2)) \ \cup \ I_n(H_{ev})/O(H_{ev}), \] which sends the pair $(X,e)$, $e\in I''_n(X)$, to the isometry class of the pair $(\widetilde{L},e)$, consisting of the primitive sub-lattice $\widetilde{L}\subset \widetilde{\Lambda}$, given in equation (\ref{eq-saturation-of-L}), and the class $e\in I_n(\widetilde{L})$. Then $f$ is a faithful monodromy-invariant function (Definition \ref{def-faithful}). \end{enumerate} \end{prop}
The proposition is proven below in Lemmas \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} and \ref{lem-non-unimodular-rank-two-lattice}. We provide an explicit and easily computable classification of the isometry classes of the pairs $(\widetilde{L},e)$ in Lemma \ref{lemma-isometry-orbits-in-rank-2}.
Let $L_0$ be a rank $2$ even lattice of signature $(1,1)$. Let $I_n(L_0)\subset L_0$ be the subset of primitive classes $e$ with $(e,e)=2-2n$. Let $I_{L_0,n}(X)\subset I''_n(X)$ be the subset consisting of classes $e$, such that the lattice $\widetilde{L}$ in equation (\ref{eq-saturation-of-L}) is isometric to $L_0$. Consider the function \begin{equation} \label{eq-f-from-I-L-0-n} f:I_{L_0,n}(X)\rightarrow I_n(L_0)/O(L_0), \end{equation} which sends the pair $(X,e)$ to the isometry class of the pair $(\widetilde{L},e)$. The faithfulness statement in Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant} follows from the following general statement.
\begin{lem} \label{lem-faithfulness-of-the-isometry-class-function-of-tilde-L-e} The function $f$, given in (\ref{eq-f-from-I-L-0-n}), is a faithful monodromy invariant. \end{lem}
\begin{proof} Let $e_1$, $e_2$ be two classes in $I_{L_0,n}(X)$. Denote by $\widetilde{L}_j$ the primitive rank $2$ sub-lattice of $\widetilde{\Lambda}$ associated to $e_j$ in equation (\ref{eq-saturation-of-L}), via a primitive embedding $\iota:H^2(X,{\mathbb Z})\rightarrow \widetilde{\Lambda}$ in the canonical $O(\widetilde{\Lambda})$-orbit, $j=1,2$. Denote $\iota(e_j)$ by $e_j$ as well.
Assume that $f(e_1)=f(e_2)$. Then there exists an isometry $g:\widetilde{L}_1\rightarrow \widetilde{L}_2$, such that $g(e_1)=e_2$. Let $v\in\widetilde{\Lambda}$ be a generator of $\iota[H^2(X,{\mathbb Z})]^\perp$. Then $v$ is orthogonal to $e_j$. Hence, $g(v)=v$ or $g(v)=-v$. If $g(v)=-v$, set $g':=-(R_{e_2}\circ g)$. Then $g':\widetilde{L}_1\rightarrow \widetilde{L}_2$ is an isometry satisfying $g'(e_1)=e_2$ and $g'(v)=v$. Hence, we may assume that $g(v)=v$.
There exists an isometry $\gamma\in O_+(\widetilde{\Lambda})$, such that $\gamma(\widetilde{L}_1)=\widetilde{L}_2$ and $\gamma$ restricts to $\widetilde{L}_1$ as $g$, by \cite{nikulin}, Theorem 1.14.4. Then $\gamma(v)=v$ and so $\gamma\circ\iota=\iota\circ \mu$, for some isometry $\mu\in O_+H^2(X,{\mathbb Z})$. The fact that the isometry $\mu$ extends to $\widetilde{\Lambda}$ implies that $\mu$ belongs to $Mon^2(X)$, by \cite{markman-monodromy-I}, Theorem 1.6 (see also Lemma 4.10 part (3) in \cite{markman-monodromy-I}). Now $\iota(\mu(e_1))=\gamma(\iota(e_1))=\iota(e_2)$. So $\mu(e_1)=e_2$. \end{proof}
\subsection{Isometry orbits in three rank two lattices} \label{sec-isometry-orbits} Set \[ M_U:=\left( \begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array} \right), \ \ \ M_{H_{ev}}:=\left( \begin{array}{cc} 2 & 0 \\ 0 & -2 \end{array} \right), \ \ \ M_{U(2)}:=\left( \begin{array}{cc} 0 & -2 \\ -2 & 0 \end{array} \right). \] Given an integer $m$, let ${\mathcal F}(m)$ be the set of unordered pairs $\{r,s\}$ of positive integers, such that $rs=m$ and $\gcd(r,s)=1$. Set \begin{eqnarray*} \Sigma_n(U) & := & {\mathcal F}(n-1), \\ \Sigma_n(U(2)) & := &
{\mathcal F}([n-1]/2), \ \mbox{if} \ n \ \mbox{is odd,}
\\ \Sigma_n(H_{ev}) & := & \left\{\begin{array}{ccl} {\mathcal F}(n-1) & \mbox{if} & n\not\equiv 1 \ \mbox{(modulo)} \ 4, \\ {\mathcal F}([n-1]/4) & \mbox{if} & n\equiv 1 \ \mbox{(modulo)} \ 4. \end{array} \right. \end{eqnarray*}
\begin{lem} \label{lemma-isometry-orbits-in-rank-2} Let $\widetilde{L}$ be $U$, $H_{ev}$, or $U(2)$, and $e\in I_n(\widetilde{L})$, $n\geq 2$.
Choose a generator $v$ of the sub-lattice of $\widetilde{L}$ orthogonal to $e$. \begin{enumerate} \item Let $\rho$ be the largest positive integer, such that $(e+v)/\rho$ is an integral class of $\widetilde{L}$. Define the integer $\sigma$ similarly using $(e-v)$. Then $\gcd(\rho,\sigma)$ is $1$ or $2$. \item \label{lemma-item-existence-and-uniqueness-of-r-s}
The integers $r:=\rho/\gcd(\rho,\sigma)$ and $s:=\sigma/\gcd(\rho,\sigma)$ have the following properties. \begin{enumerate} \item If $\widetilde{L}=U$, then $rs=n-1$, and the classes $\alpha:=\frac{e+v}{2r}$ and $\beta:=\frac{e-v}{2s}$ form a basis of $\widetilde{L}$ with Gram-matrix $M_U$. \item If $\widetilde{L}=U(2)$, then $n$ is odd, $rs=(n-1)/2$, and the classes $\alpha:=\frac{e+v}{2r}$ and $\beta:=\frac{e-v}{2s}$ form a basis of $\widetilde{L}$ with Gram-matrix $M_{U(2)}$. \item If $\widetilde{L}=H_{ev}$ and $n$ is even, then $rs=n-1$ and the classes $ \alpha:=\frac{1}{2}\left[\frac{e+v}{r}-\frac{e-v}{s}\right] \ \mbox{and} \ \beta:=\frac{1}{2}\left[\frac{e+v}{r}+\frac{e-v}{s}\right] $ form a basis of $\widetilde{L}$ with Gram-matrix $M_{H_{ev}}$. \item If $\widetilde{L}=H_{ev}$ and $n$ is odd, then $n\equiv 1$ modulo $4$, $rs\nolinebreak=\nolinebreak(n-\nolinebreak 1)/4$, and the classes $ \alpha:=\frac{1}{2}\left[\frac{e+v}{2r}-\frac{e-v}{2s}\right] \ \mbox{and} \ \beta:=\frac{1}{2}\left[\frac{e+v}{2r}+\frac{e-v}{2s}\right] $ form a basis of $\widetilde{L}$ with Gram-matrix $M_{H_{ev}}$. \end{enumerate} \item \label{lemma-item-the-affect-of-changing-the-sign-of-v} If we replace $v$ by $-v$, then $(r,s)$ gets replaced by $(s,r)$. \item \label{lemma-item-rs-induces-a-bijection} Let $rs:I_n(\widetilde{L})\rightarrow \Sigma_n(\widetilde{L})$ be the function, which assigns to a class $e\in I_n(\widetilde{L})$ the unordered pair $\{r,s\}$ occurring in the above factorization. Then $rs$ factors through a one-to-one correspondence \[ \overline{rs} \ : \ I_n(\widetilde{L})/O(\widetilde{L}) \ \ \ \longrightarrow \ \ \ \Sigma_n(\widetilde{L}). \] \end{enumerate} \end{lem}
\begin{proof} Let $\{u_1, u_2\}$ be a basis of $\widetilde{L}$ with Gram-matrix $M_{\widetilde{L}}$. Observe first that $O(\widetilde{L})$ is isomorphic to ${\mathbb Z}/2{\mathbb Z}\times {\mathbb Z}/2{\mathbb Z}$. Indeed, each of $O(U)$ and $O(U(2))$ is generated by $-id$ and the isometry, which interchanges $u_1$ and $u_2$. $O(H_{ev})$ is generated by the two commuting reflections with respect to $u_1$ and $u_2$. Write \[ e=au_1+bu_2. \] \underline{Case $\widetilde{L}=U$.} We have $n-1=-(e,e)/2=ab$ and $\gcd(a,b)=1$, since $e$ is primitive. Note also that $a$ and $b$ have the same sign. Set $v:=au_1-bu_2$. Then $\frac{e+v}{2a}=u_1$ and $\frac{e-v}{2b}=u_2$. Thus $r=\Abs{a}$ and $s=\Abs{b}$, and part \ref{lemma-item-existence-and-uniqueness-of-r-s} holds. Part \ref{lemma-item-the-affect-of-changing-the-sign-of-v} is clear. Part \ref{lemma-item-rs-induces-a-bijection} follows from part \ref{lemma-item-the-affect-of-changing-the-sign-of-v} and the identification of $O(U)$ above.
\noindent \underline{Case $\widetilde{L}=U(2)$.} We may identify the free abelian groups underlying $U$ and $U(2)$, so that the bilinear form on $U(2)$ is $2$ times that of $U$. The statement of the Lemma follows immediately from the case $\widetilde{L}=U$.
\noindent \underline{Case $\widetilde{L}=H_{ev}$.} We have $2-2n=(e,e)=2(a-b)(a+b)$. So $b-a$ and $b+a$ have the same sign, since $n\geq 2$. If $n$ is odd, then both $a$ and $b$ are odd, since $\gcd(a,b)=1$ and $(a-b)(a+b)$ is even. If $n$ is even, then $\{a,b\}$ consists of one odd and one even integer. Furthermore, \[ \gcd(b-a,b+a)=\gcd(b-a,2a)=\left\{ \begin{array}{ccl} 1, & \mbox{if} & n \ \mbox{is even,} \\ 2, & \mbox{if} & n \ \mbox{is odd.} \end{array} \right. \] Choose $v=bu_1+au_2$. We have \[ u_1=\frac{1}{2}\left[\frac{e+v}{a+b}-\frac{e-v}{b-a}\right], \ \ \ u_2=\frac{1}{2}\left[\frac{e+v}{a+b}+\frac{e-v}{b-a}\right]. \] Hence, $r=\Abs{a+b}$ and $s=\Abs{b-a}$, if $n$ is even, and $r=\Abs{a+b}/2$ and $s=\Abs{b-a}/2$, if $n$ is odd.
The rest is similar to the case $\widetilde{L}=U$. \end{proof}
The following table summarizes how the statements of Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant} and Lemma \ref{lemma-isometry-orbits-in-rank-2} determine the lattice $\widetilde{L}$ and the pair $\{r,s\}$ in terms of $(e,e)$, ${\rm div}(e,\bullet)$, $n$, and $\{\rho,\sigma\}$.
\noindent
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \hspace{1ex} & $(e,e)$ & ${\rm div}(e,\bullet)$ & $n$ & $\rho\sigma$ & $\widetilde{L}$ & $\{r,s\}$ & $r\cdot s$ \\ \hline 1) &$2-2n$ & $2n-2$ & $\geq 2$ & $4n-4$ & $U$ & $\{\frac{\rho}{2},\frac{\sigma}{2}\}$ & $n-1$ \\ \hline 2) & $2-2n$ & $n-1$ & even & $n-1$ & $H_{ev}$ & $\{\rho,\sigma\}$ & $n-1$ \\ \hline 3) & $2-2n$ & $n-1$ & odd & $2n-2$ & $U(2)$ & $\{\frac{\rho}{2},\frac{\sigma}{2}\}$ & $(n-1)/2$
\\ \hline 4) & $2-2n$ & $n-1$ & $\equiv 1$ modulo $8$ & $n-1$ & $H_{ev}$ & $\{\frac{\rho}{2},\frac{\sigma}{2}\}$ & $(n-1)/4$ \\ \hline \end{tabular}\\ In line $3$ cases where $n\equiv 1$ modulo $8$ occur as well.
\section{Monodromy-invariants of monodromy-reflective classes} \label{sec-invariant-rs} \hspace{0ex}\\ Fix $n\geq 2$. Let $X$ be a (K\"{a}hler) irreducible holomorphic symplectic manifold of $K3^{[n]}$-type. We define in this section the monodromy invariant function $rs$ of Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants} and prove that proposition. Part \ref{prop-item-Mon-orbit-in-degree-minus-2} of the Proposition was treated in \cite{markman-monodromy-I}, Lemma 8.9. We thus consider only part \ref{prop-item-Mon-orbit-in-degree-2-minus-2n}. We will relate this latter part to Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant} and prove Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant}.
It will be convenient to use the following normalization. Fix an isometry $\widetilde{\Lambda}\cong K(S)$, for some $K3$ surface $S$, and use Mukai's notation for classes in the Mukai lattice $K(S)$. The isometry group $O(\widetilde{\Lambda})$ acts transitively on the set of primitive classes in $\widetilde{\Lambda}$ of degree $2n-2$. Hence, we may choose the embedding $\iota:H^2(X,{\mathbb Z})\rightarrow \widetilde{\Lambda}$, so that $v=(1,0,1-\nolinebreak n)$ is orthogonal to the image of $\iota$. Then $v^\perp=H^2(S,{\mathbb Z})\oplus{\mathbb Z}\delta$, where $\delta:=(1,0,n-1)$. Thus \begin{equation} \label{eq-normalization-of-e-as-x-plus-t-delta} e \ \ = \ \ x+t\delta, \end{equation} for some integer $t$ and a class $x\in H^2(S,{\mathbb Z})$.
\subsection{The divisibility case ${\rm div}(e,\bullet)=(e,e)$}\hspace{0ex}\\ Let $I_n(X)\subset H^2(X,{\mathbb Z})$ be the subset of all primitive classes $e$, satisfying $(e,e)=2-2n$ and ${\rm div}(e,\bullet)=2n-2$. Recall that $\Sigma_n(U)$ is the set of unordered pairs $\{r,s\}$ of positive integers, such that $rs=n-1$ and $\gcd(r,s)=1$.
\begin{lem} \label{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} If $e$ belongs to $I_n(X)$, then $\widetilde{L}$ is isometric to the unimodular hyperbolic plane $U$. Denote by \[ rs \ : \ I_n(X) \ \ \ \longrightarrow \ \ \ \Sigma_n(U) \] the composition of the function $f:I_n(X)\rightarrow I_n(U)/O(U)$, defined in equation (\ref{eq-f-from-I-L-0-n}), with the bijection $\overline{rs}:I_n(U)/O(U)\rightarrow \Sigma_n(U)$ constructed in Lemma \ref{lemma-isometry-orbits-in-rank-2}. Then the function $rs:I_n(X)\rightarrow \Sigma_n(U)$ is surjective and a faithful monodromy-invariant (Definition \ref{def-faithful}).
\end{lem}
\begin{proof} Write $e=x+t\delta$ as in equation (\ref{eq-normalization-of-e-as-x-plus-t-delta}). The assumption that ${\rm div}(e,\bullet)=2n-2$ implies that $x=(2n-2)\xi$, for a class $\xi\in H^2(S,{\mathbb Z})$. We clearly have the equality \[ 2-2n=(e,e)=(x,x)+t^2(\delta,\delta)=(2-2n)^2(\xi,\xi)+(2-2n)t^2. \] Hence, we get the equality \[ t^2-1 \ \ \ = \ \ \ (2n-2)(\xi,\xi). \] Consequently, $4n-4$ divides $(t-1)(t+1)$. Thus $n-1$ divides $\frac{t-1}{2}\frac{t+1}{2}$. Now $\gcd\left(\frac{t-1}{2},\frac{t+1}{2}\right)=1$. We get a unique factorization $n-1=rs$, where $s$ divides $(t-1)/2$, $r$ divides $(t+1)/2$, and $\gcd(r,s)=1$. We may assume that $s$ is odd, possibly after replacing the embedding $\iota$ by $-\iota$, which replaces $t$ by $-t$.
Using the above factorization $n-1=rs$, we get \begin{eqnarray*} e+v=2r\alpha, & \mbox{where}, & \alpha:=\left(\frac{t+1}{2r},s\xi,\frac{(t-1)s}{2}\right) \\ e-v=2s\beta, & \mbox{where}, & \beta:=\left(\frac{t-1}{2s},r\xi,\frac{(t+1)r}{2}\right) \end{eqnarray*} and the classes $\alpha$ and $\beta$ belongs to $\widetilde{L}$. The Gram-matrix of $\{\alpha,\beta\}$ is\\ $ \left(\begin{array}{cc} (\alpha,\alpha) & (\alpha,\beta) \\ (\alpha,\beta) & (\beta,\beta) \end{array} \right)= \left(\begin{array}{cc} (e+v,e+v)/4r^2 & (e+v,e-v)/4rs \\ (e+v,e-v)//4rs & (e-v,e-v)/4s^2 \end{array} \right)=$ \\ $ \left(\begin{array}{cc}0 & -1\\-1&0 \end{array}\right). $
\\ We conclude that ${\rm span}\{\alpha,\beta\}$ is a unimodular sub-lattice of $\widetilde{\Lambda}$. Hence, $\widetilde{L}={\rm span}\{\alpha,\beta\}$ and $\widetilde{L}\cong U$.
The function $rs$ is shown to be surjective in Example \ref{example-any-factorization-rs-is-possible}. The faithfulness of the monodromy-invariant $rs$ was proven in Lemma \ref{lem-faithfulness-of-the-isometry-class-function-of-tilde-L-e}. \end{proof}
\begin{example} \label{example-any-factorization-rs-is-possible} (Compare with section \ref{sec-sequence-of-examples} above). Choose a factorization $n-1=rs$, with $s\geq r>0$, and $\gcd(r,s)=1$. Let $S$ be a projective $K3$ surface, $v=(r,0,-s)\in K(S)$, $H$ a $v$-generic polarization, and $X=M_H(r,0,-s)$. Let $\iota:H^2(M_H(r,0,-s),{\mathbb Z})\hookrightarrow K(S)$ be the embedding given in (\ref{eq-iota-for-a-moduli-space}). Set $e:=\theta(r,0,s)$, where $\theta$ is Mukai's isometry given in equation (\ref{eq-Mukai-isomorphism}). The class $e$ is monodromy-reflective and ${\rm div}(e,\bullet)=2n-2$. Now $(v+e)/2r=(1,0,0),$ $(e-v)/2s=(0,0,1),$ and $\widetilde{L}\cong U$. We get that $rs(e)=\{r,s\}$, by Lemma \ref{lemma-isometry-orbits-in-rank-2}.
\hide{ The class $e$ arises from a prime exceptional divisor precisely in the following two cases: \begin{enumerate} \item When $v=(1,0,1-n)$, then $2e=[E]$, for the prime exceptional divisor $E$ in Example \ref{example-diagonal-of-hilbert-scheme}. \item When $v=(2,0,[1-n]/2)$ is primitive (i.e., $n\equiv 3$ modulo $4$), then $e=[E]$, for a prime exceptional divisor $E$, by Lemma \ref{lemma-class-of-exceptional-locus} part \ref{lemma-item-L-divisible-by-2}. \end{enumerate} If $s>r>2$, and $k$ is a non-zero integer, then the class $ke$ is not effective, by Proposition \ref{prop-vanishing-in-divisibility-2n-2}.
} \end{example}
\subsection{The divisibility case ${\rm div}(e,\bullet)=(e,e)/2$} \hspace{0ex}\\ Let $n$ be an integer $\geq 2$. Let $I'_n(X)\subset H^2(X,{\mathbb Z})$ be the subset of all primitive classes $e$ satisfying $(e,e)=2-2n$, and ${\rm div}(e,\bullet)=n-1$.
Set \\ $\Sigma'_n:= \left\{\begin{array}{ccl} \Sigma_n(H_{ev})& \mbox{if} & n \ \mbox{is even,} \\ \Sigma_n(U(2)) & \mbox{if} & n \ \mbox{is odd, but} \ n\not\equiv 1 \ \mbox{modulo} \ 8. \\ \Sigma_n(U(2))\cup \Sigma_n(H_{ev}) & \mbox{if} & n\equiv 1 \ \mbox{modulo} \ 8. \end{array}\right.$ \\
In each of the above three cases, let $\IC_n$ be the union of the sets $I_n(\widetilde{L})/O(\widetilde{L})$ as $\widetilde{L}$ ranges through the one or two lattices appearing.
\begin{lem} \label{lem-non-unimodular-rank-two-lattice} Let $e$ be a class in $I'_n(X)$. \begin{enumerate} \item \label{lemma-item-case-n-is-even} If $n$ is even, then $\widetilde{L}$ is isometric to $H_{ev}$. \item \label{lemma-item-case-n-is-odd} If $n$ is odd, then $\widetilde{L}$ is isometric to $U(2)$ or to $H_{ev}$. The latter occurs only if $n\equiv 1$ modulo $8$ and $\rho\sigma=n-1$. \end{enumerate} In both cases, let \[ rs \ : \ I'_n(X) \ \ \ \longrightarrow \ \ \ \Sigma'_n \] be the composition
of the function $f:I'_n(X)\rightarrow \IC_n$, defined in equation (\ref{eq-f-from-I-L-0-n}), with the injection $\overline{rs}:\IC_n\rightarrow \Sigma'_n$, constructed in Lemma \ref{lemma-isometry-orbits-in-rank-2}. Then the function $rs:I'_n(X)\rightarrow \Sigma'_n$ is surjective and a faithful monodromy-invariant (Definition \ref{def-faithful}).
\end{lem}
\begin{proof}
Let us first observe that $\widetilde{L}$ cannot be unimodular. Assume otherwise. Then $\widetilde{\Lambda}$ decomposes as an orthogonal direct sum $\widetilde{L}\oplus\widetilde{L}^\perp$. Consequently, $v^\perp$ decomposes as the orthogonal direct sum $\widetilde{L}^\perp\oplus{\mathbb Z}\{e\}$. But then ${\rm div}(e,\bullet)=2n-2$.
We keep the normalization $e=x+t\delta$ of equation (\ref{eq-normalization-of-e-as-x-plus-t-delta}). The assumption that ${\rm div}(e,\bullet)=n-1$ implies that $x=(n-1)\xi$, for a class $\xi\in H^2(X,{\mathbb Z})$. We have the equality \begin{equation} \label{eq-t-square-1-equal-n-1-xi-xi-over-2} (t^2-1)=\frac{(n-1)(\xi,\xi)}{2}. \end{equation}
Hence, $n-1$ divides $t^2-1$.
\underline{Case $n$ is even:} Then $n-1$ is odd. Set \[ r:=\gcd(t+1,n-1), \ \ \ s:=\gcd(t-1,n-1). \] Then both $r$ and $s$ are odd and $\gcd(r,s)$ divides $\gcd(t-1,t+1)$. We conclude that $\gcd(r,s)=1$ and $rs$ divides $n-1$. Now $n-1$ divides $(t-1)(t+1)$. Thus, $n-1$ divides $rs$ and so $rs=n-1$.
Set \begin{eqnarray*} \alpha & := & \frac{1}{2}\left[\frac{e+v}{r}-\frac{e-v}{s}\right] = \frac{1}{2}\left(\frac{t+1}{r}-\frac{t-1}{s},(s-r)\xi, (s-r)t-s-r\right) \\ \beta & := & \frac{1}{2}\left[\frac{e+v}{r}+\frac{e-v}{s}\right] = \frac{1}{2}\left(\frac{t+1}{r}+\frac{t-1}{s},(s+r)\xi, (s+r)t+r-s\right). \end{eqnarray*} Note the equality $\frac{t+1}{r}-\frac{t-1}{s}=\frac{(s-r)t+s+r}{rs}$ and the fact that the denominator is odd, while the numerator is even. Hence, $\alpha$, $\beta$ are integral classes of $\widetilde{\Lambda}$ and $\Gram{\alpha}{\beta}= \left(\begin{array}{cc}2&0\\0&-2\end{array}\right)$.
\begin{claim} \label{claim-non-unimodular-sub-lattice-is-saturated} $\widetilde{L}={\rm span}\{\alpha,\beta\}$. \end{claim}
\begin{proof} Suppose otherwise. Then $\widetilde{L}$ strictly contains $L':={\rm span}\{\alpha,\beta\}$. Let $d$ be the index of $L'$ in $\widetilde{L}$. Then the determinant of the Gram-matrix of $\widetilde{L}$ is $d^2$ times the determinant of the Gram-matrix of $L'$. The latter determinant is $-4$. It follows that $\widetilde{L}$ is unimodular, a contradiction. \end{proof}
\underline{Case $n$ is odd:} Then $t$ is odd, by equation (\ref{eq-t-square-1-equal-n-1-xi-xi-over-2}). Set \[ r := \gcd\left(\frac{n-1}{2},\frac{t+1}{2}\right), \ \ \ s := \gcd\left(\frac{n-1}{2},\frac{t-1}{2}\right). \] Then $rs$ divides $(n-1)/2$, since $\gcd\left(\frac{t+1}{2},\frac{t-1}{2}\right)=1$.
\underline{Case $n$ is odd and $(\xi,\xi)/2$ is even:}\\ Then $(n-1)/2$ divides $(t+1)(t-1)/4$, by equation (\ref{eq-t-square-1-equal-n-1-xi-xi-over-2}). Hence, $rs=(n-1)/2$. Set \begin{eqnarray*} \alpha & := & \frac{e+v}{2r} = \left(\frac{t+1}{2r},s\xi,s(t-1)\right), \\ \beta & := & \frac{e-v}{2s} = \left(\frac{t-1}{2s},r\xi,r(t+1)\right). \end{eqnarray*} Then $\alpha$ and $\beta$ are
integral classes of $\widetilde{\Lambda}$ and $\Gram{\alpha}{\beta}=\left(\begin{array}{cc}0&-2\\-2&0 \end{array}\right)$. We conclude the equality $\widetilde{L}={\rm span}\{\alpha,\beta\}$, by the argument used in Claim \ref{claim-non-unimodular-sub-lattice-is-saturated}.
\underline{Case $n$ is odd and $(\xi,\xi)/2$ is odd:} Let $2^k$ be the largest power of $2$ which divides $t^2-1$. Then $k\geq 3$. Furthermore, $2^k$ is also the largest power of $2$ which divides $n-1$, by equation (\ref{eq-t-square-1-equal-n-1-xi-xi-over-2}). Thus $n\equiv 1$ (modulo $8$).
The set $\{r,s\}$ consists of one odd and one even integer. Say $s$ is odd. Then $2^{k-2}$ is the largest power of $2$, which divides $r$. We conclude that $rs=(n-1)/4$. Furthermore, both $(t+1)/2r$ and $(t-1)/2s$ are odd. Set \begin{eqnarray*} \alpha & := & \frac{1}{2}\left[\frac{e+v}{2r}+\frac{e-v}{2s}\right]= \frac{1}{2}\left(\frac{t+1}{2r}+\frac{t-1}{2s},(2s+2r)\xi,2s(t-1)+2r(t+1) \right) \\ \beta & := & \frac{1}{2}\left[\frac{e+v}{2r}-\frac{e-v}{2s}\right]= \frac{1}{2}\left(\frac{t+1}{2r}-\frac{t-1}{2s},(2s-2r)\xi,2s(t-1)-2r(t+1) \right). \end{eqnarray*} Then $\alpha$ and $\beta$ are integral classes of $\widetilde{\Lambda}$ and $\Gram{\alpha}{\beta}=\left(\begin{array}{cc}-2&0\\0&2 \end{array}\right)$. We conclude the equality $\widetilde{L}={\rm span}\{\alpha,\beta\}$, by the argument used in Claim \ref{claim-non-unimodular-sub-lattice-is-saturated}.
The function $rs$ is shown to be surjective in Examples \ref{example-rs-is-surjective-divisibility-n-1} and \ref{example-yet-another}. The faithfulness of the monodromy invariant $rs$ was proven in Lemma \ref{lem-faithfulness-of-the-isometry-class-function-of-tilde-L-e}. \end{proof}
\hide{ \begin{lem} \label{lem-faithfulness} The monodromy invariant functions $rs$, in Lemmas \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} and \ref{lem-non-unimodular-rank-two-lattice}, are both faithful. \end{lem}
\begin{proof} If $div(e,\bullet)=2n-2$ we are in the case of Lemma \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2}. If ${\rm div}(e,\bullet)=n-1$, then the parity of $n$
determines which of the two cases of Lemma \ref{lem-non-unimodular-rank-two-lattice} we are in. In each of these three cases, the value $rs(e)=\{r,s\}$ determines an unordered pair of invertible $2\times 2$ matrices $C_e=\left(\begin{array}{cc}c_{11}&c_{12}\\c_{21}&c_{22}\end{array}\right)$, and $C_{e'}$ satisfying \begin{equation} \label{eq-C-e-prime-in-terms-of-C-e} C_e'=AC_eA, \end{equation} where $A:=\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)$. $C_e=\left(\begin{array}{cc} 1/2r&1/2s\\1/2r&-1/2s\end{array}\right)$ in Lemma \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2}, as well as in Lemma \ref{lem-non-unimodular-rank-two-lattice} when $n$ is odd. $C_e=\frac{1}{2}\left(\begin{array}{cc} \frac{1}{r}-\frac{1}{s}&\frac{1}{r}+\frac{1}{s} \\ \frac{1}{r}+\frac{1}{s}&\frac{1}{r}-\frac{1}{s} \end{array}\right)$ in Lemma \ref{lem-non-unimodular-rank-two-lattice}, when $n$ is even. Interchanging $r$ and $s$ interchanges $C_e$ and $C'_e$.
For every embedding $\iota:H^2(X,{\mathbb Z})\hookrightarrow \widetilde{\Lambda}$, in the canonical $O(\widetilde{\Lambda})$-orbit, there is a choice of a sign for the generator $v$ of the orthogonal complement in $\widetilde{\Lambda}$ of the image of $\iota$, such that the matrix $C_e$ has the following property. If we set \[ \alpha:=c_{11}e+c_{21}v, \ \ \ \beta:=c_{12}e+c_{22}v, \] then $\{\alpha,\beta\}$ is an integral basis of a primitive rank $2$ sub-lattice $\widetilde{L}$ of $\widetilde{\Lambda}$. For the other choice of sign for $v$, the matrix $C'_e$ will have this property. Assume $C_e$ has this property. Then the Gram matrix $M_e:=\Gram{\alpha}{\beta}$ is given by \begin{equation} \label{eq-M-in-terms-of-C} M_e \ \ \ = \ \ \ C_e^t\left(\begin{array}{cc} 2-2n&0\\0&2n-2 \end{array}\right)C_e. \end{equation} We furthermore have the equality \begin{equation} \label{eq-Gram-matrix-is-independent-of-the-order-of-r-s} AM_eA=M_e, \end{equation} by a direct calculation, for the three Gram matrices appearing in Lemmas \ref{lem-faithful-Mon-invariant-in-case-divisibility-2n-2} and \ref{lem-non-unimodular-rank-two-lattice}.
If instead $C'_e$ has the above property, we get the same Gram matrix, by the following calculation.\\ \begin{eqnarray} \label{eq-M-e-prime-equal-M-e} M'_e&:=&(C'_e)^t\left(\begin{array}{cc}2-2n&0\\0&2n-2\end{array}\right)C'_e \\ \nonumber &\stackrel{(\ref{eq-C-e-prime-in-terms-of-C-e})}{=}& A^tC^t_eA^t\left(\begin{array}{cc}2-2n&0\\0&2n-2\end{array}\right)AC_eA \\ \nonumber &=& AC^t_e\left(\begin{array}{cc}2-2n&0\\0&2n-2\end{array}\right)C_eA \stackrel{(\ref{eq-M-in-terms-of-C})}{=} AM_eA \stackrel{(\ref{eq-Gram-matrix-is-independent-of-the-order-of-r-s})}{=}M_e. \end{eqnarray}
Assume that $\tilde{e}$ is another class with $(\tilde{e},\tilde{e})=(e,e)$, ${\rm div}(\tilde{e},\bullet)={\rm div}(e,\bullet)$, and $rs(e)=rs(\tilde{e})$.
Then $C_e=C_{\tilde{e}}$ or $C_e=C'_{\tilde{e}}$. We need to show that there exists a monodromy operator $\mu\in Mon^2(X)$, such that $\mu(e)=\tilde{e}$.
Assume first that $C_e=C_{\tilde{e}}$. Set $\tilde{\alpha}:=c_{11}\tilde{e}+c_{21}v$ and $\tilde{\beta}:=c_{12}\tilde{e}+c_{22}v$. Let $M_{\tilde{e}}$ be the Gram matrix $\Gram{\tilde{\alpha}}{\tilde{\beta}}$. Then $M_{\tilde{e}}=M_e$, by equation (\ref{eq-M-in-terms-of-C}).
Hence, there exists an isometry $\gamma\in O_+(\widetilde{\Lambda})$, such that $\gamma(\alpha)=\tilde{\alpha}$ and $\gamma(\beta)=\tilde{\beta}$, by \cite{nikulin}, Theorem 1.14.4. Then $\gamma(v)=v$ and so $\gamma\circ\iota=\iota\circ \mu$, for some isometry $\mu\in O_+H^2(X,{\mathbb Z})$. The fact that the isometry $\mu$ extends to $\widetilde{\Lambda}$ implies that $\mu$ belongs to $Mon^2(X)$, by \cite{markman-monodromy-I}, Theorem 1.6 (see also Lemma 4.10 part (3) in \cite{markman-monodromy-I}). Now $\iota(\mu(e))=\gamma(\iota(e))=\iota(\tilde{e})$. So $\mu(e)=\tilde{e}$.
Assume next that $C_e=C'_{\tilde{e}}$. Set $\tilde{\alpha}:=c'_{11}\tilde{e}+c'_{21}v$ and $\tilde{\beta}:=c'_{12}\tilde{e}+c'_{22}v$. We still have the equality $M_{\tilde{e}}=M_e$, by equation (\ref{eq-M-e-prime-equal-M-e}). Hence, there exists an isometry $\gamma'\in O_+(\widetilde{\Lambda})$, such that $\gamma'(\alpha)=\tilde{\alpha}$ and $\gamma'(\beta)=\tilde{\beta}$, by \cite{nikulin}, Theorem 1.14.4. Then $\gamma'(v)=-v$. Set $\gamma:=-\gamma'$. Then $\gamma(v)=v$ and $\gamma$ belongs to $O_+(\widetilde{\Lambda})$. We get that $\gamma\circ\iota=\iota\circ \mu$, for some isometry $\mu\in O_+H^2(X,{\mathbb Z})$. The fact that the isometry $\mu$ extends to $\widetilde{\Lambda}$, implies that $\mu$ belongs to $Mon^2(X)$, by \cite{markman-monodromy-I}, Theorem 1.6. Now $\iota(\mu(e))=\gamma(\iota(e))=\pm\iota(\tilde{e})$. So $\mu(e)=\pm\tilde{e}$. If $\mu(e)=\tilde{e}$, we are done. Otherwise, compose $\mu$ with the reflection with respect to $\tilde{e}$, which belongs to $Mon^2(X)$, by Proposition \ref{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon}. \end{proof}
}
\begin{example} \label{example-rs-is-surjective-divisibility-n-1} Let $s>r\geq 1$ be positive integers with $\gcd(r,s)=1$. Set $n:=rs+1$. Let $S$ be a projective $K3$ surface, set $v:=(r,0,-s)$, and let $H$ be a $v$-generic polarization of $S$. Set $M:=M_H(v)$. Let $A$ be a primitive isotropic class in $H^2(S,{\mathbb Z})$. Set $e:=\theta(r,(n-1)A,s)$. Then $e$ is monodromy-reflective and ${\rm div}(e,\bullet)=n-1$. If $n$ is even, then $(M,e)$ is an example of case \ref{lemma-item-case-n-is-even} of Lemma \ref{lem-non-unimodular-rank-two-lattice}, with $rs(e)=\{r,s\}$. If $n$ is odd, then $n-1=rs$ is even and precisely one of $r$ or $s$ is even. If $r$ is even, then $\rho=r$ and $\sigma=2s$. If $s$ is even, then $\rho=2r$ and $\sigma=s$. $(M,e)$ is an example of case \ref{lemma-item-case-n-is-odd} of Lemma \ref{lem-non-unimodular-rank-two-lattice}, with $\widetilde{L}\cong U(2)$ and $rs(e)=\{r/2,s\}$, if $r$ is even, and $rs(e)=\{r,s/2\}$, if $s$ is even. \end{example}
\begin{example} \label{example-yet-another} We exhibit next examples of the case of Lemma \ref{lem-non-unimodular-rank-two-lattice}, where $X=S^{[n]}$, $n\equiv 1$ modulo $8$, and $\widetilde{L}\cong H_{ev}$. Set $n=8k+1$, $k$ an integer $\geq 1$. Choose a factorization $2k=rs$, with $r$ even, $s$ odd, and $\gcd(r,s)=1$. There exists an integer $\lambda$, such that $\lambda r\equiv -1$ modulo $s$, since $\gcd(s,r)=1$. If $\lambda$ is a solution, so is $\lambda+s$. Hence, we may assume that $\lambda$ is an odd and positive integral solution. Set $g:=[r\lambda+1]/s$. Then $g$ is a positive odd integer.
Let $S$ be a $K3$ surface with a primitive class $\xi\in {\rm Pic}(S)$ of degree $(\xi,\xi)=2\lambda g$. Set \[ v:=(1,0,1-n) \ \ \ \mbox{and} \ \ \ e := (2\lambda r+1,(n-1)\xi,[2\lambda r+1](n-1)). \] Then $(e,e)=2-2n$, by following two equalities. \[ (e,e)=(n-1)[2\lambda g(n-1)-2(4\{r^2\lambda^2+r\lambda\}+1)] \] and $2\lambda g(n-1)=8r\lambda sg=8r\lambda(r\lambda+1)$. The class $e$ is primitive, since $\gcd(2\lambda r+1,n-1)=\gcd(2\lambda r+1,4rs)=\gcd(2\lambda r+1,s)= \gcd(-1,s)=1$.
The classes $\frac{e+v}{2s}=(g,2r\xi,4\lambda r^2)$ and $\frac{e-v}{2r}=(\lambda,2s\xi,4gs^2)$ are integral and primitive. We conclude that $\widetilde{L}\cong H_{ev}$, by Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant} part \ref{prop-item-three-possible-rank-2-lattices}, and $rs(e)=\{r,s\}$, by Lemma \ref{lemma-isometry-orbits-in-rank-2}.
\end{example}
\section{Numerical characterization of exceptional classes via Torelli} \label{sec-numerical-characterization-via-torelli}
The following table points to an example provided in this paper, for each possible value of the quadruple $\{n, \ (e,e), \ {\rm div}(e,\bullet), \ rs(e)\}$, for a monodromy reflective class $e$.
\noindent
\begin{tabular}{|c|c|c|c|c|c|l|} \hline \hspace{1ex} $(e,e)$ & ${\rm div}(e,\bullet)$ & $\widetilde{L}$ & $\{r,s\}$ & $n$ & Reference \\ \hline $-2$ & $1$ & NA & NA & $\geq 2$ & Examples \ref{example-case-degree-e-minus-2-div-1}, \ref{example-case-degree-e-minus-2-div-1-brill-noether} \\ \hline $-2$ & $2$ & NA & NA & $\geq 6$ and & Example \ref{example-degree-e-minus-2-div-2} \\
& & & & $\equiv 2$ mod $4$ & \\ \hline $-2$ & $2$ & NA & NA & $=2$ & Example \ref{example-diagonal-of-hilbert-scheme} \\ \hline $2-2n$ & $2n-2$ & $U$ & $\{1,n-1\}$ & $\geq 2$ & Example \ref{example-diagonal-of-hilbert-scheme} \\ \hline $2-2n$ & $2n-2$ & $U$ & $\{2,(n-1)/2\}$ & $\geq 7$ and & Lemma \ref{lemma-class-of-exceptional-locus} part \ref{lemma-item-L-divisible-by-2} \\
& & & & $\equiv 3$ mod $4$ & \\ \hline $2-2n$ & $2n-2$ & $U$ & $s>r>2$ & $=rs+1$ & Proposition \ref{prop-vanishing-in-divisibility-2n-2} \\
& & & $\gcd(r,s)=1$ & & \\ \hline $2-2n$ & $n-1$ & $H_{ev}$ & $\{1,n-1\}$ & $\geq 4$, even & Lemma \ref{lemma-class-of-exceptional-locus} part \ref{lemma-item-L-not-divisible-by-2} \\ \hline $2-2n$ & $n-1$ & $U(2)$ & $\{1,(n-1)/2\}$ & $\geq 3$, odd & Lemma \ref{lemma-class-of-exceptional-locus} part \ref{lemma-item-L-not-divisible-by-2} \\ \hline $2-2n$ & $n-1$ & $H_{ev}$ & $r\geq 3$, $s\geq 3$ & $=rs+1$ & Lemma \ref{lemma-Type-B-divisibility-n-1} \\
& & & $\gcd(r,s)=1$ & even & \\ \hline $2-2n$ & $n-1$ & $U(2)$ & $r\geq 3$, $s\geq 2$ & $=2rs+1$ & Lemma \ref{lemma-Type-B-divisibility-n-1} \\
& & & $\gcd(r,s)=1$ & & \\ \hline $2-2n$ & $n-1$ & $H_{ev}$ & $r$ even, $s$ odd & $=4rs+1$ & Example \ref{example-non-effective-divisibility-n-1-and-n-is-cong-1-mod-8} \\
& & & $\gcd(r,s)=1$ & & \\ \hline \end{tabular} \hspace{1ex}
\noindent
The congruence constraints on $n$ are necessary. If $(e,e)=-2$ and ${\rm div}(e,\bullet)=2$, then $n\equiv 2$ (modulo $4$), by \cite{markman-monodromy-I}, Lemma 8.9. If $rs(e)=\{2,(n-1)/2\}$, then $n\equiv 3$ (modulo $4$), in order for $\{2,(n-1)/2\}$ to be a pair of relatively prime integers. If $(e,e)=2-2n$, ${\rm div}(e,\bullet)=n-1$, $n>2$, and $rs(e)=\{1,n-1\}$, then $n$ must be even, since for odd $n$ the product of $r$ and $s$ is equal to $(n-1)/2$ or $(n-1)/4$, by Lemmas \ref{lem-non-unimodular-rank-two-lattice} and \ref{lemma-isometry-orbits-in-rank-2}. This explains also the value of $n$ in the last three rows.
\begin{proof} \WithoutTorelli{(Of Theorem \ref{thm-main-conjecture-follows-from-torelli}).} \WithTorelli{(Of Theorem \ref{conj-exceptional-line-bundles}).} Set $e:=c_1(L)$. The pair $(X,e)$ is deformation equivalent, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-cohomology-class}, to a pair $(M,c)$ appearing in the above table of examples, by Proposition \ref{prop-introduction-Mon-2-orbit-is-determined-by-three-invariants} and Lemma \ref{lem-monodromy-invariants-and-deformation-equivalence} part \ref{lemma-item-monodromy-invariants-and-deformation-equivalence-of-classes}. $M$ is projective and \WithoutTorelli{Conjecture} \WithTorelli{Theorem} \ref{conj-exceptional-line-bundles} holds for $(M,c)$, by the example referred to in the table. Suppose that $L$ is numerically effective. Then Part \ref{conj-item-effective} of \WithoutTorelli{Conjecture} \WithTorelli{Theorem} \ref{conj-exceptional-line-bundles} follows for $(X,L)$, by Proposition \ref{prop-main-question-on-deformation-equivalence}.
Suppose next that $L$ is not numerically effective. We prove part \ref{conj-item-vanishing} of \WithoutTorelli{Conjecture} \WithTorelli{Theorem} \ref{conj-exceptional-line-bundles} by contradiction. Assume that part \ref{conj-item-vanishing} fails. Then there exists a non-zero integer $k$, such that $h^0(X_t,L_t^k)>0$, for all $t\in D_e$.
We may assume that the absolute value $\Abs{k}$ is minimal with the above property. Now ${\rm Pic}(X_t)$ is cyclic, for a generic $t\in D_e$. Hence, the linear system $\linsys{L_t^k}$ must have a member $E_t$, which is a prime divisor, by the minimality of $\Abs{k}$. It follows that $E_t$ is the unique member of the linear system, by \cite{boucksom}, Proposition 3.13. Hence, $h^0(X_t,L_t^k)=1$, away from a closed analytic proper subset $Z\subset D_e$.
Set $U:=D_e\setminus Z$ and let ${\mathcal X}_U$ be the restriction of the semi-universal family from $Def(X)$ to $U$. There exists an irreducible divisor ${\mathcal E}\subset {\mathcal X}_U$, which does not contain the fiber $X_t$, for any $t\in U$, and which intersects $X_t$ along a divisor in $\linsys{L_t^k}$, by the argument used in the proof of Proposition \ref{prop-generic-prime-exceptional}. The argument furthermore shows, that there exists a closed analytic proper subset $Z_1\subset U$, such that the fiber $E_t$ of ${\mathcal E}$ is a prime divisor, over all points $t\in U\setminus Z_1$. We do not need the projectivity assumption, as it was used in the proof of Proposition \ref{prop-generic-prime-exceptional} only to establish that the generic dimension of $h^0(X_t,L_t)$ is $1$, a fact which was already established above.
We conclude the existence of a pair $(X_1,e_1)$, parametrized by a point in $U\setminus Z_1$, such that $X_1$ is projective, by \cite{huybrechts-norway}, Proposition 21. Let $L_2$ be the line bundle on $M$ with $c_1(L_2)=c$. Then $H^0(M,L_2^d)$ vanishes, for all non-zero integers $d$, since $L_2$ is not numerically exceptional, and the examples mentioned in the above table have this property, whenever $c$ is not numerically exceptional. Hence, $(X_1,e_1)$ and $(M,c)$ are not deformation equivalent, by Corollary \ref{cor-if-L-1-is-prime-exceptional-L-2-not-Q-effective-then-not-def-equiv}. On the other hand, $(X_1,e_1)$ is deformation equivalent to $(X,e)$, and hence to $(M,c)$, a contradiction. \end{proof}
\section{Conditions for the existence of slope-stable vector bundles} \label{sec-conditions-for-existence-ofslope-stable-vector-bundles} Let $S$ be a projective $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. We assemble in section \ref{sec-necessary-conditions} necessary conditions for the existence of locally-free $H$-slope-stable sheaves (Lemmas \ref{lem-H-slope-stability-implies-s-geq-r}, \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}, and \ref{lemma-condition-for-existence-of-H-slope-stable-sheaves-isotropic-u-case}).
In section \ref{sec-sufficient-conditions} we bound the dimension of the locus $Exc$ of $H$-stable sheaves, which are not locally free or not $H$-slope-stable. The sheaves $F$ considered all have the following involutive property: there exists an integer $t$, such that the classes in $K(S)$ of $F$ and $F^*\otimes H^t$ are equal. Equivalently, $c_1(F)=\frac{t\cdot{\rm rank}(F)}{2}h$, for some integer $t$, where $h:=c_1(H)$. I thank Kota Yoshioka for pointing out that much of the content of section \ref{sec-sufficient-conditions} is essentially proven in sections 2 and 3 of his paper \cite{yoshioka-irreducibility}. Section \ref{sec-sufficient-conditions} was not replaced by a citation, since the precise statements we need are not easily recovered from those of Yoshioka, as he was mainly concerned with proving that the locus $Exc$ has codimension $\geq 1$, while we need that in the subset of cases considered\footnote{ For some cases Yoshioka does state that the codimension is $\geq 2$ \cite[Lemma 3.1]{yoshioka-irreducibility}, but under an assumption that excludes some cases which we need.} $Exc$ has codimension $\geq 2$.
The results of this section are only lightly used in section \ref{sec-examples}, but are essential to the examples in section \ref{sec-non-effective}.
\subsection{Necessary conditions} \label{sec-necessary-conditions} Set $h:=c_1(H)\in H^2(S,{\mathbb Z})$ and $d:=\deg(H)/2$. \begin{lem} \label{lemma-locally-free-H-stable-of-rank-2-is-slope-stable} Let $F$ be a locally free $H$-stable sheaf of rank $r$ satisfying $c_1(F)=\frac{tr}{2}h$, for some integer $t$.
Then $F^*$ is $H$-stable, if and only if $F$ is $H$-slope-stable. In particular, if $r=2$, then $F$ is $H$-slope-stable. \end{lem}
\begin{proof} After tensorization by a power of $H$, we may reduce to the case where either $c_1(F)=0$, or $r=2\rho$ is even and $c_1(F)=\rho h$. Assume that we are in one of these cases. If $c_1(F)=0$, set $L:=\StructureSheaf{S}$. If $c_1(F)=\rho h$, set $L:=H$. In either case, we have the equality $[F]=[F^*\otimes L]$ of classes in $K(S)$. Furthermore, a sheaf $G$ is $H$-stable if and only if $G\otimes L$ is $H$-stable.
If $F$ is $H$-slope-stable, then so is $F^*$. Hence $F^*$ is $H$-stable as well. $F$ is $H$-slope-semi-stable, since it is $H$-stable. Suppose that $F$ is not $H$-slope-stable. then there exists a saturated subsheaf $F_1\subset F$, of rank (say) $r_1$, with $c_1(F_1)=(r_1/r)c_1(F)$ and $0<r_1<r$. Set $F_2:=F/F_1$, and $r_2:=r-r_1$. $H$-stability of $F$ yields the inequality $ \frac{\chi(F_2)}{r_2}>\frac{\chi(F)}{r}. $ We get the injective homomorphism $F_2^*\rightarrow F^*$, and the inequalities: \[ \frac{\chi(F_2^*\otimes L)}{r_2}=\frac{\chi(F_2^{**})}{r_2} \geq \frac{\chi(F_2)}{r_2}>\frac{\chi(F)}{r}=\frac{\chi(F^*\otimes L)}{r}. \] Hence, $F^*\otimes L$ is $H$-unstable. Consequently, $F^*$ is $H$-unstable.
If $r=2$, then $F^*\otimes L$ is isomorphic to $F$ and thus $F^*$ is $H$-stable. \end{proof}
\begin{lem} \label{lem-H-slope-stability-implies-s-geq-r} Let $F$ be a locally free $H$-slope-stable sheaf of class $v=(r,0,-s)$. Then $v=(1,0,1)$ or $s\geq r\geq 2$. \end{lem}
\begin{proof} If ${\rm rank}(F)=1$, then $F$ is isomorphic to $\StructureSheaf{S}$, since $F$ is locally free, and so $s=-1$. Assume that $v\neq (1,0,1)$. Then $H^0(F)$ vanishes, by the $H$-stability of $F$. Similarly, $ H^2(F)^*\cong H^0(F^*)= (0), $ by the $H$-slope-stability of $F^*$. Thus, $r-s=\chi(F)=-\dim H^1(F)\leq 0$. \end{proof}
Lemma \ref{lem-H-slope-stability-implies-s-geq-r} states a necessary cohomological condition for the existence of a locally free $H$-slope-stable sheaf of class $v$ with $c_1(v)=0$ (slope $0$). The condition states that $\chi(v)\leq 0$, unless $v$ is the class $u=(1,0,1)$ of the trivial line bundle. If $\chi(v)\leq 0$, then the locus of sheaves with non-zero global sections is expected to have positive co-dimension. The condition $\chi(v)\leq 0$ translates to $(u,v)\geq 0$. The following Lemma states a similar cohomological condition, for a class $v$ with a non-zero slope. The role of the trivial line bundle is replaced next by a simple and rigid sheaf $E$ of the same slope as $v$.
\begin{lem} \label{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half} Let $F$ be a locally free $H$-slope-stable sheaf of class $v=(2r,rh,-b)$, where $r>0$, $\gcd(r,b)=1$, $(h,h)=2d$, and $d$ is an odd integer. Set $u:=(2,h,(d+1)/2)$. If $(v,v)=-2$, then $v=u$. Otherwise, $(v,u)\geq 0$ and $(v,u)$ is even. Furthermore, $(v,u)=0$, if and only if $v=(2,h,(d-1)/2)$. \end{lem}
\begin{proof} $M_H(u)$ consists of a single isomorphism class. Let $E$ be an $H$-stable sheaf of class $u$. Then $E$ is necessarily $H$-slope-stable and locally free \cite{mukai-hodge}. $M_H(v)$ is non-empty, by assumption. Let $2n$ be its dimension. Then $2n-2=(v,v)=2dr^2+4rb$, and $b=\frac{n-1-dr^2}{2r}$. If $n=0$, then $r=1$,
since $r$ divides $(v,v)/2$. We conclude that $v=u$, if $(v,v)=-2$.
Assume that $v\neq u$. Then $n\geq 1$. We get the inequality \[ \frac{\chi(u)}{2}=\frac{5+d}{4}> \frac{(4+d)r^2+1-n}{4r^2}=\frac{\chi(v)}{2r}. \] Thus, ${\rm Hom}(E,F)=0$. Similarly, \[ {\rm Ext}^2(E,F)^*\cong {\rm Hom}(F,E)\cong {\rm Hom}(E^*,F^*)\cong {\rm Hom}(E^*\otimes H,F^*\otimes H)=0, \] since $E^*\otimes H\cong E$ and $F^*\otimes H$ is an $H$-slope-stable sheaf of class $v$. Thus, $(v,u)=-\chi(v,u)\geq 0$. Furthermore, $(v,u)=2\left[r(d-1)/2+b\right]$. If $(v,u)=0$, then $r$ divides $b$. Hence, $r=1$, since $\gcd(r,b)=1$. If $r=1$ and $(v,u)=0$, then $b=(1-d)/2$, as claimed. \end{proof}
\begin{lem} \label{lemma-condition-for-existence-of-H-slope-stable-sheaves-isotropic-u-case} Let $F$ be a locally free $H$-slope-stable sheaf of class $v=(2r,rh,-b)$, where $r>0$, $\gcd(r,b)=1$, $(h,h)=2d$, and $d$ is an even integer. Set $u:=(2,h,d/2)$. If $(v,v)=0$, then $v=u$. Otherwise, $(u,v)$ is a positive even number. \end{lem}
This lemma has a cohomological interpretation as well. $M_H(u)$ is two dimensional and it parametrizes locally-free $H$-slope-stable sheaves \cite{mukai-hodge}. Let $B\subset M_H(u)\times M_H(v)$ be the correspondence consisting of pairs $(E,F)$, with non-vanishing ${\rm Hom}(E,F)$. The lemma states that if $v\neq u$, then the expected co-dimension $(u,v)+1$ of $B$ is larger than $2$, and so $B$ is not expected to surject onto $M_H(v)$.
\begin{proof} Set $n:=(1/2)\dim_{\mathbb C} M_H(v)=1+(v,v)/2$. If $n=1$, then $(v,v)=0$, and so $v=ku$, for some positive integer $k$. If $k>1$, then the moduli space $M_H(ku)$ is the $k$-th symmetric product of $M_H(u)$ and it consists entirely of $H$-unstable but $H$-semistable sheaves. We are assuming however the existence of an $H$-slope-stable sheaf $F$ of class $v$. Hence, $k=1$ and $v=u$.
Assume that $n>1$. We get \[ \frac{\chi(v)}{2r}=\frac{(4+d)r^2-n+1}{4r^2}< \frac{4+d}{4}=\frac{\chi(u)}{2}. \] The normalized Hilbert polynomial $p$ of a sheaf $G$ of positive rank is the Hilbert polynomial divided by the rank $p(n):=\chi(G\otimes H^n)/{\rm rank}(G)$. The first two leading terms in the normalized Hilbert polynomials of $u$ and $v$ are equal, and the constant terms are related by the above inequality. Hence, ${\rm Hom}(E,F)=0$, for every $H$-slope-stable sheaf $E$ of class $u$. Such a sheaf $E$ is necessarily locally free, and so $E^*\otimes H$ is $H$-slope-stable of class $u$. We get also the vanishing of ${\rm Ext}^2(E,F)$, by the argument used in the proof of Lemma \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}. We conclude the inequality $(u,v)=-\chi(E^*\otimes F)\geq 0$. Furthermore, $(u,v)=dr+2b$ is even. If $(u,v)=0$, then $(v,v)=0$, and so $v=u$. The lemma follows. \end{proof}
\subsection{Sufficient conditions} \label{sec-sufficient-conditions}
\subsubsection{The case $c_1(v)=0$.} Let $r$, $s$ be integers satisfying $s>r>2$ and $\gcd(r,s)=1$. Set $n:=rs+1$ and $v:=(r,0,-s)$. A sheaf $F$ of class $v$ is $H$-stable, if and only if it is $H$-semi-stable.
Hence, $M_H(v)$ is smooth and projective of dimension $2n$. Let $Exc\subset M$ be the locus of $H$-stable sheaves of class $v$ that are not locally free or not $H$-slope-stable. $Exc$ is clearly a closed subset of $M_H(v)$.
\begin{lem} \label{lem-codimension-of-Exc} $Exc$ has codimension at least $2$ in $M$. \end{lem}
\begin{proof} We will use the following notation, in order for large parts of the proof to generalize to a proof of Lemma \ref{lemma-Exc-has-codimension-at-least-2-again}. Let $u:=(1,0,1)$ be the class of $\StructureSheaf{S}$. Set $\epsilon:={\rm rank}(u)=1$.
\underline{Step 1:} [Jun Li's morphism to the Uhlenbeck-Yau compactification]. Let $Y_H(w)$ be the moduli space of $H$-slope-stable locally free sheaves of class $w\in K(S)$. Let $v_1$, \dots, $v_k$ be distinct classes in $K(S)$, with $v_i=(r_i,0,-s_i)$, $r_i>0$, $(v_i,v_i)\geq -2$. Let $d_1$, \dots, $d_k$ be positive integers satisfying \[ r=\sum_{i=1}^kd_ir_i, \ \ \mbox{and} \ \ t(\vec{v},\vec{d}):=s-\left(\sum_{i=1}^kd_is_i\right)\geq 0. \] Denote the $d$-th symmetric product of $Y_H(v_i)$ by $Y_H(v_i)^{(d)}$. Set \[ Y(\vec{v},\vec{d}) \ \ := \ \ \prod_{i=1}^k Y_H(v_i)^{(d_i)}\times S^{(t(\vec{v},\vec{d}))}. \] Note that for $Y_H(v_i)$ to be non-empty, $v_i=(r_i,0,-s_i)$ should satisfy \begin{equation} \label{eq-non-emptyness-condition-on-r-s} r_i=-s_i=1, \ \ \ \mbox{or} \ \ \ s_i\geq r_i\geq 2, \end{equation} by Lemma \ref{lem-H-slope-stability-implies-s-geq-r}. If $r_i=-s_i=1$ then $v_i=u$.
Let $M^{\mu{ss}}_H(v)$ be the moduli space of $S$-equivalence classes of $H$-slope-semi-stable sheaves of class $v$ (\cite{huybrechts-lehn-book}, section 8.2). Then $M^{\mu{ss}}_H(v)$ is a projective scheme. Set theoretically, $M^{\mu{ss}}_H(v)$ is the disjoint union of all such varieties $Y(\vec{v},\vec{d})$. There exists a projective morphism \[ \bar{\phi} \ : \ M_H(v) \ \ \ \longrightarrow \ \ \ M^{\mu{ss}}_H(v) \] \cite{jun-li}.
Each irreducible component of each fiber of the morphism $\bar{\phi}$ is unirational, as it is dominated by an iterated construction of open subsets in extension bundles and bundles of punctual Quot-schemes (\cite{huybrechts-lehn-book}, Theorem 8.2.11). The morphism $\bar{\phi}$ is thus generically finite, since $M_H(v)$ is holomorphic symplectic.
It suffices to prove the inequality \[ \dim{Y}(\vec{v},\vec{d}) \ \ \ \leq \ \ \ \dim M_H(v)-4, \] for all strata $Y(\vec{v},\vec{d})\subset M^{\mu ss}_H(v)$, such that $Y(\vec{v},\vec{d})\neq Y_H(v)$. It would then follow that $Y_H(v)$ is non-empty, and the image of $\bar{\phi}$ is contained in the closure $\overline{Y}_H(v)$ of $Y_H(v)$ in $M_H^{\mu ss}(v)$. The fiber of $\bar{\phi}$ over a point of $Y_H(v)$ consists of a single point. Let $\widetilde{Y}_H(v)$ be the normalization of $\overline{Y}_H(v)$. The morphism $\bar{\phi}$ would then factor through a surjective birational morphism \[ \phi:M_H(v)\rightarrow \widetilde{Y}_H(v), \] since $M_H(v)$ is smooth and irreducible, and $Exc=M_H(v)-\nolinebreak Y_H(v)$ would be the exceptional locus of $\phi$. It would also follow that the singular locus of $\overline{Y}_H(v)$ has co-dimension $\geq 4$ in $\overline{Y}_H(v)$. It would then follow that $Exc$ has co-dimension $\geq 2$ in $M_H(v)$, by Proposition \ref{prop-dissident-locus}.
\underline{Step 2:} [Upper bounds for $\dim{Y}(\vec{v},\vec{d})$]. Fix a stratum $Y(\vec{v},\vec{d})$. Set $t:=t(\vec{v},\vec{d})$, and $v':=(r,0,t-s)$. Then $v'=\sum_{i=1}^kd_iv_i$. Set \[ c(\vec{v},\vec{d}) \ \ := \ \ \dim M_H(v)-\dim Y(\vec{v},\vec{d}). \] We compute: \begin{eqnarray} \nonumber c(\vec{v},\vec{d})&=& 2+(v,v)-\sum_{i=1}^k d_i[(v_i,v_i)+2]-2t \\ \nonumber &=& 2+2t(\epsilon r-1)+(v',v')-\sum_{i=1}^k d_i[(v_i,v_i)+2] \\ \label{eq-c-vec-r-vec-d} &=& 2+2t(\epsilon r-1)+\sum_{i=1}^k\sum_{j=1}^kd_id_j(r_is_j+r_js_i)- 2\sum_{i=1}^kd_i[r_is_i+1]. \end{eqnarray}
\underline{Case 1:} Suppose that $v_i\neq u$, for all $i$. Then $s_i\geq r_i\geq 2$, for $1\leq i\leq k$. Write $c(\vec{v},\vec{d})$ in the form \[ 2+2t(\epsilon r-1)+2\sum_{i=1}^{k-1}d_i\sum_{j=i+1}^kd_j(r_is_j+r_js_i) +2\sum_{i=1}^kd_i[(d_i-1)r_is_i-1]. \] \underline{Case 1.1:} Assume that $k=1$. Then \[ c(\vec{v},\vec{d})=2+2t(\epsilon r-1)+2d_1[(d_1-1)r_1s_1-1]. \] \underline{Case 1.1.1:} If $d_1=1$, then $c(\vec{v},\vec{d})=2t(\epsilon r-1)\geq 4t$. If $t=0$, we are in the open subset where $F$ is locally free and $H$-slope-stable. If $t>0$, we see that indeed $c(\vec{v},\vec{d})\geq 4$.
\noindent \underline{Case 1.1.2:} If $d_1>1$, then $2d_1[(d_1-1)r_1s_1-1]$ is a positive even number, so $c(\vec{v},\vec{d})\geq 4+2t(\epsilon r-1)\geq 4.$
\noindent \underline{Case 1.2:} Assume that $k>1$. Then \begin{eqnarray*} c(\vec{v},\vec{d}) & = & 2+2t(\epsilon r-1)+2(A+B), \ \ \ \mbox{where} \\ A&=&\sum_{i=1}^{k-1}d_i\left\{ \left[\sum_{j=i+1}^kd_j(r_is_j+r_js_i)\right]+ \left[(d_i-1)r_is_i-1\right] \right\}, \\ B&=& d_k[(d_k-1)r_ks_k-1]. \end{eqnarray*} We are assuming that $s_i\geq r_i\geq 2$. Hence, $(r_is_j+r_js_i)\geq 2r_ir_j\geq 8$. Hence, $A\geq 7$. Now $B>0$, if $d_k>1$, and $B=-1$, if $d_k=1$. The desired inequality $c(\vec{v},\vec{d})\geq 4$ follows.
\noindent \underline{Case 2:} Assume that $v_1=u$. Note that $r_1s_1+1=0$ and $r_1s_j+r_js_1=s_j-r_j=(u,v_j)$. Equation (\ref{eq-c-vec-r-vec-d}) becomes \begin{eqnarray*} c(\vec{v},\vec{d})&=&A+B+C, \ \ \ \mbox{where} \\ A& = & 2+2t(\epsilon r-1)-2d_1^2 \\ B& = & 2d_1\sum_{j=2}^kd_j(s_j-r_j) =2d_1(u,v'-d_1v_1) =2d_1(2d_1+s-r-\epsilon t). \\ C & = & \sum_{i=2}^k\sum_{j=2}^kd_id_j(r_is_j+r_js_i)-2\sum_{i=2}^kd_i[r_is_i+1], \end{eqnarray*} Note the equality \[ A+B=2+2t(\epsilon r-\epsilon d_1-1)+2d_1(s-r+d_1). \]
\noindent \underline{Case 2.1:} Assume that $k=1$. Then $r=d_1$, $t=(r+s)/\epsilon$, and \\ $c(\vec{v},\vec{d})=A= 2+2[(r+s)\left(r-\frac{1}{\epsilon}\right)-r^2]\geq 2+2[(s-1)(r-1)-1]\geq 2r(r-1)\geq 12$.
\noindent \underline{Case 2.2:} Assume that $k\geq 2$. Then $1\leq \epsilon d_1\leq \epsilon r-d_2r_2\leq \epsilon r-2$. So \begin{eqnarray*} A+B & \geq & 2+2t+2d_1(s-r+1)\geq 2+4d_1\geq 6. \\ C/2 & = & \sum_{i=2}^{k-1}\sum_{j=i+1}^kd_id_j(r_is_j+r_js_i)+ \sum_{i=2}^k[d_i^2r_is_i-d_i(r_is_i+1)] \\
& = & \sum_{i=2}^{k-1}d_i\left( \left[\sum_{j=i+1}^kd_j(r_is_j+r_js_i)\right]+[(d_i-1)r_is_i]-1 \right) \\ & & + d_k[(d_k-1)r_ks_k-1]. \end{eqnarray*} If $k=2$ and $d_2=1$, then $C=-2$. Otherwise, $C\geq 0$. We conclude that $c(\vec{v},\vec{d})\geq 4$. This completes the proof of Lemma \ref{lem-codimension-of-Exc}. \end{proof}
\subsubsection{The case with slope equal one half.} Let $r$ be a positive odd integer, $\sigma$ a positive integer, and set $n:=r\sigma+1$. Assume that $r\geq 3$, $\sigma\geq 3$, and $\gcd(r,\sigma)=1$. Let $S$ be a $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Set $d:=\deg(H)/2$. Choose $(S,H)$, so that $\sigma$ and $d$ have the same parity. If $d$ is odd, assume that $\sigma>r$, possibly after interchanging $r$ and $\sigma$. Set $h:=c_1(H)$ and $v := (2r,rh,-b)$, where $b:=[\sigma-rd]/2$. Note that $\gcd(r,b)=\gcd(r,\sigma)=1$. Hence, $v$ is a primitive class in $K(S)$, $(v,v)=2n-2$, and the moduli space $M_H(v)$ is smooth and projective of type $K3^{[n]}$. Let $Exc\subset M_H(v)$ be the locus parametrizing sheaves $F$ that are not locally free or not $H$-slope-stable.
\begin{lem} \label{lemma-Exc-has-codimension-at-least-2-again} $Exc$ is an algebraic subset of co-dimension\footnote{Note that the assumption $\sigma>r$, adopted above when $d$ is odd, is necessary, since otherwise $Exc=M_H(v)$, by Lemma \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}.} $\geq 2$ in $M_H(v)$. \end{lem}
\begin{proof} {\bf Proof in the case $d$ is odd:} When $d$ is odd, then $\sigma$ is odd. Set $u:=(2,h,(d+1)/2)$ and $s:=r+(v,u)$. Then $s=\sigma$ is odd. Thus, $s>r$, by assumption, and $s-r$ is even, so $s\geq r+2$.
Given an $H$-slope-stable locally free sheaf $F_i$ of class $v_i=(2r_i,r_ih,-b_i)$, set $s_i:=r_i+(v_i,u)$. If $v_i=u$, then $s_i=-1$. If $v_i\neq u$, then $s_i\geq r_i$, by Lemma \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}. Furthermore, $s_i=r_i$, if and only if $v_i=(2,h,(d-1)/2)$. If $v_i\neq u$ and $s_i\neq r_i$, then $s_i\geq r_i+2$, since $s_i-r_i$ is even, by Lemma \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}.
With the above notation of $s$ and $s_i$, the proof is almost identical to that of Lemma \ref{lem-codimension-of-Exc}. Following are the necessary changes. Replace the class $(1,0,1)$ by the class $u$ defined above. Then $\epsilon={\rm rank}(u)=2$. Set $\lambda:=(0,0,-1)$. Then $(u,\lambda)=2$ and $(\lambda,\lambda)=0$. With the above definition of $s_i$, we have $v_i=r_iu+\left(\frac{s_i+r_i}{2}\right)\lambda$. Hence, \[ (v_i,v_j) \ \ \ = \ \ \ r_is_j+r_js_i, \] as in the proof of Lemma \ref{lem-codimension-of-Exc}.
Equation (\ref{eq-non-emptyness-condition-on-r-s}) is replaced by \[ r_i=-s_i=1, \ \ \ \mbox{or} \ \ \ s_i=r_i=1, \ \ \ \mbox{or} \ \ \ s_i\geq r_i+2\geq 3, \] by Lemma \ref{lemma-non-existence-of-H-slope-stable-sheaves-with-slope-half}. Equation (\ref{eq-c-vec-r-vec-d}), for the co-dimension $c(\vec{v},\vec{d})$ of $Y(\vec{v},\vec{d})$, remains valid. The argument for case 1.1.1 remains valid. In case 1.1.2 the term $2d_1[(d_1-1)r_1s_1-1]$ vanishes, if $d_1=2$ and $r_1=s_1=1$. However, in that case $(u,v')=(u,2v_1)=0$ and $2rt=(u,v-v')=(u,v)=s-r>0$. So $t>0$ and $c(\vec{v},\vec{d})=2+2t(2r-1)\geq 4$.
In case 1.2 we are no longer assuming that $r_i\geq 2$. However, since all $v_i$ are different from $u$, and $k>1$, then at least one $v_i$, say $v_1$, is different from $(2,h,(d+1)/2)$. Then $s_1\geq r_1+2\geq 3$. Thus $r_1s_2+r_2s_1\geq 4$ and $A\geq 3$. The rest of the argument in case 1.2 is identical.
In case 2, the equations for $c(\vec{v},\vec{d})$, $A$, $B$, and $C$, remain valid. The argument in case 2.1 remains valid. The inequality $\epsilon r -d_2r_2\leq \epsilon r-2$, in the first line of case 2.2, need not hold. Nevertheless, $\epsilon r -\epsilon d_1\geq \epsilon d_2\geq 2$. So \[ A+B\geq 2 + 2d_1(s-r+1)\geq 2+6d_1\geq 8. \] The rest of the argument remains valid.
{\bf Proof\footnote{The cases where $\sigma<r$ were proven earlier in \cite[Lemma 3.1]{yoshioka-irreducibility}.} in the case $d$ is even.} When $d$ is even then $\sigma$ is even. Set $u:=(2,h,d/2)$. By assumption, $r\geq 3$, $s\geq 3$, and $\gcd(r,s)=1$.
Given an $H$-slope-stable sheaf $F_i$ of class $v_i=(2r_i,r_ih,-b_i)$, with $r_i>0$, set $\sigma_i:=(u,v_i)=2b_i+r_id$. If $v_i=u$, then $\sigma_i=0$. If $v_i\neq u$, then $\sigma_i$ is a positive even integer, by Lemma \ref{lemma-condition-for-existence-of-H-slope-stable-sheaves-isotropic-u-case}. Note also that $(u,v)=\sigma$.
The proof is again almost identical to that of Lemma \ref{lem-codimension-of-Exc}. Following are the necessary changes. Replace the class $(1,0,1)$ by the class $u$ defined above. Then $\epsilon={\rm rank}(u)=2$. Set $\lambda:=(0,0,-1)$. Then $(u,\lambda)=2$ and $(\lambda,\lambda)=0$. With the above definition of $\sigma_i$, we have $v_i=r_iu+\left(\frac{\sigma_i}{2}\right)\lambda$. Hence, \[ (v_i,v_j) \ \ \ = \ \ \ r_i\sigma_j+r_j\sigma_i, \] and we replace $s_i$ by $\sigma_i$ in the proof of Lemma \ref{lem-codimension-of-Exc}. Then Equation (\ref{eq-c-vec-r-vec-d}), for the co-dimension $c(\vec{v},\vec{d})$ of $Y(\vec{v},\vec{d})$, remains valid.
Equation (\ref{eq-non-emptyness-condition-on-r-s}) is replaced by \[ v_i=u, \ \ \ \mbox{or} \ \ \ \sigma_i \ \ \ \mbox{is a positive even integer,} \] by Lemma \ref{lemma-condition-for-existence-of-H-slope-stable-sheaves-isotropic-u-case}. The argument for case 1.1 remains valid.
In case 1.2 we are no longer assuming that $r_i\geq 2$. However, since all the $v_i$ are different from $u$, then $\sigma_i\geq 2$, for all $i$. Thus, $r_1\sigma_2+r_2\sigma_1\geq 4$ and $A\geq 3$. The rest of the argument is identical.
In case 2, $v_1=u$, $\sigma_1=0$, and $(r_1\sigma_j+r_j\sigma_1)=\sigma_j=(u,v_j)$. Then $c(\vec{v},\vec{d})=A+B+C$, where \begin{eqnarray*} A&=& 2+2t(\epsilon r-1), \\ B&=& 2d_1\sum_{j=2}^k d_j\sigma_j=2d_1(u,v'-d_1v_1)= 2d_1(\sigma-\epsilon t), \end{eqnarray*} and $C$ remains the same. Then \[ A+B \ \ \ = \ \ \ 2 + 2t(\epsilon r - \epsilon d_1 -1) +2d_1\sigma. \]
In case 2.1, $r=d_1$, $v'=ru$, $\epsilon t =(v-v',u)=\sigma$, and $c(\vec{v},\vec{d})=A=2+\sigma(\epsilon r -1)$. Hence $c(\vec{v},\vec{d})\geq 17$.
In case 2.2 we assume that $k\geq 2$ and so $\epsilon(r-d_1)\geq 2$. So $A+B\geq 2 +2d_1\sigma\geq 8$. The rest of the argument is the same. This completes the proof of Lemma \ref{lemma-Exc-has-codimension-at-least-2-again}. \end{proof}
\section{Examples of prime exceptional divisors} \label{sec-examples}
Let $e$, $E$ and $X$ be as in Theorem \ref{thm-2}. Set $n:=\dim_{{\mathbb C}}(X)/2$. The pair $(X,E)$ has the following elementary invariants: \begin{enumerate} \item \label{item-degree} $(e,e)=-2$, or $(e,e)=2-2n$. \item The divisibility ${\rm div}(e,\bullet)$ of the class $(e,\bullet)$ in $H^2(X,{\mathbb Z})^*$ is equal to $(e,e)$ or $(e,e)/2$.
\item \label{item-invariant-k} Write $[E]=ke$, where $e$ is a primitive class in $H^2(X,{\mathbb Z})$. Then $k=1$, or $k=2$.
\end{enumerate}
Set $[E]^\vee:=\frac{-2([E],\bullet)}{([E],[E])}$. We have $[E]^\vee=e^\vee$ or $[E]^\vee=2e^\vee$, where
$e^\vee$ is a primitive class in $H_2(X,{\mathbb Z})$, and the coefficient is determined by Lemma \ref{lemma-divisibility} in terms of the invariant ${\rm div}(e,\bullet)$ and the coefficient $k$ in (\ref{item-invariant-k}) above.
In particular, if ${\rm div}(e,\bullet)=(e,e)/2$, then $[E]=e$, by Lemma \ref{lemma-divisibility}.
\begin{thm} \label{thm-a-uniruled-divisor-is-exceptional} Let $X$ be a smooth projective holomorphically symplectic variety, and $E$ a prime divisor on $X$. \begin{enumerate} \item \label{thm-item-uniruled-implies-exceptional} (\cite{druel}, Theorem 1.3) Assume that through a generic point of $E$ passes a rational curve of class $\ell\in H_2(X,{\mathbb Z})$, such that $[E]\cdot \ell<0$. Then $E$ is an exceptional divisor. \item \label{thm-item-ell-is-E-vee} Let $E$ and $\ell$ be as in part \ref{thm-item-uniruled-implies-exceptional} and $\pi:X'\rightarrow Y$ the birational contraction of $E$ introduced in Proposition \ref{prop-druel}. Then $Y$ has $A_i$ singularities away from its dissident locus,\footnote{ See the paragraph preceding Proposition \ref{prop-dissident-locus} for the definition of the dissident locus.} and $i=1$ or $i=2$. Furthermore, $\ell=\left\{\begin{array}{ccc} [E]^\vee & \mbox{if} & i=1, \\ \frac{1}{2}[E]^\vee & \mbox{if} & i=2. \end{array}\right.$ \end{enumerate} \end{thm}
\begin{proof} We need only prove part \ref{thm-item-ell-is-E-vee}. $Y$ has $A_1$ or $A_2$ singularities, by Corollary \ref{cor-1}. Let $E'$ be the strict transform of $E$ in $X'$. The generic fiber of the restriction of $\pi$ to $E'$ is a rational curve, or a pair of rational curves joined at a node. The exceptional locus of the birational transformation from $X$ to $X'$ does not dominate $\pi(E')$, by the proof of Proposition 1.4 in \cite{druel}. The morphism $\pi$ thus restricts to a rational morphism from $E$ to $\pi(E')$, whose generic fiber is isomorphic to the generic fiber of $E'$ over $\pi(E')$. The class $\ell$ must be the class of an irreducible component of the generic fiber of the restriction of $\pi$ to $E$, by the uniqueness of the family of rational curves, which dominates $E$ (\cite{druel}, Proposition 4.5). The equality $\ell=\frac{1}{i}[E]^\vee$ follows from part \ref{item-integrality} of Corollary \ref{cor-1}. \end{proof}
We will say that {\em the prime exceptional divisor is of type} $A_i$, if the variety $Y$ in part \ref{thm-item-ell-is-E-vee} of Theorem \ref{thm-a-uniruled-divisor-is-exceptional} has $A_i$ singularities away from its dissident locus. All prime exceptional divisor studied in this paper are of type $A_1$.
\subsection{Brill-Noether exceptional divisors} \label{sec-Brill-Neother-exceptional-divisors} Let $S$ be a $K3$ surface, $F_0$ a simple and rigid coherent sheaf, i.e., a sheaf satisfying ${\rm End}(F_0,F_0)\cong{\mathbb C}$, and ${\rm Ext}^1(F_0,F_0)=0$. Then the class $v_0$ of $F_0$ is a primitive class in $K(S)$ with $(v_0,v_0)=-2$. Examples of exceptional divisors $E$ of degree $-2$ in moduli spaces of sheaves on $S$ seem to arise as Brill-Noether loci as follows. Let $v\in K(S)$ be a class satisfying $(v_0,v)=0$, and such that there exists a smooth and compact moduli space $M(v)$ of stable sheaves of class $v$. The locus $M(v)^1$, of points representing sheaves $F$ with non-vanishing ${\rm Ext}^1(F,F_0)$, is often an exceptional divisor of degree $-2$. The examples considered in this section are all of this type.
\begin{example} \label{example-case-degree-e-minus-2-div-1} The case $n\geq 2$, $(e,e)=-2$, and ${\rm div}(e,\bullet)= 1$.
\hide{ We provide two examples. The first example is simpler, but Druel's birational map $X\rightarrow Y$, which contracts $E$, is not a regular morphism. In the second example it is.
a)
} Let $S$ be a $K3$ surface containing a smooth irreducible rational curve $\Sigma$. Let $E\subset S^{[n]}$, $n\geq 2$, be the divisor consisting of length $n$ subschemes intersecting $\Sigma$ along a non-empty subscheme. The class $[E]$ is identified with $[\Sigma]$, under the embedding of $H^2(S,{\mathbb Z})$ as an orthogonal direct summand in the decomposition (\ref{eq-orthogonal-direct-sum}) of $H^2(S^{[n]},{\mathbb Z})$. Thus $([E],[E])=([\Sigma],[\Sigma])=-2$ and ${\rm div}([E],\bullet)=1$. $E$ is of type $A_1$. Let $F_0$ be the direct image of $\StructureSheaf{\Sigma}(-1)$ and $v_0\in K(S)$ the class of $F_0$. Then $v_0$ is orthogonal to the class of the ideal sheaf $I_Z$ of a length $n$ subscheme $Z$ of $S$, and $E$ is the Brill-Noether locus, where ${\rm Ext}^1(I_Z,F_0)$ does not vanish. \end{example}
\hide{
b) Let $S$ be a projective $K3$ surface, $H$ a polarization of degree $(H,H)=2n-2$, such that ${\rm Pic}(S)={\rm span}_{\mathbb Z}\{H\}$. Let $v\in K(S)$ be the class
$(0,H,0)$ in Mukai's notation. Let $M_H(v)$ be the moduli space of Gieseker-Simpson $H$-stable sheaves with class $v$. Then $M_H(v)$ is smooth, projective, holomorphic symplectic, and deformation equivalent to $S^{[n]}$ (see section \ref{sec-Mukai-notation}). Points of $M_H(v)$ represent torsion sheaves, with pure one-dimensional support $D$, and $D$ is a curve in the linear system $\linsys{H}$. Let $E\subset M_H(v)$ be the Brill-Noether locus of sheaves $F$ with $h^1(F)>0$. Let $v_0 \in K(S)$ be the class of the trivial line bundle. We have the equality $[E]=-\theta(v_0)$, by \cite{markman-part-two}, Lemma 4.11. Hence, $[E]=e$ and $(e,e)=-2$. Clearly, ${\rm div}(e,\bullet)=1$.
Let $sup:M_H(v)\rightarrow \linsys{H}$ be the support morphism (\cite{le-potier-coherent}, section 2.3). Let $\theta:v^\perp\rightarrow H^2(M_H(v),{\mathbb Z})$ be the Mukai isomorphism (\ref{eq-Mukai-isomorphism}). Set $e:=\theta(-v_0)$ and $f:=\theta(0,0,1)$. The following lemma is proven in the Appendix section \ref{sec-proof-of-lemma-ample-cone}. \end{example}
\begin{lem} \label{lemma-ample-cone} \begin{enumerate} \item \label{lemma-item-f-is-nef} $f=sup^*\StructureSheaf{\linsys{H}}(1)$. \item \label{lemma-item-e-and-f-generate-the-Pef-cone} The Pseudo-effective cone $Pef(M_H(v))$ is equal to $\langle e,f\rangle$. Furthermore, $E$ is the unique prime exceptional divisor in $M_H(v)$. \item \label{lemma-item-f-and-e+2f-generate-the-Nef-cone} The nef cone $Nef(M_H(v))$ is equal to $\langle e+2f,f\rangle$. Hence, the cone $Nef(M_H(v))$ is dual to $Pef(M_H(v))$ with respect to the Beauville-Bogomolov pairing. \item \label{lemma-item-extremal-class} There exists a birational morphism $\pi:M_H(v)\rightarrow Y$ onto a normal projective variety $Y$ with the following property. Exactly one of the irreducible components of the exceptional locus of $\pi$ is a divisor, and this component is $E$. \item \label{lemma-item-Y-has-A-1-singularities} The image $\pi(E)$ of $E$ has codimention $2$ in $Y$. $Y$ has $A_1$-singularities along a Zariski dense open subset of $\pi(E)$. \end{enumerate} \end{lem}
}
Let $S$ be a projective $K3$ surface, with a cyclic Picard group generated by an ample line bundle $H$ of degree $d\geq 2$. In the remainder of this section the simple and rigid sheaf $F_0$ will be $\StructureSheaf{S}$. Then ${\rm Ext}^1(F,\StructureSheaf{S})\cong H^1(F)^*$, by Serre's Duality. We will need the following results.
\begin{lem} \label{lemma-stability-of-co-kernel-for-primitive-c-1} (\cite{markman-reflections}, Lemma 3.7 part 3) Let $F$ be an $H$-stable sheaf on $S$ of rank $r$ and determinant $H$, and $U\subset H^0(F)$ a subspace of dimension $r'\leq r$. Then the evaluation homomorphism $U\otimes\StructureSheaf{S}\rightarrow F$ is injective and its co-kernel is an $H$-stable sheaf. \end{lem}
Consider the Mukai vector $v:=(r,H,s)$ and assume that $r\geq 0$ and $r+s\geq 0$. Set $v_0:=(1,0,1)$. Let $M_H(v)^t$ be the Brill-Noether locus of $H$-stable sheaves $F$ with $h^1(F)\geq t$.
\begin{thm} \label{thm-brill-noether} (\cite{markman-reflections}, Corollary 3.19, \cite{yoshioka-brill-noether}) \begin{enumerate} \item $M_H(v)^t$ is empty, if and only if $M_H(v+tv_0)$ is. \item There exists a smooth surjective projective morphism \[ f_t \ : \ \left[M_H(v)^t\setminus M_H(v)^{t+1}\right] \ \ \ \longrightarrow \ \ \ \left[M_H(v+tv_0)\setminus M_H(v+tv_0)^1\right]. \] \item \label{thm-fibers-are-grassmannians} The fiber of $f_t$, over a point representing a sheaf $E$, is naturally isomorphic to the Grassmannian $G(t,H^0(E))$. Furthermore, $H^0(E)$ is $r+s+2t$-dimensional, and the dimension $t(r+s+t)$ of the fiber is equal to the co-dimension of $M_H(v)^t$ in $M_H(v)$. \item \label{thm-item-brill-neother-divisor-has-class-minus-v0} (\cite{markman-part-two}, Lemma 4.11) If $s=-r$, then $M(v)^1$ is a prime divisor of class $\theta(-v_0)$. The class $\ell\in H^2(M_H(v),{\mathbb Z})^*$ of a ${\mathbb P}^1$-fiber of $f_1$ is $(\theta(-v_0),\bullet)$. \end{enumerate} \end{thm}
The embedding $G(t,H^0(E))\hookrightarrow M_H(v)$ in part \ref{thm-fibers-are-grassmannians} sends a $t$-dimensional subspace $U\subset H^0(E)$ to the co-kernel of the evaluation homomorphism $U\otimes \StructureSheaf{S}\rightarrow E$. The co-kernel is stable, by Lemma \ref{lemma-stability-of-co-kernel-for-primitive-c-1}.
\begin{example} \label{example-case-degree-e-minus-2-div-1-brill-noether} The case $n\geq 2$, $(e,e)=-2$, and ${\rm div}(e,\bullet)= 1$ was considered in Example \ref{example-case-degree-e-minus-2-div-1}. Additional examples of such prime exceptional divisors are provided in part \ref{thm-item-brill-neother-divisor-has-class-minus-v0} of Theorem \ref{thm-brill-noether}. $M_H(v)^1$ is exceptional, since it is prime of degree $-2$, by part \ref{thm-item-brill-neother-divisor-has-class-minus-v0} of Theorem \ref{thm-brill-noether}.
Examples of prime exceptional Brill-Noether divisors, for more general simple and rigid sheaves, can be found in the work of Yoshioka \cite{yoshioka-brill-noether}. \end{example}
\begin{example} \label{example-degree-e-minus-2-div-2} The case $(e,e)=-2$, ${\rm div}(e,\bullet)=2$, and $[E]=e$.
\\ Assume that $n$ is congruent to $2$ modulo $4$ and $n\geq 6$. Let $S$ be a projective $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$ of degree $(H,H)=\frac{n-2}{2}$. Then $h^i(H^2)=0$, for $i>0$, and $h^0(H^2)=n$. Set $X:=M_H(1,H^2,-1)\cong S^{[n]}$. Let $E:=M_H(1,H^2,-1)^1$ be the Brill-Noether divisor in $M_H(1,H^2,-1)$ of sheaves $F$ with $h^1(S,F)>0$.
We recall the explicit definition of $E$. Let $\pi_i$, $i=1,2$, be the projection from $S\times M_H(1,H^2,-1)$ onto the $i$-th factor. Let ${\mathcal Z}\subset S\times M_H(1,H^2,-1)$ be the universal subscheme, and $I_{\mathcal Z}$ its ideal sheaf. Then ${\mathcal F}:=I_{\mathcal Z}\otimes \pi_1^*H^2$ is a universal sheaf over $S\times {\mathcal M}_H(1,H^2,-1)$. We have the short exact sequence \[ 0\rightarrow {\mathcal F} \rightarrow \pi_1^*H^2\rightarrow \StructureSheaf{{\mathcal Z}}\otimes \pi_1^*H^2\rightarrow 0, \] and the homomorphism of rank $n$ vector bundles \[ g \ : \ H^0(H^2)\otimes\StructureSheaf{M_H(v)} \cong \pi_{2_*}(\pi_1^*H^2) \ \ \ \longrightarrow \ \ \ \pi_{2_*}\left(\StructureSheaf{{\mathcal Z}}\otimes\pi_1^*H^2\right). \] The homomorphism $g$ is injective, since a generic length $n$ subscheme of $S$ induces $n$ independent conditions on a linear system $\linsys{L}$, provided the line bundle $L$ on $S$ satisfies $h^0(L)\geq n$. The Brill-Noether divisor is the zero divisor of $\Wedge{n}g$. $E$ is an effective divisor of class $\theta(-v_0)$, where $v_0:=(1,0,1)$ is the class in $K(S)$ of the trivial line bundle, and $\theta$ is the Mukai isomorphism given in (\ref{eq-Mukai-isomorphism}).
\hide{ and let $E\subset S^{[n]}$ be the reduced divisor\footnote{Consider the case $n=2$ and replace the assumption that $H$ is ample by the assumption that $S$ admits an elliptic fibration $\pi:S\rightarrow {\mathbb P}^1$ and $H:=\pi^*\StructureSheaf{{\mathbb P}^1}(1)$. In that case the locus described in equation (\ref{eq-Ideal-sheaves-in-E}) is the whole of $S^{[2]}$. This explains our assumption that $n>2$. } supported by \begin{equation} \label{eq-Ideal-sheaves-in-E} \{I_Z \ : \ H^1(S,I_Z\otimes H^2)\neq 0\}. \end{equation} A length $n$ subscheme $Z$ belongs to $E$, if and only if the restriction homomorphism $H^0(S,I_Z\otimes H^2)\rightarrow H^0(Z,\StructureSheaf{Z}\otimes H^2)$ is not an isomorphism. Furthermore, the co-kernel is isomorphic to $H^1(S,I_Z\otimes H^2)$. It follows that for a generic $Z$ in $E$, $h^1(S,I_Z\otimes H^2)=1$.
If we identify $S^{[n]}$ with $M(v)$, $v=(1,H^2,-1)$, then the divisor $E$ is identified as a Brill-Noether locus in $M_H(1,H^2,-1)$, of sheaves $F$ with $h^1(S,F)>0$. The class of $E$ is equal to $-\theta(v_0)$, where $v_0:=(1,0,1)$ is the class in $K(S)$ of the trivial line bundle. The proof is similar (???) to that of Lemma 4.11 in \cite{markman-part-two}. Hence, $([E],[E])=(v_0,v_0)=-2$ and $[E]=e$. Furthermore, $(\theta(v_0),\theta(x))=(v_0,x)$, which is divisible by $2$, for all $x\in v^\perp$. Hence, ${\rm div}([E],\bullet)=2$.
Note the Serre-Duality isomorphism $H^1(S,F)\cong{\rm Ext}^1(F,\omega_S)^*$. $E$ admits a rational dominant map to the singular moduli space $M_H(2,H^2,0)$ of dimension $2n-2$. The map is well defined on the dense open subset of $E$, consisting of ideal sheaves $I_Z$, such that $h^1(S,I_Z\otimes H^2)=1$ and the unique non-trivial extension \[ 0\rightarrow \omega_S\rightarrow G\rightarrow I_Z\otimes H^2 \rightarrow 0 \] results in an $H$-semistable sheaf $G$. The map $E\rightarrow M_H(2,H^2,0)$ sends such an ideal sheaf $I_Z$ to the $S$-equivalence class of the sheaf $G$. The generic fiber is the smooth rational curve ${\mathbb P}{H}om(\omega_S,G)$. Hence, $A_i(S^{[n]},E)=A_1$.
Consider, for example, the case $n=6$. Then $S$ is a generic $K3$ surface of genus $2$, admitting a double cover $ f:S\rightarrow \linsys{H}^*\cong{\mathbb P}^2. $ We have the isomorphism $f^*:H^0({\mathbb P}^2,\StructureSheaf{{\mathbb P}^2}(2))\rightarrow H^0(S,H^2)$. The divisor $E$ in this case consists of ideal sheaves of length $6$ subschemes $Z$, such that $Z$ is contained in $f^*(D)$, for some $D\in \linsys{\StructureSheaf{{\mathbb P}^2}(2)}\cong{\mathbb P}^5$.
} \end{example}
\begin{lem} \label{lemma-brill-noether-divisor-is-exceptional} $E$ is a prime exceptional divisor of class $e:=\theta(-v_0)$. In particular, $(e,e)=-2$ and ${\rm div}(e,\bullet)=2$. \end{lem}
The rest of section \ref{sec-Brill-Neother-exceptional-divisors} is devoted to the proof of Lemma \ref{lemma-brill-noether-divisor-is-exceptional}.
\begin{lem} \label{lemma-stability-criteria} Let $F$ be an $H$-slope-stable sheaf of class $(2,H^2,0)$. \begin{enumerate} \item \label{lemma-item-Q-is-torsion-free} For every non-zero section $s\in H^0(F)$, the evaluation homomorphism $s:\StructureSheaf{S}\rightarrow F$ has a rank $1$ torsion free co-kernel sheaf. \item \label{lemma-item-G-is-H-slope-stable} If $\epsilon\in {\rm Ext}^1(\StructureSheaf{S},F)$ is a non-zero class and \[ 0\rightarrow \StructureSheaf{S}\rightarrow G_\epsilon\rightarrow F\rightarrow 0 \] the corresponding extension, then the sheaf $G_\epsilon$ is $H$-slope-stable. \end{enumerate} \end{lem}
\begin{proof} \ref{lemma-item-Q-is-torsion-free}) Denote the co-kernel of $s$ by $Q_s$. If $T$ is a subsheaf of $Q_s$ with zero-dimensional support, then ${\rm Ext}^1(T,\StructureSheaf{S})=0$. Thus, the inverse image of $T$ in $F$ would contain a subsheaf isomorphic to $T$. But $F$ is torsion free. Hence, the dimension of the support of any subsheaf of $Q_s$ is at least $1$.
If $T$ is a subsheaf of $Q_s$ of one-dimensional support, then its inverse image in $F$ is a rank one subsheaf $F'$ of $F$ with effective determinant line bundle. Hence, $\det(F')\cong H^k$, for some positive integer $k$. This contradicts the slope-stability of $F$. Hence, $Q_s$ is torsion-free.
\ref{lemma-item-G-is-H-slope-stable}) Assume that $G_\epsilon$ is $H$-slope-unstable, and let $G'\subset G_\epsilon$ be an $H$-slope-stable subsheaf of maximal slope of rank $r\leq 2$. If $G'$ maps to zero in $F$, then $G'$ is a subsheaf of $\StructureSheaf{S}$, and can not destabilize $G_\epsilon$. For the same reason, the slope of the image $\bar{G}$ of $G'$ satisfies $\mu(\bar{G})\geq \mu(G')$. Thus ${\rm rank}(\bar{G})\neq 1$, since otherwise $\bar{G}$ would destabilize $F$. Hence, ${\rm rank}(G')=2$, $G'$ maps isomorphically onto $\bar{G}$, and $\det(G')\cong H^k$, for $2/3<k\leq 2$. It follows that $k=2$.
Set $Q:=F/\bar{G}$. We get the short exact sequence \[ 0\rightarrow \bar{G}\RightArrowOf{\iota} F \rightarrow Q\rightarrow 0. \] ${\rm Ext}^1(Q,\StructureSheaf{S})$ vanishes, since $Q$ has zero-dimensional support. Hence, $\iota^*:{\rm Ext}^1(F,\StructureSheaf{S})\rightarrow {\rm Ext}^1(\bar{G},\StructureSheaf{S})$ is injective. On the other hand, the pullback $\iota^*(\epsilon)$ vanishes in ${\rm Ext}^1(\bar{G},\StructureSheaf{S})$. This contradicts the assumption that $\epsilon$ is a non-zero class. \end{proof}
The moduli space $M_H^{ss}(2,H^2,0)$, of $H$-semi-stable sheaves of class $(2,H^2,0)$, is known to be an irreducible normal projective variety of dimension $2n-2$. Furthermore, the singular locus is equal to the strictly semi-stable locus and it has co-dimension 2, if $n=6$, and $4$, if $n>6$ (\cite{kaledin-lehn-sorger}, Theorem 4.4 and Theorem 5.3). A generic $H$-stable sheaf of class $(2,H^2,0)$ is $H$-slope-stable. This is equivalent to the corresponding statement for $M_H(2,0,1-(n/2))$, and follows from the following lemma.
\begin{lem} \label{lemma-existence-of-slope-stable-vb-of-rank-2} Let $s$ be an integer $\geq 2$. Then the set of $H$-stable locally free sheaves of class $(2,0,-s)$ is Zariski dense in $M_H^{ss}(2,0,-s)$. Furthermore, any $H$-stable locally free sheaf of class $(2,0,-s)$ is $H$-slope-stable. \end{lem}
The proof of the density statement is similar to that of Lemma \ref{lem-codimension-of-Exc} and is omitted. The case $s=2$ is proven in \cite{ogrady-10}, Proposition 3.0.5. The second statement is a special case of Lemma \ref{lemma-locally-free-H-stable-of-rank-2-is-slope-stable}.
\begin{lem}
Let $U\subset M_H^{ss}(2,H^2,0)$ be the subset parametrizing $H$-slope-stable sheaves $F$ with $h^1(F)=0$. Then $U$ is a Zariski-dense open subset.
\end{lem}
\begin{proof}
Let $M^{\mu s}\subset M_H^{ss}(2,H^2,0)$ be the Zariski open subset of $H$-slope-stable sheaves. Note that $M^{\mu s}$ is a dense subset, by Lemma \ref{lemma-existence-of-slope-stable-vb-of-rank-2}. Let $t$ be the minimum of the set $\{h^1(F) \ : \ [F]\in M^{\mu s}\}$. It suffices to prove that $t=0$. Assume that $t>0$. Let $U'\subset M^{\mu s}$ be the Zariski-open subset of sheaves $F$ with $h^1(F)=t$. Let $p:{\mathbb P}\rightarrow U'$ be the projective bundle with fiber ${\mathbb P}{H}^1(F)^*$ over $F$. ${\mathbb P}$ is a Zariski open subset of the moduli space of coherent systems constructed by Le Potier in \cite{le-potier-coherent}. A point in ${\mathbb P}$ parametrizes an equivalence class of a pair $(F,\ell)$, consisting of an $H$-slope-stable sheaf $F$ of class $(2,H^2,0)$ and a one-dimensional subspace $\ell\subset {\rm Ext}^1(F,\StructureSheaf{S})$. We have $\dim({\mathbb P})=\dim(U')+t-1=2n+t-3$.
There exists a natural morphism \[ f : {\mathbb P} \ \ \ \longrightarrow \ \ \ M_H(3,H^2,1), \] sending a pair $(F,\ell)$ to the isomorphism class of the sheaf $G_\ell$ in the natural extension $ 0\rightarrow \ell^*\otimes\StructureSheaf{S}\rightarrow G_\ell\rightarrow F \rightarrow 0. $ $G_\ell$ is $H$-slope-stable, by Lemma \ref{lemma-stability-criteria} part \ref{lemma-item-G-is-H-slope-stable}.
Now $h^0(G_\ell)=h^0(F)+1=t+3$. Furthermore, the data $(F,\ell)$ is equivalent to the data $(G_\ell,\ell)$, where $\ell$ is a one-dimensional subspace of $H^0(G_\ell)$. Hence, the fiber of $f$, over the isomorphism class of $G_\ell$, has dimension at most $t+2$. The dimension of $M_H(3,H^2,1)$ is $2n-8$. Thus, $\dim({\mathbb P})\leq 2n+t-6.$ This contradicts the above computation of the dimension of ${\mathbb P}$.
\end{proof}
Let $G(1,U)$ be the moduli space of equivalence classes of pairs $(F,\lambda)$, where $F$ is an $H$-slope-stable sheaf of class $(2,H^2,0)$ with $h^1(F)=0$, and $\lambda\subset H^0(F)$ is a one-dimensional subspace. $G(1,U)$ is a Zariski open subset of the moduli space of coherent systems constructed by Le Potier in \cite{le-potier-coherent}. The forgetful morphism $G(1,U)\rightarrow U$ is a ${\mathbb P}^1$-bundle. Let \[ \psi \ : \ G(1,U) \ \ \ \longrightarrow \ \ \ M_H(1,H^2,-1) \] be the morphism, sending a pair $(F,\lambda)$ to the quotient $F/[\lambda\otimes\StructureSheaf{S}]$. The morphism $\psi$ is well-defined, by Lemma \ref{lemma-stability-criteria} part \ref{lemma-item-Q-is-torsion-free}.
\begin{lem} \label{lemma-class-of-psi-of-P-1} \begin{enumerate} \item \label{lemma-item-psi-is-an-isomorphism-onto-open-subset-of-E} The divisor $E$ is smooth along the image of $\psi$ and $\psi$ maps $G(1,U)$ isomorphically onto a Zariski open subset of $E$. \item \label{lemma-item-class-of-rational-curve-is-pairing-with-minus-v-0} Let $F$ be an $H$-slope-stable sheaf of class $(2,H^2,0)$ with $h^1(F)=0$. Then $\psi({\mathbb P}{H}^0(F))$ is a rational curve of class $(\theta(-v_0),\bullet)$ in $H_2(M_H(1,H^2,-1),{\mathbb Z})$. \end{enumerate} \end{lem}
\begin{proof} \ref{lemma-item-psi-is-an-isomorphism-onto-open-subset-of-E}) The proof is similar to that of parts 6 and 7 of Proposition 3.18 in \cite{markman-reflections}. Let us first prove that the morphism $\psi$ is injective. Let $Q$ be a sheaf represented by the point $\psi(F,\lambda)$. We know, by construction, that $H^i(F)=0$, for $i>0$, and $h^0(F)=\chi(F)=2$. Hence, $h^0(Q)=1$, $h^1(Q)=1$, and $h^2(Q)=0$. It follows that $\dim{\rm Ext}^1(Q,\StructureSheaf{S})=1$, $F$ is isomorphic to the unique non-trivial extension of $Q$ by $\StructureSheaf{S}$, and $\lambda$ is the kernel of the homomorphism $H^0(F)\rightarrow H^0(Q)$. Hence, $\psi$ is injective.
The image of $\psi$ is Zariski open in $E$, since it is characterized by $\dim {\rm Ext}^1(Q,\StructureSheaf{S})=1$, and by the $H$-slope-stability of the unique non-trivial extension. $G(1,U)$ is clearly smooth. It suffices to construct the inverse of $\psi$ as a morphism. This is done as in the proof of Proposition 3.18 in \cite{markman-reflections}.
\ref{lemma-item-class-of-rational-curve-is-pairing-with-minus-v-0}) Let $w\in K(S)$ be a class orthogonal to $(1,H^2,-1)$. The equality \[ \int_{{\mathbb P}{H}^0(F)}\psi^*(\theta(w)) \ \ \ = \ \ \ -(v_0,w) \] follows by an argument identical to the proof of Lemma 4.11 in \cite{markman-part-two}. It follows that $\psi({\mathbb P}{H}^0(F))$ has class $(\theta(-v_0),\bullet)$ in $H_2(M_H(1,H^2,-1),{\mathbb Z})$. \end{proof}
\begin{lem} The closure $E'$ of the image of $\psi$ is a prime exceptional divisor of class $\theta(-v_0)$. \end{lem}
\begin{proof} $G(1,U)$ is a ${\mathbb P}^1$-bundle over $U$. Hence $G(1,U)$ is irreducible of dimension $2n-1$. The image of $\psi$ is irreducible of dimension $2n-1$, as $\psi$ is injective.
Hence, $E'$ is irreducible. The canonical line bundle of $G(1,U)$ restricts to the fiber ${\mathbb P}{H}^0(F)$ as the canonical line bundle of the fiber, since $U$ is holomorphic-symplectic. The normal of $\psi(G(1,U))$ in $M_H(1,H^2,-1)$ is isomorphic to the canonical line bundle of $\psi(G(1,U))$, by Lemma \ref{lemma-class-of-psi-of-P-1} part \ref{lemma-item-psi-is-an-isomorphism-onto-open-subset-of-E}. Hence, $E'\cdot \psi[{\mathbb P}{H}^0(F)]=-2$, and $E'$ is exceptional, by Theorem \ref{thm-a-uniruled-divisor-is-exceptional} part \ref{thm-item-uniruled-implies-exceptional}. $E'$ is of type $A_1$, by Lemma \ref{lemma-class-of-psi-of-P-1} part \ref{lemma-item-psi-is-an-isomorphism-onto-open-subset-of-E}. The class of $E'$ is $\theta(-v_0)$, by Lemma \ref{lemma-class-of-psi-of-P-1} and Theorem \ref{thm-a-uniruled-divisor-is-exceptional} part \ref{thm-item-ell-is-E-vee}. \end{proof}
\begin{proof} {\bf (of Lemma \ref{lemma-brill-noether-divisor-is-exceptional})} $E$ is an effective divisor of class $\theta(-v_0)$, by definition of $E$. This is also the class of the reduced and irreducible divisor $E'$ supporting a component of $E$. Hence, $E$ is reduced and irreducible. We have the equality $([E],\theta(x))=(\theta(-v_0),\theta(x))=-(v_0,x)$, which is divisible by $2$, for all $x\in (1,H^2,-1)^\perp$, since $(1,H^2,-1)-v_0=2(0,H,-1)$. Hence, ${\rm div}([E],\bullet)=2$. \end{proof}
\subsection{Exceptional divisors of non-locally-free sheaves} In this section we will consider examples of prime exceptional divisors that arise as the exceptional locus for the morphism from the Gieseker-Simpson moduli space of $H$-stable sheaves to the Uhlenbeck-Yau compactification of the moduli space of $H$-slope-stable locally free sheaves. Such divisors on a $2n$-dimensional moduli space seem to have class $e$ or $2e$, where $e$ is a primitive class of degree $(e,e)=2-2n$.
\begin{example} \label{example-diagonal-of-hilbert-scheme} \cite{beauville} The case $n\geq 2$, $(e,e)=2-2n$, ${\rm div}(e,\bullet)=2n-2$, $rs(e)=\{1,n-1\}$, and $[E]=2e$.
\\
Let $S$ be a $K3$ surface, $X:=S^{[n]}$, and $E\subset X$ the big diagonal. Then $[E]=2e$, for a primitive class $e\in H^2(S^{[n]},{\mathbb Z})$, and $(e,e)=2-2n$. Hence $[E]^\vee=e^\vee$, by Corollary \ref{cor-1}.
$E$ is the exceptional locus of the Hilbert-Chow morphism $S^{[n]}\rightarrow S^{(n)}$ onto the $n$-th symmetric product. The symmetric product $S^{(n)}$ has $A_1$ singularities away from its dissident locus. The monodromy-invariant $rs(e)$ is equal to $\{1,n-1\}$, by Example \ref{example-any-factorization-rs-is-possible}. \end{example}
\hide{ \begin{example} The case \[ \{(e,e),[E],[E]^\vee,A_i\}=\{2-2n,e,e^\vee,A_1\}, \ \ \ n>2. \] Let $S$ be a $K3$ surface, $L\in{\rm Pic}(S)$ a line bundle of degree $2n-6$. If $n>3$, assume that $L$ spans ${\rm Pic}(S)$. If $n=3$ assume that we have an elliptic fibration $\pi:S\rightarrow {\mathbb P}^1$ with reduced and irreducible fibers, and $L:=\pi^*\StructureSheaf{{\mathbb P}^1}(1)$. Let $v\in K(S)$ be the class of rank $2$, with $c_1(v)=c_1(L)$, and $\chi(v)=1$. Then $v:=(2,L,-1)$ in Mukai's notation. If $n>3$, set $H:=L$. If $n=3$, choose a $v$-generic polarization $H$. Then the moduli space $M:=M_H(v)$ is smooth, projective, and $2n$-dimensional.
Let $E\subset M$ be the closure of the locus of points representing $H$-stable sheaves $F$, which are not locally free, but such that $F^{**}/F$ has length one. If $F^{**}/F$ has length one and $F$ is $H$-stable, then the reflexive hull $F^{**}$ is necessarily $H$-semi-stable\footnote{The $H$-semi-stability is proven by an easy modification of the proof of \cite{markman-constraints}, Proposition 4.10, Part 1. The assumption that the rank is $2$ is needed.} of class $(2,L,0)\in K(S)$. The irreducibility of $E$ thus follows from that of the moduli space $M_H(2,L,0)$ (for $H$ which is both $v$-generic and $(2,L,0)$-generic). The reflexive hull $F^{**}$, of a generic such $F$, is an $H$-slope-stable locally free sheaf in $M_H(2,L,0)$. Let $Y$ be the normalization of the Uhlenbeck-Yau compactification of the moduli space of locally free $H$-slope stable sheaves. Then $Y$ is a projective variety and there exists a morphism $\phi:M\rightarrow Y$, which exceptional locus contains the divisor $E$ \cite{jun-li}.
Let $U\subset M_H(2,L,0)$ be the locus of $H$-stable locally free sheaves. Choose a twisted universal sheaf ${\mathcal G}$ over $S\times U$. Then $E$ contains a Zariski dense open subset isomorphic to the projectivization of ${\mathcal G}$. We conclude that $A_i(X,E)=A_1$.
We calculate next the class $[E]^\vee\in H_2(M,{\mathbb Z})$. Recall that $[E]^\vee$ is the class of the fiber of $E\rightarrow Y$, by \cite{markman-galois}, Lemma 4.10. Fix an $H$-slope-stable locally free sheaf $G$ of class $(2,L,0)\in K(S)$. Fix a point $P\in S$ and denote by $G_P$ the fiber of $G$ at $P$. Let ${\mathbb P}{G}_P$ be the projectivization of the fiber and \[ 0\rightarrow \StructureSheaf{{\mathbb P}{G}_P}(-1)\rightarrow G_P\otimes\StructureSheaf{{\mathbb P}{G}_P}\rightarrow q_{{\mathbb P}{G}_P} \rightarrow 0 \] the short exact sequence of the tautological quotient bundle $q_{{\mathbb P}{G}_P}$. Let \[ \iota \ : \ {\mathbb P}{G}_P \ \ \ \rightarrow \ \ \ S \times {\mathbb P}{G}_P \] be the morphism given by $\iota(\ell)=(P,\ell)$. Let $\pi_i$ be the projection from $S\times {\mathbb P}{G}_P$ onto the $i$-th factor. Over $S\times {\mathbb P}{G}_P$ we get the short exact sequence \[ 0\rightarrow {\mathcal F} \rightarrow \pi_1^*G\RightArrowOf{j}\iota_*(q_{{\mathbb P}{G}_P}) \rightarrow 0, \] where $j$ is the natural homomorphism, and ${\mathcal F}$ its kernel. Given a point $\ell\in {\mathbb P}{G}_P$, we denote by $\tilde{\ell}\subset G_P$ the corresponding line. The sheaf ${\mathcal F}_\ell$, $\ell\in {\mathbb P}{G}_P$, is the subsheaf of $G$, with local sections whose values at $P$ belong to $\tilde{\ell}$. ${\mathcal F}_\ell$ is $H$-slope-stable, since $G$ is. ${\mathcal F}$ is thus a family of $H$-stable sheaves, flat over ${\mathbb P}{G}_P$. Let \[ \kappa \ : \ {\mathbb P}{G}_P \ \ \ \longrightarrow \ \ \ M_H(2,L,-1) \] be the classifying morphism associated to ${\mathcal F}$. Then $\kappa({\mathbb P}{G}_P)$ is a fiber of $E\rightarrow Y$, and we get the equality of classes in $H_2(M,{\mathbb Z})$ \[ [\kappa({\mathbb P}{G}_P)] \ \ \ = \ \ \ [E]^\vee. \]
Let us compose the Mukai isomorphism $\theta$, given in (\ref{eq-Mukai-isomorphism}), with pull-back by $\kappa$ \[ v^\perp \LongRightArrowOf{\theta} H^2(M_H(v),{\mathbb Z}) \LongRightArrowOf{\kappa^*} H^2({\mathbb P}{G}_P,{\mathbb Z}). \] The composition is given by \[ \kappa^*(\theta(x)) \ \ = \ \ c_1\left\{\pi_{2_!}\left[\pi_1^!(x^\vee)\otimes{\mathcal F}\right]\right\}. \] Let $[{\mathcal F}]$ be the class of ${\mathcal F}$ in $K(S\times{\mathbb P}{G}_P)$. Then $[{\mathcal F}]=\pi_1^![G]-\iota_![q_{{\mathbb P}{G}_P}]$.
We have the equalities \[ c_1\left\{\pi_{2_!}\left(\pi_1^!(x^\vee\otimes G)\right)\right\} \ \ = \ \ 0, \] \begin{eqnarray*} c_1\left\{\pi_{2_!}\left(\pi_1^!(x^\vee)\otimes \iota_!(q_{{\mathbb P}{G}_P}) \right)\right\} &=& c_1\left\{\pi_{2_!}\left({\rm rank}(x)\cdot \iota_!(q_{{\mathbb P}{G}_P})\right)\right\} \\ &=& {\rm rank}(x)c_1(q_{{\mathbb P}{G}_P}). \end{eqnarray*} We conclude that the following equalities hold, for all $x\in v^\perp$. \[ \int_{[E]^\vee}\theta(x)= \int_{{\mathbb P}{G}_P}\kappa^*(\theta(x))=-{\rm rank}(x). \] Now $x=(a,L',b)$ belongs to $v^\perp$, if and only if $a=2b+(L,L')$. $L$ is assumed primitive, so the rank $a$ of $x$ is one, if we choose $L'$, such that $(L,L')=2b+1$. In particular, $[E]^\vee$ is a primitive class in $H_2(M,{\mathbb Z})$.
The class $[E]$ is the unique class in $H^2(M,{\mathbb Z})$ satisfying the equality \[ \frac{-2([E],\theta(x))}{([E],[E])} \ \ = \ \ \int_{[E]^\vee}\theta(x), \] for all $x\in v^\perp$, by Lemma 4.10 in \cite{markman-galois}. We saw that the right hand side is equal to $-{\rm rank}(x)$. Set $w:=(2,L,n-2)$. The equality \[ \frac{-2(w,x)}{(w,w)} \ \ \ = \ \ \ {\rm rank}(x) \] is verified by a direct calculation, for all $x\in v^\perp$. The equality $[E]=\nolinebreak-\theta(w)$ follows. Hence, $([E],[E])=(\theta(w),\theta(w))=(w,w)=2-2n$. The class $[E]$ is primitive, since $w$ is. \end{example}
}
The following result will be needed in the next example.
\begin{lem} \label{lemma-class-of-a-contractible-rational-curve} Let $S$ be a $K3$ surface, ${\mathcal L}$ a line bundle on $S$, $v=(r,{\mathcal L},s)$ a class in $K(S)$, satisfying $(v,v)\geq 2$, and $r\geq 2$. Let $H$ be a $v$-generic polarization. Assume given an $H$-slope-stable sheaf $G$ of class $(r,{\mathcal L},s+1)$ and a point $P\in S$, such that $G$ is locally free at $P$. For each $2$-dimensional quotient $Q$ of the fiber $G_P$, there exists a natural embedding \[ \kappa \ : \ {\mathbb P}{Q} \ \ \ \longrightarrow \ \ \ M_H(v), \] whose image $C:=\kappa({\mathbb P}{Q})$ is a smooth rational curve satisfying \[ \int_C\theta(x) \ \ \ = \ \ \ (w,x) \ \ \ = \ \ \ -{\rm rank}(x), \] for all $x\in v^\perp$, where $w$ is the following rational class in $v^\perp$. \begin{equation} \label{eq-class-wich-pair-with-lambda-to-minus-rank-lambda} w \ \ := \ \ \frac{r}{(v,v)}v+(0,0,1) \ \ = \ \ \frac{1}{(v,v)}\left(r^2,r{\mathcal L},sr+(v,v)\right). \end{equation}
\end{lem}
\begin{proof} Consider the short exact sequence of the tautological quotient bundle $q_{{\mathbb P}{Q}}$ over ${\mathbb P}{Q}$ \[ 0\rightarrow \StructureSheaf{{\mathbb P}{Q}}(-1)\rightarrow Q\otimes\StructureSheaf{{\mathbb P}{Q}}\rightarrow q_{{\mathbb P}{Q}} \rightarrow 0. \] Let \[ \iota \ : \ {\mathbb P}{Q} \ \ \ \rightarrow \ \ \ S \times {\mathbb P}{Q} \] be the morphism given by $\iota(\ell)=(P,\ell)$. Let $\pi_i$ be the projection from $S\times {\mathbb P}{Q}$ onto the $i$-th factor. Over $S\times {\mathbb P}{Q}$ we get the short exact sequence \[ 0\rightarrow {\mathcal F} \rightarrow \pi_1^*G\RightArrowOf{j}\iota_*(q_{{\mathbb P}{Q}}) \rightarrow 0, \] where $j$ is the natural homomorphism, and ${\mathcal F}$ its kernel. Given a point $\ell\in {\mathbb P}{Q}$, we denote by $\tilde{\ell}\subset G_P$ the corresponding hyperplane. The sheaf ${\mathcal F}_\ell$, $\ell\in {\mathbb P}{Q}$, is the subsheaf of $G$, with local sections whose values at $P$ belong to $\tilde{\ell}$. ${\mathcal F}_\ell$ is $H$-slope-stable, since $G$ is. ${\mathcal F}$ is thus a family of $H$-stable sheaves, flat over ${\mathbb P}{Q}$. Let $\kappa:{\mathbb P}{Q}\rightarrow M_H(v)$ be the classifying morphism associated to ${\mathcal F}$. The morphism $\kappa$ is clearly injective. An elementary calculation verifies that the differential $d_\ell\kappa$ is injective.
\hide{ Indeed, the differential $d_\ell\kappa$ is the composition \[ {\rm Hom}(\ell,Q/\ell)\subset {\rm Hom}(\tilde{\ell},Q/\ell) \subset {\rm Hom}(F_\ell,(Q/\ell)_P)\rightarrow {\rm Ext}^1(F_\ell,F_\ell), \] where $(Q/\ell)_P$ is the sky-scraper sheaf supported at $P$ with fiber $Q/\ell$. The rightmost homomorphism is the connecting homomorphism in the long exact sequence obtained by applying the functor ${\rm Hom}(F_\ell,\bullet)$ to the short exact sequence $ 0\rightarrow F_\ell\rightarrow G\rightarrow (Q/\ell)_P\rightarrow 0. $ Each of the homomorphisms in the above composition is clearly injective.
}
Let us compose the Mukai isomorphism $\theta$, given in (\ref{eq-Mukai-isomorphism}), with pull-back by $\kappa$ \[ v^\perp \LongRightArrowOf{\theta} H^2(M_H(v),{\mathbb Z}) \LongRightArrowOf{\kappa^*} H^2({\mathbb P}{Q},{\mathbb Z}). \] The composition is given by \[ \kappa^*(\theta(x)) \ \ = \ \ c_1\left\{\pi_{2_!}\left[\pi_1^!(x^\vee)\otimes{\mathcal F}\right]\right\}. \] Let $[{\mathcal F}]$ be the class of ${\mathcal F}$ in $K(S\times{\mathbb P}{Q})$. Then $[{\mathcal F}]=\pi_1^![G]-\iota_![q_{{\mathbb P}{Q}}]$.
We have the equalities \[ c_1\left\{\pi_{2_!}\left(\pi_1^!(x^\vee\otimes G)\right)\right\} \ \ = \ \ 0, \] \begin{eqnarray*} c_1\left\{\pi_{2_!}\left(\pi_1^!(x^\vee)\otimes \iota_!(q_{{\mathbb P}{Q}}) \right)\right\} &=& c_1\left\{\pi_{2_!}\left({\rm rank}(x)\cdot \iota_!(q_{{\mathbb P}{Q}})\right)\right\} \\ &=& {\rm rank}(x)c_1(q_{{\mathbb P}{Q}}). \end{eqnarray*} We conclude that the following equality holds, for all $x\in v^\perp$. \[ \int_{{\mathbb P}{Q}}\kappa^*(\theta(x))=-{\rm rank}(x). \]
A direct calculation verifies that the class $w$, given in (\ref{eq-class-wich-pair-with-lambda-to-minus-rank-lambda}), is orthogonal to $v$ and satisfies $(w,x)=-{\rm rank}(x)$, for all $x\in v^\perp$. \end{proof}
\begin{example} Let $S$ be a $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Let $b$ be an odd integer, such that there exists a line bundle ${\mathcal L}\in{\rm Pic}(S)$ of degree $2n-4b-2$, where $n>2$. If $c_1({\mathcal L})$ is divisible by $2$, assume that $n>3$. Let $v\in K(S)$ be the class $(2,{\mathcal L},-b)$ in Mukai's notation. Then $(v,v)=2n-2$ and
the moduli space $M:=M_H(v)$ is smooth, projective, and $2n$-dimensional.
Let $E\subset M$ be the closure of the locus of points representing $H$-stable sheaves $F$, which are not locally free, but such that $F^{**}/F$ has length one.
Let $Y$ be the normalization of the Uhlenbeck-Yau compactification of the moduli space of locally free $H$-slope stable sheaves of class $v$. Then $Y$ is a projective variety and there exists a morphism $\phi:M\rightarrow Y$ whose exceptional locus contains $E$ \cite{jun-li}. \end{example}
\begin{lem} \label{lemma-class-of-exceptional-locus} $E$ is a prime exceptional divisor of type $A_1$. The class $[E]\in H^2(M,{\mathbb Z})$ is the primitive class $e:=\theta(2,{\mathcal L},n-b-1)$. In particular, $(e,e)=2-2n$. \begin{enumerate} \item \label{lemma-item-L-not-divisible-by-2} If the class $c_1({\mathcal L})$ is not divisible by $2$, then ${\rm div}(e,\bullet)=n-1$, \[ rs(e)=\left\{ \begin{array}{ccl} \{1,n-1\}, & \mbox{if}& n \ \mbox{is even}, \\ \{1,(n-1)/2\}, & \mbox{if}& n \ \mbox{is odd}. \end{array}\right. \]
\item \label{lemma-item-L-divisible-by-2} If the class $c_1({\mathcal L})$ is divisible by $2$, then $n\equiv 3$ (modulo $4$), ${\rm div}(e,\bullet)=2n-2$, $rs(e)=\{2,(n-1)/2\}$.
\end{enumerate} \end{lem}
\begin{proof} When the class $c_1({\mathcal L})$ is divisible by $2$ then $n\equiv 3$ (modulo $4$), since $\deg({\mathcal L})=2n-4b-2$ is divisible by $8$. In that case $n\geq 7$, by assumption.
If $F^{**}/F$ has length one and $F$ is $H$-stable, then the reflexive hull $F^{**}$ is necessarily $H$-semi-stable\footnote{The $H$-semi-stability is proven by an easy modification of the proof of \cite{markman-constraints}, Proposition 4.10, Part 1. The assumption that the rank is $2$ is needed. } of class $u:=(2,{\mathcal L},1-b)\in K(S)$. $M_H(u)$ is irreducible of dimension $2n-4$ and its generic point represents a locally free $H$-slope-stable sheaf.
This is clear if $c_1({\mathcal L})$ is not divisible by $2$. If $c_1({\mathcal L})$ is divisible by $2$, this follows from Lemma \ref{lemma-existence-of-slope-stable-vb-of-rank-2} and
the assumption that $n\geq 7$.
Let $U\subset M_H(u)$ be the locus of $H$-stable locally free sheaves. Choose a twisted universal sheaf ${\mathcal G}$ over $S\times U$. Then $E$ contains a Zariski dense open subset isomorphic to the projectivization of ${\mathcal G}$. $E$ is irreducible, since
the moduli space $M_H(u)$ is irreducible. We conclude that $E$ is a prime exceptional divisor, since it is contracted by the morphism to the Uhlenbeck-Yau compactification (also by Theorem \ref{thm-a-uniruled-divisor-is-exceptional}). Furthermore, $E$ is of type $A_1$.
We calculate
next the class $[E]^\vee\in H_2(M,{\mathbb Z})$, given in equation (\ref{eq-E-vee}). Recall that $[E]^\vee$ is the class of the fiber of $E\rightarrow Y$, by Corollary \ref{cor-1}.
Fix an $H$-slope-stable locally free sheaf $G$ of class $u\in K(S)$. Fix a point $P\in S$ and denote by $G_P$ the fiber of $G$ at $P$. Let ${\mathbb P}{G}_P$ be the projectivization of the fiber
and denote by \[ \kappa \ : \ {\mathbb P}{G}_P \ \ \ \longrightarrow \ \ \ M_H(v) \]
the morphism given in Lemma \ref{lemma-class-of-a-contractible-rational-curve}. Then $\kappa({\mathbb P}{G}_P)$ is a fiber of $E\rightarrow Y$, and we get the equality $ [\kappa({\mathbb P}{G}_P)] = [E]^\vee $ of classes in $H_2(M,{\mathbb Z})$.
We conclude that the following equalities hold, for all $x\in v^\perp$. \begin{equation} \label{eq-integral-equal-minus-rank-and-expressed-in-term-of-w} \int_{[E]^\vee}\theta(x)=
-{\rm rank}(x)=\frac{1}{n-1}(w,x), \end{equation} where $w=(2,{\mathcal L},n-b-1)$, by Lemma \ref{lemma-class-of-a-contractible-rational-curve}.
The class $[E]$ is the unique class in $H^2(M,{\mathbb Z})$ satisfying the equality \[ \frac{-2([E],\theta(x))}{([E],[E])} \ \ = \ \ \int_{[E]^\vee}\theta(x), \] for all $x\in v^\perp$, by Corollary \ref{cor-1}.
Now $(w,w)=2-2n$ and so the equality
$[E]=\nolinebreak\theta(w)$ follows from equation (\ref{eq-integral-equal-minus-rank-and-expressed-in-term-of-w}). Hence, $([E],[E])=(\theta(w),\theta(w))=2-2n$. The class $[E]$ is primitive, since $w$ is.
Let us calculate ${\rm div}(e,\bullet)$.
The class $x=(\rho,{\mathcal L}',\sigma)$ belongs to $v^\perp$, if and only if $b\rho=2\sigma-({\mathcal L},{\mathcal L}')$. Hence, $(e,\theta(x))=(w,x)=(1-n)\rho$. If $c_1({\mathcal L})$ is divisible by $2$, then every integral class $x\in v^\perp$ has even rank $\rho$, and so ${\rm div}(e,\bullet)=2n-2$. If $c_1({\mathcal L})$ is not divisible by $2$, choose a line bundle ${\mathcal L}'$, such that $({\mathcal L},{\mathcal L}')$ is odd and set $\sigma:=[({\mathcal L},{\mathcal L}')+b]/2$. Then $(1,{\mathcal L}',\sigma)$ belongs to $v^\perp$. Hence, ${\rm div}(e,\bullet)=n-1$.
The pair $(\widetilde{L},e)$, given in equation (\ref{eq-saturation-of-L}), may be chosen to consist of the saturation $\widetilde{L}$ in $K(S)$ of the lattice spanned by the classes $v$ and $w=\theta^{-1}(e)$, by Theorem
\ref{thm-item-orbit-of-inverse-of-Mukai-isom-is-natural}. The largest integer dividing $w-v=(0,0,n-1)$ is $\sigma:=n-1$. Now $w+v=(4,2{\mathcal L},n-1-2b)$. The largest integer $\rho$ dividing $w+v$ is $4$, if $c_1({\mathcal L})$ is divisible by $2$. Otherwise, $\rho=1$, if $n$ is even, and $\rho=2$, if $n$ is odd. The invariant $rs(e)$ is then calculated via the table after Lemma
\ref{lemma-isometry-orbits-in-rank-2}.
\end{proof}
\section{Examples of non-effective monodromy-reflective classes} \label{sec-non-effective}
We provide examples of monodromy-reflective classes, which are not ${\mathbb Q}$-effective. Observation \ref{observation-not-Q-effective} guides us to lift these reflections to birational self-maps. Let us first prove the observation. \begin{proof} (of Observation \ref{observation-not-Q-effective}) There exists a Zariski open subset $U\subset X$, such that $X\setminus U$ has codimension $\geq 2$, $\iota$ restricts to a regular involution of $U$, and the composition $ H^2(X,{\mathbb Z})\cong H^2(U,{\mathbb Z})\LongRightArrowOf{\iota^*} H^2(U,{\mathbb Z})\cong H^2(X,{\mathbb Z}) $ is an isometry, by \cite{ogrady-weight-two}, Proposition 1.6.2. The isometry $\iota^*$ is assumed to be the reflection $R_{e}$. Hence, $\iota^*L\cong L^{-1}$ and $L$ is not ${\mathbb Q}$-effective. \end{proof}
Let $S$ be a projective $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Set $h:=c_1(H)\in H^2(S,{\mathbb Z})$. Set $d:=\deg(H)/2$.
\subsection{Non-effective classes of divisibility ${\rm div}(e,\bullet)=2n-2$} \hspace{1ex}\\
Let $r$, $s$ be integers satisfying $s>r>2$ and $\gcd(r,s)=1$. Set $n:=rs+1$. Set $v:=(r,0,-s)$, $e:=\theta(r,0,s)$, and $M:=M_H(v)$. $M$ is smooth and projective of dimension $2n$. Let ${\mathcal L}\in{\rm Pic}(M)$ be the line bundle with class $e$.
Let $Exc\subset M$ be the locus of $H$-stable sheaves of class $v$, which are not locally free or not $H$-slope-stable. $Exc$ is a closed subset of co-dimension $\geq 2$ in $M$, by Lemma \ref{lem-codimension-of-Exc}. Let $M^0$ be the complement $M\setminus Exc$ of $Exc$ and $\eta:M^0\rightarrow M$ the inclusion. The restriction homomorphism $\eta^*:H^2(M,{\mathbb Z})\rightarrow H^2(M^0,{\mathbb Z})$ is an isomorphism.
Let $\phi:M^0\rightarrow M^0$ be the involution sending a point representing the sheaf $F$, to the point representing $F^*$. Set $\psi:=(\eta^*)^{-1}\circ \phi^*\circ \eta^*$.
\begin{prop} \label{prop-vanishing-in-divisibility-2n-2} \begin{enumerate} \item \label{lemma-item-clear} The class $e$ is monodromy-reflective, $(e,e)=2-2n$, ${\rm div}(e,\bullet)=2n-2$, and $rs(e)=\{r,s\}$. \item \label{lemma-item-R-e-is-minus-dualization} Let $R_e:H^2(M,{\mathbb Z})\rightarrow H^2(M,{\mathbb Z})$ be the reflection by $e$. Then $R_e(\theta(\lambda))=-\theta(\lambda^\vee)$, for all $\lambda\in v^\perp$. \item \label{lemma-item-psi-is-R-e} $\psi = R_e.$ \item \label{lemma-item-vanishing-in-divisibility-2n-2} $H^0(M,{\mathcal L}^k)$ vanishes, for all non-zero integral powers $k$. \end{enumerate} \end{prop}
\begin{proof} Part \ref{lemma-item-clear} was proven in Example \ref{example-any-factorization-rs-is-possible}. Set $\tilde{e}:=(r,0,s)\in v^\perp$. Part \ref{lemma-item-R-e-is-minus-dualization} follows from the fact that $\theta:v^\perp\rightarrow H^2(M,{\mathbb Z})$ is an isometry, and the equality $R_{\tilde{e}}(\lambda)=-\lambda^\vee$, for all $\lambda\in v^\perp$. Part \ref{lemma-item-vanishing-in-divisibility-2n-2} follows from part \ref{lemma-item-psi-is-R-e}, via Observation \ref{observation-not-Q-effective}. We proceed to prove part \ref{lemma-item-psi-is-R-e}. We need to prove the equality $\phi^*(\eta^*(y))=\eta^*R_e(y)$, for all $y\in H^2(M,{\mathbb Z})$.
Let $\pi_i$ be the projection from $S\times M^0$ onto the $i$-th factor, $i=1,2$. Let ${\mathcal F}$ be a universal sheaf over $S\times M$, ${\mathcal G}$ its restriction to $S\times M^0$, and $[{\mathcal G}]$ its class in $K(S\times M^0)$. The morphism $\phi:M^0\rightarrow M^0$ satisfies \[ (id\times \phi)^!{\mathcal G} \cong ({\mathcal G}\otimes \pi_2^*A)^*, \] for some line bundle $A\in {\rm Pic}(M^0)$. We have the commutative diagram \[ \begin{array}{ccc} K(S\times M)&\LongRightArrowOf{(id\times \eta)^!}& K(S\times M^0) \\ \pi_{2_!} \ \downarrow \ \hspace{2ex} & & \hspace{2ex} \ \downarrow \ \pi_{2_!} \\ K(M) & \LongRightArrowOf{\eta^!} & K(M^0), \end{array} \] by the K\"{u}nneth Theorem \cite{atiyah-book}. Hence, \[ \eta^*\theta(x)= c_1\left[\pi_{2_!}\left(\pi_1^!(x^\vee)\otimes[{\mathcal G}]\right)\right], \] for all $x\in v^\perp\subset K(S)$. This explains the first equality below. \begin{eqnarray*} \phi^*(\eta^*\theta(x))&=& c_1\left\{\phi^!\pi_{2_!}\left(\pi_1^!(x^\vee)\otimes[{\mathcal G}]\right)\right\} \\ &=& c_1\left\{\pi_{2_!} \left(\pi_1^!(x^\vee)\otimes(id\times \phi)^![{\mathcal G}]\right)\right\} \\ &=& c_1\left\{\pi_{2_!}\left(\pi_1^!(x^\vee)\otimes([{\mathcal G}]\otimes \pi_2^![A])^\vee \right)\right\} \\ &=& -c_1\left\{\pi_{2_!}\left(\pi_1^!(x)\otimes([{\mathcal G}]\otimes \pi_2^![A]) \right)\right\} \\ &=& \eta^*\theta(-x^\vee) \ \ = \ \ \eta^*(R_e(\theta(x))). \end{eqnarray*} The fourth equality follows from Grothendieck-Verdier duality, the fifth is due to the fact that $\theta$ is independent of the choice of a universal sheaf, and the sixth follows from part \ref{lemma-item-R-e-is-minus-dualization}.
\end{proof}
\subsection{Non-effective classes of divisibility ${\rm div}(e,\bullet)=n-1$} \hspace{1ex}\\ Let $r$ be a positive odd integer, $\sigma$ a positive integer, and set $n:=r\sigma+1$. Assume that $r\geq 3$, $\sigma\geq 3$, and $\gcd(r,\sigma)=1$. Let $S$ be a $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Set $d:=\deg(H)/2$. Choose $(S,H)$, so that $\sigma$ and $d$ have the same parity. If $d$ is odd, assume that $\sigma>r$, possibly after interchanging $r$ and $\sigma$. Set $h:=c_1(H)$ and \[ v \ \ \ := \ \ \ (2r,rh,-b), \] where $b:=[\sigma-rd]/2$. Note that $\gcd(r,b)=\gcd(r,\sigma)=1$. Hence, $v$ is a primitive class in $K(S)$, $(v,v)=2n-2$, and the moduli space $M_H(v)$ is smooth and projective of type $K3^{[n]}$. Let $Exc\subset M_H(v)$ be the locus parametrizing sheaves $F$, which are not locally free or not $H$-slope-stable. $Exc$ is an algebraic subset of co-dimension $\geq 2$ in $M_H(v)$, by Lemma \ref{lemma-Exc-has-codimension-at-least-2-again}.
Let $M^0$ be the complement $M\setminus Exc$ and $\eta:M^0\rightarrow M$ the inclusion. Let $\phi:M^0\rightarrow M^0$ be the involution sending a point $[F]$, representing the sheaf $F$, to the point representing $F^*\otimes H$. The homomorphism $\eta^*:H^2(M,{\mathbb Z})\rightarrow H^2(M^0,{\mathbb Z})$ is an isomorphism, by Lemma \ref{lemma-Exc-has-codimension-at-least-2-again}. Set $\psi:=(\eta^*)^{-1}\circ \phi^*\circ \eta^*$.
Set $e:=(2r,rh,\sigma-b)$. Note that $\gcd(r,\sigma-b)=\gcd(r,2\sigma-2b)=\gcd(r,\sigma+rd)= \gcd(r,\sigma)=1$. Hence, $e$ is a primitive class in $v^\perp$ of degree $(e,e)=2-2n$.
\begin{lem} \label{lemma-Type-B-divisibility-n-1} \hspace{1ex} \begin{enumerate} \item \label{lemma-item-e-is-monodromy-reflective-of-divisibility-n-1} The class $\theta(e)$ is monodromy-reflective and ${\rm div}(\theta(e),\bullet)=n-1$. \item \label{lemma-item-s-versus-sigma} $\widetilde{L}$ and $rs(e) \ \ = \ \ \left\{ \begin{array}{cl} H_{ev} \ \mbox{and} \ \{r,\sigma\} & \mbox{if} \ \sigma \ \mbox{is odd} \ (n \ \mbox{even}), \\ U(2) \ \mbox{and} \ \{r,\sigma/2\} & \mbox{if} \ \sigma \ \mbox{is even} \ (n \ \mbox{odd}). \end{array} \right. $ \item \label{lemma-item-R-e-dualize-and-then-tensorize-with-H} Let $R_e:v^\perp\rightarrow v^\perp$ be the reflection by $e$. Then $ R_e(\lambda) = -[\lambda^\vee]\otimes\nolinebreak H, $ for all $\lambda\in v^\perp$. \item \label{lemma-item-psi-equal-R-e-once-again} $\psi=R_e$. \item \label{lemma-item-e-of-type-B-again} Let ${\mathcal L}$ be the line bundle on $M_H(v)$ with $c_1({\mathcal L})=\theta(e)$. Then $H^0(M_H(v),{\mathcal L}^k)$ vanishes, for all non-zero integral powers $k$. \end{enumerate} \end{lem}
\begin{proof} \ref{lemma-item-e-is-monodromy-reflective-of-divisibility-n-1}) Let $\lambda:=(x,c,y)\in K(S)$. Then $\lambda$ belongs to $v^\perp$, if and only if $ r(h,c)-2ry+bx=0. $ In particular, $x$ is divisible by $r$, since $\gcd(r,b)=1$. Now $(\lambda,e)=(\lambda,v)-x\sigma=-x\sigma$. Thus $(e,\lambda)$ is divisible by $r\sigma=n-1$. There exists a class $c\in H^2(S,{\mathbb Z})$, satisfying $(c,h)=-b$, since the class $h$ is primitive and $H^2(S,{\mathbb Z})$ is unimodular. Then the class $\lambda:=(r,c,0)$ belongs to $v^\perp$, and $(e,\lambda)=-r\sigma=1-n$. Hence, ${\rm div}(\theta(e),\bullet)=n-1$. The class $\theta(e)$ is monodromy-reflective, by Proposition \ref{prop-reflection-by-a-numerically-prime-exceptional-is-in-Mon}.
\ref{lemma-item-s-versus-sigma}) If $\sigma$ is odd, then $n=r\sigma+1$ is even, and $\widetilde{L}\cong H_{ev}$, by Lemma \ref{lem-non-unimodular-rank-two-lattice}. Now $(e-v)/\sigma=(0,0,1)$ is primitive. Hence, $rs(e)=\{r,\sigma\}$, by Lemma \ref{lemma-isometry-orbits-in-rank-2}. If $\sigma$ is even, then $n$ is odd and $d$ is even. The classes $\alpha:=(e-v)/\sigma=(0,0,1)$ and $\beta:=(e+v)/2r=(2,h,d/2)$ are integral isotropic classes and $(\alpha,\beta)=-2$. Hence, $\{\alpha,\beta\}$ spans the primitive sublattice $\widetilde{L}\cong U(2)$. Consequently, $rs(e)=\{r,\sigma/2\}$, by Lemma \ref{lemma-isometry-orbits-in-rank-2}.
Part \ref{lemma-item-R-e-dualize-and-then-tensorize-with-H} is verified by a straightforward calculation. Part \ref{lemma-item-psi-equal-R-e-once-again} follows from part \ref{lemma-item-R-e-dualize-and-then-tensorize-with-H} by the same argument used in the proof of Proposition \ref{prop-vanishing-in-divisibility-2n-2}. Part \ref{lemma-item-e-of-type-B-again} follows from part \ref{lemma-item-psi-equal-R-e-once-again}, by Observation \ref{observation-not-Q-effective}. \end{proof}
\begin{example} \label{example-non-effective-divisibility-n-1-and-n-is-cong-1-mod-8} Let $r$ and $s$ be positive integers satisfying $s>r$, one of $r$ or $s$ is even, and $\gcd(r,s)=1$.
Set $n=4rs+1$. Note that $n\equiv 1$ (modulo $8$). Let $S$ be a $K3$ surface with a cyclic Picard group generated by an ample line bundle $H$. Set $h:=c_1(H)$ and $d:=(h,h)/2$. Assume that $d$ is odd. Then $s+rd$ is odd, since $r$ and $s$ consist of one odd and one even integer, by assumption. Set $v:=(4r,2rh,-s+rd)$ and $e:=(4r,2rh,s+rd)$. Then the classes $v$ and $e$ are primitive, $(e,v)=0$, $(v,v)=2n-2$ and $(e,e)=2-2n$. We have $(e+v)/2r=(4,2h,d)$ and $(e-v)/2s=(0,0,1)$. We claim that ${\rm div}(e,\bullet)=n-1$. Indeed, if $\lambda=(x,c,y)$ belongs to $v^\perp$, then $2r$ divides $x$. Hence, $(e,\lambda)=(v,\lambda)-2sx=-2sx$ is divisible by $4rs=n-1$. Furthermore, let $c$ be a class in $H^2(S,{\mathbb Z})$ satisfying $(c,h)=rd-s$, and set $\lambda=(2r,c,0)$. Then $(\lambda,e)=1-n$. Hence, ${\rm div}(e,\bullet)=n-1$.
Hence, $\widetilde{L}=H_{ev}$, by Proposition \ref{prop-isometry-class-of-tilde-L-e-is-a-faithful-mon-invariant}. Thus $rs(e)=\{r,s\}$, by Lemma \ref{lemma-isometry-orbits-in-rank-2}. Let $Exc\subset M_H(v)$ be the locus parametrizing sheaves which are not locally-free or not $H$-slope-stable. $Exc$ is an algebraic subset of co-dimension $\geq 2$ in $M_H(v)$, by the proof of the odd $d$ case of Lemma \ref{lemma-Exc-has-codimension-at-least-2-again} (the symbol $r$ in that proof should remain ${\rm rank}(v)/2$ and so should be set equal to twice the symbol $r$ above, and similarly the symbol $s$ in that proof should be set equal to twice the symbol $s$ above). Let ${\mathcal L}$ be the line bundle over $M_H(v)$ with $c_1({\mathcal L})=\theta(e)$. We conclude that $H^0(M_H(v),{\mathcal L}^k)$ vanishes, for all non-zero integers $k$, by the same argument used to prove Lemma \ref{lemma-Type-B-divisibility-n-1}. \end{example}
\hide{
\section{Applications of the divisorial Zariski decomposition} \label{sec-zariski-decomposition} We review the definition of the Zariski-decomposition for effective divisors on irreducible holomorphic symplectic projective manifolds. An effective divisor $D$, of negative Beauville-Bogomolov degree, which is not exceptional (Definition \ref{def-rational-exceptional}), must have a non-trivial Zariski-decomposition.
Next we provide an example of a non-exceptional effective divisor $D$, of negative Beauville-Bogomolov degree, on the Hilbert scheme $S^{[7]}$ of a $K3$ surface $S$ (Example \ref{example-effective-non-prime-divisors-of-negative-degree}). The example is particularly interesting since the reflection $R_{[D]}$, with respect to the class of $D$, is an integral isometry of $H^2(S^{[7]},{\mathbb Z})$, which does not belong to $Mon^2(S^{[7]})$.
The existence, of the divisorial Zariski decomposition, implies the emptiness of linear systems $\linsys{L}$ for line bundles $L$ with negative Beauville-Bogomolov degree, on a projective $X$ which does not contain exceptional classes. We prove a weak analogue of this result for non-projective irreducible holomorphic symplectic manifolds (Lemma \ref{lemma-generic-vanishing}).
We recall first Boucksom's divisorial Zariski decomposition in the case of a projective irreducible holomorphic symplectic manifold $X$. The {\em pseudo-effective cone} ${\rm Pef}(X)$ is the closure in $H^{1,1}(X,{\mathbb R})$ of the cone of effective divisors. \begin{defi} \label{def-divizorial-Zariski-decomposition} \cite{boucksom} Let $D$ be a rational divisor on $X$. The {\em divisorial Zariski decomposition}\footnote{This definition applies only to the special case of irreducible holomorphic symplectic manifolds. It depends on the fact that the {\em modified nef come}, introduced by Boucksom, coincides with ${\rm Pef}(X)^*$ in this case (\cite{boucksom}, Proposition 4.4). } of $D$ is the unique decomposition as a sum \[ D \ \ \ = \ \ \ P(D)+N(D), \] with $N(D)$ either zero or an exceptional ${\mathbb Q}$-divisor (Definition \ref{def-rational-exceptional}), and $P(D)$ belongs to the cone ${\rm Pef}(X)^*$, dual to the pseudo-effective cone with respect to the Beauville-Bogomolov pairing. \end{defi} The existence and uniqueness of the divisorial Zariski decomposition, of every effective ${\mathbb Q}$-divisor on $X$, is established in \cite{boucksom}. Boucksom proves an analogue of this decomposition without assuming that $X$ is projective.
\begin{example} \label{example-effective-non-prime-divisors-of-negative-degree} Let $S$ be a $K3$ surface, which is the double cover of ${\mathbb P}^2$ branched over a smooth sextic. Let $L$ be the corresponding line bundle of degree $2$ on $S$. Set $X:=S^{[7]}$ and let $d\in H^2(S^{[7]},{\mathbb Z})$ be half the class of the big diagonal. Let $e\in H^2(S^{[7]},{\mathbb Z})$ be the class \[
e \ := \ 2c_1(L)+d, \] using the orthogonal direct sum decomposition (\ref{eq-orthogonal-direct-sum}). Then $(e,e)=-4$ and so $-12<(e,e)<-2$. Thus $e$ is not monodromy-reflective. Let $R\in OH^2(S^{[7]},{\mathbb Z})$ be the reflection $ R(x) = x-\frac{2(e,x)}{(e,e)}e $ by $e$. $R$ is an integral isometry, since $(e,x)$ is even, for all $x\in H^2(S^{[7]},{\mathbb Z})$. Nevertheless, $R$ is not a monodromy operator (\cite{markman-constraints}, Example 4.8).
We observe finally that $2e$ is effective. Choose a smooth curve $C\subset S$ in $\linsys{L^4}$. Denote by $\widetilde{C}$ the divisor in $S^{[7]}$ consisting of ideal sheaves of length $7$ subschemes with non-empty intersection with $C$. Let $\Delta$ be the big diagonal in $S^{[7]}$. Then the class of $D:=\widetilde{C}+\Delta$ is $2e$. Note that this decomposition of $D$
is the divisorial Zariski decomposition. \end{example}
\begin{lem} \label{lem-vanishing-in-the-absence-of-numerically-exceptional-classes} Let $X$ be a projective\footnote{We need the projectivity assumption, since we used it in Corollary \ref{cor-1}. We do not know that a prime exceptional divisor $E\subset X$ is monodromy reflective, if $X$ is not projective. } symplectic manifold of $K3^{[n]}$-type, $n\geq 2$. Assume that ${\rm Pic}(X)$ does not contain any monodromy-reflective class (Definition \ref{def-monodromy-reflective}). Then $H^0(X,L)$ vanishes, for every line bundle $L$ on $X$ with negative Beauville-Bogomolov degree. \end{lem}
\begin{proof} A proof by contradiction. Assume that $L$ has negative Beauville-Bogomolov degree and $D$ is a non-zero divisor in $\linsys{L}$. Let $D=P(D)+N(D)$ be its Zariski decomposition.
The prime divisors $D_i$ in the support of $N(D)$ are exceptional. Hence, their classes $[D_i]\in H^2(X,{\mathbb Z})$ are multiples of monodromy-reflective classes, by Corollary \ref{cor-1}. Our assumption implies that $D=P(D)$. Then $([D],[D])=([D],[P(D)])\geq 0$. A contradiction.
\end{proof}
\begin{lem} \label{lemma-generic-vanishing} Fix a non-zero integer $k$. $H^0(Y,B^k)$ vanishes for a generic pair $(Y,B)$, of an irreducible holomorphic symplectic manifold $Y$ of $K3^{[n]}$-type and a line bundle $B$ of negative degree, such that $c_1(B)$ is a primitive class in $H^2(Y,{\mathbb Z})$ and $B$ is not monodromy-reflective. \end{lem}
\begin{proof} The proof is by contradiction. Assume that there exists a non-zero integer $k$, such that $H^0(Y,B^k)$ does not vanish, for a generic such pair $(Y,B)$. Then $H^0(X,A^k)$ does not vanish for any pair $(X,A)$, deformation equivalent to the pair $(Y,B)$, in the sense of Definition \ref{def-deformation-equivalent-pairs-with-effective-divisors}, by the Semi-Continuity Theorem.
We will prove next the vanishing for some pair $(X,A)$, deformation equivalent to the pair $(Y,B)$, obtaining a contradiction. There exists a pair $(X,A)$, deformation equivalent to $(Y,B)$, such that ${\rm Pic}(X)$ is a free abelian group of rank $2$, generated by $A$ and a line bundle $L$, with $\ell:=c_1(L)$ of positive degree $(\ell,\ell)=-(a,a)k^2$, where $a:=c_1(A)$, $k$ is an integer larger than $2n-2$, and $(a,\ell)=0$.
Such $X$ is projective, by Huybrechts projectivity criterion \cite{huybrects-basic-results}.
We claim that ${\rm Pic}(X)$ does not contain any class of degree $-2$ or $2-2n$. The degree of every class in ${\rm Pic}(X)$ is divisible by $(a,a)$, and thus ${\rm Pic}(X)$ does not contain $-2$ classes. For $xa+y\ell$ to have degree $2-2n$, the equation \begin{equation} \label{eq-inconsistent} 2-2n \ \ = \ \ (a,a)[x^2-(yk)^2] \ \ = \ \ (a,a)(x-yk)(x+yk) \end{equation} should have as integral solution $(x,y)$. If $(a,a)=2-2n$ and $(x,y)=(\pm1,0)$, then the class $xa+y\ell$ is not monodromy-reflective, by assumption. We may thus assume that $y\neq 0$. The right hand side of equation (\ref{eq-inconsistent}) is either zero, or it has absolute value larger than $2n-2$. Hence, ${\rm Pic}(X)$ does not contain any monodromy-reflective class. The vanishing of $H^0(X,A^k)$ follows from Lemma \ref{lem-vanishing-in-the-absence-of-numerically-exceptional-classes}. \end{proof}
}
\hide{
\section{Appendix: A calculation of an ample cone} \label{sec-proof-of-lemma-ample-cone} We prove Lemma \ref{lemma-ample-cone} in this section. We keep the notation of the Lemma. Part \ref{lemma-item-f-is-nef} is straightforward. We prove part \ref{lemma-item-e-and-f-generate-the-Pef-cone} next. Let $D$ be a prime divisor, whose class $xe+yf$ is not a scalar multiple of neither $e$ nor $f$. Both $e$ and $f$ are classes of prime divisors. Hence, $(xe+yf,f)=x$ and $(xe+yf,e)=y-2x$ are both non-negative, by \cite{boucksom}, Proposition 4.2. Thus $y\geq 2x\geq 0$. Hence, $Pef(M_H(v))=\langle e,f\rangle$. If $D$ is a prime exceptional divisor on $M_H(v)$ with class $d=xe+yf$, then $x\geq 0$, $y\geq 0$, and $(d,d)=2x(y-x)<0$. We have seen above, that if $y>0$, then $y\geq 2x$. Hence $x>0$ and $y=0$. Thus $D$ belongs to the linear system $\linsys{yE}$. But this linear system consists of a single divisor $yE$, by \cite{boucksom}, Proposition 3.13. Part \ref{lemma-item-e-and-f-generate-the-Pef-cone} follows.
Set $w:=v+kv_0=(k,H,k)$, $k\geq 0$. Denote by $M_H(w)^t$ the Brill-Noether locus, consisting of sheaves $F$ with $h^1(F)\geq t$. There exists a regular morphism \[ \beta_t \ : \ M_H(w)^t\setminus M_H(w)^{t+1} \ \ \ \longrightarrow \ \ \ M_H(w+tv_0)\setminus M_H(w+tv_0)^1, \] which is surjective and a $G(t+k,2t+2k)$-bundle (\cite{markman-reflections}, Theorem 3.15). $M_H(w+tv_0)\setminus M_H(w+tv_0)^1$ is a Zariski dense subset of $M_H(w+tv_0)$. In particular, $M_H(w)^t$ is non-empty, if and only if $M_H(w+tv_0)$ is non-empty, if and only if $(w+tv_0,w+tv_0)\geq -2$ (\cite{markman-reflections}, Lemma 3.17).
The proof of part \ref{lemma-item-f-and-e+2f-generate-the-Nef-cone} will be done by induction. Part \ref{lemma-item-f-and-e+2f-generate-the-Nef-cone} is a special case of following Lemma. Set $\tilde{e}:=-v_0=(-1,0,-1)$ and $\tilde{f}:=(0,0,1)$. Let $\theta_w:w^\perp\rightarrow H^2(M_H(w),{\mathbb Z})$ be the Mukai homomorphism given in (\ref{eq-Mukai-isomorphism}).
\begin{lem} \label{lemma-line-bundle-e+2f-is-nef} The class $\theta_w(\tilde{e}+2\tilde{f})$ is nef, for all $w:=v+kv_0=(k,H,k)$, with $k\geq 0$, and for which $M_H(w)$ is non-empty. \end{lem}
\begin{proof} The proof is by induction on the length of the Brill-Noether stratification. Assume that $M_H(w)^1$ is empty. Then $w=v+kv_0$, with $k>0$, since $M_H(v)^1=E$ is non-empty. If $\dim[M_H(w)]=0$, the statement is trivial. If $\dim[M_H(w)]=2$, then $M_H(w)$ is a $K3$ surface with Picard number $1$, and the statement is easily verified. Assume that $\dim[M_H(w)]>2$. Set $\sigma:=\tilde{e}+2\tilde{f}=(-1,0,1)$ and $\sigma_w:=\theta_w(\sigma)$. Note that $(\sigma,\sigma)=2$ and $(\sigma,v)=(\sigma,v_0)=0$. There exists a regular involution $\iota:M_H(w)\rightarrow M_H(w)$, which induces on $H^2(M_H(w),{\mathbb Z})$ the isometry $-R_{\sigma_w}$ (minus the reflection by the class $\sigma_w$), by \cite{markman-reflections}, Theorem 3.21. Now $\sigma_w$ spans the $\iota^*$-invariant sub-lattice of ${\rm Pic}(M_H(w))$. Hence, either $\sigma_w$ or $-\sigma_w$ is nef.
We claim that $\sigma_w$ is nef. We show this by an argument, which will be useful in the induction step as well. The closure of the graph of $\beta_t$ is a smooth correspondence \[ I_t(w) \ \ \ \subset \ \ \ M_H(w)^t\times M_H(w+tv_0), \] by \cite{le-potier-coherent}, Theorem 4.12 (see also \cite{markman-reflections}, section 3.4, where $I_t(w)$ is denoted by $G^0(2t+2k,M_H(t+k,H,t+k))$). Let $\pi_1:I_t(w)\rightarrow M_H(w)^t$ and $\pi_2:I_t(w)\rightarrow M_H(w+tv_0)$ be the two projections.
\begin{claim} \label{claim-equality-of-two-pullbacks} We have the following equality in ${\rm Pic}(I_t(w))$. \begin{equation} \label{eq-two-pullbacks-of-sigma-are-equal} \pi_1^*\sigma_w \ \ \ = \ \ \ \pi_2^*(\sigma_{w+tv_0}). \end{equation} \end{claim}
\begin{proof} A coherent system is a pair $(F,U)$, consisting of a sheaf $F$ over $S$ and a subset $U\subset H^0(S,F)$ of global sections. The correspondence $I_t(w)$ is a connected component of the moduli space of coherent systems over $S$. It parametrizes pairs $(F,U)$, consisting of an $H$-stable sheaf $F$ of class $w+tv_0$ and a $t$-dimensional subspace $U$ of its global sections (\cite{markman-reflections}, section 3.4). $I_t(w)$ thus represents a functor from the category of schemes $T$ of finite type over ${\mathbb C}$ to sets \cite{le-potier-coherent}. Associated to a scheme $T$ is the set of equivalence classes of pairs $({\mathcal F},q)$, where ${\mathcal F}$ is a coherent sheaf over $S\times T$, flat over $T$, which is a family of $H$-stable sheaves of class $w+tv_0$, and $q:R^2_{\pi_{T_*}}({\mathcal F})\rightarrow W$ is a surjective homomorphism onto a locally free $\StructureSheaf{T}$-module $W$ of rank $t$. The equivalence relation is the natural one; we refer to Le Potier for its detailed definition.
Associated to a pair $({\mathcal F},q)$ as above is its classifying morphism $\kappa:T\rightarrow I_t(w)$. We also get the short exact sequence \begin{equation} \label{eq-short-exact-sequence-associated-to-a-family-of-coherent-systems} 0\rightarrow \pi_{T_*}^*W^* \rightarrow {\mathcal F}\rightarrow Q\rightarrow 0, \end{equation} by Grothendieck-Verdier duality and the triviality of the canonical line bundle of $S$. The sheaf $Q$ is flat over $T$ and is a family of $H$-stable sheaves of class $w$, by \cite{markman-reflections}, Lemma 3.7 and section 3.4.
We have the equalities \begin{eqnarray*} \kappa^*\pi_1^*\theta_w(\lambda) & = & \det\left(R_{\pi_{T_*}}[\pi_S^*(\lambda^\vee)\otimes Q]\right), \ \mbox{for all} \ \lambda\in w^\perp, \\ \kappa^*\pi_2^*\theta_{w+tv_0}(\lambda) & = & \det\left(R_{\pi_{T_*}}[\pi_S^*(\lambda^\vee)\otimes {\mathcal F}]\right), \ \mbox{for all} \ \lambda\in (w+tv_0)^\perp. \end{eqnarray*} Note that the class $\sigma$ in $K(S)$ is the class of the ideal sheaf $G$ of a length two zero-dimensional subscheme of $S$. Equation (\ref{eq-two-pullbacks-of-sigma-are-equal}) would thus follow from the existence of an isomorphism, depending canonically on the equivalence class of $({\mathcal F},q)$, \[ \det\left(R_{\pi_{T_*}}[\pi_S^*(G^\vee)\otimes Q]\right)
\ \ \ \cong \ \ \ \det\left(R_{\pi_{T_*}}[\pi_S^*(G^\vee)\otimes {\mathcal F}]\right). \] The construction of the above isomorphism reduces to a natural trivialization of the line bundle $\det\left(R_{\pi_{T_*}}[\pi_S^*(G^\vee)\otimes \pi_T^*W^*]\right)$, by the exact sequence (\ref{eq-short-exact-sequence-associated-to-a-family-of-coherent-systems}). Note that $\chi(G)=0$ and $R_{\pi_{T_*}}[\pi_S^*(G^\vee)\otimes \pi_T^*W^*]\cong RHom(G,\StructureSheaf{S})\otimes_{\StructureSheaf{T}}W^*$, by the projection formula. Hence, \[ \det\left(R_{\pi_{T_*}}[\pi_S^*(G^\vee)\otimes \pi_T^*W^*]\right)= \det(W^*)^{\chi(G^\vee)}=\StructureSheaf{T}. \] \end{proof}
Fix a smooth curve $D\in\linsys{H}$. Then $Pic^{g(D)-1}(D)$ is a fiber of $sup:M_H(v)\rightarrow \linsys{H}$. The pullback of $\sigma_w$, to the intersection of $M_H(v)^k$ with $Pic^{g(D)-1}(D)$, is equal to the restriction of $E$, by equation (\ref{eq-two-pullbacks-of-sigma-are-equal}). Now $E$ restricts to $Pic^{g(D)-1}(D)$ as the theta divisor, which is ample. Hence, $\sigma_w$ is nef.
\underline{Induction step:} Let $C$ be a reduced and irreducible curve in $M_H(w)$. If $w=v$ and $C$ is not contained in $E=M_H(v)^1$, then $\int_C(e+2f)\geq \int_C e\geq 0$. We may thus assume that $C$ is contained in $M_H(w)^t$, $t+k>0$, but not in $M_H(w)^{t+1}$.
Let $\nu:\widetilde{C}\rightarrow C$ be the normalization.
Let $\tilde{p}:\widetilde{C}\rightarrow M_H(w+\nolinebreak tv_0)$ be the morphism extending the restriction of $\beta_t$.
The morphism $\pi_1:\nolinebreak I_t(w)\rightarrow M_H(w)^t$ is a birational morphism, which restricts as an isomorphism over $\pi_1^{-1}\left[M_H(w)^t\setminus M_H(w)^{t+1}\right]$. Hence, the morphism $\nu:\widetilde{C}\rightarrow C$ factors through a morphism $\tilde{\nu}:\widetilde{C}\rightarrow I_t(w)$, satisfying $\tilde{p}=\pi_2\circ \tilde{\nu}$ and $\nu=\pi_1\circ\tilde{\nu}$. We have the equality $\int_C\sigma_w=\int_{\widetilde{C}}\nu^*(\sigma_w)=
\deg\left(\tilde{p}^*\sigma_{w+tv_0}\right)$, by equation (\ref{eq-two-pullbacks-of-sigma-are-equal}).
If $C$ is contained in a fiber of $\beta_t$, then $\deg\left(\tilde{p}^*\sigma_{w+tv_0}\right)=0$. Otherwise, the morphism $\tilde{p}:\widetilde{C}\rightarrow M_H(w+tv_0)$ is non-constant, and $\deg\left(\tilde{p}^*\sigma_{w+tv_0}\right)\geq 0$, by the induction hypothesis.
Proof of part \ref{lemma-item-extremal-class}) The degree of the nef class $e+2f$ is $(e+2f,e+2f)=2$, which is positive, so $e+2f$ is also big (\cite{huybrects-basic-results}, Corollary 3.10). Hence $e+2f$ is semi-ample, by the Base-point-free Theorem, since the canonical line bundle of $M_H(v)$ is trivial (\cite{kollar-mori-book}, Theorem 3.3). Thus, there exists a sufficiently large integer $m$, such that the line bundle $L$ with $c_1(L)=m(e+2f)$ induces a regular morphism $\varphi_L:M_H(v)\rightarrow \linsys{L}^*$, which is birational onto its image. Let $Y$ be the normalization of the image and $\pi:M_H(v)\rightarrow Y$ the natural morphism. The line bundle $L$ restricts to the trivial line bundle on each fiber of $\beta_1: [E\setminus M_H(v)^2]\rightarrow M_H(v+v_0)$, by \cite{markman-part-two}, Lemma 4.11. Hence, the exceptional locus $Exc(\pi)$ of $\pi$ contains $E$. Now $Exc(\pi)$ can not contain any other divisor, since $E$ is the unique exceptional divisor on $M_H(v)$, by part \ref{lemma-item-e-and-f-generate-the-Pef-cone}.
Proof of Part \ref{lemma-item-Y-has-A-1-singularities}) The image $\pi(E)\subset Y$ has codimension $2$ in $Y$, by Lemma \ref{lemma-line-bundle-e+2f-is-nef} and Claim \ref{claim-equality-of-two-pullbacks}. These results show, furthermore, that the restriction of $\pi$ to $E$ factors through $\beta_1$ and a dominant birational map from $M_H(v+v_0)$ to $\pi(E)$. Hence, the generic fiber of $\restricted{\pi}{E}$ is a smooth rational curve. Part \ref{lemma-item-Y-has-A-1-singularities} thus follows from Proposition \ref{prop-dissident-locus} (see also Namikawa's classification of singularities in section 1.8 of \cite{namikawa}).
This completes the proof of Lemma \ref{lemma-ample-cone}. \end{proof}
}
{\bf Acknowledgements:} This work was influenced by S. Druel's talk at the workshop ``Holomorphically symplectic varieties and moduli spaces'', which took place at the Universite des Sciences et Technologies de Lille, in June 2009. I thank S. Druel for his clear exposition of his interesting work on the divisorial Zariski decomposition \cite{druel}, and for correcting inaccuracies in an earlier draft of this paper. I would like to express my gratitude to the organizers, Dimitri Markushevich and Valery Gritsenko, for the invitation to participate in the workshop, their hospitality, and for their wonderful work organizing this stimulating workshop. The paper is influenced by the work of Brendan Hassett and Yuri Tschinkel \cite{hassett-tschinkel-conj,hassett-tschinkel}. I thank them for communicating to me a draft of their recent preprint on the subject \cite{hassett-tschinkel-monodromy}. Section \ref{sec-deformation-equivalence} elaborates on ideas found already in \cite{hassett-tschinkel-monodromy}. I thank Brendan also for reading an early draft of the proof of Theorem \ref{thm-2}. I thank Misha Verbitsky for answering my questions about his preprint \cite{verbitsky}. I thank Valery Gritsenko for pointing out reference \cite[Corollary 3.4]{GHS-K3}. I thank Kota Yoshioka for pointing out the relationship between sections 2 and 3 of his paper \cite{yoshioka-irreducibility}, and section \ref{sec-sufficient-conditions} above.
\end{document} | arXiv |
An Introduction to Political and Social Data Analysis Using R
How to use this book
What's in this Book?
Keys to Student Success
Data Sets and Codebooks
1 Introduction to Research and Data
1.1 Political and Social Data Analysis
1.2 Data Analysis or Statistics?
1.3.1 Interests and Expectations
1.3.2 Research Preparation
1.3.3 Data Analysis and Interpretation
1.3.4 Feedback
1.4 Observational vs. Experimental Data
1.4.1 Necessary Conditions for Causality
1.5 Levels of Measurement
1.6 Level of Analysis
1.7 Next Steps
1.8 Assignments
1.8.1 Concepts and Calculations
2 Using R to Do Data Analysis
2.1 Accessing R
2.2 Understanding Where R (or any program) Fits In
2.3 Time to Use R
2.4 Some R Terminology
2.4.1 Save Your Work
2.6 Exercises
2.6.1 R Problems
3 Frequencies and Basic Graphs
3.1 Get Ready
3.3 Counting Outcomes
3.3.1 The Limits of Frequency Tables
3.4 Graphing Outcomes
3.4.1 Bar Charts
3.4.2 Histograms
3.4.3 Density Plots
3.4.4 A few Add-ons for Graphing
4 Transforming Variables
4.3 Data Transformations
4.4 Renaming and Relabeling
4.4.1 Changing Attributes
4.5 Collapsing and Reordering Catagories
4.6 Combining Variables
4.6.1 Creating an Index
4.7 Saving Your Changes
5 Measures of Central Tendency
5.2 Central Tendency
5.2.1 Mode
5.3 Median
5.4 The Mean
5.4.1 Dichotomous Variables
5.5 Mean, Median, and the Distribution of Variables
5.6 Skewness Statistic
5.7 Adding Legends to Graphs
6 Measures of Dispersion
6.3 Measures of Spread
6.3.1 Range
6.3.2 Interquartile Range (IQR)
6.3.3 Boxplots
6.4 Dispersion Around the Mean
6.4.1 Don't Make Bad Comparisons
6.5 Dichotomous Variables
6.6 Dispersion in Categorical Variables?
6.7 The Standard Deviation and the Normal Curve
6.7.1 Really Important Caveat
6.8 Calculating Area Under a Normal Curve
6.9 One Last Thing
6.10 Next Steps
6.11 Assignments
6.11.1 Concepts and Calculations
6.11.2 R Problems
7 Probability
7.3 Theoretical Probabilities
7.3.1 Large and Small Sample Outcomes
7.4 Empirical Probabilities
7.4.1 Empirical Probabilities in Practice
7.4.2 Intersection of Two Probabilities
7.4.3 The Union of Two Probabilities
7.4.4 Conditional Probabilities
7.5 The Normal Curve and Probability
8 Sampling and Inference
8.1 Getting Ready
8.2 Statistics and Parameters
8.3 Sampling Error
8.4 Sampling Distributions
8.4.1 Simulating the Sampling Distribution
8.5 Confidence Intervals
8.6 Proportions
9 Hypothesis Testing
9.1 Getting Started
9.2 The Logic of Hypothesis Testing
9.2.1 Using Confidence Intervals
9.2.2 Direct Hypothesis Tests
9.2.3 One-tail or Two?
9.3 T-Distribution
9.5 T-test in R
10 Hypothesis Testing with Two Groups
10.1 Getting Ready
10.2 Testing Hypotheses about Two Means
10.2.1 Generating Subgroup Means
10.3 Hypothesis Testing with Two means
10.3.1 A Theoretical Example
10.3.2 Returning to the Empirical Example
10.3.3 Calculating the t-score
10.3.4 Statistical Significance vs. Effect Size
10.4 Difference in Proportions
10.5 Plotting Mean Differences
10.6 What's Next?
10.7 Exercises
10.7.1 Concepts and Calculations
10.7.2 R Problems
11 Hypothesis Testing with Multiple Groups
11.2 Internet Access as an Indicator of Development
11.2.1 The Relationship between Wealth and Internet Access
11.3 Analysis of Variance
11.3.1 Important concepts/statistics:
11.4 Anova in R
11.5 Effect Size
11.5.1 Plotting Multiple Means
11.6 Population Size and Internet Access
11.7 Connecting the T-score and F-Ratio
11.8 Next Steps
11.9 Assignments
12 Hypothesis Testing with Crosstabs
12.2 Crosstabs
12.2.1 The Relationship Between Education and Religiosity
12.3 Sampling Error
12.4 Hypothesis Testing with Crosstabs
12.4.1 Regional Differences in Religiosity?
12.5 Directional Patterns in Crosstabs
12.5.1 Age and Religious Importance
12.6 Limitations of Chi-Square
13 Measures of Association
13.2 Going Beyond Chi-squared
13.3 Measures of Association for Crosstabs
13.3.1 Cramer's V
13.3.2 Lambda
13.4 Ordinal Measures of Association
13.4.1 Gamma
13.4.2 Tau-b and Tau-c
13.5 Revisiting the Gender Gap in Abortion Attitudes
13.5.1 When to Use Which Measure
14 Correlation and Scatterplots
14.1 Get Started
14.2 Relationships between Numeric Variables
14.3 Scatterplots
14.4 Pearson's r
14.4.1 Calculating Pearson's r
14.4.2 Other Independent Variables
14.5 Variation in Strength of Relationships
14.6 Proportional Reduction in Error
14.7 Correlation and Scatterplot Matrices
14.8 Overlapping Explanations
14.10 Exercises
14.10.1 Concepts and calculations
14.10.2 R Problems
15 Simple Regression
15.2 Linear Relationships
15.3 Ordinary Least Squares Regression
15.3.1 Calculation Example: Presidential Vote in 2016 and 2020
15.4 How Well Does the Model Fit the Data?
15.6 Getting Regression Results in R
15.6.1 All Fifty States
15.7 Understanding the Constant
15.8 Non-numeric Independent Variables
15.9 Adding More Information to Scatterplots
15.10 Next Steps
15.11 Assignments
16 Multiple Regression
16.1 Getting Started
16.2 Organizing the Regession Output
16.2.1 Summarizing Life Expectancy Models.
16.3.1 Assessing the Substantive Impact
16.4 Model Accuracy
16.5 Predicted Outcomes
16.5.1 Identifying Observations
17 Advanced Regression Topics
17.2 Incorporating Access to Health Care
17.3 Multicollinearity
17.4 Checking on Linearity
17.4.1 Stop and Think
17.5 Which Variables have the Greatest Impact?
17.6 Statistics vs. Substance
18 Regession Assumptions
18.2 Regression Assumptions
18.3 Linearity
18.4 Independent Variables are not Correlated with the Error Term
18.5 No Perfect Multicollinearity
18.6 The Mean of the Error Term equals zero
18.7 The Error Term is Normally Distributed
18.8 Constant Error Variance (Homoscedasticity)
18.9 Independent Errors
18.10 What's next?
Appendix: Codebooks
ANES20
County20large
Countries2
States20
Chapter 17 Advanced Regression Topics
This chapter continues the analysis of life expectancy by taking into account other factors that might be related to health conditions in general. We will also address a few important issues related to evaluating multiple regression models. If you want to follow along in R, you should load the countries2 data set, as well as the libraries for the DescTools and stargazer packages.
One set of variables that we haven't considered but that should be related to life expectancy are factors that measure access to health care. Generally speaking, countries with greater availability of health care resources (doctors, hospitals, research centers, etc) should have higher levels of life expectancy. In the table below, we expand the regression model to include two additional variables, percent of the population living in urban areas and the number of doctors per 10,000 persons. Both of these variables are expected to be positively related to life expectancy, as they are linked to the availability of health care services and facilities. Doctors per 10,000 persons is a direct measure of access to health care, and levels of access to health care are generally lower in rural areas than in urban areas.
These variables are added in the model below:
#Add % urban and doctors per 10k to the model
fit<-lm(countries2$lifexp~countries2$fert1520 +
countries2$mnschool+ log10(countries2$pop19_M)+
countries2$urban+countries2$docs10k)
#Use information in 'fit' to create table of results
stargazer(fit, type="text",
dep.var.labels=c("Life Expectancy"), #Dependent variable label
covariate.labels = c("Fertility Rate", #Indep Variable Labels
"Mean Years of Education",
"Log10 Population", "% Urban",
"Doctors per 10,000"))
Dependent variable:
Fertility Rate -3.237***
Mean Years of Education 0.413**
Log10 Population 0.082
% Urban 0.050***
Doctors per 10,000 0.049*
Constant 73.923***
Observations 179
R2 0.768
Adjusted R2 0.761
Residual Std. Error 3.606 (df = 173)
F Statistic 114.590*** (df = 5; 173)
Note: *p<0.1; **p<0.05; ***p<0.01
Here's how I would interpret the results:
The first thing to note is that four of the five variables are statistically significant (counting docs10k as significant with a one-tailed test), and the overall model fit is improved in comparison to the fit for the three-variable model from Chapter 16. The only variable that has no discernible impact is population size. The model R2 is now .768, indicating that the model explains 76.8% of the variation in life expectancy, and the adjusted R2 is .761, an improvement over the previous three-variable model (adjusted R2 was .743). Additionally, the RMSE is now 3.606, also signaling less error compared to the previous model (3.768). Interpretation of the individual variables should be something like the following:
Fertility rate is negatively and significantly related to life expectancy. For every one unit increase in the value of fertility rate, life expectancy is expected to decline by about 3.24 years, controlling for the influence of the other variables in the model.
Average years of education is positively related to life expectancy. For every unit increase in education level, life expectancy is expected to increase by about .413 years, controlling for the influence of the other variables in the model.
Percent of urban population is positively related to life expectancy. For every unit increase in the percent of the population living in urban areas, life expectancy is expected to increase by .05 years, controlling for the influence of other variables in the model.
Doctors per 10,000 population is positively related to life expectancy. For every unit increase in doctors per 10,000 population, life expectancy is expected to increase by about .05 years, controlling for the influence of the other variables in the model.
I am treating the docs10k coefficient as statistically significant even though the p-value is greater than .05. This is because the reported p-value (< .10) is a two-tailed p-value, and if we apply a one-tailed test (which makes sense in this case), the p-value is cut in half and is less than .05. The two-tailed p-value (not shown in the table) is .068, so the one-tailed value is .034. Still, in a case like this, where the level of significance if borderline, it is best to assume that the effect is pretty small. A weak effect like this is kind of surprising for this variable, especially since there are strong substantive reasons to expect greater access to health care to be strongly related to life expectancy. Can you think of an explanation for this? We explore a couple of potential reasons for this weaker-than-expected relationship in the next two sections.
Missing Data. One final thing that you need to pay attention to in this model, or whenever you work with multiple regression or any other statistical technique that involves working with multiple variables at the same time, is the number of missing cases. Missing cases occur on a given variable when cases do not have any valid outcomes. We discussed this a bit earlier in the book in the context of public opinion surveys, where missing data usually occur because people refuse to answer questions or do not have an opinion to offer. When working with aggregate cross-national data, as we are here, missing outcomes usually occur because the data are not available for some countries on some variables. For instance, some countries may not report data on some variables to the international organizations (e.g., World Bank, United Nations, etc.) that are collecting data, or perhaps the data gathering organizations collect certain types of data from certain types of countries but not for others. This is a more serious problem for multiple regression than simple regression because multiple regression uses "listwise" deletion of missing data, meaning that if a case is missing on one variable it is dropped from all variables. This is why it is important to pay attention to missing data as you add more variables to the model. It is possible that one or two variables have a lot of missing data, and you could end up making generalizations based on a lot fewer data points than you realize. At this point, using this set of five variables there are sixteen missing cases (there are 195 countries in the data set and 179 observations used in the model).50 This is not too extreme but is something that should be monitored.
One potential explanation for the tepid role of docs10k in the life expectancy model is multicollinearity. Recall from the discussions in Chapters 14 and 16 that when independent variables are strongly correlated with each other, the simple bi-variate relationships are usually overstated and the independent variables lose strength when considered in conjunction with other overlapping explanations. Normally, this is not a problem. In fact, one of the virtues of regression analysis is that the model sorts out overlapping explanations (this is why we use multiple regression). However, when the degree of overlap is very high, it can make it difficult for substantively important variables to demonstrate statistical significance. Perfect collinearity violates a regression assumption (see next chapter), but high level of collinearity can also cause problems, especially for interpreting significance levels.
Here's how this problem comes about: the t-score for a regression coefficient (\(b\)) is calculated as \(t=\frac{b}{S_b}\), and the standard error of b (\(S_b\)) for any given variable is directly influenced by how that variable is correlated with other independent variables. The formula below illustrates how collinearity can affect the standard error of \(b_1\) in a model with two independent variables:
\[S_{b_1}=\sqrt{\frac{RMSE}{\sum(x_i-\bar{x}_1)^2(1-R^2_{1\cdot2})}}\]
The key to understanding how collinearity affects the standard error, which then affects the t-score, lies in part of the denominator of the formula for the standard error: \((1-R^2_{1\cdot2})\). As the correlation (and R2) between x1 and x2 increases, the denominator of the formula decreases in size, leading to larger standard errors, and smaller t-scores. Because of this, high correlations among independent variables can make it difficult for variables to become statistically significant. This problem can be compounded when there are multiple strongly related independent variables. This problem is usually referred to as high multicollinearity, or high collinearity.
When you have good reasons to expect a variable to be strongly related to the dependent variable and it is not statistically significant in a multiple regression model, you should think about whether collinearity could be a problem. So, is this a potential explanation for the borderline statistical significance for docs10k? It does make sense that docs10k is strongly related to other independent variables, especially since most of the variables reflect an underlying level of economic development. As a first step toward diagnosing this, let's look at the correlation matrix for evidence of collinearity:
#Create "logpop" variable for exporting
countries2$logpop<-log10(countries2$pop19_M)
#Copy DV and IVs to new data set.
lifexp_corr<-countries2[,c("lifexp", "fert1520", "mnschool",
"logpop", "urban", "docs10k")]
#Restrict digits in output to fit columns together
options(digits = 3)
#Use variable in 'life_corr' to produce correlation matrix
cor(lifexp_corr, use = "complete") #"complete" drops missing data
lifexp fert1520 mnschool logpop urban docs10k
lifexp 1.0000 -0.8399 0.7684 -0.0363 0.6024 0.7031
fert1520 -0.8399 1.0000 -0.7627 0.0706 -0.5185 -0.6777
mnschool 0.7684 -0.7627 1.0000 -0.0958 0.5787 0.7666
logpop -0.0363 0.0706 -0.0958 1.0000 0.0791 -0.0168
urban 0.6024 -0.5185 0.5787 0.0791 1.0000 0.5543
docs10k 0.7031 -0.6777 0.7666 -0.0168 0.5543 1.0000
Look across the row headed by docs10k, or down the docs10k column to see how strongly it is related the other variables. First, there is a strong bivariate correlation (r=.70) between doctors per 10,000 population and the dependent variable, life expectancy. Further, docs10k is also highly correlated with fertility rates (r= -.68), mean years of education (r=.77), and percent living in urban areas (r= .55). Based on these correlations, it seems plausible that collinearity could be causing a problem for the significance of slope for docs10k.
There are two important statistics that help us evaluate the extent of the problem with collinearity: The tolerance and VIF (variance inflation) statistic. Tolerance is calculated as:
\[ \text{Tolerance}_{x_1}=1-R^2_{x_1,x_2\cdot \cdot x_k}\]
You should recognize this from the denominator of the formula for the standard error of the regression slope; it is the part of the formula that inflates the standard error. Note that as the proportion of variation in x1 that is accounted for by the other independent variables increases, the tolerance value decreases. The tolerance is the proportion of variance in an independent variable that is not explained by variation in the other independent variables.
The VIF statistic tells us how much the standard error of the slope is inflated due to inter-item correlation. The calculation of the VIF is based on the tolerance statistic:
\[\text{VIF}_{b_1}=\frac{1}{\text{Tolerance}_{b_1}}\] We can calculate tolerance for docs10k stats by regressing it on the other independent variables to get \(R^2_{x_1,x_2\cdot \cdot x_k}\) . In other words, treat docs10k as a dependent variable and see how much of its variation is accounted for by the other variables in the model. The model below does this:
#Use 'docs10k' as the DV to Calculate its Tolerance
fit_tol<-lm(countries2$docs10k~countries2$fert1520 + countries2$mnschool
+ log10(countries2$pop19_M) +countries2$urban)
#Use information in 'fit_tol' to create a table of results
stargazer(fit_tol, type="text",
title="Treating 'docs10k' as the DV to Calculate its Tolerance",
dep.var.labels=c("Doctors per 10,000"),
covariate.labels = c("Fertility Rate",
"Log10 Population", "% Urban"))
Treating 'docs10k' as the DV to Calculate its Tolerance
Doctors per 10,000
Mean Years of Education 2.860***
% Urban 0.099**
Constant -6.080
Residual Std. Error 10.200 (df = 174)
F Statistic 71.900*** (df = 4; 174)
Note that the R2=.623, so 62.3 percent of the variation in docs10k is accounted for by the other independent variables. Using this information, we get:
Tolerance: 1-.623 = .377.
VIF (1/tolerance): 1/.377 = 2.65
So, is a VIF of 2.65 a lot? In my experience, no, but it is best to evaluate this number in the context of the VIF statistics for all variables in the model. Instead of calculating all of these ourselves, we can just have R get the VIF statistics for us:
#Get VIF statistics using information from 'fit'
VIF(fit) #note uppercase VIF
countries2$fert1520 countries2$mnschool log10(countries2$pop19_M)
2.55 3.50 1.04
countries2$urban countries2$docs10k
1.63 2.65
Does collinearity explain the marginally significant slope for docs10k? Yes and no. Sure, the slope for docs10k would probably have a smaller p-value if we excluded the variables that are highly correlated with it, but some loss of impact is almost always going to happen when using multiple regression; that's the point of using it! Also, the issue of collinearity is not appreciably worse for docs10k than it is for fertility rate, and it is less severe that it is for mean education, so it's not like this variable faces a higher hurdle than the others. Based on these results I would conclude that collinearity is a slight problem for docs10k, but that there is probably a better explanation for its level of statistical significance. There is no magic cutoff point for determining that collinearity is a problem that needs to be addressed. In my experience, VIF values less than 10 are probably not worth worrying about, but that is a judgment call and you always have to consider issues like this in the context of the data and model you are using.
Even though collinearity is not a big problem here, it is important to consider what to do if there is extreme collinearity in a regression model. Some people suggest dropping a variable, possibly the one with the highest VIF. I don't generally like to do this, but it is something to consider, especially if you have several variables that are measuring different aspects of the same concept and dropping one of them would not hurt your ability to measure an important concept. For instance, suppose this model also included hospital beds per capita and nurses per capita. These variables, along with docs10k all measure different aspects of access to health care and should be strongly related to each other. If there were high VIF values for these variables, we could probably drop one or two of them to decrease collinearity, and we would still be able to consider access to health care as an explanation of differences in life expectancy.
Another alternative that would work well in this scenario would be to combine the highly correlated variables into an index that uses information from all of the variables to measure wealth with a single index. We did something like this in Chapter 3 where we created a single index of LGBTQ policy preferences based on outcomes from several separate LGBTQ policy questions. Indexing like this minimizes the collinearity problem without dropping any variables. There are many different ways to combine variables, but expanding on that here is a bit beyond the scope of this book.
The relationship between docs10k and life expectancy presents an opportunity to explore another important potential problem that can come up in regression analysis. Theoretically, there should be a relatively strong relationship between these two variables, not one whose statistical significance in the multiple regression model depends upon whether you are using one- or two-tailed test. We have already explored collinearity as an explanation for the weak showing of docs10k, but that does not appear to be the primary explanation. While the bivariate correlation (r=.70) with the dependent variable is reported above, one thing we did not do earlier was examine a simple scatterplot for this relationship. Let's do that now to see if it offers any clues.
#Scatterplot of "lifexp" by "docs10k"
plot(countries2$docs10k, countries2$lifexp, ylab="Life Expectancy",
xlab="Doctors per 10k Population")
#Add linear regression line
abline(lm(countries2$lifexp~countries2$docs10k))
This is interesting. Note that the linear prediction line does not seem to fit the data in the same way we have seen in most of the other scatterplots. Focusing on the pattern in the markers, this doesn't look like a typical linear pattern. At low levels of the independent variable there is a lot of variation in life expectancy, but, on average, it is fairly low. Increases in doctors per 10k from the lowest values to 14 or so leads to substantial increases in life expectancy but then there are diminishing returns from that point on. Looking at this plot, you can imagine that a curved line would fit the data points better than the existing straight line. One of the important assumptions of OLS regression is that all relationships are linear, that the expected change in the dependent variable for a unit change in the independent variable is constant across values of the independent variables. This is clearly not the case here. Sure, you can fit a straight line to the pattern in the data, but if the pattern in the data is not linear, then the line does not fit the data as well as it could, and we are violating an important assumption of OLS regression (see next Chapter).
Just as \(y=a + bx\) is the equation for a straight line, there are a number of possibilities for modeling a curved line. Based on the pattern in the scatterplot–one with diminishing returns–I would opt for the following model:
\[\hat{y}=a + b*log_{10}x\]
Here, we transform the independent variable by taking the logged values of its outcomes.
Log transformations are very common, especially when data show a curvilinear pattern or when a variable is heavily skewed. In this case, we are using a log(10) transformation. This means that all of the original values are transformed into their logged values, using a base of 10. A logged value is nothing more than the power to which you have to raise your log base (in this case, 10) in order to get the original raw score. For instance, log10 of 100 is 2, because 102=100, and log10 of 50 is 1.699 because 101.699=50, and so on. We have used log10 for population size since Chapter 14, and you may recall that a logged version of population density was used in one of the end-of-chapter assignments in Chapter 16. Using logged values has two primary benefits: it minimizes the impact of extreme values, which can be important for highly skewed variables, and it enables us to model relationships as curvilinear rather than linear.
One way to assess whether the logged version of docs10k fits the data better than the raw version is to just look at the bivariate relationships with the dependent variable using simple regression:
#Simple regression of "lifexp" by "docs10k"
fit_raw<-lm(countries2$lifexp~countries2$docs10k, na.action=na.exclude)
#Simple regression of "lifexp" by "log10(docs10k)"
fit_log<-lm(countries2$lifexp~log10(countries2$docs10k), na.action=na.exclude)
#Use Stargazer to create a table comparing the results from two models
stargazer(fit_raw, fit_log, type="text",
dep.var.labels = c("Life Expectancy"),
covariate.labels = c("Doctors per 10k", "Log10(Doctors per 10k)"))
(1) (2)
Doctors per 10k 0.315***
Log10(Doctors per 10k) 9.580***
Constant 66.900*** 63.400***
(0.581) (0.565)
Observations 186 186
R2 0.487 0.673
Adjusted R2 0.484 0.671
Residual Std. Error (df = 184) 5.290 4.230
F Statistic (df = 1; 184) 175.000*** 379.000***
Note: *p<0.1; **p<0.05; ***p<0.01
Note that all measures of fit, error, and relationship point to the logged (base 10) version of docs10k as the superior operationalization. Based on these outcomes, it is hard to make a case against using the log-transformed version of docs10k. You can also see this in the scatterplot in Figure 17.1, which includes the prediction lines from both the raw and logged models.
Figure 17.1: Alternative Models for the Relationship between Doctors per 10k and Life Expectancy
Note that the curved, solid line fits the data points much better than the straight, dashed line, which is what we expect based on the relative performance of the two regression models above. The interpretation of the curvilinear relationship is a bit different than for the linear relationship. Rather than concluding that life expectancy increases as doctors per capita increases, we see in the scatterplot that life expectancy increases substantially as we move from nearly 0 to 14 or so doctors per 10,000 people, but increases beyond that point have less and less impact on life expectancy.
This is illustrated further below in Figure 17.2, where the graph on the left shows the relationship between docs10k among countries below the median level (14.8) of doctors per 10,000 population, and the graph on the right shows the relationship among countries with more than the median level of doctors per 10,000 population. This is not quite as artful as Figure 17.1, with the plotted curved line, but it does demonstrate that the greatest gains in life expectancy from increased access to health care come at the low end of the scale, among countries with severely limited access. For the upper half of the distribution, there is almost no relationship between levels of docs10k and life expectancy.
Figure 17.2: A Segmented View of a Curvilinear Relationship
Finally, we can re-estimate the multiple regression model with the appropriate version of docs10k to see if this transformation has an impact on its statistical significance.
#Re-estimate five-variable model using "log10(docs10k)"
fit<-lm(countries2$lifexp~countries2$fert1520 + countries2$mnschool+
log10(countries2$pop) +countries2$urban+
log10(countries2$docs10k),na.action = na.exclude)
#Using information in 'fit' to create a table of results
stargazer(fit, type="text", dep.var.labels=c("Life Expectancy"),
covariate.labels = c("Fertility Rate","Mean Years of Education",
"Log10(Population)",
"% Urban","log10(Doctors per 10k)"))
Log10(Population) 0.050
log10(Doctors per 10k) 2.420**
Accounting for the curvilinear relationship between docs10k and life expectancy made an important difference to the model. First, we now see a statistically significant impact from the logged version of the docs10k, whether using a one- or two-tailed test (the t-score changes from 1.83 in the original model to 2.49 in the model above). We need to be a little bit careful here when discussing the coefficient for the logged variable. Although there is a curvilinear relationship between docs10k and life expectancy, there is a linear relationship between log10(docs10k) and life expectancy. So, we can say that for every unit increase in the logged value of docs10k, life expectancy increases by 2.4 years, but we need to keep in mind that when we translate this into the impact of the original version of docs10k, the relationship is not linear. In addition to doctors per 10,000 population, the fertility rate and percent urban are also statistically significant, and there is less error in predicting life expectancy when using the logged version of docs10k: the RMSE dropped from 3.606 to 3.58, and the R2 increased from .768 to .772.
These findings underscore the importance of thinking hard about results that don't make sense. There is a strong theoretical reason to expect that life expectancy would be higher in countries where there are more doctors per capita, as there is greater access to health care in those countries than in countries with fewer doctors per capita, and this greater access should lead to better outcome. However, the initial regression results suggested that doctors per capita was tangentially related to life expectancy. Had we simply accepted this result, we would have come to an erroneous conclusion. Instead, by exploring explanations for the null finding, we were able to uncover a more nuanced and powerful role for the number of doctors per 10k population. Of course, it is important to point out that one of the sources of "getting it wrong" in the first place was skipping a very important first step: examining the bivariate relationship closely, using not just the correlation coefficient but also the scatterplot. Had we started there, we would have understood the curvilinear nature of the relationship from the get go.
As a researcher, one thing you're likely to be interested in is the relative impact of the independent variables; that is which variables have the greatest and least impact on the dependent variable? If we look at the regression slopes, we might conclude that the order of importance runs from highest-to-lowest in the following order: The fertility rate (b=-2.81) and log of doctors per 10k (b=2.42) are the most important and of similar magnitude, followed by education level in distant third (b=.36), followed by log of population (b=.05) and percent urban (b=.045), neither one of which seem to matter very much. But this rank ordering is likely flawed due to differences in the way the variables are measured. A tip off that something is wrong with this type of comparison is found in the fact that the slopes for urban population and the log of population size are of about the same magnitude, even though the slope for urban population is statistically significant (p<.01) while the slope for population size is not and has not been significant in any of the analyses shown thus far. How can a variable whose effect is not distinguishable from zero have an impact equal to one that has had a significant relationship in all of the examples shown thus far? Seems unlikely.
Generally, the "raw" regression coefficients are not a good guide to the relative impact of the independent variables because the variables are measured on different scales. Recall that the regression slopes tell us how much Y is expected to change for a unit change in x. The problem we encounter when comparing these slopes is that the units of measure for the independent variables are mostly different from each other, making it very difficult to compare the "unit changes" in a fair manner.
To gain a sense of the impact of measurement scale on the size of the regression slope, consider the two plots below in Figure 17.3. Both plots show the relationship between the size of the urban population and country-level life expectancy. The plot on the left uses the percentage of a country's population living in urban areas, and the plot on the right uses the proportion of a country's population living in urban areas. These are the same variable, just using different scales. As you can see, other than the metric on the horizontal axis, both plots are identical. In both cases, there is a moderately strong positive relationship between the size of the urban population and life expectancy: as the urban population increases in size, life expectancy also increases. But look at the linear equations that summarize the results of the bi-variate regression models. The slope for the urban population variable in the second model is 100 times the size of the slope in the first model, not because the impact is 100 times greater but because of the difference in the scale of the two variables. This is the problem we encounter when comparing regression coefficients across independent variables measured on different scales.
Figure 17.3: The Impact of Scale of the Independent Variable on the Size of Regression Coefficients
What we need to do is standardize the variables used in the regression model so they are all measured on the same scale. Variables with a common metric (same mean and standard deviation) will produce slopes that are directly comparable. One common way to standardize variables is to transform the raw scores into z-scores:
\[Z_i=\frac{x_i-\bar{x}}{S}\] All variables transformed in this manner will have \(\bar{x}=0\) and \(S = 1\). If we do this to all variables (including the dependent variable), we can get the standardized regression coefficients (sometimes confusingly referred to as "Beta Weights"). Fortunately, though, we don't have to make these conversions ourselves since R provides the standardized coefficients. In order to get this information, we can use the StdCoef command from the DescTools package. The command is then simply StdCoef(fit), where "fit" is the object with the stored information from the linear model.
#Produce standardized regression coefficients from 'fit'
StdCoef(fit)
Estimate* Std. Error* df
(Intercept) 0.00000 0.0000 173
countries2$fert1520 -0.48092 0.0684 173
countries2$mnschool 0.15011 0.0687 173
log10(countries2$pop) 0.00567 0.0371 173
countries2$urban 0.13908 0.0471 173
log10(countries2$docs10k) 0.20610 0.0827 173
attr(,"class")
[1] "coefTable" "matrix"
The standardized coefficients are in the "Estimate" column. To evaluate relative importance, ignore the sign of the coefficient and only compare the absolute values. Based on the standardize coefficients, the ordering of variables from most-to-least in importance is a bit different than when we used the unstandardized coefficients: fertility rate (-.48) far outstrips all other variables, having more that twice the impact as doctors per 10,000 (.21), which is followed pretty closely level of education (.15) and urban population (.14), and population size comes in last (.006). These results give a somewhat different take on relative impact than if we relied on the raw coefficients.
We can also make more literal interpretations of these slopes. Since the variables are all transformed into z-scores, a "one unit change" in x is always a one standard deviation change in x. This means that we can interpret the standardized slopes as telling how many standard deviations y is expected to change for a one standard deviation change in x. For instance, the standardized slope for fertility (-.48) tells us that for every one standard deviation increase in the fertility rate we can expect to see a .48 standard deviation decline in life expectancy. Just to be clear, though, the primary utility of these standardized coefficients is to compare the relative impact of the independent variables.
The model we've been looking at explains about 77% of the variation in country-level life expectancy using five independent variables. This is a good model fit, but it is possible to account for a lot more variance with even fewer variables. Take, for example, the alternative model below, which explains 99% of the variance using just two independent variables, male life expectancy and maternal mortality.
#Alternative model
fit1<-lm(countries2$lifexp~countries2$malexp + countries2$infant_mort)
#Table with results from both models
stargazer(fit1, type="text",
dep.var.labels=c("Life Expectancy"),
covariate.labels = c("Male Life Expectancy",
"Infant Mortality"))
Male Life Expectancy 0.810***
Infant Mortality -0.081***
Constant 17.400***
Observations 184
R2 0.989
Adjusted R2 0.989
Residual Std. Error 0.787 (df = 181)
F Statistic 8,098.000*** (df = 2; 181)
Note: *p<0.1; **p<0.05; ***p<0.01
We had been feeling pretty good about our model that explains 77% of the variance in life expectancy, and then along comes this much simpler model that explains almost 100% of the variation. Not that it's a contest, but this does sort of raise the question, "Which model is best?" The new model definitely has the edge in measures of error: the RMSE is just .787, compared to 3.58 in the five-variable model, and as mentioned above, the R2=.99.
So, which model is best? Of course, this is a trick question. The second model provides a stronger statistical fit but at significant theoretical and substantive costs. Put another way, the second model explains more variance without providing a useful or interesting substantive explanation of the dependent variable. In effect, the second model is explaining life expectancy with other measures of life expectancy. Not very interesting. Think about it like this. Suppose you are working as an intern at the World Health Organization and your supervisor asks you to give them a presentation on the factors that influence life expectancy around the world. How impressed do you think they would be if you came back to them and said you had two main findings:
People live longer in countries where men live longer.
People live longer in countries where fewer people die really young.
Even with your impressive R2 and RMSE, your supervisor would likely tell you to go back to your desk, put on your thinking cap, and come up with a model that provides a stronger substantive explanation, something more along the lines of the first model. With the results of the first model you can report back that two factors that are most strongly related to life expectancy across countries are fertility rates and access to health care, as measured by the number of doctors per 10k population. The share of the population living in urban areas is also related to life expectancy, but this is not something that can be addressed through policies and aid programs. The fertility rate and access to health care, however, are things that policies and aid programs might be able to affect.
By now, you should be feeling comfortable with regression analysis and, hopefully, seeing how it can be applied to many different types of data problems. The last step in this journey of discovery is to examine some of the important assumptions that underlie the OLS regression. In Chapter 18, I provide a brief overview of assumptions that need to be satisfied in order to be confident in the results of the life expectancy model presented above. As you will see, in some ways, the model does a good job of satisfying the assumptions, but in other ways, the model still needs a bit of work. It is important to be familiar with these assumptions not just so you can think about them in the context of your own work, but also so you can use this information to evaluate the work of others.
Building on the regression models from the last chapter, use the states20 data set and run a multiple regression model with infant mortality (infant_mort) as the dependent variable and per capita income (PCincome2020), the teen birth rate (teenbirth), percent of low birth weight births (lowbirthwt), and southern region (south) as the independent variables.
Produce a publication-ready table using stargazer. Make sure you replace the R variable names with meaningful labels (see Chapter 16 for instructions on doing this using stargazer).
Summarize the results of this model. Make sure to address matters related to the slopes of the independent variables and how well the group of variables does in accounting for state-to-state differences in infant mortality.
Generate the standardized regression coefficients and discuss the results. Any surprises, given the findings from the unstandardized coefficients? Make it clear that you understand why you need to run this test and what these results are telling you.
Run a test for evidence of multicollinearity. Discuss the results. Any surprises? Make it clear that you understand why you need to run this test and what these results are telling you.
The number of missing cases is reported in the standard R output (not stargazer) as "16 observations deleted due to missingness."↩︎ | CommonCrawl |
Analysis of households food insecurity and its coping mechanisms in Western Ethiopia
Seid Sani ORCID: orcid.org/0000-0003-3667-23891 &
Biruk Kemaw2
This study analyzed households' food insecurity and its determinants along with the coping mechanisms opted against food insecurity and shortage in Assosa zone, western Ethiopia. The study used a primary data collected from 276 randomly selected households for 7 consecutive days from each sample using weighed records method. In addition, focus group discussions and key informants interview were also used. This study employed descriptive statistics, food insecurity index and Tobit model to analyze the data. The finding of the study revealed that, in the study area, the incidence of food insecurity was 53.62%, with the depth and severity of food insecurity being 16.84% and 7.32%, respectively. The study finding also pointed out that the mean kilocalorie intake of food insecure households was 1440.37kcal/AE/day, with the minimum and maximum being 597.65 kcal and 2048.13 kcal, respectively. Furthermore, the estimated Tobit model result revealed that age of the household head, family size and off-farm and non-farm income positively affected extent of households food insecurity; whereas access to irrigation, farm income, distance to market and access to credit negatively affected the extent of households' food insecurity. Moreover, the study also identified that reducing meal size, reducing frequency of meal served, working as a daily laborer and selling livestock's were the top four main coping mechanisms opted against food insecurity and/or shortage. Therefore, to reverse the incidence, future interventions should focus on the aforementioned factors to build the capacity of households through enhancing their access to human, financial and physical capital.
Food is both a basic need and a human right as enough food in terms of quantity and quality for all people is an important factor for a healthy and productive life as well as for a nation to sustain its development (FAO (2014); Sani and Kemaw 2017). Besides, enough food in terms of quantity and quality is a key for maintaining and promoting political stability and insuring peace among people (Idrisa et al. 2008). However, reports indicated that about 1.4 billion poor people were living on less than US$1.25 a day and 1 billion of them live in rural areas where agriculture is the main source of livelihood, especially in sub-Saharan Africa and Southern Asia (IFAD 2011). Furthermore, FAO (2015) reported that about 795 million people in the world were food insecure, with many more suffering from 'hidden hunger' caused by micronutrient or protein deficiencies. Moreover, different studies depicted that food insecurity occurred in most countries to varying degrees, and 75% of the food insecure people lived in rural areas of developing countries, in which two thirds of these lived in just seven countries (Bangladesh, China, Democratic Republic of Congo, Ethiopia, India, Indonesia and Pakistan) (Keatinge et al. 2011; Khush et al. 2012; Sani and Kemaw 2017).
As a part of Africa and developing world, Ethiopia is one of the most food-insecure and famine affected countries as large portion of the country's population has been affected by chronic and transitory food insecurity (Abduselam 2017). Over 30% of the population is below the food poverty line, unable to afford the minimum caloric intake for a healthy and active life (CSA (Central statistical agency) 2014). Furthermore, FAO (2012) finding figured out that 52% of the rural population was food insecure i.e. consume below the minimum recommended daily intake of 2100 kcal/ AE /day, which led the rural households to temporarily depend on relief food assistance. As a result, more than 8.5 million people were in need of emergency food aid and assistance (WFP 2017). Moreover, under-nutrition has been a persistent problem as 44% of children in the country were stunted, 10% of children were considered to have low weight-for-height (wasting) and 29% of children were considered to have underweight (low weight-for-age). Besides, under-nutrition was predominant in rural areas in which stunting accounts for 46%, wasting accounts for 10%, and underweight accounts for 30% of rural children in the country (CSA (Central statistical agency) 2011).
The western part of Ethiopia, Assosa zonein particular, is hit by high degree of incidence of food insecurity as agricultural production and productivity is highly vulnerable to climate variables (Sani and Kemaw 2017). In addition, Assosa zone is characterized by erratic and unreliable rainfall, land degradation, low per capita, poor infrastructure development, vulnerable groups (landless and the poor without assets, very small and fragmented land holders, female- headed households, families with large size, drought and pest affected households) which cause low agricultural production and food deficit in the area (Asfir 2016; AZBARD (Assosa Zone Agriculture and Rural Development Office) 2015). To reverse the food insecurity situation, the government has been formulating and implementing long-term strategies (such as Agricultural Development Led Industrialization, Growth & Transformation Plan I and Growth & Transformation Plan II)—which takes ensuring food security as its core objective (FAO 2012). In addition, to reduce the incidence of food insecurity households use different kinds of coping mechanisms in order to improve their livelihood. As to Gemechu et al. (2015) finding there is improvement in food security status of households in the country that shows the role of improvement in livelihood assets as well as investment strategies and policies that promoted households food security and concluded that there is still room for improvement. But, the improvement programs to be effective, they should be supported by location specific empirical evidences (Van der Veen and Tagel 2011). To this end, there is limitation of information on the issue in the study area. Hence, this study analyzes extent of food insecurity and its determinants along with the coping mechanisms opted by households against food insecurity and shortage in the study area. Thus, it addresses what factors affect households' extent of food insecurity and what coping mechanisms have been used by households against food insecurity and shortage in the study area. Various studies conducted in Ethiopia mainly focused on food availability and access dimension (Girma 2012; Mesfin 2014; Nigatu 2011; Arega 2015; Okyere et al. 2013; Hussein and Janekarnkij 2013; Motbainor et al. 2016) and others adopted 24 h or seven day recall method to capture the utilization dimension (Gemechu et al. 2015; Zemedu and Mesfin 2014; Beyene & Muche, 2010) to address households' food security and its determinants. However, considering only the food availability and access measures do not fully address the actual food energy utilization by the households and the quality of the food consumed. In addition, the drawback of relying on seven day recall method is that as a part of developing countries the majority of rural households have weak access to formal education due to that they cannot accurately respond on the types and quantities of food items consumed. The novelty of this study is that it considered households' food consumption/utilization for seven consecutive days collected using weighed records method as food energy intake is sensitive to different unforeseen factors such as religion, weather, holidays, etc., which can be captured by taking weighted data..
The rest of the paper is organized as follows: section 2 provides the methodology employed; section 3 presents and discusses the results; and section 4 concludes and infers policy implications..
The study area
Assosa zone, the study area, is one of the three administrative zones in Benishangul-Gumuz region of western Ethiopia. Administratively, the study area is divided into seven districts, namely; Assosa, Homosha, Bambasi, Menge, Kurmuk, Sherkole and Odabildi-Guli districts. The zone has a total population of 283,707 people, out of which 144,616 and 139,091 are male and female, respectively. Furthermore, 86.28% of the population lives in rural area and 13.72% lives in urban area. The population density of the study area is 28 persons per kilometer square (BGRDGA (Benishangul Gumuz Region Development Gap Assessment) 2010). Mixed farming (crop production and livestock rearing) system is the main sources of livelihood for the majority of the population in the area. Crop production is dominated by rain fed agriculture while irrigation is practiced on small scale level. The major livestock reared in the area are cattle, donkey, goats, sheep and poultry (AZBARD (Assosa Zone Agriculture and Rural Development Office) 2015).
Sampling technique and sample size
The study employed three-stage random sampling method to select sample households. In the first stage, out of 7 districts in Assosa zone, three districts (namely Assosa, Bambasi and Sherkole) were randomly selected. In the second stage, a total of 12 peasant associations (PAs) were randomly selected using probability proportional to the number of PAs in each sampled districts. The reason for selecting PAs was that, in the study area almost all the households relied on agriculture and the emphasis of this study was on assessing the extent of food insecurity of households working on agriculture and their coping mechanisms. In the third stage, a total of 276 sample household heads were randomly selected based on probability proportional to size of the households in the selected PAs. The sample size for this study was determined by using Yamane formula (Yamane 1967).
$$ n=\frac{\mathrm{N}}{1+\mathrm{N}{(e)}^2}=\frac{40530}{1+\left(40530x{0.06}^2\right)}=276 $$
Where n = designates the sample size, N = designates total number of estimated household heads in the study area (40530) and e = designates maximum variability or margin of error (6%).
Data set and collection methods
For this study, primary data collected from sample households using interview schedule through the enumerators and the researchers was used. Particularly, primary data on the types and quantities of every food item consumed by the household head and his/her family members was collected using Weighed records method for 7 consecutive days from each sampled households. The reason for collecting the data from a single household for seven consecutive days was that food security is a sensitive issue that is affected by different unforeseen factors (religious, holidays, etc.) which can be captured by taking weighed data (Muche and Esubalew 2015). In addition to this, primary data on household's socio-demographic and socio-economic factors as well as on households' food insecurity and shortage coping mechanisms was obtained through interview schedule. Besides, focus group discussions and key informants interview were also employed to supplement the research finding with qualitative information.
Method of data analysis
To analyze the collected data, the study employed descriptive statistics, food insecurity index and Tobit model. Descriptive statistics such as mean, percentage and frequency were used to describe households' food kilocalorie intake status and to explore the coping mechanisms to food insecurity in the study area. Furthermore, the study used Foster, Greer and Thorbecke (FGT) food insecurity index in the computation of the incidence, depth and severity of food insecurity. This model is widely applicable in poverty analysis. It is a class of additively decomposable measure of poverty and food insecurity. Foster and Shorrocks (1991, 1988) branded the decomposable components of FGT measures as consistent poverty indices and argued that they make analysis of the poverty dominance easier. Particularly in food security analysis, the model is essential in analyzing the sources of change in food insecurity due to changes in the components i.e. to know the change in food insecurity is due to the incidence, or increasing deprivation of the food insecure, or because of kilocalorie short-fall below the food security line have become more unequal, or some combination of the above. Thus, in this study the model enables to estimate the three food insecurity indicators, namely the number of households below the food security line (headcount), the extent of the short-fall of the kilocalorie of the food insecure from the food security line (food insecurity gap) and the exact pattern of distribution of the kilocalorie of the food insecure households (squared food insecurity gap). Accordingly, the Foster et al. (1984) measure used in estimation of food insecurity index components is given as:
$$ \mathrm{FGT}\left(\upalpha \right)=\left(1/\mathrm{n}\right){\sum}_{\mathrm{i}=1}^{\mathrm{q}}{\left[\left(\mathrm{c}-\mathrm{yi}\right)/\mathrm{c}\right]}^{\upalpha} $$
Where: FGT (α) is the FGT food insecurity index; n is the number of sample households; yi is the measure of per adult equivalent food kilocalorie intake of the ith household; c represents the cut off between food security and food insecurity households (expressed here in terms of caloric requirements of 2100 kcal*); q is the number of food-insecure households; and α is the weight attached to the severity of food insecurity. Regarding estimation of the model, when the weight attached to α = 0 the measure is simply the headcount ratio (incidence); when α = 1 the measure is food insecurity gap (depth of food insecurity); and when α = 2 the measure is squared food insecurity gap (severity of food insecurity).
Moreover, Tobit model was estimated to analyze determinants of extent of households' food insecurity in the study area. Studies confirmed that, when a particular dependent variable assumes some constant value for some observations and a continuous value for the rest observations, the appropriate model will be a Tobit model developed by Tobin (1958) (Wooldridge 2002; Sisay and Edriss 2013; Agyeman et al. 2014; Bukenya 2017). Tobit is an extension of the probit model and it is one approach to deal with the problem of censored data (Johnston and Dinardo 1997). Thus, in this study the dependent variable was a censored variable in which it assumed a constant or threshold value of 2100 kcal/AE/day* for food secure households and the actual food energy intake in kilocalorie for food insecure households. Suppose, however, that Yi is observed if the latent variable Yi* < 2100 kcal and is not observed if Yi* > 2100 kcal. Then the observed Yi will be defined as:
$$ Yi=\left\{\kern0.75em \begin{array}{c}{Yi}^{\ast }=\beta Xi+ Ui\kern3.75em if\ {Yi}^{\ast }<2100\ kcal\kern1em \\ {}2100\ kcal\kern6em if\ {Yi}^{\ast}\ge 2100\ kcal\kern1em \end{array}\right. $$
Where: Yi* is the latent (unobserved) variable, Yi is the observed variable, Xi is vector of explanatory variables, Ui is a vector of error terms and β is a vector of parameters to be estimated.
*Note that 2100 kcal/AE/day is the threshold value of food security stated by FDRE (1996).
Operational definition of variables in the study
Extent of food insecurity
It is a limited dependent variable, taking the threshold value (2100 kcal) if the total food energy intake is greater than or equal to the threshold value and assumed the actual food energy intake for those households whose energy intake level is less than the threshold value. The quantity of food items consumed was converted to gram and the caloric content was estimated by using the nutrient composition table of commonly eaten foods in Ethiopia. Moreover, the estimated food energy was converted into adult equivalent and reached at figure of food calorie in kilo calorie/day/AE. Accordingly, household food calorie intake per day per adult equivalent (HFCi) was measured as:
$$ \mathrm{HFCi}=\frac{\mathrm{Total}\ \mathrm{calorie}\ \mathrm{consumed}\ \mathrm{by}\ \mathrm{a}\ \mathrm{household}}{\mathrm{Household}\ \mathrm{size}\ \mathrm{in}\ \mathrm{Adult}\ \mathrm{equivalent}\ast 7} $$
Nature of settlement of the household heads
This is a dummy variable used to indicate origin of household's. The variable took the value of 1 if respondents were settlers and 0 if natives. As depicted in Asfir (2016), unlike settlers, native households in the study area were highly resistant to accept new technologies. However, studies argued that adoption of new technologies improves agricultural production and productivity (Tsegaye and Bekele 2012) which in turn reduces households' exposure to incidence of food shortage and insecurity. In this study, this variable was hypothesized to affect extent of households' food insecurity negatively.
Sex of head of household
It is a dummy variable taking the value 1 if the sex of household is male and 0, otherwise. As to Baten and Khan (2010) finding, female-headed households can find it difficult than men to gain access to valuable resource, which helps them to improve production and gain more income, this in turn increases their probability of being food insecure. Thus, in this study, it was expected to affect extent of households' food insecurity negatively.
Age of head of household
It is a continuous variable measured in years. Many studies argued that young households' heads are stronger and energetic than elderly households as they are expected to cultivate larger-size farm and obtain high yield (Abafita and Kim, 2014; Babatunde 2007). Hence, in this study age of the household head was expected to affect extent of food insecurity negatively.
Educational level of head of household
It is a continuous variable measured in years of schooling of the household head. Education, which is a social capital, has a positive impact on household ability to take good and well-informed production and nutritional status (Babatunde 2007). Besides, Amaza et al. (2006) argued that households with higher years of schooling are less likely to be food insecure as it enables them to produce more and consume more. Thus, higher years of schooling was expected to affect extent of food insecurity negatively.
It is a continuous variable which refers to the number of family members of the household. Studies argued that larger family size tends to exert more pressure on households consumption than the labor it contributes to production (Stephen and Samuel 2013; Muche et al. 2014). Therefore, in this study, larger household size was expected to affect extent food insecurity positively.
It refers to the proportion of economically inactive labor force (less than 15 and above 65 years old) to the active labor force (between 15 and 65 years old (Velasco 2003). Due to scarcity of resources, higher dependency ratio imposes burden on the active and inactive member of household to fulfill their immediate food demands (Muche et al. 2014). Besides, higher dependency ratio indicates that the labor force is small, with a constraint on the household per capita income and consumption, which also influences the wellbeing of the household members (Nugusse et al. 2013). In this study, it was expected to positively affect extent of households' food insecurity.
Livestock ownership (excluding oxen and donkey)
It is a continuous variable measured by the number of Tropical Livestock Unit (TLU). Livestock are important source of food and income for rural households. Households with more livestock produce more milk, milk products and meat for direct consumption. Besides, livestock enable the farm households to have better chance to earn more income from selling livestock and livestock products which assist them to purchase stable food during food shortage and invest in purchasing of farm inputs that increase food production, and ensure household food security (Mitiku et al. 2012; Gemechu et al. 2015). Livestock possession mitigates vulnerability of households during crop failures and other calamities (Abafita and Kim, 2014). Thus, this study hypothesized that owning more TLU of livestock was expected to have negative effect on the extent of food insecurity of households.
Number of oxen and donkey owned
It is a continuous variable measured in numbers owned. Oxen and donkey serve as a source of traction power in many developing countries, thereby significantly affecting household's crop production. Animal traction power enables households to cultivate their land; others land through renting, share cropping, and execute agricultural operations timely that will enhance households access to food items (Muche et al. 2014). Accordingly, in this study more number of oxen and donkeys owned by a household was expected to affect the extent of food insecurity negatively.
Cultivated land size
It is a continuous variable which refers to the total land cultivated by a household in the past one year production period. A larger size of cultivated land implies more production and availability of food grains (Mitiku et al. 2012). Therefore, higher production and the increased availability of grains produced help to insure food security status of households (Asmelash 2015). Hence, the size of cultivated land was expected to have negative impact on extent of food insecurity.
Access to irrigation
It is a dummy variable taking the value 1 if the farmers have access to irrigation and 0, otherwise. Irrigation, as one of the technology options available, enables smallholder farmers to directly produce consumable food grains and/or diversify their cropping and supplement moisture deficiency in agriculture so that it helps to increase production and food consumption (Van der Veen and Tagel 2011). Thus, in this study, it was expected to have negative impact on extent of households' food insecurity.
Farm income
This is a continuous variable which measures the amount of income obtained from crop production and livestock rearing measured in US Dollar. According to Beyene and Muche (2010) finding, higher farm income earning enables farmers to purchase different nutritious food items to satisfy their family food demand. Thus, for this study, farm income was hypothesized to affect extent of households' food insecurity negatively.
Off/non-farm income
It is a continuous variable which measures the amount of cash income obtained by any household member from off-farm and non-farm activities measured in US Dollar. Studies argued that households with higher off-farm and non-farm income are less likely to be food insecure as it enables them to purchase different food items to satisfy their family needs (Beyene & Muche, 2010; Abafita and Kim 2014). Thus, off/non-farm income was expected to affect extent of food insecurity negatively.
Cost of inputs
It is a continuous variable measured in US Dollar by converting the amount of the agricultural inputs used (such as fertilizers, seeds, pesticides, chemicals, and other agricultural implements.) into monetary value based on their market price. Investing higher amount of money on farm inputs helps farmers to increase their crop production and livestock breeding (Arene and Anyaeji, 2010). In this study, it was expected to affect extent of households' food insecurity negatively.
It is a dummy variable that takes value 1 if a household gets access to agricultural related training and 0, otherwise. Formal agricultural training on modern technologies (proper types and rates of fertilizer application, improved varieties of seeds, agro-chemicals, etc.) helps farmers to get better production, and then this most likely leads to obtain more income to fulfill their family requirements by enhancing their agricultural production skills, knowledge and experiences (Yishak et al., 2014). Therefore, in this study, it was expected to affect extent of households' food insecurity negatively.
Frequency of extension contact
It is a continuous variable measured in number of visits by extension agent per year. More frequent extension contact enhances households' access to better crop production techniques, improved input as well as other production incentives, and thishelps to improve food energy intake status of households (Hussein and Janekarnkij 2013; Nugusse et al. 2013). Accordingly, in this study more number of extension contacts were expected to affect extent of households' food insecurity negatively.
Access to credit
It is a dummy variable, which takes the value 1 if the household had access to credit and 0 otherwise. Availability of credit eases the cash constraints and allows farmers to purchase inputs such as fertilizer, improved crop varieties, and irrigation facilities; which in turn enhance food production and ultimately increase household food energy intake (Stephen and Samuel 2013). In this study, it was expected to affect extent of households' food insecurity negatively.
Remittance and aid
It is a dummy variable, which takes the value 1 if the household had access to remittance and aid in the past one year and 0 otherwise. Both remittance and aid,from governmental and non-governmental organizations are important to smooth consumption in the case of shock and shortage for the time of emergency (Okyere et al. 2013; Mesfin 2014). Thus, for this study, it was expected to negatively affect extent of households' food insecurity.
Distance to market
it is a continuous variable measured in kilometer (km). Proximity to the market may create opportunity of more income by providing off/non- farm employment opportunities, which determine income level of rural households. In addition, the closer the farmer is to the market the more likely the farmer gets valuable information, purchase agricultural inputs and final products required for family consumption. Therefore, this variable was expected to positively determine households' extent of food insecurity.
Socio-demographic characteristics of households
For this study, a primary data collected from a total of 276 sampled household heads was used. From the total samples, 89.13% of household heads were male and the rest 10.87% were female, and this figure indicates that male headed households were owners of major livelihood assets as usual. In addition, 43.84% of the sampled household heads were settlers and the rest (56.16%) were natives, and it shows that more than half of the samples were drawn from natives. Regarding the marital status of the households, the majority (85.5%) of the households were married households followed by divorced (6.89%), widowed (4.34%) and single (3.27%) households. Furthermore, the age distribution of the households range from 23 to 78 years and the majorities were in 30–40 year age group (47.83%) and the least were in the age group of below 30 years (5.79%). Moreover, the majority (58.33%) of the respondents had a family member falling between 5 and 10 members group followed by < 5 member group (37.67%) and > 10 member group (4%). As to households' literacy status, the study indicated that 46.38% of the respondents had access to formal education (Table 1).
Table 1 Socio-demographic characteristics of the sampled households
The finding of the study also figured out that the majority (74.28%) of the households were relying on combining crop and livestock production as an economic activity followed by crop production alone (21.74%) and livestock production (3.98%). In addition, it showed that 56.16% of the sampled households had access to irrigation, indicating that in the study area more than half of the samples were beneficiaries of the irrigation water. Regarding the income earning from farming activities, 44.93% of the households were earning less than 117.65 USD followed by 117.65–235.29 USDincome group (16.3%), 235.29–411.76 USD income group (15.94%), 411.76–764.71 USD income group (12.68%) and greater than 764.71 USD (10.15%). Besides, the majority (62.68%) of the households was not engaged in any type of off-farm and non-farm activities and the rest (37.32%) were earning a positive income from off-farm and non-farm activities. From the total households, 36.96% cultivated a land size of ≤0.5 ha followed by between 0.5 -1 ha (33.70%), > 1 ha (25.36%) and 0 ha (3.98%). Furthermore, the study finding showed that 80.79% of the sampled households had no access to credit service in the study area, implying that the majority of the households did not recieve any type of credit from formal and informal sources. As to households access to remittance and aid, only 5.07% of the households had obtained remittance and aid from different sources. Moreover, 47.83% of households market distance from their residence was less than 5 km followed by distance falling between 5 and 10 km (26.45%) and greater than 10 km (25.72%) (Table 1).
Households food security and energy intake in the study area
In this study, data on the type and quantity of food items consumed by the household for seven consecutive days were collected using weighed records method, and it was converted to kilocalorie and then divided to household size measured in AE and number of days. Following this, the amount of energy utilized in kilocalorie by the household was compared with the minimum subsistence requirement per adult per day (i.e. 2100 kcal). Accordingly, households in the study area were mainly consuming food items of maize products (such as white porridge, white bread, 'injera', and whole roasted, white' kitaa'), wheat products (such as bread and 'kitaa'), and teff products (such as 'injera' and porriage). Besides, vegetables such as onion, cabbage, tomato, and green pepper as well as livestock and poultry products such as milk, meat, egg, cheese and butter were also consumed by the households. Moreover, the locally known food item called 'kenkes' and oil seed products were among the food items consumed by the households. After conversion of the food items consumed to kcal/AE/day, the result of the study revealed that 148 (53.62%) of the sampled households were found to be food insecure and 128 (46.38%) of the sampled households were food secure (Table 2). This implied that more than half of the households in the study area were food insecure. Regarding the food insecurity status within each district, it is found that 58.04% of households in Assosa district were food insecure. This indicates, in the district, the incidence of food insecurity was higher i.e. there were more number of food insecure households as compared to the food secure ones and it was mainly attributed to the incidence of pest outbreak in the 2016/17 production season which led to loss of thousands of quintals of crop production in the district. Furthermore, the study revealed that 48.89% and 48.84% of the households were food insecure in Bambasi and Sherkole districts, respectively (Table 2). Though more than half of the households were food secure, the state of food insecurity was high in both districts. Generally, the incidence of food insecurity was relatively higher in Assosa district as compared to the other two districts.
Table 2 Households food security status and its breakdown between districts
Moreover, the study finding indicated that the mean calorie intake of the sampled households was 1991.42 kcal per adult equivalent per day, which was lower than the minimum calorie requirement of 2100 kcal for a healthy and productive life, with maximum and minimum level of kilocalorie energy intake being 4286.91 and 597.65, respectively. Besides, the calorie intake of food insecure households ranges from 597.65 kcal and 2048.13 kcal with mean kilocalorie energy intake of 1440.37. The finding also revealed that the mean energy intake of food secure households was 2628.56 kcal per adult equivalent per day with the maximum and minimum energy intakes being 4286.91 and 2116.67 kcal per adult equivalent per day, respectively (Table 3).
Table 3 Summary of households' energy intake in the study area
Households extent of food insecurity in the study area
FGT food insecurity index was used to assess the extent of food insecurity in the study area. Thus, the finding of head count ratio from food insecurity index indicated that the incidence of food insecurity was 53.62%, and it indicated that 53.62% of the households were actually in the state of food insecurity, that is, unable to get the minimum recommended calorie for subsistence. The food insecurity gap, which is a measure of depth of food insecurity, pointed out that each food insecure household needed 16.84% of the daily caloric requirement to bring them up to the recommended daily caloric requirement level. This means, on average, the households need to be supplied with 16.84% of the daily minimum calorie requirement to get out of the food insecurity problem. The average extent of the calorie deficiency gap for the sampled households was, therefore, 353.64Kcal/AE/day; which means, on average 353.64Kcal/AE/day of additional food energy would be needed to lift the households out of food insecurity, then at least in theory, food insecurity could be eliminated. Moreover, the result of squared food insecurity gap from food insecurity index figured out that the severity of food insecurity in the study area was 7.32% (Table 4).
Table 4 FGT food insecurity index result on extent of food insecurity in the study area
Determinants of the extent of households food insecurity in the study area
Tobit model was estimated to analyze the determinants of the extent of households' food insecurity. Accordingly, results from the Tobit model using data obtained from 276 sample households (of which 128 were censored/food secure according to the model result) are presented in Table 5. The overall model is significant at 1% as indicated by the likelihood ratio test (Prob > χ2 = 0.0001). In addition, the model estimate revealed that out of the 18 explanatory variables, 7 variables were found to have a significant impact on households' extent of food insecurity. Thus, only statistically significant variables at less than 10% probability levels were discussed.
Table 5 Tobit model result on determinants of extent of food insecurity in the study area
Age of the household head
As expected, it affected household's level of energy intake negatively (extent of food insecurity positively) and significantly at 5% significance level in the study area. The marginal effect, from of the model result, indicated that a one year increase in the age, within food insecure households, increased the likelihood of household's extent of food insecurity by 448%. This implies that old aged household heads within food insecure households were more likely to face higher degree of energy intake deficiency than younger ones. This is because as age increases households become less productive and have less courage to cultivate larger-size farm than young ones. In addition, mostly elder households have large number of families and their resources are distributed among the members, and this imposes pressure on their income to purchase consumable products. This finding is in line with the finding of Bukenya (2017).
As expected, this variable negatively and significantly affected households' intensity of energy intake at 10% significance level. From the model output, the marginal effect revealed that one extra person in the household increased the probability of household's intensity of food energy intake deficiency by 1211%. This indicates that households with larger family size tend to be more food energy deficient than households with smaller family size in the study area. This is due to the reason that, households with large family size could be composed of large number of non-productive members; which imposes high burden on the labor force and food available to each person and ultimately end up with difficulty to achieve food security. This finding supports the finding of Stephen and Samuel (2013).
It affected households' extent of food energy intake positively (extent of food insecurity negatively) and significantly at 10% significance level. From the model result, the marginal effect showed that having access to irrigation increased food insecure households' likelihood of the extent of food energy intake by 7098%. This implies that households who had irrigation access were less likely to be food energy deficient than those who had no irrigation access, and the result supports the finding of Van der Veen and Tagel (2011). This is due to the fact that, access to irrigation helps households' to produce more than once in a year through mitigating water stress and reducing risks of crop failures and obtains more yields; thereby reducing the extent of food insecurity among the households.
Total farm income
As expected, it determined households' extent of food energy intake positively (extent of food insecurity negatively) and significantly at 5% significance level. From the model output, the marginal effect pointed out that a one birr (0.0588 USD) increase in farm income, within food insecure households, decreased the probability of their energy intake deficiency by 0.76%. This indicates that higher farm income earning households were less likely to energy deficient than low farm income earning households in the study area. This is because higher farm income helps the farmers to purchase diversified and nutritious food items which in turn helps them to improve their food energy intake status (Bukenya 2017; Mitiku et al. 2012).
Off-farm and non-farm income
In contrary to the expectation, it negatively and significantly affected households' extent of energy intake at 5% significance level. From the model result, the marginal effect confirmed that a one birr (0.0588 USD) increase in the off-farm and non-farm income increased the probability of food insecure households' food energy intake deficiency by 1.02%. This indicates that food insecure households with higher off-farm and non-farm income earning were more likely to be food energy deficient than low earning households in the study area. This is because, in the study area, households engaged in off-farm and non-farm income earning activities focus on accumulating physical and financial resources to improve their future wellbeing than spending their income on purchasing food products to satisfy their current food requirement, and this result supports the finding of Indris (2012).
As expected, it affected households' extent of energy intake positively and significantly at 1% significance level. The marginal effect, from the model result, showed that having access to credit decreased food insecure household's probability of food energy deficiency by 13,921%. This implies households who had access to credit service had less chance to be food energy deficient as compared to those who had no access to credit. This is due to the reason that, in the study area, households were receiving credit mainly in kind such as in the form of fertilizer, seed, herbicide, etc., from agricultural offices, and it enabled them to use their income in purchasing diverse and nutritious food items rather than various types of inputs to reduce the risk of high degree of food insecurity. Stephen and Samuel (2013) also reported similar finding.
As expected, this variable affected extent of household's food insecurity negatively and significantly at 5% probability level in the study area. From the model output, the marginal effect indicated that a one kilometer increase in the residence of households from the nearest market decreased the probability of food energy deficiency by 818%. This implies that food insecure households living near the market center were more likely to be energy deficient than those living far from the market center. This is because, in the study area, households living far from the market center were mainly producing consumable product items as compared to those households living close to market center who were producing cash crops.
Households coping mechanisms to food insecurity and shortage in the study area
Studies conducted in Ethiopia argued that households adopt a range of coping mechanisms during food insecurity and/or food shortage (Sewnet 2015; Arega 2015). The results of the study confirmed that households in the study area adopted diversified coping mechanisms at times of food shortage and/or food insecurity. Accordingly, 81.9% of the sampled households pursued reducing frequency of meal as a coping mechanism, followed by reducing the size of meal served (78.6%) and working as a daily laborer (68.1%). This implies that the majority of the households were adopting decreasing the number of meal serving time, size of meal and working as daily laborer as their coping mechanism to cope up with the risks of food shortage and/or food insecurity. Furthermore, the study also pointed out that 49.3, 48.6, 43.5, 37.7, 37.7, 35.9, 32.2, 29.7 and 5.07% of the sampled households were using sale fire wood and charcoal, engaging in wild fruit gathering, engaging in petty trade, selling livestock's, borrowing/loan, selling different assets, mining, migrating to cities and remittance and food aid, respectively, as coping mechanisms against food shortage and food insecurity in the study area (Table 6).
Table 6 Households coping mechanisms against food insecurity and/or shortage
Moreover, the study result revealed that reducing meal size was the most effective and most important coping mechanism used by the large segment of the households (36.6%), followed by reducing frequency of meal (27.9%) and working as a daily laborer (13.77%). In addition, the finding of the study showed that 11.23, 2.9, 2.9, 2.17, 1.09, 0.72, 0.36 and 0.36% of the sampled households adopted selling livestock's, remittance and food aid, migration, wild fruit gathering, selling wood and charcoal, receiving loan, selling different assets and engaging petty trade, respectively, as their most effective and important coping mechanisms against food shortage and food insecurity (Table 7). This finding supports the findings of Sewnet (2015), Birara et al. (2015) and Woldeamanuel (2009) which concluded that rural households pursued various coping mechanisms when food crisis hits them so as to reduce the risk associated with food insecurity.
Table 7 Households most effective coping mechanisms against food insecurity and food shortage in the study area
Food insecurity and poverty are critical and persistent problems facing most Ethiopians today. In an effort to reverse the incidence of these problems, different studies recommended that improving the livelihood of the rural poor plays a key role. The improvement programs in the welfare of rural community to be effective, they need to be supported by empirical evidences that provide important input on households' food security for concerned bodies. Thus, this study assessed households' extent of food insecurity and its determinants along with the coping mechanisms opted against food insecurity and shortage in Assosa zone using a data collected from 276 sample households.
Accordingly, the findings of the study pointed out that the incidence of food insecurity (53.62%) was high in the study area, with the depth and severity of food insecurity being 16.84% and 7.32%, respectively. This implies that more than half of the households in the study area were food insecure. In addition, the estimated Tobit model results revealed that farm income, access to irrigation, access to credit and distance to market negatively affected the extent of households' food insecurity; whereas age of the household head, family size and off-farm and non-farm income positively affected households extent of food insecurity. To cope up with the food insecurity and shortage situation, households opted reducing frequency of meal, reducing the size of meal served, working as a daily laborer, selling fire wood and charcoal, engaging in wild fruit gathering and petty trade as top six coping mechanisms in the study area.
Thus, urgent actions directed towards reducing and/or eliminating rural households' food insecurity in the study area should focus on:
Awareness creation on effective family planning and the impact of large family size on ensuring food security, and awareness creation and capacity building for elder households through ensuring the availability and dissemination of accurate information should be strengthened.
Enhancing rural household's access to credit as it enables them to purchase different inputs to improve their production and consumable products and thereby helps them to reduce and/or eliminate food insecurity and improve their wellbeing.
Construction of irrigation schemes as access to irrigation enables households to produce more than once in a year through reducing water stress and the risk of crop failure and thereby helps them to reduce and/or eliminate food insecurity.
Enhancing household's farm income-earning opportunities through provision of sufficient input to enhance agricultural production and productivity; and improving households' technical skill as well as their awareness on utilization of the off-farm and non-farm income to improve households' food security situation.
Even though, better access to markets assumed to reduce transport and other market related transaction costs, the study finding indicated the opposite. Therefore, enhancing households' awareness about the importance of better access to markets on their informed decision regarding their choice of output to be produced and products to be purchased in the market that helps the households to enhance their food security status in the near future.
Generally, as a policy implication the government should exhaustively work on promoting irrigation, facilitating credit availability and subsidize the farmers to reverse the problem of food insecurity and to enhance households coping capacity to food shortage and/or insecurity. Besides, this study has attempted to come up with the result of the analysis with defined scope however a lot remained to be unanswered. To provide basic information on the determinants of food security and extent of food insecurity, the social, political, natural and environmental dimensions, descriptive data on purchasing patterns of food insecure, specific characteristics that make rural poor more vulnerable to food insecurity demands future researchers' attention.
AE:
Adult equivalent
AZBARD:
Assosa Zone Bureau of Agriculture and Rural Development
BGRDGA:
Benishangul Gumuz Region Development Gap Assessment
CSA:
Central Statistical Agency
FAO:
Food and Agriculture Organization
FDRE:
Federal Democratic Republic of Ethiopia
FGT:
Foster, Greer and Thorbecke
HFCi:
Household food calorie intake
kcal:
Kilocalorie
Peasant Association
USGAO:
United State Government Accountability Office
WFP:
World Food Program
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The authors are grateful to Assosa University for providing both financial and technical assistance in the research work. We also sincerely thank the local communities in our research area, Assosa Zone, and all the enumerators for their valuable efforts. Furthermore, we also thank Assosa university Agricultural Economics staff members for their valuable assistance during the study.
Assosa University supported this research work both financially and logistically.
The data that support the findings of this study can be obtained from the authors based on request.
Department of Agricultural Economics, College of Agriculture and Natural Resource, Wolkite University, Wolkite, Ethiopia
Seid Sani
Department of Agricultural Economics, College of Agriculture and Natural Resource, Debre Brehan University, Debre Brehan, Ethiopia
Biruk Kemaw
The idea and design of the study was generated by both authors. Both of the authors carried out the data collection and data entry. The first author carried out the data analysis and write-up. The second author read and revised the manuscript. Both authors read and approved the final manuscript.
Correspondence to Seid Sani.
Sani, S., Kemaw, B. Analysis of households food insecurity and its coping mechanisms in Western Ethiopia. Agric Econ 7, 5 (2019). https://doi.org/10.1186/s40100-019-0124-x
DOI: https://doi.org/10.1186/s40100-019-0124-x
Coping mechanism
Tobit model
Assosa zone | CommonCrawl |
\begin{document}
\title{Mixed-mode oscillations in coupled FitzHugh-Nagumo oscillators: \ blow-up analysis of cusped singularities}
\begin{abstract} In this paper, we use geometric singular perturbation theory and blowup, as our main technical tool, to study the mixed-mode oscillations (MMOs) that occur in two coupled FitzHugh-Nagumo units with \response{symmetric and} repulsive coupling. In particular, we demonstrate that the MMOs in this model are not due \response{to} generic folded singularities, but rather due to singularities at a cusp -- not a fold -- of the critical manifold. Using blowup, we determine the number of SAOs analytically, showing -- as for the folded nodes -- that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a (\response{symmetric}) saddle-node bifurcation in the desingularized reduced problem, which -- although resembling a folded saddle-node (type II) at this level -- also occurs on a cusp, and not a fold. We demonstrate that this bifurcation is associated with the emergence of an invariant cylinder, the onset of SAOs, as well as SAOs of increasing amplitude. We relate our findings with numerical computations and find excellent agreement. \end{abstract}
\begin{keywords}
Mixed-mode oscillations, cusp, folded singularities, canards, blowup, geometric singular perturbation theory. \end{keywords}
\begin{AMS}
34C15, 34D15, 37E17 \end{AMS}
\section{Introduction} Coupled nonlinear oscillators are ubiquitous in physics, chemistry, biology and many other contexts. Interestingly, the collective behavior of the population \response{of oscillators} may exhibit qualitatively different dynamics that the individual units would if uncoupled. Coupling may, e.g., lead to oscillator death \cite{bar85,ermentrout90} or, on the contrary, promote oscillatory activity \cite{gregor10,wang92,weber12}. In neurons and other cells capable of exhibiting complex {bursting} electrical activity, gap junction coupling can change the cellular behavior from a simple action potential firing to bursting \cite{sherman92,sherman94,devries00,pedersen05b,loppini15} and lead to large increases in the burst period \cite{loppini18}.
A particular kind of complex dynamics, also observed in models of cellular electrical activity, consist of mixed-mode oscillations (MMOs) where small- and large-amplitude oscillations (SAOs and LAOs, respectively) alternate \cite{brons06,desroches12}. Such dynamics is caused by cellular mechanisms operating on different time scales and can be seen, e.g., in the classical Hodgkin-Huxley model \cite{hodgkin52} for neuronal action potential generation \cite{rubin07}, and in experimental data and models of cortical neurons \cite{gutfreund95}, stellate cells \cite{dickson00,rotstein06}, neuroendocrine cells \cite{tabak07,vo10,riz14,battaglin21}, cardiac cells \cite{yaru21,kimrey20}, among others. The mathematical structure causing MMOs is increasingly well understood by Geometric Singular Perturbation Theory (GSPT henceforth) \cite{fen3,jones_1995} and often involves folded singularities, canard orbits \cite{szmolyan_canards_2001,wechselberger_existence_2005} and singular Hopf bifurcations \cite{guckenheimer08}, which are generally related to saddle-node bifurcations in the fast subsystem when treating slow variables as parameters \cite{brons06,desroches12}.
In our recent study of coupled bursting oscillators \cite{pedersen22}, we revisited the finding by Sherman \cite{sherman94} who showed that coupling of spiking cells can lead to bursting via slow desynchronization so that each burst is preceded by a large number of full action potentials (spikes). Before the transition to bursting, the averaged membrane potentials show SAOs reflecting amplitude-modulated spiking \cite{pedersen22}. We showed that the dynamical structure of the system obtained by averaging was captured by a system of two coupled FitzHugh-Nagumo (FHN) units \cite{fitzhugh61,nagumo62}: \begin{equation}\eqlab{fhn} \begin{aligned}
\dot v_1&=-v_1^3 +3v_1-w_1+g(v_2-v_1),\\
\dot v_2&=-v_2^3+3v_2-w_2+g(v_1-v_2),\\
\dot w_1 &=\epsilon(v_1-c),\\
\dot w_2 &=\epsilon(v_2-c), \end{aligned} \end{equation} with \response{symmetric and} repulsive coupling $g<0$, and that this simple system exhibits MMOs organized by a singular Hopf bifurcation related to a folded \response{singularity} \cite{guckenheimer08}. However, this \response{bifurcation was not related to a transcritical bifurcation of a folded node (the folded saddle-node \cite{krupa2010a}), as is typically seen in applications \cite{desroches12}, but rather to a cusp catastrophe in the fast subsystem. This observation motivated the current study of what we will refer to as cusped singularities.}
\subsection{Background}\label{sec:2} \response{In this paper, we continue our study of the two identical FHN units \eqref{fhn}. We start by highlighting three separate properties. Firstly, for $g=0$ then $(v_1,w_1)$ and $(v_2,w_2)$ decouple as a Lienard equation: \begin{equation}\eqlab{fhnuncp} \begin{aligned}
\dot v_i &= -v_i^3 +3v_i-w_i,\\
\dot w_i &=\epsilon (v_i-c), \end{aligned} \end{equation} for $i=1,2$, and the dynamics of each pair is identical, being oscillatory (through relaxation oscillations) for $c\in (-1,1)$ and nonoscillatory (through a globally attracting equilibrium) for $c>1$ (and $c<-1$) for all $0<\epsilon\ll 1$ \cite{fitzhugh61,izhikevich07}, see \figref{uncoupled}. }
\response{Secondly, the system \eqref{fhn} is symmetric with respect to $$\mathcal S:\,(v_1,v_2,w_1,w_2)\mapsto (v_2,v_1,w_2,w_1),$$ so that $v_1=v_2$, $w_1=w_2$, being the fixed point of this symmetry, defines an invariant subspace. This will play an important role in the following. }
Finally, there is a unique equilibrium \begin{align}\eqlab{eqtrue} q:\quad (v_1,v_2,w_1,w_2)= (c,c, -c^3+3c, -c^3+3c) , \end{align} of \eqref{fhn} and this point $q$ lies on the symmetric subspace defined by $v_1=v_2$, $w_1=w_2$. The Jacobian evaluated at $q$
has eigenvalues \begin{eqnarray} \nu_{1,2} &=& \frac{3-3c^2 \pm \sqrt{(3-3c^2)^2 -4\epsilon} }{2}\ ,\\ \nu_{3,4} &=& \frac{3-3c^2-2g \pm \sqrt{(3-3c^2-2g)^2 -4\epsilon} }{2}. \end{eqnarray} Let \begin{align} v_s(g):=\sqrt{1-\frac23g}.\eqlab{vs} \end{align}
Then a Hopf bifurcation occurs for $c=v_s(g)$ where $\nu_{3,4}$ are purely imaginary ($ \pm i\sqrt{\epsilon}$).
In fact, a direct calculation (\response{based upon center manifold and normal form theory, see specifically \cite[Equation 3.4.11]{guckenheimer97} and \remref{lyapunov} below}) shows that the associated first Liapunov number is given by \begin{align}\eqlab{lyapunov} l_1(\epsilon) = \frac{3(g-3)}{8g\sqrt{\epsilon}}(1+\mathcal O(\epsilon)). \end{align}
Seeing that $l_1(\epsilon)>0$ for $g<0$ and all $0<\epsilon\ll 1$, it follows that we have a (singular) subcritical Hopf bifurcation \cite{guckenheimer97,guckenheimer08} for all $\epsilon>0$ small enough.
In \cite{pedersen22}, it was observed numerically that the system \eqref{fhn} for \response{$c<v_s$ but $c\approx v_s$} and $0<\epsilon\ll 1$ exhibits mixed-mode oscillations (MMOs) with \response{an increasing number of} small-amplitude oscillations (SAOs) as $c$ approaches $v_s$ from below, see \figref{MMO}, \response{\figref{MMOzoom},} \figref{MMO_v1v2} and the figure captions. \response{Following large-amplitude oscillations (LAOs), the two cells almost synchronize ($v_1\approx v_2$). However, as the voltages approach $c$, they begin to diverge as they spiral apart, creating SAOs with increasing amplitudes before departing into additional large-amplitude excursions. Interestingly, for some $c$ values, e.g., $c=1.27$, there is an alternation between $v_1$ and $v_2$ being increasing (and $v_2$, respectively, $v_1$ being decreasing) at the beginning of the LAOs, whereas this does not occur for other $c$ values. We will show that this phenomenon corresponds to the system leaving the neighborhood of the cusped singularity in different directions, a behavior which is not possible for the standard folded node. We return to this point in the Conclusions.
}
\begin{figure}
\caption{SAOs near the folded singularity $(v_1,v_2,w_1,w_2)=(v_s,v_s,w_s,w_s)$ where $w_s=-v_s^3+3v_s$ (blue dot; see \propref{foldedsing}). The full-system saddle point \eqref{eqtrue} is shown as a red asterisk.
In the lower panels, the black and gray curves show, respectively, $(w_1,v_1)$ and $(w_2,v_2)$.
\response{Insets show zooms on the SAOs near the saddle point.}
Parameters as in \figref{MMO}.
}
\end{figure}
\subsection{Biophysical motivation and implications}\seclab{motivation} \response{Since the FHN model is a simplification of the Hodgkin-Huxley model for neuronal activity, our results have implication for neuroscience beyond providing insight into our previous study \cite{pedersen22}.}
\response{Negative coupling ($g<0$) resembles mutual inhibition, for example between neuronal populations, see e.g. Curtu and Rubin \cite{curtu10,curtu11}. These authors showed that MMOs can appear as a result of inhibition via a singular Hopf bifurcation.
Mutual inhibition has also been used to explain ``binocular rivalry", where
perception alternates between different images presented to each eye
\cite{laing02}.
As explained above, our choice of $c>1$ (similarly, one could consider $c<-1$) means that the FHN neurons are silent when uncoupled. Our results show that repulsive coupling can induce oscillatory activity in such otherwise silent neurons via MMOs related to cusped singularities. These results mimic previous findings for ``release'' and ``escape" mechanisms generating oscillations in a couple of inhibitory non-oscillatory neurons \cite{wang92}. We do not consider $-1<c<1$ or $g>0$ since the system in these cases does not present a cusped singularity producing SAOs, which is the main topic of the manuscript. }
\subsection{Main results}\seclab{mainres}
In this paper, we describe the origin and mechanisms underlying the MMOs in \figref{MMO}, \response{\figref{MMOzoom} and \figref{MMO_v1v2}}. In particular, we show that the coupled system \eqref{fhn} possesses a degenerate folded singularity in the singular limit $\epsilon\rightarrow 0$, and demonstrate -- through a center manifold computation -- that this singularity corresponds to a cusp, see also \figref{cusp_new} and the figure caption for details. Moreover, by performing a detailed blow-up analysis, we dissect the details of the dynamics near this new type of singularity and show that it lies at the heart of the mechanism causing SAOs. As for folded singularities \cite{szmolyan_canards_2001}, we divide our analysis into two parts: one part covering the generic case (i.e. without an additional unfolding parameter) and one covering the bifurcation in the presence of an unfolding parameter. Since, the former case resembles the folded node \cite{wechselberger_existence_2005} -- in particular, we will show that the number of SAOs is also determined by the Weber equation and the ratio of eigenvalues -- we will refer to this singularity as the ``cusped node singularity''. Similarly, our results also show that the degenerate case, which we name the ``cusped saddle-node singularity'', as the classical folded saddle-node \response{(of type II \cite{krupa2010a})}, marks the onset of SAOs in the coupled FHN system.
\response{\response{For any $b\ge 0$, let $\lfloor b \rfloor,$ denote the largest integer $n\in \mathbb N_0$ such that $n\le b$.} We then summarize our findings on the SAOs in the following theorem (we refer to \thmref{main1} and \thmref{main2} for more detailed versions). \begin{theorem}\thmlab{thm0} Consider \eqref{fhn} with $g<0$ and suppose \begin{align}
c\in \left(\frac{1-\frac12 g}{\sqrt{1-\frac{2g}{3}}},v_s\right).\eqlab{cinterval0}
\end{align}
Then we have (\textnormal{the cusped-node case}):
\begin{enumerate}[series=part1]
\item There is a desingularization of the reduced problem on the critical manifold $C$ of the slow-fast system \eqref{fhn} such that the system has an attracting singularity $f_1$, given by
\begin{align*}
v_i = v_s, \quad w_i = w_s,\quad \text{for}\quad i=1,2,
\end{align*}
for $w_s:=-v_s^3+3 v_s$,
that lies on a cusp of $C$. Moreover, the linearization around $f_1$ has the following eigenvalues
\begin{align}
\lambda_1: = -6v_s(v_s-c),\quad \lambda_2: = -\lambda_1 +2g. \eqlab{eigvals} \end{align} with $\lambda_2<\lambda_1<0$ for the values in \eqref{cinterval0}.
The point $f_1$ is therefore a stable node for the desingularized system; specifically, it (locally) attracts all points on the attracting subset of $C$.
\item \label{item2} Orbits of \eqref{fhn} that pass through $f_1$ for $0<\epsilon\ll 1$ will (in general) undergo SAOs around the symmetric subspace $v_1=v_2$, $w_1=w_2$ before leaving a neighborhood of $f_1$.
\item Suppose that $ \frac{\lambda_2}{\lambda_1}\notin \mathbb N$. Then the amplitude of these SAOs is of the order $\mathcal O(\epsilon^{\frac{\lambda_2}{2\lambda_1}})$ and the number of SAOs is given by $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor$ many full $180^\circ$ rotations around the symmetric subspace $v_1=v_2$, $w_1=w_2$, for all $0<\epsilon\ll 1$.
\end{enumerate}
Next, fix any
\begin{align}
c_2 \in \left(-\frac{1}{3v_s},0\right),\eqlab{c2interval0}
\end{align} and consider \begin{align}
c = v_s+\sqrt{\epsilon} c_2.\eqlab{c2scaling0} \end{align} Then we have (\textnormal{the cusped-saddle node case}):
\begin{enumerate}[resume*]
\item The singularity $f_1$ of the desingularized reduced problem on $C$ is a saddle-node for $c=v_s$ with $\lambda_2<\lambda_1=0$, locally attracting all points on the attracting subset of $C$.
\item \label{item5} With $c$ as in \eqref{c2scaling0} and $c_2$ fixed in \eqref{c2interval0}, the regular singularity $q$, given by \eqref{eqtrue}, is of saddle-focus type for all $0<\epsilon\ll 1$, having a two-dimensional unstable manifold with focus-type dynamics.
\item Orbits of \eqref{fhn}, with $c$ as in \eqref{c2scaling0} and $c_2$ fixed in \eqref{c2interval0}, that pass through $f_1$ for $0<\epsilon\ll 1$ will (in general) undergo SAOs around the symmetric subspace $v_1=v_2$, $w_1=w_2$, before leaving a neighborhood of $f_1$.
\item There are finitely many of the SAOs that are $\mathcal O(\epsilon^{1/4})$ in amplitude as $\epsilon\rightarrow 0$.
\item The number of SAOs with an amplitude that is exponentially small with respect to $\epsilon>0$ is unbounded as $\epsilon\rightarrow 0$.
\end{enumerate}
There are no SAOs for $c_2>0$ and all $0<\epsilon\ll 1$. \end{theorem} }
Despite the similarities between the folded and cusped versions of the singularities, we will also illuminate some differences. For example, we show that the cusped saddle-node is intrinsically related to a regular Lienard equation in the same way that the FSN is related to the canard explosion. See \lemmaref{lienard} and \propref{contractg2} for details.
\begin{remark}\label{remark1} The reference \cite{krupa14} also considers coupled oscillators in four dimensions, including systems like \eqref{fhn}. The focus is (also) on emergence of MMOs in these types of systems. However, in contrast to our work, the singularities in \cite{krupa14} are folded and not ``cusped'' and the analysis of the coupled FitzHugh-Nagumo system, see \cite[Section 4]{krupa14}, also focuses on the attractive coupling $g>0$. In the present paper, we will only consider $g<0$. \end{remark}
\subsection{Numerical results}\seclab{numerics}
\response{To illustrate \thmref{thm0}, we compare the theoretical results to numerical simulations.
We define \begin{equation}\eqlab{yuz} u=\frac12(v_1-v_2),\quad y=\frac12(w_1+w_2)-w_s, \quad z=\frac12(w_1-w_2). \end{equation}
In \thmref{thm0}, we count the number of SAOs as $180^\circ$-rotations around the symmetric subspace. In the coordinates \eqref{yuz}, the symmetric space corresponds to $u=0$, $z=0$, so when projected onto the $(u,z)$-plane, SAOs correspond to full $180^\circ$ rotations around the origin.
Therefore we have the following: \textit{The number of SAOs is one less than the number of zeros of $u=0$}. We illustrate this in \figref{c124} for $c=1.24$, $g=-1$ and $\epsilon=0.01$. Here $\frac{\lambda_2}{\lambda_1}=4.06$ and we find five simple zeros of $u$ and in agreement with \thmref{thm0} precisely four full $180^\circ$-rotations. }
\response{On the other hand, \figref{twists_ampl}A shows a typical orbit of the system \eqref{fhn} in the $(u,y,z)$-space for $c=1.28$, $g=-1$, $\epsilon=0.01$. It corresponds to the cusped saddle-node case. The orbit approaches \response{the symmetric subspace $u=0$, $z=0$ denoted by $\gamma$} (red dotted line), and moves towards and beyond the cusped singularity $f_1$ located at the origin (blue dot), coming close to the saddle-focus point $q$ (red asterisk) before spiralling outwards.
To find the number of SAOs as a function of $c$, we counted the number of zeros of $u=0$ as asymptotes of $z/u$ (see \figref{twists_ampl}B) for a range of $c$ values. These numerical results were then plotted against the theoretical values of \response{item 3~in \thmref{thm0}}
using the explicit expressions for the eigenvalues \eqref{eigvals}, see \figref{twists_ampl}C.}\response{ The correspondence is excellent with minor discrepancies for $c$ values in the interval given by \eqref{c2interval0}-\eqref{c2scaling0}, the cusped-saddle node region, where \thmref{thm0} predicts an unbounded number of exponentially small SAOs as $\epsilon\rightarrow 0$ (see \thmref{thm0}, item 8). This is not a surprise, as \thmref{thm0}, item 3, assumes that $c$ is uniformly bounded away from $v_s$. The increment in amplitude of the SAOs as $c$ increases \response{and enters the cusped-saddle node region}, due to focus dynamics near $q$, see \thmref{thm0}, items 5 and 6, is also confirmed by the simulations (\figref{twists_ampl}D).}
\subsection{Overview}\seclab{overview} In Section \ref{sec:3}, we first study \eqref{fhn} as a singular perturbation problem for $\epsilon\rightarrow 0$ using GSPT. Specifically, we present a complete analysis of the reduced problem for any \response{fixed $g<0$} and describe all bifurcations for $c>0$ at the singular level. This then leads to a local three-dimensional center manifold reduction (\response{with parameters $\epsilon,c$ and $g$}) in \propref{cmred}. \response{In \lemmaref{cuspsing}, we then show that the critical manifold of this reduced system has a cusp singularity}. In Sections \ref{sec:4} and \ref{sec:5}, we proceed to study the dynamics near the cusped node and the cusped saddle-node singularity, respectively, by using the blowup method \cite{dumortier1996a,krupa_extending_2001} as the main technical tool. \response{This leads to \thmref{main1} and \thmref{main2} describing the SAOs in the two scenarios. The two theorems imply \thmref{thm0}.} In Section \ref{sec:final}, we conclude the paper.
\section{GSPT-analysis of \eqref{fhn}}\label{sec:3} To analyze \eqref{fhn} as a slow-fast system, we first study the layer problem and the reduced problem. The layer problem is obtained by setting $\epsilon=0$ in \eqref{fhn}: \begin{equation}\eqlab{layer} \begin{aligned}
\dot v_1&=-v_1^3 +3v_1-w_1+g(v_2-v_1),\\
\dot v_2&=-v_2^3+3v_2-w_2+g(v_1-v_2),\\
\dot w_i &=0,
\end{aligned}
\end{equation} for $i=1,2$. On the other hand, the reduced problem, \response{given by}: \begin{equation}\eqlab{reduced} \begin{aligned}
0 &= -v_1^3+3v_1-w_1+g(v_2-v_1),\\
0 &=-v_2^3 +3v_2-w_2+g(v_1-v_2),\\
w_1'&=v_1-c,\\
w_2'&=v_2-c, \end{aligned} \end{equation} is obtained by setting $\epsilon=0$ in the slow time ($\tau = \epsilon t$) version of \eqref{fhn}: \begin{equation}\nonumber \begin{aligned}
\epsilon v_1'&=-v_1^3 +3v_1-w_1+g(v_2-v_1),\\
\epsilon v_2'&=-v_2^3+3v_2-w_2+g(v_1-v_2),\\
w_1' &=v_1-c,\\
w_2' &=v_2-c, \end{aligned} \end{equation} where $()'=d/d\tau$. In the following, we will analyze \eqref{layer} and \eqref{reduced} successively.
\subsection{Analysis of the layer problem \eqref{layer}} The equilibria \response{of} the layer problem are given by \begin{align*}
0 &= -v_1^3+3v_1-w_1+g(v_2-v_1),\\
0 &=-v_2^3 +3v_2-w_2+g(v_1-v_2). \end{align*} This defines a two-dimensional critical manifold \response{$C$} of \eqref{layer} in the four-dimensional phase space. The manifold $C$ can be written as a graph $w=h(v)$ over $v$ where $h=(h_1,h_2)$ with \begin{align*}
h_1(v_1,v_2):=-v_1^3+3v_1+g(v_2-v_1),\\
h_2(v_1,v_2):=-v_2^3+3v_2+g(v_1-v_2). \end{align*} We determine the stability of $C$ by linearizing the layer problem \eqref{layer} around any point $(v,h(v))\in C$. It is a basic fact, that the nontrivial eigenvalues are given by the eigenvalues of the Jacobian \begin{align*}
Dh(v_1,v_2) = \begin{pmatrix}
-3v_1^2-g+3 & g\\
g &-3v_2^2-g+3
\end{pmatrix}. \end{align*} \response{The matrix is symmetric, so the eigenvalues are real. Moreover, we have} \begin{align*}
\text{tr}\,Dh &= -3(v_1^2+v_2^2)+6-2g,\\
\text{det}\,Dh &=9 v_1^2v_2^2 \response{-}3(3-g) (v_1^2+v_2^2)+3(3-2g). \end{align*}
\response{Consequently, $\text{tr}\,Dh=0$ defines a circle centered at $(0,0)$ with radius $\sqrt{2-\frac23 g}$. On the other hand, $\text{det}\,Dh=0$ can be written in the polar coordinates $(r,\theta)$: $v_1=r\cos \theta$, $v_2=r\sin \theta$ as \begin{align}
\cos^2(\theta) \sin^2(\theta) r^4 -(1-\frac13 g)r^2 +1-2/3g=0,\eqlab{eqnr2} \end{align} which is a quadratic equation in $r^2$. } \response{ \begin{lemma}\lemmalab{mums} Consider \eqref{eqnr2} as an equation for $r>0$ and suppose that $g<0$. Then for each $\theta\ne n\pi/2$, $n\in \mathbb Z$, there exists two solutions $r=m_u(\theta)$ and $r=m_s(\theta)$ with \begin{align} 0<m_u(\theta)<\sqrt{2-\frac23 g}<m_s(\theta),\eqlab{ineq} \end{align} where
\begin{align*}
m_u:&\, \mathbb R\rightarrow \mathbb R_+,\quad m_s:\,\mathbb R\backslash \{\theta\ne n\pi/2,n\in \mathbb Z\} \rightarrow \mathbb R_+,
\end{align*}
are smooth functions. For $\theta= n\pi/2$, $n \in \mathbb Z$, there is only one solution and it is given by $r=m_u(\theta)$. Finally, for each $n\in \mathbb Z$:
\begin{align*}
m_s(\pi/4+n\pi/2) = \min m_s = v_s,\quad
m_s(\theta)\rightarrow \infty\quad \text{for}\quad \theta\rightarrow n\pi/2.
\end{align*}
\end{lemma} \begin{proof}
Follows from a direct calculation. In particular, $\theta= n\pi/2$, $n \in \mathbb Z$ are the values where the coefficient of $r^4$ vanishes. In order to obtain \eqref{ineq}, we use that
the curves defined by $\text{tr}\,Dh=0$ (a circle with radius $r=\sqrt{2-\frac23 g}$) and $\text{det}\,Dh=0$ do not intersect. To see this, one can use that
$$\text{tr}\,(Dh)^2-4\text{det}\,Dh=9(v_1^2-v_2^2)^2+4g^2>0.$$
\end{proof}}
The expressions for $m_{u,s}$ are not important and therefore left out.
\response{Following this lemma, we now define} $C_{RN}$, $C_{AN}$ and $C_S$ as the subsets of $C$ with $0\le r<m_u(\theta)$, $r>m_s(\theta)$ and $m_u(\theta)<r<m_s(\theta)$, respectively, in the polar coordinates $(r,\theta)$. Let also $F_i$ be the subset of $C$ defined by $r=m_i(\theta)$, $i=u,s$. Then \begin{align*}
C = C_{RN} \cup F_u\cup C_S \cup F_s \cup C_{AN}. \end{align*}
We then conclude the following \response{(see \figref{lemma1} for an illustration}): \begin{lemma}\lemmalab{Fi} $F_i$, $i=u,s$ are sets of loss of normal hyperbolicity, but (since $\textnormal{tr}\,Dh\gtrless 0$ on $F_i$) the linearization along $F_u$ has one positive eigenvalue whereas the nontrivial eigenvalue of the linearization along $F_s$ is negative. Moreover, we have the following classification.
\begin{itemize}
\item Suppose $(v,h(v))\in C_{RN}$. Then the eigenvalues of $Dh(v)$ are both positive and real and $v$ is therefore a \textnormal{repelling node} for the fast subsystem of \eqref{layer}. \item Suppose $(v,h(v))\in C_{AN}$. Then the eigenvalues of $Dh(v)$ are both negative and real and $v$ is therefore an \textnormal{attracting node} for the fast subsystem of \eqref{layer}. \item Suppose $(v,h(v))\in C_{S}$. Then the eigenvalues of $Dh(v)$ are real and have opposite signs and $v$ is therefore a \textnormal{saddle} for the fast subsystem of \eqref{layer}. \end{itemize} In particular, on $C_{RN}$ and $C_{AN}$, $\textnormal{det}\,Dh >0$ whereas $\textnormal{det}\,Dh<0$ on $C_S$. \end{lemma} \subsection{Analysis of the reduced problem \eqref{reduced}} The reduced problem \eqref{reduced} is defined on $C$. Since $C$ is a graph over $v$, we will write this system in terms of $v$ instead of $w$. This gives \begin{align}
Dh v' &=v-\mathbf{c},\eqlab{reducedC0} \end{align} with \begin{align}\eqlab{tbfx} \textbf{c}:=(c,c),\end{align} a notation we adopt in the following. Using the \response{adjugate} matrix \begin{align*}
\text{adj}\,Dh(v)= \begin{pmatrix}
-3v_2^2-g+3 & -g\\
-g & -3v_1^2-g+3
\end{pmatrix}, \end{align*} of $Dh$, we may write this equation in the following equivalent form: \begin{align}
\text{det}\, Dh(v)\,v' &=\text{adj}\,Dh(v)\,( v-\mathbf{c}).\eqlab{rp} \end{align} On $C_{RN}\cup C_{AN}$, where $\text{det}\,Dh>0$, recall \lemmaref{Fi}, we are therefore led to consider the equivalent system \begin{align}\eqlab{desrp}
\dot v &=\text{adj}\,Dh(v)\,( v-\mathbf{c}). \end{align} Since $\text{det}\,Dh<0$ on $C_S$, the \textit{desingularized system} \eqref{desrp} is also equivalent to the reduced problem on $C_{S}$ upon time reversal. Folded singularities, which organize SAOs and canard trajectories connecting attracting and repelling sheets of \response{the} critical manifold, are equilibria of \eqref{desrp} on $F_i$ where $\text{det}\,Dh =0$. \response{We then state and prove the following result, see also \figref{eq}}. \begin{proposition}\proplab{foldedsing}
Consider \eqref{rp} with $g<0$. \response{Then there is a regular singularity $q(c)$ at $v=\mathbf{c}$ for any $c>0$ and at most four folded singularities:
\begin{enumerate}
\item There are two folded singularities $f_1$ and $f_2$ that exist for all $c>0$, occur on the symmetric subspace defined by $v_1=v_2$, and are given by $v=\pm \mathbf{v}_s$ on $F_s$ where $$v_s(g)=\sqrt{1-\frac{2g}{3}}.$$
\item \label{cle1} For $c<1$, then there are two separate folded singularities $f_3(c)$ and $f_4(c)$ that both lie on $F_u$, but outside the symmetric subspace (i.e $v_1\ne v_2$ along these), and are given by the equations
\begin{equation}\eqlab{q3q4}
\begin{aligned}
\frac12 (v_1+v_2) &= \frac{gc}{3c^2+g-3},\\
\frac14 (v_1-v_2)^2& = \left(\frac12 (v_1+v_2)-c\right)^2 +1-c^2.
\end{aligned}
\end{equation}
\item \label{cge} For $c>\frac{1-\frac{g}{3}}{\sqrt{1-\frac{2g}{3}}}$, $c\ne \sqrt{1-\frac{g}{3}}$, then $f_3(c)$ and $f_4(c)$ (again given by the equations \eqref{q3q4}) are nonsymmetric folded singularities, now located on $F_s$. $f_3(c)$ and $f_4(c)$ go unbounded as $c\rightarrow \sqrt{1-\frac{g}{3}}$.
\end{enumerate}
The point $q(c)$ undergoes two pitchfork bifurcations of \eqref{desrp} at $c=1$ and $c=\frac{1-\frac{g}{3}}{\sqrt{1-\frac{2g}{3}}}$ (sub and super-critical, respectively, giving rise to $f_3(c)$ and $f_4(c)$ in items \ref{cle1} and \ref{cge}), and a transcritical bifurcation at $c=v_s$. }
\end{proposition} \begin{proof}
We use that folded singularities are equilibria of \eqref{desrp} where $v\ne \mathbf{c}$; $v=\mathbf{c}$ corresponds to the regular singularity $q(c)$.
Setting $v_1=v_2$, we then find the two (isolated) folded singularities \response{$f_1$ and $f_2$ given by} $v=\pm \mathbf{v}_s$ on $F_s$, recall \eqref{vs}. When $v_1\ne v_2$, we consider $u=\frac12 (v_1-v_2),x=\frac12 (v_1+v_2)$ with $u\ne 0$. This gives
\begin{align}
x=\frac{gc}{3c^2+g-3},\, u^2 =(x-c)^2+1-c^2.\eqlab{xu2fs}
\end{align}
for $c\ne \sqrt{1-\frac{g}{3}}$. \eqref{xu2fs} gives \eqref{q3q4} upon returning to $v_1,v_2$ and the existence of $f_3(c)$ and $f_4(c)$.
Setting $u=0$ in \eqref{xu2fs} gives $c=1,x=1$ and $c=\frac{1-\frac{g}{3}}{\sqrt{1-\frac{2g}{3}}},x=v_s$ for $c>0$. This gives the pitchfork bifurcations. \response{It is a direct calculation to verify the remaining claims regarding $f_3(c)$ and $f_4(c)$ of items \ref{cle1} and \ref{cge}.}
\end{proof}
The linearization of \eqref{desrp} around $v=\mathbf{v}_s(g)$ produces the following eigenvalues: \begin{align}
\lambda_1: = -6v_s(v_s-c),\quad \lambda_2: = -\lambda_1 +2g.\eqlab{eigval} \end{align}
It is possible to compute the eigenvalues of the linearization around the other singularities, but they will not be needed.
Instead, we just summarize the stability findings of singularities of \eqref{desrp} in \figref{eq}. In \figref{eqInS} we illustrate the reduced problem in the case $\frac{1-\frac{g}{3}}{\sqrt{1-\frac{2g}{3}}}<c<\sqrt{1-\frac{g}{3}}$,
where four folded singularities occur and where the regular singularity belongs to $C_S$ and is a saddle for the desingularized reduced problem \eqref{desrp}. The case $\sqrt{1-\frac{g}{3}}<c<v_s$ is similar except now the two non-symmetric folded singularities $f_3(c)$ and $f_4(c)$ occur on the $F_s$-branch located in the lower left corner.
\subsection{Center manifold reduction} We will now perform a center manifold reduction near $F_s$, which consist of partially hyperbolic points for $\epsilon=0$, recall \lemmaref{Fi}. In particular, the reduction will be based upon a local computation near the point $v=\mathbf{v}_s$, $w=\textbf w_s$, where $$w_s:=h_1(\mathbf{v}_s)=-v_s^2+3v_s.$$ (Recall the notation \eqref{tbfx}: $\textbf u=(u,u)$ for any $u$.) In further details, we consider the extended system $(\eqref{fhn},\dot \epsilon=0)$. Then $(v,w,\epsilon)=(\mathbf{v}_s,\textbf w_s,0)$ is partially hyperbolic, the linearization having one single nonzero eigenvalue given by $2g<0$. The associated eigenvector is $(1,1,0,0)$ i.e. along the ``symmetric fast'' space $v_1=v_2$. In terms of the center manifold computations, it is therefore useful to introduce \begin{align}
x= \frac12(v_1+v_2),\quad u=\frac12 (v_1-v_2),\eqlab{xueqn} \end{align} so that $u=0$ corresponds to $v_1=v_2$ in which case we also have $x=v_1=v_2$. At the same time, it is also convinient to define define similar change of coordinates on the set of slow variables:\response{ \begin{align}
y = \frac12 (w_1+w_2)-w_s,\quad z=\frac12 (w_1-w_2), \eqlab{yzeqn} \end{align} recall also \eqref{yuz}.} This gives the following system: \begin{equation}\eqlab{fhnx} \begin{aligned}
\dot x &= -x^3+3x-(y-w_s)-3xu^2,\\
\dot u&=-z-u^3+3(v_s^2-x^2)u,\\
\dot y&=\epsilon(x-c),\\
\dot z &=\epsilon u, \end{aligned} \end{equation} for which the symmetric subspace is now defined by $u=z=0$. In particular, the equations are now symmetric with respect to \begin{align}\eqlab{Sym} \mathcal S:\quad (u,z)\mapsto (-u,-z), \end{align} leaving $x$ and $y$ fixed. \response{The linearization of \eqref{fhnx} around $(v_s,0,w_s,0)$ for $\epsilon=0$ again leads to the nonzero eigenvalue $2g<0$, but now (by construction) the associated eigenvector is aligned with the $u$-axis. At the same time, we find a three-dimensional center space spanned by the vectors \begin{align*}
((2g)^{-1},0,1,0)^T,(0,1,0,0)^T,(0,0,0,1)^T. \end{align*} }By center manifold theory, we obtain the following result. \begin{proposition}\proplab{cmred} There exists an attracting four dimensional symmetric (with respect to $\mathcal S$, see \eqref{Sym}) center manifold $M_a$ of the extended system $(\eqref{fhnx},\dot \epsilon=0)$ near $(x,u,y,z,\epsilon)=(v_s,0,0,0,0)$. It is locally a graph over $(u,y,z,\epsilon)$, i.e. there is a neighborhood $N$ of $(u,y,z,\epsilon) = (0,0,0,\response{0})$ such that \begin{align}
M_a:\quad x &=v_s+\frac{1}{2g} y+\frac{3}{2g} v_s u^2+m(u,y,z,\epsilon),\quad (u,y,z,\epsilon)\in N,\eqlab{Maman}
\end{align}
where \response{the function $m:N\rightarrow \mathbb R$ is smooth and invariant with respect to $\mathcal S$}: \begin{align*}
m(u,y,z,\epsilon) = m(-u,y,-z,\epsilon), \end{align*} for all $(u, y,z,\epsilon)\in N$ (\response{where the right-hand is defined}), and satisfies: \begin{align*} m(u, y,z,\epsilon):= \mathcal O(\epsilon, uz,u^4, y^2,z^2). \end{align*}
\end{proposition} \begin{proof}
The existence of a (symmetric) center manifold follows from standard theory \cite{car1,haragus2011a}. The expansion \eqref{Maman} is also the result of a \response{direct} calculation. \end{proof}
\subsection{The reduced \response{dynamics} on $M_a$} We now proceed to study the reduced \response{dynamics} on $M_a$.
For this, we insert \eqref{Maman} into \eqref{fhnx} and obtain \begin{equation}\eqlab{cmred}
\begin{aligned}
\dot u &=-z-\frac{1}{g} \left(3v_s y +(9v_s^2+g) u^2 +n(u,y,z,\epsilon)\right)u,\\
\dot y&=\epsilon \left(v_s-c+\frac{1}{2g} y+\frac{3v_s}{2g} u^2+m(u, y,z,\epsilon)\right),\\
\dot z &= \epsilon u,
\end{aligned}
\end{equation}
on $M_a$. Here we have introduced a new smooth function \response{$n:N\rightarrow \mathbb R$} satisfying
\begin{align*}
n(u,y,z,\epsilon)&=\mathcal O(\epsilon, uz,u^2 y, u^4, y^2,z^2).
\end{align*}
The function $n$ is also invariant with respect to $\mathcal S$: $n(u, y,z,\epsilon)=n(-u, y,-z,\epsilon)$ for all $(u,y,z,\epsilon)\in N$ (\response{where the right-hand side is well-defined}).
The system \eqref{cmred} is slow-fast with one fast variable $u$ and two slow variables $y$ and $z$. We first describe the layer problem associated with \eqref{cmred}:
\begin{equation}\eqlab{cmredlay}
\begin{aligned}
\dot u &=-z-\frac{1}{g} \left(3v_s y +(9v_s^2+g) u^2 +n(u,y,z,0)\right)u,\\
\dot{ y}&=0,\\
\dot z &= 0,
\end{aligned}
\end{equation}
We then have the following result.
\begin{lemma}\lemmalab{cuspsing}
The critical manifold $S$ of \eqref{cmredlay} is locally a graph over $u$, $y$:
\begin{align}
S: \quad z = Q(u,y),\eqlab{zQ}
\end{align}
where
\begin{align*}
Q(u,v)=-\frac{u}{g}\left(3v_s y+(9v_s^2+g)u^2+\mathcal O(y^2,u^2 y,u^4)\right),
\end{align*}
with $9v_s^2+g=9-5g>0$, see \eqref{vs}. The function $Q$ is smooth and odd in $u$: $Q(-u,y)=-Q(u,y)$ for all $u,y$ sufficiently small. Moreover, locally $S$ splits into a disjoint union $S_a\cup F\cup S_r$ where
\begin{align}
S_{r,a}:=S \cap \{y\gtrless f(u^2)\},\quad F = S\cap \{y=f(u^2)\},\eqlab{Sar}
\end{align}
where \begin{align}
f(u^2):=-\frac{9v_s^2+g}{v_s}u^2+\mathcal O(u^4).\eqlab{Pu} \end{align} Finally, the point $(u,y,z)=\response{(0,0,0)}$ is a cusp singularity of $S$.
\end{lemma}
\begin{proof}
The result follows from the implicit function theorem. In particular, by implicit differentiation $S$ is non-normally hyperbolic at points where \response{$\frac{\partial Q}{\partial u}(u,y)=0$}; solving this equation, depending smoothly on $y$ and $u^2$, gives $y=f(u^2)$ with $f$ as in \eqref{Pu}, \response{or
\begin{align}
u^2 = f^{-1}(y)=-\frac{v_s}{9v_s^2+g}y+\mathcal O(y^2),\eqlab{u2finvy}
\end{align} locally by the implicit function theorem.} \response{Inserting \eqref{u2finvy} into $z=Q(u,y)$ gives
\begin{align*}
z^2 + a y^3 [1+\mathcal O(y)] =0,
\end{align*}
upon squaring both sides. Here $a= \frac{4v_s^3}{(9v_s^2+g)g^2}>0$. Setting $\bar y= y[1+\mathcal O(y)]^{1/3}$ and $\bar z= z/\sqrt{a}$ finally gives the cusp normal form $\bar z^2+\bar y^3=0$ \cite{arnold1984a}.}
\end{proof}
Notice that $S$ is symmetric; $(u,y,z)\in S$ implies that $(-u,y,-z)\in S$ for all $(u, y,z)$ sufficiently small. In particular, $(0, y,0)\in S$ for all $y\approx 0$. We illustrate the situation in \figref{MaRed}.
\response{Any point $p\in W^s(S_a)$, belongs to the stable manifold of a base point on $S_a$. We shall denote this base point by \begin{align} \pi_a(p)\in S_a,\eqlab{pia} \end{align} see \figref{MaRed}.}
\begin{figure}
\caption{Sketch of the singular dynamics of \eqref{cmred} for $\epsilon=0$ and $c\approx v_s(g)$ but $c<v_s(g)$, illustrating how the critical manifold splits into a repelling sheet $S_r$ and an attracting sheet $S_a$ along the degenerate set $F$, see also \lemmaref{cuspsing}. The purple point indicates the cusp singularity (visible in the projection onto the $(y,z)$-plane), which acts like a node for the desingularized reduced problem on $S_a$. Due to the symmetry of the problem, the set $\gamma$ (in pink) given by $u=z=0$ is invariant for all $\epsilon>0$.}
\end{figure}
Let
\begin{align*}
A(u,v) : = \begin{pmatrix}
1 & -Q'_y(u,y) \\
0 & Q'_u(u,y)
\end{pmatrix}.
\end{align*}
\response{Here $Q'_s:=\frac{\partial Q}{\partial s}$ denotes the partial derivative of $Q$ with respect to $s=u,y$.} Then the reduced problem on $S$ can be written as
\begin{align}
Q'_u(u,y)\begin{pmatrix}
\dot u \\
\dot y
\end{pmatrix} &=A(u,v)\begin{pmatrix} u,\\
v_s-c+\frac{1}{2g}y+\frac{3}{2g} v_s u^2+m(u,y,Q(u,y),0),\end{pmatrix}
,\eqlab{Sred}
\end{align}
by implicit differentiation.
\response{
\begin{lemma}\lemmalab{topconj} \eqref{Sred} is smoothly conjugated to the reduced problem \eqref{reducedC0} on $C$ in a neighborhood of $v=\mathbf{v}_s$, $w=\textbf w_s$.
\end{lemma}
\begin{proof} This is by construction: The critical $S$ within $M_a$ is the set of equilibria of \eqref{fhnx} and this set coincides with $C$ upon application of the coordinate transformation defined by \eqref{xueqn} and \eqref{yzeqn}. We therefore obtain the desired transformation through the $u$-equation in \eqref{xueqn} and the $y$-equation in \eqref{yzeqn}:
\begin{align}
u &= \frac12 (v_1-v_2),\quad y= \frac12 (h_1(v_1,v_2)+h_2(v_1,v_2))-w_s.\eqlab{uyeqn}
\end{align} We see that $v=\mathbf{v}_s$ gives $(u,y)=(0,0)$ and the Jacobian matrix of the right hand sides with respect to $v=\mathbf{v}_s$ is \begin{align*}
\begin{pmatrix}
\frac12 & -\frac12 \\
g & g
\end{pmatrix}. \end{align*} Since this matrix is regular, having determinant $g<0$, \eqref{uyeqn} defines a diffeomorphism $(v_1,v_2)\mapsto (u,y)$ on a neighborhood of $v=\mathbf{v}_s$ by the inverse function theorem and this gives the desired conjugacy between \eqref{Sred} and \eqref{reducedC0}.
\end{proof}}
\response{Consequently, our results on \eqref{reducedC0}, see e.g. \propref{foldedsing}, can (locally) be transferred to \eqref{Sred}. It will, however, be useful to perform the analysis of \eqref{Sred} in the $u,y$-plane directly nonetheless.}
To do so, we study a desingularization of \eqref{Sred}. Specifically, since $Q'_u<0$ on $S_a$, the system
\begin{equation}\eqlab{Sred2}
\begin{aligned}
\begin{pmatrix}
\dot u \\
\dot y
\end{pmatrix} &=-A(u,v)\begin{pmatrix} u,\\
v_s-c+\frac{1}{2g}y+\frac{3}{2g} v_s u^2+m(u,y,Q(u,y),0),\end{pmatrix}
\end{aligned}
\end{equation}
is equivalent to \eqref{Sred} there. Orbits of \eqref{Sred} on $S_r$ are also orbits of the desingularized system \eqref{Sred2} but the direction of the flow has changed. $(u,y)=(0,0)$ is then an equilibrium of the desingularized system \eqref{Sred2}, and a \response{direct} calculation shows that the eigenvalues of the linearization are $(-2g) \lambda_1, (-2g) \lambda_2,$ recall \eqref{eigval}. Therefore for all $c$ in the interval $\frac{1-\frac{g}{3}}{\sqrt{1-\frac{2g}{3}}}<c<v_s$, $(u,y)=(0,0)$ is a hyperbolic stable node for \eqref{Sred}, recall also \propref{foldedsing}. In particular, using \eqref{eigval} we find \response{that} $\lambda_1=\lambda_2<0$ for $c=\frac{1-\frac12 g}{\sqrt{1-\frac{2g}{3}}}$. This value of $c$ is always less than $v_s$ for $g<0$ (and greater than the value $\frac{1-\frac13 g}{\sqrt{1-\frac{2g}{3}}}$ corresponding to the second pitchfork bifurcation, recall \propref{foldedsing}). A direct calculation then gives the following.
\begin{lemma}\lemmalab{cinterval}
Consider
\begin{align}
c\in \left(\frac{1-\frac12 g}{\sqrt{1-\frac{2g}{3}}},v_s\right).\eqlab{cinterval}
\end{align} Then \begin{align}
\lambda_2<\lambda_1<0,\eqlab{ratio} \end{align} and the invariant set $u=0$ is therefore the weak direction of the stable node $(u,y)=(0,0)$.
\end{lemma}
Consequently, all points on $S_a$ approaches $(u,y)=(0,0)$ tangentially to the set $u=0$ under the forward flow of \eqref{Sred2} for these values of $c$.
In the following, we shall denote {the corresponding set $(0,y,0)$ in the $(u,y,z)$ space by $\gamma$}. For $c$ as in \eqref{cinterval}, it corresponds to a singular weak canard for the folded node \cite{wechselberger_existence_2005}. In fact, $\gamma$ is distinguished from all orbits on $S_a$ insofar that it is symmetric with respect to $\mathcal S$.
As $S$ has a cusp singularity at $(u,y,z)=0$, $v=\mathbf{v}_s,w=h(\mathbf{v}_s)$ is not a regular folded node singularity \cite{wechselberger_existence_2005} of the slow-fast system \eqref{fhn}. We refer to it as a \textbf{cusped node}.
Notice also that in the nonhyperbolic case $c=v_s$, it follows from $Q(0,y)=0$ that the invariant set $\gamma$ becomes an attracting center manifold of \eqref{Sred2} along which we have $\dot y = \frac{3v_s}{2g^2}y^2(1+ \mathcal O(y))>0$ on $S_a$. Despite the resemblance, the transcritical bifurcation at $c=v_s$ is also not a folded saddle-node (type II) \cite{krupa2010a}. We will instead call it a \textbf{cusped saddle-node}.
In the following, we describe the dynamics of \eqref{cmred} near the cusped node $(u,y,z)=0$ for $\epsilon=0$ (corresponding to $v=\mathbf{v}_s$ in \figref{eqInS}) for $c$-values fixed in the interval \eqref{cinterval}. Here we will blowup $(u,y,z,\epsilon)=0$ and describe how trajectories that start near $S_a$ will evolve as the pass the folded singularity. Subsequently, we will turn our attention to the {cusped saddle-node}. For this purpose, we will (essentially) include $c$ in the blowup transformation and blowup $(u,y,z,\epsilon,c)=\response{(0,0,0,0,v_s)}$. This will enable us to describe the onset and termination of MMOs.
\begin{remark}\remlab{strong}
\response{The point $(u,y)=(0,0)$ also has a strong eigendirection for \eqref{Sred2} along $y=0$ whenever \eqref{cinterval} holds. In fact, a direct calculations shows that
$\dot u<0$ along the fold $y=f(u^2)$, $u\ne 0$, and, as a consequence, the strong stable manifold for \eqref{Sred2} lies completely within the repelling subset $S_r$ of $S$ in this case, see the red orbit on $S$ in \figref{MaRed}. Therefore, since the direction is reversed on $S_r$, it follows that \eqref{Sred} does not have a strong canard (in contrast to the standard folded node).
The lack of a strong canard relates to another important difference between the folded node and the cusped node. Indeed, for the folded node, there is only one fast direction away from the fold. In contrast, as we see in \figref{MaRed}, there are two separate fast directions (in green, using single headed arrows to indicate the lack of hyperbolicity) away from the cusp.
}
\end{remark}
\section{Blowup analysis of the cusped node}\label{sec:4}
Consider the extended system obtained from augmenting \eqref{cmred} by $\dot \epsilon=0$ for the parameter values \eqref{cinterval} and denote the resulting right hand side by $V(u,y,z,\epsilon,c)$. Then $(u,y,z,\epsilon)=0$ is a degenerate equilibrium of the vector-field $V$, with the linearization having only zero eigenvalues. We therefore perform a spherical blowup transformation \cite{dumortier1996a,szmolyan_canards_2001} of $(u,y,z,\epsilon)=0$: \begin{align}
\Psi:\quad (r,(\bar u,\bar y,\bar z,\bar \epsilon))\mapsto \begin{cases}
u &=r\bar u,\\
y&=r^2\bar y,\\
z &=r^3 \bar z,\\
\epsilon &=r^4\bar \epsilon,
\end{cases}\eqlab{blowup1} \end{align} with $r\in [0,r_0]$, $r_0>0$ small enough, $(\bar u,\bar y,\bar z,\bar \epsilon)\in S^3$ where \begin{align*}
S^3 = \left\{(x_1,x_2,x_3,x_4)\in \mathbb R^4\,:\,\sum_{i=1}^4 x_i^2=1\right\}, \end{align*} is the unit $3$-sphere. In this way, the degenerate point $(u,y,z,\epsilon)=0$ gets blown up through the preimage of \eqref{blowup1} to the $3$-sphere with $r=0$. Let $\overline V=\Psi^* V$ denote the pull-back of $V$ under \eqref{blowup1}. Then the exponents on $r$ (also called weights) in \eqref{blowup1} have been chosen so that \begin{align} \widehat V := r^{-2} \overline V,\eqlab{hatV} \end{align} is well-defined and non-trivial for $r=0$. $\widehat V$, being equivalent with $V$ for $r>0$, will have improved hyperbolicity properties for $r=0$ and it is therefore this vector-field that we will study in the following. To do so we will use certain directional charts \cite{krupa_extending_2001}. \response{We will focus on two charts}: the ``entry chart'' obtained by setting $\bar y=-1$ in \eqref{blowup1}, and the ``scaling chart'' obtained by setting $\bar \epsilon=1$ in \eqref{blowup1}. That is, we consider local coordinates $(r_1,u_1,z_1, \epsilon_1)$ and $(r_2,u_2,y_2,z_2)$, parametrizing the subset of the sphere where $\bar y<0$ and where $\bar \epsilon>0$, respectively, such that \eqref{blowup1} takes the following local forms: \begin{align}
(r_1,u_1,z_1,\epsilon_1)\mapsto \begin{cases}
u &= r_1u_1,\\
y&=-r_1^2,\\
z &=r_1^3 z_1,\\
\epsilon &=r_1^4\epsilon_1.
\end{cases}\eqlab{blowup11} \end{align} and \begin{align}
(r_2,u_2,y_2,z_2)\mapsto \begin{cases}
u &= r_2u_2,\\
y&=r_2^2 y_2,\\
z &=r_2^3 z_2,\\
\epsilon &=r_2^4,
\end{cases}\eqlab{blowup12} \end{align} respectively. We will refer to these chart\response{s} as $\bar y=-1$ and $\bar \epsilon=1$ in the following and the dynamics in each of these are analyzed in the following sections. Notice that the charts \eqref{blowup11} and \eqref{blowup12} overlap on $\bar y<0,\bar \epsilon>0$ and the associated \response{change of coordinates} is given by the following expressions: \begin{align}\eqlab{cc1}
r_2 = r_1 \epsilon_1^{1/4},\quad z_2 = z_1 \epsilon_1^{-3/4},\quad y_2 = -\epsilon_1^{-1/2},\quad u_2 = u_1 \epsilon_1^{-1/4}, \end{align} for $\epsilon_1>0$. In the following, we analyze the dynamics in each of the two charts. \response{The analysis of the remaining charts, required to cover the sphere completely, is similar and therefore left out.} We summarize our findings in \figref{blowup2}. We refer to the figure caption for further details. In the following, we will use the convention that a set, say $P$, will be given a subscript $1$ or $2$ when expressed in the respective charts $\bar y=1$ and $\bar \epsilon=1$. When the charts overlap, $P_1$ will then be related by $P_2$ under the change of coordinates \eqref{cc1}. \begin{remark}\remlab{broercusp}
The references \cite{broer2013a,jard2016a} also describe a slow-fast cusp singularity in $\mathbb R^3$ using GSPT and blowup. However, their blowup weights differ from ours since these references consider the cusp in absence of singularities of the reduced flow. The results of \cite{broer2013a,jard2016a} therefore generalizes \cite{szmolyan2004a} on regular jump points. Moreover, at the level of the layer problem our setting corresponds to a time reversal of the system in \cite{broer2013a,jard2016a}, i.e. their $S_{a,r}$ correspond to our $S_{r,a}$, respectively. \end{remark} \begin{figure}
\caption{Illustration of the spherical blowup of the cusped node. The blowup transformation \eqref{blowup1} allows us to extend subsets of the critical manifolds $S_{a,r}$ onto the sphere $S^3$ as invariant manifolds $N_{a,r}$ of a desingularized vector-field. Since $\bar \epsilon\ge 0$, we illustrate the resulting hemi-sphere as a solid sphere (shaded and purple) with $\bar \epsilon>0$ inside. As indicated, these extended manifolds, which lie inside, intersect transversally along $\gamma$ in general (when the ratio $\frac{\lambda_2}{\lambda_1}$ of the eigenvalues is not an integer, see \lemmaref{twist}) and the number of twists of $N_a$ and $N_r$ along $\gamma$ can, as in the folded node, be directly related to the number of SAOs, see \thmref{main1}. }
\end{figure}
\subsection{Analysis in the $\bar y=-1$-chart} Inserting \eqref{blowup11} into \eqref{cmred} with $\dot \epsilon=0$ augmented gives \begin{equation}\eqlab{hatV1} \begin{aligned} \dot r_1 &=-\frac{1}{2} r_1\epsilon_1 \left[v_s-c+r_1^2\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],\\ \dot u_1 &=-z_1-\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2+\response{\mathcal O(r_1^2)}\right)u_1\\ &+\frac{ 1}{2}u_1\epsilon_1 \left[v_s-c+r_1^2\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],\\ \dot z_1 &=\epsilon_1 \left(u_1 +\frac{3}{2}z_1\left[v_s-c+r_1^2\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right]\right),\\ \dot \epsilon_1 &=2 \epsilon_1^2 \left[v_s-c+r_1^2\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],
\end{aligned} \end{equation} after division of the right hand side by $r_1^2$. All $\mathcal O$-terms are smooth functions. This is our local form of $\widehat V$, recall \eqref{hatV}. The set $r_1=\epsilon_1=0$ is invariant and on this set we find that $\dot z_1=0$ and \begin{align*}
\dot u_1 &=-z_1-\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2\right)u_1. \end{align*} There is therefore a critical manifold $S_1$ along $r_1=\epsilon_1=0$ given by \begin{align}
z_1 = -\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2\right)u_1.\eqlab{z1u1} \end{align} It is the critical manifold $S$ extended to the blowup sphere, where it has improved hyperbolicity properties. In particular, let \begin{align}\eqlab{u1p} u_{p,1}:= \sqrt{\frac{v_s}{9 v_s^2+g}}. \end{align} Then the subset $S_{a,1}$ of $S_1$ within $u_1\in (-u_{p,1},u_{p,1})$ is partially attracting, the linearization about any point in this set $(r_1,u_1,z_1,\epsilon_1)\in S_{a,1}$ having one single nonzero and negative eigenvalue. Consequently, by center manifold theory we obtain the following result. \begin{proposition}\proplab{N1a} For any $\nu>0$ small enough, consider $I(\nu):=[-u_{p,1}+\nu,u_{p,1}-\nu]$. Then there exists a three dimensional center manifold $N_{a,1}$ of points $(0,u_1,0,0)$, $u_1\in I(\nu)$ of the following graph form: \begin{align}
N_{a,1}:\quad z_1 = u_1\left(-\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2\right)+\mathcal O(r_1^2,\epsilon_1)\right),\eqlab{Na1} \end{align} for $u_1\in I(\nu),\,(r_1,\epsilon_1)\in [0,\delta]^2$ and some $\delta>0$ small enough. \end{proposition} In the expansion \eqref{Na1}, we have used that $u_1=z_1=0$ is invariant for all $r_1,\epsilon_1\ge 0$.
As usual, $N_{a,1}$ is foliated by constant $\epsilon$-values: $\epsilon=r_1^4\epsilon_1$ and $N_{a,1}\cap \{\epsilon=r_1^4\epsilon_1\}$ therefore provides an extension of Fenichel's slow manifold $S_{a,\epsilon}$, being a perturbation of a compact subset of $S_a$, up close to the blowup sphere. Specifically, at $\epsilon_1=\delta$ we have $r_1= \epsilon^{1/4}\delta^{-1/4}=\mathcal O(\epsilon^{1/4})$ and $N_{a,1}\cap\{\epsilon=r_1^4\epsilon_1\}$, upon blowing down to the $(u,y,z)$-variables, therefore extends $S_{a,\epsilon}$ as an invariant manifold up to a ``wedge-shaped'' region of $(0,0,0)$ which extends $\mathcal O(\epsilon^{1/4}),\mathcal O(\epsilon^{1/2}),\mathcal O(\epsilon^{3/4})$ in the $u,y,z$-directions, respectively, recall \eqref{blowup1}.
A \response{direct} calculation shows that the reduced problem on $N_{a,1}$ is given by \begin{equation}\eqlab{redN1a} \begin{aligned}
\dot r_1 &=-\frac12 r_1,\\
\dot u_1 &=u_1 \left(-\frac{\lambda_2}{\lambda_1}+\frac12 +\mathcal O(u_1^2,r_1^2,\epsilon_1)\right),\\
\dot \epsilon_1 &=2\epsilon_1, \end{aligned} \end{equation} recall \eqref{eigval}. Here we have used a desingularization through division by $$\epsilon_1 \left[v_s-c+\mathcal O(r_1^2)\right];$$ notice that the \response{square} bracket is positive for any $r_1\ge 0$ small enough by assumption of \eqref{cinterval}. Then for $c$ as in \eqref{cinterval}, see also \eqref{ratio}, one can show that $(r_1,u_1,\epsilon_1)=0$ is the only equilibrium on $N_{a,1}$ and it is hyperbolic for \eqref{redN1a} with eigenvalues \begin{align}\eqlab{eig1} -\frac12,-\frac{\lambda_2}{\lambda_1}+\frac12,2. \end{align} \begin{lemma}\lemmalab{linearization}
\response{ Suppose that \eqref{ratio} holds and that
\begin{align*}
\frac{\lambda_2}{\lambda_1} \ne 3.
\end{align*}
Then there is a $C^1$-linearization of \eqref{redN1a} of the form:
\begin{align}
(r_1,u_1,\epsilon_1) \mapsto \tilde u = \psi(r_1,u_1,\epsilon_1),\eqlab{tildeu1}
\end{align}
where $\psi(0,0,0)=0,\frac{\partial \psi}{\partial u_1}(0,0,0)=1$, so that
\begin{align*}
\dot{\tilde u}_1 &= \tilde u_1 \left(-\frac{\lambda_2}{\lambda_1}+\frac12\right).
\end{align*}
} \end{lemma} \begin{proof}
\response{ According to the classical work \cite{Belitskii1973}, a smooth system $\dot x = Ax+\mathcal O(x^2)$, with eigenvalues $\nu_i$, $i=1,\ldots,n$, of the matrix $A\in \mathbb R^{n\times n}$, is linearizable by a $C^1$-diffeomorphism if \begin{align*}
\nu_i \ne \operatorname{Re} (\nu_j+\nu_k), \end{align*} for all $i=1,\ldots,n$ and all $\operatorname{Re} \nu_j<0$ and $\operatorname{Re} \nu_k>0$. In the present case, this gives \begin{align*} -\frac{\lambda_2}{\lambda_1}+\frac12 \ne - \frac12 +2=\frac32, \end{align*} setting $\nu_i= -\frac{\lambda_2}{\lambda_1}+\frac12$, $\nu_j = -\frac12$ and $\nu_k=2$, and \begin{align*}
-\frac12 \ne -\frac{\lambda_2}{\lambda_1}+\frac12 + 2 = -\frac{\lambda_2}{\lambda_1}+\frac52, \end{align*} setting $\nu_i= -\frac12$, $\nu_j=-\frac{\lambda_2}{\lambda_1}+\frac12$, and $\nu_k=2$, recall \eqref{eig1}. The first inequality clearly holds since the left hand side is negative by \eqref{ratio}. Similar, the second inequality implies $\lambda_2\ne 3 \lambda_1$ and the existence of the $C^1$-linearization therefore follows. Seeing that the $r_1$- and $\epsilon_1$-equations are already linear, one can easily show that the linearization takes the form \eqref{tildeu1}. } \end{proof}
\subsection{Analysis in the $\bar \epsilon=1$-chart} Inserting \eqref{blowup12} into \eqref{cmred} with $\dot \epsilon=0$ augmented gives \begin{equation}\eqlab{hatV2} \begin{aligned}
\dot u_2 &=-z_2-\frac{1}{g} \left(3v_s y_2 +(9v_s^2+g) u_2^2 +\mathcal O(r_2^2)\right)u_2,\\
\dot y_2 &= v_s-c+r_2^2\left(\frac{1}{2g}y_2+\frac{3v_s}{2g} u_2^2+ \mathcal O(r_2^2)\right),\\
\dot z_2 &=u_2, \end{aligned} \end{equation} and $\dot r_2=0$, upon division of the right hand side by the common factor $r_2^2$. All $\mathcal O$-terms are smooth. This is our local form of $\widehat V$, recall \eqref{hatV}. Notice that since $\gamma:\,u=z=0$ is invariant for all $\epsilon\ge 0$, the set $\gamma_2$ defined by $u_2=z_2=0$ is also invariant for all $r_2\ge 0$ and $y_2$ increases for $c$ in the interval \eqref{cinterval} \response{for all $0<r_2\ll 1$}. Consider $r_2=0$. Then we obtain \begin{equation*} \begin{aligned} \frac{du_2}{dy_2} &=\frac{1}{v_s-c}\left(-z_2-\frac{1}{g} \left(3v_s y_2 +(9v_s^2+g) u_2^2\right)u_2\right),\\
\frac{dz_2}{dy_2} &=\frac{u_2}{v_s-c}. \end{aligned} \end{equation*} Linearization around $u_2=z_2=0$ gives \response{ \begin{equation}\eqlab{U2Z2var} \begin{aligned} \frac{dU_2}{dy_2} &=\frac{1}{v_s-c}\left(-Z_2-\frac{3v_s}{g}y_2U_2\right),\\
\frac{dZ_2}{dy_2} &=\frac{U_2}{v_s-c}. \end{aligned} \end{equation}} Setting \response{\begin{align*}
y_2 = \sqrt{\frac{-g(v_s-c)}{3v_s}}Y_2, \end{align*}} we can write this system as a Weber equation: \response{\begin{align}
U_2''(Y_2) - Y_2 U_2'(Y_2) + \frac{\lambda_2}{\lambda_1} U_2(Y_2) =0,\eqlab{weber} \end{align} recall \eqref{eigval}.} The implication of this is the following: Let $N_{a,2}(r_2)$ denote the center manifold obtained in the chart $\bar y=-1$ written in the $\bar \epsilon=1$\response{-coordinates} $(u_2,y_2,z_2,r_2)$. It is parametrized by $r_2=\epsilon^{1/4}$ and $N_{a,2}(0)$ denotes the intersection with $r_2=0$, i.e. with the blowup sphere.
Working in the $\bar y=1$ chart, for example, we may also obtain a repelling critical manifold $N_{2,r}(r_2)$ in much the same way. This extends the repelling slow manifold $S_{r,\epsilon}$ into scaling chart as an invariant manifold and we write $N_{2,r}(0)$ to denote the intersection with $r_2=0$. We extend each of these manifolds by the flow and denote the extended objects by the same symbol. Then $\gamma_2\subset N_{a,2}(0)\cap N_{2,r}(0)$. Using \eqref{weber}, we have the following. \begin{lemma}\lemmalab{twist}
The intersection of $N_{a,2}(0)$ and $N_{2,r}(0)$ along $\gamma_2$ is transverse whenever $\frac{\lambda_2}{\lambda_1}\notin \mathbb N$. In the affirmative case, the tangent space of $N_{a,2}(0)$ along $\gamma_2$ twists $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor$-many times, where each twist corresponds to a full rotation by $180^\circ$ degrees. \end{lemma} \begin{proof}
The proof of this is identical to the proof of \cite[Lemma 4.4]{szmolyan_canards_2001} for the folded node. \response{Basically, regarding the transversality,
we first use that the tangent spaces of $N_{a,2}(0)$ and $N_{2,r}(0)$ along $\gamma_2$ coincide with the set of solutions of \eqref{U2Z2var} having algebraic growth as $y_2\rightarrow \mp \infty$, respectively.} Next, for $\frac{\lambda_2}{\lambda_1}\notin \mathbb N$ it is standard that there are no bounded solutions of \eqref{weber}. \response{This proves the transversality. Finally, regarding the number of twists, we use that any solution of \eqref{weber} having algebraic growth as $y_2\rightarrow -\infty$ has $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor+1$ simple zeros. Two consecutive zeros correspond to a full $180^\circ$-rotation in the $(U_2,Z_2)$-plane and the result therefore follows. }
\end{proof} \begin{remark}\response{
Whenever $n:=\frac{\lambda_2}{\lambda_1}\in \mathbb N$, then $U_2(Y_2)=H_n(Y_2/\sqrt{2})$, with $H_n$ the Hermite polynomial of degree $n$, is a bounded solution of \eqref{weber}, see \cite{wechselberger_existence_2005}. This means that $TN_{a,2}(0)=TN_{r,2}(0)$ and the tangent spaces form a single band with $\frac{\lambda_2}{\lambda_1}$-many twists.} This may give rise to secondary intersections of $N_{a,2}$ and $N_{a,r}$ (like secondary canards, see \cite{kristiansen2020a,wechselberger_existence_2005}) upon perturbation. But in contrast to the folded node, the bifurcations $\frac{\lambda_2}{\lambda_1}\in \mathbb N$ do not produce additional intersections of the Fenichel slow manifolds themselves, since in our case we do not have a strong canard, \response{recall \remref{strong}}. See also \cite{kristiansen2020a}. We therefore do not pursue the description of these bifurcations any further. \end{remark}
\subsection{Completing the analysis of the cusped node } We can now state our main results on the dynamics for fixed $c$ in the interval \eqref{cinterval}. Firstly, following \lemmaref{twist} and the fact that $N_{a,2}(r_2)$ and $N_{2,r}(r_2)$ are $\mathcal O(r_2^2)$-close to $N_{a,2}(0)$ and $N_{2,r}(0)$ in the scaling chart, we conclude: \begin{proposition} The Fenichel slow manifolds $S_{a,\epsilon}$ and $S_{r,\epsilon}$ intersect transversally along $\gamma$ whenever $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor\notin \mathbb N$ \response{for all $0<\epsilon\ll 1$}. \end{proposition} \begin{remark} A similar result holds for the folded node, see e.g. \cite{szmolyan_canards_2001,wechselberger_existence_2005}. But in contrast to these results, we are here deliberately referring to the Fenichel slow manifolds, i.e. the slow manifolds obtained from perturbing compact subsets of $S_a$ and $S_r$ through Fenichel's theory \cite{fen3} and extending these by the forward flow. For the general folded node, it is only invariant manifolds -- that have been extended as center-like manifolds -- that are shown to intersect transversally along a weak canard; the Fenichel slow manifolds are only a subset of these extended manifolds. This relates to the delicacy of the weak canard and whether this object in fact ever reaches the Fenichel slow manifolds, see also \cite{kristiansen2020a} for a discussion of these technical aspects. The reason why we can be more specific in the present context is that $\gamma$, which plays the role of the weak canard, exits for all $\epsilon>0$ for our system and this set therefore (locally) belongs to $S_{a,\epsilon}$ and $S_{r,\epsilon}$. \end{remark}
\response{We now proceed to state our main result on the SAOs of the cusped node. For this, we will follow \cite{wechselberger_existence_2005} and count, in line with \lemmaref{twist}, the number of SAOs as the number of full $180^\circ$-rotations in a plane transverse to $\gamma$. More precisely, consider an orbit $O:t\mapsto (u(t),y(t),z(t))$, $t\in I:=[0,T]$, with $$(u(t),z(t))\ne (0,0),$$ for all $t\in I$. The number of SAOs is then the rotation number \begin{align*} n = \lfloor (\Theta(T)-\Theta(0))/\pi\rfloor, \end{align*} where $\Theta(t)\in \mathbb R$, $t\in [0,T]$, is the lift of the angle $\theta(t)\in \mathbb R/ 2\pi \mathbb Z$ defined by $\tan \phi(t)= \frac{z(t)}{u(t)}$, $t\in [0,T]$.
Similarly, we define the amplitude of the SAOs as $\max_{t\in [0,T]}\vert (u(t),z(t))\vert$. (In \thmref{main2}, however, we will measure the amplitude in terms of $\vert (u_2(t),z_2(t))\vert$).}
\response{In the following, we write $f=f(\mu)\sim \mu$ whenever there are positive constants $c_1<c_2$ such that \begin{align*}
c_1 \mu \le f(\mu) \le c_2 \mu, \end{align*} for all $0<\mu\ll 1$. }
\begin{theorem}\thmlab{main1} \response{Fix $c$ as in \eqref{cinterval}, any $\delta>0$ sufficiently small and suppose that \begin{align*}
\frac{\lambda_2}{\lambda_1}\notin \mathbb N. \end{align*} Consider any point $p$ so that $\pi_a(p)\in S_a\backslash \gamma$, recall \eqref{pia}. Then the following holds for all $0<\epsilon\ll 1$: The forward orbit of $p$ intersects the section defined by $y=y_{\mathrm{exit}}:=(\epsilon\delta^{-1})^{1/2}$ in a point $(u,y,z)=(u_{\mathrm{exit}},y_{\mathrm{exit}},z_{\mathrm{exit}})$ with \begin{align}
u_{\mathrm{exit}} \sim \epsilon^{\frac{\lambda_2}{2\lambda_1}},\quad z_{\mathrm{exit}} \sim \epsilon^{\frac12+\frac{\lambda_2}{2\lambda_1}},\eqlab{amplitudeuz} \end{align} and undergoes $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor$ many SAOs. The order of the amplitude of the SAOs are given by \eqref{amplitudeuz}.}
\end{theorem} \begin{proof}
We first work in the $\bar y=-1$ chart. Then the forward flow of the point $p$ can be described by the reduced problem \eqref{redN1a}. We therefore integrate these equations from $(r_{10},u_{10},\epsilon_{10})$ to $(r_{11},u_{11},\epsilon_{11})$ with $\epsilon_{11}=\delta>0$. \response{Here $r_{10},u_{10}=\mathcal O(1)$ and $\epsilon_{10}\sim \epsilon$ as $\epsilon\rightarrow 0$ by assumption on $\pi_a(p)\notin \gamma$}. To perform the integration, we apply \lemmaref{linearization} and consider
\begin{align*}
\dot r_1 &= -\frac12 r_1,\\
\dot{\tilde u}_1 &= \tilde u_1\left(-\frac{\lambda_2}{\lambda_1}+\frac12\right),\\
\dot \epsilon_1 &=2 \epsilon_1.
\end{align*} \response{Integrating these equations gives \begin{align*}
\tilde u_{11} = \left(\epsilon_{11}^{-1} \epsilon_{10}\right)^{\frac{\lambda_2}{2\lambda_1}-\frac14 }\tilde u_{10}, \end{align*} and $r_{11} = (\epsilon\delta^{-1})^{1/4}$. Therefore $u_{11} \sim \epsilon^{\frac{\lambda_2}{2\lambda_1}-\frac14 }$ using \eqref{tildeu1}. We then transform the result using \eqref{cc1} to the scaling chart. Here we apply regular perturbation theory from $y_2=-\delta^{-1/3}$, which corresponds to $\epsilon_{1}=\delta$, up to $y_2=\delta^{-1/3}$. This value of $y_2$ corresponds to $y_{\mathrm{exit}}$. Now, the order of the amplitude of $u_2$ and $z_2$ does not change during \response{this} finite time passage. Using \eqref{Na1} and \eqref{blowup11}, we therefore finally obtain \eqref{amplitudeuz}. The number of small amplitude oscillations follow from \lemmaref{twist} upon taking $\delta>0$ small enough (and subsequently $\epsilon>0$ small enough).} \end{proof}
\section{Analysis of the cusped saddle-node}\label{sec:5} Next, we consider the cusped saddle-node where $c\approx v_s$ in \eqref{cmred}. \response{For this we will use an $\epsilon$-dependent zoom near $v_s$. Looking at \eqref{hatV2} with $r_2=\epsilon^{1/4}$, we see that \begin{align} c = v_s + \sqrt{\epsilon} c_2,\eqlab{c2expr} \end{align} brings the two terms in the equation for $y_2$ to the same order. For this reason, we now consider \eqref{c2expr} before applying the blowup transformation $\Phi$.} In this way, $c=v_s$ gets blown up to $c_2\in \mathbb R$ for $\epsilon=0$.
In the following, we study each of the charts $\bar y=-1$ and $\bar \epsilon=1$ again. The results are summarized in \figref{blowup3}. \begin{figure}
\caption{Illustration of the spherical blowup of the cusped saddle-node, using the same perspective as in \figref{blowup2}. In this case, we obtain a slow-fast system on the blowup sphere with $\gamma$ as a critical manifold. The reduced problem on $\gamma$ has an equilibrium $q$ which undergoes a Hopf bifurcation for the full system. In particular, on one side of the bifurcation $q$ is of saddle-focus type (the cyan surface illustrates the unstable manifold $W^s(q)$) and this is where an increased number of SAOs occur. The fast subsystem of the slow-fast system on the blowup sphere is of Lienard-type and this gives rise to a cylinder $P$ of limit cycles on the blowup sphere (in orange).}
\end{figure} \subsection{Analysis in the $\bar y=-1$-chart} The resulting equations can be obtained from \eqref{hatV1} upon substituting \eqref{c2expr}. We have \begin{equation}\nonumber \begin{aligned} \dot r_1 &=-\frac{1}{2} r_1^3\epsilon_1 \left[-\sqrt{\epsilon_1} c_2+\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],\\ \dot u_1 &=-z_1-\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2+\mathcal O(r_1^2)\right)u_1\\ &+\frac{ 1}{2}r_1^2 u_1\epsilon_1 \left[-\sqrt{\epsilon_1} c_2+\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],\\ \dot z_1 &=\epsilon_1 \left(u_1 +\frac{3}{2}z_1r_1^2\left[-\sqrt{\epsilon_1}c_2+\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right]\right),\\ \dot{\sqrt{\epsilon_1}} &=r_1^2 \epsilon_1\sqrt{\epsilon_1} \left[-\sqrt{\epsilon_1}c_2+\left(-\frac{1}{2g}+\frac{3v_s}{2g} u_1^2+ \response{\mathcal O(r_1^2)}\right)\right],
\end{aligned} \end{equation} writing the last equation in terms of $\sqrt{\epsilon_1}$ rather than $\epsilon_1$ to indicate that the system is smooth in the former. For $r_1=\sqrt{\epsilon_1}=0$, we again find \eqref{z1u1} as a manifold of equilibria with the same stability properties. Therefore \propref{N1a} still applies, but the remainder is now smooth in $\sqrt{\epsilon_1}$. The reduced problem is then \begin{equation}\eqlab{redN1a2} \begin{aligned}
\dot r_1 &=-\frac12 r_1^3,\\
\dot u_1 &=u_1 \left(-\frac{g^2}{3v_s}+\mathcal O(u_1^2,r_1^2,\sqrt{\epsilon_1})\right),\\
\dot{\sqrt{\epsilon_1}} &=r_1^2\sqrt{\epsilon_1}, \end{aligned} \end{equation} after division of the right hand side by $\epsilon_1\left[-\frac{1}{2g}+\mathcal O(r_1,\sqrt{\epsilon_1})\right]$. Notice that the bracket is positive for all $r_1,\sqrt{\epsilon_1}\ge 0$ sufficiently small. From this we have. \begin{proposition} Fix any $c_2$ with $c$ as in \eqref{c2expr}, any $\delta>0$ sufficient small and consider any point $p$ so that $\pi_a(p)\in S_a\backslash \gamma$. Then the following holds for all $0<\epsilon\ll 1$: The forward flow of $p$ intersects the section defined by $y=y_{\mathrm{in}}:=-(\epsilon\delta^{-1})^{1/2}$ in a point $(u,y,z)=(u_{\mathrm{in}},y_{\mathrm{in}},z_{\mathrm{in}})$ with \begin{align}
u_{\mathrm{in}},z_{\mathrm{in}}=\mathcal O(e^{-\nu/\sqrt{\epsilon}}), \end{align} for some $\nu>0$.
\end{proposition} \begin{proof}
We work in the entry chart $\bar y=-1$, reduce to $N_{a,1}$, divide \eqref{redN1a2} by $\dot r_1$ and integrate from $r_1=\mathcal O(1)$ to $r_{1,\mathrm{in}}=\mathcal O(\epsilon^{1/4})$ (corresponding to the value of $y=y_{\mathrm{in}}$). \response{This leads to the estimate
\begin{align*}
\vert u_{1,\mathrm{in}}\vert\le C e^{\nu \int_{r_{1}}^{r_{1,\mathrm{in}}} s^{-3} ds} = Ce^{\frac12 \nu r_{1}^{-2}} e^{-\frac12 \nu r_{1,\mathrm{in}}^{-2}}
\end{align*} for some $C>0$ and $\nu>0$ independent of $\epsilon$. } \end{proof} Next, we notice the following: Consider the $r_1=0$ subsystem: \begin{equation}\eqlab{slowfastlienard} \begin{aligned}
\dot u_1 &=-z_1-\frac{1}{g}\left(-3v_s +(9v_s^2+g)u_1^2\right)u_1,\\ \dot z_1 &=\epsilon_1 u_1,\\ \dot{\epsilon_1} &=0.
\end{aligned} \end{equation} This system is a slow-fast Lienard system in the $(u_1,z_1)$-plane with $\epsilon_1\ge 0$ as the small parameter. The analysis is straightforward and illustrated in \figref{lienard}. In particular, the associated layer problem has the set \eqref{z1u1} as a manifold of equilibria, being attracting for $u_1\in (-u_p,u_p)$ and repelling for $u_1\notin [-u_{p,1},u_{p,1}]$, recall \eqref{u1p}. The reduced problem has a stable node at $(u_1,z_1)=0$ on the attracting branch and we are therefore in the ``relaxation regime'', but the relaxation oscillations for $\epsilon_1>0$ small enough are repelling. \response{(Notice that in contrast to \eqref{fhnuncp}, the middle branch of the critical manifold of \eqref{slowfastlienard} is attracting. Compare also \figref{uncoupled} with \figref{lienard}.)} Therefore we have the following: \begin{lemma}
On $r_1=0$ there exists an invariant cylinder $P_{1}$, contained within $\epsilon_1\in [0,\delta]$, for $\delta>0$ small enough, such that $P_1(\epsilon_{10}):=P_1\cap \{\epsilon_1=\epsilon_{10}\}$ is a repelling limit cycle for each $\epsilon_{10}\in (0,\delta]$. In particular, $P_1(0)$ is a singular slow-fast relaxation cycle. \end{lemma} \begin{figure}
\caption{The invariant cylinder $P_1$ in the \response{$\bar y=-1$}-chart within $r_1=0$. For $\epsilon_1=0$, it becomes a singular van der Pol-like relaxation cycle in the $(u_1,z_1)$-plane. }
\end{figure}
\subsection{Analysis in the $\bar \epsilon=1$-chart} The resulting equations can be obtained from \eqref{hatV2} upon substituting \eqref{c2expr}. We have \begin{equation}\eqlab{hatV21} \begin{aligned}
\dot u_2 &=-z_2-\frac{1}{g} \left(3v_s y_2 +(9v_s^2+g) u_2^2 +\mathcal O(r_2^2)\right)u_2,\\
\dot y_2 &= r_2^2\left(\response{-}c_2 +\frac{1}{2g}y_2+\frac{3v_s}{2g} u_2^2+ \mathcal O(r_2^2)\right),\\
\dot z_2 &=u_2, \end{aligned} \end{equation} and $\dot r_2=0$. This is now a slow-fast system with two fast variables, $u_2$ and $z_2$, and one single slow variable $y_2$. In particular, we notice that \begin{align*}
\gamma_2:\quad u_2=z_2=0,\,y_2\in \mathbb R, \end{align*} is now a critical manifold for $r_2=0$. In fact, the associated fast sub-system \begin{equation}\eqlab{layeru2z2} \begin{aligned}
\dot u_2 &=-z_2-\frac{1}{g} \left(3v_s y_2 +(9v_s^2+g) u_2^2\right)u_2,\\
\dot z_2 &=u_2, \end{aligned} \end{equation} with $y_2$ fixed as a parameter for $r_2=0$, is a Lienard equation. \begin{lemma}\lemmalab{lienard}
The system \eqref{layeru2z2} has a unique repelling limit cycle $P_2(y_2)$ for each $y_2<0$. \end{lemma} \begin{proof} This follows from Lienard's theorem \cite{perko2001a}. In fact, \eqref{layeru2z2} is topologically equivalent with the van-der Pol system in backward time. In particular, there is a subcritical Hopf bifurcation of \eqref{layeru2z2} at $y_2=0$.
\end{proof} By uniqueness, the set $P_2$ coincides with $P_1$ upon using the change of coordinates \eqref{cc1} where these overlap.
By Fenichel's theory \cite{fen3}, the manifold $P_2$ of repelling limit cycles of \eqref{hatV21} for $r_2=0$, perturbs as an invariant manifold $P_{2,r_2}$ within compact subsets. It creates a funnel region, \response{where trajectories inside contract towards $\gamma_2$, while trajectories outside get repelled away from the local neighborhood of the cusp}. On the perturbed cylinder, repelling limit cycles may exist. This depends upon $c_2$. Indeed, the \response{reduced problem} on $P_2$ is given by averaging: Let $T(y_2)$ be the period of $P_2(y_2)$ as a periodic orbit $(u_2(t;y_2),z_2(t;y_2))$ of \eqref{layeru2z2}. Then \begin{align}
y_2' = \response{-}c_2 +\frac{1}{2g}y_2+\frac{3v_s}{2g} \frac{1}{T(y_2)} \int_0^{T(y_2)} u_2(t;y_2)^2 dt,\eqlab{redP2} \end{align} on $P_2$. Consequently, the reduced problem has an equilibrium at $y_2$ for the parameter value $c_2$ whenever \begin{align}
c_2 =\response{ \frac{1}{2g}y_{2}+\frac{3v_s}{2g} \frac{1}{T(y_{2})} \int_0^{T(y_{2})} u_2(t;y_{2})^2 dt},\quad y_{2}<0.\eqlab{c2eq} \end{align} Notice that $y_{2}=0$ on the right hand side gives $c_2=0$. It is \response{possible} to show that $u(t,y_2) = 2\sqrt{\frac{-y_2v_s}{9v_s^2+g}} \cos(t)+\mathcal O(y_2)$ (\response{using e.g. a Melnikov computation, see \cite{kristiansen2022}, where a similar computation is performed in a related context}). This gives a linear approximation of the right hand side of \eqref{c2eq}: \begin{align}
c_2 \approx \frac{3v_s^2+g}{2g(9v_s^2+g)}y_2.\eqlab{c2lin} \end{align} Consequently, the right hand side is a decreasing function of $y_2$ for $y_2<0$ small enough for $g<0$. Numerical computations (see \figref{c2vsy2P2}) indicate that this holds for all $y_2<0$. We have not found a way to show this, but if we assume this, then we have the following result.
\begin{figure}
\caption{The right hand side of \eqref{c2eq} as a function of $y_2$ for $g=-1$. The dotted line is the linear approximation \eqref{c2lin} obtained through Melnikov. In order to compute the full line, we have first computed an accurate approximation of a limit cycle of \eqref{layeru2z2} (using shooting and Newton's method) and then subsequently computed the average. }
\end{figure}
\begin{proposition}\proplab{contractg2}
Suppose that the right hand side of \eqref{c2eq} is a strictly \response{decreasing} function of $y_2<0$. Fix any $c_{20}>0$ and let $y_{20}$ be the unique value $y_2$ such that \eqref{c2eq} holds with $c_2=c_{20}$. Then the reduced problem \eqref{redP2} on $P_2$ has a unique attracting fixed point at $y_2=y_{20}$ for the parameter value $c_2=c_{20}$.
Moreover, for all $0<r_2\ll 1$, the corresponding singular cycle $P_2(y_{20})$ then perturbs to a hyperbolic (saddle-type) limit cycle $P_{2,r_2}(y_{20})$ of \eqref{hatV21} for $c_2=c_{20}$. This limit cycle is $\mathcal O(r_2^2)$-close to $P_{2}(y_{20})$. \end{proposition} As $c_{2}$ ranges over a compact subset $I$ of $(0,\infty)$, we then obtain a family of repelling limit cycles on $P_{2,r_2}$ for all $0<\epsilon\le \epsilon_0(I)$. Recall that $r_2=\epsilon^{1/4}$. It is possible to show that the family $P_{2,r_2}$ overlaps with the repelling Hopf cycles emanating from $y_2=0$ at $c_2=0$, recall \remref{lyapunov}.
\begin{remark}\remlab{lyapunov}
\response{The Liapunov coefficient $l_1$ (recall \eqref{lyapunov}) of the Hopf bifurcation for $c_2=0$ (corresponding to $c=v_s(g)$) can be calculated from \eqref{hatV21}. Indeed, a direction calculation shows that the two-dimensional center manifold at the Hopf-point takes the following form
\begin{align*}
y_2 \approx -\frac{3v_s}{2} u_2^2 -\frac{3v_s}{2} z_2^2,
\end{align*}
for $c_2,r_2\rightarrow 0$,
up to and including quadratic order in $(u_2,z_2)$. On this center manifold, with $c_2,r_2\rightarrow 0$, we
then have that \begin{equation}\eqlab{hopflin} \begin{aligned}
\dot u_2 &=-z_2 +f(u_2,z_2),\\
\dot z_2 &=u_2, \end{aligned} \end{equation} with \begin{align*}
f(u_2,z_2) \approx -\frac{1}{g} \left(-\frac92 v_s^2 z_2^2 +\left(\frac{9}{2}v_s^2+g\right) u_2^2 \right)u_2, \end{align*} up to an including cubic order in $(u_2,z_2)$. The system \eqref{hopflin} is already in normal form and we therefore have that \begin{align*}
\hat l_1:=\frac{1}{16} \left(\frac{\partial^3 f}{\partial u_2^3}(0,0)+\frac{\partial^3 f}{\partial u_2\partial z_2^2}(0,0)\right)=\frac{3(g-3)}{8g} \end{align*} using \cite[Equation 3.4.11]{guckenheimer97}. This gives the leading order expression in \eqref{lyapunov} upon division by $r_2^2=\sqrt{\epsilon}$. This division corresponds to the desingularization in the chart $\bar \epsilon=1$, recall \eqref{hatV}. }
\end{remark}
We now proceed to study the properties of the critical manifold $$\gamma_2:\quad u_2=z_2=0,y_2\in \mathbb R,$$ of \eqref{hatV21} for $r_2=0$. The linearization of \eqref{layeru2z2} around $u_2=z_2=0$ gives \begin{align}
\begin{pmatrix}
-\frac{3v_s}{g} y_2 & -1\\
1 & 0
\end{pmatrix}.\eqlab{matrixJac} \end{align} \response{The eigenvalues are imaginary $\pm i$ for $y_2=0$ due to the Hopf.} From this we can easily deduce the stability properties. \begin{lemma}\lemmalab{L2stability}
The critical manifold $\gamma_2$ of \eqref{hatV21} for $r_2=0$ is normally hyperbolic for $y_2\ne 0$. The subset $\gamma_2^a$ with $y_2<0$ is attracting whereas the subset $\gamma_2^r$ with $y_2>0$ is repelling.
Moreover, $\gamma_2^r=\gamma_2^{rf}\cup \gamma_2^{rn}$ where $\gamma_2^{rf}$ is the subset of $\gamma_2$ with $y_2\in \left(0,-\frac{2g}{3v_s}\right)$ having normal focus stability (i.e. the eigenvalues of \eqref{matrixJac} are complex conjugated with positive real part) whereas $\gamma_2^{rn}$ is the subset of $\gamma_2^r$ with $y_2\ge -\frac{2g}{3v_s}$ having normal nodal stability (i.e. the eigenvalues of \eqref{matrixJac} are real and positive).
\end{lemma} There is a similar division of $\gamma_2^a=\gamma_2^{af}\cup \gamma_2^{an}$ for $y_2\in \left(\frac{2g}{3v_s},0\right)$ and $y_2\le \frac{2g}{3v_s}$, respectively, but this will be less important.
The reduced problem on $\gamma_2$ is given by \begin{align*}
y_2' &=-c_2 + \frac{1}{2g}y_2. \end{align*} It has a hyperbolic and attracting equilibrium at $y_2= 2c_2 g$. In combination with \lemmaref{L2stability}, we realize the following. \begin{lemma}\lemmalab{q2c2} Let $q_2$ denote the equilibrium $(u_2,y_2,z_2)=(0,2c_2g,0)$ which is hyperbolic and attracting for the reduced problem on $\gamma_2$. Then the following holds.
\begin{itemize}
\item For $c_2 > 0$, then $q_2$ sits on the attracting part of $\gamma_2$ and it perturbs to an attracting equilibrium \eqref{hatV21} for all $0<\epsilon\ll 1$.
\item For $c_2\in \left(-\frac{1}{3v_s},0\right)$, then $q_2\in \gamma_2^{rf}$ and it perturbs to a saddle-focus equilibrium of \eqref{hatV21} for all $0<\epsilon\ll 1$ with a one-dimensional stable manifold along $\gamma_2$ and a two-dimensional unstable manifold with focus-type dynamics.
\item For $c_2< -\frac{1}{3v_s}$, then $q_2\in \gamma_2^{rn}$ and it perturbs to a saddle equilibrium of \eqref{hatV21} for all $0<\epsilon\ll 1$ with a one-dimensional stable manifold along $\gamma_2$ and a two-dimensional unstable manifold with nodal-type dynamics.
\end{itemize}
\end{lemma}
We illustrate the findings in the $\bar \epsilon=1$-chart in \figref{cuspednode2}. See figure caption for further details. We are now ready to describe our main result on small amplitude oscillations for the cusped saddle-node. \begin{theorem}\thmlab{main2}
\response{Consider $c$ as in \eqref{c2expr} with $c_2\in \left(-\frac{1}{3v_s},0\right)$ fixed and any point $p$ so that $\pi_a(p)\in S_a\backslash \gamma$. Then the following holds for all $0<\epsilon\ll 1$: The forward orbit of $p$ intersects the section defined by $y=0$ in a point $(u,y,z)=(u_0,0,z_0)$ with \begin{align}\eqlab{amplitudeuzexp}
u_0,z_0=\mathcal O(e^{-c/\sqrt{\epsilon}}). \end{align} The number of SAOs of the forward orbit is unbounded as $\epsilon\rightarrow 0$, but finitely many are $\mathcal O(1)$ in amplitude in the $(u_2,z_2)$-plane. }
\end{theorem}
\begin{proof}
\response{
For $c_2\in \left(-\frac{1}{3v_s},0\right)$, $q_2$ belongs to $\gamma_2^{rf}$ and is of saddle-focus type, recall \lemmaref{q2c2}.
\eqref{amplitudeuzexp} follows directly from the exponentially contraction $e^{- c \tau/\sqrt{\epsilon}}$ towards the invariant $\gamma_2^a$ on the slow time scale $\tau$ of \eqref{hatV21}; recall that $r_2=\epsilon^{1/4}$. Due to the focus behavior of $\gamma_2$ near $y_2=0$, recall \lemmaref{L2stability}, the forward orbit will experience an unbounded number of SAOs as $\epsilon\rightarrow 0$. These will be exponentially small in amplitude. Moreover, since $\pi_a(p)\in S_a\backslash \gamma$ and $\gamma_2$ is the stable manifold of $q_2$, the forward orbit of $p$ will extend along $\gamma_2^{rf}$, remaining exponential close for all $y_2\in [0,y_{21}(c_2)]$, for some $0<y_{21}(c_2)<2c_2g$. Beyond this, the orbit will eventually be repelled away from $\gamma_2$ due to the unstable manifold of $q_2$. Since $q_2\in \gamma_2^{rf}$ for $c_2\in \left(-\frac{1}{3v_s},0\right)$, we obtain finitely many $O(1)$ SAOs due to the focus dynamics in the $(u_2,z_2)$-projection at some distance from $q_2$. This completes the proof.
}
\end{proof}
\begin{remark}\remlab{remfinal}
\response{With the assumptions of \thmref{main2}, there is a bifurcation delay along $\gamma_2$. For the statement of the theorem, we did not need to determine this delay in details. However, due to the invariance of $\gamma_2$, it can be determined by a way-in/way-out function in the following way: Let \begin{align}\eqlab{nupm} \nu_{\pm}(y_2)=-\frac{3v_s}{2g}y_2\pm \frac12 \sqrt{\frac{9v_s^2}{4g^2}y_2^2-4}, \end{align} denote the eigenvalues of \eqref{matrixJac}. Then for $c_2<0$ the exit point $y_{2,\mathrm{exit}}\in (0,2c_2g)$ is for $r_2\rightarrow 0$ determined by \begin{align}
\int_{-\infty}^{y_{2,\mathrm{exit}}} \frac{\operatorname{Re}\nu_+(y_2)}{-c_2+\frac{1}{2g}y_2} dy_2 = 0.\eqlab{entryexit} \end{align} (The integral is convergent since $\frac{\operatorname{Re}\nu_+(y_2)}{-c_2+\frac{1}{2g}y_2}\approx \frac{-4g^2}{3v_s y_2^2}$ for $y_2\rightarrow -\infty$ and $y_{2,\mathrm{exit}}>0$ exists and is unique for each $c_2<0$ since $\frac{\operatorname{Re}\nu_+(y_2)}{-c_2+\frac{1}{2g}y_2}\rightarrow \infty$ for $y_2\rightarrow 2c_2 g^-$. $y_{2,\mathrm{exit}}(c_2)$ is also continuous and $y_{2,\mathrm{exit}}(0^-)=0$.) The integral gives a lengthy expression and we have not found a way to solve for $y_{2,\mathrm{exit}}$. We therefore only present a diagram (obtained in Matlab) for $g=-1$, see \figref{entryexit} and the figure caption for further details, of $y_{2,\mathrm{exit}}$ as a function of $c_2$.
Due the invariance of $\gamma_2$, the delay for our system \eqref{fhn} is different from the bifurcation delay for the folded saddle-node, see e.g.~\cite{krupa2010a}. Indeed, for the folded saddle-node, the delay for \textit{analytic systems} depends upon (following \cite{neishtadt1987a,neishtadt1988a}) buffer points. If we were to break the symmetry of \eqref{fhn}, then one would like to rely on the same methods. But this could be problematic in this context, since the center manifold in \propref{cmred} is not expected to be analytic. } \end{remark}
\begin{figure}
\caption{Illustration of the dynamics in the $\bar \epsilon=1$-chart in the case of the cusped saddle-node. The manifold of limit cycles $P_2$ is in orange while the critical manifold $\gamma_2$ is in pink. On the positive side of $y_2=0$, we illustrate the normal dynamics on $\gamma_2^{rf}$ in cyan (focus type) and on $\gamma_2^{rf}$ in green (nodal type). When the equilibrium $q_2\in \gamma_2$ (also cyan) lies on $\gamma_2^{rf}$, SAOs of order $\mathcal O(1)$ (in the $(u_2,y_2,z_2)$-scaling) occur near $W^u(q_2)$.}
\end{figure}
A similar result holds for $c_2< -\frac{1}{3v_s}$, but
due to the normal nodal dynamics along $\gamma^{rn}$ all SAOs may be exponentially small in this case. \response{We see this in \figref{entryexit} for the value of $g=-1$. In particular, for $c_2<-0.42$ the exit point (green part of curve) is in the normal nodal regime where there are no additional $\mathcal O(1)$-oscillations when the trajectory separate from $\gamma_2$.}
For $c_2>0$, on the other hand, the forward flow of $p$ is attracted to the stable equilibrium near $q_2$.
\begin{remark}\remlab{cblowup} Formally, the scaling \eqref{c2expr} does not overlap with the regime covered by \thmref{main1} where $c$ is fixed in a compact subset of $c<v_s$. There is therefore a gap that we do not cover in this paper. However, to cover this gap, and obtain a complete description of $c$ in a full neighborhood of $v_s$, one could include $c$ in the blowup transformation \eqref{blowup1} as follows \begin{align*}
c = v_s+r^2 \bar c, \end{align*} and consider $(\bar u,\bar y,\bar z,\bar \epsilon,\bar c)\in S^4$. In particular, in this way, one could cover the gap by working in the directional chart corresponding to $\bar c=-1$. \response{Notice that the associated scaling chart $\bar \epsilon=1$ gives rise to the same coordinates $(u_2,y_2,z_2,r_2,c_2)$ where $c=v_s+r^2 c_2$ in agreement with \eqref{c2expr}. (This also motivates the use of the subscript on $c$, recall the convention before \remref{broercusp}.) } We shall not pursue this further in the present paper.
\end{remark}
\section{Conclusions}\label{sec:final}
\response{ In this paper, we have analyzed cusped singularities (cusped node and cusped saddle-node) and demonstrated that they form a mechanism for SAOs in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. As for the folded node, we showed that the number of SAOs is determined by the Weber equation and the ratio of eigenvalues of the cusped node (upon desingularization). Similarly, we showed that the cusped saddle-node marks the onset of SAOs. Although there are many similarities between the folded singularities and the cusped versions studied in the present paper, there are also several differences, see e.g. \remref{strong} and \lemmaref{lienard}. Perhaps most importantly, our cusped node does not have a strong canard and there are also two fast directions away from the cusp ($u$ increasing and $u$ decreasing in \figref{lemma1}), as opposed to just one in the case of the standard folded singularity. The latter property also has consequences on MMOs and the LAOs that we see in \figref{MMO}. For the folded node, MMOs occur if there is return to the funnel region, see \cite{brons06}. The same is true in the present case, but it is slightly more subtle. Suppose (for definiteness) that there is a return mechanism to $S_a\backslash\gamma$, leaving the cusp region along the positive $u$-direction.
Then as a consequence of \thmref{main1}, we obtain the following: {Let $\lfloor \frac{\lambda_2}{\lambda_1}\rfloor$ be even (odd) and suppose that the return to $S_a$ is on the $u$-positive side ($u$-negative side, respectively) of $\gamma$. Then we have (``one-sided'') MMOs for all $0<\epsilon\ll 1$ with $u$ always increasing upon passage through the cusp.} However, if affirmative, then the system \eqref{fhn} -- due to the symmetry $\mathcal S$ -- also has MMOs with $u$ always decreasing upon passage through the cusp. In fact, more generally, once we have a return to $S_a\backslash \gamma$ along one direction ($u$-positive or $u$-negative), then the symmetry give rise to a return along the other direction ($u$-negative or $u$-positive, respectively) too. We can then also have (``mixed'') MMOs where $u$ alternates sign upon passing through the cusp $f_1$. We see this
in \figref{MMO} for $c=1.27$. Indeed, here there is an alternation between $v_1$ and $v_2$ being increasing ($v_2$, respectively, $v_1$ decreasing) which precisely corresponds to a change in sign in $u$. The description of the return mechanism for \eqref{fhn}, and whether we have ``one-sided'' or ``mixed'' MMOs, require a careful analysis of the layer problem \eqref{layer} but also of the reduced problem \eqref{reduced} (away from the cusp). We leave such an analysis to future work. }
In future work, it would also be interesting to study the cusped singularities in a general setting without a symmetry. We already have some partial results in this direction. \response{The cusped node then becomes a co-dimension one bifurcation of a folded node that transverses the cusp upon parameter variation. In line with our findings, the number of SAOs does not change upon this passage. Within this context, it would also be interesting in future work to study the secondary canards and the role of a strong canard. }
Similarly, the cusped saddle-node becomes co-dimension two without the symmetry. However, going from a folded saddle-node to a cusped saddle-node seems slightly more involved. A folded saddle-node (type II) is accompanied by a canard-like explosion of limit cycles (due to the strong canard), see also \cite{kristiansen2022}. In our symmetric cusped saddle-node there is no explosion, but instead a cylinder on which limit cycles occur, recall \propref{contractg2}. It is unclear how this scenario unfolds without the symmetry and how it precisely connects to the folded saddle-node. Moreover, a folded saddle-node actually comes in two versions. We have only focused on type II in this manuscript \cite{krupa_extending_2001}, but there is also a type I \cite{vo15}. Future research should also uncover how the generalized cusped saddle-node relates to these.
\subsection*{Acknowledgment} The authors are thankful for the discussions they have had with Morten Br{\o}ns in preparation of this manuscript.
\end{document} | arXiv |
Properties of Triangle
1. Angle Sum Property
2. Triangle Inequality Property
3. Pythagoras Property
4. Exterior Angle Property
5. Congruence Property
6. Similarity Property
What are the basic properties of a triangle?
What is a right-angled triangle?
What is the angle sum property of a triangle?
A triangle is a polygon with three angles, three sides, and three vertices. A triangle's properties help us easily identify a triangle from a given set of figures.
Let's understand the properties of triangles that are based on their sides and angles with examples.
The properties of a triangle help us to identify relationships between different sides and angles of a triangle. Some of the important properties of a triangle are as follows.
Angle Sum Property
Triangle Inequality Property
Pythagoras Property
Exterior Angle Property
Congruence Property
Similarity Property
According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is $180^{\circ}$. The angle sum property is used to find the measure of an unknown interior angle when the values of the other two angles are known.
The angle sum property formula for any polygon is expressed as, $\text{S} = \left(n − 2 \right) \times 180^{\circ}$, where $n$ represents the number of sides in the polygon. This property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it.
In the case of a triangle, $n = 3$, therefore, the formula for triangle becomes $\text{S} = \left(3 − 2 \right) \times 180^{\circ} = 1 \times 180^{\circ} = 180^{\circ}$.
In the above figure, in $\triangle \text{LMN}$, $\angle \text{L} + \angle \text{M} + \angle \text{N} = 180^{\circ}$.
The triangle inequality theorem states, "The sum of any two sides of a triangle is greater than its third side and the difference of any two sides of a triangle is less than its third side." This theorem helps us to identify whether it is possible to draw a triangle with the given measurements or not without actually doing the construction.
For a triangle with length of sides $a$, $b$ and $c$,
$a + b \gt c$
$b + c \gt a$
$c + a \gt b$
$|a – b| \lt c$
$|b – c| \lt a$
$|c – a| \lt b$
According to the Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as $\left( \text{Hypotenuse} \right)^{2}= \left( \text{Base} \right)^{2} + \left( \text{Altitude} \right)^{2}$.
In the above figure, $\triangle \text{ABC}$ is a right-triangle, right-angled at $\text{B}$, where
CA is Hypotenuse
BC is Base
AB is Altitude
Therefore, according to Pythagoras property, $\text{CA}^{2} = \text{BC}^{2} + \text{AB}^{2}$
The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles of the triangle. The exterior angle of any interior angle and the interior angle form a linear pair.
In the above figure, the internal angles of the triangle are $\angle \text{A}$, $\angle \text{B}$, and $\angle \text{C}$, and their corresponding exterior angles are $\text{Ext} \angle \text{A}$, $\text{Ext} \angle \text{B}$, and $\text{Ext} \angle \text{C}$.
Note: The interior angle and its corresponding exterior angle form a linear pair.
According to the congruence property, two triangles are said to be congruent(equal) if all their corresponding sides and angles are equal.
In the above figure, in $\triangle \text{ABC}$ and $\triangle \text{DEF}$, if $\text{AB} = \text{EF}$, $\text{BC} = \text{FD}$, $\text{CA} = \text{DE}$ and $\angle \text{C} =\angle \text{D}$, $\angle \text{A} =\angle \text{E}$, $\angle \text{B} =\angle \text{F}$, then we say that $\triangle \text{ABC}$ is congruent to $\triangle \text{DEF}$ or $\triangle \text{ABC} \cong \triangle \text{DEF}$.
Types of Coordinate Systems
According to the similarity property, two triangles are said to be similar if all their corresponding angles are equal and the corresponding sides are in the same ratio.
In the above figure, in $\triangle \text{ABC}$ and $\triangle \text{DEF}$, if $\frac{\text{AB}}{\text{EF}} =\frac{\text{BC}}{\text{FD}} = \frac{\text{CA}}{\text{DE}}$ and $\angle \text{C} =\angle \text{D}$, $\angle \text{A} =\angle \text{E}$, $\angle \text{B} =\angle \text{F}$, then we say that $\triangle \text{ABC}$ is similar to $\triangle \text{DEF}$ or $\triangle \text{ABC} \sim \triangle \text{DEF}$.
Explain the following properties of a triangle
The basic properties of a triangle are
a) Angle Sum Property
b) Triangle Inequality Property
c) Pythagoras Property
d) Exterior Angle Property
e) Congruence Property
f) Similarity Property
A triangle that has one of the interior angles as $90^{\circ}$ is called a right-angled triangle.
According to the angle sum property of a triangle, the sum of the interior angles of a triangle is always $180^{\circ}$. For example, if the three interior angles of a triangle are given as $\angle \text{A}$, $\angle \text{B}$, and $\angle \text{C}$, then this according to this property $\angle \text{A} + \angle \text{B} + \angle \text{C} = 180^{\circ}$.
A three-sided figure commonly called a triangle has three vertices and three angles. All triangles exhibit five properties which are the angle sum property, triangle inequality property, exterior angle property, congruence property, and similarity property. The right triangles also exhibit Pythagoras property.
Difference Between Axiom, Postulate and Theorem
What Are 2D Shapes – Names, Definitions & Properties
3D Shapes – Definition, Properties & Types | CommonCrawl |
\begin{document}
\title {New bounds on the Lebesgue constants of Leja sequences on the unit disc and on $\Re$-Leja sequences.}
\author{
Moulay Abdellah CHKIFA
\footnote{UPMC Univ Paris 06, UMR 7598, Laboratoire
Jacques-Louis Lions, F-75005, Paris, France \newline CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. [email protected]} }
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\begin{abstract} In the papers \cite{Ch1,Ch2} we have established linear and quadratic bounds, in $k$, on the growth of the Lebesgue constants associated with the $k$-sections of Leja sequences on the unit disc ${\cal U}$ and $\Re$-Leja sequences obtained from the latter by projection into $[-1,1]$. In this paper, we improve these bounds and derive sub-linear and sub-quadratic bounds. The main novelty is the introduction of a ``quadratic'' Lebesgue function for Leja sequences on ${\cal U}$ which exploits perfectly the binary structure of such sequences and can be sharply bounded. This yields new bounds on the Lebesgue constants of such sequences, that are almost of order $\sqrt k$ when $k$ has a sparse binary expansion. It also yields an improvement on the Lebesgue constants associated with $\Re$-Leja sequences. \end{abstract}
\section{Introduction}
The growth of the Lebesgue constant of Leja sequences on the unit disc and $\Re$-Leja sequences was first studied in \cite{CaP1,CaP2}. The main motivation was the study of the stability of Lagrange interpolation in multi-dimension based on intertwining of block unisolvent arrays. Such sequences, more particularly $\Re$-Leja sequences, were also considered in many other works in the framework of structured hierarchical interpolation in high dimension. Although not always referred to as such, they are typically considered in the framework of sparse grids for interpolation and quadrature \cite{HJG,GWG}. Indeed, the sections of length $2^n+1$ of $\Re$-Leja sequences coincide with the Clenshaw-Curtis abscissas $\cos(2^{-n}j\pi),j=0,\dots,2^n$ which are de facto used, thanks to the logarithmic growth of their Lebesgue constant.
Motivated by the development of cheap and stable non-intrusive methods for the treatment of parametric PDEs in high dimension, we have also used these sequences in \cite{CCS1,CCPP} with a highly sparse hierarchical polynomial interpolation procedure. The multi-variate interpolation process is based on the Smolyak formula and the sampling set is progressively enriched in a structured infinite grid $\otimes_{j=0}^d Z$ together with the polynomial space by only one element at a time. The Lebesgue constant that quantifies the stability of the interpolation process depends naturally on the sequence $Z$. We have shown in \cite{Ch2} that it has quadratic and cubic bounds in the number of points of interpolation when $Z$ is a Leja sequence on ${\cal U}$ or an $\Re$-Leja sequence, thanks to the linear and quadratic bounds on the growth of the Lebesugue constant of such sequences, also established in \cite{Ch1,Ch2}. We refer to the introduction and section $2$ in \cite{Ch2} for a concise description on the construction of the interpolation process and the study of its stability.
The present paper is also concerned with the growth of the Lebesgue constant of Leja and $\Re$-Leja sequences. We improve the linear and quadratic bounds obtained in \cite{Ch2}. In particular, we show that for $\Re$-Leja sequnences, the bound is logarithmic for many values of $k$ which may be useful for proposing cheap and stable interpolation scheme in the framework of sparse grids \cite{GWG}.
\subsection{One dimensional hierarchical interpolation}
Let $X$ be a compact domain in $\mathbb{C}$ or $\mathbb{R}$, typically the complex unit disc ${\cal U}:=\{|z|\leq 1\}$ or the unit interval $[-1,1]$, and $Z=(z_j)_{j\geq0}$ a sequence of mutually distinct points in $X$. We denote by $I_{Z_k}$ the univariate interpolation operator onto the polynomials space ${\mathbb P}_{k-1}$ associated with the section of length $k$, $Z_k:=(z_0,\cdots,z_{k-1})$. The interpolation operator is given by Lagrange interpolation formula: for $f\in {\cal C}(X)$ and $z\in X$ \begin{equation} I_{Z_k} f (z) =\sum_{j=0}^{k-1} f(z_j) l_{j,k} (z),\quad\quad l_{j,k}(z):=\prod_{\substack{i=0\\infty\neq j}}^{k-1} \frac{z-z_j}{z_i-z_j}, \;\; j=0,\dots,k-1. \end{equation} Since the sections $Z_k$ are nested, it is convenient to give the operator $I_{Z_k}$ using Newton interpolation formula which amounts essentially writing: $\Delta_0 f:= I_{Z_1}f \equiv f(z_0)$ and \begin{equation} \label{newton} I_{Z_{k+1}}=I_{Z_{k}}+ \Delta_k = \sum_{l=0}^k \Delta_l, \quad\mbox{where}\quad \Delta_l(Z):=I_{Z_{l+1}}-I_{Z_l},\;\;l\geq1, \end{equation} and computing the operators $\Delta_l$ using divided differences, see \cite[Chapter 2]{Davis} or equivalently the following formula which are differently normalized, see \cite{Ch2,CCS1}, \begin{equation} \Delta_l f = \(f(z_l) - I_{Z_l}f(z_l)\)\prod_{j=0}^{l-1} \frac {(z-z_j)}{(z_l-z_j)},\quad l\geq1, \end{equation}
The stability of the operator $I_{Z_k}$ depends on the positions of the elements of $Z_k$ on $X$, in particular through the Lebesgue constant associated with $Z_k$, defined by
\begin{equation} \mathbb{L}_{Z_k}:=\max_{f\in C(X)-\{0\}}
\frac { \|I_{Z_k} f\|_{L^\infty(X)}}{\|f\|_{L^\infty(X)}} =\max_{z\in X} \lambda_{Z_k}(z), \label{leblag} \end{equation} where $\lambda_{Z_k}$ is the so-called Lebesgue function associated with $Z_{k}$ defined by \begin{equation} \label{lagrange}
\lambda_{Z_k}(z):=\sum_{i=0}^{k-1} |l_{i,k}(z)|, \quad \quad z\in X. \end{equation} We also introduce the notation \begin{equation} \label{normDeltak} \mathbb{D}_k(Z):=\max_{f\in C(X)-\{0\}}
\frac { \|\Delta_k f\|_{L^\infty(X)}}{\|f\|_{L^\infty(X)}}. \end{equation}
In the case of the unit disk or the unit interval, it is known that $\mathbb{L}_k$ the Lebesgue constant associated with any set of $k$ mutually distinct points can not grow slower than $\frac {2\log(k)}\pi$ and it is well known that such growth is fulfilled by the set of $k$-roots of unity in the case $X={\cal U}$ and the Tchybeshev or Gauss-Lobatto abscissas in the case $X= [-1,1]$, see e.g. \cite{Bern}. However such sets of points are not the sections of a fixed sequence $Z$.
In \cite{CaP1,CaP2}, the authors considered for $Z$ Leja sequences on ${\cal U}$ initiated at the boundary $\partial {\cal U}$ and $\Re$-Leja sequences obtained by projection onto $[1,1]$ of the latter when initiated at $1$. They showed that the growth of $\mathbb{L}_{Z_k}$ is controlled by ${\cal O}(k\log (k))$ and ${\cal O}(k^3\log (k))$ respectively. In our previous works \cite{Ch1,Ch2}, we have improved these bounds to $2k$ and $8\sqrt 2k^2$ respectively. We have also established in \cite{Ch2} the bound ${\mathbb D}_{k}\leq (1+k)^2$ for the difference operator, which could not be obtained directly from ${\mathbb D}_{k}\leq \mathbb{L}_{Z_{k+1}}+\mathbb{L}_{Z_k}$ and which is essential to prove that the multivariate interpolation operator using $\Re$-Leja sequences has a cubic Lebesgue constant, see \cite[formula 25]{Ch2}.
\subsection{Contributions of the paper} In this paper, we improve the bounds of the
previous paper \cite{CaP1,CaP2,Ch1,Ch2}.
Our techniques of proof share several points with those developed in \cite{Ch1,Ch2}, yet they are shorter and relies notably on the binary pattern of Leja sequences on the unit disk. The novelty in the present paper is the introduction of the ``quadractic'' Lebesgue constant \begin{equation} \label{lagrangeQuadratic} \lambda_{Z_k,2}(z):=
\(\sum_{i=0}^{k-1} |l_{i,k}(z)|^2\)^{\frac12}, \quad z\in X, \end{equation} where $l_{i,k}$ are the Lagrange polynomials as defined in \iref{lagrange}. We study this function and its maximum \begin{equation} \label{lagrangeQuadratic} \mathbb{L}_{Z_k,2} :=\max_{z\in X} \lambda_{Z_k,2}(z). \end{equation} We establish in \S 2 in the case where $Z$ is any Leja sequence on ${\cal U}$ initiated on the boundary $\partial {\cal U}$ the ``sharp'' inequality \begin{equation} \label{boundLk2Intro} \lambda_{Z_k}(z_k) \leq \mathbb{L}_{Z_k,2} \leq 3\lambda_{Z_k}(z_k), \quad{and}\quad \lambda_{Z_k}(z_k):=\sqrt {2^{\sigma_1(k)}-1}, \end{equation} where $\sigma_1(k)$ denote the number of ones in the binary expansion of $k$. Cauchy-schwatrz inequality applied to the Lebesgue function $\lambda_{Z_k}$ defined in \iref{lagrange} yields $\lambda_{Z_k} \leq \sqrt {k}~\lambda_{Z_k,2}$. This shows that we also establish \begin{equation} \mathbb{L}_{Z_k} \leq 3 \sqrt k ~\sqrt {2^{\sigma_1(k)}-1}, \end{equation} for Leja sequences on ${\cal U}$, which improves considerably the linear bound $2k$ established in \cite{Ch1} when the binary expansion of $k$ is very sparse. For example, for $k=2^n+3$ with $n$ large, we get $\mathbb{L}_{Z_k}\leq 3\sqrt{7k} <<k$. Using the bound \iref{boundLk2Intro}, we establish in \S3 a new bound on the growth of Lebesgue constants of $\Re$-Leja sequences that implies \begin{equation} \mathbb{L}_k \leq 6 \sqrt 5 ~k~ 2^{\sigma_1(l)}, \quad \mbox{where} \quad l=k-(2^n+1), \end{equation} where $n$ is the integer such that $2^n+1 \leq k <2^{n+1}+1$. Again, we remark that the previous bound improves the bound $8\sqrt 2k^2$ established in \cite{Ch2} when $l$ is small compared to $2^n$ or very sparse in the sense of binary expansion. We actually prove a bound that is logarithmic for many values of $k$ other than the values $2^n+1$, see Theorem \ref{theoLebesgueConstant}.
Finally, we provide in \S \ref{sectionDiff} new bounds on the growth of $\mathbb{D}_k$ the norm of the difference operators. We provide the bounds \begin{equation} \mathbb{D}_k \leq 1+\sqrt{k(2^{\sigma_1(k)}-1)}, \quad \quad \mathbb{D}_k \leq 2^{\sigma_1(k)} 2^{n}, \quad k\geq1. \end{equation} in the case of Leja sequences on ${\cal U}$ and the case of $\Re$-Leja sequences respectively where for the latter $n$ is defined as above.
\subsection{Notation}
In the remainder of the paper, we work with the following notation. For an infinite sequence $Z:=(z_j)_{j\geq0}$ on $X$, we introduce the section $Z_{l,m} :=(z_l,\cdots,z_{m-1})$ for any $l \leq m-1$. Given two finite sequences $A=(a_0,\dots,a_{k-1})$ and $B=(b_0,\dots,b_{l-1})$, we denote by $A\wedge B$ the concatenation of $A$ and $B$, i.e. $A \wedge B =(a_0,\dots,a_{k-1},b_0,\dots,b_{l-1})$. For any finite set $S=(s_0,\cdots,s_{k-1})$ of complex numbers and $\rho \in \mathbb{C}$, we introduce the notation \begin{equation} \label{rhoRealConjugate} \rho S:=(\rho s_0,\cdots, \rho s_{k-1}),\quad \Re (S):=(\Re(s_0),\cdots, \Re(s_{k-1})),\quad \overline{S}:=(\overline{s_0},\cdots, \overline{s_{k-1}}). \end{equation} Throughout this paper, to any finite set $S$ of numbers, we associate the polynomial \begin{equation} \label{w} w_S(z):=\prod_{s\in S}(z-s),\quad \mbox{with the convention} \quad w_{\emptyset}(z):=1 \end{equation} Any integer $k\geq1$ can be uniquely expanded according to \begin{equation} \label{binaryk} k = \sum_{j=0}^n a_j 2^j, \quad a_j \in \{0,1\}
\end{equation} We denote by $\sigma_1(k)$, $\sigma_0(k)$ the number of ones and zeros in the binary expansion of $k$ and by $p(k)$ the largest integer $p$ such that $2^p$ divide $k$.
For $k=2^n,\dots, 2^{n+1}-1$ with binary expansion as above, one has \begin{equation} \sigma_1(k) = \sum_{j=0}^n a_j \quad\mbox{and}\quad \sigma_0(k) = \sum_{j=0}^n (1-a_j) = n+1-\sigma_1(k). \label{defsigma10} \end{equation}
We should finally note that, unless stated otherwise, we only work with complex numbers $z$ belonging to the unit circle $\partial {\cal U}$. This is because in the complex setting we investigate supremums of sub-harmonic functions, $\lambda_{Z_k}$ and $\lambda_{Z_k,2}$, which is always attained on the boundary.
\section {Leja sequences on the unit disk}
Leja sequences $E=(e_j)_{j\geq0}$ on $\cal U$ considered in \cite{CaP1,CaP2,Ch1,Ch2} have all their initial value $e_0\in\partial \cal U$ the unit circle. They are defined inductively by picking $e_0 \in \partial{\cal U}$ arbitrary and defining $e_k$ for $k\geq1$ by \begin{equation} \label{defLejaC}
e_k = {\rm argmax}_{z\in \cal U} |z-e_{k-1}|\dots |z -e_0|. \end{equation} The maximum principle implies that $e_j \in \partial \cal U$ for any $j\geq1$. Also, the previous ${\rm argmax}$ problem might admit many solutions and $e_k$ is one of them. We call a $k$-Leja section every finite sequence $(e_0,\dots,e_{k-1})$ obtained by the same recursive procedure. In particular, when $E:=(e_j)_{j\geq1}$ is a Leja sequence then the section $E_k = (e_0,\dots,e_{k-1})$ is $k$-Leja section.
In contrast to the interval $[-1,1]$ where Leja sequences cannot be computed explicitly, Leja sequences on $\partial \cal U$ are much easier to compute. For instance, if $e_0=1$ then we can immediately check that $e_1=-1$ and $e_2=\pm i$. Assuming that $e_2=i$ then
$e_3$ maximizes $|z^2-1||z-i|$, so that $e_3=-i$ because
$-i$ maximizes jointly $|z^2-1|$ and $|z-i|$. Then $e_4$
maximizes $|z^4-1|$, etc. We observe a ``binary patten'' on the distribution of the first elements of $E$.
By radial invariance, an arbitrary Leja sequence $E=(e_0,e_1,\dots)$ on ${\cal U}$ with $e_0\in\partial{\cal U}$ is merely the product by $e_0$ of a Leja sequence with initial value $1$. The latter are completely determined according to the following theorem, see {\cite{BCCa,CaP1,Ch1}}. \begin{theorem} Let $n\geq0$, $2^n < k\leq 2^{n+1}$ and $l=k-2^n$. The sequence $E_k=(e_0,\dots,e_{k-1})$, with $e_0=1$, is a $k$-Leja section if and only if $E_{2^n}=(e_0,\dots,e_{2^n-1})$ and $U_l= (e_{2^n},\dots,e_{k-1})$ are respectively $2^n$-Leja and $l$-Leja sections and $e_{2^n}$ is any $2^n$-root of $-1$. \label{inversetheostruct} \end{theorem}
In the light of the previous theorem, a natural construction of a Leja sequence $E:=(e_j)_{j\geq0}$ in ${\cal U}$ follows by the recursion \begin{equation} E_1 := (e_0=1) \quad {\rm and}\quad E_{2^{n+1}} :=E_{2^n} \wedge e^{\frac {i\pi}{2^n}} E_{2^n},\quad n\geq0. \end{equation} This recursive construction of the sequence $E$ yields an interesting distribution of its elements. Indeed, by an immediate induction, see \cite{BCCa}, it can be shown that the elements $e_k$ are given by \begin{equation} \label{SimpleLejaSequence} e_k =
\exp \Big( i\pi \sum_{j=0}^n a_j 2^{-j} \Big) \quad {\rm for}\quad
k = \sum_{j=0}^n a_j 2^j, \quad a_j \in \{0,1\}. \end{equation}
The construction yields then a low-discrepancy sequence on $\partial \cal U$ based on the bit-reversal Van der Corput enumeration.
As already mentioned above, Theorem \ref{inversetheostruct} characterizes completely Leja sequences on the unit circle. It has also many implications that turn out to be very useful in the analysis of the growth of Lebesgue constants. \begin{theorem} Let $E:=(e_j)_{j\geq0}$ be a Leja sequence on $\cal U$ initiated at $e_0\in \partial {\cal U}$. We have: \begin{itemize} \item For any $n\geq0$, $E_{2^n}= e_0 {\cal U}_{2^n}$ in the set sense where ${\cal U}_{2^n}$ is the set of $2^n$-root of unity.
\item For any $k\geq1$, $|w_{E_k}(e_k)| =\sup_{z\in\partial{\cal U}} |w_{E_k}(z)| = 2^{\sigma_1(k)}$. \item For any $n\geq0$, $E_{2^n,2^{n+1}} := (e_{2^n},\cdots,e_{2^{n+1}-1})$ is a $2^n$-Leja section. \item The sequence $E^2:=(e_{2j}^2)_{j\geq0}$ is a Leja sequence on $\partial {\cal U}$. \end{itemize} \label{TheoImplications} \end{theorem}
Such properties can be easily checked for the simple sequence defined in \iref{SimpleLejaSequence} and are given in \cite{CaP1,Ch1} for more general Leja sequences.
\subsection{Analysis of the quadratic Lebesgue function}
It is proved in \cite{Ch1} that given two $k$-Leja sections $E_k$ and $F_k$, one has $F_k=\rho E_k$ in the set sense for some $\rho\in \partial {\cal U}$. This means that the sequence $F_k$ can be obtained from $E_k$ by a permutation and the product by $\rho$. By inspection of the quadratic Lebesgue function \iref{lagrangeQuadratic}, we have then that \begin{equation} \lambda_{F_k,2}(z) = \lambda_{E_k,2}(z/\rho),\;\;\;z\in {\cal U} \quad \Longrightarrow \quad \mathbb{L}_{F_k,2} = \mathbb{L}_{E_k,2}. \end{equation} In order to compute the growth of $\mathbb{L}_{E_k,2}$ for arbitrary Leja sequences $E$, it suffices then to consider $E$ to be the simple sequence given by \iref{SimpleLejaSequence}. Unless stated otherwise, for the rest of this section, $E$ is exclusively used for this notation. Let us note that
\begin{equation} \label{memoryShort} E^2 :=(e_{2j}^2)_{j\geq0}=E. \end{equation}
In order to study the functions $\lambda_{E_k,2}$, we adopt the methodology that we introduced in \cite{Ch1}. Namely, we study the implication of $E$ being a Leja sequence in general, on the growth of $\lambda_{E_k,2}$, then we use the implication of the particular binary distribution of $E$ to derive such growth.
\begin{lemma} Let $Z$ be a Leja sequence on a real or complex compact $X$. For any $k\geq 1$ and any $z\in X$, it holds \begin{equation} \label{growthLeja} \lambda_{Z_{k+1},2}(z) \leq \lambda_{Z_{k},2}(z) + \lambda_{Z_{k},2}(z_k)+1. \label{kkp1} \end{equation} \label{propertyLeja2} \end{lemma}
{\bf Proof:} We fix $k\geq 1$ and denote by $l_0,\dots,l_{k-1}$ the Lagrange polynomials associated with the section $Z_{k}$ and by $L_0,\dots,L_{k}$ the Lagrange polynomials associated with the section $Z_{k+1}$. By Lagrange interpolation formula, for $j=0,\dots,k-1$ $$ l_j = \sum_{i=0}^k l_{j}(z_i) L_i = L_j+ l_j(z_k) L_k \quad\Rightarrow\quad L_j= l_j- l_j(z_k) L_k. $$ We have then for any $z\in X$ $$
\(\sum_{j=0}^{k-1}|L_j (z)|^2\)^{1/2}
\leq\(\sum_{j=0}^{k-1}|l_j(z)|^2\)^{1/2}+
|L_k(z)|\(\sum_{j=0}^{k-1} |l_j(z_k)|^2\)^{1/2}. $$ where we have merely applied triangular inequality with the euclidean norm in $\mathbb{C}^k$. This also writes $$
\(|\lambda_{Z_{k+1},2}(z)|^2
- |L_k(z)|^2\)^{\frac 12} \leq \lambda_{Z_k,2}(z) +
|L_k(z)| \lambda_{Z_k,2}(z_k). $$ We conclude the proof using $a \leq \sqrt{a^2-b^2}+b$ for $a\geq b\geq0$, and the inequality $$
|L_k(z)| = \frac{|w_{Z_k}(z)|}{|w_{Z_k}(z_k)|} \leq 1, $$ which follows from the Leja definition \iref{defLejaC}.
$\blacksquare$ \newline
The previous result shows that given $Z$ a Leja sequence over $X$, the growth of $\mathbb{L}_{Z_k,2}$ is monitored by the growth of $\lambda_{Z_k,2}(z_k)$. In particular, it is easily checked using induction on $k$ that \begin{equation} \lambda_{Z_k,2}(z_k) = {\cal O}(\log(k))\quad \Longrightarrow \quad \mathbb{L}_{Z_k,2} = {\cal O}(k\log(k)), \end{equation} and \begin{equation} \lambda_{Z_k,2}(z_k) = {\cal O}(k^\theta) \quad \Longrightarrow \quad \mathbb{L}_{Z_k,2} = {\cal O}(k^{\theta+1}). \end{equation}
In the following, we show basically that the previous implication holds with $\theta=1/2$ for Leja sequences on ${\cal U}$. However, we use the particular structure of such sequences in order to show that the exponent $\theta=1/2$ is not deteriorated and that it is also valid for $\mathbb{L}_{E_k,2}$. We recall that we work with the simple sequence $E$ given in \iref{SimpleLejaSequence} for which $E^2=E$. The binary patten of the distribution of E on the unit disc yields the following result.
\begin{lemma} Let $E$ be as in \iref{SimpleLejaSequence}. For any $N\geq1$, one has
\begin{equation}
\lambda_{E_{2N},2}(z) = \lambda_{E_N,2} (z^2),
\quad \quad z\in \partial {\cal U}.
\label{k2nk} \end{equation} \label{TheoBoundRecurs} \end{lemma}
{\bf Proof:} Let $l_0,\dots,l_{2N-1}$ be the Lagrange polynomials associated with $E_{2N}$ and $L_0,\dots,L_{N-1}$ be the Lagrange polynomials associated with $E_N$. Since $e_{2j+1}= -e_{2j}$ for any $j\geq0$, then in view of \iref{memoryShort} $$ w_{E_{2N}}(z) = w_{E_N^2}(z^2) = w_{E_N}(z^2). $$ Deriving with respect to $z$ and using $(e_{2j+1})^2 =(e_{2j})^2= e_j$ for any $j\geq0$, we deduce that \begin{equation} \label{Deriv}
|w_{E_{2N}}'(e_{2j+1})| =
|w_{E_{2N}}'(e_{2j})| =
2 |w_{E_N}'(e_{2j}^2)| =
2 |w_{E_N}'(e_j)|,\quad j\geq0. \end{equation} We have for any $j=0,\dots,N-1$ $$
|l_{2j}(z)|
=\frac {|w_{E_{2N}}(z)|}{|w_{E_{2N}}'(e_{2j})| |z-{e_{2j}}|}, \quad \quad \quad
|l_{2j+1}(z)|
=\frac {|w_{E_{2N}}(z)|}{|w_{E_{2N}}'(e_{2j+1})| |z-{e_{2j+1}}|}. $$ Therefore in view of the previous equalities \begin{equation}
|l_{2j}(z)|^2 +
|l_{2j+1}(z)|^2 =
\frac {|w_{E_N}(z^2)|^2}{4 |w_{E_N}'(e_j)|^2 } \Big[
\frac 1{|z-e_{2j}|^2} +
\frac {1}{|z+{e_{2j}}|^2} \Big] =
\frac {|w_{E_N}(z^2)|^2}{ |w_{E_N}'(e_j)|^2 |z^2-e_j|^2}
=
|L_j(z^2)|^2, \label{sumTwoPolynomials}
\end{equation} where we have used $|a-b|^2 +|a+b|^2 = 4$ for $a,b\in\partial {\cal U}$ and $e_{2j}^2=e_j$. Summing the previous identities for the indices $j=0,\dots,N-1$, we get the result.
$\blacksquare$ \newline
We note that the previous result combined with $E_{2^n}={\cal U}_{2^n}$ in the set sense implies that \begin{equation} \label{lambda2E2n} \sum_{j=0}^{2^n-1}
\Big| \frac{z^{2^n}-1}{2^n(z-e_j)}
\Big|^2 =\lambda_{E_{2^n},2}(z) =\lambda_{E_1,2}(z^{2^n})=1, \end{equation} for any $z\in \partial {\cal U}$. We now turn to the growth of $\lambda_{E_k,2}(e_k)$, which as mentioned earlier monitor the growth of $\mathbb{L}_{E_k,2}$.
\begin{lemma} For the Leja sequence $E$ defined in \iref{SimpleLejaSequence}, we have for any $k\geq1$, \begin{equation} \label{lambdaEkk} \lambda_{E_k,2}(e_k) = \sqrt{ 2^{\sigma_1(k)}-1} \end{equation} \label{lemmaQuadraticEkek} \end{lemma}
{\bf Proof:} First, by Lemma \ref{TheoBoundRecurs} and $e_{2N}^2=e_N$, one has \begin{equation} \label{k2nkAtek}
|\lambda_{E_{2N},2}(e_{2N})|^2 =
|\lambda_{E_N,2} (e_N)|^2,\quad N\geq1. \end{equation}
Let now $k$ be an odd number and we write $k=2N+1$ with $N\geq1$. Let $l_0,\dots,l_{2N}$ be the Lagrange polynomials associated with $E_k$ and $L_0,\dots,L_{N-1}$ be the Lagrange polynomials associated with $E_N$. For any $m=0,\dots,2N$, one has $$ l_m(e_k) = \frac {w_{E_k}(e_k)}{(e_k-e_m) w_{E_k}'(e_m)} = \frac {w'_{E_{k+1}}(e_k)}{w_{E_{k+1}}'(e_m)} \quad\Rightarrow\quad
|l_m(e_k)| =
\frac {|w'_{E_{N+1}}(e_k^2)|}{|w_{E_{N+1}}'(e_m^2)|}, $$ where we have used $k+1=2(N+1)$ and \iref{Deriv}. Using $e_{k}^2=e_N$ and $(e_{2j+1})^2 =(e_{2j})^2= e_j$ for any $j$, we get for $m=2j$ or $m=2j+1$ with $j=0,\dots,N-1$ $$
|l_m(e_k)|
= \frac {|w'_{E_{N+1}}(e_N)|}{|w_{E_{N+1}}'(e_j)|}
=|L_j(e_N)| \quad\mbox{and also}\quad
|l_{2N}(e_k)| = \frac {|w'_{E_{N+1}}(e_N)|}{|w_{E_{N+1}}'(e_N)|}=1. $$
Summing the numbers $|l_m(e_k)|^2$ over $m=0,\dots,2N$, we infer \begin{equation} \label{recursionEkek}
|\lambda_{E_{2N+1},2}(e_{2N+1})|^2 = 2 |\lambda_{E_N,2}(e_N)|^2+1. \end{equation} In view of the above and $\lambda_{E_1,2}(e_1)$=1, the sequence
$\alpha:=(\alpha_k := |\lambda_{E_k,2}(e_k)|^2)_{k\geq1}$ satisfies: $$ \alpha_1=1\quad \mbox{and}\quad \alpha_{2N}=\alpha_N,\quad \alpha_{2N+1}=2\alpha_N+1, \quad N\geq1. $$ We have $\sigma_1(1)=1$ and $\sigma_1(2N)=\sigma_1(N)$, $\sigma_1(2N+1)=\sigma_1(N)+1$ for any $N\geq1$. It is then easily checked that $(2^{\sigma_1(k)}-1)_{k\geq1}$ satisfies the same recursion as $\alpha$. This shows that $\alpha_k=2^{\sigma_1(k)}-1$ for any $k\geq1$ and finishes the proof.
$\blacksquare$ \newline
We are now able to conclude the main result of this section, which states basically that for the sequence $E$ or more generally any Leja sequence on ${\cal U}$ initiated at the boundary $\partial{\cal U}$, the value of $\mathbb{L}_{E_k,2}=\max_{z\in{\cal U}}\lambda_{E_k,2}(z)$ is almost equal to $\lambda_{E_k,2}(e_k)$.
\begin{theorem} For the Leja sequence $E$ defined in \iref{SimpleLejaSequence}, we have for any $k\geq1$ \begin{equation} \label{boundLinearLambdak}
1\leq \frac {\mathbb{L}_{E_k,2}}{\lambda_{E_k,2}(e_k)} =\frac {\mathbb{L}_{E_k,2}}{\sqrt{ 2^{\sigma_1(k)}-1}} \leq 3
\end{equation} \label{theoremLEk2} \end{theorem}
{\bf Proof:} The first part of the inequality is immediate from the definition of $\mathbb{L}_{E_k,2}$. Also in view Lemma \ref{TheoBoundRecurs} and formula \iref{k2nkAtek}, we only need to show \iref{boundLinearLambdak} when $k$ is an odd number. Let $k=2N+1$ with $N\geq1$. Using Lemma \ref{propertyLeja2}, Lemma \ref{TheoBoundRecurs} and formula \iref{k2nkAtek}, we have $$ \lambda_{E_k,2}(z) \leq \lambda_{E_{2N},2}(z) +\lambda_{E_{2N},2}(e_{2N})+1 =\lambda_{E_N,2}(z^2)+ \lambda_{E_N,2}(e_N)+1. $$ If we assumes that $\lambda_{E_N,2}(z^2)\leq 3 \lambda_{E_N,2}(e_N)$, we get $$ \lambda_{E_k,2}(z) \leq 4\lambda_{E_N,2}(e_N)+1
\leq 3 \sqrt{2|\lambda_{E_N,2}(e_N)|^2+1}, $$ where we have used the elementary inequality $4t+1\leq 3\sqrt{2t^2+1}$ for any $t\geq0$. In view of \iref{recursionEkek}, one then gets $\lambda_{E_k,2}(z)\leq 3\lambda_{E_k,2}(e_k)$. The verification $\mathbb{L}_{E_1,2}= \lambda_{E_1,2}(e_1)=1$ shows that the result follows using an induction on $k\geq1$.
$\blacksquare$
\subsection{Implications on the Lebesgue constant}
The methodology we have provided so far for bounding $\mathbb{L}_{E_k,2}$ is not new, we have developed it in \cite{Ch1} in order to give linear estimate for $\mathbb{L}_{E_k}$, namely $\mathbb{L}_{E_k}\leq 2k$. Theorem \iref{theoremLEk2} has also implications on the growth of the Lebesgue constant $\mathbb{L}_{E_k}$. Indeed, Cauchy Schwartz inequality applied to the Lebesgue function $\lambda_{E_k}$ implies $\lambda_{E_k}\leq~ \sqrt k~ \lambda_{E_k,2}$, so that \begin{equation} \label{boundLkLatest} \mathbb{L}_{E_k} \leq \sqrt k~ \mathbb{L}_{E_k,2} \leq 3\sqrt {k(2^{\sigma_1(k)}-1)}. \end{equation} The Cauchy Schwartz formula $\lambda_{E_k}\leq \sqrt {k} ~\lambda_{E_{k,2}}$ is possibly not very pessimistic. It has been recently proved that the Lagrange polynomials are uniformly bounded, see \cite{Irigoyen} We shall observe in particular, see Figure, that the binary pattern observed for the exact value of $\mathbb{L}_{E_k}$ is captured by the previous bound. Moreover, we are able to provide a lower bound for $\mathbb{L}_{E_k}$, that is comparable to the previous upper bound for values of $k$ with full binary expansion.
\begin{prop} For the Leja sequence $E$ defined in \iref{SimpleLejaSequence}, we have for any $k\geq1$ \label{TheoremSharpnessLebesgue} \begin{equation}
2^{\sigma_1(k)}-1 \leq \lambda_{E_k}(e_k) \leq \mathbb{L}_{E_k}.
\end{equation}
\end{prop}
{\bf Proof:} We let $N\geq1$ and we use the notation of the proof of Lemma \ref{TheoBoundRecurs}. As for formula \iref{sumTwoPolynomials} and since
$|a-b| +|a+b| \geq2$ for any $a,b\in\partial {\cal U}$, one has $$
|l_{2j}(z)| +
|l_{2j+1}(z)| =
\frac {|w_{E_N}(z^2)|}{2 |w_{E_N}'(e_j)| }
\frac {|z-e_{2j}|+|z+e_{2j}|}{|z-e_j|} \geq
|L_j(z^2)|. $$ This implies $\lambda_{E_{2N}}(z)\geq \lambda_{E_N}(z^2)$ and more particularly $\lambda_{E_{2N}}(e_{2N})\geq \lambda_{E_N}(e_N)$. As in the proof of Lemma \ref{lemmaQuadraticEkek}, we have also $\lambda_{E_{2N+1}}(e_{2N+1}) = 2\lambda_{E_{N}}(e_N)+1$. The sequence $(b_k := \lambda_{E_k}(e_k))_{k\geq1}$ satisfies: $$ b_1=1\quad \mbox{and}\quad b_{2N}\geq b_N,\quad b_{2N+1}=2b_N+1, \quad N\geq1. $$ The sequence $b$ then satisfies $b_k \geq 2^{\sigma_1(k)}-1$ for any $k\geq1$.
$\blacksquare$ \newline
The previous theorem combined with Theorem \ref{theoremLEk2} and \iref{boundLkLatest} implies \begin{equation} \frac{\sqrt{2^{\sigma_1(k)}-1}} 3 \mathbb{L}_{E_k,2} \leq \mathbb{L}_{E_k} \leq \sqrt k~ \mathbb{L}_{E_k,2}.
\end{equation} Cauchy Schwartz inequality is then satisfactory when $k\simeq 2^{\sigma_1(k)}$, that is when $k$ has a full binary expansion. \begin{remark} For integers $k=2^n,\dots,2^{n+1}-1$, if $k=2^{n+1}-1$ in which case $\sigma_1(k)=n+1$ is the largest possible, the bound \iref{boundLkLatest} merely implies $\mathbb{L}_{E_k}\leq 3k$ which is worse than the bound $2k$ established \cite{Ch1} and the exact value $\mathbb{L}_{E_k}=k$ of this case, see \cite{CaP1}. However, since $\sigma_1(k)=n+1-\sigma_0(k)$ for any $k =2^n,\dots 2^{n+1}-1$, then by \iref{boundLkLatest} \begin{equation} \mathbb{L}_{E_k} \leq \sqrt{\frac{18}{2^{\sigma_0(k)}}}~\sqrt {2^nk} \leq \sqrt{\frac{18}{2^{\sigma_0(k)}}}~k. \end{equation} This shows in particular that $\mathbb{L}_{E_k} \leq k$ whenever $\sigma_0(k)\geq5$. This last result answers partly the conjecture raised in \cite{CaP1} and which states that $\mathbb{L}_{E_k} \leq k$ for any $k\geq1$. \end{remark}
For the purpose of the next section, we improve the bound \iref{boundLkLatest} in the case where $k$ is an even number. We recall that we have shown in \cite[Theorem 2.8]{Ch1} \begin{equation} \label{binarymemory}
\mathbb{L}_{E_{2^p l}}\leq \mathbb{L}_{2^p}\mathbb{L}_{E_l}, \quad \quad p\geq0,\;\; l \geq1, \end{equation} where $\mathbb{L}_{2^p}$ is the Lebesgue constant associated with the set of $2^p$-roots of unity. The value $\mathbb{L}_{2^p}$ can be computed easily for small values of $p$ and it grows logarithmically in $2^p$, see e.g. \cite[formula 2.25]{Ch1}, \begin{equation} \label{boundL2p} \mathbb{L}_1=1,\quad\mathbb{L}_2=\sqrt 2,\quad \mbox{and}\quad \mathbb{L}_{2^p}\leq \frac 2\pi \(\log(2^p)+ 9/4\),\quad p\geq2. \end{equation} Since $\sigma_1(k)=\sigma_1(k/2^{p(k)})$, we have then in view of \iref{boundLkLatest} and \iref{binarymemory} the following theorem
\begin{theorem} Let $E$ be the Leja sequence defined in \iref{SimpleLejaSequence} or any Leja sequence on ${\cal U}$ initiated at $\partial {\cal U}$. We have \begin{equation} \label{bestBound} \mathbb{L}_{E_k} \leq 3\sqrt {\frac {k}{2^{p(k)}}(2^{\sigma_1(k)}-1)} ~~\mathbb{L}_{2^{p(k)}},\quad k\geq1. \end{equation} \label{theoremBoundLeja} \end{theorem}
We should mention that our primary interest in studying $\lambda_{E_k,2}$ was the improvement of the results of \cite{Ch2} concerned with the Lebesgue constants of $\Re$-Leja sequences. This will be made clear in the proof of Theroem \ref{theoLebesgueConstant}. For the sake of the same theorem, we need also to provide a growth property of Leja sequences on the unit disc.
We let $E=(e_j)_{j\geq0}$ be the simple Leja sequence defined by \iref{SimpleLejaSequence}. For $m\geq0$ and $1\leq l\leq 2^{m-1}$, we introduce the notation $K=2^m+l$ and $F_{m,l} = E_{2^m,K}$ and define the quantity \begin{equation} \label{defBetaln} \gamma_{m,l} =\frac 1 {4^m} \sum_{j=0}^{K-1}
\frac {4}{|w_{F_{m,l}}(\overline{e_j})|^2}. \end{equation} The quantity $\gamma_{m,l} $ is well defined. Indeed, by the particular structure of the sequence $E$, we have $E_{2^m+2^{m-1}}=E_{2^m}\wedge e^{\frac {i\pi}{2^m}} E_{2^{m-1}}$, so that $E_{2^m+2^{m-1}}={\cal U}_{2^m}\wedge e^{\frac {i\pi}{2^m}}~{\cal U}_{2^{m-1}}$ in the set sense. We have then for $j=0,\dots,2^m+l-1$, $\overline{e_j}$ is in ${\cal U}_{2^m}\wedge e^{\frac {-i\pi}{2^m}}~{\cal U}_{2^{m-1}}$ which does not intersect with $F_{m,l}\subset e^{\frac {i\pi}{2^m}}~{\cal U}_{2^{m-1}}$. We have the following growth for $\gamma_{m,l}$.
\begin{lemma} For any $m\geq1$ and any $1\leq l\leq 2^{m-1}$, we have \begin{equation} \gamma_{m,l} \leq \frac {5}{2^{\sigma_1(l)+p(l)+1}} \end{equation} \label{lemmabetanl} \end{lemma}
{\bf Proof:} Since $(e_0,e_1,e_2)=(1,-1, i)$, it can be checked that $\gamma_{1,1}=5/4$. We then fix $m\geq2$. We define $\rho = e_{2^m}= e^{i\pi/2^m}$, so that $F_{m,1} = \{\rho\}$. We have $$ \gamma_{m,1}= \sum_{j=0}^{2^m}
\frac {4}{(2^m|e_j - \overline{\rho}|)^2} =
|\lambda_{E_{2^m},2} (\overline{\rho})|^2 +
\frac {4}{(2^m|\rho - \overline{\rho}|)^2} = 1
+\frac 1{|2^m \sin(\pi/2^m)|^2} $$ where we have used \iref{lambda2E2n} and used that $\overline\rho$ is a $2^m$-root of $-1$. Since $2^m\sin(\pi/{2^m})\geq 2$ then $\gamma_{m,1} \leq 5/4$. For the other values of $l=2,\dots,2^{m-1}$, we have \begin{itemize} \item If $l=2N$, we have for any $j\geq0$ that $w_{F_{m,l}}(\overline{e_{2j+1}})=w_{F_{m,l}}(\overline{e_{2j}}) =w_{E_{2^{m-1},2^{m-1}+N}}(\overline{e_j})$. Pairing the indices in \iref{defBetaln} as $2j$ and $2j+1$ with $j=0,\dots,2^{n-1}+N-1$, we deduce $$ \gamma_{m,l} =\frac{ \gamma_{m-1,N}}{2}. $$
\item If $l=2N+1$ with $N\geq1$, we may write $$ \gamma_{m,l} = \frac 1 {4^{m-1}} \sum_{j=0}^{K-1}
\frac {|e_K-\overline{e_j}|^2}{|w_{F_{m,l+1}}(\overline{e_j})|^2} \leq \frac 1 {4^{m-1}} \sum_{j=0}^{K}
\frac {|e_K-\overline{e_j}|^2}{|w_{F_{m,l+1}}(\overline{e_j})|^2} = \gamma_{m-1,N+1}, $$ where we have again paired the indices by $2j$ and $2j+1$ for $j=0,\dots,2^n+(N+1)-1$ and used $e_{2j+1}=-e_{2j}$ and
the identity $|a+b|^2+|a-b|^2=4$ for any $a,b\in\partial {\cal U}$. \end{itemize} Therefore $$ \gamma_{m,l} \leq \frac{~5~}4 a_{m,l}, \quad\quad 1\leq m,\;\; 1\leq l\leq2^{m-1}, $$ where $(a_{m,l})_{\substack{1\leq m\\1\leq l\leq2^{m-1}}}$ is the sequence that saturates the previous inequalities and hence is defined by the following recursion: $$ a_{m,1}=1,\;\;m\geq1 \quad\mbox{and}\quad \left\{ \begin{array}{l} a_{m,2N}\;\;\;=a_{m-1,N}{\big/} 2\quad\quad n\geq1, N=1,\dots,2^{m-2},\\ a_{m,2N+1}=a_{m-1,N+1}\quad\quad n\geq1, N=1,\dots,2^{m-2}-1. \end{array} \right. $$ The sequence $(a_{m,l})$ has no dependance on $m$ and it is equal, in the sense $a_{m,l}= a_l$, to the sequence $(a_l)_{l\geq1}$ which satisfies the recursion: $a_1=1,\;a_{2N}=a_N/2,\; a_{2N+1}=a_{N+1}$. Since $\sigma_1(1) + p(1)=1$, $\sigma_1(2N) + p(2N)=\sigma_1(N)+p(N)+1$ and $$ \sigma_1(2N+1) + p(2N+1) =\sigma_1(2N+1) =\sigma_1(N) +1 =\sigma_1(N+1) +p(N+1), $$ then an immediate induction shows that $a_l=2^{1-\sigma_1(l)-p(l)}$, which finishes the proof.
$\blacksquare$
\section{$\Re$-Leja sequences on $[-1,1]$}
$\Re$-Leja sequences were introduced and studied in \cite{CaP2}. Such sequences are simply defined as the projection, element-wise but without repetition, into [-1,1] of Leja sequences on ${\cal U}$ initiated at $1$. More precisely, given $E=(e_j)_{j\geq0}$ a Leja sequence on ${\cal U}$ initiated at $1$, the $\Re$-Leja sequence $R=(r_j)_{j\geq0}$ associated with $E$ is obtained progressively by: $r_0=\Re(e_0)=1$, $J(0)=0$ and \begin{equation} \label{algoRLeja} r_k= \Re(e_{J(k)}) \quad\mbox{where}\quad J(k)=\min\{j>J(k-1):\Re(e_j)\not\in R_k\},\quad k\geq1. \end{equation} This means one projects $e_j$ if and only if $e_j\neq\overline{e_i}$ for all $i<j$. The projection rule that prevents the repetition is provided in \cite[Theorem 2.4]{CaP2}. One has \begin{equation} R = \Re (\Xi), \quad {\rm with } \quad \Xi:=(1,-1) \wedge \bigwedge_{j=1}^\infty E_{2^j,2^j+2^{j-1}}. \label{formRLeja} \end{equation} Using a simple cardinality argument, see \cite[Theorem 2.4]{CaP2} or \cite[Formula 40]{Ch2}, this implies that the function $J$ used in \iref{algoRLeja} is given by: $J(0)=0,~J(1)=1$ and \begin{equation} \label{formulaJ} J(k) = 2^n+k-1, \quad\quad n\geq0, \;\;\; 2^n+1 \leq k <2^{n+1}+1. \end{equation} In view of \iref{formRLeja} and the properties of Leja sequences on ${\cal U}$, any $\Re$-Leja sequence $R$ satisfies $r_0=1,~r_1=-1,~r_2=0$ and $r_{2j-1} = -r_{2j}$ for any $j\geq2$. An accessible example of an $\Re$-Leja sequence is the one associated with the simple Leja sequence given by the bit-reversal enumeration \iref{SimpleLejaSequence}. We have shown in \cite{Ch1} that $R=(\cos(\phi_j))_{j\geq0}$ where the sequence of angles $(\phi_k)_{k\geq0}$ is defined recursively by $\phi_0=0$, $\phi_1=\pi$, $\phi_2 = \pi/2$ and \begin{equation} \phi_{2j-1} = \frac{\phi_j}{2},\quad\quad \phi_{2j} = \phi_{2j-1} +\pi,\;\;\;j\geq2. \end{equation} This recursion provides a simple process to compute an $\Re$-Leja sequence. We can also construct a Leja sequence by simply using the recursion $r_0=1$, $r_1=-1$, $r_2=0$ and \begin{equation} r_{2j-1} = \sqrt{\frac{r_j+1}{2}},\quad\quad r_{2j} = - r_{2j-1},\;\;\;j\geq2. \end{equation} One can check that the last sequence is obtained from the Leja sequence $F$ which is constructed recursively by $F_1=\{1\}$ and $F_{2^{n+1}}=F_{2^n}\wedge e^{\frac {i\pi}{2^n}} \overline{F_{2^n}}$. Both $\Re$-Leja sequences $R$ satisfies $2r_0^2-1=1$, $2r_2^2-1=-1$ and more generally $2r_{2j}^2-1 = 2r_{2j-1}^2-1 = r_j$ for any $j\geq2$, thanks to the trigonometric identity $2\cos^2(\theta/2)-1=\cos(\theta)$. This shows that in both cases $R$ satisfies the property \begin{equation} R^2=R \quad \mbox{where}\quad R^2 := (2r_{2j}^2-1)_{j\geq0}, \label{defR2} \end{equation}
In general, given a Leja sequence $E$ in ${\cal U}$ initiated at $1$ and $R$ the associated $\Re$-Leja sequence, we have that $R^2$ is an $\Re$-Leja sequence and it is associated with $E^2$ which, in view of Theorem \ref{TheoImplications}, is also a Leja sequence initiated at $1$. This result is given in \cite[Lemma 3.4]{Ch2} and it has many useful implications that we have exploited in order to prove that $\mathbb{D}_{k}(R)$ grows at worse quadratically.
For all Leja sequences $E$ on ${\cal U}$ initiated at $1$, the section $E_{2^{n+1}}$ is equal in the set sense with the set of $2^{n+1}$-roots of unity, therefore for all $\Re$-Leja sequences $R$, the section $R_{2^n+1}$ is equal to the set of Gauss-Lobatto abscissas of order $2^n$, i.e. \begin{equation} \label{GaussLobatto} R_{2^n+1} = \Big\{\cos(j\pi/2^n): j=0,\dots,2^n\Big\}, \end{equation} in the set sense. This set of abscissas is optimal as far as Lebesgue constant is concerned, in the sense $\mathbb{L}_{R_{2^n+1}}\simeq\frac {2\log (2^n+1)}\pi$. More precisely, we have the bound \begin{equation} \label{lebesgueGaussLobatto} \mathbb{L}_{R_{2^n+1}}\leq 1+\frac 2 \pi \log(2^n), \end{equation} see \cite[Formulas 5 and 13]{DzI}. This suggests that the sequence $R$ might have a moderate growth of the Lebesgue constant of its section $R_k$.
In the paper \cite{CaP2}, it has been proved that $\mathbb{L}_{R_k} = {\cal O}(k^3\log (k))$. We have improved this bound in \cite{Ch1,Ch2} and showed that $\mathbb{L}_{R_k} \leq 8\sqrt 2~k^2$ for any $k\geq2$. Here we again exploit our approach of \cite{Ch2} which, using simple calculatory arguments, relate the analysis of the Lebesgue function associated with $R_k$ to that of the Lebesgue function associated with the smaller Leja section that yields $R_k$ by projection. This approach allows us to circumvent cumbersome real trigonometric functions which arise in the study $\lambda_{R_k}$, see \cite{CaP2,Ch1}, and to take full benefit from the machinery developed for Leja sequence on ${\cal U}$.
\begin{remark} Without loss of generality, we assume for the remainder of this section that $E$ is the simple Leja sequence in \iref{SimpleLejaSequence} and $R$ the associated $\Re$-Leja sequence. All our arguments hold in the more general case, the assumption is essentially for notational clearness. It allows us, in view of \iref{memoryShort}, to use $E$ instead for $E^2$ and more generally instead of $E^{2^p}$ which is defined by $E^{2^p}:=((e_{2^p j})^{2^p})_{j\geq0}$. \end{remark}
The bound \iref{lebesgueGaussLobatto} is sharp and we are only interested in bounding $\mathbb{L}_{R_k}$ when $k-1$ is not a power of $2$. For the remainder of this section, we use the notation \begin{equation} \label{valuesNKKprime} \begin{array}{l} n\geq 0, \quad\quad 2^n < k-1 < 2^{n+1}, \quad\quad 0< l := k-(2^n+1) < 2^n\\%\in\{1,\dots,2^n-1\},\\ \quad K:=2^{n+1}+l, \quad\quad\quad G_k:= E_K, \quad\quad\quad F_K :=E_{2^{n+1},K}. \end{array} \end{equation} We should note that in \cite{Ch2} we have used $k'$ and $F_k$ to denote $l$ and $F_K$. In view of \iref{formulaJ}, we have $K=J(k)$, so that $E_K$ is the smallest section that yields $R_k$ by projection into $[-1,1]$. We denote by $L_0,L_1,L_2,\cdots,L_{K-1}$ the Lagrange polynomials associated with $E_K$. The inspection of the the proof of \cite[Lemma 6]{Ch2} shows that for $z\in \partial {\cal U}$ and $x=\Re(z)$, \begin{equation} \label{boundLRk} \lambda_{R_k}(x) \leq \gamma_K(z) + \gamma_K(\overline z),\quad\quad \gamma_K(z):=
|w_{F_K}(\overline{z})| \sum_{j=0}^{K-1}
\frac {|L_j(z)|}
{|w_{F_K}(\overline{e_j})|}. \end{equation} In the proof of \cite[Lemma 6]{Ch2}, we have bounded
the functions $|w_{F_k}|/{|w_{F_k}(\overline{e_j})|}$ in the previous sum by $2^{n+\frac 12 -p(l)}$. This implied the result of \cite[Theorem 5]{Ch2}, namely $\mathbb{L}_{R_k}\leq 2^{n+\frac 32 -p(l)} \mathbb{L}_{E_K}$. In view of the new bound \iref{bestBound}
and the facts that $p(K)=p(l)$, $\sigma_1(K)=1+\sigma_1(l)$ and $K=2^n+k-1 \leq 3 \times2^n$, the previous bound implies
\begin{equation} \label{boundGood} \mathbb{L}_{R_k} \leq 12\sqrt {3} ~ 2^{\frac {3n-3p(l)+\sigma_1(l)} 2} ~\mathbb{L}_{2^{p(l)}},\quad k\geq1, \end{equation} where $\mathbb{L}_{2^p}$ is bounded as in \iref{boundL2p}. We propose to improve slightly the previous inequality by applying rather Cauchy Schwartz inequality when bounding the function $\gamma_K$.
\begin{theorem} Let $R$ be an $\Re$-Leja sequence and $n,~k$ and $l$ as in \iref{valuesNKKprime}. We have \begin{equation} \label{boundBestRLeja} \mathbb{L}_{R_k} \leq 6\sqrt{5}~ 2^{n+\sigma_1(l)-p(l)} \mathbb{L}_{2^{p(l)}}, \end{equation} where $\mathbb{L}_{2^p}$ is bounded as in \iref{boundL2p}. \label{theoLebesgueConstant} \end{theorem}
{\bf Proof:} In order to lighten the notation, we use the shorthand $p$ in order to denote $p(l)$. We introduce $l'$ and $K'$ and $F_{K'}$ defined by $$ l' := l/2^p, \quad \quad K' :=K/2^p=2^{n-p+1}+l', \quad\quad F_{K'} := E_{2^{n-p+1},K'}. $$ The sequence $E$ satisfies $E^2=E$ and one can check that $w_{F_K}(z) = w_{F_{K'}}(z^{2^p})$. Also by $ e_{2j}^2 = e_{2j+1}^2 =e_j$, one has $(e_{2^p j+q})^{2^p}=e_j$ for any $q=0,\dots,2^p-1$. Moreover, if $M_1,\dots,M_{K'-1}$ are the Lagrange polynomials associated with $E_{K'}$, then $$
\sum_{q=0}^{2^p-1}|L_{2^pj+q}(z)| \leq \mathbb{L}_{2^p} M_j(z^{2^p}),\quad\quad j=0,\dots,K'-1, $$ see the proof of \cite[Theorem 2.8]{Ch1}. Therefore by pairing the indices in the sum giving $\gamma_{K}$ by $2^p j+q$ for $j=0,\dots,K'-1$ and $q=0,\dots,2^p-1$, we infer $$ \gamma_K(z) \leq \(
|w_{F_{K'}}(\overline{\xi})| \sum_{j=0}^{K'-1}
\frac {|M_j(\xi)|}
{|w_{F_{K'}}(\overline{e_j})|} \) \mathbb{L}_{2^p} = \mathbb{L}_{2^p} \gamma_{K'}(\xi),\quad\mbox{with}\quad \xi=z^{2^p}. $$ In view of \iref{boundLRk}, this implies that $\mathbb{L}_{R_k} \leq 2 \mathbb{L}_{2^p} \sup_{\xi\in{\cal U}}\gamma_{K'}(\xi)$. Applying Cauchy Schwatrz inequality to $\gamma_{K'}$ and using that $F_{K'}$ is an $l'$-Leja sequence, we have for any $\xi\in{\cal U}$ $$ \gamma_{K'}(\xi) \leq 2^{\sigma_1(l')} \(\sum_{j=0}^{K'-1}
\frac 1{|w_{F_{K'}}(\overline{e_j})|^2}\)^{1/2}
\(\sum_{j=0}^{K'-1} |M_j(\xi)|^2\)^{1/2} = 2^{\sigma_1(l')+n-p} \sqrt{\gamma_{n-p+1,l'}}~\lambda_{E_{K'},2}(\xi), $$ where $\gamma_{n-p+1,l'}$ is defined as in \iref{defBetaln} with $m=n-p+1$ and $\lambda_{E_{K'},2}$ is the quadratic Lebesgue function associated with $E_{K'}$. In view of the bounds we have for these quantities and in view of $\sigma_1(K')=1+\sigma_1(l')$ and $\sigma_1(l')=\sigma_1(l)$, we get $$ \gamma_{K'}(\xi) \leq 2^{\sigma_1(l)+n-p} \sqrt{\frac {5}{2^{\sigma_1(l')+1}}} ~3\sqrt {2^{1+\sigma_1(l')}-1} \leq 3\sqrt5 ~ 2^{\sigma_1(l)+ n- p}. $$
The proof is then complete.
$\blacksquare$ \newline
The bound in \iref{boundBestRLeja} improves the bound in \iref{boundGood} by $2^{\frac{\sigma_1(l)+p(l) -n}2}$. The bound can also yield linear estimates for $\mathbb{L}_{R_k}$, for instance when $l$ is such that $ 2^{\sigma_1(l)-p(l)} \mathbb{L}_{2^{p(l)}} \leq 1$, which is the case if for example $p(l) \geq 2 \sigma_1(l)$. However, if $0<l<2^n$ is the integer with the most number of ones in the binary expansion, i.e. $\sigma_1(l)=n$ or $l=2^n-1$ and $k=2^{n+1}$, we merely get the quadratic bound \begin{equation} \label{pessimisticQuadratic} \mathbb{L}_{R_k} \leq 6\sqrt 5~ 2^{2n}= \frac {3\sqrt 5}2 k^2. \end{equation} In \cite{CaP2}, section 3.4, it is shown that for the values $k=2^{n}$, in other words $R_k$ is the set of Gauss-Lobatto abscissas \iref{GaussLobatto} missing one abscissa, one has $\mathbb{L}_{R_k}\geq \lambda_{R_k}(r_k)= k-1$. As a consequence, the growth of $\mathbb{L}_{R_k}$ for $k\geq1$ can not be slower than $k$. However, for this case, we can prove $\mathbb{L}_{R_k}\leq 3 k$, see \iref{finalRemark}, showing that \iref{pessimisticQuadratic} is rather pessimistic.
The estimate in \iref{boundBestRLeja} is logarithmic for many values of the integer $k$. For instance, if $k=(2^n+1)+2^{n-p} k'$ for some $p=1,\dots,n$ and some $k'=0,\dots,2^p-1$, then we have $l=2^{n-p} k'$, so that $n-p \leq p(l)\leq n$ and $\sigma_1(l)=\sigma_1(k')\leq p$ implying that \begin{equation} \mathbb{L}_{R_k} \leq 6 \sqrt 5 ~2^{2p}~ \mathbb{L}_{2^{p(l)}} \leq 6 \sqrt 5 ~2^{2p}~ \mathbb{L}_{2^n} \leq 6 \sqrt 5 ~2^{2p} \frac 2\pi \(\log(2^n)+ 9/4\). \end{equation} For a small value of $p$, the previous estimate is as good as the optimal logarithmic estimate $\frac {2\log(k)}\pi$ for large values of $n$. Given then $p$ fixed, one has $2^p$ intermediate values between $2^n+1$ and $2^{n+1}+1$, which are the numbers $k=(2^n+1)+2^{n-p} k'$ for $k'=0,\dots,2^p-1$, for which the Lebesgue constant is logarithmic. This observation can be used in order to modify the doubling rule with Clemshaw-Curtis abscissas in the framework of sparse grids, see \cite{GWG}.
\section{Growth of the norms of the difference operators} \label{sectionDiff}
In this section, we discuss the growth of the norms of the difference operators $\Delta_0 = I_{Z_1}$ and $\Delta_{k} = I_{Z_{k+1}} - I_{Z_k}$ for $k\geq1$, associated with interpolation on Leja or $\Re$-Leja sequences. We are interested in estimating their norms $\mathbb{D}_k$ defined in \iref{normDeltak}. Elementary arguments, see \cite{Ch2}, show that \begin{equation} \label{normDiff} \mathbb D_k (Z)= \Big(1+\lambda_{Z_k} (z_k) \Big)
\sup_{z\in X} \frac {|w_{Z_k}(z)|}{|w_{Z_k}(z_k)|},\quad k\geq1. \end{equation} In particular if $Z$ is a Leja sequence on the compact $X$, then \begin{equation} \mathbb{D}_k(Z)=1+\lambda_{Z_k} (z_k). \end{equation} In \cite{Ch1}, we have established that $\lambda_{E_k} (e_k)\leq k$ if $E$ is a Leja sequence on ${\cal U}$ initiated at $\partial {\cal U}$, which implies $\mathbb{D}_k(E)\leq1+k$. Here, we improve slightly this bound. As for the improvement of \iref{boundLkLatest} into \iref{bestBound}, we have \begin{theorem} Let $E$ be a Leja sequence on the unit disk initiated at $e_0\in \partial {\cal U}$, One has $\mathbb{D}_0(E)=1$ and \begin{equation} \mathbb{D}_k(E) \leq 1+\sqrt {\frac {k}{2^{p(k)}}(2^{\sigma_1(k)}-1)} ~~\mathbb{L}_{2^{p(k)}} \end{equation} \end{theorem}
For $\Re$-Leja sequences $R$ on $[-1,1]$, we have shown in \cite{Ch2} using a recursion argument based on the fact that $R^2$ defined as in \iref{defR2} is also an $\Re$-Leja sequence, that \begin{equation} \mathbb{D}_k(R) \leq (1+k)^2,\quad\quad k\geq0. \end{equation} In view of the new bounds obtained in this paper for Lebesgue constant of $\Re$-Leja sections, the previous bound is not sharp. Indeed, we have $\mathbb{D}_k\leq\mathbb{L}_k+\mathbb{L}_{k-1} \leq 12\sqrt{5}~ k^{3/2}$, for $k$ such that $l=k-(2^n+1)\leq 2^{n/2}$. We give here a sharper bound for $\mathbb{D}_k(R)$.
We recall that up to a rearrangement in the formula \iref{normDiff}, see \cite{Ch2} for justification, we may write the quantities $\mathbb{D}_k(R)$ in a more convenient form for $\Re$-Leja sequences. We introduce the polynomial $W_{R_k} := 2^k w_{R_k}$, we have \begin{equation} \label{deltabeta} \mathbb D_k(R) = 2 \beta_k(R)
\sup_{x\in [-1,1]} |W_{R_k}(x)|, \;\; \;\; \beta_k(R):=
\frac {1+\lambda_{R_k}(r_k)}{2|W_{R_k}(r_k)|}, \end{equation} We have already proved in \cite[Lemma 7]{Ch2} that \begin{equation} \label{beta2nk} \beta_{2^n}(R)=1/4 \quad\mbox{and}\quad \beta_k(R) \leq 2^{\sigma_0(k)-p(k)-1}, \quad\mbox{for}\quad k\neq 2^n. \end{equation} Here we provide a sharper bound for $\mathbb{D}_k(R)$ by slightly improving the estimate $4^{\sigma_1(k)+p(k)-1}$ that we have established in \cite{Ch2} for
$\sup_{x\in [-1,1]} |W_{R_k}(x)|$.
\begin{lemma} Let $R$ be an $\Re$-Leja sequence in $[-1,1]$, $n\geq1$,
$2^n+1\leq k<2^{n+1}+1$ and $l=k-(2^n+1)$. One has $\sup_{x\in [-1,1]} |W_{R_{k}}(x)|\leq 2^{n+3}$ if $k=2^{n+1}$, else \begin{equation}
\sup_{x\in [-1,1]} |W_{R_k}(x)| \leq 2^{2\sigma_1(k)+p(k)-1}. \end{equation} \end{lemma}
{\bf Proof:} We use the notation $K$, $G_k$ and $F_K$ as in \iref{valuesNKKprime} and introduce $G_{k+1}:=E_{K+1}$ and $F_{K+1}:=E_{2^{n+1},K+1}$. In view of \cite[Lemma 5]{Ch2}, one has for $z\in\partial {\cal U}$ and $x=\Re(z)$ $$
|W_{R_k}(x)| =
|z^2-1||w_{G_k}(z)| |w_{F_K}(\overline z)| =
|z-\overline z||w_{G_k}(z)| |w_{F_K}(\overline z)|. $$
Also since $|z-\overline{z}|\leq |z-e_K|+|\overline z-e_K|$, then $$
|W_{R_k}(x)| \leq
|w_{G_{k+1}}(z)||w_{F_K}(\overline z)|
+|w_{G_k}(z)||w_{F_{K+1}}(\overline z)|. $$ In the two previous inequalities, one has $F_K=\emptyset$ and $w_{F_K}\equiv1$ in the case $k=2^n+1$.
We have that $G_k$, $G_{k+1}$, $F_K$ and $F_{K+1}$ are all Leja sections with length $K,~K+1,~l$ and $l+1$ respectively. Therefore, by the second property in Theorem \ref{TheoImplications} $$
|W_{R_k}(x)| \leq \min\(2^{1+\sigma_1(K)+\sigma_1(l)} , 2^{\sigma_1(K+1)+\sigma_1(l)} + 2^{\sigma_1(K)+\sigma_1(l+1)} \) =2^{2+\sigma_1(l)} \min(2^{\sigma_1(l)},2^{\sigma_1(l+1)}), $$ where we have used $\sigma_1(K)=1+\sigma_1(l)$ and $\sigma_1(K+1)=1+\sigma_1(l+1)$ since $K=2^{n+1}+l$ and $l<2^n$. If $k=2^{n+1}$, i.e. $l=2^n-1$, then
$|W_{R_k}(x)|\leq 2^{3+n}$. Else by $k=2^n+(l+1)$ and $0\leq l<2^n-1$, $$ \sigma_1(k) -1= \sigma_1(l+1)\quad\mbox{and}\quad \sigma_1(k)-2+p(k)=\sigma_1(k-1)-1 = \sigma_1(l). $$ Therefore $$
|W_{R_k}(x)| \leq 2^{2\sigma_1(k)+p(k)-1} \min(2^{-1+p(k)},1), $$ which completes the proof.
$\blacksquare$
\newline
By injecting the estimate of the previous lemma and the estimate of \iref{beta2nk} in formula \iref{deltabeta} and by using the identity $\sigma_0(k)+\sigma_1(k)=n+1$ for $2^n\leq k<2^{n+1}$, we are able to conclude the following result. \begin{cor} Let $R$ be an $\Re$-Leja sequence in $[-1,1]$. The norms of the difference operators satisfy, $\mathbb D_0(R) =1$ and for $2^n \leq k<2^{n+1}$ \begin{equation} \mathbb D_k(R) \leq 2^{\sigma_1(k)} 2^{n} \end{equation} \end{cor}
The previous estimates can be used in order to provide estimates for $\mathbb{L}_{R_k}$ that can be sharper than \iref{boundBestRLeja}. We have $\Delta_k=I_k - I_{k-1}$, therefore \begin{equation}
|\mathbb{L}_{R_{k+1}}-\mathbb{L}_{R_k}|\leq \mathbb{D}_k(R),\quad k\geq1. \end{equation} In particular, the estimate in the previous corollary combined with the sharp bound \iref{lebesgueGaussLobatto} implies that for the value $k=2^n$, we get \begin{equation} \mathbb{L}_{R_k} \leq 1+\frac 2 \pi \log(2^n)+ 2^{n+1} \leq 3k \end{equation} This shows that in the case $k=2^n$ which corresponds to $R_k$ being the set of Gauss-Lobatto abscissas \iref{GaussLobatto} missing one abscissas and for which $\mathbb{L}_{R_k}\geq k$, the previous bound is satisfactory. This also confirm that the estimates \iref{boundBestRLeja} is indeed pessimistic in this case, see the inequality \iref{pessimisticQuadratic}. This added to the observed growth of $\mathbb{L}_{R_k}$ for values $k\leq 128$, Figure \ref{fig}, suggests that the bound \begin{equation} \label{finalRemark} \mathbb{L}_{R_k} \leq 3k,\quad\quad k\geq1, \end{equation} might be valid for any $\Re$-Leja sequence $R$. We conjecture its validity.
In Figure \ref{fig}, we also represent for the values $k\leq 128$, the growth of the Lebesgue constant $\mathbb{L}_{E_k}$ (in blue) and the estimate $\sqrt {k(2^{\sigma_1(k)}-1)}$ (in red) which multiplied by $3$ bounds $\mathbb{L}_{E_k}$, see \iref{boundLkLatest}. We observe that the regular patterns in the graph of $k \mapsto \mathbb{L}_{E_k}$, which reveals the particular role of divisibility by powers of 2 in $k$, is caught by the estimate. The worst values of $\mathbb{L}_{E_k}$ appear for the values $k = 2^n-1$ for which it was proved in \cite{CaP1} that $\mathbb{L}_{E_k} = k$ and which is also equal to $\sqrt {k(2^{\sigma_1(k)}-1)}$ since $\sigma_1(k)=n$.
\begin{figure}
\caption{Exact Lebesgue constants associated to the $k$-sections of the Leja sequence $E$ and the assciated $\Re$-Leja sequence $R$ for $k=1,3,\dots,129$.}
\label{fig}
\end{figure}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi
\end{document} | arXiv |
Path optimization in a DAG: maximizing number of least cost arcs
I've got the following problem.
I've a graph $G=(V,E)$ as in the picture and I have to calculate the optimal path from $R$ to $S$.
The optimal path has to maximize the number of least cost arcs. In the example:
R->B->S is the least cost path but it's not the optimal. R->-A->D->S costs less than R->A->C->E->S , but the latter is the optimum as it goes through more, least cost arcs.
My current problem, which constitute the object this question, is to find a way of defining formally how to compare two paths according to my intuitive understanding, as comparing them comes before choosing the optimal one.
Of two paths, the better one is the one that traverses the highest number of least cost arcs.
I have been considering three ways, some that seem not fully adequate for my purpose, but would welcome other suggestions.
My three ideas are the following:
Minimizing the average of the cost of arcs on the path.
Minimizing the average is not fully ok: imagine to have a path with cost [3,12], and a path with cost [3,10,10]. The average of the first path is less than the average of the second, but the second is the optimal as it traverses more paths.
Comparing individually the arcs of the paths
Given two paths $A$ and $B$:
Let $A$ = [$a_1,\ldots,a_n$] and $B$ = [$b_1,\ldots,b_m$] where $a_i$ and $b_i$ are the arc costs of $A$ and $B$ respectively.
Then $A$ is better than $B$ if $\sum_{i=0}^n card(B < a_i)$ < $\sum_{i=0}^m card(A < b_i)$ where $card$ represents the cardinality of the set, and $B < a_i$ is defined as $\{b_j \in B \mid b_j < a_i\}$, the comparison being on the weight of the concerned arcs.
However, with this second approach I get many ties between different paths, increasing the risk of getting a suboptimal path.
Comparing the min-max of the arcs weights for each path.
Another possibility is to compute the min-max of the arcs of each path, and choose the smaller one. It is intrinsically different from the other approaches.
But it is not fully what I seek to achieve. For example, min-max over [1,1,1,3] and [2,2,2,2] chooses the latter one, while the best one is the first.
To motivation behind my intuition is the following: each node is a "checkpoint" that the solution may traverse on its way from the node $R$ to node $S$. The goal is to output a sequence $Q$ with an associated "precision"
if the chosen path is R->A->D->S, then $\;Q\;$ is $\;\;$A $\pm$ 10, D $\pm$ 5, S $\pm$ 7
Clearly, the optimal solution has to be as precise as possible, so the optimal path has to emit the maximum number of least cost arcs.
Once the comparison of paths has been formalized precisely, my problem will be to compute the optimal path on a given graph. But right now the question is only to find a proper way of comparing paths. The role of the graph structure is only, possibly, to help convey the intuition of my problem.
algorithms graph-traversal weighted-graphs partial-order order-theory
babou
Charles G.Charles G.
$\begingroup$ Note that, in the special case where all edges have the same cost, you're trying to find the longest path between two vertices, which is NP-hard. As such, you shouldn't expect there to be an efficient algorithm for the general problem. $\endgroup$
$\begingroup$ @DavidRicherby : $\;\;\;$ How is that shown? $\:$ (Is it by using a definition of "path" that either doesn't allowing revisiting vertices or doesn't allow retracing edges?) $\;\;\;\;\;\;\;\;$ $\endgroup$
$\begingroup$ @DavidRicherby according to the definition (with ∑ and unusual notations $B<a_i$), if all edges have the same cost, all paths are equivalent since their associated conmparative values are 0. If he had used "≤" rather than "<", then these values would be |B|×|A|. and all paths would still be equivalent. So your conclusion may be right, but it requires more to be established,. $\endgroup$
$\begingroup$ Am I correct in interpreting $B<a_i$ as $\{b_j\in B\mid b_j<a_i\}$ ? $\endgroup$
$\begingroup$ I edited your question so as to clarify as much as I can, and remove the second part (only one problem in a question). I also tried to remain compatible with existing answers, - - - It is not yet clear what you are after. You should give us a brief example of the real problem you are working on, because we will never get your intuition otherwise. Where do these approximations come from? What do they mean? What really happens on an arc of the graph (some kind of unprecise processing step?) What is the process you want to optimize? Give us the real problem, not your vision. $\endgroup$
I've not thought completely through it, but some thoughts on 1) which might be useful:
If you add a constant value k to each edge traversed, you could take into account the number of nodes traversed. In your case, this constant value k should be negative (e.g. the negative value of the maximum edge weight, in your example 11).
Then you might be able to state the following:
The path maximizes the number of least links if $\sum_{i=0}^n (a_i + k)$ < $\sum_{i=0}^m (b_i + k)$.
For your example: (let $P_i$ denote the $i$-th path and $C_i$ denote the costs of path $P_i$)
$P_A=[R, B, S], C_A = -2$
$P_B=[R, A, D, S], C_B = -11$
$P_C=[R, A, C, E, S], C_C = -20$
So $C_C < C_B < C_A$ and hence $P_C$ is your optimal path.
As mentioned, I've not spent too much time to look if there is an example where this approach does not work, but maybe it's a good point to start with.
Of course you get some ties with this approach as well but I think all of these ties are equivalent in the sense that the least cost arcs are traversed and the number of traversed vertices are equivalent. Correct me, if I'm wrong.
SX.SX.
The question is proposing 3 definitions for comparing paths. The first and third are apparently not satisfactory for the author of the question. We show that the second definition is not usable either.
The second formalization seems more sophisticated, and is precisely given. It is not a metric but the definition of a "better" relation between paths, that is noted here "$\prec$".
$$A\prec B\;\;\;\mbox{ iff }\;\;\;\sum_{i=0}^n |B < w(a_i)|$ < $\sum_{i=0}^m |A < w(b_i)|$$ where $w(a)$ is the weight of an arc $a$,
and $B < w(a_i)$ is defined as $\{b_j \in B \mid w(b_j) < w(a_i0\}$
Actually, this can be redefined, I think more intuitively, as it helped me find the example below, as:
$\begin{align} A\prec B\;\;\text{ iff }\;\;|\{(a_i,b_j)\mid a_i\in A, b_j\in& B, w(b_j)<w(a_i)\}|\;<\; \\ &|\{(a_i,b_j)\mid a_i\in A, b_j\in B, w(b_j)>w(a_i)\}| \end{align}$
However, the problen is that this relation is not an order on paths, not even a preorder, since it is not transitive:
A counter example is given by the following three paths, here just given as sets of weights:
$$A=\{9,5,4\},\;\;\; B= \{6\},\;\;\; C= \{8,7,3\}$$
It is easily verified that
$$A\prec B,\;\;\; B\prec C,\;\;\; C\prec A$$
Thus, despite the notation chosen here, the relation $\prec$ cannot be used to order solutions. Hence this relation cannot be used to define the choice of an optmal path, independently of how paths are defined or the graph explored.
None of the three modes of comparison seems to meet the need of the author of the question. The motivation given by the author for his intuition is too vague to help. A full example of the real problem he is adressing should be given to have a chance for further progress.
baboubabou
$\begingroup$ Thank for your time and effort. You pointed out correctly the problem: I'm still trying to define what makes a path better than another one. I edited the question in the final part of the question $\endgroup$
– Charles G.
$\begingroup$ Thank you. So I should just ask for a proper definition of "optimal"? $\endgroup$
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\begin{definition}[Definition:Bilinear Mapping]
Let $\left({R, +_R, \times_R}\right)$ be a commutative ring.
Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \left({A_3, +_3, \circ_3}\right)_R$ be $R$-modules.
Let $\oplus: A_1 \times A_2 \to A_3$ be a binary operator with the property that: $\forall \left({a_1, a_2}\right) \in A_1 \times A_2$:
: $a_1 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_1$ to $A_3$
: $a_2 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_2$ to $A_3$
Then $\oplus$ is a '''bilinear mapping'''.
That is, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$:
: $\left({\left({a \circ_1 x}\right) +_1 \left({y \circ_1 b}\right)}\right) \oplus z = \left({a \circ_3 \left({x \oplus z}\right)}\right) +_3 \left({\left({y \oplus z}\right) \circ_3 b}\right)$
and for all $z \in A_1, x,y \in A_2$:
: $z \oplus \left({\left({a \circ_2 x}\right) +_2 \left({y \circ_2 b}\right)}\right) = \left({a \circ_3 \left({z \oplus x}\right)}\right) +_3 \left({\left({z \oplus y}\right) \circ_3 b}\right)$
Equivalently, this can be expressed:
: $\left({x +_1 y}\right) \oplus z = \left({x \oplus z}\right) +_3 \left({y \oplus z}\right)$
: $z \oplus \left({x +_2 y}\right) = \left({z \oplus x}\right) +_3 \left({z \oplus y}\right)$
: $\left({a \circ_1 x}\right) \oplus z = a \circ_3 \left({x \oplus z}\right)$
: $z \oplus \left({y \circ_2 b}\right) = \left({z \oplus y}\right) \circ_3 b$
If $\left({A, +, \circ}\right)_R = A_1 = A_2 = A_3$, the notation simplifies considerably:
: $\left({\left({a \circ x}\right) + \left({b \circ y}\right)}\right) \oplus z = \left({a \circ \left({x \oplus z}\right)}\right) + \left({b \circ \left({y \oplus z}\right)}\right)$
: $z \oplus \left({\left({a \circ x}\right) + \left({y \circ b}\right)}\right) = \left({a \circ \left({z \oplus x}\right)}\right) + \left({\left({z \oplus y}\right) \circ b}\right)$
or equivalently, more easily digested:
: $\left({x + y}\right) \oplus z = \left({x \oplus z}\right) + \left({y \oplus z}\right)$
: $z \oplus \left({x + y}\right) = \left({z \oplus x}\right) + \left({z \oplus y}\right)$
: $\left({a \circ x}\right) \oplus z = a \circ \left({x \oplus z}\right)$
: $z \oplus \left({y \circ b}\right) = \left({z \oplus y}\right) \circ b$
\end{definition} | ProofWiki |
\begin{definition}[Definition:Regular Tessellation]
A '''regular tessellation''' is a tessellation such that:
:Its tiles consist of congruent regular polygons
:No vertex of one tile may lie against the side of another tile.
\end{definition} | ProofWiki |
\begin{document}
\thanks{$^\ast$Corresponding author, E-mail address: [email protected]}
\keywords{Subordination; univalent functions; starlike functions; lemniscate of Bernoulli}
\subjclass[2010]{30C45; 30C80}
\begin{abstract}
For an analytic function $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ satisfying $f(0)=0=f'(0)-1,$ we obtain sufficient conditions so that $f$ satisfies $|(zf'(z)/f(z))^2-1|<1.$ The technique of differential subordination of first or second order is used. The admissibility conditions for lemniscate of Bernoulli are derived and employed in order to prove the main results.
\end{abstract} \title{Starlikeness associated with lemniscate of Bernoulli}
\section{Introduction}
The set of analytic functions $f$ on the unit disk $\mathbb{D}=\{z:|z|<1\}$ normalized as $f(0)=0$ and $f'(0)=1$ will be denoted by $\mathcal{A}$ and $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of univalent functions. A function $f\in\mathcal{SL}$ if $zf'(z)/f(z)$ lies in the region bounded by the right half of lemniscate of Bernoulli given by $\{w:|w^2-1|=1\}$ and such a function will be called \textit{lemniscate starlike}. Evidently, the functions in class $\mathcal{SL}$ are univalent and starlike i.e. $\operatorname{Re}(zf'(z)/f(z))>0$ in $\mathbb{D}.$ The set $\mathcal{H}[a,n]$ consists of analytic functions $f$ having Taylor series expansion of the form $f(z)=a+a_nz^n+a_{n+1}z^{n+1}+\ldots$ with $\mathcal{H}_1:=\mathcal{H}[1,1].$ For two analytic functions $f$ and $g$ on $\mathbb{D},$ the function $f$ is said to be \textit{subordinate} to the function $g,$ written as $f(z)\prec g(z)$ (or $f\prec g$), if there is a Schwarz function $w$ with $w(0)=0$ and $|w(z)|<1$ such that $f(z)=g(w(z)).$ If $g$ is a univalent function, then $f(z)\prec g(z)$ if and only if $f(0)=g(0)$ and $f(\mathbb{D})\subset g(\mathbb{D}).$ In terms of subordination, a function $f\in\mathcal{A}$ is lemniscate starlike if $zf'(z)/f(z)\prec\sqrt{1+z}.$ The class $\mathcal{SL}$ was introduced by Sok\'{o}l and Stankiewicz \cite{MR1473947}.
The class $\mathcal{S}^*(\varphi)$ of \textit{Ma-Minda starlike functions} \cite{MR1343506} is defined by \[\mathcal{S}^*(\varphi):=\left\{f\in\mathcal{S}:\dfrac{zf'(z)}{f(z)}\prec\varphi(z)\right\}, \] where $\varphi$ is analytic and univalent on $\mathbb{D}$ such that $\varphi(\mathbb{D})$ is starlike with respect to $\varphi(0)=1$ and is symmetric about the real axis with $\varphi'(0)>0.$ For particular choices of $\varphi,$ we have well known subclasses of starlike functions like for $\varphi(z):=\sqrt{1+z},\ \mathcal{S}^*(\varphi):=\mathcal{SL}.$ If $\varphi(z):=(1+Az)/(1+Bz),$ where $-1\leq B<A\leq1$, the class $\mathcal{S}^*[A,B]:=\mathcal{S}^*((1+Az)/(1+Bz))$ is called the class of \textit{Janowski starlike functions} \cite{MR0267103}. If for $0\leq\alpha<1,\ A=1-2\alpha$ and $B=-1,$ then we obtain $\mathcal{S}^*(\alpha):=\mathcal{S}^*[1-2\alpha,-1],$ the class of \textit{starlike functions of order $\alpha.$} The class $\mathcal{S}^*(\alpha)$ was introduced by Robertson \cite{MR0783568}. The class $\mathcal{S}^*:=\mathcal{S}^*(0)$ is simply the class of \textit{starlike functions}. If the function $\varphi_{PAR}:\mathbb{D}\to\mathbb{C}$ is given by \[ \varphi_{PAR}(z):=1+\frac{2}{\pi^2}\left(\log\frac{1+\sqrt{z}}{1-\sqrt{z}}\right)^2,\ \operatorname{Im}\sqrt{z}\geq0 \]
then $\varphi_{PAR}(\mathbb{D}):=\{w=u+iv:v^2<2u-1\}=\{w:\operatorname{Re}w>|w-1|\}.$ Then the class $\mathcal{S}_P:=\mathcal{S}^*(\varphi_{PAR})$ of parabolic functions, introduced by R\o{}nning\cite{MR1128729}, consists of the functions $f\in\mathcal{A}$ satisfying
\[ \operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right)>\left|\frac{zf'(z)}{f(z)}-1\right|,\ z\in\mathbb{D}. \] Sharma \emph{et. al} \cite{MR3536076} introduced the set $\mathcal{S}^*_C:=\mathcal{S}^*(1+4z/3+2z^2/3)$ which consists of functions $f\in\mathcal{A}$ such that $zf'(z)/f(z)$ lies in the region bounded by the cardioid
\[ \Omega_C:=\{w=u+iv:(9u^2+9v^2-18u+5)-16(9u^2+9v^2-6u+1)=0 \}. \] The class $\mathcal{S}^*_e:=\mathcal{S}^*(e^z),$ introduced by Mendiratta \emph{et. al} \cite{MR3394060}, contains functions $f\in\mathcal{A}$ that satisfy $|\log (zf'(z)/f(z)|<1.$
For $b\geq1/2$ and $a\geq 1,$ Paprocki and Sok\'{o}l \cite{MR1473960} introduced a more general class $\mathcal{S}^*[a,b]$ for the functions $f\in\mathcal{A}$ satisfying $|(zf'(z)/f(z))^a-b|<b.$ Evidently, the class $\mathcal{SL}:=\mathcal{S}^*[2,1].$ Kanas \cite{MR2209585} used the method of differential subordination to find conditions for the functions to map the unit disk onto region bounded by parabolas and hyperbolas. Ali \emph{et al.} \cite{MR2917253} studied the class $\mathcal{SL}$ with the help of differential subordination and obtained some lower bound on $\beta$ such that $p(z)\prec\sqrt{1+z}$ whenever $1+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=0,1,2),$ where $p$ is analytic on $\mathbb{D}$ with $p(0)=1.$ Kumar \emph{et al.} \cite{MR3063215} proved that whenever $\beta>0,\ p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=0,1,2)$ implies $p(z)\prec\sqrt{1+z}$ for $p$ as mentioned above.
Motivated by work in \cite{MR2917253,MR2209585,MR3496681,MR3063215,MR3394060,MR1128729,MR3536076,MR3518217}, the method of differential subordination of first and second order has been used to obtain sufficient conditions for the function $f\in\mathcal{A}$ to belong to class $\mathcal{SL}.$ Let $p$ be an analytic function in $\mathbb{D}$ with $p(0)=1.$ In Section \ref{fods}, using the first order differential subordination, conditions on complex number $\beta$ are determined so that $p(z)\prec\sqrt{1+z}$ whenever $p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=3,4)$ or whenever $p^2(z)+\beta zp'(z)/p^n(z)\prec 1+z\ (n=-1,0,1,2)$ and alike. Also, conditions on $\beta$ and $\gamma$ are obtained that enable $p^2(z)+zp'(z)/(\beta p(z)+\gamma)\prec 1+z$ imply $p(z)\prec\sqrt{1+z}.$ Section \ref{sods} deals with obtaining sufficient conditions on $\beta$ and $\gamma,$ using the method of differential subordination which implies $p(z)\prec\sqrt{1+z}$ if $\gamma zp'(z)+\beta z^2p''(z)\prec z/(8\sqrt{2})$ and others. Section \ref{al} admits alternate proofs for the results proved in \cite{MR2917253} and \cite{MR3063215}. The proofs are based on properties of admissible functions formulated by Miller and Mocano \cite{MR1760285}.
\section{The admissibility condition}
Let $\mathcal{Q}$ be the set of functions $q$ that are analytic and injective on $\overline{\mathbb{D}} \setminus \mathbf{E}(q),$ where
\[ \mathbf{E}(q)=\left\{ \zeta \in \partial \mathbb{D} : \underset{z \rightarrow \zeta} \lim q(z)=\infty \right\} \]
and are such that $q'(\zeta) \neq 0$ for $\zeta \in \partial \mathbb{D}\setminus\mathbf{E}(q)$.
\begin{definition}
Let $\Omega$ be a set in $\mathbb{C} , q \in \mathcal{Q}$ and $n$ be a positive integer. The class of admissible functions $\Psi_n[\Omega,q],$ consists of those functions $\psi: \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ that satisfy the admissiblity condition $\psi(r,s,t;z)\not\in\Omega$ whenever $r=q(\zeta)$ is finite, $s =m\zeta q'(\zeta)$ and $\operatorname{Re} \left( \dfrac{t}{s}+1 \right) \geq m \operatorname{Re} \left( \dfrac{\zeta q''(\zeta)}{q'(\zeta)}+1 \right),$ for $z\in\mathbb{D},\zeta\in\partial\mathbb{D}\setminus E(q)$ and $m\geq n\geq 1.$ The class $\Psi_1[\Omega,q]$ will be denoted by $\Psi[\Omega,q].$
\end{definition}
\begin{theorem}\cite[Theorem 2.3b, p.\,28]{MR1760285}\label{main thm}
Let $\psi\in\Psi_n[\Omega,q]$ with $q(0)=a.$ Thus for $p\in\mathcal{H}[a,n]$ such that \begin{equation}\label{adm basic}
\psi(p(z),zp'(z),z^2p''(z);z)\in\Omega\ \ \Rightarrow \ \ p(z)\prec q(z).
\end{equation} \end{theorem} If $\Omega$ is a simply connected region which is not the whole complex plane, then there is a conformal mapping $h$ from $\mathbb{D}$ onto $\Omega$ satisfying $h(0)=\psi(a,0,0;0).$ Thus, for $p\in\mathcal{H}[a,n],$ \eqref{adm basic} can be written as \begin{equation}\label{adm h}
\psi(p(z),zp'(z),z^2p''(z);z)\prec h(z) \ \ \Rightarrow \ \ p(z)\prec q(z). \end{equation}
The univalent function $q$ is said to be the \textit{dominant of the solutions} of the second order differential equation \eqref{adm h}. The dominant $\tilde{q}$ that satisfies $\tilde{q}\prec q$ for all the dominants of \eqref{adm h} is said to be the \textit{best dominant} of \eqref{adm h}.
Consider the function $q:\mathbb{D} \to \mathbb{C}$ defined by $q(z)=\sqrt{1+z},\ z\in\mathbb{D}.$ Clearly, the function $q$ is univalent in $\overline{\mathbb{D}}\setminus\{-1\}.$ Thus, $q\in\mathcal{Q}$ with $E(q)=\{-1\}$ and $q(\mathbb{D})=\{w:|w^2-1|<1\}.$ We now define the admissibility conditions for the function $\sqrt{1+z}.$ Denote $\Psi_n[\Omega,\sqrt{1+z}]$ by $\Psi_n[\Omega,\mathcal{L}].$ Further, the case when $\Omega=\Delta=\{w:|w^2-1|<1\,,\operatorname{Re}w>0\},\ \Psi_n[\Omega,\sqrt{1+z}]$ is denoted by $\Psi_n[\mathcal{L}].$
If $|\zeta|=1,$ then \[
q(\zeta)\in q(\partial\mathbb{D})=\partial q(\mathbb{D})=\{w:|w^2-1|=1\}
=\left\{\sqrt{2\cos{2\theta}}e^{i\theta}:-\frac{\pi}{4}<\theta<\frac{\pi}{4}\right\}.\]
Then, for $\zeta= 2\cos{2\theta} e^{2i\theta} -1,$ we have
\[ \zeta q'(\zeta)=\frac{1}{2}\left(\sqrt{2\cos 2 \theta} e^{i\theta}-\frac{1}{\sqrt{2\cos 2 \theta} e^{i\theta}} \right) \quad
\text{and}\quad q''(\zeta)=\dfrac{-1}{4(2\cos{2\theta}e^{2i\theta})^{3/2}}\]
and hence \[\operatorname{Re} \left(\frac{\zeta q''(\zeta)}{q'(\zeta)}+1 \right)=\operatorname{Re} \left( \frac{e^{-2 i \theta }}{4 \cos 2 \theta}+\frac{1}{2} \right)=\frac{3}{4}.\]
Thus, the condition of admissibility reduces to $\psi(r,s,t;z)\not \in \Omega$ whenever $(r,s,t;z)\in \operatorname{Dom}\psi$ and
\begin{equation}\label{adm for q}
\begin{split}
&r=\sqrt{2\cos 2 \theta}e^{i \theta},\\ &s=\displaystyle{\frac{m}{2}\left(\sqrt{2\cos 2 \theta} e^{i\theta}-\frac{1}{\sqrt{2\cos 2 \theta} e^{i\theta}} \right)=\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}},\\ &\displaystyle{\operatorname{Re}\left( \frac{t}{s}+1\right)\geq\frac{3m}{4}}
\end{split}
\end{equation}
where $\theta\in(-\pi/4,\pi/4)$ and $m\geq n\geq 1.$
As a particular case of Theorem \ref{main thm}, we have
\begin{theorem}\label{lem thm}
Let $p\in \mathcal{H}[1,n]$ with $p(z)\not\equiv 1$ and $n\geq 1.$ Let $\Omega\subset\mathbb{C}$ and $\psi:\mathbb{C}^3\times\mathbb{D}\to\mathbb{C}$ with domain $D$ satisfy \[\psi(r,s,t;z)\not\in\Omega\ \text{ whenever } z\in\mathbb{D}, \]
for $r=\sqrt{2\cos 2\theta}e^{i\theta},\ s=me^{3i\theta}/(2\sqrt{2\cos2\theta})$ and $\operatorname{Re}(t/s+1)\geq 3m/4$ where $m\geq n\geq1$ and $-\pi/4<\theta<\pi/4.$ For $z\in\mathbb{D},$ if $(p(z),zp'(z),z^2p''(z);z)\in D,$ and $\psi(p(z),zp'(z),z^2p''(z);z)\in \Omega,$ then $p(z)\prec\sqrt{1+z}.$
\end{theorem}
The case when $\psi\in\Psi_n[\mathcal{L}]$ with domain $D,$ the above theorem reduces to the case: For $z\in \mathbb{D},$ if $(p(z),zp'(z),z^2p''(z);z)\in D$ and $\psi(p(z),zp'(z),z^2p''(z);z)\prec \sqrt{1+z},$ then $p(z)\prec\sqrt{1+z}.$
We now illustrate the above result for certain $\Omega.$ Throughout $r,s,t$ refer to as mentioned in \eqref{adm for q}.
\begin{example}
Let $\displaystyle{\Omega=\{w:|w-1|<1/(2\sqrt{2})\}}$ and define $\psi:\mathbb{C}^3\times\mathbb{D}\to \mathbb{C} $ by $\psi(a,b,c;z)=1+b.$
For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$
Then, $\psi(r,s,t;z)$ is given by
\begin{align*}
\psi(r,s,t;z)&=1+\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}\\
\intertext{and therefore we have that}
|\psi(r,s,t;z)-1|&=\left|\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}\right|=\frac{m}{2\sqrt{2\cos2\theta}}\geq\frac{m}{2\sqrt{2}}\geq\frac{1}{2\sqrt{2}}.
\end{align*}
Thus, $\psi\in\Psi[\Omega,\mathcal{L}].$
Hence, whenever $p\in\mathcal{H}_1$ such that $|zp'(z)|< 1/(2\sqrt{2}),$ then $p(z)\prec \sqrt{1+z}.$
\end{example}
\begin{example}
Let $\Omega=\{w:\operatorname{Re}w<1/4\}$ and define $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}^2\times\mathbb{D}\to\mathbb{C}$ by
$\psi(a,b,c;z)=b/a.$
For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$
Now, consider $\psi(r,s,t;z)$ given by
\begin{align*}
\psi(r,s,t;z)&=\frac{s}{r}=\frac{me^{2i\theta}}{4\cos2\theta}.\\
\intertext{Then}
\operatorname{Re}\psi(r,s,t;z)&=\frac{m}{4\cos2\theta}\operatorname{Re}(e^{2i\theta})=\frac{m}{4} \geq\frac{1}{4}.
\end{align*}
That is $\psi(r,s,t;z)\not\in \Omega.$ Hence, we see that $\psi \in \Psi[\Omega,\mathcal{L}].$
Therefore, for $p(z)\in \mathcal{H}_1$ if \[\operatorname{Re}\left(\frac{zp'(z)}{p(z)}\right)<\frac{1}{4},\]
then $p(z)\prec \sqrt{1+z}.$ Moreover, the result is sharp as for $p(z)=\sqrt{1+z},$ we have \[ \operatorname{Re}\left(\frac{zp'(z)}{p(z)}\right)= \operatorname{Re}\left(\frac{z}{2(1+z)}\right)\to \frac{1}{4} \text{ as } z\to 1. \]
That is $\sqrt{1+z}$ is the best dominant.
\end{example}
\begin{example}
Let $\Omega=\{w:|w-1|<1/(4\sqrt{2})\}$ and define $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}^2\times\mathbb{D}\to \mathbb{C} $ by $\psi(a,b,c;z)=1+b/a^2.$
For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$
Then, $\psi(r,s,t;z)$ is given by
\begin{align*}
\psi(r,s,t;z)&=1+\frac{me^{i\theta}}{2(2\cos2\theta)^{3/2}}\\
\intertext{and so}
|\psi(r,s,t;z)-1|&=\left|\frac{me^{i\theta}}{2(2\cos2\theta)^{3/2}}\right|=\frac{m}{4\sqrt{2}(\cos2\theta)^{3/2}}\geq\frac{m}{4\sqrt{2}}\geq\frac{1}{4\sqrt{2}}.
\end{align*}
Thus, $\psi\in\Psi[\Omega,\mathcal{L}].$
Hence, whenever $p\in\mathcal{H}_1$ such that \[ \left|\frac{zp'(z)}{p^2(z)}\right|< \frac{1}{4\sqrt{2}}, \] then $p(z)\prec \sqrt{1+z}.$
\end{example} \section{First Order Differential Subordination}\label{fods} In case of first order differential subordination, Theorem \ref{lem thm} reduces to: \begin{theorem}
Let $p\in \mathcal{H}[1,n]$ with $p(z)\not\equiv 1$ and $n\geq 1.$ Let $\Omega\subset\mathbb{C}$ and $\psi:\mathbb{C}^2\times\mathbb{D}\to\mathbb{C}$ with domain $D$ satisfy \[\psi(r,s;z)\not\in\Omega\ \text{ whenever } z\in\mathbb{D}, \]for $r=\sqrt{2\cos 2\theta}e^{i\theta}$ and $s=me^{3i\theta}/(2\sqrt{2\cos2\theta})$ where $m\geq n\geq1$ and $-\pi/4<\theta<\pi/4.$ For $z\in\mathbb{D},$ if $(p(z),zp'(z);z)\in D$ and $\psi(p(z),zp'(z);z)\in \Omega,$ then $p(z)\prec\sqrt{1+z}.$ \end{theorem} Likewise for an analytic function $h,$ if $\Omega=h(\mathbb{D}),$ then the above theorem becomes \[ \psi(p(z),zp'(z);z)\prec h(z) \Rightarrow p(z)\prec \sqrt{1+z}. \]
Using the above theorem, now some sufficient conditions are determined for $p\in\mathcal{H}_1$ to satisfy $p(z)\prec\sqrt{1+z}$ and hence sufficient conditions are obtained for function $f\in\mathcal{A}$ to belong to the class $\mathcal{SL}.$
Kumar \emph{et al.} \cite{MR3063215} proved that for $\beta>0$ if $p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=0,1,2),$ then $p(z)\prec\sqrt{1+z}.$ Extending this, we obtain lower bound for $\beta$ so that $p(z)\prec\sqrt{1+z}$ whenever $p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=3,4).$
\begin{lemma}\label{extended version}
Let $p$ be analytic in $\mathbb{D}$ and $p(0)=1$ and $\beta_0=1.1874.$ Let \[p(z)+\frac{\beta zp'(z)}{p^3(z)}\prec\sqrt{1+z} \ (\beta>\beta_0), \] then \[p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1,\operatorname{Re}w>0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times \mathbb{D}\to \mathbb{C}$ be defined by $\psi(a,b;z)=a+\beta b/a^3.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$
Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=\sqrt{2\cos2\theta}e^{i\theta}+\frac{\beta m}{8\cos^2 2\theta},
\intertext{so that}
|\psi(r,s;z)^2-1|^2&=1+\frac{\beta^2m^2}{32}(4\sec^3 2\theta+2\sec^2 2\theta-\sec^4 2\theta)+\frac{\beta m}{\sqrt{2}}\sec^{3/2}2\theta\cos3\theta\\
&\quad{}+\frac{\beta^4m^4}{4096}\sec^8 2\theta+\frac{\beta^3m^3}{64\sqrt{2}}\sec^{11/2}2\theta\cos\theta=:g(\theta)
\end{align*}
Observe that $g(\theta)=g(-\theta)$ for all $\theta\in(-\pi/4,\pi/4)$ and the second derivative test shows that the minimum of $g$ occurs at $\theta=0$ for $\beta m>1.1874.$
For $\beta>1.1874,$ we have $\beta m>1.1874.$ Thus, $g(\theta)$ attains its minimum at $\theta=0$ for $\beta>\beta_0.$
For $\psi\in \Psi[\mathcal{L}],$ we must have $g(\theta)\geq 1$ for every $\theta \in (-\pi/4,\pi/4)$ and since
\begin{align*}
\min g(\theta)&=1+\frac{\beta m}{\sqrt{2}}+\frac{5\beta^2m^2}{32}+\frac{\beta^3m^3}{64\sqrt{2}}+\frac{\beta^4m^4}{4096}\\
&\geq 1+\frac{\beta }{\sqrt{2}}+\frac{5\beta^2}{32}+\frac{\beta^3}{64\sqrt{2}}+\frac{\beta^4}{4096}>1.
\end{align*}
Hence for $\beta>\beta_0,\ \psi\in\Psi[\mathcal{L}]$ and therefore, for $p(z)\in \mathcal{H}_1,$ if \[ p(z)+\frac{\beta zp'(z)}{p^3(z)}\prec\sqrt{1+z}\ (\beta>\beta_0), \]
we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} \begin{lemma}
Let $p$ be analytic in $\mathbb{D}$ and $p(0)=1$ and $\beta_0=3.58095.$ Let \[p(z)+\frac{\beta zp'(z)}{p^4(z)}\prec\sqrt{1+z} \ (\beta>\beta_0), \] then \[p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1,\operatorname{Re}w>0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times \mathbb{D}\to \mathbb{C}$ be defined by $\psi(a,b;z)=a+\beta b/a^4.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$
Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=\sqrt{2\cos2\theta}e^{i\theta}+\frac{\beta me^{-i\theta}}{8\cos^2 2\theta\sqrt{2\cos2\theta} },
\intertext{so that}
|\psi(r,s;z)^2-1|^2&=1+\beta m(1-\frac{1}{2}\sec^2 2\theta)+\frac{\beta^2m^2}{64}(\sec^4 2\theta+4\sec^2 2\theta)\\
&\quad{}+\frac{\beta^3m^3}{256}\sec^6 2\theta+\frac{\beta^4m^4}{128^2}\sec^{10}2\theta=:g(\theta)
\end{align*}
Observe that $g(\theta)=g(-\theta)$ for all $\theta\in(-\pi/4,\pi/4)$ and the second derivative test shows that $g$ attains its minimum at $\theta=0$ if $\beta m>3.58095.$
For $\beta>3.58095,$ we have $\beta m> 3.58095.$ Thus, $g(\theta)$ attains its minimum at $\theta=0$ for $\beta>\beta_0.$
For $\psi\in \Psi[\mathcal{L}],$ we must have $g(\theta)\geq 1$ for every $\theta \in (-\pi/4,\pi/4)$ and since
\begin{align*}
\min g(\theta)&=1+\frac{\beta m}{2}+\frac{5\beta^2m^2}{64}+\frac{\beta^3m^3}{256}+\frac{\beta^4m^4}{128^2}\\
&\geq 1+\frac{\beta}{2}+\frac{5\beta^2}{64}+\frac{\beta^3}{256}+\frac{\beta^4}{128^2}>1.
\end{align*}
Hence for $\beta>\beta_0,\ \psi\in\Psi[\mathcal{L}]$ and therefore, for $p(z)\in \mathcal{H}_1,$ if \[ p(z)+\frac{\beta zp'(z)}{p^4(z)}\prec\sqrt{1+z}\ (\beta>\beta_0), \]
we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} On the similar lines, one can find lower bound for $\beta_n$ such that $p(z)+\beta_n zp'(z)/p^n(z)\prec\sqrt{1+z},\ n\in\mathbb{N}$ implies $p(z)\prec\sqrt{1+z}.$
Now, the conditions on $\beta$ and $\gamma$ are discussed so that $p^2(z)+zp'(z)/(\beta p(z)+\gamma)\prec 1+z$ implies $p(z)\prec\sqrt{1+z}.$ \begin{lemma}
Let $\beta,\gamma>0$ and $p$ be analytic in $\mathbb{D}$ such that $p(0)=1.$ If \[ p^2(z)+\frac{zp'(z)}{\beta p(z)+\gamma}\prec 1+z, \] then \[ p(z)\prec \sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=1+z$ and let $\Omega=h(\mathbb{D})=\{w:|w-1|<1\}.$ Let $\psi:(\mathbb{C}\setminus\{-\gamma/\beta\})\times\mathbb{C}\times\mathbb{D}\to\mathbb{C}$ be defined by \[\psi(r,s;z)=r^2+\frac{s}{\beta r+\gamma}.\] For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{me^{3i\theta}}{(2\sqrt{2\cos2\theta})(\beta\sqrt{2\cos2\theta}e^{i\theta}+\gamma)},
\intertext{and so}
|\psi(r,s;z)-1|^2&=\left[\cos\theta+\frac{m\beta\sqrt{2\cos2\theta}\cos\theta+\gamma m}{2\sqrt{2\cos2\theta}d(\theta)}\right]^2\\
&\quad{}+\left[\sin\theta-\frac{m\beta\sqrt{2\cos2\theta}\sin\theta}{2\sqrt{2\cos2\theta}d(\theta)}\right]^2,
\intertext{where $d(\theta)=|\beta\sqrt{2\cos2\theta} e^{i\theta}+\gamma|^2=\cos2\theta(2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1}).$}
\intertext{Hence on solving, we get that}
|\psi(r,s;z)-1|^2&=1+\frac{\beta^2m^2\sec^2 2\theta}{4(2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1})^2}\\
&\quad{}+\frac{\gamma^2m^2\sec^3 2\theta}{8(2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1})^2}\\
&\quad{}+\frac{\beta\gamma m^2\sqrt{\sec2\theta+1}\sec^2 2\theta}{4(2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1})^2}\\
&\quad{}+\frac{\beta m}{2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1}}\\
&\quad{}+\frac{\gamma m \sqrt{\sec2\theta+1}\sec2\theta}{2(2\beta^2+\gamma^2\sec2\theta+2\beta\gamma\sqrt{\sec2\theta+1})}=:g(\theta)
\end{align*}
Using the second derivative test, we get that minimum of $g$ occurs at $\theta=0.$
For $\psi\in\Psi[\Omega,\mathcal{L}],$ we must have $g(\theta)\geq1$ for every $\theta\in(-\pi/4,\pi/4)$ and since
\begin{align*}
\min g(\theta)&=1+\frac{\beta^2m^2}{4(\beta\sqrt{2}+\gamma)^4}+\frac{\gamma^2m^2}{8(\beta\sqrt{2}+\gamma)^4}+\frac{\beta\gamma m^2}{2\sqrt{2}(\beta\sqrt{2}+\gamma)^4}\\
&\quad{}+\frac{\beta m}{(\beta\sqrt{2}+\gamma)^2}+\frac{\gamma m}{\sqrt{2}(\beta\sqrt{2}+\gamma)^2}\\
&\geq 1+\frac{\beta^2}{4(\beta\sqrt{2}+\gamma)^4}+\frac{\gamma^2}{8(\beta\sqrt{2}+\gamma)^4}+\frac{\beta\gamma}{2\sqrt{2}(\beta\sqrt{2}+\gamma)^4}+\frac{\beta}{(\beta\sqrt{2}+\gamma)^2}\\
&\quad{}+\frac{\gamma}{\sqrt{2}(\beta\sqrt{2}+\gamma)^2}>1.
\end{align*}
Hence, for $\beta,\gamma>0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and therefore, for $p\in\mathcal{H}_1,$ if \[p^2(z)+\frac{zp'(z)}{\beta p(z)+\gamma}\prec 1+z,\] then \[p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} Now, conditions on $\beta$ are derived so that $p^2(z)+\beta zp'(z)/p^n(z)\prec 1+z\ (n=-1,0,1,2)$ implies $p(z)\prec\sqrt{1+z}.$ \begin{lemma}
Let $p$ be analytic in $\mathbb{D}$ with $p(0)=1.$ Let $\beta$ be a complex number such that $\operatorname{Re}\beta>0.$ If \[ p^2(z)+\beta zp'(z)p(z)\prec 1+z, \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=1+z$ and let $\Omega=h(\mathbb{D})=\{w:|w-1|<1\}.$ Let $\psi:\mathbb{C}^2\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b;z)=a^2+\beta ab.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{\beta me^{4i\theta}}{2}
\intertext{and we see that}
|\psi(r,s;z)-1|&=\left|1+\frac{m\beta}{2}\right|\geq1+\frac{m\operatorname{Re}\beta}{2}\geq1+\frac{\operatorname{Re}\beta}{2}> 1.
\end{align*}
Hence, for $\beta$ such that $\operatorname{Re}\beta>0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and therefore, for such complex number $\beta$ and for $p\in\mathcal{H}_1,$ if \[p^2(z)+\beta zp(z)p'(z)\prec 1+z,\] then \[p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} \begin{lemma}
Let $\beta>0$ and $p$ be analytic in $\mathbb{D}$ with $p(0)=1.$ If \[p^2(z)+\beta zp'(z)\prec 1+z, \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=1+z$ and let $\Omega=h(\mathbb{D})=\{w:|w-1|<1\}.$ Let $\psi:\mathbb{C}^2\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b;z)=a^2+\beta b.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{\beta me^{3i\theta}}{2\sqrt{2\cos2\theta}},
\intertext{and so}
|\psi(r,s;z)-1|^2&=1+\frac{\beta^2m^2}{8}\sec2\theta+\frac{\beta m}{2}\sqrt{\sec2\theta+1}\\
&\geq 1+\frac{\beta^2m^2}{8}+\frac{\beta m}{\sqrt{2}}\geq1+\frac{\beta^2}{8}+\frac{\beta}{\sqrt{2}}>1.
\end{align*}
Hence, for $\beta>0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and therefore, for $p(z)\in\mathcal{H}_1,$ if \[p^2(z)+\beta zp'(z)\prec 1+z, \] then \[ p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} \begin{lemma}
Let $\beta>0$ and $p$ be analytic in $\mathbb{D}$ with $p(0)=1.$ If \[p^2(z)+\frac{\beta zp'(z)}{p(z)}\prec 1+z, \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=1+z$ and let $\Omega=h(\mathbb{D})=\{w:|w-1|<1\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b;z)=a^2+\beta b/a.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{\beta me^{2i\theta}}{4\cos2\theta},
\intertext{and so}
|\psi(r,s;z)-1|^2&
=1+\frac{\beta^2m^2}{16\cos^2 2\theta}+\frac{\beta m}{2}
\geq1+\frac{\beta^2m^2}{16}+\frac{\beta m}{2}\\
&\geq1+\frac{\beta^2}{16}+\frac{\beta}{2}> 1.
\end{align*}
Hence for $\beta>0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and therefore, for $p(z)\in\mathcal{H}_1,$ if \[p^2(z)+\frac{\beta zp'(z)}{p(z)}\prec 1+z, \] then \[ p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} \begin{lemma}
Let $\beta_0=2\sqrt{2}.$ Let $p$ be analytic in $\mathbb{D}$ with $p(0)=1.$ If \[p^2(z)+\frac{\beta zp'(z)}{p^2(z)}\prec 1+z \ (\beta>\beta_0), \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=1+z$ and let $\Omega=h(\mathbb{D})=\{w:|w-1|<1\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b;z)=a^2+\beta b/a^2.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{\beta me^{i\theta}}{4\sqrt{2}\cos^{3/2}2\theta},
\intertext{and so}
|\psi(r,s;z)-1|^2
&=1+\frac{\beta^2m^2}{32\cos^3 2\theta}+\frac{\beta m\cos3\theta}{2\sqrt{2}\cos^{3/2} 2\theta}=:g(\theta)
\end{align*}
It is clear using the second derivative test that for $\beta m>2\sqrt{2},$ minimum of $g$ occurs at $\theta=0.$
For $\beta>2\sqrt{2},\ \beta m> 2\sqrt{2}$ which implies that minimum of $ g(\theta)$ is attained at $\theta=0$ for $\beta>\beta_0.$ Hence
\[ \min g(\theta)=1+\frac{\beta^2m^2}{32}+\frac{\beta m}{2\sqrt{2}}\geq1+\frac{\beta^2}{32}+\frac{\beta}{2\sqrt{2}}>1.\]
Hence for $\beta>\beta_0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and therefore, for $p(z)\in\mathcal{H}_1,$ if \[p^2(z)+\frac{\beta zp'(z)}{p^2(z)}\prec 1+z\ (\beta>\beta_0), \] then \[ p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} Next result depicts some sufficient conditions so that $p(z)\prec\sqrt{1+z}$ whenever $p^2(z)+\beta zp'(z)p(z)\prec(2+z)/(2-z).$ \begin{lemma}
Let $\beta_0=2$ and $p$ be analytic in $\mathbb{D}$ with $p(0)=1.$ If \[ p^2(z)+\beta zp'(z)p(z)\prec \frac{2+z}{2-z}\ \ (\beta\geq\beta_0), \] then \[ p(z)\prec\sqrt{1+z}. \] The lower bound $\beta_0$ is best possible. \end{lemma} \begin{proof}
Let $\beta>0.$ Let $h$ be the analytic function defined on $\mathbb{D}$ by $h(z)=(2+z)/(2-z)$ and let $\Omega=h(\mathbb{D})=\{w:|2(w-1)/(w+1)|<1\}.$ Let $\psi:\mathbb{C}^2\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b;z)=a^2+\beta ab.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=2\cos2\theta e^{2i\theta}+\frac{\beta me^{4i\theta}}{2}
\intertext{then}
\left|\frac{2(\psi(r,s;z)-1)}{\psi(r,s;z)+1}\right|^2&=\frac{4(1+m\beta/2)^2}{(1+\beta m/2)^2+4+4(1+\beta m/2)\cos4\theta}=:g(\theta)
\end{align*}
Using the second derivative test, one can verify that minimum of $g$ occurs at $\theta=0.$
Thus \[\min g(\theta)=\frac{4(1+\beta m/2)^2}{(1+\beta m/2)^2+4(1+\beta m/2)+4} . \]
Now, the inequality
\begin{align*}
&\frac{4(1+\beta/2)^2}{(1+\beta/2)^2+4(1+\beta/2)+4}\geq 1
\intertext{holds if}
&3\left(1+\frac{\beta}{2}\right)^2-4-4\left(1+\frac{\beta}{2}\right)\geq 0
\end{align*}
or equivalently if $\beta\geq2.$
Since, $m\geq1,\ \beta m\geq2$ implies that \[\frac{4(1+\beta m/2)^2}{(1+\beta m/2)^2+4(1+\beta m/2)+4}\geq1\] and therefore $\left|\dfrac{2(\psi(r,s;z)-1)}{\psi(r,s;z)+1}\right|^2\geq 1$.
Hence, for $\beta\geq\beta_0,\ \psi\in\Psi[\Omega,\mathcal{L}]$ and for $p\in\mathcal{H}_1,$ if \[p^2(z)+\beta zp'(z)p(z)\prec \frac{2+z}{2-z}\ (\beta\geq\beta_0),\] then \[p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} \begin{remark}
All of the above lemmas give a sufficient condition for $f$ in $\mathcal{A}$ to be lemniscate starlike. This can be seen by defining a function $p:\mathbb{D}\to\mathbb{C}$ by $p(z)=zf'(z)/f(z).$ \end{remark} \section{Second Order Differential Subordinations}\label{sods} This section deals with the case that if there is an analytic function $p$ such that $p(0)=1$ satisfying a second order differential subordination then $p(z)$ is \textit{subordinate} to $\sqrt{1+z}.$ Now, for $r,s,t$ as in \eqref{adm for q}, we have $\displaystyle{\operatorname{Re}\left(\frac{t}{s}+1\right)\geq \frac{3m}{4}}$ for $m\geq n\geq 1.$ On simplyfying, \begin{align}
\operatorname{Re}(te^{-3i\theta})&\geq \frac{m(3m-4)}{8\sqrt{2\cos2\theta}}.\label{re t} \intertext{If $m\geq2,$ then}\operatorname{Re}(te^{-3i\theta}) &\geq \frac{1}{2\sqrt{2\cos2\theta}}\geq \frac{1}{2\sqrt{2}}.\nonumber \end{align} \begin{lemma}\label{lem5.1}
Let $p$ be analytic in $\mathbb{D}$ such that $p(0)=1.$ If \[ zp'(z)+z^2p''(z)\prec\frac{3z}{8\sqrt{2}}, \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h(z)=3z/(8\sqrt{2}),$ then $\Omega=h(\mathbb{D})=\{w:|w|<3/(8\sqrt{2})\}$ and let $\psi:\mathbb{C}^3\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b,c;z)=b+c.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s,t;z)$ is given by
\begin{align*}
\psi(r,s,t;z)&=\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}+t.
\intertext{So, we have that}
|\psi(r,s,t;z)|&=\left|\frac{m}{2\sqrt{2\cos2\theta}}+te^{-3i\theta}\right|\geq\frac{3m^2}{8\sqrt{2\cos2\theta}}.
\intertext{Since $m\geq1,$ so}
|\psi(r,s,t;z)|&\geq \frac{3}{8\sqrt{2\cos2\theta}} \geq \frac{3}{8\sqrt{2}}.
\end{align*}
Therefore, $\psi\in\Psi[\Omega,\mathcal{L}].$ Hence, for $p\in\mathcal{H}_1$ if \[ zp'(z)+z^2p''(z)\prec\frac{3z}{8\sqrt{2}}, \] then \[ p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} We obtain the following theorem by taking $p(z)=zf'(z)/f(z)$ in Lemma \ref{lem5.1}, where $p$ is analytic in $\mathbb{D}$ and $p(0)=1.$ \begin{theorem}
Let $f$ be a function in $\mathcal{A}.$ If $f$ satisfies the subordination
\begin{align*}
&\frac{zf'(z)}{f(z)}\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)+\frac{zf'(z)}{f(z)}\Bigg(\frac{z^2f'''(z)}{f'(z)}-\frac{3z^2f''(z)}{f(z)}\\
&\quad{}+\frac{2zf''(z)}{f'(z)}+2\left(\frac{zf'(z)}{f(z)}\right)^2-\frac{2zf'(z)}{f(z)}\Bigg)\prec\frac{3z}{8\sqrt{2}} ,
\end{align*}
then $f\in\mathcal{SL}.$ \end{theorem}
\begin{lemma}\label{lem5.3}
Let $p$ be analytic in $\mathbb{D}$ such that $p(0)=1$ and let $p\in\mathcal{H}[1,2].$ If \[ p^2(z)+zp'(z)+z^2p''(z)\prec1+\left(1+\frac{3}{2\sqrt{2}}\right)z, \] then \[ p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $h(z)=1+(1+3/(2\sqrt{2}))z$ then $\Omega=h(z)=\{w:|w-1|<1+3/(2\sqrt{2})\}$. Let $\psi:\mathbb{C}^3\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b,c;z)=a^2+b+c.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s,t;z)$ is given by
\begin{align*}
\psi(r,s,t;z)&=2\cos2\theta e^{2i\theta}+\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}+t.\\
\intertext{So, we have}
|\psi(r,s,t;z)-1|&=\left|e^{i\theta}+\frac{m}{2\sqrt{2\cos2\theta}}+te^{-3i\theta}\right|\\
&\geq \operatorname{Re}\left(e^{i\theta}+\frac{m}{2\sqrt{2\cos2\theta}}+te^{-3i\theta}\right)\\
&=\cos\theta+\frac{3m^2}{8\sqrt{2}}\sec^{1/2}2\theta =: g(\theta)
\end{align*}
The second derivative test shows that minimum of $g$ occurs at $\theta=0$ if $m\geq2.$ Therefore, $\psi\in\Psi[\Omega,\mathcal{L}].$ Hence, for $p\in\mathcal{H}[1,2]$ if \[ p^2(z)+zp'(z)+z^2p''(z)\prec1+\left(1+\frac{3}{2\sqrt{2}}\right)z,\] then \[ p(z)\prec\sqrt{1+z}. \qedhere \] \end{proof} The following theorem holds by taking $p(z)=zf'(z)/f(z)$ in Lemma \ref{lem5.3}, where $p$ is analytic in $\mathbb{D}$ and $p(0)=1.$ \begin{theorem}
Let $f$ be a function in $\mathcal{A}$ such that $zf'(z)/f(z)$ has Taylor series expansion of the form $1+a_2z^2+a_3z^3+\ldots.$ If $f$ satisfies the subordination
\begin{align*}
&\left(\frac{zf'(z)}{f(z)}\right)^2+\frac{zf'(z)}{f(z)}\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)+\frac{zf'(z)}{f(z)}\Bigg(\frac{z^2f'''(z)}{f'(z)}-\frac{3z^2f''(z)}{f(z)}\\
&\quad{}+\frac{2zf''(z)}{f'(z)}+2\left(\frac{zf'(z)}{f(z)}\right)^2-\frac{2zf'(z)}{f(z)}\Bigg)\prec1+\left(1+\frac{3}{2\sqrt{2}}\right)z,
\end{align*}
then $f\in\mathcal{SL}.$ \end{theorem}
The next result admits some conditions on $\beta$ and $\gamma$ for $p(z)\prec\sqrt{1+z}$ whenever $\gamma z p'(z)+\beta z^2p''(z)\prec z/(8\sqrt{2}).$ \begin{lemma}\label{lem5.5}
Let $\gamma\geq\beta>0$ be such that $4\gamma-\beta\geq 1.$ Let $p$ be analytic in $\mathbb{D}$ such that $p(0)=1$ and \[ \gamma z p'(z)+\beta z^2p''(z)\prec\frac{z}{8\sqrt{2}} \text{ for }\gamma\geq \beta>0 \text{ and } 4\gamma-\beta\geq 1, \] then\[p(z)\prec\sqrt{1+z}.\] \end{lemma} \begin{proof}
Let $h(z)=z/(8\sqrt{2})$ for $z\in\mathbb{D}$ and $\Omega=h(\mathbb{D})=\{w:|w|<1/(8\sqrt{2})\}.$ Let $\psi:\mathbb{C}^3\times\mathbb{D}\to\mathbb{C}$ be defined by $\psi(a,b,c;z)=\gamma b+\beta c.$ For $\psi$ to be in $\Psi[\Omega,\mathcal{L}],$ we must have $\psi(r,s,t;z)\not\in\Omega$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s,t;z)$ is given by
\begin{align*}
\psi(r,s,t;z)&=\frac{\gamma me^{3i\theta}}{2\sqrt{2\cos2\theta}}+\beta t.
\intertext{Hence, we see that}
|\psi(r,s,t;z)|&=\left|\frac{\gamma m}{2\sqrt{2\cos2\theta}}+\beta te^{-3i\theta}\right|\geq \frac{\gamma m}{2\sqrt{2\cos2\theta}}+\beta\operatorname{Re}(te^{-3i\theta}).
\intertext{Using \eqref{re t},}|\psi(r,s,t;z)|&\geq \frac{4m(\gamma-\beta)+3\beta m^2}{8\sqrt{2\cos2\theta}}.
\intertext{Since $m\geq 1,$ so}
|\psi(r,s,t;z)|&\geq \frac{4(\gamma-\beta)+3\beta}{8\sqrt{2\cos2\theta}}=\frac{4\gamma-\beta}{8\sqrt{2\cos2\theta}}.
\intertext{Given that $4\gamma-\beta\geq 1,$}
|\psi(r,s,t;z)|&\geq \frac{1}{8\sqrt{2\cos2\theta}} \geq \frac{1}{8\sqrt{2}}.
\end{align*}
Therefore, $\psi\in\Psi[\Omega,\mathcal{L}].$ Hence for $p\in\mathcal{H}_1$ satisfying \[ \gamma z p'(z)+\beta z^2p''(z)\prec\frac{z}{8\sqrt{2}} \text{ for }\gamma\geq \beta>0 \text{ and } 4\gamma-\beta\geq 1, \] we have\[p(z)\prec\sqrt{1+z}.\qedhere\] \end{proof} By taking $p(z)=zf'(z)/f(z)$ in Lemma \ref{lem5.5}, where $p$ is analytic in $\mathbb{D}$ and $p(0)=1,$ the following theorem holds. \begin{theorem}
Let $f$ be a function in $\mathcal{A}.$ Let $\gamma,\beta$ be as stated in Lemma \ref{lem5.5}. If $f$ satisfies the subordination
\begin{align*}
&\gamma \frac{zf'(z)}{f(z)}\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)+\beta \frac{zf'(z)}{f(z)}\Bigg(\frac{z^2f'''(z)}{f'(z)}\\
&\quad{}-\frac{3z^2f''(z)}{f(z)}+\frac{2zf''(z)}{f'(z)}+2\left(\frac{zf'(z)}{f(z)}\right)^2-\frac{2zf'(z)}{f(z)}\Bigg)\prec\frac{z}{8\sqrt{2}},
\end{align*}
then $f\in\mathcal{SL}.$ \end{theorem}
\section{Further results}\label{al} Now, we discuss alternate proofs to the results proven in \cite{MR2917253} where lower bounds for $\beta$ are determined for the cases where $1+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=0,1,2)$ imply $p(z)\prec\sqrt{1+z}.$ The method of admissible functions provides an improvement over the results proven in \cite{MR2917253}. \begin{lemma}\label{lem-1-0}
Let $p$ be analytic function on $\mathbb{D}$ and $p(0)=1.$ Let $\beta_0=2\sqrt{2}(\sqrt{2}-1)\approx 1.17.$ If \[ 1+\beta zp'(z) \prec \sqrt{1+z}\ (\beta\geq\beta_0), \] then \[ p(z) \prec \sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1,\operatorname{Re}w>0\}.$ Let us define $\psi:\mathbb{C}^2\times \mathbb{D}\to \mathbb{C}$ by $\psi(a,b;z)=1+\beta b.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$
Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=1+\frac{\beta m}{2\sqrt{2\cos{2\theta}}}e^{3i\theta}
\intertext{and so}
|\psi(r,s;z)^2-1|^2
&=\frac{\beta^4m^4}{64}\sec^2{2\theta}+\frac{\beta^3m^3}{4\sqrt{2}}\sec^{3/2}{2\theta}\cos3\theta+\frac{\beta^2m^2}{2}\sec{2\theta}=:g(\theta)
\end{align*} Observe that $g(\theta)=g(-\theta)$ for all $\theta\in(-\pi/4,\pi/4)$ and the second derivative shows that the minimum of $g$ occurs at $\theta=0$ when $\beta>2\sqrt{2}(\sqrt{2}-1).$
For $\psi\in \Psi[\mathcal{L}],$ we must have $g(\theta)\geq 1$ for every $\theta \in (-\pi/4,\pi/4)$ and since \[ \min g(\theta)=\frac{\beta^4m^4}{64}+\frac{\beta^3m^3}{4\sqrt{2}}+\frac{\beta^2m^2}{2}\geq \frac{\beta^4}{64}+\frac{\beta^3}{4\sqrt{2}}+\frac{\beta^2}{2}.\] The last term is greater than or equal to 1 if \begin{align*} &(\beta+2\sqrt{2})^2(\beta-4+2\sqrt{2})(\beta+4+2\sqrt{2})\geq 0 \intertext{ or equivalently if } &\beta\geq 4-2\sqrt{2}=2\sqrt{2}(\sqrt{2}-1)=\beta_0. \end{align*}
Hence, for $\beta\geq\beta_0,\ \psi\in\Psi[\mathcal{L}]$ and therefore for $p(z)\in \mathcal{H}_1,$ if \[ 1+\beta zp'(z) \prec \sqrt{1+z}\ (\beta\geq\beta_0), \] then, we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} As in \cite[Theorem 2.2]{MR2917253}, using above lemma, we deduce the following. \begin{theorem}
Let $\beta_0=2\sqrt{2}(\sqrt{2}-1)\approx1.17$ and $f\in \mathcal{A}.$
\begin{enumerate}
\item If $f$ satisfies the subordination \[ 1+\beta\frac{zf'(z)}{f(z)}\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)\prec \sqrt{1+z}\ \ (\beta\geq \beta_0), \]
then $f\in\mathcal{SL}$.
\item If $1+\beta zf''(z)\prec\sqrt{1+z}\ (\beta\geq\beta_0),$ then $f'(z)\prec\sqrt{1+z}.$
\end{enumerate} \end{theorem}
\begin{lemma}\label{lem-1-1}
Let $p$ be analytic function on $\mathbb{D}$ and $p(0)=1.$ Let $\beta_0=4(\sqrt{2}-1)\approx1.65$. If \[ 1+\beta \frac{zp'(z)}{p(z)}\prec\sqrt{1+z}\ (\beta\geq\beta_0), \] then \[ p(z)\prec \sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1\,, \operatorname{Re} w >0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times \mathbb{C}\times \mathbb{D} \to \mathbb{C}$ be defined by $\psi(a,b;z)=1+\beta b/a.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=1+\beta \frac{m}{2}\left(1-\frac{e^{-2i\theta}}{2\cos{2\theta}}\right)
\intertext{so that}
|\psi(r,s;z)^2-1|^2
&= \frac{\beta^4m^4}{256}\sec^4{2\theta}+\left(\frac{\beta^2m^2}{4}+\frac{\beta^3m^3}{16}\right)\sec^2{2\theta}\\
&\geq \frac{\beta^4m^4}{256}+\left(\frac{\beta^2m^2}{4}+\frac{\beta^3m^3}{16}\right)
\geq\frac{\beta^4}{256}+\frac{\beta^2}{4}+\frac{\beta^3}{16}
\intertext{The last term is greater than or equal to 1 if}
&(\beta+4)^2(\beta+4+4\sqrt{2})(\beta+4-4\sqrt{2})\geq 0,
\end{align*}
which is same is $\beta \geq 4\sqrt{2}-4=\beta_0.$
Therefore, for $p(z)\in\mathcal{H}_1,$ if \[1+\beta \frac{zp'(z)}{p(z)}\prec\sqrt{1+z} \ (\beta\geq\beta_0), \] we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} As in \cite{MR2917253}, Theorem 2.4, we get the following. \begin{theorem}
Let $\beta_0=4(\sqrt{2}-1)\approx1.65$ and $f\in\mathcal{A}.$
\begin{enumerate}
\item If $f$ satisfies the subordination \[ 1+\beta\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)\prec \sqrt{1+z}\ \ (\beta\geq\beta_0), \]then $f\in\mathcal{SL}.$
\item If $1+\beta zf''(z)/f'(z)\prec\sqrt{1+z}\ (\beta\geq\beta_0),$ then $f'(z)\prec\sqrt{1+z}.$
\item If $f$ satisfies the subordination \[ 1+\beta\left(\frac{(zf(z))''}{f'(z)}-\frac{2zf'(z)}{f(z)}\right)\prec\sqrt{1+z}\ \ (\beta\geq\beta_0), \] then $z^2f'(z)/f^2(z)\prec\sqrt{1+z}.$
\end{enumerate} \end{theorem}
\begin{lemma}\label{lem-1-2}
Let $p$ be analytic function on $\mathbb{D}$ and $p(0)=1.$ Let $\beta_0=4\sqrt{2}(\sqrt{2}-1)\approx2.34.$ If \[ 1+\beta \frac{zp'(z)}{p^2(z)}\prec\sqrt{1+z} \ (\beta\geq\beta_0), \] then \[ p(z)\prec \sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1\,, \operatorname{Re} w >0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times \mathbb{C}\times \mathbb{D} \to \mathbb{C}$ be defined by $\psi(a,b;z)=1+\beta b/a^2.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*} \psi(r,s;z)&=1+\beta\frac{me^{i\theta}}{4\sqrt{2}\cos^{3/2}2\theta}
\intertext{so that}
|\psi(r,s,t;z)^2-1|^2
&=\frac{\beta^4m^4}{1024}\sec^6 2\theta+\frac{\beta^2m^2}{8}\sec^3 2\theta+\frac{\beta^3m^3}{64}\sec^4{2\theta}\sqrt{\sec2\theta+1}\\
&\geq\frac{\beta^4m^4}{1024}+\frac{\beta^2m^2}{8}+\frac{\beta^3m^3}{32\sqrt{2}}\geq\frac{\beta^4}{1024}+\frac{\beta^2}{8}+\frac{\beta^3}{32\sqrt{2}}.
\end{align*}
The last term is greater than or equal to 1 if
\begin{align*}
&(\beta+4\sqrt{2})^2(\beta-4\sqrt{2}(\sqrt{2}-1))(\beta+4\sqrt{2}(\sqrt{2}+1))\geq 0
\intertext{equivalently}
&\beta \geq 4\sqrt{2}(\sqrt{2}-1)=\beta_0.
\end{align*}
Thus, for $\beta\geq \beta_0,$ we have $\psi\in\Psi[\mathcal{L}].$
Therefore, for $p(z)\in\mathcal{H}_1,$ if \[1+\beta \frac{zp'(z)}{p^2(z)}\prec\sqrt{1+z} \ (\beta\geq\beta_0), \] we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} By taking $p(z)=\dfrac{zf'(z)}{f(z)}$ as in \cite{MR2917253}, we obtain the following. \begin{theorem}
Let $\beta_0=4\sqrt{2}(\sqrt{2}-1)\approx2.34$ and $f\in\mathcal{A}.$ If $f$ satisfies the subordination \[ 1-\beta+\beta\left(\frac{1+zf''(z)/f'(z)}{zf'(z)/f(z)}\right)\prec\sqrt{1+z}\ (\beta\geq\beta_0), \] then $f\in\mathcal{SL}.$ \end{theorem} Kumar \emph{et al.} introduced that for every $\beta>0,\ p(z)\prec\sqrt{1+z}$ whenever $p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\ (n=0,1,2).$ Using admissibility conditions \eqref{adm for q}, alternate proofs to the mentioned results are discussed below. \begin{lemma}\label{lem-4-0}
Let $\beta>0$ and $p$ be analytic in $\mathbb{D}$ and $p(0)=1$ such that \[p(z)+\beta zp'(z)\prec\sqrt{1+z}, \] then \[p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1\,, \operatorname{Re} w >0\}.$ Let $\psi:\mathbb{C}^2\times \mathbb{D} \to \mathbb{C}$ be defined by $\psi(a,b;z)=a+\beta b.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=\sqrt{2\cos2\theta}e^{i\theta}+\beta\frac{me^{3i\theta}}{2\sqrt{2\cos2\theta}}
\intertext{so that}
|\psi(r,s;z)^2-1|^2&=1+2\beta m+\frac{5\beta^2 m^2}{4}+\frac{\beta^3 m^3}{4}+\frac{\beta^4 m^4}{64}\sec^2 2\theta\\
&\geq 1+2\beta m+\frac{5\beta^2 m^2}{4}+\frac{\beta^3 m^3}{4}+\frac{\beta^4 m^4}{64}\\
&\geq 1+2\beta+\frac{5\beta^2}{4}+\frac{\beta^3}{4}+\frac{\beta^4}{64}>1.
\end{align*}
Thus $\psi\in\Psi[\mathcal{L}].$ Therefore, for $p(z)\in\mathcal{H}_1,$ if \[p(z)+\beta zp'(z)\prec\sqrt{1+z} \ (\beta>0), \] we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} Taking $p(z)=zf'(z)/f(z)$ and $p(z)=f'(z),$ we get the following. \begin{theorem}
Let $\beta>0$ and $f$ be a function in $\mathcal{A}.$
\begin{enumerate}
\item If $f$ satisfies the subordination \[\frac{zf'(z)}{f(z)}+\beta\frac{zf'(z)}{f(z)}\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)\prec \sqrt{1+z}, \] then $f\in\mathcal{SL}.$
\item If $f'(z)+\beta zf''(z)\prec\sqrt{1+z},$ then $f'(z)\prec\sqrt{1+z}.$
\end{enumerate} \end{theorem}
\begin{lemma}\label{lem-4-1}
Let $\beta>0$ and $p$ be analytic in $\mathbb{D}$ and $p(0)=1$ such that \[p(z)+\frac{\beta zp'(z)}{p(z)}\prec\sqrt{1+z}, \] then \[p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1\,, \operatorname{Re} w >0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times \mathbb{D} \to \mathbb{C}$ be defined by $\psi(a,b;z)=a+\beta b/a.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=\sqrt{2\cos2\theta}e^{i\theta}+\beta\frac{me^{2i\theta}}{4\cos2\theta}
\intertext{so that}
|\psi(r,s,t;z)^2-1|^2
&=1+\frac{\beta^4m^4}{256}\sec^4 2\theta+\frac{\beta^2m^2}{8}\sec^2 2\theta+\frac{\beta^2m^2}{2}\sec2\theta\\
&\quad{}+\beta m\sqrt{\sec2\theta+1}+\frac{\beta^3m^3}{16}\sqrt{\sec2\theta+1}\sec^2\theta\\
&\geq 1+\sqrt{2}\beta m+\frac{5\beta^2m^2}{8}+\frac{\beta^3m^3}{8\sqrt{2}}+\frac{\beta^4m^4}{256}\\
&\geq 1+\sqrt{2}\beta+\frac{5\beta^2}{8} +\frac{\beta^3}{8\sqrt{2}}+\frac{\beta^4}{256}> 1.
\end{align*}
Thus, $\psi\in\Psi[\mathcal{L}].$ Therefore, for $p(z)\in\mathcal{H}_1,$ if \[p(z)+\beta \frac{zp'(z)}{p(z)}\prec\sqrt{1+z} \ (\beta>0), \] we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} For $p(z)=zf'(z)/f(z)$ and $p(z)=z^2f'(z)/f^2(z),$ we have \begin{theorem}
Let $\beta>0$ and $f$ be a function in $\mathcal{A}.$
\begin{enumerate}
\item If $f$ satisfies the subordination \[\frac{zf'(z)}{f(z)}+\beta\left(1+\frac{zf''(z)}{f'(z)}-\frac{zf'(z)}{f(z)}\right)\prec \sqrt{1+z}, \] then $f\in\mathcal{SL}.$
\item If $f$ satisfies the subordination \[ \frac{z^2f'(z)}{f^2(z)}+\beta\left(\frac{(zf(z))''}{f'(z)}-\frac{2zf'(z)}{f(z)}\right)\prec\sqrt{1+z}, \] then $z^2f'(z)/f^2(z)\prec\sqrt{1+z}.$
\end{enumerate} \end{theorem}
\begin{lemma}\label{lem-4-2}
Let $\beta>0$ and $p$ be analytic in $\mathbb{D}$ and $p(0)=1$ such that \[p(z)+\frac{\beta zp'(z)}{p^2(z)}\prec\sqrt{1+z}, \] then \[p(z)\prec\sqrt{1+z}. \] \end{lemma} \begin{proof}
Let $\beta>0.$ Let $\Delta=\{w:|w^2-1|<1\,, \operatorname{Re} w >0\}.$ Let $\psi:(\mathbb{C}\setminus\{0\})\times\mathbb{C}\times \mathbb{D} \to \mathbb{C}$ be defined by $\psi(a,b;z)=a+\beta b/a^2.$
For $\psi$ to be in $\Psi[\mathcal{L}],$ we must have $\psi(r,s;z)\not\in\Delta$ for
$z\in\mathbb{D}.$ Then, $\psi(r,s;z)$ is given by
\begin{align*}
\psi(r,s;z)&=\sqrt{2\cos2\theta}e^{i\theta}+\beta\frac{me^{i\theta}}{4\sqrt{2}\cos^{3/2}2\theta},
\intertext{so that}
|\psi(r,s;z)^2-1|^2&=1+\beta m+\frac{5\beta^2m^2}{16}\sec^2 2\theta+\frac{\beta^3 m^3}{32}\sec^4 2\theta+\frac{\beta^4m^4}{1024}\sec^6 2\theta\\
&\geq 1+\beta m+\frac{5\beta^2m^2}{16}+\frac{\beta^3 m^3}{32}+\frac{\beta^4m^4}{1024}\\
&\geq 1+\beta +\frac{5\beta^2}{16}+\frac{\beta^3}{32}+\frac{\beta^4}{1024}>1.
\end{align*} Thus, $\psi\in\Psi[\mathcal{L}].$ Therefore, for $p(z)\in\mathcal{H}_1,$ if \[p(z)+\beta \frac{zp'(z)}{p^2(z)}\prec\sqrt{1+z} \ (\beta>0), \] we have \[ p(z)\prec\sqrt{1+z}. \qedhere\] \end{proof} Taking $p(z)=\dfrac{zf'(z)}{f(z)},$ we obtain the following. \begin{theorem}
Let $\beta>0$ and $f$ be a function in $\mathcal{A}.$ If $f$ satisfies the subordination \[ \frac{zf'(z)}{f(z)}-\beta+\beta\left(\frac{1+zf''(z)/f'(z)}{zf'(z)/f(z)}\right)\prec\sqrt{1+z}, \] then $f\in\mathcal{SL}.$ \end{theorem}
\end{document} | arXiv |
\begin{document}
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\setcounter{page}{347} \author{Jan van Mill} \title[Loc. conn. continuum w/o convergent sequences]{A locally connected continuum without convergent sequences} \address{Faculty of Sciences\\ Division of Mathematics and Computer Science\\ Vrije Universiteit\\ De Boelelaan 1081\\ 1081 HV Amsterdam\\ The Netherlands} \email{[email protected]} \thanks{Reprinted from Topology and its Applications, in press, Jan van Mill, A locally connected continuum without convergent sequences, Copyright (2002), with permission from Elsevier Science \cite{vm}.} \thanks{Jan van Mill, {\em A locally connected continuum without convergent sequences}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp.~347--352, Topology Atlas, Toronto, 2002} \keywords{continuum, Fedorchuk space, convergent sequence, Continuum Hypothesis} \subjclass[2000]{54A20, 54F15} \begin{abstract} We answer a question of Juh\'asz by constructing under \CH\ an example of a locally connected continuum without nontrivial convergent sequences. \end{abstract} \maketitle
\section{Introduction}
During the Ninth Prague Topological Symposium, Juh\'asz asked whe\-ther there is a locally connected continuum without nontrivial convergent sequences. This question arose naturally in his investigation in~\cite{juhasz2001} with Gerlits, Soukup, and Szentmikl\'ossy on characterizing continuity in terms of the preservation of compactness and connectedness. The aim of this note is to answer this question in the affirmative under the Continuum Hypothesis (abbreviated: \CH).
Fedorchuk~\cite{vitaly77} constructed a consistent example of a compact space of cardinality $\cont$ containing no nontrivial convergent sequences. See also van Douwen and Fleissner~\cite{metfleissner} for a somewhat simpler construction under the Definable Forcing Axiom. These constructions yield zero-dimensional spaces. As a consequence, our construction has to be somewhat different. As in~\cite{vitaly77} and~\cite{metfleissner}, we `kill' all possible nontrivial convergent sequences in a transfinite process of length $\omega_1$. However, our `killing' is done in the Hilbert cube $Q=\prod_{n=1}^\infty [-1,1]_n$ instead of the Cantor set.
For all undefined notions, see \cite{engelking:gentop} and \cite{vm:book}.
\section{The Hilbert cube}\label{Hilbert}
A \emph{Hilbert cube} is a space homeomorphic to $Q$. Let $M^Q$ denote an arbitrary Hilbert cube.
A closed subset $A$ of $M^Q$ is a \emph{$Z$-set} if for every $\varepsilon>0$ there is a continuous function $f\from M^Q\to M^Q\setminus A$ which moves the points less than $\varepsilon$. It is clear that a closed subset of a $Z$-set is a $Z$-set. We list some other important properties of $Z$-sets. \begin{enumerate} \item[(1)] Every singleton subset of $M^Q$ is a $Z$-set. \item[(2)] A countable union of $Z$-sets is a $Z$-set provided it is closed. \item[(3)] A homeomorphism between $Z$-sets can be extended to a homeomorphism of $M^Q$. \item[(4)] If $X$ is compact and $f\from X\to M^Q$ is continuous then $f$ can be approximated arbitrarily closely by an imbedding whose range is a $Z$-set. \end{enumerate} See~\cite[Chapter~6]{vm:book} for details.
Observe that by (1) and (2), every nontrivial convergent sequence with its limit is a $Z$-set in $M^Q$.
A \emph{near homeomorphism} between compacta $X$ and $Y$ is a continuous surjection $f\from X\to Y$ which can be approximated arbitrarily closely by homemorphisms. This means that for every $\varepsilon > 0$ there is a homeomorphism $g\from X\to Y$ such that for every $x\in X$ we have that the distance between $f(x)$ and $g(x)$ is less than $\varepsilon$.
A closed subset $A\con M^Q$ has \emph{trivial shape} if it is contractible in any of its neighborhoods. A continuous surjection $f$ between Hilbert cubes $M^Q$ and $N^Q$ is \emph{cell-like} provided that $f^{-1}(q)$ has trivial shape for every $q\in N^Q$. The following fundamental result is due to Chapman~\cite{chapman:lectures} (see also~\cite[Theorem 7.5.7]{vm:book}).
\begin{enumerate} \item[(5)] Let $f\from M^Q\to N^Q$ be cell-like, where $M^Q$ and $N^Q$ are Hilbert cubes. Then $f$ is a near homeomorphism. \end{enumerate}
It is easy to see that if $f\from M^Q\to N^Q$ is a near homeomorphism between Hilbert cubes then $f$ is cell-like. So within the framework of Hilbert cubes the notions `near homeomorphism' and `cell-like' are equivalent.
A continuous surjection $f$ between Hilbert cubes $M^Q$ and $N^Q$ is called a $Z^*$-map provided that for every $Z$-set $A\con N^Q$ we have that $f^{-1}[A]$ is a $Z$-set in $M^Q$.
\begin{lemma}\label{eerstelemma} Let $M^Q$ and $N^Q$ be Hilbert cubes, and let $f\from M^Q\to N^Q$ be a continuous surjection for which there is a $Z$-set $A\con M^Q$ which contains all nondegenerate fibers of $f$. Then $f$ is a $Z^*$-map. \end{lemma}
\begin{proof} Let $B\con N^Q$ be an arbitrary $Z$-set, and put $B_0= B\setminus f[A]$. Write $B_0$ as $\bigcup_{n=1}^\infty E_n$, where each $E_n$ is compact. It follows from~\cite[Theorem 7.2.5]{vm:book} that for every $n$ the set $f^{-1}[E_n]$ is a $Z$-set in $M^Q$. As a consequence, $$ f^{-1}[B] \con A\cup\bigcup_{n=1}^\infty f^{-1}[E_n] $$ is a countable union of $Z$-sets and hence a $Z$-set by (2). \end{proof}
\begin{theorem}\label{eerstestelling} Let $(Q_n,f_n)_n$ be an inverse sequence of Hilbert cubes such that every $f_n$ is cell-like as well as a $Z^*$-map. Then \begin{enumerate} \item[(A)] $\varprojlim (Q_n,f_n)_n$ is a Hilbert cube. \item[(B)] The projection $f^{\infty}_n\from \varprojlim (Q_n,f_n)_n \to Q_n$ is a cell-like $Z^*$-map for every $n$. \end{enumerate} \end{theorem}
\begin{proof} It will be convenient to let $Q_\infty$ denote $\varprojlim (Q_n,f_n)_n$.
By (5), every $f_n$ is a near homeomorphism. Hence we get (A) from Brown's Approximation Theorem for inverse limits in \cite{brown:inverse}. It follows from~\cite[Theorem 6.7.4]{vm:book} that every projection $f^{\infty}_{n}\from Q_\infty \to Q_n$ is a near homeomorphism, hence is cell-like.
For every $n$ let $\varrho_n$ be an admissible metric for $Q_n$ which is bounded by $1$. The formula $$ \varrho(x,y)=\sum_{n=1}^\infty 2^{-n} \varrho_n(x_n,y_n) $$ defines an admissible metric for $Q_\infty$. With respect to this metric we have that $f^{\infty}_{n}$ is a $2^{-(n-1)}$-mapping (\cite[Lemma 6.7.3]{vm:book}).
For (B) it suffices to prove that $f^{\infty}_{1}$ is a $Z^*$-map. To this end, let $A\con Q_1$ be a $Z$-set, and let $\varepsilon > 0$. Pick $n\in{\mathbb N}$ so large that $2^{-(n-1)} < \varepsilon$. It follows that for every $x\in Q_n$ we have that the diameter of the fiber $(f^{\infty}_{n})^{-1}(x)$ is less than $\varepsilon$. An easy compactness argument gives us an open cover $\mathcal{U}$ of $Q_n$ such that for every $U\in \mathcal{U}$ we have that $$ \operatorname{diam} (f^{\infty}_{n})^{-1}[U] < \varepsilon.\eqno{(*)} $$ Let $\gamma > 0$ be a Lebesgue number for this cover (\cite[Lemma 1.1.1]{vm:book}). Since $f^{\infty}_{n}$ is a near homeomorphism, there is a homeomorphism $\varphi\from Q_\infty\to Q_n$ such that for every $x\in Q_\infty$ we have $$ \varrho_n(f^{\infty}_{n}(x),\varphi(x))<\leukfrac{1}{2}\gamma. $$ Observe that $A_n=(f^{n}_{1})^{-1}[A]$ is a $Z$-set in $Q_n$. There consequently is a continuous function $\xi\from Q_n\to Q_n\setminus A_n$ which moves the points less than $\leukfrac{1}{2}\gamma$. Now define $\eta\from Q_\infty \to Q_\infty$ by $$ \eta = \varphi^{-1} \circ \xi \circ f^{\infty}_{n}. $$ It is clear that $\eta [Q_\infty]$ misses $(f^{\infty}_{1})^{-1}[A]$. In order to check that $\eta$ is a `small' move, pick an arbitrary element $x\in Q_\infty$. By construction, $$ \varrho_n{\big (}x_n,\xi(x_n){\big )}< \leukfrac{1}{2}\gamma. $$ Since $\eta(x) = \varphi^{-1}{\big (} \xi(x_n) {\big )}$, clearly $$ \varrho_n{\big (}\eta(x)_n, \xi(x_n){\big )} < \leukfrac{1}{2}\gamma. $$ We conclude that $\varrho_n(\eta(x)_n,x_n)<\gamma$. Pick an element $U\in \mathcal{U}$ which contains both $\eta(x)_n$ and $x_n$. By $(*)$ it consequently follows that $\varrho(\eta(x),x)<\varepsilon$, which is as required. \end{proof}
\begin{theorem}\label{tweedestelling} If $(A_n)_n$ is a relatively discrete sequence of closed subsets of $Q$ such that $\CL{\bigcup_{n=1}^\infty A_n}$ is a $Z$-set then there are a Hilbert cube $M$ and a continuous surjection $f\from M\to Q$ such that \begin{enumerate} \item[(A)] $f$ is a cell-like $Z^*$-map. \item[(B)] The closures of the sets $\bigcup_{n=1}^\infty f^{-1}[A_{2n}]$ and $\bigcup_{n=0}^\infty f^{-1}[A_{2n+1}]$ are disjoint. \end{enumerate} \end{theorem}
\begin{proof} Consider the subspace $A=\CL{\bigcup_{n=1}^\infty A_n}$ of $Q$, and the `remainder' $R= A\setminus \bigcup_{n=1}^\infty A_n$. Observe that $R$ is compact since the sequence $(A_n)_n$ is relatively discrete. Let $T$ denote the product $A\times {\mathbb I}$; put $$ S = (R\times {\mathbb I}) \cup {\Big(} \bigcup_{n=1}^\infty A_{2n}\times\{0\} {\Big)} \cup {\Big(} \bigcup_{n=0}^\infty A_{2n+1}\times \{1\} {\Big)} . $$ Then $S$ is evidently a closed subspace of $T$. Let $\pi\from R\times{\mathbb I} \to R$ denote the projection. It is clear that the adjunction space (cf., \cite[Page~507]{vm:book:twee}) $S\cup_\pi (R\times{\mathbb I})$ is homeomorphic to $A$. By (4), any constant function $S\to Q$ can be approximated by an imbedding whose range is a $Z$-set. So we may assume without loss of generality that $S$ is a $Z$-subset of some Hilbert cube $M^Q$. Now consider the space $N = M^Q\cup_\pi (R\times{\mathbb I})$ with natural decomposition map $f$. It is clear that $f$ is cell-like, each non-degenerate fiber of $f$ being an arc (\cite[Corollary 7.1.2]{vm:book}). We will prove below that $N\approx Q$. Once we know that, we also get by Lemma~\ref{eerstelemma} that $f$ is a $Z^*$-map. Observe that the projection $\pi\from R\times {\mathbb I}\to R$ is a hereditary shape equivalence. So by a result of Kozlowski~\cite{kozlow:anr} (see also~\cite{ancel:ce}), it follows that $N$ is an $\mathsf{AR}$. Since $S$ is a $Z$-set in $M^Q$ it consequently follows from~\cite[Proposition 7.2.12]{vm:book} that $f[S]\approx A$ is a $Z$-set in $N$. But $N\setminus f[S]$ is obviously a $Q$-manifold, and consequently has the disjoint-cells property. But this implies that $N$ has the disjoint-cells property, i.e., $N\approx Q$ by Toru\'nczyk's topological characterization of $Q$ in~\cite{tor:hilbertcube} (see also~\cite[Corollary 7.8.4]{vm:book}). So we conclude that $f[S]\approx A$ is a $Z$-set in the Hilbert cube $N$. By (3) there is a homeomorphism of pairs $(Q,A)\approx (N,f[S])$. This homeomorphism may be chosen to be the `identity' on $A$. This shows that we are done by Lemma~\ref{eerstelemma} and the obvious fact that the sets $$ \bigcup_{n=1}^\infty A_{2n}\times \{0\},\quad \bigcup_{n=0}^\infty A_{2n+1}\times \{1\} $$ have disjoint closures in $M^Q$. \end{proof}
\section{The construction}
We will now construct our example under \CH. After the preparatory work in \S\ref{Hilbert}, the construction is very similar to known constructions in the literature (see e.g., Kunen~\cite{kunen:Lspace}).
Consider the `cube' $Q^{\omega_1}$. For every $1\le\alpha < \omega_1$ let $\{S_\xi^\alpha : \xi < \omega_1\}$ list all nontrivial convergent sequences in $Q^\alpha$ that do not contain their limits. For all $\alpha,\xi<\omega_1$ pick disjoint complementary infinite subsets $A_\xi^\alpha$ and $B_\xi^\alpha$ of $S_\xi^\alpha$.
We shall construct for $1\le\alpha \le \omega_1$ a closed subspace $M_\alpha\con Q^\alpha$. The space we are after will be $M_{\omega_1}$.
Let $\tau\from\omega_1\to\omega_1\times\omega_1$ be a surjection such that $\tau(\beta) =\orpr{\alpha}{\xi}$ implies $\alpha\le\beta$.
For $\alpha \le \beta\le\omega_1$ let $\pi^\beta_\alpha$ be the natural projection from $Q^\beta$ onto $Q^\alpha$. The following conditions will be satisfied: \begin{enumerate} \item[(A)] $M_\alpha\approx Q$ for every $1\le \alpha<\omega_1$, and if $\alpha \le \beta$ then $\pi^\beta_\alpha[M_\beta] = M_\alpha$.
\noindent We put $\rho^\beta_\alpha = \pi^\beta_\alpha\restriction M_\beta\from M_\beta\to M_\alpha$. \item[(B)] If $\alpha\le\beta$ then $\rho^\beta_\alpha\from M_\beta\to M_\alpha$ is a cell-like $Z^*$-map. \item[(C)] If $\beta<\omega_1$, $\tau(\beta)=\orpr{\alpha}{\xi}$, and $S^\alpha_\xi\con M_\alpha$ then $(\rho^{\beta+1}_\alpha)^{-1}[A^\alpha_\xi]$ and $(\rho^{\beta+1}_\alpha)^{-1}[B^\alpha_\xi]$ have disjoint closures in $M^{\beta+1}$. \end{enumerate} Observe that the construction is determined at all limit ordinals $\gamma$. By compactness and (A) we must have $$ M_\gamma = \{x\in Q^\gamma : (\forall \alpha < \gamma)(\pi^\gamma_\alpha(x)\in M_\alpha)\}. $$ Also, if $(\gamma_n)_n$ is any strictly increasing sequence of ordinals with $\gamma_n\nearrow\gamma$ then $M_\gamma$ is canonically homeomorphic to $$ \varprojlim {\big (}M_{\gamma_n},\rho^{\gamma_{n+1}}_{\gamma_{n}}{\big )}_n. $$ By Theorem~\ref{eerstestelling} this implies that $M_\gamma\approx Q$ and also that $\rho^\gamma_{\gamma_n}$ is a cell-like $Z^*$-map for every $n$. Since $\gamma_1$ can be \emph{any} ordinal smaller than $\gamma$, the same argument yields that $\rho^\gamma_\alpha$ is a cell-like $Z^*$-map for \emph{every} $\alpha < \gamma$. So in our construction we need only worry about successor steps.
Put $M_1 = Q^{\{0\}}$, and let $1\le\beta<\omega_1$ be arbitrary. We shall construct $M_{\beta+1}$ assuming that $M_\beta$ has been constructed. To this end, let $\tau(\beta)=\orpr{\alpha}{\xi}$. We make the obvious identification of $Q^{\beta+1}$ with $Q^\beta\times Q$. If $S^\alpha_\xi\not\con M_\alpha$ then there is nothing to do. We then fix any element $q\in Q$, and put $$ M_{\beta+1} = M_\beta \times \{q\}. $$ So assume that $S^\alpha_\xi\con M_\alpha$. By Theorem~\ref{tweedestelling} there exists a cell-like $Z^*$-map $f\from Q\to M_\beta$ such that $$ f^{-1}{\big [}(\rho^\beta_\alpha)^{-1}[A^\alpha_\xi]{\big ]}, \quad f^{-1}{\big [}(\rho^\beta_\alpha)^{-1}[B^\alpha_\xi]{\big ]} $$ have disjoint closures in $Q$. Put $$ M_{\beta+1} = \{\orpr{f(x)}{x}\in Q^{\beta}\times Q : x\in Q\}. $$ So $M_{\beta+1}$ is nothing but the graph of $f$. It is clear that $M_{\beta+1}$ is as required.
Now put $M = M_{\omega_1}$. Observe that $M$ is a locally connected continuum, being the inverse limit of an inverse system of locally continua with monotone surjective bonding maps (see e.g., \cite[6.3.16 and 6.1.28]{engelking:gentop}). Assume that $T$ is a nontrivial convergent sequence with its limit $x$ in $M$. Since $T\cup \{x\}$ is countable, there exists $\alpha < \omega_1$ such that $\rho^{\omega_1}_\beta\restriction (T\cup\{x\})$ is one-to-one and hence a homeomorphism for every $\beta \ge \alpha$. Pick $\xi < \omega_1$ such that $S^\alpha_\xi = \rho^{\omega_1}_\alpha[T]$, and $\beta \ge\alpha$ such that $\tau(\beta) = \orpr{\alpha}{\xi}$. Then $\rho^{\omega_1}_{\beta+1}[T\cup\{x\}]$ is a nontrivial convergent sequence with its limit in $M_{\beta+1}$ which is mapped by $\rho^{\beta+1}_\alpha$ onto $S^\alpha_\xi$ with its limit. But this is clearly in conflict with (C).
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January 2015, 14(1): 229-244. doi: 10.3934/cpaa.2015.14.229
Optimal matching problems with costs given by Finsler distances
J. M. Mazón 1, , Julio D. Rossi 2, and J. Toledo 1,
Departament d'Anàlisi Matemàtica, U. de València, Valencia, Spain, Spain
Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante
Received January 2014 Revised April 2014 Published September 2014
In this paper we deal with an optimal matching problem, that is, we want to transport two commodities (modeled by two measures that encode the spacial distribution of each commodity) to a given location, where they will match, minimizing the total transport cost that in our case is given by the sum of the two different Finsler distances that the two measures are transported. We perform a method to approximate the matching measure and the pair of Kantorovich potentials associated with this problem taking limit as $p\to \infty$ in a variational system of $p-$Laplacian type.
Keywords: MongeKantorovichs mass transport theory, $p$Laplacian systems., Optimal matching problem.
Mathematics Subject Classification: Primary: 49J20, 35J9.
Citation: J. M. Mazón, Julio D. Rossi, J. Toledo. Optimal matching problems with costs given by Finsler distances. Communications on Pure & Applied Analysis, 2015, 14 (1) : 229-244. doi: 10.3934/cpaa.2015.14.229
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Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675
Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1
Yuxiang Zhang, Shiwang Ma. Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 251-260. doi: 10.3934/dcdsb.2009.12.251
Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
J. M. Mazón Julio D. Rossi J. Toledo | CommonCrawl |
Solve the congruence $6n \equiv 7 \pmod{13}$, as a residue modulo 13. (That is, find the value of $n$ that satisfies the congruence such that $0\le n \le 12$.)
Note that $7 \equiv -6 \pmod{13}$, so we can write the given congruence as $6n \equiv -6 \pmod{13}$. Since 6 is relatively prime to 13, we can divide both sides by 6, to get $n \equiv -1 \equiv \boxed{12} \pmod{13}$. | Math Dataset |
Why is $\mathbb{Z}_4\times\mathbb{Z}_6 / \langle (0,2)\rangle$ isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_2$?
The subgroup generated by $\langle(0,2)\rangle$ has order $3$ and is given by $H=\{(0,2), (0,4), (0,6)\}$, so there must be $8$ elements in factor group.
In my book, it says that "$\mathbb{Z}_6$ factor is collapsed by a subgroup of order $3$, giving a factor group in the second factor of order $2$ isomorphic to $\mathbb{Z}_2$."
I don't understand this explanation. Can anyone help me?
abstract-algebra group-theory
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$\begingroup$ Any element of the quotient group is a coset whose representative can be taken to be (a,b) where b is 0 or 1, since we can always adjust an element in the full group by an element of the subgroup to get something into this form. I like the book's explanation. The quotient leaves the first factor alone and in the second factor we're taking Z_6/<2>, which is isomorphic to Z_2. $\endgroup$ – John Brevik Mar 14 '15 at 22:31
$\begingroup$ Is there reason why the identity element of $H$ is written as $(0,6)$ and not $(0,0)$? $\endgroup$ – Karl Mar 14 '15 at 22:56
$\begingroup$ other than my mistake, no $\endgroup$ – hho Mar 14 '15 at 23:08
Note that $\Bbb Z_4 \times \Bbb Z_6$ has $24$ elements, and $H = \{(0,0),(0,2),(0,4)\}$ has $3$, so $(\Bbb Z_4 \times \Bbb Z_6)/H$ has $8$ elements.
Note as well that $H = \{0\} \times \langle 2\rangle$, so it seems plausible that:
$(\Bbb Z_4 \times \Bbb Z_6)/H \cong (\Bbb Z_4/\{0\}) \times (\Bbb Z_6/\langle 2\rangle)$
But rather than prove the general theorem this is a special case of, let's just exhibit a surjective abelian group homomorphism:
$\phi: \Bbb Z_4 \times \Bbb Z_6 \to \Bbb Z_4 \times \Bbb Z_2$
with kernel $H$.
Specifically, let $\phi(a,b) = (a,b\text{ (mod }2))$, This is clearly onto, and we see at once that $H \subseteq \text{ker }\phi$.
On the other hand, if $\phi(a,b) = (0,0)$, we must have $a = 0$ (since $\phi$ is just the identity map on the first coordinate), and $b$ must be even, that is $b = 0,2,4$. This shows that $\text{ker }\phi \subseteq H$, and thus the two sets are equal.
So by the Fundamental Isomorphism Theorem, $(\Bbb Z_4 \times \Bbb Z_6)/H \cong \Bbb Z_4 \times \Bbb Z_2$.
A word about the explanation you were given: a homomorphism essentially "shrinks" its kernel to an identity. Given that all cosets of a subgroup are "the same size", the size of the kernel is "the shrinkage factor" (if the kernel had two elements, the size of the quotient group would be half the size of the original group). Note how $\phi$ in what I wrote above acts on each factor group of our direct product: it does nothing to $\Bbb Z_4$, and it identifies all the "even" elements of $\Bbb Z_6$-there are $3$ of these, so we get a cyclic subgroup of order $2$ (because $6/3 = 2$) in the quotient.
David WheelerDavid Wheeler
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Compute (Z4 * Z6) / <(0,2)>
How to find all elements of $\mathbb{Z}_{4} \times\mathbb{Z}_{4}/\langle(1,1)\rangle$?
Calculate the Factor Group $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$
Compute this factor group: $\mathbb Z_4\times\mathbb Z_6/\langle (0,2) \rangle$
Classify $\mathbb{Z}\times\mathbb{Z}/\langle(2,2)\rangle$
$\mathbb{Z} \times \mathbb{Z}/\langle (1,2)\rangle$ is isomorphic to $\mathbb{Z}$?
Finding subgroups of $G=\displaystyle\normalsize{\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}}\LARGE_{/}\large_{\langle(1,0)\rangle}$
$(\mathbb{Z}_2\times \mathbb{Z}_4)/\langle (0,1)\rangle$ and $(\mathbb{Z}_2\times \mathbb{Z}_4)/\langle (1,2) \rangle$ isomorphisms
Order of coset is greater than number of cosets in $\mathbb{Z}_2\times\mathbb{Z}_4/\langle(0,1)\rangle$
What's the meaning of factors 'collapsing' in quotients like $\mathbb{Z}_4\times\mathbb{Z}_6/\langle(0,1)\rangle$?
Prove $\langle k \rangle \big/ \langle n\rangle$ is isomorphic to $\mathbb{Z}_{(n/k)}$. | CommonCrawl |
\begin{document}
\title{Observing a Quantum Measurement} \author{Jay Lawrence} \affiliation{Department of Physics and Astronomy, Dartmouth
College, Hanover, NH 03755, USA} \affiliation{The James Franck Institute, University of Chicago,
Chicago, IL 60637} \date{revised \today}
\begin{abstract}
With the example of a Stern-Gerlach measurement on a spin-1/2 atom, we show that a superposition of both paths may be observed compatibly with properties attributed to state collapse - for example, the {\it singleness} (or mutual exclusivity) of outcomes. This is done by inserting a quantum two-state system (an ancilla) in each path, capable of responding to the passage of the atom, and thus acting as a virtual detector. We then consider real measurements on the compound system of atomic spin and two ancillae. Nondestructive measurements of a set of compatible joint observables can be performed, one for a superposition and others for collapse properties. A novel perspective is given as to why, within unitary quantum theory, ordinary measurements are blind to such superpositions. Implications for the theory of measurement are discussed. \end{abstract}
\pacs{03.67-a, 03.65.Ta, 03.65.Ud}
\maketitle
\section{Introduction}
Opinions differ on whether or not there is a quantum measurement problem, and if so, exactly what it is \cite{Schlossbook,survey}. A more focused question regards the collapse of the state vector in projective measurements, as formalized by von Neumann \cite{VN55}. By collapse we mean the observed process described by one of the textbook postulates, which states (in its simplest form) \cite{preMGM,MGM}:
\noindent{{\it Measurement Postulate: A measurement of the observable A yields one of its eigenvalues, $a_n$, and, in an ideal measurement \cite{Ideal,Pokorny}, places the measured object in the corresponding eigenstate, $\phi_n$ (where $A \phi_n = a_n\phi_n$)}}.
\noindent Included with this, or stated as a separate postulate, is the Born rule probability of this outcome, $|\langle \phi_n | \psi \rangle |^2$, where $\psi$ is the initial state of the object being measured. All interpretations must agree on the above as a statement of fact, but they disagree on its status - {\it i.e.}, is it independent of the other postulates, which stipulate unitary evolution in the appropriate Hilbert space, or is it derivable from them?
Regarding attitudes on this more focused issue, it seems reasonable to identify a small number of broad categories. Here is a grouping into three: The most conservative takes collapse as axiomatic - that is, it cannot be derived from the other axioms - suggesting that the collapse process itself is not subject to quantum analysis \cite{Bohr1958,Weinberg1}. This position is consistent with most textbooks written over more than the last half century, which list collapse among the axioms. It is intended to include those who apply quantum theory according to these textbook axioms, without (however) adopting any particular interpretation philosophically. It also includes the epistemic and information-based approaches \cite{Cabellomap},
whose intellectual roots extend back to the Copenhagen interpretation \cite{Bohr1963,Wigner1961}. We refer to this general position as Standard Quantum Theory (SQT), interpreted broadly.
A contrasting position is that the collapse phenomenon, as observed, is in fact derived from the other axioms, following unitary evolution of an appropriate closed system which includes the apparatus and the relevant environment \cite{relevant,Zurek.81}, as well as the object of study. And indeed unitary evolution describes what we see, but it also describes what we do {\it not} see - namely, that all branches of the state vector (representing all possible measurement outcomes) survive the measurement process. This is nevertheless consistent because it also predicts that an observer can be aware of only one such outcome \cite{observer}. We shall refer to this position as unitary quantum theory (UQT). It includes the Many Worlds Interpretation \cite{Weinberg2,Everett,DeWitt1973}, which asserts that the unobservable branches are just as real as the branch we experience, but it is broader. It includes orthodox decoherence theory \cite{Zurek2003,Schloss.19}, whose practitioners represent a variety of interpretations \cite{Joos.et.al,Zurek2009}, and other operational approaches which assert independence from interpretations \cite{MGM,CerfAdami}, while assuming unitarity.
A third position holds that the unobserved branches are removed from the theory by a mechanism of yet unknown origin, which takes effect in sufficiently large systems, and which is, in principle, subject to quantum analysis. The mechanism is
represented by adding a nonlinear stochastic term to the Hamiltonian, whose effect is to remove all but a single branch \cite{GRW, Pearle,Bassi1,JZchapter8}. This approach, in effect, replaces the collapse postulate with an expansion of the dynamics postulate beyond its otherwise unitary and deterministic character. This has consequences, which are measurable in principle, but to date undetected. Predicted effects are difficult to separate from decoherence and other random influences. There are proposals to utilize molecular interferometry and optomechanical phenomena, as well as particle diffusion \cite{diffusion}, and it is hoped that over the next decade or two, definitive tests will be possible \cite{Bassi2}. We refer to this general position as objective collapse theory (OCT). It has fewer adherents than the other two \cite{survey}, but it provides an important alternative.
In this paper we will make several points about the measurement process, mostly interpretation independent, although the blindness of ordinary measurements to superpositions calls for specific justification within the UQT approach. For all points it will be useful to distinguish two stages of the measurement process. First comes the reversible premeasurement stage, where the object of interest becomes entangled (unitarily) with an ancillary system in the apparatus (in our case, two paths). Second is the detection stage, where the ancillary system transfers its entanglement to the detector system (in our case, two detectors), which then act irreversibly and record a result. Perhaps surprisingly, two signature collapse properties are established in the premeasurement stage, as properties of the object/ancillary system, and survive through the detection stage as correlations among the object and the two detectors. These are implicit in the postulate stated above: (i) {\it singleness} (or mutual exclusivity of outcomes $a_n$); and (ii) {\it projection} (the correlations between detector readings and the post-measurement spin state). Note that the singleness of outcomes implies randomness. Randomness is {\it not} a property of the premeasurement state - it only shows up at the detection stage as a result of local (``which path'') measurement, which breaks the entanglement while preserving the correlations. There is a {\it third} property of the premeasurement state ({\it not} a collapse property!), namely (iii) {\it superposition} (the state vector is a superposition of distinct collapse scenarios). In the ancilla model, this property can be detected, but in an ordinary apparatus it cannot be, so that its survival in the state vector is open to interpretation.
The ancilla model expands the premeasurement stage by adding a physical realization (qubits) to the ancillary system. The three properties are represented by Hermitian operators in the Hilbert space of the object/ancilla system. The operators commute, and all three properties are observable at the detection stage. In an ordinary apparatus without the ancillae, we will show that all three properties are again present in the premeasurement stage, but that only the two collapse properties are observable at the detection stage.
Interpretations differ on the reason behind this blindness of an ordinary apparatus to superpositions. In two of the approaches outlined above, only one branch of the state vector survives the detection stage - in SQT this is axiomatic; in OCT it is by construction of the model interaction. In UQT, on the other hand, the superposition extends to the detectors and persists through the detection stage. We will offer a physically intuitive explanation why it is nonetheless undetectable, prompted by comparison with the ancilla model. This will provide a useful perspective on a more conventional explanation in decoherence theory (Sec. III and Appendix A).
In the next section we introduce the Stern-Gerlach measurement model with ancilla qubits as virtual detectors, and we then show how this atom/ancilla system may be ``observed'' in a real experiment. This observation demonstrates the compatibility of the two collapse properties with the superposition property for this system. In Section III, we compare the analogous measurements made with two ordinary detectors. We show that the collapse properties are identical to those of the ancilla model, while the superposition that persists in the UQT approach is now undetectable. We discuss the reason for this blindness and compare with the decoherence perspective.
Results are summarized in Sec. IV.
\section{ A Model Measurement with Virtual Detectors}
\begin{figure}
\caption{ Stern-Gerlach interferometer showing the evolution described by Eqs. \ref{state1} - \ref{state4}.}
\label{fig1}
\end{figure}
Consider the compound system consisting of a spin-1/2 atom and two quantum two-state systems (ancillae, $A_\uparrow$ and $A_\downarrow$) serving as virtual detectors in the Stern-Gerlach interferometer pictured in Fig. 1 \cite{SG.interf.19}. Each ancilla interacts locally with the atom, and it makes a transition from its 0 to its 1 state if and only if the atom passes through it. The interaction is spin-independent, preserving the spin state of the atom on its path. We assume that the process is reversible, so that the ancilla by itself does not perform a measurement - hence we call it a virtual detector.
Let us trace the evolution of entanglement as the atom passes through the device from points 1 - 4. The atom enters the picture at time $t_1$ with the spatial wavefunction $\phi(\vec{r},t_1)$, in an arbitrary pure spin state, $(\alpha,\beta)$, \begin{equation}
\ket{\psi(t_1)} = \phi(\vec{r},t_1) \bigg(\alpha \ket{\uparrow}_s +\beta \ket{\downarrow}_s
\bigg) \ket{0}_{A\uparrow} \ket{0}_{A\downarrow}, \label{state1} \end{equation} with ancillae in their 0 states. By the time $t_2$, the Stern-Gerlach magnetic field gradient has separated the spin components into two ideally nonoverlapping paths,
\begin{equation}
\ket{\psi(t_2)} = \bigg( \alpha \phi_{\uparrow}(\vec{r},t_2) \ket{\uparrow}_s
+ \beta \phi_{\downarrow} (\vec{r},t_2) \ket{\downarrow}_s \bigg) \ket{0}_{A\uparrow}
\ket{0}_{A\downarrow}, \label{state2} \end{equation} entangling the atom's spatial and spin degrees of freedom. By $t_3$, the atom has passed through an ancilla, $A_{\uparrow}$ {\it or} $A_{\downarrow}$, conditioned on its path, so that \begin{equation}
\ket{\psi(t_3)} = \alpha \phi_{\uparrow}(\vec{r},t_3) \ket{\uparrow}_s \ket{1}_{A\uparrow}
\ket{0}_{A\downarrow} + \beta \phi_{\downarrow}(\vec{r},t_3) \ket{\downarrow}_s
\ket{0}_{A\uparrow} \ket{1}_{A\downarrow}. \label{state3} \end{equation} Finally, a reversed magnetic field gradient brings the two paths back together at $t_4$. Assuming that there is no net phase difference between the paths, the result is \begin{equation}
\ket{\psi(t_4)} = \phi (\vec{r},t_4) \bigg( \alpha \ket{\uparrow}_s
\ket{1}_{A\uparrow} \ket{0}_{A\downarrow} + \beta \ket{\downarrow}_s
\ket{0}_{A\uparrow} \ket{1}_{A\downarrow} \bigg). \label{state4} \end{equation} The last step allows us to ignore the spatial part and study the remaining entanglement between the spin and the two ancillae. We further simplify by setting $\alpha = \beta = 1/\sqrt{2}$, leaving the three-qubit state, \begin{equation}
\ket{\psi(t_4)} \rightarrow \frac{1}{\sqrt{2}} \bigg(\ket{110} + \ket{001} \bigg), \label{ESG4} \end{equation} where the spin states ($\uparrow,\downarrow$) are relabeled as ($1,0$), and the ordering of the indices identifies with (spin,$A_ {\uparrow},A_{\downarrow}$).
This is a Greenberger-Horne-Zeilinger (GHZ) state \cite{GHZ,Mermin.90}, an entangled state of three qubits. It was first realized experimentally in 1999 (\cite{Bouwmeester}) as a polarization state of three photons. Analogous states (and their generalizations to more than three particles) have been produced and documented in other systems - for example, trapped ions \cite{Monz}, superconducting circuits \cite{Schoelkopf, Song}, and Rydberg atoms \cite{Rydberg}. The original goal was to demonstrate non-locality \cite{nonlocality}; more practical goals involve quantum error correction \cite{Shor} and quantum communication \cite{QSS}, and in general the manipulation of entanglement.
Let us discuss the observables that characterize the state, and then the question of how to measure them. The three-qubit system lives in a Hilbert space of dimension eight, and $\ket{\psi(t_4)}$ is an eigenstate of three tensor product operators, whose eigenvalues determine it completely. The choice is not unique \cite{CSCO}; the most revealing in the present context is $ZZI$, $ZIZ$, and $XXX$. Recalling the definitions of the individual Pauli matrices $Z$, $X$, and $Y$, as given in Table I, one may confirm that $\ket{\psi(t_4)}$ is indeed the (unique) simultaneous eigenstate of the three tensor products with the eigenvalues quoted in Table II.
Each observable may be characterized by a statement about its physical meaning, with eigenvalue ($\pm1$) giving the truth value \cite{Zeilinger}. With GHZ entanglement, all such statements concern either two-particle or three-particle correlations, and none concerns a property of an individual particle. The combination of statements appearing in Table II is an example. These statements may appear contradictory, because the product of the first two tells us that $IZZ = -1$, whose clear physical meaning is that the atom can be found on one and only one path (the ``singleness'' property, or mutual exclusivity). On the other hand, the definiteness of $XXX$ implies that the state vector (\ref{ESG4}) is a superposition of two classically inconsistent scenarios. We will explain in detail why the singleness and superposition statements are {\it not} contradictory. But first we must describe the measurements, made with real (irreversible) detecting devices \cite{Wheeler78}. We will describe two modes (called local and joint) which differ in the acquisition of local information.
\begin{table} \caption{Multiplication table for Pauli matrices acting on kets ($\ket{1},\ket{0}$)}
\begin{tabular}{|c|cc|} \hline \ Pauli \ & \ $\ket{1}$ \ & \ $\ket{0}$ \ \\ \hline $I$ & \ $\ket{1}$ \ & \ $\ket{0}$ \ \\ $Z$ & \ $\ket{1}$ \ & $-\ket{0}$\ \\ $X$ & \ $\ket{0}$ \ & \ $\ket{1}$ \ \\ $Y$ & $ \ i\ket{0}$ \ & \ $-i\ket{1}$ \ \\ \hline \end{tabular}
\end{table}
\begin{table} \caption{Tensor product operators, eigenvalues and physical interpretations.}
\begin{tabular}{|c|c|c|} \hline \ (spin,$A\uparrow$,$A\downarrow$) \ & \ eigenvalue \ & \ meaning \ \\ \hline $ZZI$ & \ $+1$ \ & \ atom takes upper path if and only if spin is up \ \\ $ZIZ$ & \ $-1$ \ & \ atom takes lower path if and only if spin is down \ \\ $XXX$ & \ $+1$ \ & \ state is invariant under simultaneous flips ($0 \leftrightarrow 1$) \ \\ \hline \end{tabular}
\end{table}
\centerline{\bf A. Local measurements}
\begin{figure}
\caption{ Local measurements of $Z$ factors (a), and $X$ factors (b). In both parts, $M$ denotes standard readout devices, and $M_z$ ($M_x$) denote the measurement of the atom's spin component $Z_s$ ($X_s$) with a ``downstream'' Stern-Gerlach apparatus. Wavy lines denote the atom's wave packet at $t_4$}
\label{fig2}
\end{figure}
We begin with the simpler and more typical mode of GHZ experiments \cite{Bouwmeester}, in which one measures the local factors and multiplies the results together to obtain the eigenvalue of the desired joint observable. First consider the local $Z$ factors, each of which can take the value 1 or $-1$. We measure the ancilla factors, $Z_{A\uparrow}$ and $Z_{A\downarrow}$, with a standard readout device \cite{readout} for each ancilla qubit. We measure the atomic spin component, $Z_s$, with another ``downstream'' Stern-Gerlach device, similarly oriented, with a single-atom detector placed in the $\uparrow$ path to register the arrival (or not) of the atom and thus record the value of $Z_s$. The results are as follows: The outcome of each local measurement is random because no local observable has $\Psi(t_4) \sim \ket{110} + \ket{001}$ as an eigenstate \cite{randomness}, but the product of {\it any two} $Z$ factors is definite, as shown in the top two panels in Table III). These are collapse properties, and they are properties of the state $\Psi(t_4)$ prior to measurement. They characterize the observed collapse phenomenon that we would see in an ordinary measurement without the ancillae. We emphasize that $IZZ = -1$ represents the {\it singleness}, or {\it mutual exclusivity} of measurement outcomes, while $ZZI = 1$ and $ZIZ = -1$ represent the {\it projection property} - the perfect correlation between detector readouts and the (remeasured) value of the atomic spin downstream.
It is notable that the two-way correlations above persist in the face of local measurements ($ZII$, $IZI$, and $IIZ$), whose random outcomes indicate that the measurement changed the state of the system. The specific ``which path'' information thus obtained is {\it not} a property of $\Psi(t_4)$; it enters only at the detection stage. Its randomness, a feature shared with ordinary measurements, is a necessary consequence of the singleness property.
\begin{table} \caption{Observables characterizing the premeasurement state, and the properties they represent. These properties and their measurements are described in the text.}
\begin{tabular}{|c|c|} \hline \ observable \ & \ property \ \\ \hline $IZZ \rightarrow -1$ \ & \ singleness of outcome \ \\ $ZZI \rightarrow 1$ and $ZIZ \rightarrow -1$ \ & \ projection property \ \\ $XXX \rightarrow 1$ and above \ & \ superposition of potential outcomes \ \\ \hline \end{tabular}
\end{table}
Now consider the measurements of the individual $X$ factors. Clearly, to measure $X_s$, we simply reorient the ``downstream'' Stern-Gerlach system. To measure $X_{A\uparrow}$ and $X_{A\downarrow}$, noting that the readout devices are keyed to the ancillas' $Z$-bases, we apply a Hadamard transformation ($H$) \cite{Hadamard} to each ancilla before the readout. The readout value ($\pm 1$) then indicates in which linear combination of $Z$-basis states ($\ket{1} \pm \ket{0}$) each ancilla has been ``found.'' The results of these $X$-measurements are the following: The individual outcomes are again random, but the product of all three is always +1. The definiteness of the product indicates a coherent superposition, and its positivity confirms the sign in \Eq{ESG4}. Note that this measurement distinguishes the pure entangled state from a mixed state of the same two components. The mixed state would duplicate the $Z$-measurement results but reveal itself through random outcomes for the product $XXX$.
To better fill out the picture of how the same state, $\Psi(t_4)$, can accommodate both a collapse scenario and a superposition, consider the local measurement of the spin component $X_s$, which ``finds'' the atom to have taken both paths. The outcome is random: When it is ($+1$), then the {\it product} $X_{A\uparrow} X_{A\downarrow}$ must also be ($+1$). Knowing the compatible product $Z_{A\uparrow} Z_{A\downarrow} = -1$ (Table III), it is easy to see that the state of the two ancillae is the Bell state, \begin{equation}
\ket{\Psi}_{AA} \rightarrow {1\over \sqrt{2}} \big( \ket{10} + \ket{01} \big)_{AA}. \label{ESG5} \end{equation} On the other hand, when $X_s$ is ($-1$), then $X_{A\uparrow} X_{A\downarrow}$ must also be ($-1$), and this, combined again with $Z_{A\uparrow} Z_{A\downarrow} = -1$, indicates another Bell state, \begin{equation}
\ket{\Psi}_{AA} \rightarrow {1\over \sqrt{2}} \big( \ket{10} - \ket{01} \big)_{AA}. \label{ESG6} \end{equation} Each of these Bell states represents perfect anticorrelations between the ancilla's $Z$-values, which are individually random. This shows how the ancillae are able to respect the singleness property, while (in correlation with the atomic spin) enabling the coherent superposition of both paths.
It may be worth noting that the observation of either superposition state is just another form of apparent collapse - instead of finding ``which path,'' one is finding ``which superposition of paths.''
So far, we have appealed to local measurements to verify the joint eigenvalues which define $\Psi(t_4)$. But since these measurements change the state, each new set of them requires another identically prepared state of the system. One can circumvent this need and achieve a more direct demonstration of compatibility, as follows.
\centerline{\bf B. Joint measurements and compatibility}
\begin{figure}
\caption{ (a) Joint measurement of $IZZ$ using ancilla $B$, and (b) of $XXX$ using $B$ and $C$.}
\label{fig3}
\end{figure}
One can measure compatible joint observables nondestructively by refusing to acquire local information. For the singleness property ($IZZ$), one couples $A_{\uparrow}$ and $A_{\downarrow}$ to an added ancilla qubit ($B$) through CNOT gates (Fig. 3a), so that the state of $B$ changes if and only if $Z_{A\uparrow}$ and $Z_{A\downarrow}$ are opposite. Thus, the readout will show that $IZZ = -1$ without revealing the value of either $Z_{A\uparrow}$ or $Z_{A\downarrow}$. Similar measurements hold for the projection property ($ZZI$ or $ZIZ$), by moving one of the CNOT connections to ancilla $C$, located on the ($\uparrow$) leg of the downstream Stern-Gerlach device. To measure $XXX$, one couples three ancillae ($A_{\uparrow}$, $A_{\downarrow}$, and $C$) to $B$ through Hadamard and CNOT gates (Fig. 3b). The readout will show that $XXX = 1$, and this, together with the above, establishes the superposition property.
While these joint measurements provide the (logically) most direct demonstration of compatibility, the local measurements have the virtue of demonstrating the randomness characteristic of ordinary measurements. So the ancilla model, subjected to local $Z$ measurements only, is reduced to an ordinary apparatus. But the $X$ measurements give it access to the superposition property, which is {\it not} accessible to ordinary measurements.
It should be noted in passing that the ancilla setup in either form can be realized starting with a photon propagating through a Mach-Zender interferometer and two ancillae, one associated with each arm. This is an extension of the so-called delayed choice quantum eraser (DCQE), which employs a single ancilla and is capable of realizing Wheeler's proposed delayed choice experiment \cite{Wheeler.DC} (the choice between Òwhich pathÓ and a superposition of paths), as closely realized experimentally by V. Jacques et. al. \cite{Jacques.07}. A different realization, and a comprehensive discussion of DCQE setups, is provided in \Ref{DCQE}. With a single ancilla, one has a two qubit system and is able to detect the superposition and projection properties (XX and ZZ). Superposition (XX) is manifested by interference as a function of the difference in path lengths, and the action of the ancilla X factor corresponds to quantum erasure of Òwhich pathÓ information. But this system cannot detect singleness as the third independent observable. With two ancillae, on the other hand, one has a three qubit system which accommodates this observable. Furthermore, the two-ancillae system forms a parallel with an ordinary apparatus which employs two detectors. We show below that the singleness property must hold here as a correlation between the detectors.
\section{Apparati with Real Detectors}
Reconsider the Stern-Gerlach setup of Fig. 1 with ancillae replaced by real detectors, $D_{\uparrow}$ and $D_{\downarrow}$. These act like the ancillae together with their readout devices: Like the ancillae, they transmit the atom without changing its spin state (this is in keeping with the von Neumann measurement formalism \cite{VN55,Ideal} and the stated postulate). And, like the readout devices, they record (irreversibly) the passage (or not) of the atom.
In this section, first, we show that the collapse properties of ordinary apparati are equivalent to those of the ancilla model, in the sense that the singleness and projection properties are represented by observables and established at the premeasurement stage, while the randomness of ``which path'' information enters only at the detection stage. Then second, we discuss the superposition property, which survives the detection stage in UQT but is nonetheless undetectable. We offer a physical explanation why this is the case.
\centerline{{\bf A. Equivalence of Collapse Properties}}
Recall that the ancilla-based premeasurement state, $\Psi(t_4)$, exhibits nonrandom collapse properties represented by joint observables: (i) the singleness property, by $Z_{A\uparrow} Z_{A\downarrow} = -1$, and (ii) the projection property, by $Z_s Z_{A\uparrow} = +1$ and $Z_s Z_{A\downarrow} = -1$. To show that these properties are similarly (pre)established in ordinary measurements, note that the ancilla operators, $Z_{A\uparrow}$ and $Z_{A\downarrow}$, act as stand-ins for path occupation variables. And we can define such variables for the ordinary setup as $Z_{P\uparrow} = 2{\cal P}_{\uparrow} - 1$ and $Z_{P\downarrow} = 2{\cal P}_{\downarrow} - 1$, where ${\cal P}_k$ is a projector onto that
subvolume of the path $k = (\uparrow,\downarrow)$ occupied by the spatial wave packet $\phi_k(\vec{r},t_2)$ at time $t_2$. So $Z_{Pk} \rightarrow \pm 1$ tells us whether or not the atom will enter the detector $D_k$. The path occupation {\it states} analogous to the ancilla states may be written as $\ket{1}_{P\uparrow}\ket{0}_{P\downarrow}$ and $\ket{0}_{P\uparrow}\ket{1}_{P\downarrow}$. These convey all of the necessary information about the spatial wave packets $\phi_{\uparrow}(t_2)$ and $\phi_{\downarrow}(t_2)$, so that $\ket{\psi(t_2)}$ may be written as \begin{equation}
\ket{\psi(t_2)} = \frac{1}{ \sqrt{2}} \bigg( \ket{1}_s
\ket{1}_{P\uparrow} \ket{0}_{P\downarrow} + \ket{0}_s
\ket{0}_{P\uparrow} \ket{1}_{P\downarrow} \bigg), \label{state8} \end{equation} which is analogous to \Eq{state4} of the ancilla system prior to readout. This is clearly an eigenstate of $Z_{P\uparrow} Z_{P\downarrow}$, $Z_s Z_{P\uparrow}$, and $Z_s Z_{P\downarrow}$, with eigenvalues -1, 1, and -1, respectively, thus establishing the singleness and projection properties in the premeasurement state. These properties arise here from the entanglement of just the spin and the path, with path occupation alone playing the ancillary role.
By the time $t_3$, the path occupation variables have mediated correlations between the atom's spin and the detectors - analogous to the readout state in the ancilla case. The main difference is that here, a detector readout follows closely the passage of the atom - one cannot delay this readout as was done with the ancilla system. But the ancilla system can duplicate this situation by moving the ancilla readouts to an earlier time, say $t_3$, ahead of the atomic spin remeasurement (at some $t > t_4$). The time ordering has no effect on the final result \cite{ordering}.
As an aside on the projection property - although this property is axiomatic in SQT, it is nonetheless conditioned on the above assumption that a detector transmits the atom without changing its spin state \cite{Ideal}. Clearly this assumption fails for detectors which work by absorbing the atom. In this case the projection property still holds, but it takes the form of a correlation between the reading of a detector and the angular momentum imparted to it. Only one detector can receive the impulse (the singleness property), and that impulse ($\pm \hbar$) is delivered to $D_{\uparrow}$ or $D_{\downarrow}$, respectively, depending on which detector reads 1.
\centerline{{\bf B. Blindness to Superpositions}}
While the collapse properties of the two systems are equivalent, the superposition property is not, being detectable in the ancilla system but not in the ordinary system. All approaches outlined at the beginning of the paper must agree on the facts: the observation of a single definite outcome (with its associated probability), and the invisibility of superpositions to ordinary apparati. But the approaches differ fundamentally on the physics behind these facts. In the SQT and OCT approaches, because the detectors are macroscopic, one of the terms existing in the premeasurement state is removed from the theory at detection. In SQT the removal is axiomatic; in OCT it is dynamical, a result of the nonlinear stochastic model. The evolution is nonunitary in both cases. In the UQT approach, on the other hand, the evolution including the detectors is unitary, and therefore, the state of the atom/detectors system must reflect the superposition in \Eq{state8}. This approach is viable, as we have said, only if the superposition of distinct detector states is undetectable. Decoherence theory shows that it is (see Appendix A), but the ancilla analogy suggests a more fundamental explanation involving irreversibility \cite{Raimond.97}.
In UQT, the state of the atom/detectors system, at time $t_4$, has the same GHZ form as \Eq{state8} (or, for that matter, \Eq{ESG4} of the ancilla system). The crucial difference is that the detector states involve microscopic internal degrees of freedom \cite{micro-degrees}, whose states are labeled by $\mu$ and $\mu'$ in addition to the necessary readout variables, 0 and 1 respectively. So \Eq{state8} becomes \begin{equation}
\ket{\psi(t_4)} = {1\over \sqrt{2}} \bigg( \ket{1}_s
\ket{1,\mu'}_{D\uparrow} \ket{0,\mu}_{D\downarrow} + \ket{0}_s
\ket{0,\mu}_{D\uparrow} \ket{1,\mu'}_{D\downarrow} \bigg), \label{state9} \end{equation} where we have dropped the common spatial factor at $t_4$ as done in \Eq{ESG4}. One can imagine (0) to be a metastable configuration of a detector, which would make a transition to a final stable configuration (1) if triggered by the passage of the atom. Equation \ref{state9} represents just a single element of an ensemble in which each initial state $\mu$ of the (0) configuration evolves unitarily into the state $\mu'$ of the (1) configuration \cite{caveat}. The corresponding density matrix is written in Appendix A. In the text, for clarity, we shall continue to refer to state vectors.
Again we ask - how can one observe a superposition involving both paths? One must measure, among other things, the spin component $X_s$. Since this measurement by itself produces random outcomes, one must measure a correlation of which $\Psi(t_4)$ is an eigenstate, of which the simplest is $XXX$. There are three other options, such as $XYY$, but these offer nothing further. So, supposing that the $X_s$ measurement produces the outcome $+1$, we must then show that the product of detector $X$ values is also $+1$. Given that the product of detector $Z$ values is $-1$, this would demonstrate that the detectors are in the Bell state analogous to \Eq{ESG5}: \begin{equation}
\ket{\Psi}_{DD} \rightarrow {1\over \sqrt{2}} \bigg( \ket{1,\mu'}_{D_{\uparrow}}
\ket{0,\mu}_{D_{\downarrow}} + \ket{0,\mu}_{D_{\uparrow}} \ket{1,\mu'}_{D_{\downarrow}} \bigg). \label{ESG7} \end{equation} A detector operator $X_k$ connects its 0 and 1 states, that is, $\ket{1,\mu'}_{Dk} = X_{Dk} \ket{0,\mu}_{Dk}$ and $\ket{0,\mu}_{Dk} = X_{Dk} \ket{1,\mu'}_{Dk}$. Since the first of these represents the natural evolution of the detector, $\ket{1,\mu'} = U(t_3,t_2) \ket{0,\mu}$, the $X$ operators must be \begin{equation}
X_{Dk} = P_k(1) U_k (t_3,t_2) P_k(0) + P_k(0) U_k^{-1}(t_3,t_2) P_k(1), \label{Xoperator} \end{equation} where $P_k(i)$ are projection operators onto the $i = 0$ or 1 configurations of detector $D_k$. Now $U_k (t_3,t_2)$ represents the time evolution of a complex many-body system, and while this is reversible {\it in principle}, it is not reversible
{\it thermodynamically} \cite{Batal.15}. That is, we do not have control over the microscopic degrees of freedom required to implement $U_k^{-1}(t_3,t_2)$.
So the detector operators $X_k$ are not accessible to us, and without them we cannot detect a superposition of states in the 0 and 1 configurations of $D_k$ - and we cannot access $XXX$, which would demonstrate a Schr{\"o}dinger~cat-like superposition of the two collapse scenarios of the spin/detectors system. In short, we only have access to the detector variables $Z_k$ which the detectors record, and in these there can be no evidence for the existence of the superposition which (in UQT) continues to exist in the state vector. A different but related argument on thermodynamic irreversibility in measurement was given by Peres \cite{Peres80}.
The demonstration of blindness changes very little with detectors which absorb the atom rather than transmitting it, as is essentially the case in the original Stern-Gerlach experiment \cite{Stern Gerlach}. This case is discussed in Appendix B.
The above arguments suggest that our inability to detect superpositions of detector output states is a manifestation of the second law of thermodynamics. One can imagine a quantum Maxwell Demon who possesses the microscopic control that we lack, and is capable of detecting superpositions which are invisible to us. Thus, the quantum measurement process, by its construction, employs the thermodynamic arrow of time. There is no need to invoke a different ``measurement'' arrow.
\centerline{{\bf C. A Decoherence Perspective}}
A decoherence argument for blindness involves the concept of the pointer; Brasil \cite{Brasil15} defines ``pointer states (as) eigenstates of the observable of the measuring apparatus that represents the possible positions of the display pointer of the equipment.'' The concept was introduced in the present context by Zeh \cite{Zeh.70} and developed by Zurek \cite{Zurek.81,Zurek.82}, who showed that interactions with the environment select the pointer's preferred basis (the states we observe). In our system these states are associated with the {\it pair} of detectors; they are denoted by $\ket{1}_{D\uparrow} \ket{0}_{D\downarrow}$ and $\ket{0}_{D\uparrow} \ket{1}_{D\downarrow}$ (or in shorthand, $\ket{10}_{DD}$ and $\ket{01}_{DD}$), by simply dropping the environmental variables $\mu$ and $\mu'$.
An argument specific to our system is written out in Appendix A. In brief, one traces over the environmental degrees of freedom ($\mu$ and $\mu'$) within each detector, and derives the reduced density matrix of the spin/pointer system, which is diagonal in the basis $\ket{110}_{sDD}$ and $\ket{001}_{sDD}$, where the second and third indices refer to the pointer. In the case of detectors which absorb the atom, this basis reduces to just that of the pointer, $\ket{10}_{DD}$ and $\ket{01}_{DD}$. In either case, blindness to superpositions is represented by the diagonality of the ($2 \times 2$) density matrix.
There are interesting consequences when the pointer is realized within the system of two detectors. Environmental interactions determine the preferred basis of an
individual detector, $\ket{1}_{Dk}$ and $\ket{0}_{Dk}$ (corresponding to {\it atom/no atom}, or $Z_{Dk} = \pm 1$). But they do {\it not} determine the preferred basis of the full pointer associated with the {\it pair} of detectors, because they do not exclude the possibility of $\ket{11}_{DD}$ and $\ket{00}_{DD}$ - exclusion resulting from the singleness property. In fact, the environment is not responsible for the singleness or the projection property, or implicitly, for the choice of which spin component is measured - all of which are established in premeasurement. So, while the environment is crucial for enforcing the blindness to superpositions, its role (whether internal or expanded to include the external) is limited to the proper functioning of the individual detectors.
This separation between the premeasurement stage (governed by reversible unitary evolution), and the detection stage (where the environment enters bringing practical irreversibility), is the defining characteristic of controlled von Neumann-type measurements. The roles of these stages are in some sense complementary: While blindness to superpositions may be seen as an emergent classical property induced by interactions with the environment, the singleness and projection properties are quantum entanglement properties, represented by observables and manifested in correlations between noninteracting macroscopic objects.
\section{Conclusions}
We studied an ancilla-aided Stern-Gerlach experiment allowing delayed-choice measurements on the three-qubit system of atomic spin and two ancillae acting as virtual detectors. We first considered local measurements, and showed that one choice ($Z_i$) reproduces the collapse properties of ordinary (unaided) measurements, while another ($X_i$) demonstrates a superposition of the two collapse-like scenarios, involving both paths. Both choices require repeated measurements on identically-prepared states of the system, since local measurements destroy the state.
So secondly, we showed that nondestructive measurements can be made by avoiding the acquisition of local information. Thus relinquishing only the specific ``which path'' information, one can still measure a complete set of commuting
joint observables - these represent the superposition property and the two (nonrandom) collapse properties, namely the singleness and projection properties.
In Sec. III we applied the above ideas to an ordinary apparatus. First, we showed that the collapse properties occur with the same status as in the ancilla model: The singleness and projection properties are represented by operators and established at the premeasurement stage, while the randomness of local measurement outcomes enters only at the detection stage, as a consequence of the singleness property. The various approaches mentioned in the introduction - the standard, unitary, and objective collapse approaches - agree on the observed collapse phenomenon itself, but they differ on its unobservable underpinnings - the existence/nonexistence of unobserved branches in the state vector - and the nature of the observed randomness of outcomes (is it objective or subjective?). It is possible, but far from certain, that future experiments alone will resolve these differences.
The viability of the UQT approach rests upon the invisibility of the alternate (unobserved) branches in the state vector. The ancilla system points to a simple explanation: Detectors are irreversible, and this makes the required complementary local observables ($X_{D\uparrow}$ and $X_{D\downarrow}$) inaccessible. The absence of a known fundamental mechanism of irreversibility acting in typical measurements suggests that the irreversibility is thermodynamic, so that (at least within UQT), the observed collapse phenomenon is a manifestation of the second law.
\centerline{{\bf Appendix A: Density Matrix of Spin-Detectors System}}
Here we write out the density matrix of the spin/detectors system, and we show how the trace over the unobserved states of the detectors' internal degrees of freedom yields the appropriate reduced density matrix, which expresses the blindness of the apparatus to superpositions of output states.
The initial mixed state of the spin/detectors system, assuming probabilities $p_{\mu_\uparrow}$ and $p_{\mu_\downarrow}$ for the microstates, $\ket{0,\mu_\uparrow}_{D_{\uparrow}}$ and $\ket{0,\mu_\downarrow}_{D_{\downarrow}}$ of the two detectors, is \begin{equation}
\rho(t_1) = \sum_{\mu_\uparrow,\mu_\downarrow} p_{\mu_\uparrow} p_{\mu_\downarrow}
\bigg( \alpha \ket{1}_s +\beta \ket{0}_s \bigg) \ket{0,\mu_\uparrow}_{D_{\uparrow}}
\ket{0,\mu_\downarrow}_{D_{\downarrow}} \bigg( \alpha^* \bra{1}_s +\beta^* \bra{0}_s \bigg)
\bra{0,\mu_\uparrow}_{D_{\uparrow}} \bra{0,\mu_\downarrow}_{D_{\downarrow}}. \label{AppB1} \end{equation} After the atom passes through the detectors and the paths are recombined at $t_4$, this becomes \begin{eqnarray}
\rho(t_4) = \sum_{\mu_\uparrow,\mu_\downarrow} & p_{\mu_\uparrow} p_{\mu_\downarrow}
\bigg( \alpha \ket{1}_s \ket{1,\mu_\uparrow'}_{D_{\uparrow}} \ket{0,\mu_\downarrow}_{D_{\downarrow}} +
\beta \ket{0}_s \ket{0,\mu_\uparrow}_{D_{\uparrow}} \ket{1,\mu_\downarrow'}_{D_{\downarrow}} \bigg) \cdot \nonumber \\
& \cdot \bigg( \alpha^* \bra{1}_s \bra{1,\mu_\uparrow'}_{D_{\uparrow}} \bra{0,\mu_\downarrow}_{D_{\downarrow}} +
\beta^* \bra{0}_s \bra{0,\mu_\uparrow}_{D_{\uparrow}} \bra{1,\mu_\downarrow'}_{D_{\downarrow}} \bigg). \label{AppB2} \end{eqnarray} Since we only read the detectors' outputs and do not monitor the microscopic degrees of freedom (considered as the ``environment'' $E$), we trace over the latter to define the reduced density matrix describing the state of the spin and the detector displays, which is called the spin/pointer system \cite{Brasil15}). To be more precise, each detector consists of its own pointer (with readout states 0 and 1), and its own environment (with associated states $\mu$ and $\mu'$, respectively). The trace consists of independent traces over the environments within each detector, {\it i.e.}, $\rho^r(t_4) \equiv Tr_{E_\uparrow,E_\downarrow} \rho(t_4)$. To evaluate each of these, it is convenient to sum over $\mu$ ({\it i.e.}, $\sum_\mu \bra{\mu} ... \ket{\mu}$ in those terms where the pointer state 0 appears, and over $\mu'$ where it does not. The latter choice is legitimate because the two sets are related unitarily. It is straightforward then to show that \begin{eqnarray}
& \rho^r(t_4) = |\alpha|^2 \ket{1}_s \ket{1}_{D_{\uparrow}} \ket{0}_{D_{\downarrow}}
\bra{1}_s \bra{1}_{D_{\uparrow}} \bra{0}_{D_{\downarrow}} \nonumber \\
& + \alpha \beta^* \ket{1}_s \ket{1}_{D_{\uparrow}} \ket{0}_{D_{\downarrow}} \bra{0}_s
\bra{0}_{D_{\uparrow}} \bra{1}_{D_{\downarrow}} \sum_{\mu_\uparrow,\mu_\downarrow} p_{\mu_\uparrow} p_{\mu_\downarrow}
\braket{\mu_\uparrow}{\mu_\uparrow'}_{D_{\uparrow}} \braket{\mu_\downarrow'}{\mu_\downarrow}_{D_{\downarrow}} \nonumber \\
& + \alpha^* \beta \ket{0}_s \ket{0}_{D_{\uparrow}} \ket{1}_{D_{\downarrow}} \bra{1}_s
\bra{1}_{D_{\uparrow}} \bra{0}_{D_{\downarrow}} \sum_{\mu_\uparrow,\mu_\downarrow} p_{\mu_\uparrow} p_{\mu_\downarrow}
\braket{\mu_\uparrow'}{\mu_\uparrow}_{D_{\uparrow}} \braket{\mu_\downarrow}{\mu_\downarrow'}_{D_{\downarrow}} \nonumber \\
& + |\beta|^2 \ket{0}_s \ket{0}_{D_{\uparrow}} \ket{1}_{D_{\downarrow}}
\bra{0}_s \bra{0}_{D_{\uparrow}} \bra{1}_{D_{\downarrow}} \label{reduction} \end{eqnarray} The environmental sums in the second and third terms essentially vanish (they are undetectably small) because the inner product factors, none greater than unity in magnitude, have random phases, in contrast with analogous factors $\big(\braket{\mu}{\mu}\braket{\mu'}{\mu'} = 1\big)$ which appeared in the first and fourth terms and summed to unity. Thus $\rho^r$ is diagonal in the spin-pointer basis, which consists of $\ket{1}_s \ket{1}_{D_{\uparrow}} \ket{0}_{D_{\downarrow}}$ and $\ket{0}_s \ket{0}_{D_{\uparrow}} \ket{1}_{D_{\downarrow}} $. The surviving singleness and projection correlations result from the entanglement generated between \Eqs{AppB1}{AppB2} by the passage of the atom. In fact it should be noted that, except for the remaining summations in the off-diagonal terms, \Eq{reduction} is equivalent in form to the density matrix of the spin-ancilla system (see \Eq{state4} in Sec. II). Thus, the environmental factors in \ref{reduction} neatly summarize how the superposition of outcomes becomes undetectable with real detectors.
\centerline{{\bf Appendix B: Detectors that Absorb}}
In the original Stern-Gerlach experiment \cite{Stern Gerlach}, silver atoms were directed at a glass plate and formed two separated deposits, with segments of the glass acting as the two detectors. Imagining ideally a pair of absorbing single-atom detectors, their state at time $t_3$ could still be written as in \Eq{ESG7}, but the 1 states now represent the absorbed atom as well as excitations created by the absorption event. Natural evolution produces these states from the 0 states of the detectors multiplied by the corresponding path occupation states $\ket{1}_{Pk}$ of the atom: $\ket{1,\mu'}_{Dk} = U(t_3,t_2) \ket{0,\mu}_{Dk} \ket{1}_{Pk}$. So the $X_{Dk}$ operator analogous to \Eq{Xoperator} is \begin{equation}
X_{Dk} = P_k(1) U_k (t_3,t_2) \ket{1}_{Pk} P_k(0) \bra{1}_{Pk}
+ \ket{1}_{Pk} P_k(0) \bra{1}_{Pk} U_k^{-1}(t_3,t_2) P_k(1), \label{Xoperator2} \end{equation} and the subsequent blindness argument is unchanged.
The decoherence approach of Appendix A is similarly adapted: Since the $\ket{\mu'}$ states include the absorbed atom, the inner product factors in \ref{reduction} are replaced by $\bra{1}_{Pk} \braket{\mu}{\mu'}$ or its complex conjugate. The set \{$\mu'$\} is not complete because it refers to more particles than $\{\mu\}$, but it includes all states generated unitarily from \{$\mu$\} and the incident atom.
\end{document} | arXiv |
Bayesian history matching
Bayesian history matching is a statistical method for calibrating complex computer models. The equations inside many scientific computer models contain parameters which have a true value, but that true value is often unknown; history matching is one technique for learning what these parameters could be.
The name originates from the oil industry, where it refers to any technique for making sure oil reservoir models match up with historical oil production records.[1] Since then, history matching has been widely used in many areas of science and engineering, including galaxy formation,[2] disease modelling,[3] climate science,[4] and traffic simulation.[5]
The basis of history matching is to use observed data to rule-out any parameter settings which are ``implausible’’. Since computer models are often too slow to individually check every possible parameter setting, this is usually done with the help of an emulator. For a set of potential parameter settings ${\boldsymbol {\theta }}$, their implausibility $I({\boldsymbol {\theta }})$ can be calculated as:
$I({\boldsymbol {\theta }})={\frac {|E[f({\boldsymbol {\theta }})]-y|}{\sqrt {Var[f({\boldsymbol {\theta }})]}}}$
where $E[f({\boldsymbol {\theta }})]$ is the expected output of the computer model for that parameter setting, and $Var[f({\boldsymbol {\theta }})]$ represents the uncertainties around the computer model output for that parameter setting. In other words, a parameter setting is scored based on how different the computer model output is to the real world observations, relative to how much uncertainty there is.
For computer models that output only one value, an implausibility of 3 is considered a good threshold for rejecting parameter settings.[6] For computer models which output more than one output, other thresholds can be used.[7] A key component of history matching is the notion of iterative refocussing,[8] where new computer model simulations can be chosen to better improve the emulator and the calibration, based on preliminary results.
References
1. Craig, Peter S.; Goldstein, Michael; Seheult, Allan H.; Smith, James A. (1997). Gatsonis, Constantine; Hodges, James S.; Kass, Robert E.; McCulloch, Robert; Rossi, Peter; Singpurwalla, Nozer D. (eds.). "Pressure Matching for Hydrocarbon Reservoirs: A Case Study in the Use of Bayes Linear Strategies for Large Computer Experiments". Case Studies in Bayesian Statistics. Lecture Notes in Statistics. New York, NY: Springer. 121: 37–93. doi:10.1007/978-1-4612-2290-3_2. ISBN 978-1-4612-2290-3.
2. Vernon, Ian; Goldstein, Michael; Bower, Richard (February 1, 2014). "Galaxy Formation: Bayesian History Matching for the Observable Universe". Statistical Science. 29 (1): 81–90. arXiv:1405.4976. doi:10.1214/12-STS412. S2CID 18315892 – via Project Euclid.
3. Andrianakis, Ioannis; Vernon, Ian R.; McCreesh, Nicky; McKinley, Trevelyan J.; Oakley, Jeremy E.; Nsubuga, Rebecca N.; Goldstein, Michael; White, Richard G. (January 1, 2015). "Bayesian history matching of complex infectious disease models using emulation: a tutorial and a case study on HIV in Uganda". PLOS Computational Biology. 11 (1): e1003968. Bibcode:2015PLSCB..11E3968A. doi:10.1371/journal.pcbi.1003968. PMC 4288726. PMID 25569850.
4. Williamson, Daniel; Goldstein, Michael; Allison, Lesley; Blaker, Adam; Challenor, Peter; Jackson, Laura; Yamazaki, Kuniko (October 1, 2013). "History matching for exploring and reducing climate model parameter space using observations and a large perturbed physics ensemble". Climate Dynamics. 41 (7): 1703–1729. Bibcode:2013ClDy...41.1703W. doi:10.1007/s00382-013-1896-4. S2CID 120737289.
5. Boukouvalas, Alexis; Sykes, Pete; Cornford, Dan; Maruri-Aguilar, Hugo (June 1, 2014). "Bayesian Precalibration of a Large Stochastic Microsimulation Model" (PDF). IEEE Transactions on Intelligent Transportation Systems. 15 (3): 1337–1347. doi:10.1109/TITS.2014.2304394. S2CID 16209605.
6. Pukelsheim, Friedrich (May 1, 1994). "The Three Sigma Rule" (PDF). The American Statistician. 48 (2): 88–91. doi:10.1080/00031305.1994.10476030.
7. Vernon, Ian; Goldstein, Michael; Bower, Richard G. (December 1, 2010). "Galaxy formation: a Bayesian uncertainty analysis". Bayesian Analysis. 5 (4): 619–669. doi:10.1214/10-BA524 – via Project Euclid.
8. Salter, James M.; Williamson, Daniel B.; Scinocca, John; Kharin, Viatcheslav (October 2, 2019). "Uncertainty Quantification for Computer Models With Spatial Output Using Calibration-Optimal Bases". Journal of the American Statistical Association. 114 (528): 1800–1814. doi:10.1080/01621459.2018.1514306. hdl:10871/33707.
| Wikipedia |
\begin{definition}[Definition:Finite Measure]
Let $\mu$ be a measure on a measurable space $\struct {X, \Sigma}$.
Then $\mu$ is said to be a '''finite measure''' {{iff}}:
:$\map \mu X < \infty$
\end{definition} | ProofWiki |
Entropically engineered formation of fivefold and icosahedral twinned clusters of colloidal shapes
Binary icosahedral clusters of hard spheres in spherical confinement
Da Wang, Tonnishtha Dasgupta, … Alfons van Blaaderen
How to design an icosahedral quasicrystal through directional bonding
Eva G. Noya, Chak Kui Wong, … Jonathan P. K. Doye
A Brownian quasi-crystal of pre-assembled colloidal Penrose tiles
Po-Yuan Wang & Thomas G. Mason
Superstructures generated from truncated tetrahedral quantum dots
Yasutaka Nagaoka, Rui Tan, … Ou Chen
Formation of a single quasicrystal upon collision of multiple grains
Insung Han, Kelly L. Wang, … Ashwin J. Shahani
Synthesis and assembly of colloidal cuboids with tunable shape biaxiality
Yang Yang, Guangdong Chen, … Zhihong Nie
Self-templating assembly of soft microparticles into complex tessellations
Fabio Grillo, Miguel Angel Fernandez-Rodriguez, … Lucio Isa
Quantitative 3D real-space analysis of Laves phase supraparticles
Da Wang, Ernest B. van der Wee, … Alfons van Blaaderen
Emergent tetratic order in crowded systems of rotationally asymmetric hard kite particles
Zhanglin Hou, Yiwu Zong, … Kun Zhao
Sangmin Lee ORCID: orcid.org/0000-0002-1145-47081 nAff3 &
Sharon C. Glotzer ORCID: orcid.org/0000-0002-7197-00851,2
Colloids
Fivefold and icosahedral symmetries induced by multiply twinned crystal structures have been studied extensively for their role in influencing the shape of synthetic nanoparticles, and solution chemistry or geometric confinement are widely considered to be essential. Here we report the purely entropy-driven formation of fivefold and icosahedral twinned clusters of particles in molecular simulation without geometric confinement or chemistry. Hard truncated tetrahedra self-assemble into cubic or hexagonal diamond colloidal crystals depending on the amount of edge and vertex truncation. By engineering particle shape to achieve a negligible entropy difference between the two diamond phases, we show that the formation of the multiply twinned clusters is easily induced. The twinned clusters are entropically stabilized within a dense fluid by a strong fluid-crystal interfacial tension arising from strong entropic bonding. Our findings provide a strategy for engineering twinning behavior in colloidal systems with and without explicit bonding elements between particles.
Twinning arises from a type of grain boundary called a twin boundary, where two separate crystals sharing the same lattice plane intergrow with a certain symmetry. Fivefold and icosahedral symmetries, which are generally known to be incompatible with long-range order, can be induced in relatively large atomic crystal clusters (from several nanometers to a few microns) via multiple twinning1,2,3,4. For instance, materials that form face-centered cubic (fcc)2,3,5,6,7 or cubic diamond8,9 crystals can form twin boundaries toward the (111) or its equivalent lattice directions, promoting the formation of a \(\sim 70^\circ\) twin angle that can produce multiply twinned structures with fivefold or icosahedral symmetry. Twinning has been widely used to obtain interesting properties in synthetic nanomaterials, such as enhancing the mechanical properties of nanowires10 and increasing oxidation resistance11, and to synthesize noble metal nanostructures with decahedral or icosahedral shape3,12, which are useful as catalysts13. Many studies have been conducted to control the growth mechanism12,14,15 and stability of multiply twinned structures, e.g., tailoring solution chemistry5,6,16,17,18 and using multicomponent materials6,11. However, the multiplicity of variables in multiply twinned crystals makes it difficult to know which variables are responsible for, or have the most influence on, the formation of fivefold and icosahedral twins.
Compared to most systems, hard particle systems are regarded as relatively simple because the phase behavior is driven solely by entropy maximization19,20. The diversity and complexity of entropy-driven phase behavior, however, are enormously rich. The self-assembly of complex crystals and quasicrystals20,21, as well as complex assembly pathways such as multistep crystallization, have been reported in hard particle systems22. The spontaneous formation of icosahedral twins has been observed in hard sphere systems both in experiments and in computer simulations, but only under spherical confinement4,7,15,23, which produces an artificial "surface tension" to overcome the internal strain within the cluster arising from local particle packing. That, and the negligible free energy difference between fcc and hexagonal close-packed (hcp), two competing crystal structures of hard spheres24, conspire to produce twins. However, the formation and stabilization of fivefold and icosahedral twinned clusters without geometrical confinement or in any other hard particle systems or have yet to be reported. We hypothesized that artificial surface tension provided by spherical confinement could be achieved naturally, without any confinement, through the judicious choice of particle shape.
Here, we show the purely entropy-driven, confinement-free assembly of fivefold and icosahedral twinned clusters from multiply twinned diamond crystals in equilibrium with a dense fluid phase. Monte Carlo (MC) simulations show that hard truncated tetrahedrons (TTs) self-assemble into either cubic and hexagonal diamond crystals depending on the amount of edge and vertex truncation (Fig. 1). We tuned the TT shape to have a negligible free energy difference between the two diamond crystal phases, and found the formation of twin boundaries is easily induced in a fluid phase (Fig. 2). This finding demonstrates that the stability of twin boundaries in a fluid can be controlled by particle shape design, even entropically. Using this strategy, we induce the formation of fivefold and icosahedral twins through seed-assisted growth of a single-crystalline seed of cubic diamond. Since the formation of twin boundaries is easily induced toward multiple directions during crystal growth, multiply twinned structures are easily formed. We show that, through an error-and-repair process, a multiply twinned structure with defects transforms into a fivefold twinned cluster in a dense fluid phase (Fig. 3). We also show the formation of an icosahedral twin in a dense fluid upon further growth of the fivefold twin. The icosahedral twin of hard TTs is entropically stabilized within the fluid without spherical confinement (Fig. 3), unlike the icosahedral cluster of hard spheres that rapidly destabilizes and falls apart (Fig. 4). We show that icosahedral clusters of hard TTs have twice the fluid-solid interfacial free-energy (or entropy) compared to icosahedral clusters of hard spheres as a natural consequence of stronger entropic bonding25 in the former system. Our study isolates the essential role of surface tension in stabilizing the inherently strained icosahedral twinned cluster in a dense fluid, and suggests approaches for engineering the surface tension.
Fig. 1: Twinned crystals of truncated tetrahedra.
a Two types of pairwise contacts (left) and the local environment of particles in cubic diamond (upper right) and hexagonal diamond (bottom right). b, c Unit cells of cubic diamond (b, left and middle) and hexagonal diamond (c, left and middle) crystals. The (111) plane of cubic diamond (b, right) and the (0001) plane of hexagonal diamond (c, right) have the same structure. d–j Twinned structures of TTs. Red particles are \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\), blue are \({{{{{{\rm{S}}}}}}}_{3}{{{{{{\rm{E}}}}}}}_{1}\), green are \({{{{{{\rm{S}}}}}}}_{2}{{{{{{\rm{E}}}}}}}_{2}\) and purple are \({{{{{{\rm{S}}}}}}}_{1}{{{{{{\rm{E}}}}}}}_{3}\). d A twin boundary is formed when the (111) plane of cubic diamond (magenta arrow) and the (0001) plane of hexagonal diamond (cyan arrow) meet. e, Five-fold twinned structure of TTs. The angle between twin boundaries is ~72o. f–j Structure of icosahedral twinned crystal. The bond orientational order diagram (f, upper right) and diffraction pattern (f, bottom right) along the 10-fold symmetry axis. The ico-twin crystal consists of 20 cubic diamond clusters with a tetrahedron Wulff shape (red in f, g), 30 twin planes between the tetrahedral clusters (blue in f, h), and 12 columns with 5-fold symmetry (green in f, i). j The center of the ico-twin is a dodecahedron super-cluster of 100 TTs.
Fig. 2: Stability control of diamond crystals via particle shape design.
a Per-particle Helmholtz free-energy (\(F/N{k}_{B}T\)) plot, in units of kBT, of cubic (red) and hexagonal (blue) diamond crystals of hard TTs as a function of vertex and edge truncation parameters at constant particle volume fraction (\(\phi=0.62\)). b Phase diagram of hard TTs in the shape space determined by the free-energy calculation. c Free-energy difference between cubic and hexagonal diamond (\(\triangle F={F}_{{{{{{\rm{H}}}}}}}-{F}_{{{{{{\rm{C}}}}}}}\)) in shape space. d, When the TT is designed \((a=1.20,\,{c}=2.16)\) to have negligible \(\triangle F \sim+0.007\), the initial cubic diamond (left) single-crystalline cluster forms multiple stacking faults (right) in fluid, suggesting that the free energy loss from the twin boundaries is small. Each TT is represented by a tiny sphere at the TT center of mass. Red and blue spheres represent cubic and hexagonal diamonds, respectively. e When the TT is designed (\(a=1.22,\,{c}=2.16\)) to have \(\triangle F \sim -0.028\), the initial cubic diamond cluster (left) completely transforms into single-crystalline hexagonal diamond (right) cluster in a relatively short simulation time. f The change of the number ratio of particles in cubic diamond over time for three different \(\triangle F=+0.138,+0.007\) and \(-0.028\) (black X markers in c). Orange and cyan circles indicate each snapshot in d and e, respectively. g, Local volume fraction distribution plots calculated from the last snapshot from d (upper) and e (lower). Two peaks in the distribution show that the system is in coexistence between fluid and solid.
Fig. 3: Growth process of fivefold and icosahedral twinned crystals from seed in fluid.
Seed assisted growth of twinned crystals of hard TTs (\(a=1.20,\,{c}=2.16\)) at coexistence between crystal and fluid (\(\phi=0.58\)). a–h Simulation snapshots showing the error-and-repair mechanism of the five-fold twinned crystal. Each TT is represented by a tiny sphere at the TT center of mass. a When a spherical seed of cubic diamond crystal (\(N=500\)) grows, b twin boundaries are formed along (111) or its equivalent plane directions. Because the growth of each direction is independent, c, d an error can occur when stacking sequences of two growth directions mismatch. e The growing direction with the error is re-melted toward its opposite direction (f) until the boundary with its adjacent plane matches. Once the boundary matches, g, h) the five-fold twinned crystal grows to fully form the five-fold twinned crystal with truncated pentagonal dipyramid (PD) crystal shape. The final crystal is fully surrounded by fluid and stabilized. i–l Simulation snapshots of icosahedral twinned crystal formation from PD seed. All particles are represented by spheres showing centers of mass. m The icosahedral twinned crystal exposes (111) surfaces of cubic diamond crystal when stabilized in fluid. n Local volume fraction distribution plot of the system at coexistence shows a bimodal shape at \(\phi=0.57\) and \(0.64\). o The change of pressure over time for the ico-twin crystal-forming system. Pressure decreases during the growth of the ico-twin crystal and is constant (\({P}^{*} \sim 12.6\)) after the growth, indicating that the ico-twin crystal is stable in fluid. The red dashed lines indicate the simulation time when the snapshots in i–l are taken. The inset snapshots and diffraction patterns show that the ico-twin crystal structure surrounding by fluid is maintained during pressure stabilization.
Fig. 4: Fluid-solid interfacial energy calculation.
a Icosahedral twinned crystal of FCC of hard spheres quickly destabilizes in fluid (\( < {10}^{6}\) MC steps). b (111) surfaces of a cubic diamond crystal of hard TTs (left) and an FCC crystal of hard spheres (right), and their side views (middle). Simulation setup of the capillary fluctuation method for calculating the fluid-solid interfacial stiffness \(\widetilde{\gamma }\) of (c) cubic diamond and (d) FCC. e–g The change of interfacial profiles of the (111) direction of (e) cubic diamond and (g) FCC. f–h Fluid-solid interfacial free energy \(\gamma\) of (f) cubic diamond and (h) FCC. For all three lattice directions, the hard TT system has more than twice the value of \(\gamma\) of the hard sphere system, indicating its stronger fluid-solid surface tension.
Cubic diamond is different from hexagonal diamond (or lonsdaleite) in the conformation of local tetrahedral bonds26. Each atom has four tetrahedral bonds with nearest neighbors, and every atom in cubic diamond has four staggered bonds, while every atom in hexagonal diamond has three staggered bonds and one eclipsed bond. For a hard particle with a tetrahedron shape, the two types of chemical bond conformations can be mapped to the staggered and eclipsed entropic bond conformations associated with face-to-face contacts. (Fig. 1a). An appropriate degree of tip truncation of hard tetrahedra allows them to exclusively form staggered contacts, resulting in the self-assembly of the cubic diamond phase27,28 (Fig. 1b). However, the self-assembly of the hexagonal diamond phase, where the two bond conformation types are mixed, has not been reported in hard particle systems. We hypothesized that, through judicious truncation of a regular tetrahedron, we can find an appropriate shape that introduces both staggered and eclipsed contacts and favors hexagonal diamond (Fig. 1c), and, intermediate between the cubic diamond-forming and hexagonal diamond forming shapes, one that produces both diamond structures with negligible difference in entropy. Controlling the relative stability of cubic and hexagonal diamond will give us a route to control twinning and stacking behaviors of diamond crystals (Fig. 1b–d). Cubic and hexagonal diamond share an equivalent plane along the \((111)\) and \((0001)\) directions (Fig. 1b, c); fcc and hcp share the same equivalent plane. If the entropy difference between the two crystals can be engineered to be sufficiently small, the crystal will be prone to form twin boundaries.
The twinning of cubic diamond occurs towards \((111)\) and its equivalent planes such as \((\bar{1}11)\), \((1\bar{1}1)\) and \((11\bar{1})\); thus when twinning occurs in multiple directions, two twin planes meet with a specific angle, \({{{\cos }}}^{-1}(1/3)=70.5^\circ\), which is close to \(72^\circ\), the same angle that promotes the formation of multiply twinned structures with fivefold symmetry2 (Fig. 1e). The structure of the multiply twinned crystals made by TTs is easily analyzed by classifying particles based on the number of staggered and eclipsed contacts (Supplementary Figs. 1, 2), following the notation \({{{{{{\rm{S}}}}}}}_{n}{{{{{{\rm{E}}}}}}}_{m}\), where \(n+m=4\). For instance, the cubic diamond crystal consists of \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) particles (Fig. 1a, b), and the hexagonal diamond crystal consists of \({{{{{{\rm{S}}}}}}}_{3}{{{{{{\rm{E}}}}}}}_{1}\) particles (Fig. 1a, c). In a fivefold twin, five cubic diamond clusters with a tetrahedron shape made by \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) particles together form a pentagonal bipyramid super-cluster with five twin planes of \({{{{{{\rm{S}}}}}}}_{3}{{{{{{\rm{E}}}}}}}_{1}\) particles between the faces of the tetrahedron clusters (Fig. 1e). At the point where the five twin planes meet, \({{{{{{\rm{S}}}}}}}_{2}{{{{{{\rm{E}}}}}}}_{2}\) particles form a column where small pentagonal bipyramids (\(N=5\)) are linearly stacked. An icosahedral twin (Fig. 1f) comprises twenty cubic diamond clusters (\({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) particles), each with a tetrahedron shape (Fig. 1g). Thirty twin planes (\({{{{{{\rm{S}}}}}}}_{3}{{{{{{\rm{E}}}}}}}_{1}\)) exist between the faces of the tetrahedron clusters (Fig. 1h), and there twelve fivefold columns made by \({{{{{{\rm{S}}}}}}}_{2}{{{{{{\rm{E}}}}}}}_{2}\) particles where five tetrahedron clusters meet (Fig. 1i). At the center, there is a dodecahedron super-cluster \((N=100)\) comprising three different shells with icosahedral symmetry: 60 \({{{{{{\rm{S}}}}}}}_{2}{{{{{{\rm{E}}}}}}}_{2}\) particles comprising the outer shell form a rhombicosidodecahedron, 20 \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) particles comprising the intermediate shell form a dodecahedron and 20 \({{{{{{\rm{S}}}}}}}_{1}{{{{{{\rm{E}}}}}}}_{3}\) particles comprising the smallest shell form a dodecahedron. This hierarchical structure is equivalent to the super-cluster of water \({({{{{{{\rm{H}}}}}}}_{2}{{{{{\rm{O}}}}}})}_{100}\) when connecting oxygen atoms29.
A competition between staggered and eclipsed contracts arises from many-body interactions responsible for the entropic forces producing the effective attraction between neighboring particles, and preferences to form a certain type of face-to-face TT contact—which are determined by the strength of this attraction—can be controlled by varying the amount of edge and vertex truncation28. We know that regular tetrahedra prefer to form a dodecagonal quasicrystal21 in which every particle has eclipsed alignments (\({{{{{{\rm{S}}}}}}}_{0}{{{{{{\rm{E}}}}}}}_{4}\)). Truncation introduces additional facets, weakening the strength of the entropic bond between two primary eclipsed faces, and thereby weakening the preference for eclipsed alignment. The entropic bonding between primary eclipsed faces is weakened with increasing truncation because less and less free volume is gained by maintaining this alignment (though this alignment still maximizes free volume and thus system entropy). Beyond a certain amount of truncation, however, the system has more entropy (more free volume) if the neighbors align in a staggered way, and thus the staggered configuration becomes preferred over the eclipsed configuration (\({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\))27. Motivated by this, we searched the shape space28 between \({{{{{{\rm{S}}}}}}}_{0}{{{{{{\rm{E}}}}}}}_{4}\) and \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) to find a shape intermediate between the two that favors the hexagonal diamond phase \(({{{{{{\rm{S}}}}}}}_{3}{{{{{{\rm{E}}}}}}}_{1})\). To quantify the relative thermodynamic stability of cubic and hexagonal diamond as a function of TT shape, we constructed free energy surfaces in TT shape space varying the edge (\(1.14\le a\le 1.30\)) and vertex (\(2.10\le c\le 2.30\)) truncation at constant volume fraction (Fig. 2a), using the Frenkel–Ladd free energy calculation method24,30 ("Methods"). We found a region where hexagonal diamond is thermodynamically more stable than cubic diamond, as shown in the phase diagram generated from the bottom view of the free energy landscape (Fig. 2b). We also confirmed those shapes self-assemble hexagonal diamond (Supplementary Fig. 3).
To assess the stability of the twin boundaries depending on the TT shape, we chose three systems with different relative stabilities between the two diamond phases, \(\Delta F={F}_{H}-{F}_{C}=0.138,0.007,\,{{{{{\rm{and}}}}}}-0.028\) (Fig. 2c). For each system, we prepared a cubic diamond cluster fully surrounded by a fluid phase at coexistence and equilibrated the system ("Methods"). We first distinguished particles in fluid and crystal based on the local volume fraction ("Methods"), which shows two peaks in distribution plots that indicate the two coexisting phases (Fig. 2g). Then, we colored particles in the crystal phase based on their face-to-face contact type \(({{{{{{\rm{S}}}}}}}_{n}{{{{{{\rm{E}}}}}}}_{m})\) and analyzed the crystal structures. When the free energy difference is large enough \(\left(\Delta F \sim+0.138\right)\), the cubic diamond cluster is stable without any notable changes in the crystal structure (Fig. 2f). However, when \(\Delta F\) is negligible (\(\Delta F \sim 0.007\)), the initial cubic diamond cluster repeatedly developed stacking faults (Fig. 2d) during the simulation, rather than stabilizing either diamond phase. This shows that the formation of twin boundaries between the two diamond phases is easily induced when the thermodynamic stability of the two crystals is comparable. On the other hand, when we further increased the stability of hexagonal diamond (\(\Delta F=-\!\!0.028\)), a complete phase transformation into a single-crystalline hexagonal diamond was observed (Fig. 2e). The phase transition behavior of the three systems was quantitatively compared by the change of the number ratio of \({{{{{{\rm{S}}}}}}}_{4}{{{{{{\rm{E}}}}}}}_{0}\) particles in a solid phase (\({N}_{{{{{{\rm{cubic}}}}}}}/{N}_{{{{{{\rm{solid}}}}}}}\)) (Fig. 2f). We confirmed a more gradual decrease of the \({N}_{{{{{{\rm{cubic}}}}}}}/{N}_{{{{{{\rm{solid}}}}}}}\) in the \(\Delta F=0.007\) system compared to that of the \(\Delta F=-\!\!0.028\) system, due to frequent formation of stacking faults.
Based on these findings, we hypothesized that the formation of a multiply twinned structure might be easily induced during the growth of the cubic diamond when \(\Delta F\) is negligible because of the easy formation of twin boundaries in multiple directions, which kinetically promotes the formation of ~\(72^\circ\) twin angles. To confirm the hypothesis, we conducted seed-assisted growth simulations ("Methods") with a system \((a=1.20,\,{c}=2.16)\) where \(\Delta F\) is negligible \((\Delta F=0.007)\). We placed a small spherical seed of cubic diamond crystal (\(N=500\)) in a fluid and allowed the seed to grow at the coexistence volume fraction \(\phi=0.58\) where the resulting crystal cluster is fully surrounded by a fluid phase in equilibrium. In the early stage of growth, we observed the formation of twin boundaries in multiple directions (Fig. 3a, b). Through subsequent multiple twinning events, a fivefold center was formed (Fig. 3c). Although the fivefold centers form a new local structure, \({{{{{{\rm{S}}}}}}}_{2}{{{{{{\rm{E}}}}}}}_{2}\) (Supplementary Fig. 2)—a pentagonal column —it has a similar local structure to that of hexagonal diamond, resulting in minimal entropy loss. Note that an error or defect can easily occur during the growth process because the formation of twin boundaries independently in multiple directions can cause a mismatch in the stacking sequence between two adjacent directions (Fig. 3d). However, interestingly, we observed a self-repair process during growth, where a growing direction with stacking errors re-melts until the boundary with its adjacent plane matches (Fig. 3e, f). Once the boundary matches, the fivefold twin crystal fully grows and is stabilized in fluid with a truncated pentagonal dipyramid shape (Fig. 3g, h). This error-and-repair mechanism of fivefold twinned crystals occurs via particle migration on the crystal surface, differently from the error-and-repair mechanism recently reported, in which particle rearrangement occurs throughout the fivefold twinned crystal during growth14, and differently from that reported recently for a dodecagonal quasicrystal of hard tetrahedra31. The observation of this new mechanism demonstrates that there exist multiple types of error-and-repair mechanisms for the formation of fivefold twinned clusters as well as for entropically stabilized colloidal crystals.
Next, we investigated the formation of an icosahedral twinned cluster within a fluid phase. Experiments have shown that an icosahedral twinned cluster can be formed from additional multiple twinning from a decahedral cluster12,32. Simulations have shown it can be formed in a purely entropy-driven system provided there is artificial confinement of the cluster. Our simulation results show that this is possible also in a purely entropy-driven system, but without confinement. We conducted a growth simulation of a seed with a fivefold twinned structure \((N=1020)\) in a system where \(\Delta F\) is negligible (\(\Delta F \sim 0.007\) at \(a=1.20,\,{c}=2.16\)) ("Methods"), and where the solid cluster is fully surrounded by fluid throughout the simulation. We observed the additional twinning from the surface of the seed, which eventually grew into an icosahedral twin (Fig. 3i–l), a process reminiscent of a simultaneous successive twinning of a palladium icosahedral cluster32. The fluid-solid coexistence of our ico-twin was confirmed by the local volume fraction distribution (Fig. 3n) showing two peaks. We classified particles into three types: fluid, interface and crystal. After removing particles at the interface, we confirmed that the icosahedral cluster exposes twenty (111) planes of the cubic diamond crystal toward the fluid (Fig. 3m). This implies that the interfacial entropy difference between fluid and the (111) plane plays an important role in the stability of the icosahedral cluster17.
The stability of the icosahedral cluster in the fluid can be checked by tracking the change in reduced system pressure (\({P}^{*}=P{v}_{0}/{k}_{B}T\)) during a simulation run (Fig. 3o). In the early stage (\(\le 50\times {10}^{5}\) MC steps), we observed a rapid decrease in pressure due to the growth of the icosahedral cluster. After the growth (\( > 50\times {10}^{5}\) MC steps), the pressure stabilized at a constant value (\({P}^{*} \sim 12.6\)) with small fluctuations, indicating that the ico-twin structure had stabilized in the fluid. From the simulation snapshots and diffraction patterns at \(100\times {10}^{5}\) and \(175\times {10}^{5}\) MC steps, we confirmed that the icosahedral symmetry of the solid cluster was maintained (Fig. 3o, insets).
Structurally, an icosahedral twin has internal and surface strain33, thus an internal or external effect to compensate for the strain energy is required to stabilize the cluster. For instance, we know that hard sphere systems require spherical confinement to entropically stabilize an icosahedral cluster4,7,23, otherwise, the icosahedral cluster rapidly destabilizes (Fig. 4a). This indicates that the fluid-solid interfacial tension of hard spheres is not strong enough to overcome the entropy loss from the strain within the icosahedral cluster. On the other hand, the icosahedral cluster of hard TTs is stable in a fluid without geometric confinement, suggesting that the fluid-solid interfacial tension is enhanced by the shape of the particle. Figure 4b shows the crystal structure of the (111) plane in the clusters of hard TTs and hard spheres, which is the largest surface of the icosahedral cluster exposed to the fluid. The (111) surface of the hard TT crystal is flat because the faces of the TTs are exposed toward the plane. On the other hand, the (111) surface of the hard sphere crystal is rough due to the spherical shape of the particles, implying that the roughness of the surface can affect the surface tension of the crystal.
To quantify the difference in the surface effect, we calculated the fluid-solid interfacial free energy of hard TTs and hard spheres using the capillary fluctuation method34 ("Methods"). We exposed a specific lattice plane of a crystal toward a fluid in a thin slab (\(z\)-axis in Fig. 4c, d) and monitored the fluctuation of the interface in equilibrium. The interfacial profile normalized by the circumscribed radius of the particle, \((h-{h}_{0})/{r}_{{{{{{\rm{circ}}}}}}}\), shows that the fluctuations of the (111) plane of the hard sphere system are larger than that of the hard truncated tetrahedra system (Fig. 4e and Supplementary Movies 1, 2). From 200 samples of the interfacial profiles obtained in equilibrium, we calculated the interfacial stiffness (\(\widetilde{\gamma }\)) of the crystal planes (Supplementary Fig. 6 and "Methods") and measured the fluid-solid interfacial free energy \((\gamma )\) of several major crystal orientations (Supplementary Table 2). The interfacial stiffness of the (111) plane of the fcc crystal of hard spheres, \({\widetilde{\gamma }}_{{{{{{\rm{HS}}}}}\_}(111)}=0.684{k}_{B}T{\sigma }^{-2}\), is \(56\%\) lower than that of the cubic diamond crystal of hard truncated tetrahedra, \({\widetilde{\gamma }}_{{{{{{\rm{TT}}}}}\_}(111)}=1.547{k}_{B}T{\sigma }^{-2}\) ("Methods"). In addition, we confirmed that the fluid-solid interfacial free energy difference (surface tension) of the (111) plane of the hard TT crystal, \({\gamma }_{{{{{{\rm{TT}}}}}\_}(111)}=1.347{k}_{B}T{\sigma }^{-2}\), is \(\sim 2.4\) times higher than that of the hard sphere crystal, \({\gamma }_{{{{{{\rm{HS}}}}}\_}(111)}=0.560{k}_{B}T{\sigma }^{-2}\) (Supplementary Table 2). This indicates that a strong fluid-solid interfacial tension of the hard TT system stabilizes the icosahedral twinned cluster in fluid without geometric confinement.
The dependence of the fluid-solid interfacial tension on particle shape is indirectly indicated by the change in local volume fraction distributions arising from TT shape change (Supplementary Fig. 5). The volume fraction of fluid and crystal at fluid-solid coexistence depends on the fluid-solid surface tension; thus, the local volume fraction change due to TT truncation suggests a change in fluid-solid interfacial energy from the TT shape change. To confirm this hypothesis, we compared three different hard particle systems with very different local volume fraction distributions at fluid-solid coexistence (Supplementary Fig. 7): (1) Hard triangular bipyramids (TBP) forming clathrate type-1 crystal22, (2) hard TTs with \(a=1.20\), \(c=2.16\) forming cubic diamond, and (3) hard spheres (HS) forming FCC crystal. The three systems show different local volume fraction difference between crystal and fluid (\({\phi }_{{{{{{\rm{Crystal}}}}}}}-{\phi }_{{{{{{\rm{Fluid}}}}}}}=\triangle \phi\)): \(\triangle {\phi }_{{{{{{\rm{TBP}}}}}}}=0.116\), \(\triangle {\phi }_{{{{{{\rm{TT}}}}}}}=0.077\) and \(\triangle {\phi }_{{{{{{\rm{HS}}}}}}}=0.05\) (Supplementary Fig. 7). Next, we conducted capillary fluctuation simulations for the three systems to calculate the interfacial stiffness of each system. We observed that the TBPs with the largest local volume fraction difference (\(\triangle \phi=0.116\)) show the smallest fluctuations of the interface, resulting in the strongest interfacial stiffness (\(\widetilde{\gamma } \sim 7.21{k}_{B}T{\sigma }^{-2}\)) (Supplementary Fig. 7b, c). On the other hand, the HSs with the smallest local volume fraction difference (\(\triangle \phi=0.05\)) show the largest fluctuations of the interface, resulting in the weakest interfacial stiffness (\(\widetilde{\gamma } \sim 0.74{k}_{B}T{\sigma }^{-2}\)) (Supplementary Fig. 7h, i). This comparison provides clear evidence that particle shape determines the volume fraction of fluid and solid at coexistence, which in turn determines the interfacial energy.
In summary, we studied the entropy-driven assembly and stabilization of fivefold and icosahedral twinned clusters in a one-component fluid of hard truncated tetrahedra (TT). We showed that the thermodynamic stability of the cubic and hexagonal diamond phases can be entropically controlled by designing the TT shape. If the particle shape is designed to have a negligible free energy difference between the two diamond crystals, the formation of twin boundaries is easily induced. This strategy can be used to induce the formation of fivefold and icosahedral twinned clusters in fluid via seed-assisted growth. We found that the formation of a fivefold cluster follows an error-and-repair mechanism that removes mismatches of the stacking sequence of adjacent twin planes through particle rearrangement at the cluster surface. We showed the formation of an icosahedral cluster by additional growth of twinned structures from a fivefold twinned seed. We showed that the icosahedral cluster of TTs can be entropically stabilized in a fluid, which is not possible for hard spheres without confinement. The capillary fluctuation method showed that a hard TT system has a much higher fluid-solid interfacial free energy (here, entropy) difference than that of a hard sphere system, directly implicating fluid-solid interfacial tension in stabilizing the TT ico-twin.
Our findings provide a quantitative understanding of the formation and stabilization of fivefold and icosahedral twinned clusters in hard particle systems. Importantly, we showed that the twinning behavior and the interfacial free energy difference may be entropically engineered by particle shape design. Experimentally, colloidal tetrahedral particles are synthesizable26,35,36,37,38 with tunable shape37. Non-complementary DNA could be used to make non-interacting particles that might closely approximate those studied here. Fivefold and icosahedral twinned clusters should be also attainable with attractive TT particle shapes, provided the explicit attraction favors face-to-face alignment. Such interparticle attraction could be realized through organic ligands36,38 via van der Waals forces or self-complementary DNA26, both of which are commonly used to drive nanoparticle assembly.
Particle geometry
The truncated tetrahedron (TT) shape is a member of the spherical triangle invariant 323 family, with truncation parameters \(\left(a,b,c\right)\) according to previous convention39. In this study, \(b=1.0\) for all systems, and \(a\) and \(b\) vary in a range: \(1.14\le a\le 1.30\) and \(2.10\le c\le 2.30\).
Identification of staggered and eclipsed pairs
Each TT in a crystal phase has four nearest neighbors that form face-to-face contacts. We identified the type of each face-to-face contact (staggered or eclipsed), following a similar identification protocol described in ref. 28. Briefly, for each face of the particle, we assigned three vectors that are defined from the face center to each tip of the face (Supplementary Fig. 1a). Then, for each pair contact (\(i,j\)), we can define three vectors for \(i\) particle and three other vectors for \(j\) particle (Supplementary Fig. 1b, c). We calculated every pair combination between the \(i\) and \(j\) vectors, which is nine, and found the minimum angle (\(\theta\)). All the \(\theta\) from every pair contact accumulated from the entire system give an angle distribution. For instance, Supplementary Fig. 1d was calculated from hexagonal diamond crystal. The angle distribution shows a clear bimodal distribution, and the height of the peak at \({\theta }_{1}\) and \({\theta }_{2}\) are around a 1:3 ratio, indicating that the angles around \({\theta }_{1}\) and \({\theta }_{2}\) come from eclipsed pairs and staggered pairs, respectively. This allows us to select the range of the pair contact angle to identify the eclipsed pair (\({\theta }_{1}\pm 10^\circ\)) and the staggered pair (\({\theta }_{2}\pm 10^\circ\)).
Monte Carlo simulation
Simulations were performed by the hard particle Monte Carlo (HPMC) implemented in the HOOMD-blue simulation package40,41, which is available at https://github.com/glotzerlab/hoomd-blue. The system size for the self-assembly simulations (Fig. 2) is \(N={{{{\mathrm{2000}}}}}\), and all simulations were performed under periodic box condition. The translational and rotational movement of particles is decided to have around a \(20\%\) acceptance ratio for every simulation. The self-assembly simulations were initialized from a dilute isotropic fluid \(\phi\) \(=N{v}_{0}/V < 0.01\) and compressed until the desired thermodynamic condition (\(\phi\) or reduced pressure \({P}^{*}=P{v}_{0}/{k}_{B}T\)) was reached. Here, \({v}_{0}\) and \(V\) are the volume of a particle and the volume of the simulation box, respectively. The unit length of simulation is defined by \(\sigma\), and the particle volume of a TT is set to \({{v}_{0}=1.0\sigma }^{3}\) regardless of its truncation amount. After initialization, each run was continued in the isochoric ensemble (NVT) at constant particle volume fraction \(\phi\) until equilibration was reached. For the most systems, crystallization occurs within \(1.0\times {10}^{8}\) Monte Carlo steps.
Stability of diamond crystals in a dense fluid
We checked the stability of a cubic diamond crystal in a dense fluid for three different systems: (1) \(a=1.16,\,{c}=2.16\) at \(\phi=0.57\), (2) \(a=1.20,\,{c}=2.16\) at \(\phi=0.57\)5 (Fig. 2d), and (3) \(a=1.24,\,{c}=2.16\) at \(\phi=0.58\) (Fig. 2e). All three systems are in a fluid-solid coexistence state, where a single-crystalline cluster is fully surrounded by a dense fluid phase. System size is \(N={{{{\mathrm{20,000}}}}}\) for all three systems. For the MC simulation, we first placed a spherical cubic diamond single-crystalline cluster (\(N \sim {{{{\mathrm{8000}}}}}\)) at the center of a large simulation box \((\phi=0.01)\) and put other particles around the solid cluster without overlapping each other. Then, holding the particles of the crystal phase (no movements), we rapidly compressed the system to the target particle volume fraction. Then, we released the particles of the crystal and equilibrated the whole system.
Free-energy calculation
The reduced Helmholtz free energy per particle \(F/N{k}_{B}T\) of the cubic and hexagonal diamond phases were calculated using the Frenkel–Ladd method24,30 at a constant volume. The details of the calculation method can be found in ref. 30. Briefly, we first constructed an Einstein lattice (i.e., reference state) of each phase, for which the free energy is analytically solvable. Then, we applied strong translational and rotational harmonic potentials between the hard TTs and the Einstein lattice to tether each particle to its reference lattice site, which makes the hard TT system a reference state. Then we gradually released the harmonic potential until the strength of the potential became nearly zero, yielding the real system without correlation to the reference state. This process allows us to set up a continuous and reversible path between the reference state and a real state. Thus, by integrating the potential energy along the path, we can calculate the free energy (or equivalently, entropy) difference between the reference state and the real system. Because that the free energy of the reference crystal is analytically solvable, we can calculate the absolute free energy of the real crystal.
Using this method, we calculated the free energy of each phase with different shape parameters \((a,{c})\) to construct a free energy surface (Fig. 2a) in the shape space: \(1.14\le a\le 1.30\) with 0.02 interval and \(2.10\le c\le 2.30\) with 0.02 interval, for a total of 99 systems with different shape parameters \((a,{c})\). The system size is \(N=512\) for the cubic diamond and \(N=992\) for the hexagonal diamond. For each phase, we calculated the free energy of the 99 systems at \(\phi=0.64\), \(0.66\), \(0.68\), \(0.70\) and \(0.72\), and for each \(\phi\), the free energy surface was constructed by interpolating the 99 data points (Supplementary Fig. 4). The free energy of the crystal at \(\phi=0.62\) (Fig. 2a) was estimated by extrapolating the free energy surface plots of the five particle volume fractions studied.
Seed-assisted growth MC simulation
We conducted seed-assisted growth simulation at a fluid-solid coexistence state (\(\phi=0.58\) for \(a=1.20,\,{c}=2.16\) system). For the growth of the fivefold twinned cluster, we used a spherical cluster of a cubic diamond crystal (\(N=500\)) as a seed, and the total system size including the seed is \(N={{{{\mathrm{20,000}}}}}\). At a very dilute condition (\(\phi=0.01\)), we placed the seed at the center of simulation box and placed other particles around the seed without overlap. Fixing the seed particles in place, we compressed the simulation box to the target particle volume fraction. Once it reached the target particle volume fraction, we released the seed particles and equilibrated the system. For the growth of the icosahedral twinned cluster, we used a fivefold twinned cluster (\(N \sim {{{{\mathrm{1000}}}}}\)) as a seed, and the total system size including the seed is \(N={{{{\mathrm{20,000}}}}}\). The compression and equilibration protocol is the same in both cases.
Local volume fraction calculation
Seed-assisted growth simulations for fivefold and icosahedral twinned clusters were performed in at coexistence between a fluid phase and a crystal phase, where the solid clusters are fully surrounded by fluid. To distinguish the two phases, we calculated the local particle volume fraction \({\phi }_{{{{{{\rm{loc}}}}}}}\), which is defined as the particle volume fraction around a particle within a certain radius, \({r}_{{{{{{{\rm{cut}}}}}}}}\). The value of \({r}_{{{{{{{\rm{cut}}}}}}}}\) in this study was taken to be 3 times the distance from a particle center to its nearest neighbor particle center, in order to properly average the local environment. TTs are identified as belonging to the fluid region if \({\phi }_{{{{{{\rm{loc}}}}}}}\le 0.60\) and to the crystal region if \({\phi }_{{{{{{\rm{loc}}}}}}} > 0.60\). We utilized the freud python library for this calculation42.
Stability check of an icosahedral twinned cluster of hard spheres
We performed MC simulations of a hard sphere system to check the stability of an icosahedral twinned cluster of hard spheres at fluid-solid coexistence (\(\phi=0.515\)). We first constructed an icosahedral cluster (\(N=5971\)) with an ideal structure, following the method described in ref. 4. At a very dilute condition (\(\phi=0.01\)), we placed the ico-twin cluster at the center of simulation box and placed other hard spheres around the cluster without overlap. Temporarily "freezing" the ico-twin particles in place, we compressed the simulation box to the target particle volume fraction. Once it reached the target particle volume fraction, we released the ico-twin cluster particles and equilibrated the system. To check if the ico-twin cluster is stable in the coexistence phase, we monitored the bond-order diagram and the morphology of the ico-twin cluster during the simulation. We observed the icosahedral structure rapidly destabilize within \(5\times {10}^{5}\) MC steps (Fig. 4a).
Capillary fluctuation method
The fluid-solid interfacial free energy of hard TT (\(a=1.20,\,{c}=2.16\)) and hard sphere systems was calculated by the capillary fluctuation method34. We prepared systems in a fluid-solid coexistence state (\(\phi=0.582\) for hard TTs and \(\phi=0.515\) for hard spheres) containing a crystal, where the crystal of each system exposes different orientations toward a fluid phase: \((111)[\bar{1}10]\), \((100)[001]\), \((100)[\bar{1}10]\) and \((110)[001]\). Here, the crystal orientation is defined as \(({ijk})[{lmn}]\), where \(({ijk})\) is a crystal plane toward the fluid phase (\(z\)-direction in Fig. 4c, d) and \([{lmn}]\) is a crystal plane perpendicular to the fluid phase along the short direction (\(y\)-direction in Fig. 4c, d). The system size is \(N \sim {{{{\mathrm{40,000}}}}}\) for every system, and the shortest direction of the simulation box contains 2–3 unit cells (\(y\)-direction). At coexistence, we distinguished fluid and solid phases using the local volume fraction \({\phi }_{{{{{{\rm{loc}}}}}}}\) for the hard TT system (\({\phi }_{{{{{{\rm{loc}}}}}}} > 0.6\) for crystal phase) and the average bond-orientational order parameter42,43 \({\bar{q}}_{6}\) for the hard sphere system (\({\bar{q}}_{6} > 0.3\) for crystal phase).
For the calculation of the fluid-solid interfacial free energy, we followed the calculation process described in ref. 34. Briefly, we obtained the interfacial profile \(h(x)\) as a function of MC steps (Fig. 4e–g), which is then normalized by the circumscribed sphere radius of a particle. The values of \(h(x)\) were measured at a discrete set of points \({x}_{n}=n\Delta\), where \(n=1,\ldots,{N}\) and \(\Delta={{L}}/{{N}}\), where L is the length of crystal along \(x\)-direction. The Fourier modes \({h}_{q}\) are defined as \({h}_{q}=\frac{1}{N}{\sum }_{n=1}^{N}h({x}_{n}){{{{{{\rm{e}}}}}}}^{{{{{{{\rm{i}}}}}}q}{x}_{n}}\), where wave number \(q=\frac{2\pi k}{L}\) with \(k=1,\ldots,{N}\). From the equipartition theorem, the size of the capillary fluctuation modes is related to the interfacial stiffness \(\widetilde{\gamma }\) as follows44:
$$\left\langle {\left|{h}_{q}\right|}^{2}\right\rangle={k}_{B}T/A\widetilde{\gamma }{q}^{2}$$
where \(A\) is the area of the fluid-crystal interface. The \(\widetilde{\gamma }\) for different crystal orientations were calculated and averaged for various \(q\) and \(\varDelta x\) (Supplementary Fig. 6). The interfacial stiffness is related to the interfacial free energy \(\gamma\) by the formula:
$$\widetilde{\gamma }\left(\theta \right)=\gamma \left(\theta \right)+\frac{{{{{{{\rm{d}}}}}}}^{2}\gamma }{{{{{{\rm{d}}}}}}{\theta }^{2}}$$
where \(\theta\) is the angle between the average orientation of the interface and the local normal of the interface. Based on this relationship, the interfacial free energy can be obtained by comparing the following two equations:
$$\gamma \left({{{{{\boldsymbol{n}}}}}}\right)/{\gamma }_{0}=1+{A}_{1}{\epsilon }_{1}+{A}_{2}{\epsilon }_{2}$$
$$\widetilde{\gamma }\left({{{{{\boldsymbol{n}}}}}}\right)/{\gamma }_{0}=1+{B}_{1}{\epsilon }_{1}+{B}_{2}{\epsilon }_{2}$$
where \({{{{{\boldsymbol{n}}}}}}\) is the unit vector normal to the interfacial plane. The values of \({A}_{1}\), \({A}_{2}\), \({B}_{1}\) and \({B}_{2}\) for each interface direction is listed in Supplementary Table 1. Then, the interfacial free energy \(\gamma\) for each crystal plane was obtained using the coefficients (Supplementary Table 1) and the interfacial stiffness (Supplementary Fig. 6). The calculation results are listed in Supplementary Table 2.
All data needed to evaluate the conclusions in this manuscript are present in the main text or the supplementary materials. Additional data are available upon request to the corresponding author.
Source code for HOOMD-Blue is available at https://github.com/glotzerlab/hoomd-blue.
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This work was supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award # DE-SC0000989. This work used the Extreme Science and Engineering Discovery Environment (XSEDE)45, which is supported by National Science Foundation grant number ACI-1053575; XSEDE award DMR 140129. Computational resources and services also supported by Advanced Research Computing at the University of Michigan, Ann Arbor.
Sangmin Lee
Present address: Department of Biochemistry, University of Washington, Seattle, WA, USA
Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA
Sangmin Lee & Sharon C. Glotzer
Biointerfaces Institute, University of Michigan, Ann Arbor, MI, USA
Sharon C. Glotzer
S.C.G. directed the research. S.L. designed the study and performed all simulations. Both authors contributed to data analysis and manuscript preparation.
Correspondence to Sharon C. Glotzer.
Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work.
Description of Additional Supplementary Files
Supplementary Movie 1
Lee, S., Glotzer, S.C. Entropically engineered formation of fivefold and icosahedral twinned clusters of colloidal shapes. Nat Commun 13, 7362 (2022). https://doi.org/10.1038/s41467-022-34891-5 | CommonCrawl |
The origin of very wide binary stars [PDF]
M. B. N. Kouwenhoven,S. P. Goodwin,Melvyn B. Davies,Richard J. Parker,P. Kroupa,D. Malmberg
Abstract: A large population of fragile, wide (> 1000 AU) binary systems exists in the Galactic field and halo. These wide binary stars cannot be primordial because of the high stellar density in star forming regions, while formation by capture in the Galactic field is highly improbable. We propose that these binary systems were formed during the dissolution phase of star clusters (see Kouwenhoven et al. 2010, for details). Stars escaping from a dissolving star cluster can have very similar velocities, which can lead to the formation of a wide binary systems. We carry out N-body simulations to test this hypothesis. The results indicate that this mechanism explains the origin of wide binary systems in the Galaxy. The resulting wide binary fraction and semi-major axis distribution depend on the initial conditions of the dissolving star cluster, while the distributions in eccentricity and mass ratio are universal. Finally, since most stars are formed in (relatively tight) primordial binaries, we predict that a large fraction of the wide "binary stars" are in fact higher-order multiple systems.
Formation of the Galactic stellar halo. Origin of the metallicity-eccentricity relation [PDF]
Kenji Bekki,Masashi Chiba
Physics , 2000, DOI: 10.1086/312636
Abstract: Motivated by the recently improved knowledge on the kinematic and chemical properties of the Galactic metal-poor stars, we present the numerical simulation for the formation of the Galactic stellar halo to interpret the observational results. As a model for the Galaxy contraction, we adopt the currently standard theory of galaxy formation based on the hierarchical assembly of the cold dark matter fluctuations. We find, for the simulated stars with [Fe/H]$\le-1.0$, that there is no strong correlation between metal abundances and orbital eccentricities, in good agreement with the observations. Moreover, the observed fraction of the low eccentricity stars is reproduced correctly for [Fe/H]$\le-1.6$ and approximately for the intermediate abundance range of $-1.6<$[Fe/H]$\le-1.0$. We show that this successful reproduction of the kinematics of the Galactic halo is a natural consequence of the hierarchical evolution of the subgalactic clumps seeded from the cold dark matter density fluctuations.
Orbital Eccentricity Distribution of Solar-Neighbour Halo Stars [PDF]
Kohei Hattori,Yuzuru Yoshii
Physics , 2011, DOI: 10.1111/j.1365-2966.2011.19639.x
Abstract: We present theoretical calculations for the differential distribution of stellar orbital eccentricity for a sample of solar-neighbour halo stars. Two types of static, spherical gravitational potentials are adopted to define the eccentricity e for given energy E and angular momentum L, such as an isochrone potential and a Navarro-Frenk-White potential that can serve as two extreme ends covering in-between any realistic potential of the Milky Way halo. The solar-neighbour eccentricity distribution \Delta N(e) is then formulated, based on a static distribution function of the form f(E,L) in which the velocity anisotropy parameter \beta monotonically increases in the radial direction away from the galaxy center, such that beta is below unity (near isotropic velocity dispersion) in the central region and asymptotically approaches \sim 1 (radially anisotropic velocity dispersion) in the far distant region of the halo. We find that \Delta N(e) sensitively depends upon the radial profile of \beta, and this sensitivity is used to constrain such profile in comparison with some observational properties of \Delta N_{obs}(e) recently reported by Carollo et al. (2010). Especially, the linear e-distribution and the fraction of higher-e stars for their sample of solar-neighbour inner-halo stars rule out a constant profile of \beta, contrary to the opposite claim by Bond et al. (2010). Our constraint of \beta \lesssim 0.5 at the galaxy center indicates that the violent relaxation that has acted on the inner halo is effective within a scale radius of \sim 10 kpc from the galaxy center. We discuss that our result would help understand the formation and evolution of the Milky Way halo.
A new test for the Galactic formation and evolution -- prediction for the orbital eccentricity distribution of the halo stars [PDF]
Abstract: We present theoretical calculations for the differential distribution of stellar orbital eccentricity in a galaxy halo, assuming that the stars constitute a spherical, collisionless system in dynamical equilibrium with a dark matter halo. In order to define the eccentricity e of a halo star for given energy E and angular momentum L, we adopt two types of gravitational potential, such as an isochrone potential and a Navarro-Frenk-White potential, that could form two ends covering in-between any realistic potential of dark matter halo. Based on a distribution function of the form f(E,L) that allows constant anisotropy in velocity dispersions characterized by a parameter \beta, we find that the eccentricity distribution is a monotonically increasing function of e for the case of highly radially anisotropic velocity dispersions (\beta > 0.6), while showing a hump-like shape for the cases from radial through tangential velocity anisotropy (\beta < 0.6). We also find that when the velocity anisotropy agrees with that observed for the Milky Way halo stars (\beta = 0.5-0.7), a nearly linear eccentricity distribution of N(e) \alpha e results at e < 0.7, largely independent of the potential adopted. Our theoretical eccentricity distribution would be a vital tool of examining how far out in the halo the dynamical equilibrium has been achieved, through comparison with kinematics of halo stars sampled at greater distances. Given that large surveys of the SEGUE and Gaia projects would be in progress, we discuss how our results would serve as a new guide in exploring the formation and evolution of the Milky Way halo.
Super Massive Black Holes and the Origin of High-Velocity Stars [PDF]
Roberto Capuzzo-Dolcetta,Giacomo Fragione
Abstract: The origin of high velocity stars observed in the halo of our Galaxy is still unclear. In this work we test the hypothesis, raised by results of recent high precision $N$-body simulations, of strong acceleration of stars belonging to a massive globular cluster orbitally decayed in the central region of the host galaxy where it suffers of a close interaction with a super massive black hole, which, for these test cases, we assumed $10^8$ M$_\odot$ in mass.
Cosmological Origin of the Lowest Metallicity Halo Stars [PDF]
Xavier Hernandez,Andrea Ferrara
Abstract: We explore the predictions of the standard hierarchical clustering scenario of galaxy formation, regarding the numbers and metallicities of PopIII stars likely to be found within our Galaxy today. By PopIII we shall be referring to stars formed at large redshift ($z>4$), with low metallicities ($[Z/Z_{\odot}]<-2.5$) and in small systems (total mass $\simlt$ $2\times 10^{8} M_{\odot}$) that are extremely sensitive to stellar feedback, and which through a prescribed merging history (Lacey & Cole 1993) end up becoming part of the Milky Way today. An analytic, extended Press-Schechter formalism is used to get the mass functions of halos which will host PopIII stars at a given redshift, and which will end up in Milky Way sized systems today. Each of these is modeled as a mini galaxy, with a detailed treatment of the dark halo structure, angular momentum distribution, final gas temperature and disk instabilities, all of which determine the fraction of the baryons which are subject to star formation. Use of new primordial metallicity stellar evolutionary models allows us to trace the history of the stars formed, give accurate estimates of their expected numbers today, and their location in $L/L_{\odot}$ vs. $T/K$ HR diagrams. A first comparison with observational data suggests that the IMF of the first stars was increasingly high mass weighted towards high redshifts, levelling off at $z\simgt 9$ at a characteristic stellar mass scale $m_s=10-15 M_\odot$.
On the origin of high-velocity runaway stars [PDF]
V. V. Gvaramadze,A. Gualandris,S. Portegies Zwart
Abstract: We explore the hypothesis that some high-velocity runaway stars attain their peculiar velocities in the course of exchange encounters between hard massive binaries and a very massive star (either an ordinary 50-100 Msun star or a more massive one, formed through runaway mergers of ordinary stars in the core of a young massive star cluster). In this process, one of the binary components becomes gravitationally bound to the very massive star, while the second one is ejected, sometimes with a high speed. We performed three-body scattering experiments and found that early B-type stars (the progenitors of the majority of neutron stars) can be ejected with velocities of $\ga$ 200-400 km/s (typical of pulsars), while 3-4 Msun stars can attain velocities of $\ga$ 300-400 km/s (typical of the bound population of halo late B-type stars). We also found that the ejected stars can occasionally attain velocities exceeding the Milky Ways's escape velocity.
Galactic Halo Stars in Phase Space :A Hint of Satellite Accretion? [PDF]
Chris B. Brook,Daisuke Kawata,Brad K. Gibson,Chris Flynn
Abstract: The present day chemical and dynamical properties of the Milky Way bear the imprint of the Galaxy's formation and evolutionary history. One of the most enduring and critical debates surrounding Galactic evolution is that regarding the competition between ``satellite accretion'' and ``monolithic collapse''; the apparent strong correlation between orbital eccentricity and metallicity of halo stars was originally used as supporting evidence for the latter. While modern-day unbiased samples no longer support the claims for a significant correlation, recent evidence has been presented by Chiba & Beers (2000,AJ,119,2843) for the existence of a minor population of high-eccentricity metal-deficient halo stars. It has been suggested that these stars represent the signature of a rapid (if minor) collapse phase in the Galaxy's history. Employing velocity- and integrals of motion-phase space projections of these stars, coupled with a series of N-body/Smoothed Particle Hydrodynamic (SPH) chemodynamical simulations, we suggest an alternative mechanism for creating such stars may be the recent accretion of a polar orbit dwarf galaxy.
Oxygen abundances in low- and high-alpha field halo stars and the discovery of two field stars born in globular clusters [PDF]
I. Ramirez,J. Melendez,J. Chaname
Physics , 2012, DOI: 10.1088/0004-637X/757/2/164
Abstract: Oxygen abundances of 67 dwarf stars in the metallicity range -1.6<[Fe/H]<-0.4 are derived from a non-LTE analysis of the 777 nm O I triplet lines. These stars have precise atmospheric parameters measured by Nissen and Schuster, who find that they separate into three groups based on their kinematics and alpha-element (Mg, Si, Ca, Ti) abundances: thick-disk, high-alpha halo, and low-alpha halo. We find the oxygen abundance trends of thick-disk and high-alpha halo stars very similar. The low-alpha stars show a larger star-to-star scatter in [O/Fe] at a given [Fe/H] and have systematically lower oxygen abundances compared to the other two groups. Thus, we find the behavior of oxygen abundances in these groups of stars similar to that of the alpha elements. We use previously published oxygen abundance data of disk and very metal-poor halo stars to present an overall view (-2.3<[Fe/H]<+0.3) of oxygen abundance trends of stars in the solar neighborhood. Two field halo dwarf stars stand out in their O and Na abundances. Both G53-41 and G150-40 have very low oxygen and very high sodium abundances, which are key signatures of the abundance anomalies observed in globular cluster (GC) stars. Therefore, they are likely field halo stars born in GCs. If true, we estimate that at least 3+/-2% of the local field metal-poor star population was born in GCs.
Formation of the Galactic Halo [PDF]
Ortwin Gerhard
Abstract: Recent observational and theoretical work suggests that the formation of the Galactic stellar halo involved both dissipative processes and the accretion of subfragments. With present data, the fraction of the halo for which an accretion origin can be substantiated is small, of order 10 percent. The kinematics of the best halo field star samples show evidence for both dissipative and dissipationless formation processes. Models of star-forming dissipative collapse, in a cosmological context and including feedback from star formation, do not confirm the simple relations between metallicity, rotation velocity, and orbital eccentricity for halo stars as originally predicted. The new model predictions are much closer to the observed distributions, which have generally been interpreted as evidence for an accretion origin. These results are broadly consistent with a hierarchical galaxy formation model, but the details remain to be worked out. | CommonCrawl |
Harold F. Dodge
Harold French Dodge (January 23, 1893 in Lowell, Massachusetts – December 10, 1976) was one of the principal architects of the science of statistical quality control. His father was the photographer William H. Dodge. Harold Dodge is universally known for his work in originating acceptance sampling plans for putting inspection operations on a scientific basis in terms of controllable risks. Dodge earned his B.S. in Electrical Engineering from M.I.T. in 1916 and his A.B. (Master's degree) in Physics from Columbia University in 1917.
From 1917 to 1958 worked at quality assurance department at Bell Laboratories with Walter Shewhart, George Edwards, Harry Romig, R. L. Jones, Paul Olmstead, E.G.D. Paterson, and Mary N. Torrey. At that time the basic concepts of acceptance sampling was developed, such as:
• Consumer's risk
• Producer's risk
• Double sampling
• Lot tolerance percent defective (LTPD)
• Average outgoing quality limit (AOQL)
Also he originated several types of:
• Acceptance sampling schemes
• CSP type continuous sampling plans
• Chain sampling plans
• Skip-lot sampling plans
During World War II, Dodge had an office in the Pentagon and served as a consultant to the Secretary of War, to NASA and the Sandia Corporation. He was also chairman of the American Standards Association (now the American National Standards Institute) War Committee Z1, which prepared the Z1.1, Z1.2, and Zl.3 quality control standards.
After he retired from Bell Labs in 1958, Dodge became a professor of applied mathematical statistics at Rutgers.
The American Society for Testing and Materials honors Harold Dodge's memory with the Harold F. Dodge Award.
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| Wikipedia |
\begin{document}
\thispagestyle{empty} \begin{flushright} \begin{spacing}{0} \textit{\footnotesize key words:} \\ \textit{\footnotesize measure, category, decomposition, full subset, Luzin, nonmeasurable set} \end{spacing}
\end{flushright}
\begin{flushleft} Marcin MICHALSKI\footnote{Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław. The author was supported by grant S50129/K1102 (0401/0086/16), Faculty of Fundamental Problems of Technology.} \end{flushleft}
\begin{center} \textbf{{\fontsize{13}{13}\selectfont REDISCOVERED THEOREM OF LUZIN}} \end{center}
\begin{adjustwidth}{0.5cm}{0pt} \footnotesize In 1934 N. N. Luzin proved in his short (but dense) paper \textit{Sur la decomposition des ensembles} that every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. We will try to (at least partially) decipher the reasoning of Luzin and prove this result following his idea. \end{adjustwidth}
{\normalsize \section{INTRODUCTION AND PRELIMINARIES} } \hspace{0.50cm} Although there are more recent and stronger results on topic of decompositions, involving advanced methods (e.g. Gittik-Shelah theorem in \cite{G-S}), there is a unique charm in classic results. In this paper we will provide a detailed proof of one of such results, namely, the following theorem of Luzin (\cite{Luzin1})
\begin{myth} Every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. \end{myth}
A need for such a endeavor arose in us mainly for two reasons. The first is curiosity since the formulation of the theorem gives an impression that it can be proved in a quite simple way. As it has occurred it is not the case. The second reason is that the paper of Luzin is not, let us say, reader -friendly. It has many shortcomings of works published at that time - it lacks precise definitions, most claims come without a proof and key reasonings are hard to interpret due to a relatively frivolous language. In the case of the category Luzin lost us right after defining the set $T$ and sets $H_n$. We took off from there on our own with the Lemma \ref{Lemma for Luzin} and the rest of of our reasoning seems not to correspond to that of Luzin. In the case of measure we managed to follow Luzin quite well, although parts where Lemma \ref{Fubini for Lebesgue outer measure} are needed, as well, as the finish, where one has to invoke the ccc property and Borel hulls, came originally without the proof. \\ It should be also noted that we are not the only ones that found this result captivating. In the same time and independently from us E. Grzegorek and I. Labuda published a marvelous work \cite{GL} on the topic, including the result of Luzin (calling it "forgotten"), with a wide historical overview. They also took different approach - they were much more patient with the lecture of the work of Luzin and managed to translate the reasoning literally to the letter. We strongly recommend their paper for any reader interested in the historical context of the topic and analysis of similar results from that time.
We will use the standard set theoretic notation, e.g. as in \cite{JECH}. We denote the real line by $\mathbb{R}$. By Greek alphabet letters $\alpha, \beta, \gamma, \kappa, … $we denote ordinal numbers with an exception of $\lambda$ ($\lambda^*$) which denotes the Lebesgue (outer) measure on $\mathbb{R}$ and $\mathbb{R}^2$ (it will be clear from the context). A cardinality of any set $A$ will be denoted by $|A|$. If $|A|\leq \omega$ then we say that $A$ is countable. In the other case we shall say it is uncountable. A $\sigma$-ideal of sets of Lebesgue measure zero (linear or planar- it will be clear from the context) will be denoted by $\mc{N}$. In the same fashion by $\mc{M}$ we will denote a $\sigma$-ideal of sets of the first category (also: meager). Sets not belonging to $\mc{M}$ are called of the second category. We shall denote a $\sigma$-algebra of Borel sets by $\mc{B}$. We say that a Borel set $B$ is $\mc{I}$-positive with respect to a $\sigma$-ideal $\mc{I}$ (or simply: positive) if $B\notin \mc{I}$. Let us recall some notions regarding $\sigma$-ideals and Borel sets. \begin{defi} We say that a $\sigma$-ideal $\mc{I}$ \begin{itemize} \item has a Borel base if $(\forall A\in\mc{I}) (\exists B\in\mc{B}\cap\mc{I} (A\subseteq B)$; \item has a Borel hull property if for $(\forall A) (\exists B\in\mc{B}) (A\subseteq B \textnormal{ and } (\forall B'\in\mc{B}) (A\subseteq B'\subseteq B) (B\backslash B'\in\mc{I}))$. We call such a set $B$ a Borel hull of $A$, $B=[A]_\mc{I}$. \end{itemize} \end{defi} Both $\mc{M}$ and $\mc{N}$ have Borel bases, since every meager set is contained in some $F_\sigma$ meager set and each null set is contained in a $G_\delta$ set of measure zero. Also both $\mc{M}$ and $\mc{N}$ satisfy countable chain condition (briefly: ccc) property (there is no uncountable family of positive pairwise disjoint Borel sets), so they have the Borel hull property. We give a short proof of this fact.
\begin{prop} If a $\sigma$ $\mathcal{I}$ is ccc and has a Borel base, then it has the Borel hull property. \end{prop} \begin{proof} Let $A$ be a set. If $A\in\mathcal{I}$ we are done. If not, let us take Borel $B_0\supseteq A$. If $B_0\backslash A$ contains some $\mathcal{I}$-positive Borel set $C_1$, then let $B_1=B_0\backslash C_1$. Let say that we are at the step $\alpha<\omega_1$. If $\bigcap_{\beta<\alpha}B_{\beta}\backslash A$ does not contain any $\mathcal{I}$-positive Borel set, we are done. Otherwise let $C_\alpha$ be that set and set $B_\alpha=\bigcap_{\beta<\alpha}B_{\beta}\backslash C_\alpha$. By ccc a family $\{C_\alpha: \alpha<\omega_1\}$ is countable, which means that the above procedure stabilizes at some countable step $\kappa$ and $\bigcap_{\alpha<\kappa}B_{\alpha}$ is the set. \end{proof} Let us recall that a set $M$ is $\mc{I}$-measurable if it belongs to $\sigma(\mc{B}\cup\mc{I})$- a $\sigma$-algebra of sets generated by Borel sets and $\mc{I}$. Now let us specify a notion of fullness. \begin{defi} We say that a set $A\subseteq B$ is full in $B$ with respect to $\mc{I}$, if for each $\mc{I}$-measurable set $M$ we have $M\cap A\in\mc{I} \Leftrightarrow M\cap B\in\mc{I}$. \end{defi} We will call a set $A$ comeager in $B$ if $B\backslash A$ is meager. Let us give somewhat more intuitive characterization of full subsets with respect to measure. \begin{prop}\label{being full with respect to measure} Let $A\subseteq B$. $(\forall M\in\sigma(\mc{B}\cup\mc{N}))(\lambda^*(A\cap M)=\lambda^*(B\cap M))\Leftrightarrow$ $A$ is full in $B$ with respect to $\mc{N}$. \end{prop} \begin{proof} Let us consider the nontrivial implication "$\Leftarrow$". Suppose that there is a measurable set $M$ such that $0\neq\lambda^*(A\cap M)=\delta <\lambda^*(B\cap M)$. Let $G_1$ and $G_2$ be $G_\delta$ sets covering $A\cap M$ and $B\cap M$ respectively such that $\lambda(G_1)=\lambda^*(A\cap M)$ and $\lambda(G_2)=\lambda^*(B\cap M)$. Then $\lambda^*(G_2\backslash G_1\cap A)=0$, but $\lambda^*(G_2\backslash G_1\cap B)\geq \lambda^*(G_2\cap B)-\lambda^*(G_1)>0$. \end{proof} It is easy to check that for sets $A$ and $B$, $\lambda^*(B)<\infty$, $A$ is full in $B$ $\Leftrightarrow$ $A$ and $B$ have the same outer measure. \\ Let us conclude this section with the two following facts which we will find useful later. \begin{prop}\label{countable union of full subsets of decomposition is full} If $A=\bigcup_{n\in\omega} A_n$ and we have sets $B_n, n\in\omega$, full in $A_n$, then $\bigcup_{n\in\omega}B_n$ is full in $A$. \end{prop} \begin{prop}\label{Continuity from below of outer Lebesgue measure} Lebesgue outer measure $\lambda^*$ is continuous from below. \end{prop} \begin{proof} Let $(A_n: n\in\omega)$ be a sequence of sets, $\bigcup_{n\in\omega}A_n=A$. Let $\widetilde{A}$ be a Borel hull of $A$ and $\widetilde{A}_n$ be a Borel hull of $A_n$, $n\in\omega$. Then $\lambda^*(\bigcup_{k<n}A_k)=\lambda(\bigcup_{k<n}\widetilde{A}_k)\rightarrow^{n\rightarrow\infty}\lambda(\bigcup_{n\in\omega}\widetilde{A}_n)=\lambda(\widetilde{A})=\lambda^*(A)$ by continuity of $\lambda$. \end{proof}
{\normalsize\section{DECOMPOSITIONS}} \begin{center} \footnotesize{2.1 THE CASE OF THE CATEGORY} \end{center} Let $X\subseteq\mathbb{R}$ be of the second category (otherwise we have nothing to do), enumerated $(X=\{x_\alpha: \alpha<\kappa\})$, and let $\preceq$ be an order on $X$ given by the enumeration. Thanks to Proposition \ref{countable union of full subsets of decomposition is full} we may assume that $X\subseteq [0, 1]$. \\ Without loss of generality we may assume that every initial segment of $X$ is meager. Otherwise we take the smallest initial segment $I_0$ of $X$ which is of the second category and remove its hull $[I_0]$ from $X$. We repeat the procedure for $X\backslash [I_0]$, find $I_1$ - the smallest initial segment of $X\backslash [I_0]$ and remove its hull from $X\backslash [I_0]$ and so on. By ccc we will stop after countably many steps. Now if we have a collection $\{I_n: n\in\omega\}$ of these smallest initial segments of the second category, then it is sufficient to find decompositions of sets $I_n$, $n\in\omega$. The rest follows by Proposition \ref{countable union of full subsets of decomposition is full}. \\ Let us fix a set $T\subseteq\mathbb{R}^2$ as follows \begin{linenomath*} \[T=\{(x,y)\in X\times X: y\prec x\},\] \end{linenomath*} where $\prec$ is the strict well order determined by the enumeration of $X$. Let us observe that each vertical slice $T_x$ is meager and each horizontal slice $T^y$ is equal to $X$ except some set of the first category. For every $x\in X$ and $n\in \omega$ let us denote nowhere dense sets $F_x^n$ such that $\bigcup_{n\in\omega}F^n_x=T_x$. We may assume that for fixed $x\in X$ sets $F_x^n, n\in\omega,$ are pairwise disjoint. Clearly \begin{linenomath*} \[ X=\bigcup_{x\in X}\bigcup_{n\in\omega}(\{x\}\times F^n_x)=\bigcup_{n\in\omega}\bigcup_{x\in X}(\{x\}\times F^n_x). \] \end{linenomath*} For every $n\in\omega$ let us denote $\bigcup_{x\in X}(\{x\}\times F^n_x)$ by $H_n$. Let $\{I_n: n\in\omega\}$ be an enumeration of intervals with rational endpoints which have intersection with $X$ of the second category. Before we continue we will prove the following lemma. \begin{lem}\label{Lemma for Luzin} Let a set $X\subseteq\mathbb{R}$ be nonmeager and $A\subseteq\mathbb{R}^2$ such that for every $x\in\mathbb{R}$ $A_x$ is nowhere dense. Then a set \begin{linenomath*} \[ Y=\{y: X\backslash A^y\in\mc{M}\} \] \end{linenomath*} is nowhere dense. \end{lem} \begin{proof} Suppose that it is not. Then there exists a countable set $Q\subseteq Y$ dense in $Y$. Then $\bigcup_{y\in Q}X\backslash A^y\in\mc{M}$, so \begin{linenomath*} \[ X\backslash (\bigcup_{y\in Q}X\backslash A^y)=\bigcap_{y\in Q}A^y \] \end{linenomath*} is nonmeager and thus nonempty. It means that there is $x\in \bigcap_{y\in Q}A^y$ and \begin{linenomath*} \[ (\forall y\in Q)(x\in A^y)\equiv (\forall y\in Q)(y\in A_x) \] \end{linenomath*} which means $Q\subseteq A_x$, a contradiction with $A_x$ being nowhere dense. \end{proof} Now, let us consider a set \begin{linenomath*} \[ Y=\{y\in X: (\forall k\in\omega) (\exists^{\infty}n)(H^y_n\cap I_k\notin\mc{M})\}. \] \end{linenomath*} We claim that $Y$ is comeager in $X$. Suppose that $B=X\backslash Y$ is nonmeager. \begin{linenomath*} \[ B=\{y\in X: (\exists k\in\omega) (\forall^{\infty}n)(H^y_n\cap I_k\in\mc{M})\} \] \end{linenomath*} Then there exist $n_0, k_0\in\omega$ such that a set \begin{linenomath*} \[ B_{n_0,k_0}=\{y:\in X: (\forall n>n_0)(H_n^y\cap I_{k_0}\in\mc{M})\} \] \end{linenomath*} is nonmeager. Let us denote $H=\bigcup_{n\leq n_0}H_n$. Clearly, for every $x\in X$ the slice $H_x$ is nowhere dense. For every $y\in X$ we have that $\bigcup_{n\in\omega}H_n^y$ is comeager in $X$ and, since for each $y\in B_{n_0,k_0}$ a set $\bigcup_{n> n_0}H_n^y\cap I_{k_0}$ is meager, $H^y$ is comeager in $X\cap I_{k_0}$ for nonmeager many $y$. On the other hand by Lemma \ref{Lemma for Luzin} we have that $\{y: X\cap I_{k_0}\backslash H^y\in\mc{M}\}$ is nowhere dense which brings a contradiction. \\ Therefore $Y$ is nonempty so let us pick $y\in Y$. Let us construct by induction two sequences of sets $(X^1_k: k\in\omega)$ and $(X^2_k: k\in\omega)$. For $k=0$ we have infinitely many $n's$ such that $H^y_n\cap I_0\notin\mc{M}$, so let us pick two, say, $n_0^1$ and $n_0^2$ and let us set $X_0^1=H^y_{n_0^1}$ and $X_0^2=H^y_{n_0^2}$. Notice that they are disjoint, since sets $F^n_x$ were pairwise disjoint. Let us assume that at the step $k\in\omega$ we have already sequences $(X^1_m: m<k)$ and $(X^2_m: m<k)$, for which $X^1_m=H^y_{n_m^1}$ and $X^2_m=H^y_{n_m^2}$ for all $m<k$. We still have infinitely many natural numbers $n$ distinct from each of $n_m^i$, $i\in\{1, 2\}, m<k,$ such that $H^y_n\cap I_k\notin\mc{M}$, so let us pick $n^1_k$ and $n^2_k$ such that $H^y_{n^i_k}\cap I_k\notin\mc{M}, i\in\{1, 2\}$ and set $X^i_k=H^y_{n^i_k}, i\in\{1, 2\}$. Sets $X_1=\bigcup_{n\in\omega}X^1_n$ and $X_2=\bigcup_{n\in\omega}X^2_n$ constitute the desired decomposition. \begin{center} \footnotesize{2.2 THE CASE OF MEASURE} \end{center}
Similarly to the previous case let $X\subseteq[0,1]$ be a set of positive outer measure, $X=\{x_\alpha: \alpha<\kappa\}$, such that its initial segments are null. Again, let us fix a set $T\subseteq\mathbb{R}^2$ as follows \begin{linenomath*} \[ T=\{(x,y)\in X\times X: y\prec x\}, \] \end{linenomath*} where $\prec$ is the strict well order determined by the enumeration of $X$. Let us observe that each vertical slice $T_x$ is null and each horizontal slice $T^y$ is equal to $X$ except some set of measure zero. Let $\epsilon>0$. For every $x\in X$ and $n\in \omega$ let us denote intervals with rational endpoints $I_x^n$ such that $\bigcup_{n\in\omega}I^n_x\supseteq T_x$ and $\lambda(\bigcup_{n\in\omega}I_x^n)<\epsilon$. Clearly \begin{linenomath*} \[ X=\bigcup_{x\in X}\bigcup_{n\in\omega}(\{x\}\times (I^n_x\cap T_x))=\bigcup_{n\in\omega}\bigcup_{x\in X}(\{x\}\times (I^n_x\cap T_x)). \] \end{linenomath*} For every $n\in\omega$ let us denote $\bigcup_{x\in X}(\{x\}\times I^n_x)$ by $H_n$. For every $y\in X$ we have that $\bigcup_{n\in\omega}H_n^y=X$ (mod $\mc{N}$), therefore by Proposition \ref{Continuity from below of outer Lebesgue measure} for every $y\in X$ there exists $n_y\in\omega$ such that $\lambda^*(\bigcup_{k<n_y}H_k^y)>\lambda^*(X)-\epsilon$. Let us denote \begin{linenomath*} \[ Y_n=\{y\in X: \lambda^*(\bigcup_{k<n}H_k^y)>\lambda^*(X)-\epsilon\}. \] \end{linenomath*} Clearly $\bigcup_{n\in\omega}Y_n=X$, so again by Proposition \ref{Continuity from below of outer Lebesgue measure} there exists $N\in\omega$ such that $\lambda^*(\bigcup_{k<N}Y_k)>\lambda^*(X)-\epsilon$. For $y\in\bigcup_{k<N}Y_k$ we have $\lambda^*(\bigcup_{k<N}H_k^y)>\lambda^*(X)-\epsilon$. Let us denote $\bigcup_{k<N}H_k$ by $H$. Before we continue we will prove the following lemma.
\begin{lem}\label{Fubini for Lebesgue outer measure} Let $X\subseteq[0,1]^2$. Assume that there is a set $A\subseteq [0,1]$ such that $\lambda^*(A)=a$ and $(\forall x\in A)( \lambda^*(X_x)\geq b)$. Then $\lambda^*(X)\geq ab$. \end{lem} \begin{proof} Let $G\supseteq X$ be such that $\lambda(G)=\lambda^*(X)$. Without loss of generality let us assume that $G\subseteq \widetilde{A}\times [0,1]$, $\widetilde{A}$ - the Borel hull of $A$ with respect to measure. Let us consider a function $f: [0,1]\rightarrow [0,1]$, $f(x)=\lambda(G_x)$. Since $f$ is measurable we may write \begin{linenomath*} \begin{align*} \lambda(G)&=\int_{[0,1]}f(x)d \lambda(x)=\int_{\widetilde{A}}f(x)d \lambda(x)=\\ &=\int_{\{x\in\widetilde{A}: f(x)\geq b\}}f(x)d \lambda(x)+\int_{\{x\in\widetilde{A}: f(x)< b\}}f(x)d \lambda(x). \end{align*} \end{linenomath*} Since $\{x\in\widetilde{A}: f(x)\geq b\}\supseteq A$ we have that $\lambda(\{x\in\widetilde{A}: f(x)\geq b\})=a$ and $\lambda(\{x\in\widetilde{A}: f(x)< b\})=0$, so \[ \lambda^*(X)=\lambda(G)=\int_{\{x\in\widetilde{A}: f(x)\geq b\}}f(x)d \lambda(x)\geq b\int_{\{x\in\widetilde{A}: f(x)\geq b\}}1 \,d \lambda(x)=ab. \] \end{proof} Now, by Lemma \ref{Fubini for Lebesgue outer measure} we have that $\lambda^*(H)>(\lambda^*(X)-\epsilon)^2$. Let $\{G_n: n\in\omega\}$ be a family of finite unions of intervals with rational endpoints such that $\{G_n: n\in\omega\}=\{\bigcup_{k<N}I_x^k: x\in X\}$. For every $n\in\omega$ let $A_n=\{x\in X: G_n=\bigcup_{k<N}I_x^k \}$. We see that we cannot separate sets $A_n$ with their Borel hulls $[A_n]$, otherwise we would have $H\subseteq \bigcup_{n\in\omega}([A_n]\times G_n)$ which has a measure $<\lambda^*(X)\cdot\epsilon$. So there are $n_0\neq m_0$ such that $[A_{n_0}]\cap[A_{m_0}]$ is of positive measure. Sets $X^1_0=A_{n_0}\cap[A_{n_0}]\cap[A_{m_0}]$ and $X^2_0=A_{m_0}\cap[A_{n_0}]\cap[A_{m_0}]$ constitute a desired decomposition of $X\cap [A_{n_0}]\cap[A_{m_0}] $. We repeat the whole procedure for $X_0=X\backslash [A_{n_0}]\cap[A_{m_0}]$ and so forth; by ccc we shall have a decomposition of $X$ after countably many steps.
\end{document} | arXiv |
Examples of Innumeracy
I read Innumeracy by John Allen Paulos and would like to share more up-to-date and relevant examples of innumeracy to motivate my grade 8, 9 & 10 students. Are there any websites, blogs, books, etc. with lots of examples of innumeracy in the form of pictures, reporting, news articles, etc.?
Here are just 2 examples of what I'm thinking about, I just want to find LOTS more:
http://johnquiggin.com/2011/05/08/two-billion-examples-of-innumeracy/
http://i.imgur.com/T7KThEy.jpg
UPDATE: Some MESE members recently started a new site devotedd to innumeracy: http://innumeracy.net/welcome/
examples student-motivation
David Ebert
David EbertDavid Ebert
$\begingroup$ I feel pretty bad about pasting this but youtube.com/watch?v=Qhm7-LEBznk $\endgroup$ – Steven Gubkin Apr 12 '14 at 19:26
$\begingroup$ The "infographic" on the cover of USA Today every day is frequently nonsensical or misleading. $\endgroup$ – Eric Lippert Apr 12 '14 at 19:57
$\begingroup$ Let your students read a random newspaper, and check whether the percentages, fractions and the like are right. It's an interesting exercise, and very rewarding towards their self-esteem, since they will find mistakes for sure. $\endgroup$ – Quora Feans Apr 13 '14 at 8:11
$\begingroup$ Reactions to Marilyn Vos Savant's presentation of The Monty Hall Problem: wwwp.cord.edu/faculty/andersod/TaxicabWorksheets.pdf $\endgroup$ – David Ebert Apr 13 '14 at 15:13
$\begingroup$ The problem with the Monty Hall problem is that tiny differences in the way that the problem is posed make significant differences to what the correct answer is, and most people are not quite clever enough to handle that. They read the problem and the answer, then try to test someone who is supposedly good at maths, but pose a slightly different problem and feel good about themselves when they get a different answer than what they read as the correct answer. $\endgroup$ – gnasher729 Apr 13 '14 at 17:01
A recent Times article titled Americans Are Bad at Math, but It's Not Too Late to Fix offered an example -
A&W's "Third Pounder hamburger failed to catch on because
During focus groups, the company discovered that customers believed they were getting less meat. Because the "3" in ⅓ was smaller than "4" in ¼, "customers believed they were being overcharged."
If this is not classic innumeracy, I don't know what is.
Ben Crowell
JTP - Apologise to MonicaJTP - Apologise to Monica
There's the Verizon "0.002 cents versus 0.002 dollars" mishap, wherein an unhappy customer calls to complain that he was billed 0.002 $/kB after being told the rate is 0.002 cents/kB. The confusion is perhaps deeper than expected.
David SteinbergDavid Steinberg
$\begingroup$ ...So this is pretty funny, but I'm not sure whether this is "innumeracy" in action or just a misunderstanding of terminology. It reminds me of confusion around 0.05% being interpreted as 5% (since 0.05 = 5%). But perhaps all of these are examples of what is considered innumeracy. $\endgroup$ – Benjamin Dickman Apr 14 '14 at 11:56
$\begingroup$ @BenjaminDickman Is this any better: publicshaming.tumblr.com/post/36857566279/… $\endgroup$ – David Steinberg Apr 14 '14 at 15:41
$\begingroup$ Was Verizon found at fault or the guy? $\endgroup$ – Jeff-Inventor ChromeOS Aug 11 '14 at 3:24
$\begingroup$ @Jeff-InventorChromeOS: Verizon admitted the error, and credited back the fees in question: verizonmath.blogspot.com/2006/12/… $\endgroup$ – Daniel R. Collins Jan 16 '16 at 1:36
Re-Re-Edit (May 2019): Found in a selection of tweets here but pasted as images to preserve.
Credit for the first one goes to @lizardbill and to the rest to @GeneticJen:
Re-Edit (Jan 2016): Perhaps this does not quite qualify, but I was rather surprised to spot the following question (#6 in the image below) in a recent airplane Mensa quiz:
(Side-note: #2 part 1 has two answers, as does part 4.)
Take a moment to solve #6 before reading on.
Here are the "answers" from that same page (American Way, Jan 2016):
The description is not wrong, but it is a bit different from the phrasing I had in mind (adding 18).
Edit (Jan. 2015): After seeing MESE 7200, a quick google yielded the following from a blog called 360:
Looking at your examples, I recall a story that had been trending online for quite some time: It alleged that Samsung paid Apple about \$1 billion (USD) in nickels (\$0.05 coins) as carried by "30 trucks."
After seeing this shared on facebook far too many times, I posted the following:
It seems the story goes back at least to August of 2012, and was also debunked by Snopes.
(The Snopes estimate is 2,755 eighteen wheelers, but is based off of the judgement being \$1.05 billion, as opposed to the \$1 billion claim in the image above. Scaling my estimation up by 5% gives 2,625 eighteen wheelers. So, the numbers are pretty close. In any event: There is some "innumeracy" here.)
Benjamin DickmanBenjamin Dickman
$\begingroup$ Yes! More of this! $\endgroup$ – David Ebert Apr 12 '14 at 17:49
$\begingroup$ Fact is, the lawsuit in question is now at the next higher court and no money has exchanged hands at all at this point in time. $\endgroup$ – gnasher729 Apr 13 '14 at 16:58
$\begingroup$ @gnasher729 Mathematically, though, the salient point is that 30 trucks (even huge ones) would not be able to deliver that many nickels without numerous (around 90...) trips back and forth. $\endgroup$ – Benjamin Dickman Apr 13 '14 at 19:26
$\begingroup$ There's also the material limitation of needing the entire production of nickel metal in Madagaskar in 2011 and the entire production of copper in Portugal from 2006 to manufacture this. Also, the treasury only minted about 500,000,000 coins in 2010 (and less than 1/5th of that in 2009), so you'd easily need over 40 years of nickel production to just make that many coins. $\endgroup$ – Nzall Apr 14 '14 at 11:05
$\begingroup$ About the deliciousness example: Maybe the original (and final) deliciousness was $0$. Or maybe each pack used to contain only $2/3$ of a bar. $\endgroup$ – Andreas Blass Jan 17 '16 at 15:45
Let me offer a different type of response, a student's answer to a problem.
The question offered the height of a building, the equation for distance of a falling object, and asked to calculate the time till a rock dropped off the building would fall to the ground. The student used his calculator and the answer was 900 seconds. I asked if that was right, and tried to get him to apply common sense. 900 sec = 15 minutes. Do you think you can see your friend drop the rock, go to Starbuck's, get a coffee, and step back out before it hits the ground? Of course not. His answer was off by a factor of 100.
I'd read Innumeracy a long time ago, but recall that this was one of the author's lessons, the ability to estimate orders of magnitude as being correct or way off. Part of my goal is to ask students if the answer makes sense, in cases where it's not just numbers but real life situations.
edited Aug 6 '14 at 2:45
True story: I ordered new carpet flooring for a room in my house. The length of the room was 13 foot 11 inches. The employee took his calculator and typed "13.11 x 30.48 =" to convert into centimetres. I didn't actually manage to convince him of his mistake, but had to ask for a more experienced colleague. Would have been a nasty surprise if I hadn't noticed and they had delivered a piece of carpet 10 inch too short.
$\begingroup$ "How would you type in 13 feet and 12 inches?" $\endgroup$ – Chris Cunningham♦ Apr 13 '14 at 17:46
$\begingroup$ 13.12 :-( Seriously, I told him that 13 foot 11 inch is almost 14 foot which is more than 4.20 metre, not under four metre. Didn't get more than a blank stare. $\endgroup$ – gnasher729 Apr 13 '14 at 22:47
$\begingroup$ Well, 13.11*2 = 26.22, But isn't 'just under 14' * 2 'just under 28' and not 'just over 26'? Meters? Never heard of them. $\endgroup$ – JTP - Apologise to Monica Apr 14 '14 at 1:01
Common accounts in popular press and TV and on-line about "the rate of increase of X is slowing", with varying interpretations, all too often mistaking this for X itself decreasing, etc.
As in "unemployment" or "inflation" or "debt" or ...
paul garrettpaul garrett
A lot of the xkcd "what if" posts, for example this one about hitting golf balls off a spaceship in order to reach escape velocity, seem surprising to me in part, I suspect, because of my own innumeracy. (It turns out, in this case, you might well need a bag of golf balls about 100 billion miles in diameter...)
Adam BjorndahlAdam Bjorndahl
A YouTube video of an Illustrious Senator Talking about the cost of health care, 500 trillion dollars. This is more than all the world's wealth, and nearly 8 times all the wealth in the US. I guess he meant Billion, but in Washington, no one is listening anyway.
Recent meme that spending 360 million dollars to give 317 million people health care means that you're spending over 1 million dollars person:
http://www.dailymail.co.uk/femail/article-3180161/Maddening-online-debate-cost-Obamacare-turns-simple-math-question-complex-equation-able-solve.html
Daniel R. CollinsDaniel R. Collins
$\begingroup$ see also: publicshaming.tumblr.com/post/36857566279/… $\endgroup$ – David Steinberg Oct 7 '15 at 2:55
$\begingroup$ @David Steinberg: Good lord! $\endgroup$ – Daniel R. Collins Oct 7 '15 at 3:05
This was making the rounds on Twitter this week. There's so much wrong with this, I don't know where to start. It certainly goes back to the manipulation of large numbers, which people in general tend to struggle with.
(Note - in the US, our lottery hit a record $1.5 billion dollar prize. Our billion has 9 zeros.)
There was this story from 2009, where a town mistakenly believed it had passed a measure which required support from two-thirds of the voters, when it only had support from 66%. (The town accountant (!) calculated two-thirds by multiplying by .66.)
This seems to be a common problem. One of my colleagues was consulted by a representative of the Pennsylvania judicial system to sort out a standing dispute about how to calculate fees which were capped by law at one-third: some judges multiplied by .33, others by .333, and they wanted to know what was right. (They were allegedly impressed by the suggestion that they could simply divide by three.)
Henry TowsnerHenry Towsner
One example that annoys me is when science stories in newspapers (especially stories about high energy physics or astronomy) insist on writing out large numbers, such writing $1,000,000,000,000,000,000,000,000,$ or writing things like trillion trillion, when neither is very useful to a reader with a high school education and neither would make any sense to anyone else. Why not write $10^{24}?$ Scientific notation is taught (in the U.S.) to students who have not yet begun the study of school algebra, and it is used in high school science classes.
Below are two other examples that I've previously posted about in the past.
Example 1: Atlanta mayor: In resettling evacuees, FEMA no help, CNN news article, 14 October 2005. [I previously posted a different version in this 15 October 2005 sci.math post.]
CNN anchor Miles O'Brien on Friday spoke about the challenges facing one city with Atlanta Mayor Shirley Franklin. Atlanta took in 42,000 families fleeing the disaster.
O'BRIEN: All right, let's talk about this, 42,000 families. You're a big city. It's a prosperous city, but that still puts a burden on the city, doesn't it?
FRANKLIN: Well, it certainly does, but I don't think it's a burden that FEMA [Federal Emergency Management Agency] can't help us to address.
The Congress and the president have allocated 62 billion [dollars]. Our estimates are that a family needs assistance for about six months in order to stabilize themselves and that would cost about 11,000 [dollars] per family. The city of Atlanta can't absorb that cost, but we can certainly work with FEMA, if they were willing, to help families get resettled in the city and the metropolitan area.
O'BRIEN: So 11,000 times 42,000. I can't do that kind of math on the fly here. But how much of that money have you seen?
Note: This is trivially estimated via $1$ times $4$ followed by $4+4=8$ zeros, or $400$ million. By missing this, and hence also the fact that $60$ billion divided by $400$ million is $(1.5)(100)=150,$ Miles O'Brien (a well known broadcast news journalist who, incidentally, specializes in science, technology, and aerospace reporting) missed a good opportunity to make a point I suspect he would have liked to make.
Example 2: Formerly posted in this 8 February 2009 math-teach post at Math Forum:
I wonder if the author of the article below has any awareness of just how mathematically illiterate one of the comments below makes professional news writers sound. The author writes "CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos" for something that any college-bound middle school student should be able to do, even without a calculator. Although some of Paulos' comments are nice, especially his speculation "People tend to lump [million, billion, trillion] together, perhaps because they rhyme", going to him in order to check McConnell's numbers is like asking a university linguist for the correct spelling of the word "especially".
I mention this because I've seen many examples of this over the years, especially newspaper writers consulting mathematicians for something that is nothing more than an easy high school level probability or combinatorics problem (easier than many of the problems in standard precalculus and college algebra texts). I don't know whether the reporters really don't know how to work these problems (like checking McConnell's numbers below) or whether they are just using the occasion to get some possibly interesting remarks from someone well known and don't realize how stupid their rationale sounds to a large percentage of their readership.
"Numb and number: Is trillion the new billion?" by Christine Romans CNN's American Morning
"To put a trillion dollars in context, if you spend a million dollars every day since Jesus was born, you still wouldn't have spent a trillion," McConnell said.
CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos.
"A million dollars a day for 2,000 years is only three-quarters of a trillion dollars. It's a big number no matter how you slice it," Paulos said. Here's another way to look at it.
"A million seconds is about 11.5 days. A billion seconds is about 32 years, and a trillion seconds is 32,000 years," Paulos said. "People tend to lump them together, perhaps because they rhyme, but if you think of it in terms of a jail sentence, do you want to go to jail for 11.5 days or 32 years or maybe 32,000 years? So, they're vastly different, and people generally don't really have a real visceral grasp of the differences among them."
Dave L RenfroDave L Renfro
$\begingroup$ A police (TV) show recently talked about suicide. The policeman said in the US there was a suicide every 40 seconds. Knowing there are 525,600 minutes (from a song in the Broadway musical Rent)in a year, I paused the TiVo and told my wife the statistic was wrong, there aren't 750,000 suicides a year. In fact, the number is closer to 40,000. The show's writers were off by nearly a factor of 20. $\endgroup$ – JTP - Apologise to Monica Dec 14 '14 at 1:12
$\begingroup$ @JoeTaxpayer: Here's something I came across this last Friday (today is Monday). (For future readers, see here also.) Now I can somewhat sympathize with the cube root of 125 question, since if you haven't heard or dealt with cube roots in 30 or more years you'll likely not know, especially when put on the spot like this, but look further down for where England's Chancellor George Osborne did the same thing with a child asking him what $7 \times 8$ is. $\endgroup$ – Dave L Renfro Dec 15 '14 at 14:34
$\begingroup$ The author of the New Yorker's article earlier this year on the work of Yitang Zang starts like this: "I don't see what difference it can make now to reveal that I passed high-school math only because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x's and y's. On test days, I sat next to... smart boys whose handwriting I could read—and divided my attention between his desk and the teacher's eyes." I rant more about this here: angrymath.com/2015/02/yitang-zhang-article-in-new-yorker.html $\endgroup$ – Daniel R. Collins Oct 7 '15 at 3:03
$\begingroup$ @Daniel R. Collins: After reading your blog post, I thought you might enjoy my 2010 pi day rant. $\endgroup$ – Dave L Renfro Oct 7 '15 at 15:02
$\begingroup$ @Dave L Renfro: That's great! If you don't mind, I'm going to link to that from my blog when I get a chance. $\endgroup$ – Daniel R. Collins Oct 7 '15 at 22:22
Let me mention a bit of (somewhat) good news in this area. Some years ago, Wal-Mart sold cardboard shipping boxes, labeled as 14" by 14" by 14". They were also labeled in metric units; since an inch is 2.54 centimeters, the dimensions were printed as 35.56 cm. I used to make fun of that --- dimensions of a shipping box accurate to a tenth of a millimeter. The good news is that now those boxes, still 14" on a side, are labeled 35.5 cm. It still seems to claim an unwarranted level of precision, but not as absurdly unwarranted as before.
Andreas BlassAndreas Blass
A general answer of "every time the unit pricing makes no sense." The specific example:
The product is identical. Presale, 12 for 2.29, 24 for 3.99. Fair enough, a bit of a discount to buy twice the number of pencils. Now, the unit price for the 24 pack is 50% more than the unit price of the 12 pack. I wonder how many of the large pack they'll sell this summer.
All too often I've seen a 1lb package of a grocery item sell for 1.99, and the 2lb package of the identical item, 3.99. But this is the first I've seen the side by side unit price so off.
$\begingroup$ Note that technically, the store is not selling individual pencils, it is selling boxes of pencils. If there is an over-supply of boxes of 12 and an under-supply of boxes of 24, I think it makes good business sense to sell the boxes of 12 at a lower "unit price" than the boxes of 24. $\endgroup$ – Joel Reyes Noche Aug 11 '14 at 1:15
$\begingroup$ Also, it seems you are assuming that one box of 24 contains exactly the same type of pencils as two boxes of 12. This may not be the case. For example, the box of 12 could have pencils of 12 different colors and the box of 24 could have pencils of 24 different colors. That is, it is possible that the box of 24 has a, say, gold colored pencil and the box of 12 does not. (It may not be so in this particular case, but it may be so in other cases of differing unit prices.) $\endgroup$ – Joel Reyes Noche Aug 11 '14 at 1:19
$\begingroup$ Joel - great points. In this case, I happen to have both a 12 and 24 box, bought, presale, and grabbed the last boxes. The first 12 are identical, but the back row in the 24 box does have different colors. There's no added value to me, as I offer these to my students who are trying to produce 3 or 4 color graphs, but I can see other uses where the extra colors are needed. Good call. $\endgroup$ – JTP - Apologise to Monica Aug 11 '14 at 2:36
$\begingroup$ Thanks. Also, +1 because I agree that in other cases the objects being sold will all be similar and so there will be cases where very different unit prices won't make sense. $\endgroup$ – Joel Reyes Noche Aug 11 '14 at 3:06
$\begingroup$ With an item of food I used to buy, the larger packet was significantly worse value than the smaller one. So much so, that even when the bigger bag was discounted and the smaller one wasn't, it was still better value to buy the smaller one. And here smaller was better, as an open packet would go off, where an unopened one wouldn't. $\endgroup$ – Jessica B May 30 '15 at 12:01
I would check out some of the work of Edward Tufte (http://www.edwardtufte.com). His book The Visual Display of Quantitative Information is replete with examples of deliberate and accidental mis-use of graphs which mislead readers. I know this is going to sound like a commercial. I have no connection to the author in any way. Unfortunately, I'm only aware of his hardbound books. It's not easy to find online examples for you to look through. You may be able to find copies at a local library. In the book mentioned above, chapter 2 "Graphical Integrity," shows many practical examples of misleading graphs taken from pictures, newspapers, articles, etc.
Randall StewartRandall Stewart
Paulos defines innumeracy as "an inability to deal comfortably with the fundamental notions of number and chance" (Paulos, 1988, p. 3). I read (and enjoyed) his book back when I was primarily a software developer and didn't know enough to ask why he wrote the definition that way.
The first example he gives of innumeracy is a newscaster concluding that the weekend had a 100% chance of rain if both Saturday and Sunday had a 50% chance. But he also found that a fellow viewer of the weather report found no problem with the conclusion. Today I would ask Paulos: "What evidence do you have of the level of comfort felt by the participants in your example?" And if he told me "That's not what I meant" I would ask why he wrote it that way if he didn't mean it.
Beside the point? It is not. Definitions are important in our understanding of mathematics as well as in our understanding people (and education).
The book is really mostly about the types of errors similar to this weathercaster's. In this case, conclusions that don't follow mathematically from the available data. Some people collect examples of this sort of thing. I credit user Michael-E2 for bringing this collection to my attention: Collected Forsooths of the newsletter of Royal Statistics Society.
But there is a problem here. So little context is given with these examples that it is impossible to determine whether the errors are truly the result of mathematical errors, some different sort of error, or no error at all.
How bad a problem is this? For education purposes, I say it is no problem at all, depending on the lesson and the point you are trying to make. In my view, mathematics in the world is a sense-making endeavor. An education gives your students a greater capability to make sense of more of the world, allowing the world to actually make more sense to them. And that includes the things people say -- also including the occasional nonsensical-sounding mathematical claim.
How useful is it to be given a list of known mathematical errors? It is certainly of some use. But is not it also of good use to be given a possible error and to have to argue for what it could mean? In some cases, these errors may not reflect realty, but they could have some meaning. How do we use our mathematical understanding to make sense of the situation?
At some point, someone said something; at another point, someone used mathematical knowledge to determine "that was a problematic statement." How do we choose sides? Can we narrow the choices of what the statement could mean? What do we rely on? What arguments can we make?
I think this would be a valuable educational activity: give students statements and have them argue about the mathematical reasoning used. And what they can make sense of. And how they support their own view of the situation. These are types of mathematical reasoning that appear in standards documents, but also are what they will have to rely on if they use math to make sense of their world. And of course, what they will do to convince themselves and others of what they know.
JPBurkeJPBurke
$\begingroup$ @JoelReyesNoche - Thanks. I must have gotten interrupted. $\endgroup$ – JPBurke Jan 28 '16 at 23:15
$\begingroup$ JP - on re-reading your answer, I'd make two points. It was Douglas Hofstadter that coined the term, and I think that Paulos first book on the topic was written in a way that was a bit scattered. Random examples, some of which that were meta-equivalent to errant apostrophes and commas, which are not quite symptoms of illiteracy. $\endgroup$ – JTP - Apologise to Monica Jan 30 '16 at 19:15
$\begingroup$ I addressed Paulos' definition because the original question was framed in terms of Paulos' book, so his definition seems at least somewhat relevant for talking about the types of examples that David was looking for. But thanks for the additional detail! Are you aware of where Hofstadter coined the term? I'm only familiar with one of his works. $\endgroup$ – JPBurke Feb 1 '16 at 6:29
$\begingroup$ Got it. He had a series of articles in Scientific American titled Mathematical Games. A May 1982 article "On Number Numbness" was included in a 1985 compilation (Metamagical Themas). "It is fashionable for people to decry the appalling illiteracy of this generation, particularly its supposed inability to write grammatical English. But, what of the appalling innumeracy of most people, old and young, when it comes to making sense of the numbers that, in point of fact, and whether they like it or not, run their lives?" I have the book on my shelf. Innumeracy was just a small bit of the book. $\endgroup$ – JTP - Apologise to Monica Feb 2 '16 at 12:53
$\begingroup$ @JoeTaxpayer Excellent, thank you. I am familiar with that quotation, but I didn't know the origin. I appreciate the reference! $\endgroup$ – JPBurke Feb 3 '16 at 15:46
This was told in an interview I read with a successful businessman (a dane). He told that he learned business from his grandfather. They sold roses on the street, his grandfather told him to offer "a rose for 75 øre (that is, cent of a krone), three for only 2.50 kroner".
kjetil b halvorsenkjetil b halvorsen
$\begingroup$ You offer a price break to poor people who can only afford to buy one. $\endgroup$ – Gerald Edgar Jan 26 '16 at 13:22
$\begingroup$ This reminds me of a story - a woman sees a beautiful chair in a woodworkers shop and asks how much a beautiful chair costs. \$500. How much for four? $2500 the owner says. She's a bit incredulous, until he explains - it was fun to build that first one, another 3 of that design is just work. $\endgroup$ – JTP - Apologise to Monica Jan 27 '16 at 1:01
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Acharya, Shreyasi et al.
Accessed 599 times Last updated on 2019-04-26 16:28
$\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV
The ALICE collaboration
Acharya, Shreyasi Calcutta, VECC , Torales - Acosta, Fernando UC, Berkeley , Adamova, Dagmar Rez, Nucl. Phys. Inst. , Adhya, Souvik Priyam Calcutta, VECC , Adler, Alexander Goethe U., Frankfurt, Inst. Inform. , Adolfsson, Jonatan Lund U. , Aggarwal, Madan Mohan Panjab U. , Aglieri Rinella, Gianluca CERN , Agnello, Michelangelo Turin Polytechnic , Agrawal, Neelima Indian Inst. Tech., Mumbai
Phys.Lett., 2018
https://doi.org/10.17182/hepdata.89397 https://doi.org/None
http://inspirehep.net/record/1696315 INSPIREhttp://inspirehep.net/record/1696315
A measurement of the production of prompt $\Lambda_{\rm {c}}^{+}$ baryons in Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV with the ALICE detector at the LHC is reported. The $\Lambda_{\rm {c}}^{+}$ and $\Lambda_{\rm {c}}^{-}$ were reconstructed at midrapidity ($|y| < 0.5$) via the hadronic decay channel $\Lambda_{\rm {c}}^{+} \to {\rm {p}} {\rm {K}}^{0}_{\rm {S}}$ (and charge conjugate) in the transverse momentum and centrality intervals 6 < $p_{\rm {T}}$ <12 GeV/${\it {c}}$ and 0-80%. The $\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio, which is sensitive to the charm quark hadronisation mechanisms in the medium, is measured and found to be larger than the ratio measured in minimum-bias pp collisions at $\sqrt{s} = 7$ TeV and in p-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV. In particular, the values in p-Pb and Pb-Pb collisions differ by about two standard deviations of the combined statistical and systematic uncertainties. The $\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio is also compared with model calculations including different implementations of charm quark hadronisation. The measured ratio is reproduced by models implementing a pure coalescence scenario, while adding a fragmentation contribution leads to an underestimation. The $\Lambda_{\rm {c}}^{+}$ nuclear modification factor, $R_\mathrm{AA}$, is also presented. The measured values of the $R_\mathrm{AA}$ of $\Lambda_{\rm {c}}^{+}$, ${\rm D}_{\rm {s}}$ and non-strange D mesons are compatible within the combined statistical and systematic uncertainties. They show, however, a hint of a hierarchy $R_\mathrm{AA}^{{\rm D}^{0}}<R_\mathrm{AA}^{{\rm D}_{\rm {s}}}<R_\mathrm{AA}^{\Lambda_{\rm {c}}^{+}}$, conceivable with a contribution of recombination mechanisms to charm hadron formation in the medium.
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Data from Figure 2
10.17182/hepdata.89397.v1/t1
$\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio in 0-80% most central Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV in the transverse momentum...
https://www.hepdata.net/record/data/413690yaml https://www.hepdata.net/record/data/413690/csvcsv
The nuclear modification factor $R_\mathrm{AA}$ of prompt $\Lambda_{\rm {c}}^{+}$ baryons in 0-80% most central Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} =...
Table 1 Table 2
Citing this record
When using this data, please cite the original publication:
The ALICE collaboration (2018).
$\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV. Phys.Lett..
https://doi.org/None
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Additionally, you should also cite the record:
$\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV. HEPData.
https://doi.org/10.17182/hepdata.89397
You can also cite the 2 data tables individually:
Table 1 of "$\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV". HEPData.
https://doi.org/10.17182/hepdata.89397.v1/t1
@misc { 1696315, title = "{$\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV}", abstract = "A measurement of the production of prompt $\Lambda_{\rm {c}}^{+}$ baryons in Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV with the ALICE detector at the LHC is reported. The $\Lambda_{\rm {c}}^{+}$ and $\Lambda_{\rm {c}}^{-}$ were reconstructed at midrapidity ($|y| < 0.5$) via the hadronic decay channel $\Lambda_{\rm {c}}^{+} \to {\rm {p}} {\rm {K}}^{0}_{\rm {S}}$ (and charge conjugate) in the transverse momentum and centrality intervals 6 < $p_{\rm {T}}$ <12 GeV/${\it {c}}$ and 0-80%. The $\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio, which is sensitive to the charm quark hadronisation mechanisms in the medium, is measured and found to be larger than the ratio measured in minimum-bias pp collisions at $\sqrt{s} = 7$ TeV and in p-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV. In particular, the values in p-Pb and Pb-Pb collisions differ by about two standard deviations of the combined statistical and systematic uncertainties. The $\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio is also compared with model calculations including different implementations of charm quark hadronisation. The measured ratio is reproduced by models implementing a pure coalescence scenario, while adding a fragmentation contribution leads to an underestimation. The $\Lambda_{\rm {c}}^{+}$ nuclear modification factor, $R_\mathrm{AA}$, is also presented. The measured values of the $R_\mathrm{AA}$ of $\Lambda_{\rm {c}}^{+}$, ${\rm D}_{\rm {s}}$ and non-strange D mesons are compatible within the combined statistical and systematic uncertainties. They show, however, a hint of a hierarchy $R_\mathrm{AA}^{{\rm D}^{0}}<R_\mathrm{AA}^{{\rm D}_{\rm {s}}}<R_\mathrm{AA}^{\Lambda_{\rm {c}}^{+}}$, conceivable with a contribution of recombination mechanisms to charm hadron formation in the medium.", doi = "10.17182/hepdata.89397", year = "2018", type = "Data Collection", collaboration = "ALICE", authors = "Acharya, Shreyasi and others" }
@misc { 1696315/t1, title = "{Table 1 of $\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV}", abstract = "$\Lambda_{\rm {c}}^{+}$/${\rm D}^{0}$ ratio in 0-80% most central Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} = 5.02$ TeV in the transverse momentum...", doi = "10.17182/hepdata.89397.v1/t1", year = "2018", type = "Data Set", collaboration = "ALICE", authors = "Acharya, Shreyasi and others" }
@misc { 1696315/t2, title = "{Table 2 of $\Lambda_\mathrm{c}^+$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV}", abstract = "The nuclear modification factor $R_\mathrm{AA}$ of prompt $\Lambda_{\rm {c}}^{+}$ baryons in 0-80% most central Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} =...", doi = "10.17182/hepdata.89397.v1/t2", year = "2018", type = "Data Set", collaboration = "ALICE", authors = "Acharya, Shreyasi and others" }
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\begin{document}
\title{Stochastic Subspace Correction in Hilbert Space} \begin{abstract} We consider an incremental approximation method for solving variational problems in infinite-dimensional separable Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is solved. We show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in \cite{GrOs2016}.
\keywords{infinite space splitting \and subspace correction \and multiplicative Schwarz \and block coordinate descent \and greedy \and randomized \and convergence rates \and online learning} \subclass{65F10 \and 65N22\and 49M27} \end{abstract}
\section{Introduction}\label{sec1} The fast solution of quadratic minimization problems or, correspondingly, of large linear systems of equations is an important topic in many application areas of numerical simulation. To this end, iterative algorithms play a major role. They can be formalized by means of subspace correction methods, either in the so-called additive or the multiplicative variant, see \cite{Xu,GrOs1995}. There, a set of variational problems in appropriate subspaces is chosen and the current approximation is iteratively (either collectively or successively) improved by means of the respective solutions of these subproblems. The subspaces can be one-dimensional or, in a block type fashion, they can have arbitrary (finite) dimension as well. Examples are the well-known Jacobi and Gauss-Seidel algoritms from linear algebra and their block-wise variants, but also domain decomposition methods or multigrid and multilevel techniques from scientific computing.
For a (finite or infinite) set of subproblems at hand, the question arises in which order the incremental updates should be made and what the convergence behavior of the associated subspace correction algorithm will be. Most conventionally, the order is a priorily fixed in a deterministic fashion. This is the case for basically all the classical methods, like Gauss-Seidel, domain decomposition or multigrid methods. The order of traversal through the subproblems is prescribed by the method itself. Examples are lexicographical or so-called red-black orderings for systems stemming from finite element or finite difference discreitizations and, additionally, level wise traversal orderings in multigrid algorithms. Besides, for the multiplicative subspace correction approach, there are greedy methods where the next subspace is identified according to an optimization criterion such that the actual error is reduced by the following incremental update as much as possible. This may substantially improve the convergence of the overall algorithm. A detailed analysis of various greedy approximation methods is given in the seminal book \cite{Tem2012}. A simple example from linear algebra is the so-called Gauss-Southwell approach, where the next update variable is that with the largest residuum. Usually, in the case of finitely many subspaces, the determination of the optimal next subspace can be done exactly, but it involves additional costs. In the case of infinitely many subspaces, this is not possible any more, and heuristic choices are employed there in practical methods.
Besides a deterministic or greedy pick, we may also choose the next subproblem in a random fashion according to a probability distribution $\rho$ on the set of subspaces, see \cite{GrOs2012} and the references cited therein. The analysis of such stochastic iterations has been a very active research topic in large-scale convex optimization, see \cite{FerRic2016} for a recent survey, but also in the area of machine learning and compressed sensing.
Compared to the greedy approach, the cost for determining the next subspace is dramatically reduced to the cost of sampling the underlying probability distribution $\rho$. Moreover, the random pick is feasible also for infinite sets of subspaces. But the question is now what the associated convergence rate (in expectation) will be. For finitely many subspaces, the answer is very encouraging \cite{GrOs2012}: Both greedy and stochastic iterations yield the same exponential rates of convergence, although with different constants, and in the latter case almost surely and in expectation only.
In this article, we deal with the case of an infinite number of subspaces for which a first comparison of greedy and stochastic subspace correction methods was carried out in \cite{GrOs2016} for countable sets of subspaces. It was shown that the (much more involved and costly) greedy method converges at an algebraic rate for solutions from a certain class ${\mathcal{A}}_1$ while basically the same convergence rate can be achieved in expectation by a stochastic subspace correction method on a class ${\mathcal{A}}_\infty^\rho\subset {\mathcal{A}}_1$ depending on $\rho$. Details will be given in the next sections.
The aim of this paper is to show that convergence rates for the expectation of the squared error can be guaranteed under weaker conditions than previously established in \cite{GrOs2016}, namely for solutions from a class $A_2$ still depending on $\rho$, where ${\mathcal{A}}_\infty^\rho \subset A_2 \subset {\mathcal{A}}_1$. This result reveals some connection to the theory of approximation algorithms in reproducing kernel Hilbert spaces, and may also allow for a wider range of applications of incremental, multiplicative subspace correction methods with randomly picked orderings which may have interesting applications in numerical linear algebra, scientific computing, quadratic optimization, machine learning and compressed sensing.
The remainder of this paper is organized as follows: In Section \ref{sec2} we give basic notation and introduce our multiplicative subspace correction/approximation algorithm with random picking in the case of a family of subspaces $\{V_\omega:\,\omega\in \Omega\}$ with an infinite (possibly uncountable) index set $\Omega$. Moreover, we give in Theorem \ref{theo1} and Theorem \ref{theo2} sharp bounds of its error and thus of its convergence rate in expectation for the class $A_2$. In Section \ref{sec3} we discuss various examples of our abstract theory. First, we consider the case of a countable index set $\Omega$ and discrete probability measures $\rho$ on it. Moreover, in Lemma \ref{lem1} we also relate our new function class $A_2$ to the classes ${\mathcal A}_\infty^\rho$ and ${\mathcal{A}}_1$, previously used in \cite{GrOs2016}. Then, we consider the case of stochastic approximation in reproducing kernel Hilbert spaces and show that our theory can be applied there as well. Next, we study the case of general unit norm dictionaries and approximation with these, and provide in Theorem \ref{theo3} a version of our main results from Section \ref{sec2} with simplified proof. Finally, we deal with a collective approximation problem from \cite{AJOP2017} and show how our theory applies. We conclude in Section \ref{sec4} with some further remarks on our convergence results.
\section{Details and Proofs}\label{sec2}
Throughout this paper, let $V$ be a separable real Hilbert space. For a given continuous and coercive Hermitian form $a(\cdot,\cdot)$ on $V$ and a bounded linear functional $F$ on $V$, we consider the variational problem of finding the unique element $u\in V$ such that \begin{equation}\label{VP} a(u,v)=F(v)\qquad\forall v\in V. \end{equation} Equivalently, (\ref{VP}) can be formulated as quadratic minimization problem in $V$ or as linear operator equation in the dual space of $V$. In the following, we use the fact that
$a(\cdot,\cdot)$ defines a spectrally equivalent scalar product on $V$, equip $V$ with it, and write $\|v\|={a(v,v)}^{1/2}$.
Our aim is to study a particular instance of an incremental subspace correction (or Schwarz iterative) method for solving (\ref{VP}). Let $\Omega$ be a fixed index set equipped with a probability measure $\rho$ (compared to \cite{GrOs2016}, we also allow for uncountable $\Omega$, see below for an example). Consider a family $\{V_\omega\}_{\omega\in\Omega}$ of separable real Hilbert spaces, each equipped with a spectrally equivalent scalar product
$a_\omega(\cdot,\cdot)$ and norm $\|v_\omega\|_\omega:=a_\omega(v_\omega,v_\omega)^{1/2}$, and linear operators $R_\omega:\,V_\omega\to V$ such that \begin{equation}\label{La}
\|R_\omega\|_{V_\omega\to V} = \sup_{\|v_\omega\|_\omega=1} \|R_\omega v_\omega\| \le \Lambda < \infty,\qquad \omega\in \Omega. \end{equation} Finally, we introduce another family of linear operators $T_\omega:\,V\to V_\omega$ by the solution of auxiliary variational problems in $V_\omega$: \begin{equation}\label{VPo} a_\omega(T_\omega v,v_\omega)=a(v,R_\omega v_\omega)\qquad \forall\;v_\omega\in V_\omega,\qquad \omega\in \Omega.
\end{equation} It is easy to see that $\|T_\omega\|_{V\to V_\omega} \le \Lambda$ as well. Without loss of generality, we can assume that $\mathrm{Ker}(R_\omega)=\{0\}$ for all $\omega$ (otherwise replace $V_\omega$ by $V_\omega \ominus_\omega \mathrm{Ker}(R_\omega)$).
With these preparations at hand, the algorithm under consideration has the form \begin{equation} \label{Rec} u^{(m+1)} = \alpha_m u^{(m)} + \xi_m R_{\omega_m}r^{(m)}_{\omega_m},\qquad r^{(m)}_{\omega_m}=T_{\omega_m}(u-u^{(m)}),\quad m=0,1,\ldots,\quad u^{(0)}=0, \end{equation} where $\{\omega_m\}$ is a sequence of independent samples from $\Omega$ which are identically distributed according to $\rho$. Furthermore $\alpha_m=1-(m+2)^{-1}$, and $\xi_m$ is such that the error $$
\delta_{m+1}^2:=\|u-u^{(m+1)}\|^2 $$ is minimized. This gives the explicit formula \begin{equation}\label{Xi}
\xi_m=\mathrm{argmin}_\xi \|u-\alpha_m u^{(m)}-\xi R_{\omega_m}r^{(m)}_{\omega_m}\|^2 =\frac{F(R_{\omega_m}r^{(m)}_{\omega_m})-\alpha_m a(u^{(m)},R_{\omega_m}r^{(m)}_{\omega_m})}{a(R_{\omega_m}r^{(m)}_{\omega_m},R_{\omega_m}r^{(m)}_{\omega_m})}. \end{equation} Since $r^{(m)}_{\omega_m}$ is defined via (\ref{VP}) and (\ref{VPo}) by the variational problem \begin{equation}\label{VPoe} a_\omega(r^{(m)}_{\omega_m} ,v_{\omega_m})=a(u-u^{(m)},R_{\omega_m} v_{\omega_m})=F(R_{\omega_m} v_{\omega_m})-a(u^{(m)},R_{\omega_m} v_{\omega_m})\qquad \forall\;v_\omega\in V_\omega, \end{equation} we see that (\ref{Rec}) can be executed once $u^{(m)}$ and $\omega_m$ are available.
We note that this way $u^{(m)}$ and thus $\delta_m^2$ become random variables on $\Omega^m$ equipped with the product measure $\rho^m$. To provide estimates for the expected squared error $\mathbb{E}(\delta_m^2)$, we need the notion of Bochner integrals \cite{Bo}. Given any Bochner-measurable $V$-valued function $\phi:\, \omega\in \Omega \to \phi_\omega \in V$, its Bochner integral \begin{equation}\label{E} \mathbb{E}_\rho (\phi):=\int_{\Omega} \phi_\omega\,d\rho_\omega \end{equation} is well-defined with value in $V$ iff the scalar integral \begin{equation}\label{En}
\mathbb{E}_\rho (\|\phi\|):=\int_{\Omega} \|\phi_\omega\| \,d\rho_\omega < \infty \end{equation} exists. The Bochner integral is similarly well-defined if $V$ is replaced by a separable Banach space. In the case of a discrete probability measure on a countable index set $\Omega$, measurability of $\phi$ is not an issue, in other situations, it needs to be checked. For the following, we assume that for any fixed $e\in V$ the function \begin{equation}\label{A1} \tilde{\psi}:\, \omega\in \Omega \to \tilde{\psi}_\omega \in R_{\omega}(V_\omega)\subset V,\qquad \tilde{\psi}_\omega := \left\{
\begin{array}{ll}\frac{R_{\omega}T_\omega e}{\|R_{\omega}T_\omega e\|},& R_{\omega}T_\omega e\neq 0,\\ 0,& R_{\omega}T_\omega e = 0,\end{array}\right. \end{equation} is Bochner-measurable.
Next, we introduce the class $A_2\equiv A_{2,\rho}\subset V$ which will play a central role in the convergence theory for (\ref{Rec}). We say that $u\in V$ belongs to $A_2$ if there exists
a Bochner-measurable function $\phi: \,\omega\to R_\omega v_\omega$ with $v_\omega\in V_\omega$ for all $\omega\in\Omega$ such that the scalar-valued function $\omega\to \|v_\omega\|_\omega$ is also measurable, and \begin{equation}\label{A2}
u=\mathbb{E}_\rho(\phi)=\int_{\Omega} R_\omega v_\omega\,d\rho_\omega,\qquad \mathbb{E}_\rho(\|v_\omega\|_\omega^2)=\int_{\Omega} \|v_\omega\|_\omega^2\,d\rho_\omega<\infty, \end{equation} Define a norm on $A_2$ by \begin{equation}\label{NA2}
\|u\|_{{A}_2}:= \inf\, \mathbb{E}_\rho(\|v_\omega\|_\omega^2)^{1/2}, \end{equation} where the infimum is taken with respect to all admissible representations of $u$ in (\ref{A2}). How this class is related to the classes $\mathcal{A}_p^\gamma$ introduced in \cite{GrOs2016} for discrete measures $\rho$ on countable index sets $\Omega$ and other classes used in similar context in the literature will be elaborated on in Section \ref{sec3}.
The central result of this note is the following: \begin{theo}\label{theo1} If (\ref{A1}) holds and if $u$ belongs to the linear space ${A}_2$ induced by the condition (\ref{A2}) then, for the incremental approximation algorithm (\ref{Rec}), we have \begin{equation}\label{EC2}
\mathbb{E}(\delta_{m}^2) \le \frac{(\Lambda\|u\|_{{A}_2}+\|u\|)^2}{m+1},\qquad m=0,1,\ldots. \end{equation} \end{theo}
{\bf Proof }. We start with an analysis of the error reduction in one recursion step, i.e., with an estimate of $\mathbb{E}_\rho (\delta_{m+1}^2|u^{(m)})$. By (\ref{Xi}) and with the notation $$ e^{(m)}:=u-u^{(m)}, \qquad \bar{\alpha}_m:=1-\alpha_m=(m+2)^{-1}, \qquad w:=\alpha_m e^{(m)}+\bar{\alpha}_m u, $$ we have \begin{eqnarray*}
\delta_{m+1}^2&=&\min_\xi\, \|\alpha_m(u-u^{(m)})+\bar{\alpha}_m u - \xi R_{\omega_m}r^{(m)}_{\omega_m}\|^2\\
&=& \|w\|^2 - a(w,\tilde{\psi}_{\omega_m})^2 = \alpha_m^2 (\delta_m^2-a(e^{(m)},\tilde{\psi}_{\omega_m})^2)\\ &&\quad +2\alpha_m\bar{\alpha}_m(a(e^{(m)},u)-a(e^{(m)},\tilde{\psi}_{\omega_m})a(u,\tilde{\psi}_{\omega_m}))
+\bar{\alpha}_m^2(\|u\|^2-a(u,\tilde{\psi}_{\omega_m})^2). \end{eqnarray*} Here and throughout the proof $\tilde{\psi}$ stands for the function defined in (\ref{A1}) for $e=e^{(m)}$. The measurability assumption for this $\tilde{\psi}$ allows us to take expectations with respect to the choice of $\omega_m$ in the above error representation: \begin{equation}\label{I1} \begin{array}{l}
\mathbb{E}_\rho(\delta_{m+1}^2|u^{(m)})= \alpha_m^2 (\delta_m^2-\mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})^2))\\ \\ \qquad +2\alpha_m\bar{\alpha}_m(a(e^{(m)},u)-\mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})a(u,\tilde{\psi}_{\omega})))
+\bar{\alpha}_m^2(\|u\|^2-\mathbb{E}_\rho(a(u,\tilde{\psi}_{\omega})^2)). \end{array} \end{equation} By the definition of $r_\omega^{(m)}=T_\omega e^{(m)}$ via (\ref{VPo}) we have \begin{equation}\label{I0}
\frac{\|R_\omega r_\omega^{(m)}\|}{\|r_\omega^{(m)}\|_\omega} a(e^{(m)},\tilde{\psi}_\omega) = a_\omega(r_\omega^{(m)},\frac{r_\omega^{(m)}}{\|r_\omega^{(m)}\|_\omega})\ge a_\omega(r_\omega^{(m)},\frac{v_\omega}{\|v_\omega\|_\omega}) = \frac{a(e^{(m)},R_\omega v_\omega)}{\|v_\omega\|_\omega} \end{equation} for any $v_\omega\in V_\omega$ and $\omega\in \Omega$. Together with (\ref{La}) and (\ref{A2}), this implies $$
a(e^{(m)},u) = \mathbb{E}_\rho(a(e^{(m)},R_\omega v_\omega)) \le \Lambda \mathbb{E}_\rho(\|v_\omega\|_\omega a(e^{(m)},\tilde{\psi}_{\omega})). $$
Thus, we can apply the Cauchy-Schwarz inequality to the second term in the right-hand side of (\ref{I1}): \begin{eqnarray*} && 2\alpha_m\bar{\alpha}_m( a(e^{(m)},u)-\mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})\,a(u,\tilde{\psi}_{\omega}))\\
&&\qquad \le 2\alpha_m\bar{\alpha}_m \mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})(\Lambda \|v_\omega\|_\omega - a(u,\tilde{\psi}_{\omega})))\\
&&\qquad \le 2\alpha_m\bar{\alpha}_m \mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})^2)^{1/2}\mathbb{E}_\rho((\Lambda\|v_\omega\|_\omega- a(u,\tilde{\psi}_{\omega}))^2)^{1/2}\\
&&\qquad \le \alpha_m^2 \mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})^2) + \bar{\alpha}_m^2(\Lambda^2\mathbb{E}_\rho(\|v_\omega\|_\omega^2)-2\Lambda
\mathbb{E}_\rho(\|v_\omega\|_\omega\, a(u,\tilde{\psi}_{\omega})) + \mathbb{E}_\rho(a(u,\tilde{\psi}_{\omega})^2))\\
&&\qquad \le \alpha_m^2 \mathbb{E}_\rho(a(e^{(m)},\tilde{\psi}_{\omega})^2) + \bar{\alpha}_m^2(\Lambda^2\|u\|_{A_2}^2+2\Lambda \|u\|_{A_2} \|u\| + \mathbb{E}_\rho(a(u,\tilde{\psi}_{\omega})^2)), \end{eqnarray*} where we have used that by (\ref{A1}) $$
|\mathbb{E}_\rho(\|v_\omega\|_\omega a(u,\tilde{\psi}_{\omega}))|\le \mathbb{E}_\rho(\|v_\omega\|_\omega)\|u\|\le
\mathbb{E}_\rho(\|v_\omega\|_\omega^2)^{1/2}\|u\| = \|u\|_{A_2}\|u\|. $$ After substitution into (\ref{I1}) some terms cancel, and we arrive at the estimate \begin{equation}\label{I2}
\mathbb{E}_\rho(\delta_{m+1}^2|u^{(m)})\le \alpha_m^2 \delta_m^2 + \bar{\alpha}_m^2(\Lambda \|u\|_{A_2}+\|u\|)^2 \end{equation} for the expectation of the squared error $\delta_{m+1}^2$ conditioned on $u^{(m)}$. Because of the independence assumption, this gives the recursion for the expected error \begin{equation}\label{ERec}
\mathbb{E}(\delta_{m+1}^2) \le \alpha_m^2 \mathbb{E}(\delta_m^2) + \bar{\alpha}_m^2(\Lambda \|u\|_{A_2}+\|u\|)^2,\qquad m=0,1,\ldots,
\end{equation} with $\mathbb{E}_\rho(\delta_{0}^2)=\|u\|^2$ since we set $u^{(0)}=0$. Due to the specific choice of $\alpha_m$, for the sequence $b_m:=(m+1)\mathbb{E}(\delta_m^2)$ this yields the recursion $$
b_{m+1} \le \alpha_m b_m + \bar{\alpha}_m(\Lambda\|u\|_{A_2}+\|u\|)^2, \qquad m=0,1,\ldots,\qquad b_0=\|u\|^2, $$
which implies $b_m\le (\Lambda\|u\|_{A_2}+\|u\|)^2$ uniformly in $m$ (note $\alpha_m+\bar{\alpha}_m=1$). This is equivalent to (\ref{EC2}), and concludes the proof of Theorem \ref{theo1}.
$\Box$
As in \cite{GrOs2016}, the proof of Theorem \ref{theo1} can be modified to yield an estimate valid for arbitrary $u\in V$. This results in the following:
\begin{theo}\label{theo2} If the functions defined in (\ref{A1}) are Bochner-measurable then for arbitrary $u\in V$ the algorithm (\ref{Rec}) satisfies \begin{equation}\label{ECV}
\mathbb{E}(\delta_{m}^2)^{1/2} \le 2\left(\|u-h\| +\frac{((\Lambda\|h\|_{{A}_2}+\|h\|)^2+\|u\|^2)^{1/2}}{(m+1)^{1/2}}\right),\qquad m=0,1,\ldots, \end{equation} where $h\in {A}_2$ is arbitrary. As a consequence, we have $\mathbb{E}(\delta_{m}^2)\to 0$ if $u$ belongs to the closure of $A_2$ in $V$. \end{theo}
{\bf Proof}. To see (\ref{ECV}), write $$ a(e^{(m)},u)-a(e^{(m)},\tilde{\psi}_{\omega_m})\,a(u,\tilde{\psi}_{\omega_m})\le a(e^{(m)},h)-a(e^{(m)},\tilde{\psi}_{\omega_m})\,a(h,\tilde{\psi}_{\omega_m})
+\|u-h\|\|e^{(m)}\|, $$ and proceed as above for the first term in the right-hand side, using the assumption $h\in {A}_2$. Instead of (\ref{ERec}), this yields \begin{equation}\label{ERecV}
\mathbb{E}(\delta_{m+1}^2) \le \alpha_m^2 \mathbb{E}(\delta_m^2) + 2\alpha_m\bar{\alpha}_m \mathbb{E}(\delta_m)\|u-h\|+\bar{\alpha}_m^2((\Lambda\|h\|_{A_2}+\|h\|)^2+\|u\|^2), \end{equation} $m=0,1,\ldots$. The rest of the argument leading to (\ref{ECV}) is the same as in the proof of \cite[Theorem 2]{GrOs2016}.
$\Box$
The bounds in Theorem \ref{theo1} and Theorem \ref{theo2} carry over to the stochastic version of orthogonal matching pursuit (OMP), where the recursion (\ref{Rec}) is replaced by \begin{equation} \label{OMP} u^{(m+1)} = P_{W_{m}}u,\qquad r^{(m)}_{\omega_m}=T_{\omega_m}(u-u^{(m)}),\quad m=0,1,\ldots,\quad u^{(0)}=0, \end{equation} with $P_{W_{m}}$ denoting the orthogonal projection onto the subspace $$ W_{m}:=\mathrm{span}(\{R_{\omega_0}r^{(0)}_{\omega_0},\ldots,R_{\omega_m}r^{(m)}_{\omega_m}\}) $$ in $V$. This is because, for given $u^{(k)}$ and $\omega_k$, $k=0,\ldots,m$, the error of the stochastic OMP algorithm after the update step satisfies $$
\|u-u^{(m+1)}\|=\|u-P_{W_{m}}u\|\le \|u-\alpha_m u^{(m)}-\xi_m R_{\omega_m}r^{(m)}_{\omega_m}\| $$
for any choice of $\xi_m$. Consequently, the estimates for one step of (\ref{Rec}) can be applied, and we obtain the same recursions for the expectations of the squared error $\mathbb{E}(\|u-P_{W_{m}}u\|^2)$ of stochastic OMP as in (\ref{ERec}) and (\ref{ERecV}) for our algorithm (\ref{Rec}). Thus, the bounds in Theorem \ref{theo1} and Theorem \ref{theo2} hold for stochastic OMP as well. In practice, stochastic OMP (\ref{OMP}) is expected to converge slightly faster than our algorithm (\ref{Rec}), at the expense of a more costly evaluation of the projections $P_{W_{m}}u$ in each step.
Finally, note that the class ${A}_2\subset V$ delicately depends on the choices for $\{V_\omega\}_{\omega\in \Omega}$ and the probability measure $\rho$. It can be made more explicit in some cases which we outline in the next section.
\section{Examples}\label{sec3}
\subsection{Countable $\Omega$ and discrete measures}\label{sec31} The most common situation in which our results can be made more explicit is the case of a discrete measure $\rho$ on a countable $\Omega$. To allow for a direct comparison with the results in \cite{GrOs2016}, set without loss of generality $\Omega=\mathbb{N}$ and denote $\rho_i:=\rho(\{i\})>0$. Then the measurability assumptions for (\ref{A1}) and (\ref{A2}) are irrelevant. Thus, $u\in {A}_2$ is, according to (\ref{A2}), equivalent to the existence of $v_i\in V_i$ such that $$
u=\sum_i \rho_i R_iv_i,\qquad \sum_i \rho_i \|v_i\|^2_i < \infty, $$
where $\|\cdot\|_i$ is the norm in $V_i$. Moreover, $$
\|u\|_{A_2}^2 = \inf_{u=\sum_{i} R_iv_i}\, \sum_i \rho_i \|v_i\|^2_i . $$ In \cite{GrOs2016}, for any sequence $\gamma:=\{\gamma_i>0\}$ and $0<q\le \infty$, classes $\mathcal{A}_q^\gamma$ were introduced by the requirement that $u\in \mathcal{A}_q^\gamma$ if there are $w_i\in V_i$ such that $$
u=\sum_i R_iw_i,\qquad \|\{\gamma_i^{-1}\|w_i\|_i \}\|_{\ell_q}< \infty. $$ The (quasi-)norm on $\mathcal{A}_q^\gamma$ is given by $$
\|u\|_{\mathcal{A}_q^\gamma} = \inf_{u=\sum_{i} R_iw_i}\, \|\{\gamma_i^{-1}\|w_i\|_i \}\|_{\ell_q}. $$ In particular, for the sequence $\gamma=\mathbf{1}$ given by $\gamma_i=1$, we simply use the notation $\mathcal{A}_q=\mathcal{A}_q^{\mathbf{1}}$.
\begin{lem}\label{lem1} For any given $\{V,a\}$ and $\{V_i,a_i\}_{i\in\mathbb{N}}$ and any discrete probability measure $\rho$ on $\mathbb{N}$, the following continuous embbedings hold with norm $\le 1$: $$ \mathcal{A}^\rho_\infty \subset A_2=\mathcal{A}_2^{\sqrt{\rho}} \subset \mathcal{A}_1. $$ \end{lem}
{\bf Proof}. Since $$
\|u\|_{\mathcal{A}^{\sqrt{\rho}}_2}^2=\inf_{u=\sum_{i} R_iw_i} \sum_i \rho_i^{-1}\|w_i\|_i^2 =
\inf_{u=\sum_{i} \rho_i R_iv_i} \sum_i \rho_i\|v_i\|_i^2 =\|u\|^2_{A_2}, $$ the equality $A_2=\mathcal{A}_2^{\sqrt{\rho}}$ is obvious. Take any $u$ of the form $u=\sum_i R_i w_i$. The inequalities $$
\sum_i \rho_i^{-1}\|w_i\|_i^2 \le \sum_i \rho_i \sup_i (\rho_i^{-1}\|w_i\|_i)^2=(\sup_i \rho_i^{-1}\|w_i\|_i)^2 $$ and $$
\sum_i \|w_i\|_i = \sum_i \rho_i^{1/2}(\rho^{-1/2}\|w_i\|_i) \le (\sum_i \rho_i)^{1/2}(\sum_i \rho_i^{-1}\|w_i\|_i^2)^{1/2}
=(\sum_i \rho_i^{-1}\|w_i\|_i^2)^{1/2} $$ imply the embeddings $\mathcal{A}^\rho_\infty \subset \mathcal{A}_2^{\sqrt{\rho}}$ and $\mathcal{A}_2^{\sqrt{\rho}} \subset \mathcal{A}_1$, respectively.
$\Box$
In \cite{GrOs2016}, the condition $u\in \mathcal{A}^\rho_\infty$ was shown to be sufficient for estimates essentially identical with (\ref{ERec}) and (\ref{ERecV}) to hold, therefore the present paper improves the results from \cite{GrOs2016} (and extends them to uncountable $\Omega$). On the other hand, as shown in \cite{BCDD2008,GrOs2016} the condition $u\in \mathcal{A}_1$ is sufficient for proving convergence rates similar to (\ref{ERec}) and (\ref{ERecV}) for the weak greedy version of our algorithm (\ref{Rec}), where the random choice of $\omega_m$ is replaced by a residual-based search for a $\omega_m\in \Omega$ such that \begin{equation}\label{Gr}
\|r_{\omega_m}^{(m)}\|_{\omega_m} \ge \beta \sup_{\omega\in\Omega}\|r_{\omega}^{(m)}\|_{\omega}. \end{equation} Here, $\beta\in (0,1]$ is a fixed parameter. In other words, for the specific algorithm (\ref{Rec}) the greedy rule (\ref{Gr}) of picking the $\omega_m$ yields the same convergence bound on a larger class of $u$ than any of the stochastic search algorithms. The drawback of greedy algorithms is the cost of implementing (\ref{Gr}) which typically requires the computation of residuals $r_\omega^{(m)}$ for many $\omega\in \Omega$.
\subsection{Stochastic approximation in RHKS}\label{sec32} Another case where the above theory can be substantiated is the approximation of functions in a reproducing kernel Hilbert space from randomly selected point evaluations. The standard setting \cite{RKHS,Bog} is as follows: Let $\Omega$ be a compact metric space, and let $K:\,\Omega\times \Omega \to \mathbb{R}$ be a continuous positive-definite kernel. This kernel defines a Hilbert space $H_K$ with scalar product $(\cdot,\cdot)_K$ whose elements are continuous functions $f:\;\Omega\to \mathbb{R}$ such that \begin{equation}\label{RK} (K_\omega,f)_K = f(\omega) \qquad \forall\;f\in H_K\quad \forall \;\omega\in\Omega. \end{equation} Here, $K_\omega\in H_K$ is given by $K_\omega(\eta)=K(\omega,\eta)$, $\eta\in \Omega$. Now, choose $V=H_K$ with the scalar product $a(\cdot,\cdot)=(\cdot,\cdot)_K$ and consider the family of one-dimensional subspaces $V_\omega\subset V$ spanned by $K_\omega$, $\omega\in \Omega$. In particular, $a_\omega(\cdot,\cdot)$ is the restriction of $(\cdot,\cdot)_K$ to $V_\omega$, and $R_\omega$ is the natural injection ($\Lambda=1$). With this, we compute $$ R_\omega T_\omega f = T_\omega f = \frac{(K_\omega,f)_K}{(K_\omega,K_\omega)_K}K_\omega =\frac{f(\omega)}{K(\omega,\omega)}K_{\omega}, $$ where in the last step we have used the reproducing kernel property (\ref{RK}). Thus, our algorithm (\ref{Rec}) turns into an incremental approximation process, requiring in each step the evaluation of $$ e^{(m)}(\omega_m)=f(\omega_m)-u^{(m)}(\omega_m), $$ where $\omega_m$ is chosen randomly and independently from $\Omega$ according to a certain probability distribution $\rho$. This scenario is typical in learning theory \cite{SmZh}, where the samples $(\omega_m,y_m)\in \Omega\times \mathbb{R}$, which are drawn according to an (unknown) joint probability distribution $\tilde{\rho}$ on $\Omega\times \mathbb{R}$, become incrementally available, and one tries to recover the regression function $$
f(\omega)=\mathbb{E}_{\tilde{\rho}}(y|\omega). $$
In the "no-noise" case ($\mathbb{E}_{\tilde{\rho}}((y-f(\omega))^2|\omega) = 0$ a.e. on $\Omega$), we would have $y_m=f(\omega_m)$ almost surely, while the $\omega_m$ are independent samples drawn from $\Omega$ according to the marginal distribution $\rho=\tilde{\rho}_\omega$.
To apply our theory, i.e., to obtain rates for the expectation of the squared error from (\ref{EC2}) and (\ref{ECV}), we need to check (\ref{A1}) and have to examine the condition $u\in A_2$ and the approximability of $u\in H_K$ by elements from $A_2$, respectively. The measurability assumptions for (\ref{A1}) and (\ref{A2}) follow from the uniform continuity of the kernel which implies the uniform continuity of the function $\omega\to K_{\omega}$, and the measurability of the function $\omega \to R_\omega v_\omega = c_\omega K_\omega$ for any measurable scalar-valued function $\omega\to c_\omega$. Thus, $u\in A_2$ if $$ u(\eta)=(u,K_\eta)_K=\mathbb{E}_\rho( (c_\omega K_\omega,K_\eta)_K)=\int_\Omega c_{\omega}K(\omega,\eta)\,d\rho_\omega,\qquad \int_\Omega c_\omega^2 \,d\rho_\omega < \infty, $$ i.e., if $u$ is in the image of $L_2(d\rho)$ under the action of the integral operator $L_K$ with kernel $K$ given by the formula $$ (L_Kf)(\eta) := \int_\Omega K(\omega,\eta)f(\omega)\,d\rho_\omega. $$ It is well known that the operator $L_K$ is also well-defined on $V=H_K$, that it is trace-class positive semi-definite on $H_K$, and that $A_2=L_K(L_2(d\rho))=L_K^{1/2}(H_K)$. Thus, our result recovers rates for the noiseless case analogous to those known in online learning with kernels for similar approximation algorithms \cite{TaYa2013,LiZh2015,DeBa2016}, where the spaces defined in terms of the spectral decomposition of $L_K$ often serve as smoothness classes.
\subsection{General unit norm dictionaries}\label{sec33} As a third, slightly different but also slightly more general example, let us consider the case when, for a given separable Hilbert space $V=H$ with scalar product $a(\cdot,\cdot)=(\cdot,\cdot)$, we choose a Borel measure $\rho$ concentrated on the unit sphere $\Omega=S_H=
\{\omega\in H:\;\|\omega\|=1\}$ of $H$. Then, we consider the algorithm (\ref{Rec}) with the family $V_\omega :=\mathrm{span}(\{\omega\})$ of one-dimensional subspaces of $H$ (again, $a_\omega(\cdot,\cdot)=(\cdot,\cdot)$ on $V_\omega$, $R_\omega$ are the natural injections, and $\Lambda=1$ ). Since any function of the form $\omega \in S_H \to v_\omega=c_\omega \omega$ is Bochner-measurable if the scalar-valued function $\omega \to c_\omega$ is measurable, we have $u\in A_2$ iff \begin{equation}\label{A2a} u=\int_{S_H} c_\omega \omega d\rho, \qquad \int_{S_H} c_\omega^2\, d\rho_\omega < \infty. \end{equation} In this case, the proof of (\ref{EC2}) can be carried out directly, using the covariance operator $L:\,H\to H$ given by \begin{equation}\label{CoOp} L v = \mathbb{E}_\rho((v,\omega)\omega) =\int_{S_H} (v,\omega)\omega \,d\rho_\omega, \qquad v\in H. \end{equation} This operator is positive semi-definite and trace-class, i.e., there is a complete orthonormal system of eigenfunctions $\psi_k$ of $L$ for the subspace $$ \tilde{H}:= H\ominus \mathrm{Ker}(L) $$ with associated eigenvalues $\mu_k>0$ satisfying $\sum_k \mu_k=1$. The powers $L^s$, $s>0$, are well defined on $H$ and act as isometries between $\tilde{H}$ and the Hilbert spaces $$
H^s_L=L^s(H):=\{v=\sum_k \mu_k^s c_s \psi_k: \; \|v\|_{H^s_L}:= (\sum_k c_k^2)^{1/2}<\infty\}. $$ The latter serve as smoothness spaces and, as we will see, $u\in H^{1/2}_L$ implies an analog of (\ref{EC2}). Indeed, since $\omega\in S_H$ we have $\tilde{\psi}_{\omega}=\omega$ in (\ref{A1}) for any $e$. Taking into account (\ref{CoOp}) the counterpart of (\ref{I1}) reads as follows: \begin{eqnarray*} &&\mathbb{E}_\rho(\delta_{m+1}^2)= \alpha_m^2 (\delta_m^2-\mathbb{E}_\rho((e^{(m)},\omega)^2))\\ &&\qquad\qquad\qquad\qquad +2\alpha_m\bar{\alpha}_m((e^{(m)},u)-\mathbb{E}_\rho((e^{(m)},\omega)(u,\omega)))
+\bar{\alpha}_m^2(\|u\|^2-\mathbb{E}_\rho((u,\omega)^2))\\
&&\qquad\qquad \,= \alpha_m^2 (\delta_m^2-(Le^{(m)},e^{(m)}))+2\alpha_m\bar{\alpha}_m((e^{(m)},u)-(Le^{(m)},u))+\bar{\alpha}_m^2(\|u\|^2-(Lu,u)).
\end{eqnarray*} Assuming $u\in H^{1/2}_L$, i.e., $u=L^{1/2}v$ for some $v\in \tilde{H}\subset H$ with $\|u\|_{H^{1/2}_L}=\|v\|$, we estimate the second term in the right-hand side by \begin{eqnarray*} 2\alpha_m\bar{\alpha}_m((e^{(m)},u)-(Le^{(m)},u))&=&2\alpha_m\bar{\alpha}_m(L^{1/2}e^{(m)},(L^{-1/2}-L^{1/2})u)\\
&\le& 2\alpha_m\bar{\alpha}_m \|L^{1/2}e^{(m)}\|\|(L^{-1/2}-L^{1/2})u\|\\
&\le& \alpha_m^2 (Le^{(m)},e^{(m)}) +\bar{\alpha}_m^2 (\|v\|^2 -2\|u\|^2 + (Lu,u)). \end{eqnarray*} Substitution and cancellation of several terms yields the following analog of (\ref{I2}): $$
\mathbb{E}_\rho(\delta_{m+1}^2)\le \alpha_m^2 \delta_m^2 + \bar{\alpha}_m^2 \|u\|_{H^{1/2}_L}^2. $$ The rest is as in the above proof of Theorem \ref{theo1}. This results in the following estimate with slightly improved constant.
\begin{theo}\label{theo3} In the setting described in this subsection, the algorithm (\ref{Rec}) converges in expectation for arbitrary $u\in H_L^{1/2}$: \begin{equation}\label{EC2a}
\mathbb{E}(\delta_{m}^2)\le \frac{\|u\|_{H^{1/2}_L}^2}{m+1},\qquad m=0,1,\ldots . \end{equation} The analog of (\ref{ECV}) is \begin{equation}\label{ECVa}
\mathbb{E}(\delta_{m}^2)^{1/2}\le 2(\|u-h\|+\frac{(\|h\|_{H^{1/2}_L}^2+\|u\|^2)^{1/2}}{(m+1)^{1/2}}),\qquad m=0,1,\ldots , \end{equation} valid for any $u\in H$ and $h\in H_L^{1/2}$. Convergence in expectation $\mathbb{E}(\delta_{m}^2)\to 0$ holds for any $u\in \tilde{H}$.\\
Moreover, the classes $A_2$ and $H_L^{1/2}$ coincide, with equality of norms $\|u\|_{A_2}=\|u\|_{H_L^{1/2}}$ for any $u\in H_L^{1/2}$. \end{theo}
{\bf Proof}. The estimate (\ref{EC2a}) was already established, the modification leading to (\ref{ECVa}) is similar to the one in the proof of Theorem \ref{theo2}: Since \begin{eqnarray*} (e^{(m)},u)-(Le^{(m)},u)&=&(e^{(m)},h)-(Le^{(m)},h) + (e^{(m)},(I-L)(u-h))\\
&\le& (e^{(m)},h)-(Le^{(m)},h) + \|e^{(m)}\|\|u-h\|, \end{eqnarray*} we can proceed for the first term as above, with $u$ replaced by $h\in A_2$, to arrive at $$
\mathbb{E}(\delta_{m+1}^2)\le \alpha_m^2 \mathbb{E}(\delta_m^2)+2\alpha_m\bar{\alpha}_m\mathbb{E}(\delta_m)\|u-h\| +\bar{\alpha}_m^2(\|h\|^2_{A_2}+\|u\|^2). $$ The last term results from a rough estimate of the collection of all terms with forefactor $\bar{\alpha}_m^2$ remaining after substitution, namely \begin{eqnarray*}
&&\|h\|_{A_2}^2-2\|h\|^2 +(Lh,h) +\|u\|^2-(Lu,u)=\|h\|_{A_2}^2+\|u\|^2-\|h\|^2 -((I-L)h,h) -(Lu,u)\\
&&\qquad\qquad \le \|h\|_{A_2}^2+\|u\|^2. \end{eqnarray*}
For the rest of the argument, we again refer to the proof of Theorem 2 b) in \cite{GrOs2016}.
It remains to check that $A_2=H_L^{1/2}$. For $u\in A_2$ satisfying (\ref{A2a}) we can write $$
\|u\|_{H_L^{1/2}}^2 =\sum_k \frac{(u,\psi_k)^2}{\mu_k} = \sum_k \left(\int_{S_H} c_\omega (\omega,\mu_k^{-1/2}\psi_k)\, d\rho_\omega\right)^2
= \sum_k (c_\omega,f_{k,\omega})^2_{L_2(d\rho)} \le \|c_\omega\|_{L_2(d\rho)}^2 <\infty. $$ The last step follows because the functions $f_{k,\omega}:=(\omega,\mu_k^{-1/2}\psi_k)$ form an orthonormal system in $L_2(d\rho)$: $$ (f_{k,\omega},f_{l,\omega})^2_{L_2(d\rho)} = \int_{S_H} \frac{(\omega,\psi_k))(\omega,\psi_l)}{\mu_k^{1/2}\mu_l^{1/2}}\,d\rho_\omega =\frac{(L\psi_k,\psi_l)}{\mu_k^{1/2}\mu_l^{1/2}} = \delta_{kl}. $$ Moreover, for similar reasons any $u\in A_2$ must be orthogonal to $\mathrm{Ker}(L)$, i.e., belongs to $\tilde{H}$ and is thus in the closure in $H$ of the orthonormal system $\{\psi_k\}$ of eigenfunctions of $L$. Indeed, if $v\in \mathrm{Ker}(L)$ then we have $(\omega,v)=0$ almost everywhere on $\Omega$ since $$ \int_{S_H} (\omega,v)^2 \,d\rho_\omega = (Lv,v)=0. $$ This implies the desired orthogonality $$ (u,v)=\int_{S_H} c_\omega (\omega,v) \,d\rho_\omega = 0, $$
and shows $u\in H^{1/2}_L$ and $\|u\|_{H^{1/2}_L}\le \|u\|_{A_2}$ for all $u\in A_2$.
Now, take $u\in H^{1/2}_L$, i.e., $$
u=\sum_k c_k \psi_k, \qquad \|u\|_{H^{1/2}_L}^2 = \sum_k \mu_k^{-1}c_k^2 < \infty. $$ We will check that (\ref{A2a}) holds with $c_\omega = \sum_k \mu_k^{-1/2}c_k f_{k,\omega}$, which immediately implies
$u\in A_2$ and the opposite inequality $\|u\|_{A_2}\le \|u\|_{H^{1/2}_L}$. This is done by verifying that the moments $(u,\psi_l)$ coincide for both representations of $u$: On the one hand, we have $(u,\psi_l)=c_l$, on the other hand, we have $$ \left(\int_{S_H} c_\omega\omega\,d\rho_\omega,\psi_l\right)=\int_{S_H} \left(\sum_k \mu_k^{-1/2}c_k f_{k,\omega}\right) (\omega,\psi_l) \,d\rho_\omega =\sum_k (\mu_l/\mu_k)^{1/2} c_k (f_{k,\omega},f_{l,\omega})_{L_2(d\rho)} = c_l $$ by the orthonormality of the system $\{f_{k,\omega}\}$ in $L_2(d\rho)$.
$\Box$
\subsection{Collective approximation}\label{sec34} To demonstrate the versatility of the abstract scheme developed in Section \ref{sec2}, we consider a problem raised in \cite{AJOP2017}: Given an $n$-dimensional subspace $V_n$ of a Hilbert space $H$ and a dictionary $D$ of unit norm elements in $H$ (the condition $D\subset S_H$ is silently kept througout this subsection), construct, by incrementally selecting dictionary elements $\omega_0,\omega_1,\ldots$, subspaces $W_{m-1}=\mathrm{span}\{\omega_0,\ldots,\omega_{m-1}\}$ which approximate $V_n$ well, i.e., for which estimates for the approximation quantities $$
\sigma_m=\sup_{v\in V_n:\,\|v\|=1} \inf_{w\in W_{m-1}} \|v-w\|_H = \sup_{v\in V_n:\,\|v\|=1} \|v-P_{W_{m-1}}v\|_H $$ hold. The collective OMP algorithm proposed in \cite{AJOP2017} uses greedy selection of $\omega_m\in D$ based on computations involving the ortho-projections $P_{W_{m-1}}$ onto $W_{m-1}$ which become more costly for larger $m$. It comes with a convergence rate for the quantity $$
\epsilon_m({\Phi})= \left(\sum_{i=1}^n \|\phi_i -P_{W_{m-1}}\phi_i\|_H^2\right)^{1/2} = \|\Phi-P_{W_{m-1}}\Phi\|_{H^n} \ge \sigma_m,\qquad m=1,2,\ldots, $$ where ${\Phi}=(\phi_1,\ldots,\phi_n)$ is a given orthonormal basis in $V_n$.
We apply our results and design algorithms avoiding the projections $P_{W_{m-1}}$ while still guaranteeing similar convergence rates. To set the scene, identify $V$ with $H^n$ equipped with the usual scalar product $$ a(\mathbf{u},\mathbf{v}):=\sum_{i=1}^n (u_i,v_i),\qquad \mathbf{u},\mathbf{v}\in V \quad (\;\mathbf{u}=(u_1,\ldots,u_n)\;). $$ Let $\Omega=D$, and consider the family $$ V_\omega := \{\mathbf{v}_\omega=\mathbf{c}\omega:\; \mathbf{c}\in\mathbb{R}^n\},\qquad \omega\in \Omega, $$ of $n$-dimensional subspaces of $V$ (again, $R_\omega$ are the natural injections, $\Lambda=1$). The problem we want to solve is $\mathbf{u}=\Phi$ or, in variational form, $$ a(\mathbf{u},\mathbf{v})=a(\Phi,\mathbf{v})\qquad \forall\; \mathbf{v}\in \mathbf{u}. $$ With this, we have $$ R_\omega T_\omega \mathbf{v} = T_\omega \mathbf{v} =a (\mathbf{v},\omega)\omega, $$ where $a(\mathbf{v},\omega):=((v_1,\omega),\ldots,(v_n,\omega))\in \mathbb{R}^n$.
Independently of the method of choosing $\omega_m$ (randomly or greedy), our algorithm (\ref{Rec}) $$ \mathbf{u}^{(m+1)} = \alpha_m \mathbf{u}^{(m)} + \xi_m r^{(m)}_{\omega_{m}},\qquad r^{(m)}_{\omega_{m}}=T_{\omega_{m}}\mathbf{e}^{(m)}= (\Phi-\mathbf{u}^{(m)},\omega_m)\omega_{m}, \qquad m=0,1,\ldots, $$
when started with $\mathbf{u}^{(0)}=\mathbf{0}$, produces a sequence of $\mathbf{u}^{(m)}$ whose components belong to $W_{m-1}$ if $m>0$. Thus, we have upper estimates $$
\epsilon_m({\Phi})\le \delta_m:=\|\Phi-\mathbf{u}^{(m)}\|,\qquad m=1,2,\ldots. $$
If we choose the $\omega_m$, $m=0,1,\ldots$, randomly and independently according to a Borel measure $\rho$ on $S_H$ with support on $D$, then Theorems \ref{theo1} and \ref{theo2} are applicable, and they imply rates (in expectation) for $\Phi\in A_2$ and general $\Phi$ in terms of its approximability by elements $\mathbf{h}\in A_2$. Moreover, it is easy to see that the proof of Theorem \ref{theo3} remains valid if the application of the operators $L$ and $L^s$, respectively, which are defined on $H$ and depend on $\rho$, is extended componentwise to $V=H^n$. This way, we obtain the estimate \begin{equation}\label{EC2b}
\sigma_m^2\le \epsilon_m^2\le \mathbb{E}(\delta_{m}^2)\le \frac{\|\Phi\|_{A_2}^2}{m+1},\qquad m=1,2,\ldots , \end{equation} if $\Phi\in A_2=(H^{1/2}_L)^n$ with norm in $A_2$ defined as $$
\|\mathbf{v}\|_{A_2}^2 =\sum_{i=1}^n \|u_i\|^2_{H^{1/2}_L}. $$ The counterpart of (\ref{ECVa}) holds, too: If $\Phi\in H^n$ then for arbitrary $\Psi\in A_2$ we have \begin{equation}\label{ECVb}
\mathbb{E}(\delta_{m}^2)^{1/2}\le 2(\|\Phi-\Psi\| + \frac{(\|\Psi\|_{A_2}^2+\|\Phi\|^2)^{1/2}}{(m+1)^{1/2}}), \qquad m=1,2,\ldots. \end{equation}
These estimates for the expected error decay of our randomized algorithm are qualitatively the same as for the more expensive collective OMP algorithm with weak greedy selection of the $\omega_m$
proposed in \cite{AJOP2017}. However, the class $A_2$ is smaller then the class $\mathcal{A}_1(D)$ appearing in the convergence theory in \cite{AJOP2017}, and depends on the choice for $\rho$.
The weak greedy version of our algorithm was already analyzed in \cite{GrOs2016} by generalizing earlier results from \cite{BCDD2008}. For completeness, we repeat it here in the setting and notation of Section \ref{sec2}.
Define the class $A_1$ as the set of all $u\in V$ for which a representation of the form
\begin{equation}\label{A11}
u=\sum_j R_{\omega^j}v_{\omega^j}, \qquad \sum_j\|v_{\omega^j}\|_{\omega^j} <\infty,\quad \omega^j\in \Omega,
\end{equation}
holds, and set
$$
\|u\|_{A_1}:= \inf_{u=\sum_j R_{\omega^j}v_{\omega^j}}\; \sum_j\|v_{\omega^j}\|_{\omega^j} .
$$ For countable $\Omega$, $A_1$ coincides with the class $\mathcal{A}_1$ defined before.
\begin{theo}\label{theo4} If $u\in A_1$, the algorithm (\ref{Rec}) with $\omega_m$ chosen according to the weak greedy rule (\ref{Gr}) possesses the error bound \begin{equation}\label{CG1}
\delta_m^2 \le \frac{2((\Lambda/\beta)^2\|u\|_{A_1}^2 +\|u\|^2)}{m+1},\qquad m=0,1,\ldots. \end{equation} \end{theo} {\bf Proof}. The proof is almost identical to that of Theorem \ref{theo1}. Indeed, using
(\ref{Gr}) in (\ref{I0}), we have
$$
\frac{\Lambda}{\beta} a(e^{(m)},\tilde{\psi}_{\omega_m}) \ge\frac1{\beta} a_{\omega_m}(r_{\omega_m}^{(m)},\frac{r_{\omega_m}^{(m)}}{\|r_{\omega_m}^{(m)}\|_{\omega_m}})\ge a_{\omega}(r_{\omega}^{(m)},\frac{r_{\omega}^{(m)}}{\|r_{\omega}^{(m)}\|_{\omega}})\ge \frac{a(e^{(m)},R_\omega v_\omega)}{\|v_\omega\|_\omega}, $$ for any $\omega\in \Omega$ (as before $\tilde{\psi}_{\omega_m}$ is defined in (\ref{A1}) with $e=e^{(m)}$). Thus, representing $u\in A_1$ as in (\ref{A11}), we arrive at $$
a(e^{(m)},u)= \sum_j (a(e^{(m)},R_{\omega^j} v_{\omega^j}) \le \frac{\Lambda a(e^{(m)},\tilde{\psi}_{\omega_m}) }{\beta} \sum_j \|v_{\omega^j}\|_{\omega^j}, $$ and, after taking the infimum over all such representations of $u$, we get $$
a(e^{(m)},u) \le \frac{\Lambda \|u\|_{A_1} }{\beta}a(e^{(m)},\tilde{\psi}_{\omega_m}). $$ For the corresponding term of the error representation for $\delta_{m+1}^2$, this yields \begin{eqnarray*} && 2\alpha_m\bar{\alpha}_m(a(e^{(m)},u)-a(e^{(m)},\tilde{\psi}_{\omega_m})a(u,\tilde{\psi}_{\omega_m})\\
&& \qquad \le 2\alpha_m\bar{\alpha}_m a(e^{(m)},\tilde{\psi}_{\omega_m})((\Lambda/\beta) \|u\|_{A_1}-a(u,\tilde{\psi}_{\omega_m}))\\
&& \qquad \le \alpha_m^2 a(e^{(m)},\tilde{\psi}_{\omega_m})^2+\bar{\alpha}_m^2((\Lambda/\beta)\|u\|_{A_1}-a(u,\tilde{\psi}_{\omega_m}))^2, \end{eqnarray*} and after substitution and cancellation of terms we have \begin{eqnarray*} \delta_{m+1}^2&\le& \alpha_m^2 \delta_m^2 +\bar{\alpha}_m^2
(((\Lambda/\beta) \|u\|_{A_1}-a(u,\tilde{\psi}_{\omega_m}))^2+\|u\|^2-a(u,\tilde{\psi}_{\omega_m})^2)\\ &\le& \alpha_m^2 \delta_m^2 +2\bar{\alpha}_m^2
((\Lambda/\beta)^2\|u\|_{A_1}^2+\|u\|^2). \end{eqnarray*} The rest is as before.
$\Box$
\section{Concluding remarks} \label{sec4} We conclude with three further remarks.
{\bf Remark 1.} In the generality considered here, the obtained convergence rates for the expectation of the squared error $\delta_m^2$ of the algorithm (\ref{Rec}) for $u\in A_2$ cannot be improved without additional assumptions on $\rho$ or $u$. To see this, consider the case of a discrete measure $\rho$ concentrated on a complete orthonormal system $\{e_j\}\subset S_H$ in a Hilbert space $V=H$ with scalar product $a(\cdot,\cdot)=(\cdot,\cdot)$, and denote $\rho_j=\rho(\{e_j\}) >0$, $j\in \Omega=\mathbb{N}$. This is within the setting of Section \ref{sec33}. Obviously, we have $$ Lv=\sum_j \rho_j(v,e_j) e_j,\qquad v\in H\quad (\psi_j=e_j, \;\mu_j=\rho_j,\; j\in \mathbb{N}), $$ and $\mathrm{Ker}(L)=\{0\}$. In other words, $\tilde{H}=H$, and $$
H^s_L = \{u=\sum_j \rho_j^{s} c_j e_j:\quad \|u\|_{H^s_L}^2:= \sum_j c_j^2<\infty\},\qquad s\in \mathbb{R}. $$ Following the reasoning in Remark 5 in \cite{GrOs2016}, for any algorithm that produces the iterates $u^{(m)}$ as linear combinations of at most $m$ elements $e_j$ drawn randomly and independently according to $\rho$, we then have the lower estimate $$
\mathbf{E}(\|u-u^{(m)}\|^2)\ge \sum_j (u,e_j)^2(1-\rho_j)^m. $$ We mention as a side note that this lower bound is achieved for the stochastic OMP method (\ref{OMP}). The condition $u \in H^{r}_L$ is for $r>0$ equivalent to $(u,e_j)=\rho_j^r (v,e_j)$, $j\in \Omega$, for some $v\in H$. Thus, the worst case behavior of the expected squared error of any such algorithm for recovering $u\in H^{r}_L$ is characterized by $$
\epsilon_{m,r}:=\sup_{0\neq u\in H^{r}_L} \frac{\mathbf{E}(\|u-u^{(m)}\|^2)}{\|u\|^2_{H^r_L}}\ge \sup_{0\neq v\in H} \frac{\sum_j (v,e_j)^2\rho_j^{2r}(1-\rho_j)^m}{\sum_j (v,e_j)^2}=\sup_j \rho_j^{2r}(1-\rho_j)^m. $$ Since the function $f(t)=t^{2r}(1-t)^m$ takes its maximum for $t\in [0,1]$ at $t_0=(m+2r)^{-1}$, we see that in general no rate better than $\mathrm{O}(m^{-2r})$ can be expected on the class $H^{r}_L$. If we take $r=1/2$, we see that Theorem \ref{theo3} provides an optimal result, in the sense that the upper limit of $m^{2r}\epsilon_{m,r}$ for $m\to \infty$ is finite \emph{and} strictly positive for any $\rho$.
However, with other assumptions on $u$ or on the spectral properties of $L$ (as it is custom in learning with kernel methods \cite{DeBa2016,LiZh2015}), one may expect better results.
\noindent
{\bf Remark 2.} The choice for the parameters $\alpha_m$ and $\xi_m$ in the algorithm (\ref{Rec}) is appropriate if the evaluations in (\ref{Rec}) and (\ref{Xi}) (in particular, the functional evaluation $F(u^{(m)}$)) are exact. If one attempts to analyze the same algorithm with, e.g., an independent additive noise term $\varepsilon_m$ the update formula (\ref{Rec}) (in addition to independence, assume $\mathbb{E}(\varepsilon_m)=0$, and $\sigma^2:=\mathbb{E}(\|\varepsilon_m\|^2)=\mathrm{const.}>0$) then, in the formulas for $\delta_{m+1}^2$ and subsequently in (\ref{ERec}), an additional term $\sigma^2$ appears in the right-hand side, i.e., $$
\mathbf{E}(\delta_{m+1}^2)\le \alpha_m^2 \mathbf{E}(\delta_{m}^2)+ \bar{\alpha}_m^2(\Lambda \|u\|_{A_2}+\|u\|)^2+\sigma^2,\qquad m=0,1,\ldots. $$ Now, the term $\sigma^2$ renders any attempt of proving $\mathbf{E}(\delta_m^2)\to 0$ meaningless. At the $m$-th step of the recursion, an additional term of the order $(m+1) \sigma^2/3$ would appear in the final estimate for $\mathbf{E}(\delta_m^2)$ which is subdominant only for small $\sigma^2$ and at the initial stages of the iteration. A crude calculation shows that under these assumptions the best possible bound for the expectation of the squared error is $$
\mathbf{E}(\delta_m^2) \approx \frac{\sigma}{(\Lambda\|u\|_{A_2}+\|u\|)} \quad \mbox{if } \; m\approx \frac{\Lambda \|u\|_{A_2}+\|u\|}{\sigma}. $$ This says that, on average, the squared error $\delta_m^2$ cannot be approximated better than the standard deviation of the additive noise $\varepsilon_m$ relative to the size of $u$ which is unsatisfactory. A possible repair is to give up the minimization requirement for $\xi_m$, and to execute (\ref{Rec}) with some suitably chosen sequence $\xi_m\to 0$. This is well understood for kernel methods in online learning, and represents a significant difference between the noisy and noiseless case.
\noindent {\bf Remark 3.}
The right-hand sides in the estimates (\ref{ECV}) and (\ref{ECVa}) in Theorem \ref{theo2} and Theorem \ref{theo3} have the form of a $K$-functional for the pairs $(V,{A}_2)$ and $(H,H_L^{1/2})$, respectively, This implies that rates of the form $$ \mathbb{E}(\delta_{m}^2)=\mathrm{O}(m^{-\theta}),\qquad m\to \infty, $$ with exponent $\theta\in (0,1)$ hold for spaces obtained by real interpolation. E.g., in the setting of Theorem \ref{theo3}, we obtain \begin{equation}\label{ECVr}
\mathbb{E}(\delta_{m}^2)\le C (m+1)^{-2s} \|u\| _{H_L^s}^2, \qquad m=0,1,\ldots, \end{equation} valid for all $u\in H^s_L$ and $0<s<1/2$ with a certain fixed constant $C$. Indeed, $u\in H_L^s$ can be represented as $$
u=\sum_k c_k\mu_k^s \psi_k, \qquad \|u\|_{H^s_L}^2 =\sum_k c_k^2, $$ and setting $$ h=\sum_{k:\,\mu_k\ge (m+1)^{-1}} c_k\mu_k^s \psi_k, $$ we have \begin{eqnarray*}
\|h\|_{H^{1/2}_L}^2 &=& \sum_{k:\,\mu_k\ge (m+1)^{-1}} c_k^2\mu_k^{2s-1}\le (m+1)^{1-2s}\|u\|_{H^s_L}^2,\\
\|u-h\|^2 &=& \sum_{k:\,\mu_k < (m+1)^{-1}} c_k^2\mu_k^{2s}\le (m+1)^{-2s}\|u\|_{H^s_L}^2,\\
\|u\|^2 &=& \sum_{k} c_k^2\mu_k^{2s}\le \|u\|_{H^s_L}^2. \end{eqnarray*} Thus, after substitution into (\ref{ECVa}), we get (\ref{ECVr}) with $C=(2(1+\sqrt{2}))^2<24$.
\end{document} | arXiv |
Arakelov theory
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals ${\mathfrak {p}}\in {\text{Spec}}(\mathbb {Z} )$ and finite places $v_{p}:\mathbb {Q} ^{*}\to \mathbb {R} $, but there also exists a place at infinity $v_{\infty }$, given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying ${\text{Spec}}(\mathbb {Z} )$ into a complete space
${\overline {{\text{Spec}}(\mathbb {Z} )}}$
which has a prime lying at infinity. Arakelov's original construction studies one such theory, where a definition of divisors is constructor for a scheme ${\mathfrak {X}}$ of relative dimension 1 over ${\text{Spec}}({\mathcal {O}}_{K})$ such that it extends to a Riemann surface $X_{\infty }={\mathfrak {X}}(\mathbb {C} )$ for every valuation at infinity. In addition, he equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of $X$. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.
Note that other techniques exist for constructing a complete space extending ${\text{Spec}}(\mathbb {Z} )$, which is the basis of F1 geometry.
Original definition of divisors
Let $K$ be a field, ${\mathcal {O}}_{K}$ its ring of integers, and $X$ a genus $g$ curve over $K$ with a non-singular model ${\mathfrak {X}}\to {\text{Spec}}({\mathcal {O}}_{K})$, called an arithmetic surface. Also, we let
$\infty :K\to \mathbb {C} $
be an inclusion of fields (which is supposed to represent a place at infinity). Also, we will let $X_{\infty }$ be the associated Riemann surface from the base change to $\mathbb {C} $. Using this data, we can define a c-divisor as a formal linear combination
$D=\sum _{i}k_{i}C_{i}+\sum _{\infty }\lambda _{\infty }X_{\infty }$
where $C_{i}$ is an irreducible closed subset of ${\mathfrak {X}}$ of codimension 1, $k_{i}\in \mathbb {Z} $, and $\lambda _{\infty }\in \mathbb {R} $, and the sum
$\sum _{\infty }$
represents the sum over every real embedding of $K\to \mathbb {C} $ and over one embedding for each pair of complex embeddings $K\to \mathbb {C} $. The set of c-divisors forms a group ${\text{Div}}_{c}({\mathfrak {X}})$.
Results
Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields. Gerd Faltings (1984) extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context.
Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings (1991) in his proof of Serge Lang's generalization of the Mordell conjecture.
Pierre Deligne (1987) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov.
Arakelov's theory was generalized by Henri Gillet and Christophe Soulé to higher dimensions. That is, Gillet and Soulé defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soulé is the arithmetic Riemann–Roch theorem of Gillet & Soulé (1992), an extension of the Grothendieck–Riemann–Roch theorem to arithmetic varieties. For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, and defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups. The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé.
Arakelov's intersection theory for arithmetic surfaces was developed further by Jean-Benoît Bost (1999). The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space $L_{1}^{2}$. In this context, Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.
Arithmetic Chow groups
An arithmetic cycle of codimension p is a pair (Z, g) where Z ∈ Zp(X) is a p-cycle on X and g is a Green current for Z, a higher-dimensional generalization of a Green function. The arithmetic Chow group ${\widehat {\mathrm {CH} }}_{p}(X)$ of codimension p is the quotient of this group by the subgroup generated by certain "trivial" cycles.[1]
The arithmetic Riemann–Roch theorem
The usual Grothendieck–Riemann–Roch theorem describes how the Chern character ch behaves under pushforward of sheaves, and states that ch(f*(E))= f*(ch(E)TdX/Y), where f is a proper morphism from X to Y and E is a vector bundle over f. The arithmetic Riemann–Roch theorem is similar, except that the Todd class gets multiplied by a certain power series. The arithmetic Riemann–Roch theorem states
${\hat {\mathrm {ch} }}(f_{*}([E]))=f_{*}({\hat {\mathrm {ch} }}(E){\widehat {\mathrm {Td} }}^{R}(T_{X/Y}))$
where
• X and Y are regular projective arithmetic schemes.
• f is a smooth proper map from X to Y
• E is an arithmetic vector bundle over X.
• ${\hat {\mathrm {ch} }}$ is the arithmetic Chern character.
• TX/Y is the relative tangent bundle
• ${\hat {\mathrm {Td} }}$ is the arithmetic Todd class
• ${\hat {\mathrm {Td} }}^{R}(E)$ is ${\hat {\mathrm {Td} }}(E)(1-\epsilon (R(E)))$
• R(X) is the additive characteristic class associated to the formal power series
$\sum _{m>0 \atop m{\text{ odd}}}{\frac {X^{m}}{m!}}\left[2\zeta '(-m)+\zeta (-m)\left({1 \over 1}+{1 \over 2}+\cdots +{1 \over m}\right)\right].$
See also
• Hodge–Arakelov theory
• Hodge theory
• P-adic Hodge theory
• Adelic group
Notes
1. Manin & Panchishkin (2008) pp.400–401
References
• Arakelov, Suren J. (1974), "Intersection theory of divisors on an arithmetic surface", Math. USSR Izv., 8 (6): 1167–1180, doi:10.1070/IM1974v008n06ABEH002141, Zbl 0355.14002
• Arakelov, Suren J. (1975), "Theory of intersections on an arithmetic surface", Proc. Internat. Congr. Mathematicians Vancouver, vol. 1, Amer. Math. Soc., pp. 405–408, Zbl 0351.14003
• Bost, Jean-Benoît (1999), "Potential theory and Lefschetz theorems for arithmetic surfaces" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 4, 32 (2): 241–312, doi:10.1016/s0012-9593(99)80015-9, ISSN 0012-9593, Zbl 0931.14014
• Deligne, P. (1987), "Le déterminant de la cohomologie", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) [The determinant of the cohomology], Contemporary Mathematics, vol. 67, Providence, RI: American Mathematical Society, pp. 93–177, doi:10.1090/conm/067/902592, MR 0902592
• Faltings, Gerd (1984), "Calculus on Arithmetic Surfaces", Annals of Mathematics, Second Series, 119 (2): 387–424, doi:10.2307/2007043, JSTOR 2007043
• Faltings, Gerd (1991), "Diophantine Approximation on Abelian Varieties", Annals of Mathematics, Second Series, 133 (3): 549–576, doi:10.2307/2944319, JSTOR 2944319
• Faltings, Gerd (1992), Lectures on the arithmetic Riemann–Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton, NJ: Princeton University Press, doi:10.1515/9781400882472, ISBN 0-691-08771-7, MR 1158661
• Gillet, Henri; Soulé, Christophe (1992), "An arithmetic Riemann–Roch Theorem", Inventiones Mathematicae, 110: 473–543, doi:10.1007/BF01231343
• Kawaguchi, Shu; Moriwaki, Atsushi; Yamaki, Kazuhiko (2002), "Introduction to Arakelov geometry", Algebraic geometry in East Asia (Kyoto, 2001), River Edge, NJ: World Sci. Publ., pp. 1–74, doi:10.1142/9789812705105_0001, ISBN 978-981-238-265-8, MR 2030448
• Lang, Serge (1988), Introduction to Arakelov theory, New York: Springer-Verlag, doi:10.1007/978-1-4612-1031-3, ISBN 0-387-96793-1, MR 0969124, Zbl 0667.14001
• Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
• Soulé, Christophe (2001) [1994], "Arakelov theory", Encyclopedia of Mathematics, EMS Press
• Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, doi:10.1017/CBO9780511623950, ISBN 0-521-41669-8, MR 1208731
• Vojta, Paul (1991), "Siegel's Theorem in the Compact Case", Annals of Mathematics, Annals of Mathematics, Vol. 133, No. 3, 133 (3): 509–548, doi:10.2307/2944318, JSTOR 2944318
External links
• Original paper
• Arakelov geometry preprint archive
| Wikipedia |
\begin{document}
\subjclass[2000]{ 32A25} \keywords{Bergman kernel, weighted Bergman kernel, Fock-Bargmann space, polylogarithm function, Lu Qi-Keng problem, Forelli-Rudin construction} \email{[email protected]} \begin{abstract} We consider a certain Hartogs domain which is related to the Fock-Bargmann space. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover we solve the Lu Qi-Keng problem of the domain in some cases. \end{abstract} \maketitle \section{Introduction} In this paper we consider a Hartogs domain in $\mathbb{C}^{n+m}$ defined by the inequality
$||\zeta||^2<e^{-\mu||z||^2 }, $ where $(z,\zeta)\in\Bbb{C}^n\times\Bbb{C}^m$ and $\mu>0$. Our aim is to show that the Bergman kernel of this domain can be written explicitly in terms of the polylogarithm functions.\par
The polylogarithm function appears in many different areas of mathematics. For example it appears in analysis of the Riemann zeta function, algebraic geometry and mathematical physics (cf. \cite{d2010}, \cite{Hirzebruch2008}). The polylogarithm function is a rational function under a certain condition. Our formula is expressed in terms of these rational cases of the polylogarithm functions and their derivatives.\par
It is usually hard to obtain an explicit formula of the Bergman kernel of a complex domain. Only few domains with an explicit Bergman kernel are known until now. In this situation, it is fundamental and important to find a domain with explicit Bergman kernel.\par
There are two basic approaches for obtaining explicit Bergman kernels. One is to construct a complete orthonormal basis of the Bergman space explicitly. For the unit disk, one can find a complete orthonormal basis $\{\pi^{-1/2}(n+1)^{1/2}z^n \}_{n=1}^\infty$. However this approach faces difficulty in general. If the domain does not have symmetry, a computation of integral on the domain is unexecutable or extremely difficult.\par
If the automorphism group of a domain contains enough information (e.g. transitivity) to obtain explicit formula, then we can use it. For example, the Lie group $SU(n,m)$ acts transitively on the classical domain $D=\{z\in M_{n,m}(\mathbb{C}) ;I-\overline{z}{^t}z>0 \}$ of type I by linear fractional transformation. It is known that the Bergman kernel $K$ of a classical domain has the property that $K(z,0)$ is a non-zero constant. These facts and a transformation rule of the Bergman kernel (cf. \cite{Boas2000}) imply that the computation of the Bergman kernel is reduced to the computation of the Jacobian of the linear fractional map. L. K. Hua \cite{Hua1964} computed the Bergman kernels for the classical domains in this way. As above, this approach faces difficulty in general.\par
The approach in this article is different from the above two. Our method is based on Ligocka's theorem \cite{Ligocka1989} which relates the Bergman kernel of a Hartogs domain to weighted Bergman kernels of the base domain. Thanks to this theorem one can find that our domain and the Fock-Bargmann space are closely related. We will see that Ligocka's theorem and an explicit formula of the Fock-Bargmann kernel lead to an explicit formula of the Bergman kernel of our domain.\par
As an application of our formula, we solve the Lu Qi-Keng problem for our domain in some cases. The Lu Qi-Keng problem asks whether the Bergman kernel has zeros or not. This problem was investigated for various domains by many authors in this decade. Yin \cite{WeiPing1999} obtained explicit form of the Bergman kernel of the Cartan-Hartogs domain. The Lu Qi-Keng problem for the Cartan-Hartogs domain was studied by several authors (cf. \cite{Demmand2009}). In \cite{Wang2009}, the authors obtained an explicit formula of the Bergman kernel of some Hartogs domains and solved the Lu Qi-Keng problem for the domains in some cases. Recently Lu Qi-Keng himself studied the location of the zeros of Bergman kernel in \cite{Lu2009}. Further information about the Lu Qi-Keng problem can be found in \cite{Boas2000},\cite{Boas1999},\cite{Jarnicki2005} and \cite{WeiPing2008}.
\section{Preliminaries} Let $\Omega$ be a domain in $\Bbb{C}^n$, $L^2_a(\Omega)$ the Hilbert space of square integrable holomorphic functions on $\Omega$ with the inner product: $$ \langle f,g \rangle=\int _{\Omega} f(z)\overline{g(z)} dz,\mbox{\quad for all $f,g\in L^2_a(\Omega)$. }$$ The Bergman kernel $K(z,w)=\overline{K_z(w)}$ is the reproducing kernel for $L_a^2(\Omega)$, i.e. if $f\in L_a^2(\Omega)$ then $$f(z)= \langle f,K_z \rangle=\int _{\Omega} f(w)K(z,w) dw, \mbox{\quad for all $z\in\Omega$. }$$ Let $\{\phi_k\}$ be a complete orthonormal basis of $L_a^2(\Omega)$. Then the Bergman kernel can be also defined by $$ K(z,w)=\sum_k\phi_k(z)\overline{\phi_k(w)}. $$ Let $p$ be a positive continuous function on $\Omega$ and $L^2_a(\Omega,p)$ the Hilbert space of square integrable holomorphic functions with respect to the weight function $p$ on $\Omega$ with the inner product $$ \langle f,g \rangle=\int _{\Omega} f(z)\overline{g(z)} p(z) dz,\mbox{\quad for all $f,g\in L^2_a(\Omega)$. }$$ The weighted Bergman kernel $K_{\Omega, p}$ of $\Omega$ with respect to the weight $p$ is the reproducing kernel of $L^2_a(\Omega,p)$.\par We define the Hartogs domain $\Omega_{m,p}$ by \begin{align*}
\Omega_{m,p}:= \{ (z,\zeta)\in\Omega\times\Bbb{C}^m; ||\zeta||^2 < p(z)\}. \end{align*} E. Ligocka \cite[Proposition 0]{Ligocka1989} showed that the Bergman kernel of $\Omega_{m,p}$ is expressed as infinite sum of weighted Bergman kernels of the base domain $\Omega$. \begin{theo} Let $K_m$ be the Bergman kernel of $\Omega_ {m ,p }$ and $K_{\Omega,p^k}$ the weighted Bergman kernel of $\Omega$ with respect to the weight function $p^k$. Then \begin{align*}
K_m((z,\zeta),(z',\zeta') ) =\dfrac{m!}{\pi^m}\sum_{k=0}^\infty \dfrac{ (m+1)_k}{k!} K_{\Omega,p^{k+m}}(z,z')\langle\zeta,\zeta'\rangle^k. \end{align*} Here $(a)_k$ denotes the Pochhammer symbol $(a)_k =a(a+1)\cdots (a+k-1). $ \end{theo} M. Engli\v{s} and G. Zhang generalized this theorem for wider class of domains in \cite{Englivs2006}. Since theorem of this type was first proved by F. Forelli and W. Rudin \cite{FR} for
$\Omega$ the unit disk and $p(z)=1-|z|^2$, some authors call it the Forelli-Rudin construction. \par We introduce the polylogarithm function which is necessary to state our main theorem. The polylogarithm function is defined by \begin{align}
Li_s(z)=\sum_{k=1}^\infty k^{-s}z^k ,
\end{align}
which converges for $|z|<1$ and any $s\in\Bbb C$. If $s$ is a negative integer, say $s=-n$, then the polylogarithm function has the following closed form: \begin{align*}
Li_{-n}(z)&= \dfrac{ z }{(1-z)^{n+1}}
\sum_{j=0}^{n-1} A(n,j+1)z^{j}
\end{align*} where $A(n,m)$ is the Eulerian number \cite[eq.(2.17)]{d2010}\begin{align*}
A(n,m)=\sum_{\ell=0}^{m} (-1)^\ell \binom{n+1}{\ell} (m-\ell)^n. \end{align*} The first few are $$\begin{array}{ll}
Li_{-1}(z)=\dfrac{z}{(1-z)^2}, & Li_{-2}(z)=\dfrac{z^2 + z}{(1-z)^3}, \\
Li_{-3}(z)=\dfrac{z^3 + 4z^2+ z}{(1-z)^4}, & Li_{-4}(z)=\dfrac{z^4 +11z^3 +11z^2 +z}{(1-z)^5}. \end{array}$$
The polynomial $A_n(z) = \sum_{j=0}^{n-1} A(n,j+1)z^{j}$ is called the Eulerian polynomial. More information about the polylogarithm function and the Eulerian polynomial can be found in \cite{Carlitz},\cite{d2010} and \cite{Hirzebruch2008}.
\section{The Bergman kernel of $D_{n,m}$}
Let $\mu>0$. Define $D_{n,m}$ by
$$D_{n,m}:= \{ (z,\zeta)\in\Bbb{C}^n\times\Bbb{C}^m; ||\zeta||^2 <
e^{-\mu ||z||^2}\}.$$ This section is devoted to the study of the Bergman
kernel of $D_{n,m}$. We shall begin with the Fock-Bargmann space and its reproducing kernel.\par The Fock-Bargmann space $L_a^2(\Bbb{C}^n,e^{-\mu||z||^2})$ is
the Hilbert space of square integrable entire functions on $\Bbb{C}^n$ with the inner product
$$\langle f,g\rangle =\int_{\Bbb{C}^n }f(z)\overline{g(z)} e^{-\mu||z||^2} dz .$$
The reproducing kernel $K_{n,\mu}$ of $L_a^2(\Bbb{C}^n,e^{-\mu|| z||^2})$ is expressed explicitly as \begin{align}
K_{n,\mu}(z,w)=\dfrac{ \mu^n e^{\mu \langle z,w\rangle}}{\pi^n}. \end{align} The kernel function $K_{n,\mu}$ is called the Fock-Bargmann kernel (see \cite{bargmann}). We are now ready to state our main result. \begin{theo} The Bergman kernel of $D_{n,m}$ is given by \begin{align}
K_{D_{n,m}}((z,\zeta),(z',\zeta') ) &= \dfrac{\mu^n}{\pi^{n+m}} e^{m\mu \langle z,z'\rangle }\dfrac{d^m}{d t^m} Li_{-n}(t)\lvert_{t=e^{\mu\langle z,z'\rangle}\langle \zeta ,\zeta' \rangle }\\ &=\dfrac{\mu^n}{\pi^{n+m}} \dfrac{d^{m-1}}{d t^{m-1}} \dfrac{Li_{-(n+1)} (e^{\mu\langle z,z'\rangle }t) }{t} \lvert_{t=\langle \zeta ,\zeta' \rangle }. \end{align} \end{theo} \begin{proof} By Ligocka's theorem and the formula $(2)$, we have \begin{align*}
K_{D_{n,m}}((z,\zeta),(z',\zeta') ) & =\dfrac{ m! }{ \pi^{m} } \sum_{k=0}^\infty
\dfrac{(m+1)_k}{k!} \dfrac{(k+m)^n\mu^n}{\pi^n} e^{\mu(k+m)\langle z,z' \rangle} \langle \zeta,\zeta'\rangle^k\\ &=\dfrac{ m!\mu^n }{ \pi^{n+m} } e^{\mu m\langle z,z' \rangle} \sum_{k=0}^\infty \dfrac{(m+1)_k}{k!} (k+m)^n
e^{\mu k\langle z,z' \rangle}\langle \zeta,\zeta'\rangle^k. \end{align*} Using a simple identity $(m+1)_k/k!=(k+1)_m/m!$, we get \begin{align*}
K_{D_{n,m}}((z,\zeta),(z',\zeta') ) & =\dfrac{\mu^n}{ \pi^{n+m} } e^{\mu m\langle z,z' \rangle} \sum_{k=0}^\infty
(k+1)_m (k+m)^n e^{\mu k\langle z,z' \rangle} \langle \zeta,\zeta'\rangle^k. \end{align*} Here we remark that \begin{align}
|e^{\mu \langle z,z'\rangle} \langle\zeta,\zeta'\rangle |<1 \end{align} for all $(z,\zeta),(z',\zeta') \in D_{n,m}$. Indeed, from the definition of $D_{n,m}$ and the Cauchy-Schwartz inequality, we see that
$ |\left\langle \zeta,\zeta' \right\rangle |^2 \leq ||\zeta||^2||
\zeta'||^2<
e^{-\mu (||z||^2 +||z'||^2)}, $
for any $(z,\zeta),(z',\zeta') \in D_{n,m}$. Combining this and a simple inequality $||z||^2+||z'||^2 \geq 2\mbox{Re}\left\langle z,z' \right\rangle $, we have $|\left\langle \zeta,\zeta' \right\rangle |^2 < |e^{-\mu\left\langle z,z' \right\rangle }|^2$. Hence $|e^{\mu \langle z,z'\rangle}
\langle\zeta,\zeta'\rangle |<1$. \par Let us evaluate the series
\begin{align} H_{m,n}((z,\zeta),(z',\zeta'))= \sum_{k=0}^\infty
(k+1)_m (k+m)^n e^{\mu k\langle z,z' \rangle} \langle \zeta,\zeta'\rangle^k. \end{align} It is easy to see from (1) that the $m$-th derivative of the polylogarithm function has the following series representation:
\begin{align} \dfrac{d^m Li_s (z) }{d z^m} &=\sum_{k=m}^\infty (k-m+1)_m k^{-s} z^{k-m}\\ &= \sum_{k=0}^\infty (k+1)_m (k+m)^{-s}z^k, \end{align}
for $|z|<1$. Comparing (6) and (8), we obtain $$H_{m,n}((z,\zeta),(z',\zeta'))= \dfrac{d^m}{d t^m} Li_{-n}(t) \lvert_{t=e^{\mu\langle z,z'\rangle}\langle \zeta ,\zeta' \rangle }.$$ This proves the formula $(3)$. The formula $(4)$ follows from $(3)$ and a well-known property of the polylogarithm function \cite[eq. 2.1]{d2010}: $$ \dfrac{d}{d t} Li_s (t)=\dfrac{Li_{s-1}(t)}{t} .$$ We have just completed the proof of Theorem 2. \end{proof} \begin{rem} There is a following closed form of the $m$-th derivative of the polylogarithm function: \begin{align} \dfrac{d^m Li_{-n} (t) }{d t^m} = \frac{m!\sum_{j=0}^{n} (-1)^{n+j} (m+1)_j S(1+n,1+j)(1-t)^{n-j} }{(1-t)^{n+m+1}} , \end{align} where $S(\cdot,\cdot)$ denotes the Stirling number of the second kind (see \cite{d2010}).\par \end{rem}
\section{ An application}
As an application of Theorem 2 we solve the Lu Qi-Keng problem for $D_{n,m}$ in some cases. The Lu Qi-Keng problem asks whether the Bergman kernel has zeros or not. He posed this problem in connection with the global well-definedness of the representative coordinates (see \cite{Boas2000}). Lu Qi-Keng's recent result \cite{Lu2009} implies that the zero of the Bergman kernel has a geometric interpretation.\par We begin with the following lemma which together with the inequality (5) tells us that the image of the map $D_{n,m}\times D_{n,m}\ni ((z,\zeta),(z',\zeta')) \mapsto e^{\mu \langle z,z' \rangle} \langle \zeta,\zeta' \rangle \in \mathbb{C} $ is the unit disk.
\begin{lem}
For any $\alpha\in\mathbb{C}$ such that $|\alpha|<1$, there exist $(z,\zeta) ,(z',\zeta')\in D_{n,m}$ such that
$\alpha=e^{\mu\langle z,z' \rangle}\langle \zeta,\zeta'\rangle$. \end{lem} \begin{proof} Let $\alpha=re^{i\theta}, r<1.$ For any fixed $z\in\mathbb{C}^n$, we can choose $\zeta\in\mathbb{C}^m$ such that
$|| \zeta||^2=re^{-\mu ||z||^2} $. Then $(z,\zeta),(z, e^{-i\theta}\zeta) \in D_{n,m}$ and
$e^{\mu\left\langle z,z \right\rangle }\left\langle \zeta,e^{-i \theta}\zeta \right\rangle =r e^{i\theta}$. \end{proof} We next discuss the location of zeros of the polylogarithm function and its derivative. \begin{lem}
The function $Li_{-n}(z)/z$ has a zero $z_0$ such that $|z_0|<1$ for all $n \geq 3$. \end{lem}
\begin{proof}
It is well-known that the Eulerian polynomial has only negative real, simple roots (see \cite[p. 292, Exercise 3]{comtet}). Since $n\geq3$, there exists a root $\alpha$ such that $|\alpha|\neq1$. If $|\alpha|<1$, then $\alpha$ is a desired zero. Now we assume that
$|\alpha|>1$. Then the following formula \cite[eq.(2.2)]{d2010} $$ Li_{-n}\left( \dfrac{1}{z} \right) =(-1)^{n+1}Li_{-n}(z) \quad(n\in \mathbb{N}),$$ implies that $\alpha^{-1}$ is a desired zero.
\end{proof} Further information of the location of zeros of $Li_{-n}(z)/z$ is found in \cite{Peyerimhoff1966}. This short proof was obtained by private communication with Prof. Ochiai and Dr. Shiomi (compare with the proof in \cite{Peyerimhoff1966}).\par The following is immediate from a straightforward computation. \begin{lem} For any $m\in\Bbb{N}$, the $(m-1)$-th derivative of $Li_{-2}(t)/t$ is expressed as
$$\dfrac{d^{m-1}}{dt^{m-1}} \dfrac{Li_{-2}(t)}{t}=\dfrac{(m+1)!(t+m)}{(1-t)^{m+2}}. $$ \end{lem} From this lemma, we see that $t=-m$ is the zero of $\frac{d^{m-1}}{dt^{m-1}} \frac{Li_{-2}(t)}{t}$.
Summarizing, we get: \begin{theo} The Bergman kernel $K_{D_{n,m}}$ is zero-free if $n=1$ and $m\geq1$. If $m=1$ and $n\geq 2$ then $K_{D_{n,m}}$ has a zero. \begin{rem} For our domain $D_{n,m}$, the solution of the Lu Qi-Keng problem depends only on the value $(m,n)$. In general, the solutions of the Lu Qi-Keng problem for the Hartogs domains
$ \{ (z,\zeta)\in\Omega\times\Bbb{C}^m; ||\zeta||^2 <p(z)^\mu\}$ depend not only on $(m,n)$ but also on $\mu$ (cf. \cite{Demmand2009}). \end{rem}
\end{theo}
\end{document} | arXiv |
CABI Agriculture and Bioscience
Journal Videos
IITA's genebank, cowpea diversity on farms, and farmers' welfare in Nigeria
Abel-Gautier Kouakou ORCID: orcid.org/0000-0003-4048-29751,
Ademola Ogundapo2,
Melinda Smale3,
Nelissa Jamora1,
Julius Manda4 &
Michael Abberton2
CABI Agriculture and Bioscience volume 3, Article number: 14 (2022) Cite this article
Cowpea or black-eyed pea (Vigna unguiculata L.) is one of the preferred food crops in Nigeria, as expressed in land area and production. The popularity of the crop is in part related to the successful development and adoption of improved cowpea varieties. Although the genebank of the International Institute of Tropical Agriculture (IITA) has contributed to cowpea conservation and improvement efforts by breeding programs internationally and in Nigeria, few studies have attempted to link the genebank to the management of cowpea genetic resources (CGRs) on farms. This study explores the linkage between IITA's genebank and cowpea variety diversity on farms and other measures of farmers' welfare in Nigeria.
A multistage stratified sampling was used to select the sample households. A cross-sectional household survey was conducted to collect data from 1524 cowpea-producing households. In addition, "Helium", a multi-platform pedigree visualization tool with phenotype display was used to gather information about improved cowpea breeding lines and their pedigrees. For data analysis, ecological indices of spatial diversity were employed, and a conditional recursive mixed-process model and a multinomial endogenous treatment effect model were developed.
We found that growing an improved variety with genebank ancestry is not significantly associated with lower spatial diversity among cowpea varieties. While they may introduce new traits through ancestry, improved varieties do not displace other cowpea varieties or landraces. We also found that genebank ancestry is positively and significantly associated with cowpea yield and farmers' welfare.
These findings show additional benefits from IITA's genebank in Nigeria and that adoption of improved varieties with genebank ancestry does not contribute to the erosion of CGRs on smallholder farms in Nigeria. Policymakers and practitioners should consider these findings when analyzing the benefits of conserving crop genetic diversity in genebanks and on farms.
Cowpea or black-eyed pea (Vigna unguiculata L.) is a food legume that provides food and fodder as well as improving soil fertility and contributes to the sustainability of food production in marginal areas of the dry tropics (Singh 1997). It is one of the preferred food crops in Nigeria, in terms of land area and production. For instance, land areas of cowpea were estimated at 0.117 million ha in 1981 and rose to 3.2 million ha and 4.3 million ha in 2012 and 2019, respectively (FAO 2020). The North West and North East regions of Nigeria are the most productive, including Borno, Bauchi, Gombe, Jigawa, Kaduna, Kano, Katsina, Kebbi, Sokoto, and Zamfara States, which represent 75% of the total cowpea production in Nigeria (Manda et al. 2019). Likewise, the national production of cowpea has increased by 165% from 1980 to 1990 and by 50% from 2009 to 2019 (FAO 2020; Singh 2005; See also in the Additional file 1: Fig. S1).
Rising land area and production of cowpea are partially related to cowpea conservation and improvement efforts at the International Institute of Tropical Agriculture (IITA) as well as the adoption of improved cowpea varieties in Nigeria (Ogundapo et al. 2020). Research on cowpea conservation and improvement was initiated at IITA in 1970, and over 50 countries, including Nigeria, have identified and released improved cowpea varieties from IITA for general cultivation (Singh 1997). Some recent studies have indicated that IITA's genebank houses over 17,000 accessions of cowpea (Genebank Platform 2020), which have been used for the development of over 800 improved cowpea cultivars, including lines and varieties. A substantial number of the released improved cowpea varieties have been adopted by Nigerian farmers (IITA 2013; Ogundapo 2016).
Although the genebank of IITA has contributed to cowpea conservation and improvement efforts by breeding programs worldwide and in Nigeria, few studies have attempted to link the genebank to on-farm management of cowpea genetic resources (CGRs). This may be explained by the fact that the primary role of IITA's genebank is the maintenance of crop diversity outside its natural environment. Linking IITA's genebank to on-farm management of crop genetic resources is important because it can reveal benefits from conservation of genetic materials under ex situ conditions in the context of scarce funding (Wale et al. 2011).
To our knowledge, only two studies have tried to investigate the impact on farms of the cowpea collection held in IITA's genebank or released from IITA. Ogundapo et al. (2020) used a combination of DNA fingerprinting and an economic surplus model to demonstrate the outcomes of CGR conservation and improvement efforts on smallholder farms in Kano State, Nigeria. The authors found increased productivity of low-income cowpea farmers who adopted improved cowpea varieties and increased net present value for cowpea germplasm conservation. They estimated that productivity changes lifted 487,219 persons out of poverty between 1980 and 2015. Manda et al. (2019) rigorously estimated the poverty impacts of crop genetic improvement on the income and poverty of farmers in Nigeria using an endogenous switching regression model and nationally representative data. Their results indicated that adoption of improved cowpea varieties raised per capita household income and asset ownership, also reducing income and asset poverty. However, the second study did not capture the link between the IITA genebank and its potential contribution to the development of cultivated cowpea varieties in Nigeria.
We built on these two studies and utilized the same data as Manda et al. (2019) to explore the linkage to the genebank and to the variety diversity on farms and other measures of farmers' welfare. Our objective was twofold. First, we established the link between the IITA's genebank and the development of improved cowpea varieties. We related this link to measures of on-farm diversity of cowpea varieties in Nigeria. Second, we examined the impact of IITA's genebank on cowpea yield and farmers' welfare in Nigeria through the adoption of improved varieties.
Our study contributes to the empirical literature on the valuation of genebanks, especially those based in Africa. Smale and Jamora (2020) reviewed earlier work on genebank valuation and assembled a set of current empirical studies that document some of the values associated with the international genebanks coordinated by the CGIAR (formerly the Consultative Group on International Agricultural Research). Two recent studies attempted to value international genebanks in Africa. Sellitti et al. (2020) analyzed the contribution of the genebank of the International Center for Tropical Agriculture (CIAT) to the development of iron-biofortified bean varieties and impacts among farming households in Rwanda. Their study showed the role of CIAT's genebank in the improvement of bean varieties and in generating benefits for farmers. Kitonga et al. (2020) explored the benefits of using the two most popular fodder tree species among smallholder farmers, sourced from the genebank of the World Agroforestry (ICRAF). The authors traced the benefits of ICRAF's genebank germplasm distributions to smallholder farmers. However, neither of these studies related their results specifically to in situ, on-farm conservation. On-farm agrobiodiversity has both potential private benefits to smallholder farmers and public benefits to the world's producers and consumers. We contribute to previous literature by testing the linkage from the genebank to spatial diversity of varieties grown on farms, and ultimately to the welfare of smallholder farmers. We present the case of the IITA's genebank and cowpea production in Nigeria as an example.
Description of the study area and data sources
The study was conducted in the framework of the Tropical Legumes III project and the Genebank Impacts project. The Tropical Legumes III project is an international initiative supported by the Bill & Melinda Gates Foundation and implemented by ICRISAT, CIAT, IITA and national agriculture research system partners from Africa and India (Varshney et al. 2019). Data were collected through a household survey conducted in Northern Nigeria in 2017 and used by Manda et al. (2019). The survey was conducted in ten states (Borno, Bauchi, Gombe, Jigawa, Kaduna, Kano, Katsina, Kebbi, Sokoto, and Zamfara), which represent about 75% of the total cowpea production in Nigeria (Manda et al. 2019). Figure 1 shows the location of the states selected for the household survey. Enumerators collected information from 1524 cowpea-producing households. A multistage stratified sampling was used to select the surveyed households, based on a sampling frame of local government areas and villages, and households, provided by the National Population Commission and the extension agents from the Agricultural Development Program, respectively. The survey was administered electronically using "Surveybe" and covered household composition and characteristics; knowledge of improved crop varieties; input use and crop production, including cowpea varieties grown and area allocated to each; adoption of improved cowpea varieties; crop utilization and household food security; marketing of crops; household assets; livestock production and marketing; sources of income; access to credit; household expenditure; social capital; and networking.
Map of the surveyed states for the IITA's Tropical Legumes III project in Nigeria
The quality of data was checked electronically. The data were uploaded in an electronic format immediately after collection. Supervisors were able to automatically record each interview's start time, end time and GPS location, validating the interview and comparing its time and GPS location to that of other interviews during which a supervisor was present. Considering that some areas where data were collected had limited electricity connectivity, each enumerator was given a battery pack to ensure that the tablets had the power to complete the interviews without problems.
The second source of data was the genebank of IITA, supported by the Genebank Impacts project. We gathered information about improved cowpea breeding lines and their pedigrees through key expert consultations and reports from IITA's cowpea breeding program (Singh 1997). We also consulted the database (or information management system) of the cowpea program of IITA and "Helium", a multi-platform pedigree visualization tool with phenotype display (Shaw et al. 2014).
Description of variables
Our choice of variables is motivated by the case of the farm household model in which production and consumption decisions cannot be separated because of missing markets, leading to endogenous decision prices (de Janvry et al. 1991). Benin et al. (2004) adapted the model to analyze the determinants of crop diversity as an outcome of cropland allocation by Ethiopian smallholders. Other empirical examples are found in Smale (2006). In this approach, diversity "outcomes" are not an explicit choice but a result of optimizing choices over goods consumed from production or purchase given the constraints imposed by farm physical conditions and labor availability, market features, and the household-specific characteristics that influence transactions costs. Definitions of the variables used for our analysis are presented in Table 1.
Table 1 List and description of variables used in regression models
Following Magurran (2004) and Smale (2006), we measured the varietal diversity of cowpeas on farms by adapting ecological indices of spatial diversity: the Menhinick index, the Shannon index, the Berger-Parker index, and the Herfindahl index. The choice of these indices was motivated by their use in the existing literature reported above and the fact that they represent various diversity dimensions and fit the information collected (cowpea varieties grown and percentage of area under cowpea varieties grown).
As explained by Magurran (2004), the Menhinick index \(d^{r}\) is a richness index that represents the number of distinct plant populations (varieties or crops) in a defined geographical area, such as a region, community, or in our case, a farm. The applied economics literature cited above adapts this concept using crop or variety area planted by farmers as a proxy for plant populations. Thus, the Menhinick index was computed as follows:
$$d^{r} = S/\sqrt A ,$$
where S is the number of cowpea varieties and A is the total cowpea area on a farm.
The Shannon index \(d^{e}\) is an evenness (or heterogeneity) measure, which takes the relative abundance of the plant populations into account and is defined as:
$$d^{e} = - \mathop \sum \limits_{i = 1}^{n} p_{i} lnp_{i} \ldots p_{i} \ge 0.$$
In our case, \(p_{i}\) is the cowpea area share planted to variety i.
The Berger-Parker index \(d^{d}\) expresses the inverse of the degree to which the most abundant plant population dominates the geographical area. We computed the Berger-Parker index as follows:
$$d^{d} = 1/max\left( {p_{i} } \right),$$
where \(max\left( {p_{i} } \right)\) is the maximum cowpea area share planted to any of the farmer's cowpea varieties.
The Herfindahl index, \(d^{c}\), is derived from the better known Herfindahl–Hirschman index of concentration that is widely applied in economic analysis of industrial organizations. As applied here, it expresses specialization and tells us whether a single variety occupies most of the planted area. We calculated the Herfindahl index as follows:
$$d^{c} = \mathop \sum \limits_{i = 1}^{n} p_{i}^{2} ,$$
where \(p_{i}\) is cowpea area share occupied by variety i.
We accounted for the impact of genebank ancestry using "Anc", a binary variable that measures the adoption of an improved cowpea variety that has a genebank ancestor. Anc takes the value 1 if the farmer is cultivating an improved cowpea variety that has a genebank ancestor and 0 otherwise. This variable helped establish the link between the genebank and improved cowpea varieties grown by farmers.
We also considered cowpea yield. Yield is obtained by dividing the total cowpea harvested on a farm by the farm size, expressed in kg/ha. Two variables were used for measuring farmers' welfare: consumption and sale. Consumption is a nutrition indicator, which refers to the quantity of cowpea used for home food consumption by the household, expressed in kg. Sale is a market (or revenue) indicator, referring to the quantity of cowpea grain sold by the household, expressed in kg.
The other variables that were used for our econometric analysis were vectors of independent variables that represent household characteristics (age of the household head, sex of the household head, education of the household head, household size, household's need of credit, and household's experience growing an improved cowpea variety), pedigree information (whether the household is growing a cowpea variety that has a genebank ancestor), farm characteristics (size, number of plots that are perceived as flat, number of plots that are perceived as poor, and distance to field from residence), market characteristics (distance to the nearest seed dealer, distance to the village market and distance to the district market), and geographical zone (North West and North East).
Methods of data analysis
We used both descriptive and econometric analyses to analyze the data. Primary data from the household survey and secondary data from the genebank of IITA were analyzed using measures of central tendency (means), dispersion (standard deviations), and frequency (percentages). We also used parametric and non-parametric tests (t-test, Fisher test, and Chi-squared test) to compare these measures between populations and regions.
To measure the impact of IITA's genebank on the on-farm diversity of cowpea varieties, we applied a system of two equations. The system helps capture: (1) farmers' decisions to grow a cowpea variety that has a genebank ancestor, and (2) the impact of growing this variety on the spatial diversity of cowpea varieties. We hypothesized that growing an improved cowpea variety that has a genebank ancestor generates benefits from the decision, such as the introduction of new traits or attributes through diverse ancestry.Footnote 1 If the farmer favors the improved variety with genebank ancestry over others, growing it may lead to abandonment or a reduction in area allocated to other cowpea varieties—reducing the spatial diversity of cowpea varieties. The model was formulated for the ith farmer as follows:
$$G_{i}^{*} = \alpha {\varvec{X}}_{{\varvec{i}}} + e_{i}$$
$$I_{i}^{*} = \beta G_{i} + \gamma {\varvec{Z}}_{{\varvec{i}}} + u_{i}$$
Equation 5 describes a farmer's decision to grow a cowpea variety with genebank ancestry. The farmer i compares the expected utility from growing an improved cowpea variety with genebank ancestry, \(U_{genebank}\), with the expected utility from growing other cowpea varieties, \(U_{others}\). She grows an improved cowpea variety with genebank ancestry if \(G_{i}^{*} = U_{genebank} - U_{others} > 0\). \(G_{i}^{*}\) is a latent variable that captures the expected benefits from the decision and is determined by a set of exogenous variables \({\varvec{X}}_{{\varvec{i}}}\) and the error term \(e_{i}\). The farmer's observed decision is a binary variable:
$$G_{i} = \left\{ {\begin{array}{*{20}c} {1\, if\,G_{i}^{*} > 0} \\ {0\, otherwise} \\ \end{array} } \right.$$
Equation 6 describes the impact of growing a cowpea variety with genebank ancestry on the spatial diversity of cowpea varieties. \(I_{i}^{*}\) is an unobservable variable that captures farmer's diversification strategy, which is determined by the decision to grow a cowpea variety with genebank ancestry and a set of exogenous variables \({\varvec{Z}}_{{\varvec{i}}} \user2{ }\) and the error term \(u_{i}\). However, on farms, this diversification strategy may be approached by an index \(I_{i}\), which has the minimum value \(\underline {I}\) and the maximum value \(\overline{I}\):
$$I_{i} = \left\{ {\begin{array}{*{20}c} {\underline {I} \, if \, I_{i}^{*} \le \underline {I} } \\ {I_{i}^{*} \, if \, \underline {I} < I_{i}^{*} < \overline{I}} \\ {\overline{I} \, if \, I_{i}^{*} \ge \overline{I}} \\ \end{array} } \right.$$
We used the conditional recursive mixed-process (CMP) framework (Roodman 2011) to estimate the parameters of the two-equation system. The use of the CMP approach was motivated by the following reasons. First, our system is a multiequation mixed model (the two equations have different forms of dependent variables), with Eqs. 5 and 6 being probit/logit and tobit models, respectively. Second, our system may be perceived as recursive, in the sense that we have clearly defined stages. Stage 1 (Eq. 5), the probit/logit model, captured a farmer's decision to grow a cowpea variety that had a genebank ancestor, whereas Stage 2 (Eq. 6), the tobit model, captured the effect of a farmer's decision on spatial diversity of cowpea varieties. The system of equations was estimated using a maximum likelihood (ML) approachFootnote 2 (Roodman 2011). As our recursive system is fully observed, meaning that the endogenous variable \(G_{i}^{*}\) appears on the right-hand side as observed, the CMP framework provided consistent estimates (Roodman 2011).
To measure the impact of IITA's genebank on Nigerian farmers' welfare, we applied a multinomial endogenous treatment effect model. The multinomial endogenous treatment effect model helps analyze the effects of an endogenous multinomial treatment (when exactly one treatment is chosen from a set of more than two choices) on a specific outcome (Deb and Trivedi 2006a, b). In settings with potential selection on unobservable characteristics and a treatment variable that has more than two categories, both the multinomial endogenous treatment effect model and the multinomial endogenous switching regression model may be used to measure a treatment effect. Our choice of the multinomial endogenous treatment model was motivated by our interest in the average treatment effect and testing the significance of selection effects.
We assumed that farmers were growing one of the three types of cowpea varieties as the main cropFootnote 3: (1) an (improved) cowpea variety that has a genebank ancestor, (2) an (improved) cowpea variety that does not have a genebank ancestor, and (3) a cowpea landrace. We hypothesized that each of the three types of cowpea varieties has a different impact on farmers' welfare, the improved cowpea variety with genebank ancestor having the highest impact on farmers' welfare.Footnote 4 The farmer i selects one of the three types of cowpea varieties mentioned above. Following Deb and Trivedi (2006a), let \(EV_{ij}^{*}\) denotes the indirect utility that farmer i would obtain by selecting the jth cowpea variety type (the jth treatment), \(j = 0,1,2\) and
$$EV_{ij}^{*} = z_{i}^{^{\prime}} \alpha_{j} + \delta_{j} l_{ij} + \eta_{ij}$$
where \(z_{i}\) is a vector of exogenous covariates with associated parameters \(\alpha_{j}\), and \(\eta_{ij}\) are independently and identically distributed error terms.\(l_{ij}\) are unobserved characteristics common to farmer i's cowpea variety choice (treatment choice) and outcome, with associated parameters \(\delta_{j}\).
Let \(j = 0\) denote the control group, farmers who are growing a cowpea landrace, and \(EV_{i0}^{*} = 0\). While \(EV_{ij}^{*}\) was not observed, we observed farmer i's cowpea variety choice (treatment choice). Let \(d_{j}\) refer to binary variables representing the observed cowpea variety choice (observed treatment choice) and \({\varvec{d}}_{{\varvec{i}}} = \left( {d_{i0} , d_{i1} , d_{i2} } \right)\). Also let \({\varvec{l}}_{{\varvec{i}}} = \left( {l_{i0} , l_{i1} , l_{i2} } \right)\). Then the probability of growing a specific type of cowpea variety (the probability of treatment) can be represented with a mixed multinomial logit structure (MMNL)Footnote 5:
$$Pr\left( {{\varvec{d}}_{{\varvec{i}}} {|}{\varvec{z}}_{{\varvec{i}}} , {\varvec{l}}_{{\varvec{i}}} } \right) = \frac{{exp\left( {z_{i}^{^{\prime}} \alpha_{j} + \delta_{j} l_{ij} } \right)}}{{1 + \mathop \sum \nolimits_{k = 1}^{2} exp\left( {z_{i}^{^{\prime}} \alpha_{k} + \delta_{k} l_{ik} } \right)}}$$
The second stage of the model assessed the impact of growing a specific type of cowpea variety on three outcome variables: (1) cowpea yield, (2) cowpea consumption, and (3) cowpea sale. The expected outcome equation for farmer i was formulated as follows:
$$E\left( {y_{i} {|}{\varvec{d}}_{{\varvec{i}}} , {\varvec{x}}_{{\varvec{i}}} , {\varvec{l}}_{{\varvec{i}}} } \right) = {\varvec{x}}_{i}^{^{\prime}} \beta + \mathop \sum \limits_{j = 1}^{2} \gamma_{j} d_{ij} + \mathop \sum \limits_{j = 1}^{2} \lambda_{j} l_{ij}$$
where \({\varvec{x}}_{{\varvec{i}}}\) is a set of exogenous covariates with associated parameter vectors \(\beta\), and \(\gamma_{j}\) denoting the treatment effects relative to the control. \(\lambda_{j}\) shows the impacts of unobserved characteristics (common to farmer i's cowpea variety choice and outcome) on the outcome. We also assumed that the outcome variables were continuousFootnote 6 and followed a normal (Gaussian) probability distribution.Footnote 7 The model was estimated using a maximum simulated likelihood (MSL) approach. Provided that the number of draws is sufficiently large, the maximization of the simulated log-likelihood is equivalent to maximizing the log-likelihood (Deb and Trivedi 2006a). Regarding the identification of the model, in principle, the parameters of the model are identified even if the regressors in the treatment equation are identical to those used in the outcome equation (Deb and Trivedi 2006a). However, in practice, we followed Deb and Trivedi's (2006a) recommendation, which consists of using exclusion restrictions (or instruments) through the inclusion of regressors in the treatment equations that do not enter the outcome equation. We used the geographical zone as the exclusion restriction (or instrument), in the sense that it affected the treatments (growing an improved cowpea variety with genebank ancestry and growing an improved cowpea variety without genebank ancestry) significantly and had no partial effect on the outcomes.Footnote 8
Our analysis was based on the surveyed households and the characteristics of their household heads. Table 2 presents a summary of descriptive statistics of independent variables. We found significant differences between the North East and North West regions, where the survey was conducted. For socioeconomic characteristics, we found that household heads from the North West regions were older and needed less credit compared with households from the North East region. In addition, in the North West region, fewer women (4%) were heads of cowpea-producing households than in the North East region (10%). However, the two regions were similar in terms of the level of education of the household head. On average, household heads of both regions had 5 years of education.
Table 2 Descriptive statistics for independent (control) variables
Regarding the pedigree information of cowpea varieties grown by farmers, we found that more households from the North West region were growing improved cowpea varieties that had a genebank ancestor. In the North West region, 44% of households were growing improved cowpea varieties that had a genebank ancestor, compared with 36% in the North East region. In addition, on average, the improved cowpea varieties grown in the North West region had more genebank ancestors (12) than those grown in the North East region (8). Finally, regarding farm and market characteristics, farms were larger and had lands with more variation in elevation in the North East region, but households living in this region were farther away from village and district markets.
Spatial diversity of cowpea varieties on farms
The first research objective consisted of measuring the spatial diversity of cowpea varieties on farms, and testing their association with genebank ancestry in the pedigrees of improved cowpea varieties. The survey on the 1524 cowpea-producing households was able to identify and name 16 improved cowpea varieties and 6 cowpea landraces grown by farmers. Other improved cowpea varieties and cowpea landraces grown by farmers were also identified. The other improved cowpea varieties had been developed by the Institute of Agricultural Research (IAR), affiliated to the Ahmadu Bello University (Nigeria), and the Institute of Agricultural Research and Training (IAR&T), affiliated to the Obafemi Awolowo University (Nigeria).
On average, each household dedicated 2 plots to cowpea growing, which covered an area of 1.938 ha per farm. The distribution of cowpea variety types on these plots was as follows: 62.34% of these plots were dedicated to cowpea landraces, 37.19% were dedicated to improved cowpea varieties that have a genebank ancestor, and 0.47% were dedicated to other improved cowpea varieties.Footnote 9 In addition, 41.29% of households were growing at least one improved cowpea variety as the main crop,Footnote 10 whereas 40.91% were growing at least one improved cowpea variety that has a genebank ancestor as the main crop. Finally, 68.50% of households were growing at least one cowpea landrace as the main crop.
Table 3 presents descriptive statistics for spatial diversity indices of cowpea varieties grown in the North East and North West regions of Nigeria. The average value of the Menhinick index (2.896) was higher in the North West region (two-sample two-sided t-test: p = 0.000), than in the North East region (2.204), suggesting greater richness of cowpea varieties on farms in the North West region, when standardized by area. For instance, Table 4, which shows the repartition of households over main cowpea varieties grown, indicates that some improved cowpea varietiesFootnote 11 (UAM09-1046-6-1 and other improved cowpea varieties) were not grown as main cowpea varieties by households from the North East region, whereas 15 households were growing them as main cowpea varieties in the North West region.
Table 3 Descriptive statistics for indices of the spatial diversity of the cowpea varieties grown in Nigeria
Table 4 Repartition of households over main cowpea varieties grown
The average value of the Shannon index (0.343) in the North West region was not statistically different from that in the North East region (0.345) (two-sample two-sided t-test: p = 0.275), indicating that cowpea varieties grown are equally abundant in both regions.
Finally, in terms of inverse dominance, we found that the difference between the average values of the Berger-Parker index in the North East region and the North West region was not statistically significant (two-sample two-sided t-test: p = 0.417). While Kananado White/Dan Bokolo was the most widely grown cowpea variety in both regions, farmers cultivated other varieties too.
IITA's contribution to the ancestry of improved cowpea varieties grown by farmers
Before investigating the impact of IITA's genebank on varietal diversity of cowpeas on farms, we provide some results on the link between IITA's genebank and improved cowpea varieties grown in Nigeria. Research on cowpea improvement was initiated at IITA in 1970 and over 50 countries, including Nigeria, have identified and released improved cowpea varieties from IITA for general cultivation (Singh 1997). Table 5 presents the contribution of IITA's genebank to the ancestry of the improved cowpea varieties grown in Nigeria. Additional file 1: Fig. S2 also shows diagrams depicting the pedigrees of improved cowpea varieties. We found that most of the improved cowpea varieties grown by Nigerian farmers were released recently (between 2005 and 2015) and had a genebank ancestor.
Table 5 Contribution of IITA's genebank to the ancestry of the adopted improved cowpea varieties in Nigeria
Earlier breeding activities (from the1970s to early 2000s) focused on the development of insect and multiple disease resistances, varieties characteristics by white rough seed coat, extra-early maturity (60–70 days) and late maturity (85–120 days). Photo-insensitive, dual-purpose varieties were developed, along with photosensitive early to late maturing varieties and high yielding, bush-type vegetable varieties (Singh et al. 1997; Boukar et al. 2019). The uniqueness of recently released improved cowpea varieties reflects advances in cowpea genetics, genomics, and the deployment of integrated breeding approaches (Boukar et al. 2019; Varshney et al. 2019).
On average, the pedigree of an improved cowpea variety grown by Nigerian farmers included 9 unique IITA ancestors, which have been incorporated 39 times during the breeding process. For instance, the most recently released improved cowpea variety, UAM09-1055-6, had 8 unique IITA ancestors that have been incorporated 56 times in the breeding process. The oldest improved cowpea variety, IT90K-277-2 (Sasakawa), had 7 unique IITA ancestors that have been incorporated 14 times during the breeding process. UAM09-1055-6 is the result of a single cross between Borno Brown and IT97K-499-35. IT90K-277-2 (Sasakawa) is the result of breeding IITA's genebank accession IT87F-1777-2 with IT84S-2246-4, crossed with TVx3236.
The improved cowpea variety, IT89KD-288/Sampea-11, released in 2009 was the most widely adopted by Nigerian farmers, in terms of the main cowpea variety grown on plots (it was grown as the main cowpea variety on 7.24% of plots). IT07K-318-33/Sampea 17, released in 2015, was the least adopted by Nigerian farmers, in terms of the main cowpea variety grown on plots (it was grown on 0.08% of plots). IT89KD-288/Sampea-11 was the result of the combination between an IITA's genebank accession, IT87F-1777-2, and IT84s-2246-4, whereas IT07K-318-33/Sampea 17 was developed through the cross of IT98K-131-2 with IT95K-238-3. Additional file 1: Fig. S2 shows the IT89KD-288/Sampea-11 and IT07K-318-33/Sampea 17 pedigree trees.
Overall, we confirm the use of germplasm from the IITA genebank by scientists to develop improved cowpea varieties grown by Nigerian farmers.
IITA's genebank and spatial diversity of cowpea varieties on farms
To measure the effect of IITA's genebank on the spatial diversity of cowpea varieties on farms, we ran the recursive mixed-process model (Eqs. 5 and 6), using an ML estimation approach.Footnote 12 Table 6 presents the estimates of the model, where the spatial diversity index is a richness index (the Menhinick index). Based on the results, we found that growing an improved cowpea variety that had a genebank ancestor was not significantly associated with richness—either positively or negatively.
Table 6 Recursive mixed-process model estimates, first specification with the richness index
Farm characteristics, including farm size, distance to the farm from residence, the number of plots that are perceived as flat, and the geographical zone (being part of the North West region), were important determinants of the richness of cowpea varieties on farms in Nigeria. The richness of cowpea varieties on farms was higher in the North West region and in households with smaller sizes and farms, a higher number of flat plots, or whose members reside not far from their plots. Results also showed that farmers who either belong to the North West region, have been exposed to (or have experience of) improved cowpea varieties, or do not need credit for their farming activities were more likely to grow improved cowpea varieties that have a genebank ancestor. The result concerning experience aligns with that presented by Manda et al. (2019), who found that the number of years a farmer has been exposed to improved cowpea varieties is an important determinant of adoption.
Table 7 presents the estimates of the model, using an inverse dominance index (the Berger-Parker index) as a spatial diversity index. The results indicate that the adoption of an improved cowpea variety that had a genebank ancestor had a positive and significant effect on the inverse dominance index. Farm characteristic, especially the slope, was a determinant of the inverse dominance index. Farmers who had a high number of plots perceived as flat devote more area to their preferred variety. A new important determinant of growing a cowpea variety that has a genebank ancestor is soil fertility. Farmers were more likely to grow an improved cowpea variety that had a genebank ancestor when they had a higher number of plots with perceived poor soil quality. This suggests a possible association with traits conferred through diverse ancestry. These traits may be adapted to the poor soil quality conditions.
Table 7 Recursive mixed-process model estimates, second specification with the inverse dominance index
Finally, Table 8, which presents the estimates of the model, using a concentration index (the Herfindahl index) as spatial diversity index, indicates that growing an improved cowpea variety that has a genebank ancestor has a negative and significant effect on the concentration index. In other words, growing an improved cowpea variety that had a genebank ancestor decreased the specialization in a single cowpea variety. This is consistent with the results for the Berger-Parker index.
Table 8 Recursive mixed-process model estimates, third specification with the concentration index
To sum up, genebank ancestry did not contribute to more specialization or dominance of any particular cowpea variety on farms or lead to the displacement of other cowpea varieties; in fact, it was consistent with less concentration and less dominance by the main cowpea variety. The results also indicated that certain farm characteristics were more important determinants of greater richness among cowpea varieties.
IITA's genebank, cowpea yield and farmers' welfare
To investigate the effect of IITA's genebank on cowpea yield and farmers' welfare, we ran a multinomial endogenous treatment effect model, using an MSL approach.Footnote 13 Table 9 presents the results of multinomial endogenous treatment effects model estimates of impacts on cowpea yield. We found a positive and significant treatment effect of growing an improved cowpea variety that had a genebank ancestor on cowpea yield. Growing an improved cowpea variety that had a genebank ancestor increased by 177.042% the yield of cowpea,Footnote 14 compared to growing a cowpea landrace. However, the significant value (− 1.194) of the coefficient on the latent factor indicated significant selection on unobservables. In other words, farmers who were more likely to grow a cowpea variety that had a genebank ancestor relative to a cowpea landrace, based on their unobserved characteristics,Footnote 15 experienced a decline in cowpea yield more often, which might upset this effect on the yield of cowpea for some of them. Other factors like household characteristics (sex of the household head and need of credit), farm characteristics (size, distance to farm from residence and soil fertility), and market characteristics (distance to village market from residence and distance to district market from residence) had significant effects on cowpea yield. For instance, a household which either has a man as head, has a small farm size, has plots not farm from the residence, or has plots not far from the village/district market is likely to have higher cowpea yields.
Table 9 Multinomial endogenous treatment effects model estimates
Table 9 also presents the results of multinomial endogenous treatment effects model estimates of impacts on cowpea consumption. We found a positive and significant treatment effect of growing an improved cowpea variety with a genebank ancestor on cowpea consumption as food by the household. Growing an improved cowpea variety that had a genebank ancestor increased household consumption of cowpea as food by 46.375%,Footnote 16 compared to growing a cowpea landrace. However, the significant value (− 0.419) of the coefficient on the latent factor indicated significant selection on unobservables. In other words, farmers who were more likely to grow a cowpea variety having a genebank ancestor relative to a cowpea landrace, based on their unobserved characteristics, reduced their level of cowpea consumption as food more often, which might upset this effect on cowpea consumption for some of them. Other factors like farm characteristics (size, slope, and soil fertility) had significant effects on cowpea consumption as food.Footnote 17
Finally, Table 9 also presents the results of multinomial endogenous treatment effects model estimates of impacts on cowpea sale. We did not find a significant treatment effect of growing a cowpea variety that had a genebank ancestor on cowpea sale. However, the need for credit and the distance to village market (from residence) did have significant effects on cowpea sale. Farmers who either did not need credit for their farming activities or were not far from the village market increased their levels of cowpea sales.
In summary, growing a cowpea variety with genebank ancestry had a positive and significant impact on cowpea yield and cowpea consumption at home, but not on cowpea sale.
The evidence presented in this research indicates that Nigerian farmers are growing improved cowpea varieties that have genebank ancestors in their pedigree trees, showing the contribution of IITA's genebank to the development and release of improved cowpea varieties in Nigeria. Some recent studies confirmed that genebanks in Africa contribute to the development of improved crop varieties and the conservation and distribution of tree germplasm (Kitonga et al. 2020; Sellitti et al. 2020).
Evidence also shows that adoption of a cowpea variety with genebank ancestry does not contribute to the specialization or dominance of any particular variety, and has no significant association with richness of cowpea varieties grown. Therefore, although IITA's genebank accessions are used for the development of improved cowpea varieties that have been widely adopted by farmers in Nigeria, we see no evidence that their adoption contributes to fewer varieties grown by smallholder farmers. Other factors like household size, farm characteristics and geographical zone are more important in explaining the pattern of cowpea varieties grown on farms. Our findings are consistent with the empirical literature on the determinants of crop diversity on farms, which finds that household characteristics, farm characteristics, and geographical zone have significant effects on the diversity within crops (Smale et al. 2003; Benin et al. 2004; Bellon et al. 2020).
Regarding the decision to grow a cowpea variety that has a genebank ancestor, farmers' experience is an important determinant, meaning that the exposition to/adoption of a former agricultural technology is a predictor of the adoption of a new agricultural technology. This is line with a recent study on the poverty impacts of improved cowpea varieties in Nigeria (Manda et al. 2019). We found that education of the household head is not a significant determinant of a farmer's decision to grow a cowpea variety that has a genebank ancestor, whereas education has been cited as an important determinant of the adoption of agricultural technologies in Africa in other studies (Alene and Manyong 2007; Foster and Rosenzweig 2010). A possible explanation is that education does not matter when geographical factors incentivize the farmer's decision to grow a cowpea variety that has a genebank ancestor. We found that the geographical zone is an important determinant of farmers' decision to grow a cowpea variety that has a genebank ancestor. This is also in line with Manda et al. (2019), who found that the adoption of improved cowpea varieties was lower in the North East region compared to the North West region, reflecting the unobservable differences in terms of the resources and weather patterns between the two regions of Nigeria. Finally, as expected, growing a cowpea variety that has a genebank ancestor affects cowpea yield and farmers' welfare. Evidence showed a positive and significant effect on cowpea yield and cowpea consumption. Numerous studies have demonstrated that improved crop varieties or agricultural technologies have a positive and significant impact on agricultural productivity in Africa (for example, Duflo et al. 2008; Kassie et al. 2008; Pender and Gebremedhin 2007; Abdulai and Huffman 2014).
Cowpea is an important food legume that provides food and fodder, improves soil fertility and contributes to the sustainability of food production in marginal areas of the dry tropics (Singh 1997). Using data from a household survey conducted in Northern Nigeria in 2017, and data from IITA's cowpea breeding program, we measured varietal diversity, linked improved cowpea varieties grown to IITA's genebank and investigated the effect of IITA's genebank on varietal diversity of cowpeas on farms. We also examined the impact of IITA's genebank on cowpea yield and farmers' welfare.
Our spatial diversity indices show that richness of cowpea varieties is higher in the North West region than the North East region (when standardized by area). The pedigree analyses confirm the use of germplasm from the IITA genebank by scientists to develop improved cowpea varieties grown by Nigerian farmers. Regarding the effect of IITA's genebank on varietal diversity of cowpeas on farms, our recursive mixed-process model indicates that genebank ancestry does not lead to the displacement of other cowpea varieties. In addition, it does not contribute to specialization or dominance of any particular variety. Finally, our multinomial endogenous treatment effect model indicates that growing a cowpea variety that has a genebank ancestor has a positive and significant impact on cowpea yield and cowpea consumption.
These findings show additional benefits from IITA's genebank, through the adoption of improved cowpea varieties that have a genebank ancestor. Benefits are threefold. First, we find no negative effects of growing improved varieties on the spatial diversity of cowpea varieties grown on farms. Second, IITA's genebank helps increase cowpea yield on farmers, showing a contribution to agricultural productivity in smallholder farms in Nigeria. Finally, IITA's genebank contributes to increased household consumption of cowpea as food, contributing to farmers' welfare in Nigeria. Policymakers and practitioners should consider these findings when analyzing the benefits of conserving crop genetic diversity in genebanks and on farms.
Several caveats are in order when considering the results. Farmers' welfare is only measured by cowpea consumption and cowpea sale. Further empirical research could explore other welfare dimensions. For instance, in the context of climate change, reduced vulnerability to drought and reduced soil erosion could be added to farmers' welfare dimensions. Linkages between genebank ancestry and traits conferred to cowpea varieties grown on farms have not been clearly established. Understanding these linkages is needed to draw inferences about their value on farms and in varietal portfolios. Possible non-use benefits from IITA's genebank may also be found.
The datasets used and analyzed during the current study are available from the corresponding authors on request.
This is confirmed by the general strategy for cowpea breeding at IITA, which combines multiple disease and insect resistance and broad adaptability to meet the varied requirements of different countries and regions, including Nigeria (Singh 1997; Singh et al. 1997). For instance, the IT90K-277–2 (Sasakawa), which is an improved cowpea variety grown by Nigerian farmers, combines disease (Brown Blotch and Anthracnose) and insect (Aphid) resistance and intercropping characteristics (Singh 1997; Singh et al. 1997).
Please note that Stata's ML approach in the CMP framework is fundamentally an ML seemingly unrelated regression (SUR) estimation program (Roodman 2011).
This is confirmed by the dataset in which farmers are growing a specific cowpea variety as the main crop.
This hypothesis is motivated by the fact that the improved cowpea variety with genebank ancestor combines multiple disease and insect resistance and broad adaptability to meet the varied requirements of the region, which may have a higher impact on yield, home food consumption, and sale.
Please note that the MMNL structure is an assumption. Other multinomial probability distributions could also be considered.
Please note that in other contexts the outcome variable may be a count variable. In this case, the negative binomial-2 density could be a good choice.
Using ln(.) helps have normal distributions of outcome variables.
We establish the admissibility of the exclusion restriction (or instrument) by performing a simple test, which shows that the geographical zone affects the treatments significantly, whereas it does not affect the outcomes.
According to the household survey questionnaire, a field is a piece of land physically separated from others and a plot is a subunit of a field. Some farmers may be intercropping on a plot. Only nine households (0.58%) were growing more than one cowpea variety per plot.
Main crop refers to the crop that occupies the largest share of farm area.
A possible explanation is that some of these varieties have not been released officially. Few farmers may have had access to the seeds through evaluation trials.
The model is estimated using the Stata command cmp. The same analysis was conducted for improved varieties that do not have a genebank ancestor. We found no significant effects on diversity indices.
The model is estimated using the Stata command mtreatreg. We used 1000 simulation draws.
It is computed, using the value 1.090, the estimate of the average treatment effect: \(\left( {e^{1.090} - 1} \right) \times 100\).
The unobserved characteristics are common to the farmer's/household's adoption of improved cowpea varieties that have a genebank ancestor and outcomes (cowpea yields). For instance, Abdulai and Huffman (2014) show that the management and technical ability of the farmers to understand new technology may affect outcomes, including crop yields.
This is understandable, in the sense that, ceteris paribus, a bigger or more fertile farm may mean a greater use of cowpea as food.
CGR:
Cowpea genetic resource
CIAT:
International Center for Tropical Agriculture
CMP:
Conditional recursive mixed-process
DNA:
Deoxyribonucleic Acid
IAR:
Institute of Agricultural Research
IAR&T:
Institute of Agricultural Research and Training
ICRAF:
World Agroforestry
ICRISAT:
International Crops Research Institute for the Semi-Arid Tropics
IITA:
International Institute of Tropical Agriculture
ML:
Maximum likelihood
MMNL:
Mixed multinomial logit structure
MSL:
Maximum simulated likelihood
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The authors gratefully acknowledge the CGIAR Genebank Platform, the International Institute of Tropical Agriculture (IITA), and the Crop Trust for providing the financial support for the research work. The first author acknowledges the hosting support of the IITA Genetic Resources Center and the facilitation by Tahirou Abdoulaye. We also thank Trushar Shah for his assistance on the Helium program.
Funding for this research was provided by the CGIAR Genebank Platform, the International Institute of Tropical Agriculture (IITA), and the Crop Trust through the 2020 Genebank Impacts Fellowship.
Global Crop Diversity Trust, Platz der Vereinten Nationen 7, 53113, Bonn, Germany
Abel-Gautier Kouakou & Nelissa Jamora
International Institute of Tropical Agriculture, PMB 5320, Oyo Road, Ibandan, 200001, Oyo State, Nigeria
Ademola Ogundapo & Michael Abberton
Michigan State University, 220 Trowbridge Rd, East Lansing, MI, 48824, USA
Melinda Smale
International Institute of Tropical Agriculture-Tanzania, Plot 25 Light Industrial Area, CocaCola Rd, Dar es Salaam, Tanzania
Julius Manda
Abel-Gautier Kouakou
Ademola Ogundapo
Nelissa Jamora
Michael Abberton
A-GK contributed to the research conceptualization and design, data gathering, data analysis, writing, and editing. AO contributed to data gathering and data analysis. MS and NJ contributed to research conceptualization and design, writing, and editing. JM and MA contributed to data gathering. All authors read and approved the final manuscript.
Correspondence to Abel-Gautier Kouakou.
Additional file 1: Fig. S1.
Land area harvested (ha) and production quantity (tons) of cowpea in Nigeria (1980–2020). Fig. S2. Diagrams of adopted improved cowpea varieties pedigree trees.
Kouakou, AG., Ogundapo, A., Smale, M. et al. IITA's genebank, cowpea diversity on farms, and farmers' welfare in Nigeria. CABI Agric Biosci 3, 14 (2022). https://doi.org/10.1186/s43170-022-00083-w
Genebank
On-farm cowpea diversity
Farmers' welfare
The Value of Genebanks on Farms in Developing Agriculture | CommonCrawl |
Abstract: We study the detailed evolution of the fine-structure constant $\alpha$ in the string-inspired runaway dilaton class of models of Damour, Piazza and Veneziano. We provide constraints on this scenario using the most recent $\alpha$ measurements and discuss ways to distinguish it from alternative models for varying $\alpha$. For model parameters which saturate bounds from current observations, the redshift drift signal can differ considerably from that of the canonical $\Lambda$CDM paradigm at high redshifts. Measurements of this signal by the forthcoming European Extremely Large Telescope (E-ELT), together with more sensitive $\alpha$ measurements, will thus dramatically constrain these scenarios. | CommonCrawl |
David Wolfe (mathematician)
David Wolfe is a mathematician and amateur Go player.
David Wolfe
NationalityAmerican
Alma materCornell University
UC Berkeley College of Engineering
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
Gustavus Adolphus College
Dalhousie University
Education and career
Wolfe graduated from Cornell University in 1985, with a bachelor's degree in electrical engineering.[1] He obtained a Ph.D. in computer science from the University of California, Berkeley in 1994, with a dissertation Mathematics of Go: Chilling Corridors combining both subjects and supervised by Elwyn Berlekamp.[2]
After working as a lecturer at the University of California, Berkeley from 1991 to 1996, as an associate professor at Gustavus Adolphus College from 1996 to 2008, and then as an adjunct faculty member at Dalhousie University, he moved from academia to the software industry.[1] Wolfe was a fan of Martin Gardner and in 2009 he teamed up with Tom M. Rodgers to edit a Gardner tribute book.[3]
Books
Wolfe is the author of books on combinatorial game theory, including:
• Mathematical Go: Chilling Gets the Last Point (with Elwyn Berlekamp, A K Peters, 1994; also published as Mathematical Go Endgames: Nightmares for the Professional Go Player, Ishi Press, 1994)[4]
• Lessons in Play: An Introduction to Combinatorial Game Theory (with Michael H. Albert and Richard Nowakowski, A K Peters, 2007; 2nd ed., CRC Press, 2019)[5]
References
1. "David Wolfe", LinkedIn, retrieved 2020-07-07
2. David Wolfe at the Mathematics Genealogy Project
3. Puzzlers' Tribute: A Feast for the Mind (AK Peters) ISBN 9781568811215
4. Reviews of Mathematical Go:
• Hentzschel, J., zbMATH, Zbl 0852.90149{{citation}}: CS1 maint: untitled periodical (link)
• Campbell, Paul J. (December 1994), Mathematics Magazine, 67 (5): 391–393, doi:10.2307/2691006, JSTOR 2691006{{citation}}: CS1 maint: untitled periodical (link)
• Loeb, Daniel Elliott (1995), Mathematical Reviews, MR 1274921{{citation}}: CS1 maint: untitled periodical (link)
• Guy, Richard K.; Nowakowski, Richard J. (1995), Bulletin of the American Mathematical Society, New Series, 32 (4): 437–441, doi:10.1090/S0273-0979-1995-00601-4, MR 1568191{{citation}}: CS1 maint: untitled periodical (link)
• Hayes, Brian (July–August 1995), American Scientist, 83 (4): 381–382, JSTOR 29775500{{citation}}: CS1 maint: untitled periodical (link)
• You, Zhiping; Yorke, James A. (September 1996), SIAM Review, 38 (3): 543–544, doi:10.1137/1038104, JSTOR 2132518{{citation}}: CS1 maint: untitled periodical (link)
5. Reviews of Lessons in Play:
• Borchers, Brian (May 2007), "Review", MAA Reviews, Mathematical Association of America; updated August 2019 for 2nd ed.
• Campbell, Paul J. (December 2007), Mathematics Magazine, 80 (5): 399–400, doi:10.1080/0025570X.2007.11953519, JSTOR 27643070, S2CID 218541649{{citation}}: CS1 maint: untitled periodical (link)
• Ward, Michael (July 2009), The Mathematical Gazette, 93 (527): 382–383, doi:10.1017/S0025557200185080, JSTOR 40378761, S2CID 166040709{{citation}}: CS1 maint: untitled periodical (link)
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
Xubo Tang1 &
Yanni Sun ORCID: orcid.org/0000-0003-1373-80231
There are many different types of microRNAs (miRNAs) and elucidating their functions is still under intensive research. A fundamental step in functional annotation of a new miRNA is to classify it into characterized miRNA families, such as those in Rfam and miRBase. With the accumulation of annotated miRNAs, it becomes possible to use deep learning-based models to classify different types of miRNAs. In this work, we investigate several key issues associated with successful application of deep learning models for miRNA classification. First, as secondary structure conservation is a prominent feature for noncoding RNAs including miRNAs, we examine whether secondary structure-based encoding improves classification accuracy. Second, as there are many more non-miRNA sequences than miRNAs, instead of assigning a negative class for all non-miRNA sequences, we test whether using softmax output can distinguish in-distribution and out-of-distribution samples. Finally, we investigate whether deep learning models can correctly classify sequences from small miRNA families.
We present our trained convolutional neural network (CNN) models for classifying miRNAs using different types of feature learning and encoding methods. In the first method, we explicitly encode the predicted secondary structure in a matrix. In the second method, we use only the primary sequence information and one-hot encoding matrix. In addition, in order to reject sequences that should not be classified into targeted miRNA families, we use a threshold derived from softmax layer to exclude out-of-distribution sequences, which is an important feature to make this model useful for real transcriptomic data. The comparison with the state-of-the-art ncRNA classification tools such as Infernal shows that our method can achieve comparable sensitivity and accuracy while being significantly faster.
Automatic feature learning in CNN can lead to better classification accuracy and sensitivity for miRNA classification and annotation. The trained models and also associated codes are freely available at https://github.com/HubertTang/DeepMir.
Non-coding RNAs (ncRNAs) refer to the RNAs that do not encode proteins and function directly as RNAs. Genome annotation of many different genomes show that ncRNAs are ubiquitous and have various important functions [1]. Besides commonly seen house-keeping ncRNAs such as transfer RNAs (tRNAs), ribosome RNAs (rRNAs), many small ncRNAs play important roles in gene regulation. This work is mainly concerned with a type of small ncRNA, microRNA (miRNA), which act as key regulators of gene expression at post-transcriptional level in different species [2–5]. In metazoans, mature miRNAs bind to the 3'-UTR of target mRNAs and can repress translation or promote mRNA degradation. As an miRNA can bind to multiple mRNA transcripts, a large number of protein-coding genes can be regulated by miRNAs [6, 7].
Because miRNAs' important functions and their associations with complicated diseases in human, there are intensive research about miRNA gene annotation, target search, function identification etc. A fundamental step in miRNA research is the identification of miRNA genes in genomes. In the canonical miRNA biogenesis pathway, miRNAs are processed from longer transcripts named as primary miRNAs (pri-miRNAs) [3]. The hairpin structures of pri-miRNAs are cleaved by a member of RNase II family of enzymes, Drosha and produce precursor miRNA (pre-miRNA) in the nucleus [8, 9]. Pre-miRNAs are then exported to the cytoplasm, where Dicer cleaves off the loop region of the hairpin and further processes it to mature miRNA(s) of about 21 nucleotides [10, 11]. MiRNA gene annotation usually refers to identification of pre-miRNAs and mature miRNAs.
Existing miRNA annotation tools can be generally divided into two groups depending on whether reference miRNA genes are used. Homology-based miRNA search identifies pre-miRNAs by conducting sequence and/or secondary structural similarity search against existing miRNA genes. Like other ncRNAs, pre-miRNAs preserve strong secondary structures [2]. Thus, homology search models [12, 13] that can explicitly encode both sequence and structural similarities usually achieve high sensitivity and accuracy in classifying query sequences into their originating homologous families. However, the high sensitivity comes with a price of high computational cost. For example, structural homology search models based on context-free grammar have cubic running time complexity [14]. Even with various heuristic filtration techniques, it can be still very time-consuming to conduct large-scale sequence classification using both sequence and structural alignments. Sequence similarity-based homology search tools such as BLAST [15] can be also applied to classify pre-miRNAs to their native families. However, remote homologs with high structural but low sequence conservation tend to be missed. Another group of tools [16–18] do not use reference sequences for pre-miRNA search. These de novo miRNA search methods mainly use features such as hairpin structures of pre-miRNAs to identify putative pre-miRNAs in genomes. As a large number of regions in a genome can form hair-pin structures, features from RNA-Seq [19] data such as expression levels and read mapping patterns are often used to reduce the false positive rate of miRNA search [20–23]. Both types of tools are useful for miRNA search and annotation. De novo methods have the advantage of identifying possibly novel miRNAs but additional processing is needed to validate the findings.
Homology search-based miRNA search methods can take advantage of accumulating characterized miRNAs. For example, MiRBase [24] is an online database for miRNA sequences and annotation. The current release 22 contains 1983 miRNA families from 271 organisms, including 38,589 pre-miRNAs and 48,860 mature miRNAs. Rfam [25] is a comprehensive ncRNA family database with over 3,000 ncRNA families. The release 14.1 contains 529 pre-miRNA families and 215,122 precursor sequences.
These classified pre-miRNA sequences can be used as training data for deep learning based models. Depending on the choice of the training sequences and the design of the model architecture, deep learning-based miRNA search can be applied to distinguish miRNAs from other types of ncRNAs and also to conduct finer scale classification for different types of miRNAs. In this work, we explore whether using convolutional neural network (CNN) has advantages in distinguishing different types of miRNAs over powerful covariance models. In particular, we investigated how the input sequence encoding and training set construction affect the performance of miRNA characterization using CNN.
We choose CNN as the deep learning model because of its recent success in other sequence classification studies [26–29]. Empirical analyses have shown that CNN can be applied to extract "motifs" from a set of homologous sequences. Motifs are essential features to distinguishing different groups of sequence families including miRNAs. DeepBind [26] used a single convolution layer to capture the motif from protein binding sites. DeepFam [29] applied the CNN on the protein classification and found that the frequently activated convolution filters are consistent with known motifs. As different miRNA families tend to have different conserved sequences, the convolution layers in CNN are expected to capture distinctive features for fine-grained classification. DanQ [30], proposed by Qiang et al., added additional long short term memory (LSTM) layers above the convolution layers to capture the dependency between the separated motifs extracted by convolution layers. But as miRNAs are relatively short, the sequential features within a filter are sufficient for classification.
In this section, we summarize related work on homology search-based miRNA identification. Some homology search tools are designed for comprehensive ncRNA search and can divide miRNAs into different types. For example, there are hundreds of different miRNA families in Rfam. The associated tool, Infernal [12], conducts homology search by incorporating both sequence and secondary structure similarities in context-free grammar based models. Input sequences can be classified into different miRNA families for functional inference. For identifying miRNAs with high sequence similarity, generic homology search tools such as BLASTn [15] can be applied as well.
Most tools designed specifically for miRNA search aim to distinguish miRNAs from other types of sequences [31–33]. The most successful ones usually employ transcriptomic data to improve the identification accuracy. When the reference genomes are available, reads from small RNA-Seq data are mapped to the reference genomes to locate possible pre-miRNA genes. Features such as the conserved hairpin structure, read mapping patterns on the mature miRNA vs. other regions, expression levels across multiple samples are utilized to screen miRNAs in those candidate regions. From the perspective of machine learning, distinguishing miRNAs from other regions can be formulated as a binary classification problem. Pre-miRNAs have the positive label and all others have the negative label. Classification models such as SVM [34, 35], Random Forest [36], and CNN [37] have been applied for miRNA search. Being different from these binary classification tools, ours focuses on classifying input sequences into different miRNA families for more detailed function annotation. Unrelated sequences including other types of ncRNAs are rejected using a threshold in the softmax value.
CNN was also employed by Genta Aoki [38] for ncRNA classification. The authors took ncRNA pairwise alignments and associated features as input to CNN and got 98% accuracy for 6 types of ncRNA.
Advances of feature selection and classification models in machine learning have enhanced the sensitivity and precision for miRNA search. However, highly unbalanced training set is still a challenge for various learning models [39]. Being formulated as a binary classification problem, there are significantly more negative samples (non-miRNAs) than positive samples (miRNAs). In addition, there are many different types of non-miRNA sequences. It is not clear how to compose the negative training data from such large and highly diverse sequences.
In this study, we intend to formulate miRNA search as a multi-label classification problem. Instead of using non-miRNAs as training data, we reject those un-relevant sequences using methods from open set problem [40]. In addition, we implemented two types of encoding methods based on whether we explicitly encode the secondary structure information.
The deep learning model we choose is Convolutional Neural Network (CNN), which has demonstrated some success in ncRNAs classification [38]. We implemented and compared two different encoding methods for CNN-based miRNA classification. In the first encoding method, we explicitly encode secondary structure information into matrices and use these matrices as training/testing data. In the second method, we use one-hot encoding matrix to represent the input sequences and do not take into account predicted secondary structures.
Explicitly encode secondary structures into matrices
We implemented three types of matrix to encode the secondary structure information from sequences: probability matrix, pair matrix, and mixed matrix. The first two are inspired from adjacency matrix for modeling secondary structures. The structural information is derived from the sequences using RNAfold [41], which is one module in the ViennaRNA [41] package. As the optimal structure predicted based on Minimum Free Energy (MFE) is often not accurate, we use RNAfold to output both the optimal and suboptimal structures. In addition, we also use the base pairing probabilities computed by the software.
Probability matrix simply contains the values of the base pairing probability outputted by RNAfold. For a sequence s, the size of the matrix is |s|×|s|. Pi,j is the predicted base pairing probability between the ith and jth base in s if the probability p is above a given threshold T. The equation for defining the value of each cell can be found below.
$$P_{i,j(probability\ matrix)} =\left\{ \begin{array}{rcl} p & & {if\ p\ \geq\ T}\\ 0 & & {if\ p\ <\ T.} \end{array} \right. $$
Being different from probability matrix, pair matrix distinguishes different base pairs including Watson-Crick pairs and G-U pair. If the base pairing probability is above a given threshold, we will record this base pair using its ID number, which is used to distinguish different base pairs. Depending on whether we take into account the order of the bases in a base pair, different base pairs can be converted into 6 or 3 different values. The conversion rules are summarized in the following equations. Xi,j refers to an element at position (i,j) in a pair matrix. si refers to the ith base in sequence s. T is a given threshold.
$$ X_{i,j(pair\ matrix\ with\ order)} = \left\{\begin{array}{ll} 0, &\text{if } {p} < \mathrm{T} \\ 1/6, &\text{if}\ (s_{i} s_{j}=AU)\ \text{and}\ p \geq \mathrm{T} \\ 2/6, &\text{if}\ (s_{i} s_{j}=UA)\ \text{and}\ p \geq \mathrm{T} \\ 3/6, &\text{if}\ (s_{i} s_{j}=CG)\ \text{and}\ p \geq \mathrm{T} \\ 4/6, &\text{if}\ (s_{i} s_{j}=GC)\ \text{and}\ p \geq \mathrm{T} \\ 5/6, &\text{if}\ (s_{i} s_{j}=GU)\ \text{and}\ p \geq \mathrm{T} \\ 6/6, &\text{if}\ (s_{i} s_{j}=UG)\ \text{and}\ p \geq \mathrm{T} \end{array}\right. $$
$${\begin{aligned} X_{i,j(pair\ matrix\ without\ order)}\! =&\\ &\left\{\begin{array}{ll} \!0, &\text{if } {p} < \mathrm{T} \\ \!1/3, &\text{if}\ (s_{i} s_{j}\,=\,AU \text{or}\ s_{i} s_{j}\,=\,UA)\!\ \text{and}\ p \geq \mathrm{T} \\ \!2/3, &\text{if}\ (s_{i} s_{j}\,=\,CG \text{or}\ s_{i} s_{j}\,=\,GC)\!\ \text{and}\ p \geq \mathrm{T} \\ \!3/3, &\text{if}\ (s_{i} s_{j}\,=\,GU \text{or}\ s_{i} s_{j}\,=\,UG)\!\ \text{and}\ p \geq \mathrm{T} \end{array}\right. \end{aligned}} $$
Combining these two features together, the original 2D matrix will become a 3D matrix with two layers, which is called mixed matrix, as shown in Fig. 1c. One layer of size |s|×|s| is the probability matrix and another layer of the same size is the pair matrix. Essentially, this matrix integrates different base pairs with the predicted pairing intensities.
Examples of different encoding matrices. (a) Probability matrix; (b) Pair matrix; (c) Mixed matrix; (d) One-hot encoding matrix
The pair and mixed matrices can be conveniently visualized as images. We presented the corresponding images for one miRNA and one tRNA in Fig. 2. The threshold T is 0.0001 in all the matrices. It is not hard to observe the stacking base pairs of the hairpin and cloverleaf structures of the miRNA and tRNA, respectively. The secondary structures are less obvious in the pair matrix because the cell values in the pair matrix are decided by the base pairs rather than the base pairing probabilities. Given a small T, cells with low pairing probabilities might still get a relatively big value because of the conversion rules.
The probability, pair and mixed matrix images of miRNA and tRNA. (a), (b), (c) correspond to probability matrix, ordered pair matrix, mixed matrix of a miRNA sequence respectively. (d), (e), (f) correspond to probability matrix, ordered pair matrix, mixed matrix of a tRNA sequence respectively. For the mixed matrices, the color green is from the layer of probability matrix while blue represents the layer of the pair matrix
CNN architecture for the matrices containing base pairing information
The CNN model contains two convolutional layers, followed by max pooling layers and three fully connected layers. Figure 3 sketches this architecture. To prevent overfitting, dropout is also applied. During the training of the CNN model, several hyperparameters were tuned within the given ranges, which are shown in Table 1. The parameters with best performance were selected. Finally, the hyperparameters were set as follows: number of convolution layers = 2, kernel size for each convolution layer = 2, the number of kernels in the two convolution layer = 64: 128, pooling method = max pooling, number of units in two fully connected layer = 256: 128, learning algorithm = Adam, dropout rate = 0.5, learning rate = 0.001, batch size = 32. The CNN model was implemented in Keras [42].
CNN structure of the probability/pair/mixed matrix
Table 1 The list of the tuned hyperparameters
Encoding the sequence using one-hot matrix
One-hot encoding matrix has been successfully used in encoding genomic sequences for deep learning models. Essentially, the sequence is converted to a |s|×4 one-hot encodidng matrix, where |s| is the length of an input sequence and 4 is the number of different bases. Let the matrix be M, where Mi,j is 1 if the ith base in the input sequence is the jth character in the alphabet. For any other characters, Mi,j is 0 (k≠j). An example one-hot encoding matrix is given in Fig. 1d.
The CNN architecture for one-hot encoding matrices
Inspired by Yoon Kim's work in sentence classification [43], a similar model is used in this work. Several convolution layers with different size of kernels, followed by global max pooling layer, are connected to input layer directly. The outputs of all pooling layers are concatenated together and then fed into two fully connected layers. Dropout is also employed to overcome overfitting. Tuned parameters are shown in Table 1. Finally, the hyperparameters are set as follow: the number of convolution layers = 1, the size of the convolution filters = [2, 4, 6, 8, 10, 12, 14, 16], the number of kernel in convolutional layer = 512, the number of units in first fully connected layer = 1024, dropout rate = 0.7, learning rate = 0.001, learning algorithm = Adam, batch size = 64. Figure 4 shows the architecture.
The CNN architecture of the one-hot encoding matrix encoding method
Excluding other ncRNA sequences using softmax probability threshold
As next-generation sequencing data such as small RNA-Seq data have become the major source of new miRNA discovery, useful miRNA search tools should be able to distinguish miRNAs from other types of ncRNAs, which usually co-exist with miRNAs in RNA-Seq data. Identifying miRNAs in RNA-Seq data is open set and thus any useful system must reject unknown/unseen classes in test set [40]. Existing binary classification tools often treat all the non-miRNA sequences as negative and need to choose non-miRNAs as the negative training samples. This often creates a highly unbalanced training set because there are significantly more non-miRNAs than miRNAs. In addition, it is not clear how to sample negative training sequences from many different types of ncRNAs. Our CNN model does not use an extra label for other ncRNAs. Instead, we reject out-of-distribution samples using the probability output of the softmax layer [44].
There are previous studies showing that the softmax probabilities of out-of-distribution samples are smaller than the probabilities of targeted samples [44]. Intuitively, out-of-distribution queries tend to produce a softmax probability vector with similar (small) values while an in-distribution query often yields a large softmax probability for one class. Thus, we will use carefully chosen softmax probability threshold to reject out-of-distribution samples, which in our case can be other types of ncRNAs in small RNA-Seq data. In addition, not all miRNA families are used in our training data. Any unseen miRNA families are also out-of-distribution samples. The softmax probability threshold should be used to reject them as well. We will use ROC curves to empirically choose a threshold.
We will first compare the classification accuracy of the two types of encoding methods. In particular, we will examine whether explicitly encoding the structural information in input matrices can improve the performance of miRNA classification. As real data such as small RNA-Seq data contain different types of transcripts, we will examine whether the softmax output can be used to reject non-miRNA sequences. Then, we will compare the performance of the CNN-based miRNA classification with other ncRNA classification tools.
Experimental data and pre-processing
For most of our training process, we use pre-miRNA families from Rfam as the training and testing data because we would like to compare our method with Infernal [12], which can conveniently use trained covariance models from Rfam. The current release of Rfam contains 529 pre-miRNA families and 215,122 precursor sequences. Another popular miRNA database is miRBase [24], which currently contains 1983 miRNA families from 271 organisms, including 38,589 pre-miRNAs and 48,860 mature miRNAs. In the experiment where we only use the mature miRNAs as the training data, we use miRBase because miRBase provides easy access to collect all the mature miRNAs.
We noticed that some of the pre-miRNA families in Rfam contain repeated sequences. Thus, in our pre-processing step, we will remove all the redundant sequences from the 529 pre-miRNA families in Rfam. As a result, 17.6% sequences were removed and 177,160 sequences were kept for downstream analysis. Each family contained different number of sequences (from 1 to 95,247) with different length. The distribution of the family size is shown in Fig. 5.
Rfam characteristics. Percentage of families in family size
To train in mini-batch, a fixed size of the input matrix should be set. Although there are a few pre-miRNA families with particularly long sequences, 96.88% miRNAs in Rfam were less than 200nt. Thus, we only keep the families with size at most 200nt. Although commonly seen pre-miRNAs are about 70nt, we did not exclude the long ones, such as those occurring in plant genomes, before pre-processing. The input matrix has size 200. All the shorter sequences were converted into 200nt sequences by inserting zero padding at the end. These padded zeros will lead to zero during the scanning of a convolution filter and thus won't affect the downstream layers after maxpooling.
Classification performance of probability and pair matrix
Following our definition of the probability and pair matrix, a threshold T is needed to decide the values of these matrices. In this experiment, we evaluate the change of T on the classification performance. At the same time, we also compare the performance of ordered and unordered pair matrices. These experiments were conducted using 30 randomly selected pre-miRNA families with at least 100 member sequences.
Considering that the probabilities may not be linearly distributed from 0 to 1, we sorted all the pairing probabilities (greater than 0.0001) of each miRNA sequence in Rfam and then used the values of different percentiles as the thresholds. The 0th, 10th, 20th, 30th and 40th percentile are selected; the corresponding values are 0.0001, 0.00487, 0.00772, 0.01307, and 0.02411.
For the 30 pre-miRNA families, 100 sequences were randomly selected from all member sequences. Then we used 5-fold cross validation so that there were 80 training sequences vs. 20 test sequences. CNN models with 30 classes are trained using different types of encoding methods. As there are 10 different types of matrices using 5 thresholds combined with two types of base pairs (ordered vs. unordered), 10 CNNs are trained. Note that the test sequences are encoded using the same method as the corresponding training data. We first compared the classification accuracy of using different thresholds with boxplot in Fig. 6a. For each threshold, there are 10 classification accuracy values for 5-fold cross validation results of both ordered and unordered cases. The comparison shows that allowing small base pairing probabilities yields higher average accuracy but also a slightly larger deviation. Overall, because of the higher average accuracy, we set the default threshold T as 0.0001 in all the following experiments. Figure 6b compares the classification accuracy of ordered vs. unordered matrices. The results show that they have very similar accuracy, with median accuracy around 0.92. By default, we use ordered base pairs in the pair matrix.
Performance comparison on classification accuracy using different secondary structure encoding methods. a 5 different thresholds (T) of base pairing probabilities. b ordered vs. unordered base pairs
Performance on pre-miRNAs classification
One-hot encoding matrix has been widely adopted for converting genomic data as inputs to deep learning models. Although it does not explicitly incorporate any structure information from the sequences, it has successful applications in protein homology search [29]. Thus, we will conduct a comprehensive experiment to compare the performance of one-hot encoding matrix and probability/pair matrix using pre-miRNA families from Rfam.
As different pre-miRNA families have different numbers of sequences, which can affect the performance of classification, we built 4 different datasets based on the size of families. Each dataset has different number of "classes" or "labels". The details about the four groups can be found in Table 2. Taken the Rfam-300 dataset as an example, there are 47 families in this dataset and each family contains 300 sequences (including 250 training sequences and 50 testing sequences). The model trained using this dataset needs to classify queries into one of the 47 families (or classes). We will compare the classification performance of CNNs on the four groups of training data and examine how the training set size affects the accuracy.
Table 2 Four groups of pre-miRNA families with different training set sizes
In order to quantify the prediction performance, we use two metrics: accuracy and F-score \(\left (F-score = \frac {2 \times Precision \times Recall}{Precision+Recall}\right)\). Classification accuracy quantifies the percentage of the correct predictions in all the test sequences. For each family, we also computed the recall \(\left (Recall=\frac {TP}{TP+FN}\right)\) and precision \(\left (Precision=\frac {TP}{TP+FP}\right)\). Here, TP, TN, FP, and FN correspond to the numbers of true positive, true negative, false positive, and false negative, respectively. The average F-score for all different families for one trained CNN is reported in Table 3. We evaluated the performance by the average accuracy of 5 independent experiments, each of which was measured with randomly selected testing sequences.
Table 3 Prediction accuracy(%) and F-score(%) of CNNs trained on families of different sizes
The results show that using one-hot encoding matrix led to much better performance than other methods even though it does not integrate base pairing information. In addition, it was less susceptible to the reduction of training data size. On the other hand, matrices focusing on base pairs need bigger training data to achieve better classification accuracy. These comparisons indicate that using one-hot encoding matrices is able to distinguish different types of miRNA families. One possible reason behind the inferior performance of using base pairing information is that all these pre-miRNA families have similar secondary structures and thus it is more difficult to conduct finer scale classification within the big family of miRNAs. For using one-hot matrix is less vulnerable to the decreased size of the training dataset, one possible reason is that one-hot matrix model has much fewer trainable parameters. For example, inputting the same sequence of length 200nt, one-hot model can update 4,485,255 parameters while the pair matrix model can update 78,748,399 parameters. Fewer parameters can help the model maintain high accuracy even if the training set is relatively small.
However, our additional experiments (next section) showed that these matrices cannot distinguish miRNAs from C/D box snoRNAs with high accuracy either, probably because of the similarity in the secondary structures, indicating that it is more difficult to train effective CNNs for matrices encoding base pairs. Larger training data are needed to improve the classification accuracy, which may not be always available for some miRNA families.
Use softmax probability threshold to reject other types of ncRNA sequences
Transcriptomic data such as small RNA-seq data can contain reads from other types of ncRNAs or miRNA families that are different from the many data. In this experiment, we will show that appropriate softmax probability value can be chosen as the threshold to distinguish targeted miRNAs from out-of-distribution samples.
As an example, we demonstrate the softmax output using the CNN model trained on Rfam-60 dataset (including 165 miRNA families). The positive set includes 155,392 test sequences from the Rfam-60 dataset while the negative (i.e. out-of-distribution) set contains all sequences from untrained miRNA families and randomly selected sequences from all other types of ncRNA in Rfam. There are 186,112 sequences in the out-of-distribution set. For each test sequence, the softmax layer will output a vector of normalized probabilities for all the 165 classes. The test sequence is assigned to the class with the the highest probability in the vector. We will set a threshold on this value so that a test sequence with maximum softmax output below this threshold will be rejected. We empirically determined the threshold by analyzing the distribution of the maximum softmax values for each input sequences.
We first plot the distribution of softmax values of the targeted miRNAs and other ncRNAs. Then we show the receiver operating characteristic (ROC) curve, which is constructed using false positive rate\(\left (FPR=\frac {FP}{FP+TN}\right)\) and true positive rate\(\left (TPR=\frac {TP}{TP+FN}\right)\) computed under different thresholds. Figure 7a and c show the distribution of the softmax probabilities for targeted miRNAs and negative samples. The comparison of (a) and (c) shows that using one-hot encoding matrix leads to smaller overlaps between the two distributions, which is consistent to the comparison of the ROC curves in Fig. 7b and d. Most of softmax values of the targeted miRNAs are greater than 0.9 and the area under the ROC curve for one-hot encoding matrix is very close to 1. By using one-hot encoding matrix, we can find an appropriate probability threshold to reject a majority of the negative samples (high precision) while still keeping targeted pre-miRNAs (high sensitivity). According to Fig. 7b, we choose the threshold leading to a large F-score. The default softmax value threshold for our trained CNNs is 0.977, with associated FPR of 0.05. Any test sequence with maximum softmax probability below 0.977 will be rejected.
Choosing appropriate softmax probability threshold to reject out-of-distribution samples.
We hypothesized that using pair and probability matrix cannot distinguish different pre-miRNA families because of their similar secondary structures. These matrices should thus be able to distinguish different types of ncRNAs with different secondary structures. Thus, we constructed a smaller negative data set containing tRNA, C/D box snoRNA, and other unseen miRNA families, including 20,000, 60,000 and 6,500 sequences, respectively. The secondary structure of tRNA is cloverleaf, which is very different from miRNA's hairpin structure. But the C/D box's stem box structure is somewhat similar to miRNA's. According to Fig. 8b, probability/pair matrix can distinguish tRNA from miRNA well, but still has difficulty rejecting C/D box snoRNAs. Considering that different types of ncRNAs might share globally or locally similar structures, pair and probability matrices have limited utilities in ncRNA classification.
Distribution of softmax values for unseen miRNAs, tRNAs, and C/D box snoRNAs. In both plots, the bin width is 0.01. (a) uses the one-hot encoding matrix model; (b) uses the pair matrix model
Directly classifying mature miRNAs
As many small RNA-seq datasets contain only mature miRNA, we evaluated whether deep learning could be used to directly classify mature miRNAs. As mature miRNAs in the same family can be well conserved because of their binding preference, using either mature miRNAs or pre-miRNAs as the training data may lead to similar classification accuracy for mature miRNAs. We again conduct the comparison using Rfam-60 set, where 50 sequences are used for training and 10 for testing. As we cannot conveniently obtain the mature miRNA annotation in the pre-miRNA families in Rfam, we downloaded the mature miRNAs from MiRBase. Thus, two CNN models are trained on pre-miRNAs and mature miRNAs, respectively. All the test sequences are mature miRNAs. For all the sequences, only one-hot matrix is used because of its superior performance. The mature miRNA classification accuracy of using pre-miRNAs and mature miRNAs as training data is 65.26% and 92.43%, respectively. Thus, when there are no reference genomes and read mapping cannot be used to identify possible pre-miRNAs, mature miRNAs should be used as training data for CNNs.
Performance on the input sequences with extra bases
Determining the exact boundary of pre-miRNAs in genomes is still challenging. For example, reads from small-RNA seq data can be mapped to reference genomes to identify possible mature miRNAs. Then those regions plus possibly mapped miRNA regions will be extended to identify candidate pre-miRNAs. The extension can go beyond the true pre-miRNA boundaries. Thus, we investigate whether having extra bases affects the classification accuracy. We still use Rfam-60 as our dataset, but 5, 10, 15 or 20 random nucleotides are added around each test sequence. The results can be found in Table 4.
Table 4 Classification performance on the test sequences with added bases
Comparison with other tools
In addition to the classification accuracy, the running time is also an important consideration for practical applications, especially when identifying miRNAs from next-generation sequencing data. Here, we compared the classification accuracy and running time of our trained CNNs with Infernal and miRClassify [45]. We also evaluated the performance of each method as the number of miRNA families (i.e. classes) increased. Four testing dataset were constructed by randomly selecting 1000 sequences from Rfam-300, Rfam-120, Rfam-60, and Rfam-30 respectively. Note that all these testing sequences are chosen from the set excluding training sequences and thus have no overlap with the training data for our CNN models. This experiment was repeated for five times and the average performance was reported in Table 5. The variance of each experiment in one-hot matrix method and Infernal is very small (less than 5e- 3). And for the miRClassify, the variance is slightly bigger and the biggest variance is 0.02. In order to run Infernal, we directly downloaded the covariance models associated with the corresponding dataset from Rfam. Thus, it is possible that some of these test sequences were used for training the covariance models. MiRClassify uses a hierarchical random forest model to classify the miRNAs into different families. The models of MiRClassify were downloaded from their website and they were constructed from miRBase version 16.0.
Table 5 Comparison with Infernal and miRClassify
To ensure a fair comparison in the running time, we used single core for all the three tools because miRClassify is single-threaded. For Infernal, we set the option '–cpu' as 1. All other options for Infernal are the default parameters. The command is:
>cmscan –cpu 1 rfam_60.cm rfam_60.fa
Here, 'rfam_60.cm' contained all the required covariance models and 'rfam_60.fa' is the test sequence set. For each query sequence, Infernal might generate several hits. In that case, we only kept the one with the lowest E-value. CNN model was implemented by Keras so we added extra commands to make sure only one core was used. In addition, the mini-batch size used in CNN was 64. Table 5 summarized the results.
The result in Table 5 shows that despite the possible overlaps between training and testing data for Infernal and MiRClassify, our trained CNN models still have high accuracy with minimum running time. We then conducted the χ2-test between the 20 accuracy values output by the three methods. The p-value between the one-hot matrix method and Infernal was very close to 1 (0.999), indicating that their accuracy is comparable. On the other hand, the p-value between ours and miRClassify is 4.59e- 275. The running time comparison also shows that Infernal took more time as the number of families increased. The other two methods were not affected by the number of families.
Frequently activated filters represent part of mature miRNAs
To interpret why the one-hot encoding method performed well, we visualized some motifs extracted by our CNN model. Employing the method used in DeepFam [29], we utilized the most frequently activated filters in trained Rfam-300 model to extract motifs from the RF00247 training sequences. We compared the motifs obtained by CNN with the motifs produced by MEME on training sequences, as shown in Fig. 9. Because the convolution layer used filters of different sizes, this model can identify motifs with various lengths. We found that the identified motifs represented part of the mature miRNA. We tested other families and had the same observation. This is consistent to the findings by DeepFam.
Visualizing and comparing the motifs extracted by MEME [46] and CNN model in RF00247. (a) Motifs extracted by MEME and CNN and the corresponding convolution filter of length 8. (b) Motifs extracted by MEME and CNN and the corresponding convolution filter of length 16. (c) The secondary structure of RF00247 with highlighted mature miRNA
We evaluated and compared the classification performance using different encoding methods and CNN architectures. Based on the experimental results, simple one-hot matrix performed much better than other encoding methods that explicitly incorporate predicted secondary structures. This could be caused by similar secondary structures among different types of pre-miRNA families. As shown by Do et al. [37], it is possible that encoding secondary structures will benefit distinguishing miRNAs from other ncRNAs in the binary classification problem.
In practice, input data such as small RNA-Seq can contain sequences from other types of ncRNAs. Useful miRNA classification must be able to reject out-of-distribution samples. Our experiments demonstrated that using softmax output can achieve an optimal trade-off between sensitivity and precision in distinguishing targeted miRNAs from other sequences. Thus, the designed classification models are practically useful in conducting finer scale miRNA analysis. By comparing our tool with a general ncRNA classification tool Infernal and also another machine learning based miRNA classification tool, we conclude that ours can achieve high sensitivity and accuracy with significantly reduced running time.
In this work, we developed CNN-based classification models for identifying different types of miRNAs. By using the output of the softmax probability as a threshold, our model can reject other types of ncRNAs and out-of-distribution miRNAs with high precision. Comparing with two existing methods, our one-hot encoding method takes much less time and still has high accuracy.
Although this work only concerns miRNAs, the trained CNNs can be extended to classify other types of ncRNAs. The method holds the promise to achieve comparable performance while achieving significant speedups compared to Infernal. It is our future work to extend and optimize our model for other types of ncRNAs.
The source code and datasets used during the current study are available at https://github.com/HubertTang/DeepMir
CNN:
Convolution neural network
FPR:
False position rate
LSTM:
Long short term memory
MFE:
Minimum free energy
pre-miRNA:
precursor microRNA
pri-miRNA:
primary microRNA
rRNA:
ribosome RNA
ROC:
Receiver Operating characteristic
tRNA:
transfer RNA
TPR:
True positive rate
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About this supplement
This article has been published as part of BMC Bioinformatics Volume 20 Supplement 23, 2019: Proceedings of the Joint International GIW & ABACBS-2019 Conference: bioinformatics. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-20-supplement-23.
This work and the publication costs were supported by City University of Hong Kong (Hong Kong, China SAR) project 7200620. The funding did not play any role in design/conclusion.
Department of Electronic Engineering, City University of Hong Kong, Kowloon Tong, Hong Kong SAR
Xubo Tang & Yanni Sun
Xubo Tang
Yanni Sun
YS initiated the project. Both YS and XT designed the methods. XT conducted the experiments. Both YS and XT contributed to the writing of this manuscript. Both YS and XT read and approved the final manuscript.
Correspondence to Yanni Sun.
Tang, X., Sun, Y. Fast and accurate microRNA search using CNN. BMC Bioinformatics 20, 646 (2019). https://doi.org/10.1186/s12859-019-3279-2
Convolution neural network (CNN)
Open set problem | CommonCrawl |
Uncovering the internal structure of Boko Haram through its mobility patterns
Rafael Prieto Curiel1,2,
Olivier Walther3 &
Neave O'Clery4
Boko Haram has caused nearly 40,000 casualties in Nigeria, Niger, Cameroon and Chad, becoming one of the deadliest Jihadist organisations in recent history. At its current rate, Boko Haram takes part in more than two events each day, taking the lives of nearly 11 people daily. Yet, little is known concerning Boko Haram's internal structure, organisation, and its mobility.
Here, we propose a novel technique to uncover the internal structure of Boko Haram based on the sequence of events in which the terrorist group takes part. Data from the Armed Conflict Location & Event Data Project (ACLED) gives the location and time of nearly 3,800 events in which Boko Haram has been involved since the organisation became violent 10 years ago. Using this dataset, we build an algorithm to detect the fragmentation of Boko Haram into multiple cells, assuming that travel costs and reduced familiarity with unknown locations limit the mobility of individual cells.
Our results suggest that the terrorist group has a very high level of fragmentation and consists of at least 50–60 separate cells. Our methodology enables us to detect periods of time during which Boko Haram exhibits exceptionally high levels of fragmentation, and identify a number of key routes frequently travelled by separate cells of Boko Haram where military interventions could be concentrated.
Boko Haram is one of the deadliest armed organisations in recent history. Since the Jihadist group became violent in 2009, it has caused nearly 40,000 casualties and displaced 2.4 million people around Lake Chad, an impoverished region divided between Nigeria, Niger, Cameroon and Chad in West Africa (UNHCR, 2019). Boko Haram has adopted a strategy of violence against Sufi and Salafi religious movements, traditional leaders, the wider civilian population, and the Nigerian state, which the organisation regards as corrupted and illegitimate (Matfess 2017). The organisation, which declared its own "state among the states of Islam" and sworn allegiance to the Islamic State in March 2015 (Pieri 2019), adheres to a literal interpretation of the religious texts of Islam and enforces a strict adherence to religious law. Its goal is to overthrow secular governments, cut their ties with the West and destroy the social and political order of the Lake Chad region.
Over the years, Boko Haram has been torn apart by internal rivalries that have their origins in the balance of power between the various leaders and factions that compose the main organisation (Zenn 2019). Boko Haram is now split between a faction led by Abubakar Shekau that controls parts of Borno State around Gwoza and the Cameroon-Nigeria border, and another faction led by Abu Mus'ab al-Barnawi, that is mainly active in the islands of Lake Chad, West of Maiduguri and along the Niger border in the Diffa region (Seignobos 2017).
Yet, due to the secretive nature of Boko Haram, the internal structure of the organisation remains largely unknown. Of particular importance is whether Boko Haram is a centralised organisation structured around a few key leaders or a network of decentralised cells (Anugwom 2018). Centralised organisations in which decisions and resources flow from the top down are theoretically more efficient than decentralised ones but also less resilient to counter-terrorism measures (Cunningham et al., 2016). Decentralised organisations in which individual cells are relatively independent from the core are more difficult to dismantle but also much more challenging to coordinate than centralised ones (Everton 2013; Price 2019). The issue of whether Boko Haram fighters tend to operate locally or travel extensively between their historical bases in northern Nigeria and their new sanctuaries in neighbouring countries also remains under-explored. Terrorist organisations capable of coordinating attacks over long distance are a much greater threat to African states and the international community that local organisations whose attacks are isolated in one particular region (Walther et al., 2020).
The paper proposes a novel technique to uncover the internal structure of Boko Haram, based on the sequence of events in which a terrorist cell takes part, using disaggregated data from the Armed Conflict Location & Event Data Project (Raleigh et al. 2010) on political violence in the region. We develop an algorithm to detect the fragmentation of Boko Haram into several cells, assuming that travel costs and reduced familiarity with unknown locations limit the mobility of the organisation. Shedding light on both the social structure and spatial organisation of Boko Haram, our analysis suggests that Boko Haram has a very high level of fragmentation and consists of at least 50–60 separate active cells. The method also identifies a number of key routes frequently travelled by separate Boko Haram cells, including international border crossings, where military interventions could be concentrated.
Complex networks and quantitative models of crime and terrorism
Network analysis can yield powerful insights into the latent structure of spatial and temporal data, as it is often the case with violent events (Yuan et al. 2019). Yet, as noted by Malcolm Sparrow in one of the earliest studies of crime and network analysis: "It would be enormously gratifying, therefore, if we could simply throw the existing network analysis toolkit at criminal intelligence databases, and come away with a set of valuable new insights. Of course it is not that easy" (Sparrow 1991). One of the main challenges of network-based studies of crime and terrorism is usually data incompleteness, dynamic behaviour (Gera et al. 2017) and the fact that "dark" networks tend to be covert and illegal (Bakker et al., 2012; Raab and Milward 2003; Gerdes 2015), which makes the identification of key nodes and links more difficult than with other networks.
Despite potential "hidden" data limitations, network-based studies of crime and terrorism have rapidly expanded since the beginning of the 2000s due to the availability of new data sources and the development of complex networks and quantitative models. Spatial networks, which are usually constructed by connecting crimes and potential criminal's address or connecting pairs of crimes (Oliveira et al. 2015) have helped identify crime pattern motifs (Davies and Marchione 2015), and have been used to predict crime, considering a street network (Rosser et al. 2017). The analysis of social networks has expanded to study organised crime networks, drug production (Malm et al., 2008), cybercrime and extremist networks (Morselli 2013). Social networks have also been used to model the diffusion of fear of crime as a reaction to direct and indirect victimisation (Prieto Curiel and Bishop 2017), providing a potential explanation as to why fear of crime can increase even if crime rates are being reduced (Prieto Curiel and Bishop 2018).
Networks are also increasingly used to visualise, model and counter terrorist organisations (Bakker et al., 2012; Krebs 2002; Carley 2006). The study of terrorist social networks usually looks at the network topology and identifies which actors are the most central (Everton 2009). Extant literature shows that terrorist organisations tend to find a balance between efficiency and security (Gerdes 2015; Morselli et al., 2007). Centralised networks, such as the Provisional Irish Republican Army (IRA) in Northern Ireland, are theoretically more efficient than decentralised ones but also less resilient to external threats, while decentralised networks are more difficult to detect and disrupt but also much less efficient at communicating resources and orders (Chuang and D'Orsogna 2019; Price 2019).
Beyond the social dimension of terrorism, space is now recognised as a fundamental dimension of both criminal and terrorist networks (Radil 2019; Bahgat and Medina 2013; Medina and Hepner 2008). Space provides the physical framework upon which crime and terrorist attacks are conducted. It shapes the strategies of covert organisations by acting as a facilitating or constraining factor in their fight against government forces or civilian populations. Geographical distance plays a critical role, as attacks are frequently executed near important areas or the city centre (Savitch 2014). Therefore, a frequent approach in terrorism studies is to detect spatially dependent events and self-reinforcing hotspots (Bahgat and Medina 2013). This approach focuses on how different events are linked or how spatial proximity can influence the formation of social networks (Skillicorn et al. 2019).
Another approach is to use exponential random graph models to explore the spatial and social network causes of violence. In Africa, recent research using exponential random graph models suggests that rebel groups whose turfs overlap are more likely to fight each other (Cunningham and Everton 2017). Space can also enable criminal and terrorist organisation to spread geographically by using border regions as sanctuaries (Arsenault and Bacon 2015), as in the Lake Chad region today (Walther et al., 2020).
Additional variables can be added to shed light on the social and spatial dynamics of terrorist networks, including ideology, tactics, weapons, targets and active regions (Gera et al. 2017; Campedelli et al., 2019a, b). A recent analysis of the terrorist attacks which occurred from 1997 to 2016 around the world shows, for instance, that groups with opposite ideologies can share very common behaviours (Campedelli et al., 2019a, b). In recent years, particular emphasis has been given to radical Islamist organisations, whose structure has been found to be resilient even if important social nodes were removed (Medina 2014). In West Africa, network studies have shown that Islamist organisations were capable of travelling long distances (Skillicorn et al. 2019), relied on a limited number of key brokers able to establish links with other rebel groups (Walther and Christopoulos 2015), and had a destabilising effect on regional political stability (Dorff et al., 2020).
Internal structure and mobility patterns of Boko Haram
Boko Haram (which means "Western education is a sin") is the name given by external observers to the organisation founded by Mohammed Yusuf in 2002. The organisation has used several different names since the Nigerian police killed Yusuf in Maiduguri in July 2009. From 2010 to March 2015 and from August 2016 onwards, the organisation led by Yusuf's successor, Abubakar Shekau, was known as Jama'at Ahl al-Sunna li-l-Da'wa wa-l-Jihad (Group of the People of Sunnah for Preaching and Jihad). The organisation adopted the name Wilayat Gharb Ifriqiya (Islamic State in West in Africa Province, ISWAP) after it pledged allegiance to the Islamic State under the leadership of Shekau in March 2015. In August 2016, Islamic State announced that it had appointed Abu Mus'ab al-Barnawi as the new leader of ISWAP.
There is little agreement as to the organisational structure of Boko Haram. For some scholars, Boko Haram is a "centralized and nominally unified organization" in which Abubakar Shekau exercises a high degree of strategic and operational control (Zenn 2019). According to this perspective, Shekau's ruthless leadership allowed him to build a strongly unified organisation in which opponents were either killed, expelled or forced to follow his orders. While not particularly effective in winning battles and holding territories, this centralised leadership was instrumental in limiting the number of splinter groups, with the exception of the short-lived group Ansar al-Muslimin fi Bilad al-Sudan, better known as Ansaru, founded in 2012 and largely dormant since 2013 (Zenn and Pieri 2018).
Another strand of literature argues that Boko Haram is "organised under a loose federation of operating cells under the broad umbrella headship of the Islamic standard 'Shura Council'" , a consultative assembly (Anugwom 2019). According to this view, Boko Haram operates more as "a collection of loosely linked cells and bands than as a tightly disciplined hierarchical army" (Thurston 2017). For some authors (Weeraratne 2017), Boko Haram has adopted a "cell-like structure" since the execution of its leader Muhammad Yusuf in 2009. This structure, in which individual cells maintain little direct contact with the central leadership, allows local and regional commanders to enjoy a significant level of autonomy in their operations against governmental and civilian targets. The number of decentralised cells that composes Boko Haram, however, remains a matter of speculation. Local informants report that while Boko Haram is divided internally, "no one can pinpoint precisely how many these cells are and how far connected to the apex leadership these were" (Anugwom 2019). Fragmentation, however, has a cost as different cells might antagonise and compete against each other (Chuang and D'Orsogna 2019).
Boko Haram is known for its high mobility. Since it became violent in 2009, the organisation has been able to conduct an average of two attacks each day, taking on average the lives of nearly 11 people daily. The Boko Haram insurgency, which initially focused on cities, has mainly been active in rural areas since 2013, where it relies on cheap Chinese motorcycles to conduct its attacks (Agbiboa 2019). The move to rural areas has allowed Boko Haram to challenge the Nigerian military and to exploit agricultural and natural resources around Lake Chad. While Boko Haram had focused its attacks on northeastern Nigeria until 2014, increasing pressure from government forces and vigilante groups has led the terrorist organisation to conduct an increasing number of attacks in neighbouring Chad, Cameroon and Niger. Focusing on the organisation's diffusion across the region, Dowd (Dowd 2017) shows, for example, that Boko Haram has contracted subnationally, suggesting that the organisation is relocating to neighbouring countries instead of expanding. The mobility patterns that sustain these attacks remain largely under-reported.
Thus far, the debate on the organisational structure and mobility of Boko Haram primarily relies on qualitative data collected through interviews with former members of the Jihadist organisation, evaluation of tactics, court transcript, letters written between Boko Haram commanders and other extremist organisations, and propaganda videos (Kassim and Nwankpa 2018). Studies using quantitative approaches to detect and describe the social networks and spatial patterns of Boko Haram have mainly focused on relationships between the organisation and its enemies rather than on its internal dynamics (Walther et al., 2020). An exponential random graph model approach has shown that the emergence of Boko Haram in northern Nigeria led to an increase in the number of conflicts, even between pairs of actors that did not include Boko Haram (Dorff et al., 2020). Finally, some attempts have been made to create a multi-layer network of Boko Haram based on open-source data that includes shared events, collaborations, membership and financial ties (Cunningham 2014). That network is extremely sparse due to its relatively young cell-like structure and its lack of collective leadership (Gera et al. 2017).
Due to the secretive nature of terrorist groups, the internal structure of Boko Haram and whether it is a centralised organisation is still unknown. Whether Boko Haram cells tend to operate locally or have a high degree of mobility also remains under-explored. And, in that vein, paths which are frequently travelled by Boko Haram members and whether international borders work as frictions to the group or as safety structures is still an open question with potential policy implications.
The method used in our paper to understand the internal structure of Boko Haram differs from existing approaches. Building on a comprehensive dataset that includes all violent events in northern Nigeria and the neighbouring countries since 1997, we provide an estimate of the fragmentation of Boko Haram based on an agent-based model that identifies cells which move between Boko Haram events (Epstein 2002; Moon and Carley 2007; Park et al. 2012). Our approach requires two input parameters (the maximum cell speed and distance between events), whose impact on the results of the model (e.g., the number of cells detected) is analysed.
To analyse the mobility of Boko Haram cells, the locations of events are clustered and a spatial undirected weighted network is constructed based on those clusters, which captures how violent events are spatially linked and how cells move between different locations.
Our study uses data from the Armed Conflict Location & Event Data project (ACLED) (Raleigh et al. 2010). To date, ACLED has recorded approximately half a million individual events and contains information about all reported political violence and protest events across Africa, South and Southeast Asia, the Middle East, Europe, and Latin America, mainly from local and regional media, reports from NGOs and social media accounts. Reports are separated into individual events that took place in different locations, have different types of violence, and involve different actors. For each event, the dataset records the date, actors, types of violence, locations, fatalities, and it also includes a space and time precision estimate.
All events in which Boko Haram was involved as an actor or associate actor were selected from the ACLED dataset including all Boko Haram factions, which in total gives 3795 events. Because our goal is to analyse the most recent mobility patterns of Boko Haram, a small number of isolated events involving Boko Haram before May 21st, 2012 were excluded from the analysis. This is the only filter applied to the 3795 events, and it removes 29.8% of the days since the first Boko Haram event but only drops 7.3% events. Two major events were dismissed, however: the July 2009 uprising of Boko Haram in Maiduguri against the police and military which resulted in 800 casualties, and the suicide attacks that took place in Kano in January 2012, which resulted in 185 casualties. Other events during the omitted period were less violent and resulted in fewer casualties.
In total, our dataset comprises 3,517 events and 36,775 casualties recorded by ACLED from May 2012 to May 2019, which represents 92.7% of the events and 94.4% of the total casualties attributed to Boko Haram since 2009.
Algorithm to detect fragmentation
Boko Haram has been most active around Lake Chad, a swampy region which has lost 90% of its surface water since 1960 (Policelli et al. 2018; Itno et al. 2015). The road infrastructure around the Lake and in northern Nigeria is in very poor condition, which results in limited, slow or costly mobility. Due to the lack of roads, it takes nearly 10 hours and 600 road kilometres to travel between Maiduguri (Nigeria) and Bol (Chad), two cities located on opposite sides of Lake Chad and only separated by 250 kilometres as the crow flies. It is roughly the same linear distance as between Lagos and Benin City, two Nigerian cities that can be travelled in 5.2 hours by road.
Some authors have argued that Boko Haram intensifies its attacks in rural areas during the rainy season (June–September), a period during which the mobility of government forces is limited by water-logged roads (Agbiboa 2019). ACLED data does not confirm this assumption. The highest number of events is recorded in January (with nearly two events each day of the month since 2016) and the highest number of casualties is recorded in February (with 12.3 casualties each day of the month since 2016), during the dry season.
Since 2014, there has been at least one Boko Haram event in 75% of the days and in 92% of any two consecutive days. If a single Boko Haram group (which we call a "cell") was responsible for all of these events, they would have travelled on average 216 kilometres each day for the past 7 years, the equivalent of travelling around the Earth twice each year. Since this is highly unrealistic and improbable, we assume that Boko Haram is fragmented into an unknown number of cells responsible for the observed patterns of attacks in the region.
Our model (algorithm) for constructing different Boko Haram cells is based on the principle of least action which assumes that the mobility of Boko Haram is constrained by environmental (distance, lack of roads) and security factors (presence of government forces) that reduce familiarity with unknown locations and limit the impact of its attacks. Boko Haram events are analysed in sequential order in a manner similar to that used previously to detect crime pattern motifs (Davies et al. 2016). Specifically, the algorithm assesses each event, assuming that cells move as little as needed. The first event is assigned a cell. The location and the date of the event is considered to be the last known location of that cell. For each subsequent event:
If the event takes place at a "reasonable distance" and within "reasonable time" from the last known location of a cell (from the set of existing cells), then we assume that the cell has moved between the two locations and is also responsible for the event. The location and time of the cell is updated. If the event could have been conducted by multiple cells, then one is selected at random.
However, if the event takes place either too far away or too soon after the last event (from the set of existing cells), then we assume that the event was conducted by a different cell. Hence, a new cell is created.
This approach thus also uses the principle of least group size (Thelen 1949), which assumes that if Boko Haram had more cells, it would be capable of committing more attacks and with a higher frequency than is observed.
In order to quantify "reasonable distance" and "reasonable time", let di, j be the distance between events i and j and ti, j the number of days between them. Let ν > 0 be the maximum daily speed of a cell (in kilometres per day) and let μ > 0 be the maximum distance between two consecutive events (in kilometres) such that if:
$$ \frac{d_{i,j}}{t_{i,j}}>\nu, $$
or if
$$ {d}_{i,j}>\mu, $$
we assume that the two events were executed by a different cell. In other words, Equation (1) restricts the maximum daily speed of a cell (ν), and Equation (2) restricts the total distance that a cell can move between two consecutive events (μ). Figure 1 illustrates the cell assignment process outlined above.
Schematic representation of the methodology. Events are analysed in sequential order and a unique Boko Haram cell is assigned to each one. For each event, the algorithm decides if an existing cell is involved in the attack or if a new (or not previously identified cell) is responsible. In the figure, an event took place during the first day, which means that a cell is created. The location of that event is its last known location and the date of the event is its last known date. The potential location of that cell increases each day according to its daily speed, . After a few days (four in the example) a cell has reached the maximum distance between consecutive events and so it is assumed that it remains within that region (in the example, μ = 4 ν). Then, during days 2 and 4, there is no cell nearby who could have been involved in the new events and so new cells are identified. During day 4, there is an event for which an existing cell is potentially responsible, so its last known location and date are updated
Since Boko Haram attacks spans over 10 years, we presume that some of its cells will disappear, either because its members are killed or unable to coordinate their activities any longer. We therefore assume that a cell which has not been active for 1 year has dissolved and is no longer responsible for any future events. We also treat the main known Boko Haram factions identified by ACLED (Barnawi and Shekau) separately in our analysis. We assume that Barnawi cells do not take part in Shekau's events and Shekau's cells do not take part in Barnawi events.
The total number of cells, Tτ(ν, μ) which counts all cells which existed up to time τ, and the active number of cells, Aτ(ν, μ) which counts only the ones that are still active at a certain time τ, are identified and reported, as a function of the parameters ν and μ. In the example of Fig. 1, four events lead us to identify three cells. We write T2019(ν, μ) and A2019(ν, μ) to represent the latest known number of cells and active cells for some values of ν and μ and Tτ(ν, μ) and Aτ(ν, μ) if the period under consideration is different.
Parameter space and sensitivity analysis
The restrictions of maximum distance that a cell could have moved (μ), and their maximum daily speed (ν) are input model parameters. The range of what it is considered to be a "reasonable" daily speed and maximum distance is thus the parameter space. We consider that a cell can move at a maximum daily speed of up to 200 kilometres per day (and so values of ν range between 0 and 200) and the distance between any two consecutive events is, at most, 400 kilometres (and so values of μ range between 0 and 400).
Notice that with very large values of ν and μ, we get cells that could be "almost everywhere" as they move very fast and over long distances. This results in a small Tτ(ν, μ) and Aτ(ν, μ) since the same cell could have been responsible for most of the events (except for the ones which happen simultaneously). With μ = 0 or ν = 0, we obtain cells with no mobility and so, except for events which took place in the same location, the procedure assigns a different cell to each unique location. In that case, we get that T2019(0, 0) = 900, which means that Boko Haram has been active in roughly 900 unique locations, and that A2019(0, 0) = 233, meaning that they have been active only in 233 different locations during the past year and so many cells would be considered to be dissolved by now. Different values of ν and μ yield different numbers of total and active cells. We analyse Tτ(ν, μ) and Aτ(ν, μ) to illustrate the impact of the two parameters.
Our model consists of two parameters, ν and μ. The parameter space, which corresponds to values of the maximum distance between two events, μ between 0 and 400 kilometres and values of maximum daily speed, ν between 0 and 200 kilometres per day, was analysed first, by randomly choosing a value of ν and μ and then analysing the consecutive Boko Haram events as described in the text. This procedure was computed 100,000 times for different values of ν and μ before the corresponding T2019(ν, μ) and A2019(ν, μ) were reported. Also, since we are interested in detecting when has Boko Haram been more or less fragmented, we also computed Tτ(ν, μ) and Aτ(ν, μ) for values of τ from 2012 to 2019, for some fixed values ν and μ.
Spatial network of Boko Haram events
Although it would be possible to observe the mobility of cells by looking directly at the location of their corresponding events, the spatial grouping of locations into n clusters enables us to consolidate very short-distance movements. It also limits the possible journeys between distinct locations by n(n − 1)/2 and make it possible to analyse the most frequent journeys. Note that the construction of the network depends on our choice of parameters μ and ν. In other words, we will get a different network for alternative choices of μ and ν.
Event locations were clustered into nodes using Partitioning Around Medoids (Reynolds et al. 2006) (a procedure similar to K-means) with the restriction that locations inside a node are at a distance smaller than 20 kilometres. The result is a spatial network with 420 nodes: 294 of the nodes (70%) are in Nigeria, 80 nodes (19%) in Cameroon, 27 nodes (6%) in Niger and 19 nodes (5%) in Chad. Each event is assigned to its corresponding medoid. The medoids (or the nodes of the network) are located such that 99.4% of the events occurred in the same country as the corresponding medoid (except for 23 events where the medoid is located in a different country than the event).
We examine specific parts of the parameter space. To do so, we take pairs of values of ν0 and μ0 and selected all the realisations for which the values ν and μ are close to ν0 and μ0. Formally, from all the realisations, if ∣ν − ν0 ∣ < 3.5 kilometres per day and if ∣μ − μ0 ∣ < 3.5 kilometres, a realisation is considered to be "close" and it is used to construct the spatial network around ν0 and μ0. Instead of assuming that one realisation is the "true" network for a set of parameters ν0 and μ0, we consider many realisations with a slight parameter change, in case a small perturbation changes the structure of the network completely. For a specific set of parameters ν0 and μ0, the link ij is added to the network if our algorithm introduced above detects that a cell moved from node i to node j or from j to i. The corresponding weight of the edge is the number of journeys that is made by any cell in the set of realisations around ν0 and μ0 between i and j or between j and i (more details on the Supplementary materials) 5.1.
Therefore, the edge weights wij are the likelihood of one journey between i and j undertaken by a Boko Haram cell with maximum distance μ0 and daily speed ν0. We measure the percentage of trips completed inside the same node, the percentage of trips which happen within the top 1% of the edges and the percentage of present edges for different values of μ0 and ν0.
Boko Haram, a highly fragmented organisation
The results of our mobility pattern analysis suggest that Boko Haram is a highly fragmented terrorist organisation. The estimate of the number of cells depends on whether we believe that Boko Haram is rather mobile or not: highly mobile cells are capable of committing more attacks than immobile ones. If a high mobility scenario is selected, then there are at least 40 active cells in 2019 (Fig. 2). If a low mobility scenario is selected, then Boko Haram should have at least 150 active cells. An analysis of the total number of cells and the ratio between active and total cells in the parameter space is in the Supplementary materials 5.2.
The number of Boko Haram active cells A2019(ν, μ), varies depending on two model input parameters: their daily speed ν and their maximum distance between two events μ. The smallest number of active cells is obtained when each cell travels ν > 90 kilometres each day and μ > 250 kilometres between consecutive events, which appears unlikely, considering the poor road conditions in the region. A more realistic hypothesis is that cells travel at most ν = 60 kilometres each day and μ = 180 kilometres between every pair of events, which would mean that Boko Haram is fragmented in roughly 53 active cells and 83 total cells
Are Boko Haram cells specialised?
Very few datasets, besides ACLED, can be used as a source of validation of these results. Measuring the mobility of Boko Haram cells, estimating their daily speed and the maximum distance between events is almost impossible due to the risks of doing fieldwork in the region. Although mobility studies have rapidly evolved due to the development of new techniques and the use of new sources of data, such as mobile phone data (Wilson et al. 2016; Widhalm et al. 2015; Schneider et al. 2013) or credit card data (Clemente et al. 2018), this type of data simply does not exist for Boko Haram. Similarly, there is too little evidence about the internal structure of Boko Haram to validate the number of groups that we observed, apart from the fact that the organisation has adopted a "cell-like structure" (Weeraratne 2017), and some speculation around the number of decentralised cells (Anugwom 2019).
Due to the lack of an alternative validation exercise, we analyse whether some of the cells are more violent than others, or more specialised on certain types of events, as might be expected. We propose that evidence of such behaviour increases our confidence in the set of cells identified.
Boko Haram has participated in six main types of events. The majority of attacks (41.3%) are classified as armed clashes against state actors. Roughly a third (29.7%) of the attacks are committed against civilians, 8.9% are suicide bombs, 6.1% relate to governmental territorial gains, 4.4% are remote explosives and 3.7% are air or drone strikes. Sexual violence, abduction, violent demonstrations and other violent events are far less represented.
A metric of specialisation S for different regions in the parameter space ν, μ is constructed in order to measure how homogeneous or heterogeneous Boko Haram cells are. For each value of ν and μ, we look at the distribution of events by type within each cell. We then use the "distance" between the distribution of events by type of the most specialised cell and the distribution of events by type across all events (for more details, see the Supplementary materials 5.3).
Our results suggest that most specialised cells correspond to those found with parameters ν = 60 kilometres per day and 180 kilometres each day, and between μ = 170 and 200 kilometres between two consecutive events (Fig. 3). The number of cells Tτ(ν, μ), and active cells Aτ(ν, μ), in that parameter range is roughly the same, which means that it is likely that Boko Haram cells have a daily speed just below ν = 60 kilometres, and that they move around μ = 180 kilometres between any two events. These estimates are based on the hypothesis that mobility is reduced by environmental and security factors, that cells are specialised in certain types of attacks, and that the highest level of specialisation in the parameter space is an indication of the accuracy of our method. In what follows, we present results for the whole parameter space with a slight emphasis around the ν = 60 kilometres each day, and μ = 180 kilometres values.
Metric of specialisation S(ν, μ) of Boko Haram cells according to their speed ν and their maximum distance between two consecutive events μ. Cells are more specialised if they commit more attacks, armed clashes, suicide bombs or other types of events than expected. The region inside the white frame had the highest level of specialisation of the whole parameter space
Finally, the number of casualties per event was also considered as a potential way to distinguish between cells. The idea is that some cells could be more violent than others. Results show, however, that the number of casualties per cell is proportional to the number of events except for the most deadly events. This result is in line with the terrorism literature that suggests that the casualties or severity of terrorist events follows a power-law distribution (Clauset et al., 2007; Guo 2019). In the case of Boko Haram there is indeed a high concentration of casualties in some events. The 1% most violent events have caused 24% of the total casualties while the top 5% and top 10% events caused 47% and 61% of the casualties respectively (see the Supplementary materials 5.4 for further discussion on the number of casualties per cell).
Boko Haram has been restructured a few times
For a specific set of parameters ν0, μ0, the method suggests that Tτ(ν0, μ0) and Aτ(ν0, μ0) are not constant for different values of τ, that is the number of cells and the number of active cells changes according to the context. Particularly, the method shows that both Tτ(ν0, μ0) and Aτ(ν0, μ0) increased rapidly since the early 2010s, particularly in 2013, 2015 and 2019, 3 years during which Boko Haram has experienced internal changes (Fig. 4) and these breaking points are observed across the whole parameter space.
With low mobility, either small ν or small μ (left part) there is a very large number of total cells (more than 150 cells) and roughly half of them are still active in 2019. With high mobility, with large ν and μ (right part), there is a smaller number of total cells and the number of active cells in 2019 is roughly 70% of the total cells. Notice that with low, medium or high mobility, results show that in early 2015 (shaded year) there is a structural change in the structure of Boko Haram, as there is an increase of more than 30% in the number of active cells
In 2013, Boko Haram expanded its activities to neighbouring countries, committing a number of attacks and kidnappings often associated with its splinter group Ansaru (Zenn 2014). This period is synonymous with major internal tensions between Boko Haram leader Shekau and two of his senior commanders, Khalid al-Barnawi and Mamman Nur, who condemn Shekau's strategy of indiscriminate violence against Muslim civilians and defectors. The year 2015 is another turning point in the war against Boko Haram. After several years of unsuccessful counter-insurgency operations, the Nigerian forces launched a series of attacks with the Multinational Joint Task Force (MNJTF), a regional initiative from Benin, Cameroon, Chad, Niger, and Nigeria, against the terrorist organisation. Boko Haram was defeated in a number of strategic locations and pushed back to remote or mountainous regions, around Lake Chad and the Cameroon border (Zenn 2019). The new cells observed in 2019 have been linked to the Barnawi faction of Boko Haram, which became active in 2016, but has committed an increasing number of attacks recently.
Some Boko Haram cells are more active than others
Our results suggest that a large fraction of Boko Haram events are committed by a few cells. If we assume that Boko Haram cells are highly mobile, with a speed of ν = 60 kilometres per day and a maximum distance of μ = 180 kilometres, then 12 cells are responsible for 70% of the events. Even with a low mobility scenario, 30 cells concentrate 70% of the events (Fig. 5). However, our model does not indicate that the cell of Boko Haram leader Shekau is significantly more active or more deadly than other highly active cells.
Each one of the 3,517 Boko Haram events are assigned to a unique cell. For each subfigure, the horizontal axis represents the 3,517 events, grouped by their corresponding cell, sorted in decreasing order according to their number of events. Wider (and taller) bars represent cells with a higher number of events
Boko Haram attacks are clustered in a few regions
The network formed by the mobility patterns of Boko Haram is very sparse. If we construct a composite network formed of cells generated for a range of μ and ν, only 4% of potential edges are actually present. This means that most pairs of nodes are not connected and, therefore, journeys between most pairs of locations were not identified by the model. Furthermore, 13% of sequential events are inside one of the nodes (e.g., the cells involved stayed within a 20 kilometres radius).
Boko Haram moves between a few regions
Our methodology identifies regions that are frequently traversed by Boko Haram cells. Results show that most of the movements take place between the capital of Borno State Maiduguri and the cities of Damaturu and Potiskum in Yobe State, along the major A3 Highway. Numerous movements are also recorded between Maiduguri and the Sambisa Forest, where Boko Haram has found a safe haven, and between the capital of Borno and the Cameroon border, where the headquarters of the organisation (Gwoza) was located until March 2015 (Fig. 6). For all types of mobility, the road between Maiduguri and Bama is very frequently travelled by Boko Haram cells. The most highly frequented paths are found within Nigeria, with the exception of the Mandara Mountains in Cameroon and the Diffa region in Niger.
The 420 nodes and the top 1% edges according to their weight for different mobility scenarios. Certain paths (between Maiduguri, which is the largest node) and the border with Cameroon and the border with Niger are travelled frequently by Boko Haram cells. Also, the journey between Maiduguri and Damaturu and Potiskum (both located west of Maiduguri) are frequently travelled with high mobility cells. With low mobility cells, the movement of Boko Haram cells is most frequently between Maiduguri and Bama, Damboa, Gwoza and Kondua, four urban agglomerations south and south-east part of Maiduguri
For a high mobility scenario, the frequency at which a cell is active twice in the same location (or node) is less than 6% of all its events (Fig. 7). Even with medium and low mobility, 10 and 30% of any two consecutive attacks take place in the same location. With a speed of ν = 60 kilometres per day and a maximum distance of μ = 180 kilometres, only 8% of any two consecutive events committed by the same cell are in the same location. These results suggests that Boko Haram fighters most probably leave the region they have attacked immediately and plan another attack from a different location.
For each combination of the daily speed ν and the maximum distance between two events μ, a spatial network was constructed. We measure the percentage of trips completed inside the same node (top), the percentage of trips which happen within the top 1% of the edges (middle) and the percentage of present edges (bottom). With a speed of ν = 60 kilometres and a maximum distance of μ = 180 kilometres, roughly 8% of the trips happen inside the nodes, nearly 50% of the trips happen within the top 1% of the edges and only 12% of the edges are present
Some of the journeys between two specific locations are very frequently travelled by Boko Haram cells (Fig. 7). Assuming that Boko Haram cells are highly mobile, the top 1% of the edges concentrate more than 40% of all the Boko Haram journeys. With low mobility, this increases to more than 70%. With a speed of ν = 60 kilometres per day and a maximum distance of μ = 180 kilometres, the top 1% edges concentrate roughly 50% of the journeys. Therefore, although cells bounce between different locations, most of their journeys are through very specific and repeated routes. Similar results are observed if we take the top 5% or other concentration units.
Finally, as we saw with the composite network, journeys between pairs of locations are not that frequent (Fig. 7). Depending on the mobility scenario, only 10–12% of node pairs are connected (e.g., at least one trip was detected).
Boko Haram is a regional problem
Boko Haram started as an insurgency primarily focused on attacking the Nigerian government and for many years the vast majority of its attacks were conducted within Nigeria (Dowd 2017). In recent years, Boko Haram has relocated to Chad, Cameroon and Niger (Matfess 2019). Our results confirm this trend by showing that cross-border crossings have become more frequent (Skillicorn et al. 2019).
It is possible then to measure the number of cross-border trips by Boko Haram, this is simply the number of times that a cell was active in two consecutive events in nodes located at a different side of a country border. For a high mobility scenario, roughly 35% of the journeys of a Boko Haram cell cross an international border. Fewer crossings are observed under a lower mobility scenario (Fig. 8). With a speed of ν = 60 kilometres per day and a maximum distance of μ = 180 kilometres, roughly a third of the cells move across borders.
The fraction of times that a cell crosses an international border depending on daily speed ν, and the distance between two events μ. With a speed of ν = 60 kilometres and a maximum distance of μ = 180 kilometres, Boko Haram cells cross a border 30% of the time between any two consecutive events
The objective of this study was to uncover the internal structure of a terrorist organisation through its mobility patterns. Our method identifies cells which move between Boko Haram events at a certain speed and for a certain maximum distance. Once cells and their mobility patterns have been extracted, a spatial network is constructed. Our study suggests that the terrorist organisation Boko Haram is structured around 50–60 cells active around Lake Chad in West Africa. Our work contributes to a long-lasting and often heated debate about the internal structure of Boko Haram. It suggests that Boko Haram is a rather fragmented organisation in which decentralised cells are capable of committing numerous and repetitive attacks against government and civilian targets in northern Nigeria and the surrounding countries. This result corresponds to earlier qualitative studies that noted that Boko Haram was organised around a loose federation of cells (Thurston 2017; Anugwom 2019). Due to the speed and spatial dispersion of attacks, it seems unlikely that Boko Haram is a strongly unified organisation, despite being formally ruled by a ruthless leader. While previous studies argue that the main unit led by Abubakar Shekau is responsible for more attacks than others, it appears to be less dominant in terms of casualties and geographical extent than some studies claim (Zenn 2019).
The fact that Boko Haram is fragmented into numerous cells that operate along specific routes, possibly across borders, can be used to inform counter-insurgency strategies. Firstly, dismantling one of the 50 presumed cells is unlikely to significantly reduce violence in the region, as each cell is on average responsible for only 2–3% of the casualties related to Boko Haram. Secondly, some paths are more frequently travelled by Boko Haram cells than others and so prevention interventions can be oriented to stopping cells when they move between two consecutive events, rather than a reactive strategy, targeting specific locations, such as where a cell previously has attacked. Thirdly, the large number of cross-border movements reported in our study suggests that Boko Haram has been able to operate regionally despite the multinational task force established by Nigeria and the neighbouring countries to secure the borders of the Lake Chad region (OECD/SWAC 2020). Cross-border cooperation remains a crucial factor in countering the Boko Haram insurgency and preventing its transnational spread in the region.
Our study of the mobility of Boko Haram suggests that members of the Jihadist organisation are capable of travelling over long-distance repeatedly. Based on the level of specialisation, we estimate that each cell of Boko Haram travels at most 60 kilometres per day on average, which is a significant distance considering the local road infrastructure and the need to avoid detection. Our model enables us to detect the main locations and paths travelled by Boko Haram cells, and indicates that both are highly concentrated in a number of cities (notably Maiduguri) and major road corridors.
In recent years, Boko Haram has been able to relocate (rather than spread) to remote places that are difficult to access to government troops, such as the Mandara mountains in Cameroon, the Sambisa Forest in Nigeria and the islands of Lake Chad. Our results show that more than a third of the journeys of Boko Haram cells cross an international border. Thus, security is not a national issue but a regional one and cross-border cooperation will play a fundamental role in the region.
Despite being composed of highly mobile cells, Boko Haram is nevertheless a rather territorial terrorist organisation which concentrates most of its attacks in what used to be the western part of the Kanem-Bornu, a pre-colonial empire that ruled from the 1380s to 1893. As such the spatial patterns of Boko Haram differ from those of Al Qaeda in the Islamic Maghreb, who does not seek to hold territory and is capable of conducting attacks thousands of kilometres apart (Walther et al., 2020).
Our method is capable of detecting known internal changes across time, particularly the 2015 turning point in the war against Boko Haram and the newly incorporated Barnawi events in 2019. The study confirms that the counter-offensive led by Nigerian forces, the Multinational Joint Task Force and vigilante groups in 2015 has contributed to further fragment Boko Haram and limit the spatial reach of its cells. By forcing Boko Haram to leave numerous cities and villages in northeast Nigeria, the counter-offensive is a turning point in the war against the organisation. We observe that since 2015 the number of armed clashes has increased much more rapidly than violence against civilians, which remains a modus operandi of Boko Haram. We also detect a series of major changes in the internal composition of Boko Haram during this period, with the creation of new cells, including but not limited to the faction led by Abu Mus'ab al-Barnawi. The fragmentation process observed during this period also leads to significant changes in the spatial patterns of Boko Haram, with a higher fragmentation but with cells which are less mobile and therefore less capable of conducting attacks far from their safe havens.
Constructing a spatial metric using the location of events
Boko Haram has committed attacks in roughly 900 different locations across northern Nigeria and the neighbouring countries. However, the Euclidean distance between some events is very small, for instance, when two attacks place in distinct parts of Maiduguri, a city in Nigeria which, according to Africapolis (OECD/SWAC 2018), has a population of more than one million inhabitants and a surface of 139 kilometres2. In large cities, two events can be separated by more than 10 kilometres and still be inside the same metropolitan area. To address this issue, we aggregated events that took place in nearby regions.
We used Partitioning Around Medoids (Reynolds et al. 2006) to construct our spatial metric. This approach considers the location of all the events and a number k which corresponds to the number of clusters that the algorithm will produce. It then takes k representative objects (or medoids) among the observations of the locations and identifies several clusters of locations by assigning each observation to the nearest medoid. The algorithm tries different combinations of medoids by swapping between the options, with the objective to minimise the sum of the dissimilarities between the locations of each group and its medoid.
There are ways to find an optimal number of clusters, for example by looking at the quality of the clustering with different values of k (Rousseeuw and Kaufman 1990). Following a similar idea, we clustered the locations into k groups, with k = 2, 3, … and searched for the smallest number of clusters such that the Euclidean distance between any two locations of each cluster was smaller than 20 kilometres. This choice is justified by the idea that events separated by small distances might belong to the same metropolitan area or a similar region. The procedure was executed in R (R Core Team 2018) using the cluster package (Maechler et al. 2017). The distance between any two locations is smaller than 20 kilometres with k = 420 clusters.
Active cells, total cells and cells which remain active
Results show that with low daily speed ν or with a small maximum distance μ between events, there are at least 300 total cells, from which less than 50% are still active by 2019. The ratio between the total cells and those that are still active in 2019 is not uniformly distributed, but it ranges between 40 and 80% (Fig. 9). With a daily speed of ν = 60 kilometres per day and a maximum distance of μ = 180 kilometres, the total number of cells T2019(ν, μ) = 83, and the active number of cells A2019(ν, μ) = 53, meaning that 63% of the cells are still active in 2019.
Total number of cells T2019(ν, μ) (left) and the ratio between the active number of cells and the total number of cells A2019(ν, μ)/T2019(ν, μ) (right) for a range of daily speeds and maximum distance between consecutive events
Specialisation per cell
Our measure of specialisation was constructed by examining the type of events in which each cell had been involved, using the categories of events provided by ACLED (armed clash, attacks, suicide bombs, government regains territory; remote explosives; air or drone strikes and others). Then, we calculated the Euclidean distance between the distribution of the type of events of each cell and the distribution of the type of events across all observed events. We then weighted this distance by the number of events of the cell and reported the maximum value (across all cells) as the level of specialisation.
The level of specialisation for a realisation of the algorithm S(ν0, μ0) is defined as
$$ S\left({\nu}_0,{\mu}_0\right)=\underset{i}{\max}\left\{{e}_i\parallel {D}_i-{D}_{\nu_0,{\mu}_0}\parallel \right\}, $$
where ei is the number of events of cell i, Di is the distribution of events per type of cell i, \( {D}_{\nu_0,{\mu}_0} \) is the average distribution and ∥ ∘ ∥ means the Euclidean distance.
Since taking only the most specialised cell potentially leads to biased results (or could be the result of randomness), the average level of specialisation of the top 3, 9 and 27 cells was also considered (Fig. 10).
The average metric of specialisation under different values of the daily speed ν and the maximum distance μ according to the number of cells which are considered for the metric
Results show that the level of specialisation of the most specialised cell is highly correlated to the average of three most specialised cells, as are the results for the 9 or the 27 most specialised cells. Therefore, the section of the parameter space which is identified as very specialised with 1, 3, 9 or 27 cells are also very similar and we keep, as metric of specialisation, the level for just one cell.
A signal in the number of casualties?
It is possible to incorporate more information into each event to detect whether they were committed by the same cell or by different cells. The methodology could be extended by considering that two events are part of the same cell if they satisfy certain restrictions, and part of different cells otherwise. For example, if two events are executed by a substantially different number of people, or with distinct weapons, then the model could assign the events to different cells or assume the existence of new ones. See, for instance Campedelli et al. (2019a).
The number of casualties is another variable that could potentially have provided useful information to distinguish between cells. Unfortunately, this variable provides very little information that can be used in our analysis. Boko Haram has killed tens of thousands of people in northern Nigeria over the last 10 years but many of the events in which the organisation is involved have a small number of casualties, while a few events concentrate a disproportionate number of casualties. The top 5% most violent events concentrate 47% of the casualties of Boko Haram, whereas the 50% least violent events concentrate just 4.8% of the casualties. Using only the most violent events to discriminate between cells is also problematic because most of them happened during the first 2 months of 2015 when Nigerian forces launched a major military offensive. This period during which Boko Haram was the deadliest, with 5% of its events and 17% of its fatalities, is too short to study long term changes within the organisation.
Figure 11 shows the cumulative number of events per cell on the horizontal axis and the corresponding cumulative number of casualties per cell on the vertical axis for different mobility scenarios. Since few events are highly violent, the cells responsible for them are much more deadlier than the rest. All simulations have a similar structure in terms of the number of casualties (even with more or fewer cells) and so the number of casualties does not provide a signal to differentiate between cells.
Four sections of the parameter space showing the cumulative number of events (horizontal axis) and the cumulative number of casualties (vertical axis) of the cells, sorted in decreasing order, from the cell with the highest number of events to the cell with the lowest number of events
The data used in the manuscript is produced by the Armed Conflict Location & Event Data project (ACLED) (Raleigh et al. 2010) and it is available at their website https://www.acleddata.com/.
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We would like to acknowledge Laurent Bossard, Marie Trémolières, Philipp Heinrigs, Inhoi Heo and Sarah Lawan from the OECD Sahel and West Africa Club Secretariat (SWAC) and Alex Thurston for their insightful suggestions and comments.
This article was completed with support from the PEAK Urban programme, funded by UKRI's Global Challenge Research Fund, Grant Ref: ES/P011055/1.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK
Rafael Prieto Curiel
Research in Spatial Economics, Universidad Eafit, Carrera 49, número 7 sur 50, Medellín, Antioquia, 050022, Colombia
Department of Geography, University of Florida, Gainesville, Florida, 32611, USA
Olivier Walther
Centre for Spatial Analysis, University College London, Gower Street, London, WC1E 6BT, UK
Neave O'Clery
RPC designed the study. OW and RPC analysed the results. All authors wrote the manuscript. The author(s) read and approved the final manuscript.
Correspondence to Rafael Prieto Curiel.
Prieto Curiel, R., Walther, O. & O'Clery, N. Uncovering the internal structure of Boko Haram through its mobility patterns. Appl Netw Sci 5, 28 (2020). https://doi.org/10.1007/s41109-020-00264-4
Structure and dynamics of crime | CommonCrawl |
communications chemistry
Coherent vibrations in methanol cation probed by periodic H3+ ejection after double ionization
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H2 roaming chemistry and the formation of H3+ from organic molecules in strong laser fields
Nagitha Ekanayake, Travis Severt, … Marcos Dantus
Attosecond coupled electron and nuclear dynamics in dissociative ionization of H2
L. Cattaneo, J. Vos, … U. Keller
Toshiaki Ando1,
Akihiro Shimamoto1,
Shun Miura1,
Atsushi Iwasaki1,
Katsunori Nakai ORCID: orcid.org/0000-0002-2172-53131 &
Kaoru Yamanouchi1
Communications Chemistry volume 1, Article number: 7 (2018) Cite this article
Reaction kinetics and dynamics
When hydrocarbon molecules are exposed to an intense laser field, triatomic hydrogen molecular ion, H3+, is ejected. Here we describe pump–probe measurements of the ejection of H3+ from methanol dication with high temporal resolution using intense few-cycle laser pulses and find a long-lasting periodic increase in the yield of H3+. We show that H3+ ejection is the lowest energy decomposition channel and that its yield is enhanced each time when the vibrational wave packet coming back to the inner turning point of methanol cation is projected onto the dication potential energy surface. We also show that the time-resolved measurement of the yield of H3+ is an efficient tool not only for probing ultrafast nuclear dynamics of hydrocarbon cations but also for deriving vibrational frequencies of hydrocarbon cations with high precision.
It has been shown that triatomic hydrogen molecular ion, H3+, plays an important role in interstellar chemistry1. For example, H3+ formed through a bimolecular reaction, H2++H2→H3++H, can protonate a neutral molecule X by a subsequent bimolecular collision, H3++X→HX++H2, which may lead to further subsequent interstellar chemical reactions. It has also been shown that H3+ can be emitted from hydrocarbon molecular dication prepared by electron impact ionization2, by extreme ultraviolet photoionization3, and by irradiation of intense femtosecond laser pulses4,5,6. The formation of H3+ from allene7,8,9 and the formation of HD2+ from CD3OH10 showed that migration of hydrogen atoms can proceed in the formation of H3+. Recently, we interpreted the mechanism of H3+ formation from dication of methanol theoretically as a unimolecular decomposition process in which a long-lived neutral moiety of H2, formed within a dication molecule, abstracts a proton in the other moiety having the charge of +211. Through a series of our studies on the formation of H3+ from hydrocarbon molecules by the irradiation of ultrashort intense laser pulses7,10,12,13,14, we showed that the hydrocarbon dications from which H3+ is ejected are all long lived, and that their lifetimes are comparable to or longer than a period of the overall rotation of the dications. These findings mean that the intramolecular vibrational-energy redistribution proceeds to a large extent within the energized dications prior to the ejection of H3+.
When hydrocarbon dication is prepared by sequential ionization from neutral by two ultrashort laser pulses, the vibrational energy of the dication is varied depending on the delay time of the second ionization pulse with respect to the first ionization pulse that prepares a vibrational wave packet on the potential energy surface of the monocation, reflecting the temporal evolution of the wave packet. Therefore, if we monitor the ejection of H3+ from the dication in real time, we will be able to explore ultrafast nuclear dynamics of hydrocarbon molecules at the monocation stage by taking advantage of the existence of the long-lived dication, acting as an energized state in the unimolecular dissociation.
In the present study, we record the yield of H3+ produced from methanol dication,
$${\mathrm{CH}}_3{\mathrm{OH}}^{2 + } \to {\mathrm{CHO}}^ + + {\mathrm{H}}_3^ + ,$$
in time domain by the pump–probe coincidence momentum imaging (CMI) method using near-infrared few-cycle laser pulses whose pulse duration is 6 fs and monitor the vibrational motion of CH3OH+ with high temporal resolution of as high as ~6 fs.
Origin of periodic ejection of H3 +
Figure 1a shows the distribution of the released kinetic energy, Ekin, and the ion yield of the two-body Coulomb explosion pathway (1) as a function of the pump–probe time delay Δt. The ion yield exhibits a periodic increase with the period of ~38 fs starting from Δt ~ 58 fs, and this periodic structure continues beyond 500 fs as seen in Fig. 1b. The temporal pulse profiles of the few-cycle laser pulses employed in the present experiment are shown in Supplementary Figure 1. In order to extract the frequency of the ion yield oscillation, we perform the Fourier transform of the yield in the range of 120 < Δt < 500 fs in Fig. 1b, and obtain Fig. 1c, in which a distinct peak appears at ~26.3 THz, corresponding to the period of ~38.0 fs. This long-lasting oscillation can only be ascribed to the motion of the vibrational wave packet on the electronic ground state of CH3OH+ because even the lowest electronically excited state of CH3OH+ is located 0.5 eV above the dissociation threshold for CH2OH++H15,16 and the wave packet prepared on these electronically excited states is expected to decay into the dissociation continuum. This frequency is closed to the C-O stretching frequency 26.8 THz (895 cm−1) of the electronic ground state of CH3OH+15.
The ion yield of H3+ ejection pathway. a The Ekin distribution of the H3+ ejection Coulomb explosion pathway (1). b The ion yields of the H3+ ejection Coulomb explosion pathway (1). Black curve: ion yield as a function of the Δt. Blue broken curve: optimized exponential curve to which the ion yield data are fitted in the time range between Δt = 120 fs and 500 fs. Red curve: residual ion yield after the subtraction of the optimized exponential curve. c The Fourier-transformed ion yields of the H3+ ejection pathway (1) using the data in the range of 120 < Δt < 500 fs
It is known from photoion–photoion coincidence measurement17 that the H3+ ejection pathway (1) and the H+ ejection pathway
$${\mathrm{CH}}_3{\mathrm{OH}}^{2 + } \to {\mathrm{CH}}_3{\mathrm{O}}^ + + {\mathrm{H}}^ + ,$$
have the lowest threshold energies among all the possible decomposition pathways of CH3OH2+. The decomposition pathway having the next lowest threshold energy is the H2+ ejection pathway,
$${\mathrm{CH}}_3{\mathrm{OH}}^{2 + } \to {\mathrm{CH}}_2{\mathrm{O}}^ + + {\mathrm{H}}_2^ + ,$$
whose threshold energy is located at ~0.5 eV higher than the pathways (1) and (2)17, and the decomposition pathway having the second next lowest threshold energy is the three body explosion pathway,
$${\mathrm{CH}}_3{\mathrm{OH}}^{2 + } \to {\mathrm{CHO}}^ + + {\mathrm{H}}^ + + {\mathrm{H}}_2,$$
whose threshold energy is located at ~ 1 eV higher than the pathways (1) and (2)17.
From the number of events at Δt ~ 500 fs in the high Ekin region above 3.5 eV, the relative yields of the pathways (1), (2), (3), and (4) in our measurements are derived to be 1:1:0.07:3.6, respectively, showing that the three body explosion pathway (4) has the largest ion yield and the pathway (3) is a minor pathway.
Considering that the period of ~38 fs identified in Fig. 1c in time domain is in good agreement with the vibrational frequency of the CO stretching mode of the electronic ground state of CH3OH+, we plot the schematic potential energy curves of the electronic ground state of CH3OH (\({\tilde{\mathrm X}}{}^{\mathrm{1}}{\mathrm{A{\prime}}}\)), CH3OH+ ((2a'')−1 \({\tilde{\mathrm X}}{}^{\mathrm{2}}{\mathrm{A{\prime\prime}}}\)), and CH3OH2+ ((2a'')−2\({\tilde{\mathrm X}}{}^{\mathrm{1}}{\mathrm{A{\prime}}}\)) along the CO stretching coordinate in Fig. 2. According to previous experimental and theoretical studies, the equilibrium C-O internuclear distances of CH3OH, CH3OH+, and CH3OH2+ are 1.428(3) Å18 1.36 Å19, and 1.19 Å19, reflecting the fact that the 2a'' orbital has an anti-bonding character, and therefore, upon the first ionization, the vibrational wave packet is expected to be prepared at the outer turning point of the potential energy curve. Consequently, as the delay time increases, the energy of the dication decreases and takes the minimum value at Δt = 19 fs, a half of the vibrational period of the CO stretch of the monocation, and then the energy becomes maximum at Δt = 38 fs. The delay times at which the yield of H3+ became maximum, 58 fs, 96 fs, 134 fs…, shown in Fig. 1a, correspond to the timings when the vibrational wave packet reached the inner turning point of the potential energy curve of the monocation, i.e., the timing when the energy of the dication takes the minimum value. Therefore, it is probable that, at the lowest energy of the dication, only the lowest energy channels (1) and (2) are open energetically, and that once the other dissociation channels are opened energetically, they dominate over the lowest energy channels, leading to the decrease in the yield of H3+ through the channel (1).
Schematics of the potential energy curves along the CO internuclear distance. The vibrational wave packet of CH3OH+ prepared by the first laser pulse at the outer turning point starts moving towards the shorter internuclear distance region and the oscillation continues. The wave packet prepared in CH3OH2+ by the second laser pulse has the lowest energy every time when the wave packet of CH3OH+ comes back to the inner turning point periodically with the interval of 38 fs starting from at Δt = 19 fs
Unimolecular decomposition of CH3OH2+
In order to discuss more precisely the unimolecular decomposition leading to the periodic ejection of H3+, we explored the multi-dimensional potential energy surface of CH3OH2+ by density functional theory calculations to find the transition states for the dissociation pathways. As plotted in Fig. 3, there are two transition states, TS1a (0.56 eV) and TS1b (0.85 eV), in the H3+ ejection pathway (1). At TS1a and TS1b, CH3OH2+ takes two different characteristic geometrical structures, both of which show that neutral H2 is attached to CHOH2+ moiety. These two transition state structures at TS1a and TS1b, representing a loosely bound complex in which the charge induced dipole moment of the H2 moiety is bound by the positive charge of the CHOH2+ moiety. When the unimolecular reaction proceeds through TS1a, H3+ is expected to be ejected from methyl group, and, when it proceeds through TS1b, H3+ is expected to be ejected from hydroxyl group after the migration of neutral H2 from methyl group as was discussed in ref. 11. The energy of the transition state of the pathway (2) (TS2), 0.65 eV, is located between the energies of TS1a and TS1b.
Theoretical energy diagram of CH3OH2+. The zero point energies at the three transition states (TS1a, TS1b, TS2, and TS4) measured from the zero point energy at the equilibrium structure (EQ) are shown
As shown in Fig. 4, the yield of the pathway (2) as a function of the pump–probe delay also exhibits the oscillatory structure similar to the one shown in Fig. 1b for the pathway (1). Even though the yield of the pathway (2) is almost the same as the yield of the pathway (1), the amplitude of the oscillation of the pathway (2) is found to be only one third of that of the pathway (1).
The ion yield of H+ ejection pathway. a The Ekin distribution of the H+ ejection Coulomb explosion pathway (2). b The ion yields of the H+ ejection Coulomb explosion pathway (2). Black curve: ion yield in the high Ekin region above 3.5 eV as a function of the Δt. Blue broken curve: optimized exponential curve to which the ion yield data are fitted in the time range between 120 and 500 fs. Red curve: residual ion yield after the subtraction of the optimized exponential curve. c The Fourier-transformed ion yields of the H+ ejection pathway (2) using the data in the range of 120 < Δt < 500 fs
This larger oscillation amplitude in the pathway (1) can be discussed by the Rice–Ramsperger–Kassel–Marcus (RRKM) unimolecular reaction theory. The spontaneous reaction rate k2(E) in the RRKM theory20 can be written as
$$k_2(E) = \frac{{W^{\mathrm{\ddagger }}\left( {E - E_0} \right)}}{{h\rho (E)}},$$
where E and E0 are the internal energy for the parent dication and the zero point energy for the TS, respectively, W‡ is the number of the energetically allowed vibrational levels for the vibrational motion along the directions perpendicular to the reaction coordinate at the transition state, and ρ(Ε) is the density of states of the parent dication. Since ρ(Ε) is common in these two pathways (1) and (2), the difference in the yield can be ascribed to the difference in W‡. This means that when the difference between the threshold energy for H3+ production and that for the H+ production is small, their relative yield is governed by the W‡ values.
We calculated the vibrational frequencies of the vibrational modes at TS1a, TS1b, and TS2 as shown in Supplementary Table 1, and evaluated W‡ as a function of E−E0. The results are plotted in Fig. 5, showing that W‡ at TS1a and TS1b increases more rapidly than W‡ at TS2, which can be ascribed to the difference in the numbers of the low-frequency vibrational modes at these three transition states. The numbers of the vibrations modes having the lower wavenumber than 600 cm−1 are six and five at TS1a and TS1b, respectively, while the number at TS2 is only two. The larger number of the low-frequency vibrational modes contribute to the increase in W‡ at TS1a and TS1b. Among these low-frequency vibrational modes, the vibrational mode (215 cm−1) assigned to the rotation of the H2 loosely bound neutral moiety at TS1a and the corresponding vibrational mode (242 cm−1) at TS1b contribute largely to the more rapid increase in W‡ at these two transition states than the increase in W‡ at TS2.
The numbers of energetically allowed vibrational levels at the TSs of CH3OH2+. The W‡ at TS1a (black), TS1b (blue), and TS2 (red) are counted by the Beyer Swinhart algorithm29 as a function of the internal energy, which is measured from the zero point energy at the EQ of CH3OH2+
It may be said that TS1b could not contribute in the competition between the pathway (1) and the pathway (2) because the energy of TS1b is 0.31 eV higher than the energy of TS1a. However, within the accuracy of the theoretical calculations, it is difficult to judge if such a small energy difference is accurately evaluated. Therefore, we regard that both TS1a and TS1b are located closely in energy to TS2 in this threshold region of the two decomposition pathways and that the more rapid increase in W‡ at TS1a and TS1b than W‡ at TS2 results in the more rapid increase in the spontaneous reaction rate for the H3+ ejection pathway (1) than that for the H+ ejection pathway (2), which explains the reason why the observed amplitude for the oscillation in the H3+ ejection pathway is about three times as large as that in the H+ ejection pathway, i.e., the pathway (2).
It is known that the ejection of H+ through the pathway (2) proceeds very rapidly within the time range of 70–290 fs13. Therefore, it is probable that the large portion of H+ are produced via the pathway (2) from the precursor dications prepared in a much higher energy region above TS2, resulting in the non-oscillating components seen in Fig. 4, and that the small portion of the wave packet prepared in the low energy region slightly above TS2 contribute to the oscillating component in the pathway (2).
Mechanism of H3 + and H2D+ ejections from CH3OD2+
In order to confirm our discussion above, we have also performed pump–probe CMI measurements of partially deuterated methanol, CH3OD, for the H3+ ejection pathway from CH3OD2+,
$${\mathrm{CH}}_3{\mathrm{OD}}^{2 + } \to {\mathrm{COD}}^ + + {\mathrm{H}}_3^ + ,$$
proceeding via TS1a, and the H2D+ ejection pathway,
$${\mathrm{CH}}_3{\mathrm{OD}}^{2 + } \to {\mathrm{CHO}}^ + + {\mathrm{H}}_{\mathrm{2}}{\mathrm{D}}^ + ,$$
proceeding via TS1b.
As shown in Fig. 6, both of the ion yield of the pathway (6) and that of the pathway (7) exhibit periodic peak structures with a common period of 26.3(26) THz, which agrees well with the frequency of the C-O stretching that is 27.8(9) THz (928(30) cm−1)21. As shown in Fig. 6e, f, the oscillation amplitude of the pathway (7) via TS1b is around one fourth of the pathway (6) via TS1a, suggesting that the two pathways compete and their threshold energies are located closely to each other even though TS1b is expected to be somewhat higher in energy than TS1a.
The ion yield of H3+ and H2D+ ejection pathway from CH3OD2+. The Ekin distribution of the H3+ ejection pathway (6) (a) and the H2D+ ejection pathway (7) (b) from CH3OD2+. The ion yields of the H3+ ejection pathway (6) (c) and the H2D+ ejection pathway (7) (d). Black curve: ion yield as a function of the Δt. Blue broken curve: optimized exponential curve to which the ion yield data are fitted in the time range between Δt = 100 fs and 480 fs. Red curve: residual ion yield after the subtraction of the optimized exponential curve. The Fourier-transformed ion yields of the H3+ ejection pathway (6) (e) and the H2D+ ejection pathway (7) (f) using the data in the range of 100 < Δt < 480 fs
The number of the energetically allowed levels W‡ of CH3OD2+ at TS1a and that at TS1b are calculated as shown in the Fig. 7. Even though the W‡ values at the two transition states increases rapidly in a similar manner, the threshold energy for TS1b is about 0.3 eV higher than that for TS1a. The observation that the dissociation pathways (6) and (7) compete shows the energy difference between TS1a and TS1b is overestimated by about 0.3 eV in the calculation. The experiment using the partially deuterated methanol, CH3OD, has revealed that H3+ ejection can proceed through both of the transition states, TS1a and TS1b, and that TS1b is only slightly (<0.1 eV) higher in energy than TS1a.
The numbers of energetically allowed vibrational levels at the TSs. The W‡ at TS1a (black), TS1b (blue), and TS2 (red) are counted by the Beyer Swinhart algorithm29 as a function of the internal energy, which is measured from the zero point energy at the EQ of CH3OD2+
The pathway (4) has the largest yield among these pathways, indicating that a large portion of CH3OH2+ is prepared in the higher internal energy than the threshold energy of the pathway (4). As shown in Fig. 8, no long-lasting periodic peak structure can be seen in the yield of the pathway (4), indicating that the periodic dip in the yield of the pathway (4) that should be associated with the increase in the yields of the pathways (1) and (2) is so small in its magnitude compared with the total yield of the pathway (4).
The three body explosion pathway (4). The distribution of the kinetic energy of H+ generated through the three body explosion pathway (4). The strong signal with a vertical stripe at Δt ~ 70 fs is ascribed to the optical interference between the satellite structure of one of the pump and probe laser pulses and the main central part of the other laser pulse
We have found the periodic increase in the yield of H3+ from CH3OH2+ and that of H3+ and H2D+ from CH3OD2+, lasting for as long as 500 fs or more by the pump–probe measurement with the temporal resolution as high as 4–6 fs, and demonstrated that the period (~38 fs) of the C-O stretching vibration of methanol monocation can be obtained from the Fourier transform of the time-resolved signals.
Because methanol dication from which H3+ is ejected is long lived, unimolecular reaction theory have been applied to the decomposition process. By evaluating the spontaneous reaction rate for the H3+ decomposition pathway by the RRKM unimolecular reaction theory, we have found that the loose transition states having the low-frequency vibrational modes play a crucial role in guiding the dication towards the H3+ ejection pathway.
The H3+ ejection from hydrocarbon dications can be regarded as a general phenomenon for hydrocarbon molecules having more than two hydrogen atoms4. It is possible that H3+ ejection from other hydrocarbon dication species proceeds also through this type of loose transition state. Because the energized state is regarded as the dense manifold of the vibrational levels to which the wave packet motion of monocation is to be projected, time-resolved monitoring of the very slow H3+ ejection process and its Fourier transform can be a useful spectroscopic method for determining the vibrational frequencies of hydrocarbon cations that are difficult to be obtained by conventional spectroscopic methods. The time-resolved detection of H3+ from methanol dication induced by the intense field sequential ionization have given us an opportunity to investigate the vibrational dynamics of methanol cation.
Experimental apparatus
The details of our experimental setup have been described in the previous report22. Briefly, few-cycle laser pulses were generated by a hollow-core fiber compression technique using an output of a chirped pulse amplification femtosecond Ti:sapphire laser system (800 nm, 5 kHz, 0.6 mJ, 30 fs). The few-cycle laser pulses were introduced into a Michelson interferometer to generate pump and probe pulses. The optical time delay Δt between the pump and probe laser pulses was varied in the range between around −20 fs and 500 fs using a piezo-controlled optical stage. The pulse duration was measured to be 6.0 fs and the center wavelength was 770 nm for CH3OH. In the experiment of CH3OD, the pulse was compressed so that the duration became 4.4 fs and the center wavelength was 770 nm. Both the pump and probe laser pulses were focused onto an effusive molecular beam of methanol in a vacuum chamber by a concave mirror (f = 150 mm). The laser polarization direction of the pump laser pulses and that of the probe laser pulses were set to be parallel to the propagation axis of the molecular beam. The focal intensity was estimated to be 2.1 × 1014 W cm−2 for CH3OH and 2.7 × 1014 W cm−2 for CH3OD. The temporal shapes of the few-cycle laser pulses were measured by the method of two-dimensional spectral shearing interferometry23 and shown in Supplementary Figure 1.
The fragment ions generated from methanol were guided by static electric fields toward a two-dimensional position sensitive detector (HEX120, RoentDek) in the velocity map imaging configurations. From the flight times of the fragment ions and the positions on the detector plane where the fragment ions hit, the momentum vectors of the fragment ions were determined. Events of the two-body Coulomb explosion pathways were extracted by the CMI method24. The released kinetic energy Ekin was plotted as a function of the time delay Δt in the delay time range of −20 to 500 fs with delay increment step of 4 fs. Events of the three body explosion pathway (4) were extracted by the covariance mapping method25.
Extraction of pump–probe signals
The row data of Ekin distributions of the dissociation pathways include the signals generated by the pump laser pulse only and those generated by the probe laser pulse only because methanol dication can also be generated by a single laser pulse. The Ekin distribution Ydiff(Ekin,Δt) generated from methanol dication prepared exclusively by the sequentially ionization by the pump and probe laser pulses, which is shown for example in Fig. 1a for the dissociation pathway (1), was derived by subtracting the signals generated by the pump laser pulse only Ypump(Ekin) and those generated by the probe laser pulse only Yprobe(Ekin) from the Ekin distributions Ypump-probe(Ekin, Δt) as
$$Y_{\rm diff}(E_{\rm kin},\triangle t) = Y_{\rm pump - probe}(E_{\rm kin},\triangle t){\mathrm{ }}-{\mathrm{ }}(Y_{\rm pump}\left( {E_{\rm kin}} \right){\mathrm{ }}-\alpha Y_{\rm probe}\left( {E_{\rm kin}} \right)),$$
where α is a correction factor describing a depletion of sample molecules by irradiating the pump laser pulse. As the value of α, α = 0.79(5) was adopted for CH3OH, which was obtained in ref. 22 by comparing the total ion yield obtained by the pump laser pulse only, by the probe laser pulse only, and by the pump and probe laser pulses. For CH3OD, the lower value α, α = 0.4, was adopted for CH3OD because the sample is depleted to a larger extent associated with the larger laser pulse intensity.
Geometrical structures of CH3OH2+ at TSs and EQ
The geometry optimization of the transition states and the equilibrium structures were performed using the global reaction route mapping (GRRM1.2) program26,27,28 using energies and gradient vectors computed by the Gaussian09 program with the density functional theory at the UB3LYP/aug-cc-pVTZ level after exploring transition states and equilibrium structures with UB3LYP/6-31G(d) level.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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The present research was supported by the following three grants from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan: two Grant-in-Aid for Specially Promoted Research (#19002006 and #15H05696) and a Grant-in-Aid for Scientific Research A (#24245003).
Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
Toshiaki Ando, Akihiro Shimamoto, Shun Miura, Atsushi Iwasaki, Katsunori Nakai & Kaoru Yamanouchi
Toshiaki Ando
Akihiro Shimamoto
Shun Miura
Atsushi Iwasaki
Katsunori Nakai
Kaoru Yamanouchi
T.A., A.S., S.M. and A.I., developed the experimental method. T.A. performed the experiments. T.A. and K.Y. analyzed the data. K.N. performed the GRRM calculations. The manuscript was prepared by T.A. and K.Y., and was discussed among all authors.
Correspondence to Kaoru Yamanouchi.
Ando, T., Shimamoto, A., Miura, S. et al. Coherent vibrations in methanol cation probed by periodic H3+ ejection after double ionization. Commun Chem 1, 7 (2018). https://doi.org/10.1038/s42004-017-0006-7
Ester Livshits
Itamar Luzon
Daniel Strasser
Communications Chemistry (2020)
Formation of H3+ from ethane dication induced by electron impact
Baihui Ren
Baoren Wei
Nagitha Ekanayake
Travis Severt
Marcos Dantus
Communications Chemistry (Commun Chem) ISSN 2399-3669 (online) | CommonCrawl |
A periodic and diffusive predator-prey model with disease in the prey
DCDS-S Home
Condensing operators and periodic solutions of infinite delay impulsive evolution equations
June 2017, 10(3): 463-473. doi: 10.3934/dcdss.2017022
Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales
Yongkun Li , and Pan Wang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
* Corresponding author: Yongkun Li
Received November 2015 Revised December 2016 Published February 2017
Fund Project: The first author is supported by the National Natural Sciences Foundation of China under Grant 11361072.
We first propose a concept of almost periodic functions in the sense of Stepanov on time scales. Then, we consider a class of neutral functional dynamic equations with Stepanov-almost periodic terms on time scales in a Banach space. By means of the contraction mapping principle, we establish some criteria for the existence and uniqueness of almost periodic solutions for this class of dynamic equations on time scales. Finally, we give an example to illustrate the effectiveness of our results.
Keywords: Neutral functional dynamic equation, Stepanov-almost periodic function, the contraction mapping principle, time scales.
Mathematics Subject Classification: Primary: 34K40, 34K14; Secondary: 34N05.
Citation: Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022
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Yongkun Li Pan Wang | CommonCrawl |
\begin{document}
\setcounter{secnumdepth}{6} \setcounter{tocdepth}{2} \newcounter{temp}\newcounter{kk}\newcounter{iter}\newcounter{Vh}\newcounter{Ch} \newcounter{tempx}\newcounter{tempy}\newcounter{Lx}\newcounter{Ly}\newcounter{Lslope} \newcounter{tx1}\newcounter{tx2}\newcounter{tx3}\newcounter{ty1}\newcounter{ty2}\newcounter{ty3} \newcounter{x1}\newcounter{x2}\newcounter{x3}\newcounter{y1}\newcounter{y2}\newcounter{y3}
\title{Enumeration of uni-singular algebraic hypersurfaces
} \begin{abstract} We enumerate complex algebraic hypersurfaces in $\mP^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in details the generalized Newton-non-degenerate singularities. We give also examples of enumeration in some other cases. \end{abstract} \tableofcontents
\section{Introduction} \subsubsection{Preface} This paper is a generalization of the previous one \cite{Ker}, where enumeration of plane singular curves (with one singular point) was done. The results for curves were as follows: ~\\ $\bullet$ for a large class of singularity types (the so-called linear singularities) we gave a method to write immediately
explicit formulae solving enumeration problem. ~\\ $\bullet$ for all other singularity types we gave an algorithm (which is quite efficient and for every particular singularity type gives the final answer in a bounded number of steps).
The goal of this work is to generalize the method to the case of (uni-singular, complex, algebraic) hypersurfaces in $\mP^n$.
The theory of singular hypersurfaces is much richer and complicated than that of curves. Correspondingly the enumeration is much more difficult both technically and conceptually. It seems that there does not exist one (relatively) easy method applicable to all types of singularities. We propose a method of calculation applicable to a class of the {\it generalized Newton-non-degenerate} singularities (this includes in particular the $A,D,E$ types and all the singularities with $\mu\le14$). The method can also be applied to some other singularity types, we consider examples in Appendix A.
\subsubsection{General settings} We work with (complex) algebraic hypersurfaces in the ambient space $\mP^n$. A hypersurface is defined by a polynomial equation $f(x)=0$ of degree $d$ in the homogeneous coordinates of $\mP^n$. The parameter space of such hypersurfaces (the space of homogeneous polynomials of total degree $d$ in $(n+1)$ variables) is a projective space. We denote it by $\mPN$ (here \mbox{$N_d={d+n\choose{n}}-1$} and $d$ is assumed to be sufficiently high).
The {\it discriminant}, $\Sigma\subset\mPN$, is the (projective) subvariety of the parameter space, whose points correspond to the singular hypersurfaces (generic points of the discriminant correspond to hypersurfaces with one node). Everywhere in this paper (except for Appendix A) we restrict consideration {\it to isolated} singularities. Even more, we consider only hypersurfaces with just {\it one} singular point.
When working with a singular point (specifying its type, parameters etc.) we usually pass from the category of projective hypersurfaces to that of hypersurface germs and consider everything in a small neighborhood in classical topology.
Consider the classification by (local embedded) topological type: two hypersurface germs $(V_i,0)\subset(\mC^n,0)$ are of the same type if there exists a homeomorphism $\mC^n\stackrel{\phi}{\rightarrow}\mC^n$ such that $\phi(V_1)=V_2$. Two singular projective hypersurfaces are said to be of the same (local embedded topological) singularity type $\mS$, if the corresponding germs are. For a given topological type $\mS$, consider the stratum of (projective) hypersurfaces $\Si_\mS$ with a singular point of this type.
Unlike the case of curves the notions of the topological equivalence and the corresponding equisingular stratification are quite complicated for hypersurfaces (as is discussed shortly in $\S$ \ref{SecOnEquisingularity}). For the purposes of enumeration we use a more restricted equivalence: by the Newton diagram. We work always with {\it commode} (convenient) diagrams.
A singular hypersurface germ is called {\it generalized Newton-non-degenerate} if it can be brought to a Newton-non-degenerate form by locally analytic transformations. A hypersurface with one singular point is called \gNnd if the corresponding germ is \gNnd.
(For the precise definitions and discussion of the relevant notions from singularities cf. $\S$ \ref{SecTypesSingul}). Everywhere in this paper (except for Appendix A) we consider \gNnd hypersurfaces.
\bed\label{DefNDEquisingularStratum} Two (generalized Newton-non-degenerate) hypersurface-germs are called ND-equivalent if they can be brought by locally analytic transformations to \Nnd forms with the same Newton diagram.
For a given Newton diagram $\mD$, the equisingular family $\Sigma_{\mD}^{d,n}$ is defined as the set of all the points in the parameter space $\mPN$, corresponding to \gNnd hypersurfaces of degree $d$ that can be brought by locally analytic transformations to $\mD$. \eed Note that for \gNnd hypersurfaces this equivalence implies equivalence by the embedded topological type. In general this equivalence is weaker than the (contact) analytic equivalence. Therefore we call this equivalence (and the corresponding families) {\it ND-topological}. From now on (except for $\S$ \ref{SecOnEquisingularity}), by singularity type we mean the ND-topological type $\mD$.
As the degree of the hypersurfaces $d$ and the dimension $n$ are always fixed we omit them.
For the enumeration purposes we always work with the {\it topological closure} of the strata
($\bar\Si_\mD\subset\mPN$), to simplify the formulae we usually omit the closure sign
(e.g. $\Sigma_{A_k}$, $\Sigma_{D_k}$, $\Sigma_{E_k}$ etc.).
The so defined closures sometimes coincide with the closures of the topological equisingular strata (section \ref{SecOnEquisingularity}). For example this is the case for $A,D,E$ singularities \cite{AVGL,Var}.
To specify the ND-topological singularity type (i.e. to construct the corresponding diagram) we usually give
a representative. In the simplest cases this representative (the normal form) is classically fixed
(cf. tables in \cite[chapter 1]{AVGL}). \bex The normal forms of some simplest singularities (since we do not consider analytical equivalence the moduli are omitted): \beq\ber\label{NormalForms} A_k:~z_1^{k+1},~~D_k:~z_1^2z_2+z_2^{k-1},~~E_{6k}:~z_1^3+z_2^{3k+1},~~E_{6k+1}:~z_1^3+z_1z_2^{2k+1},~~E_{6k+2}:~z_1^3+z_2^{3k+2} \\ P_8:~z_1^3+z_2^3+z_3^3,~~X_9:~z_1^4+z_2^4,~~J_{10}:~z_1^3+z_2^6,~~T_{p,q,r}:~z_1^p+z_2^q+z_3^r+z_1z_2z_3,~~ \frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1 \\ Q_{10}:~z^3_1+z^4_2+z_2z^2_3,~~S_{11}:~z^4_1+z^2_2z_3+z_1z^2_3,~~U_{12}:~z^3_1+z^3_2+z^4_3 \eer\eeq Here we consider the normal form up to stable equivalence, i.e. up to the (non~degenerate) quadratic forms: $f(z_1,\dots,z_k)+\sum_{i=k+1}^nz_i^2\sim f(z_1,\dots,z_k)$. \eex
Everywhere in the paper we assume that the degree of the hypersurfaces, $d$, is high. A sufficient condition is: $d$ is bigger than the degree of determinacy for a given singularity type (the later is often the maximal coordinate of the Newton diagram). This condition is not necessary, e.g. in the case of curves the method works also in the irregular region (of small $d$), the algorithm should be slightly modified \cite[section 5]{Ker}.
The so defined equsingular strata are (almost by construction): non-empty, algebraic, pure dimensional
and irreducible (cf. proposition \ref{ClaimNDtopStratumIsAlgVariety}).
Therefore the enumerative problem is well defined.
Every equisingular stratum is (by construction) embedded into $\mPN$, we identify the stratum with its embedding. Its closure has the homology class in
the corresponding integer homology group: \beq [\overline{\Sigma}_\mD]\in H_{2dim(\Si_\mD)}(\mPN,\mZ)\approx\mZ \eeq The degree of this class is the degree of the stratum. The {\it cohomology class of the stratum}, $[\overline{\Sigma}_\mD]\in H^{2N_d-2dim(\Si_\mD)}(\mPN,\mZ)$, is the class dual (by Poincare duality) to the above homology class.
We denote the homology and cohomology class by the same letter, no confusion should arise.
\subsubsection{The goal of the paper, motivation and main results} The goal of this paper is to provide a method to enumerate the hypersurfaces, i.e. to calculate the cohomology classes, for the strata corresponding to hypersurfaces with one singular point of a given ND-topological type (for generalized Newton-non-degenerate hypersurface-germs). \\ \\ \\ The discriminant and more generally, varieties of equisingular hypersurfaces (Severi-type varieties), have been a subject of study for a long time. Already in 19'th century it was known, that the (closure of the) variety of nodal hypersurfaces of degree d in $\mP^n$ (i.e. the discriminant) is an irreducible subvariety of $\mPN$ of codimension 1 and degree: \beq (n+1)(d-1)^n \eeq Any further progress happens to be difficult. The work was mainly concentrated on the enumeration of curves on surfaces \cite{KleiPien1} with many simple singularities.
The present situation in enumeration of singular hypersurfaces seems to be as follows. (This is not a complete/historical review, for a much better description cf: \cite{Klei1,Klei2,Kaz4}.) ~\\ $\bullet$ {In 1998 P.Aluffi\cite{Alufi}} has calculated the degrees of the strata of cuspidal and bi-nodal hypersurfaces
($\Si_{A_2}$, $\Si_{2A_1}$) ~\\ $\bullet$ {In 2001 R.Hern\'aández and M.J.V\'aázquez-Gallo\cite{HV}} enumerated most of the singularities
of cubic surfaces in $\mP^3$. ~\\ $\bullet$ {In 2003 I.Vainsencher\cite{Vain2}} has calculated the degrees of some strata of multi-nodal hypersurfaces ($\Si_{rA_1}$ for $r\leq6$). ~\\ $\bullet$ {In 2000-2003 M.Kazarian in the series of papers \cite{Kaz1,Kaz2,Kaz3,Kaz4}} used topological approach to prove that there exist {\it universal} formulae for the degrees of equisingular strata. In the spirit of Thom \cite{Thom54}, they are (unknown) polynomials in some combinations of the Chern classes of the ambient space and the linear family. (In our case these are the Chern classes of $\mP^n$, and $\mPN$.) The coefficients of those polynomials depend on the singularity type only (and not on the degree of hypersurfaces or their dimension). The enumerative answer for a particular question is obtained just by substitution of the corresponding Chern classes into the universal polynomial.
Kazarian has developed a method for calculation of these Thom polynomials. From them one gets the degrees for strata of singular hypersurfaces. In particular, he gives the answers for all the possible combinations of types up to codimension 7.
His method enumerates all the singularity types of the given codimension simultaneously. Therefore it needs a preliminary classification of the singularities of a given codimension. Even if one does this, the computations are non-effective when one needs the answer for just one type (e.g. $A_k$)
Our motivation was to show that the approach suggested in \cite{Ker} (and used there to completely solve the problem for plane uni-singular curves) generalizes to the case of hypersurfaces. In particular, from the theorems \ref{TheoremEnumLinearSings},\ref{TheoremEnumNonLinearSings} it follows \bcor\label{ClaimMainOfTheMethod} The proposed method of degenerations (the algorithm) allows enumeration of any (generalized Newton non-degenerate) singularity (in a bounded number of steps). \ecor
The result of the enumeration procedure is the cohomology class of a stratum $Si_\mD$: a polynomial of degree $n$ (the dimension of the ambient space, $\mP^n$) in $d$ (the degree of hypersurfaces). The coefficients are functions of $n$ and of the singularity type. As an example of calculations we have \bprop The cohomology classes of the lifted strata in the following cases are given in Appendix \ref{SecApendCohomologClasses}:
ordinary multiple points, reducible multiple points (see the definition in Appendix \ref{SecReducibleForms}), $A_{k\leq4}$, $D_{k\leq6}$, $E_{6}$, $P_8, X_9$, $Q_{10}$, $S_{11}$, $U_{12}$
\eprop From our results one can obtain some restrictions on the universal Thom polynomials. We should note, however, that from our results it is impossible to recover Thom polynomials completely when the $\mu\ge7$ (cf. Appendix C).
\subsubsection{Description of the method} Here we briefly describe the method. It is considered in more details in $\S$ \ref{SecMainIssues}. We start from the ingredients and then formulate the enumeration theorems.
We go in a naive way, trying to work with (locally) complete intersections of hypersurfaces defined by explicit equations. The resulting cohomology class is obtained as the product of the classes of hypersurfaces, with various corrections subtracted. Repeat that we always work with the closed strata.
\paragraph{\hspace{-0.3cm}Liftings.\hspace{-0.3cm}}\label{SecMinimalLifting} To write explicit equations,
lift a given equisingular stratum (which initially lies in $\mPN$) to a bigger space ($Aux\times\mPN$). Here $Aux$ is an auxilliary space that traces the parameters of the singularity (singular point, tangent cone etc.). \bex The {\it minimal lifting (partial desingularization)} is just the universal hypersurface \beq\label{ExampleLiftings}
\tSi_\mD(x):=\overline{\left\{(x,f)\Big|\ber \mbox{The~hypersurface~defined~by~}f(x)=0~\mbox{has}\\ \mbox{singularity type}~\mD~\mbox{at~the point~}x\eer\right\}}\subset\mP^n_x\times\mPN \eeq Here $\mP^n_x$ is the ambient space of singular hypersurfaces (the subscript $x$ emphasizes that the point of the space
is denoted by $x$). \eex The cohomology class of the lifted version ($\tSi_\mD$) is easier to calculate (e.g. for ordinary multiple point the lifted stratum is just a complete intersection, cf. $\S$ \ref{SecOrdinaryMultiplePoint}). The (co)homology class is now not just a number, but a polynomial (in the generators of the cohomology ring of the bigger ambient space). Hence we have the multidegree of $\tSi$. This provides, of course, much more information about a particular stratum. \\\\ Once the class $[\tSi_\mD]$ has been calculated, the cohomology class of the
original stratum ($[\Si_\mD]$) is obtained using the Gysin homomorphism. Namely, the projection $Aux\times\mPN\stackrel{\pi}{\rightarrow}\mPN$ induces the projection on homology: $H_{i}(Aux\times\mPN)\stackrel{\pi_*}{\rightarrow}H_i(\mPN)$. By Poincare duality this gives a projection in cohomology: $H^{i+2dim(Aux)}(Aux\times\mPN)\stackrel{\pi_*}{\rightarrow}H^i(\mPN)$. It sends the component $H^k(\mPN)\otimes H^{2dim(Aux)}(Aux)$ isomorphically to $H^k(\mPN)$ and sends all other cohomology classes to zero.
From the calculational point of view, we should just extract from the cohomology class $[\tSi_\mD]$ (which is a polynomial in the cohomology ring of $Aux\times\mPN$) the coefficient of the maximal powers of the generators of $H^*(Aux)$.
In the above example of minimal lifting this homomorphism is $H^{k+2n}(\mP^n_x\times\mPN)\mapsto H^{k}(\mPN)$, and from $[\tSi_\mD]$ one should extract the coefficient of the $n$'th power of the generator of $H^*(\mP^n_x)$.
Summarizing: the cohomology class of a stratum $\Sigma_\mD$ is completely fixed by that of its lifted version $\tSi_\mD$.
\paragraph{\hspace{-0.3cm}Degenerations from simple to complicated.\hspace{-0.3cm}}The lifted stratum can often be globally defined by some explicit equations (the case of {\it linear singularity}, cf. definition \ref{DefLinearSingularities}
in $\S$ \ref{SecLinearSingularDefinitions}).
Unfortunately it is usually only a locally complete intersection (but not globally). So, if one chooses a locally defining set of hypersurfaces, their intersection contains some residual (unnecessary) pieces, whose contribution to the cohomology class should be subtracted. The serious difficulty is that these residual pieces can be of dimension {\it bigger} than the true stratum.
In this case we proceed as follows. Let $\mD$ be the singularity under consideration, let $\mD_0$ be some singularity type to which $\mD$ is adjacent and for which the enumeration is already done (the trivial choice for $\mD_0$ is just an ordinary multiple point of the same multiplicity as $\mD$). Represent $\mD$ as a chain of successive degenerations (i.e. adjacencies), starting from $\mD_0$. At each step the codimension of the variety grows by 1, the stratum being intersected by a hypersurface. Each intersection can be non-transversal somewhere, the resulting variety of the intersection is usually reducible. In addition to the needed (true) variety it contains some residual varieties. We emphasize, that at each step the intersection is with a hypersurface, therefore the dimensions of residual pieces are not bigger than that of the true variety. Thus, at each step the contribution of residual pieces can be removed from the cohomology class of the intersection. In this process one should check: ~\\ $\bullet$ where the non-transversality happens? This question is considered in $\S$ \ref{SecPossibleCyclesOfJump} using proposition \ref{ClaimTransversalityUnderNewtonDiagram}. ~\\ $\bullet$ what are the residual pieces produced in the intersection? This question is considered in sections \ref{SecLiftings} and \ref{SecIdeologyOfDegenerations}. ~\\ $\bullet$ how "to remove" their contributions from the answer? This amounts to calculation of the cohomology classes of residual varieties and is considered in sections \ref{SecCohomologyClassesOfCyclesofJump} and \ref{SecCohomologyClassOfRestrictionFibration}.
All above can be formulated as a proposition (proved in \ref{SecIdeologyOfDegenerations}): \bthe\label{TheoremEnumLinearSings} ~\\ $\bullet$ For a given linear type $\mD$ and an auxiliary type $\mD_0$ the algorithm forms the chain of degenerations $\mD_0\rightarrow..\rightarrow\mD$. All the vertices correspond to linear types (fixed by the choice of $\mD,\mD_0$). The number of vertices equals $codim(\Si_{\mD})-codim(\Si_{\mD_0})$. ~\\ $\bullet$ Each edge $\mD_i\rightarrow\mD_{i+1}$ (a codimension 1 degeneration) provides a linear expression
for the class $[\mD_{i+1}]$ in terms of $[\mD_i]$ and the classes of residual cycles. The residual cycles are
fixed by the geometry of the degeneration. ~\\ $\bullet$ The number of steps, needed to achieve the result, is not bigger than the number of points under the Newton diagram. \ethe
\paragraph{\hspace{-0.3cm}Degenerations from complicated to simple (simplifying degenerations).\hspace{-0.3cm}}\label{SecSimplifyingDegenerations}
In most cases, even the lifted stratum is difficult to define explicitly. This is the case of {\it non-linear} singularities (defined in $\S$ \ref{SecLinearSingularDefinitions}). Then, instead of trying to arrive at the needed singularity $\mD$ by degenerations of some
simpler singularity, we degenerate the $\mD$ itself. The goal is to arrive at some singularity ($\mD'$) of higher
codimension (or higher Milnor number),
which is however simple to work with (e.g. linear singularity).
Or, geometrically, we intersect the lifted stratum ($\tSi_{\mD}$) with a cycle in the ambient space so that the cohomology class of the resulting stratum ($\tSi_{\mD'}$) is easier to calculate. Then (if the intersection is transversal) we have the equation for the cohomology classes: \beq [\tSi_{\mD}]\times[\mbox{degenerating cycle}]=[\tSi_{\mD'}]\in H^*(Aux\times\mPN) \eeq We choose the degenerating cycle in the appropriate manner, so that the above equation of cohomology classes fixes the class of $\tSi_{\mD}$ uniquely (cf. $\S$ \ref{SecInvertibilityofDegeneration}).
In fact the situation is more complicated: ~\\ $\bullet$ The resulting stratum in general is reducible and non reduced. Its reduced components enter with different multiplicities (since the initial stratum $\tSi_{\mD}$ is singular at these loci). The resulting strata and their multiplicities are obtained from the defining ideal, by explicit check of the equations. ~\\ $\bullet$ The needed stratum $\tSi_{\mD'}$ usually is not a complete intersection. Thus, on the right hand side in the above equation there can appear some residual pieces. In this case one should also remove their contributions.
The result for non-linear (\gNnd) singularities is (proved in $\S$ \ref{SecIdeologyOfDegenerations}): \bthe\label{TheoremEnumNonLinearSings} ~\\ $\bullet$ For each (non-linear) singularity type $\mD$ a tree of degenerations is constructed. The root of the tree is the original type $\mD$, the leaves are some linear singularities, adjacent to the original stratum. This tree is constructed from the Newton diagram of the given singularity type, without any preliminary classification or preliminary knowledge of adjacent strata. ~\\ $\bullet$ Every edge of the tree corresponds to a degeneration, resulting in a pure dimensional variety. The corresponding equation for cohomology classes is of the form $[\tSi_{\mD_i}]\times[\bet degenerating\\divisor\eet]$=$\sum a_j[\tSi_{\mD_{i+1,j}}]+[\bet residual\\piece\eet]$. The cohomology class of the residual piece is calculated by a standard procedure. The cohomology class of the stratum $[\tSi_{\mD_i}]$ is restored uniquely from this equation. ~\\ $\bullet$ If the initial non-linear singularity has order of determinacy $k$ and multiplicity $p$, then the number of vertices in this tree is not bigger than ${k+n\choose{n}}-{p-1+n\choose{n}}$. \ethe
\paragraph{\hspace{-0.3cm}Some special simple cases.\hspace{-0.3cm}} In some (very special) cases the enumeration is almost immediate. These are the cases of (mostly) Newton degenerate singularities with reducible jets (section \ref{SecReducibleJets}), i.e.$jet_p(f)$ defines a reducible hypersurface. The simplest (nontrivial) example is the degenerate multiple point of order $p$ with hypersurfaces of a form: $f(z_1,\dots,z_n)=\prod_i \Omega^{(p_i)}_i+higher~order~terms$. Here $\Omega^{(p_i)}_i$ are non-degenerate mutually generic homogeneous forms of orders $p_i$ such that $\sum_ip_i=p$.
In this case, the equisingular family is defined by reducibility of the tensor of derivatives (Appendix A). As reducibility is not invariant under topological transformations, the equsingular family in this case does not coincide with the ND-topological stratum. The enumeration goes in the same way as the enumeration of curves \cite{Ker}.
\paragraph{\hspace{-0.3cm}Computer calculations and efficiency.\hspace{-0.3cm}} As we are working with polynomials of high degree in many variables, computer is used. The calculations are essentially polynomial algebra: add/subtract polynomials, multiply, open the brackets, eliminate variables, solve a big system linear equations etc. Therefore a restriction arises from a computer's speed and memory. We discuss some aspects of this step in $\S$ \ref{SecComputerCalcul}.
The restrictions are not severe for linear singularities, but are quite tough for non-linear ones. In particular for $A_8$ (a "very non-linear" case) there was just not enough memory even in the case of curves. We emphasize, however that this is a purely computer restriction.
\subsubsection{The simplest examples} The case of ordinary multiple point is elementary because the defining conditions of the lifted stratum are globally transversal. Usually the best we can hope for is to obtain the locally transversal conditions.
In this case, every time we degenerate, we should check for possible residual varieties and remove their contributions if necessary. This technique is most simply demonstrated by the case of a node, naively defined in affine coordinates.
\paragraph{Globally complete intersections: ordinary multiple point, $f=\sum z^{p+1}_i$.}\label{SecOrdinaryMultiplePoint}
We work with hypersurfaces $\{f(x)=0\}\subset\mP^n_x$ of degree $d$. The defining condition here is: all the derivatives up to order $p$ should vanish. This can be written as $f|_x^{(p)}=0$ (tensor of derivatives of order $p$, in homogeneous coordinates, calculated at the point $x$). The lifted variety in this case is: \beq\label{node}
\tSi(x)=\{(f,x)|~f|_x^{(p)}=0\}\subset\mPN\times\mP^n_x \eeq (Recall, we always speak about topological closure of $\Si_\mD$.) This variety is defined by ${p+n\choose{n}}$ transversal conditions. The transversality is proven in general in $\S$ \ref{SecDefinCondLinearQuasiHomog}. For pedagogical reasons we check it here explicitly.
Note, that $\mP GL(n+1)$ acts freely and transitively on $\mP^n_x$, therefore it is sufficient to check the transversality, at some particular point. For example
fix $x=(1,0,\dots,0)\in\mP^n_x$. Then the conditions of (\ref{node}) are just linear equations in the space $\mPN$ of all polynomials of the given degree, so the transversality is equivalent to linear independence. And it is checked directly (note that $d$ is sufficiently high).
Thus the variety is a globally complete intersection and its cohomology class is just the product of the classes of defining hypersurfaces. Since all the hypersurfaces have the same class, we obtain\: \beq\label{nodefull} [\tSi(x)]=\left((d-p)X+F\right)^{n+p\choose{p}} \eeq Here $F,X$ are the generators of the cohomology ring of $\mPN\times\mP^n_x$.
To obtain the cohomology class of $\Sigma$ we apply the Gysin homomorphism (as explained in section \ref{SecMinimalLifting}). From the expression in (\ref{nodefull}) one should extract the maximal non-vanishing power of $X$, i.e. $X^n$. The coefficient of this term is the cohomology class of the needed stratum. This gives the degree: \beq\label{DegMultPoint} deg(\Sigma)={{n+p\choose{p}}\choose{n}}(d-p)^n \eeq
\paragraph{Locally complete intersections: nodal hypersurfaces defined in affine coordinates.} Let $x=(z_0\dots z_n)$ be the homogeneous coordinates in $\mP^n_x$. Choose the affine part: $(z_0\neq0)\subset\mP^n_x$. In local coordinates a hypersurface has a node if the corresponding function vanishes together with its derivatives. Thus we define: \beq\label{nodeaff}
\widetilde\Xi(x)=\{(f,x)|\partial_1f|_x=\dots=\partial_nf|_x=0,~f|_x=0\}\subset\mPN\times\mP^n_x \eeq Over the affine part $\{z_0\ne0\}\subset\mP^n_x$ this variety coincides with the lifted stratum of nodal curves, $\tSi_{A_1}(x)$.
However at infinity ($z_0=0$) one can expect some additional pieces.
Indeed, the Euler formula (\ref{Euler}) shows that the equations of (\ref{nodeaff}), when translated to
the neighborhood of infinity, are:
\beq z_0\partial_0f|_x=0=\partial_1f|_x=\dots=\partial_nf|_x \eeq That is, the (projective closure of the) variety of (\ref{nodeaff}) is reducible: it is the union of $\tSi_{A_1}(x)$ and some residual variety (at $z_0=0$), taken with multiplicity one (since $z_0$ appears in the first degree).
In terms of co-homology classes: \beq [\widetilde\Xi(x)]=[\tSi_{A_1}(x)]+1[z_0=0,~\partial_1f=\dots=\partial_nf=0] \eeq So, to calculate the (co)homological class $[\tSi_{A_1}(x)]$ one should subtract from $[\widetilde\Xi(x)]$ the (co)homological class of the variety defined by: $\{z_0=0,~\partial_1f=\dots=\partial_nf=0\}$.
Explicit calculation gives the degree of the discriminant (i.e. the result of (\ref{DegMultPoint}) in the case $p=1$).
\subsubsection{Organization of material} The main body of the paper gives the proof of the corollary \ref{ClaimMainOfTheMethod}.
In $\S$ 2 we recall some important definitions, fix the notations and introduce some auxiliary notions used throughout the paper. We discuss the singularity types and the strata (topological and ND-topological, section \ref{SecOnEquisingularity}) and clarify their relation. In $\S$ \ref{SecLinearSingularDefinitions} we introduce
{\it linear singularities}. Then (in \ref{SecDefiningConditionsGeneral}) we formulate {\it covariant defining conditions} for linear singularities.
In $\S$ \ref{SecLiftings} we define the {\it liftings} of the strata and consider related questions. Then discuss the problem of non-transversality (\ref{SecPossibleCyclesOfJump}) and obtain the characterization of points of non-transversal intersection.
In $\S$ \ref{SecIdeologyOfDegenerations} we prove the main theorems (\ref{TheoremEnumLinearSings} and \ref{TheoremEnumNonLinearSings}). Essentially just collecting all the results from sections \ref{SecDefinitionsAuxilliaryResults}, \ref{SecMainIssues} and Appendix.
In $\S$ \ref{SecExamples} we demonstrate the algorithm by simple examples: the cusp $A_2$ and the tacnode $A_3$. In $\S$ \ref{SecFurtherCalculations} we consider some higher singularities. We start from the double points of a given co-rank, this corresponds to $A_2,D_4,P_8..$ Then, further degenerations are considered (e.g. $A_4,D_5,E_6$).
In most part of the paper we work with \gNnd singularities. In Appendix A we consider a special subclass of Newton-degenerate singularities: with reducible jets.
Appendix B is devoted to some results from intersection theory, intensively used throughout the paper (multiplicity of intersections, cohomology classes of some special varieties and restrictions of fibrations).
In Appendix C we give some explicit results (cohomology classes) and discuss the issues of computer calculations and some consistency checks of the formulae.
\section{Some definitions and auxiliary results}\label{SecDefinitionsAuxilliaryResults} \subsection{On variables and notations}\label{SectVar} \subsubsection{On coordinates.\hspace{-0.3cm}} In this paper we deal with many varieties, embedded into various (products of) projective spaces. Adopt the following convention. If we denote a point in a projective space by the letter $x$, the corresponding projective space is denoted by $\mP^n_x$. The points of projective spaces will be typically denoted by $x,y$ or $y_i$. For enumerative purposes we use homogeneous coordinates: $x=(z_0:\dots:z_n)\in\mP^n_x$. When considering a particular singular hypersurface-germ we use local coordinates, centered at the singular point, e.g. $(z_1,\dots,z_n)$ (assuming $z_0=1$).
When working with multi-projective space, the point $(x_1,\dots,x_k)\in\mP^n_{x_1}\times\dots\times\mP^n_{x_k}$ is called generic if no subset of the points $x_{i_1},\dots,x_{i_l}$ lies in a $(l-2)$-plane. The points $(x_1,\dots,x_k)$ will always be assumed mutually generic, unless a restriction is explicitly specified. By identifying $\mPN=\mP roj(V)$, we will often consider a point $x\in\mP^n_x$ as a vector of $(n+1)$ dimensional
vector space $V$ (defined up to a scalar multiplication). For example, the above condition of genericity can be formulated as: the vectors $x_1,\dots,x_k$ are linearly independent.
A hyperplane in $\mP^n$ is defined by a 1-form: $l\in(\mP^n_l)^*$. So, e.g. the incidence variety of hyperplanes and their points is defined as: \beq
\{(l,x)|l(x)=0\}\subset\mP^n_x\times(\mP^n_l)^* \eeq
\subsubsection{On the monomial order.\hspace{-0.3cm}}\label{SecOnTheMonomialOrder} For the purpose of degeneration we should fix an order on monomials $\bf{z}^{\bf I}$. So, we say that $\bf{z}^{\bf I}<\bf{z}^{\bf J}$ if $|I|<|J|$ (the total degrees). For $|I|=|J|$ the order could be defined quite arbitrarily, we chose the lexicographic: $z_1>z_2>...z_n$.
\subsubsection{On symmetric forms.\hspace{-0.3cm}}We will often work with symmetric $p-$forms $\Omega^{p}\!\in\! S^p V^*$ (here $S^p V$ is a symmetric power of $(n+1)$ dimensional vector space). Thinking of a form as being a symmetric tensor with $p$ indices ($\Omega^{(p)}_{i_1,\dots,i_p}$), we often write $\Omega^{(p)}(\underbrace{x,\dots,x}_{k})$ as a shorthand for the tensor multiplied $k$ times by a point $x=(z_0,\dots,z_n)\in\mP^n$ (considered here as a vector in $V$): \beq \Omega^{(p)}(\underbrace{x,\dots,x}_{k}):=\sum_{0\le i_1,\dots,i_k\le n}\Omega^{(p)}_{i_1,\dots,i_p}z_{i_1}\dots z_{i_k} \eeq So, for example, the expression $\Omega^{(p)}(x)$ is a $(p-1)$ form. Unless stated otherwise, we assume the symmetric form $\Omega^{(p)}$ to be generic (in particular non-degenerate, i.e. the corresponding hypersurface $\{\Omega^{(p)}(\underbrace{x,\dots,x}_{p})=0\}\subset\mP_x^n$ is smooth).
Symmetric forms will typically occur as tensors of derivatives of order $p$: e.g. $f^{(p)}$ (here $f$ is a homogeneous polynomial defining a hypersurface). Sometimes, to emphasize the point at which the derivative is calculated we assign it. So, e.g.
$f|_x^{(p)}(\underbrace{y,\dots,y}_k)$ means: the tensor of derivatives of $p$'th order, calculated at the point $x$, and contracted $k$ times with $y$.
Throughout the paper we tacitly assume the Euler identity for a homogeneous polynomial of degree $d$ \beq\label{Euler} \sum_{i=0}^nz_i\partial_if=df \eeq and its consequences (e.g. $\sum_iz_i\partial_i\partial_jf=(d-1)\partial_jf$). So, for example, the nodal point,
defined by $f|_x^{(1)}=0$, can also be defined by $f|_x^{(p)}(\underbrace{x,\dots,x}_{p-1})=0$.
\subsubsection{On cohomology classes.\hspace{-0.3cm}}The generator of the cohomology ring of $\mP^n_x$ are denoted by the corresponding upper case letter $X$, so that $H^*(\mP^n_x)=\mZ[X]/(X^{n+1})$. Alternatively, $X=c_1({\cal O}_{\mP^n_x}(1))$. By the same letter we also denote the hyperplane class in homology of $\mP^n_x$. Since it is always clear, where we speak about coordinates and where about (co)homology classes, no confusion should arise.
To demonstrate this, consider the hypersurface: \beq
\Si=\{(x,y,f)|~f(x,y)=0\}\subset\mP^n_{x}\times\mP^n_{y}\times\mPN \eeq Here $f$ is a polynomial of bi-degree $d_x,d_y$ in homogeneous coordinates $x=(z_0:\dots:z_n),y=(w_0:\dots:w_n)$, the coefficients of $f$ are the homogeneous coordinates of the parameter space $\mPN$. The cohomology class of this variety is: \beq [\Si]=d_xX+d_yY+F\in H^2(\mP^n_{x}\times\mP^n_{y}\times\mPN) \eeq
The formulae for the cohomology classes of the lifted strata are polynomials in the generators of the cohomology rings of the products of projective spaces ($X$ for $\mP^n_{x}$, $F$ for $\mPN$ etc..) The polynomials depend actually on some combinations of the generators. For example, $F$ always enters as $F+(d-k)X$ (for some $k\in\mN$, which depends on the singularity type only). Correspondingly, we always use the (relative) class \beq Q:=(d-k)X+F \eeq (the value of $k$ will be specified or evident from the context).
\subsubsection{On the strata.\hspace{-0.3cm}} We denote an ND-topological stratum by $\Sigma$ (it will be always clear from the context, which singularity type is meant). The lifted stratum is denoted by $\tSi$. Usually there will be many liftings for one stratum, to distinguish between them, we assign the auxiliary parameters. So, e.g. the stratum defined in (\ref{ExampleLiftings}) is denoted by $\tSi(x)$.
We always work with the topological closures of the strata. To simplify the formulae we
write just $\Sigma$ (or $\tSi$) for the closure of a (lifted) equisingular stratum. The lifted stratum $\tSi$ is often considered as a fibration over the auxiliary space. For a cycle $C$ in the auxiliary space the (scheme-theoretic) restriction of the fibration to the cycle is denoted by: $\tSi|_C$. For example,
if $x$ is a point of the auxiliary space, then $\tSi|_x$ is the fibre over $x$.
\subsection{On the residual varieties and cycles of jump}\label{SecResidualVarieties}
We try to represent a stratum as an explicit intersection of hypersurfaces. The intersection will be usually non-transversal. The resulting variety, being reducible, will contain (except for the needed stratum) some additional pieces. We call these pieces {\bf residual varieties}.
The intersection process occurs in the space: $\mPN\times Aux$. Here $\mPN$ is the parameter space of hypersurfaces, while $Aux$ is the {\bf auxiliary space}, used to define
the lifted stratum explicitly. It will be typically a multi-projective space $Aux=\mP^{n_1}\times\dots\times\mP^{n_k}$ or a projective irreducible subvariety of it.
We often consider the lifted stratum as a fibration over the auxiliary space.
The fibres correspond to hypersurfaces with some specified parameters of the singularity (e.g. chosen singular point, tangent cone etc.) The fibration will be generically locally trivial (in Zariski topology) and this local triviality induces the stratification on the auxiliary space. \bed Let $R_1\subset\bar{R}_1=Aux$ be the maximal (open, dense) subvariety of $Aux$ over which the fibration $\tSi\rightarrow Aux$ is locally trivial. By induction, let
$R_k\subset\bar{R}_{k-1}\backslash R_{k-1}=Aux\backslash(\bigcup_{i=1}^{k-1}R_i)$ be the maximal subvariety such that the (scheme) restriction $\tSi|_{R_k}\rightarrow R_k$ is a locally trivial fibration (over each connected component). The set $\{R_i\}_i$ is called: {\bf the stratification of the auxiliary space by local triviality}. \eed Let $\bar{R}_k\backslash R_k=\bigcup m_i C_i$ be the decomposition to a union of closed irreducible subvarieties of $Aux$, not containing each other. We call these irreducible subvarieties: {\bf the cycles of jump}. The generic fibres of the projection $\tSi\rightarrow Aux$ will be linear spaces, therefore the dimension of fibers jumps over the cycles of jump (cf. $\S$ \ref{SecTypesSingul}).
Various cycles of jump can contain or intersect other cycles of jump (that appeared from $R_k$ with higher $k$). It is useful to introduce grading on these cycles. \bed\label{DefCycleOfJumpGrading}
The cycle of jump $C_i$ is assigned grading 1 if it does not contain other (proper) cycles of jump (equivalently if the fibration $\tSi|_{C_i}\rightarrow C_i$ is locally trivial). A cycle of jump is assigned grading $k$ if it contains a (proper) cycle of grading $(k-1)$ and no cycles of higher grading. \eed
\bex Quadratic forms of co-rank $r$ (this example is important for enumeration of singularities as $A_2$,$D_4$,$P_8\dots$).
Consider the variety of degenerate symmetric matrices (quadratic forms) of co-rank $r\geq2$ acting on $\mP^n$. It is a classical determinantal variety \cite{Ful}. Its lifted version is the incidence variety of degenerate quadratic forms with $r$ vectors of the kernel specified. \beq\ber \tSi(x_1,\dots,x_r):=
\overline{\Bigg\{(\!\!\!\underbrace{x_1,\dots,x_r}_{\ber\mbox{do not lie in an}\\(r-2)\mbox{-plane}\eer}\!\!\!,\Omega^{(2)})\Big| \Omega^{(2)}(x_1)=\dots=\Omega^{(2)}(x_r)=0\Bigg\}}\subset\mP^n_{x_1}\times\dots\times\mP^n_{x_r}\times\mP_\Omega
\eer\eeq (here $\mP_\Omega$ is the parameter space of quadratic forms, $dim(\mP_\Omega)={n+2\choose{2}}-1$, $Aux=\mP^n_{x_1}\times\dots\times\mP^n_{x_r}$).
The projection: $(x_1,\dots,x_r,\Omega^{(2)})\stackrel{\pi}{\mapsto}(x_1,\dots,x_r)$ is generically locally trivial fibration. The dimension of the generic fiber is ${n+2\choose{2}}-1-\frac{2n+3-r}{2}r$. The {\it cycles of jump} here are all the diagonals: $\{(x_{i_1},\dots,x_{i_k})$ are linearly dependent$\}$. The cycles of jump of minimal and maximal grades are: \beq\ber C_{min}=\{x_1=\dots=x_r\}\subset Aux~~~\rm{codim}_{Aux}(C_{min})=n(r-1) \\C_{max}=\{(x_1\dots,x_r)\mbox{ lie in an }(r-2)\rm{-plane}\}\subset Aux~~~\rm{codim}_{Aux}(C_{max})=n+2-r \eer\eeq
\eex The first important question about cycles of jump is the jump in the dimension of fibers. \bed Let $c\in C,~x\in Aux$ be generic points of the cycle of jump and of the auxiliary space. The {jump of fiber dimension}
for the cycle $C$ is $\Delta\rm{dim}_C:=\rm{dim}(\tSi|_c)-\rm{dim}(\tSi|_x)$. \eed In the example above the jumps of dimension are: \beq \Delta\rm{dim}_{Cmin}:=\frac{(2n+2-r)(r-1)}{2}-1,~~~~~\Delta\rm{dim}_{Cmax}:=n+1-r \eeq The total variety, $\tSi$, will be always irreducible, in particular of pure dimension, therefore we have immediate: \bcor\label{ClaimJumpOfDimensionVsCodimension} The jump of dimension over a cycle of jump is less than the codimension of the cycle of jump: $\Delta\rm{dim}_{C}<codim_{Aux}(C)$ \ecor
The restriction of the fibration $\tSi\rightarrow Aux$ to the cycles of jump will be the source of residual varieties, therefore we are interested in the cohomology classes of such restrictions: $\tSi|_C$. This question is considered in $\S$ \ref{SecCohomologyClassOfRestrictionFibration}. By now we need a simple technical result: \bprop\label{ClaimCycleOfJump&Hypersurf} Let $C$ be a cycle of jump and $\{C_i\}_i$ all the cycles of jump that are not contained in $C$. There exists a hypersurface in the auxiliary space that contains $C$ and does not contain any of $C_i$ (though it can intersect them). \eprop
\subsection{On singularities}\label{SecTypesSingul} For completeness we recall some notions related to singularities of hypersurfaces \cite{AVGL,Dim,Shustin}. \\ \parbox{13cm} {
For a given (singular) hypersurface ($f=\sum a_{\bf I}{\bf z}^{\bf I}$) the Newton polytope is defined as the convex hull of the support of $f$ in $\mZ^n$, namely: $conv({\bf I}\in\mZ^n|~a_{\bf I}\ne0)$. We always take a generic representative for a given singularity type. Therefore we can assume that the polynomial $f$ contains the monomials of the form $x^{d_i}_i,$ for $d_i\gg0$, and the polytope intersects all the coordinate axes. Such a germ is called {\it commode} (or convenient). In addition we assume (due to a high degree) that the hypersurface does not contain any line. The upper part of the Newton diagram is defined as $\Gamma_+:=conv\Big(\bigcup ({\bf I}+\mR^n_+)|~a_{\bf I}\ne0\Big)$. The Newton diagram is defined by $\Gamma_f:=\partial\Gamma_+$. By the above assumption it is compact and consists of a finite number of top dimensional faces. } \begin{picture}(0,0)(-60,0) \put(0,0){\vector(0,1){60}}\put(0,0){\vector(-1,-1){50}}\put(0,0){\vector(1,0){80}} \put(5,55){$\hat{e}_1$} \put(-45,-55){$\hat{e}_2$} \put(70,-10){$\hat{e}_3$} \thicklines \put(0,40){\line(-1,-2){40}} \put(-40,-40){\line(5,2){100}} \put(0,40){\line(3,-2){60}} \put(-2.5,38){$\bullet$} \put(3,42){$p$} \put(-41,-41){$\bullet$} \put(-45,-35){$q$} \put(58,-2.5){$\bullet$} \put(60,-10){$r$} \put(5,-10){$\bullet$} \put(8,-8){\line(-1,6){8}} \put(7,-7){\line(-3,-2){47}} \put(8,-8){\line(6,1){53}} \put(-15,-40){\tiny The Newton diagram of $T_{pqr}$} \put(30,35){$\Gamma_f$} \put(30,28){\vector(-1,-1){10}} \put(-15,-50){\tiny $f=z^p_1+z^q_2+z^r_3+z_1z_2z_3$} \end{picture} \\
The restriction of a polynomial $f$ to its Newton diagram: $f|_{\Gamma_f}$ is called {\it the principal part}.
\subsubsection{On the singularity types and strata}\label{SecOnEquisingularity} The embedded topological type and ND-topological type were defined in the introduction (definition \ref{DefNDEquisingularStratum}).
\bed\cite[section 3.4]{Shustin}\label{DefEquiStratumNormalForm} ~\\ $\bullet$ The singular hypersurface $\{f=0\}$ is called Newton-non-degenerate if the restriction of the polynomial $f$ to every face (of every dimension) of its Newton diagram is non-degenerate (i.e. the truncated polynomial has no singular points in the torus $(\mC^*)^k$). ~\\ $\bullet$ The singular hypersurface $f=0$ is \gNnd if it can be brought to a \Nnd form by a locally analytic transformation. \eed For example, while the hypersurface $(x-y^2)^2+y^5$ is not Newton-non-degenerate, it is certainly generalized Newton-non-degenerate. \\ A {\it topological type} is called \Nnd if at least one of its representatives is \Nnd (or \gNnd). Otherwise it is called Newton-degenerate. The following natural question seems to be open (as always, we assume the topological strata to be algebraic and irreducible):
Let $\{f=0\}$ be the {\it generic} representative of a \Nnd topological type. Is $f$ \gNnd? \\\\ All the simple and the uni-modal singularity types are Newton-non-degenerate \cite[chapter 1]{AVGL}. The simplest examples of not \gNnd hypersurfaces are $W_{1,p\geq1}^\sharp$ with Milnor number $\mu=15+p$ and $S^\sharp_{1,p\geq1}$ with Milnor number $\mu=14+p$. Even if a topological type is represented by not \gNnd hypersurface (e.g. as $W_{1,p\geq1}^\sharp$ before), it is not clear whether {\it every} representative of this type is not \gNnd (this is the converse of the previous question).
A way to check this in particular cases was pointed to me by G.M.Greuel. \bprop\label{ClaimNormalFamily} Let $\{f_\al=0\}$ be a family of hypersurfaces (with one singular point) with the following properties: ~\\ $\bullet$ They all have the same (local embedded) topological type $\mS=\mS_{f_\al}$. ~\\ $\bullet$ The family covers all the moduli. Namely, for every value of moduli for the type $\mS$, there is a representative $\{f_\al=0\}$ in this family with this value of moduli. ~\\ $\bullet$ Every member of this family is {\it not} \gNnd. \\Then every representative of the type $\mS$ is not \gNnd. \eprop The proof is immediate by observation that the whole stratum is a collection of equi-modular orbits under the group of locally analytic transformations and our family intersects every orbit.
For example, from Arnol'd's classification it follows that the topological types $W_{1,p\geq1}^\sharp$ and $S^\sharp_{1,p\geq1}$ are Newton-degenerate (since the modality in both cases is 2 and the families of normal forms indeed cover all the moduli). \\\\ \bed ~\\ $\bullet$ The singular hypersurface $\{f=0\}$ is called semi-quasi-homogeneous (SQH) if by a locally analytic transformation it can be brought to a \Nnd form whose principal part is quasi-homogeneous. ~\\ $\bullet$ The singularity type is called quasi-homogeneous if it has a SQH representative \eed In this case there is a strong result by \cite{Saito71}: \bprop Let $f$ be a quasi-homogeneous polynomial of degree $d$ and weights $w_1..w_n$, defining a singular hypersurface, of topological singularity type $\mS_f$. Let another representative of this type (algebraic hypersurface) be defined by a SQH polynomial $g$. Then $g$ is semi-quasi-homogeneous of the same degree $d$ and weights $w_1..w_n$ as $f$. \eprop
~\\\\ In case of curves the embedded topological type and its strata possess all the good properties. For example, the generic representative of the \Nnd type is \Nnd and can be brought to the given Newton diagram by locally analytic transformations. Therefore the ND-topological type often coincides with the embedded topological type (the same for the closures of the strata). For high enough degrees of curves the strata are irreducible and smooth in their interior.
For hypersurfaces the situation is much more complicated. By choosing big degree $d$ of hypersurfaces, the non-emptiness of the strata is assured. But the strata of embedded topological type can behave quite badly. Just to mention, the algebraicity of the strata has not yet been proven in general (though it is known for quite a broad class of types \cite{Var}). Even assumed to be algebraic, the topological strata can be singular and reducible \cite{Luengo}
(for large degrees of hypersurfaces). The associated Newton diagram can be non-constant along the equisingular stratum (in the sense that the transformation needed to achieve it, is not locally analytic but a homeomorphism) \cite[chapter 1, example 2.14]{Dim}. The constancy of multiplicity along the equisingular stratum has up to now been proved for semi-quasi-homogeneous singularities only \cite{Greuel,GreuelPfister1} (see also \cite{EG} for recent results). \\ \\ The equisingular strata we work with (the ND-topological strata) are chosen especially to possess all the good geometric properties. \bprop\label{ClaimNDtopStratumIsAlgVariety} For a given Newton diagram $\mD$ the stratum $\Si_\mD$ is a (non-empty) irreducible algebraic variety \eprop
Indeed, the family of hypersurfaces with a specific diagram (the fiber over the diagram) is defined by linear equations in the parameter space $\mPN$. And then the ND stratum is obtained by the action of algebraic group (locally analytic transformations) on this fiber.
From the irreducibility we get that every invariant, defined in an algebraic way (e.g. sectional Milnor numbers $\mu^*$) is semi-continuous along the ND-topological strata. \\ \\ By construction, to every ND-topological type $\mD$ an embedded topological type $\mS$ is associated, with the inclusion of the (closures of the) strata: $\Si_\mD\subset\Si_\mS$. A natural question is: when the two types of strata coincide?
\beR The simplest example of non-coincidence (or non-uniqueness of ND-topological type for a given embedded topological)
is just the case of curves: $z^p_1+z_1x^p_2+z^d_2,~~d\ge p+2$. \eeR So, to get the equality of strata $\Si_\mS=\Si_\mD$ one should demand, at least, the minimality of Newton diagram. In more details, introduce the partial order on the set of Newton diagrams with the same topological type: by inclusion. Call a diagram {\it minimal} if it is not bigger than some other diagram corresponding to the same topological type. In general, it is not clear whether the minimal Newton diagram is unique (up to a permutation of axes). Even if it is unique, it is still unclear whether the two strata coincide.
A constructive way to compare the strata is by codimension. \bed ~\\ $\bullet$ The codimension of a local embedded topological type $\mS$ is the codimension of the topological stratum $\Si_\mS$ in the space of its semi-universal deformation. It equals: $\tau-\sharp(moduli)$ (here $\tau$ is the Tjurina number). ~\\ $\bullet$ The codimension of an ND-topological type $D$, is the codimension of the stratum $\Si_\mD$ in the parameter space of the hypersurfaces $\mPN$. \eed Restrict the topological type to be \Nnd and assume that the topological stratum is an irreducible algebraic variety. From algebraicity and irreducibility we get: \bcor For a \Nnd type $\mS$, if the topological stratum $\Si_\mS$ is algebraic, irreducible and the codimension of the topological type $\mS$ equals to that of the ND-topological type $\mD(\mS)$ then the (closure of the) strata coincide: $\Si_\mS=\Si_\mD$. So, in this case the generic representative of the topological type $\mS$ is \gNnd. \ecor Another observation is the following. \bprop Let $f_\al$ be the family as in the proposition \ref{ClaimNormalFamily}. Assume also that they all have the same Newton diagram. Then the corresponding topological and ND-topological strata coincide. \eprop Using this criterion we get the coincidence of the strata for all singularities with number of moduli $\le2$ (in this case the corresponding families are classified in \cite{AVGL}).
\subsubsection{On the vector spaces associated to the Newton diagram}\label{SecVectorSpacesOfNewtonDiagram} The Newton diagram $\mD$ defines a stratification of the tangent space at the origin $T_0\mC^n$ as follows. Let $\{f=0\}$ be the generic (\gNnd) hypersurface with diagram $\mD$, let $l$ be a line through the origin, let the degree of their intersection be $k_l:=deg(k\cap \{f=0\})$. (We assume the hypersurface to be generic, in particular it does not contain lines.)
The tangent space is stratified: $T_0\mC^n=\bigsqcup_k\cU_k$, according to the intersection degree $\cU_k:=\{l\in T_0\mC^n|k_l=k\} $. Take the topological closures of $\cU_k$ and consider the irreducible components: $\bar\cU_k=\cup_j \tV_{k,j}$. Call the collection of these components $\tcV$. \bex For a SQH hypersurface, restrict to the principal (quasihomogeneous) part. Then the so obtained varieties $\tV_{kj}$ are just the vector spaces of a flag of $T_0\mC^n$. The flag may be non-complete, if some weight of (quasi-homogeneous) variables coincide. \eex \bprop The varieties $\tV_{kj}$ are vector subspaces of $\mC^n$ of the form$\{z_{i_1}\!=\!..\!=\!z_{i_j}\!=\!0\}$ (i.e. some coordinate planes). They satisfy the property: if $\tV_{kj}\subsetneqq\tV_{k'j'}$ then $k>k'$. \eprop \bpr Let $z_i=\al_it$ be the parametrization of a line through the origin. Consider its intersection with the hypersurface $\{f=\sum a_{\bf I}{\bf z}^{\bf I}=0\}$, whose Newton diagram is $\mD$. So, we study the function restricted to the line: $\sum a_{\bf I}({\bf \al t})^{\bf I}$. If the line is contained in $\cU_k$ then $\sum_{I_1+..+I_n<k} a_{\bf I}({\bf \al t})^{\bf I}=0$ This equation must be satisfied for arbitrary (generic) coefficients of $f$ and for all small values of $t$. Thus it causes the system of monomial equations of the form: ${\bf\al}^{\bf I}=0$. So, the irreducible components of $\bar\cU_k$ are coordinate planes (which do not include one another). \epr
\parbox{13.5cm} {We need a more refined stratification. Every top dimensional face $\si$ of the Newton diagram defines a flag of vector spaces as follows. Let $k_i$ denote the non-zero coordinate of the intersection points of the $(n-1)$ plane $Span(\si)$
with the coordinate axes (so $k_i$ are not necessarily integers). Apply permutation $\sigma\in S^n\subset GL(n)$ on the coordinate axes to arrange: $k_n\le\dots\le k_1$. Define vector spaces
$\{\tilde{V}_i\}$ inductively: \beq \tilde{V}_n:=span(\hat{e}_1,\dots,\hat{e}_n),~~~\tilde{V}_{i-1}:= \Big\{\ber \tilde{V}_i,~\rm{if}~k_{i-1}=k_i\\span(\hat{e}_1,\dots,\hat{e}_{i-1}),~\rm{if}~k_{i-1}>k_i\eer \eeq } \begin{picture}(0,0)(-45,0) \put(0,0){\vector(0,1){50}}\put(0,0){\vector(-1,-1){40}}\put(0,0){\vector(1,0){65}} \put(5,50){$\hat{e}_n$} \put(-30,-40){$\hat{e}_1$} \put(55,-13){$\hat{e}_2$} \put(0,-25){$Span(\si)$} \put(0,30){\line(-1,-2){30}} \put(-30,-30){\line(5,2){75}} \put(0,30){\line(3,-2){45}} \put(-3,27){$\bullet$} \put(4,29){$k_n$} \put(42,-3){$\bullet$} \put(40,10){$k_2$} \put(-33,-33){$\bullet$} \put(-45,-30){$k_1$} \end{picture} \\ Take now the inverse permutation of axes ($\sigma^{-1}\in S^n\subset GL(n)$) that restores the initial values of all $k_i$'s. Define: $V_i:=\sigma^{-1}(\tilde{V}_i)$. \bed\label{DefFlagOfNewtonDiagrQuasiHomog} The sequence of vector spaces: $\mC^n=V_n\supseteq\dots\supseteq V_1\supset\{0\}$ is called {\bf the flag} of the face $\si$. \eed For a general (not SQH) singularity the Newton diagram consists of several top-dimensional faces, each of them defines the corresponding flag. Now combine the flags together into the collection of vector spaces $\{V_\al\}_{\al\in\cV}$. (The coinciding spaces are identified.) We call $\cV$: {\it the collection of vector spaces associated to the Newton diagram}. \bex ~\\ $\bullet$ For SQH hypersurface the collection is just a flag (as in the previous example). ~\\ $\bullet$ The hypersurface $z_1^{p_1}+z_2^{p_2}+z_3^{p_3}+z_1z_2z_3+\sum_{i\ge4}z^2_i$ with $\frac{1}{p_1}+\frac{1}{p_1}+\frac{1}{p_1}<1$, $p_1\lneqq p_2\lneqq p_3$ and $p_i>3$. \\The top dimensional faces are $Span(z_1^{p_1},z_2^{p_2},z_1z_2z_3,z^2_4...,z^2_n)$ and similarly for $(2,3)$ and $(1,3)$. \\The flag of the top dimensional face is \beq 0\subset Span(\hat{e}_2)\subset Span(\hat{e}_1,\hat{e}_2)\subset Span(\hat{e}_1,\hat{e}_2,\hat{e}_3)\subset\mC^n \eeq similarly for $(2,3)$ and $1,3$. The collection of vector spaces is: \beq \{Span(\hat{e}_i)\}_{i=1,2,3},~~ \{Span(\hat{e}_i\hat{e}_j)\}_{\ber i\ne j\\i,j\le3\eer},~~ Span(\hat{e}_1\hat{e}_2\hat{e}_3),~~ \mC^n \eeq \eex \beR The latter stratification is finer than that by the degree of the intersection. As an example, for the hypersurface $z_1^{2p}+z_2^{2p}+z_3^{2p}+z^{q-1}_1z^{q-1}_2z^{q-1}_3(z_1^3+z_2^3+z_3^3)+z_1^{p-1}z_2^{p-1}$ with $2p-2>3q$ and $q>1$ we have $\tcV=\bigcup_i\{z_i=0\}$ while $\cV=\bigcup_i\{z_i=0\}\cup\{z_2=0=z_3\}\cup\{z_3=0=z_1\}$. Therefore in the following we work with the collection $\cV$. \eeR As we consider the hypersurfaces of arbitrary dimensions, it is important to check how the collection $\cV$ varies with $n$. More precisely, suppose $f_1$ (with $mult(f_1)>2$) is stably equivalent to $f_2$, i.e. $f_2(z_1..z_{n+k})=f_1(z_1..z_n)+\sum_{i>n}z_i^2$. The relation between $\cV_{f_1}$ and $\cV_{f_2}$ is described by the simple proposition: \bprop ~\\ $\bullet$ Let $\si_1..\si_r$ be the top dimensional faces of $\Gamma_{f_1}$. Then the top dimensional faces of $\Gamma_{f_2}$ are: $\Big\{conv(\si_i,\hat{e}_{n+1}.,\hat{e}_{n+r})\Big\}_{i=1..r}$. ~\\ $\bullet$ Let $\{V_\al\}_{\al\in\cV_{f_1}}$ be the collection of vector spaces associated to $\Gamma_{f_1}$. Then the collection $\cV_{f_2}$ is obtained as: $\{V_\al\}_{\al\in\cV_{f_1}}\cup \Big\{Span(V_\al,\hat{e}_{n+1}.,\hat{e}_{n+r})\Big\}_{\al\in\cV_{f_1}}$. \eprop \beR All the vector spaces above are defined in local coordinates. For enumerative purposes we need their counterparts in homogeneous coordinates. For this, embed every $V\subset\mC^n$ into $\mC^{n+1}$, by $\mC^n\ni(z_1,\dots,z_n)\stackrel{i}{\mapsto}(0,z_1,\dots,z_n)\in\mC^{n+1}$ and define: \beq \mC^{n+1}\supset\mV:=span(i(V),\hat{e}_0) \eeq The corresponding flag $\{\mV_i\}$ (or the collection of vector spaces) will be of key importance for writing down the covariant defining equations of the strata. \eeR
\subsubsection{Linear singularities}\label{SecLinearSingularDefinitions} Fix a Newton diagram and the corresponding ND-topological type $\mD$. Consider a (hypersurface) representative of this type. In general, to bring it to the fixed Newton diagram a locally analytic transformation is applied. Split it into steps are done (in local coordinates): \\$\bullet$ Move the singular point to the origin. Rotate around the origin to fix the needed tangent cone. \\$\bullet$ Make the (purely) quadratic transformation: $\vec{z}\rightarrow\vec{z}+\vec\Omega^{(2)}$ (here $\vec\Omega^{(2)}$ is an $n$-tuple of homogeneous quadratic forms), to remove some monomials. ~\\ $\bullet$ Make the (purely) cubic transformation \dots \dots
For some singularity classes the needed Newton diagram is achieved by just the first step (linear transformations). As in the case of curves \cite[section 3]{Ker}, such singularities are much simpler for enumeration purposes. As is shown later, these strata can be lifted to varieties, defined by equations linear in function or its derivatives, and therefore easy to work with. \bed\label{DefLinearSingularities} ~\\ $\bullet$ For a given ND-topological type $\mD$, a hypersurface singularity (with this singularity type) is called linear, if it can be brought to the needed Newton diagram by linear transformations only. ~\\ $\bullet$ An ND-topological type/stratum is called linear if its generic representative is linear. Otherwise the type/stratum is called non-linear. \eed The simplest linear singularity is the ordinary multiple point (here the needed diagram is achieved just by translation of the singular point to the origin). \\ \parbox{15cm} {There is an easy characterization of linear singularities via their Newton diagrams: \bel\label{ClaimCriterionLinearity} The Newton-non-degenerate singularity type is linear iff all the angles between any face of the Newton diagram
and the coordinate hyperplanes have the slope $\frac{1}{2}\leq\mbox{tg}(\alpha)\leq2$. \eel } \begin{picture}(0,0)(-10,20) \put(0,0){\vector(0,1){60}}\put(0,0){\vector(1,0){70}} \put(-3,50){$\bullet$} \put(0,52){\line(1,-2){15}} \multiput(13,23)(-4,0){4}{\line(-1,0){2}} \put(2,30){\tiny$\alpha_1$} \qbezier(12,28)(8,25)(8,23) \put(12,20){$\bullet$} \put(13,24){\line(1,-1){15}} \multiput(30,8)(-4,0){5}{\line(-1,0){2}} \put(7,12){\tiny$\alpha_2$} \qbezier(23,14)(21,13)(19,8) \put(27,5){$\bullet$} \put(30,7){\line(3,-1){20}} \put(26,2){\tiny$\alpha_k$} \qbezier(37,5)(36,3)(35,0) \put(20,45){An example}\put(30,30){for n=2} \end{picture}\\ \bpr$\Leftarrow$ Suppose all the slopes are bounded as above and a hypersurface germ has been brought to the given Newton diagram by a chain of locally analytic transformations. Start undoing these transformations to achieve the initial germ. Immediate check shows that any nonlinear analytic transformation (without linear part) has no effect on the points under the Newton diagram. Correspondingly the monomials of the initial polynomials that lie under the Newton diagram are restored by linear transformation only. But it means that the germ could be brought to the Newton diagram by linear transformations only. \\$\Rightarrow$ Suppose at least one of the angles (of the diagram) does not satisfy the condition $\frac{1}{2}\leq\mbox{tg}(\alpha)\leq2$. Then there exists a quadratic shift of coordinates that changes the Newton diagram. Of course, such shift cannot be undone by linear transformations. \epr \beR\label{CorollarLinearSingMultPOrderDeterm2p} As follows from the lemma, every Newton-non-degenerate singularity of multiplicity $p$ and order of determinacy $k$, with $k\le 2p$, is linear. \eeR In case of plane curves there is only one angle for every segment of the diagram, correspondingly the condition on the singularity to be linear is not too restrictive. In the low modality cases the curve singularities brought to a Newton diagram by projective (linear) transformation are (all the notations are from \cite{AVGL}): \\$\bullet$ Simple singularities (no moduli): $A_{k\leq3},~~D_{k\leq6},~~E_{k\leq8}$ \\$\bullet$ Unimodal singularities: $X_9(=X_{1,0}),~J_{10}(=J_{2,0}),~Z_{k\leq13},~~W_{k\leq13}$ \\$\bullet$ Bimodal: $Z_{1,0},~W_{1,0},~W_{1,1},~W_{17},~W_{18}$
In case of hypersurfaces (of dimension$\geq3$) only a few singularities in each series can be linear. The low modality cases are: \\$\bullet$ Simple singularities (no moduli): $A_{k\leq3},~~D_{k\leq5},~~E_{6}$ \\$\bullet$ Unimodal singularities: $P_8,~X_9(=X_{1,0}),~Q_{10},~S_{11},~U_{12},~T_{p,q,r}~\{p,q,r\}\leq4$
For surfaces, there are some additional unimodal linear singularities: $Q_{11},Q_{12}$,$S_{12},Q_{2,0}$,$S_{1,p\leq1}$, $U_{1,q\leq2}$, $Q_{17}$, $Q_{18}$, $S_{16}$,$S_{17}$
We emphasize, that there is infinity of linear singularities. More precisely: for every singularity type $\mD_1$, there is a linear singularity $\mD_2$, which is adjacent to $\mD_1$ (e.g. one could take as $\mD_2$ an ordinary multiple point of sufficiently high multiplicity).
Even if the singularity is non-linear, one can consider the collection of singular hypersurfaces that can be brought to the given Newton diagram by {\it linear transformations} only (or projective transformations in homogeneous coordinates). This defines a subvariety of the non-linear stratum: {\it the linear substratum}. Our method, of course, enables to calculate the cohomology classes of both the true strata and their linear sub-strata.
\subsection{Defining conditions of the singularities}\label{SecDefiningConditionsGeneral} For Newton-non-degenerate singularities the defining conditions of a singular point are read from the Newton diagram. Consider the points under the Newton diagram. The corresponding monomials should be absent i.e. the corresponding derivatives should vanish. To define the stratum, one has to write these conditions in a covariant form. For linear singularities this can be done in especially simple way, since one should achieve the covariance under the group of projective transformations (or linear in local coordiantes). Every condition has a form $\{f^{(p)}_{i_1,\dots,i_p}=0\}$ and is transformed by $\mP GL(n+1)$ to $\{f^{(p)}(y_1,\dots,y_p)=0\}$, where $\{y_i\}$ are some points of $\mP^n_{y_i}$ (regarded here as $(n+1)-$vectors).
First consider some simple examples.
\bex $\bullet$ An ordinary point of multiplicity $p$. Here in local coordinates we have: $f|_x=0$, $\partial_1f|_x=\dots=\partial_nf|_x=0$,
$\dots$, $\Big\{\partial_{i_1}\dots\partial_{i_{p-1}}f|_x=0\Big\}_{i_1\dots i_{p-1}}$
Passing to the homogeneous coordinates (and using Euler formula (\ref{Euler}) and its consequences) we get the defining conditions in a covariant form: $f|_x^{(p-1)}=0$. \\ \parbox{14cm} {$\bullet$\label{ExamFlagOfKernelsHomogeneousForms} An ordinary point of corank $r$. Consider the singularity with the normal form $\sum_{i=1}^{r}z^{p+1}_i+\sum^n_{i=r+1}z^p_i$. (For $p=2$ these singularities are $A_2,D_4,P_8..$). The defining conditions are read directly from the Newton diagram:
\beq\ber f|_x=0,~\partial_1f|_x=\dots=\partial_nf|_x=0,~\dots,~\Big\{\partial_{i_1}\dots\partial_{i_{p-1}}f|_x=0\Big\}_{i_1\dots i_{p-1}},
\\\Big\{\partial_{i_1}\dots\partial_{i_{p-1}}\partial_jf|_x=0\Big\}_{\ber 1\le i_1\dots i_{p-1}\le n\\ j\le r\eer} \eer\eeq} \begin{picture}(0,0)(-50,0) \put(0,0){\vector(0,1){50}} \put(5,45){$\hat{e}_n$} \put(-2,31){$\bullet$} \put(-10,30){$p$} \put(0,0){\vector(1,2){20}} \put(25,40){$\hat{e}_{n-1}$} \put(11,24){$\bullet$} \put(20,27){$p$} \put(0,0){\vector(2,1){45}} \put(50,30){$\dots$} \put(23,10){$\bullet$} \put(33,9){$p$} \put(0,0){\vector(1,0){70}} \put(50,-10){$\hat{e}_{r+1}$} \put(32,-2.2){$\bullet$} \put(35,-10){$p$} \put(0,0){\vector(1,-1){35}} \put(37,-43){$\hat{e}_{r}$} \put(21,-26){$\bullet$} \put(35,-27){$p+1$} \put(0,-30){$\dots$} \put(0,0){\vector(-1,-1){33}} \put(-45,-38){$\hat{e}_1$} \put(-26,-27){$\bullet$} \put(-45,-16){$p+1$} \put(-25,-23.5){\line(1,0){50}} \put(-23,-23.5){\line(2,5){23}} \put(0,34){\line(2,-1){15}} \put(13,28){\line(5,-6){13}} \put(24,15){\line(4,-5){12}} \put(35,0){\line(-1,-2){12}} \end{picture} \\ To transform them to covariant form, introduce the flag of the Newton diagram (as defined in \ref{DefFlagOfNewtonDiagrQuasiHomog}): \beq \mC^n=V_n=\dots=V_{r+1}\supset V_{r}=span(\hat{e}_1,\dots,\hat{e}_r)=\dots=V_{1}\supset\{0\} \eeq and the corresponding flag $\{\mV_i\}_i$ in $\mC^{n+1}$. Then the conditions in homogeneous coordinates are:
\beq f|_x^{(p-1)}=0,~~f|_x^{(p)}(y)=0,~~\forall y\in \mV_{r}\subset \mC^{n+1} \eeq \parbox{15.3cm}{$\bullet$ The $A_k$ point: $z^{k+1}_1+\sum^n_{i=2}z^2_i$. Here the flag is: $\mC^n=V_n=\dots=V_2\supset V_1=span(\hat{e}_1)\supset\{0\}$. The conditions are:
\beq f|_x=0,~\partial_1f|_x=\dots=\partial_nf|_x=0,~\Big\{\partial^i_1\partial_*f|_x=0\Big\}_{0<i<\frac{k+1}{2}+1},~
\Big\{\partial^i_1f|_x=0\Big\}_{\frac{k+1}{2}+1\le i\le k} \eeq} \begin{picture}(0,0)(-40,0) \put(0,0){\vector(0,1){40}} \put(5,40){$\hat{e}_2$} \put(-2,26){$\bullet$} \put(-10,30){$2$} \put(0,0){\vector(1,1){30}} \put(32,28){$\hat{e}_3$} \put(16,16){$\bullet$} \put(16,9){$\dots$}
\put(0,0){\vector(1,0){45}} \put(35,-12){$\hat{e}_{n}$} \put(26,-2.2){$\bullet$} \put(25,-15){$2$} \put(0,0){\vector(-2,-1){33}} \put(-25,-25){$\hat{e}_1$} \put(-26,-15){$\bullet$} \put(-40,-3){\small$k+1$}
\put(-24,-13){\line(3,5){25}} \put(-24,-13){\line(4,1){53}}
\put(0,28){\line(2,-1){20}} \put(28,0){\line(-1,2){10}} \end{picture} \\ $\bullet$ The $D_k$ point: $z^{k-1}_1+z^2_2z_1+\sum^n_{i=3}z^2_i$. We assume for simplicity that $k$ is even. The flag is: $\mC^n=V_n=\dots=V_3\supset V_{2}=span(\hat{e}_1,\hat{e}_2)\supset V_1=span(\hat{e}_1)\supset\{0\}$. The conditions are:
\beq f|_x=0,~\partial_*f|_x=0,~~
\Big\{\partial^i_2\partial^{l-i-1}_1\partial_*f|_x=0\Big\}_{\ber 0\le i<\frac{k+1-2l}{k-4}\\l<\frac{k+1}{2}\eer},~
\Big\{\partial^i_2\partial^{l-i}_1f|_x=0\Big\}_{\ber i<\frac{2(k-1-l)}{k-4}\\l<k-1\eer} \eeq \eex We describe now the general procedure of formulating the covariant conditions for linear singularities. Recall that in $\S$ \ref{SecVectorSpacesOfNewtonDiagram} the collection of vector spaces associated to a given Newton diagram was defined. Start from the SQH case, where the collection is just a flag in $\mC^n$ (definition \ref{DefFlagOfNewtonDiagrQuasiHomog}).
\subsubsection{The case of semi-quasi-homogeneous linear singularities}\label{SecDefinCondLinearQuasiHomog} \parbox{13.5cm}{ Let $\hat{e}_1,\dots,\hat{e}_n$ be the coordinate axes of the lattice (by the same letters we also denote the corresponding unit vectors). Consider the points of intersection of the hyperplane $\Gamma_f$ with the coordinates axes: $\{k_i\}_i$. By renumbering the axes we can assume: $k_1\ge k_2\ge\dots\ge k_n$.
To write the conditions of the Newton diagram explicitly, consider the points of the lattice lying under the Newton diagram. As in the examples above, at each step we consider points corresponding to partial derivatives of a given order. } \begin{picture}(0,0)(-60,0) \put(0,0){\vector(0,1){60}}\put(0,0){\vector(-1,-1){45}}\put(0,0){\vector(1,0){70}} \put(5,50){$\hat{e}_n$} \put(-35,-45){$\hat{e}_1$} \put(60,-12){$\hat{e}_{n-1}$} \put(-15,40){$\Gamma_f$} \put(-15,40){\vector(1,-2){10}} \put(0,40){\line(-1,-2){40}} \put(-40,-40){\line(5,2){100}} \put(0,40){\line(3,-2){60}} \put(-20,-20){\line(3,1){60}} \put(-20,-20){\line(1,5){6.5}} \put(40,0){\line(-1,2){10}}
\put(-13.5,13){\line(6,1){43}} \put(35,20){$\Delta_r$} \put(35,20){\vector(-1,-1){10}} \put(-3,36){$\bullet$} \put(12,36){$k_n$} \put(56,-3){$\bullet$} \put(56,5){$k_{n-1}$} \put(-42,-42){$\bullet$} \put(-50,-35){$k_1$} \end{picture} \\ Namely, for every $0\le r\le k_1$ define an $(n-1)$ dimensional simplex: \beq
\Delta_r:=\{x=(m_1,\dots,m_n)|~m_i\ge0,~\sum^n_{i=1} m_i=r,~~x\mbox{ lies strictly below~}\Gamma_f\} \eeq Every integral point of $\Delta_r$ corresponds to a vanishing derivative (in local coordinates): $\partial^{m_1}_1\dots\partial^{m_n}_nf\equiv f^{(r)}_{\underbrace{\tinyM1\dots1}_{m_1},\dots,\underbrace{\scriptstyle n\dots n}_{m_n}}=0$.
We need a more precise relation between the axes of the diagram and the vector spaces of the flag of the diagram. \bed Let $\{V_i\}_i$ be the flag of the Newton diagram. For each axis $\hat{e}_i$ define the {\bf associated vector space} as $V(\hat{e_i}):=V_j$, such that $\hat{e_i}\in V_j$ and $\hat{e_i}\notin V_{j-1}$ . \eed Recall also that each vector space $V_i\subset\mC^n$ has its (homogeneous) version $\mV_i\subset\mC^{n+1}$.
The transition from local conditions (that arise from the given Newton diagram) to the conditions covariant under $PGL(n+1)$ is done by the following \bel\label{ClaimDefiningConditionsQasHomSing} For a SQH singularity the local set of conditions $\Big\{\partial^{m_1}_1\dots\partial^{m_n}_nf=0\Big\}$ for $\Big\{(m_1\dots m_n)\in\Delta_r\Big\}$, corresponding to the points under the Newton diagram, can be written in a covariant way as: \beq \forall y_i\in \mV(\hat{e}_i)~~~\Big\{\forall (m_1,\dots,m_n)\in \Delta_r\Big\}^{k_1}_{r=0}~~~~ f^{(r)}(\underbrace{y_1\dots y_1}_{m_1},\dots,\underbrace{y_n\dots y_n}_{m_n})=0 \eeq \eel \bpr $\Rightarrow$ We should check that the covariant conditions imply the local conditions. This is immediate (just take: $y_i=\hat{e}_i$). \\$\Leftarrow$ Without loss of generality, can assume $k_1\ge k_2\ge\dots\ge k_n$. Introduce the lexicographic order on the points of $\Delta_r$ (for a fixed $r$): \beq (m_1,\dots,m_n)<(\tilde{m}_1,\dots,\tilde{m}_n)~~\rm{if}~\Big\{\ber m_i=\tilde{m}_i:~\rm{for}~i<j\\m_j<\tilde{m}_j\eer \eeq We have: if the point $(\tilde{m}_1..\tilde{m}_n)$ lies under the Newton diagram and $(m_1..m_n)\leq(\tilde{m}_1..\tilde{m}_n)$, then the point $(m_1..m_n)$ also lies under the Newton diagram (because: $\sum \frac{m_i}{k_i}\leq\sum \frac{\tilde{m}_i}{k_i}<1$).
Now, expanding: $y_i=\sum \alpha_i\hat{e_i}$ and substituting into $f^{(r)}(\underbrace{y_1..y_1}_{m_1}..\underbrace{y_n..y_n}_{m_n})$ we get the sum of terms, corresponding to the points $(\tilde{m}_1..\tilde{m}_n)$, satisfying: $(m_1..m_n)\ge(\tilde{m}_1..\tilde{m}_n)$. Therefore, if $(m_1..m_n)\in\Delta_r$ then $f^{(r)}(\underbrace{y_1..y_1}_{m_1}..\underbrace{y_n..y_n}_{m_n})=0$. \epr
\bex\label{ExamFlagOfKernelsTacnode} Below we write the conditions in several cases, considered in the example \ref{ExamFlagOfKernelsHomogeneousForms}. ~\\ $\bullet$ Linear singularities of $A_k$ series are $A_{k\le 3}$. In the $A_2$-case, the covariant conditions are:
$f|_x^{(1)}=0$, $\forall y\in \mV_1: f|_x^{(2)}(y)=0$. In the tacnodal case ($A_3$) the additional condition is $f|_x^{(3)}(y,y,y)=0$.
~\\ $\bullet$ Linear singularities of $D_k$ series are $D_{k\le 5}$.The $D_4$-case was considered in example \ref{ExamFlagOfKernelsHomogeneousForms}. Here we consider $D_5$. The covariant conditions are: $f|_x^{(1)}=0$, $\forall~y_1\in\mV_1,~y_2\in\mV_2:$ $f|_x^{(2)}(y_1)=0=f|_x^{(2)}(y_2)$,
$f|_x^{(3)}(y_1,y_1,y_1)=0=f|_x^{(3)}(y_1,y_1,y_2)$. ~\\ $\bullet$ The singularities of ND-topological type with the representative: $\sum^{r_1}_{i=1}z^{p+2}_i+\sum^{r_2}_{i=r_1+1}z^{p+1}_i+\sum^{n}_{i=r_2+1}z^p_i$. For $p=2$ this class contains: $A_{k\le 3},$ $D_4,$ $E_6~(r_1+1=r_2=2),$ $P_8~(r_1=0,~r_2=3)$ $X_9~(r_1=r_2=2),$ $U_{12}~(r_1+2=r_2=3)$ etc. \\ The flag is: $\mC^{n}=V_n=\dots=V_{r_2+1}\supset V_{r_2}=span(\hat{e}_{1},\dots,\hat{e}_{r_2})=\dots= V_{r_1+1}\supset V_{r_1}=span(\hat{e}_{1},\dots,\hat{e}_{r_1})=\dots=V_1\supset\{0\}$. \\ The point $(m_1,\dots,m_n)\in\Delta_r$ lies under the Newton diagram provided: \beq \sum_i m_i=r,~~~~~\sum^{r_1}_{i=1}\frac{m_i}{p+2}+\sum^{r_2}_{i=r_1+1}\frac{m_i}{p+1}+ \sum^{n}_{i=r_2+1}\frac{m_i}{p}<1 \eeq The covariant equations are:
\beq\ber f|_x^{(p-1)}=0,~\forall~y_1\in V_{r_1},~\forall~y_2\in V_{r_2}:~f|_x^{(p)}(y_2)=0,~
f^{(p+1)}|_x(\underbrace{y_1,\dots,y_1}_{k},\underbrace{y_2,\dots,y_2}_{l})=0,~\rm{for}~\frac{2k}{p+2}+\frac{l}{p+1}>1 \eer\eeq ~\\ $\bullet$ The singularities of ND-topological type with the representative $\sum^{r_1}_{i=1}z^{4}_i+z_{r_1}z^2_{r_1+1}+\sum^{r_2}_{i=r_1+2}z^{3}_i+\sum^{n}_{i=r_2+1}z^2_i$. This class contains: $D_4,$ $P_8,$ $Q_{10},$ $V_{1,0}$ etc. \\ The flag is: $\mC^{n}=V_n=\dots=V_{r_2+1}\supset V_{r_2}=span(\hat{e}_{1},\dots,\hat{e}_{r_2})=\dots= V_{r_1+2}\supset V_{r_1+1}=span(\hat{e}_{1},\dots,\hat{e}_{r_1+1})\supset V_{r_1}=span(\hat{e}_{1},\dots,\hat{e}_{r_1}) =\dots=V_1\supset\{0\}$. \\ The point $(m_1,\dots,m_n)\in\Delta_r$ lies under the Newton diagram provided: \beq \sum_i m_i=r,~~~~~\sum^{r_1}_{i=1}\frac{m_i}{4}+\frac{3m_{r_1+1}}{8}+\sum^{r_2}_{i=r_1+2}\frac{m_i}{3}+ \sum^{n}_{i=r_2+1}\frac{m_i}{2}<1 \eeq The covariant equations are:
\beq\ber f|_x^{(1)}=0,~\forall~y_1\in V_{r_1},~\forall~y_2\in V_{r_2},~\forall~y_3\in V_{r_1+1}:~f|_x^{(2)}(y_2)=0,~
f|_x^{(3)}(y_1,y_2,y_3)=0 \eer\eeq \eex
\subsubsection{The general linear case}\label{SecDefinCondLinearNonQuasHom} In the linear non-SQH case we have the collection of vector spaces for the top-dimensional faces of the Newton diagram. So, perform the procedure for each top dimensional face separately (i.e. for the hyperplane that contains it). Thus the lemma \ref{ClaimDefiningConditionsQasHomSing} is translated verbatim to a collection of lemmas, for each face. Now combine all the conditions (the coinciding conditions should be identified).
So, this generalizes the method of obtaining defining conditions to the case of non-quasi-homogeneous singularities.
\bex The singularity of the type $T_{pqr}$ with the normal form: $z^p_1+z^q_2+z^k_3+z_1z_2z_3+\sum_{i=4}^nz^2_i$ $(p\ge q\ge k)$. It is linear for $p,q,r\le5$. Here the Newton diagram consists of the three planes: \beq\ber \scriptsize\scriptstyle L_{pq}:=\Big\{\!\!\frac{m_1}{p}+\frac{m_2}{q}+m_3(1-\frac{1}{p}-\frac{1}{q})+\sum_{i\ge4}\frac{m_i}{2}=1\!\!\Big\},~ \scriptstyle L_{pk}:=\Big\{\!\!\frac{m_1}{p}+m_2(1-\frac{1}{p}-\frac{1}{k})+\frac{m_3}{k}+\sum_{i\ge4}\frac{m_i}{2}=1\!\!\Big\},~ \scriptstyle L_{qk}:=\Big\{\!\!m_1(1-\frac{1}{k}-\frac{1}{q})+\frac{m_2}{q}+\frac{m_3}{k}+\sum_{i\ge4}\frac{m_i}{2}=1\!\!\Big\} \eer\eeq So, one considers the three flags: $(V_{i,pq},V_{i,qk},V_{i,pk})$ and the three sets of polytopes: $(\Delta_{r,pq},\Delta_{r,qk},\Delta_{r,pk})$. We get a set of local conditions, which is transformed to the set of covariant conditions, by the prescription of lemma \ref{ClaimDefiningConditionsQasHomSing}. We omit the calculations. \eex
\subsubsection{On the transversality of covariant conditions} We would like to address here the issue of transversality. All the initial conditions (corresponding to the points under the Newton diagram) are just linear equations on the (derivatives of the) function $f$, the transversality is equivalent to the linear independence. And this is obvious for conditions that are read from the Newton diagram. The conditions are made covariant by introducing vectors from the collection of vector spaces of the diagram. As the linear independence is preserved under (invertible) linear transformations we obtain that the covariant conditions remain transversal as far as these vectors are mutually generic. More precisely: \bprop\label{ClaimTransversalityUnderNewtonDiagram} The set of conditions $\{f^{(p_j)}(y_{i_{1,j}},\dots,y_{i_{p_j,j}})=0\}_{i,j}$ (that correspond to the points under the Newton diagram) is transversal if by $PGL(n+1)$ the points $\{y_{i,j}\}_i$ can be brought to the coordinate axes. \eprop In particular one demands that no subset of $k$ variables lies in a $(k-2)$ hyperplane (for any $k$).
\subsubsection{The non-linear singularities and defining conditions}\label{SecDefinCondNonLinearSing} Singularity types/strata for which a Newton diagram cannot be achieved by projective (linear) transformations are non-linear. Most types/strata are nonlinear. \bex\label{ExampleDefiningEqsA4} Consider a hypersurface with the $A_4$ point. Try to bring it to the Newton diagram of $A_4$ (i.e. that of $z^4_1+z^2_2+..z^2+n$). The best we can do by projective transformations is to bring it to a form of $A_3$: \beq f=\sum_{i=2}^n\alpha_iz_i^2+z_1^2\sum_{i=2}^n\beta_iz_i+\gamma z^4_1+\dots \eeq To achieve the Newton diagram of $A_4$ we must do the non-linear shift: $z_i\rightarrow z_i+\delta_iz^2_1$ (to kill the monomials $z^4_2,z^2_1z_i$). Elimination of the parameters of the transformation gives the non-linear equation: $\gamma=\sum\frac{\beta^2_i}{4\alpha_i}$. \eex
In general, to obtain the locally defining equations of a non-linear singularity one considers the locally analytic transformation of coordinates: \beq z_i\rightarrow z_i+\sum_{k_i=1}^\infty a_{i,k}z_{k_i},~~~\rm{for}~~i=1\dots n \eeq Then, demanding that some derivatives (corresponding to the points under the Newton diagram) vanish, one eliminates the parameters of the transformation $\{a_{ik}\}_{i,k}$ to obtain the system of (non-linear) equations. This forms the locally defining ideal of the stratum The ideal can be quite complicated (since the stratum can be not a locally complete intersection). It is important to trace the whole ideal, i.e. all of its generators. Of course the whole procedure is done by computer and is time consuming.
Note that, as we enumerate the non-linear strata by the {\it simplifying degenerations} (defined in \ref{SecSimplifyingDegenerations}), we do not need to convert the defining conditions to a covariant form.
Among the generators of the ideal there are (a finite number of) non-linear expressions in the coefficients of $f$. The goal of degeneration is to turn these non-linear equations into monomial equations. In this case the ideal would correspond to a collection of linear strata (with some multiplicities).
For enumeration of a non-linear singularity we will be interested in the set of linear singularities to which the given singularity is adjacent. \bed The linear singularity type $L$ is called assigned to a given singularity type $N$ if $N$ is adjacent to $L$ (i.e. $\Si_N\subset\bar{\Si_L}$) and the adjacency is minimal. Namely there is no other linear type $L'$ such that $\Si_N\subset\bar{\Si_L'}\subsetneqq\bar{\Si_L}$. \eed The assigned type is non-unique in general. As singularities always appear in series, which start from linear singularities, we have a natural way to fix an assigned linear singularity. In the simplest cases the assigned linear singularities are:
$A_3$ for $A_{k\ge4}$, $D_5$ for $D_{k\ge6}$, $E_6$ for $E_{k>6}$ etc.
\subsubsection{On the invertibility of degenerations}\label{SecInvertibilityofDegeneration} At each step of the degeneration procedure we get an equation in the cohomology ring of $Aux\times\mPN$: \beq [\tSi_1][\mbox{degenerating divisor}]=[\tSi_2] \eeq An important issue is to check that the degeneration is "invertible", i.e. this equation
fixes the cohomology class of the original stratum uniquely. The generators of the cohomology ring are nilpotent (e.g. $X^{n+1}=0=F^{D+1}$). Thus the solution for $[\tSi_1]$ is unique, provided $dim(\Si_1)+1\le D$ and the class of degenerating divisor depends in essential way on $F$. The first condition is always satisfied, while the second means that the degeneration must involve (in essential way) the function $f$ (or its derivatives). In particular, conditions involving the parameters of the auxiliary space only (e.g. coincidence of points) are non-invertible and will not be used for the degenerations.
\section{Enumeration} \subsection{Main issues of the method}\label{SecMainIssues} \subsubsection{Liftings (desingularizations)}\label{SecLiftings} We lift the strata to a bigger ambient space to define them by explicit equations.
The simplest example (minimal lifting), consisting of pairs (the function, the singular point), was considered in \ref{SecMinimalLifting}. This lifting is sufficient for ordinary multiple points only. In general one should lift further to the space $Aux\times\mP_f^D$. We will consider the objects: (the function, the singular point, hyperplanes in the tangent cone, some special linear subspaces of the hyperplanes).
In $\S$ \ref{SecVectorSpacesOfNewtonDiagram} we constructed for a given Newton diagram $\mD$ the collection of vector spaces (in homogeneous coordinates) $\{\mV_\al\}\in\cV(\mD)$. Taking their projectivization we arrive at a collection $\{\mP_{y_\al}^{n_\al}\}_{\al\in\cV}$ (with the conditions that some are subspaces of others). This defines the lifting: \bed\label{DefLifting} For linear ND-topological type the lifting is defined by \beq \tSi(x,\{y_\al\}_i):=\overline{\Bigg\{\!\!\!
\ber (x,\{y_\al\},f)\\\mbox{generic}\eer\!\!{\Bigg|} \!\!\ber\Big\{y_\al\in\mP_{y_\al}^{n_\al}\Big\}_\al\in{\cV}(\mD),~~f
\mbox{ has the prescribed}\\\mbox{ singularity with } \mD~\mbox{as its Newton diagram}\eer\Bigg\}}\subset \mP^n_x\times\prod_\al\mP_{y_\al}^{n_\al}\times\mPN \eeq\eed \bex In the SQH case the collection of vector spaces is just the flag therefore \beq \tSi(x,\{y_i\}_i):=\overline{\Bigg\{\!\!\!
\ber (x,\{y_i\}_i,f)\\\mbox{generic}\eer\!\!{\Bigg|} \!\!\ber\Big\{\mV_i=span(y_1,\dots,y_{dim(\mV_i)})\Big\}_i\!\!\!\!\\\ f \mbox{ has the prescribed singularity with}\\ \mV_n\supset\dots\supset\mV_{1}~\mbox{as the flag of the Newton diagram}\eer\Bigg\}}\subset \mP^n_x\times\prod_i\mP^n_{y_i}\times\mPN \eeq \eex We emphasize that the tuples $(x,\{y_i\}_i)$ are always taken to be generic (in particular, no $k$ of the points span a $(k-2)$-plane). For the sake of exposition we will often omit this, but it is always meant. \beR As follows from lemma \ref{ClaimDefiningConditionsQasHomSing} the so defined stratum $\tSi$ is indeed a lifting. That is: every point of $\tSi$ projects to a point of $\Si$. This definition reduces the enumerative problem for linear singularities to the intersection theory. The intersection theory step has still many complications, related questions are considered in Appendix B. \eeR \bex\label{ExampleNumerousLiftings} For many cases the defining covariant conditions were given in example \ref{ExamFlagOfKernelsTacnode}. Therefore, we immediately have: \\$\bullet$ For multiple point of co-rank $r$ (with normal form $f=\sum_{i=r+1}^n x_i^p+higher~order~terms$). \beq\ber\label{CorankLiftings}
\tSi(x,y_1,\dots,y_r)=\overline{\Big\{\ber(x,y_1,\dots,y_r)\\~~generic\eer|
f|_x^{(p-1)}=0=\Big(f|_x^{(p)}(y_i)\Big)_{i=1}^r\Big\}}\subset\mP^n_x\times\prod_{i=1}^r\mP^n_{y_i}\times\mPN \eer\eeq \\$\bullet$ For singularities with the normal form: $\sum^{r_1}_{i=1}z^{p+2}_i+\sum^{r_2}_{i=r_1+1}z^{p+1}_i+\sum^{n}_{i=r_2+1}z^p_i$: \beq
\tSi(x,y_1,\dots,y_{r_2})=\overline{\Big\{(x,y_1,\dots,y_{r_2)}\Big|\ber f|_x^{(p-1)}=0~f|_x^{(p)}(y_1)=0~\rm{for}~i\le r_2\\
f^{(p+1)}|_x(y_{i_1},\dots,y_{i_k},y_{j_1},\dots,y_{j_l})=0\\ ~\rm{for}~i_1..i_k\le r_1,~j_1..j_l\le r_2~~\frac{2k}{p+2}+\frac{l}{p+1}>1\eer\Big\}} \subset\mP^n_x\times\prod_{i=1}^{r_2}\mP^n_{y_i}\times\mPN \eeq $\bullet$ For singularities with the normal form: $\sum^{r_1}_{i=1}z^{4}_i+z_{r_1}z^2_{r_1+1}+\sum^{r_2}_{i=r_1+2}z^{3}_i+\sum^{n}_{i=r_2+1}z^2_i$: \beq
\tSi(x,y_1,\dots,y_{r_2})=\overline{\Big\{(x,y_1,\dots,y_{r_2)}\Big|\ber f|_x^{(1)}=0~f|_x^{(2)}(y_i)=0~\rm{for}~i\le r_2\\
f|_x^{(3)}(y_i,y_j,y_k)=0~\rm{for}~i\le r_1,~j\le r_1+1,~k\le r_2\eer\Big\}} \subset\mP^n_x\times\prod_{i=1}^{r_2}\mP^n_{y_i}\times\mPN \eeq \eex
\paragraph{\hspace{-0.3cm}Lifted varieties as fibrations over the auxiliary space.}\hspace{-0.3cm}~ \\ Note that for a given singular germ $(f=0)$ only the flag $\mV_n\supset\dots\supset \mV_{1}$ and the assigned polytopes are defined uniquely. The points $\{y_i\}_i$ of the auxiliary space (in definition \ref{DefLifting}) can vary freely as far as the flag and the polytopes are preserved. So we obtain an important result: \bprop The projection: $\tSi\stackrel{\pi}{\rightarrow}\Sigma$ is a fibration with generic fiber the multi-projective space. \eprop In the example of multiple point of co-rank $r$ the generic fiber is $\mP^r_{y_1}\times\dots\times\mP^r_{y_r}$.
Note, that the nonzero dimensionality of the fibers already restricts the possible cohomology class of the lifted variety. In fact, let $(X,Y_1,\dots,Y_r,F)$ be the generators of the cohomology rings of $\mP^n_x,\mP^n_{y_i},\mPN$ (as defined in $\S$ \ref{SectVar}). The cohomology class of $\tSi$ is a polynomial in $(X,Y_1,\dots,Y_r,F)$. \bel\label{ClaimFirstConsistencyCondition} {\bf (The first consistency condition).}\\ \parbox{12cm} {Let the fibration with generic fiber $\mP^r_{y_k}\subset\mP^n_{y_k}$ be given as in the diagram. Then the variable $Y_k$ that appears in monomials of the polynomial $[\tSi(x,y_1,\dots,y_k)]$ has powers not bigger than $(n-r)$.}~~~~~~ $\ber \tSi(x,y_1,\dots,y_k)\subset Aux\times\mP^n_{y_k}\times\mP^D \\~~~~~~~~~~ \downarrow~~~~~~~~~~~~~~~~\downarrow\\ \tSi(x,y_1,\dots,y_{k-1})\subset Aux\times\mP^D \eer$ \eel \parbox{15cm} {\bpr We give the proof in the general case. Consider projective fibration $E\rightarrow B$ with the generic fiber $\mP^r_y$. Assume that the generic fiber is linearly embedded into $\mP^n_y$. Write the cohomology class $[E]\in H^*(A\times \mP^n_y)$ as: }~~~ $\ber E\hookrightarrow A\times \mP^n_y\\\downarrow~~~~~\downarrow\\B\hookrightarrow A \eer$ \beq [E]=\sum_i Y^iQ_{dim(A)+n-dim(E)-i} \eeq here $Y$ is a generator of $H^*(\mP^n_y)$, while $Q_{dim(A)+n-dim(E)-i}\in H^{2(dim(A)+n-dim(E)-i)}(A)$. We want to show that terms appearing in the above sum have: $i\le n-r$. To see this, multiply $[E]$ by $Y^{n-i}\tilde{Q}_{dim(E)-n+i}$ (for some arbitrary $\tilde{Q}$). By the duality between homology and cohomology this product corresponds to the intersection of $E$ with (generic) cycle of the form: $\mL^{(i)}_{\mP^n}\times C^{dim(A)-dim(E)+n-i}_A$ (here $\mL^{(i)}_{\mP^n}$ is a linear $i-$dimensional subspace of $\mP^n$, $C-$a cycle in $A$). Then (by the dimensional consideration in $A$) the intersection is empty unless: \beq dim(A)-dim(E)+n-i+dim(B)\ge dim(A) \eeq which amounts to $n-i\ge r$. So we have: $Q_{dim(A)+n-dim(E)-i}\tilde{Q}_{dim(E)+i-n}=0$ for any $\tilde{Q}$, so $Q_{dim(A)+n-dim(E)-i}=0$ for $i>n-r$. \epr Another restriction comes from the symmetry of the definition \ref{DefLifting} with respect to $y_i$. $\tSi$ is invariant with respect to a subgroup $G$ of group of permutations of $y_i$ that preserves the
flag structure. The group has the orbits: \beq (y_1\dots y_{dim(V_{n-1})})~~(y_{dim(V_{n-1})+1}\dots y_{dim(V_{n-2})})\dots(y_{dim(V_{2})+1}\dots y_{dim(V_{1})}) \eeq Therefore we have: \bcor\label{ClaimSeconConsistencCondition} {\bf (The second consistency condition)}\\ The cohomology class of~ $\tSi(x,y_1,\dots,y_r)$ is invariant under the action of the group $G$ (i.e. is a polynomial symmetric with respect to relevant subsets of $Y_i$). \ecor \bex In particular for a multiple point of co-rank $r$, equation (\ref{CorankLiftings}), we have:
The cohomology class of the lifted stratum of multiple point of co-rank $r$, expressed in terms of
$(Y_1,\dots,Y_r,X)$ and~\mbox{$Q=(d-p)X+F$}, is symmetric with respect to $(Y_1,\dots,Y_r,X)$ and the maximal
powers of variables $(Y_1,\dots,Y_r,X)$ are not higher than $(n-r)$. \eex So we have rather restrictive conditions on the possible class of $\tSi$. These conditions enable us to avoid lengthy calculations of some parameters (as will be demonstrated in section: \ref{SecFormOfCorank}).
One could also consider the second projection: from $\tSi$ to the auxiliary space, in definition \ref{DefLifting} this space is: $\mP^n_x\times\prod_i\mP^n_{y_i}$. \bex For a multiple point of co-rank $r$, the variety defined in (\ref{CorankLiftings}) is a locally trivial fibration over the auxiliary space $\mP^n_x\times\prod_{i=1}^r\mP^n_{y_i}$ outside the "diagonals" (two coinciding points, three points on a line, four points in the plane etc.). \eex
This situation happens in general case: the projection is a locally trivial fibration, outside the cycles of jump.
\paragraph{On the possible cycles of jumps}\label{SecPossibleCyclesOfJump} The cycles of jump were discussed in general in $\S$ \ref{SecResidualVarieties}. Here we describe the possible cycles of jump for linear singularities. The lifted stratum is intersection of hypersurfaces, each being defined by vanishing of a particular derivative. As follows from the discussion in $\S$ \ref{SecDefinCondLinearQuasiHomog} the defining equations of the hypersurfaces are of the form: \beq\label{EqHypersurfCyclHump}
f^{(k)}|_x(\underbrace{y_{i_1},\dots,y_{i_{l_k}}}_{l_k},\hat{e}_{j_1},\dots,\hat{e}_{j_{k-l_k}})=0~~~ 0\le j_1\le\dots\le j_{k-l_k}\le n \eeq (Here $\hat{e}_{j_1},\dots,\hat{e}_{j_{k-l_k}}$ are the vectors of the standard basis, defined in section \ref{SecDefinCondLinearQuasiHomog}).
The cycles of jumps consist of points of $\mP^n_{x}\times\mP^n_{y_1}\times\dots\times\mP^n_{y_r}$ over which the intersection is non-transversal. Every equation of the type (\ref{EqHypersurfCyclHump}) is linear in the coefficients of $f$ (which are the homogeneous coordinates of the parameter space $\mPN$). Therefore the non-transversality can happen only when the vectors $y_{i_1},\dots,y_{i_{l_k}},\hat{e}_{i_1},\dots,\hat{e}_{i_n-l_k}$ are non-generic with respect to the vectors of other equations. More precisely: the non-transversality can happen only when some of the vectors $y_{i_1},\dots,y_{i_{l_k}},\hat{e}_{i_1},\dots,\hat{e}_{i_n-l_k}$ of one equation belong to the span of the vectors of other equations.
This condition can be nicely written using projections of vector space. Let $I=\{i_1,\dots,i_k\}$ be an arbitrary (non-empty) subset of $\{0,1,\dots,n\}$. Represent the point of a projective space by its homogeneous coordinates: $y=(z_0,\dots,z_n)\in\mP^n_y$. The projection is defined by: \beq \pi_I(y):=(z_{i_1},\dots,z_{i_k}) \eeq Note that $\pi_I(y)$ is defined up to a scalar multiplication and can have all the entries zero. Immediate check shows that the above condition of non-transversality corresponds to one of the following: \bei \item $\pi_I(y)\in$Span$(\pi_I(x),\pi_I(y_{i_1}),\dots,\pi_I(y_{i_k}))$ \item $(\pi_I(x),\pi_I(y_{i_1}),\dots,\pi_I(y_{i_k}))$ are linearly dependent. \eei (In particular if all the entries of $\pi_I(y)$ are zero, both conditions are trivially satisfied.) Note that the second condition is the closure of the first. Summarizing: \bprop\label{ClaimPossibleCyclesOfJump} The possible cycles of jump in the auxiliary space $Aux=\mP^n_x\times\mP^n_{y_1}\times\dots\times\mP^n_{y_k}$ are of the form: $\Bigg((\pi_I(x),\{\pi_I(y_j\}_{j\in J})$ are linearly dependent for some $I\subseteq\{0,\dots,n\},~~J\subseteq\{1,\dots,k\}\Bigg)$ \eprop \beR While the set $I$ is nonempty, $J$ can be empty. In this case the cycle of jump is defined as $\pi_I(x)=(0\dots0)$. \eeR
\paragraph{On the adjacency}\label{SecOnTheAdjacency} To each equisingular stratum some strata of higher singularities are adjacent (i.e.the strata of higher singularities are included in the closure of the given stratum). For example $\Sigma_{D_{k+1}}\subset\overline\Sigma_{A_k}$.
We constantly use the codimension one adjacency, i.e. when $\Si_{\mD'}$ is a divisor in $\Si_\mD$ (or the same for the lifted versions, cf. $\S$ \ref{SecLiftings}). In other words, these are the strata that can be reached by just one degeneration. Many tables of adjacencies are given in \cite{AVGL}. In each particular case the adjacency can be checked by the analysis of Newton diagram of the singularity or of the defining ideal of the singular germ.
The adjacency can depend on moduli \cite{Pham}. However, if one chooses moduli generically (and we always do that), the adjacency is completely fixed by the topological type.
A more important feature is: the set of relevant adjacent strata can depend on the dimensionality of the ambient space. As an example consider the enumeration of $A_4$ (cf. \ref{SecFurtherCalculations}). It is degenerated by increasing the corank of the quadratic form. Then:\\ $\bullet n=2$, the case of curves $A_4\rightarrow D_5$\\ $\bullet n\geq3$, the case of hypersurfaces, $A_4\rightarrow D_5\cup P_8$ \\ As will be shown in $\S$ \ref{SecHigherSing}, in the second case both $D_5$ and $P_8$ are relevant.
In our approach we enumerate separately for each fixed dimension $n$. By universality, the final answer is given by a unique polynomial (in Cherna classes that depend on $n$) of a known degree. Therefore it suffices to calculate just for a few needed dimensions and by universality we get the complete answer.
\subsubsection{The ideology of degenerating process}\label{SecIdeologyOfDegenerations} Here we prove the main theorems stated in Introduction (\ref{TheoremEnumLinearSings} and \ref{TheoremEnumNonLinearSings}). We must prove
the three statements:\\ $\bullet$the degenerating step is always possible and the degeneration is invertible.\\ $\bullet$for linear singularities we achieve the cohomology class of the lifted stratum in a finite number of steps and all the intermediate types are linear.\\ $\bullet$for non-linear singularities we reduce the problem to enumeration of linear ones (in a finite number of steps).
\paragraph{The degenerating step}consists of intersection of a lifted stratum with a hypersurface and subtraction of the residual pieces. In more details, let $\tSi$ be the initial lifted stratum and $\tSi_{degen}$ the stratum we want to reach. The degenerating hypersurface is defined by the equation $f^{(p)}(y_{i_1},\dots,y_{i_k})_{j_1\dots j_{p-k}}=0$. For linear singularities this is just the derivative corresponding to a point under the Newton diagram. For non-linear singularities (as is explained in $\S$ \ref{SecDefinCondNonLinearSing}) the derivative corresponds to a point on the Newton diagram with the minimal distance to the origin. Such a point can be non-unique, in this case we use monomial order from $\S$ \ref{SecOnTheMonomialOrder}.
The intersection is in general non-transversal and the resulting variety is reducible (containing residual pieces in addition to the needed degenerated stratum). \beq \tSi\cap \{f^{(p)}(y_{i_1},\dots,y_{i_k})_{j_1\dots j_{p-k}}=0\}=\tSi_{degen}\cup\{\mbox{Residual pieces}\} \eeq The non-transversality happens over cycles of jump (described in $\S$ \ref{SecPossibleCyclesOfJump}). The procedure to calculate the cohomology classes of the residual pieces is explained in section \ref{SecCohomologyClassOfRestrictionFibration}. Note that in the above equation the degenerated stratum
$\tSi_{degen}$ can be reducible and non-reduced (but is always pure dimensional), this happens for non-linear singularities. The multiplicity
of the intersection of $\tSi$ with the degenerating hypersurface along $\tSi_{degen}$ is calculated
in the classical way (section \ref{SecMultiplicityOfIntersection}).
Summarizing, the degenerating step produces the equation in cohomology: \beq [\tSi][f^{(p)}(y_{i_1},\dots,y_{i_k})_{j_1\dots j_{p-k}}=0]=[\tSi_{degen}]+[\mbox{Residual pieces}] \eeq By $\S$ \ref{SecInvertibilityofDegeneration} the degeneration is invertible, so the equation enables the calculation of either $[\tSi]$ or $[\tSi_{degen}]$.
\paragraph{Degenerations for linear singularities (from simple to complicated).} The defining set of conditions for linear singularities was described in $\S$ \ref{SecDefiningConditionsGeneral}. Starting from the stratum of ordinary multiple point (of the relevant multiplicity), lifted to the space $Aux\times\mPN$, we apply the defining conditions one-by-one. Each condition means the absence of a particular monomial, arrange the conditions by the monomial order (defined in \ref{SecOnTheMonomialOrder}). This guarantees that at each step we get a Newton-non-degenerate type.
The degenerating step was described above. After a restricted number of degenerations (not bigger than the number of points under the Newton diagram) we arrive at the lifted stratum of the needed singularity.
At each intermediate step of the process the singularity type is linear. Indeed by the criterion
\ref{ClaimCriterionLinearity} all the initial and final slopes are bounded in the interval $[\frac{1}{2},2]$.
And in the process of degeneration the slopes change monotonically. \beR Usually it is simpler to start not from the stratum of ordinary multiple point, but from a stratum of a higher (linear) singularity to which the given singularity is adjacent and for which the enumeration problem is already solve. For example to enumerate the tacnode ($A_3$) we start from the cusp ($A_2$). \eeR
\paragraph{Degenerations for non-linear singularities (from complicated to simple)} As was explained in $\S$ \ref{SecSimplifyingDegenerations} the original non-linear type is degenerated to a combination of linear ones.
The goal of degenerating process is to convert the defining non-linear equations to monomial ones (i.e. of the form $\vec{z}^{\vec{m}}=0$). For this, at each step of the process, we consider a (non-vanishing) monomial of the lowest monomial order (section \ref{SecOnTheMonomialOrder}).
Suppose the multiplicity of the singularity is $p$ while the order of determinacy is $k$. The process goes by first demanding that the derivatives $f^{(p)}_{i_1\dots i_p}$ vanish (the conditions are applied one-by-one, at each step we have just the degeneration by a hypersurface). Then one arrives at the singularity of multiplicity $(p+1)$ and order of determinacy $k$. If the so obtained singularity (or a collection of singularities) is still non-linear, the process is continued. In the simplest case (Newton-non-degenerate singularity), once we have $k\le 2p$, we necessarily have a collection of linear singularities (by corollary \ref{CorollarLinearSingMultPOrderDeterm2p}). In the worst case (Newton-degenerate singularity) one continues up to the ordinary point of multiplicity $k$.
This process transforms a non-linear singularity to a collection of linear singularities whose enumeration was described above. As all the degenerations are invertible, this solves the enumerative problem for non-linear singularities.
\subsection{The simplest examples}\label{SecExamples} We consider here some simplest typical examples to illustrate the method. First we consider the case of cusp. Having enumerated the cusp, one enumerates the tacnode by just one additional degeneration.
\subsubsection{Quadratic forms of co-rank 1 (cuspidal hypersurfaces)} Here we consider singularity with the normal form $\sum_{i=2}^nz^2_i+z^3_1$. The lifted stratum was defined in example \ref{ExampleNumerousLiftings}: \beq
\tSi_{A_2}(x,y)=\overline{\{(x,y,f)|~(x\ne y),~f|_x^{(2)}(x)=0=f|_x^{(2)}(y)\}}\subset\mP^n_x\times\mP^n_y\times\mPN \eeq
(here the tensor of second derivatives is calculated at the point $x$, therefore: $f|_x^{(2)}(x)\sim f|_x^{(1)}$).
We want to represent $\tSi_{A_2}(x,y)$ as a (possibly) transversal intersection of hypersurfaces. The $(n+1)$ conditions: $f^{(2)}(x)=0$ are transversal. Suppose we add to them one condition: $(f^{(2)}(y))_0=0$. Then, the non-transversality occurs over two cycles of jump in $Aux=\mP^n_x\times\mP^n_y$: \bei \item{$x=y$} (co-dimension $n$). The jump in the dimension of the fiber is 1. \item{$x=(1,0,\dots,0)$} (co-dimension $n$). In this case, since $f^{(2)}(x)=0$, we already know that the form $f^{(2)}(y)$ annihilates $x$ (i.e. $f^{(2)}(y)_0=0$). The jump in the dimension of the fiber is 1. \eei In both cases the dimension of the jump of fiber is less than the codimension of the cycle of jump, so the resulting variety is irreducible. So, for the cohomology classes we have: \beq [\overline{f^{(2)}(x)=0,f^{(2)}(y)_0=0,~x\neq y}]=[f^{(2)}(x)=0]\times[f^{(2)}(y)_0=0] \eeq We continue to degenerate by the conditions in such a way up to $(f^{(2)}(y)_{n-1}=0)$. At this point for generic (non-coinciding) $x,y$ we have: $f^{(2)}(x)=0=f^{(2)}(y)$. \\ \parbox{13cm}{Additional pieces that arose are: \\$\bullet$ over $x=y$, the condition of codimension $n$. Over this diagonal the jump in the dimension of fibers is $n$. \\$\bullet$ over $x=(*\dots*,0)$, the condition of codimension 1. Over this subvariety the jump in the dimension of fibers is $1$.\\ On the picture the result of the intersection is shown in a "log-scale", i.e. the dimensionality is transformed to the relative hight.} \begin{picture}(0,0)(0,20) \ellipse{0}{15}{30}{0}{120}{20}{80}{40} \put(10,2){\line(3,1){100}}\put(10,42){\line(3,1){100}}\put(15,15){$x=y$} \put(10,2){\line(0,1){40}} \put(107,34){\vector(0,1){40}}\put(107,74){\vector(0,-1){40}} \put(115,50){$n$}
\put(110,10){\line(-3,1){40}}\put(110,20){\line(-3,1){40}}\multiput(65,24)(-6,2){6}{.}\multiput(65,34)(-6,2){6}{.} \put(107,13){$\updownarrow$} \put(110,0){$x_n=0$}\put(112,12){$1$} \put(45,-15){$\mP^n_x\times\mP^n_y=Aux$}
\end{picture} \\ So, the resulting variety is reducible, containing residual pieces over the cycles of jump: \beq
\Big\{f|_x^{(2)}(x)=0=f|_x^{(2)}(y)\Big\}=\tSi_{A_2}(x,y)\cup \Big\{Res_{x=y}\Big\}\cup \Big\{Res_{z_n=0}\Big\} \eeq (By direct check we obtain that the multiplicity in both cases is 1.) To remove the contributions from additional pieces we should calculate the classes of the restrictions to the cycles of jump. The method of calculation of the cohomology classes of residual pieces is described in section \ref{SecCohomologyClassOfRestrictionFibration}. \\ $\bullet$ The residual piece over the diagonal can be described explicitly in a simple manner. As all the intersections over the points of the diagonal are nontransversal, a point over the diagonal corresponds to a nodal hypersurface. So, the residual piece is $Res_{x=y}=\{x=y\}\cap\Big\{\tSi_{A_1}(x)\times\mP^n_y\Big\}$, and its cohomology class $[Res_{x=y}]$=$[x=y]\times$$[\tSi_{A_1}(x)]$. \\ $\bullet$ Over the generic point of the cycle $z_n=0$ (i.e. the point for which $z_0\dots. z_{n-1}\ne 0$) all the intersections, except for the last were transversal. Therefore we have: \beq
\Big\{f|_x^{(2)}(x)=0,~~z_n=0\Big\}\cap^{n-2}_{i=0}\Big\{f|_x^{(2)}(y)_i=0\Big\}=Res_{z_n=0}\cup Res_{z_n=0=z_{n-1}} \eeq where $Res_{z_n=0=z_{n-1}}$ is a "secondary" residual piece. Its description is the same as that of $Res_{z_n=0}$ and so one has a recursion: \beq
\Big\{f|_x^{(2)}(x)=0,~~z_n=0\Big\}\cap^{j-2}_{i=0}\Big\{f|_x^{(2)}(y)_i=0\Big\}=Res_{z_n=0=\dots=z_j} \cup Res_{z_n=0=\dots=z_{j-1}} \eeq After completion of the recursion we get the formula: \beq [\tSi_{A_2}(x,y)]=[f^{(2)}(x)=0]\Bigg(\sum_{i=0}^{n-1}(-1)^i[\prod_{j=n+1-i}^n\{x_j=0\}] [\prod_{j=0}^{n-1-i}\{f^{(2)}(y)_j=0\}]-[x=y]\Bigg) \eeq Or, in terms of cohomology classes: \beq\label{CuspCoefficient} [\tSi_{A_2}(x,y)]=(Q+X)^{n+1}\left(\sum^n_{i=0}(-1)^i(Q+Y)^{n-i}X^i-\sum^n_{i=0}X^iY^{n-i}\right),~~Q=(d-2)X+F \eeq Here $(X,Y,F)$ are the generators of the corresponding cohomology rings. Note, that this polynomial (if written in variables $(Q,X,Y)$)is symmetric in $X,Y$ (eventhough it was obtained in a very non-symmetric way) as it should be. Also, the terms $X^n,Y^n$ cancel in the polynomial.
To obtain the degree of the variety we should extract the coefficient of $X^nY^{n-1}$ (after the substitution: $Q=(d-2)X+F$). We get: \beq\label{cuspdeg} \mbox{deg}(\Sigma_{A_2})=(d-1)^{n-1}(d-2)\frac{n(n+1)(n+2)}{2} \eeq (which of course coincides with the result obtained by P.Aluffi in \cite{Alufi}).
\subsubsection{The use of adjacency: tacnodal hypersurfaces}\label{SecTacnodalHypersurf} The tacnodal singularity ($A_3$) has the normal form $f=z^4_1+\sum_{i=2}^nx_i^2$. The corresponding lifted variety was defined in example \ref{ExampleNumerousLiftings}. We represent the tacnode as a degeneration of the cusp. Correspondingly, we think of the (lifted) stratum of tacnodal hypersurfaces as a subvariety of the cuspidal stratum: \beq
\tSi_{A_3}(x,y)=\overline{\{(x,y,f)\in\tSi_{A_2}(x,y),~x\ne y~~f|_x^{(3)}(y,y,y)=0\}}\subset\tSi_{A_2}(x,y)\subset\mP^n_x\times\mP^n_y\times\mPN \eeq For generic $x,y$ the intersection is $\tSi_{A_2}(x,y)\cap\{f^{(3)}(y,y,y)=0\}$ transversal. The possible non-transversality can occur only when $x=y$. So for the cohomology classes we have: \beq [\tSi_{A_3}(x,y)]=[\tSi_{A_2}(x,y)][f^{(3)}(y,y,y)=0]-[\mbox{Residual piece over }x=y] \eeq The method to calculate the cohomology class of the residual piece over $(x=y)$, is given in Appendix B. By the Corollary \ref{CorCohomologyClassResidualPieceOverDiagonal} we have: \beq [\mbox{Residual piece over }x=y]=[x=y]\frac{1}{(n-1)!}\frac{\partial^{n-1}[\tSi_{A_2}(x,y)]}{\partial^{n-1}Y} \eeq In this simple case the residual piece can be also easily described explicitly: every point of it corresponds to a hypersurface with a cusp at a given point, but with arbitrary tangent line. So the variety over the diagonal ($x=y$), is just $\tSi_{A_2}(x)$ taken with some multiplicity. The multiplicity can be computed in two ways: ~\\ $\bullet$ {directly,} as the degree of tangency of the nontransversal intersection. ~\\ $\bullet$ {via consistency condition} (as was explained in $\S$ \ref{SecLiftings}). Since the lifted stratum $\tSi_{A_3}$ is a fibration over $\Sigma_{A_3}$ with fiber $\mP^1_y$, the corresponding cohomology class should not include terms with $Y^n$).
Both methods give the multiplicity 3. Thus the cohomology class of the cuspidal stratum is: \beq\ber [\tSi_{A_3}(x,y)]=[\tSi_{A_2}(x,y)][f^{(3)}(y,y,y)=0]-3[x=y][\tSi_{A_2}(x)]= \\ =(Q+X)^{n+1}\Bigg(\Big(\sum^n_{i=0}(Q+Y)^{n-i}(-X)^i-\sum^n_{i=0}X^iY^{n-i}\Big)(Q+3Y-X)- 3(nQ-2X)\sum^n_{i=0}X^iY^{n-i}\Bigg) \eer\eeq Here, as always $Q=(d-2)X+F$. Note again, that (in full accordance with the fibration conditions) the answer (if written in terms of $X,Y,Q$) is symmetric in $X,Y$ and no terms with $X^n$ or $Y^n$ appear. Finally, the degree is: \beq [\Sigma_{A_3}]={n+2\choose{3}}(d-1)^{n-2}\Bigg(\frac{(3n-1)(n+3)}{2}(d-2)^2+2(n-1)(d-2)-4\Bigg) \eeq which for $n=2$, curves, coincides with the result of Aluffi \cite{Alufi}.
\subsection{Further calculations}\label{SecFurtherCalculations} We start from homogeneous forms of some (co-)rank (section \ref{SecFormOfCorank}). For quadratic forms ($p=2$) the rank fixes the degeneracy class completely. For higher forms one can impose various additional degeneracy conditions (e.g. for $p=3$ the form: $z_1^2z_2+\sum_{i=3}^nz_i^3$ is of full rank, but the corresponding singularity is not an ordinary triple point). We consider examples of such singularities in section \ref{SecHigherSing}.
Having calculated the classes of the lifted strata for $2-$forms of some rank, we can start enumeration of other singularities. This is done by further degenerations. For example, the tacnode $A_3$ is the cusp with some degeneracy of the tensor of order-3 derivatives (it was enumerated in $\S$ \ref{SecExamples}). We consider here the simplest examples: $A_4,D_5,E_6$.
\subsubsection{The forms of co-rank$\geq1$}\label{SecFormOfCorank} We first recall: \bed The homogeneous symmetric form of order $p$, in $n$ variables, $\Omega^{p}(z_1,\dots,z_n)$, is called {\bf of rank $(n-r)$} (of co-rank $r$) if by linear transformation of $GL(n)$ in the space of variables ($\mC^n=\{z_1,\dots,z_n\}$) it can be brought to a homogeneous form in $n-r$ variables: $\tilde\Omega^{(p)}(z_1,\dots,z_{n-r})$. \eed The collections of such forms are natural generalizations of classical determinantal varieties (symmetric matrices of a given co-rank) \cite[chapter 14.3]{Ful}. To emphasize the co-rank of the form we often assign it as a subscript: $\Omega^{p}_r$.
There are (at least) two approaches to calculate the cohomology classes of the stratum of a given co-rank forms: ~\\ $\bullet$ {Start from the ordinary multiple point (it corresponds to the non-degenerate form) treated before}. Apply the degenerating conditions (one-by-one) to get the form of the needed co-rank. At each step it is necessary to remove the residual pieces. This approach works well for the forms of low co-rank. ~\\ $\bullet$ {Degenerate the given form to a form of rank 2 or 1}. (The forms of rank$=2,1$ are particular cases of reducible forms, their enumeration is immediate and is treated in $\S$ \ref{SecReducibleForms}) Then we will have equation in cohomology of the form: \beq [\Omega^{(p)}_r][\mbox{degenerating cycle}]=[\Omega^{(p)}_2]+[\mbox{Residual variety}] \eeq And from this equation the class $[\Omega^{(p)}_r]$ is restored uniquely.
We describe here the first approach. As the computations are extremely involved we solve explicitly only the case of the quadratic forms of arbitrary corank.
The lifted stratum was defined in example \ref{ExampleNumerousLiftings}: \beq\label{CuspVector}
\tSi^n_r(x,(y_i)_{i=1}^r)=\overline{\Big\{(x,\{y_i\}_{i=1}^r,f)\Big| \Big(\!\!\ber x,y_1,\dots,y_r~\rm{are}\\\rm{linearly~independent}\eer\!\!\Big), \Big(\!\!\ber~~~f^{(2)}(x)=0\\(f^{(2)}(y_i)=0)_{i=1}^r\eer\!\!\Big)\Big\}}\subset\mP^n_x\times\prod_{i=1}^r\mP^n_{y_i}\times\mPN \eeq As was explained in $\S$ \ref{SecLiftings}, the cohomology class of $\tSi^n_r$ is a polynomial in $(X,Y_1,\dots,Y_r$, $Q=(d-2)X+F)$, symmetric in $(X,Y_1,\dots,Y_r)$ and does not contain powers of $Y_i$, greater than $(n-r)$. The cohomology class of $\Sigma^n_r$ is just the coefficient of the monomial $Y_1^{n-r}\dots Y_k^{n-r}X^n$ in the cohomology class of $\tSi^n_r$.
The enumeration of singularities with quadratic form of co-rank $r$ is completed by the claim: \bel The cohomology class of the lifted variety $\tSi^n_r(x,(y_i)_{i=1}^r)$ can be calculated by successive degenerations starting from $\tSi^n_0(x)$ (the nodal hypersurfaces). In particular, the cohomology class of the minimal lifting is: \beq\ber [\tSi^n_r(x)]=C_{n,r}Q^{r\choose{2}} \sum^r_{i=0}\frac{{n-i\choose{r-i}}{r\choose{i}} }{{2r\choose{r+i}}}Q^{r-i}(-X)^i~~~~~~Q=F+(d-2)X\\ C_{n,1}=2,~~C_{n,2}=2{n+1\choose{1}},~~C_{n,3}=2{n+2\choose{3}},~~C_{n,4}=2{n+3\choose{5}}\frac{n+1}{3}\\ C_{n,n}=2{2n\choose{n}},~~C_{n,n-1}=\frac{2^r{2r\choose{r}}}{n},~~ C_{n,n-2}=\frac{{2(r+1)\choose{r+1}}{2r\choose{r}}}{{r+2\choose{2}}} \eer\eeq \eel Note that here we give the constant for some specific values of $(n,r)$ only. This is due to computer limitations: each time we calculate for a specific value of $r$ and $n$. So, for every specific $n,r$ one can get the answer (provided the computer is strong enough), but it is not clear how to combine these values into one nice expression. \bpr The case $r=1$ corresponds to cuspidal hypersurfaces and was considered in $\S$ \ref{SecExamples}. The general case is done recursively. Suppose we have obtained the cohomology class of $\tSi^n_r(x,y_1,\dots,y_r)$, as in equation (\ref{CuspCoefficient}). Intersect the variety $\tSi^n_r(x,y_1,\dots,y_r)$ with $(n-r)$ hypersurfaces: $f^{(2)}(y_{r+1})_0=0,\dots,f^{(2)}(y_{r+1})_{n-r-1}=0$. The possible (significant) non-transversality can occur in two cases, either: $y_{r+1}\in$Span($x,y_1,\dots,y_r$), or $(*,\dots,*,\underbrace{0,\dots,0}_k)$$\in$Span($x,y_1,\dots,y_r$). In both cases one continues as in the case of $r=1$. In such a way we get the cohomology class: \beq [\tSi^n_{r+1}]=[\tSi^n_r]\Big(\sum_{i=0}^{n-r}(Q+Y_{r+1})^{n-r-i}(-1)^i \hspace{-0.3cm}\sum_{j_1+\dots+j_r\leq i}\hspace{-0.3cm}Y_1^{j_1}\dots Y_r^{j_r}X^{i-(j_1+\dots+j_r)}- \hspace{-0.9cm} \sum_{i_1+\dots+i_{r+1}=n-r-j}\hspace{-0.9cm}X^{j}Y_1^{i_1}\dots Y_{r+1}^{i_{r+1}}\Big)+\rm{residual~terms} \eeq Here the residual terms correspond to varieties which occur over the diagonals: $x=y_1$, $y_2\in$span($x,y_1$),
\dots, $y_r\in$span($x,y_1,\dots,y_{r-1}$). These residual pieces can be calculated by classical intersection theory (as explained in $\S$ \ref{SecCohomologyClassOfRestrictionFibration}). However (as happens in all other cases) the classes are actually completely fixed by the consistency conditions (lemma \ref{ClaimFirstConsistencyCondition} and \ref{ClaimSeconConsistencCondition}). In this case they read:
{\it The final expression (polynomial in $Q,X,Y_1,\dots,Y_{k+1}$) should be symmetric in $(X,Y_1,\dots,Y_{k+1})$ and should not contain the powers of $X,Y_1,\dots,Y_{k+1}$ that are greater than $n-k-1$}.
The explicit calculations are extremely complicated and can be done by computer only. The cohomology classes of the lifted strata are awkward polynomials in many variables.
For example the cohomology class in the co-rank 2 case ($D_4$ singularity) is: \beq\ber [\tSi^2_{2}(x,y_1,y_2)]=\scriptstyle(Q+X)^{n+1} \sum_{i=0}^n(Q+Y_1)^{n-i}(-X)^i\Big(\sum_{j=0}^{n-1}(Q+Y_2)^{n-1-j}(-1)^j \sum_kX^{j-k}Y_1^k-\sum_{j,k}X^jY_1^kY_2^{n-1-j-k}\Big) -\\-\scriptstyle (Q+X)^{n+1}\sum_{i=0}^nX^iY_1^{n-i}\sum^n_{i=1}\Big(\sum^{n-i}_{j=0}{n-j\choose{i}}Q^{n-i-j}(-X)^j-X^{n-i}\Big) \Big( \sum_{j=0}^{i-1}(-1)^{n-j-1}(Q+Y_2)^{j}X^{i-j-1}-\sum_{j=0}^{i-1}X^{i-1-j}Y_2^{j}\Big) \eer\eeq Extracting the coefficients of $Y_1^{n-r-1}\dots Y_{k+1}^{n-r-1}$ we get the classes of the strata $\tSi^n_r(x)$. \epr
\subsubsection{Some linear singularities}\label{SecHigherSing}
\paragraph{Hypersurfaces with a $D_5$ point,} the normal form: $f=z^4_1+z^2_2z_1+\sum_{i=3}^nz_i^2$. The lifted stratum was defined in example \ref{ExampleNumerousLiftings}. We represent the stratum $\tSi_{D_5}$ as a subvariety of $\tSi_{D_4}$: \beq
\tSi_{D_5}(x,y_1,y_2)=\overline{\{(x,y_1,y_2,f)\in\tSi_{D_4}(x,y_1,y_2),~~f|_x^{(3)}(y_1,y_1,y_1)=0=f|_x^{(3)}(y_1,y_1,y_2)\}} \subset\tSi_{D_4}(x,y_1,y_2) \eeq For generic $x,y_1,y_2$ the intersection is transversal. The non-transversality occurs over diagonals: \mbox{$y_2\in$Span$(x,y_1)$} or $x=y_1$. Note, that the first variety is non-closed. We approximate it by the variety: ($x,y_1,y_2$ are linearly dependent), the two varieties coincide for $x\neq y_1$. Correspondingly, over $x=y_1$ we have additional (secondary) residual piece. Thus the cohomology class is: \beq [\tSi_{D_5}(x,y_1,y_2)]=\Big(\ber[\tSi_{D_4}(x,y_1,y_2)][f^{(3)}(y_1,y_1,y_1)=0][f^{(3)}(y_1,y_1,y_2)=0]-\\- [x=y_1]A(X,Y_2,Q)-[\rm{rk}\Big(\begin{smallmatrix}x\\y_1\\y_2\end{smallmatrix}\Big)<3]B(X,Y_1,Y_2,Q) \eer\Big) \eeq Here $A,B$ are some (homogeneous) polynomials in the generators of the cohomology ring. By the identity in the cohomology ring: $(X-Y)\sum_{i=0}^n X^iY^{n-i}=X^{n+1}-Y^{n+1}\equiv0$, we can assume that $A$ does not depend on $Y$. The only additional condition on $A,B$ is the consistency condition from $\S$ \ref{SecLiftings}:
{\it The cohomology class $[\tSi_{D_5}(x,y_1,y_2)]$ should not contain monomials with $Y^n_1,Y^{n-1}_2,Y^n_2$.}
As always, this condition itself fixes the polynomials completely. The final cohomology class is given in Appendix.
\paragraph{Hypersurfaces with an $E_6$ point,}the normal form: $f=z^4_1+z^3_2+\sum_{i=3}^nz_i^2$. We represent $E_6$ as a degeneration of $D_5$: \beq
\tSi_{E_6}(x,y_1,y_2)=\overline{\{(x,y_1,y_2,f)\in\tSi_{D_5}(x,y_1,y_2),~~f|_x^{(3)}(y_1,y_2,y_2)=0\}} \subset\tSi_{D_5}(x,y_1,y_2) \eeq Again, instead of describing the residual varieties explicitly, we use the consistency conditions, which completely fix the class. The final answer is in Appendix.
\subsubsection{Some non-linear singularities} \paragraph{The $A_4$ case} Here we consider the simplest non-linear case. By linear transformation, the singularity germ can be brought to the Newton diagram of $A_3$: $f=\sum_{i=2}^n\alpha_iz_i^2+z_1^2\sum_{i=2}^n\beta_iz_i+\gamma z^4_1+\dots$
To achieve the Newton diagram of $A_4$ we must do the non-linear shift: $z_i\rightarrow z_i-\frac{\beta_i}{2\al_i}z^2_1$ (to get rid of the monomials $z^4_1,z^2_1z_i$). Elimination gives $\gamma=\sum\frac{\beta^2_i}{4\alpha_i}$.
Therefore, degenerating $\alpha_2=0$ we get: $\beta_2=0$ or $\prod_{i\ne 2}\alpha_i=0$, corresponding to adjacency $\bar\Si_{A_4}\supset\Si_{D_5},\Si_{P_8}$
In this way we obtain the cohomology class of $\tSi_{A_4}(x,y)$. So, we get the equation for cohomology classes: \beq [\tSi_{A_4}(x,y_1)][degeneration]=2[\tSi_{D_5}(x,y_1)]+2[\tSi_{P_8}(x,y_1,y_2)]+[residual~~piece] \eeq The final result is in Appendix.
\paragraph{The $D_6$ case} By linear transformation, the singularity germ can be brought to the Newton diagram of $D_5$: $f=\sum_{i=3}^n\alpha_iz_i^2+z_1^2\sum_{i=3}^n\beta_iz_i+\gamma z^4_1+z_1z^2_2\dots$ To achieve the Newton diagram of $D_6$ we must do the non-linear shift: $z_i\rightarrow z_i+\delta_iz^2_1$ (to get rid of the monomials $z^4_1,z^2_1z_i$). Elimination gives: $\gamma=\sum\frac{\beta^2_i}{4\alpha_i}$. We degenerate in the same way as in the $A_4$ case and get: \beq [\tSi_{D_6}(x,y_1,y_2)][degeneration]=2[\tSi_{P_8}(x,y_1,y_2,y_3)_{degenerated}]+ [corank~4]+[residual~~piece] \eeq We omit the calculations.
\appendix \section{Singularities with reducible jets}\label{SecReducibleForms} Here we consider singular polynomials whose low order jet is reducible. The simplest such case is that of reducible form: \beq\label{ReducibForm} \Omega^{(p)}=\prod_{i=1}^k\Bigg(\Omega^{(p_i)}_i\Bigg)^{r_i},~~~\sum_{i=1}^k r_ip_i=p \eeq (here the homogeneous forms $\Omega^{(p_i)}_i$ are irreducible, though they can be degenerate and mutually non-generic). These singularity types are of high codimension, therefore extremely rare, nevertheless they deserve some attention, being sometimes the final goal of degenerating process.
Since reducibility is in general not invariant under the topological transformations we (in general) cannot define the corresponding stratum as a topological one. We define the stratum as the collection of hypersurfaces that can be brought (by locally analytic transformation) to {\it of a given form}.
This stratum is included into the topological stratum. On the other side it usually contains families of
analytical strata, since by the Newton diagram we do not specify the moduli.
In case of curves ($n=2$) every singularity of multiplicity $p$ has reducible $p-$jet (being homogeneous polynomial of two variables). So the corresponding stratum coincides with the topological equisingular stratum. Some singularities with reducible $p-$form are: $A_1,A_2,D_4$,$E_6,X_9$,$Z_{11},W_{12}\dots$.
For surfaces ($n=3$) some singularities of this type are: $A_2,D_4,P_8,S_{11}$, $U_{12},T_{3,4,4},T_{4,4,4}$,$V_{1,0},V'_1\dots$.
Another example of reducible p-form (for any $n$) is the p-form of rank 2 or 1.
For singularities with reducible jets, the lifted stratum can be explicitly defined by conditions of a very standard type: proportionality of tensors.
\subsubsection{Reducible homogeneous forms}\label{SecDegenerateHomogenForms}
We consider here the case of mutually generic forms $\Omega^{(p_i)}_i$ in (\ref{ReducibForm}). Every such form defines a hypersurface, by the mutual generality of the forms the hypersurfaces intersect in a generic way, however each hypersurface can be singular. We restrict to the case of ordinary multiple point of maximal multiplicity (so for each hypersurface the condition is: $\Omega^{(p_i)}_i(x)=0$). The stratum of hypersurfaces with this type of reducibility is defined as: \beq\label{VarietyReducForm}
\tSi\Big(x,(\Omega^{(p_i)}_i)_{i=1}^k\Big)=\Bigg\{\Big(x,(\Omega^{(p_i)}_i)_{i=1}^k,f\Big)\Big| ~f^{(p)}\sim\rm{SYM}\Big((\Omega^{(p_1)}_1)^{r_1},\dots,(\Omega^{(p_k)}_k)^{r_k}\Big), ~(\Omega^{(p_i)}_i(x)=0)_{i=1}^k\Bigg\} \eeq Here SYM means symmetrization of indices. Note that the sets of defining conditions are mutually transversal, e.g. $f$ appears in the first proportionality condition only. Therefore the cohomology class of the lifted stratum is just the product of classes of conditions: \beq [\tSi\Big(x,(\Omega^{(p_i)}_i)_{i=1}^k\Big)]=[f^{(p)}\sim\rm{SYM} \Big((\Omega^{(p_1)}_1)^{r_1},\dots,(\Omega^{(p_k)}_k)^{r_k}\Big)]\prod_i [\Omega^{(p_i)}_i(x)=0] \eeq The condition of proportionality of two tensors is considered in $\S$ \ref{SecCohomologyClassesOfCyclesofJump} (equation (\ref{EqProportionalityTwoTensors})). In terms of the cohomology ring generators of the ambient space ($X,Q=(d-p)X+F,\Omega_i$) we have: \beq\label{EqClassesForReducibleForms} [\tSi\Big(x,(\Omega^{(p_i)}_i)_{i=1}^k\Big)]=(\Omega_1+X)^{p_1-1+n\choose{n}}\dots(\Omega_k+X)^{p_k-1+n\choose{n}} \sum_{i=0}^{{p+n\choose{n}}-1}Q^i(r_1\Omega_1+\dots+r_k\Omega_k)^{{p+n\choose{n}}-1-i} \eeq Note, that depending on the singularity type, the projection: $\tSi\rightarrow\Sigma$ can be not 1:1. The permutation group of forms of equal multiplicity and degeneracy acts on fibers. Thus, to obtain the cohomology class of
$\Sigma$ one should divide the corresponding coefficient by the order of this group: $|Aut|$.
\subsubsection{Singularities with reducible jets}\label{SecReducibleJets} Here we consider singularities of the type: \beq\label{ReducibJet} f=\prod_{i=1}^k f_i(z_1,\dots,z_n)+higher~order~terms~~~~~\rm{deg}(f_i)=p_i~~\sum_{i=1}^kp_i=p \eeq The polynomials $f_i$ are (non-homogeneous) of fixed degrees. We assume that the singularities $f_i=0$ are linear, in particular they satisfy: $2\times multiplicity\ge p_i$.
In particular, we can assume that all the hypersurfaces $\{f_i=0\}$ pass through the origin. Introducing factors that do not vanish at the origin leads to hypersurfaces with flexes, (the property which is not invariant under local diffeomorphism/homeomorphism).
If the hypersurface ($f_i=0$) is smooth and generic with respect to other hypersurfaces
(e.g. all the normals are in general position, intersection is along generic subvarieties etc.) then deg($f_i$)$=p_i=1$.
The procedure of enumeration is as in $\S$ \ref{SecReducibleForms}: the problem is reduced to enumeration of particular singularities ($f_i$), if some of the hypersurfaces are in a mutually special position this should be also taken into account.
We consider some typical situations: ~\\ $\bullet$ {\bf Mutually generic smooth hypersurfaces.} As was explained above, in this case all the degrees are necessarily equal to 1. So, all the factors are linear, this case was treated in section \ref{SecReducibleForms}. ~\\ $\bullet$ {\bf Mutually generic singular hypersurfaces}. In this case every singular hypersurface is treated separately, then the results are combined. The simplest case is: \beq f=g\prod_{i=1}^k f_i+higher~order~terms \eeq Here $f_i$ are homogeneous polynomials, while $g$ is not necessarily homogeneous, with the condition: the lowest order part of $g$ is completely reducible. This kind of singularity occurs e.g. for curves ($n=2$) as $Z_{11}~jet_5(f)=z_2(z_1^3+z_2^4)$, for surfaces ($n=3$) as $T_{455}~jet_4(f)=z_1(z_2z_3+z_1^3)$.
The lifted variety is: \beq
\tSi(x,(l_i)_{i=1}^{q-1},\Omega^{*})=\Big\{(x,(l_i)_{i=1}^{q-1},\Omega^{*},f)\Big|\ber f^{(p)}\sim\rm{SYM}(\Omega^{(p_1)},\dots,\Omega^{(p_k)},\Omega^{(q)})\\ \Omega^{(q)}(x)\sim\rm{SYM}(l_1,\dots,l_{q-1}),~l_i(x)=0=\Omega^{(p_i)}(x)\eer\Big\} \eeq ~\\ $\bullet$ {\bf The normals to some of the hypersurfaces are in special position.} Some simple cases: \bei \item{several coinciding normals: $f=\Big(\prod_{i=1}^k f_i\Big)z^r_1\Omega^{(q)},~~\Omega^{(q)}(x)\sim x^{q-1}_1$} For example this occurs for curves ($n=2$) as $A_3,D_6,E_7,E_8,W_{13},W_{1,0},W_{17},W_{18}$, for surfaces ($n=3$) as $X_9\dots$.
The lifted variety: \beq
\tSi(x,l,\Omega^{*})=\Big\{(x,l,\Omega^{*},f)\Big|\ber f^{(p)}\sim\rm{SYM}(\Omega^{(p_1)},\dots,\Omega^{(p_k)},l^r,\Omega^{(q)})\\~~ \Omega^{(q)}(x)\sim\rm{SYM}(l,\dots,l)~l(x)=0=\Omega^{(p_i)}(x)\eer\Big\} \eeq ~\\ $\bullet$ {several normals in one plane: $f=l_1\dots l_k\Omega^{(p)}$} For $n=2$ it is e.g. an ordinary multiple point. For $n=3$ $U_{12}:~z^3_1+z^3_2+z^4_3$ \beq
\tSi(x,(l_i)_{i=1}^k,\Omega^{(p)})=\Bigg\{(x,(l_i)_{i=1}^k,f)\Big| \ber f^{(p+k)}\sim\rm{SYM}(l_1,\dots,l_k,\Omega^{(p)})\\ (l_i(x)=0)_{i=1}^3~~~\Omega^{(p)}(x)=0~~\rm{rk}\Bigg(\begin{smallmatrix}l_1\\\dots\\l_k\end{smallmatrix}\Bigg)<3\eer\Bigg\} \eeq \eei ~\\ $\bullet$ {\bf The intersection of two hypersurfaces lies in the singular locus of one of them}
For example: $f=z_1(z^2_2\Omega^{(p)}+z_1\Omega^{(p+1)})$. The simplest case is $n=3:~S_{11}~jet_3(f)=z_3(z^2_2+z_1z_3)$. We omit the calculations.
\section{Some results from intersection theory}
\subsubsection{The multiplicity of intersection}\label{SecMultiplicityOfIntersection} At each step of the degenerating process we intersect the lifted stratum $\tSi$ with a hypersurface. As the intersection is in general non-transversal the resulting variety will be typically reducible: in addition to the needed (degenerated) stratum it will contain a residual variety over a cycle of jump. This residual variety will (in general) enter with a non-trivial multiplicity. The multiplicity is calculated in the classical way. Suppose the hypersurface is defined by the equation $\{g=0\}$. Restrict the function $g$ to the stratum $\tSi$ and find the vanishing order along the residual variety.
We illustrate this procedure in a typical: \bex Degeneration of the ordinary multiple point. Start from the lifted stratum: \beq
\tSi=\{(x,f)|~~f^{(m)}|_x=0\}\subset\mP^n_x\times\mPN \eeq Suppose we want to degenerate by intersection with the hypersurface:
\beq S=\{f|_x^{(p)}(\underbrace{y,\dots,y}_k)_{i_1,\dots,i_{p-k}}=0\}~~~~~m+1\le p\le l+k \eeq This case occurs for example in enumeration of multiple point of co-rank $r$ (in particular $A_2$ point).
The intersection $S\cap \tSi$ is non-transversal over the diagonal $x=y$. To calculate the multiplicity, i.e. to find the vanishing order we expand $y=x+\Delta y$. Then restricting to $\tSi$ we have (neglecting the numerical coefficients since we are interested in the vanishing order only): \beq\ber
\scriptstyle\scriptsize f|_x^{(p)}(\underbrace{x+\Delta y,\dots,x+\Delta y}_k)_{i_1,\dots,i_{p-k}} \sim
\sum_{i=0}^k f|_x^{(p)}(\underbrace{x \dots x}_{k-i}\underbrace{\Delta y\dots\Delta y}_i)_{i_1,\dots,i_{p-k}} \sim
\scriptstyle\scriptsize f|_x^{(p)}(\underbrace{x,\dots,x}_{p-m-1},\underbrace{\Delta y,\dots,\Delta y}_{k+1-p+m})_{i_1,\dots,i_{p-k}}+ \bet higher\\order\\terms\eet
\eer\eeq So, the function $f|_x^{(p)}(\underbrace{y,\dots,y}_k)_{i_1,\dots,i_{p-k}}$ has over the diagonal $x=y$ zero of the generic order $(k+m+1-p)$. Therefore: \beq
\tSi\cap X=\tSi_{degenerated}\cup(k+m+1-p)\tSi|_{x=y} \eeq \eex
\subsubsection{On cohomology classes of cycles of jump}\label{SecCohomologyClassesOfCyclesofJump} The possible cycles of jump are described in $\S$ \ref{SecPossibleCyclesOfJump}. Here we present their cohomology classes in the cohomology ring of their ambient space (which is the auxiliary space). The corresponding varieties (degeneracy loci) are known classically, in particular the cohomology classes are given in
\cite[section 14.5]{Ful}.
We define the incidence correspondence: \beq
\Sigma=\{(y_1,\dots,y_k)|y_k\in\rm{Span}(y_1,\dots,y_{k-1})\} \subset\mP^n_{y_1}\times\dots\times\mP^n_{y_k} \eeq Note, that for $k>2$ this variety is not closed. Its closure is: \beq \overline\Sigma=\{\mbox{The points }(y_1,\dots,y_k)\mbox{ lie in a }(k-2)\mbox{-plane}\} \subset\mP^n_{y_1}\times\dots\times\mP^n_{y_k} \eeq The cohomology class of such variety is: \beq [\overline\Sigma]=\sum_{i_1+\dots+i_k=n+1-k}Y_1^{i_1}Y_2^{i_2}\dots Y_k^{i_k} \eeq The points of $\overline\Sigma\setminus\Sigma$ correspond to configuration $\Big\{y_1,\dots,y_{k-1}$ are linearly dependent$\Big\}$. This subvariety (of codimension 1) will be also important in the calculations.
In case the cycle is defined by projection: $\pi_I$ (i.e. $(\{\pi_I(y^{(j)}\}_{j\in J})$ are linearly dependent), one continues similarly (thinking of $\pi_I(y^{(j)}$ as being a point in $\pi_I(\mP^n)$).
We will often face another condition of a special type: proportionality of symmetric tensors. Let $a,b$ be two symmetric tensors of rank $p$. By writing their independent components in a row we can think of each of them as being a point (in homogeneous coordinates) of some big projective space $\mP^N$. Then the proportionality of the tensors means that $a$ and $b$ coincide as the points in $\mP^N$. This condition was considered above. Its cohomology class is: \beq\label{EqProportionalityTwoTensors} [a\sim b]=\sum_{i=0}^{{p+n\choose{n}}-1}A^iB^{{p+n\choose{n}}-1-i} \eeq Here on the right hand side $A,B$ are the cohomology classes of the elements of $a,b$ i.e. the classes of the corresponding hypersurfaces. Equivalently, they are the first Chern classes of the corresponding line bundles.
\subsubsection{The cohomology class of a restriction of fibration}\label{SecCohomologyClassOfRestrictionFibration}
As was explained in $\S$ \ref{SecResidualVarieties} the fibration $\tSi\rightarrow Aux$ is generically locally trivial, it is not locally trivial over the cycles of jump ($C_i\subset Aux$). The key to degenerating procedure is the calculation of the cohomology class of the restriction $\tSi|_{C_i}\subset Aux\times\mPN$.
The first naive idea is to represent it as a product: $[C]R_{codim\tSi-codimC}$, here $R$ is a polynomial representing a class in $H^{2(codim\tSi-codimC)}(Aux\times\mPN)$. This happens only in special cases. \bel ~\\ $\bullet$ If $C$ is defined by a set of monomial equations in $Aux$ (i.e. by a set of the form $\vec{x}^{\vec{m_1}}=\dots=\vec{x}^{\vec{m_k}}=0$, for $\vec{m_i}$ multi-degrees), then
$[\tSi|_C]=[C]R_{codim\tSi-codimC}$.
~\\ $\bullet$ If $C$ is a "diagonal" in $Aux$ (i.e. $C_k=\{(y_1\dots y_k)|\mbox{linearly dependent}\}$) then, for a
"flag of sub-diagonals" ($C_i=\{(y_1\dots y_i)|\mbox{linearly dependent}\}$, $2\le i<k$), we have:
$[\tSi|_C]=[C_k]R_{codim\tSi-codimC}+[C_{k-1}]R_{codim\tSi-1-codimC}+\dots+ [C_2]R_{codim\tSi-n}$. \eel
\bpr The first claim is immediate since it follows that (up to the multiplicity) the restriction $\tSi|_C$ lies in a linear subspace of $Aux\times\mPN$. To prove the second, note that $\tSi$ is defined by a collection of conditions: $\{f^{(p)}(y_{i_1}\dots y_{i_r})_{***}\}=0$ (section \ref{SecDefinCondLinearQuasiHomog}). Thus over the open subset of $C$:
$y_r\in span(y_1\dots y_{r-1})$ the variable $y_r$ can be eliminated from the conditions. So, the two sets of conditions
(conditions of $C\subset Aux$ and conditions of $\tSi$ over the open subset of $C$) are explicitly transversal.
The non-transversality can happen only over the 'infinity': $(y_1\dots y_{r-1})|\mbox{linearly dependent}$. So, we can write:
$$[\tSi|_C]=[C_k]R_{codim\tSi-codimC}+[\mbox{a piece over }C_{k-1}].$$ Then by recursion we get the
statement of the lemma. \epr
An important case of the above lemma is the simplest case $C=\{y_1=y_2\}\subset Aux=\mP^n_{y_1}\times\mP^n_{y_2}$. In this case the residual piece can be written explicitly: \bcor\label{CorCohomologyClassResidualPieceOverDiagonal}
Suppose the projection $\tSi(y_1,y_2)|_{y_1=y_2}\mapsto\tSi(y_1)$ has the generic fiber $\mP^r,~0\le r<n$. Then
$[\tSi(y_1,y_2)|_C]=\frac{[y_1=y_2]}{(n-r)!}\frac{\partial^{n-r}[\tSi(y_1,y_2)]}{\partial Y^{n-r}_2}$. \ecor \bpr Over the diagonal $(y_1=y_2)$ the variable $y_2$ can be completely eliminated from the defining conditions of $\tSi$. This corresponds to projection: $Aux=\mP^n_{y_1}\times\mP^n_{y_2}\rightarrow \mP^n_{y_1}$. Then the class of the image is obtained by the Gysin homomorphism (section \ref{SecMinimalLifting}) from the initial class. \epr
In general, the calculation of the cohomology class of the restriction is done as follows. As will be explained later we can assume $\tSi|_{C_i}$ to be irreducible (reduced). The calculation is in fact a typical procedure from intersection theory and does not use any property of $\tSi$ related to the singularity theory.
So, let $\tSi\subset Aux\times\mPN$ be an irreducible (reduced) projective variety. In general $\tSi$ is not a globally complete intersection, however we assume that we can calculate the cohomology class $[\tSi]\in H^*(Aux\times\mPN)$ by the classical intersection theory (i.e. by intersecting various hypersurfaces and subtracting the contributions of the residual pieces). Let $\{C_i\}_i$ be all the cycles of jump.
\bel The classes $[\tSi|_{C_i}]\in H^*(Aux\times\mPN)$ can all be calculated recursively using the following data: \bei \item The cohomology class of $\tSi$, obtained by the classical intersection theory \item The cohomology classes of $C_i\subset Aux$, obtained by the classical intersection theory \eei \eel \bpr The proof goes by the induction on the grading of the cycles of jump and by the recursion on the dimensionality of the auxiliary space $Aux$.
We first calculate $[\tSi|_{C}]$ for $C$ the cycle of jump of grading 1 (see the definition \ref{DefCycleOfJumpGrading}). As follows from proposition \ref{ClaimCycleOfJump&Hypersurf} for such a cycle there exists a hypersurface $\{g=0\}$ containing $C$ and do not containing any other cycle. So, consider the intersection $\tSi\cap\{g=0\}$. There can be two possibilities for the jump of dimension of fibre
and the codimension of the cycle in $Aux=Aux_0$ (by corollary \ref{ClaimJumpOfDimensionVsCodimension}): \bei \item $\Delta dim_{C_i}<codim_{Aux}(C)-1.$ In this case the intersection gives just the restriction of the fibration
$\tSi\cap(g=0)=\tSi|_{g=0}$, without any residual terms. Then decomposing the polynomial into irreducible factors $g=\prod_i g_i^{n_i}$ we get the union of restrictions:
$\tSi|_{g=0}=\bigcup_i n_i\tSi|_{g_i}$, each restriction again being irreducible.
Consider now the cycle $C$ as a subvariety of the new (smaller) auxiliary space $Aux_1=\{g=0\}\subset Aux_0$ intersect it with the next hypersurface and so on. \item $\Delta dim_{C}=codim_{Aux}(C)-1$ In this case the intersection brings residual piece (of the same dimension): \beq\label{ClassOfRestriction}
\tSi\cap(g=0)=\tSi|_{g=0}\cup\alpha\tSi|_C \eeq Here $\alpha$ is the multiplicity with which the residual piece enters. Note that the residual piece consists of the restriction to the cycle $C$ only, because we have chosen $C$ to be of grade 1 (and then all other restrictions are excluded by codimension). In this case we actually obtain the needed restriction as a residual piece, so the problem is reduced to the calculation of
$[\tSi|_{g=0}]$. \eei Thus in both cases the calculation is reduced to enumeration in a smaller auxiliary space: $Aux_1=\{g=0\}\subset Aux_0$
By the assumption of the lemma, the class $[\tSi]$ is obtained by the classical intersection procedure (consisting of intersections and removal of residual varieties contribution). It follows that the class
$[\tSi|_{g=0}]$ can be obtained by the same procedure. In the course of calculation there will appear new cycles of jump, however the dimension of the auxiliary space has been reduced by 1. In this way by recursion we obtain the class of restriction: $[\tSi|_C]$ for any cycle $C$ of grade 1.
The case of general grade is treated by induction. Suppose we have calculated the classes of restriction
$[\tSi|_{C_i}]$ for all the cycles of grades up to $k$. When doing the procedure for a cycle $C$ of grade $(k+1)$ the only difference will be that the equation (\ref{ClassOfRestriction}) is replaced by a more general: \beq
\tSi\cap(g=0)=\tSi|_{g=0}\cup\alpha\tSi|_{C}\bigcup \alpha_i\tSi|_{C_i} \eeq that is on the right hand side there appear restrictions to other cycles. However (as was noted above), by lemma \ref{ClaimCycleOfJump&Hypersurf} other cycles will be of grade at most $k$, the case already solved. So, from the above equation we get the class of the needed restriction. \epr
\subsubsection{On use of consistency conditions}\label{SecUseOfConsistenCondition} In the preceding sections we have described how to calculate the cohomology classes of residual varieties. The method is recursive and often is quite cumbersome (though it is always possible to perform the calculations using computer). It happens, that one often can avoid lengthy calculations by using (heavily) the consistency conditions.
The consistency conditions were stated in $\S$ \ref{SecLiftings} (lemma \ref{ClaimFirstConsistencyCondition} and \ref{ClaimSeconConsistencCondition}). An "experimental" observation is that they are very restrictive and in fact often {\it fix the cohomology classes of residual varieties}. (This happens for {\bf all} the examples considered in the paper).
In general, the consistency conditions fix the cohomology class in the following equation: \beq
[\tSi][degeneration]=[\tSi_{degenerated}]+[(y_1\dots y_k)|\mbox{linearly dependent}][R_k]+\dots+
[(y_1,y_2)|\mbox{linearly dependent}][R_2] \eeq Here the classes of the initial stratum ($\tSi$) and degenerating divisor/cycle are known, while the class of degenerated stratum ($\tSi_{degenerated}$) satisfies some consistency conditions (symmetric in some variables, with no terms of $Y^i,~~i>n-k$). The "experimental fact" ({\it which happens in all the examples considered in the paper}) is: {\bf the above equation, together with consistency conditions has unique solution}.
The general formal way of calculations was already explained in details (sections \ref{SecPossibleCyclesOfJump}, \ref{SecIdeologyOfDegenerations}, \ref{SecCohomologyClassOfRestrictionFibration}). Thus we do not consider the above equation in details and do not prove any general statement of uniqueness of solution. We emphasize, however, that all the results of this paper can be (and in fact were) obtained using the consistency conditions only.
\section{Some explicit formulae for cohomology classes of singular strata}\label{SecApendCohomologClasses} \subsubsection{Computer calculations}\label{SecComputerCalcul} As was already mentioned, except for the simplest cases (ordinary multiple points, $A_{k\leq3}$) the computation should be done on computer (systems as Singular or Mathematica can be used). The specific programs can be obtained from the author.
Here we meet the following difficulty of purely software nature. The calculation consists of addition/subtraction and multiplication of polynomials of indefinite degree and indefinite number of variables (i.e. both the degree and the number of variables are parameters). For example, when enumerating the double point of corank $r$ (section \ref{SecFormOfCorank}) the multidegree of $\tSi^n_r$ is a polynomial in $(f,x,y_1,\dots,y_r)$ of degree $n+1+r\frac{2n-1+r}{2}$. Here both $r$ and $n$ are {\it parameters}. Another task is elimination and solution of systems of big linear equations. To the best of our knowledge, neither Singular nor Mathematica can in general process such expressions (i.e. open the brackets, simplify, extract the coefficient of, say, $y^{n-r}_1\dots y^{n-r}_k$).
However the programs solve perfectly the problem for any fixed values of $n,r$. Thus to obtain the final answer (which is a polynomial in $n,r$) one should calculate separately for a sufficient number of pairs $(n,r)$ and then interpolate. For the interpolation to be rigorous, one must know the degree of the polynomial we want to recreate. This degree is known by universality \cite{Kaz2}.
We emphasize that this problem is due to the current state of software only, and not of any mathematical origin.
\subsubsection{On the possible checks of numerical results} As in every problem, whose answer is explicit numerical formula, it is important to have some ways to check the numerical results. Our results "successfully pass" the following checks: \\ $\bullet$Comparison to the already known degrees. The most extensive "database" in this case is Kazarian's tables of Thom classes for singularities of codimension up to 7. Our results are obtained by specializations from the general case to the case of complete linear system hypersurfaces of degree $d$ in $\mP^n$.
Very few of Kazarian's results were known before. The degree for ordinary multiple point is a classical result (known probably from the 19'th century). Another check is for the cusp ($A_2$), enumerated by P.Aluffi.\\ $\bullet$Comparison to the known results for curves ($n=2$). By universality, the substitution $n=2$ to the formulae must give the degrees of the strata for curves. This enables to check, for example: $A_3,A_4,D_4,D_5,$$E_6,X_9,Z_{11},W_{12}$\\ $\bullet$Comparison to the case when the jet is reducible. For example, for singularity with degenerate quadratic form of co-rank $k$, the substitution $k=n-2$ or $k=n-1$ gives singularities with reducible two-jet. And in these cases the enumeration is immediate.
\subsubsection{Cohomology classes for hypersurfaces} We present here the cohomology classes of the (minimally) lifted strata: \beq \tSi(x)=
\overline{\{(x,f)|~\mbox{the hypersurface defined by}~f~\mbox{has singularity of the given type at the point}~x\}}\subset \mP^n_x\times\mPN \eeq The classes $[\tSi(x)]$ are expressed in terms of the generators of the cohomology ring of the ambient space ($H^*(\mP^n_x)=\mZ[X]/X^{n+1},~~H^*(\mPN)=\mZ[F]/F^{D+1}$). So, the polynomials represent the multi-degrees of the lifted strata. The degree of the stratum itself $[\Sigma]$ is the coefficient of $X^n$.
All the notations are from \cite{AVGL}. Here (as anywhere in the paper): $d$ is the degree of singular hypersurfaces, $n$ is the dimensionality of the ambient space (thus hypersurfaces are of dimension $n-1$). \bprop The cohomology classes of the lifted strata and the degrees of the strata in several simplest cases are: \eprop $\bullet${\bf Ordinary multiple point:} $f=\sum_i z^{p+1}_i$. Includes, for $n=2$: $A_1,D_4,X_9,\dots$; for $n=3$: $A_1,P_8,\dots$. $$[\tSi(x)]=Q^{n+p\choose{p}},~~Q=(d-p)X+F,~~~~~[\Sigma]={{n+p\choose{p}}\choose{n}}(d-p)^n.$$
$\bullet${\bf Degenerate multiple point (with reducible defining form):} (Defined in Appendix A)\\ $jet_p(f)=\prod_{i=1}^k\Big(\Omega^{(p_i)}_i\Big)^{r_i}$ $\sum_{i=1}^k r_ip_i=p$. Includes: \\$\star$ {curves $n=2$}: $A_1(k=2,r_i=1,~p_i=1),$ $A_2(k=1,r_1=2,~p_1=1),$ $D_4(k=3,r_i=1,~p_i=1),$ $E_6(k=1,r_1=3,~p_1=1),$ $X_9(k=4,r_i=1,~p_i=1),$ $Z_{11}(k=2,r_1=1,r_2=3,~p_i=1),$ $W_{12}(k=1,r_1=4,~p_1=1),\dots$ \\$\star$ {surfaces $n=3$}: $A_2(k=2,r_i=1,~p_i=1),$ $D_4(k=1,r_1=2,~p_1=1),$ $T_{3,4,4}(k=2,r_i=1,~p_1=1,p_2=2),$ $T_{4,4,4}(k=3,r_i=1,~p_i=1),$ $V_{1,0}(k=2,r_1=1,r_2=2,~p_i=1),$ $V'_1(k=1,r_1=3,~p_1=1)\dots$.
Here the forms $\Omega^{(p_i)}_i$ are mutually generic (i.e. the corresponding hypersurfaces intersect generically near the singular point). The cohomology (multi-)class was given in equation (\ref{EqClassesForReducibleForms}). To obtain the answer we should extract from the equation the coefficient of maximal (non-zero) powers of $\Omega^{(p_1)}_1\dots\Omega^{(p_k)}_k$. \beq\hspace{-1cm}
[\tSi(x)]=\frac{1}{\rm{|Aut|}}\sum_{i=0}^{{p+n\choose{n}}-1}Q^{{p+n\choose{n}}-1-i} X^{k+i-\sum_{j=1}^k{p_j-1+n\choose{p_j}}} \hspace{-2.5cm}\sum_{\ber ~~~~~~~~~~~~i_1+\dots+i_k=i\\{p_j-1+n\choose{p_j-1}}-1\le i_j\le{p_j+n\choose{n}}-1\eer} \hspace{-2.5cm}{i\choose{i_1\dots i_k}}\prod_{j=1}^k r_j^{i_j}{{p_j-1+n\choose{n}}\choose{{p_j+n\choose{n}}-1-i_j}} \eeq Here $Aut$ is the group of automorphisms of the branches, $i\choose{i_1~\dots~i_k}$ is the multinomial coefficient from expansion of $(\dots)^i$ and $Q=(d-p)X+F$. \\ $\bullet${\bf Singularity with degenerate quadratic form $\Sigma^n_k:$} $f=\sum^k_{i=1}z^3_i+\sum^n_{i=k+1}z^2_i$. ($k=1:A_2,~~k=2:D_4,~~k=3:P_8\dots$). $$[\tSi^n_k(x)]=C_{n,k}(Q+X)^{n+1}Q^{k\choose{2}} \sum^k_{i=0}\frac{{n-i\choose{k-i}}{k\choose{i}} }{{2k\choose{k+i}}}Q^{k-i}(-x)^i~~~~~~Q=F+(d-2)X$$\\ $$C_{n,1}=2,~~C_{n,2}=2{n+1\choose{1}},~~C_{n,3}=2{n+2\choose{3}},~~C_{n,4}=2{n+3\choose{5}}\frac{n+1}{3}$$ \\ $$C_{n,n}=2{2n\choose{n}},~~C_{n,n-1}=\frac{2^k{2k\choose{k}}}{n},~~C_{n,n-2}=\frac{{2(k+1)\choose{k+1}}{2k\choose{k}}}{{k+2\choose{2}}}$$ In particular: \\$\star$ $[\Sigma_{A_2}]=3{n+2\choose{3}}(d-1)^{n-1}(d-2)$ \\$\star$ $[\Sigma_{D_4}]=\frac{(n+1)}{8}{n+1\choose{3}}(d-1)^{n-3}(d-2)^2\Big((d-2)(n^3+n^2+10n+8)+4(n^2+6)\Big)$ \\$\star$ $[\Sigma_{P_8}]={n+2\choose{3}}{n+4\choose{7}}(d-1)^{n-6}(d-2)^3\Big(\ber {n+7\choose{3}}\frac{(d-2)^3}{10}+{n+6\choose{2}}(d-2)^2+(n+5)\frac{9(d-2)}{2}+12\eer\Big)$
~\\ $\bullet$ {\bf $A_3:$} $f=z^4_1+\sum_{i=2}^nz^2_i,~~~~Q=F+(d-2)X,~~~~ [\tSi_{A_3}(x)]=(Q+X)^{n+1}(nQ-2X)(Q\frac{3n-1}{2}-4X)$. $$[\Sigma_{A_3}]={n+2\choose{3}}(d-1)^{n-2}\Bigg(\frac{(3n-1)(n+3)}{2}(d-2)^2+2(n-1)(d-2)-4\Bigg)$$
$\bullet${\bf $A_4:$} $f=z^5_1+\sum_{i=2}^nz^2_i$. $$[\tSi_{A_4}(x)]=(Q+X)^{n+1}\Big(\frac{n(5n^2-5n+2)}{2}Q^3-4(5n^2-3n+1)Q^2x+4(13n-5)Qx^2-48x^3\Big)~~~~Q=F+(d-2)X$$ $$[\Sigma_{A_4}]=\frac{1}{8}{n+2\choose{3}}(d-1)^{n-3} \Bigg(\ber d^3(24-46n+27n^2+30n^3+5n^4)-d^2(+96-184n+138n^2+160n^3+30n^4)+\\+d(144-316n+192n^2+280n^3+60n^4) -136+224n-48n^2-160n^3-40n^4\eer\Bigg)$$
$\bullet${$D_5$:} $f=z^4_1+z_1z^2_2+\sum_{i=3}^nz^2_i,~~~~Q=(d-2)X+F$. $$[\tSi_{D_5}(x)]=(Q+X)^{n+1}\frac{Q(n+1)}{6}\Bigg((3n-2)Q-10X\Bigg)\Bigg((n^2-n)Q^2-6(n-1)QX+12X^2\Bigg)$$ $\bullet${$D_6$:} $f=z^5_1+z_1z^2_2+\sum_{i=3}^nz^2_i,~~~~Q=(d-2)X+F$ $$[\tSi_{D_6}(x)]=(Q+X)^{n+1}(1+n)Q \Bigg(\ber\frac{4(n-1)n(3n^2-5n+3)}{15}Q^4-\frac{2(n-1)(16n^2-19n+9)}{3}Q^3x+\\ \frac{2(83n^2-140n+69)}{3}Q^2x^2-136(n-1)Qx^3+136x^4\eer\Bigg)$$
$\bullet${\bf $E_6$:} $f=z^4_1+z^3_2+\sum_{i=3}^nz^2_i,~~~~Q=(d-2)X+F$. $$[\tSi_{E_6}(x)]=(Q+X)^{n+1}Q(n+1)\Bigg(\ber\frac{n(n-1)(12n^2-15n+2)}{40}Q^4-\frac{(n-1)n(15n-14)}{4}Q^3X+\\+ \frac{37n^2-52n+12}{2}Q^2X^2-6(7n-5)QX^3+36X^4\eer\Bigg)$$
$\bullet${\bf $X_9$:} $f=z^4_1+z^4_2+\sum_{i=3}^nz^2_i,~~~~Q=(d-2)X+F$. $$[\tSi_{X_9}(x)]=(Q+X)^{n+1}Q(n+1)\Bigg(\ber\frac{n(n-1)(3n+1)(33n^3-102n^2+102n-20)}{1680}Q^6- \frac{(n-1)n(138n^3-309n^2+153n+86)}{120}Q^5X\\+ \frac{(379n^4-1004n^3+719n^2+166n-160)}{40}Q^4X^2-\frac{126n^3-247n^2+58n+72}{2}Q^3X^3+\\+ \frac{625n^2-670n-216}{6}Q^2x^4-16(8n-1)QX^5+48X^6\eer\!\!\!\!\Bigg)\\ $$
$\bullet${\bf $Q_{10}$:} $z_1^4+z_2^3+z_1z_3^2+\sum_{i=4}^n z_i^2,~~~~Q=(d-2)X+F$. $$[\tSi_{Q_{10}}(x)]=(Q+X)^{n+1}{n+2\choose{3}}Q^3\Big((n-1)Q-4X\Big)^2\Bigg(\frac{3}{5}{n\choose{3}}Q^3- \frac{12}{5}{n-1\choose{2}}Q^2X+6(n-2)QX^2-12X^3\Bigg)$$
$\bullet${\bf $S_{11}$:} $z_1^4+z_2^2z_3+z_1z_3^2+\sum_{i=4}^n z_i^2,~~~~Q=(d-2)X+F$. $$[\tSi_{S_{11}}(x)]=(Q+X)^{n+1}6{n+2\choose{3}}Q^3\Big((n-1)Q-4x\Big) \Bigg(\ber\frac{(n-2)(n-1)n(51n^2-98n+31)}{3360}Q^5+\frac{(56n-79)}{2}Qx^4 \\-\frac{(n-2)(n-1)(253n^2-392n+75)}{840}Q^4x-27x^5 \\+\frac{(n-2)(103n^2-190n+67)}{40}Q^3x^2-\frac{(47n^2-130n+75)}{4}Q^2x^3\eer\Bigg)$$
$\bullet${\bf $U_{12}$:} $z_1^3+z_2^3+z_3^4+\sum_{i=4}^n z_i^2,~~~~Q=(d-2)X+F$. $$[\tSi_{U_{12}}(x)]=(Q+X)^{n+1}{n+2\choose{3}}Q^3 \Bigg(\ber {n\choose{3}}\frac{(n-1)(117n^3-328n^2+207n+12)}{1120}Q^7- {n-1\choose{2}}\frac{(323n^4-1057n^3+1021n^2-231n-72)}{336}Q^6x\\ +{n-1\choose{2}}\frac{(991n^3-2545n^2+1525n-27)}{84}Q^5x^2+ \frac{(n-1)(2161n^2-5606n+2553)}{12}Q^3x^4 \\ -\frac{(3491n^4-16290n^3+25396n^2-14838n+2577)}{84}Q^4x^3+28(25n-31)Qx^6-\\ (475n^2-1150n+631)Q^2x^5-448x^7 \eer \Bigg)$$
{\it Address}: School of Mathematical Sciences, \mbox{Tel Aviv} University, \mbox{Ramat Aviv}, 69978 \mbox{Tel Aviv}, Israel.
{\it E-mail}: [email protected]
\end{document} | arXiv |
Integral linear operator
An integral bilinear form is a bilinear functional that belongs to the continuous dual space of $X{\widehat {\otimes }}_{\epsilon }Y$, the injective tensor product of the locally convex topological vector spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form.
These maps play an important role in the theory of nuclear spaces and nuclear maps.
Definition - Integral forms as the dual of the injective tensor product
See also: Injective tensor product and Projective tensor product
Let X and Y be locally convex TVSs, let $X\otimes _{\pi }Y$ denote the projective tensor product, $X{\widehat {\otimes }}_{\pi }Y$ denote its completion, let $X\otimes _{\epsilon }Y$ denote the injective tensor product, and $X{\widehat {\otimes }}_{\epsilon }Y$ denote its completion. Suppose that $\operatorname {In} :X\otimes _{\epsilon }Y\to X{\widehat {\otimes }}_{\epsilon }Y$ denotes the TVS-embedding of $X\otimes _{\epsilon }Y$ into its completion and let ${}^{t}\operatorname {In} :\left(X{\widehat {\otimes }}_{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }$ :\left(X{\widehat {\otimes }}_{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }} be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of $X\otimes _{\epsilon }Y$ as being identical to the continuous dual space of $X{\widehat {\otimes }}_{\epsilon }Y$.
Let $\operatorname {Id} :X\otimes _{\pi }Y\to X\otimes _{\epsilon }Y$ denote the identity map and ${}^{t}\operatorname {Id} :\left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\pi }Y\right)_{b}^{\prime }$ :\left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\pi }Y\right)_{b}^{\prime }} denote its transpose, which is a continuous injection. Recall that $\left(X\otimes _{\pi }Y\right)^{\prime }$ is canonically identified with $B(X,Y)$, the space of continuous bilinear maps on $X\times Y$. In this way, the continuous dual space of $X\otimes _{\epsilon }Y$ can be canonically identified as a vector subspace of $B(X,Y)$, denoted by $J(X,Y)$. The elements of $J(X,Y)$ are called integral (bilinear) forms on $X\times Y$. The following theorem justifies the word integral.
Theorem[1][2] — The dual J(X, Y) of $X{\widehat {\otimes }}_{\epsilon }Y$ consists of exactly those continuous bilinear forms c on $X\times Y$ that can be represented in the form of a map
$b\in B(X,Y)\mapsto v(b)=\int _{S\times T}b{\big \vert }_{S\times T}\left(x',y'\right)\mathrm {d} \mu \!\left(x',y'\right)$
where S and T are some closed, equicontinuous subsets of $X_{\sigma }^{\prime }$ and $Y_{\sigma }^{\prime }$, respectively, and $\mu $ is a positive Radon measure on the compact set $S\times T$ with total mass $\leq 1.$ Furthermore, if A is an equicontinuous subset of J(X, Y) then the elements $v\in A$ can be represented with $S\times T$ fixed and $\mu $ running through a norm bounded subset of the space of Radon measures on $S\times T.$
Integral linear maps
A continuous linear map $\kappa :X\to Y'$ is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by $(x,y)\in X\times Y\mapsto (\kappa x)(y)$.[3] It follows that an integral map $\kappa :X\to Y'$ is of the form:[3]
$x\in X\mapsto \kappa (x)=\int _{S\times T}\left\langle x',x\right\rangle y'\mathrm {d} \mu \!\left(x',y'\right)$
for suitable weakly closed and equicontinuous subsets S and T of $X'$ and $Y'$, respectively, and some positive Radon measure $\mu $ of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every $y\in Y$, $ \left\langle \kappa (x),y\right\rangle =\int _{S\times T}\left\langle x',x\right\rangle \left\langle y',y\right\rangle \mathrm {d} \mu \!\left(x',y'\right)$.
Given a linear map $\Lambda :X\to Y$, one can define a canonical bilinear form $B_{\Lambda }\in Bi\left(X,Y'\right)$, called the associated bilinear form on $X\times Y'$, by $B_{\Lambda }\left(x,y'\right):=\left(y'\circ \Lambda \right)\left(x\right)$. A continuous map $\Lambda :X\to Y$ is called integral if its associated bilinear form is an integral bilinear form.[4] An integral map $\Lambda :X\to Y$ is of the form, for every $x\in X$ and $y'\in Y'$:
$\left\langle y',\Lambda (x)\right\rangle =\int _{A'\times B''}\left\langle x',x\right\rangle \left\langle y'',y'\right\rangle \mathrm {d} \mu \!\left(x',y''\right)$
for suitable weakly closed and equicontinuous aubsets $A'$ and $B''$ of $X'$ and $Y''$, respectively, and some positive Radon measure $\mu $ of total mass $\leq 1$.
Relation to Hilbert spaces
The following result shows that integral maps "factor through" Hilbert spaces.
Proposition:[5] Suppose that $u:X\to Y$ is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings $\alpha :X\to H$ and $\beta :H\to Y$ such that $u=\beta \circ \alpha $.
Furthermore, every integral operator between two Hilbert spaces is nuclear.[5] Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.
Sufficient conditions
Every nuclear map is integral.[4] An important partial converse is that every integral operator between two Hilbert spaces is nuclear.[5]
Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that $\alpha :A\to B$, $\beta :B\to C$, and $\gamma :C\to D$ are all continuous linear operators. If $\beta :B\to C$ is an integral operator then so is the composition $\gamma \circ \beta \circ \alpha :A\to D$.[5]
If $u:X\to Y$ is a continuous linear operator between two normed space then $u:X\to Y$ is integral if and only if ${}^{t}u:Y'\to X'$ is integral.[6]
Suppose that $u:X\to Y$ is a continuous linear map between locally convex TVSs. If $u:X\to Y$ is integral then so is its transpose ${}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }$.[4] Now suppose that the transpose ${}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }$ of the continuous linear map $u:X\to Y$ is integral. Then $u:X\to Y$ is integral if the canonical injections $\operatorname {In} _{X}:X\to X''$ (defined by $x\mapsto $ value at x) and $\operatorname {In} _{Y}:Y\to Y''$ are TVS-embeddings (which happens if, for instance, $X$ and $Y_{b}^{\prime }$ are barreled or metrizable).[4]
Properties
Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If $\alpha :A\to B$, $\beta :B\to C$, and $\gamma :C\to D$ are all integral linear maps then their composition $\gamma \circ \beta \circ \alpha :A\to D$ is nuclear.[5] Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection $u:X\to X$ cannot be an integral operator.
See also
• Auxiliary normed spaces
• Final topology
• Injective tensor product
• Nuclear operators
• Nuclear spaces
• Projective tensor product
• Topological tensor product
References
1. Schaefer & Wolff 1999, p. 168.
2. Trèves 2006, pp. 500–502.
3. Schaefer & Wolff 1999, p. 169.
4. Trèves 2006, pp. 502–505.
5. Trèves 2006, pp. 506–508.
6. Trèves 2006, pp. 505.
Bibliography
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• Dubinsky, Ed (1979). The Structure of Nuclear Fréchet Spaces. Lecture Notes in Mathematics. Vol. 720. Berlin New York: Springer-Verlag. ISBN 978-3-540-09504-0. OCLC 5126156.
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
• Hogbe-Nlend, Henri; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. Vol. 52. Amsterdam New York New York: North Holland. ISBN 978-0-08-087163-9. OCLC 316564345.
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• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
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External links
• Nuclear space at ncatlab
Functional analysis (topics – glossary)
Spaces
• Banach
• Besov
• Fréchet
• Hilbert
• Hölder
• Nuclear
• Orlicz
• Schwartz
• Sobolev
• Topological vector
Properties
• Barrelled
• Complete
• Dual (Algebraic/Topological)
• Locally convex
• Reflexive
• Reparable
Theorems
• Hahn–Banach
• Riesz representation
• Closed graph
• Uniform boundedness principle
• Kakutani fixed-point
• Krein–Milman
• Min–max
• Gelfand–Naimark
• Banach–Alaoglu
Operators
• Adjoint
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• Unbounded
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Algebras
• Banach algebra
• C*-algebra
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• Operator algebra
• Group algebra of a locally compact group
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Open problems
• Invariant subspace problem
• Mahler's conjecture
Applications
• Hardy space
• Spectral theory of ordinary differential equations
• Heat kernel
• Index theorem
• Calculus of variations
• Functional calculus
• Integral operator
• Jones polynomial
• Topological quantum field theory
• Noncommutative geometry
• Riemann hypothesis
• Distribution (or Generalized functions)
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Interesting insights on math.
Sigma fields are Venn diagrams
The starting point for probability theory will be to note the difference between outcomes and events.
An outcome of an experiment is a fundamentally non-empirical notion, about our theoretical understanding of what states a system may be in -- it is, in a sense, analogous to the "microstates" of statistical physics. The set of all outcomes $x$ is called the sample space $X$, and is the fundamental space to which we will give a probabilistic structure (we will see what this means).
Our actual observations, the events, need not be so precise -- for example, our measurement device may not actually measure the exact sequence of heads and tails as the result of an experiment, but only the total number of heads, or something -- analogous to a "macrostate". But these measurements are statements about what microstates we know are possible for our system to be in -- i.e. they correspond to sets of outcomes. These sets of outcomes that we can "talk about" are called events $E$, and the set of all possible events is called a field $\mathcal{F}\subseteq 2^X$.
For instance: if our sample space is $\{1,2,3,4,5,6\}$ and our measurement apparatus is a guy who looks at the reading and tells us if it's even or odd, then the field is $\{\varnothing, \{1,3,5\},\{2,4,6\},X\}$. We simply cannot talk about sets like $\{1,3\}$ or $\{1\}$. Our information just doesn't tell us anything about sets like that -- when we're told "odd", we're never hinted if the outcome was 1 or 3 or 5, so we can't even have prior probabilities -- we can't even give probabilities to whether a measurement was a 1 or a 3.
Well, what kind of properties characterise a field? There's actually a bit of ambiguity in this -- it's clear that a field should be closed under negation and finite unions (and finite intersections follow via de Morgan) -- if you can talk about whether $P_1$ and $P_2$ are true, you can check each of them to decide if $P_1\lor P_2$ is true (and since a proposition $P$ corresponds to a set $S$ in the sense that $P$ says "one of the outcomes in $S$ is true", $\lor$ translates to $\cup$). But if you have an infinite number of $P_i$'s, can you really check each one of them so that you can say without a doubt that a field is closed under arbitrary union?
Well, this is (at this point) really a matter of convention, but we tend to choose the convention where the field is closed under negation and countable unions. Such a field is called a sigma-field. We will actually see where this convention comes from (and why it is actually important) when we define probability -- in fact, it is required for the idea that one may have a uniform probability distribution on a compact set in $\mathbb{R}^n$.
A beautiful way to understand fields and sigma fields is in terms of venn diagrams -- in fact, as you will see, fields are precisely a formalisation of Venn diagrams. I was pretty amazed when I discovered this (rather simple) connection for myself, and you should be too.
Suppose your experiment is to toss three coins, and make "partial measurements" on the results through three "measurement devices":
A: Lights up iff the number of heads was at least 2.
B: Lights up iff the first two coins landed heads.
C: Lights up iff the third coin landed heads.
What this means is that $A$ gives you the set $\{HHT, HTH, THH, HHH\}$, $B$ gives you the set $\{HHH, HHT\}$, $C$ gives you the set $\{HHH, HTH, THH, TTH\}$. Based on precisely which devices light up, you can decide the truth values of $\lnot$'s and $\lor$'s of these statements, i.e. complements and unions of these sets -- this is the point of fields, of course.
Or we could visualise things.
Well, the Venn diagram produces a partition of $X$ corresponding to the equivalence relation of "indistinguishability", i.e. "every event containing one outcome contains the other"? The field consists precisely of any set one can "mark" on the Venn diagram -- i.e. unions of the elements of the partition.
A consequence of this becomes immediately obvious:
Given a field $\mathcal{F}$ corresponding to the partition $\sim$, the following bijection holds: $\mathcal{F}\leftrightarrow 2^{X/\sim}$.
Consequences of this include: the cardinalities of finite sigma fields are precisely the powers of two; there is no countably infinite finite field.
Often, one may want to some raw data from an experiment to obtain some processed data. For example, let $X=\{HH,HT,TH,TT\}$ and the initial measurement is of the number of heads:
$$\begin{align}
\mathcal{F}=&\{\varnothing, \{TT\}, \{HT, TH\}, \{HH\},\\
& \{TT, HT, TH\}, \{TT, HH\}, \{HT, TH, HH\}, X \}
\end{align}$$
What kind of properties of the outcome can we talk about with certainty given the number of heads? For example, we can talk about the question "was there at least one heads?"
$$\mathcal{G}=\{\varnothing, \{TT\}, \{HT, TH, HH\}, X\}$$
There are two ways to understand this "processing" or "re-measuring". One is as a function $f:\frac{X}{\sim_\mathcal{F}}\to \frac{X}{\sim_\mathcal{G}}$. Recall that:
\frac{X}{\sim_\mathcal{F}}&=\{\{TT\},\{HT,TH\},\{HH\}\}\\
\frac{X}{\sim_\mathcal{G}}&=\{\{TT\},\{HT,TH,HH\}\}
Any such $f$ is a permissible "measurable function", as long as $\sim_\mathcal{G}$ is at least as coarse a partition as $\sim_\mathcal{F}$. In other words, a function from $X/\sim_1$ to $(X/\sim_1)/\sim_2$ is always measurable.
But there's another, more "natural", less weird and mathematical way to think about a re-measurement -- as a function $f:X\to Y$, where in this case $Y=\{0,1\}$ where an outcome maps to 1 if it has at least one heads, and 0 if it does not.
But there's a catch: knowing that an event $E_Y$ in $Y$ occurred is equivalent to knowing that an outcome in $X$ mapping to $E_Y$ occurred -- i.e. that the event $\{x\in X\mid f(x)\in Y\}$ occurred. Such an event must be in the field on $X$, i.e.
$$\forall y\in\mathcal{F}_Y,f^{-1}(y)\in\mathcal{F}_X$$
This is the condition for a measurable function, also known as a random variable.
One may observe certain analogies between the measurable spaces outlined above, and topology -- in the case of countable sample spaces, there actually is a correspondence. The similarity between a Venn diagram and casual drawings of a topological space is not completely superficial.
The key idea behind fields is mathematically a notion of "distinguishability" -- if all we can measure is the number of heads, $HHTTH$ and $TTHHH$ are identical to us. For all practical purposes, we can view the sample space as the partition by this equivalence relation. They are basically the "same point".
It's this notion that a measurable function seeks to encapsulate -- it is, in a sense, a generalisation of a function from set theory. A function cannot distinguish indistinguishable points -- in set theory, "indistinguishability" is just equality, the discrete partition; a measurable function cannot distinguish indistinguishable points -- but in measurable spaces, "indistinguishability" is given by some equivalence relation.
Let's see this more precisely.
Given sets with equivalence relations $(X,\sim)$, $(Y,\sim)$, we want to ensure that some function $f:X\to Y$ "lifts" to a function $f:\frac{X}{\sim}\to\frac{Y}{\sim}$ such that $f([x])=[f(y)]$.
(Exercise: Show that this (i.e. this "definition" being well-defined) is equivalent to the condition $\forall E\in\mathcal{F}_Y, f^{-1}(E)\in \mathcal{F}_X$. It may help to draw out some examples.)
Well, this expression of the condition -- as $f([x])=[f(y)]$ -- even if technically misleading (the two $f$'s aren't really the same thing) give us the interpretation that a measurable function is one that commutes with the partition or preserves the partition.
While homomorphisms in other settings than measurable spaces do not precisely follow the "cannot distinguish related points" notion, they do follow a generalisation where equivalence relations are replaced with other relations, operations, etc. -- in topology, a continuous function preserves limits; in group theory, a group homomorphism preserves the group operation; in linear algebra, a linear transformation preserves linear combinations; in order theory, an increasing function preserves order, etc. In any case, a homomorphism is a function that does not "break" relationships by creating a "finer" relationship on the target space.
Written by Abhimanyu Pallavi Sudhir on October 17, 2019
Tags -- measurable function, probability, probability theory, random variables, sigma field, venn diagram
Copyright © 2016-2019, The Winding Number by Abhimanyu Pallavi Sudhir. Picture Window theme. Powered by Blogger. | CommonCrawl |
Ultrafast all-optical tuning of direct-gap semiconductor metasurfaces
Maxim R. Shcherbakov ORCID: orcid.org/0000-0001-7198-54821,
Sheng Liu2,
Varvara V. Zubyuk1,
Aleksandr Vaskin3,
Polina P. Vabishchevich1,
Gordon Keeler2,
Thomas Pertsch ORCID: orcid.org/0000-0003-4889-08693,
Tatyana V. Dolgova1,
Isabelle Staude3,
Igal Brener2 &
Andrey A. Fedyanin1
Nature Communications volume 8, Article number: 17 (2017) Cite this article
103 Altmetric
Metamaterials
Ultrafast photonics
Optical metasurfaces are regular quasi-planar nanopatterns that can apply diverse spatial and spectral transformations to light waves. However, metasurfaces are no longer adjustable after fabrication, and a critical challenge is to realise a technique of tuning their optical properties that is both fast and efficient. We experimentally realise an ultrafast tunable metasurface consisting of subwavelength gallium arsenide nanoparticles supporting Mie-type resonances in the near infrared. Using transient reflectance spectroscopy, we demonstrate a picosecond-scale absolute reflectance modulation of up to 0.35 at the magnetic dipole resonance of the metasurfaces and a spectral shift of the resonance by 30 nm, both achieved at unprecedentedly low pump fluences of less than 400 μJ cm–2. Our findings thereby enable a versatile tool for ultrafast and efficient control of light using light.
Optical metasurfaces, which consist of designed subwavelength building blocks arranged in two dimensions, provide a versatile platform for the manipulation of light fields1,2,3,4. While comprehensive spatial, spectral and polarisation control of the optical response was demonstrated by a large body of research, achieving fast and efficient temporal control remains an open challenge. Such control would dramatically enhance the scope of optical metasurfaces, as their functionalities would no longer be permanently encoded during the fabrication process but could be tuned on demand. Numerous ways of actively modulating the optical properties of metasurfaces were reported, including phase-change materials5, 6, mechanical tuning7, 8, liquid-crystal-based tuning9, and all-optical modulation10,11,12,13. However, none of these approaches allows for fast and efficient modulation at the same time.
A new generation of tunable metasurfaces was recently enabled by high-index semiconductor materials in their transparency windows14,15,16,17,18. The interest in Mie-resonant semiconductor metasurfaces is triggered by their low absorption losses as compared to plasmonic metasurfaces as well as by the diversification of optical engineering options by adding magnetic dipole (MD) Mie-type resonances to the toolbox19, 20. Furthermore, semiconductors provide the possibility of modulating the refractive index via the injection of free carriers (FCs); this effect is quite subtle in metals due to the very high initial electron density. Nonlinear-optical properties of semiconductor metasurfaces were recently found to be improved by orders of magnitude when compared with the respective constituent materials21,22,23,24,25,26, paving the way to efficient all-optical tuning.
Silicon has been employed to demonstrate all-optical modulation in Mie-resonant nanostructures23, 27,28,29. Although silicon is the most obvious choice of material for semiconductor metasurfaces, due to its widespread use in microelectronics and on-chip photonics, its indirect bandgap does not allow for efficient generation and recombination of FCs. Gallium arsenide is a material vastly utilised for all-optical modulation; examples include coupled waveguides30, microring cavities31, and photonic crystal cavities32, 33, to name a few. Although there are clear advantages for using direct-gap semiconductors for ultrafast all-optical tuning of metasurfaces, an experimental demonstration of this concept is still missing.
In this work, we report on ultrafast and efficient all-optical tuning of Mie-resonant GaAs metasurfaces. By means of femtosecond wide-band pump–probe spectroscopy, we demonstrate, for the first time to our knowledge, free-carrier-induced absolute reflectance modulation of up to 0.35 under low pump fluence values (not exceeding 400 μJ cm–2) and with recovery times of about 6 ps mainly determined by surface-mediated recombination processes. The observed reflectance modulation is explained by ultrafast tuning of the spectral position of the MD resonance by up to ∆λ = 30 nm, as well as its broadening, and agrees well with full-wave numerical simulations of the metasurface response that is based on a model for the FC dynamics.
The device schematic is depicted in Fig. 1a. The GaAs metasurfaces were fabricated using a recently published procedure34 that involves mask-etching of GaAs/AlGaAs heterostructures to nanopillars, and subsequent oxidation of an AlGaAs layer to AlGaO. A scanning electron micrograph of a typical sample is shown in Fig. 1b. The residual silica cap on top of the GaAs nanodisk provides for the index matching between the top and the bottom interfaces of the GaAs nanodisks, resulting in more pronounced resonances. The pillars are situated on a bulk GaAs substrate that we used for control measurements. The dimensions of the metasurface are as follows: both diameter and height of GaAs nanodisks are 300 nm, the structure period is 620 nm, the oxide layer thickness is 300 nm, the silica cap thickness is 200 nm. The MD mode of a GaAs nanodisk is visualised in Fig. 1c, where the local electric field map is given; see Methods and Supplementary Note 1 for the calculation details. The field amplitude and direction form a vortex-type profile, which is characteristic to the MD resonance in nanodisks15.
Tuning GaAs metasurfaces with femtosecond laser pulses. a Illustration of ultrafast tuning of the MD mode at low pump fluences. The resonance position is tuned within a 6-ps time window due to free carrier injection and subsequent recombination. b Scanning electron micrograph of a metasurface sample. The scale bar is 500 nm. c Electric fields of the MD mode in the vertical cross-section of a nanodisks, as excited by the probe beam
Pump–probe spectroscopy
Active ultrafast control over light propagation can be accomplished via femtosecond-pulse-induced FC generation, which is an efficient modulation process utilised, for example, in nonlinear integrated photonics35, 36, computational photonics37, and terahertz metasurfaces38. In order to explore the tuning possibilities in GaAs metasurfaces, we employ pump–probe transient reflectance spectroscopy using supercontinuum radiation as a probe. The setup is schematically shown in Fig. 2a. Amplified femtosecond laser pulses were split into pump and probe beams; the former consisted of a 50-fs, 800-nm-centred, 1-kHz pulse train and the latter was an 850–1300 nm supercoutinuum generated in a sapphire plate and conditioned by a longpass filter. Both beams were focused at the sample plane; the p-polarised pump was normally incident, and the p-polarised probe was reflected at an angle of 11.5 ± 0.5° from the normal; for more details, refer to Methods.
Experimental ultrafast all-optical tuning of GaAs metasurfaces. a Setup for broadband pump–probe spectroscopy. Amplified femtosecond pulses are used both for pumping metasurfaces and for generation of the supercontinuum (SC) probe. Transients are measured as a function of both time delay between the pump and probe pulses and probe wavelength. b Experimental reflectance of the sample at different pump–probe delays. The MD mode blueshift of 30 nm is observed at a pump fluence of 310 μJ cm−2. c Transient reflectance spectra that reveal the ultrafast modulation of the resonance within a 6-ps time window. Note that the reflectance values higher than 0.56 are clipped with the false-colour scale for the sake of better visualisation of the post-pump processes
In our pump–probe measurements, we focus on the MD mode. The MD mode is centred at 1018 nm and has a Q-factor of approximately 20, as shown in Fig. 2b with the blue curve. In the same panel, the reflectance is shown for two pump–probe delay values of τ = 1 ps, where the MD mode is blue-shifted by ∆λ = 30 nm due to generated FCs, and τ = 6 ps, where its central wavelength position is recovered. This is a situation opposite to that observed in Si nanodisks28, where heating was shown to dominate the resonance behaviour, causing it to red-shift. In Fig. 2c, the transient dynamics of the mode is shown as a function of time delay in the range of τ from −2 to 17 ps. The analysis shows that the blue shift is attributed to reduction of the real part of the refractive index of GaAs via the band filling effect and the Drude term39:
$$\Delta n=\Delta {n}_{{\rm{BF}}}+\Delta {n}_{{\rm{D}}} < 0$$
The detailed analysis of these contributions will be given below.
The metasurface provides all-optical modulation superior to that of the bulk GaAs. Two typical pump–probe transients are shown in Fig. 3a as recorded for the sample excited at its MD mode (red curve) and for the substrate (black curve; the modulation values are multiplied by 20 for comparison). We find that the modulation depth is enhanced by a factor of 50 at the metasurface when compared to the substrate. Second, it is apparent that the initially high modulation of the metasurface has a short relaxation time, although there is a residue after 6–7 ps, which will be discussed below. In contrast, the initial relaxation of the substrate happens on the scale of 100 ps. Though an additional discussion is needed to clarify the observations, a subwavelength-scale nanodisk metasurface acts as an efficient active element for all-optical modulation.
Transient optical properties of GaAs metasurfaces. a Transient reflectance measurement results for the sample (red curve) and for the substrate (black curve; ordinate values are multiplied by 20 for comparison) at a pump fluence of 380 μJ cm−2. An enhancement factor of about 50 is attained for the modulation depth at the metasurface sample. b Measured differential reflectance as a function of probe wavelength and time delay between the pump and probe pulses. White dashed lines denote the spectral positions of the ED and MD modes, while the black one indicates the area of low reflectance situated in between the resonances. c Power-dependent transient reflectance traces. The low-perturbation regime (black curve) reveals exponential decay with a time constant of τ relax ≈ 2.5 ps, whereas higher fluences (red and green curves) provide more complicated traces owing to the blue shift of the MD resonance
A comprehensive study of all-optical modulation as a function of experimental parameters is presented in Fig. 3b,c. Transients were measured as a function of both pump–probe delay and probe wavelength in the spectral vicinity of the electric dipole (ED) and MD resonances. A low-Q regime allows for ultrafast cavity buildup time that does not affect the "on"–"off" cycle the way it does in high-Q systems like photonic crystal cavities40. Fig. 3b reveals both positive and negative all-optical modulation, as well as a variety of relaxation behaviours observed for the metasurface. Even for fluence values as low as 310 μJ cm–2, we observe a considerable relative all-optical modulation spanning from ∆R/R = −50 to 73%, where ∆R = R pump − R, R is reflectance in the absence of the pump beam and R pump is reflectance in the presence of the pump beam; see Methods for further details. The maximum relative modulation of 90% was achieved at a fluence of 380 μJ cm–2 at λ probe = 975 nm; further fluence increase brought irreversible changes to the pump–probe traces.
Relaxation dynamics analysis
On each pump–probe trace, we point out two characteristic periods: a FC-dominated part (0–6 ps) and a phonon-dominated part (>6 ps); the latter is identified by referring to characteristic time scales of the relaxation processes in semiconductors41. Here, we will focus on the origins of the ultrafast component, that is, related to FCs, and leave the consideration of the phonon relaxation out of the scope of the paper. After the dense FC plasma is generated, because of the high initial temperature of FCs, thermalisation and cooling of carriers happen within 1 ps41. FC temperature dynamics is included in the model describing the band filling term; refer to Supplementary Note 4. Assuming the FC density is N, the dynamics of a dense e-h plasma is defined by the following expression42:
$$\frac{{\rm d}N}{{\rm d}t}=-{{AN}}-{{{BN}}}^{2}-{C}_{{\rm{eff}}}{N}^{3},$$
where A is the monomolecular coefficient responsible for radiationless decay through defects and surface states, B is the bimolecular recombination with radiation emission, and C eff is the effective Auger recombination coefficient that takes into account all the three-body processes, including scattering of excess carriers to the side valleys. We estimate the density of injected FCs as N 0 = 2F(1−e−αh)/hE pump, where h = 300 nm is the metasurface thickness, α = 1.4 × 104 cm–1 is the GaAs linear absorption coefficient at E pump = 1.55 eV, and F is the pump fluence. This estimate does not take into account possible absorption increase due to resonances at 800 nm; however, as full-wave simulations show, the metasurface is not resonant at the pump wavelength. While the bulk value of monomolecular relaxation rate is as low as A bulk = 7 × 107 s–1 (see ref. 42) and cannot be connected to the ultrafast relaxation we observe, it was reported repeatedly that nanostructuring affects monomolecular recombination via surface states with energy levels within the band gap33, 43. We verify that in our case, in the low-density regime—i.e., at a fluence value of 45 μJ cm–2, or estimated plasma density of N 0 ≈ 4·1018 cm–3—the time constant of the FC recombination process is approximately τ relax ≈ 2.5 ps, as derived from the black curve in Fig. 3c. At plasma densities this low, the fast relaxation cannot be attributed to high order processes, as the characteristic times obtained through two-body and three-body contributions combined are 340 ps, as calculated by numerically solving Eq. (2); see Supplementary Note 2. Therefore, from the black curve in Fig. 3c, we can extract the effective monomolecular coefficient, which is A eff = 1/τ relax ≈ 4.0 × 1011 s–1. The second-order and third-order recombination processes with B = 1.7 × 10–10 cm3 s–1 (see ref. 44) and C eff = 7 × 10–30 cm6 s–1 (see ref. 42) provide a slight reduction of the FC relaxation constant. The overall initial relaxation rate is expressed as follows:
$$\frac{1}{\it{\Gamma}}={A}_{{\rm{eff}}}+B{N}_{0}+{C}_{{\rm{eff}}}{N}_{0}^{2},$$
and for the densest initial plasmas of N 0 = 7.5 × 1019 cm–3 that we estimate from calculations, the second term of the right-hand side is 0.13 of the first, and the third one is 0.10 of the first. Putting all the contributions together, the lowest value of Γ is estimated at ≈1.9 ps, with the dominant contribution from the surface recombination processes.
Refractive index modulation by FCs
Here, we will consider two main mechanisms responsible for the observed tuning of the MD mode: the Drude and band filling terms. The Drude term implies a negative addition to the refractive index39:
$$\Delta {n}_{{\rm{D}}}(N,E)=-\left(\frac{{N}_{{\rm{e}}}}{{m}_{{\rm{e}}}}+{N}_{{\rm{h}}}\frac{{m}_{{\rm{hh}}}^{0.5}+{m}_{{\rm{lh}}}^{0.5}}{{m}_{{\rm{hh}}}^{1.5}+{m}_{{\rm{lh}}}^{1.5}}\right)\frac{{\hslash }^{2}{e}^{2}}{2{n}{\varepsilon }_{0}({E}^{2}+{\hslash }^{2}{\gamma }^{2})},$$
where N e and N h are the densities of electrons and holes, respectively (we assume N e = N h = N/2); m e, m hh and m lh are the effective masses of electrons, light holes and heavy holes, respectively; n is the initial index of GaAs, γ is the inverse collision time of the carriers (the damping factor), and ε 0 is the permittivity of vacuum. The band filling term is derived from the FC-modified interband absorption spectrum through the Kramers–Kronig relation:
$$\begin{matrix} \Delta \Delta \!{n}_{{\rm{BF}}}(N,E,T)=\frac{2c\hslash }{{e}^{2}}{\rm PV}{\int }_{\!\!\!\!0}^{\infty }\frac{\Delta \alpha (N,E{\prime},T)}{{E^{\prime} }^{2}-{E}^{2}}{\rm d}E{\prime} ,\end{matrix}$$
where ∆α(N, E, T) is the modulation of absorption for photon energies E lying above the band gap, and "PV" denotes the principal value. The derivation of ∆α as a function of N, E and the FC temperature T is given in Supplementary Note 4. Given Eqs. (4) and (5) and ∆λ ≈ λ 0∆n/n, the experimental value of ∆λ = 30 nm at τ = 1 ps can be reproduced using the FC density of N = 4.9 × 1019 cm–3 (the corresponding value at τ = 0 ps is N 0 = 7.5 × 1019 cm–3); the maximum refractive index change is ∆n = −0.14.
In order to verify the experimental data and the physics behind the metasurface tunability, we perform full-wave simulations using COMSOL Multiphysics; see Methods for details. In the simulations, the reflectance spectra of the metasurface are calculated as a function of time with the dielectric function of the GaAs modified according to Eqs. (2), (4), and (5) at the initial plasma density of N 0 = 7.5 × 1019 cm–3. For better correspondence between the calculations and experiment, we increased the damping factor γ of the Drude term by an order of magnitude; this correction might come from an increase of the damping in nanostructures45. Calculated reflectance is plotted in Fig. 4 as a function of time and wavelength, along with the corresponding cross sections at times <0, 1 and 6 ps. In the inset of Fig. 4b, the calculated dynamics of the real part of the refractive index is given. The non-exponential behaviour of ∆n is in agreement with the experimental data and is attributed to the temperature dependence of the band filling term. With the parameters used, we also observe excellent agreement with the experiment in terms of the MD mode dynamics: 30 nm shift at 1 ps delay and characteristic recovery of the resonance central wavelength in about 6 ps.
Full-wave simulation results. Transient reflectance spectra obtained using COMSOL Multiphysics with Eqs. (2), (4), and (5) describing the refractive index dynamics. a Reflectance of the sample at τ < 0 ps (pre-pump, blue curve), τ = 1 ps (carriers cooled, purple curve) and τ = 6 ps (carriers recombined, yellow curve). b Reflectance as a function of probe wavelength and time. The inset shows the time-dependent addition to the index of GaAs used in calculations
Analysing the efficiency of the observed all-optical modulation, it becomes evident that using direct-gap semiconductors as the constituent material for metasurfaces is key to developing active nonlinear-optical nanodevices. Indeed, if compared to indirect-gap materials like silicon, all the FC-related microscopic processes are in favour of direct-gap nanostructures: larger linear absorption and more efficient recombination lead to absolute modulation as high as 0.35 at a fluence of 310 μJ cm–2 and relaxation constant Γ < 2 ps, whereas in silicon, the most pronounced reported modulation was 0.2 at >1 mJ cm–2 (ref. 28) at a 30-ps relaxation. Another recent approach to all-optical modulation in nanostructures is using conductive transparent oxides46,47,48. However, their performance is still limited by high fluences of 1–10 mJ cm–2 needed to observe considerable modulation, and material properties that constrain their operation to the near-IR. In contrast, samples based on Mie-type resonances and direct-gap semiconductors can be utilised in the visible with judicious choice of semiconductors. For further comparison of GaAs metasurfaces with other types of metasurfaces—including silicon, metallic, and hybrid approaches,—please refer to Supplementary Note 3. For any subwavelength technology presented in the literature, we find that direct-gap semiconductor metasurfaces presented in our study allow for the most efficient, ultrafast and low-power all-optical modulation solution reported thus far.
To conclude, we have demonstrated an actively tunable metasurface based on ultrafast injection and relaxation of FCs in a direct-gap semiconductor. Arrays of Mie-resonant GaAs nanodisks provide for low-power, efficient and ultrafast all-optical modulation enabled by improved absorption together with rapid recombination likely enabled by surface states. As an outlook, the proposed concept of actively and efficiently tuning the metasurface optical properties may become the basis of a wide class of novel ultrafast metadevices. Application of our approach to metasurfaces may open a range of possibilities for ultrafast wavefront control, including for example beam steering49, 50, beam shaping49, 51, magnetic mirrors52, polarisation manipulation53, holography54, imaging55, 56, and spectroscopy57, as well as enabling novel integrated devices. The suggested direct-gap semiconductor metasurfaces may find applications as spatial light modulators with orders-of-magnitude reduced switching times as compared to common liquid-crystal based solutions, and could for example allow ultrahigh bit-rate spatial multiplexing or stimulate the development of new schemes for coherent control of light-matter interactions.
Sample fabrication and characterisation
The GaAs metasurface sample was fabricated starting from metal-organic chemical vapour deposition of a 300-nm thick GaAs layer on top of a 300-nm thick Al0.85Ga0.15As layer on a semi-insulating GaAs substrate. Etch masks were created using standard electron-beam lithography that converted the exposed negative tone hydrogen silsesquioxane (HSQ Fox-16) to SiO x . The unexposed HSQ was then developed using tetramethylammonium hydroxide leaving SiO x nanodisks as etch masks. The mask shape was transferred onto the GaAs and Al0.85Ga0.15As layer using inductively coupled-plasma etch. The refractive index isolation between the top GaAs dielectric resonators and lower layers (including the AlGaAs layer and the GaAs substrate) was realized by a selective wet oxidisation process that converts the Al0.85Ga0.15As into its oxide (Al x Ga1–x )2O3, which has a low refractive index of ≈1.6.
Femtosecond laser pulses from a Ti:Sa regenerative amplifier (Coherent Libra USPHE) with 50-fs-long pulses at a photon energy of 1.55 eV and a repetition rate of 1 kHz were used to investigate the ultrafast carrier dynamics in the samples under study. The fundamental pulse with controllably variable average power from 10 to 500 μW (from 0.01 to 0.5 μJ per pulse) was used as a pump. It was focused at the sample into a radius spot with a diameter of ≈350 μm, where a transient carrier distribution in GaAs nanodisks was excited. The probe beam was a white light supercontinuum that was generated in a 4-mm-thick sapphire plate. After a longpass filter, the probe beam covered the spectral range of 850–1300 nm. It had an average power P = 0.3 μW and was focused on the sample into the spot with a diameter of ≈80 μm. The time delay between the pump and the probe pulses was varied by a computer-controlled mechanical stage (Sigma Koki OSMS series) in the pump arm. Upon reflection from the perturbed sample, the probe beam was brought to the input slit of a monochromator (Avesta ASP-IR) coupled to an InGaAs photodiode; the spectral resolution was set to 5 nm. The pump beam was chopped at 500 Hz by a trigger-controlled chopper (ThorLabs MC2000B). The pump-induced variation of reflectance was software lock-in detected at the frequency of the optical chopper (I 500 Hz) along with the amplitude of the reflected signal itself (I 1 kHz). However, since the relation I 500 Hz ≪ I 1 kHz does not hold, I 1 kHz is affected by I 500 Hz, having ∆R/R = I 500 Hz/(I 1 kHz – I 500 kHz/2). Note that I 500 kHz is a phase-sensitive signal that reveals both positive and negative transients. The measured data is a series of differential reflectance spectra ∆R/R for different time delays between pump and probe. A variable optical density filter was used to attenuate the pump beam fluences down to a range of 10–500 μJ cm–2. Data in Fig. 2b,c was obtained by combining the measured ∆R/R spectra and the linear spectra as provided by angle-resolved reflectance spectroscopy.
For the numerical calculation of the transient GaAs metasurface reflectance, we used the commercial software package COMSOL Multiphysics with Floquet periodic boundary conditions and plane wave excitation from the top at an angle of incidence of 12°. Perfect magnetic conductor boundary conditions were used at the symmetry plane to reduce the size of the computational volume by a factor of two. At the GaAs substrate side, a perfectly matched layer mimicking the response of a GaAs half space terminated the computational volume. The dimensions of the structure were as follows: the period of the structure was set to 620 nm, the silica cap was 200 nm in height, the GaAs nanodisk was 300 nm in height, the AlGaO layer was 330 nm in height, and the diameter of the whole pillar was taken to be 236 nm. The discrepancy between the diameter taken from SEM images and the one used in simulations may be due to sample imperfections, such as deviations of the fabricated pillars from a perfect cylindrical shape. For the complex dielectric permittivity function of GaAs we used experimental data from Palik58, to which we added the transient response of the induced dense plasma. The AlGaO oxide pedestal and the cap layer were modelled as non-dispersive dielectric media with n = 1.6 and n = 1.45, respectively.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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The authors acknowledge fruitful discussions with B. Luk'yanchuk. Financial support from the Russian Foundation for Basic Research (16-29-11811), Thuringian State Government through its ProExcellence Initiative (ACP2020) and by the German Research Foundation through the Emmy Noether Programme (STA 1426/2-1) are gratefully acknowledged. S.L. and I.B. acknowledge the support from the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the US DOE Office of Science. Sandia National Laboratories is a multi-programme laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US DOE's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991, Russia
Maxim R. Shcherbakov, Varvara V. Zubyuk, Polina P. Vabishchevich, Tatyana V. Dolgova & Andrey A. Fedyanin
Center for Integrated Nanotechnologies, Sandia National Laboratories, Albuquerque, New Mexico, 87185, USA
Sheng Liu, Gordon Keeler & Igal Brener
Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, Jena, 07743, Germany
Aleksandr Vaskin, Thomas Pertsch & Isabelle Staude
Maxim R. Shcherbakov
Sheng Liu
Varvara V. Zubyuk
Aleksandr Vaskin
Polina P. Vabishchevich
Gordon Keeler
Thomas Pertsch
Tatyana V. Dolgova
Isabelle Staude
Igal Brener
Andrey A. Fedyanin
M.R.S., S.L., I.B., I.S. and A.A.F. conceived the idea. S.L. and G.K. fabricated the sample. V.V.Z., M.R.S., P.P.V. and T.V.D. conducted the pump–probe measurements. P.P.V., I.S. and M.R.S. conducted the reflectance measurements. A.V., M.R.S. and I.S. performed the numerical simulations. M.R.S. prepared the figures and the initial draft of the manuscript. All the co-authors contributed to the discussion of the results and preparation of the manuscript.
Correspondence to Maxim R. Shcherbakov.
Supplementary Figures, Supplementary Notes and Supplementary References
Shcherbakov, M.R., Liu, S., Zubyuk, V.V. et al. Ultrafast all-optical tuning of direct-gap semiconductor metasurfaces. Nat Commun 8, 17 (2017). https://doi.org/10.1038/s41467-017-00019-3
Visualization of dynamic polaronic strain fields in hybrid lead halide perovskites
Burak Guzelturk
, Thomas Winkler
, Tim W. J. Van de Goor
, Matthew D. Smith
, Sean A. Bourelle
, Sascha Feldmann
, Mariano Trigo
, Samuel W. Teitelbaum
, Hans-Georg Steinrück
, Gilberto A. de la Pena
, Roberto Alonso-Mori
, Diling Zhu
, Takahiro Sato
, Hemamala I. Karunadasa
, Michael F. Toney
, Felix Deschler
& Aaron M. Lindenberg
Nature Materials (2021)
Single-walled carbon nanotube membranes as non-reflective substrates for nanophotonic applications
Denis M Zhigunov
, Daniil A Shilkin
, Natalia G Kokareva
, Vladimir O Bessonov
, Sergey A Dyakov
, Dmitry A Chermoshentsev
, Aram A Mkrtchyan
, Yury G Gladush
, Andrey A Fedyanin
& Albert G Nasibulin
Nanotechnology (2021)
All-dielectric multifunctional transmittance-tunable metasurfaces based on guided-mode resonance and ENZ effect
Xiaoming Qiu
, Jian Shi
, Yanping Li
& Fan Zhang
Broadband Liquid Crystal Tunable Metasurfaces in the Visible: Liquid Crystal Inhomogeneities Across the Metasurface Parameter Space
James A. Dolan
, Haogang Cai
, Lily Delalande
, Xiao Li
, Alex B. F. Martinson
, Juan J. de Pablo
, Daniel López
& Paul F. Nealey
ACS Photonics (2021)
Active metasurfaces for manipulatable terahertz technology
Jing-Yuan Wu
, Xiao-Feng Xu
& Lian-Fu Wei
Chinese Physics B (2020)
Editors' Highlights
Nature Communications ISSN 2041-1723 (online) | CommonCrawl |
The positive difference between two consecutive perfect squares is 35. What is the greater of the two squares?
Call the greater of the two squares $x^2$. Because the squares are consecutive, we can express the smaller square as $(x-1)^2$. We are given that $x^2 - (x-1)^2 = 35$. Expanding yields $x^2 - x^2 + 2x - 1 = 35$, or $2x = 36$. Therefore, $x = 18$, so the larger square is $18^2 = \boxed{324}$. | Math Dataset |
Burt Totaro
Burt James Totaro, FRS (b. 1967), is an American mathematician, currently a professor at the University of California, Los Angeles, specializing in algebraic geometry and algebraic topology.
Burt Totaro
Born
Burt James Totaro
1967 (age 55–56)
Alma materPrinceton University
University of California, Berkeley
AwardsWhitehead Prize (2000)
Prix Franco-Britannique (2001)
Scientific career
InstitutionsUniversity of California, Los Angeles
University of Cambridge
University of Chicago
ThesisK-Theory and Algebraic Cycles (1989)
Doctoral advisorShoshichi Kobayashi
Website
• burttotaro.wordpress.com
• www.math.ucla.edu/~totaro/
Education and early life
Totaro participated in the Study of Mathematically Precocious Youth while in grade school and enrolled at Princeton University at the age of thirteen, becoming the youngest freshman in its history.[1] He scored a perfect 800 on the math portion and a 690 on the verbal portion of the SAT-I exam at the age of 12.[2] He graduated in 1984 and went on to graduate school at the University of California, Berkeley, receiving his Ph.D. in 1989.[3]
Career and research
Since 2009, he has been one of three managing editors of the journal Compositio Mathematica;[4] he is also on the editorial boards of Forum of Mathematics, Pi and Sigma, the Journal of the American Mathematical Society, and the Bulletin of the American Mathematical Society. In 2012, he became a Professor in the UCLA Department of Mathematics.[5]
Totaro's work is influenced by the Hodge conjecture, and is based on the connections and application of topology to algebraic geometry. His work has applications in a number of diverse areas of mathematics, from representation theory to Lie theory and group cohomology.[6]
Selected works
• Totaro, Burt (1996). "Configuration spaces of algebraic varieties". Topology. 35 (4): 1057–1067. doi:10.1016/0040-9383(95)00058-5. MR 1404924.
• Totaro, Burt (2014). Group Cohomology and Algebraic Cycles. Cambridge: Cambridge University Press. ISBN 978-1-107-01577-7.
Recognition
In 2000, he was elected Lowndean Professor of Astronomy and Geometry at the University of Cambridge. In the same year, he was awarded the Whitehead Prize by the London Mathematical Society.[7]
In 2009, Totaro was elected Fellow of the Royal Society.[8] He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to algebraic geometry, Lie theory and cohomology and their connections and for service to the profession".[9]
References
1. "Princeton Alumni Weekly". 1980.
2. "Princeton Alumni Weekly". 1980.
3. Burt Totaro at the Mathematics Genealogy Project
4. "Editorial Board of Compositio Mathematica".
5. "UCLA Department of Mathematics newsletter" (PDF).
6. Cambridge academics elected as Fellows of the Royal Society
7. Citation for Burt Totaro Archived October 12, 2007, at the Wayback Machine
8. "Cambridge academics elected as Fellows of the Royal Society". Cambridge University news. 2009-05-16. Archived from the original on 2009-05-21. Retrieved 2009-05-16.
9. 2019 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2018-11-07
Fellows of the Royal Society elected in 2009
Fellows
• Robert Ainsworth
• Ross J. Anderson
• Michael Ashfold
• Michael Batty
• Martin Buck
• Peter Buneman
• Michel Chrétien
• Jenny Clack
• Michael Duff
• Richard Ellis
• Jeff Ellis
• James Gimzewski
• David Glover
• Chris Goodnow
• Wendy Hall
• Nicholas Harberd
• John Hardy
• Brian Hemmings
• Christine Holt
• Christopher Hunter
• Graham Hutchings
• Peter Isaacson
• Jonathan Keating
• Dimitris Kioussis
• Stephen Larter
• David Leigh
• David MacKay
• Arthur B. McDonald
• Angela McLean
• David Owen
• Richard Passingham
• Guy Richardson
• Wolfram Schultz
• Keith Shine
• Henning Sirringhaus
• Maurice Skolnick
• Karen Steel
• Malcolm Stevens
• Jesper Svejstrup
• Jonathan Tennyson
• John Todd
• Burt Totaro
• John Vederas
• John Wood
Foreign
• John Holdren
• H. Robert Horvitz
• Thomas Kailath
• Roger D. Kornberg
• Yakov Sinai
• Joseph Stiglitz
• Rashid Sunyaev
• Steven D. Tanksley
Royal
• William, Prince of Wales
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
Academics
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
Other
• IdRef
| Wikipedia |
Compute $\dbinom{1293}{1}$.
$\dbinom{1293}{1} = \dfrac{1293!}{1!1292!}=\dfrac{1293}{1}=\boxed{1293}.$ | Math Dataset |
Availability and requirements
MetaMeta: integrating metagenome analysis tools to improve taxonomic profiling
Vitor C. Piro1, 2,
Marcel Matschkowski1 and
Bernhard Y. Renard1Email author
Microbiome20175:101
Many metagenome analysis tools are presently available to classify sequences and profile environmental samples. In particular, taxonomic profiling and binning methods are commonly used for such tasks. Tools available among these two categories make use of several techniques, e.g., read mapping, k-mer alignment, and composition analysis. Variations on the construction of the corresponding reference sequence databases are also common. In addition, different tools provide good results in different datasets and configurations. All this variation creates a complicated scenario to researchers to decide which methods to use. Installation, configuration and execution can also be difficult especially when dealing with multiple datasets and tools.
We propose MetaMeta: a pipeline to execute and integrate results from metagenome analysis tools. MetaMeta provides an easy workflow to run multiple tools with multiple samples, producing a single enhanced output profile for each sample. MetaMeta includes a database generation, pre-processing, execution, and integration steps, allowing easy execution and parallelization. The integration relies on the co-occurrence of organisms from different methods as the main feature to improve community profiling while accounting for differences in their databases.
In a controlled case with simulated and real data, we show that the integrated profiles of MetaMeta overcome the best single profile. Using the same input data, it provides more sensitive and reliable results with the presence of each organism being supported by several methods. MetaMeta uses Snakemake and has six pre-configured tools, all available at BioConda channel for easy installation (conda install -c bioconda metameta). The MetaMeta pipeline is open-source and can be downloaded at: https://gitlab.com/rki_bioinformatics.
Taxonomic profiling
A large and increasing number of metagenome analysis tools are presently available aiming to characterize environmental samples [1–4]. Motivated by the large amounts of data produced from whole metagenome shotgun (WMS) sequencing technologies, profiling of metagenomes has become more accessible, faster and applicable in real scenarios and tends to become the standard method for metagenomics analysis [5–7]. Tools which perform sequence classification based on WMS sequencing data come in different flavors. One basic approach is the de novo sequence assembly [8–10], which aims to reconstruct complete or near complete genomes from fragmented short sequences without any reference or prior knowledge. It is the method which provides the best resolution to assess the community composition. However, it is very difficult to produce meaningful assemblies from metagenomics data due to short read length, insufficient coverage, similar DNA sequences, and low abundant strains [11].
More commonly, methods use the WMS reads directly without assembly and are in general reference-based, meaning that they rely on previously obtained genome sequences to perform their analysis. In this category of applications, two standard definitions are employed: taxonomic profiling and binning tools. Profilers aim to analyze WMS sequences as a whole, predicting organisms and their relative abundances based on a given set of reference sequences. Binning tools aim to classify each sequence in a given sample individually, linking each one of them to the most probable organism of the reference set. Regardless of their conceptual differences, both groups of tools could be used to characterize microbial communities. Yet binning tools produce individual classification for each sequence and should be converted and normalized to be used as a taxonomic profiler.
Methods available among these two categories make use of several techniques, e.g. read mapping, k-mer alignment, and composition analysis. Variations on the construction of the reference databases, e.g., complete genome sequences, marker genes, protein sequences, are also common. Many of those techniques were developed to overcome the computational cost of dealing with the high throughput of modern sequencing technologies as well as the large number of reference genome sequences available.
The availability of several options for tools, parameters, databases, and techniques create a complicated scenario to researchers to decide which methods to use. Different tools provide good results in different scenarios, being more or less precise or sensitive in multiple configurations. It is hard to rely on their output for every study or sample variation. In addition, when more than one method is used, inconsistent results between tools using different reference sets are difficult to be integrated. Furthermore, installation, parameterization, and database creation as well as the lack of standard outputs are challenges not easily overcome.
We propose MetaMeta, a new pipeline for the joint execution and integration of metagenomic sequence classification tools. MetaMeta has several strengths: easy installation and set-up, support for multiple tools, samples and databases, improved final profile combining multiple results, out-of-the-box parallelization and high performance computing (HPC) integration, automated database download and set-up, custom database creation, integrated pre-processing step (read trimming, error correction, and sub-sampling) as well as standardized rules for integration of new tools. MetaMeta achieves more sensitive profiling results than single tools alone by merging their correct identifications and properly filtering out false identifications. MetaMeta was built with SnakeMake [12] and is open-source. The pipeline has six pre-configured tools that are automatically installed using Conda through the BioConda channel (https://bioconda.github.io). We encourage the integration of new tools, making it available to the community through a participative Git repository (via pull request). MetaMeta source-code is available at: https://github.com/pirovc/metameta.
MetaMeta executes and integrates metagenomic sequence classification tools. The integration is based on several tools' output profiles and aims to improve organism identification and quantification. An optional pre-processing and sub-sampling step is included. The pipeline is generalized for binning and profiling tools, categories that were previously described in the CAMI (Critical Assessment of Metagenome Interpretation) challenge (http://www.cami-challenge.org). MetaMeta provides a pre-defined set of standardized rules to facilitate the integration of tools, easy parallelization and execution in high performance computing infrastructure. The pre-configured tools are available at the BioConda channel to facilitate download and installation, avoiding set-up problems and broken dependencies.
The pipeline accepts one or multiple WMS samples as well as one or more databases and the output is an integrated taxonomic profile for each sample per database (as well as a separated output from each executed tool). The MetaMeta pipeline can be described in four modules: database generation, pre-processing, tool execution, and integration (Fig. 1).
MetaMeta Pipeline. The MetaMeta Pipeline: one or more WMS read samples and a configuration file are the input. The pipeline consists of four main modules: Database Generation (only on the first run), Pre-processing (optional), Tool Execution and Integration. The output is a unified taxonomic profile integrating the results from all configured tools for each sample, generated by the MetaMetaMerge module
Database generation
On the first run, the pipeline downloads and builds the databases for each of the configured tools. Pre-configured databases (Additional file 1: Table S1) are provided as well as a custom database creation option based on reference sequences. Since each tool has its own database with a specific version of reference sequences, database profiles are generated, collecting which taxonomic groups each tool can identify. Given a list of accession version identifiers for each sequence on the reference set, MetaMeta automatically generates a taxonomic profile for each tool's database.
Pre-processing
An optional pre-processing step is provided to remove errors and improve sequence classification: Trimommatic [13] for read trimming and BayesHammer [14] for error correction. A sub-sampling step is also included, allowing the sub division of large read sets among several tools by equally dividing them or by taking smaller random samples with or without replacement, to reduce overall run-time.
Tool execution
In this step, the pre-processed reads are analyzed by the configured tools. Tools can be added to the pipeline if they follow a minimum set of requirements. They should output their results based on the NCBI Taxonomy database [15] (by name or taxonomic id). Profiling tools should output a rank separated taxonomic profile with relative abundances while binning tools should provide an output with sequence id, length used in the assignment and taxon. The BioBoxes [16] data format for binning and profiling (https://github.com/bioboxes/rfc/tree/master/data-format) is directly accepted. Tools which provide non-standard output should be configured with an additional step converting their output to be correctly integrated into the pipeline (More details are given in the Additional file 1: File Formats).
The integration step will merge identified taxonomic groups and abundances and provide a unified profile for each sample. MetaMeta aims to improve the final results based on the assumption that the more identifications of the same taxon by different tools are reported, the higher its chance to be correct. This task is performed by the MetaMetaMerge module. This module accepts binning and profiling results and relies on previously generated database profiles. Taxonomic classification can change over time and each tool can use a different version/definition of it. For that reason, a recent taxonomy database version is used to solve name and rank conflicts (e.g., changing name specification, species turning into sub-species, etc.).
Abundance estimation - binning tools
Binning tools provide a single classification for each sequence in the dataset instead of relative abundances for taxons. An abundance estimation step is necessary for a correct interpretation of such data and posterior integration. The lengths of the binned sequences are summed up for each identified taxonomic group and normalized by the length of their respective reference sequences, estimating the abundance for each identified taxon n as:
$$ {abundance}_{n} = \sum_{i=1}^{r} \frac{\sum_{j=1}^{t_{i}}b_{j}}{l_{i}} $$
where r is the number of reference sequences belonging to the taxonomic group n, t i is the total of reads classified to the reference i, b j is the number of aligned bases of a read j and l i is the length of the reference i. The abundance of the parent nodes is based on the cumulative sum of their children nodes' abundance.
Merging approach
The first step on the merging approach is to normalize estimated abundances to 100% for each taxonomic level. That is necessary because some tools do account for the unclassified reads and others do not. MetaMetaMerge only considers classified reads. Once normalized, all profiles are then integrated to a single profile. In this step, MetaMetaMerge saves the number of occurrences of each taxon among all profiles. This occurrence count is used to better select taxons that are more often identified, assuming that they have higher chances of being a correct identification. MetaMetaMerge also calculates an integrated value for the relative abundance estimation, defined as the harmonic mean of all normalized abundances for each taxon, avoiding outliers and obtaining a general trend among the estimated abundances. All steps taken in the merging process are performed for each taxonomic level independently, from super kingdom to species by default.
Since tools use different databases of reference sequences it is necessary to account for this bias. Previously generated database profiles provide which taxons are available for each tool. By merging all database profiles, it is possible to anticipate how many times each taxon could be identified among all tools used. The number of occurrences of each taxon from the tools' output and the database presence number are integrated to generate a score S for each taxon, defined as:
$$ S_{ij} = \frac{(i+1)^{2}}{j+1} $$
where i is the number of times the taxon was identified and j the number of times it is contained in the databases. This score calculation accounts for the presence/absence of taxonomic groups on different databases. It gives higher scores to the most identified taxons present in more databases. At the same time, lower scores are assigned to taxons present in many databases but not identified too many times. The score calculation is purposely biased for higher scores when i=j (Additional file 1: Figure S1), given the benefit of the doubt for taxons with low identification that are available only in few databases.
Commonly, metagenome analysis methods have to deal with a moderate to high number of false positive identifications at lower taxonomic levels. That occurs mainly because metagenomes can contain very low abundant organisms with similar genome sequences. This problem is even extended in our merged profile by collecting all false positives from different methods, generating a long tail of false positives with lower scores mixed together with true identifications. A filtering step is therefore necessary to avoid wrong assignments. This step is usually performed by an abundance cutoff value. Setting up this value is subject to uncertainty since the real abundances are usually not known and the separation between low abundant organisms and false identifications is not clear [17]. A simple cutoff would not provide a good separation between true and false results in this scenario.
To overcome this problem, MetaMetaMerge classifies each taxon in a set of bins (four by default) based on the calculated score (Eq. 2). Bins are defined by equally dividing the range of scores in the numbers of bins selected. Now each taxon has a score and a bin assigned to it. Taxons with higher scores are more likely to be true identifications and are going to be grouped together in the same bin. With this strategy it is possible to obtain a general separation among taxons which are prone to be true or false identifications.
Within each taxon grouped in a bin (sorted by relative abundance) a cutoff is applied to remove possible false identifications with low abundance. Here, the cutoff value is a percentile relative to the number of taxons on each bin and it is selected based on predefined functions, which can achieve more sensitive or precise results (Additional file 1: Mode functions). Each bin will have a different cutoff value depending on the chosen function.
If precision is chosen, a gradually more stringent cutoff will be used, selecting only the most abundant taxa for each bin. If sensitivity is selected, cutoffs will be set higher, allowing more identifications to be kept. Sensitive results have an increased chance of containing more true positives but at the same time they will likely have more false identifications due to less strict cutoffs.
Based on this percentile cutoff, MetaMetaMerge keeps only the top abundant taxa on each bin and removes taxons below it. After this step, the remaining taxons on each bin are re-grouped and sorted by relative abundance to generate the final profile.
At the end, MetaMeta will provide a final taxonomic profile, integrating all tools results, a detailed profile with co-occurrence and individual abundances, an interactive Krona pie chart [18] to easily compare taxonomic abundances among the tools as well as single profiles for each executed tool.
Tool selection
MetaMeta was evaluated with a set of six tools: CLARK [19], DUDes [20], GOTTCHA [21], Kraken [22], Kaiju [23], and mOTUs [24]. The choice was partially motivated by recent publications comparing the performance of such tools [3, 4, 25]. CLARK, GOTTCHA, Kraken, and mOTUs achieved very low false positive numbers according to [4]. DUDes was an in-house developed tool which achieves good trade-off between precision and sensitivity according to [25]. Kaiju uses a translated database, bringing diversity to the current whole genome-based methods. We also considered the amount of data/run time performance for each tool, selecting only the ones that can handle large amounts of data as commonly used today in metagenomics analysis in an acceptable time (less than 1 day for our largest CAMI dataset −7.4 Gbp). MetaPhlAn [26] a widely used metagenomics tool could not be included due to taxonomic incompatibility. Any other sequence classification tool could be configured and used in MetaMeta, as long as it fits with our pipeline requirements described in the Methods - Tool execution section. We selected an equal number of tools for each category: DUDes, GOTTCHA, and mOTUs are taxonomic profiling tools, while CLARK, Kraken, and Kaiju are binning tools. Databases were created following the default guidelines for each tool, considering only bacteria and archaea as targets (Additional file 1: Table S1).
Datasets and evaluation
The pipeline was evaluated with a set of simulated and real samples (Table 1). The simulated data were provided as part of the CAMI Challenge (toy samples) and the real samples were obtained from the Human Microbiome Project (HMP) [27, 28]. MetaMeta was compared to each single result from each tool configured in the pipeline. Although the pipeline can work on the strain level, we evaluate the results until species levels since most of the tools still do not provide strain level identifications. We compare the results to the ground truth in a binary (true and false positives, sensitivity, and precision) and quantitative way with the L 1 norm, which is the sum of absolute differences between predicted and real abundances, when abundance profiles are available. Computer specifications and parameters can be found on the Additional file 1.
Samples used in this study and run-time (based on the computer specifications on Additional file 1)
# Samples
# Species
Cpu time/sample
Estimated wall time/sample
CAMI toy low
CAMI toy medium
CAMI toy high
HMP stool
1.44 Tbp
cpu time/sample stands for the mean cpu time for each sample without paralellization. Estimated wall time/sample considers a double speed-up by using 12 threads and concurrently running all six tools (when computational resources are available the pipeline can run all tools/samples/databases at the same time). *expected number of species from isolated genomes from the gastrointestinal tract
CAMI data
The CAMI challenge provided three toy datasets of different complexity (Table 1) with known composition and abundances. From low to high complexity, they provide an increasing number of organisms and samples. The samples within a complexity group contain the same organisms with variable abundances among samples. The sets contain real and simulated strains from complete and draft bacterial and archaeal genome sequences. The simulated CAMI datasets, especially those of medium and high complexity, provide a very challenging and realistic data in terms of complexity and size.
In Fig. 2, it is possible to observe the tools performance in terms of true and false positives for the CAMI high complexity set. All configured tools perform similarly in the true positive identifications but vary among the false positives. Binning tools have a higher number of false positive identifications due to the fact that even single classified reads are considered. The MetaMetaMerge profile surpassed all other methods in true positive identifications while keeping the false positive number low. The same trend occurs in the other complexity sets (Additional file 1: Figures S3–S8). Figure 3 shows the trade-off between precision and sensitivity for all high complexity samples. MetaMetaMerge achieved the best sensitivity while GOTTCHA the best precision among the compared tools with default parameters. Those results show how the merging module of the MetaMeta pipeline is capable of better selecting and identifying true positives based on the co-occurrence information. MetaMetaMerge also has the flexibility to provide more precise or sensitive results (Fig. 3) just by changing the mode parameter (details are given in the Additional file 1: Mode functions). In the very precise mode, the merged profile outperformed all tools in terms of precision, but with the cost of losing sensitivity. In the very sensitive mode, the merged profile could improve the sensitivity compared to the run with default parameters, with some loss of precision. It is important to notice that the trade-off between precision and sensitivity could also be explored by the cutoff parameter (default 0.0001), depending on what is expected to be the lowest abundant organism in the sample. The MetaMetaMerge mode parameter will give more precise or sensitive results based on this cutoff value.
True and False Positives - CAMI high complexity set. In blue (left y axis): True Positives. In red (right y axis): False Positives. Results at species level. Each marker represents one out of five samples from the CAMI high complexity set
Precision and Sensitivity - CAMI high complexity set. Dotted black linemarks the maximum possible sensitivity value (0.57) that could be achieved with the given tools and databases. Results at species level. Each marker represents one out of five samples from the CAMI high complexity set
In terms of relative abundance, MetaMetaMerge provides the most reliable predictions with smaller difference from the real abundances, as shown in Fig. 4 with regard to the L 1 norm measure. By taking the harmonic mean, we succeed in reducing the effect of outliers that occur among the tools and capture the trend of the estimated relative abundances, providing a new, more robust estimate (Additional file 2).
L 1 norm error. Mean of the L 1 norm measure at each taxonomic level for five samples from the high complexity CAMI set
Pre-processing and sub-sampling effects
We explore here the effects of pre-processing and sub-sampling on the CAMI toy sets. Results shown in this section were trimmed and sub-sampled in several sizes, with and without replacement and executed five times for each sub-sample. Trimming effects were small on this set, slightly increasing precision (data not shown). Figure 5 shows the effects of sub-sampling in terms of sensitivity and run-time (wall time for the full pipeline) for one of the high complexity CAMI sets. Sub-sampling provides a high decrease on run-time for every tool and consequently for the whole pipeline. However, only below 5% it is possible to see a significant but still small decrease on sensitivity. All tools behave similarly on the sub-sampled sets, with GOTTCHA and mOTUs having a high decrease of sensitivity when using only 1% of the data. With the same sub-sample configuration (1%), MetaMetaMerge achieved a sensitivity higher than any other tool alone using 100% of the set. It also runs the whole pipeline approximately 17 times faster than with the full set (from 05 h 41 min 36 s to 20 min 19 s on average), being faster than the fastest tool with 100% of the data (kraken 29 min 26 s on average) and the second best sensitive tool (kaiju 1 h 47 min 44 s on average). As expected, precision is slightly increased in small sub-samples due to less data (Additional file 1: Figure S9).
Sub-sampling. Sensitivity (left y axis) and run-time (right y axis) at species level for one randomly selected CAMI high complexity sample. Each sub-sample was executed five times. Linesrepresent the mean and the area around it the maximum and minimum achieved values. Run-time stands for the time to execute the MetaMeta pipeline. The evaluated sample sizes are 100, 50, 25, 16.6, 10, 5, and 1%. 16.6% is the exact division among six tools, using the the whole sample. Sub-samples above that value were taken with replacement and below without replacement. The plot is limited to a value of 0.57 (left y axis) that is the maximum possible sensitivity value that could be achieved with the given tools and databases
Human Microbiome Project data
The HMP provided several resources to characterize the microbial communities at different sites of the human body. MetaMeta was tested on stool samples to evaluate the performance of the pipeline on real data. For evaluation we used a list of reference genome sequences that were isolated from specific body sites and sequenced as part of the HMP. They do not represent the complete content of microbial diversity in each community but serve as a guide to check how well the tools are performing. Stool samples were compared against the isolated genomes obtained from the gastrointestinal tract.
Figure 6 shows the results for 147 samples. In sensitive mode, MetaMetaMerge achieved the highest number of true positive identifications with a moderate number of false positives, below all binning tools but above all taxonomic profilers. mOTUs produced good results in the selected samples mainly because its database is based on the isolated genomes from the HMP (the same as the ground truth used here). Since mOTUs is the only tool with a distinct set of reference sequences that could classify this set, the scores (from Eq. 2) attributed to mOTUs' unique identifications were low. Still, MetaMetaMerge could improve the true identifications keeping a lower rate of false positives by incorporating the true identifications from other methods.
True and False Positives - HMP stool samples. In blue (left y axis): True Positives. In red (right y axis): False Positives. Results at species level. Each marker represents one out of 147 stool samples from the HMP
MetaMeta is a complete pipeline for classification of metagenomic datasets. It provides improved profiles over the tested tools by merging their results. In addition, the pipeline provides easy configuration, execution and parallelization. With simulated and real data, MetaMeta is capable to achieve higher sensitivity. That is possible due to the MetaMetaMerge module, which extracts information of co-occurrence of taxons on databases and profiles, collecting complementary results from different methods. Further, the guided cutoff approach avoids false positives and keeps most of the true identifications, enhancing final sensitivity and exploring the complementarity of currently available methods.
By running several tools, MetaMeta has an apparently prohibitive execution time. In reality, the parallelization provided by Snakemake makes the pipeline run in a reasonable time using most of the computational resources (Table 1). That is possible by the way the rules are chained and executed among several cores, lasting not more than the slowest tool plus pre- and post-processing time, which are very small in comparison to the analysis time. In addition, sub-sampling allows the reduction of input data and a high decrease of execution time with small if any impact on the final result. That is viable due to redundant data contained in many metagenomic samples as well as redundant execution by several tools provided in the MetaMeta environment. However sub-sampling should be used with caution, taking in consideration the coverage of low abundant organisms.
All tools presented here are available at the BioConda channel and are automatically installed in the first MetaMeta run, working out-of-the-box for several computer environments and avoiding conflicts and broken dependencies. MetaMeta can also handle multiple large samples and databases at the same time, with options to delete intermediate files and keep only necessary ones, being well suited to large scale projects. It also reduces idle computational time by smartly parallelizing samples among one or more tools (Additional file 1: Figures S10–S13). The parallelization noticeably decreases the run time when computational power is available and manages to serialize and control the run when access to computational power is limited. Integration into HPC systems is also possible and we provide a pre-configured file for queuing systems (e.g., slurm). As stated by Lee et al. [29], solid-state drives accelerate the run time of many bioinformatics tools. Such drives were used in some evaluations shown in this paper and are beneficial for the MetaMeta pipeline.
MetaMeta makes it easier for the user to obtain more precise or sensitive results by providing a single default parameter as well as advanced options for more refined results. This parameter when set towards sensitivity tends to output an extensive list of taxons, being at the same time less stringent with the minimum abundance cutoff. When set towards precision it will apply a more strict abundance cutoff and provide a smaller but more accurate list of predicted taxons. Since all tools were used in default mode, it is possible to obtain problem-centric optimized results only by changing the way MetaMeta works. That facilitates and simplifies the task for researchers that are in search for a specific goal.
MetaMeta supports strain level identification. Nevertheless all evaluations were made at species level due to lack of support to strain identification in some tools. Also the lack of standard was a limiting factor. Taxonomic IDs are no longer assigned to strain levels [30] and tools output them in different ways. With standard output definitions, the use of strain classification on the pipeline is straight forward.
Related in parts, a method called WEVOTE was developed in parallel and recently published [31] where five classification tools were used to generate a merged taxonomic profile. Although the two methods present distinct ways of achieving better taxonomic profiling, they are not built for the same use case. WEVOTE relies on BLAST based tools and thereby is not suited for the large scale WMS applications, since the dataset sizes practically prohibit analyses via BLAST based approaches. Differently, MetaMeta was built accounting for high throughput data. Moreover, we supply an easy way to install tools and MetaMeta provides a complete pipeline which can configure databases and run classification tools with an integration module at the end, where WEVOTE provides only the integration method. As a result a comparison among the pipelines is hard to perform and interpret since they both use a different set of tools and databases.
In conclusion, MetaMeta is an easy way to execute and generate improved taxonomic profiles for WMS samples with multiple tool support. We believe the method can be very useful for researchers that are dealing with multiple metagenomic samples and want to standardize their analysis. The MetaMeta pipeline was built in a way to facilitate the execution in many computational environments using Snakemake and BioConda. That diminishes the burden of installing and configuring multiple tools. The pipeline also gives control over the storage of the results and has an easy set of parameters which makes it possible to obtain more precise or sensitive results. MetaMeta was coded in a standardized manner, allowing easy expansion to more tools, also collectively in the MetaMeta git repository (https://github.com/pirovc/metameta). We believe that the final profile could be even further improved with novel tools configured into the pipeline.
Project name: MetaMetaProject home page: https://github.com/pirovc/metametaOperating systems: LinuxProgramming language: PythonOther requirements: SnakemakeLicence: MIT
CAMI:
Critical assessment of metagenome interpretation
HMP:
Human Microbiome Project
HPC:
WMS:
Whole metagenome shotgun
We thank Enrico Seiler for proof-reading the manuscript and for technical support and Martin S. Lindner for fruitful discussions.
This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Ciencia sem Fronteiras [BEX 13472/13-5 to VCP] and by the German Federal Ministry of Health [IIA5-2512-FSB-725 to BYR].
VCP and BYR conceived the project and designed the methods. VCP developed the pipeline and MM led the sub-sampling analysis. VCP and BYR interpreted the data. VCP drafted the manuscript with contributions by MM and BYR. All authors read and approved the final manuscript.
This manuscript does not report data collected from humans or animals.
This manuscript does not contain any individual person's data in any form.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Additional file 1 Additional file with supplementary figures and information. (PDF 1024 kb)
Additional file 2 Additional File with interactive charts for all CAMI toy set results on default, very-precise and very-sensitive mode. File prefix S, M, and H for low, medium and high complexity, respectively. (TAR 3573 kb)
The software presented in this manuscript is available at: https://gitlab.com/rki_bioinformatics.
Source-code available at: https://github.com/pirovc/metameta and https://github.com/pirovc/metametamerge.
CAMI data sets are available at: http://data.cami-challenge.org/.
HMP data sets are avaiable at NCBI Sequence Read Archive: https://www.ncbi.nlm.nih.gov/sraAccession numbers available at the Additional file 1.
Research Group Bioinformatics (NG4), Robert Koch Institute, Nordufer 20, Berlin, 13353, Germany
CAPES Foundation, Ministry of Education of Brazil, Brasília, 70040-020, DF, Brazil
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(Redirected from Minimum wages)
lowest wage which can be paid legally in a state for working
Protesters call for an increased minimum wage as part of the "Fight for $15" effort to require a $15 per hour minimum wage in 2015.
A minimum wage is the lowest remuneration that employers can legally pay their workers—the price floor below which workers may not sell their labor. Most countries had introduced minimum wage legislation by the end of the 20th century.[1]
Supply and demand models suggest that there may be welfare and employment losses from minimum wages. However, if the labor market is in a state of monopsony (with only one employer available who is hiring), minimum wages can increase the efficiency of the market. There is debate about the full effects of minimum wages.[2][3][4]
The movement for minimum wages was first motivated as a way to stop the exploitation of workers in sweatshops, by employers who were thought to have unfair bargaining power over them. Over time, minimum wages came to be seen as a way to help lower-income families. Modern national laws enforcing compulsory union membership which prescribed minimum wages for their members were first passed in New Zealand and Australia in the 1890s.
Although minimum wage laws are in effect in many jurisdictions, differences of opinion exist about the benefits and drawbacks of a minimum wage. Supporters of the minimum wage say it increases the standard of living of workers, reduces poverty, reduces inequality, and boosts morale.[5] In contrast, opponents of the minimum wage say it increases poverty, increases unemployment because some low-wage workers "will be unable to find work...[and] will be pushed into the ranks of the unemployed"[6][7][8] and is damaging to businesses, because excessively high minimum wages require businesses to raise the prices of their product or service to accommodate the extra expense of paying a higher wage.[9]
2 Minimum wage laws
2.1 Informal minimum wages
2.2 Setting minimum wage
3 Economic models
3.1 Supply and demand model
3.2 Monopsony
3.3 Criticisms of the supply and demand model
3.4 Mathematical models of the minimum wage and frictional labor markets
3.4.1 Welfare and labor market participation
3.4.2 Job search effort
4 Empirical studies
4.1 Card and Krueger
4.2 Research subsequent to Card and Krueger's work
4.3 Statistical meta-analyses
5 Debate over consequences
6 Surveys of economists
7 Alternatives
7.1 Basic income
7.2 Guaranteed minimum income
7.3 Refundable tax credit
7.4 Collective bargaining
7.5 Wage subsidies
7.6 Education and training
8 US movement
9 Countries by minimum wage to average wage ratio
"It is a serious national evil that any class of his Majesty's subjects should receive less than a living wage in return for their utmost exertions. It was formerly supposed that the working of the laws of supply and demand would naturally regulate or eliminate that evil [...and...] ultimately produce a fair price. Where... you have a powerful organisation on both sides... there you have a healthy bargaining.... But where you have what we call sweated trades, you have no organisation, no parity of bargaining, the good employer is undercut by the bad, and the bad employer is undercut by the worst... where those conditions prevail you have not a condition of progress, but a condition of progressive degeneration."
Winston Churchill MP, Trade Boards Bill, Hansard House of Commons (28 April 1909) vol 4, col 388
Modern minimum wage laws trace their origin to the Ordinance of Labourers (1349), which was a decree by King Edward III that set a maximum wage for laborers in medieval England.[10][11] King Edward III, who was a wealthy landowner, was dependent, like his lords, on serfs to work the land. In the autumn of 1348, the Black Plague reached England and decimated the population.[12] The severe shortage of labor caused wages to soar and encouraged King Edward III to set a wage ceiling. Subsequent amendments to the ordinance, such as the Statute of Labourers (1351), increased the penalties for paying a wage above the set rates.[10]
While the laws governing wages initially set a ceiling on compensation, they were eventually used to set a living wage. An amendment to the Statute of Labourers in 1389 effectively fixed wages to the price of food. As time passed, the Justice of the Peace, who was charged with setting the maximum wage, also began to set formal minimum wages. The practice was eventually formalized with the passage of the Act Fixing a Minimum Wage in 1604 by King James I for workers in the textile industry.[10]
By the early 19th century, the Statutes of Labourers was repealed as increasingly capitalistic England embraced laissez-faire policies which disfavored regulations of wages (whether upper or lower limits).[10] The subsequent 19th century saw significant labor unrest affect many industrial nations. As trade unions were decriminalized during the century, attempts to control wages through collective agreement were made. However, this meant that a uniform minimum wage was not possible. In Principles of Political Economy in 1848, John Stuart Mill argued that because of the collective action problems that workers faced in organisation, it was a justified departure from laissez-faire policies (or freedom of contract) to regulate people's wages and hours by the law.
It was not until the 1890s that the first modern legislative attempts to regulate minimum wages were seen in New Zealand and Australia.[13] The movement for a minimum wage was initially focused on stopping sweatshop labor and controlling the proliferation of sweatshops in manufacturing industries.[14] The sweatshops employed large numbers of women and young workers, paying them what were considered to be substandard wages. The sweatshop owners were thought to have unfair bargaining power over their employees, and a minimum wage was proposed as a means to make them pay fairly. Over time, the focus changed to helping people, especially families, become more self-sufficient.[15]
Minimum wage laws[edit]
Hourly minimum wages in select developed economies in 2013. For a complete list of global wages see: List of minimum wages by country. Wages are given in US$ exchange rates.[16]
Main articles: Minimum wage law and List of minimum wages by country
The first modern national minimum wages were enacted by the government recognition of unions which in turn established minimum wage policy among their members, as in New Zealand in 1894, followed by Australia in 1896 and the United Kingdom in 1909.[13] In the United States, statutory minimum wages were first introduced nationally in 1938,[17] and they were reintroduced and expanded in the United Kingdom in 1998.[18] There is now legislation or binding collective bargaining regarding minimum wage in more than 90 percent of all countries.[19][1] In the European Union, 22 member states out of 28 currently have national minimum wages.[20] Other countries, such as Sweden, Finland, Denmark, Switzerland, Austria, and Italy, have no minimum wage laws, but rely on employer groups and trade unions to set minimum earnings through collective bargaining.[21][22]
Minimum wage rates vary greatly across many different jurisdictions, not only in setting a particular amount of money—for example $7.25 per hour ($14,500 per year) under certain US state laws (or $2.13 for employees who receive tips, which is known as the tipped minimum wage), $11.00 in the US state of Washington,[23] or £7.83 (for those aged 25+) in the United Kingdom[24]—but also in terms of which pay period (for example Russia and China set monthly minimum wages) or the scope of coverage. Currently the United States federal minimum wage is $7.25 per hour. However, some states do not recognize the minimum wage law, such as Louisiana and Tennessee.[25] Other states operate below the federal minimum wage such as Georgia and Wyoming. Some jurisdictions allow employers to count tips given to their workers as credit towards the minimum wage levels. India was one of the first developing countries to introduce minimum wage policy in its law in 1948. However, it is rarely implemented, even by contractors of government agencies. In Mumbai, as of 2017, the minimum wage was Rs. 348/day.[26] India also has one of the most complicated systems with more than 1,200 minimum wage rates depending on the geographical region.[27]
Informal minimum wages[edit]
Customs and extra-legal pressures from governments or labor unions can produce a de facto minimum wage. So can international public opinion, by pressuring multinational companies to pay Third World workers wages usually found in more industrialized countries. The latter situation in Southeast Asia and Latin America was publicized in the 2000s, but it existed with companies in West Africa in the middle of the 20th century.[28]
Setting minimum wage[edit]
Among the indicators that might be used to establish an initial minimum wage rate are ones that minimize the loss of jobs while preserving international competitiveness.[29] Among these are general economic conditions as measured by real and nominal gross domestic product; inflation; labor supply and demand; wage levels, distribution and differentials; employment terms; productivity growth; labor costs; business operating costs; the number and trend of bankruptcies; economic freedom rankings; standards of living and the prevailing average wage rate.
In the business sector, concerns include the expected increased cost of doing business, threats to profitability, rising levels of unemployment (and subsequent higher government expenditure on welfare benefits raising tax rates), and the possible knock-on effects to the wages of more experienced workers who might already be earning the new statutory minimum wage, or slightly more.[30] Among workers and their representatives, political considerations weigh in as labor leaders seek to win support by demanding the highest possible rate.[31] Other concerns include purchasing power, inflation indexing and standardized working hours.
In the United States, the minimum wage have been set under the Fair Labor Standards Act of 1938. According to the Economic Policy Institute, the minimum wage in the United States would have been $18.28 in 2013 if the minimum wage had kept pace with labor productivity.[32] To adjust for increased rates of worker productivity in the United States, raising the minimum wage to $22 (or more) an hour has been presented.[33][34][35][36]
Economic models[edit]
See also: Labour economics
Supply and demand model[edit]
Graph showing the basic supply and demand model of the minimum wage in the labor market.
Main article: Supply and demand
According to the supply and demand model of the labor market shown in many economics textbooks, increasing the minimum wage decreases the employment of minimum-wage workers.[8] One such textbook states:[4]
If a higher minimum wage increases the wage rates of unskilled workers above the level that would be established by market forces, the quantity of unskilled workers employed will fall. The minimum wage will price the services of the least productive (and therefore lowest-wage) workers out of the market. … the direct results of minimum wage legislation are clearly mixed. Some workers, most likely those whose previous wages were closest to the minimum, will enjoy higher wages. Others, particularly those with the lowest prelegislation wage rates, will be unable to find work. They will be pushed into the ranks of the unemployed.
A firm's cost is an increasing function of the wage rate. The higher the wage rate, the fewer hours an employer will demand of employees. This is because, as the wage rate rises, it becomes more expensive for firms to hire workers and so firms hire fewer workers (or hire them for fewer hours). The demand of labor curve is therefore shown as a line moving down and to the right.[37] Since higher wages increase the quantity supplied, the supply of labor curve is upward sloping, and is shown as a line moving up and to the right.[37] If no minimum wage is in place, wages will adjust until quantity of labor demanded is equal to quantity supplied, reaching equilibrium, where the supply and demand curves intersect. Minimum wage behaves as a classical price floor on labor. Standard theory says that, if set above the equilibrium price, more labor will be willing to be provided by workers than will be demanded by employers, creating a surplus of labor, i.e. unemployment.[37] The economic model of markets predicts the same of other commodities (like milk and wheat, for example): Artificially raising the price of the commodity tends to cause an increase in quantity supplied and a decrease in quantity demanded. The result is a surplus of the commodity. When there is a wheat surplus, the government buys it. Since the government does not hire surplus labor, the labor surplus takes the form of unemployment, which tends to be higher with minimum wage laws than without them.[28]
The supply and demand model implies that by mandating a price floor above the equilibrium wage, minimum wage laws will cause unemployment.[38][39] This is because a greater number of people are willing to work at the higher wage while a smaller number of jobs will be available at the higher wage. Companies can be more selective in those whom they employ thus the least skilled and least experienced will typically be excluded. An imposition or increase of a minimum wage will generally only affect employment in the low-skill labor market, as the equilibrium wage is already at or below the minimum wage, whereas in higher skill labor markets the equilibrium wage is too high for a change in minimum wage to affect employment.[40]
Monopsony[edit]
Modern economics suggests that a moderate minimum wage may increase employment as labor markets are monopsonistic and workers lack bargaining power.
Main article: Monopsony
The supply and demand model predicts that raising the minimum wage helps workers whose wages are raised, and hurts people who are not hired (or lose their jobs) when companies cut back on employment. But proponents of the minimum wage hold that the situation is much more complicated than the model can account for. One complicating factor is possible monopsony in the labor market, whereby the individual employer has some market power in determining wages paid. Thus it is at least theoretically possible that the minimum wage may boost employment. Though single employer market power is unlikely to exist in most labor markets in the sense of the traditional 'company town,' asymmetric information, imperfect mobility, and the personal element of the labor transaction give some degree of wage-setting power to most firms.[41]
Modern economic theory predicts that although an excessive minimum wage may raise unemployment as it fixes a price above most demand for labor, a minimum wage at a more reasonable level can increase employment, and enhance growth and efficiency. This is because labor markets are monopsonistic and workers persistently lack bargaining power. When poorer workers have more to spend it stimulates effective aggregate demand for goods and services.[42][43]
Criticisms of the supply and demand model[edit]
The argument that a minimum wage decreases employment is based on a simple supply and demand model of the labor market. A number of economists (for example Pierangelo Garegnani,[44] Robert L. Vienneau,[45] and Arrigo Opocher & Ian Steedman[46]), building on the work of Piero Sraffa, argue that that model, even given all its assumptions, is logically incoherent. Michael Anyadike-Danes and Wynne Godley[47] argue, based on simulation results, that little of the empirical work done with the textbook model constitutes a potentially falsifiable theory, and consequently empirical evidence hardly exists for that model. Graham White[48] argues, partially on the basis of Sraffianism, that the policy of increased labor market flexibility, including the reduction of minimum wages, does not have an "intellectually coherent" argument in economic theory.
Minimum wage levels in OECD countries as a share of average full-time wage, 2013.[49]
Gary Fields, Professor of Labor Economics and Economics at Cornell University, argues that the standard textbook model for the minimum wage is ambiguous, and that the standard theoretical arguments incorrectly measure only a one-sector market. Fields says a two-sector market, where "the self-employed, service workers, and farm workers are typically excluded from minimum-wage coverage... [and with] one sector with minimum-wage coverage and the other without it [and possible mobility between the two]," is the basis for better analysis. Through this model, Fields shows the typical theoretical argument to be ambiguous and says "the predictions derived from the textbook model definitely do not carry over to the two-sector case. Therefore, since a non-covered sector exists nearly everywhere, the predictions of the textbook model simply cannot be relied on."[50]
An alternate view of the labor market has low-wage labor markets characterized as monopsonistic competition wherein buyers (employers) have significantly more market power than do sellers (workers). This monopsony could be a result of intentional collusion between employers, or naturalistic factors such as segmented markets, search costs, information costs, imperfect mobility and the personal element of labor markets.[citation needed] In such a case a simple supply and demand graph would not yield the quantity of labor clearing and the wage rate. This is because while the upward sloping aggregate labor supply would remain unchanged, instead of using the upward labor supply curve shown in a supply and demand diagram, monopsonistic employers would use a steeper upward sloping curve corresponding to marginal expenditures to yield the intersection with the supply curve resulting in a wage rate lower than would be the case under competition. Also, the amount of labor sold would also be lower than the competitive optimal allocation.
Such a case is a type of market failure and results in workers being paid less than their marginal value. Under the monopsonistic assumption, an appropriately set minimum wage could increase both wages and employment, with the optimal level being equal to the marginal product of labor.[51] This view emphasizes the role of minimum wages as a market regulation policy akin to antitrust policies, as opposed to an illusory "free lunch" for low-wage workers.
Another reason minimum wage may not affect employment in certain industries is that the demand for the product the employees produce is highly inelastic.[52] For example, if management is forced to increase wages, management can pass on the increase in wage to consumers in the form of higher prices. Since demand for the product is highly inelastic, consumers continue to buy the product at the higher price and so the manager is not forced to lay off workers. Economist Paul Krugman argues this explanation neglects to explain why the firm was not charging this higher price absent the minimum wage.[53]
Three other possible reasons minimum wages do not affect employment were suggested by Alan Blinder: higher wages may reduce turnover, and hence training costs; raising the minimum wage may "render moot" the potential problem of recruiting workers at a higher wage than current workers; and minimum wage workers might represent such a small proportion of a business's cost that the increase is too small to matter. He admits that he does not know if these are correct, but argues that "the list demonstrates that one can accept the new empirical findings and still be a card-carrying economist."[54]
Mathematical models of the minimum wage and frictional labor markets[edit]
The following mathematical models are more quantitative in orientation, and highlight some of the difficulties in determining the impact of the minimum wage on labor market outcomes.[55] Specifically, these models focus on labor markets with frictions.
Welfare and labor market participation[edit]
Assume that the decision to participate in the labor market results from a trade-off between being an unemployed job seeker and not participating at all. All individuals whose expected utility outside the labor market is less than the expected utility of an unemployed person V u {\displaystyle V_{u}} decide to participate in the labor market. In the basic search and matching model, the expected utility of unemployed persons V u {\displaystyle V_{u}} and that of employed persons V e {\displaystyle V_{e}} are defined by:
r V e = w + q ( V u − V e ) r V u = z + θ m ( θ ) ( V e − V u ) {\displaystyle {\begin{aligned}rV_{e}&=w+q(V_{u}-V_{e})\\rV_{u}&=z+\theta m(\theta )(V_{e}-V_{u})\end{aligned}}}
Let w {\displaystyle w} be the wage, r {\displaystyle r} the interest rate, z {\displaystyle z} the instantaneous income of unemployed persons, q {\displaystyle q} the exogenous job destruction rate, θ {\displaystyle \theta } the labor market tightness, and θ m ( θ ) {\displaystyle \theta m(\theta )} the job finding rate. The profits Π e {\displaystyle \Pi _{e}} and Π v {\displaystyle \Pi _{v}} expected from a filled job and a vacant one are:
r Π e = y − w + q ( Π v − Π e ) , r Π v = − h + m ( θ ) ( Π e − Π v ) {\displaystyle r\Pi _{e}=y-w+q(\Pi _{v}-\Pi _{e}),\quad r\Pi _{v}=-h+m(\theta )(\Pi _{e}-\Pi _{v})}
where h {\displaystyle h} is the cost of a vacant job and y {\displaystyle y} is the productivity. When the free entry condition Π v = 0 {\displaystyle \Pi _{v}=0} is satisfied, these two equalities yield the following relationship between the wage w {\displaystyle w} and labor market tightness θ {\displaystyle \theta } :
New Unionism
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Eight-hour day
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h m ( θ ) = y − w r + q {\displaystyle {h \over {m(\theta )}}={y-w \over {r+q}}}
If w {\displaystyle w} represents a minimum wage that applies to all workers, this equation completely determines the equilibrium value of the labor market tightness θ {\displaystyle \theta } . There are two conditions associated with the matching function:
m ′ ( θ ) < 0 , [ θ m ( θ ) ] ′ > 0 {\displaystyle m'(\theta )<0,\quad [\theta m(\theta )]'>0}
This implies that θ {\displaystyle \theta } is a decreasing function of the minimum wage w {\displaystyle w} , and so is the job finding rate α = θ m ( θ ) {\displaystyle \alpha =\theta m(\theta )} . A hike in the minimum wage degrades the profitability of a job, so firms post fewer vacancies and the job finding rate falls off. Now let's rewrite r V u {\displaystyle rV_{u}} to be:
r V u = ( r + q ) z + θ m ( θ ) w r + q + θ m ( θ ) {\displaystyle rV_{u}={(r+q)z+\theta m(\theta )w \over {r+q+\theta m(\theta )}}}
Using the relationship between the wage and labor market tightness to eliminate the wage from the last equation gives us:
r V u = θ m ( θ ) y + ( r + q ) z − θ ( r + q ) h r + q + θ m ( θ ) {\displaystyle rV_{u}={\theta m(\theta )y+(r+q)z-\theta (r+q)h \over {r+q+\theta m(\theta )}}}
If we maximize r V u {\displaystyle rV_{u}} in this equation, with respect to the labor market tightness, we find that:
[ 1 − η ( θ ) ] ( y − z ) r + q + η ( θ ) θ m ( θ ) = h m ( θ ) {\displaystyle {[1-\eta (\theta )](y-z) \over {r+q+\eta (\theta )\theta m(\theta )}}={h \over {m(\theta )}}}
where η ( θ ) {\displaystyle \eta (\theta )} is the elasticity of the matching function:
η ( θ ) = − θ m ′ ( θ ) m ( θ ) ≡ − θ d d θ log m ( θ ) {\displaystyle \eta (\theta )=-\theta {m'(\theta ) \over {m(\theta )}}\equiv -\theta {d \over {d\theta }}\log m(\theta )}
This result shows that the expected utility of unemployed workers is maximized when the minimum wage is set at a level that corresponds to the wage level of the decentralized economy in which the bargaining power parameter is equal to the elasticity η ( θ ) {\displaystyle \eta (\theta )} . The level of the negotiated wage is w ∗ {\displaystyle w^{*}} .
If w < w ∗ {\displaystyle w<w^{*}} , then an increase in the minimum wage increases participation and the unemployment rate, with an ambiguous impact on employment. When the bargaining power of workers is less than η ( θ ) {\displaystyle \eta (\theta )} , an increases in the minimum wage improves the welfare of the unemployed - this suggests that minimum wage hikes can improve labor market efficiency, at least up to the point when bargaining power equals η ( θ ) {\displaystyle \eta (\theta )} . On the other hand, if w ≥ w ∗ {\displaystyle w\geq w^{*}} , any increases in the minimum wage entails a decline in labor market participation and an increase in unemployment.
Job search effort[edit]
In the model just presented, we found that the minimum wage always increases unemployment. This result does not necessarily hold when the search effort of workers in endogenous.
Consider a model where the intensity of the job search is designated by the scalar ϵ {\displaystyle \epsilon } , which can be interpreted as the amount of time and/or intensity of the effort devoted to search. Assume that the arrival rate of job offers is α ϵ {\displaystyle \alpha \epsilon } and that the wage distribution is degenerated to a single wage w {\displaystyle w} . Denote φ ( ϵ ) {\displaystyle \varphi (\epsilon )} to be the cost arising from the search effort, with φ ′ > 0 , φ ″ > 0 {\displaystyle \varphi '>0,\;\varphi ''>0} . Then the discounted utilities are given by:
r V e = w + q ( V u − V e ) r V u = max ϵ z − φ ( ϵ ) + α ϵ ( V e − V u ) {\displaystyle {\begin{aligned}rV_{e}&=w+q(V_{u}-V_{e})\\rV_{u}&=\max _{\epsilon }\;z-\varphi (\epsilon )+\alpha \epsilon (V_{e}-V_{u})\end{aligned}}}
Therefore, the optimal search effort is such that the marginal cost of performing the search is equation to the marginal return:
φ ′ ( ϵ ) = α ( V e − V u ) {\displaystyle \varphi '(\epsilon )=\alpha (V_{e}-V_{u})}
This implies that the optimal search effort increases as the difference between the expected utility of the job holder and the expected utility of the job seeker grows. In fact, this difference actually grows with the wage. To see this, take the difference of the two discounted utilities to find:
( r + q ) ( V e − V u ) = w − max ϵ [ z − φ ( ϵ ) + α ϵ ( V e − V u ) ] {\displaystyle (r+q)(V_{e}-V_{u})=w-\max _{\epsilon }\left[z-\varphi (\epsilon )+\alpha \epsilon (V_{e}-V_{u})\right]}
Then differentiating with respect to w {\displaystyle w} and rearranging gives us:
d d w ( V e − V u ) = 1 r + q + α ϵ ∗ > 0 {\displaystyle {d \over {dw}}(V_{e}-V_{u})={1 \over {r+q+\alpha \epsilon ^{*}}}>0}
where ϵ ∗ {\displaystyle \epsilon ^{*}} is the optimal search effort. This implies that a wage increase drives up job search effort and, therefore, the job finding rate. Additionally, the unemployment rate u {\displaystyle u} at equilibrium is given by:
u = q q + α ϵ {\displaystyle u={q \over {q+\alpha \epsilon }}}
A hike in the wage, which increases the search effort and the job finding rate, decreases the unemployment rate. So it is possible that a hike in the minimum wage may, by boosting the search effort of job seekers, boost employment. Taken in sum with the previous section, the minimum wage in labor markets with frictions can improve employment and decrease the unemployment rate when it is sufficiently low. However, a high minimum wage is detrimental to employment and increases the unemployment rate.
Empirical studies[edit]
Estimated minimum wage effects on employment from a meta-study of 64 other studies showed insignificant employment effect (both practically and statistically) from minimum-wage raises. The most precise estimates were heavily clustered at or near zero employment effects (elasticity = 0).[56]
Economists disagree as to the measurable impact of minimum wages in practice. This disagreement usually takes the form of competing empirical tests of the elasticities of supply and demand in labor markets and the degree to which markets differ from the efficiency that models of perfect competition predict.
Economists have done empirical studies on different aspects of the minimum wage, including:[15]
Employment effects, the most frequently studied aspect
Effects on the distribution of wages and earnings among low-paid and higher-paid workers
Effects on the distribution of incomes among low-income and higher-income families
Effects on the skills of workers through job training and the deferring of work to acquire education
Effects on prices and profits
Effects on on-the-job training
Until the mid-1990s, a general consensus existed among economists, both conservative and liberal, that the minimum wage reduced employment, especially among younger and low-skill workers.[8] In addition to the basic supply-demand intuition, there were a number of empirical studies that supported this view. For example, Gramlich (1976) found that many of the benefits went to higher income families, and that teenagers were made worse off by the unemployment associated with the minimum wage.[57]
Brown et al. (1983) noted that time series studies to that point had found that for a 10 percent increase in the minimum wage, there was a decrease in teenage employment of 1–3 percent. However, the studies found wider variation, from 0 to over 3 percent, in their estimates for the effect on teenage unemployment (teenagers without a job and looking for one). In contrast to the simple supply and demand diagram, it was commonly found that teenagers withdrew from the labor force in response to the minimum wage, which produced the possibility of equal reductions in the supply as well as the demand for labor at a higher minimum wage and hence no impact on the unemployment rate. Using a variety of specifications of the employment and unemployment equations (using ordinary least squares vs. generalized least squares regression procedures, and linear vs. logarithmic specifications), they found that a 10 percent increase in the minimum wage caused a 1 percent decrease in teenage employment, and no change in the teenage unemployment rate. The study also found a small, but statistically significant, increase in unemployment for adults aged 20–24.[58]
CBO table illustrating projections of the effects of minimum wage increases on employment and income, under two scenarios
Wellington (1991) updated Brown et al.'s research with data through 1986 to provide new estimates encompassing a period when the real (i.e., inflation-adjusted) value of the minimum wage was declining, because it had not increased since 1981. She found that a 10% increase in the minimum wage decreased the absolute teenage employment by 0.6%, with no effect on the teen or young adult unemployment rates.[59]
Some research suggests that the unemployment effects of small minimum wage increases are dominated by other factors.[60] In Florida, where voters approved an increase in 2004, a follow-up comprehensive study after the increase confirmed a strong economy with increased employment above previous years in Florida and better than in the US as a whole.[61] When it comes to on-the-job training, some believe the increase in wages is taken out of training expenses. A 2001 empirical study found that there is "no evidence that minimum wages reduce training, and little evidence that they tend to increase training."[62]
Some empirical studies have tried to ascertain the benefits of a minimum wage beyond employment effects. In an analysis of census data, Joseph Sabia and Robert Nielson found no statistically significant evidence that minimum wage increases helped reduce financial, housing, health, or food insecurity.[63] This study was undertaken by the Employment Policies Institute, a think tank funded by the food, beverage and hospitality industries. In 2012, Michael Reich published an economic analysis that suggested that a proposed minimum wage hike in San Diego might stimulate the city's economy by about $190 million.[64]
The Economist wrote in December 2013: "A minimum wage, providing it is not set too high, could thus boost pay with no ill effects on jobs....America's federal minimum wage, at 38% of median income, is one of the rich world's lowest. Some studies find no harm to employment from federal or state minimum wages, others see a small one, but none finds any serious damage. ... High minimum wages, however, particularly in rigid labour markets, do appear to hit employment. France has the rich world's highest wage floor, at more than 60% of the median for adults and a far bigger fraction of the typical wage for the young. This helps explain why France also has shockingly high rates of youth unemployment: 26% for 15- to 24-year-olds."[65]
The restaurant industry is commonly studied because of its high number of minimum wage workers. A 2018 study from the Center on Wage and Employment Dynamics at the University of California, Berkeley focusing on food services showed that minimum wage increases in Washington, Chicago, Seattle, San Francisco, Oakland, and San Jose gave workers higher pay without hampering job growth.[66] A 2017 study of restaurants in the San Francisco Bay Area examined the period 2008-2016 and the effect that a minimum wage increase had on the probability of restaurants going out of business, and broke out results based on the restaurant's rating on the review site Yelp. The study found no effect for 5-star (highest rated) restaurants (regardless of the expensiveness of the cuisine) but those with increasingly lower ratings were increasingly likely to go out of business (for example a 14% increase at 3.5 stars for a $1 per hour minimum wage increase). It also noted that the Yelp star rating was correlated with likelihood of minority ownership and minority customer base. Importantly, it noted that restaurants below 4 star in rating were proportionally more likely to hire low-skilled workers. The minimum wage increases during this period did not prevent growth in the industry overall – the number of restaurants in San Francisco went from 3,600 in 2012 to 7,600 in 2016.[67] An August 2019 study from The New School's Center for New York City Affairs and the think tank National Employment Law Project, which advocates for raising the minimum wage,[68] found that the restaurant industry in New York City has been "thriving" following an increase in the minimum wage to $15 an hour.[69]
A 2019 study in the Quarterly Journal of Economics found that minimum wage increases did not have an impact on the overall number of low-wage jobs in the five years subsequent to the wage increase. However, it did find disemployment in 'tradeable' sectors, defined as those sectors most reliant on entry level or low skilled labor.[70]
In another study, which shared authors with the above, published in the American Economic Review found that a large and persistent increase in the minimum wage in Hungary produced some disemployment with the large majority of additional cost being passed on to consumers. The authors also found that firms began substituting capital for labor over time.[71]
Card and Krueger[edit]
In 1992, the minimum wage in New Jersey increased from $4.25 to $5.05 per hour (an 18.8% increase), while in the adjacent state of Pennsylvania it remained at $4.25. David Card and Alan Krueger gathered information on fast food restaurants in New Jersey and eastern Pennsylvania in an attempt to see what effect this increase had on employment within New Jersey. A basic supply and demand model predicts that relative employment should have decreased in New Jersey. Card and Krueger surveyed employers before the April 1992 New Jersey increase, and again in November–December 1992, asking managers for data on the full-time equivalent staff level of their restaurants both times.[72] Based on data from the employers' responses, the authors concluded that the increase in the minimum wage slightly increased employment in the New Jersey restaurants.[72]
Card and Krueger expanded on this initial article in their 1995 book Myth and Measurement: The New Economics of the Minimum Wage.[73] They argued that the negative employment effects of minimum wage laws are minimal if not non-existent. For example, they look at the 1992 increase in New Jersey's minimum wage, the 1988 rise in California's minimum wage, and the 1990–91 increases in the federal minimum wage. In addition to their own findings, they reanalyzed earlier studies with updated data, generally finding that the older results of a negative employment effect did not hold up in the larger datasets.[74]
Research subsequent to Card and Krueger's work[edit]
A 2010 study published in the Review of Economics and Statistics compared 288 pairs of contiguous U.S. counties with minimum wage differentials from 1990 to 2006 and found no adverse employment effects from a minimum wage increase. Contiguous counties with different minimum wages are in purple. All other counties are in white.[75]
In 1996, David Neumark and William Wascher reexamined Card and Krueger's result using administrative payroll records from a sample of large fast food restaurant chains, and reported that minimum wage increases were followed by decreases in employment. An assessment of data collected and analyzed by Neumark and Wascher did not initially contradict the Card and Krueger results,[76] but in a later edited version they found a four percent decrease in employment, and reported that "the estimated disemployment effects in the payroll data are often statistically significant at the 5- or 10-percent level although there are some estimators and subsamples that yield insignificant—although almost always negative" employment effects.[77][78] Neumark and Wascher's conclusions were subsequently rebutted in a 2000 paper by Card and Krueger.[79] A 2011 paper has reconciled the difference between Card and Krueger's survey data and Neumark and Wascher's payroll-based data. The paper shows that both datasets evidence conditional employment effects that are positive for small restaurants, but are negative for large fast-food restaurants.[80] A 2014 analysis based on panel data found that the minimum wage reduces employment among teenagers.[81]
In 1996 and 1997, the federal minimum wage was increased from $4.25 to $5.15, thereby increasing the minimum wage by $0.90 in Pennsylvania but by just $0.10 in New Jersey; this allowed for an examination of the effects of minimum wage increases in the same area, subsequent to the 1992 change studied by Card and Krueger. A study by Hoffman and Trace found the result anticipated by traditional theory: a detrimental effect on employment.[82]
Further application of the methodology used by Card and Krueger by other researchers yielded results similar to their original findings, across additional data sets.[83] A 2010 study by three economists (Arindrajit Dube of the University of Massachusetts Amherst, William Lester of the University of North Carolina at Chapel Hill, and Michael Reich of the University of California, Berkeley), compared adjacent counties in different states where the minimum wage had been raised in one of the states. They analyzed employment trends for several categories of low-wage workers from 1990 to 2006 and found that increases in minimum wages had no negative effects on low-wage employment and successfully increased the income of workers in food services and retail employment, as well as the narrower category of workers in restaurants.[83][84]
However, a 2011 study by Baskaya and Rubinstein of Brown University found that at the federal level, "a rise in minimum wage have [sic] an instantaneous impact on wage rates and a corresponding negative impact on employment", stating, "Minimum wage increases boost teenage wage rates and reduce teenage employment."[85] Another 2011 study by Sen, Rybczynski, and Van De Waal found that "a 10% increase in the minimum wage is significantly correlated with a 3−5% drop in teen employment."[86] A 2012 study by Sabia, Hansen, and Burkhauser found that "minimum wage increases can have substantial adverse labor demand effects for low-skilled individuals", with the largest effects on those aged 16 to 24.[87]
A 2013 study by Meer and West concluded that "the minimum wage reduces net job growth, primarily through its effect on job creation by expanding establishments ... most pronounced for younger workers and in industries with a higher proportion of low-wage workers."[88] This study by Meer and West was later critiqued for its trends of assumption in the context of narrowly defined low-wage groups.[89] The authors replied to the critiques and released additional data which addressed the criticism of their methodology, but did not resolve the issue of whether their data showed a causal relationship.[90][91] A 2019 paper published in the Quarterly Journal of Economics by Cengiz, Dube, Lindner and Zipperer argues that the job losses found using a Meer and West type methodology "tend to be driven by an unrealistically large drop in the number of jobs at the upper tail of the wage distribution, which is unlikely to be a causal effect of the minimum wage."[92] Another 2013 study by Suzana Laporšek of the University of Primorska, on youth unemployment in Europe claimed there was "a negative, statistically significant impact of minimum wage on youth employment."[93] A 2013 study by labor economists Tony Fang and Carl Lin which studied minimum wages and employment in China, found that "minimum wage changes have significant adverse effects on employment in the Eastern and Central regions of China, and result in disemployment for females, young adults, and low-skilled workers".[94][95]
A 2017 study found that in Seattle, increasing the minimum wage to $13 per hour lowered income of low-wage workers by $125 per month, due to the resulting reduction in hours worked, as industries made changes to make their businesses less labor intensive. The authors argue that previous research that found no negative effects on hours worked are flawed because they only look at select industries, or only look at teenagers, instead of entire economies.[96]
Finally, a study by Overstreet in 2019 examined increases to the minimum wage in Arizona. Utilizing data spanning from 1976 to 2017, Overstreet found that a 1% increase in the minimum wage was significantly correlated with a 1.13% increase in per capita income in Arizona. This study could show that smaller increases in minimum wage may not distort labor market as significantly as larger increases experienced in other cities and states. Thus, the small increases experienced in Arizona may have actually led to a slight increase in economic growth.[97]
In 2019, economists from Georgia Tech published a study that found a strong correlation between incresases to the minimum wage and detectable harm to the financial conditions of small businesses, including a higher rate of bankruptcy, lower hiring rates, lower credit scores, and higher interest payments. The researchers noted that these small businesses were also correlated with minority ownership and minority customer bases. [98]
In July 2019, the Congressional Budget Office published the impact on proposed national $15/hour legislation. It noted that workers who retained full employment would see a modest improvement in take home pay offset by a small decrease in working conditions and non-pecuniary benefits. However, this benefit is offset by three primary factors; the reduction in hours worked, the reduction in total employment, and the increased cost of goods and services. Those factors result in a decrease of about $33 Billion in total income and nearly 1.7-3.7 million lost jobs in the first three years (the CBO also noted this figure increases over time). [99]
In response to an April 2016 Council of Economic Advisers (CEA) report advocating the raising of the minimum wage to deter crime, economists used data from the 1998-2016 Uniform Crime Reports (UCR), National Incident-Based Reporting System (NIBRS), and National Longitudinal Study of Youth (NLSY) to assess the impact of the minimum wage on crime. They found that increasing the minimum wage resulted in increased property crime arrests among those ages 16-to-24. They estimated that an increase of the Federal minimum wage to $15/hour would "generate criminal externality costs of nearly $2.4 billion." [100]
Economists in Denmark, relying on a discontinuity in wage rates when a worker turns 18, found that employment fell by 33% and total hours fell by 45% when the minimum wage law was in effect. [101]
Statistical meta-analyses[edit]
Several researchers have conducted statistical meta-analyses of the employment effects of the minimum wage. In 1995, Card and Krueger analyzed 14 earlier time-series studies on minimum wages and concluded that there was clear evidence of publication bias (in favor of studies that found a statistically significant negative employment effect). They point out that later studies, which had more data and lower standard errors, did not show the expected increase in t-statistic (almost all the studies had a t-statistic of about two, just above the level of statistical significance at the .05 level).[102] Though a serious methodological indictment, opponents of the minimum wage largely ignored this issue; as Thomas Leonard noted, "The silence is fairly deafening."[103]
In 2005, T.D. Stanley showed that Card and Krueger's results could signify either publication bias or the absence of a minimum wage effect. However, using a different methodology, Stanley concluded that there is evidence of publication bias and that correction of this bias shows no relationship between the minimum wage and unemployment.[104] In 2008, Hristos Doucouliagos and T.D. Stanley conducted a similar meta-analysis of 64 U.S. studies on disemployment effects and concluded that Card and Krueger's initial claim of publication bias is still correct. Moreover, they concluded, "Once this publication selection is corrected, little or no evidence of a negative association between minimum wages and employment remains."[105] In 2013, a meta-analysis of 16 UK studies found no significant effects on employment attributable to the minimum wage.[106] a 2007 meta-analyses by David Neumark of 96 studies found a consistent, but not always statistically significant, negative effect on employment from increases in the minimum wage.[107]
Debate over consequences[edit]
Minimum wage laws affect workers in most low-paid fields of employment[15] and have usually been judged against the criterion of reducing poverty.[108] Minimum wage laws receive less support from economists than from the general public. Despite decades of experience and economic research, debates about the costs and benefits of minimum wages continue today.[15]
Various groups have great ideological, political, financial, and emotional investments in issues surrounding minimum wage laws. For example, agencies that administer the laws have a vested interest in showing that "their" laws do not create unemployment, as do labor unions whose members' finances are protected by minimum wage laws. On the other side of the issue, low-wage employers such as restaurants finance the Employment Policies Institute, which has released numerous studies opposing the minimum wage.[109][110] The presence of these powerful groups and factors means that the debate on the issue is not always based on dispassionate analysis. Additionally, it is extraordinarily difficult to separate the effects of minimum wage from all the other variables that affect employment.[28]
The following table summarizes the arguments made by those for and against minimum wage laws:
Arguments in favor of minimum wage laws
Arguments against minimum wage laws
Supporters of the minimum wage claim it has these effects:
Improves functioning of the low-wage labor market which may be characterized by employer-side market power (monopsony).[111][112]
Raises family incomes at the bottom of the income distribution, and lowers poverty.[113][114]
Positive impact on small business owners and industry.[115]
Removes financial stress[116] and encourages education,[117] resulting in better paying jobs.
Increases the standard of living for the poorest and most vulnerable class in society and raises average.[118]
Increases incentives to take jobs, as opposed to other methods of transferring income to the poor that are not tied to employment (such as food subsidies for the poor or welfare payments for the unemployed).[119]
Stimulates consumption, by putting more money in the hands of low-income people who spend their entire paychecks. Hence increases circulation of money through the economy.[118]
Increased job growth and creation.[120][121]
Encourages efficiency and automation of industry.[122]
Removes low paying jobs, forcing workers to train for, and move to, higher paying jobs.[123][124]
Increases technological development. Costly technology that increases business efficiency is more appealing as the price of labor increases.[125]
Increases the work ethic of those who earn very little, as employers demand more return from the higher cost of hiring these employees.[118]
Decreases the cost of government social welfare programs by increasing incomes for the lowest-paid.[118]
Encourages people to join the workforce rather than pursuing money through illegal means, e.g., selling illegal drugs[126][127]
Opponents of the minimum wage claim it has these effects:
Minimum wage alone is not effective at alleviating poverty, and in fact produces a net increase in poverty due to disemployment effects.[128]
As a labor market analogue of political-economic protectionism, it excludes low cost competitors from labor markets and hampers firms in reducing wage costs during trade downturns. This generates various industrial-economic inefficiencies.[129]
Hurts small business more than large business.[130]
Reduces quantity demanded of workers, either through a reduction in the number of hours worked by individuals, or through a reduction in the number of jobs.[131][132]
May cause price inflation as businesses try to compensate by raising the prices of the goods being sold.[133][134]
Benefits some workers at the expense of the poorest and least productive.[135]
Wage/price spiral
Encourages employers to replace low-skilled workers with computers, such as self-checkout machines.[136]
Increases property crime and misery in poor communities by decreasing legal markets of production and consumption in those communities;[137]
Can result in the exclusion of certain groups (ethnic, gender etc.) from the labor force.[138]
Small firms with limited payroll budgets cannot offer their most valuable employees fair and attractive wages above unskilled workers paid the artificially high minimum, and see a rising hurdle-cost of adding workers.[130]
Is less effective than other methods (e.g. the Earned Income Tax Credit) at reducing poverty, and is more damaging to businesses than those other methods.[139]
Discourages further education among the poor by enticing people to enter the job market.[139]
Discriminates against, through pricing out, less qualified workers (including newcomers to the labor market, e.g. young workers) by keeping them from accumulating work experience and qualifications, hence potentially graduating to higher wages later.[6]
Slows growth in the creation of low-skilled jobs[88]
Results in jobs moving to other areas or countries which allow lower-cost labor.[140]
Results in higher long-term unemployment.[141]
Results in higher prices for consumers, where products and services are produced by minimum-wage workers[142] (though non-labor costs represent a greater proportion of costs to consumers in industries like fast food and discount retail)[143][144]
A widely circulated argument that the minimum wage was ineffective at reducing poverty was provided by George Stigler in 1949:
Employment may fall more than in proportion to the wage increase, thereby reducing overall earnings;
As uncovered sectors of the economy absorb workers released from the covered sectors, the decrease in wages in the uncovered sectors may exceed the increase in wages in the covered ones;
The impact of the minimum wage on family income distribution may be negative unless the fewer but better jobs are allocated to members of needy families rather than to, for example, teenagers from families not in poverty;
Forbidding employers to pay less than a legal minimum is equivalent to forbidding workers to sell their labor for less than the minimum wage. The legal restriction that employers cannot pay less than a legislated wage is equivalent to the legal restriction that workers cannot work at all in the protected sector unless they can find employers willing to hire them at that wage.[108]
In 2006, the International Labour Organization (ILO) argued that the minimum wage could not be directly linked to unemployment in countries that have suffered job losses.[1] In April 2010, the Organisation for Economic Co-operation and Development (OECD) released a report arguing that countries could alleviate teen unemployment by "lowering the cost of employing low-skilled youth" through a sub-minimum training wage.[145] A study of U.S. states showed that businesses' annual and average payrolls grow faster and employment grew at a faster rate in states with a minimum wage.[146] The study showed a correlation, but did not claim to prove causation.
Although strongly opposed by both the business community and the Conservative Party when introduced in the UK in 1999, the Conservatives reversed their opposition in 2000.[147] Accounts differ as to the effects of the minimum wage. The Centre for Economic Performance found no discernible impact on employment levels from the wage increases,[148] while the Low Pay Commission found that employers had reduced their rate of hiring and employee hours employed, and found ways to cause current workers to be more productive (especially service companies).[149] The Institute for the Study of Labor found prices in the minimum wage sector rose significantly faster than prices in non-minimum wage sectors, in the four years following the implementation of the minimum wage.[150] Neither trade unions nor employer organizations contest the minimum wage, although the latter had especially done so heavily until 1999.
In 2014, supporters of minimum wage cited a study that found that job creation within the United States is faster in states that raised their minimum wages.[120][151][152] In 2014, supporters of minimum wage cited news organizations who reported the state with the highest minimum-wage garnered more job creation than the rest of the United States.[120][153][154][155][156][157][158]
In 2014, in Seattle, Washington, liberal and progressive business owners who had supported the city's new $15 minimum wage said they might hold off on expanding their businesses and thus creating new jobs, due to the uncertain timescale of the wage increase implementation.[159] However, subsequently at least two of the business owners quoted did expand.[160][161]
The dollar value of the minimum wage loses purchasing power over time due to inflation. Minimum wage laws, for instance proposals to index the minimum wage to average wages, have the potential to keep the dollar value of the minimum wage relevant and predictable.[162]
With regard to the economic effects of introducing minimum wage legislation in Germany in January 2015, recent developments have shown that the feared increase in unemployment has not materialized, however, in some economic sectors and regions of the country, it came to a decline in job opportunities particularly for temporary and part-time workers, and some low-wage jobs have disappeared entirely.[163] Because of this overall positive development, the Deutsche Bundesbank revised its opinion, and ascertained that "the impact of the introduction of the minimum wage on the total volume of work appears to be very limited in the present business cycle".[164]
A 2019 study published in the American Journal of Preventive Medicine showed that in the United States, those states which have implemented a higher minimum wage saw a decline in the growth of suicide rates. The researchers say that for every one dollar increase, the annual suicide growth rate fell by 1.9%. The study covers all 50 states for the years 2006 to 2016.[165]
Surveys of economists[edit]
According to a 1978 article in the American Economic Review, 90% of the economists surveyed agreed that the minimum wage increases unemployment among low-skilled workers.[166] By 1992 the survey found 79% of economists in agreement with that statement,[167] and by 2000, 46% were in full agreement with the statement and 28% agreed with provisos (74% total).[168][169] The authors of the 2000 study also reweighted data from a 1990 sample to show that at that time 62% of academic economists agreed with the statement above, while 20% agreed with provisos and 18% disagreed. They state that the reduction on consensus on this question is "likely" due to the Card and Krueger research and subsequent debate.[170]
A similar survey in 2006 by Robert Whaples polled PhD members of the American Economic Association (AEA). Whaples found that 47% respondents wanted the minimum wage eliminated, 38% supported an increase, 14% wanted it kept at the current level, and 1% wanted it decreased.[171] Another survey in 2007 conducted by the University of New Hampshire Survey Center found that 73% of labor economists surveyed in the United States believed 150% of the then-current minimum wage would result in employment losses and 68% believed a mandated minimum wage would cause an increase in hiring of workers with greater skills. 31% felt that no hiring changes would result.[172]
Surveys of labor economists have found a sharp split on the minimum wage. Fuchs et al. (1998) polled labor economists at the top 40 research universities in the United States on a variety of questions in the summer of 1996. Their 65 respondents were nearly evenly divided when asked if the minimum wage should be increased. They argued that the different policy views were not related to views on whether raising the minimum wage would reduce teen employment (the median economist said there would be a reduction of 1%), but on value differences such as income redistribution.[173] Daniel B. Klein and Stewart Dompe conclude, on the basis of previous surveys, "the average level of support for the minimum wage is somewhat higher among labor economists than among AEA members."[174]
In 2007, Klein and Dompe conducted a non-anonymous survey of supporters of the minimum wage who had signed the "Raise the Minimum Wage" statement published by the Economic Policy Institute. 95 of the 605 signatories responded. They found that a majority signed on the grounds that it transferred income from employers to workers, or equalized bargaining power between them in the labor market. In addition, a majority considered disemployment to be a moderate potential drawback to the increase they supported.[174]
In 2013, a diverse group of 37 economics professors was surveyed on their view of the minimum wage's impact on employment. 34% of respondents agreed with the statement, "Raising the federal minimum wage to $9 per hour would make it noticeably harder for low-skilled workers to find employment." 32% disagreed and the remaining respondents were uncertain or had no opinion on the question. 47% agreed with the statement, "The distortionary costs of raising the federal minimum wage to $9 per hour and indexing it to inflation are sufficiently small compared with the benefits to low-skilled workers who can find employment that this would be a desirable policy", while 11% disagreed.[175]
Alternatives[edit]
Economists and other political commentators have proposed alternatives to the minimum wage. They argue that these alternatives may address the issue of poverty better than a minimum wage, as it would benefit a broader population of low wage earners, not cause any unemployment, and distribute the costs widely rather than concentrating it on employers of low wage workers.
Basic income[edit]
Main article: Basic Income
A basic income (or negative income tax - NIT) is a system of social security that periodically provides each citizen with a sum of money that is sufficient to live on frugally. Supporters of the basic-income idea argue that recipients of the basic income would have considerably more bargaining power when negotiating a wage with an employer, as there would be no risk of destitution for not taking the employment. As a result, jobseekers could spend more time looking for a more appropriate or satisfying job, or they could wait until a higher-paying job appeared. Alternatively, they could spend more time increasing their skills (via education and training), which would make them more suitable for higher-paying jobs, as well as provide numerous other benefits. Experiments on Basic Income and NIT in Canada and the USA show that people spent more time studying while the program[which?] was running.[176][need quotation to verify]
Proponents argue that a basic income that is based on a broad tax base would be more economically efficient than a minimum wage, as the minimum wage effectively imposes a high marginal tax on employers, causing losses in efficiency.[citation needed]
Guaranteed minimum income[edit]
A guaranteed minimum income is another proposed system of social welfare provision. It is similar to a basic income or negative income tax system, except that it is normally conditional and subject to a means test. Some proposals also stipulate a willingness to participate in the labor market, or a willingness to perform community services.[177]
Refundable tax credit[edit]
A refundable tax credit is a mechanism whereby the tax system can reduce the tax owed by a household to below zero, and result in a net payment to the taxpayer beyond their own payments into the tax system. Examples of refundable tax credits include the earned income tax credit and the additional child tax credit in the US, and working tax credits and child tax credits in the UK. Such a system is slightly different from a negative income tax, in that the refundable tax credit is usually only paid to households that have earned at least some income. This policy is more targeted against poverty than the minimum wage, because it avoids subsidizing low-income workers who are supported by high-income households (for example, teenagers still living with their parents).[178]
In the United States, earned income tax credit rates, also known as EITC or EIC, vary by state—some are refundable while other states do not allow a refundable tax credit.[179] The federal EITC program has been expanded by a number of presidents including Jimmy Carter, Ronald Reagan, George H.W. Bush, and Bill Clinton.[180] In 1986, President Reagan described the EITC as "the best anti poverty, the best pro-family, the best job creation measure to come out of Congress."[181] The ability of the earned income tax credit to deliver larger monetary benefits to the poor workers than an increase in the minimum wage and at a lower cost to society was documented in a 2007 report by the Congressional Budget Office.[182]
The Adam Smith Institute prefers cutting taxes on the poor and middle class instead of raising wages as an alternative to the minimum wage.[183]
Collective bargaining[edit]
Italy, Sweden, Norway, Finland, and Denmark are examples of developed nations where there is no minimum wage that is required by legislation.[20][22] Such nations, particularly the Nordics, have very high union participation rates.[184] Instead, minimum wage standards in different sectors are set by collective bargaining.[185]
Wage subsidies[edit]
Some economists such as Scott Sumner[186] and Edmund Phelps[187] advocate a wage subsidy program. A wage subsidy is a payment made by a government for work people do. It is based either on an hourly basis or by income earned. Advocates argue that the primary deficiencies of the EITC and the minimum wage are best avoided by a wage subsidy.[188][189] However, the wage subsidy in the United States suffers from a lack of political support from either major political party.[190][191]
Education and training[edit]
Providing education or funding apprenticeships or technical training can provide a bridge for low skilled workers to move into wages above a minimum wage. For example, Germany has adopted a state funded apprenticeship program that combines on-the-job and classroom training.[192] Having more skills makes workers more valuable and more productive, but having a high minimum wage for low-skill jobs reduces the incentive to seek education and training.[193] Moving some workers to higher-paying jobs will decrease the supply of workers willing to accept low-skill jobs, increasing the market wage for those low skilled jobs (assuming a stable labor market). However, in that solution the wage will still not increase above the marginal return for the role and will likely promote automation or business closure.
US movement[edit]
Main article: Minimum wage in the United States
Protest calling for raising the Minneapolis minimum wage to $15/hour. 12 September 2016
In January 2014, seven Nobel economists—Kenneth Arrow, Peter Diamond, Eric Maskin, Thomas Schelling, Robert Solow, Michael Spence, and Joseph Stiglitz—and 600 other economists wrote a letter to the US Congress and the US President urging that, by 2016, the US government should raise the minimum wage to $10.10. They endorsed the Minimum Wage Fairness Act which was introduced by US Senator Tom Harkin in 2013.[194][195] U.S. Senator Bernie Sanders introduced a bill in 2015 that would raise the minimum wage to $15, and in his 2016 campaign for president ran on a platform of increasing it.[196][197] Although Sanders did not become the nominee, the Democratic National Committee adopted his $15 minimum wage push in their 2016 party platform.[198]
Reactions from former McDonald's USA Ed Rensi about raising minimum wage to $15 is to completely push humans out of the picture when it comes to labor if they are to pay minimum wage at $15 they would look into replacing humans with machines as that would be the more cost-effective than having employees that are ineffective. During an interview on FOX Business Network's Mornings with Maria, he stated that he believes an increase to $15 an hour would cause job loss at an extraordinary level. Rensi also believes it does not only affect the fast food industry, franchising he sees as the best business model in the United States, it is dependent on people that have low job skills that have to grow and if you cannot pay them a reasonable wage then they are going to be replaced with machines.[199]
In late March 2016, Governor of California Jerry Brown reached a deal to raise the minimum wage to $15 by 2022 for big businesses and 2023 for smaller businesses.[200]
In contrast, the relatively high minimum wage in Puerto Rico has been blamed by various politicians and commentators as a highly significant factor in the Puerto Rican government-debt crisis.[201][202][203] One study concluded that "Employers are disinclined to hire workers because the US federal minimum wage is very high relative to the local average".[204]
As of December 2014[update], unions were exempt from recent minimum wage increases in Chicago, Illinois, SeaTac, Washington, and Milwaukee County, Wisconsin, as well as the California cities of Los Angeles, San Francisco, Long Beach, San Jose, Richmond, and Oakland.[205]
Scholars found in 2019 that, in America, "Between 1990 and 2015, raising the minimum wage by $1 in each state might have saved more than 27,000 lives, according to a report published this week in the Journal of Epidemiology & Community Health. An increase of $2 in each state's minimum wage could have prevented more than 57,000 suicides."[206] The researchers stated, "The effect of a US$1 increase in the minimum wage ranged from a 3.4% decrease (95% CI 0.4 to 6.4) to a 5.9% decrease (95% CI 1.4 to 10.2) in the suicide rate among adults aged 18–64 years with a high school education or less. We detected significant effect modification by unemployment rate, with the largest effects of minimum wage on reducing suicides observed at higher unemployment levels."[207] They concluded, "Minimum wage increases appear to reduce the suicide rate among those with a high school education or less, and may reduce disparities between socioeconomic groups. Effects appear greatest during periods of high unemployment."[207]
Countries by minimum wage to average wage ratio[edit]
Countries by minimum wage to mean wage ratio[208]
Australia 0.45 0.45 0.44 0.44 0.44 0.44 0.45 0.46
Belgium 0.42 0.42 0.43 0.43 0.41 0.41 0.42 0.40
Canada 0.39 0.40 0.40 0.39 0.40 0.40 0.40 0.41
Chile 0.44 0.43 0.43 0.45 0.45 0.46 0.47 0.49
Czech Republic 0.32 0.31 0.31 0.31 0.32 0.33 0.34 0.35
Estonia 0.34 0.33 0.32 0.33 0.34 0.35 0.35 0.35
France 0.50 0.50 0.51 0.51 0.51 0.50 0.50 0.50
Greece 0.38 0.36 0.30 0.31 0.32 0.33 0.33 0.33
Hungary 0.35 0.36 0.40 0.40 0.40 0.40 0.39 0.40
Ireland 0.38 0.37 0.38 0.37 0.37 0.37 0.39 0.38
Israel 0.41 0.41 0.41 0.42 0.41 0.42 0.43 0.44
Japan 0.33 0.33 0.33 0.34 0.34 0.34 0.35 0.36
Korea 0.36 0.36 0.34 0.35 0.36 0.38 0.40 0.41
Luxembourg 0.46 0.47 0.47 0.47 0.45 0.45 0.44 0.43
Mexico 0.27 0.27 0.27 0.27 0.29 0.29 0.29 0.31
Netherlands 0.41 0.40 0.40 0.40 0.39 0.39 0.39 0.39
New Zealand 0.51 0.50 0.51 0.51 0.51 0.51 0.51 0.52
Poland 0.37 0.37 0.39 0.40 0.41 0.41 0.43 0.44
Portugal 0.36 0.36 0.36 0.36 0.39 0.40 0.42 0.43
Slovakia 0.37 0.36 0.36 0.36 0.37 0.37 0.38 0.38
Slovenia 0.48 0.49 0.50 0.52 0.49 0.49 0.48 0.48
Spain 0.32 0.32 0.32 0.32 0.31 0.31 0.31 0.34
Turkey 0.39 0.39 0.40 0.40 0.39 0.40 0.42 0.42
United Kingdom 0.38 0.38 0.39 0.39 0.40 0.41 0.41 0.44
United States 0.28 0.28 0.27 0.27 0.27 0.25 0.25 0.24
Latvia 0.38 0.41 0.39 0.37 0.39 0.41 0.41 0.39
Lithuania 0.40 0.39 0.38 0.44 0.41 0.40 0.45 0.43
Romania 0.32 0.33 0.33 0.35 0.38 0.40 0.41 0.44
Countries by minimum wage to median wage ratio[208]
Belgium 0.5 0.51 0.51 0.51 0.49 0.49 0.49 0.47
Czech Republic 0.38 0.37 0.36 0.37 0.37 0.39 0.4 0.41
Estonia 0.4 0.39 0.38 0.4 0.4 0.41 0.41 0.41
Japan 0.37 0.38 0.38 0.39 0.39 0.4 0.4 0.42
Korea 0.45 0.45 0.43 0.44 0.46 0.49 0.5 0.53
Mexico 0.35 0.36 0.36 0.37 0.37 0.37 0.37 0.4
New Zealand 0.59 0.59 0.59 0.59 0.6 0.6 0.61 0.6
Poland 0.45 0.45 0.48 0.5 0.51 0.51 0.53 0.54
Slovenia 0.59 0.61 0.62 0.64 0.6 0.6 0.59 0.58
Spain 0.38 0.38 0.38 0.38 0.37 0.37 0.37 0.4
Turkey 0.7 0.71 0.73 0.72 0.69 0.7 0.74 0.74
Lithuania 0.5 0.48 0.48 0.55 0.51 0.5 0.56 0.54
Romania 0.43 0.45 0.45 0.48 0.51 0.55 0.56 0.6
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Burkhauser, R. V. (2014). Why minimum wage increases are a poor way to help the working poor (No. 86). IZA Policy Paper, Institute for the Study of Labor (IZA).
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Uri buys two burgers and a soda for $\$2.10$, and Gen buys a burger and two sodas for $\$2.40$. How many cents does a soda cost?
Let's work with this problem in cents, not dollars, because the answer calls for a number in cents. So, Uri's two burgers and a soda cost 210 cents and Gen's food costs 240 cents. Let a burger cost $b$ cents and a soda cost $s$ cents. We are trying to find the value of $s$. We can set up a system of two equations to represent the given information. These equations are:
\begin{align*}
2b + s &= 210 \\
b + 2s &= 240 \\
\end{align*}
We are solving for $s$, so we want to eliminate $b$ from the equations above. Multiplying both sides of the second equation by 2, we get $2b+4s = 480$, or $2b = 480 - 4s$. Substituting this equation into the first equation above to eliminate $b$, we get that $(480 - 4s) + s = 210$, or $s=90$. Thus, a soda costs $\boxed{90}$ cents. | Math Dataset |
\begin{document}
\title[Sums of four prime cubes]{Sums of four prime cubes in short intervals} \date{} \author{Alessandro Languasco \lowercase{and} Alessandro Zaccagnini}
\subjclass[2010]{Primary 11P32; Secondary 11P55, 11P05} \keywords{Waring-Goldbach problem, Hardy-Littlewood method}
\begin{abstract} We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known. \end{abstract}
\maketitle
\section{Introduction} Let $N$ be a sufficiently large integer and $1\le H \le N$ an integer.
Let \begin{equation} \label{r-def} \sum_{n=N+1}^{N+H} r (n), \quad \textrm{where} \quad r (n) = \sum_{n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}} \!\!\!\!\!\!\!\! \log p_{1} \log p_{2} \log p_{3}\log p_{4}, \end{equation} be a suitable short interval average of the number of representation of an integer as a sum of four prime cubes. The problem of representing integers as sum of prime cubes is quite an old one; we recall that Hua \cite{Hua1938}-\cite{Hua1965} stated that almost all positive integers satisfying some necessary congruence conditions are the sum of five prime cubes and that Daveport \cite{Davenport1939} proved that almost all positive integers are the sum of four positive cubes. More recent results on the positive proportions of integers that are the sum of four prime cubes were obtained by Roth \cite{Roth1951}, Ren \cite{Ren2003} and Liu \cite{Liu2012}. In fact, see Br\"udern \cite{Brudern1995}, it is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four prime cubes.
Here we prove that
\begin{Theorem} \label{thm-uncond}
Let $N\ge 2$, $1\le H \le N$ be integers.
Then, for every $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that
\begin{align*} \sum_{n=N+1}^{N+H} & r (n) = \Gamma \Bigl(\frac{4}{3}\Bigr)^3 H N^{1/3} + \Odig{ H N^{1/3} \exp \Bigl( -C \Bigl( \frac{ \log N}{\log \log N} \Bigr)^{1/3} \Bigr) }
\quad \text{as} \ N \to \infty, \end{align*} uniformly for $N^{13/18 +\varepsilon}\le H \le N^{1-\varepsilon}$, where $\Gamma$ is Euler's function. \end{Theorem}
This should be compared with a recent result about the positive proportion of such integers in short intervals by Liu-Zhao \cite{LiuZ2017} which holds for $H=N^{17/18}$. As an immediate consequence of Theorem \ref{thm-uncond} we can say that, for $N$ sufficiently large, every interval of size larger than $N^{13/18 +\varepsilon}$ contains the expected amount of integers which are a sum of four prime cubes. We remark that this level is essentially optimal given the known density estimates for the non trivial zeroes of the Riemann zeta function.
Assuming the Riemann Hypothesis (RH) holds we can further improve the size of $H$.
\begin{Theorem} \label{thm-RH}
Let $\varepsilon>0$, $N\ge 2$, $1\le H \le N$ be integers and
assume the Riemann Hypothesis (RH) holds.
Then there exists a constant $B>3/2$ such that
\begin{align*} \sum_{n=N+1}^{N+H} & r (n) = \Gamma \Bigl(\frac{4}{3}\Bigr)^3 H N^{1/3} + \Odig{ \frac{H^2}{N^{2/3}} + H^{3/4}N^{5/12+\varepsilon} + H^{1/2} N^{2/3}(\log N)^B
+N(\log N)^3} \end{align*}
as $N \to \infty$, uniformly for $\infty(N^{2/3}L^{2B}) \le H \le \odi{N}$, where $f=\infty(g)$ means $g=\odi{f}$ and $\Gamma$ is Euler's function. \end{Theorem}
As an immediate consequence of Theorem \ref{thm-RH} we can say that, for $N$ sufficiently large, every interval of size larger than $N^{2/3+\varepsilon}$ contains the expected amount of integers which are a sum of four prime cubes. We remark that this level is essentially optimal given the spacing of the cubic sequence.
In both the proofs of Theorems \ref{thm-uncond}-\ref{thm-RH} we will use the original Hardy-Littlewood generating functions to exploit the easier main term treatment they allow (comparing with the one which would follow using Lemmas 2.3 and 2.9 of Vaughan \cite{Vaughan1997}). \begin{comment} \footnote{\`E sempre cruciale usare la stima di Hua della potenza ottava con i logaritmi (Lemma \ref{Hua-lemma-series}) che si sa per le somme finite e non per le serie; per mezzo di un taglio della somma a $(NL)^{1/3}$, la formula di somma per parti per trattare $e^{-n^3/N}$ e poi la disuguaglianza di H\"older,
si ottiene la stima di Hua (potenza ottava) con i logs anche per le serie. Notare che comunque tutta la dimostrazione del caso incondizionale pu\`o essere rifatta con le somme finite. In questo caso il trattamento del main term \`e molto pi\`u complicato.} \end{comment}
\section{Setting} Let $e(\alpha) = e^{2\pi i\alpha}$, $\alpha \in [-1/2,1/2]$, $L=\log N$, $z= 1/N-2\pi i\alpha$,
\begin{equation*}
\widetilde{S}_\ell(\alpha) = \sum_{n=1}^{\infty} \Lambda(n) e^{-n^{\ell}/N} e(n^{\ell}\alpha) \quad \textrm{and} \quad \widetilde{V}_\ell(\alpha) = \sum_{p=2}^{\infty} \log p \, e^{-p^{\ell}/N} e(p^{\ell}\alpha). \end{equation*}
We remark that
\begin{equation} \label{z-estim} \vert z\vert^{-1} \ll \min \bigl(N, \vert \alpha \vert^{-1}\bigr). \end{equation}
We further set
\begin{equation} \notag
U(\alpha,H)
=
\sum_{m=1}^H e(m \alpha) \end{equation}
and, moreover, we also have the usual numerically explicit inequality
\begin{equation} \label{UH-estim} \vert U(\alpha,H) \vert \le \min \bigl(H; \vert \alpha \vert^{-1}\bigr), \end{equation}
see, \emph{e.g.}, on page 39 of Montgomery \cite{Montgomery1994}.
We list now the needed preliminary results.
\begin{Lemma}[Lemma 3 of \cite{LanguascoZ2016b}] \label{tilde-trivial-lemma}
Let $\ell\ge 1$ be an integer. Then
\( \vert \widetilde{S}_{\ell}(\alpha)- \widetilde{V}_{\ell}(\alpha) \vert \ll_{\ell}
N^{1/(2\ell)} . \)
\end{Lemma}
\begin{Lemma} \label{Linnik-lemma} Let $\ell \ge 1$ be an integer, $N \ge 2$ and $\alpha\in [-1/2,1/2]$. Then
\begin{equation*}
\widetilde{S}_{\ell}(\alpha) = \frac{\Gamma(1/\ell)}{\ell z^{1/\ell}} - \frac{1}{\ell}\sum_{\rho}z^{-\rho/\ell}\Gamma (\rho/\ell ) + \Odip{\ell}{1}, \end{equation*}
where $\rho=\beta+i\gamma$ runs over the non-trivial zeros of $\zeta(s)$.
\end{Lemma} \begin{Proof} It follows the line of Lemma 2 of \cite{LanguascoZ2016a}; we just correct an oversight in its proof. In eq. (5) on page 48 of \cite{LanguascoZ2016a} the term \( -
\sum_{m=1}^{\ell \sqrt{3}/4} \Gamma (- 2m/\ell ) z^{2m/\ell} \) is missing. Its estimate is trivially $\ll_{\ell} \vert z \vert^{\sqrt{3}/2} \ll_{\ell} 1$. Hence such an oversight does not affect the final result of Lemma 2 of \cite{LanguascoZ2016a}. \end{Proof}
\begin{Lemma} [Lemma 4 of \cite{LanguascoZ2016a}]
\label{Laplace-formula} Let $N$ be a positive integer and $\mu > 0$. Then \[
\int_{-1 / 2}^{1 / 2} z^{-\mu} e(-n \alpha) \, \mathrm{d} \alpha
=
e^{- n / N} \frac{n^{\mu - 1}}{\Gamma(\mu)}
+
\Odipg{\mu}{\frac{1}{n}}, \] uniformly for $n \ge 1$. \end{Lemma}
\begin{Lemma}
\label{LP-Lemma-gen} Let $\varepsilon$ be an arbitrarily small positive constant, $\ell \ge 1$ be an integer, $N$ be a sufficiently large integer and $L= \log N$. Then there exists a positive constant $c_1 = c_{1}(\varepsilon)$, which does not depend on $\ell$, such that
\[ \int_{-\xi}^{\xi} \, \Bigl\vert \widetilde{S}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{1/\ell}} \Bigr\vert^{2} \mathrm{d} \alpha \ll_{\ell}
N^{2/\ell-1} \exp \Big( - c_{1} \Big( \frac{L}{\log L} \Big)^{1/3} \Big) \]
uniformly for $ 0\le \xi < N^{-1 +5/(6\ell) - \varepsilon}$. Assuming RH we get
\[ \int_{-\xi}^{\xi} \, \Bigl\vert \widetilde{S}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{1/\ell}} \Bigr\vert^{2} \mathrm{d} \alpha \ll_{\ell} N^{1/\ell}\xi L^{2} \] uniformly for $0 \le \xi \le 1/2$. \end{Lemma} \begin{Proof} It follows the line of Lemma 3 of \cite{LanguascoZ2016a} and Lemma 1 of \cite{LanguascoZ2016b}; we just correct an oversight in their proofs which is based on Lemma \ref{Linnik-lemma} above. Both eq. (8) on page 49 of \cite{LanguascoZ2016a} and eq. (6) on page 423 of \cite{LanguascoZ2016b} should read as \[ \int_{1/N}^{\xi} \Big \vert\sum_{\rho\colon \gamma > 0}z^{-\rho/\ell}\Gamma (\rho/\ell ) \Big \vert^2 \mathrm{d} \alpha \le \sum_{k=1}^K \int_\eta^{2\eta} \Big \vert\sum_{\rho\colon \gamma > 0}z^{-\rho/\ell}\Gamma (\rho/\ell ) \Big \vert^2 \mathrm{d} \alpha, \] where $\eta=\eta_k= \xi/2^k$, $1/N\le \eta \le \xi/2$ and $K$ is a suitable integer satisfying $K=\Odi{L}$. The remaining part of the proofs are left untouched. Hence such oversights do not affect the final result of Lemma 3 of \cite{LanguascoZ2016a} and Lemma 1 of \cite{LanguascoZ2016b}. \end{Proof}
\begin{comment}
\begin{Lemma} \label{zac-lemma-series}
Let $\ell\ge 2 $ be an integer and $0<\xi\le 1/2$. Then
\[ \int_{-\xi}^{\xi}
|\widetilde{S}_{\ell}(\alpha)|^2 \ \mathrm{d}\alpha \ll_{\ell} \xi N^{1/\ell} L + \begin{cases} L^{2} & \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2 \end{cases} \ \text{and} \ \int_{-\xi}^{\xi}
|\widetilde{V}_{\ell}(\alpha)|^2 \ \mathrm{d}\alpha \ll_{\ell} \xi N^{1/\ell} L + \begin{cases} L^{2} & \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2. \end{cases} \]
\end{Lemma} \begin{Proof} The first part was proved in Lemma 2 of \cite{LanguascoZ2016b}. For the second part we argue analogously. We use Corollary 2 of Montgomery-Vaughan \cite{MontgomeryV1974} with $T=\xi$,
$a_r=\log(r) \exp(-r^\ell/N)$ if $r$ is prime, $a_r= 0$ otherwise
and $\lambda_r= 2\pi r^\ell$. By the Prime Number Theorem we get
\begin{align*} \int_{0}^{\xi} \vert \widetilde{V}_{\ell}(\alpha) \vert^2\, \mathrm{d} \alpha &= \sum_{p} \log^2 (p) e^{-2p^\ell/N} \bigl(\xi +\Odim{\delta_p^{-1}}\Bigr) \ll_\ell \xi N^{1/\ell} L + \sum_{p} \log^2 (p) p^{1-\ell} e^{-2p^\ell/N} \end{align*}
since $\delta_{r} = \lambda_r - \lambda_{r-1} \gg_{\ell} r^{\ell-1}$. The last term is $\ll_{\ell}1$ if $\ell >2$ and $\ll L^{2}$ otherwise. The second part of Lemma \ref{zac-lemma-series} follows. \end{Proof} \end{comment}
In the unconditional case a crucial role is played by \begin{Lemma}[Hua] \label{Hua-lemma-series}
Let $N$ be sufficiently large, $\ell,k$ integers, $\ell \ge 1$, $1\le k \le \ell$. There exists a suitable positive constant $A=A(\ell,k)$ such that
\[ \int_{-1/2}^{1/2} \vert \widetilde{S}_{\ell}(\alpha)\vert ^{2^k} \ \mathrm{d}\alpha \ll_{k,\ell}
N^{(2^k-k)/\ell} L^A
\quad
\text{and}
\quad
\int_{-1/2}^{1/2} \vert \widetilde{V}_{\ell}(\alpha)\vert^{2^k} \ \mathrm{d}\alpha \ll_{k,\ell}
N^{(2^k-k)/\ell} L^A.
\]
\end{Lemma}
\begin{Proof}
We just prove the first part since the second one follows immediately by remarking that
the primes are supported on a thinner set than the prime powers. Let $P=(2NL/\ell)^{1/\ell}$. A direct estimate gives $\widetilde{S}_{\ell}(\alpha)= \sum_{n\le P} \Lambda(n) e^{-n^\ell/N} e(n^\ell\alpha) + \Odipm{\ell}{L^{1/\ell}}$.
Recalling that the Prime Number Theorem implies $S_{\ell}(\alpha;t) := \sum_{n\le t} \Lambda(n) e(n^{\ell}\alpha) \ll t$, a partial integration argument gives \[ \sum_{n\le P} \Lambda(n) e^{-n^\ell/N} e(n^\ell\alpha) = -\frac{\ell}{N} \int_1^P t^{\ell-1} e^{-t^\ell/N} S_{\ell}(\alpha;t) \ \mathrm{d} t + \Odipm{\ell}{L^{1/\ell}}. \]
Using the inequality $(\vert a\vert + \vert b \vert)^{2^k} \ll_k \vert a\vert^{2^k} + \vert b \vert^{2^k}$, H\"older's inequality and interchanging the integrals, we get that
\begin{align*}
\int_{-1/2}^{1/2} &\vert \widetilde{S}_{\ell}(\alpha)\vert ^{2^k} \ \mathrm{d}\alpha \ll_{k,\ell}
\int_{-1/2}^{1/2} \Bigl\vert \frac{1}{N} \int_1^P t^{\ell-1} e^{-t^\ell/N} S_{\ell}(\alpha;t)\ \mathrm{d} t \Bigr\vert ^{2^k} \ \mathrm{d}\alpha + L^{2^k/\ell} \\ &\ll_{k,\ell} \frac{1}{N^{2^k}} \Bigl( \int_1^P t^{\ell-1} e^{-t^\ell/N} \ \mathrm{d} t \Bigr)^{2^k-1} \Bigl( \int_1^P t^{\ell-1} e^{-t^\ell/N} \int_{-1/2}^{1/2} \vert S_{\ell}(\alpha;t)\vert ^{2^k} \mathrm{d}\alpha \ \mathrm{d} t \Bigr) + L^{2^k/\ell}. \end{align*}
Theorem 4 of Hua \cite{Hua1965} implies, remarking that the von Mangoldt function is supported on a thinner set than the integers and inserts a logarithmic weight whose total contribution can be inserted in the power of $L$, that there exists a positive constant $B=B(\ell,k)$ such that $\int_{-1/2}^{1/2} \vert S_{\ell}(\alpha;t)\vert^{2^k}\ \mathrm{d}\alpha \ll_{k,\ell} t^{2^k-k} (\log t)^B$. Using such an estimate and remarking that $ \int_1^P t^{\ell-1} e^{-t^\ell/N} \ \mathrm{d} t \ll_\ell N$, we obtain that
\begin{align*}
\int_{-1/2}^{1/2} \vert \widetilde{S}_{\ell}(\alpha)\vert ^{2^k} \ \mathrm{d}\alpha &\ \ll_{k,\ell} \frac{1}{N}
\int_1^P t^{\ell-1+2^k-k} e^{-t^\ell/N} (\log t)^B \ \mathrm{d} t + L^{2^k/\ell}
\ll_{k,\ell} N^{(2^k-k)/\ell} L^{B+(2^k-k)/\ell} \end{align*}
by a direct computation. This proves the first part of the lemma. \end{Proof}
In fact the argument used in the proof of Lemma \ref{Hua-lemma-series} can be used to derive other estimates on $\widetilde{S}_{\ell}(\alpha)$ from the ones on $S_{\ell}(\alpha;t)$. Another instance of this fact is the following lemma about the truncated fourth-mean average of $ \widetilde{S}_{\ell}(\alpha)$ which is based on a result by Robert-Sargos \cite{RobertS2006}.
\begin{Lemma} \label{S-ell_quarta_tau} Let $N\in \mathbb{N}$, $\varepsilon>0$, $\ell>1$ and $\tau>0$. Then we have
\[ \int_{-\tau}^{\tau}\vert \widetilde{S}_{\ell}(\alpha)\vert ^4\, \mathrm{d} \alpha \ll \bigl(\tau N^{2/\ell}+N^{4/\ell-1}\bigr)N^{\varepsilon}
\quad
\text{and}
\quad \int_{-\tau}^{\tau}\vert \widetilde{V}_{\ell}(\alpha)\vert ^4\, \mathrm{d} \alpha \ll \bigl(\tau N^{2/\ell}+N^{4/\ell-1}\bigr)N^{\varepsilon}. \] \end{Lemma} \begin{Proof} We can argue as in the proof of Lemma \ref{Hua-lemma-series} using Lemma 4 of \cite{GambiniLZ2018} on $S_{\ell}(\alpha;t) = \sum_{n\le t} \Lambda(n) e(n^\ell\alpha)$ instead of Theorem 4 of Hua \cite{Hua1965}. \end{Proof}
The last lemma is a consequence of Lemma \ref{S-ell_quarta_tau}.
\begin{Lemma} \label{S-ell_quarta_coda} Let $N\in \mathbb{N}$, $\varepsilon>0$, $\ell>2$, $c\ge 1$ and $N^{-c}\le \omega \le N^{2/\ell-1}$. Then we have
\[ \Bigl(\int_{-1/2}^{-\omega} +\int_{\omega}^{1/2} \Bigr) \vert \widetilde{S}_{\ell}(\alpha)\vert ^4 \frac{\mathrm{d} \alpha}{\vert \alpha\vert} \ll \frac{ N^{4/\ell-1+\varepsilon}}{\omega}
\quad
\text{and}
\quad \Bigl(\int_{-1/2}^{-\omega} +\int_{\omega}^{1/2} \Bigr) \vert \widetilde{V}_{\ell}(\alpha)\vert ^4 \frac{\mathrm{d} \alpha}{\vert \alpha\vert} \ll \frac{ N^{4/\ell-1+\varepsilon}}{\omega}. \] \end{Lemma} \begin{Proof} By partial integration and Lemma \ref{S-ell_quarta_tau} we get that \begin{align*} \int_{\omega}^{1/2} \vert \widetilde{S}_{\ell}(\alpha)\vert ^4 \frac{\mathrm{d} \alpha}{ \alpha} &\ll \frac{1}{\omega} \int_{-\omega}^{\omega} \vert \widetilde{S}_{\ell}(\alpha) \vert^{4}\ \mathrm{d} \alpha + \int_{-1/2}^{1/2} \vert \widetilde{S}_{\ell}(\alpha) \vert^{4}\ \mathrm{d} \alpha + \int_{\omega}^{1/2} \Bigl( \int_{-\xi}^{\xi} \vert \widetilde{S}_{\ell}(\alpha) \vert^{4}\ \mathrm{d} \alpha \Bigr) \frac{\mathrm{d} \xi}{\xi^2} \\& \ll \frac{1}{\omega}\bigl(\omega N^{2/\ell}+N^{4/\ell-1}\bigr)N^{\varepsilon} +N^{2/\ell+\varepsilon} + N^{\varepsilon} \int_{\omega}^{1/2} \frac{\xi N^{2/\ell}+N^{4/\ell-1}}{\xi^2}\mathrm{d} \xi \\& \ll N^{2/\ell+\varepsilon} \vert \log (2\omega) \vert + \frac{ N^{4/\ell-1+\varepsilon}}{\omega} \ll \frac{ N^{4/\ell-1+\varepsilon}}{\omega} \end{align*} since $N^{-c}\le \omega \le N^{2/\ell-1}$. A similar computation proves the result in $[-1/2,-\omega]$ too. The estimate on $\widetilde{V}_{\ell}(\alpha)$ can be obtained analogously. \end{Proof}
\section{The unconditional case}
\label{unconditional}
Let $H>2B$, where \begin{equation} \label{B-def} B= N^{2\varepsilon}. \end{equation}
Letting $I(B,H):=[-1/2,-B/H]\cup [B/H, 1/2]$, and recalling \eqref{r-def}, we have
\begin{align}
\notag \sum_{n=N+1}^{N+H} e^{-n/N} r(n) & = \int_{-1/2}^{1/2}\widetilde{V}_{3}(\alpha)^4 U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
\\
\notag
&
= \int_{-B/H}^{B/H} \widetilde{S}_{3}(\alpha)^4
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
+ \int\limits_{I(B,H)} \widetilde{S}_{3}(\alpha)^4
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
\\
\label{main-dissection-series}
& \hskip1cm
+
\int_{-1/2}^{1/2} \bigl( \widetilde{V}_{3}(\alpha)^4 -\widetilde{S}_{3}(\alpha)^4 \bigr)
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
=
I_1+I_2+I_3,
\end{align}
say.
Now we evaluate these terms.
\subsection{Estimation of $I_2$}
Using \eqref{UH-estim} and Lemma \ref{S-ell_quarta_coda} with $\omega = B/H$ and $\ell=3$, we obtain \begin{equation} \label{I2-estim} I_2 \ll \int_{B/H}^{1/2} \vert \widetilde{S}_{3}(\alpha) \vert^{4} \frac{\mathrm{d} \alpha}{\alpha}
\ll \frac{H }{B } N^{1/3+\varepsilon}, \end{equation}
provided that $H\gg N^{1/3}B$.
\subsection{Estimation of $I_3$} \label{I3-estim} Clearly \begin{align*} \vert \widetilde{V}_{3}(\alpha)^4 -\widetilde{S}_{3}(\alpha)^4 \vert &= \vert\widetilde{V}_{3}(\alpha) -\widetilde{S}_{3}(\alpha) \vert
\vert \widetilde{V}_{3}(\alpha)^3+\widetilde{V}_{3}(\alpha)^2\widetilde{S}_{3}(\alpha) +\widetilde{V}_{3}(\alpha) \widetilde{S}_{3}(\alpha)^2+\widetilde{S}_{3}(\alpha)^3\vert \\ & \ll
\vert\widetilde{V}_{3}(\alpha) -\widetilde{S}_{3}(\alpha) \vert
( \vert \widetilde{V}_{3}(\alpha) \vert + \vert \widetilde{S}_{3}(\alpha) \vert ) ^3 \\ & \ll
\vert\widetilde{V}_{3}(\alpha) -\widetilde{S}_{3}(\alpha) \vert
\max ( \vert \widetilde{V}_{3}(\alpha)\vert ^3 ; \vert \widetilde{S}_{3}(\alpha)\vert^3 ). \end{align*}
Hence by Lemma \ref{tilde-trivial-lemma} we have
\begin{equation}
\label{I3-estim-1} I_3 \ll N^{1/6} \int_{-1/2}^{1/2} \bigl( \vert \widetilde{V}_{3}(\alpha)\vert^3 + \vert \widetilde{S}_{3}(\alpha)\vert^3 \bigr) \vert U(-\alpha,H) \vert \, \mathrm{d} \alpha = N^{1/6}(K_1+K_2), \end{equation} say.
By \eqref{UH-estim} we get \begin{equation} \label{K2-split} K_2 \ll H \int_{-1/H}^{1/H}
\vert \widetilde{S}_{3}(\alpha)\vert^3 \, \mathrm{d} \alpha
+
\Bigl( \int_{-1/2}^{-1/H} + \int_{1/H}^{1/2} \Bigr)
\vert \widetilde{S}_{3}(\alpha)\vert^3 \frac{\mathrm{d} \alpha}{\vert \alpha\vert}
= K_{2,1}+K_{2,2},
\end{equation}
say. Using the H\"older inequality and Lemma \ref{S-ell_quarta_tau} with $\tau = 1/H$ and $\ell=3$ we get \begin{equation} \label{K21-estim} K_{2,1} \ll H^{3/4}
\Bigl( \int_{-1/H}^{1/H}
\vert \widetilde{S}_{3}(\alpha)\vert^4 \, \mathrm{d} \alpha
\Bigr)^{3/4} \ll H^{3/4}N^{1/4+\varepsilon}, \end{equation}
provided that $H\gg N^{1/3}$.
Using the H\"older inequality and Lemma \ref{S-ell_quarta_coda} with $\omega = 1/H$ and $\ell=3$ we get \begin{equation}
\label{K22-estim} K_{2,2} \ll
\Bigl( \int_{1/H}^{1/2}
\vert \widetilde{S}_{3}(\alpha)\vert^4 \frac{\mathrm{d} \alpha}{\alpha}
\Bigr)^{3/4}
\Bigl( \int_{1/H}^{1/2} \frac{\mathrm{d} \alpha}{\alpha}
\Bigr)^{1/4}
\ll
\bigl( H N^{1/3+\varepsilon}
\bigr)^{3/4}
L^{1/4}
\ll H^{3/4}N^{1/4+\varepsilon}, \end{equation}
provided that $H\gg N^{1/3}$. Combining \eqref{K2-split}-\eqref{K22-estim} we obtain \begin{equation} \label{K2-estim} K_2 \ll H^{3/4}N^{1/4+\varepsilon}, \end{equation} provided that $H\gg N^{1/3}$.
An analogous computation gives
\begin{equation} \label{K1-estim} K_1 \ll H^{3/4}N^{1/4+\varepsilon}, \end{equation}
and, by \eqref{I3-estim-1} and \eqref{K2-estim}-\eqref{K1-estim}, we can finally write
\begin{equation}
\label{I3-estim-series} I_3 \ll H^{3/4}N^{5/12+\varepsilon}, \end{equation}
provided that $H\gg N^{1/3}$.
\subsection{Evaluation of $I_1$}
Since $\Gamma(4/3)= (1/3)\Gamma(1/3)$,
we have that
\begin{align} \notag I_1 &= \int_{-B/H}^{B/H} \frac{\Gamma(4/3)^4}{z^{4/3}}
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
+ \int_{-B/H}^{B/H} \Bigl(\widetilde{S}_{3}(\alpha)^4 - \frac{\Gamma(4/3)^4}{z^{4/3}} \Bigr)
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
\\&
\label{I1-split}
= J_1+J_2,
\end{align} say. By \eqref{z-estim}-\eqref{UH-estim} and Lemma \ref{Laplace-formula}, a direct calculation gives
\begin{align} \notag J_1 & = \Gamma\Bigl(\frac{4}{3}\Bigr)^3\ \sum_{n=N+1}^{N+H} e^{-n/N} n^{1/3} +\Odig{\frac{H}{N}} + \Odig{\int_{B/H}^{1/2} \frac{\mathrm{d} \alpha}{\alpha^{7/3}}} \\ \notag & = \frac{\Gamma(4/3)^3}{e}\ \sum_{n=N+1}^{N+H} n^{1/3} +\Odig{\frac{H}{N}+\frac{H^2}{N^{2/3}} + \frac{H^{4/3}}{B^{4/3}}} \\ \label{J1-eval-series} & = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e} +\Odig{ \frac{H^{4/3}}{B^{4/3}}+N^{1/3}}.
\end{align}
From now on, we denote \( \widetilde{E}_{3}(\alpha) : =\widetilde{S}_3(\alpha) - \frac{\Gamma(4/3)}{z^{1/3}}. \)
By $f^2 - g^2 = 2g (f-g) + (f-g)^2$, \eqref{z-estim} and $\widetilde{S}_{3}(\alpha) \ll N^{1/3}$ we get
\begin{align} \notag \widetilde{S}_{3}(\alpha)^4 - \frac{\Gamma(4/3)^4}{z^{4/3}} &= \Bigl( \widetilde{S}_{3}(\alpha) ^2 + \frac{\Gamma(4/3)^2}{z^{2/3}} \Bigr) \Bigl( \widetilde{S}_{3}(\alpha) ^2 - \frac{\Gamma(4/3)^2}{z^{2/3}} \Bigr) \\ \notag &= \Bigl( \widetilde{S}_{3}(\alpha) ^2 + \frac{\Gamma(4/3)^2}{z^{2/3}} \Bigr) \Bigl( 2\frac{\Gamma(4/3)}{z^{1/3}} \widetilde{E}_{3}(\alpha) +
\widetilde{E}_{3}(\alpha) ^2 \Bigr) \\ \label{S-tilde-uncond-approx} & \ll \vert \widetilde{S}_{3}(\alpha) \vert^2 \frac{\vert \widetilde{E}_{3}(\alpha) \vert}{\vert z \vert^{1/3}} + \frac{\vert \widetilde{E}_{3}(\alpha) \vert}{\vert z \vert} + N^{2/3}\vert \widetilde{E}_{3}(\alpha) \vert^2. \end{align}
Using \eqref{S-tilde-uncond-approx} and \eqref{z-estim} we get
\begin{align}
\notag J_2 & \ll H \int_{-B/H}^{B/H} \vert \widetilde{S}_{3}(\alpha) \vert^2 \frac{\vert \widetilde{E}_{3}(\alpha) \vert}{\vert z \vert^{1/3}} \, \mathrm{d} \alpha + H \int_{-B/H}^{B/H} \frac{\vert \widetilde{E}_{3}(\alpha) \vert}{\vert z \vert} \, \mathrm{d} \alpha + HN^{2/3} \int_{-B/H}^{B/H} \vert \widetilde{E}_{3}(\alpha) \vert^2 \, \mathrm{d} \alpha \\ \label{J2-uncond-split}& = H( E_1+ E_2 +N^{2/3}E_3 ),
\end{align}
say.
By \eqref{UH-estim} and Lemma \ref{LP-Lemma-gen} we obtain
that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that
\begin{equation}
\label{E3-estim-series} E_3 \ll
N^{-1/3} \exp \Big( - c_{1} \Big( \frac{L}{\log L} \Big)^{1/3} \Big) \end{equation} provided that $B/H \le N^{-13/18 - \varepsilon}$, \emph{i.e.}, $H \ge BN^{ 13/18+ \varepsilon}$.
By the Cauchy-Schwarz inequality, \eqref{z-estim}
and \eqref{E3-estim-series} we obtain
that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that
\begin{equation} \label{E2-estim-series} E_2
\ll
E_3^{1/2} \Bigl(\int_{-B/H}^{B/H} \frac{\mathrm{d} \alpha}{\vert z \vert^2} \Bigr)^{1/2}
\ll
E_3^{1/2} N^{1/2}
\ll
N^{1/3} \exp \Big( - \frac{c_{1}}{2} \Big( \frac{L}{\log L} \Big)^{1/3} \Big), \end{equation}
provided that $H \ge BN^{ 13/18+ \varepsilon}$.
By using twice the Cauchy-Schwarz inequality, Lemma \ref{Hua-lemma-series},
\eqref{z-estim}
and \eqref{E3-estim-series} we obtain
that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that
\begin{align} \notag E_1 & \ll
E_3^{1/2} \Bigl(\int_{-B/H}^{B/H} \frac{\vert \widetilde{S}_{3}(\alpha) \vert^4}{\vert z \vert^{2/3}} \ \mathrm{d} \alpha \Bigr)^{1/2} \ll
E_3^{1/2} \Bigl(\int_{-1/2}^{1/2} \vert \widetilde{S}_{3}(\alpha) \vert^8 \ \mathrm{d} \alpha \Bigr)^{1/4} \Bigl(\int_{-B/H}^{B/H} \frac{\mathrm{d} \alpha}{\vert z \vert^{4/3}} \Bigr)^{1/4} \\& \label{E1-estim-series}
\ll
E_3^{1/2} N^{1/2}L^{A/4}
\ll
N^{1/3} \exp \Big( - \frac{c_{1}}{4} \Big( \frac{L}{\log L} \Big)^{1/3} \Big), \end{align}
provided that $H \ge BN^{ 13/18+ \varepsilon}$. Hence by \eqref{J2-uncond-split}-\eqref{E1-estim-series} we finally can write that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that
\begin{equation} \label{J2-estim-series} J_2 \ll H N^{1/3} \exp \Big( - \frac{c_{1}}{4} \Big( \frac{L}{\log L} \Big)^{1/3} \Big) ,
\end{equation}
provided that
$H \ge BN^{ 13/18+ \varepsilon}$.
Summing up, by \eqref{I1-split}-\eqref{J1-eval-series} and \eqref{J2-estim-series} we have that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that \begin{equation}
\label{I1-final-eval} I_1 = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e} +\Odig{ H N^{1/3} \exp \Big( - \frac{c_{1}}{4} \Big( \frac{L}{\log L} \Big)^{1/3} \Big) } \end{equation}
provided that
$H \ge BN^{ 13/18+ \varepsilon}$. \subsection{Final words}
Summing up, by \eqref{main-dissection-series}-\eqref{I2-estim}, \eqref{I3-estim-series} and \eqref{I1-final-eval} we have that, for every $\varepsilon>0$, there exists $c_1=c_1(\varepsilon)>0$ such that
\begin{align*} \sum_{n=N+1}^{N+H} e^{-n/N} r(n) &= \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e}
+\Odig{H N^{1/3} \exp \Big( - \frac{c_{1}}{4} \Big( \frac{L}{\log L} \Big)^{1/3} \Big) + \frac{H }{B } N^{1/3+\varepsilon}} \end{align*}
provided that $H \ge BN^{ 13/18+ \varepsilon}$. The second error term is dominated by the first one since $B=N^{2\varepsilon}$ by \eqref{B-def}. Hence we can write that, for every $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that
\begin{equation} \label{almost-done-2} \sum_{n=N+1}^{N+H} e^{-n/N} r(n) = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e} +\Odig{H N^{1/3} \exp \Big( - C\Big( \frac{L}{\log L} \Big)^{1/3} \Big)} \end{equation} provided that $H \ge N^{13/18+ 3\varepsilon}$.
{}From $e^{-n/N}=e^{-1}+ \Odi{H/N}$ for $n\in[N+1,N+H]$, $1\le H \le N$, we get that, for every $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that
\begin{align*}
\sum_{n = N+1}^{N + H} r(n) &= \Gamma\Bigl(\frac{4}{3}\Bigr)^3 HN^{1/3} + \Odig{H N^{1/3} \exp \Big( - C\Big( \frac{L}{\log L} \Big)^{1/3} \Big)} +
\Odig{\frac{H}{N}\sum_{n = N+1}^{N + H} r(n) } \end{align*}
provided that $H \ge N^{13/18+ 3\varepsilon}$ and $H \le N$.
Using $e^{n/N}\le e^{2}$ and \eqref{almost-done-2},
the last error term is $\ll H^2N^{-2/3}$. Hence we get that, for every $\varepsilon>0$, there exists $C=C(\varepsilon)>0$ such that
\begin{equation*}
\sum_{n = N+1}^{N + H} r(n) = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 HN^{1/3} +\Odig{H N^{1/3} \exp \Big( - C \Big( \frac{L}{\log L} \Big)^{1/3} \Big)}, \end{equation*}
provided that $N^{13/18+ 3\varepsilon} \le H \le N^{1-\varepsilon}$. Theorem \ref{thm-uncond} follows by rescaling $\varepsilon$.
\section{The conditional case} \label{conditional} From now on we assume the Riemann Hypothesis holds. Comparing with section \ref{unconditional} we can simplify the setting. Recalling \eqref{r-def} and $\Gamma(4/3)= (1/3)\Gamma(1/3)$, we have
\begin{align}
\notag \sum_{n=N+1}^{N+H} e^{-n/N} r(n) & = \int_{-1/2}^{1/2}\widetilde{V}_{3}(\alpha)^4 U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
\\
\notag
&
= \Gamma \Bigl(\frac{4}{3}\Bigr)^4
\int_{-1/2}^{1/2}\frac{U(-\alpha,H)}{z^{4/3}}e(-N\alpha) \, \mathrm{d} \alpha
\\
\notag
& \hskip1cm + \int_{-1/2}^{1/2} \Bigl( \widetilde{S}_{3}(\alpha)^4 - \frac{\Gamma(4/3)^4}{z^{4/3}}\Bigr)
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha
\\
\notag
& \hskip1cm + \int_{-1/2}^{1/2} \bigl( \widetilde{V}_{3}(\alpha)^4 -\widetilde{S}_{3}(\alpha)^4 \bigr)
U(-\alpha,H)e(-N\alpha) \, \mathrm{d} \alpha \\
\label{main-dissection-RH-series}
& = \mathcal{J}_1 +\mathcal{J}_2 + \mathcal{J}_3,
\end{align}
say.
Now we evaluate these terms.
\subsection{Evaluation of $\mathcal{J}_1$}
By Lemma \ref{Laplace-formula}, a direct calculation gives
\begin{align} \notag \mathcal{J}_1 & = \Gamma\Bigl(\frac{4}{3}\Bigr)^3\ \sum_{n=N+1}^{N+H} e^{-n/N} n^{1/3} +\Odig{\frac{H}{N}} = \frac{\Gamma(4/3)^3}{e}\ \sum_{n=N+1}^{N+H} n^{1/3} +\Odig{\frac{H}{N}+\frac{H^2}{N^{2/3}}} \\ \label{J1-eval-RH-series} & = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e} +\Odig{\frac{H^2}{N^{2/3}}+N^{1/3}}.
\end{align}
\subsection{Estimate of $\mathcal{J}_2$}
Recall that
\( \widetilde{E}_{3}(\alpha) : =\widetilde{S}_3(\alpha) - \frac{\Gamma(4/3)}{z^{1/3}}. \)
Using $ f^2 - g^2 = 2g (f-g) + (f-g)^2$
we can write that
\begin{align*} \widetilde{S}_{3}(\alpha)^4 - \frac{\Gamma(4/3)^4}{z^{4/3}} &= \Bigl( \widetilde{S}_{3}(\alpha) ^2 + \frac{\Gamma(4/3)^2}{z^{2/3}} \Bigr) \Bigl( \widetilde{S}_{3}(\alpha) ^2 - \frac{\Gamma(4/3)^2}{z^{2/3}} \Bigr) \\& \ll (\vert \widetilde{S}_{3}(\alpha) \vert^2 + \vert z \vert^{-2/3}) (
\vert z \vert^{-1/3} \vert \widetilde{E}_{3}(\alpha) \vert +
\vert \widetilde{E}_{3}(\alpha) \vert^2
). \end{align*}
Hence \begin{align} \notag \mathcal{J}_2 &\ll \int_{-1/2}^{1/2} \frac{\vert \widetilde{S}_{3}(\alpha) \vert^2}{\vert z \vert^{1/3}} \vert \widetilde{E}_{3}(\alpha) \vert \vert U(-\alpha,H)\vert \, \mathrm{d} \alpha + \int_{-1/2}^{1/2} \frac{\vert \widetilde{E}_{3}(\alpha) \vert}{\vert z \vert}\vert U(-\alpha,H)\vert \, \mathrm{d} \alpha \\ \notag & \hskip 1cm+ \int_{-1/2}^{1/2} (\vert \widetilde{S}_{3}(\alpha) \vert^2 + \vert z \vert^{-2/3})
\vert \widetilde{E}_{3}(\alpha) \vert^2 \vert U(-\alpha,H)\vert \, \mathrm{d} \alpha \\ \label{J2-RH-split} &
= \mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3, \end{align}
say. Let \[ \mathcal{E} := \int_{-1/2}^{1/2} \vert \widetilde{E}_{3}(\alpha) \vert^2 \vert U(-\alpha,H) \vert \, \mathrm{d} \alpha. \]
By \eqref{UH-estim}, Lemma \ref{LP-Lemma-gen} and a partial integration we obtain
\begin{align} \notag \mathcal{E} &\ll H \int_{-1/H}^{1/H} \vert \widetilde{E}_{3}(\alpha) \vert^2 \, \mathrm{d} \alpha + \int_{1/H}^{1/2} \frac{\vert \widetilde{E}_{3}(\alpha) \vert^2}{\alpha} \, \mathrm{d} \alpha \\& \label{E-estim} \ll H \frac{N^{1/3} L^2}{H} + N^{1/3} L^2 + \int_{1/H}^{1/2} \frac{ N^{1/3} L^2} {\xi} \ \mathrm{d} \xi \ll N^{1/3} L^3. \end{align}
By \eqref{z-estim}, $\widetilde{S}_{3}(\alpha) \ll N^{1/3}$ and \eqref{E-estim} we obtain
\begin{equation}
\label{I3-estim-RH-series} \mathcal{I}_3 \ll N^{2/3} \mathcal{E} \ll N L^3. \end{equation}
By the Cauchy-Schwarz inequality, \eqref{z-estim}-\eqref{UH-estim}
and \eqref{E-estim}, we obtain
\begin{align}
\notag \mathcal{I}_2 & \ll \mathcal{E}^{1/2} \Bigl( \int_{-1/2}^{1/2}
\frac{\vert U(-\alpha,H) \vert }{\vert z \vert^{2}} \, \mathrm{d} \alpha \Bigr)^{1/2} \ll \mathcal{E}^{1/2} \Bigl(
H N^{2} \int_{-1/N}^{1/N} \mathrm{d} \alpha
+
H \int_{1/N}^{1/H}
\frac{ \mathrm{d} \alpha}{ \alpha ^{2}}
+
\int_{1/H}^{1/2}
\frac{ \mathrm{d} \alpha}{ \alpha ^{3}} \Bigr)^{1/2} \\ \label{I2-estim-RH-series} & \ll H^{1/2} N^{2/3} L^{3/2}. \end{align}
By the Cauchy-Schwarz inequality we obtain
\[
\mathcal{I}_1 \ll \mathcal{E}^{1/2} \Bigl( \int_{-1/2}^{1/2} \vert \widetilde{S}_3(\alpha) \vert^4 \frac{\vert U(-\alpha,H) \vert }{\vert z \vert^{2/3}} \, \mathrm{d} \alpha \Bigr)^{1/2}.
\]
Again by the Cauchy-Schwarz inequality, \eqref{z-estim}-\eqref{UH-estim}
and \eqref{E-estim}, we obtain
\begin{align}
\notag \mathcal{I}_1 & \ll \mathcal{E}^{1/2} \Bigl( \int_{-1/2}^{1/2} \vert \widetilde{S}_3(\alpha) \vert^8 \, \mathrm{d} \alpha \Bigr)^{1/4} \Bigl( \int_{-1/2}^{1/2} \frac{\vert U(-\alpha,H) \vert^2 }{\vert z \vert^{4/3}} \, \mathrm{d} \alpha \Bigr)^{1/4} \\ \notag & \ll N^{1/6} L^{3/2} N^{5/12} L^{A/4} \Bigl(
H^2 N^{4/3} \int_{-1/N}^{1/N} \mathrm{d} \alpha
+
H^2 \int_{1/N}^{1/H}
\frac{ \mathrm{d} \alpha}{\alpha^{4/3}}
+
\int_{1/H}^{1/2}
\frac{ \mathrm{d} \alpha}{\alpha^{10/3}} \Bigr)^{1/4} \\ \label{I1-estim-RH-series} & \ll H^{1/2} N^{2/3} L^{3/2+A/4}. \end{align}
Summing up by \eqref{J2-RH-split} and \eqref{I3-estim-RH-series}-\eqref{I1-estim-RH-series}, we can finally write that
\begin{equation} \label{J2-estim-RH-series} \mathcal{J}_2 \ll H^{1/2} N^{2/3} L^{3/2+A/4} + N L^3.
\end{equation}
\subsection{Estimate of $\mathcal{J}_3$}
\begin{comment} Clearly \[ \widetilde{V}_{3}(\alpha)^4 -\widetilde{S}_{3}(\alpha)^4 =
(\widetilde{V}_{3}(\alpha)^2 + \widetilde{S}_{3}(\alpha)^2) (\widetilde{V}_{3}(\alpha)^2 -\widetilde{S}_{3}(\alpha)^2), \] and using $f^2-g^2= 2f(f-g)- (f-g)^2$, we also have \[ \widetilde{V}_{3}(\alpha)^2 -\widetilde{S}_{3}(\alpha)^2 \ll \vert \widetilde{V}_{3}(\alpha) -\widetilde{S}_{3}(\alpha) \vert^2 +
\vert \widetilde{V}_{3}(\alpha) -\widetilde{S}_{3}(\alpha) \vert \vert \widetilde{S}_{3}(\alpha) \vert. \]
By Lemma \ref{tilde-trivial-lemma},
$\widetilde{V}_{3}(\alpha) \ll N^{1/3}$ and $\widetilde{S}_{3}(\alpha) \ll N^{1/3}$
we get
\begin{equation} \label{V-tilde-approx}
\vert \widetilde{V}_{3}(\alpha)^4 -\widetilde{S}_{3}(\alpha)^4
\vert \le \bigl( \vert \widetilde{V}_{3}(\alpha)\vert^2 + \vert \widetilde{S}_{3}(\alpha)\vert^2 \Bigr) \bigl( N^{1/3}+ N^{1/6}\vert \widetilde{S}_{3}(\alpha)\vert \Bigr) \ll N^{7/6}. \end{equation}
Hence, by \eqref{UH-estim}, we can finally write
\begin{equation}
\label{J3-estim-RH-series} \mathcal{J}_3 \ll N^{7/6} \int_{-1/2}^{1/2} \vert U(-\alpha,H) \vert \, \mathrm{d} \alpha \ll N^{7/6} \Bigl( \int_{-1/H}^{1/H} H \ \mathrm{d} \alpha + \int_{1/H}^{1/2} \frac{\mathrm{d} \alpha}{\alpha} \Bigr) \ll N^{7/6} L. \end{equation} \end{comment}
It is clear that $\mathcal{J}_3=I_3$ of section \ref{I3-estim}. Hence by \eqref{I3-estim-series} we obtain \begin{equation}
\label{J3-estim-RH-series} \mathcal{J}_3 \ll H^{3/4}N^{5/12+\varepsilon}. \end{equation}
\subsection{Final words}
Summing up, by \eqref{main-dissection-RH-series}-\eqref{J1-eval-RH-series}, \eqref{J2-estim-RH-series} and \eqref{J3-estim-RH-series}, there exists $B=B(A)>0$ such that we have
\begin{equation} \label{almost-done} \sum_{n=N+1}^{N+H} e^{-n/N} r(n) = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 \frac{HN^{1/3}}{e} +\Odig{\frac{H^2}{N^{2/3}} + H^{3/4}N^{5/12+\varepsilon} + H^{1/2} N^{2/3} L^{B}
+ NL^3} \end{equation}
which is an asymptotic formula $\infty(N^{2/3}L^{2B}) \le H \le \odi{N}$.
{}From $e^{-n/N}=e^{-1}+ \Odi{H/N}$ for $n\in[N+1,N+H]$, $1\le H \le N$, we get
\begin{align*}
\sum_{n = N+1}^{N + H} r(n) &= \Gamma\Bigl(\frac{4}{3}\Bigr)^3 HN^{1/3} +\Odig{\frac{H^2}{N^{2/3}} + H^{3/4}N^{5/12+\varepsilon} + H^{1/2} N^{2/3} L^{B}
+NL^3} +
\Odig{\frac{H}{N}\sum_{n = N+1}^{N + H} r(n) }. \end{align*}
Using $e^{n/N}\le e^{2}$ and \eqref{almost-done},
the last error term is $\ll H^2N^{-2/3}+ H^{7/4}N^{-7/12+\varepsilon} + H^{3/2} N^{-2/3} L^{B} + HL^3$. Hence we get
\begin{equation*}
\sum_{n = N+1}^{N + H} r(n) = \Gamma\Bigl(\frac{4}{3}\Bigr)^3 HN^{1/3} +\Odig{\frac{H^2}{N^{2/3}}+ H^{3/4}N^{5/12+\varepsilon} + H^{1/2} N^{2/3} L^{B}
+NL^3} , \end{equation*}
uniformly for $\infty(N^{2/3}L^{2B}) \le H \le \odi{N}$, $B>3/2$. Theorem \ref{thm-RH} follows.
\renewcommand{\normalsize}{\normalsize}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \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}
\vskip0.5cm \noindent \begin{tabular}{l@{\hskip 20mm}l} Alessandro Languasco & Alessandro Zaccagnini\\ Universit\`a di Padova & Universit\`a di Parma\\
Dipartimento di Matematica & Dipartimento di Scienze Matematiche,\\
``Tullio Levi-Civita'' & Fisiche e Informatiche \\ Via Trieste 63 & Parco Area delle Scienze, 53/a \\ 35121 Padova, Italy & 43124 Parma, Italy\\ {\it e-mail}: [email protected] & {\it e-mail}: [email protected] \end{tabular}
\end{document} | arXiv |
Incorporating nonlinearity into mediation analyses
George J. Knafl1,
Kathleen A. Knafl1,
Margaret Grey2,
Jane Dixon2,
Janet A. Deatrick3 &
Agatha M. Gallo4
Mediation is an important issue considered in the behavioral, medical, and social sciences. It addresses situations where the effect of a predictor variable X on an outcome variable Y is explained to some extent by an intervening, mediator variable M. Methods for addressing mediation have been available for some time. While these methods continue to undergo refinement, the relationships underlying mediation are commonly treated as linear in the outcome Y, the predictor X, and the mediator M. These relationships, however, can be nonlinear. Methods are needed for assessing when mediation relationships can be treated as linear and for estimating them when they are nonlinear.
Existing adaptive regression methods based on fractional polynomials are extended here to address nonlinearity in mediation relationships, but assuming those relationships are monotonic as would be consistent with theories about directionality of such relationships.
Example monotonic mediation analyses are provided assessing linear and monotonic mediation of the effect of family functioning (X) on a child's adaptation (Y) to a chronic condition by the difficulty (M) for the family in managing the child's condition. Example moderated monotonic mediation and simulation analyses are also presented.
Adaptive methods provide an effective way to incorporate possibly nonlinear monotonicity into mediation relationships.
Mediation is an important issue considered in the behavioral, medical, and social sciences, addressing situations where the effect of a predictor variable X on an outcome (or dependent or response) variable Y is explained to some extent by an intervening, mediator variable M. Methods for addressing mediation have existed for some time [1–3]. Since then, they have undergone a variety of refinements [4–28].
Relationships underlying mediation are commonly treated as linear in Y, X, and M. Mediating relationships, however, can be nonlinear. A few authors have addressed nonlinearity in the mediation context [29–32]. Pearl [30, 31] has developed a general approach for quantifying direct, indirect, and total effects allowing for nonlinearity as well as for categorical variables. Standard polynomials sometimes can be effectively used to address nonlinearity in predictors, but not in general [33]. To fully address nonlinearity requires more complex methods like nonparametric regression. Methods are needed to assess linearity of mediation relationships and to conduct mediation analyses when the underlying relationships are nonlinear.
The purpose of this paper is 1. To present an approach for assessing linearity of mediation relationships and for conducting nonlinear mediation analyses, based on the adaptive regression methods of Knafl and Ding [34], and 2. To provide example analyses using these methods. Adaptive methods were originally developed for nonlinear modeling in the Poisson regression context with Poisson distributed count outcomes [35], but the methods extend to linear regression with continuous outcomes treated as normally distributed, as often used in mediation analyses, as well as logistic regression with discrete outcomes [36]. SAS® (SAS Institute, Inc., Cary, NC) macros have been developed to support adaptive regression. Knafl [37] presents examples of the use of one of these macros for nonlinear growth curve modeling. These methods address not only nonlinearity in predictors but also nonlinearity in outcomes when those outcomes are continuous and positive valued. They also allow for correlated outcomes and/or non-constant variances.
Behavioral, medical, and social science theories usually do not hypothesize that relationships are nonlinear but implicitly assume they are linear. However, Hayes and Preacher [29] provide a variety of examples of behavioral theories that incorporate nonlinearity. In any case, hypothesized relationships are usually stated in terms of directionality: as X increases, Y increases or decreases. Such statements are not inherently linear but can be represented more generally by possibly nonlinear monotonic relationships. Monotonicity is operationalized in this paper with single power transforms. Moderated monotonic mediation is also addressed in the paper.
This section starts by formulating standard linear mediation relationships, then nonlinear mediation relationships followed by monotonic mediation relationships. Adaptive regression methods are described next along with how they can be used to assess monotonic mediation and moderated monotonic mediation. The section ends with a description of the two data sets used in example analyses.
Linear mediation relationships
As commonly considered, regression models underlying mediation are formulated as linear in Y, X, and M. As an example, work pressure X could result in increased work stress M leading to increased alcohol consumption Y. The following models are considered relating Y to X (e.g., alcohol consumption to work pressure), M to X (e.g., work stress to work pressure), and Y to M and X (e.g., alcohol consumption to work pressure controlling for work stress), respectively (Fig. 1).
$$ Y = {i}_{Y,1}+ c\cdot X+{U}_{Y,1} $$
$$ M = {i}_M+ a\cdot X+{U}_M $$
$$ Y = {i}_{Y,2}+{c}^{\mathit{\hbox{'}}}\cdot X+ b\cdot M+{U}_{Y,2} $$
The random variables U Y,1, U M , and U Y,2 are assumed to have mean zero. They can be thought of as omitted factors (as in [31]) or as random errors.
The slope c for X in model (1) is the total effect of X on Y (e.g., of work pressure on alcohol consumption) while the slope c ' for X in model (3) is the direct effect of X on Y controlling for M (e.g., of work pressure on alcohol consumption controlling for work stress). The indirect effect Δ satisfies
$$ \varDelta = c-{c}^{\mathit{\hbox{'}}}= a\cdot b $$
as demonstrated by substituting model (2) for the mediator M into model (3), and so the total effect is the sum of the direct and indirect effects. In the Baron and Kenny [1] approach, a significant total effect c was considered necessary for mediation to hold. However, this no longer is considered necessary [11]. There can be a nonzero indirect effect even when the total effect is non-significant [38]. However, model (1) is often investigated in practice. In any case, a crucial issue for mediation to hold is a significantly nonzero indirect effect Δ.
A variety of tests for a zero indirect effect are available. Sobel's test [21, 39] is the best known, but MacKinnon et al. [14] considered 14 alternatives including Sobel's test. Difference in coefficients approaches capitalize on the first equality for Δ while product of coefficients approaches including Sobel's test capitalize on the second equality. The assumptions underlying these tests can be questionable, and so the test based on the bootstrapped confidence interval (CI) for Δ has been proposed as a robust alternative [21, 29, 40, 41], especially when bias-corrected [42]. However, bias-corrected bootstrapped CIs can generate inflated Type I errors [43].The strength of the indirect effect [21] can be measured by the relative indirect effect \( {\varDelta}^{\hbox{'}}=\raisebox{1ex}{$\varDelta $}\!\left/ \!\raisebox{-1ex}{$ c$}\right. \) (often denoted as P M ).
Since mediation addresses causality, temporal precedence is an important issue. This can be addressed by measuring X before measuring M and that before measuring Y [44]. However, mediation sometimes is addressed using cross-sectional data. Equations (1)-(3) are the same for these two cases. Random assignment of the settings of X provides support for causality for the X to Y relationship and the X to M relationship, but the M to Y relationship can still be confounded, and even when X, M, and Y have been measured longitudinally [45]. Nonexperimental or observational studies need theoretical justification for hypothesized causal relationships [21]. Pearl [31] has provided sufficient conditions for identifying direct, indirect, and total effects for observational studies.
Nonlinear mediation relationships
General approach for addressing nonlinearity
Nonlinear mediation involves the following two regression models (4)-(5) generalizing the linear regression models (2)-(3) while no longer considering model (1).
$$ M = {i}_M+{T}_{M, X}(X)+{U}_M $$
$$ Y = {i}_Y+{T}_{Y, X}(X)+{T}_{Y, M}(M)+{U}_Y $$
T M,X (X) and T Y,X (X) are possibly nonlinear transforms of the predictor X while T Y,M (M) is a possibly nonlinear transform of the mediator M. The parameters i M and i Y are standard intercepts. When one of the variables X or M is discrete, only its identity transform is considered.
As an example, Hayes and Preacher [29] considered the special case with
$$ {T}_{M, X}( X)= a\cdot \log X,\kern0.5em {T}_{Y, X}( X)={c}^{\mathit{\hbox{'}}}\cdot X,\kern0.47em \mathrm{and}\kern0.37em {T}_{Y, M}( M)={b}_1\cdot M+{b}_2\cdot {M}^2 $$
where log denotes the natural log transform. Following Stolzenberg [32], they defined the instantaneous indirect effect, generalizing the product of coefficients approach, as the product of the derivatives \( \frac{\partial M}{\partial X} \) and \( \frac{\partial Y}{\partial M} \) replacing M by model (4) and assuming U M and U Y have no impact on these derivatives. For their example (6), this equals
$$ \begin{array}{l}\\ {}\left({b}_1+2\cdot {b}_2\cdot \left({i}_M+ a\cdot \log X\right)\right)\cdot \frac{a}{X}\end{array} $$
(as also reported in their eq. (10)).
Pearl's more general approach
Pearl [31] provided the following general model for addressing nonlinearity in the mediation context, also allowing for categorical variables,
$$ m={f}_M\left( x,{u}_M\right)\ \mathrm{and}\ y={f}_Y\left( x, m,{u}_y\right) $$
where x, m, y, u M , and u Y are possible values for the random variables X, M, Y, U M , and U Y while f M and f Y are general transforms. For example, the values m of work stress M could depend nonlinearly on the values x of work pressure X and the values u M of the omitted factor or error variable U M while the values y of alcohol consumption Y could depend nonlinearly on the values x of work pressure X, the values m of work stress M, and the values u y of the omitted factor or error variable U Y . Pearl also included the relationship x = f X (u X ), but that is not needed in what follows. He assumed that variables have been standardized, but that is not assumed here. He provided definitions of total, natural direct, and natural indirect effects under model (8), which generalize those effects for the linear case of models (1)-(3).
The natural direct effect for a change in value of X from x to x ' is defined as the expected value NDE(x ', x) = E(nde(x ', x)) where
$$ n d e\left( x\mathit{\hbox{'}}, x\right)={f}_Y\left( x\mathit{\hbox{'}},{f}_M\left( x,{u}_M\right),{u}_Y\right)-{f}_Y\left( x,{f}_M\left( x,{u}_M\right),{u}_Y\right). $$
In other words, nde(x ', x) is the change in y when x changes to x ' while m is held fixed at its value for the initial value x. Expectations, here and in what follows, are with respect to U M and U Y . The natural indirect effect for a change in value of x to x ' is the expected value NIE(x ', x) = E(nie(x ', x)) where
$$ n i e\left( x\mathit{\hbox{'}}, x\right)={f}_Y\left( x,{f}_M\left( x\mathit{\hbox{'}},{u}_M\right),{u}_Y\right)-{f}_Y\left( x,{f}_M\left( x,{u}_M\right),{u}_Y\right). $$
In other words, nie(x ', x) is the change in y when x is held fixed at its initial value while m changes from its value for x to its value for x '. The total effect for a change in value of x to x ' is the expected value TE(x ', x) = E(te(x ', x)) where
$$ t e\left( x\mathit{\hbox{'}}, x\right)={f}_Y\left( x\mathit{\hbox{'}},{f}_M\left( x\mathit{\hbox{'}},{u}_M\right),{u}_Y\right)-{f}_Y\left( x,{f}_M\left( x,{u}_M\right),{u}_Y\right). $$
In other words, te(x ', x) is the change in y when x changes to x ' and m changes from its value for x to its value for x '.
Adding and subtracting f Y (x ', f M (x, u M ), u Y ) to te(x ', x) gives that
$$ T E\left( x\mathit{\hbox{'}}, x\right)= N D E\left( x\mathit{\hbox{'}}, x\right)- N I E\left( x, x\mathit{\hbox{'}}\right) $$
(also eq. (14) of [31]). Thus, the total effect is only the sum of the natural direct and natural indirect effects in the special case that NIE(x ', x) = − NIE(x, x '), which holds for the linear mediation case of models (1)-(3). This result indicates the shortcoming of trying to generalize the indirect effect using a difference of coefficients approach.
The definitions of NDE(x ', x), NIE(x ', x), and TE(x ', x) are sufficient for handling categorical predictors X. For the case of continuous predictors X, they can be used to define instantaneous natural direct, natural indirect, and total effects using limits as follows:
$$ \frac{dNDE(x)}{dx}=\underset{x\hbox{'}\to x}{ \lim}\frac{NDE\left( x\mathit{\hbox{'}}, x\right)}{x\mathit{\hbox{'}}- x}, $$
$$ \frac{dNIE(x)}{dx}=\underset{x\hbox{'}\to x}{ \lim}\frac{NIE\left( x\mathit{\hbox{'}}, x\right)}{x\mathit{\hbox{'}}- x}, $$
$$ \frac{dTE(x)}{dx}=\underset{x\hbox{'}\to x}{ \lim}\frac{TE\left( x\mathit{\hbox{'}}, x\right)}{x\mathit{\hbox{'}}- x}. $$
The relative instantaneous natural indirect effect function can be computed as
$$ \Delta \mathit{\hbox{'}}=\raisebox{1ex}{$\frac{dNIE(x)}{dx}$}\!\left/ \!\raisebox{-1ex}{$\frac{dTE(x)}{dx}.$}\right. $$
For the Hayes and Preacher [29] nonlinear example, m = f M (x, u M ) and y = f Y (x, m, u Y ) are determined by eqs. (4)-(5) evaluated at (6). Thus,
$$ n d e\left( x\mathit{\hbox{'}}, x\right)= c\mathit{\hbox{'}}\cdot \left( x\mathit{\hbox{'}}- x\right)= N D E\left( x\mathit{\hbox{'}}, x\right) $$
so that \( \frac{dNDE(x)}{dx}= c\mathit{\hbox{'}} \) as would be expected for a linear model for Y in X. Also,
$$ \begin{array}{c} nie\left( x\mathit{\hbox{'}}, x\right)={b}_1\cdot {f}_M\left( x\mathit{\hbox{'}},{u}_M\right)+{b}_2\cdot {f}_M^2\left( x\mathit{\hbox{'}},{u}_M\right)-{b}_1\cdot {f}_M\left( x,{u}_M\right)-{b}_2\cdot {f}_M^2\left( x,{u}_M\right)\\ {}={b}_1\cdot a\cdot \left( \log x\mathit{\hbox{'}}- \log x\right)+{b}_2\cdot \left({\left({i}_M+ a\cdot \log x\mathit{\hbox{'}}+{u}_M\right)}^2-{\left({i}_M+ a\cdot \log x+{u}_M\right)}^2\right)\\ {} = {b}_1\cdot a\cdot \left( \log x\mathit{\hbox{'}}- \log x\right)+{b}_2\cdot \left(2\cdot \left({i}_M+{u}_M\right) + a\cdot \left( \log x\mathit{\hbox{'}}+ \log x\right)\right)\cdot a\cdot \left( \log x\mathit{\hbox{'}}- \log x\right)\end{array} $$
$$ \frac{dNIE(x)}{dx}=\left({b}_1+2\cdot {b}_2\cdot \left({i}_M+ a\cdot \log x\right)\right)\cdot \frac{a}{x}, $$
which agrees with result (7).
As a second example, consider the following case with M = X + U M (i.e., the linear model (2) with i m = 0 and a = 1) and Y = i Y + c ' ⋅ X + b ⋅ M 3 + U Y . Since \( \frac{\partial M}{\partial X}=1 \) and \( \frac{\partial Y}{\partial M}= b\cdot 3\cdot {M}^2, \) the instantaneous indirect effect for a fixed value x of X using the product of coefficients approach of Hayes and Preacher [29] would be b ⋅ 3 ⋅ x 2 (i.e., substituting x for M based on the relationship M = X + U M and ignoring the associated value u M for U M ). On the other hand, the general approach of Pearl [31] gives
$$ n i e\left( x\mathit{\hbox{'}}, x\right)= b\cdot {f}_M^3\left( x\mathit{\hbox{'}},{u}_M\right)- b\cdot {f}_M^3\left( x,{u}_M\right)= b\cdot \left({\left( x\mathit{\hbox{'}}+{u}_M\right)}^3-{\left( x+{u}_M\right)}^3\right) $$
$$ \underset{x\mathit{\hbox{'}}\to x}{ \lim}\frac{nie\left( x\mathit{\hbox{'}}, x\right)}{x\mathit{\hbox{'}} - x}= b\cdot 3\cdot {\left( x+{u}_M\right)}^2= b\cdot 3\cdot \left({x}^2+2\cdot x\cdot {u}_M+{u}_M^2\right). $$
Since E(u M ) = 0, the instantaneous natural indirect effect satisfies
$$ \frac{dNIE(x)}{dx}= b\cdot 3\cdot \left({x}^2+ E\left({U}_M^2\right)\right). $$
The expected value E(U M 2 ) > 0 except in trivial cases. Consequently, this is not the same value as obtained by the Hayes-Preacher product of coefficients approach. The problem with that approach is that U M cannot always be ignored in computing indirect effects. Similar problems could occur if f Y is nonlinear in u Y . This result indicates the shortcoming of trying to generalize the indirect effect using a product of coefficients approach.
Monotonic mediation
Theorized relationships are reasonably operationalized as monotonic relationships. Monotonic mediation can be formulated with models (9)-(10).
$$ {M}^q={i}_M+ a\cdot {X}^{q\mathit{\hbox{'}}}+{U}_M $$
$$ {Y}^p={i}_Y+ c\mathit{\hbox{'}}\cdot {X}^{p\mathit{\hbox{'}}}+ b\cdot {M}^{q\cdot p\mathit{\hbox{'}}\mathit{\hbox{'}}}+{U}_Y $$
For example, transformed work stress M q could depend nonlinearly on transformed work pressure X q ' in model (9) while transformed alcohol consumption Y p could depend nonlinearly on transformed work pressure X p ' controlling for transformed work stress M q ⋅ p" in model (10). When Y > 0, M > 0, and X > 0, the power transforms Y p, X p ', X q ', M q, and M q" are well-defined for arbitrary real valued powers p, p ', q, q ', and q" = q ⋅ p ". An approach for extending this to arbitrary valued variables is presented later. The power p = 0 represents the natural log transform rather than the constant transform. These are Box-Tidwell transformations [46], although Box and Tidwell considered them only for predictors. Models (9)-(10) provide for transformation of outcomes as well as predictors as opposed to just predictors as in models (4)-(5) and (8). Fractional polynomials of degree 1 in X, M, and Y have been used in (9)-(10) to guarantee that relationships are monotonic.
Models (9)-(10) can be represented in the general form of model (8) replacing m and y by m q and y p giving
$$ {m}^q={f}_M\left( x,{u}_M\right)={i}_M+ a\cdot {x}^{q\hbox{'}}+{u}_M $$
$$ {y}^p={f}_Y\left( x, m,{u}_Y\right)={i}_Y+ c\mathit{\hbox{'}}\bullet {x}^{p\mathit{\hbox{'}}}+ b\cdot {m}^{q\cdot p\mathit{\hbox{'}}\mathit{\hbox{'}}}+{u}_M. $$
Hence, nde(x ', x) = c ' ⋅ ((x ')p ' − x p) = NDE(x ', x) so that the instantaneous natural direct effect satisfies
$$ \frac{dNDE(x)}{dx}= c\mathit{\hbox{'}}\cdot p\mathit{\hbox{'}}\cdot {x}^{p^{\hbox{'}}-1} $$
for p' ≠ 0 while \( \frac{dNDE(x)}{dx}= c\mathit{\hbox{'}}\cdot {x}^{-1} \) for p' = 0. Also,
$$ \begin{array}{c} nie\left( x\mathit{\hbox{'}}, x\right)= b\cdot {f}_M^{p\hbox{'}\hbox{'}}\left( x\mathit{\hbox{'}},{u}_M\right)- b\cdot {f}_M^{p\hbox{'}\hbox{'}}\left( x,{u}_M\right)\\ {}= b\cdot \left({\left({i}_M+ a\cdot {\left( x\mathit{\hbox{'}}\right)}^{q^{\hbox{'}}}+{u}_M\right)}^{p^{\hbox{'}\hbox{'}}}-{\left({i}_M+ a\cdot {(x)}^{q\mathit{\hbox{'}}}+{u}_M\right)}^{p^{\hbox{'}\hbox{'}}}\right).\end{array} $$
Note that nie(x, x ') = − nie(x ', x) so that te(x ', x) = nde(x ', x) + nie(x ', x), and so the total effect is the sum of the natural direct and natural indirect effects as for the linear mediation case of models (1)-(3).
For the special case with p'' = 1,
$$ n i e\left( x\mathit{\hbox{'}}, x\right)= b\cdot a\cdot \left({\left( x\mathit{\hbox{'}}\right)}^{q\mathit{\hbox{'}}}-{x}^{q\mathit{\hbox{'}}}\right)= N I E\left( x\mathit{\hbox{'}}, x\right) $$
$$ \frac{dNIE(x)}{dx}= b\cdot a\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1} $$
for q ' ≠ 0 while \( \frac{dNIE(x)}{dx}= b\cdot a\cdot {x}^{-1} \) for q ' = 0. Figure 2 represents this model (assuming q ' ≠ 0 and p ' ≠ 0). The paths are labeled with derivatives of expected values for M q and Y p relative to X or to M q generalizing the slopes used in Figure 1. Note that as in the linear case of Figure 1, the instantaneous natural indirect effect equals the product b ⋅ a ⋅ q ' ⋅ x q ' − 1 of the labels for the two upper paths and the instantaneous natural direct effect equals the label c ' ⋅ p ' ⋅ x p ' − 1 for the lower path.
Monotonic mediation relationships (assuming \( q^{\prime}\ne 0 \) and \( p^{\prime}\ne 0 \)).
For the special case with p" = 2 and q " ≠ 0,
$$ \begin{array}{c} nie\left( x\mathit{\hbox{'}}, x\right)= b\cdot \left({\left({i}_M+ a\cdot {\left( x\mathit{\hbox{'}}\right)}^{q\mathit{\hbox{'}}}+{u}_M\right)}^2-{\left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}+{u}_M\right)}^2\right)\\ {}= b\cdot \left(2\cdot \left({i}_M+{u}_M\right)+ a\cdot {\left( x\mathit{\hbox{'}}\right)}^{q\mathit{\hbox{'}}}+ a\cdot {x}^{q\mathit{\hbox{'}}}\right)\cdot a\cdot \left({\left( x\mathit{\hbox{'}}\right)}^{q\mathit{\hbox{'}}}-{x}^{q\mathit{\hbox{'}}}\right)\end{array} $$
$$ \frac{dNIE(x)}{dx}= b\cdot 2\cdot \left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}\right)\cdot a\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}. $$
For the special case with p" = 3 and q ≠ 0,
$$ n i e\left( x\mathit{\hbox{'}}, x\right)= b\cdot \left({\left({i}_M+ a\cdot {\left( x\mathit{\hbox{'}}\right)}^{q\mathit{\hbox{'}}}+{u}_M\right)}^3-{\left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}+{u}_M\right)}^3\right) $$
$$ \begin{array}{c}\underset{x\hbox{'}\to x}{ \lim}\frac{nie\left( x\mathit{\hbox{'}}, x\right)}{x\mathit{\hbox{'}} - x}= b\cdot 3\cdot {\left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}+{u}_M\right)}^2\cdot a\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}\\ {}= b\cdot 3\cdot \left({\left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}\right)}^2+2\cdot \left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}\right)\cdot {u}_M+{u}_M^2\right)\cdot a\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}.\end{array} $$
Hence, the instantaneous natural indirect effect satisfies
$$ \frac{dNIE(x)}{dx}= b\cdot 3\cdot \left({\left({i}_M+ a\cdot {x}^{q\mathit{\hbox{'}}}\right)}^2+\mathrm{E}\left({U}_M^2\right)\right)\cdot a\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}. $$
We recommend using the special case with p " = 1 due to its desirable properties. It provides a natural generalization of the linear case of models (1)-(3) to account for monotonicity. In what follows, p " = 1 is assumed unless otherwise stated.
Adaptive regression
Methods are needed for estimating the relationships of models (4)-(5) and (9)-(10) and for assessing whether those relationships are non-constant in X or M and whether they are reasonably treated as linear in Y, X, and/or M or are distinctly nonlinear in any of those variables. We use adaptive regression modeling [34] for these purposes. Methods are needed as well for assessing whether the instantaneous natural indirect effect \( \frac{dNIE(x)}{dx} \) is nonzero. This can be addressed as in linear mediation with bootstrapped CIs, but now computed for a grid of possible values x for X.
Knafl et al. [35] developed adaptive regression methods for nonlinear modeling of Poisson distributed count outcome variables Y. Knafl et al. [36] extended this to generalized linear modeling including adaptive linear regression with outcomes treated as normally distributed, as often used in mediation analyses (but they used these methods to address modeling of electronically monitored medication adherence data rather than mediation). How adaptive regression analyses are conducted is described next.
Adaptive regression methods use likelihood cross-validation (LCV) scores (as defined later) to evaluate and compare models. These scores generalize to contexts where estimation is based on maximizing likelihood-like functions such as extended quasi-likelihood functions [47]. Heuristic (i.e., rule-based) search techniques guided by LCV scores are used to identify effective fractional polynomial models [33] in primary predictor variables including predictors\( X \) and mediators M as needed for mediation. Fractional polynomials generalize standard polynomials, which use only nonnegative integer powers, to allow for the possibility of negative and fractional powers. Fractional polynomial models for continuous outcomes are linear in associated parameter vectors (consisting of the intercept and slopes for power transforms of predictors), and so they are linear models estimated using linear regression methods. However, these models are based on relationships that are in general nonlinear (or curvilinear) in the predictor X for models (4)-(5) and (9)-(10) as well as in the mediator M for models (5) and (10).
Power transforms f(X, p) are defined for arbitrary real valued primary predictors X and powers p by setting f(X, p) to X p when X > 0, to 0 when X = 0, and to cos(π ⋅ p) ⋅ |X|p when X < 0 where\( \cos \) denotes the standard cosine function, π is the usual constant, and |X| denotes the absolute value of X. Note that for X < 0, the sign of f(X, p) oscillates between ±1 as the power \( p \) varies. For simplicity of notation, f(X, p) is denoted as X p.
Likelihood cross-validation
LCV scores are computed by randomly partitioning observations into k disjoint sets called folds, calculating likelihoods for folds using parameter estimates computed from the remaining data in the complement of the fold, and combining these deleted likelihoods into a geometric mean deleted likelihood score. Formally, let S = {s : 1 ≤ s ≤ n} denote the index set for the \( n \) subjects (or observations or cases) in the data set under analysis, θ the vector of model parameters, and L(⋅; θ) a likelihood function or a likelihood-like function (e.g., the extended quasi-likelihood function used with generalized linear models [47]). Randomly partition S into k > 1 disjoint nonempty subsets S(h), called folds, for h ∈ H = {h : 1 ≤ h ≤ k}. The LCV score is defined as
$$ \mathrm{L}\mathrm{C}\mathrm{V}={\displaystyle \prod_{h\in H}}{L}^{\frac{1}{n}}\left( S(h);\boldsymbol{\theta} \Big( S\backslash S(h)\right) $$
where L(S(h); θ(S\S(h))) denotes the joint likelihood for the observations with indexes in S(h) using the maximum likelihood estimate θ(S\S(h)) of the parameter vector θ computed using the data in the complement S\S(h) of the fold S(h). For a given data set, the same random fold assignment is used for all models so that the LCV scores for those models are comparable. LCV scores for multivariate data are normalized by the number of outcome measurements for all subjects rather than by the number of subjects.
Larger LCV scores indicate better models, but not necessarily substantially better models. This issue is assessed with LCV ratio tests, analogous to likelihood ratio tests in being based on the χ2 distribution. LCV ratio tests are expressed in terms of a cutoff for a substantial (or distinct or significant) percent decrease in the LCV score, changing with the sample size. The formula for the cutoff is provided in Section 4.4.2 of Knafl and Ding [34] and in eq. (6) of Knafl et al. [36]. LCV ratio tests are more conservative than standard tests for zero coefficients (examples are provided in [48–50]) in the sense that removal from the model of a predictor with a significant slope does not always generate a substantial percent decrease in the LCV score. Consequently, LCV ratio tests are similar in effect to adjustments for multiple comparisons.
As an example, suppose that a model\( {M}_1 \) generates a score \( \mathrm{L}\mathrm{C}\mathrm{V}\left({M}_1\right) \) smaller than the score \( \mathrm{L}\mathrm{C}\mathrm{V}\left({M}_2\right) \) for another model \( {M}_2 \). If the percent decrease in these LCV scores, that is,
$$ \frac{\mathrm{LCV}\left({M}_2\right)-\mathrm{LCV}\left({M}_1\right)}{\mathrm{LCV}\left({M}_2\right)}\cdot 100\%, $$
is larger than the cutoff for a substantial percent decrease, then model M 2 substantially improves on model M 1. On the other hand, if the percent decrease is less than or equal to the cutoff, model M 1 is a competitive alternative to model M 2. If model M 1 is also simpler (e.g., based on fewer parameters or containing no versus some interactions) than model M 2, then it would be preferable to model M 2 as a parsimonious, competitive alternative.
Overview of the adaptive modeling process for a fixed outcome
Adaptive fractional polynomial models for a fixed outcome in terms of primary predictors are identified using a heuristic search process beginning with an expansion phase, systematically adding power transforms of those primary predictors into the model, followed by a contraction phase, systematically removing any extraneous power transforms and adjusting the powers of the remaining transforms to improve the LCV score. The search process is controlled by tolerance parameters indicating the change in LCV scores that can be tolerated for each step in the search process. For example, the contraction stopping tolerance parameter setting is based on a LCV ratio test so that the final selected model is parsimonious. The tolerance parameter settings change with the number of measurements in the sample, thereby adjusting the search process by the sample size. The full adaptive modeling process is formulated in Chapter 20 of [34]. That full process is needed for estimating models (4)-(5) with arbitrary nonlinearity. Estimation of the monotonic models of (9)-(10) requires a simpler search process as addressed later.
The adaptive modeling process can generate adaptive models for variances (or more generally dispersions [47]) as well as for means. For example, when M and/or Y are continuous, the omitted factors or errors U M and U Y can be treated as normally distributed (or at least approximately so) with non-constant variances that are functions of X, M, and/or some other primary predictors. The log of the variances is modeled as linear in the coefficient parameters for possibly power transformed primary predictors. Coefficient parameters for the means and variances are estimated jointly using maximum likelihood with likelihoods based on the normal distribution. For correlated continuous outcomes, for example, outcomes measured over clusters such as family members or patients of the same provider, likelihoods are based on the multivariate normal distribution.
Adaptive nonlinear moderation can be addressed simply by including interactions as primary predictors. More generally, the adaptive modeling process can automatically generate geometric combinations of two or more primary predictors, that is, products of power transforms of distinct subsets of the primary predictors using possibly different powers to transform those primary predictors. The subset of transforms in the geometric combinations and their powers are generated adaptively.
Searching for power transforms
A base model M 0 is expanded to include a transform of a predictor X as follows. Let M 0(p) denote the model M 0 with the power transform X p added to it. A grid search is conducted first to maximize LCV(M 0(p)) over an initial set of powers. By default, the initial powers p = −3, −2.5, ⋯, −0.5, 0.5, ⋯, 2.5, 3 are used, but this set can be adjusted if desired. The power 0 is purposely not considered. For the case with X > 0, the effect of p = 0 on M 0 depends on whether or not M 0 includes an intercept. When there is an intercept, the power 0 corresponds to the natural log transform (demonstrated by taking the limit as p → 0); otherwise it corresponds to the constant transform adding in an intercept parameter to M 0. When X has zero or negative values, the effect is more complex. Not considering the power 0 avoids slowing the adaptive modeling process to check M 0 to see what the effect of that power is. In any case, powers p close to 0 approximate the appropriate case without having to check to see which one it is. By default, the smallest powers in absolute value that are considered are ±0.0001, but this can be adjusted.
Let p 0 denote the power which maximizes LCV(M 0(p)) for this initial set of powers. When p 0 = −3, a search is conducted over powers p = p 0 + i ⋅ δ for integer multiples i of δ = −1 until
$$ \mathrm{L}\mathrm{C}\mathrm{V}\left({M}_0\left({p}_0+\left( i+1\right)\cdot \delta \right)\right)<\mathrm{L}\mathrm{C}\mathrm{V}\left({M}_0\left({p}_0+ i\cdot \delta \right)\right)>\mathrm{L}\mathrm{C}\mathrm{V}\left({M}_0\left({p}_0+\left( i-1\right)\cdot \delta \right)\right) $$
When p 0 = 3, δ is set to +1 instead. These produce powers p 1 = p 0 + i ⋅ δ with i ≥ 0 for these two cases. For −3 < p 0 < 3, p 1 = p 0. The choice of an initial power for X is given by p 1.
Next the choice of a power p 2 for changes in powers of δ = ±0.1 is identified through a similar search on either side of p 1. Then, this is iterated searching around p 2 over changes δ = ±0.01, then δ = ±0.001, and so on. By default, changes of at most δ = ±0.0001 are considered, but this can be adjusted. At any stage of this process, if the smallest of the three LCV scores analogous to those of inequality (11) compared to the largest of those three scores generates a percent decrease greater than the associated tolerance parameter (indicating these three LCV scores are not close to each other), continue the search with one more decimal digit; otherwise stop the process. When the process stops at the j th stage, the selected model is M 0(p j ).
By default, the expansion stopping tolerance is set generously to 2.5 times the cutoff for a LCV ratio test, and so it is likely a transform of X would be added to the base model. However, it is also possible that the expansion might not add a transform of X to the base model, supporting the conclusion that X does not have an effect on the outcome Y after controlling for the transforms of the base model.
Identification of a transform of a predictor to add to a base model is a small part of the full adaptive modeling process. Multiple predictors need to be considered as part of the expansion; the best transform to remove from a base model needs to be determined as part of the contraction. With each such removal, the powers of the remaining transforms need to be adjusted to improve the LCV score. However, models (9)-(10) with p " = 1 and p and q fixed require identification of only the single power q ' for X in (9) and the single power p ' for X in (10). An algorithm for identifying choices for these two powers is defined later.
For the general adaptive modeling process, tests for zero coefficients for transforms in selected models are inappropriate to conduct since these tests are usually significant. Due to the contraction heuristics, the removal of each transform of the selected model generates a substantial percent decrease in the LCV score. However, this is not the case for models (9)-(10) based on single transforms of X. While the selected power transform for X in one of these models generates an optimal LCV score, the LCV score for the associated constant base model might not generate a substantial percent decrease in the LCV score, and so the slope for that selected power transform might not always be significantly nonzero. However, a LCV ratio test can also be used to assess the impact of including X in the model versus not including it and is likely to be more conservative than the test for a zero slope for X.
Setting the number of folds
Knafl et al. [35] used\( 10 \) folds for estimating nonlinear individual-subject medication adherence curves on the recommendation of Kohavi [51]. However, the data sets they used had limited sample sizes at most \( 100. \) The choice of the number k of folds may need to be adjusted for different sample sizes. Knafl and Grey [52] investigated this issue for exploratory factor analysis models. They found that for three different sets of items the same number of factors was selected by maximizing LCV scores as long as the value of k was not too small. Consequently, they recommended using the first local maximum in k over multiples of 5 folds for some benchmark analysis, in their case the selection of the number of factors extracted through maximum likelihood. This choice balances the need for a sufficiently large number of folds while limiting the amount of computation. Section 2.8 of [34] provides a more complete assessment of this issue.
The composite mediation model
Models (9)-(10) with p " = 1 can be combined into a single bivariate outcome model as follows. With superscript T denoting the transpose operator, let Y ' = (M q Y p)T, I 1 = (1 0)T, I 2 = (0 1)T, X 1 = (X 0)T, X 2 = (0 X)T, M = (0 M)T, and U = (U M U Y )T. The model
$$ {\boldsymbol{Y}}^{\hbox{'}}={\beta}_1\cdot {\boldsymbol{I}}_1+{\beta}_2\cdot {\boldsymbol{I}}_2+{\beta}_3\cdot {\boldsymbol{X}}_1^{q\hbox{'}}+{\beta}_4\cdot {\boldsymbol{X}}_2^{p\hbox{'}}+{\beta}_5\cdot {\boldsymbol{M}}^q+\boldsymbol{U}, $$
where a power transform of a vector is the vector of its entries transformed by that power, satisfies β 1 = i M , β 2 = i Y , β 3 = a, β 4 = c ', and β 5 = b, and so provides a single model for the five coefficient parameters for the means of models (9)-(10). Assume that U is bivariate normally distributed with covariance matrix Σ, thereby allowing for possibly correlated omitted factors or errors. Model (12) is a path model nonlinear in the outcome Y, the mediator M, and the predictor X with Y and M measured with error and X measured without error.
Let θ be the vector of model parameters, including β j 1 ≤ j ≤ 5 and all the parameters for modeling the covariance matrix Σ. Using subscripts s for s ∈ S as defined earlier, the likelihood L(S; θ) satisfies \( \log L\left( S;\boldsymbol{\theta} \right)={\displaystyle \prod_{s\in S}}{\ell}_s \) where
$$ {\ell}_s=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\cdot {\boldsymbol{u}}_s^T\cdot {\boldsymbol{\varSigma}}_s^{-1}\cdot {\boldsymbol{u}}_s-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\cdot \log \left(\left|{\boldsymbol{\varSigma}}_s\right|\right)-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\cdot 2\cdot \log \left(2\cdot \pi \right), $$
$$ {\boldsymbol{\mu}}_s={\beta}_1\cdot {\boldsymbol{I}}_1+{\beta}_2\cdot {\boldsymbol{I}}_2+{\beta}_3\cdot {\boldsymbol{X}}_{1, s}^{q^{\hbox{'}}}+{\beta}_4\cdot {\boldsymbol{X}}_{2, s}^{p^{\hbox{'}}}+{\beta}_5\cdot {\boldsymbol{M}}_s^q, $$
Σ s is the covariance matrix for the s th observation, |Σ s | its determinant, and u s = Y ' s − μ s the associated residual vector. For model (12), there is only one correlation parameter ρ that is the same for all s, but each of the variances for U M,s and for U Y,s might change with s or be the same for all s.
When the omitted factors or errors are independent, that is, when ρ = 0,
$$ L\left( S;\boldsymbol{\theta} \right)={L}_M\left( S;\boldsymbol{\theta} \right)\cdot {L}_Y\left( S;\boldsymbol{\theta} \right) $$
separates into two terms corresponding to the likelihoods L M (S; θ) and L Y (S; θ) for models (9) and (10), respectively. The LCV score for model (12) also separates, but into
$$ \mathrm{L}\mathrm{C}\mathrm{V}={\mathrm{LCV}}_M^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cdot {\mathrm{LCV}}_Y^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
where LCV M and LCV Y are LCV scores for models (9) and (10), respectively. This holds because the LCV score for model (12) is normalized by the number 2 ⋅ n of outcome measurements while LCV M and LCV Y each are normalized by the number n of subjects. If q is fixed (e.g., with q = 1 so models (9)-(10) are linear in M), maximum likelihood estimation and adaptive modeling of models (9) and (10) separately generate the same results as for modeling them in combination using model (12), assuming consistent fold assignments for these two cases. However, this only holds when q is fixed and ρ = 0. Even when ρ = 0, identification of an appropriate value for q requires comparing LCV scores for the composite model (12) since the same value of q is used in its submodels (9)-(10).
Identifying power transforms for \( \mathrm{X} \) in the composite model
For fixed choices for the powers p and q, the adaptive expansion process is constrained to generate a single transform for X 1 and for X 2 as follows. Let M 0 be the base model with means based only on I 1 and I 2 along with a fixed choice for the covariance matrix of U M and U Y . Let M 0(q ' 1) be this base model expanded to include a power transform of X 1 with selected power q ' 1 and M 0(p ' 1) expanded to include a power transform of X 2 with selected power p '1. If
$$ \mathrm{L}\mathrm{C}\mathrm{V}\left({M}_0\left( p{\mathit{\hbox{'}}}_1\right)\right)>\mathrm{L}\mathrm{C}\mathrm{V}\left({M}_0\left( q{\mathit{\hbox{'}}}_1\right)\right), $$
then expand M 0(p ' 1) to include a power transform of X 1 with power q ' 2, possibly different than q ' 1. Otherwise, expand M 0(q ' 1) to include a power transform of X 2 with power p ' 2, possibly different than p ' 1. This is the standard adaptive expansion process constrained to include a single power transform of primary predictors X 1 and X 2 for the means.
Since power transforms are added to the model without adjusting the powers of previously added transforms, there might be an improvement if these powers are adjusted. The formal power transform process is defined in Section 20.4.4 of [34]. Informally, each power is adjusted using the power adjustment process described earlier but starting at its current value holding the other powers fixed. Stop if no adjusted powers provide an improvement in the LCV score. Otherwise, continue the process using the adjusted power generating the best improvement in the LCV score. However, since the generated model (12) for fixed p and q has only two transforms, the improvement if any is unlikely to be substantial. Consequently, power adjustment is treated as an optional feature of the adaptive monotonic mediation process.
The model for the variances (technically, the model for the log of the variances) based on I 1 and I 2 results in separate but constant variances for U M and U Y . The associated covariance matrix also allowing for the single general correlation ρ is called compound symmetry heterogeneous (CSH). Non-constant variances can be generated by considering X 1, X 2, and/or M as primary predictors for modeling the variances. The full adaptive modeling process can be used for modeling the variances since these need not be monotonic. This is achieved by including I 1 and I 2 in the base model for the variances, not restricting the expansion of the model for the variances, and restricting the contraction to contract only transforms from the model for the variances. By default, the contraction considers removal of the intercepts corresponding to I 1 and I 2 in the model for the variances, but this can be overridden.
Covariates can be included in model (12). For each covariate Z, include Z 1 = (Z 0)T and Z 2 = (0 Z)T as primary predictors for the means and/or variances, respectively, addressing models (9)-(10). The full adaptive modeling process can be used to generate power transforms of Z 1 and Z 2 for modeling the means and/or variances.
Power-adjusted likelihood cross-validation
Assume that Y > 0 so that transforms Y p are well-defined for all p. Model (10) for Y p for a fixed power\( p \) can be estimated using existing adaptive regression methods as described above. Standard LCV scores for these models, not accounting for the power p, as defined earlier are based on the normal density function. The power \( p \) can be chosen by maximizing an alternative power-adjusted LCV score that also accounts for power transformation of the outcome Y using the power-adjusted likelihood function as defined next.
If Y p is normally distributed for p > 0, the distribution function F(y; p, θ) for Y satisfies
$$ F\left( y; p,\boldsymbol{\theta} \right)= P\left( Y\le y; p,\boldsymbol{\theta} \right)= P\left({Y}^p\le {y}^p;\boldsymbol{\theta} \right) $$
where θ is the vector of model parameters. Consequently, the power-adjusted density function f(y; p, θ) for Y satisfies
$$ f\left( y; p,\boldsymbol{\theta} \right)=\frac{dP\left( Y\le y; p,\boldsymbol{\theta} \right)}{dy}= p\cdot {y}^{p-1}\frac{dP\left({Y}^p\le {y}^p;\boldsymbol{\theta} \right)}{dy} $$
where \( \frac{dP\left({Y}^p\le {y}^p;\boldsymbol{\theta} \right)}{dy} \) is the usual univariate normal density function φ(v, θ) evaluated at v = y p with mean and variance based on the parameter vector θ. When p < 0,
$$ F\left( y; p,\boldsymbol{\theta} \right)= P\left( Y\le y; p,\boldsymbol{\theta} \right)= P\left({Y}^p\ge {y}^p;\boldsymbol{\theta} \right) $$
and the power-adjusted density function f(y; p, θ) for Y satisfies
$$ f\left( y; p,\boldsymbol{\theta} \right)=\frac{dP\left( Y\le y; p,\boldsymbol{\theta} \right)}{dy}=- p\cdot {y}^{p-1}\cdot \varphi \left({y}^p;\boldsymbol{\theta} \right). $$
Thus, for p ≠ 0,
$$ f\left( y; p,\boldsymbol{\theta} \right)=\left| p\right|\cdot {y}^{p-1}\cdot \varphi \left({y}^p;\boldsymbol{\theta} \right). $$
A similar argument for p = 0 corresponding to the natural log transform gives
$$ f\left( y; p,\boldsymbol{\theta} \right)={y}^{-1}\cdot \varphi \left( \log y;\boldsymbol{\theta} \right). $$
The power-adjusted LCV score is defined as
$$ \mathrm{L}\mathrm{C}\mathrm{V}(p)={\displaystyle \prod_{h\in H}}{f}^{\frac{1}{n}}\left( S(h); p,\boldsymbol{\theta} \left( S\backslash S(h); p\right)\right) $$
where f(S(h); p, θ(S\S(h); p)) denotes the joint power-adjusted likelihood for the subjects with indexes in fold S(h) computed with estimates θ(S\S(h); p) of the parameter vector θ generated using the data in the complement S S(h) of the fold S(h) and with the outcome Y transformed to Y p. The LCV(p) score can be maximized in the power p using a grid search to choose an appropriate power transform for the outcome.
The assumption that Y > 0 can be relaxed by extending Y p in the same way as for X p given earlier, that is, by setting it to 0 when Y = 0 and to cos(π ⋅ p) ⋅ |Y|p when Y < 0. Then, the sign is not always reversed for the case Y < 0, affecting the computation of f(y; p, θ). However, the derivative f(0; p, θ) is not always well-defined. For example, when p < 1 and y > 0,
$$ f\left( y; p,\boldsymbol{\theta} \right)= p\cdot {y}^{p-1}\cdot \varphi \left({y}^p;\boldsymbol{\theta} \right)\uparrow \infty\ \mathrm{a}\mathrm{s}\ \mathrm{y}\downarrow 0. $$
It seems better to add a constant to Y to make it positive valued, which is the approach recommended by Royston and Altman [33] for transforming non-positive predictors.
The univariate outcome transformation process can be applied as well to model (9). The formulation also extends readily to multivariate data. When those data are based on repeatedly measuring the same outcome Y at different times or over different conditions (such as members of the same family or patients of the same provider), it is reasonable to transform each such outcome measurement with the same power p. However, model (12) requires consideration of different powers p and q for Y and M with associated power-adjusted LCV scores LCV(p, q).
Identification of appropriate choices for \( p \) and q can be achieved by starting with p = q = 1 and using grid searches to adjust p with q = 1 fixed giving LCV(p 1, 1) and to adjust q with p = 1 fixed giving LCV(1, q 1). If LCV(p 1, 1) > LCV(1, q 1), use a grid search in q with p = p 1; otherwise use a grid search in p with q = q 1. This generates powers p 2 and q 2. In example analyses, the grid searches are first conducted over changes of ±0.5 generating the powers p 2 and q 2, then over changes of ±0.1 starting at the powers p 2 and q 2 generating the powers p 3 and q 3, and then stops identifying powers for Y and M to within one decimal digit.
Monotonic mediation analysis
In what follows, unless otherwise stated, base models for means and covariances are based on the predictors I 1 and I 2, and so with only intercepts i M and i Y for the means along with CSH covariance structure. Also assume p " = 1 unless otherwise indicated.
Selecting the number k of folds.
The benchmark analysis is the generation of model (12) constrained so that p = q = 1, that is, with untransformed M and Y, by adaptively expanding the model for the means in X 1 and X 2 limiting the number of power transforms for each of these predictors to one. The number of folds to use in all subsequent analyses is the one generating the first local maximum in the standard LCV score over multiples of 5.
Selecting the powers p and q.
Use the search through alternative values for p and q based on power-adjusted LCV scores LCV(p,q) described earlier to generate the full model (12) without changing the CSH covariance structure. Use the selected powers p and q in subsequent analyses unless otherwise indicated.
Assessing the need for transforming M and Y.
Use a LCV ratio test to compare LCV(1,1) (the same as its standard LCV score) generated in Step 1 to LCV(p,q) generated in Step 2.
Assessing the need for transforming X.
For given values of p and q, use a LCV ratio test to compare LCV(p,q) as generated in Step 2, with X 1 and X 2 adaptively transformed, to the LCV(p,q) score for the associated model linear in X 1 and X 2.
When the power q ' = 1, the instantaneous natural indirect effect \( \frac{dNIE(x)}{dx}= a\cdot b \) is constant. Whether the instantaneous natural indirect effect is reasonably treated as constant can be assessed with a LCV ratio test comparing a given model with q ' ≠ 1 to that model adjusted to satisfy q ' = 1.
Assessing mediation relationships.
In the linear mediation case of models (1)-(3), mediation requires significantly nonzero slopes a and b. In the monotonic mediation context of model (12), these issues can be addressed with LCV ratio tests. To assess for a dependence of M q on X in the component model (9), compare the LCV(p,q) score for the full model (12) to the associated model with X 1 removed (or with a = 0) and the transform for X 2 adaptively generated. To assess the dependence of Y p on M q in the component model (10) with p " = 1, compare the LCV(p,q) score for the full model (12) to the associated model with M q removed (or with b = 0) and the transforms for X 1 and X 2 adaptively generated. It is also possible to assess the dependence of Y q on \( X \) in the component model (10) by comparing the LCV(p,q) score for the full model (12) to the associated model with X 2 removed (or with c ' = 0) and the transform for X 1 adaptively generated.
Considering non-constant variances.
With base model the adaptive model (12) with CSH covariance structure generated in Step 2 with initial powers p ' and q ' for X 1 and X 2, adaptively expand and then contract the model for the variances in X 1, X 2, and M, allowing for possible adjustment of the powers p ' and q ' of the base model as part of the contraction, but leaving M q in the base model untransformed (so that p " = 1).
Considering covariates.
Covariates can be considered for inclusion in the component models (9)-(10) of model (12). With base model the adaptive model of Step 2, adaptively expand and then contract the model for the means and the variances in Z 1 and Z 2 for all covariates Z. As part of the contraction, allow for possible adjustment of the powers p ' and q ' of the base model, leave M q in the base model untransformed (so that p " = 1), and only contract transforms of covariates from the model for the means. This can be combined with Step 6 to allow the variances to depend as well on X 1, X 2, and M.
Assessing ρ = 0.
For any model generated earlier, compare its LCV score using a LCV ratio test to the associated model constrained to satisfy ρ = 0.
Assessing p " = 1
With base model the adaptive model of Step 2, adaptively retransform the model allowing the powers of the single transforms of X 1, X 2, and M q to be changed to improve the LCV score. The assumption p " = 1 is supported if the associated model generated with p " = 1 in Step 2 is a competitive alternative to this latter model. Similar assessments can be conducted allowing for non-constant variances as in Step 6, for covariates as in Step 7, and/or ρ = 0 as in Step 8.
Assessment of Model Assumptions
Yuan and MacKinnon [26] provide a detailed discussion of the impact of violations of the constant variances and normality assumptions of standard regression models. Modeling of variances can address the first problem. Data transformation, applied to outcomes and/or predictors, as considered here can sometimes resolve normality problems, but not always. Consequently, model assumption assessments are important to conduct in general regression contexts including the special case of mediation analyses, whether treated as linear or nonlinear. Should data transformation not resolve such problems, then quantile regression methods as described by Yuan and MacKinnon [26] are more appropriate to use. However, if data transformation resolves model assumption problems, then normality-based methods are optimal [26] when applied to the transformed data and so would likely generate more efficient and powerful estimates.
In the case of mediation analyses, the use of bootstrapped CIs on indirect effects circumvents distributional assumption problems for parametric estimates of those effects. However, as demonstrated by the simulations of [26], bootstrap methods cannot fully address distributional assumption problems for the data. Moreover, bootstrapped CIs are likely to be relatively narrower, and so more precise, when data are transformed to be as close as possible to normal, than when untransformed. The example analyses reported later provide two examples supporting this conclusion. This, of course, assumes that the indirect effect has been consistently estimated so that the true value is in the confidence interval.
For composite model (12), the constant variances assumption can be addressed with a LCV ratio test comparing constant and non-constant variances models to assess whether variances are reasonably close to constant or are distinctly non-constant. If a sufficiently broad set of primary predictors for the variances are considered, the associated non-constant variances model should provide an appropriate depiction of the variances, thereby relaxing the constant variances assumption if necessary. These variances estimates combined with the estimated correlation provide estimates Σ s (S) of the covariance matrices for observed outcome vectors \( {\boldsymbol{Y}}_{\boldsymbol{s}}^{\mathit{\hbox{'}}} \) for subjects with indexes s in the set S. Associated residual vectors are u s (S) = \( {\boldsymbol{Y}}_{\boldsymbol{s}}^{\mathit{\hbox{'}}} \) − μ s (S) where μ s (S) are estimated mean vectors for subjects \( s \) based on X 1s , X 2s , M s, and possibly covariates. Associated standardized or scaled residuals are given by stdu s (S) = (V s T (S))− 1 · u s (S) where V s (S) is the square root of Σ s (S) determined by its Cholesky decomposition. The combined standardized residuals over all s can be used to assess the normality assumption by visually checking for linearity in the associated normal (probability) plot and with the Shapiro-Wilk test for normality of the standardized residuals.
Assessing natural indirect effects
Once an appropriate choice for composite model (12) with p " = 1 has been identified, possibly including non-constant variances and/or covariates, the assessment of whether the instantaneous natural indirect effect function \( \frac{dNIE(x)}{dx} \) for this model is nonzero needs to be assessed. This can be addressed with bootstrapped CIs [21, 29, 40, 41], but computed for a grid of possible values x for the predictor X. For models with a constant instantaneous natural indirect effect (i.e., with q ' = 1), only one value for X need be considered. The bias-corrected version [42] is recommended by MacKinnon et al. [40], and so is used in example analyses unless otherwise indicated. All reported CIs are based on 1,000 resamples.
The powers p, q, p ', and q ' for a composite model (12) as well as powers for all covariate predictors and variance predictors are held fixed with associated slope parameters estimated for resamples of the composite data. The generated 95% CI for the instantaneous natural indirect effect function at each nonzero value of \( x \) has lower and upper bounds
$$ L(x)={b}_L\cdot {a}_L\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}< U(x)={b}_U\cdot {a}_U\cdot q\mathit{\hbox{'}}\cdot {x}^{q\mathit{\hbox{'}}-1}. $$
Define the normalized width W of these intervals as
$$ W=\frac{ \max \left(\left| L(x)\right|,\left| U(x)\right|\right)- \min \left(\left| L(x)\right|,\left| U(x)\right|\right)}{ \max \left(\left| L(x)\right|,\left| U(x)\right|\right)}=1-\frac{ \min \left(\left|{b}_L\cdot {a}_L\right|,\left|{b}_U\cdot {a}_U\right|\right)}{ \max \left(\left|{b}_L\cdot {a}_L\right|,\left|{b}_U\cdot {a}_U\right|\right)}, $$
which is constant in nonzero \( x \) with a value between 0 and 1. Models for the data generating smaller values for W provide more precise predictions of the instantaneous natural indirect effect function.
Moderated monotonic mediation
One of the covariates Z might be considered as a moderator. There are a variety of ways that models (9)-(10) can be adjusted to accommodate moderation. For example, Preacher, Rucker, and Hayes [53] propose five alternatives. Under their fifth alternative, Z moderates the effect of X on M, X on Y, and M on Y. Under this alternative, models (9)-(10) become possibly with other covariates included where is the set of all possible values z for Z and I(Z = z) the indicator for Z = z (i.e., it equals 1 when Z = z and 0 otherwise). The dependence of M on X and Y on X and M have been defined separately for the values z of Z taking an analysis of variance approach. This formulation allows associated intercepts, slopes, and powers to change with the values z of Z while preserving monotonicity. However, it requires estimation of model parameters for each value z of Z. This requirement is reasonable for moderators Z with discrete numbers of possible values, but problematic for continuous moderators Z with many possible values z but sparse numbers of observations for some values z. In this latter case, the moderator Z could be replaced by a split based on its tertiles, quartiles, etc.
Under (13)-(14), the instantaneous natural direct effect \( {c}^{\mathit{\hbox{'}}}( z)\cdot {p}^{\mathit{\hbox{'}}}( z)\cdot {X}^{p^{\mathit{\hbox{'}}}(z)-1} \) (assuming for simplicity that p ' (z) ≠ 0) changes with the values z of Z. When p " (z)≡1, the instantaneous natural indirect effect b(z) ⋅ a(z) ⋅ q′(z) ⋅ X (q ' (z) − 1) (assuming for simplicity that q′(z) ≠ 0) also changes with the values z of Z. The associated normalized widths W(z) change with z as well.
Model (12) generalizes to where
$$ \boldsymbol{H}( z)={\beta}_1( z)\cdot {\boldsymbol{I}}_1+{\beta}_2( z)\cdot {\boldsymbol{I}}_2+{\beta}_3( z)\cdot {\boldsymbol{X}}_1^{q^{\prime }(z)}+{\beta}_4( z)\cdot {\boldsymbol{X}}_2^{p^{\prime }(z)}+{\beta}_5( z)\cdot {\boldsymbol{M}}^q. $$
The adaptive modeling process can be used to identify the powers q ' (z) and p ' (z) for all z combined.
The individual moderation components of model (15) can be assessed by comparing model (15) to the model with each of those moderation component removed using LCV ratio tests. Specifically, moderation of the effect of X on M in model (13) can be assessed by replacing in (15) with β 3 ⋅ X 1 q ' . Moderation of the effect of X on Y in model (14) can be assessed by replacing in (15) with β 4 ⋅ X 2 p ' . Moderation of the effect of M on Y in model (14) can be assessed by replacing in (15) with β 5 ⋅ M q.
It is also possible to test effects for specific values z ' of Z. Specifically, the effect of X on M in model (13) for the value z′ of Z can be assessed by replacing in (15) with . The effect of X on Y in model (14) for the value\( z^{\prime } \) of Z can be assessed by replacing in (15) with . The effect of M on Y in model (14) for the value z′ of Z can be assessed by replacing in (15) with .
Data on family management of childhood chronic conditions
Example analyses are reported later using a subset of data from a cross-sectional study on family management of childhood chronic conditions [54] reported by 187 partnered mothers. General family functioning is measured using the General Functioning Scale of the McMaster Family Assessment Device [55], coded so that larger values indicate better family functioning with range 1–4. Difficulty managing the child's condition is measured by the family life difficulty scale of the Family Management Measure [54], coded so that larger scores mean more difficulty. This scale measures the extent to which having a child with a chronic condition makes family life more difficult. Child adaptation, in terms of the intensity of the child's conduct-disordered behavior, is measured using the Eyberg Child Behavior Inventory [56], coded so that larger values indicate better child adaptation or less conduct-disordered behavior.
Example analyses use these family management data to demonstrate nonlinear mediation analyses by considering mediation of the impact of family functioning X on child adaptation Y by difficulty M in managing the condition. The cutoff for a substantial percent decrease for these data using models (12) and (15) with 374 = 2 ⋅ 187 measurements is 0.51%. Example analyses assume p " = 1 or p " (z) = 1 for all values z of a moderator Z unless otherwise stated.
The proposed mediation relationships for these observational data can be justified on the following basis. General family functioning would be developed by a family prior to the diagnosis of the child's chronic condition, which would affect how difficult that chronic condition makes family life which would then affect the child's adaption to the condition. However, the purpose of these analyses is to provide example mediation analyses not to establish mediation in this context.
Simulated mediation data
Data were simulated for 101 observations with equally spaced values for the predictor X sim between 1 and 10 (i.e., 0.09 units apart) with mediator M sim = 1 + X sim + U M , outcome Y sim = Y ' sim 0.4 , and
$$ Y{\mathit{\hbox{'}}}_{sim}=\frac{5+{X}_{sim}+{M}_{sim}+{U}_Y}{25}, $$
where U M and U Y are independent standard normal random variables. Y sim was computed by raising Y ' sim to the power 2.5; the normalizing value 25 used in computing Y ' sim was chosen to be the smallest integer value larger than the maximum generated value for the unnormalized Y ' sim values. Normalizing the values of Y ' sim avoids generating very large values for Y sim . The true values for the powers are p = 0.4, q = 1, p ' = 1, and q ' = 1 with true constant instantaneous natural indirect effect \( a\cdot b=1\cdot \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$25$}\right.=0.04 \). The cutoff for a substantial percent decrease for these data using model (12) with 202 = 2 ⋅ 101 measurements is 0.95%.
Analyses at the beginning of this section consider mediation of the effect of family functioning on child adaptation to a childhood chronic condition by the difficulty in managing that condition. Intercept parameters for the means and variances are constrained in all analyses not to be removed as part of contractions. In computing LCV scores, Y p and M q measurements for the same mother are randomly assigned to the same fold. Analyses are also conducted at the end of this section using the simulated mediation data.
Selecting the number of folds
For model (12) applied to the family management data with p = q = 1 and CSH covariance structure, the adaptive model in X 1 and X 2 generates a first local maximum at k = 10 with \( 10 \)-fold LCV score \( 0.015024 \) and selected powers p ' = −3 and q ' = 2. Using k = 5, the selected powers are p ' = 3 and q ' = 1.5 with \( 10 \)-fold LCV \( 0.015018 \) and insubstantial percent decrease in the LCV score compared to the model selected with k = 10 of 0.04% (i.e., less than the cutoff of 0.51% for the data). Using k = 15, the selected powers are p ' =−1 and q ' = 1.5 with \( 10 \)-fold LCV \( 0.015005 \) and insubstantial percent decrease in the LCV score of 0.13%. Consequently, the generated model is reasonably robust to the choice of the number of folds. Subsequent analyses all use k = 10 folds for computing standard as well as power-adjusted LCV scores.
Using\( 10 \) folds, the number of measurements per fold ranges from \( 26 \) to \( 56 \) for \( 13 \) to \( 28 \) mothers. Consequently, fold complements contain at least \( 318 \) or \( 85.0\% \) of the \( 374 \) available measurements, and so deleted parameter estimates should be reasonably reliable.
Selecting the powers \( p \) and \( q \) for \( Y \) and \( M \)
For model (12) with CSH covariance structure, the adaptive search for models in \( {\boldsymbol{X}}_1 \) and \( {\boldsymbol{X}}_2 \) with varying choices for \( p \) and \( q \), first selects the powers p = 1 and q = 0 (i.e., the log transform) over changes in these powers of ±0.5 with LCV(1, 0) = 0.015683 and then the powers p = 1.3 and q = 0 over changes in these powers of ±0.1 with LCV(1.3,0) = 0.015776. For untransformed Y and M with p = q = 1, LCV(1, 1) = 0.015024 (the same as its standard LCV score reported above), and so the percent decrease is substantial at \( 4.77\% \). Consequently, the mediation relationships are distinctly nonlinear in \( Y \) and \( M \). Model (12) for the standard linear mediation model with p = q = p ' = q ' = 1 has even smaller LCV(1,1) score 0.014958, and so is substantially improved upon by consideration of monotonicity.
Furthermore, the adaptively generated model with\( Y \) transformed and \( M \) held untransformed, that is, with q = 1, generates the powers p = 1.4, p ' = −5, and q ' = 2 with LCV(1.4,1) score 0.015143 and substantial percent decrease compared to the model with p = 1.3 and q = 0 of 4.01%. Consequently, relationships (9)-(10) are distinctly nonlinear in \( M \). Moreover, the adaptively generated model with \( Y \) untransformed, that is, with p = 1, generates the powers q = 0, p ' = −2, and q ' = 2.4 with LCV(1,0) score 0.15683 and substantial percent decrease compared to the model with p = 1.3 and q = 0 of 0.59%. Hence, relationship (10) is distinctly nonlinear in Y.
Assessing ρ = 0
Using p = 1.3, q = 0, p ' = −2.5, and q ' = 3.5 as selected with the CSH covariance structure, the estimated correlation is −0.08. Rerunning this model with ρ = 0, the LCV(1.3,0) score is 0.015804. Using p = 1.3 and q = 0 as selected with CSH covariance structure, but with ρ = 0, the adaptively generated model (12) has somewhat different powers p ' = −3 and q ' = 3 and LCV(1.3,0) score that also rounds to 0.015804. Since this score is larger than the LCV(1.3,0) score under CSH, the omitted variables or errors U M and U Y are reasonably treated as independent. Using the values of p and q selected under CSH reduces the computations compared to identifying adaptive values for p and q under ρ = 0. However, in this case, the same powers p = 1.3 and q = 0 are adaptively identified with ρ = 0 starting from p = q = 1. An alternative approach would be to start the search for p and q with ρ = 0 at the values generated for CSH while searching over grids of \( \pm 0.1 \) to reduce the computations. This also generates the same solution \( p=1.3 \) and \( q=0 \).
For the case \( p= q=1 \) with selected powers \( {p}^{\hbox{'}}=-3 \) and \( {q}^{\hbox{'}}=2 \), a similar result holds. The estimated correlation is \( -0.05 \) and the model with \( \rho =0 \) has LCV score \( 0.015043 \), larger than the score \( 0.015024 \) reported earlier for the associated model with CSH covariance structure. Subsequent analyses use \( \rho =0 \) since it provides an improvement in these two cases.
Assessing a = 0
The adaptively generated model with \( p=1.3 \), \( q=0 \), \( \rho =0 \), and constrained not to include a transform of \( {\boldsymbol{X}}_1 \) in the model for the means has model for the means depending on \( {\boldsymbol{X}}_2 \) transformed by the power p' = −3 with LCV(1.3,0) score \( 0.015077 \) and substantial percent decrease \( 4.60\% \) compared to the model also including a transform of \( {\boldsymbol{X}}_1. \) Consequently, transformed difficulty \( \log M \) in model (9) is reasonably assumed to depend substantially on family functioning \( X \).
Assessing \( b=0 \)
The adaptively generated model with \( p=1.3 \), \( q=0 \), \( \rho =0 \), and constrained not to include the transform \( {\boldsymbol{M}}^q \) in the model for the means has model for the means depending on \( {\boldsymbol{X}}_1 \) and \( {\boldsymbol{X}}_2 \) transformed by the powers p′ = −0.4 and q′ = 3, respectively, with LCV(1.3,0) and substantial percent decrease \( 1.07\% \) compared to the model also including the transform of M q. Consequently, transformed adaptation \( {Y}^{1.3} \) in model (10) is reasonably assumed to depend substantially on transformed difficulty \( \log M \) corresponding to \( q=0 \).
Assessing c ' = 0
The adaptively generated model with \( p=1.3 \), \( q=0 \), \( \rho =0 \), and constrained not to include a transform of \( {\boldsymbol{X}}_2 \) in the model for the means has model for the means depending on \( {\boldsymbol{X}}_1 \) transformed by the power q' = 3 with LCV(1.3,0) and substantial percent decrease \( 0.57\% \) compared to the model also including a transform of \( {\boldsymbol{X}}_2 \). Consequently, in model (10), transformed adaptation \( {Y}^{1.3} \) is reasonably assumed to depend substantially on family functioning \( X \), and so the instantaneous natural direct effect is substantial.
Using the model (12) selected with \( p=1.3 \), \( q=0 \), and \( \rho =0 \) (i.e., with p' = −3 and q' = 3), an adaptive transformation of this model allowing for p" ≠ 1 (and also possible adjustments to p' and q') is the same model generated with p" = 1. Consequently, the simplifying assumption p" = 1 is reasonable in this case.
Assessing non-constant variances
As described earlier, adaptive models can be generated allowing for variances of the omitted factors or errors \( {U}_M \) and \( {U}_Y \) to depend on transforms of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \). Starting from the adaptively generated model for \( p=1.3 \) and \( q=0 \) with \( \rho =0 \) (i.e., with p' = −3 and q' = 3), the expansion adds in transforms of these predictors to the model for the variances, but the contraction removes all of them, leaving the base constant variances model. This result indicates that the variances are reasonably treated as constant in \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \).
Also considering a covariate
The family management study enrolled parents of children with a variety of chronic conditions. One childhood chronic condition type was Crohn's disease or a bowel disorder with \( 54 \) (\( 28.9\% \)) children having this condition. The indicator \( Z \) for having this condition can be considered as a possible covariate for models (9)-(10). Adaptive modeling starts with the adaptively generated model for \( p=1.3 \), \( q=0 \), and \( \rho =0 \) (i.e., with p' = −3 and q' = 3) along with constant variances (and so based on \( {\boldsymbol{I}}_1 \) and \( {\boldsymbol{I}}_2 \)). The model for the means is expanded considering the indicators \( {\boldsymbol{Z}}_1 \) and \( {\boldsymbol{Z}}_2 \) (as defined similarly to \( {\boldsymbol{X}}_1 \) and \( {\boldsymbol{X}}_2 \)) while the model for the variances is expanded considering arbitrary transforms of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \) along with \( {\boldsymbol{Z}}_1 \) and \( {\boldsymbol{Z}}_2 \). The contraction is constrained so that \( {\boldsymbol{M}}^q \) is not retransformed in the model for the means (so \( {p}^{\hbox{'}\hbox{'}}=1 \)) while \( {\boldsymbol{X}}_1 \) and \( {\boldsymbol{X}}_2 \) are not removed from the model for the means, but associated powers are allowed to be changed.
The generated model has the same powers p' = −3 and q' = 3 as without consideration of \( {\boldsymbol{Z}}_1 \) and \( {\boldsymbol{Z}}_2 \) along with the covariate \( {\boldsymbol{Z}}_2 \) added to both models for the means and the variances and no transforms of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \) in the model for the variances. The LCV(1.3,0) score is \( 0.016062 \), which is a substantial improvement on the score \( 0.015804 \) for the associated model not considering covariates with percent decrease \( 1.61\% \). Consequently, the indicator for having Crohn's disease or a bowel disorder substantially influences the means and variances for model (10), but not for model (9) (since only \( {\boldsymbol{Z}}_2 \) is included in the generated model and not \( {\boldsymbol{Z}}_1 \)).
Model assumptions
A standard linear mediation analysis, that is, with model (12) based on p = q = p' = q' = 1 and \( \rho =0 \), generates the standardized residuals plotted in Fig. 3. While this plot is reasonably close to linear for most of the data, there are exceptions at the low and high ends of the plot. There are also three outliers (i.e., with values outside of \( \pm 3\Big) \) with standardized residual values of \( -3.44 \), \( 3.03 \), and \( 3.07 \). The Shapiro-Wilk test for normality of the standardized residuals is significant at \( p=0.037 \). Consequently, the normality assumption is questionable for the linear mediation model. The corresponding adaptive non-constant variances model with no changes to the model for the means but possible inclusion of transforms of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \) in the model for the variances contains the single transform \( {\boldsymbol{X}}_1^{19} \) in the model for the variances. The LCV(1,1) score for this adjusted model is \( 0.015046 \) with substantial improvement over the score \( 0.014958 \) for the associated constant variances model (as reported earlier) with percent decrease \( 0.58\% \). Consequently, the constant variance assumption is questionable for the component model (2) of the standard linear moderation model.
Normal plot for the linear mediation model for child adaptation as a function of family functioning as mediated by difficulty with independent omitted factors or errors
Figure 4 contains the normal plot generated by the monotonic mediation model (with \( p=1.3 \), \( q=0 \), p' = −3, q' = 3, and \( \rho =0 \)) adjusted for the covariate having Crohn's disease or a bowel disorder with the best power-adjusted LCV score generated so far. This plot is reasonably close to linear, the standardized residuals range for \( -3.01 \) to \( 2.51 \) with only one observation having a value \( -3.01 \) outside of \( \pm 3 \), but very close to the boundary value of \( -3 \). The Shapiro-Wilk test for normality is now non-significant (\( p=0.475 \)), and so the normality assumption seems reasonable for this case. Figure 5 contains the plot of the standardized residuals in terms of family functioning. The assumption of constant variances is reasonable at least for all but a few exceptional, low family functioning values, suggesting that the standardized residuals are reasonable close to having constant variances. These results indicate that monotonic transformation can resolve distributional assumption problems for mediation models with continuous positive valued outcomes and mediators.
Normal plot for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder with independent omitted factors or errors
Standardized residual plot for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder with independent omitted factors or errors
Results for the selected model
Using this monotonic mediation model with the best LCV score so far, estimated instantaneous total, natural direct, natural indirect, and relative natural indirect effects are presented in Table 1 for the grid of family functioning values 1, 2, 3, and 4 (1 is the smallest possible value while 4 is the largest possible value for the scale). The instantaneous natural indirect effect of family functioning on child adaptation increases with increasing or improving family functioning values while the instantaneous natural direct effect of family functioning on child adaptation decreases in such a way that the relative instantaneous natural indirect effect of family functioning on child adaptation increases. Family functioning needs to be relatively high in order for the relative instantaneous natural indirect effect to be relatively strong; in other words, mediation in this case is quite weak for values of family functioning between 1 and 2 In contrast, the standard linear mediation model (i.e., with p = q = p' = q' = 1 and \( \rho =0\Big) \) generates a total effect of 18.5, a natural direct effect of 12.3, a natural indirect effect of 6.2, and a relative natural indirect effect of 0.34, quite different results.
Table 1 Estimated instantaneous total, natural direct, natural indirect, and relative natural indirect effects for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder with independent omitted factors or errors
Bootstrapped CIs
The standard linear mediation model (i.e., with p = q = p' = q' = 1 and \( \rho =0 \)) generates the same bootstrapped CI of \( 2.00 \) – \( 11.96 \) for the instantaneous natural indirect effect at each value of family functioning. The normalized width \( W \) for this CI is \( 0.83 \).
Table 2 contains bias-corrected bootstrapped \( 95\% \) CIs for the natural indirect effect at a range of values of family functioning under the monotonic mediation model with the best LCV score so far, that is, model (12) with powers \( p=1.3 \), \( q=0 \), p' = −3, and q' = 3, controlling for the covariate having Crohn's disease or a bowel disorder with independent omitted factors or errors (i.e., \( \rho =0 \)). The lower and upper bounds on the instantaneous natural indirect effects of family functioning on child adaptation increase with increasing values of family functioning. The normalized width \( W \) for these CIs is \( 0.72 \). Since this is smaller than the value \( 0.83 \) (about \( 13\% \) smaller) for the standard linear mediation model, the monotonic mediation model generates more precise estimates of the instantaneous natural indirect effects.
Table 2 Bias-corrected bootstrapped \( 95\% \) confidence intervals for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder with independent omitted factors or errors
The bootstrapped CI of \( 2.00 \) – \( 11.96 \) for the linear moderation model (as reported earlier) overlaps with the bootstrapped CIs for \( x=1 \) and \( x=2 \), but is entirely below the bootstrapped CIs for \( x=3 \) and \( x=4 \), suggesting that the linear moderation model in this case generates biased estimates of natural indirect effects for larger values of family functioning.
Another childhood chronic condition type for the family management study was diabetes with \( 51 \) (\( 27.3\% \)) children having this condition. The indicator \( Z \) for having this condition can be considered as a possible moderator for models (13)-(15). As before, the indicator for the child having Crohn's disease or a bowel disorder is considered as a covariate to include in both the models for the means and the variances as well as the model for the variances to depend on transforms of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \).
The generated model for the case of children with a chronic condition other than diabetes (\( Z=0 \)) has powers p'(0) = −1.7 and q'(0) = 6. The generated model for the case of children with diabetes (\( Z=1 \)) has the power q′(1) = 0.5 with no transform of family functioning (so a missing p′(1) power). As before the covariate Crohn's disease or a bowel disorder is included in the model for the means and variances, but only in the component of model (15) addressing the outcome variable, that is, submodel (14). The LCV(1.3,0) score is \( 0.016215 \), which is a substantial improvement on the score \( 0.016062 \) for the associated model not considering moderation with percent decrease \( 0.94\%. \) Consequently there is distinct moderated mediation.
The linear moderated mediation model (i.e., with p = q = 1, p'(0) = q(0) = 1, p'(1) = q(1) = 1, and \( \rho =0 \)) allowing for effects of the covariate having Crohn's disease or a bowel disorder on the means and variances as well as possibly nonlinear effects of \( {\boldsymbol{X}}_1 \), \( {\boldsymbol{X}}_2 \), and \( \boldsymbol{M} \) on the variances has means and variances depending on the covariate having Crohn's disease of a bowel disorder for only the submodel (14) as for the moderated monotonic mediation model. Its LCV(1,1) score is \( 0.015460 \) with substantial percent decrease \( 4.66\% \). Consequently, the moderated mediation is distinctly nonlinear.
Model (15) adjusted to remove moderation of the effect of transformed \( X \) on transformed \( M \) in submodel (13) has LCV(1.3,0) score \( 0.016112 \) with substantial percent decrease \( 0.64\% \) compared to the full model (15). Moreover, model (15) adjusted to remove moderation of the effect of transformed \( X \) on transformed \( Y \) in submodel (14) has LCV(1.3,0) score \( 0.016112 \) (same score as above but a different model) with substantial percent decrease \( 0.64\% \) compared to the full model (15). Consequently, there is distinct moderation of both effects of the predictor \( X \) in model (15).
On the other hand, model (15) adjusted to remove moderation of the effect of transformed M on transformed Y in submodel (14) has LCV(1.3,0) score \( 0.016155 \) with insubstantial percent decrease \( 0.37\% \) compared to the full model (15). Consequently, this is a parsimonious, competitive alternative model, and the effect of transformed \( M \) on transformed \( Y \) is reasonably considered not to be moderated by having diabetes.
Model (15) with the term β 3(0) ⋅ X 1 q ' (0) ⋅ I(Z = 0) removed has powers q ' (1) = 0.5 and p ' (0) = −1 with LCV(1.3,0) score \( 0.015846 \) and substantial percent decrease \( 2.28\% \). Thus, there is a distinct effect of transformed \( X \) on transformed \( M \) for children with a chronic condition other than diabetes in submodel (13). With the term β 5(0) ⋅ M q ⋅ I(Z = 0) removed, the model has powers q'(0) = 6, q ' (1) = 0.5, and p ' (0) = 0.5 with LCV(1.3,0) score \( 0.015949 \) and substantial percent decrease \( 1.64\% \). Thus, there is a distinct effect of transformed \( M \) on transformed \( Y \) for children with a chronic condition other than diabetes in submodel (14).
Model (15) with the term β 3(1) ⋅ X 1 q ' (1) ⋅ I(Z = 1) removed has powers q'(0) = 6 and p'(0) = −1 with LCV(1.3,0) score \( 0.015674 \) and substantial percent decrease \( 3.34\% \). Thus, there is a distinct effect of transformed \( X \) on transformed \( M \) for children with diabetes in submodel (13). On the other hand, with the term β 5(1) ⋅ M q ⋅ I(Z = 1) removed, the model has powers q'(0) = 6, q'(1) = 0.5, and p'(0) = −1 with LCV(1.3,0) score \( 0.016137 \) and insubstantial percent decrease \( 0.48\% \). Thus, the effect of transformed \( M \) on transformed \( Y \) for children with diabetes in submodel (14) is not distinct. This latter result indicates that the effect of transformed \( X \) on transformed \( Y \) is not distinctly mediated by transformed \( M \) for children with diabetes.
With the term \( {\beta}_4(0)\cdot {\boldsymbol{X}}_2^{p^{\hbox{'}}(0)}\cdot I\left( Z=0\right) \) removed, the model has powers q'(0) = 6 and q'(1) = 0.5 with LCV(1.3,0) score \( 0.016058 \) and substantial percent decrease \( 0.97\% \). This result indicates that there is a distinct effect of transformed \( X \) on transformed \( Y \) for children with a chronic condition other than diabetes in submodel (14). On the other hand, the fact that \( {\beta}_4(1)\cdot {\boldsymbol{X}}_2^{p^{\hbox{'}}(1)}\cdot I\left( Z=0\right) \) is not in the generated model indicates that the effect of transformed \( X \) on transformed \( Y \) for children with diabetes in submodel (14) is not distinct.
Using the parsimonious, competitive model with the term β 5(1) ⋅ M q ⋅ I(Z = 1) removed, the standardized residuals range from \( -2.89 \) to \( 2.48 \) with nonsignificant (\( p=0.430 \)) Shapiro-Wilk normality test and normal plot reasonably close to linear (not displayed). Table 3 contains the associated estimated instantaneous total, natural direct, natural indirect, and relative natural indirect effects for the grid of family functioning values 1, 2, 3, and 4, but just for families having children with chronic conditions other than diabetes. The instantaneous natural indirect effect of family functioning on child adaptation increases with increasing or improving family functioning values while the instantaneous natural direct effect of family functioning on child adaptation decreases in such a way that the relative instantaneous natural indirect effect of family functioning on child adaptation increases. Compared to Table 1, instantaneous natural indirect effects and instantaneous total effects are smaller for low values of family functioning (1–\( 3) \) and larger than for the highest value of family functioning (4). Relative instantaneous natural indirect effects are all smaller.
Table 3 Estimated instantaneous total, natural direct, natural indirect, and relative natural indirect effects for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder and as moderated by having diabetes with independent omitted factors or errors
Table 4 contains bias-corrected bootstrapped \( 95\% \) CIs for associated estimated instantaneous natural indirect effects of family functioning on child adaptation for the grid of family functioning values 1, 2, 3, and 4, also just for families having children with chronic conditions other than diabetes. Values for the lower and upper bounds increase with family functioning. Compared to Table 2, widths of the CIs are wider in absolute value for the highest value for family functioning (4) and narrower in absolute value otherwise (1–\( 3) \). However, the relative width \( W \) is larger, \( 0.84 \) versus \( 0.72. \) In contrast, the associated linear model (i.e., with powers p = q = 1 and p'(0) = q'(0) = 1, \( \rho =0 \), and having Crohn's disease or a bowel disorder covariate effects on the means and variances for submodel (14)) has estimated constant instantaneous natural indirect effect \( 5.5 \) with bias-corrected bootstrapped \( 95\% \) CI \( 1.4 \) – \( 11.7 \) and normalized width \( W \) \( 0.88 \). Hence, the moderated monotonic mediation model generates more precise bootstrapped CIs than the standard moderated linear mediation model.
Table 4 Bias-corrected bootstrapped \( 95\% \) confidence intervals for the monotonic mediation model for child adaptation as a function of family functioning as mediated by difficulty controlling for having Crohn's disease or a bowel disorder and as moderated by having diabetes with independent omitted factors or errors
In summary, moderated mediation for the family management data is distinctly nonlinearly monotonic. However, mediation only occurs for mothers of children with a chronic condition other than diabetes and not for mothers of children with diabetes.
Example analyses of the simulated mediation data
Using the CSH covariance structure with \( p= q=1 \), the first local maximum in the LCV score occurs at \( k=15 \) with LCV(1,1) \( =0.41542 \), and so \( k=15 \) folds are used to compute subsequent LCV scores for the simulated data. The generated CSH model allowing for arbitrary \( p \) and \( q \) has \( p=0 \) and \( q=0.3 \) with distinctly improved \( \mathrm{L}\mathrm{C}\mathrm{V}\left(0,0.3\right)=0.93577 \) (i.e., the percent decrease for the \( p= q=1 \) model is 55.6%, much larger than the cutoff of 0.95% for the data). Also, the estimated correlation for this model is the substantial value \( 0.92 \), suggesting highly dependent omitted factors or errors \( {U}_M \) and \( {U}_Y. \) However, the generated model allowing for arbitrary \( p \) and \( q \) with uncorrelated omitted factors or errors (i.e.., with \( \rho =0 \)), has \( p=0.4 \) and \( q=1 \) (i.e., the true values for these powers) with distinctly improved LCV(0.4,1) \( =1.08555 \) (i.e., the percent decrease for the model with \( p=0 \) and \( q=0.3 \) is 13.8%), indicating that the omitted factors or errors are reasonably treated as independent as simulated. Under this model, p' = 1.1 and q' = 1.2, close to their simulated values of p' = q' = 1.
The constant instantaneous indirect effect model associated with this latter model (i.e., with q′ = 1) has LCV(0.4,1) \( =1.08074 \) and insubstantial percent decrease \( 0.44\% \). Consequently, the instantaneous natural indirect effect is reasonably considered to be constant as simulated. Also, the true model as simulated (i.e., also setting \( {p}^{\hbox{'}}=1\Big) \), has LCV(0.4,1) \( =1.07928 \) and insubstantial percent decrease \( 0.58\% \), indicating that the true model is a competitive alternative for the simulated data.
Using the constant instantaneous natural indirect effect model (i.e., with \( p=0.4 \), \( q=1 \), p' = 1.1, and q' = 1), the \( 95\% \) bootstrapped CI with bias correction for that effect is \( 0.0358 \) – \( 0.0563 \), and so contains the true constant instantaneous natural indirect effect \( 0.04 \). The associated \( 95\% \) bootstrapped CI without bias correction is \( 0.0363 \) – \( 0.0567 \), and so is quite similar, suggesting that bias correction has not inflated the Type I error in this case.
Using the standard linear mediation model (i.e., with \( p=1 \), \( q=1 \), p' = 1, q' = 1, and \( \rho =0 \)), the \( 95\% \) bootstrapped CI with bias correction for the constant instantaneous indirect effect is \( 0.0480 \) – \( 0.0974 \), and so does not contain the true constant instantaneous natural indirect effect \( 0.04 \). The associated \( 95\% \) bootstrapped CI without bias correction is \( 0.0481 \) – \( 0.0986 \), and so is quite similar and also does not contain the true value.
We formulated and demonstrated an approach for conducting possibly moderated monotonic mediation analyses based on adaptively selected fractional polynomial models. This formulation considers transformation of outcomes, predictors, and mediators, not just predictors and mediators as previously considered [29–32]. Results of the example analyses of the family management data indicated that transformation of positive valued continuous outcomes can provide distinct improvements over leaving those outcomes untransformed and can resolve problems with model assumptions.
Other nonlinear regression methods could have been used instead to estimate relationships. An advantage of fractional polynomial models is that they are based on linear regression models and so have no more limitations, assumptions, and requirements than models used in linear mediation analyses. Associated derivatives are also readily computed as needed for estimating monotonic instantaneous natural indirect, natural direct, and total effects.
The example analyses used likelihood cross-validation (LCV) for evaluating models, both unadjusted and adjusted for transformation of the outcome. By assessing how a model performs on randomly selected subsets, model evaluation using LCV scores is robust to effects of chance variation compared to using likelihoods based on the complete data. LCV ratio tests, generalizing likelihood ratio tests, can be used to assess whether mediation relationships are constant, linear, or nonlinear as well as a variety of other issues.
The example analyses of the family management data demonstrated the need to address nonlinearity in the context of mediation. Relationships considered in mediation analyses can be distinctly nonlinear. Even when mediation relationships are reasonably treated as linear, consideration of nonlinear alternatives is needed to determine that this assumption holds. It is conventional to treat mediation relationships as linear without checking this assumption. However, like any assumption, it should be checked.
While the example analyses of the family management data demonstrated that distinct nonlinear monotonic mediation can be identified using composite model (12), linear mediation also held. However, the normality assumption was questionable for the linear mediation case and also the constant variances assumption. There are likely to be data sets where mediation can be identified only by consideration of monotonicity. However, there are also likely to be data sets where consideration of monotonicity does not resolve problems related to the normality and constant variances assumptions. Quantile regression methods [26] can be used in such cases.
The example data analyses of the family management data also demonstrated that standard linear mediation analyses can provide a misleading impression that the natural indirect effect is constant in the predictor \( X \) when the indirect effect can actually vary quite a bit from this constant value. Moreover, the single \( 95\% \) bootstrapped CI generated by the linear moderation analysis can be less precise than the CIs generated by a monotonic mediation analysis and can even not overlap for some values of the predictor \( X \), suggesting that a linear approach can generate biased indirect effects. Furthermore, the example analyses of the family management data demonstrated that moderated mediation analyses are important to consider because mediation may be weaker for some subpopulations than others and even not hold for some subpopulations (e.g., mothers of children with diabetes compared to mothers of children with other chronic conditions).
The example analyses of the simulated mediation data support the effectiveness of adaptive mediation modeling since the true model was a competitive alternative to the adaptively selected model and since bootstrapped \( 95\% \) CIs for the constant instantaneous indirect effect contained the true value. However, these analyses also demonstrate that allowing for correlated omitted factors or errors when in fact they are independent can generate models quite different from the true model, indicating the importance of conducting analyses of both cases. These analyses also demonstrate that conducting a standard linear mediation analysis when in fact the relationship (9) is nonlinear in the outcome (i.e., \( p\ne 1 \)) but when the true instantaneous natural indirect effect is constant (i.e., \( q^{\prime }=1 \)) can result in a biased \( 95\% \) CI for that constant effect. This is also likely to hold when relationships (8)-(9) are nonlinear in the mediator (i.e., \( q\ne 1 \)).
The example mediation analyses of the family management data were limited since they were based on cross-sectional data, and so the timing of measurements for predictors, mediators, and outcomes could not be controlled to reflect precedence as needed to support causality [11, 44]. These analyses were also limited by the absence of control over the predictor variable given the non-experimental design of the study [21]. However, the primary purpose of those analyses was to demonstrate nonlinear mediation and the need for such analyses. This purpose was effectively achieved by the example analyses. Further work, though, is needed to address monotonic mediation in situations where the timing of measurement of variables has been controlled and where the predictor is experimentally controlled. An advantage of an experimentally controlled predictor is that it would be categorical and hence not require transformation. However, the outcome \( Y \) and the mediator \( M \) might benefit from transformation. There is also a need to replicate these analyses using a wider variety of data sets.
Nonlinearity was addressed here for mediation involving univariate outcomes. However, mediation is also conducted using repeated measurements, either over clusters or longitudinally over time, analyzed with multilevel modeling, linear mixed modeling, or structural equation modeling [6, 13, 57–61]. However, linear relationships are usually assumed in these analyses. Further research is needed to extend monotonic mediation to these cases. The example analyses used a continuous outcome and mediator, but outcomes and mediators can sometimes be categorical [13]. Further research is needed to extend monotonic mediation to address categorical mediators and/or outcomes. The analyses also only addressed the case with a single mediator. An extension to monotonic mediation is needed that accounts for multiple mediators.
Mediation relationships are commonly assumed to be linear without assessing the validity of this assumption. Reported example analyses demonstrate that mediation relationships can be nonlinear. Moreover, standard linear mediation analyses can generate models that violate model assumptions and generate biased estimates of indirect effects, but this can in some cases be resolved through more general monotonic mediation analyses. Adaptive methods as extended here to the monotonic mediation and moderated monotonic mediation contexts can effectively account for nonlinearity in mediation relationships. The advantage of restricting to monotonic relationships is that no adjustments are needed to underlying theory about directionality between changes in pairs of variables.
CSH:
Compound symmetry heterogeneous
LCV:
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The collection of the family management data used in the example analyses was funded by the National Institute of Nursing Research, National Institutes of Health (R01 NR08048).
The family management data used in example analyses are available from Yale University but restrictions apply to the availability of these data, which were used under a data use agreement for the current study, and so are not publicly available. These data are, however, available from the corresponding author upon reasonable request and with permission of Yale University. The simulated mediation data are available at [62].
SAS macros have been developed to support adaptive (moderated) monotonic mediation analyses. Models for means and variances for fixed values of \( p \) and \( q \) are generated with the genreg (for general regression) macro. Adaptive composite models (12) and (15) are generated using the compmed (for composite mediation) macro. Bootstrapped \( 95\% \) CIs, either bias-corrected or not, are also generated using this macro, generalizing the macro of [41], over a grid of possible values for the predictor \( X \) between its lowest and highest observed values and over alternative values for a moderator \( Z \) if included. These macros as well as code for generating example analyses are available at [62].
The family management data used in the example analyses were collected as part of an instrument development project [54]. All authors were involved in the development and conduct of this project including data collection and data analysis. For the current project, GJK developed the methods, generated the simulated data and the results, and wrote the initial description of those methods, data, and results. All authors reviewed drafts of the manuscript, rewrote sections, offered critical comments, and approved the final version.
The Institutional Review Board of the University of North Carolina at Chapel Hill approved secondary analyses of the family management data used in example analyses.
School of Nursing, University of North Carolina at Chapel Hill, 5014 Carrington Hall, Campus Box 7460, Chapel Hill, NC, 27599-7460, USA
George J. Knafl
& Kathleen A. Knafl
School of Nursing, Yale University, New Haven, USA
Margaret Grey
& Jane Dixon
School of Nursing, University of Pennsylvania, Philadelphia, USA
Janet A. Deatrick
College of Nursing, University of Illinois at Chicago, Chicago, USA
Agatha M. Gallo
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Correspondence to George J. Knafl.
Knafl, G.J., Knafl, K.A., Grey, M. et al. Incorporating nonlinearity into mediation analyses. BMC Med Res Methodol 17, 45 (2017). https://0-doi-org.brum.beds.ac.uk/10.1186/s12874-017-0296-6
Childhood chronic conditions
Fractional polynomials
Nonlinearity | CommonCrawl |
\begin{document}
\title[Ambrosetti-Malchiodi-Ni conjecture: clustering concentration layers]{On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains:
clustering concentration layers}
\author{Suting Wei$^\ddag$} \thanks{$\ddag\, $ Department of Mathematics, South China Agricultural University, Guangzhou, 510642, P. R. China. Email: [email protected]}
\author{Jun Yang$^\S$} \thanks{$\S\, $ School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China. Email: [email protected]}
\thanks{Corresponding author: Jun Yang, [email protected]}
\begin{abstract} We consider the problem $$
\varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }\Omega,
\qquad
\nabla_{{\mathfrak a}(y)} u\cdot \nu\, =\, 0\quad\mbox{on } \partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p$ is greater than $1$, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. For two positive smooth functions ${\mathfrak a}_1(y), {\mathfrak a}_2(y)$ on $\bar\Omega$, the operator $\nabla_{{\mathfrak a}(y)}$ is given by $$ \nabla_{{\mathfrak a}(y)} u=\Bigg({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, \, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2}\Bigg). $$
(1). Let $\Gamma\subset{\bar\Omega}$ be a smooth curve intersecting orthogonally with
$\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts.
Moreover, $\Gamma$ is a {\it non-degenerate geodesic} embedded in the Riemannian manifold ${\mathbb R}^2$ with metric $V^{2\sigma}(y)\big[{\mathfrak a}_2(y){\mathrm d}y_1^2+{\mathfrak a}_1(y){\mathrm d}y_2^2\big]$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. By assuming some additional constraints on the functions ${\mathfrak a}(y)$, $V(y)$ and the curves $\Gamma$, $\partial\Omega$, we prove that there exists a sequence of $\varepsilon$ such that the problem has solutions $u_\varepsilon$ with clustering concentration layers directed along $\Gamma$, exponentially small in $\varepsilon$ at any positive distance from it.
(2). If ${\tilde\Gamma}$ is a simple closed smooth curve in $\Omega$ (not touching the boundary $\partial\Omega$), which is also a {\it non-degenerate geodesic} embedded in the Riemannian manifold ${\mathbb R}^2$ with metric $V^{2\sigma}(y)\big[{\mathfrak a}_2(y){\mathrm d}y_1^2+{\mathfrak a}_1(y){\mathrm d}y_2^2\big]$, then a similar result of concentrated solutions is still true.
{\textbf{Keywords: }}{ Ambrosetti-Malchiodi-Ni conjecture, Clustering concentration layers, Toda-Jacobi system, Resonance phenomena}
{\textbf{MSC 2020: }}{35B25, 35J25, 35J61}
\end{abstract}
\date{}\maketitle
\section{Introduction}\label{section1}
We consider the following problem for the existence of solutions with concentration phenomena
\begin{align}
\label{originalproblem}
\varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0\quad\mbox{in } \Omega, \qquad\ \nabla_{{\mathfrak a}(y)} u\cdot \nu\, =\, 0\quad\mbox{on } \partial \Omega,
\end{align} where $\Omega$ is a bounded domain in $\mathbb R^{\mathbf d}$ with smooth boundary, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$, the exponent $p>1$. For ${\mathfrak a}(y)=\big({\mathfrak a}_1(y), \cdots, {\mathfrak a}_{\mathbf{d}}(y)\big)$, the operator is defined in the form $$ \nabla_{{\mathfrak a}(y)} u=\big({\mathfrak a}_1(y)u_{y_1}, \cdots, {\mathfrak a}_{\mathbf{d}}(y)u_{y_{\mathbf{d}}}\big), $$ where ${\mathfrak a}_1(y), \cdots, {\mathfrak a}_{\mathbf{d}}(y)$ are positive smooth functions on $\bar\Omega$.
\noindent $\bullet$ In the case ${\mathfrak a}\equiv 1$ and $V\equiv 1$, problem \eqref{originalproblem} takes the form \begin{equation}\label{original equationconstant}
\varepsilon^2\Delta u-u+u^p\, =\, 0, \quad u>0 \quad{\rm in}\quad \Omega,
\ \quad \nabla u\cdot \nu\, =\, 0\quad \;\;{\rm on} \;\; \partial \Omega,
\end{equation} which is known as the stationary equation of Koeller-Segel system in chemotaxis \cite{LinNiTak1988}. It can also be viewed as a limiting stationary equation of Gierer-Meinhardt system in biological pattern formation \cite{GierMein1972}.
In the pioneering papers \cite{LinNiTak1988, NiTaka1991, NiTaka1993}, under the condition that $p$ is a subcritical Sobolev exponent, C.-S. Lin, W.-M. Ni and I. Takagi established, for $\varepsilon$ sufficiently small, the existence of a least-energy
solution $U_\varepsilon$ of (\ref{original equationconstant}) with only one local maximum point locating at the most curved point of $\partial\Omega$. Such a solution is called a spike-layer, which has concentration phenomena at interior or boundary points. For the existence of interior spikes, we refer the reader to the articles
\cite{BatesFusc, DancerYan19992, dPFW1, GrosPistWei2000, GuiWei1999, Wei1998} and the references therein.
On the other hand, boundary spikes related to the mean curvature
of $\partial \Omega$ can be found in
\cite{BatesDanShi, DancerYan19991, dPFW, GuiWeiWint2000, Li1998, Wei1997, WeiWinter1998},
and the references therein. The coexistence of interior and boundary spikes was due to C. Gui and J. Wei \cite{GuiWei2000}.
A good review of the subject up to 2004 can be found in \cite{Ni2}.
There is a conjecture on higher-dimensional concentration by W.-M. Ni \cite{Ni1} (see also \cite{Ni2}): \\ \noindent {\bf Conjecture 1.} {\em For any integer $1\leq \mathbf{k}\leq {\mathbf d}-1$, there exists $p_\mathbf{k}\in (1, \infty)$ such that for all $1<p<p_\mathbf{k}$, problem (\ref{original equationconstant}) has a solution $U_\varepsilon$ which concentrates on a $\mathbf{k}$-dimensional subset of $\bar{\Omega}$, provided that $\varepsilon$ is sufficiently small. \qed } \\ We here mention some results for the existence of higher dimensional boundary concentration phenomena in the papers \cite{MahMal2007, Malchi2004, Malchi2005, MalMont2002, MalMont2004}. The papers \cite{JWeiYang2007, JWeiYang2008} set up the existence of concentration on an interior line, which connects the boundary $\partial\Omega$ and is non-degenerate in the sense of variation of arc-length. There are also some other results \cite{AoMussoWeiJDE, AoMussoWeiSIAM, dAprile2011, dAprilePist20101, dAprilePist20102} to exhibit concentration phenomena on interior line segments connecting the boundary of $\Omega$. For higher dimensional extension, the reader can refer to \cite{AoYang, DancerYan2007, delPKowParWei, Guoyang, LiPengYan2007}. The reader can also refer to the survey paper by J. Wei \cite{Wei2008}.
\noindent $\bullet$ In the case ${\mathfrak a}\equiv 1$ and $V{\not\equiv}$ constant, problem (\ref{originalproblem}) on the whole space corresponds to the following problem
\begin{align}
\label{originalproblem-space}
\varepsilon^2 \Delta u- V(y)u+u^p\, =\, 0, \quad u>0\qquad \mbox{in } {\mathbb R}^{\mathbf d},
\end{align} where $\varepsilon>0$ is a small parameter, the exponent $p>1$, and $V$ is a smooth function with
\begin{equation*}
\displaystyle {\inf_{y\in {\mathbb R}^{\mathbf d}}} V(y)>0. \end{equation*} Started by \cite{FloerWein1986}, solutions exhibiting concentration around one or more points of space under various assumptions on the potential and the nonlinearity were given by many authors \cite{{AmbBadCing1997}, AmbMalSecc2001, CingLazzo, delPFelmer1996, delPFelmer1997, delPFelmer1998, delPFelmer2002, FelmTorr2002, wangx}. On the other hand, radially symmetric solutions with concentration on sphere of radius $r_0$ can be constructed in \cite{AMNconjecture1}, whenever $V$ is a radial function and $r_0 >0$ is a non-degenerate critical point of the function \begin{equation*}
M(r)\, =\, r^{{\mathbf d}-1}V^{\sigma}(r), \quad {\rm where}\quad \sigma\, =\, \frac {p+1}{p-1}-\frac 12. \end{equation*} Based on heuristic arguments, in 2003, A. Ambrosetti, A. Malchiodi and W.-M. Ni raised the following conjecture (p.465, \cite{AMNconjecture1}): \\ \noindent {\bf Conjecture 2.} {\em Let ${\mathcal K}$ be a non-degenerate $\mathbf{k}$-dimensional stationary manifold of the following functional \[ \int_{\mathcal K} V^{\frac{p+1}{p-1} -\frac{1}{2} ({\mathbf d}-\mathbf{k})}, \] where the exponent $p$ is subcritical w.r.t. ${\mathbb R}^{\mathbf{d}-\mathbf{k}}$. Then there exists a solution to (\ref{originalproblem-space}) concentrating near ${\mathcal K}$, at least along a subsequence $\varepsilon_j \to 0$. \qed } \\ The validity of this conjecture was confirmed in two-dimensional general case with concentration on stationary and non-degenerate curves for $\varepsilon$ satisfying a gap condition due to the resonance character of the problem, see \cite{delPKowWei2007}. More results can be found in \cite{bartschpeng-1, bartschpeng-2, WangWeiYang, MahMalMont2009}.
Let us go back to the problem on smooth bounded domains with homogeneous Neumann boundary condition, \begin{align}
\label{originalproblem-domain}
\varepsilon^2 \Delta u - V(y)u+u^p\, =\, 0, \quad u>0\quad\mbox{in } \Omega, \qquad\ \nabla u\cdot \nu\, =\, 0\quad\mbox{on } \partial \Omega,
\end{align} where $\Omega$ is a bounded domain in $\mathbb R^{\mathbf d}$ with smooth boundary, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth
potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$ and the exponent $p>1$. If $\Omega$ is a unit ball $B_1(0)$, the existence of radial solutions to (\ref{originalproblem-domain}) was shown in \cite{AMNconjecture2}, where the concentration lying on spheres in $B_1(0)$ will approach the boundary with speed $O(\varepsilon|\log\varepsilon|)$ as $\varepsilon\rightarrow 0$. For the interior concentration phenomena connecting the boundary $\partial\Omega$, there is a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni in 2004 (p. 327, \cite{AMNconjecture2}), which can be stated as: \\ \noindent {\bf Conjecture 3.} {\em Let ${\mathcal K}$ be a $\mathbf{k}$-dimensional manifold intersecting $\partial\Omega$ perpendicularly, which is also stationary and non-degenerate with respect to the following functional \[ \int_{\mathcal K} V^{\frac{p+1}{p-1} -\frac{1}{2} ({\mathbf d}-\mathbf{k})}, \] where the exponent $p$ is subcritical w.r.t. ${\mathbb R}^{\mathbf{d}-\mathbf{k}}$. Then there exists a solution to (\ref{originalproblem-domain}) concentrating near ${\mathcal K}$, at least along a subsequence $\varepsilon_j \to 0$. \qed } \\ In \cite{weixuyang}, S. Wei, B. Xu and J. Yang considered (\ref{originalproblem-domain}) and provided an affirmative answer to the mentioned conjecture only in the case: ${\mathbf d}=2$ and $\mathbf{k}=1$ for the existence of solutions with single concentration layer connecting the boundary $\partial\Omega$.
\noindent $\bullet$ In present paper, we will consider a little bit more general case, i.e., problem \eqref{originalproblem}, and investigate first the existence of clustering phenomena of multiple concentration layers, which connect the boundary $\partial\Omega$. As the descriptions in ¡°Concluding remarks¡± of \cite{delPKowWei2010}, the difficulties arise from the multiple resonance phenomena, see also Remark \ref{remark14}. On the other hand, the much more complicated situation is the balance between neighbouring layers plus the interaction among the interior concentration layers, the boundary of $\Omega$ and the competition between $\mathcal{{\mathfrak a}}$ and $V$. It is natural to introduce the new ingredient (\ref{boundaryadmissibility}) to handle this delicate thing. Whence, in the present paper we focus on the two dimensional case of problem \eqref{originalproblem}. We make the following assumptions: \begin{itemize}
\item[ \bf(A1).] {\em
Let $\Omega$ be a smooth and bounded domain in $\Bbb R^2, $\;
$\Gamma$ be a curve intersecting $\partial \Omega$ at exactly two points, saying $P_1, P_2$, and, at these points $\Gamma\bot \partial \Omega$.
In the small neighborhoods of $P_1, P_2$, the boundary $\partial \Omega$ are two curves, say $\mathcal{C}_1$ and $\mathcal{C}_2$, which can be represented
by the graphs of two functions respectively:
$$
{y}_2=\varphi_1({y}_1)\quad\mbox{with } (0, \varphi_1(0))=P_1,
$$
$$
{y}_2=\varphi_2({y}_1)\quad\mbox{with } (0, \varphi_2(0))=P_2.
$$ Without loss of generality, we can assume that $\Gamma$ has length $1$, and then denote $k_1$, $k_2$ the signed curvatures of ${\mathcal C}_1$ and ${\mathcal C}_2$ respectively, also $k$ the curvature of $\Gamma$. }
\item[ \bf(A2).] {\em
$\Gamma$ separates the domain $\Omega$ into two disjoint components $\Omega_1$ and $\Omega_2$. }
\item[ \bf(A3).] {\em The functions ${\mathfrak a}_1(y)$ and ${\mathfrak a}_2(y)$ satisfy the condition \begin{align}\label{a1=a2} {\mathfrak a}_1={\mathfrak a}_2\quad \mbox{at the points } P_1 \mbox{ and } P_2. \end{align} The curve $\Gamma$ is a non-degenerate geodesic embedded in the Riemannian manifold ${\mathbb R}^2$ with the following metric \begin{equation}\label{sigma}
V^{2\sigma}(y) \big[ {\mathfrak a}_2(y){\mathrm d}y_1^2 +{\mathfrak a}_1(y) {\mathrm d}y_2^2 \big],
\quad\mbox{with}\ \sigma\, =\, \frac{p+1}{p-1}-\frac{1}{2}. \end{equation} This will be clarified in the next section (see (\ref{stationary}) and (\ref{nondegeneracy})). \qed } \end{itemize}
Let $w$ denote the unique positive solution of the problem
\begin{equation}
\label{wsolution}
w{''}-w+w^{p}\, =\, 0, \quad w>0\quad\mbox{in }{\mathbb R}, \quad w{'}(0)\, =\, 0, \quad w(\pm \infty)\, =\, 0.
\end{equation} We can formulate the first result.
\begin{theorem} \label{theorem 1.1} Let ${\mathbf d}=2$, $p>1$ and recall the assumptions in {\bf (A1)}-{\bf (A3)} as well as the modified Fermi coordinates $(t, \theta)$ in \eqref{Fermicoordinates-modified}. Moreover, we assume that \begin{equation}\label{taupositivity} \begin{split} \tau_2(\theta) \, \equiv\, \mathcal{H}_2'(\theta)-\mathcal{H}_3(\theta)
+2\frac{|\beta'(\theta)|^2}{\beta^2(\theta)}\mathcal{H}_1(\theta) -\frac{\beta''(\theta)}{\beta(\theta)}\mathcal{H}_1(\theta) -\frac{\beta'(\theta)}{\beta(\theta)}\mathcal{H}_1'(\theta) \, >\, 0, \end{split} \end{equation} and also the validity of the admissibility conditions \begin{align}\label{boundaryadmissibility} \frac{{\mathfrak b}_2}{{\mathfrak b}_1}+\frac{\beta'(0)}{\beta(0)}=0, \qquad \frac{{\mathfrak b}_7}{{\mathfrak b}_6}+\frac{\beta' (1)}{\beta(1)}=0, \end{align}
where the function $\beta>0$ is defined in (\ref{alpha-beta}), the functions $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are given in \eqref{mathcalH1}-\eqref{mathcalH3}, and the constants ${\mathfrak b}_1$, ${\mathfrak b}_2$, ${\mathfrak b}_6$, ${\mathfrak b}_7$ are given in \eqref{b1b2} and \eqref{b6b7}. Then for each $N$, there exists a sequence of $\varepsilon$, say $\{\varepsilon_l \}$, such that problem \eqref{originalproblem} has a positive solution $u_{\varepsilon_l}$ with exactly N concentration layers at mutual distances $O(\varepsilon_l|\ln\varepsilon_l|)$.
In addition, the center of mass for N concentration layers collapses to $\Gamma$ at speed $O(\varepsilon_l^{1+\mu})$ for some small positive constant $\mu$. More precisely, $u_{\varepsilon_l}$ has the form
\begin{equation}
\label{taketheform}
u_{\varepsilon_l}( y_1, y_2)\, \approx\, \sum_{j=1}^N V(0, \theta)^{\frac {1}{p-1}}
\, w\left(
\sqrt{\frac{V(0, \theta) \Big({\mathfrak a}_1(0, \theta)|n_1(\theta)|^2+{\mathfrak a}_2(0, \theta)|n_2(\theta)|^2\Big)}
{|{\mathfrak a}_1(0, \theta)|^2+|{\mathfrak a}_2(0, \theta)|^2} }
\frac {\, t-\varepsilon_l\, f_j(\theta)\, } {\varepsilon_l}
\right),
\end{equation}
where $n(\theta)=(n_1(\theta), n_2(\theta))$ is the unit normal to $\Gamma$. The functions $f_j$'s satisfy \begin{align}
\|f_j\|_{\infty}\leq\, C|\ln\varepsilon_l|^2,
\qquad
\sum_{j=1}^Nf_j=O(\varepsilon_l^\mu)\quad\mbox{with } \mu>0, \label{fproperties1} \end{align} \begin{align} \min\limits_{1\leq j\leq N-1}(f_{j+1}-f_j)
\approx 2 |\ln\varepsilon_l|\,\sqrt{
\frac {|{\mathfrak a}_1(0, \theta)|^2+|{\mathfrak a}_2(0, \theta)|^2}
{V(0, \theta) \Big({\mathfrak a}_1(0, \theta)|n_1(\theta)|^2+{\mathfrak a}_2(0, \theta)|n_2(\theta)|^2\Big)} } \, , \label{fproperties2} \end{align} and solve the Jacobi-Toda system, for $j=1, \cdots, N$, \begin{align} &\varepsilon_l^2\, \varsigma \Big[\, \mathcal{H}_1f''_j +\mathcal{H}_1'f'_j + \big(\mathcal{H}_2'-\mathcal{H}_3\big)f_j \, \Big]
-e^{-\beta(f_j-f_{j-1})}+e^{-\beta(f_{j+1}-f_j)}\approx0\quad\mbox{in } {(0, 1)}, \label{fJacobiToda} \end{align} with boundary conditions \begin{align} {\mathfrak b}_6f_j'(1)-{\mathfrak b}_7f_j(1)\approx 0, \qquad {\mathfrak b}_1f_j'(0)-{\mathfrak b}_2f_j(0)\approx 0, \qquad\forall\, j=1, \cdots, N, \label{fboundary} \end{align} where the function $\varsigma>0$ is defined in (\ref{varsigma}) with the conventions $ f_0=-\infty, \ f_{N+1}=\infty. $ \qed \end{theorem}
Here is the second result for the existence of interior clustering concentration layers, which do not touch the boundary $\partial\Omega$.
\begin{theorem} \label{theorem 1.2} Let ${\mathbf d}=2$ and $p>1$. Suppose that $({\hat t}, {\hat\theta})$ are the modified Fermi coordinates given in \eqref{Fermicoordinates-modified-tilde}. Assume that ${\hat\Gamma}$ is a simple closed smooth curve with unit length in $\Omega$, which is also a non-degenerate geodesic embedded in the Riemannian manifold ${\mathbb R}^2$ with the following metric \begin{equation}\label{sigma22222222}
V^{2\sigma}(y) \big[ {\mathfrak a}_2(y){\mathrm d}y_1^2 +{\mathfrak a}_1(y) {\mathrm d}y_2^2 \big],
\quad\mbox{with}\ \sigma\, =\, \frac{p+1}{p-1}-\frac{1}{2}, \end{equation} see (\ref{stationary4}) and (\ref{nondegeneracy4}). Moreover, we assume that \begin{equation}\label{taupositivity2} \begin{split} {\hat\tau}_2({\hat\theta}) \, \equiv\, \widehat{\mathcal H}_2'({\hat\theta})-\widehat{\mathcal H}_3({\hat\theta})
+2\frac{|\beta'({\hat\theta})|^2}{\beta^2({\hat\theta})}\widehat{\mathcal H}_1({\hat\theta}) -\frac{\beta''({\hat\theta})}{\beta({\hat\theta})}\widehat{\mathcal H}_1({\hat\theta}) -\frac{\beta'({\hat\theta})}{\beta({\hat\theta})}\widehat{\mathcal H}_1'({\hat\theta}) \, >\, 0, \end{split} \end{equation} where the functions $\beta>0$, $\widehat{\mathcal H}_1$, $\widehat{\mathcal H}_2$ and $\widehat{\mathcal H}_3$
are given in \eqref{alpha-beta}, \eqref{mathcalH1tilde}-\eqref{mathcalH3tilde}. Then for each $N$, there exists a sequence of $\varepsilon$, say $\{{\hat\varepsilon}_l \}$, such that problem \eqref{originalproblem} has a positive solution $u_{{\hat\varepsilon}_l}$ with exactly N concentration layers at mutual distances $O({\hat\varepsilon}_l|\ln{\hat\varepsilon}_l|)$.
In addition, the center of mass for N concentration layers collapses to ${\hat\Gamma}$ at speed $O({\hat\varepsilon}_l^{1+\mu})$ for some small positive constant $\mu$. More precisely, $u_{{\hat\varepsilon}_l}$ has the form \begin{equation}
\label{taketheform2}
u_{{\hat\varepsilon}_l}( y_1, y_2)\, \approx\, \sum_{j=1}^N V(0, {\hat\theta})^{\frac {1}{p-1}}\,
w\left(\sqrt{\frac{V(0, {\hat\theta}) \Big({\mathfrak a}_1(0, {\hat\theta})|{\hat n}_1({\hat\theta})|^2+{\mathfrak a}_2(0, {\hat\theta})|{\hat n}_2({\hat\theta})|^2\Big)}
{|{\mathfrak a}_1(0, {\hat\theta})|^2+|{\mathfrak a}_2(0, {\hat\theta})|^2} }
\frac {\, {\hat t} -{\hat\varepsilon}_l\, {\hat f}_j({\hat\theta})\, } {{\hat\varepsilon}_l}\right), \end{equation} where ${\hat n}({\hat\theta})=({\hat n}_1({\hat\theta}), {\hat n}_2({\hat\theta}))$ is the unit normal to ${\hat\Gamma}$. The functions ${\hat f}_j$'s satisfy \begin{equation}
\|{\hat f}_j\|_{\infty}\leq\, C|\ln{\hat\varepsilon}_l|^2, \qquad
\sum_{j=1}^N {\hat f}_j=O({\hat\varepsilon}_l^\mu)\quad\mbox{with } \mu>0, \end{equation} \begin{equation} \min\limits_{1\leq j\leq N-1}({\hat f}_{j+1}-{\hat f}_j)
\approx 2|\ln{\hat\varepsilon}_l|\, \sqrt{
\frac{|{\mathfrak a}_1(0, {\hat\theta})|^2+|{\mathfrak a}_2(0, {\hat\theta})|^2}
{V(0, {\hat\theta}) \Big({\mathfrak a}_1(0, {\hat\theta})|{\hat n}_1({\hat\theta})|^2+{\mathfrak a}_2(0, {\hat\theta})|{\hat n}_2({\hat\theta})|^2\Big)}
}
\, , \end{equation} and solve the Jacobi-Toda system, for $j=1, \cdots, N$, \begin{align} &{\hat\varepsilon}_l^2\, \varsigma \Big[\, \widehat{\mathcal H}_1 {\hat f}''_j +\widehat{\mathcal H}_1' {\hat f}'_j + \big(\widehat{\mathcal H}_2'-\widehat{\mathcal H}_3\big){\hat f}_j \, \Big] -e^{-\beta({\hat f}_j-{\hat f}_{j-1})} +e^{-\beta({\hat f}_{j+1}-{\hat f}_j)}\approx0\quad\mbox{in } {(0, 1)}, \end{align} with boundary conditions $$ {\hat f}_j'(0)={\hat f}_j'(1), \qquad {\hat f}_j(0)={\hat f}_j(1), \qquad\forall\, j=1, \cdots, N, $$ where $\varsigma>0$ is defined in (\ref{varsigma}) with the conventions $ {\hat f}_0=-\infty, \ {\hat f}_{N+1}=\infty. $ \qed \end{theorem}
Here are some words for further discussions. Since the solutions have exponential decaying as $y$ leaves away the curve ${\tilde\Gamma}$, the proof of Theorem \ref{theorem 1.2} is much more simpler than that of Theorem \ref{theorem 1.1}. Whence, in the present paper, we will only provide the details to show the validity of Theorem \ref{theorem 1.1}. Based on the same reason, a same result as in Theorem \ref{theorem 1.2} also holds for the first equation of \eqref{originalproblem}
in the whole space $\mathbb R^2$ under the condition $u(y)\rightarrow 0$ as $|y|\rightarrow\infty$. Whence, Theorems \ref{theorem 1.1} and \ref{theorem 1.2} for the existence of cluster of multiple concentration layers can be concerned as the extensions of the results in \cite{weixuyang} and \cite{delPKowWei2007}, where solutions with single concentration layer were constructed for partial confirmation of the two dimensional cases of Conjectures 2 and 3.
However, in addition to the interaction between neighbouring layers in the cluster of multiple concentration layers, the new ingredient is the role of the term ${\mathfrak a}(y)=\big({\mathfrak a}_1(y), {\mathfrak a}_2(y)\big)$. This is the reason that we shall set up the new local coordinates, see \eqref{Fermicoordinates2} together with \eqref{Fermicoordinates-modified} and also \eqref{Fermicoordinates-modified-tilde}. In the procedure of variational calculus, the deformations of the curves $\Gamma$ and ${\tilde\Gamma}$ are no longer directed along their normal directions, see \eqref{deformation1}-\eqref{weightedlength1} and \eqref{deformation2}-\eqref{weightedlength2}. The term ${\mathfrak a}$ will play an effect in the variational properties of the curves, see the notions of non-degenerate stationary curves in Section \ref{Stationary and non-degenerate curves}.
\begin{remark}\label{remark12} {\it The Toda system was used first in \cite{delPKowWei2008} to construct the clustered interfaces for Allen-Cahn model in a two dimensional bounded domain. Later, M. del Pino, M. Kowalczyk, J. Wei and J. Yang \cite{delPKowWeiYang} used the Jacobi-Toda system in the construction of clustered phase transition layers for Allen-Cahn model on general Riemannian manifolds. The reader can refer to \cite{delPKowWei2010, delPKowWei2013, JWeiYang2008, JWeiYang2010, weiyang20201, weiyang20202, YangYang2013}for more results.
For a ${\mathbf d}$-dimensional smooth compact Riemannian manifold $(\tilde{\mathcal M}, {\mathfrak g})$, M. del Pino, M. Kowalczyk, J. Wei and J. Yang \cite{delPKowWeiYang} considered the singularly perturbed Allen-Cahn equation $$ \varepsilon^2\Delta_{ {\mathfrak g}} {u}\, +\, (1 - {u}^2)u \, =\, 0\quad \mbox{in } \tilde{\mathcal M}, $$ where $\varepsilon$ is a small parameter. We let in what follows ${\mathcal K}$ be a minimal $({\mathbf d}-1)$-dimensional embedded submanifold of $\tilde{\mathcal M}$, which divides $\tilde{\mathcal M}$ into two open components $\tilde{\mathcal M}_\pm$. (The latter condition is not needed in some cases.) Assume that ${\mathcal K}$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that \begin{align}\label{posivity}
|\mathcal{A}_{{\mathcal K}}|^2+\mbox{Ric}_{\mathfrak g}(\nu_{{\mathcal K}}, \nu_{{\mathcal K}})>0 \quad \mbox{along } {\mathcal K}. \end{align}
Then for each integer $N\geq 2$, they established the existence of a sequence $\varepsilon = \varepsilon_j\to 0$, and solutions $u_{\varepsilon}$ with $N$-transition layers near ${\mathcal K}$, with mutual distance $O(\varepsilon|\ln \varepsilon|)$.
As the above geometric language, we consider ${\mathbb R}^2$ as a manifold with the metric $$ {\tilde{\mathfrak g}}=V^{2\sigma}(y) \big[ {\mathfrak a}_2(y){\mathrm d}y_1^2 +{\mathfrak a}_1(y) {\mathrm d}y_2^2 \big], $$ with $\sigma$ in (\ref{sigma}) and $\Gamma$ as its submanifold with boundary. In the manifold $({\mathbb R}^2, {\tilde{\mathfrak g}})$, $\Gamma$ is a non-degenerate geodesic with endpoints on $\partial\Omega$. In other words, in order to construct the clustering phase transition layers connecting the boundary $\partial\Omega$ in Theorem \ref{theorem 1.1}, we need the condition (\ref{taupositivity}), which is similar as (\ref{posivity}) in \cite{delPKowWeiYang}. The reader can refer to Section \ref{section6.2}. \qed } \end{remark}
\begin{remark}\label{remark13} {\it At $P_1$ and $P_2$ (the intersection points of $\Gamma$ and $\partial\Omega$), the conditions in (\ref{boundaryadmissibility}) set up relations between the terms ${\mathfrak a}$, $V$ and the geometric properties of the curves $\partial\Omega$ and $\Gamma$. For example, by recalling the unit normal $n(\theta)=(n_1(\theta), n_2(\theta))$ to $\Gamma$ and also the curvatures $k_1$ and $k_2$ of $\partial\Omega$ at $P_1$ and $P_2$, the first one in \eqref{boundaryadmissibility} can be exactly expressed in the following form \begin{align} \frac{\sqrt{2}}{{\mathfrak a}_1(0, 0)}\Big[\partial_t{\mathfrak a}_1(0, 0)- \partial_t{\mathfrak a}_2(0, 0) \Big]n_1(0)n_2(0) \, +\,
\sqrt{2}\Big({\tilde{\mathfrak a}'}_1(0)|n_1(0)|^2+{\tilde{\mathfrak a}'}_2(0) |n_2(0)|^2\Big) \, +\, k_1 =\frac{\beta'(0)}{\beta(0)}, \end{align} with the conventions $$
{\tilde{\mathfrak a}}_1(\theta)=\frac{{\mathfrak a}_1(0, \theta)}{\sqrt{|{\mathfrak a}_1(0, \theta)|^2+|{\mathfrak a}_2(0, \theta)|^2}\, }, \qquad
{\tilde{\mathfrak a}}_2(\theta)=\frac{{\mathfrak a}_2(0, \theta)}{\sqrt{|{\mathfrak a}_1(0, \theta)|^2+|{\mathfrak a}_2(0, \theta)|^2}\, }, $$
$$ \beta(\theta)
=\sqrt{\frac{ V(0, \theta)\big(a_1(0, \theta)|n_1(\theta)|^2+a_2(0, \theta)|n_2(\theta)|^2\big)}
{|a_1(0, \theta)|^2+|a_2(0, \theta)|^2}}, $$ where we have used \eqref{tildea1a2}, \eqref{alpha-beta}, \eqref{b1=mathfrakw0}, \eqref{b2=mathfrakw1}. These conditions will be used to decompose the interaction of neighbouring layers on the boundary $\partial\Omega$, see Remark \ref{remark61}. On the other hand, we still have to deal with the delicate boundary terms for the reduced equations in Section \ref{section6.2}. \qed } \end{remark}
\begin{remark}\label{remark14} We construct solutions with multiple clustering concentration layers only for a sequence of $\varepsilon$ due to the coexistence of two types of resonances, see also the fourth open question in "Concluding Remarks" of \cite{delPKowWei2010}. The first one is due to the instability of the profile function $w$, see Proposition \ref{proposition7point1}, in which we can impose the following gap condition for $\varepsilon$ \begin{align}\label{gapconditionofve}
|\lambda_*-j^2\varepsilon^2|\geq {\tilde c}\, \varepsilon, \quad \forall\, j \in \mathbb{N}, \end{align} where ${\tilde c}$ is a given small positive constant. In the above, $\lambda_{*}$ is a positive constant given by
\begin{align} \label{definenumber} \lambda_{*}\, =\, \frac {\lambda_0 \ell^2}{\pi^2}, \end{align} where $\lambda_0$ and $\ell$ are the positive constants given in \eqref{lambda0} and \eqref{ell}. More details about this resonance phenomena were described in \cite{delPKowWei2007}. The other one comes from the Jacobi-Toda system which was concerned in \cite{delPKowWeiYang}. In this case, we shall choose a sequence of $\varepsilon$ from those satisfying (\ref{gapconditionofve}), see Proposition \ref{proposition7point2}. \qed \end{remark}
By the rescaling \begin{equation}\label{rescaling}
y\, =\, \varepsilon {\tilde y} \end{equation} in ${\mathbb R}^2$, problem (\ref{originalproblem}) will be rewritten as \begin{align} \label{problemafterscaling} {\mathrm {div}}\big( \nabla_{{\mathfrak a}(\varepsilon \tilde y)} u\big)- V(\varepsilon \tilde y)u+u^p\, =\, 0\quad{\rm in }\ \Omega_\varepsilon, \qquad \nabla_{{\mathfrak a}(\varepsilon \tilde y)} u \cdot\nu_{\varepsilon}\, =\, 0\quad\mbox{on }\ \partial \Omega_\varepsilon, \end{align} where $\Omega_\varepsilon=\Omega/\varepsilon$ and $\Gamma_\varepsilon=\Gamma/\varepsilon$, $\nu_\varepsilon$ is the unit outer normal of $\partial \Omega_\varepsilon$. The remaining part of this paper is devoted to the proof of Theorem \ref{theorem 1.1}, which will be organized as follows: \begin{itemize} \item[1.] In Section \ref{section2}, we will set up a coordinate system in a neighborhood of $\Gamma$. Next we write down the local form of (\ref{problemafterscaling}), especially the differential operators the differential operators ${\rm div}\big(\nabla_{\mathfrak a(y)}u\big)$ and $\nabla_{{\mathfrak a}(y)}u\cdot\nu$. This local coordinate system also help us set up the stationary and non-degeneracy conditions for the curve $\Gamma$, see (\ref{stationary}) and (\ref{nondegeneracy}).
\item[2.] We will set up an outline of the proof in Section \ref{thegluingprocedure}, which involves the gluing procedure from \cite {delPKowWei2007}, so that we can transform \eqref{problemafterscaling} into a projected form, see \eqref{system-1}-\eqref{system-4}.
\item[3.] In Section \ref{section4}, we are devoted to the constructing of a local approximate solution in such a way that it solves the nonlinear problem locally up to order $O(\varepsilon^2)$.
\item[4.] To get a real solution, the well-known infinite dimensional reduction method \cite{delPKowWei2007} will be needed in Sections \ref{section5}-\ref{sectionsolvingreducedequation}. In fact, the reduced problem involves a Toda-Jacobi system and inherits the resonance phenomena, which will be handled by complicated Fourier analysis. \end{itemize}
\section{Geometric preliminaries} \label{section2} \setcounter{equation}{0} In this section, we will set up a coordinate system in a neighborhood of $\Gamma$. This system is similar to the modified Fermi coordinates in \cite{weixuyang}. However, some adaptions should be introduced due to the existence of the term ${\mathfrak a}$ in \eqref{originalproblem}, which make the geometric computations much more complicated. The differential operators in \eqref{originalproblem} will be then derived in the local coordinates. The notion of a stationary and non-degenerate curve $\Gamma$ will be also derived in the last part of this section.
\subsection{Modified Fermi coordinates}\label{Fermi}\
Recall the assumptions {\bf (A1)}-{\bf (A3)} in Section \ref{section1} and notation therein. For basic notions of curves, such as the signed curvature, the reader can refer to the book by do Carmo \cite{docarmo}. \\[2mm] \noindent {\bf Step 1.} Let the natural parameterization of the curve $\Gamma$ be as follows. $$ \gamma_0:[0, 1]\rightarrow \Gamma\subset \bar\Omega\subset {\mathbb R}^2. $$ For some small positive number $\sigma_0$, one can make a smooth extension and define the mapping $$ \gamma=(\gamma_1, \gamma_2):(-\sigma_0, 1+\sigma_0)\rightarrow {\mathbb R}^2, $$ such that $$ \gamma(\tilde {\theta})\, =\, \gamma_0({\tilde\theta}), \quad \forall\, {\tilde\theta}\in [0, 1]. $$ There holds the Frenet formula \begin{align}\label{Frenet} \gamma''\, =\, kn \quad\textrm{and}\quad n'\, =\, -k\gamma', \end{align} where $k$, $n=(n_1, n_2)$ are the curvature and the normal of $\gamma$. The relations \begin{align}
|\gamma_1'({\tilde\theta})|^2\, +\, |\gamma_2'({\tilde\theta})|^2=1, \qquad
|n_1({\tilde\theta})|^2+|n_2({\tilde\theta})|^2=1, \qquad \gamma_1'({\tilde\theta})n_1({\tilde\theta})+ \gamma_2'({\tilde\theta})n_2({\tilde\theta})=0, \label{perpendicular} \end{align} will give that \begin{equation}\label{relationofga&n}
\big(\gamma_1'({\tilde\theta}), \gamma_2'({\tilde\theta})\big)\, =\, \big(-n_2({\tilde\theta}), n_1({\tilde\theta})\big), \end{equation} and \begin{align}\label{relationofga&n2} -\gamma_1'({\tilde\theta})n_2({\tilde\theta})+\gamma_2'({\tilde\theta})n_1({\tilde\theta})\, =\, 1. \end{align}
Choosing $\delta_0>0$ very small, and setting $$ {\mathfrak S}_1\, \equiv\, (-\delta_0, \delta_0)\times (-\sigma_0, 1+\sigma_0), $$ we construct the following mapping \begin{align} {\hat{\mathbb H}}: {\mathfrak S}_1\rightarrow {\hat{\mathbb H}}({\mathfrak S}_1)\, \equiv\, {\hat\Omega}_{\delta_0, \sigma_0} \quad \mbox{with} \quad {\hat{\mathbb H}}(\tilde {t}, \tilde {\theta})\, =\, \gamma({\tilde\theta})+{\tilde t}\, n({\tilde\theta}). \label{Fermicoordinates0} \end{align} Note that ${\hat{\mathbb H}}$ is a diffeomorphism (locally) and ${\hat{\mathbb H}}(0, {\tilde\theta})=\gamma({\tilde\theta})$. By this, we will write the functions ${\mathfrak a}_1$, ${\mathfrak a}_2$ in the forms ${\mathfrak a}_1({\tilde t}, {\tilde\theta})$ and ${\mathfrak a}_2({\tilde t}, {\tilde\theta})$ and then set \begin{align}\label{tildea1a2}
{\tilde{\mathfrak a}}_1({\tilde\theta})=\frac{{\mathfrak a}_1(0, {\tilde\theta})}{\sqrt{|{\mathfrak a}_1(0, {\tilde\theta})|^2+|{\mathfrak a}_2(0, {\tilde\theta})|^2}\, }, \qquad
{\tilde{\mathfrak a}}_2({\tilde\theta})=\frac{{\mathfrak a}_2(0, {\tilde\theta})}{\sqrt{|{\mathfrak a}_1(0, {\tilde\theta})|^2+|{\mathfrak a}_2(0, {\tilde\theta})|^2}\, }. \end{align} Note that \begin{align}\label{tildea1=a2} {\tilde{\mathfrak a}}_1(0)={\tilde{\mathfrak a}}_2(0)={\tilde{\mathfrak a}}_1(1)={\tilde{\mathfrak a}}_2(1)=\frac{1}{\sqrt 2}, \end{align} due to the assumptions in \eqref{a1=a2}. After that, we construct another mapping \begin{align} {\mathbb H}: {\mathfrak S}_1\rightarrow {\mathbb H}({\mathfrak S}_1)\, \equiv\, \Omega_{\delta_0, \sigma_0} \quad \mbox{with} \quad \mathbb{H}(\tilde {t}, \tilde {\theta})\, =\, \gamma({\tilde\theta}) \, +\, {\tilde t}\, \big({\tilde{\mathfrak a}}_1({\tilde\theta})n_1({\tilde\theta}), \, {\tilde{\mathfrak a}}_2( {\tilde\theta})n_2({\tilde\theta})\big), \label{Fermicoordinates2} \end{align} in such a way that it is a local diffeomorphism and ${\mathbb H}(0, {\tilde\theta})=\gamma({\tilde\theta})$. This is due to the fact that ${\mathfrak a}_1$ and ${\mathfrak a}_2$ are positive functions.
\noindent {\bf Step 2}. Recall ${\mathcal C}_1$, ${\mathcal C}_2$ given in the assumptions {\bf (A1)}-{\bf (A3)} in Section \ref{section1} and then denote the preimages $$ {\tilde{\mathcal C}}_1\, \equiv\, {\mathbb H}^{-1}\left(\mathcal C_1\right) \quad\textrm{and}\quad {\tilde{\mathcal C}}_2\, \equiv\, {\mathbb H}^{-1}\left(\mathcal C_2\right), $$ which are two smooth curves in $({\tilde t}, {\tilde\theta})$ coordinates of \eqref{Fermicoordinates2} and can be parameterized respectively by $\left({\tilde t}, {\tilde \varphi}_1({\tilde t})\right)$ and $ \left({\tilde t}, {\tilde \varphi}_2({\tilde t})\right)$ for some smooth functions ${\tilde \varphi}_1({\tilde t})\, {\rm and}\; {\tilde \varphi}_2({\tilde t})$ with the properties \begin{align} {\tilde \varphi}_1(0)=0, \qquad {\tilde \varphi}_2(0)=1. \label{fact0} \end{align} We define a mapping $$ \tilde{{\mathbb H}}:\, {\mathfrak S}_1\rightarrow {\mathfrak S}_2\, \equiv\, \tilde{{\mathbb H}}({\mathfrak S}_1)\subset \Bbb R^2, $$ such that $$ t={\tilde t}, \quad \theta=\frac {{\tilde\theta}-{\tilde \varphi}_1({\tilde t})}{{\tilde \varphi}_2({\tilde t})-{\tilde \varphi}_1({\tilde t})}. $$ \noindent This transformation will straighten up the curves ${\tilde{\mathcal C}}_1$ and ${\tilde{\mathcal C}}_2$. It is obvious that \begin{align*} {\tilde {\mathbb H}}^{-1}(0, \theta)\, =\, (0, {\theta}), \ \theta\in [0, 1]. \end{align*} Moreover, we have
\begin{lemma} \label{lemma2.1} There hold \begin{align}\label{fact2} \tilde\varphi_1'(0)=0, \quad \tilde\varphi_2'(0)=0, \quad \tilde{\varphi}_1''(0)={\tilde k}_1, \quad \tilde{\varphi}_2''(0)={\tilde k}_2, \end{align} where \begin{equation}
{\tilde k}_1
=
\frac
{\big(|{\tilde{\mathfrak a}}_1(0)n_1(0)|^2
\, +\,
|{\tilde{\mathfrak a}}_2(0)n_2(0)|^2\big)^{3/2}}
{{\tilde{\mathfrak a}}_1(0) |n_1(0)|^2
\, +\,
{\tilde{\mathfrak a}}_2(0) |n_2(0)|^2}
\, k_1
=\frac{1}{2}k_1,
\label{tildek1} \end{equation} \begin{equation}
{\tilde k}_2 =
\frac
{\big(|{\tilde{\mathfrak a}}_1(1)n_1(1)|^2
\, +\,
|{\tilde{\mathfrak a}}_2(1)n_2(1)|^2\big)^{3/2}}
{{\tilde{\mathfrak a}}_1(1) |n_1(1)|^2
\, +\,
{\tilde{\mathfrak a}}_2(1) |n_2(1)|^2}
\, k_2
=\frac{1}{2}k_2.
\label{tildek2} \end{equation} \end{lemma} \proof In fact, the curves can be expressed in the following forms \begin{align*} {\mathcal C}_1:&\ {\mathbb H}\big({\tilde t}, {\tilde \varphi}_1({\tilde t})\big) \, =\, \gamma\big({\tilde \varphi}_1({\tilde t})\big) \, +\, {\tilde t}\, \Big({\tilde{\mathfrak a}}_1\big( {\tilde \varphi}_1({\tilde t})\big) n_1\big({\tilde \varphi}_1({\tilde t})\big), \, {\tilde{\mathfrak a}}_2\big( {\tilde \varphi}_1({\tilde t})\big) n_2\big({\tilde \varphi}_1({\tilde t})\big)\Big), \\[2mm] {\mathcal C}_2:&\ {\mathbb H}\big({\tilde t}, {\tilde \varphi}_2({\tilde t})\big) \, =\,\gamma\big({\tilde \varphi}_2({\tilde t})\big) \, +\, {\tilde t} \, \Big({\tilde{\mathfrak a}}_1\big( {\tilde \varphi}_2({\tilde t})\big) n_1\big({\tilde \varphi}_2({\tilde t})\big), \, {\tilde{\mathfrak a}}_2\big( {\tilde \varphi}_2({\tilde t})\big) n_2\big({\tilde \varphi}_2({\tilde t})\big)\Big), \\[2mm] \Gamma:&\ {\mathbb H}({\tilde\theta})\, =\, \gamma({\tilde\theta}). \end{align*} It follows that the tangent vectors of ${\mathcal C}_1$ at $P_1$ can be written as \begin{align*}
\frac {{\mathrm d} {\mathcal C}_1}{{\mathrm d}{\tilde t}}\Big|_{{\tilde t}=0} \, =\,
&\frac {\partial \gamma}{\partial {\tilde\theta}}\Big|_{{\tilde\theta}\, =\, {{\tilde\varphi}_1(0)}}
\cdot \frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big|_{{\tilde t}=0} \, +\, \Big({\tilde{\mathfrak a}}_1\big( {\tilde \varphi}_1({\tilde t})\big) n_1\big({\tilde \varphi}_1({\tilde t})\big), \,
{\tilde{\mathfrak a}}_2\big( {\tilde \varphi}_1({\tilde t})\big) n_2\big({\tilde \varphi}_1({\tilde t})\big)\Big)\Big|_{{\tilde t}=0} \\[2mm] \, =\, & \gamma'(0){\tilde\varphi}_1'(0) \, +\, \big({\tilde{\mathfrak a}}_1(0) n_1(0), \, {\tilde{\mathfrak a}}_2(0) n_2(0)\big) , \end{align*} and the tangent vector of $\Gamma$ at $P_1$ is $\gamma'(0)$. According to the condition: $\Gamma\bot \partial \Omega$ at $P_1$, we have that $$
\left\langle\frac {{\mathrm d}{\mathcal C}_1}{{\mathrm d}{\tilde t}}\Big|_{{\tilde t}=0}, \, \, \gamma'(0) \right\rangle\, =\, 0. $$ By \eqref{perpendicular} and \eqref{tildea1=a2}, we have $$ {\tilde{\mathfrak a}}_1(0) n_1(0)\gamma_1'(0)\, +\, {\tilde{\mathfrak a}}_2(0) n_2(0)\gamma_2'(0)=0, $$ and then drive from the above to get $$ \tilde{\varphi}_1'(0)=0.$$ Similarly, we can show ${\tilde \varphi}_2'(0)=0$.
The curve ${\mathcal C}_1$ can be expressed in the following form \begin{equation*} \begin{split}
{\mathcal C}_1: {\mathbb H}\big({\tilde t}, {\tilde \varphi}_1({\tilde t})\big) \, &=\Big(\gamma_1\big({\tilde \varphi}_1({\tilde t})\big)+{\tilde t} {\tilde{\mathfrak a}}_1\big( {\tilde \varphi}_1({\tilde t})\big)n_1\big({\tilde \varphi}_1({\tilde t})\big), \, \, \gamma_2\big({\tilde \varphi}_1({\tilde t})\big)+{\tilde t} {\tilde{\mathfrak a}}_2\big( {\tilde \varphi}_1({\tilde t})\big) n_2\big({\tilde \varphi}_1({\tilde t})\big)\Big) \\[2mm] &\equiv\, \big( y_1({\tilde t})\, , \, y_2({\tilde t})\big). \end{split} \end{equation*} The calculations \begin{align*} y_1'({\tilde t}) =&\, \gamma_1'({\tilde\varphi}_1)\cdot \frac {{\mathrm d}{{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +{\tilde{\mathfrak a}}_1({\tilde\varphi}_1) n_1({\tilde\varphi}_1) +{\tilde t}\cdot{\tilde{\mathfrak a}}_1({\tilde\varphi}_1)\cdot n_1'({\tilde\varphi}_1)\cdot \frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +{\tilde t}\cdot{\tilde{\mathfrak a}}_1'({\tilde\varphi}_1)\cdot n_1({\tilde\varphi}_1)\cdot \frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}, \\[2mm] y_2'({\tilde t}) =&\, \gamma_2'({\tilde\varphi}_1)\cdot \frac {{\mathrm d}{{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +{\tilde{\mathfrak a}}_2({\tilde\varphi}_1) n_2({\tilde\varphi}_1) +{\tilde t}\cdot{\tilde{\mathfrak a}}_2({\tilde\varphi}_1)\cdot n_2'({\tilde\varphi}_1)\cdot \frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +{\tilde t}\cdot{\tilde{\mathfrak a}}_2'({\tilde\varphi}_2)\cdot n_2({\tilde\varphi}_1)\cdot \frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}, \end{align*} and \begin{align*} y_1''({\tilde t})=&\, \gamma_1''({\tilde\varphi}_1)\cdot \Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +\gamma_1'({\tilde\varphi}_1)\cdot \frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2} +2\, {\tilde{\mathfrak a}}_1'({\tilde\varphi}_1)\cdot n_1({\tilde\varphi}_1)\cdot\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} \\[2mm] & +2\, {\tilde{\mathfrak a}}_1({\tilde\varphi}_1)\cdot n_1'({\tilde\varphi}_1)\cdot\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +2{\tilde t}\, {\tilde{\mathfrak a}}_1'({\tilde\varphi}_1) n_1'({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +{\tilde t}\, {\tilde{\mathfrak a}}_1({\tilde\varphi}_1) n_1''({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 \\[2mm] & +{\tilde t}\, {\tilde{\mathfrak a}}_1({\tilde\varphi}_1) n_1'({\tilde\varphi}_1)\cdot\frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2} +{\tilde t}\, {\tilde{\mathfrak a}}_1''({\tilde\varphi}_1) n_1({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +{\tilde t}\, {\tilde{\mathfrak a}}_1'({\tilde\varphi}_1) n_1({\tilde\varphi}_1)\cdot\frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2},
\end{align*} \begin{align*} y_2''({\tilde t})=&\, \gamma_2''({\tilde\varphi}_1)\cdot \Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +\gamma_2'({\tilde\varphi}_1)\cdot \frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2} +2\, {\tilde{\mathfrak a}}_2'({\tilde\varphi}_1)\cdot n_2({\tilde\varphi}_1)\cdot\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} \\[2mm] & +2\, {\tilde{\mathfrak a}}_2({\tilde\varphi}_1)\cdot n_2'({\tilde\varphi}_1)\cdot\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}} +2{\tilde t}\, {\tilde{\mathfrak a}}_2'({\tilde\varphi}_1) n_2'({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +{\tilde t}\, {\tilde{\mathfrak a}}_2({\tilde\varphi}_1) n_2''({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 \\[2mm] & +{\tilde t}\, {\tilde{\mathfrak a}}_2({\tilde\varphi}_1) n_2'({\tilde\varphi}_1)\cdot\frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2} +{\tilde t}\, {\tilde{\mathfrak a}}_2''({\tilde\varphi}_1) n_2({\tilde\varphi}_1)\cdot\Big(\frac {{\mathrm d} {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}}\Big)^2 +{\tilde t}\, {\tilde{\mathfrak a}}_2'({\tilde\varphi}_1) n_2({\tilde\varphi}_1)\cdot\frac {{\mathrm d}^2 {{\tilde\varphi}_1}}{{\mathrm d} {\tilde t}^2},
\end{align*} imply that
\begin{align*}
|y_1'(0)|^2+|y_2'(0)|^2
\, =\,
|{\tilde{\mathfrak a}}_1(0)n_1(0)|^2
\, +\,
|{\tilde{\mathfrak a}}_2(0)n_2(0)|^2,
\end{align*} and \begin{align*} y_1'(0)\, y_2''(0)-y_1''(0)\, y_2'(0) =&\, \big({\tilde{\mathfrak a}}_1(0)n_1(0)\, \gamma_2'(0) \,-\, {\tilde{\mathfrak a}}_2(0)n_2(0)\, \gamma_1'(0)\big)\, {\tilde\varphi}_1''(0) \\[2mm] \, =&\,
\big({\tilde{\mathfrak a}}_1(0) |n_1(0)|^2 \, +\,
{\tilde{\mathfrak a}}_2(0) |n_2(0)|^2\big)\, {\tilde\varphi}_1''(0). \end{align*} Therefore, the signed curvature of the curve $\mathcal{C}_1$ at the point $P_1$ is \begin{equation*}
k_1=\frac{\, y_1'(0)y_2''(0)-y_1''(0)y_2'(0)\, }{\big[\, |y_1'(0)|^2+|y_2'(0)|^2\, \big]^{\frac{3}{2}}}
=
\frac{{\tilde{\mathfrak a}}_1(0) |n_1(0)|^2
\, +\,
{\tilde{\mathfrak a}}_2(0) |n_2(0)|^2}
{\big(|{\tilde{\mathfrak a}}_1(0)n_1(0)|^2
\, +\,
|{\tilde{\mathfrak a}}_2(0)n_2(0)|^2\big)^{3/2}}
\, \tilde{\varphi}_1''(0)
=2\tilde{\varphi}_1''(0). \end{equation*} Similarly, we can show \begin{equation*} k_2=
\frac{{\tilde{\mathfrak a}}_1(1) |n_1(1)|^2 \, +\,
{\tilde{\mathfrak a}}_2(1) |n_2(1)|^2}
{\big(|{\tilde{\mathfrak a}}_1(1)n_1(1)|^2
\, +\,
|{\tilde{\mathfrak a}}_2(1)n_2(1)|^2\big)^{3/2}} \, \tilde{\varphi}_2''(0) =2\tilde{\varphi}_2''(0). \end{equation*}
\noindent {\bf Step 3}. We define the {\bf modified Fermi coordinates} \begin{equation} \label{Fermicoordinates-modified} (y_1, y_2)\, =\, F(t, \theta) \, =\, {\mathbb H}\circ{\tilde {\mathbb H}}^{-1}(t, \theta)\, :\, (-\delta_0, \delta_0)\times(-\sigma_0, 1+\sigma_0)\rightarrow{\mathbb R}^2 \end{equation} for given small positive constants $\sigma_0$ and $\delta_0$. More precisely, we write $F(t, \theta)=\big(F_1(t, \theta), \, F_2(t, \theta)\big)$ with \begin{equation}\label{F1} F_i(t, \theta) \, =\, \gamma_i\big(\Theta(t, \theta)\big) \, +\, t\, {\tilde{\mathfrak a}}_i\big(\Theta(t, \theta)\big) n_i\big(\Theta(t, \theta)\big), \quad i=1,2, \end{equation} where \begin{equation}\label{Theta} \Theta(t, \theta) \, \equiv\, \big({\tilde \varphi}_2(t)-{\tilde \varphi}_1(t)\big)\theta \, +\, {\tilde \varphi}_1(t). \end{equation} From \eqref{fact0}-\eqref{tildek2}, we have \begin{align} \Theta(0, \theta)\, =\, \theta, \qquad \Theta_t(0, \theta)\, =\, 0, \qquad \Theta_\theta(0, \theta)\, =\, 1, \qquad \Theta_{\theta t}(0, \theta)\, =\, 0, \label{Thetaderivative1} \\[2mm] \Theta_{tt}(0, \theta)\, =\, \big({\tilde k}_2-{\tilde k}_1\big)\theta +{\tilde k}_1, \qquad \Theta_{tt\theta}(0, \theta)\, =\, {\tilde k}_2-{\tilde k}_1. \label{Thetaderivative2} \end{align} These quantities will play an important role in the further settings.
We now derive the asymptotical behaviors of the coordinates. For given $i=1$ or $i=2$, consider the derivative of first order \begin{align} \begin{aligned} \frac{{\partial} F_i}{{\partial} t} &\, =\, \gamma_i'(\Theta)\cdot\Theta_t \, +\, {\tilde{\mathfrak a}}_i(\Theta) n_i(\Theta) \, +\, t\, {\tilde{\mathfrak a}}_i'(\Theta) n_i(\Theta)\cdot {\Theta}_t \, +\, t\, {\tilde{\mathfrak a}}_i(\Theta) n_i'(\Theta)\cdot {\Theta}_t, \end{aligned} \label{partialFipartialt} \end{align} and also the derivative of second order \begin{equation} \begin{split} \frac{{\partial}^2 F_i}{{\partial} t^2}
&\, =\, \gamma_i''(\Theta)\cdot |\Theta_t|^2 \, +\, \gamma_i'(\Theta)\cdot \Theta_{tt} \, +\, 2{\tilde{\mathfrak a}}_i'(\Theta) n_i(\Theta)\cdot \Theta_{t} \, +\, 2{\tilde{\mathfrak a}}_i(\Theta) n_i'(\Theta)\cdot \Theta_{t} \\[2mm] &\quad
\, +\, 2t\, {\tilde{\mathfrak a}}_i'(\Theta) n_i'(\Theta)\cdot|\Theta_t|^2
\, +\, t\, {\tilde{\mathfrak a}}_i''(\Theta) n_i(\Theta)\cdot|\Theta_t|^2 \, +\, t\, {\tilde{\mathfrak a}}_i'(\Theta) n_i(\Theta)\cdot\Theta_{tt} \\[2mm] &\quad
\, +\, t\, {\tilde{\mathfrak a}}_i(\Theta) n_i''(\Theta)\cdot|\Theta_t|^2 \, +\, t\, {\tilde{\mathfrak a}}_i(\Theta) n_i'(\Theta)\cdot\Theta_{tt}. \end{split} \label{partialFipartialtt} \end{equation} These imply that \begin{equation} \frac{{\partial} F_i}{{\partial} t}(0, \theta)\, =\, {\tilde{\mathfrak a}}_i(\theta)n_i(\theta), \end{equation} and \begin{equation}
\frac{{\partial}^2F_i}{{\partial} t^2}(0, \theta) \, =\, \gamma_i'(\theta)\cdot \Theta_{tt}(0, \theta) \, =\, \big({\tilde k}_2-{\tilde k}_1\big)\theta +{\tilde k}_1 \, \equiv\, q_i(\theta). \label{q_i} \end{equation} Hence \begin{equation}
\Theta_{tt}(0, \theta)\, =\, q_1 \gamma_1'+q_2 \gamma_2'\, =\, -q_1n_2+q_2n_1.
\label{2.22} \end{equation} Here comes the derivative of third order \begin{align}\label{m_i} \frac{{\partial}^3F_i}{{\partial} t^3}(0, \theta) &\, =\, \bigg[ \gamma_i'''(\Theta)\cdot (\Theta_t)^3 +3\gamma_i''(\Theta)\cdot \Theta_t\cdot \Theta_{tt} +\gamma_i'(\Theta)\cdot \Theta_{ttt} \nonumber\\[2mm] &\qquad
+3{\tilde{\mathfrak a}}_i''(\Theta) n_i(\Theta)\cdot |\Theta_t|^2
+8{\tilde{\mathfrak a}}_i'(\Theta) n_i'(\Theta)\cdot |\Theta_t|^2 +3{\tilde{\mathfrak a}}_i'(\Theta) n_i(\Theta)\cdot \Theta_{tt} \nonumber\\[2mm] &\qquad
+3{\tilde{\mathfrak a}}_i(\Theta) n_i''(\Theta)\cdot |\Theta_t|^2 +3{\tilde{\mathfrak a}}_i(\Theta) n_i'(\Theta)\cdot \Theta_{tt}
\bigg]\Big|_{(t, \theta)=(0, \theta)} \nonumber\\[2mm] &\, =\, \gamma_i'(\theta)\cdot \Theta_{ttt}(0, \theta) +3{\tilde{\mathfrak a}}_i'(\theta) n_i(\theta)\cdot \Theta_{tt}(0, \theta) +3{\tilde{\mathfrak a}}_i(\theta) n_i'(\theta)\cdot \Theta_{tt}(0, \theta) \nonumber\\[2mm] &\, \equiv\, m_i(\theta). \end{align} These results will be collected in the following way. \begin{lemma}\label{derivativeofF} The mapping $F$ has the following properties: \\
\noindent {\rm\textbf{(1).}} $\, \, F(0, \theta)\, =\, \gamma(\theta)$, \qquad $\, \, \frac{{\partial} F}{{\partial} t}(0, \theta)\, =\, \big({\tilde{\mathfrak a}}_1(\theta)n_1(\theta), {\tilde{\mathfrak a}}_2(\theta)n_2(\theta)\big)$, \\
\noindent {\rm\textbf{(2).}} $\, \, \, \frac{{\partial}^2 F}{{\partial} t^2}(0, \theta)=q(\theta) \quad\mbox{with}\quad q(\theta)\, =\, \big(q_1(\theta), q_2(\theta)\big)\quad \bot\, n(\theta) $, \\
\noindent {\rm\textbf{(3).}} $\, \, \frac{{\partial}^3F}{{\partial} t^3}(0, \theta)=m(\theta) \quad\mbox{with}\quad m(\theta)\, =\, \big(m_1(\theta), m_2(\theta)\big) $. \\
\noindent Here $m_i$'s and $q_i$'s are given in \eqref{q_i}-\eqref{m_i}. \qed \end{lemma}
As a conclusion, as $t$ is small enough, there holds the expansion \begin{align} F(t, \theta) \, =\, & \gamma(\theta) +t\, \big({\tilde{\mathfrak a}}_1(\theta)n_1(\theta), {\tilde{\mathfrak a}}_2(\theta)n_2(\theta)\big) +\frac {t^2}{2}q(\theta) \nonumber\\[2mm] &+\frac {t^3}{6}m(\theta) +O(t^4), \quad \forall\, \theta\in [0, 1], t\in (-\delta_0, \delta_0), \label{expansionofFermi} \end{align} where $\delta_0>0$ is a small constant. This gives us that \begin{align} \frac{{\partial} F}{{\partial} t}(t, \theta) \, =\, & \big({\tilde{\mathfrak a}}_1(\theta)n_1(\theta), {\tilde{\mathfrak a}}_2(\theta)n_2(\theta)\big) +t q(\theta) +\frac{t^2}{2}m(\theta) +O(t^3), \label{F_t} \\[2mm] \frac{{\partial} F_1}{{\partial} \theta}(t, \theta) \, =\, &\gamma_1'(\theta) +t{\tilde{\mathfrak a}}_1'(\theta)n_1(\theta) -tk{\tilde{\mathfrak a}}_1(\theta)\gamma_1'(\theta) +\frac{t^2}{2}q_1'(\theta)+O(t^3) \nonumber\\[2mm] \, =\, &-n_2(\theta) +t{\tilde{\mathfrak a}}_1'(\theta)n_1(\theta) +tk{\tilde{\mathfrak a}}_1(\theta)n_2(\theta) +\frac{t^2}{2}q_1'(\theta)+O(t^3), \label{F1_theta} \\[2mm] \frac{{\partial} F_2}{{\partial} \theta}(t, \theta) \, =\, & \gamma_2'(\theta) +t{\tilde{\mathfrak a}}_1'(\theta)n_2(\theta) -tk{\tilde{\mathfrak a}}_1(\theta)\gamma_2'(\theta) +\frac{t^2}{2}q_2'(\theta)+O(t^3) \nonumber\\[2mm] \, =\, & n_1(\theta) +t{\tilde{\mathfrak a}}_1'(\theta)n_2(\theta) -tk{\tilde{\mathfrak a}}_1(\theta)n_1(\theta) +\frac{t^2}{2}q_2'(\theta)+O(t^3), \label{F2_theta} \end{align} where we have used \eqref{relationofga&n} and the Frenet formula \eqref{Frenet}.
Moreover, there hold \begin{align}\label{qi'} q_i'(\theta) \, =\, \gamma_i''(\theta)\cdot\Theta_{tt}(0, \theta) \, +\, \gamma_i'(\theta)\Theta_{tt\theta}(0, \theta), \end{align} and specially, \begin{align*} q'(0)\, =\, \gamma{''}(0){\tilde k}_1\, +\, \gamma'(0)({\tilde k}_2-{\tilde k}_1), \qquad q'(1)\, =\, \gamma{''}(1){\tilde k}_2\, +\, \gamma'(1)({\tilde k}_2-{\tilde k}_1). \end{align*}
\subsection{The metric} In the local coordinates $(t, \theta)$ in \eqref{Fermicoordinates-modified}, here are the preparing computations for the metric matrix: \begin{align} \frac{{\partial} F_i}{{\partial} t}\frac{{\partial} F_i}{{\partial} t}
&\, =\, ({\tilde{\mathfrak a}}_i)^2|n_i|^2 +2t\, {\tilde{\mathfrak a}}_i n_i q_i +t^2\, {\tilde{\mathfrak a}}_i n_i m_i
+t^2|q_i|^2 +O(t^3), \label{Fit-Fit} \end{align} \begin{align}\label{Fit-Fitheta} \frac{{\partial} F_1}{{\partial} t} \frac{{\partial} F_1}{{\partial} \theta}
\, =\, & -{\tilde{\mathfrak a}}_1 n_1 n_2
+t {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1 |n_1|^2 +tk {\tilde{\mathfrak a}}_1^2 n_1 n_2 -t q_1n_2 +\frac{t^2}{2}{\tilde{\mathfrak a}}_1 q_1'n_1 \nonumber\\[2mm] &+t^2{\tilde{\mathfrak a}' }_1q_1n_1 +t^2k {\tilde{\mathfrak a}}_1 q_1 n_2 -\frac{t^2}{2}m_1 n_2+O(t^3), \\[2mm] \frac{{\partial} F_2}{{\partial} t} \frac{{\partial} F_2}{{\partial} \theta}
\, =\, & {\tilde{\mathfrak a}}_2 n_2 n_1
+t {\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2 -tk {\tilde{\mathfrak a}}_2^2 n_2n_1 +t q_2n_1 +\frac{t^2}{2}{\tilde{\mathfrak a}}_2 q_2'n_2 \nonumber\\[2mm] &+t^2{\tilde{\mathfrak a}' }_2q_2n_2 -t^2k {\tilde{\mathfrak a}}_2 q_2n_1 +\frac{t^2}{2}m_2n_1+O(t^3), \end{align} \begin{align} \frac{{\partial} F_1}{{\partial} \theta} \frac{{\partial} F_1}{{\partial} \theta}
\, =\, & |n_2|^2 -2t {\tilde{\mathfrak a}'}_1 n_1n_2
-2t k{\tilde{\mathfrak a}}_1 |n_2|^2 -t^2 q_1'n_2
+t^2({\tilde{\mathfrak a}'}_1)^2|n_1|^2 \nonumber\\[2mm] &+2t^2k{\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}'}_1 n_1n_2
+t^2k^2 ({\tilde{\mathfrak a}}_1)^2 |n_2|^2 +O(t^3), \label{Fitheta-Fitheta} \end{align} and \begin{align} \frac{{\partial} F_2}{{\partial} \theta} \frac{{\partial} F_2}{{\partial} \theta}
\, =\, & |n_1|^2 +2t {\tilde{\mathfrak a}'}_2 n_1 n_2
-2t k{\tilde{\mathfrak a}}_2 |n_1|^2 +t^2 q_2'n_1
+t^2({\tilde{\mathfrak a}'}_2)^2|n_2|^2 \nonumber\\[2mm] &-2t^2k{\tilde{\mathfrak a}}_2 {\tilde{\mathfrak a}'}_2 n_1n_2
+t^2k^2 ({\tilde{\mathfrak a}}_2)^2 |n_1|^2 +O(t^3). \label{F2theta-F2theta} \end{align}
The elements of the metric matrix are: \begin{align} g_{11}\, =\, & \frac{{\partial} F_1}{{\partial} t}\frac{{\partial} F_1}{{\partial} t} \, +\, \frac{{\partial} F_2}{{\partial} t}\frac{{\partial} F_2}{{\partial} t} \nonumber\\[2mm]
\, =\, &\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] +2t\, \big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big] \nonumber\\[2mm] &+t^2\, \big[ {\tilde{\mathfrak a}}_1 n_1 m_1 + {\tilde{\mathfrak a}}_2 n_2 m_2 \big]
+t^2\big[|q_1|^2+|q_2|^2\big]+O(t^3), \nonumber\\[2mm] g_{12}\, =\, & \frac{{\partial} F_1}{{\partial} t}\frac{{\partial} F_1}{{\partial} \theta} \, +\, \frac{{\partial} F_2}{{\partial} t}\frac{{\partial} F_2}{{\partial} \theta} \nonumber\\[2mm]
=&\big[-{\tilde{\mathfrak a}}_1n_1n_2+{\tilde{\mathfrak a}}_2n_1n_2\big]
+t\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big] -tk\big[-{\tilde{\mathfrak a}}_1^2n_1n_2+{\tilde{\mathfrak a}}_2^2n_1n_2\big] \nonumber\\[2mm] &+t \big[-q_1n_2+q_2n_1\big] +\frac{t^2}{2}\big[{\tilde{\mathfrak a}}_1q_1'n_1+{\tilde{\mathfrak a}}_2q_2'n_2\big] +t^2\big[{\tilde{\mathfrak a}'}_1q_1n_1+{\tilde{\mathfrak a}'}_2q_2n_2\big] \nonumber\\[2mm] &-t^2k\big[-{\tilde{\mathfrak a}}_1q_1n_2+{\tilde{\mathfrak a}}_2q_2n_1\big] +\frac{t^2}{2}\big[-m_1n_2+m_2n_1\big]+O(t^3), \end{align} and \begin{align} g_{22}\, =\, & \frac{{\partial} F_1}{{\partial} \theta}\frac{{\partial} F_1}{{\partial} \theta} \, +\, \frac{{\partial} F_2}{{\partial} \theta}\frac{{\partial} F_2}{{\partial} \theta} \nonumber\\[2mm]
=& 1+2t \big[-{\tilde{\mathfrak a}'}_1n_1 n_2+{\tilde{\mathfrak a}'}_2n_1 n_2\big]
-2tk\big[{\tilde{\mathfrak a}}_1|n_2|^2+{\tilde{\mathfrak a}}_2|n_1|^2\big] +t^2\big[-q_1'n_2+q_2'n_1\big] \nonumber\\[2mm]
&+t^2\big[({\tilde{\mathfrak a}'}_1)^2|n_1|^2+({\tilde{\mathfrak a}'}_2)^2|n_2|^2\big] -2t^2k\big[-{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1n_1n_2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2n_1n_2\big] \nonumber\\[2mm]
&+t^2k^2 \big[{\tilde{\mathfrak a}}_1^2 |n_2|^2+{\tilde{\mathfrak a}}_2^2 |n_1|^2\big]+O(t^3). \end{align}
So the determinant of the metric matrix is $$g=g_{11}g_{22}-g_{12}g_{12}$$ where \begin{align*} g_{11}g_{22}
=&\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] +2t\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big]
+2t\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[-{\tilde{\mathfrak a}'}_1n_1 n_2+{\tilde{\mathfrak a}'}_2n_1 n_2\big] \\[2mm]
&-2tk\big[{\tilde{\mathfrak a}}_1|n_2|^2+{\tilde{\mathfrak a}}_2|n_1|^2\big]\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] +t^2\, \big[ {\tilde{\mathfrak a}}_1 n_1 m_1 + {\tilde{\mathfrak a}}_2 n_2 m_2 \big]
+t^2\big[|q_1|^2+|q_2|^2\big] \\[2mm] &+4t^2\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big]\big[-{\tilde{\mathfrak a}'}_1+{\tilde{\mathfrak a}'}_2\big]n_1n_2
+t^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[-q_1'n_2+q_2'n_1\big] \\[2mm]
&+t^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[({\tilde{\mathfrak a}'}_1)^2|n_1|^2+({\tilde{\mathfrak a}'}_2)^2|n_2|^2\big]
-4t^2k\big[{\tilde{\mathfrak a}}_1|n_2|^2+{\tilde{\mathfrak a}}_2|n_1|^2\big]\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big] \nonumber\\[2mm]
&-2t^2k\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[-{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1n_1n_2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2n_1n_2\big] \nonumber\\[2mm]
&+t^2k^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] \big[{\tilde{\mathfrak a}}_1^2 |n_2|^2+{\tilde{\mathfrak a}}_2^2 |n_1|^2\big]+O(t^3), \end{align*} and \begin{align*}
g_{12}g_{12}
=&\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big]^2
+2t\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big]\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big] \\[2mm] &+2t \big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \big[-q_1n_2+q_2n_1\big] -2tk\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big] \\[2mm]
&+t^2 \big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]^2
+2t^2 \big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]\big[-q_1n_2+q_2n_1\big] +t^2 \big[-q_1n_2+q_2n_1\big]^2 \\[2mm] &+t^2\big[{\tilde{\mathfrak a}}_1q_1'n_1+{\tilde{\mathfrak a}}_2q_2'n_2\big]\big[-{\tilde{\mathfrak a}}_1 n_1n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] +2t^2\big[{\tilde{\mathfrak a}'}_1q_1n_1+{\tilde{\mathfrak a}'}_2q_2n_2\big]\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \\[2mm] &+t^2\big[-m_1n_2+m_2n_1\big]\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big]
-2t^2k \big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big] \\[2mm] &-2t^2k\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big] \big[-q_1n_2+q_2n_1\big] -2t^2k \big[-{\tilde{\mathfrak a}}_1q_1n_2+{\tilde{\mathfrak a}}_2q_2n_1\big]\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \\[2mm] &+t^2k^2\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big]^2+O(t^3). \end{align*}
We now make a rearrangement of the terms in $g$ and then consider the following terms.
\noindent $\clubsuit$ Term 1: \begin{align*}
&\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]
- \big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big]^2
=\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big]^2. \end{align*}
\noindent $\clubsuit$ Term 2: \begin{align*}
&2t\, \big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big]
- 2t \big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_1 n_2\big]\big[-q_1n_2+q_2n_1\big]
\\[2mm]
&=2t \big[ {\tilde{\mathfrak a}}_1 n_1 q_1 \big( 1- |n_2|^2 \big)
+{\tilde{\mathfrak a}}_2 n_2q_2\big( 1- |n_1|^2 \big)
+{\tilde{\mathfrak a}}_2q_1 n_1|n_2|^2
+{\tilde{\mathfrak a}}_1q_2 |n_1|^2n_2 \big] \\[2mm]
&=2t \big[ {\tilde{\mathfrak a}}_1 n_1 q_1 |n_1|^2
+{\tilde{\mathfrak a}}_2 n_2q_2|n_2|^2
+{\tilde{\mathfrak a}}_2q_1 n_1|n_2|^2
+{\tilde{\mathfrak a}}_1q_2 |n_1|^2n_2 \big] \\[2mm]
&=2t \big[ {\tilde{\mathfrak a}}_1 |n_1|^2 \big( q_1n_1+q_2n_2\big)
-{\tilde{\mathfrak a}}_2 |n_2|^2\big( -q_2n_2- q_1n_1 \big) \big] =0, \end{align*} due to the fact that $q\bot n$ given in Lemma \ref{derivativeofF}.
\noindent $\clubsuit$ Term 3: \begin{align*}
&2t \big[-{\tilde{\mathfrak a}'}_1n_1n_2+{\tilde{\mathfrak a}'}_2n_1n_2\big]\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]
- 2t\big[-{\tilde{\mathfrak a}}_1 n_1n_2+{\tilde{\mathfrak a}}_2 n_1n_2\big]\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]
\\[2mm]
&=2t \big[
{\tilde{\mathfrak a}'}_2{\tilde{\mathfrak a}}_1^2 n_2|n_1|^3
-{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2^2 n_1|n_2|^3
-{\tilde{\mathfrak a}}_2 {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1n_2|n_1|^3
+{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 n_1 |n_2|^3 \big] \\[2mm]
&=2t \big(
{\tilde{\mathfrak a}'}_2{\tilde{\mathfrak a}}_1 -{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2\big) \big( {\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big) n_1n_2. \end{align*}
\noindent $\clubsuit$ Term 4: \begin{align*}
&-2tk\big[{\tilde{\mathfrak a}}_1|n_2|^2+{\tilde{\mathfrak a}}_2|n_1|^2\big]\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] +2tk\big[-{\tilde{\mathfrak a}}_1 n_1n_2+{\tilde{\mathfrak a}}_2 n_1n_2\big]\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_1n_2\big] \\[2mm]
&=-2tk \Big[
{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_1^2 |n_1|^2|n_1|^2
+{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2^2 |n_2|^2|n_2|^2
+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_1^2 |n_1|^2|n_2|^2
+{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2^2 |n_1|^2|n_2|^2 \Big] \\[2mm]
&=-2tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2\big[{\tilde{\mathfrak a}}_1|n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big]. \end{align*}
\noindent $\clubsuit$ Term 5: \begin{align*} &t^2\, \big[ {\tilde{\mathfrak a}}_1 n_1 m_1 + {\tilde{\mathfrak a}}_2 n_2 m_2 \big] -t^2\big[-m_1n_2+m_2n_1\big]\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \\[2mm]
&=t^2\, \big[ {\tilde{\mathfrak a}}_1 n_1 m_1 \big( 1- |n_2|^2\big)
+{\tilde{\mathfrak a}}_2 n_2 m_2 \big( 1- |n_1|^2\big) +{\tilde{\mathfrak a}}_1m_2n_1^2n_2 +{\tilde{\mathfrak a}}_2m_1n_1n_2^2 \big] \\[2mm]
&=t^2\big( m_1n_1+ m_2n_2\big)\big[ {\tilde{\mathfrak a}}_1 |n_1|^2
+{\tilde{\mathfrak a}}_2 |n_2|^2 \big]. \end{align*}
\noindent $\clubsuit$ Term 6: \begin{align*}
t^2\big[|q_1|^2+|q_2|^2\big]-t^2 \big[-q_1n_2+q_2n_1\big]^2
&=t^2\big[|q_1|^2+|q_2|^2\big]
-t^2 \big(q_1^2|n_2|^2
-2q_2q_1n_1n_2+q_2^2|n_1|^2 \big) \\[2mm]
&=t^2\big[|q_1|^2 \big( 1- |n_2|^2\big)
+|q_2|^2 \big( 1-|n_1|^2 \big) +2q_2q_1n_1n_2 \big] \\[2mm] &=t^2\big[q_1n_1+q_2n_2\big]^2 =0, \end{align*} due to the fact that $q\bot n$ given in Lemma \ref{derivativeofF}.
\noindent $\clubsuit$ Term 7: \begin{align*} &4t^2 \big[-{\tilde{\mathfrak a}'}_1n_1n_2+{\tilde{\mathfrak a}'}_2n_1 n_2\big]\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big] -2t^2\big[{\tilde{\mathfrak a}'}_1q_1n_1+{\tilde{\mathfrak a}'}_2q_2n_2\big]\big[-{\tilde{\mathfrak a}}_1 n_1 n_2+{\tilde{\mathfrak a}}_2 n_1 n_2\big] \\[2mm]
&-2t^2 \big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]\big[-q_1n_2+q_2n_1\big] \\[2mm]
&=t^2 \big[ 2{\tilde{\mathfrak a}'}_2{\tilde{\mathfrak a}}_1 n_1n_2 \big( q_1n_1+q_2n_2 \big) -2{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2 n_1n_2 \big( q_2n_2+q_1n_1 \big) \\[2mm] &\quad
+2{\tilde{\mathfrak a}'}_2q_1n_2 {\tilde{\mathfrak a}}_1|n_1|^2
+2{\tilde{\mathfrak a}'}_2q_1n_2 {\tilde{\mathfrak a}}_2|n_2|^2
-2{\tilde{\mathfrak a}'}_1q_2n_1 {\tilde{\mathfrak a}}_2|n_2|^2
-2{\tilde{\mathfrak a}'}_1q_2n_1 {\tilde{\mathfrak a}}_1|n_1|^2 \big] \\[2mm]
&= 2t^2\big[{\tilde{\mathfrak a}}_2'q_1n_2-{\tilde{\mathfrak a}}_1'q_2n_1\big]\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big], \end{align*} where we have used the fact that $q\bot n$ given in Lemma \ref{derivativeofF}.
\noindent $\clubsuit$ Term 8: \begin{align*}
&t^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2
+{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[-q_1'n_2+q_2'n_1\big] -t^2\big[{\tilde{\mathfrak a}}_1q_1'n_1+{\tilde{\mathfrak a}}_2q_2'n_2\big] \big[-{\tilde{\mathfrak a}}_1 n_1n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \\[2mm]
&= t^2 \big[
-{\tilde{\mathfrak a}}_2^2 |n_2|^3q_1'
+{\tilde{\mathfrak a}}_1^2 |n_1|^3q_2' +{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2q_2'n_1n_2^2 -{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_1q_1'n_1^2 n_2\big] \\[2mm]
&=t^2 \big[{\tilde{\mathfrak a}}_1q_2'n_1-{\tilde{\mathfrak a}}_2q_1'n_2\big]\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big]. \end{align*}
\noindent $\clubsuit$ Term 9: \begin{align*}
&t^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[({\tilde{\mathfrak a}'}_1)^2|n_1|^2+({\tilde{\mathfrak a}'}_2)^2|n_2|^2\big]
-t^2 \big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]^2 \\[2mm]
&=t^2 \big[
{\tilde{\mathfrak a}}_2^2 ({\tilde{\mathfrak a}'}_1)^2|n_1|^2|n_2|^2
+{\tilde{\mathfrak a}}_1^2 ({\tilde{\mathfrak a}'}_2)^2|n_1|^2|n_2|^2
-2{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_1|^2|n_2|^2 \big] \\[2mm]
&=t^2\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2'-{\tilde{\mathfrak a}}_1'{\tilde{\mathfrak a}}_2\big]^2|n_1|^2|n_2|^2. \end{align*}
\noindent $\clubsuit$ Term 10: \begin{align*}
&-4t^2k\big[{\tilde{\mathfrak a}}_1|n_2|^2+{\tilde{\mathfrak a}}_2|n_1|^2\big]\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big] +2t^2k\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big]\big[-q_1n_2+q_2n_1\big] \\[2mm] &+2t^2k \big[-{\tilde{\mathfrak a}}_1q_1n_2+{\tilde{\mathfrak a}}_2q_2n_1 \big] \big[-{\tilde{\mathfrak a}}_1 n_1n_2+{\tilde{\mathfrak a}}_2 n_2n_1\big] \\[2mm]
&=-t^2k \Big[
2{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_1 q_1 n_1 \big(|n_1|^2+|n_2|^2\big)
+2{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2q_2 n_2\big(|n_2|^2+|n_1|^2\big) \\[2mm]
&\qquad \qquad+2{\tilde{\mathfrak a}}_2q_1n_1 \big( {\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big)
+2{\tilde{\mathfrak a}}_1 q_2 n_2 \big( {\tilde{\mathfrak a}}_2 |n_2|^2 +{\tilde{\mathfrak a}}_1|n_1|^2\big) \Big] \\[2mm] &=-t^2k \Big[ 2{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_1 \big( q_1 n_1 +q_2 n_2 \big) +2 \big( {\tilde{\mathfrak a}}_2q_1n_1+{\tilde{\mathfrak a}}_1 q_2 n_2 \big)
\big( {\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big) \Big] \\[2mm]
&=-2t^2k\big( {\tilde{\mathfrak a}}_2q_1n_1+{\tilde{\mathfrak a}}_1 q_2 n_2\big) \big( {\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big), \end{align*} where we have used the fact that $q\bot n$ given in Lemma \ref{derivativeofF}.
\noindent $\clubsuit$ Term 11: \begin{align*}
&-2t^2k\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big]\big[-{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1n_1n_2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2n_2n_1\big]
+2t^2k\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big]\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big] \\[2mm]
&=-2t^2k \big[
-{\tilde{\mathfrak a}}_2^2 {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1n_1|n_2|^3
+{\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2|n_1|^3n_2
+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 {\tilde{\mathfrak a}}_1^2 n_1|n_2|^3
-{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2^2|n_1|^3 n_2 \big] \\[2mm]
&= 2t^2k \big[{\tilde{\mathfrak a}}_2^2 {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1-{\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2\big]n_1n_2. \end{align*}
\noindent $\clubsuit$ Term 12: \begin{align*}
&t^2k^2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] \big[{\tilde{\mathfrak a}}_1^2 |n_2|^2+{\tilde{\mathfrak a}}_2^2 |n_1|^2\big] -t^2k^2\big[-{\tilde{\mathfrak a}}_1^2 n_1n_2+{\tilde{\mathfrak a}}_2^2 n_2n_1\big]^2 \\[2mm]
&=t^2k^2 \big[ {\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2^2 |n_1|^2|n_1|^2
+{\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2^2 |n_2|^2|n_2|^2 +2{\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2^2 n_1^2n_2^2\big] \\[2mm] &=t^2k^2{\tilde{\mathfrak a}}_1^2{\tilde{\mathfrak a}}_2^2 \big( n_1^2+n_2^2 \big)^2 =t^2k^2{\tilde{\mathfrak a}}_1^2{\tilde{\mathfrak a}}_2^2. \end{align*}
Therefore, we obtain that \begin{align}\label{mathfrakh1} g\, =\, &\mbox{det}(g_{ij})=g_{11}g_{22}-g_{12}g_{12} \nonumber\\[2mm]
=&\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big]^2 +
t \Big\{ 2 \big({\tilde{\mathfrak a}'}_2{\tilde{\mathfrak a}}_1 -{\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2\big)\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big] n_1n_2 -2k{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2\big[{\tilde{\mathfrak a}}_1n_1^2+{\tilde{\mathfrak a}}_2n_2^2\big] \Big\} \nonumber\\[2mm] &+
t^2\Big\{ 2\big[{\tilde{\mathfrak a}}_2'q_1n_2-{\tilde{\mathfrak a}}_1'q_2n_1\big]\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big]
+ \big[{\tilde{\mathfrak a}}_1q_2'n_1-{\tilde{\mathfrak a}}_2q_1'n_2\big]\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big] \nonumber\\[2mm] &\qquad
+\big(m_1n_1+m_2n_2\big)\big[ {\tilde{\mathfrak a}}_1 |n_1|^2 +{\tilde{\mathfrak a}}_2 |n_2|^2\big]
+\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2'-{\tilde{\mathfrak a}}_1'{\tilde{\mathfrak a}}_2\big]^2|n_1|^2|n_2|^2 \nonumber\\[2mm] &\qquad +2 k \big[{\tilde{\mathfrak a}}_2^2 {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1-{\tilde{\mathfrak a}}_1^2 {\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2\big]n_1n_2
-2 k\big( {\tilde{\mathfrak a}}_2q_1n_1+{\tilde{\mathfrak a}}_1 q_2n_2\big)\big[{\tilde{\mathfrak a}}_1 |n_1|^2+{\tilde{\mathfrak a}}_2|n_2|^2\big] + k^2{\tilde{\mathfrak a}}_1^2{\tilde{\mathfrak a}}_2^2 \Big\} +O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak h}_1(\theta) \, +\, t{\mathfrak h}_2(\theta) \, +\, t^2{\mathfrak h}_3(\theta)+O(t^3). \end{align}
We now compute the inverse of the metric matrix. By the formula $$ \big[1+at+bt^2+O(t^3)\big]^{-1}\, =\, 1-at+(a^2-b)t^2+O(t^3), $$ we have \begin{align} \frac{1}{g}\, =\, &\frac{1}{{\mathfrak h}_1} -\frac{{\mathfrak h}_2}{{\mathfrak h}_1^2}t +\Big[\frac{{\mathfrak h}_2^2}{{\mathfrak h}_1^3} -\frac{{\mathfrak h}_3}{{\mathfrak h}_1^2}\Big]t^2 +O(t^3) \nonumber\\[2mm] \, \equiv\, &\frac{1}{{\mathfrak h}_1} +{\mathfrak g}_1t +{\mathfrak g}_2t^2 +O(t^3). \label{g-inverse} \end{align} Note that if the function has the following asymptotic expansion $$f(s)\, =\, 1+as+bs^2+O(s^3), \quad f(0)\, =\, 1, $$ then for $s$ close to zero $$ \sqrt{f(s)}\, =\, 1+\frac{a}{2}s+\frac12\left(b-\frac14a^2\right)s^2+O(s^3). $$ We can get the following formulas \begin{equation}
\sqrt{g}\, =\, \sqrt{{\mathfrak h}_1} +\frac{1}{2}\frac{{\mathfrak h}_2}{\sqrt{{\mathfrak h}_1}} t +\Big[\frac{1}{2}\frac{{\mathfrak h}_3}{\sqrt{{\mathfrak h}_1}}t^2 -\frac{1}{8}\frac{{\mathfrak h}_2^2}{(\sqrt{{\mathfrak h}_1})^3}\Big] +O(t^3), \label{g-inverse1} \end{equation} and also \begin{align} \frac{1}{\sqrt{g}}&\, =\, \frac{1}{\sqrt{{\mathfrak h}_1}} -\frac{1}{2}\frac{{\mathfrak h}_2}{(\sqrt{{\mathfrak h}_1})^3} t +\Big[\frac{3}{8}\frac{{\mathfrak h}_2^2}{(\sqrt{{\mathfrak h}_1})^5} -\frac{1}{2}\frac{{\mathfrak h}_3}{(\sqrt{{\mathfrak h}_1})^3}\Big]t^2+O(t^3) \nonumber\\[2mm] &\, \equiv\, \frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2+O(t^3). \label{g-inverse2} \end{align} Whence \begin{align} g^{12}\, =\, &-\frac{g_{12}}{ g} \nonumber\\
\, =\, &\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}}_1-{\tilde{\mathfrak a}}_2\big]n_1n_2 \nonumber\\[2mm]
&+t\Big\{-\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big] +k\frac{1}{{\mathfrak h}_1}\big[-{\tilde{\mathfrak a}}_1^2+{\tilde{\mathfrak a}}_2^2\big]n_1n_2 \nonumber\\[2mm] &\qquad-\frac{1}{{\mathfrak h}_1}\big[-q_1n_2+q_2n_1\big] +{\mathfrak g}_1\big[{\tilde{\mathfrak a}}_1-{\tilde{\mathfrak a}}_2\big]n_1n_2\Big\} \nonumber\\[2mm] & +t^2\Big\{-\frac{1}{2}\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}}_1q_1'n_1+{\tilde{\mathfrak a}}_2q_2'n_2\big] -\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}'}_1q_1n_1+{\tilde{\mathfrak a}'}_2q_2n_2\big] +k\frac{1}{{\mathfrak h}_1}\big[-{\tilde{\mathfrak a}}_1q_1n_2+{\tilde{\mathfrak a}}_2q_2n_1\big] \nonumber\\[2mm] &\qquad+\frac{1}{2}\frac{1}{{\mathfrak h}_1}\big[-m_1n_2+m_2n_1\big]
+{\mathfrak g}_1\big[{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2+{\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2 |n_2|^2\big] \nonumber\\[2mm] &\qquad+k{\mathfrak g}_1\big[-{\tilde{\mathfrak a}}_1^2+{\tilde{\mathfrak a}}_2^2\big]n_1n_2 -{\mathfrak g}_1\big[-q_1n_2+q_2n_1\big] +{\mathfrak g}_2\big[{\tilde{\mathfrak a}}_1-{\tilde{\mathfrak a}}_2\big]n_1n_2\Big\}+O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak g}_3+{\mathfrak g}_4t+{\mathfrak g}_5t^2+O(t^3), \label{g12inverse} \end{align} and \begin{align} g^{22}\, =\, \frac{g_{11}}{g}
\, =\, &\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2 \big] +t\Big\{2\frac{1}{{\mathfrak h}_1}\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big]
+{\mathfrak g}_1\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2\big]\Big\} \nonumber\\[2mm] & +t^2\Big\{\frac{1}{{\mathfrak h}_1}\big[ {\tilde{\mathfrak a}}_1 n_1 m_1+{\tilde{\mathfrak a}}_2 n_2 m_2 \big]
+\frac{1}{{\mathfrak h}_1}\big[|q_1|^2+|q_2|^2\big] \nonumber\\[2mm] &\qquad \quad+2{\mathfrak g}_1\big[{\tilde{\mathfrak a}}_1 n_1 q_1 + {\tilde{\mathfrak a}}_2 n_2 q_2\big]
+{\mathfrak g}_2\big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +{\tilde{\mathfrak a}}_2^2 |n_2|^2\big]\Big\}+O(t^3) \nonumber\\[2mm] \, \equiv\, &\, {\mathfrak g}_6+{\mathfrak g}_7t+{\mathfrak g}_8t^2+O(t^3). \label{g22inverse} \end{align} Similar asymptotic expression holds for the term $g^{11}=g_{22}/g$.
\subsection{Local forms of the differential operators in \eqref{originalproblem}}\ \label{section2.3}
In this section, we are devoted to presenting the expressions of the differential operators ${\rm div}\big(\nabla_{\mathfrak a(y)}u\big)$ and $\nabla_{{\mathfrak a}(y)}u\cdot\nu$ in problem \eqref{originalproblem}.
\noindent{\bf{Part 1: the operator ${\rm div}\big(\nabla_{\mathfrak a(y)}u\big)$}}
We recall the relation of \eqref{Fermicoordinates-modified} and then have $$ {\mathrm d}y_1=\frac{\partial F_1}{\partial t}{\mathrm d}t\, +\, \frac{\partial F_1}{\partial \theta}{\mathrm d}\theta, \qquad {\mathrm d}y_2=\frac{\partial F_2}{\partial t}{\mathrm d}t\, +\, \frac{\partial F_2}{\partial \theta}{\mathrm d}\theta. $$ This implies that \begin{align} {\mathrm d}t=\frac{1}{\, \frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta}\,-\, \frac{\partial F_2}{\partial t}\frac{\partial F_1}{\partial \theta}\, } \left[\frac{\partial F_2}{\partial \theta}{\mathrm d}y_1\,-\, \frac{\partial F_1}{\partial \theta}{\mathrm d}y_2\right], \label{dt} \end{align} \begin{align} {\mathrm d}\theta=\frac{1}{\, \frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta}\,-\, \frac{\partial F_2}{\partial t}\frac{\partial F_1}{\partial \theta}\, } \left[\frac{\partial F_1}{\partial t}{\mathrm d}y_2\,-\, \frac{\partial F_2}{\partial t}{\mathrm d}y_1\right]. \label{dtheta} \end{align} On the other hand, there hold \begin{align} \frac{\partial u}{\partial t}=\frac{\partial u}{\partial y_1}\frac{\partial F_1}{\partial t}\, +\, \frac{\partial u}{\partial y_2}\frac{\partial F_2}{\partial t}, \qquad \frac{\partial u}{\partial \theta}=\frac{\partial u}{\partial y_1}\frac{\partial F_1}{\partial \theta}\, +\, \frac{\partial u}{\partial y_2}\frac{\partial F_2}{\partial \theta}, \end{align} which give that \begin{align} \frac{\partial u}{\partial y_1} =\frac{1}{\, \frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta}\,-\, \frac{\partial F_2}{\partial t}\frac{\partial F_1}{\partial \theta}\, } \left[ \frac{\partial u}{\partial t}\frac{\partial F_2}{\partial \theta}\,-\, \frac{\partial u}{\partial \theta}\frac{\partial F_2}{\partial t} \right], \label{partialuy1} \end{align} \begin{align} \frac{\partial u}{\partial y_2} =\frac{1}{\, \frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta}\,-\, \frac{\partial F_2}{\partial t}\frac{\partial F_1}{\partial \theta}\, } \left[ \frac{\partial u}{\partial \theta}\frac{\partial F_1}{\partial t}\,-\, \frac{\partial u}{\partial t}\frac{\partial F_1}{\partial \theta} \right]. \label{partialuy2} \end{align}
We now compute \begin{align}\label{udiri} \nabla_{{\mathfrak a}(y)}u \, =\, &{\mathfrak a}_1\frac{\partial u}{\partial y_1}\frac{\partial }{\partial y_1}\, +\, {\mathfrak a}_2\frac{\partial u}{\partial y_2}\frac{\partial }{\partial y_2} \nonumber\\[2mm] \, =\, &{\mathfrak a}_1\frac{\partial u}{\partial y_1}\left[ \frac{\partial t}{\partial y_1}\frac{\partial }{\partial t}\, +\, \frac{\partial \theta}{\partial y_1}\frac{\partial }{\partial \theta}\right] \, +\, {\mathfrak a}_2\frac{\partial u}{\partial y_2}\left[ \frac{\partial t}{\partial y_2}\frac{\partial }{\partial t}\, +\, \frac{\partial \theta}{\partial y_2}\frac{\partial }{\partial \theta}\right] \nonumber\\[2mm] \, =\, &\left[ {\mathfrak a}_1\frac{\partial t}{\partial y_1}\frac{\partial u}{\partial y_1}\, +\, {\mathfrak a}_2\frac{\partial t}{\partial y_2}\frac{\partial u}{\partial y_2}\right] \frac{\partial }{\partial t} \, +\, \left[ {\mathfrak a}_1\frac{\partial \theta}{\partial y_1}\frac{\partial u}{\partial y_1}\, +\, {\mathfrak a}_2\frac{\partial \theta}{\partial y_2}\frac{\partial u}{\partial y_2}\right] \frac{\partial }{\partial \theta}. \end{align} By substituting \eqref{dt}, \eqref{dtheta}, \eqref{partialuy1} and \eqref{partialuy2} in \eqref{udiri}, we obtain \begin{align} \nabla_{{\mathfrak a}(y)}u \, =\, &\frac{1}{g}\left[{\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial \theta}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial \theta}\right]\frac{\partial u}{\partial t}\frac{\partial }{\partial t} \,-\, \frac{1}{g}\left[{\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial t}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial t}\right]\frac{\partial u}{\partial \theta}\frac{\partial }{\partial t} \nonumber\\[2mm] &\,-\, \frac{1}{g}\left[{\mathfrak a}_1\frac{\partial F_2}{\partial t}\frac{\partial F_2}{\partial \theta}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial t}\frac{\partial F_1}{\partial \theta}\right]\frac{\partial u}{\partial t}\frac{\partial }{\partial \theta} \, +\, \frac{1}{g}\left[{\mathfrak a}_1\frac{\partial F_2}{\partial t}\frac{\partial F_2}{\partial t}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial t}\frac{\partial F_1}{\partial t}\right]\frac{\partial u}{\partial \theta}\frac{\partial }{\partial \theta}. \end{align} Furthermore, by setting \begin{align} {\tilde g}^{11}\, =\, {\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial \theta}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial \theta}, \\ {\tilde g}^{21}\, =\,{\tilde g}^{12}\, =\, {\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial t}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial t}, \\ {\tilde g}^{22}\, =\, {\mathfrak a}_1\frac{\partial F_2}{\partial t}\frac{\partial F_2}{\partial t} \, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial t}\frac{\partial F_1}{\partial t}, \end{align}
the definition of ${\rm div}$ operator will give that \begin{align} {\rm div}\big(\nabla_{{\mathfrak a}(y)} u\big) \, =\, &\frac{1}{\sqrt{g}}\frac{\partial}{\partial t} \left[ \frac{1}{\sqrt{g}}{\tilde g}^{11}\frac{\partial u}{\partial t} \right] \,-\, \frac{1}{\sqrt{g}}\frac{\partial}{\partial t} \left[ \frac{1}{\sqrt{g}}{\tilde g}^{12}\frac{\partial u}{\partial \theta} \right] \nonumber\\[2mm] &\,-\, \frac{1}{\sqrt{g}}\frac{\partial}{\partial \theta} \left[ \frac{1}{\sqrt{g}}{\tilde g}^{21}\frac{\partial u}{\partial t} \right] \, +\, \frac{1}{\sqrt{g}}\frac{\partial}{\partial \theta} \left[ \frac{1}{\sqrt{g}}{\tilde g}^{22}\frac{\partial u}{\partial \theta} \right].
\label{divoperator} \end{align}
Here are the computations of all coefficients in \eqref{divoperator}. By using the Taylor expansion \begin{equation}\label{expressionofai}
{\mathfrak a}_i(t, \theta)= {\mathfrak a}_i(0, \theta)+t \partial_{t} {\mathfrak a}_i(0, \theta)
+\frac{t^2}{2}\partial_{tt} {\mathfrak a}_i (0, \theta)+O(t^3), ~~\forall\, \theta\in [0, 1], ~t\in (-\delta_0, \delta_0), \end{equation} and recalling \eqref{Fit-Fit}-\eqref{Fitheta-Fitheta}, we obtain \begin{align} {\tilde g}^{11}\, =\, &{\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial \theta}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial \theta} \nonumber\\[2mm]
\, =\, &{\mathfrak a}_1\Big[|n_1|^2 +2t {\tilde{\mathfrak a}'}_2 n_1 n_2
-2t k{\tilde{\mathfrak a}}_2 |n_1|^2 +t^2 q_2'n_1
+t^2({\tilde{\mathfrak a}'}_2)^2|n_2|^2 -2t^2k{\tilde{\mathfrak a}}_2 {\tilde{\mathfrak a}'}_2 n_2n_1
+t^2k^2 {\tilde{\mathfrak a}}_2^2 |n_1|^2\Big] \nonumber\\[2mm]
&+{\mathfrak a}_2\Big[ |n_2|^2 -2t {\tilde{\mathfrak a}'}_1 n_2 n_1
-2t k{\tilde{\mathfrak a}}_1 |n_2|^2 -t^2 q_1'n_2
+t^2({\tilde{\mathfrak a}'}_1)^2|n_1|^2 +2t^2k{\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}'}_1 n_1n_2
+t^2k^2 {\tilde{\mathfrak a}}_1^2 |n_2|^2\Big]+O(t^3) \nonumber\\[2mm] \, =\, &
\Big[{\mathfrak a}_1(0, \theta) |n_1|^2 +{\mathfrak a}_2(0, \theta) |n_2|^2\Big] \nonumber\\[2mm] &+t\Big\{ 2\big[{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}'}_2 n_1 n_2 -{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}'}_1 n_2 n_1\big]
-2k\big[{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}}_2|n_1|^2 +{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 |n_2|^2\big] \nonumber\\[2mm] &\qquad
+\big[\partial_{t} {\mathfrak a}_1(0, \theta) |n_1|^2 +\partial_{t} {\mathfrak a}_2(0, \theta) |n_2|^2\big] \Big\} \nonumber\\[2mm] &+t^2\Big\{ \big[{\mathfrak a}_1(0, \theta) n_1q_2' -{\mathfrak a}_2(0, \theta) n_2q_1'\big]
+ \big[{\mathfrak a}_1(0, \theta) ({\tilde{\mathfrak a}'}_2)^2|n_2|^2+{\mathfrak a}_2(0, \theta) ({\tilde{\mathfrak a}'}_1)^2|n_1|^2\big] \nonumber\\[2mm] &\qquad -2 k\big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2 {\tilde{\mathfrak a}'}_2 n_2n_1-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}'}_1n_1n_2\big]
+ k^2\big[{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}}_2|n_1|^2 +{\mathfrak a}_2(0, \theta) {\tilde{\mathfrak a}}_1|n_2|^2\big] \nonumber\\[2mm] &\qquad +2\big[\partial_{t} {\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}'}_2n_1n_2 -\partial_{t} {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}'}_1n_2 n_1\big]
-2 k\big[\partial_{t} {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2 |n_1|^2 +\partial_{t} {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 |n_2|^2\big] \nonumber\\[2mm] &\qquad
+\frac{1}{2} \big[\partial_{tt} {\mathfrak a}_1(0, \theta) |n_1|^2 +\partial_{tt} {\mathfrak a}_2(0, \theta) |n_2|^2\big] \Big\} + O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak f}_0 +t{\mathfrak f}_1 +t^2{\mathfrak f}_2+ O(t^3). \label{m22} \end{align} Similarly, there hold \begin{align}
{\tilde g}^{12}\, =\, &{\mathfrak a}_1\frac{\partial F_2}{\partial \theta}\frac{\partial F_2}{\partial t}\, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial \theta}\frac{\partial F_1}{\partial t} \nonumber\\[2mm]
\, =\, &{\mathfrak a}_1\Big[{\tilde{\mathfrak a}}_2 n_2n_1
+t {\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2|n_2|^2 -tk {\tilde{\mathfrak a}}_2^2 n_2n_1 +t q_2n_1 +\frac{t^2}{2}{\tilde{\mathfrak a}}_2q_2'n_2
\Big] \nonumber\\[2mm] &+{\mathfrak a}_2\Big[-{\tilde{\mathfrak a}}_1 n_1 n_2
+t {\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1 |n_1|^2 +tk {\tilde{\mathfrak a}}_1^2 n_1n_2 -t q_1n_2
\Big]+O(t^2) \nonumber\\[2mm] \, =\, & t\Big\{
\big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}'}_2|n_2|^2+{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_1|n_1|^2\big] -k\big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2^2 n_1n_2 -{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1^2 n_2n_1 \big] \nonumber\\[2mm] &\quad + \big[{\mathfrak a}_1(0, \theta)n_1q_2-{\mathfrak a}_2(0, \theta)n_2q_1 \big] + \big[\partial_{t} {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2n_1n_2 -\partial_{t} {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1n_2n_1 \big] \Big\} +O(t^2) \nonumber\\ \, \equiv\, &t{\mathfrak l}_1 +O(t^2), \label{m12} \end{align} and \begin{align} {\tilde g}^{22}\, =\, &{\mathfrak a}_1\frac{\partial F_2}{\partial t}\frac{\partial F_2}{\partial t} \, +\, {\mathfrak a}_2\frac{\partial F_1}{\partial t}\frac{\partial F_1}{\partial t} \nonumber\\[2mm]
\, =\, &{\mathfrak a}_1\Big[{\tilde{\mathfrak a}}_2^2 |n_2|^2 +2t\, {\tilde{\mathfrak a}}_2 n_2 q_2 +t^2\, {\tilde{\mathfrak a}}_2 n_2 m_2
+t^2|q_2|^2\Big] \nonumber\\[2mm]
&+{\mathfrak a}_2\Big[{\tilde{\mathfrak a}}_1^2 |n_1|^2 +2t\, {\tilde{\mathfrak a}}_1n_1q_1 +t^2\, {\tilde{\mathfrak a}}_1n_1m_1
+t^2|q_1|^2\Big]+O(t^3) \nonumber\\[2mm]
\, =\, &\Big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2^2 |n_2|^2+{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1^2 |n_1|^2\Big] +t\Big\{ 2\, {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2n_2q_2 +2\, {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1n_1q_1 \nonumber\\[2mm]
&+\partial_{t} {\mathfrak a}_1 (0, \theta){\tilde{\mathfrak a}}_2^2 |n_2|^2
+\partial_{t} {\mathfrak a}_2 (0, \theta){\tilde{\mathfrak a}}_1^2 |n_1|^2 \Big\}
+ O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak w}_0 +t{\mathfrak w}_1
+ O(t^2). \label{m11} \end{align}
By recalling \eqref{g-inverse}-\eqref{g-inverse2} and \eqref{m22}-\eqref{m11}, we can obtain that \begin{align*}
\frac{{\tilde g}^{11}}{g} \, =\, &\Big[\frac{1}{{\mathfrak h}_1} +{\mathfrak g}_1t +{\mathfrak g}_2t^2 +O(t^3)\Big]\times \Big[{\mathfrak f}_0 +t{\mathfrak f}_1 +t^2{\mathfrak f}_2+O(t^3)\Big] \\[2mm] \, =\, & \frac{1}{{\mathfrak h}_1}{\mathfrak f}_0 +t\frac{1}{{\mathfrak h}_1}{\mathfrak f}_1 +t{\mathfrak g}_1{\mathfrak f}_0 +t^2\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +t^2{\mathfrak g}_1{\mathfrak f}_1 +t^2{\mathfrak g}_2{\mathfrak f}_0 +O(t^3), \\[2mm] \frac{{\tilde g}^{11}}{\sqrt{g}} \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2+O(t^3)\Big] \times\Big[{\mathfrak f}_0 +t{\mathfrak f}_1 +t^2{\mathfrak f}_2+O(t^3)\Big] \\ \, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_0 +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_1 +t{\mathfrak r}_1{\mathfrak f}_0 +t^2\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_2 +t^2{\mathfrak r}_1{\mathfrak f}_1 +t^2{\mathfrak r}_2{\mathfrak f}_0 +O(t^3), \\[2mm]
\frac{{\tilde g}^{22}}{g} \, =\, &\Big[\frac{1}{{\mathfrak h}_1} +{\mathfrak g}_1t +{\mathfrak g}_2t^2+O(t^3)\Big] \times \Big[{\mathfrak w}_0 +t{\mathfrak w}_1 + O(t^2)\Big] \\[2mm] \, =\, &\frac{1}{{\mathfrak h}_1}{\mathfrak w}_0 +t\frac{1}{{\mathfrak h}_1}{\mathfrak w}_1 +t{\mathfrak g}_1{\mathfrak w}_0 +O(t^2), \\[2mm] \frac{{\tilde g}^{22}}{\sqrt{g}} \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2+O(t^3)\Big] \times\big[{\mathfrak w}_0 +t{\mathfrak w}_1 +t^2{\mathfrak w}_2+O(t^3)\big] \\ \, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0 +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_1 +t{\mathfrak r}_1{\mathfrak w}_0
+O(t^2), \\[2mm] \frac{{\tilde g}^{12}}{g} \, =\, &\Big[\frac{1}{{\mathfrak h}_1} +{\mathfrak g}_1t +{\mathfrak g}_2t^2 +O(t^3)\Big] \times \big[t{\mathfrak l}_1 +O(t^2)\big] \, =\, t\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1 + O(t^2), \\[2mm] \frac{{\tilde g}^{12}}{\sqrt{g}} \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2 +O(t^3)\Big] \times \Big[t{\mathfrak l}_1 +O(t^2)\Big] \, =\, t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 + O(t^2). \end{align*} Then, there hold \begin{align*} \partial_t \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] =\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 + O(t), \qquad
\partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] =t \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big]' + O(t^2), \end{align*} and \begin{align*} \partial_t \Big[\frac{1}{\sqrt{g}} {\tilde g}^{11} \Big] \, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_1 +{\mathfrak r}_1{\mathfrak f}_0 +2t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_2 +2t{\mathfrak r}_1{\mathfrak f}_1 +2t{\mathfrak r}_2{\mathfrak f}_0 +O(t^2), \\[2mm] \partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{22} \Big] \, =\, &\Big[ \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big]' +t \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_1\Big]' +t \big[{\mathfrak r}_1{\mathfrak w}_0\big]' + O(t^2). \end{align*} Combining the expression of $(\sqrt{g})^{-1}$ as in \eqref{g-inverse2}, we can obtain that
\begin{align*} \frac{1}{\sqrt{g}}\partial_t \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2 +O(t^3)\Big] \times \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 + O(t) \Big] \, =\, \frac{1}{{\mathfrak h}_1}{\mathfrak l}_{1} + O(t), \end{align*} \begin{align*} \frac{1}{\sqrt{g}} \partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}} +{\mathfrak r}_1t +{\mathfrak r}_2t^2 +O(t^3)\Big] \times \Big[t\, \Big(\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big)' + O(t^2) \Big] \, =\, t\, \frac{1}{\sqrt{{\mathfrak h}_1}}\, \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big]'+ O(t^2), \end{align*} and \begin{align*} \frac{1}{\sqrt{g}}\partial_t\Big[\frac{1}{\sqrt{g}} {\tilde g}^{11} \Big]
\, =\, &\frac{1}{{\mathfrak h}}{\mathfrak f}_1 +{\mathfrak r}_1\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_0 +2t\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +3t\frac{1}{\sqrt{{\mathfrak h}_1}} {\mathfrak r}_1 {\mathfrak f}_1 +2t \frac{1}{\sqrt{{\mathfrak h}_1}} {\mathfrak r}_2 {\mathfrak f}_0 +t{\mathfrak r}_1{\mathfrak r}_1{\mathfrak f}_0 +O(t^2), \end{align*}
\begin{align*} \frac{1}{\sqrt{g}} \partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{22} \Big]
\, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[ \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big]' +t\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_1\Big]' +t\frac{1}{\sqrt{{\mathfrak h}_1}} \big[{\mathfrak r}_1{\mathfrak w}_0\big]' +t{\mathfrak r}_1 \Big[ \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big]' + O(t^2). \end{align*}
\noindent{\bf{Notation 1: }} {\em By collecting all the computations in the above, we set the following conventions. \begin{equation}\label{h1} h_1(\theta)=\frac{{\mathfrak f}_0(\theta)}{{\mathfrak h}_1(\theta)}
=\frac{|{\mathfrak a}_1(0, \theta)|^2+|{\mathfrak a}_2(0, \theta)|^2} {{\mathfrak a}_1(0, \theta)|n_1(\theta)|^2+{\mathfrak a}_2(0, \theta)|n_2(\theta)|^2}, \end{equation}
\begin{equation}\label{h2}
h_2(\theta)=\frac{{\mathfrak w}_0(\theta)}{{\mathfrak h}_1(\theta)}
=\frac{{\mathfrak a}_1(0, \theta){\mathfrak a}_2(0, \theta)} {{\mathfrak a}_1(0, \theta)|n_1(\theta)|^2+{\mathfrak a}_2(0, \theta)|n_2(\theta)|^2}, \end{equation}
\begin{equation}\label{h3h4}
h_3(\theta)=\frac{1}{{\mathfrak h}_1}{\mathfrak f}_1 +{\mathfrak r}_1\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_0 =\frac{1}{{\mathfrak h}_1}{\mathfrak f}_1 -\frac{1}{2}\frac{{\mathfrak h}_2}{{\mathfrak h}_1^{2}}{\mathfrak f}_0, \qquad h_4(\theta)=\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[ \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big]' -\frac{1}{{\mathfrak h}_1}{\mathfrak l}_{1}, \end{equation}
\begin{align} h_5(\theta) \, =\, & 2\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +3{\mathfrak r}_1\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_1 +2{\mathfrak r}_2\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak f}_0 +{\mathfrak r}_1{\mathfrak r}_1{\mathfrak f}_0 -\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big]' \nonumber\\[2mm]
\, =\, &2\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 -\frac{3}{2}\frac{{\mathfrak h}_2}{{\mathfrak h}_1^{2}}{\mathfrak f}_1 +\frac{{\mathfrak h}_2^2}{{\mathfrak h}_1^{3}}{\mathfrak f}_0 -\frac{{\mathfrak h}_3}{{\mathfrak h}_1^{2}}{\mathfrak f}_0 -\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big]', \label{h5} \end{align}
\begin{equation}\label{h6h7}
h_6(\theta)=-2\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1, \qquad h_7(\theta)\, =\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +{\mathfrak g}_1{\mathfrak f}_1 +{\mathfrak g}_2{\mathfrak f}_0 \, =\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 -\frac{{\mathfrak h}_2}{{\mathfrak h}_1^2}{\mathfrak f}_1 +\Big[\frac{{\mathfrak h}_2^2}{{\mathfrak h}_1^3} -\frac{{\mathfrak h}_3}{{\mathfrak h}_1^2}\Big]{\mathfrak f}_0, \end{equation} \begin{equation} h_8(\theta)\, =\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_1 +{\mathfrak g}_1{\mathfrak f}_0 \, =\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_1 -\frac{{\mathfrak h}_2}{{\mathfrak h}_1^2}{\mathfrak f}_0. \label{h8} \end{equation} \qed }
Hence, the term ${\rm div}\big(\nabla_{{\mathfrak a}(y)} u\big) $ in \eqref{divoperator} has the following form in the modified Fermi coordinate system \begin{align} {\rm div}\big(\nabla_{{\mathfrak a}(y)} u\big) \, =\, &\frac{{\tilde g}^{11}}{g} u_{tt} \, +\, \frac{1}{\sqrt{g}}\partial_t \Big[\frac{1}{\sqrt{g}} {\tilde g}^{11} \Big] u_{t} \,-\, 2\frac{{\tilde g}^{12}}{g} u_{\theta t} \,-\, \frac{1}{\sqrt{g}} \partial_t \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] u_{\theta} \nonumber\\[2mm] &\,-\, \frac{1}{\sqrt{g}} \partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{12} \Big] u_{t} \, +\, \frac{{\tilde g}^{22}}{g} u_{\theta\theta} \, +\, \frac{1}{\sqrt{g}} \partial_{\theta} \Big[\frac{1}{\sqrt{g}} {\tilde g}^{22} \Big]u_{\theta} \nonumber\\[2mm]
\, =\, &h_1(\theta) u_{tt}+h_2(\theta)u_{\theta\theta} +h_3(\theta)u_{t}+h_4(\theta) u_\theta+\bar{B}_1(u)+\bar{B}_0(u), \label{laplacelocal} \end{align} where \begin{equation} \bar{B}_1(u)\, =\, h_5(\theta)tu_{t}+h_6(\theta) tu_{t\theta} +h_7(\theta)t^2 u_{tt}+h_8(\theta)tu_{tt}, \label{B1bar} \end{equation} and \begin{equation} \bar{B}_0(u) \, =\, h_9(t, \theta)t u_{\theta\theta} +h_{10}(t, \theta)t^2 u_{\theta\theta} +h_{11}(t, \theta)t^2u_{\theta t} +h_{12}(t, \theta)t u_{\theta}. \label{B0bar} \end{equation} Here, $h_9, \cdots, h_{12}$ are smooth functions.
\noindent{\bf{Part 2: the operator $\nabla_{{\mathfrak a}(y)}u\cdot\nu$}}
We finally show the local expression of $\nabla_{{\mathfrak a}(y)}u\cdot\nu$ in (\ref{originalproblem}). Suppose that, in the local coordinates $(t, \theta)$ of \eqref{Fermicoordinates-modified}, the unit outer normal of $\partial\Omega$ is expressed in the form $$ \nu\, =\, \sigma_1\frac{{\partial} F}{{\partial} t}+\sigma_2\frac{{\partial} F}{{\partial} \theta}. $$ If $\theta=0$ or $\theta=1$, the expression of $F(t, \theta)$ in \eqref{Fermicoordinates-modified} gives the curves $\mathcal{C}_1$ or $\mathcal{C}_2$. We then have $$ \langle {{\partial} F}/{{\partial} t}, \, \nu\rangle=0\quad \mbox{at } \theta=0, 1. $$ For the convenience of notation, in the following lines of this part, we will always take $\theta=0$ or $\theta=1$ without any further announcement. Hence $$ \sigma_2\neq 0, \quad \frac{{\partial} F}{{\partial} \theta}\neq 0, \quad {\rm and}\quad \sigma_1g_{11}+\sigma_2g_{12}\, =\, 0.$$ On the other hand, $\langle \nu, \nu \rangle=1$, that is $$ \left<\sigma_1\frac{{\partial} F}{{\partial} t}+\sigma_2\frac{{\partial} F}{{\partial} \theta}, \, \sigma_1\frac{{\partial} F}{{\partial} t}+\sigma_2\frac{{\partial} F}{{\partial} \theta}\right>\, =\, 1, $$ which implies that $$ \sigma_1^2g_{11}+\sigma_2^2g_{22}+2\sigma_1\sigma_2g_{12}\, =\, 1. $$ Combining above two equations, one can get $$\sigma_1\, =\, \pm\frac{g^{12}}{\sqrt{g^{22}}}, \qquad \sigma_2\, =\, \pm\sqrt{g^{22}}.$$ By choosing the sign $"+"$ and using \eqref{g12inverse}-\eqref{g22inverse}, it is easy to check that \begin{align} \sigma_1 \, =\, &\Big[\, {\mathfrak g}_3+{\mathfrak g}_4t+{\mathfrak g}_5t^2+O(t^3)\, \Big] \times\Big\{\, \frac{1}{{\mathfrak g}_6}-\frac{{\mathfrak g}_7}{{\mathfrak g}_6^2}t +\Big[\frac{1}{2}\frac{{\mathfrak g}_7^2}{{\mathfrak g}_6^{3}}-\frac{{\mathfrak g}_8}{{\mathfrak g}_6^{2}}\Big]t^2+O(t^3)\, \Big\} \nonumber\\[2mm] \, =\, &\frac{{\mathfrak g}_3}{{\mathfrak g}_6} +t\, \Big[\, \frac{{\mathfrak g}_4}{{\mathfrak g}_6}-\frac{{\mathfrak g}_7{\mathfrak g}_3}{{\mathfrak g}_6^2}\, \Big] +t^2\, \Big[\, \frac{{\mathfrak g}_5}{{\mathfrak g}_6} -\frac{{\mathfrak g}_7{\mathfrak g}_4}{{\mathfrak g}_6^2} +{\mathfrak g}_3{\mathfrak g}_5\, \Big]+O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak y}_1(\theta)+t{\mathfrak y}_2(\theta)+t^2{\mathfrak y}_3(\theta)+O(t^3), \label{sigma1} \end{align} and \begin{align}\label{sigma2} \sigma_2\, =\, \sqrt{g^{22}} \, =\, &\sqrt{{\mathfrak g}_6} +\frac{1}{2}\frac{{\mathfrak g}_7}{\sqrt{{\mathfrak g}_6}}t +\frac{1}{2}\Big[\frac{{\mathfrak g}_8}{\sqrt{{\mathfrak g}_6}} -\frac{1}{4}\frac{{\mathfrak g}_7^2}{(\sqrt{{\mathfrak g}_6})^3}\Big]t^2+O(t^3) \nonumber\\[2mm] \, \equiv\, &{\mathfrak y}_4(\theta)+t{\mathfrak y}_5(\theta)+t^2{\mathfrak y}_6(\theta)+O(t^3). \end{align}
In the modified Fermi coordinates $(t, \theta)$ in (\ref{Fermicoordinates-modified}), the normal derivative $\nabla_{{\mathfrak a}(y)}u\cdot\nu$ has a local form as follows \begin{align*}
\nabla_{{\mathfrak a}(y)}u\cdot\nu \, =\, &\Big({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2} \Big)\Big( \sigma_1 \frac{\partial F_1}{\partial t}+\sigma_2 \frac{\partial F_1}{\partial \theta}, \sigma_1 \frac{\partial F_2}{\partial t}+\sigma_2 \frac{\partial F_2}{\partial \theta}\Big)
\\[2mm] \, =\, &{\mathfrak a}_1(y)\frac{\partial u}{\partial y_1} \Big( \sigma_1 \frac{\partial F_1}{\partial t}+\sigma_2 \frac{\partial F_1}{\partial \theta} \Big) \, +\, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2} \Big(\sigma_1 \frac{\partial F_2}{\partial t}+\sigma_2 \frac{\partial F_2}{\partial \theta}\Big) \\[2mm] \, =\, & {\mathfrak a}_1(y)\frac{1}{\sqrt{g}} \left[ \frac{\partial u}{\partial t}\frac{\partial F_2}{\partial \theta} \,-\, \frac{\partial u}{\partial \theta}\frac{\partial F_2}{\partial t} \right] \Big( \sigma_1 \frac{\partial F_1}{\partial t}+\sigma_2 \frac{\partial F_1}{\partial \theta} \Big) \\[2mm] &\, +\, {\mathfrak a}_2(y)\frac{1}{\sqrt{g}} \left[ \frac{\partial u}{\partial \theta}\frac{\partial F_1}{\partial t} \,-\, \frac{\partial u}{\partial t}\frac{\partial F_1}{\partial \theta} \right] \Big(\sigma_1 \frac{\partial F_2}{\partial t}+\sigma_2 \frac{\partial F_2}{\partial \theta}\Big) \\[2mm]
\, =\, & \frac{\sigma_1}{\sqrt{g}} \left[{\mathfrak a}_1(y)\frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta} -{\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t}\right]\frac{\partial u}{\partial t} +\frac{\sigma_2 }{\sqrt{g}} \left[{\mathfrak a}_1(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta} -{\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta} \right]\frac{\partial u}{\partial t} \\[2mm] &+\frac{\sigma_1}{\sqrt{g}} \left[ {\mathfrak a}_2(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t} -{\mathfrak a}_1(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t}\right] \frac{\partial u}{\partial \theta} +\frac{\sigma_2 }{\sqrt{g}} \left[{\mathfrak a}_2(y) \frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial \theta} -{\mathfrak a}_1(y) \frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t}\right] \frac{\partial u}{\partial \theta}. \end{align*}
According to the expressions of $\frac{\partial F_i}{\partial t}$ and $\frac{\partial F_i}{\partial \theta}$ as in \eqref{F_t}-\eqref{F2_theta}, it is easy to derive that \begin{align} \frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t}
\, =\, &{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2n_1n_2 +t \big( {\tilde{\mathfrak a}}_2q_1n_2+{\tilde{\mathfrak a}}_1q_2n_1 \big) + t^2q_1q_2+\frac{t^2}{2}\big( {\tilde{\mathfrak a}}_2m_1n_2+{\tilde{\mathfrak a}}_1m_2n_1 \big) +O(t^3), \label{FtFt} \\[2mm] \frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta}
\, =\, &-n_1n_2
\, +\, t \big(- {\tilde{\mathfrak a}' }_2|n_2|^2+ {\tilde{\mathfrak a}' }_1|n_1|^2\big) \, +\, tk\big({\tilde{\mathfrak a}}_1+{\tilde{\mathfrak a}}_2 \big) n_1n_2 \, +\, \frac{t^2}{2}\big(-q_2'n_2+ q_1'n_1\big) \nonumber\\[2mm] &\, +\, t^2{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}' }_2n_1n_2
\,-\, t^2k \big( {\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2 |n_1|^2- {\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}' }_2|n_2|^2 \big) \,-\, t^2k^2{\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}}_2 n_1n_2 \, +\, O(t^3), \label{FteFte} \end{align} and \begin{align} \frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial \theta}
\, =\, &{\tilde{\mathfrak a}}_1|n_1|^2 +t{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}' }_2n_1n_2
\,-\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2 \, +\, t q_1n_1 \, +\, t^2q_1{\tilde{\mathfrak a}' }_2n_2 \nonumber\\[2mm] &\,-\, t^2k {\tilde{\mathfrak a}}_2q_1n_1 \, +\, \frac{t^2}{2}{\tilde{\mathfrak a}}_1n_1q_2'\, +\, \frac{t^2}{2}m_1n_1+O(t^3), \label{FtFte} \\ \frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t}
\, =\, &-{\tilde{\mathfrak a}}_2|n_2|^2 \, +\, t{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2n_1n_2
\, +\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2 \,-\, t q_2n_2 \, +\, t^2{\tilde{\mathfrak a}' }_1n_1 q_2 \nonumber\\[2mm] &\, +\, t^2k{\tilde{\mathfrak a}}_1 q_2n_2 \, +\, \frac{t^2}{2}{\tilde{\mathfrak a}}_2n_2q_1' \,-\, \frac{t^2}{2}m_2n_2\, +\, O(t^3). \label{FteFt} \end{align}
By using the Taylor expansion of ${\mathfrak a}_i(y)$ as in \eqref{expressionofai} and \eqref{FtFt}-\eqref{FteFt}, it is easy to derive that \begin{align} &{\mathfrak a}_1(y)\frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial \theta} \,-\, {\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t} \nonumber\\[2mm]
&\, =\, {\mathfrak a}_1(y)\big[{\tilde{\mathfrak a}}_1|n_1|^2 +t{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}' }_2n_1n_2
\,-\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2 \, +\, t q_1n_1 +O(t^2)\big] \nonumber\\[2mm]
&\qquad\,-\, {\mathfrak a}_2(y)\big[-{\tilde{\mathfrak a}}_2|n_2|^2 \, +\, t{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2n_1n_2
\, +\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2 \,-\, t q_2n_2 \, +\, O(t^2)\big] \nonumber\\[2mm]
&\, =\, \big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1|n_1|^2\, +\, {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_2|n_2|^2 \big] \nonumber\\[2mm] &\qquad+t\Big\{\big[ {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}'}_2n_1n_2-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}'}_1{\tilde{\mathfrak a}}_2n_1n_2 \big] \nonumber\\[2mm] &\qquad \qquad
- k\big[ {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2 \big] +\big[{\mathfrak a}_1(0, \theta)q_1n_1+{\mathfrak a}_2(0, \theta)q_2n_2 \big] \nonumber\\[2mm] &\qquad \qquad
+\big[\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1|n_1|^2+\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_2|n_2|^2\big]\Big\} \, +\, O(t^2) \nonumber\\[2mm] &\, \equiv\, \mathfrak{p}_1(\theta)+ t\mathfrak{p}_2(\theta)\, +\, O(t^2), \label{p1p2} \end{align} \begin{align} &{\mathfrak a}_1(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta}\,-\, {\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta} \nonumber\\[2mm] &\, =\, \Big[{\mathfrak a}_1(y)-{\mathfrak a}_2(y)\Big]\times\Big[-n_1n_2
+t \big({\tilde{\mathfrak a}' }_1|n_1|^2- {\tilde{\mathfrak a}' }_2|n_2|^2\big) +tk\big({\tilde{\mathfrak a}}_1+{\tilde{\mathfrak a}}_2 \big) n_1n_2 +\frac{t^2}{2}\big(q_1'n_1-q_2'n_2\big) \nonumber\\[2mm] &\qquad \qquad \qquad \qquad \qquad +t^2{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}' }_2n_1n_2
-t^2k \big( {\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2 |n_1|^2- {\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}' }_2|n_2|^2 \big) -t^2k^2{\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}}_2 n_1n_2 +O(t^3)\Big] \nonumber\\[2mm] &\, =\, -\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]n_1n_2 \nonumber\\[2mm]
&\quad\, \, +t\Big\{\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big] \big(- {\tilde{\mathfrak a}' }_2|n_2|^2+ {\tilde{\mathfrak a}' }_1|n_1|^2\big) \nonumber\\[2mm] &\qquad\qquad+k\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]\big({\tilde{\mathfrak a}}_1+{\tilde{\mathfrak a}}_2 \big) n_1n_2 -\big[\partial_t{\mathfrak a}_1(0, \theta)- \partial_t{\mathfrak a}_2(0, \theta) \big]n_1n_2\Big\} \nonumber\\[2mm] &\quad\, \, +t^2\Big\{\frac{1}{2}\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]\big(-q_2'n_2+ q_1'n_1\big) +\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}' }_2n_1n_2 \nonumber\\[2mm]
&\qquad \qquad -k\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big] \big( {\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2 |n_1|^2- {\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}' }_2|n_2|^2 \big) -k^2\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big] {\tilde{\mathfrak a}}_1 {\tilde{\mathfrak a}}_2 n_1n_2 \nonumber\\[2mm] &\qquad \qquad
+\big[\partial_t{\mathfrak a}_1(0, \theta)- \partial_t{\mathfrak a}_2(0, \theta) \big] \big(- {\tilde{\mathfrak a}' }_2|n_2|^2+ {\tilde{\mathfrak a}' }_1|n_1|^2\big) \nonumber\\[2mm] &\qquad \qquad+k\big[\partial_t{\mathfrak a}_1(0, \theta)- \partial_t{\mathfrak a}_2(0, \theta) \big]\big({\tilde{\mathfrak a}}_1+{\tilde{\mathfrak a}}_2 \big) n_1n_2\Big\} +O(t^3) \nonumber\\[2mm] &\, \equiv\, \mathfrak{p}_3(\theta)+t \mathfrak{p}_4(\theta)\, +\, t^2 \mathfrak{p}_5(\theta)+O(t^3), \label{p3p4} \end{align} \begin{align} {\mathfrak a}_2(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t} \,-\, {\mathfrak a}_1(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t} \, =\, &\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2n_1n_2 \nonumber\\[2mm] &+t \Big\{\big[{\mathfrak a}_1(0, \theta)- {\mathfrak a}_2(0, \theta) \big]\big( {\tilde{\mathfrak a}}_2q_1n_2+{\tilde{\mathfrak a}}_1q_2n_1 \big) \nonumber\\[2mm] &\qquad+\big[\partial_t{\mathfrak a}_1(0, \theta)- \partial_t{\mathfrak a}_2(0, \theta) \big]{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2n_1n_2 \Big\} +O(t^2) \nonumber\\[2mm] \, \equiv\, &\mathfrak{p}_6(\theta)\, +\, t\mathfrak{p}_7(\theta)+O(t^2), \label{p6p7} \end{align} and \begin{align} &{\mathfrak a}_2(y) \frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial \theta} \,-\, {\mathfrak a}_1(y) \frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t} \nonumber\\[2mm]
&\, =\, {\mathfrak a}_2(y) \big[{\tilde{\mathfrak a}}_1|n_1|^2 +t{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}' }_2n_1n_2
\,-\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2 \, +\, t q_1n_1 \, +\, t^2{\tilde{\mathfrak a}' }_2n_2q_1 \nonumber\\[2mm] &\qquad \qquad\,-\, t^2k {\tilde{\mathfrak a}}_2q_1n_1 \, +\, \frac{t^2}{2}{\tilde{\mathfrak a}}_1n_1q_2' \, +\, \frac{t^2}{2}m_1n_1+O(t^3)\big] \nonumber\\
&\quad\,-\, {\mathfrak a}_1(y)\big[-{\tilde{\mathfrak a}}_2|n_2|^2 \, +\, t{\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2n_1n_2
\, +\, tk{\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2 \,-\, t q_2n_2 \, +\, t^2{\tilde{\mathfrak a}' }_1n_1 q_2 \nonumber\\[2mm] &\qquad \qquad \quad\, +\, t^2k{\tilde{\mathfrak a}}_1 q_2n_2 \, +\, \frac{t^2}{2}{\tilde{\mathfrak a}}_2n_2q_1' \,-\, \frac{t^2}{2}m_2n_2\, +\, O(t^3)\big] \nonumber\\[2mm]
&\, =\, \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_1|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_2|^2 \big] \nonumber\\[2mm]
&\qquad+t\Big\{\big[\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_1|^2 +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_2|^2\big] +\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}' }_2 -{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2\big]n_1n_2 \nonumber\\[2mm]
&\qquad \qquad\,-\, k\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2\big] +\big[{\mathfrak a}_2(0, \theta)q_1n_1+{\mathfrak a}_1(0, \theta)q_2n_2 \big]\Big\} \nonumber\\[2mm] &\qquad+t^2\Big\{\big[\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}' }_2 -\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}' }_1{\tilde{\mathfrak a}}_2\big]n_1n_2 \nonumber\\[2mm]
&\qquad \qquad\,-\, k\big[ \partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_1|^2+\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_2|n_2|^2 \big] \, +\, \big[\partial_t{\mathfrak a}_2(0, \theta)q_1n_1+\partial_t{\mathfrak a}_1(0, \theta)q_2n_2 \big] \nonumber\\[2mm] &\qquad \qquad+\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}' }_2n_2q_1-{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}' }_1n_1q_2\big] \,-\, k\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_2q_1n_1-{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2q_2n_2 \big] \nonumber\\[2mm] &\qquad \qquad\, +\, \frac{1}{2} \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1n_1q_2'-{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2n_2q_1' \big] \, +\, \frac{1}{2} \big[{\mathfrak a}_2(0, \theta)m_1n_1+{\mathfrak a}_1(0, \theta)m_2n_2 \big]\Big\} +O(t^3) \nonumber\\[2mm] &\, \equiv\, \mathfrak{p}_8(\theta)\, +\, t\mathfrak{p}_9(\theta) \, +\, t^2\mathfrak{p}_{10}(\theta)+O(t^3). \label{p8p9p10} \end{align}
On the other hand, by recalling \eqref{g-inverse2}, \eqref{sigma1} and \eqref{sigma2}, we can obtain that \begin{align*} \frac{1}{\sqrt{g}} \sigma_1 \, =\, & \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}+{\mathfrak r}_1t+{\mathfrak r}_3t^2+O(t^3) \Big]\times\big[{\mathfrak y}_1(\theta)+t{\mathfrak y}_2(\theta)+t^2{\mathfrak y}_3(\theta)+O(t^3)\big] \\[2mm] \, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta) +t{\mathfrak r}_1{\mathfrak y}_1(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta) +t^2\Big[{\mathfrak r}_1{\mathfrak y}_2(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_3(\theta) +{\mathfrak c}_3{\mathfrak y}_1(\theta) \Big]+O(t^3) \\[2mm] \, \equiv\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta) +t{\mathfrak r}_1{\mathfrak y}_1(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta) +t^2{\mathfrak k}_1+O(t^3), \end{align*} \begin{align*}
\frac{1}{\sqrt{g}} \sigma_2
\, =\, & \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}+{\mathfrak r}_1t+{\mathfrak r}_3t^2+O(t^3) \big]\times \big[{\mathfrak y}_4(\theta)+t{\mathfrak y}_5(\theta)+t^2{\mathfrak y}_6(\theta)+O(t^3)\Big] \\[2mm] \, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta) +t{\mathfrak r}_1{\mathfrak y}_4(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta) +t^2\Big[{\mathfrak r}_3{\mathfrak y}_4(\theta) +{\mathfrak y}_5(\theta){\mathfrak r}_1 +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_6(\theta) \Big]+O(t^3) \\[2mm] \, \equiv\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta) +t{\mathfrak r}_1{\mathfrak y}_4(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta) +t^2{\mathfrak k}_2+O(t^3). \end{align*}
Therefore, we obtain that \begin{align*} &\frac{1}{\sqrt{g}} \sigma_1 \left[{\mathfrak a}_1(y)\frac{\partial F_1}{\partial t}\frac{\partial F_2}{\partial\theta} \,-\, {\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t}\right]u_t \, +\, \frac{1}{\sqrt{g}}\sigma_2 \left[{\mathfrak a}_1(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta} \,-\, {\mathfrak a}_2(y)\frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial \theta} \right]u_t \\[2mm] \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta) +t{\mathfrak r}_1{\mathfrak y}_1(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta) +t^2{\mathfrak k}_1+O(t^3)\Big]\times \Big[\mathfrak{p}_1(\theta)+ t\mathfrak{p}_2(\theta)\, +\, O(t^2)\Big]u_t \\[2mm] &\, +\, \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta) +t{\mathfrak r}_1{\mathfrak y}_4(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta) +t^2{\mathfrak k}_2+O(t^3)\Big]\times \Big[\mathfrak{p}_3(\theta)+t \mathfrak{p}_4(\theta)+ t^2 \mathfrak{p}_5(\theta)+O(t^3)\Big]u_t \\[2mm] \, \equiv\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_1(\theta)u_t +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_3(\theta)u_t \\[2mm] &+\Big[{\mathfrak r}_1{\mathfrak y}_1(\theta)\mathfrak{p}_1(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta)\mathfrak{p}_1(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_2(\theta) \\[2mm] &\quad+\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_4(\theta) +{\mathfrak r}_1{\mathfrak y}_4(\theta)\mathfrak{p}_3(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta)\mathfrak{p}_3(\theta)\Big]tu_t +b_1(\theta)t^2u_t+O(t^3), \end{align*} and \begin{align*} &\frac{1}{\sqrt{g}} \sigma_1 \left[ {\mathfrak a}_2(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t} \,-\, {\mathfrak a}_1(y)\frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial t}\right]u_{\theta} \, +\, \frac{1}{\sqrt{g}}\sigma_2 \left[{\mathfrak a}_2(y) \frac{\partial F_1}{\partial t} \frac{\partial F_2}{\partial \theta} \,-\, {\mathfrak a}_1(y) \frac{\partial F_1}{\partial \theta}\frac{\partial F_2}{\partial t}\right] u_{\theta} \\[2mm] \, =\, &\Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta) +t{\mathfrak r}_1{\mathfrak y}_1(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta) +t^2{\mathfrak k}_1+O(t^3)\Big]\times \Big[\mathfrak{p}_6(\theta)\, +\, t\mathfrak{p}_7(\theta)+O(t^2)\Big]u_{\theta} \\[2mm] &\, +\, \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta) +t{\mathfrak r}_1{\mathfrak y}_4(\theta) +t\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta) +t^2{\mathfrak k}_2+O(t^3)\Big]\times \Big[\mathfrak{p}_8(\theta)+t\mathfrak{p}_9(\theta) +t^2\mathfrak{p}_{10}(\theta)+O(t^3)\Big]u_{\theta} \\[2mm]
\, \equiv\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_6(\theta)u_{\theta} +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_8(\theta)u_{\theta} +b_3(\theta)tu_{\theta} +b_2(\theta)t^2u_{\theta}+O(t^3). \end{align*}
Whence, the terms in $\nabla_{{\mathfrak a}(y)}u\cdot\nu$ will be rearranged in the form \begin{align*} \nabla_{{\mathfrak a}(y)}u\cdot\nu
\, =\, &\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_1(\theta)u_t +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_3(\theta)u_t +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_6(\theta)u_{\theta} +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_8(\theta)u_{\theta} \\[2mm] &+\Big[{\mathfrak r}_1{\mathfrak y}_1(\theta)\mathfrak{p}_1(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_2(\theta)\mathfrak{p}_1(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_1(\theta)\mathfrak{p}_2(\theta) \\[2mm] &\qquad+\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_4(\theta)\mathfrak{p}_4(\theta) +{\mathfrak r}_1{\mathfrak y}_4(\theta)\mathfrak{p}_3(\theta) +\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak y}_5(\theta)\mathfrak{p}_3(\theta)\Big]tu_t \\[2mm] &+b_3(\theta)tu_{\theta}\, +\, b_1(\theta)t^2u_t+b_2(\theta)t^2u_{\theta}+O(t^3). \end{align*} More precisely, we shall derive the boundary operator in the following way. According to \eqref{tildea1=a2}, we can obtain that \begin{align*} {\mathfrak y}_1(0)\, =\, {\mathfrak y}_1(1)\, =\, 0, \quad {\mathfrak p}_3(0)\, =\, {\mathfrak p}_3(1)\, =\, 0. \end{align*} For $\theta\, =\, 0$, the boundary operator becomes \begin{equation} \begin{split} & {\mathfrak b}_1u_{\theta} \, +\, {\mathfrak b}_2t u_t +{\mathfrak b}_3t^2 u_t +{\mathfrak b}_4tu_{\theta} +{\mathfrak b}_5t^2 u_{\theta}+{\bar D}_0^0(u), \end{split} \label{boundaryoriginal0} \end{equation} where \begin{align} {\mathfrak b}_1=\frac{1}{\sqrt{{\mathfrak h}_1(0)}}{\mathfrak y}_4(0)\mathfrak{p}_8(0), \qquad {\mathfrak b}_2=\frac{1}{\sqrt{{\mathfrak h}_1(0)}}\big[{\mathfrak y}_4(0)\mathfrak{p}_4(0)+{\mathfrak y}_2(0)\mathfrak{p}_1(0)\big], \label{b1b2} \\[2mm] {\mathfrak b}_3=b_1(0), \qquad {\mathfrak b}_4=b_3(0), \qquad {\mathfrak b}_5=b_2(0), \qquad {\bar D}_0^0(u)\, =\, \sigma_3(t)\, u_t\, +\, \sigma_4(t)u_\theta. \end{align} On the other hand, for $\theta=1$, it has the form \begin{equation} \begin{split} {\mathfrak b}_6u_{\theta} \, +\, {\mathfrak b}_7 t u_t +{\mathfrak b}_8t^2 u_t +{\mathfrak b}_9tu_{\theta} +{\mathfrak b}_{10}t^2 u_{\theta}+{\bar D}_0^1(u), \end{split} \label{boundaryoriginal1} \end{equation} with the notation \begin{align} {\mathfrak b}_6=\frac{1}{\sqrt{{\mathfrak h}_1(1)}}{\mathfrak y}_4(1)\mathfrak{p}_8(1), \qquad {\mathfrak b}_7=\frac{1}{\sqrt{{\mathfrak h}_1(1)}}\big[{\mathfrak y}_4(1)\mathfrak{p}_4(1)+{\mathfrak y}_2(1)\mathfrak{p}_1(1)\big], \label{b6b7} \\[2mm] {\mathfrak b}_8=b_1(1), \qquad {\mathfrak b}_9=b_3(1), \qquad {\mathfrak b}_{10}=b_2(1), \qquad {\bar D}_0^1(u)\, =\, \sigma_5(t)\, u_t\, +\, \sigma_6(t)u_\theta. \end{align} In the above, the functions $\sigma_3, \cdots, \sigma_6$ are smooth functions of $t$ with the properties $$
|\sigma_i(t)|\leq C|t|^3, \quad i=3, 4, 5, 6. $$
\subsection{Stationary and non-degenerate curves}\label{Stationary and non-degenerate curves}\
In the following, for a simple smooth curve $\Gamma$ connecting the boundary $\partial\Omega$, we will make precisely the notion of a non-degenerate geodesic with respect to the metric ${\mathrm d}s^2= V^{2\sigma}(y)\big[{\mathfrak a}_2(y){\mathrm d}{y}_1^2+{\mathfrak a}_1(y){\mathrm d}{y}_2^2\big]$. Consider the deformation of $\Gamma$ in the form \begin{equation} \label{deformation1} \Gamma_{{\mathfrak t}}(\theta)=\big(\Gamma_{{\mathfrak t}1}(\theta), \Gamma_{{\mathfrak t}2}(\theta)\big): \gamma\big(\Theta({\mathfrak t}(\theta), \theta)\big)
\, +\, {\mathfrak t}(\theta)\, \Big({\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big),
{\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big), \end{equation} where $\theta\in [0, 1]$ and ${\mathfrak t}$ is a smooth function of $\theta$ with small $L^{\infty}$-norm. Note that the end points of $\Gamma_{{\mathfrak t}}$ stay on $\partial \Omega$. We denote ${\mathfrak a}_1(F(t, \theta)), {\mathfrak a}_2(F(t, \theta))$ and $ V(F(t, \theta))$ as ${\mathfrak a}_1(t, \theta)$, ${\mathfrak a}_2(t, \theta)$, $ V(t, \theta)$. The weighted length of the curve $\Gamma_{{\mathfrak t}}$ is given by the functional of ${\mathfrak t}$ \begin{equation} \begin{split} {\mathcal J}({\mathfrak t})&\equiv \int_0^1 V^{\sigma}\big(\Gamma_{{\mathfrak t}}(\theta)\big)
\sqrt{{\mathfrak a}_2\left(\Gamma_{{\mathfrak t}}(\theta)\right)\big|\Gamma'_{{\mathfrak t}1}(\theta)\big|^2
+ {\mathfrak a}_1\left(\Gamma_{{\mathfrak t}}(\theta)\right)\big|\Gamma'_{{\mathfrak t}2}(\theta)\big|^2}\, {\rm d}\theta \\[2mm] &=\int_0^1 V^{\sigma}\big(\Gamma_{{\mathfrak t}}(\theta)\big) \sqrt{{\mathcal W}\big( {\mathfrak t}(\theta)\big)} \, {\rm d}\theta. \end{split} \label{weightedlength1} \end{equation}
where we have denoted \begin{align} {\mathcal W}\big( {\mathfrak t}(\theta)\big) =& \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big)^2 \nonumber\\[2mm]
&\qquad\times\Big\{\, \big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]^2{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm]
&\qquad\qquad+\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]^2{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2\, \Big\} \nonumber\\[2mm]
&\, +\, |{\mathfrak t}'(\theta)|^2\bigg[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1(\theta)\big|^2\big|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2(\theta)\big|^2\big|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2\bigg] \nonumber\\[2mm]
&+2{\mathfrak t}'(\theta){\mathfrak t}(\theta)\bigg[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta){\tilde{\mathfrak a}}_1'(\theta)|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta){\tilde{\mathfrak a}}_2'(\theta)|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2\bigg] \nonumber\\[2mm]
&+|{\mathfrak t}(\theta)|^2\bigg[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1'(\theta)\big|^2|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2'(\theta)\big|^2|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2\bigg] \nonumber\\[2mm] &\, +\, 2{\mathfrak t}'(\theta)\, \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times
\Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad+\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big\} \nonumber\\[2mm] &\, +\, 2{\mathfrak t}(\theta)\Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big]\Big\}. \label{mathcalM} \end{align}
\noindent{\bf{Step 1. }} The first variation of ${\mathcal J}$ at ${\mathfrak t}$ along the direction $h$ is given by \begin{align} {\mathcal J}'({\mathfrak t})[h]
=\, \frac {\mathrm d}{{\mathrm d}s}{\mathcal J}({\mathfrak t}+sh)\Big|_{s=0} &=\, \int_0^1 \sigma V^{\sigma-1}\big(\Gamma_{{\mathfrak t}}(\theta)\big)\, V_t\big(\Gamma_{{\mathfrak t}}(\theta)\big)\, h\, \, \sqrt{{\mathcal W}\big( {\mathfrak t}(\theta)\big)} \, {\rm d}\theta \nonumber\\[2mm] &\quad\ +\ \frac{1}{2}\int_0^1\frac{ V^{\sigma}\big(\Gamma_{{\mathfrak t}}(\theta)\big)}{\sqrt{\mathcal W\big( {\mathfrak t}(\theta)\big)}}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}({\mathfrak t}+sh)\Big|_{s=0} \, {\rm d}\theta, \end{align} where \begin{align}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}({\mathfrak t}+sh)\Big|_{s=0} \, =\, &{\mathcal W}_1\big( {\mathfrak t}(\theta)\big)[h] \ +\ {\mathcal W}_2\big( {\mathfrak t}(\theta)\big)[h] \ +\ {\mathcal W}_3\big( {\mathfrak t}(\theta)\big)[h] \nonumber\\[2mm] &\, +\, {\mathcal W}_4\big( {\mathfrak t}(\theta)\big)[h] \, +\, {\mathcal W}_5\big( {\mathfrak t}(\theta)\big)[h] \, +\, {\mathcal W}_6\big( {\mathfrak t}(\theta)\big)[h], \end{align} in which ${\mathcal W}_i\big( {\mathfrak t}(\theta)\big)[h], (i=1, \cdots, 6)$ are given by \begin{align} {\mathcal W}_1\big( {\mathfrak t}(\theta)\big)[h] &= \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big)^2 \nonumber\\[2mm]
&\qquad\quad\times\Big\{\, 2\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]\, (-k{\tilde{\mathfrak a}}_1)\, {\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm]
&\qquad \qquad+2\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]\, (-k{\tilde{\mathfrak a}}_2)\, {\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2\, \Big\}h \nonumber\\[2mm] &\quad\, +\, 2\Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \Big( \Theta_{tt}({\mathfrak t}(\theta), \theta)\, {\mathfrak t}'(\theta)\, h \, +\, \Theta_{t}({\mathfrak t}(\theta), \theta)\, h' \, +\, \Theta_{\theta t}({\mathfrak t}(\theta), \theta)\, h \Big) \nonumber\\[2mm]
&\qquad\quad\times\Big\{\, \big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]^2{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm]
&\qquad \qquad \quad+\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]^2{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2\, \Big\} \nonumber\\[2mm] &\quad+\Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big)^2 \nonumber\\[2mm]
&\qquad\quad\times\Big\{\, \big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]^2\partial_t{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm] &\qquad\quad\quad\, +\, \big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]^2 2{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1''\big(\Theta({\mathfrak t}(\theta), \theta)\big) \Theta_t({\mathfrak t}(\theta), \theta) \nonumber\\[2mm] &\qquad\quad\quad\, +\,
\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]^2\partial_t{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm] &\qquad\quad\quad\, +\, \big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]^22{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2''\big(\Theta({\mathfrak t}(\theta), \theta)\big) \Theta_t({\mathfrak t}(\theta), \theta)\Big\}h, \label{mathcalM1} \end{align} \begin{align} {\mathcal W}_2\big( {\mathfrak t}(\theta)\big)[h]
&=2{\mathfrak t}'(\theta)\, \bigg[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1(\theta)\big|^2\cdot\big|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2(\theta)\big|^2\cdot\big|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2\bigg]\, h' \nonumber\\[2mm] &\quad\, +\,
|{\mathfrak t}'(\theta)|^2\Big\{\partial_t{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1(\theta)\big|^2\cdot\big|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2
\, +\, \partial_t{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2(\theta)\big|^2\cdot\big|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\big|^2 \nonumber\\[2mm]
&\qquad\qquad\qquad\, +\, {\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1(\theta)\big|^2\,2 n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_{t}({\mathfrak t}(\theta), \theta) \nonumber\\[2mm]
&\qquad\qquad\qquad\, +\, {\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2(\theta)\big|^2\,2 n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_{t}({\mathfrak t}(\theta), \theta)\Big\}h, \label{mathcalM2} \end{align} \begin{align} {\mathcal W}_3\big( {\mathfrak t}(\theta)\big)[h]
&=2\, {\mathfrak t}(\theta)\Big[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta){\tilde{\mathfrak a}}_1'(\theta)|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta){\tilde{\mathfrak a}}_2'(\theta)|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2\Big]h' \nonumber\\[2mm]
&\quad\, +\, 2{\mathfrak t}'(\theta)\Big[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta){\tilde{\mathfrak a}}_1'(\theta)|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta){\tilde{\mathfrak a}}_2'(\theta)|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2\Big]h \nonumber\\[2mm] &\quad\, +\, 2{\mathfrak t}'(\theta){\mathfrak t}(\theta)
\Big\{\partial_t{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta){\tilde{\mathfrak a}}_1'(\theta)|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
\, +\, \partial_t{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta){\tilde{\mathfrak a}}_2'(\theta)|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2 \nonumber\\[2mm] &\qquad \qquad \qquad \quad\, +\, {\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta){\tilde{\mathfrak a}}_1'(\theta)2n_1\big(\Theta({\mathfrak t}(\theta), \theta)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\Theta_t({\mathfrak t}(\theta)) \nonumber\\[2mm] &\qquad \qquad \qquad \quad\, +\, {\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta){\tilde{\mathfrak a}}_2'(\theta)2n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta))\Big\}h, \label{mathcalM3} \end{align} \begin{align} {\mathcal W}_4\big( {\mathfrak t}(\theta)\big)[h]
&=2{\mathfrak t}(\theta) \Big[{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1'(\theta)\big|^2|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
+{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2'(\theta)\big|^2|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2\Big]h \nonumber\\[2mm]
&\quad\, +\, |{\mathfrak t}(\theta)|^2
\Big\{\partial_t{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1'(\theta)\big|^2|n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2
\, +\, \partial_t{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2'(\theta)\big|^2|n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)|^2 \nonumber\\[2mm]
&\qquad \qquad \qquad \, +\, {\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_1'(\theta)\big|^22n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta) \nonumber\\[2mm]
&\qquad \qquad \qquad \, +\, {\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big)\big|{\tilde{\mathfrak a}}_2'(\theta)\big|^22n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta) \Big\}h, \label{mathcalM4} \end{align} \begin{align} {\mathcal W}_5\big( {\mathfrak t}(\theta)\big)[h]
&=2\, \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big\}h' \nonumber\\[2mm] &\quad\, +\, 2{\mathfrak t}'(\theta)\, \Big( \Theta_{tt}({\mathfrak t}(\theta), \theta)\, {\mathfrak t}'(\theta)\, h \, +\, \Theta_{t}({\mathfrak t}(\theta), \theta)\, h' \, +\, \Theta_{\theta t}({\mathfrak t}(\theta), \theta)\, h \Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big\}h \nonumber\\[2mm] &\quad\, +\, 2{\mathfrak t}'(\theta)\, \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[-k(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad\qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]\partial_t{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad\qquad -\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta))n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad\qquad -\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta) \nonumber\\[2mm] &\qquad \qquad +\big[-k(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\partial_t\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta({\mathfrak t}(\theta), \theta) \Big\}h, \label{mathcalM5} \end{align} and \begin{align} {\mathcal W}_6\big( {\mathfrak t}(\theta)\big)[h] &=2\, \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big\}h \nonumber\\[2mm] &+2{\mathfrak t}(\theta)\, \Big( \Theta_{tt}({\mathfrak t}(\theta), \theta)\, {\mathfrak t}'(\theta)\, h \, +\, \Theta_{t}({\mathfrak t}(\theta), \theta)\, h' \, +\, \Theta_{\theta t}({\mathfrak t}(\theta), \theta)\, h \Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Big\} \nonumber\\[2mm] &+2{\mathfrak t}(\theta)\, \Big(\Theta_t({\mathfrak t}(\theta), \theta){\mathfrak t}'(\theta)+\Theta_{\theta}({\mathfrak t}(\theta), \theta)\Big) \nonumber\\[2mm] &\qquad\times \, \Big\{\big[-k(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\partial_t\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad -\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_1\big]{\mathfrak a}_2\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_1'(\theta)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta) \nonumber\\[2mm] &\qquad \qquad +\big[-k(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]\partial_t{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)\gamma_2'\big(\Theta({\mathfrak t}(\theta), \theta)\big) \nonumber\\[2mm] &\qquad \qquad +\big[1-k(\theta){\mathfrak t}(\theta){\tilde{\mathfrak a}}_2\big]{\mathfrak a}_1\big(\Gamma_{{\mathfrak t}}(\theta)\big){\tilde{\mathfrak a}}_2'(\theta)n_2\big(\Theta({\mathfrak t}(\theta), \theta)\big)n_1'\big(\Theta({\mathfrak t}(\theta), \theta)\big)\Theta_t({\mathfrak t}(\theta), \theta)\Big\}h. \label{mathcalM6} \end{align}
By substituting the relations in \eqref{Thetaderivative1}-\eqref{Thetaderivative2} into \eqref{mathcalM1}-\eqref{mathcalM6}, we get \begin{align}
{\mathcal W}(0)\, =\, &{\mathfrak a}_1(0, \theta)\big|n_1\big|^2+{\mathfrak a}_2(0, \theta)\big|n_2\big|^2={\mathfrak f}_0(\theta), \label{mathcalM(0)} \\[2mm]
{\mathcal W}_1(0)[h]\, =\, &-2\, k \bigg[\, {\tilde{\mathfrak a}}_1\, {\mathfrak a}_2(0, \theta)\big|n_2\big|^2
+{\tilde{\mathfrak a}}_2\, {\mathfrak a}_1(0, \theta)\big|n_1\big|^2\, \bigg]h \nonumber\\[2mm]
&+\bigg[\partial_t{\mathfrak a}_2(0, \theta)\big|n_2\big|^2
\, +\, \partial_t{\mathfrak a}_1(0, \theta)\big|n_1\big|^2\bigg]\, h, \\[2mm] {\mathcal W}_2(0)[h]\, =\, &0, \qquad {\mathcal W}_3(0)[h]=0, \qquad {\mathcal W}_4(0)[h]=0, \label{mathcalM23(0)} \\[2mm] {\mathcal W}_5(0)[h] \, =\, &2h'\big[-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1(\theta) +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2(\theta)\big]n_1n_2\, =\, 0, \\[2mm] {\mathcal W}_6(0)[h]\, =\, & 2h\, \big[-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1'(\theta) +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2'(\theta)\big]n_1n_2. \label{mathcalM6(0)} \end{align} From the definitions of ${\mathfrak f}_1$ as in \eqref{m22}, we know that \begin{equation*} {\mathcal W}_1(0)[h]+{\mathcal W}_2(0)[h]+{\mathcal W}_3(0)[h]+{\mathcal W}_4(0)[h]+{\mathcal W}_5(0)[h]+{\mathcal W}_6(0)[h]= {\mathfrak f}_1. \end{equation*} The above computations give that \begin{align} {\mathcal J}'(0)[h]
=&\int_0^1 \Bigg\{ \frac{\sigma V_t(0, \theta)}{ V^{1-\sigma}(0, \theta)}\, \sqrt{{\mathfrak f}_0} +\frac{1}{2}\frac{ V^{\sigma}(0, \theta)}{\, \sqrt{{\mathfrak f}_0}} {\mathfrak f}_1\Bigg\} \, h\, {\rm d}\theta. \end{align}
The curve $\Gamma$ is said to be {\bf stationary} with respect to the weighted length in \eqref{weightedlength1} if the first variation of ${\mathcal J}$ at ${\mathfrak t}=0$ is equal to zero. That is, for any smooth function $h(\theta)$ defined at $[0, 1]$ there holds
\begin{equation*}
\begin{split} {\mathcal J}'(0)[h]=0.
\end{split}
\end{equation*} This is equivalent to the relation \begin{equation} \label{stationary} \frac{\sigma V_t(0, \theta)}{ V^{1-\sigma}(0, \theta)}\, \sqrt{{\mathfrak f}_0} +\frac{1}{2}\frac{ {\mathfrak f}_1}{\, \sqrt{{\mathfrak f}_0}} V^{\sigma}(0, \theta) =0, \quad\forall\, \theta\in(0, 1), \end{equation} where ${\mathfrak f}_0$ and ${\mathfrak f}_1$ are given in \eqref{m22}. Specially, if the parameter $\sigma$ in \eqref{weightedlength1} is $\frac{p+1}{p-1}-\frac{1}{2}$, then \eqref{stationary} has the form \begin{equation} \label{stationary1} \Big(\frac{p+1}{p-1}-\frac{1}{2}\Big) V_t(0, \theta)\, \sqrt{{\mathfrak f}_0} +\frac{1}{2}\frac{ V(0, \theta)}{\, \sqrt{{\mathfrak f}_0}} {\mathfrak f}_1 =0, \quad\forall\, \theta\in(0, 1), \end{equation} i.e., \begin{align}\label{stationary2}
k\, =\, &\frac{{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}'}_2 n_1 n_2 -{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}'}_1 n_2 n_1}{{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}}_2|n_1|^2 +{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 |n_2|^2}
+\frac{1}{2}\frac{\partial_{t} {\mathfrak a}_1(0, \theta) |n_1|^2 +\partial_{t} {\mathfrak a}_2(0, \theta) |n_2|^2}{{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}}_2|n_1|^2 +{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 |n_2|^2} \nonumber\\[2mm]
&+\sigma\frac{ V_t(0, \theta)}{ V(0, \theta)} \frac{{\mathfrak a}_1(0, \theta) |n_1|^2 +{\mathfrak a}_2(0, \theta) |n_2|^2}{{\mathfrak a}_1(0, \theta) {\tilde{\mathfrak a}}_2|n_1|^2 +{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 |n_2|^2}, \qquad \forall\, \theta\in(0, 1). \end{align}
\noindent{\bf{Step 2. }} We now consider the second variation of ${\mathcal J}$ \begin{align} {\mathcal J}''(0)[h, f]
=\, &\frac {\mathrm d}{{\mathrm d}s}{\mathcal J}'(0+sf)[h]\Big|_{s=0} \nonumber\\[2mm] =\, &\int_0^1 \left[\, \frac{\sigma V_{tt}(0, \theta)\, }{ V^{1-\sigma}\big(0, \theta\big)} +
\frac{\sigma(\sigma-1)\big| V_t(0, \theta)\big|^2\, }{ V^{2-\sigma}(0, \theta)} \, \right] \, \sqrt{{\mathcal W}(0)} \, hf\, {\rm d}\theta \nonumber\\[2mm] &\, +\, \frac{\sigma}{2}\int_0^1\frac{ V_t(0, \theta) h\, }{ V^{1-\sigma}\big(0, \theta\big)} \, \frac{1}{\sqrt{{\mathcal W}(0)}}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sf)\Big|_{s=0} \, {\rm d}\theta \nonumber\\[2mm] & \, +\,
\frac{\sigma}{2}\int_0^1\frac{\, V_t(0, \theta)f\, }{ V^{1-\sigma}\big(0, \theta\big)} \, \frac{1}{\sqrt{{\mathcal W}(0)}}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sh)\Big|_{s=0}
\, {\rm d}\theta \nonumber\\[2mm] & \,-\,
\frac{1}{4}\int_0^1\frac{ V^{\sigma}\big(0, \theta\big)\, }{\ \big(\sqrt{{\mathcal W}(0)}\big)^3\ }
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sf)\Big|_{s=0}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sh)\Big|_{s=0}
\, {\rm d}\theta \nonumber\\[2mm] &\, +\, \frac{1}{2}\int_0^1\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{\mathcal W(0)}}
\frac {\mathrm d}{{\mathrm d}s}\big[ \sum_{i=1}^{6}{\mathcal W}_i\big(sf\big)[h]\big]\Big|_{s=0} \, {\rm d}\theta. \end{align}
From the definitions of ${\mathcal W}(f)$ and ${\mathfrak f}_1$ in \eqref{mathcalM}, \eqref{m22}, we can obtain that \begin{align*}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sf)\Big|_{s=0}
\, =\, &-2k \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\big]f
+\big[\partial_t{\mathfrak a}_2(0, \theta)|n_2|^2+\partial_t{\mathfrak a}_1(0, \theta)|n_1|^2\big]f \\[2mm] &+2\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 n_1\gamma_1' +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2 n_2\gamma_2' \big]f' +2\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1' n_1\gamma_1' +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2' n_2\gamma_2' \big]f \\[2mm]
\, =\, &-2k \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\big]f
+ \big[\partial_t{\mathfrak a}_2(0, \theta)|n_2|^2+\partial_t{\mathfrak a}_1(0, \theta)|n_1|^2\big]f \\[2mm] &+2\big[-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1' +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2' \big]n_1n_2f \\[2mm] \, =\, &{\mathfrak f}_1f. \end{align*} Moreover, \begin{align*}
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sh)\Big|_{s=0} ={\mathfrak f}_1h, \qquad
\frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sf)\Big|_{s=0} \frac {\mathrm d}{{\mathrm d}s}{\mathcal W}(sh)\Big|_{s=0} ={\mathfrak f}_1^2hf. \end{align*} On the other hand, we use \eqref{mathcalM1}-\eqref{mathcalM6} together with the relations in \eqref{Thetaderivative1}-\eqref{Thetaderivative2} to derive the following \begin{align}
&\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_1\big(sf\big)[h] \Big|_{s=0} \nonumber\\[2mm]
&= 2\, k^2\big[\, {\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\, \big]\, fh
-4\, k\big[\, \partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\, \big]\, fh \nonumber\\[2mm] &\quad\, +\, 2\big[\Theta_{tt}(0, \theta)f'h+\Theta_{tt}(0, \theta)fh'+\Theta_{\theta tt}(0, \theta)fh\big]
\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big] \nonumber\\[2mm]
&\quad\, +\, \big[\partial_{tt}{\mathfrak a}_2(0, \theta)|n_2|^2+\partial_{tt}{\mathfrak a}_1(0, \theta)|n_1|^2\big]fh \, +\, 2\Theta_{tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)\gamma_1'\gamma_1''+{\mathfrak a}_1(0, \theta)\gamma_2'\gamma_2''\big]fh, \label{mathcalM1deri} \end{align} \begin{align}
\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_2\big(sf\big)[h] \Big|_{s=0}
\, =\, 2f'\, h'\, \big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2^2 |n_2|^2+{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1^2 |n_1|^2\big] \, =\, 2f'h'\, {\mathfrak w}_0, \label{mathcalM2deri} \end{align} \begin{align}
\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_3\big(sf\big)[h] \Big|_{s=0} \, =\, &2\, (h\, f'+h'f)
\, \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1'|n_1|^2
+ {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'|n_2|^2\big], \label{mathcalM3deri} \\[3mm]
\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_4\big(sf\big)[h] \Big|_{s=0}
\, =\, &2fh\big[{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1'|^2|n_1|^2
+ {\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2'|^2|n_2|^2\big], \label{mathcalM4deri} \\[3mm]
\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_5\big(sf\big)[h] \Big|_{s=0} \, =\, &-2k\, (h\, f'+h'f)
\, \big[{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1|^2\gamma_1'n_1
+ {\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2|^2\gamma_2'n_2\big] \nonumber\\[2mm] &\, +\, 2(h\, f'+h'f)\, \big[\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1\gamma_1'n_1 \, +\, \partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2\gamma_2'n_2\big] \nonumber\\[2mm] \, =\, &-2k\, (h\, f'+h'f)
\, \big[-{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1|^2
+ {\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2|^2\big]n_1n_2 \nonumber\\[2mm] &\, +\, 2(h\, f'+h'f)\, \big[-\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 + \partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2\big]n_1n_2, \label{mathcalM5deri} \end{align} and \begin{align}
\frac {\mathrm d}{{\mathrm d}s} {\mathcal W}_6\big(sf\big)[h] \Big|_{s=0} &=-4k\, \big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1'\gamma_1'n_1 + {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'\gamma_2'n_2\big]fh \nonumber\\[2mm] &\quad\, +\, 4\big[\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1'\gamma_1'n_1 +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2'\gamma_2'n_2\big]fh \nonumber\\ &=-4k\, \big[-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1' + {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'\big]n_1n_2fh \nonumber\\[2mm] &\quad\, +\, 4\big[-\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1' +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2'\big]n_1n_2 fh, \label{mathcalM6deri} \end{align} where ${\mathfrak w}_0$ is given in \eqref{m11}. Hence, we obtain \begin{align}
&\frac {\mathrm d}{{\mathrm d}s}\big[ \sum_{i=1}^{6}{\mathcal W}_i\big(sf\big)[h] \big]\Big|_{s=0} \nonumber\\[2mm] &=\Big\{ -4k\big[-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1' +{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'\big]n_1n_2 \, +\, 4\big[-\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1' +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2'\big]n_1n_2 \nonumber\\[2mm]
&\qquad+2\big[{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1'|^2|n_1|^2
+ {\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2'|^2|n_2|^2\big]
+2\, k^2\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\, \big]
\nonumber\\[2mm]
&\qquad-4k\big[\, \partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\, \big]
\, +\, 2\Theta_{\theta tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big] \nonumber\\[2mm]
&\qquad+\big[\partial_{tt}{\mathfrak a}_2(0, \theta)|n_2|^2+\partial_{tt}{\mathfrak a}_1(0, \theta)|n_1|^2\big] +2\Theta_{tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)\gamma_1'\gamma_1''+{\mathfrak a}_1(0, \theta)\gamma_2'\gamma_2''\big]
\Big\}fh \nonumber\\[2mm]
&\quad+\Big\{2\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big]\Theta_{tt}(0, \theta)
+2k\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1'|n_1|^2
+ {\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'|n_2|^2\big]
\nonumber\\[2mm]
&\qquad \quad-2k\big[-{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1'|^2
+{\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2'|^2\big]n_1n_2
\nonumber\\[2mm] &\qquad \quad+2\big[-\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2\big]n_1n_2 \Big\}(f'h+fh')\, +\, 2f'h'\, {\mathfrak w}_0
\nonumber\\[2mm] &\equiv {\mathfrak f}_3 fh \, +\, {\mathfrak f}_4 (f' h +fh') \, +\, 2f'h'\, {\mathfrak w}_0. \end{align} The combinations of the above computations together with \eqref{mathcalM(0)}-\eqref{mathcalM6(0)} will give that \begin{align} {\mathcal J}''(0)[h, f] =\, &\int_0^1 \left[\, \frac{\sigma V_{tt}(0, \theta)\, }{ V^{1-\sigma}\big(0, \theta\big)}
+\frac{\sigma(\sigma-1)\big| V_t(0, \theta)\big|^2\, }{ V^{2-\sigma}(0, \theta)}\, \right]\, \sqrt{{\mathfrak f}_0}\, hf\, {\rm d}\theta \nonumber\\[2mm] &\, +\, \sigma\int_0^1\frac{ V_t(0, \theta) \, }{ V^{1-\sigma}\big(0, \theta\big)}\, \frac{1}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_1hf\, {\rm d}\theta \,-\, \frac{1}{4}\int_0^1\frac{ V^{\sigma}\big(0, \theta\big)}{\ \big(\sqrt{{\mathfrak f}_0}\, \big)^3\ }{\mathfrak f}_1^2hf \, {\rm d}\theta \nonumber\\[2mm] &\, +\, \frac{1}{2}\int_0^1\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}
\big[ {\mathfrak f}_3 fh \, +\, {\mathfrak f}_4 (f' h +fh') \, +\, 2f'h'\, {\mathfrak w}_0 \big]\Big|_{s=0} \, {\rm d}\theta. \label{secondvariation} \end{align}
\noindent{\bf{Notation 2: }}\label{notation2} By recalling ${\mathfrak w}_0, {\mathfrak l}_1$ and ${\mathfrak f}_0, {\mathfrak f}_2, {\mathfrak f}_3$ in \eqref{m11}, \eqref{m12} and \eqref{m22}, we introduce following functions:
\begin{align}\label{mathcalH1} {\mathcal H}_1(\theta)=& \frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}\, {\mathfrak w}_0, \end{align}
\begin{align}\label{mathcalH2} {\mathcal H}_2(\theta) =&\frac{ V^{\sigma}\big(0, \theta\big)}{2\sqrt{{\mathfrak f}_0}}
\Big\{2\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big]\Theta_{tt}(0, \theta)
+2\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1'|n_1|^2
+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'|n_2|^2\big] \nonumber\\[2mm]
&\qquad \qquad -2k\big[-{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1'|^2
+ {\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2'|^2\big]n_1n_2 +2\big[-\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1 +\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2\big]n_1n_2 \Big\} \nonumber\\ =&\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}\Big\{{\mathfrak l}_1 -\big[{\mathfrak a}_1(0, \theta)n_1q_2+{\mathfrak a}_2(0, \theta)n_2q_1\big]
+\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big]\Theta_{tt}(0, \theta)\Big\} \nonumber\\[2mm] =&\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}{\mathfrak l}_1, \end{align} and \begin{align}\label{mathcalH3} {\mathcal H}_3(\theta) =&\frac{ V^{\sigma}\big(0, \theta\big)}{2\sqrt{{\mathfrak f}_0}} \Big\{-4k\, \big[{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'-{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1' \big]n_1n_2 +4\big[\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2' -\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1' \big]n_1n_2 \nonumber\\[2mm]
&\qquad\qquad \quad+2\big[{\mathfrak a}_2(0, \theta)|{\tilde{\mathfrak a}}_1'|^2|n_1|^2
+{\mathfrak a}_1(0, \theta)|{\tilde{\mathfrak a}}_2'|^2|n_2|^2\big]
+2k^2\big[{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\big]
\nonumber\\[2mm]
&\qquad\qquad \quad-4k\big[\partial_t{\mathfrak a}_2(0, \theta){\tilde{\mathfrak a}}_1|n_2|^2+\partial_t{\mathfrak a}_1(0, \theta){\tilde{\mathfrak a}}_2|n_1|^2\big]
+\big[\partial_{tt}{\mathfrak a}_2(0, \theta)|n_2|^2+\partial_{tt}{\mathfrak a}_1(0, \theta)|n_1|^2\big] \nonumber\\[2mm]
&\qquad\qquad \quad+2\Theta_{\theta tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big] \nonumber\\[2mm] &\qquad\qquad \quad+2\Theta_{tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)\gamma_1'\gamma_1''+{\mathfrak a}_1(0, \theta)\gamma_2'\gamma_2''\big]
\Big\} \nonumber\\[2mm] &+\left[\, \frac{\sigma V_{tt}(0, \theta)\, }{ V^{1-\sigma}\big(0, \theta\big)}
+\frac{\sigma(\sigma-1)\big| V_t(0, \theta)\big|^2\, }{ V^{2-\sigma}(0, \theta)} \, \right]\sqrt{{\mathfrak f}_0} \, +\, \sigma\frac{ V_t(0, \theta) \, }{ V^{1-\sigma}\big(0, \theta\big)}\frac{1}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_1 \,-\, \frac{1}{4}\frac{ V^{\sigma}\big(0, \theta\big)}{\ \big(\sqrt{{\mathfrak f}_0}\, \big)^3\ }{\mathfrak f}_1^2 \nonumber\\[2mm] \, =\, &\frac{ V^{\sigma}\big(0, \theta\big)}{2\sqrt{{\mathfrak f}_0}} \Big\{2{\mathfrak f}_2-\big[{\mathfrak a}_1(0, \theta) n_1q_2' -{\mathfrak a}_2(0, \theta) n_2q_1'\big]
+2\Theta_{\theta tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)|n_2|^2+{\mathfrak a}_1(0, \theta)|n_1|^2\big] \nonumber\\[2mm] &\qquad\qquad \quad+2\Theta_{tt}(0, \theta)\big[{\mathfrak a}_2(0, \theta)\gamma_1'\gamma_1''+{\mathfrak a}_1(0, \theta)\gamma_2'\gamma_2''\big]\Big\} \nonumber\\[2mm] &+\left[\, \frac{\sigma V_{tt}(0, \theta)\, }{ V^{1-\sigma}\big(0, \theta\big)}
+\frac{\sigma(\sigma-1)\big| V_t(0, \theta)\big|^2\, }{ V^{2-\sigma}(0, \theta)} \, \right]\, \sqrt{{\mathfrak f}_0} \, +\, \sigma\frac{ V_t(0, \theta) \, }{ V^{1-\sigma}\big(0, \theta\big)} \, \frac{1}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_1 \,-\, \frac{1}{4}\frac{ V^{\sigma}\big(0, \theta\big)}{\ \big(\sqrt{{\mathfrak f}_0}\, \big)^3\ }{\mathfrak f}_1^2 \nonumber\\[2mm] =&\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_2 +\left[\, \frac{\sigma V_{tt}(0, \theta)\, }{ V^{1-\sigma}\big(0, \theta\big)}
+\frac{\sigma(\sigma-1)\big| V_t(0, \theta)\big|^2\, }{ V^{2-\sigma}(0, \theta)} \, \right]\, \sqrt{{\mathfrak f}_0} \nonumber \\[2mm] &\, +\, \sigma\frac{ V_t(0, \theta) \, }{ V^{1-\sigma}\big(0, \theta\big)} \, \frac{1}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_1 \,-\, \frac{1}{4}\frac{ V^{\sigma}\big(0, \theta\big)}{\ \big(\sqrt{{\mathfrak f}_0}\, \big)^3\ }{\mathfrak f}_1^2. \end{align} In the above, we have used the facts \eqref{q_i} and \eqref{qi'}.
The terms in \eqref{secondvariation} can be rearranged in the following way \begin{align} {\mathcal J}''(0)[h, f] =\, &\int_0^1\Big( {\mathcal H}_1f' \, +\, {\mathcal H}_2 f \Big) h'\, {\rm d}\theta \, +\, \int_0^1\Big( {\mathcal H}_2f' \, +\, {\mathcal H}_3 f \Big) h\, {\rm d}\theta \nonumber\\[2mm]
=&\Big( {\mathcal H}_1f' \, +\, {\mathcal H}_2 f \Big) h\Big|^1_{\theta=0} \,-\, \int_0^1\big[ {\mathcal H}_1f'' \, +\, {\mathcal H}_1'f' \, +\, \big({\mathcal H}_2'-{\mathcal H}_3\big) f \big]h\, {\rm d}\theta. \end{align} Note that \begin{align} {\mathcal H}_1(1)f'(1) \, +\, {\mathcal H}_2(1) f(1) \, =\, &\frac{ V^{\sigma}\big(0, 1\big)}{\sqrt{{\mathfrak f}_0(1)}} \big[ {\mathfrak w}_0(1)f'(1) \ +\ {\mathfrak l}_1(1)f(1) \big] \label{boundaryoffunctional1000} \\[2mm] \, =\, &\frac{ V^{\sigma}\big(0, 1\big)}{2\sqrt{{\mathfrak f}_0(1)}} \big[ {\mathfrak b}_6f'(1) \ -\ {\mathfrak b}_7f(1) \big], \label{boundaryoffunctional1} \end{align} and \begin{align} {\mathcal H}_1(0)f'(0) \, +\, {\mathcal H}_2(0) f(0) \, =\, &\frac{ V^{\sigma}\big(0, 0\big)}{\sqrt{{\mathfrak f}_0(0)}} \big[ {\mathfrak w}_0(0)f'(0) \ +\ {\mathfrak l}_1(0)f(0) \big] \label{boundaryoffunctional2000} \\[2mm] \, =\, &\frac{ V^{\sigma}\big(0, 0\big)}{2\sqrt{{\mathfrak f}_0(0)}} \big[ {\mathfrak b}_1f'(0) \ -\ {\mathfrak b}_2f(0) \big]. \label{boundaryoffunctional2} \end{align} For more details of the derivation of the equalities \eqref{boundaryoffunctional1} and \eqref{boundaryoffunctional2}, the reader can refer to the computations in Appendix \ref{appendixE}.
For a stationary curve $\Gamma$, we say that it is {\bf non-degenerate} in the sense that if
\begin{equation}
\label{2-relation}
{\mathcal J}{''}(0)[h, f]=0, \quad \forall\, h\in H^1(0, 1),
\end{equation} then $f\equiv 0$. It is equivalent to that the boundary problem
\begin{equation}
\label{nondegeneracy}
\begin{split} {\mathcal H}_1f'' \, +\, {\mathcal H}_1'f' \, +\, \big({\mathcal H}_2'-{\mathcal H}_3\big) f=0 \quad \mbox{in}\ (0, 1), \\[2mm] {\mathfrak b}_1f'(0) \ -\ {\mathfrak b}_2f(0)=0, \qquad {\mathfrak b}_6f'(1) \ -\ {\mathfrak b}_7f(1)=0, \end{split} \end{equation} has only the trivial solution.
Paralleling with the above arguments, we finally give the following Remark. \begin{remark} For a simple closed curve ${\hat\Gamma}$ in ${\mathbb R}^2$ with unit length, we construct another {\bf modified Fermi coordinates} \begin{align} y\, =\, {\hat\gamma}({\hat\theta}) \, +\, {\hat t}\, \big({\tilde{\mathfrak a}}_1({\hat\theta}){\hat n}_1({\hat\theta}), \, {\tilde{\mathfrak a}}_2( {\hat\theta}){\hat n}_2({\hat\theta})\big), \label{Fermicoordinates-modified-tilde} \end{align} where ${\hat\gamma}({\hat\theta})=({\hat\gamma}_1({\hat\theta}), {\hat\gamma}_2({\hat\theta}))$ is a natural parametrization of ${\hat\Gamma}$ in ${\mathbb R}^2$, ${\hat n}({\hat\theta})=({\hat n}_1({\hat\theta}), {\hat n}_2({\hat\theta}))$ is the unit normal of ${\hat\Gamma}$, and the functions ${\tilde{\mathfrak a}}_1$ and ${\tilde{\mathfrak a}}_2$ have similar expressions as in \eqref{tildea1a2} for ${\hat\theta}\in [0, 1]$. Consider the deformation of ${\hat\Gamma}$ in the form \begin{equation} \label{deformation2} {\hat\Gamma}_{{\mathfrak t}}({\hat\theta})=\big({\hat\Gamma}_{{\mathfrak t}1}({\hat\theta}), {\hat\Gamma}_{{\mathfrak t}2}({\hat\theta})\big) \, :\, {\hat\gamma}({\hat\theta})
\, +\,
{\mathfrak t}({\hat\theta})\, \Big({\tilde{\mathfrak a}}_1({\hat\theta}){\hat n}_1({\hat\theta}), \, {\tilde{\mathfrak a}}_2({\hat\theta}){\hat n}_2({\hat\theta})\Big), \end{equation} where ${\mathfrak t}$ is a smooth function of ${\hat\theta}$ with small $L^{\infty}$-norm, and then the length functional \begin{equation} \begin{split} {\hat {\mathcal J}}({\mathfrak t})&\equiv \int_0^1 V^{\sigma}\big({\hat\Gamma}_{{\mathfrak t}}({\hat\theta})\big)
\sqrt{{\mathfrak a}_2\big({\hat\Gamma}_{{\mathfrak t}}({\hat\theta})\big)\big|{\hat\Gamma}'_{{\mathfrak t}1}({\hat\theta})\big|^2
+ {\mathfrak a}_1\big({\hat\Gamma}_{{\mathfrak t}}({\hat\theta})\big)\big|{\hat\Gamma}'_{{\mathfrak t}2}({\hat\theta})\big|^2}\, {\rm d}{\hat\theta}.
\end{split} \label{weightedlength2} \end{equation} We can do the same variational calculations to the functional ${\hat {\mathcal J}}$ and then derive the following notions. If the curvature ${\hat k}$ of ${\hat\Gamma}$ satisfies \begin{align} {\hat k}=&\frac{{\mathfrak a}_1(0, {\hat\theta}) {{\tilde{\mathfrak a}}'}_2 {\hat n}_1 {\hat n}_2 -{\mathfrak a}_2(0, {\hat\theta}){{\tilde{\mathfrak a}}'}_1 {\hat n}_2 {\hat n}_1}
{\, {\mathfrak a}_1(0, {\hat\theta}) {\tilde{\mathfrak a}}_2|{\hat n}_1|^2 +{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1 |{\hat n}_2|^2\, } \, +\,
\frac{1}{2}\frac{\partial_{\hat t} {\mathfrak a}_1(0, {\hat\theta}) |{\hat n}_1|^2 +\partial_{\hat t} {\mathfrak a}_2(0, {\hat\theta}) |{\hat n}_2|^2}
{\, \big[{\mathfrak a}_1(0, {\hat\theta}) {\tilde{\mathfrak a}}_2|{\hat n}_1|^2 +{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1 |{\hat n}_2|^2\big]\, } \nonumber\\[2mm] &\, +\, \sigma\frac{ V_{\hat t}(0, {\hat\theta})}{ V(0, {\hat\theta})}
\frac{{\mathfrak a}_1(0, {\hat\theta}) |{\hat n}_1|^2 +{\mathfrak a}_2(0, {\hat\theta}) |{\hat n}_2|^2}
{\, \big[{\mathfrak a}_1(0, {\hat\theta}) {\tilde{\mathfrak a}}_2|{\hat n}_1|^2 +{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1 |{\hat n}_2|^2\big]\, }, \quad\forall\, {\hat\theta}\in(0, 1), \label{stationary4} \end{align} then the curve ${\hat\Gamma}$ is said to be {\bf stationary}. Set the notation \begin{align}
{\widehat{\mathcal H}}_1=\frac{ V^{\sigma}\big(0, {\hat\theta}\big)}{\sqrt{\hat{\mathfrak f}_0}}\, \big[{\mathfrak a}_1(0, {\hat\theta}){\tilde{\mathfrak a}}_2^2 |{\hat n}_2|^2+{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1^2 |{\hat n}_1|^2\big] \label{mathcalH1tilde} \end{align} \begin{align} {\widehat{\mathcal H}}_2 =&\frac{ V^{\sigma}\big(0, {\hat\theta}\big)}{2\sqrt{\hat{\mathfrak f}_0}}
\Big\{2\big[{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1'|{\hat n}_1|^2
+{\mathfrak a}_1(0, {\hat\theta}){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'|{\hat n}_2|^2\big]
-2{\hat k}\big[{\mathfrak a}_1(0, {\hat\theta})|{\tilde{\mathfrak a}}_2'|^2
-{\mathfrak a}_2(0, {\hat\theta})|{\tilde{\mathfrak a}}_1'|^2 \big]{\hat n}_1{\hat n}_2 \nonumber\\[2mm] &\qquad \qquad \quad +2\big[-\partial_{\hat t}{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1 +\partial_{\hat t}{\mathfrak a}_1(0, {\hat\theta}){\tilde{\mathfrak a}}_2\big]{\hat n}_1{\hat n}_2 \Big\} \label{mathcalH2tilde} \end{align} \begin{align} {\widehat{\mathcal H}}_3 =&\frac{ V^{\sigma}\big(0, {\hat\theta}\big)}{2\sqrt{\hat{\mathfrak f}_0}} \Big\{-4{\hat k}\, \big[{\mathfrak a}_1(0, {\hat\theta}){\tilde{\mathfrak a}}_2{\tilde{\mathfrak a}}_2'-{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1{\tilde{\mathfrak a}}_1' \big]{\hat n}_1{\hat n}_2 +4\big[\partial_{\hat t}{\mathfrak a}_1(0, {\hat\theta}){\tilde{\mathfrak a}}_2' -\partial_{\hat t}{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1' \big]{\hat n}_1{\hat n}_2 \nonumber\\[2mm]
&\qquad\qquad+2\big[{\mathfrak a}_2(0, {\hat\theta})|{\tilde{\mathfrak a}}_1'|^2|{\hat n}_1|^2
+{\mathfrak a}_1(0, {\hat\theta})|{\tilde{\mathfrak a}}_2'|^2|{\hat n}_2|^2\big]
+2{\hat k}^2\big[{\mathfrak a}_2\big(0, {\hat\theta}\big){\tilde{\mathfrak a}}_1|{\hat n}_2|^2+{\mathfrak a}_1\big(0, {\hat\theta}\big){\tilde{\mathfrak a}}_2|{\hat n}_1|^2\big]
\nonumber\\[2mm]
&\qquad\qquad-4{\hat k}\big[\partial_{\hat t}{\mathfrak a}_2\big(0, {\hat\theta}\big){\tilde{\mathfrak a}}_1|{\hat n}_2|^2+\partial_{\hat t}{\mathfrak a}_1\big(0, {\hat\theta}\big){\tilde{\mathfrak a}}_2|{\hat n}_1|^2\big]
+\big[\partial_{{\hat t}{\hat t}}{\mathfrak a}_2\big(0, {\hat\theta}\big)|{\hat n}_2|^2
+\partial_{{\hat t}{\hat t}}{\mathfrak a}_1\big(0, {\hat\theta}\big)|{\hat n}_1|^2\big]\Big\} \nonumber\\[2mm] &+\left[\, \frac{\sigma V_{{\hat t}{\hat t}}(0, {\hat\theta})\, }{ V^{1-\sigma}\big(0, {\hat\theta}\big)} +
\frac{\sigma(\sigma-1)\big| V_{\hat t}(0, {\hat\theta})\big|^2\, }{ V^{2-\sigma}(0, {\hat\theta})} \, \right] \, \sqrt{\hat{\mathfrak f}_0} +\sigma \frac{ V_{\hat t}(0, {\hat\theta}) \, }{ V^{1-\sigma}\big(0, {\hat\theta}\big)} \, \frac{1}{\sqrt{\hat{\mathfrak f}_0}}\hat{\mathfrak f}_1 \,-\,
\frac{1}{4}\frac{ V^{\sigma}\big(0, {\hat\theta}\big)}{\ \big(\sqrt{\hat{\mathfrak f}_0}\, \big)^3\ }\hat{\mathfrak f}_1^2, \label{mathcalH3tilde} \end{align} where $$
\hat{\mathfrak f}_0({\hat\theta})={\mathfrak a}_1(0, {\hat\theta}) |{\hat n}_1({\hat\theta})|^2 +{\mathfrak a}_2(0, {\hat\theta}) |{\hat n}_2({\hat\theta})|^2, $$ \begin{align*} \hat{\mathfrak f}_1({\hat\theta}) =&\,2\bigg[{\mathfrak a}_1(0, {\hat\theta}) {\tilde{\mathfrak a}'}_2({\hat\theta}) {\hat n}_1({\hat\theta}) {\hat n}_2({\hat\theta}) \,-\, {\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}'}_1({\hat\theta}) {\hat n}_2({\hat\theta}) {\hat n}_1({\hat\theta})\bigg] \nonumber\\[2mm]
&\,-\, 2{\hat k}({\hat\theta})\bigg[{\mathfrak a}_1(0, {\hat\theta}) {\tilde{\mathfrak a}}_2({\hat\theta})|{\hat n}_1({\hat\theta})|^2 \,+\,
{\mathfrak a}_2(0, {\hat\theta}){\tilde{\mathfrak a}}_1({\hat\theta}) |{\hat n}_2({\hat\theta})|^2\bigg] \nonumber\\[2mm] &
\,+\,\bigg[\partial_{\hat t} {\mathfrak a}_1(0, {\hat\theta}) |{\hat n}_1({\hat\theta})|^2 +\partial_{\hat t} {\mathfrak a}_2(0, {\hat\theta}) |{\hat n}_2({\hat\theta})|^2\bigg]. \end{align*} For a simple closed curve ${\hat\Gamma}$ satisfying the stationary condition, if the boundary problem
\begin{equation}
\label{nondegeneracy4}
\begin{split} {\widehat{\mathcal H}}_1f'' \, +\,
{\widehat{\mathcal H}}_1'f' \, +\, \big({\widehat{\mathcal H}}_2'-{\widehat{\mathcal H}}_3\big) f&=0 \quad \mbox{in}\ (0, 1), \\[2mm] f'(1)=f'(0), \qquad f(1)&=f(0), \end{split} \end{equation} has only the trivial solution, we call ${\hat\Gamma}$ a {\bf non-degenerate} stationary curve. \qed \end{remark}
\section{Outline of the proof}\label{thegluingprocedure}\label{section3} \setcounter{equation}{0}
Recall that $w$ is the solution to \eqref{wsolution}. In fact, $w$ is an even function defined in the form \begin{equation} w(x)=C_p\big[\, e^{\frac{(p-1)x}{2}}\, +\, e^{-\frac{(p-1)x}{2}}\, \big]^{-\frac{2}{p-1}}, \quad \forall\, x\in {\mathbb R}. \end{equation} We consider the associated linearized eigenvalue problem, \begin{equation} \label{eigenvalue} h{''}-h+pw^{p-1}h\, =\, \lambda h\quad\mbox{in } {\mathbb R}, \quad h(\pm \infty)\, =\, 0. \end{equation} It is well known that this equation possesses a unique positive eigenvalue $\lambda_0$, with associated eigenfunction $Z$ (even and positive) in $H^1({\mathbb R})$, which can be normalized so that $\int_{\mathbb R} Z^2=1$. In fact, a simple computation shows that \begin{align} \label{lambda0} \lambda_0\, =\, \frac 14(p-1)(p+3), \quad Z\, =\, \frac {1}{\sqrt{\int_{\mathbb R} w^{p+1}}\, }w^{\frac {p+1}{2}}. \end{align}
In this section, the strategy to prove Theorem \ref{theorem 1.1} will be provided step by step. \subsection{The gluing procedure}
Recall that $\delta_0>0$ is a small constant in (\ref{Fermicoordinates-modified}) and define a cut-off function $\eta_{3\delta}^{\varepsilon}(s)=\eta_{3\delta}(\varepsilon |s|)$ where $\eta_{\delta}(t)$ is also a smooth cut-off function defined as $$ \eta_{\delta}(t)=1, \, \forall\, 0\leq t\leq \delta \quad\mbox{and}\quad \eta_{\delta}(t)=0, \, \forall\, t>2\delta, $$ for a fixed number $\delta<\delta_0/100$.
For any given approximate solution ${\mathbf w}$ (to be chosen later, cf. (\ref{globalapproximation})) and a perturbation term $ \tilde{\phi}({\tilde y})\, =\, \eta_{3\delta}^{\varepsilon}(s){\check\phi}({\tilde y})\, +\, {\check\psi}({\tilde y}) $ on $\Omega_\varepsilon$, the function $u({\tilde y})=\mathbf{w}({\tilde y})+\tilde{\phi}({\tilde y})$ satisfies (\ref{problemafterscaling}) if $({\check\phi}, {\check\psi})$ satisfies the following coupled system: \begin{align}\label{equivalent system-1} \eta_{3\delta}^{\varepsilon}\, {\mathbf L}({\check\phi}) \, &=\, \eta_{\delta}^{\varepsilon}\, \big[-{\mathbf N}(\eta_{3\delta}^{\varepsilon}\, {\check\phi}+{\check\psi})
\,-\, {\mathbf E}
\,-\, p\, \mathbf{w}^{p-1}{\check\psi}\big]\quad\mbox{in } \Omega_\varepsilon, \end{align} \begin{align} \label{on the boundary-1} \eta_{3\delta}^{\varepsilon}\nabla_{{\mathfrak a}(\varepsilon {\tilde y})} {\check\phi}\cdot \nu_\varepsilon \, +\, \eta_{\delta}^{\varepsilon}\nabla_{{\mathfrak a}(\varepsilon {\tilde y})} \mathbf{w}\cdot \nu_\varepsilon\, =\, 0 \quad\mbox{on } \partial\, \Omega_\varepsilon, \end{align} and \begin{align} \label{equivalent system-2} &\sum_{i=1}^2\partial_{{\tilde y}_i}\big({\mathfrak a}_i(\varepsilon{\tilde y}){\check\psi}_{{\tilde y}_i}\big) \,-\, V(\varepsilon \tilde y) \, {\check\psi} \, +\, (1-\eta_{\delta}^{\varepsilon})\, p\, \mathbf{w}^{p-1}\, {\check\psi} \nonumber\\[2mm] &\, =\,-\, (1-\eta_{\delta}^{\varepsilon})\, {\mathbf E} \,-\, \sum_{i=1}^2{\mathfrak a}_i(\varepsilon{\tilde y}){\check\phi}_{{\tilde y}_i}\partial_{{\tilde y}_i}\big(\eta_{3\delta}^{\varepsilon}(s)\big) \,-\, \sum_{i=1}^2\partial_{{\tilde y}_i}\Big[{\mathfrak a}_i(\varepsilon{\tilde y}){\check\phi}\partial_{{\tilde y}_i}\big( \eta_{3\delta}^{\varepsilon}(s) \big) \Big] \nonumber\\[2mm] &\quad\, \,-\, (1-\eta_{\delta}^{\varepsilon})\, {\mathbf N}\, (\eta_{3\delta}^{\varepsilon}\, {\check\phi}+{\check\psi}) \quad\mbox{in } \Omega_\varepsilon, \end{align} \begin{align}\label{on the boundary-2} \nabla_{{\mathfrak a}(\varepsilon {\tilde y})} {\check\psi}\cdot \nu_\varepsilon \, +\, \big(1-\eta_{\delta}^{\varepsilon}\big)\nabla_{{\mathfrak a}(\varepsilon {\tilde y})} \mathbf{w}\cdot \nu_\varepsilon \, +\, \varepsilon \nabla_{{\mathfrak a}(\varepsilon {\tilde y})} \eta_{3\delta}^{\varepsilon}\cdot \nu_\varepsilon {\check\phi}\, =\, 0 \quad\mbox{on } \partial\, \Omega_\varepsilon, \qquad \end{align} where \begin{equation}\label{globalerror} {\mathbf E}\, =\, \sum_{i=1}^2\partial_{{\tilde y}_i}\big({\mathfrak a}_i(\varepsilon{\tilde y}){\mathbf w}_{{\tilde y}_i}\big) \,-\, V(\varepsilon \tilde y) {\mathbf w} + {\mathbf w}^p, \end{equation} \begin{equation}
{\mathbf L}({\check\phi})\, =\, \sum_{i=1}^2\partial_{{\tilde y}_i}\big({\mathfrak a}_i(\varepsilon{\tilde y}){\check\phi}_{{\tilde y}_i}\big) \,-\, V(\varepsilon \tilde{y}){\check\phi} \, +\, p\, \mathbf{w}^{p-1}{\check\phi}, \qquad {\mathbf N}(\tilde{\phi})\, =\, (\mathbf{w}+\tilde{\phi})^p \,-\, \mathbf{w}^p \,-\, p\, \mathbf{w}^{p-1}\tilde{\phi}. \end{equation}
Assume now that ${\check\phi}$ satisfies the following decay property \begin{align} \label{decay property}
\big |\nabla {\check\phi}(\tilde{y})\big|+\big|{\check\phi}(\tilde{y})\big|\, \leq\, Ce^{-\rho/\varepsilon} \quad\mbox{if } \mathrm{dist}({\tilde y}, \Gamma_\varepsilon)>\delta/\varepsilon, \end{align} for a certain constant $\rho>0$, and also that \begin{align}\label{constraint condition} \mathbf{w}({\tilde y}) \ \mbox{is exponentially small if } \mathrm{dist}({\tilde y}, \Gamma_\varepsilon)>\delta/\varepsilon. \end{align}
Since ${\mathbf N}$ is power-like with power greater than one, a direct application of Contraction Mapping Principle yields that (\ref{equivalent system-2})-(\ref{on the boundary-2}) has a unique (small) solution ${\check\psi}={\check\psi}({\check\phi})$ with \begin{align} \label{contraction-1}
\|{\check\psi}({\check\phi})\|_{L^{\infty}}\, \leq\, C\varepsilon\big[\, \|{\check\phi}\|_{L^{\infty}(|s|>\delta/\varepsilon)}+
\|\nabla {\check\phi}\|_{L^{\infty}(|s|>\delta/\varepsilon)}+e^{-\delta/\varepsilon}\, \big], \end{align}
where $|s|>\delta/\varepsilon$ denotes the complement in $\Omega_{\varepsilon}$ of $\delta/\varepsilon$-neighborhood of $\Gamma_\varepsilon$. Moreover, the nonlinear operator ${\check\psi}$ satisfies a Lipschitz condition of the form \begin{align} \label{psi-lip}
\|{\check\psi}({\check\phi}_1)-{\check\psi}({\check\phi}_2)\|_{L^{\infty}}\, \leq\, C\varepsilon \big[\, \|{\check\phi}_1-{\check\phi}_2\|_{L^{\infty}(|s|>\delta/\varepsilon)} \, +\,
\|\nabla {\check\phi}_1-\nabla {\check\phi}_2\|_{L^{\infty}(|s|>\delta/\varepsilon)}\, \big]. \end{align}
The key observation is that, after solving (\ref{equivalent system-2})-(\ref{on the boundary-2}), we can concern (\ref{equivalent system-1})-(\ref{on the boundary-1}) as a local nonlinear problem involving ${\check\psi}\, =\, {\check\psi}({\check\phi})$, which can be solved in local coordinates in the sense that we can decompose the interaction among the boundary, the concentration set and the terms ${\mathfrak a}$, $ V$, and then construct a good approximate solution and also derive the resolution theory of the nonlinear problem by delicate analysis. This whole procedure is called a {\bf gluing technique} in \cite{delPKowWei2007}.
\subsection{Local formulation of the problem}\label{section3.2} As described in the above, the next step is to consider (\ref{equivalent system-1})-(\ref{on the boundary-1}) in the neighbourhood of $\Gamma_\varepsilon$ so that by the relation ${\tilde y}=y/\varepsilon$ in (\ref{rescaling}) close to $\Gamma_\varepsilon$, the variables $y$ can be represented by the modified Fermi coordinates, say $(t, \theta)$ in (\ref{Fermicoordinates-modified}), which have been set up in Section \ref{section2}.
\subsubsection{Local forms of the problem}\label{section22}
By recalling the local coordinates $(t, \theta)$ in (\ref{Fermicoordinates-modified}), we can also define the local rescaling, \begin{equation}\label{coordinatessz} (t, \theta)=\varepsilon (s, z), \end{equation} and then use the results in \eqref{laplacelocal}, \eqref{boundaryoriginal0} and \eqref{boundaryoriginal1} to give local expressions of the problem. The equation in (\ref{equivalent system-1})
can be locally recast in $(s, z)$ coordinate system as follows \begin{align} \label{s-z-laplace} \eta_{3\delta}^{\varepsilon}\, \check{L}({\check\phi})=\eta_{\delta}^{\varepsilon}\, \big[-{\mathbf N}(\eta_{3\delta}^{\varepsilon}\, {\check\phi}+{\check\psi}) -{\mathbf E}-p\, \mathbf{w}^{p-1}{\check\psi}\big],
\quad \forall\, (s, z)\in (-\delta_0/\varepsilon, \delta_0/\varepsilon)\times(0, 1/\varepsilon). \end{align}
\noindent $\bullet$ The linear operator is \begin{align} \label{tilde L check phi} \check{L}({\check\phi})\, =\, &h_1(\varepsilon z){\check\phi}_{ss}+h_2(\varepsilon z){\check\phi}_{zz}+\varepsilon h_3(\varepsilon z){\check\phi}_{s}+\varepsilon h_4(\varepsilon z) {\check\phi}_{z} \nonumber\\[2mm] &+\hat{B}_{1}({\check\phi})+\hat{B}_0({\check\phi})- V(\varepsilon s, \varepsilon z)\, {\check\phi}+p{\mathbf w}^{p-1}{\check\phi}, \end{align} where \begin{equation*} \hat{B}_1(\cdot)= \varepsilon^2 sh_5(\varepsilon z)\partial_s +\varepsilon sh_6(\varepsilon z) \partial^2_{sz} +\varepsilon^2 s^2h_7(\varepsilon z) \partial^2_{ss} +\varepsilon sh_8(\varepsilon z)\partial^2_{ss}, \end{equation*} and \begin{equation}\label{tilteB00} \hat{B}_0(\cdot)= \varepsilon sh_9(\varepsilon s, \varepsilon z)\, \partial^2_{zz} +\varepsilon^2 s^2h_{10}(\varepsilon s, \varepsilon z)\, \partial^2_{zz} +\varepsilon^2 s^2h_{11}(\varepsilon s, \varepsilon z)\, \partial^2_{sz} +\varepsilon^2 sh_{12}(\varepsilon s, \varepsilon z)\partial_z. \end{equation}
\noindent $\bullet$ The error is then expressed in the form \begin{align} {\mathbf E}\, =\, &h_1(\varepsilon z){\mathbf w}_{ss}+h_2(\varepsilon z){\mathbf w}_{zz}+\varepsilon h_3(\varepsilon z){\mathbf w}_{s}+\varepsilon h_4(\varepsilon z) {\mathbf w}_{z} \nonumber\\[2mm] &+\hat{B}_{1}({\mathbf w})+\hat{B}_0({\mathbf w})- V(\varepsilon s, \varepsilon z)\, {\mathbf w}+{\mathbf w}^{p}. \end{align}
On the other hand, the boundary condition in (\ref{on the boundary-1}) can also be expressed precisely in local coordinates. If $z=0$, \begin{equation} \label{boundarycondition11} \eta_{3\delta}^{\varepsilon}\, \mathbb{D}_0({\check\phi})=-\eta^\varepsilon_\delta\, {\mathbf G}_0\qquad{\rm with}\qquad {\mathbf G}_0=\mathbb{D}_0({\mathbf w}). \end{equation} And, at $z=1/\varepsilon$ there holds \begin{equation} \label{boundarycondition22} \eta_{3\delta}^{\varepsilon}\, \mathbb{D}_1({\check\phi})=-\eta^\varepsilon_\delta\, {\mathbf G}_1\qquad{\rm with}\qquad {\mathbf G}_1=\mathbb{D}_1({\mathbf w}). \end{equation} The operators on the boundary are \begin{equation} \label{boundarycondition1} \mathbb{D}_0\, =\, {\mathfrak b}_1\partial_z +{\mathfrak b}_2\varepsilon\, s\, \partial_s +{\mathfrak b}_3\varepsilon^2 s^2\, \partial_s +{\mathfrak b}_4\varepsilon s\partial_z +{\mathfrak b}_5\varepsilon^2 s^2 \partial_z +{\hat D}_0^0(\cdot), \end{equation}
and \begin{equation} \label{boundarycondition2} \mathbb{D}_1\, =\, {\mathfrak b}_6 \, \partial_z +{\mathfrak b}_7\varepsilon s\, \partial_s +{\mathfrak b}_8\varepsilon^2 s^2\, \partial_s +{\mathfrak b}_9\varepsilon s \, \partial_z +{\mathfrak b}_{10}\varepsilon^2 s^2 \, \partial_z+{\hat D}_0^1(\cdot), \end{equation} where $\hat{D}^0_0\big(\cdot(s, z)\big)=\varepsilon\, \bar{D}^0_0\big(\cdot(t, \theta)\big)$ and $\hat{D}^1_0\big(\cdot(s, z)\big)=\varepsilon\, \bar{D}^1_0\big(\cdot(t, \theta)\big)$.
\subsubsection{Further changing of variables}\label{further change} A further change of variables in equation (\ref{s-z-laplace}) will be chosen in the forms \begin{equation}\label{vdefine} {\check\phi}(s, z)\, =\, \alpha(\varepsilon z)\phi(x, z), \quad\mbox{with}\quad x\, =\, \beta(\varepsilon z)s, \end{equation} where \begin{equation} \label{alpha-beta} \alpha(\theta)\, =\, V(0, \theta)^{\frac {1}{p-1}}, \qquad \beta(\theta)\, =\, \sqrt{\frac{ V(0, \theta)}{h_1(\theta)}}
=\sqrt{\frac{ V(0, \theta)\big(a_1(0, \theta)|n_1(\theta)|^2+a_2(0, \theta)|n_2(\theta)|^2\big)}
{|a_1(0, \theta)|^2+|a_2(0, \theta)|^2}}. \end{equation} It is also convenient to expand \begin{equation} \label{Vexpan}
V(\varepsilon s, \varepsilon z)\, =\, V(0, \varepsilon z)+ V_t(0, \varepsilon z)\cdot \varepsilon s+\frac 12 V_{tt}(0, \varepsilon z)\cdot \varepsilon^2s^2+a_6(\varepsilon s, \varepsilon z)\, \varepsilon^3\, s^3, \end{equation} for a smooth function $a_6(t, \theta)$. In order to express (\ref{tilde L check phi}) and \eqref{boundarycondition11}-(\ref{boundarycondition22}) in terms of these new coordinates, the following identities will be prepared
\begin{align*} {\check\phi}_s=&\, \alpha\beta \phi_x,
\qquad {\check\phi}_{ss}=\alpha\beta^2 \phi_{xx},
\qquad
{\check\phi}_z=\, \varepsilon \alpha' \phi
+ \varepsilon \alpha \frac{\beta'}{\beta}x\phi_x+\alpha \phi_z,
\\[2mm] {\check\phi}_{sz}\, =\, &\, \varepsilon \alpha'\beta \phi_{x}+ \varepsilon\alpha\beta'\phi_{x} +\varepsilon\alpha\beta'x\phi_{xx}+\alpha\beta \phi_{xz}, \end{align*} and \begin{align*} {\check\phi}_{zz}
\, =\, &\varepsilon^2 \alpha''\phi+2\varepsilon^2\alpha'\frac{\beta'}{\beta}x\phi_{x}+\varepsilon^2\alpha\frac{\beta''}{\beta}x\phi_{x} \\[2mm] &+\varepsilon^2\alpha\Big( \frac{\beta'}{\beta} \Big)^2x^2\phi_{xx} +2\varepsilon\alpha\frac{\beta'}{\beta}x\phi_{xz}+2\varepsilon\alpha'\phi_{z}+\alpha \phi_{zz}. \end{align*}
We can deduce that \begin{align} \frac{1}{\alpha\beta^2}{\check L}({\check\phi})
=\frac{h_2}{\beta^2}\phi_{zz}+h_1 \phi_{xx}-h_1 \phi+\beta^{-2}p\mathbf{w}^{p-1}\phi+B_2(\phi)+B_3(\phi)\equiv L(\phi), \end{align} where $B_3(\phi)$ is a linear differential operator defined by \begin{align} B_3(\phi)\, =\, & \varepsilon \frac{h_3}{\beta} \phi_x +\varepsilon \frac{h_4}{\alpha \beta^2}\big[\, \varepsilon \alpha' \phi
+ \varepsilon \alpha \frac{\beta'}{\beta}x\phi_x+\alpha \phi_z\big] \nonumber\\[2mm]
&+\frac{h_2}{\beta^2}\left[\, \varepsilon ^2\Big|\frac {\beta'}{\beta}\Big|^2x^2\phi_{xx}+2\varepsilon \frac {\beta'}{\beta}x \phi_{xz} +\varepsilon^2\frac {\beta{''}}{\beta}x\phi_x\, \right] \nonumber\\[2mm] &+\varepsilon^2 \frac{h_2\alpha''}{\alpha \beta^2}\phi +2\varepsilon^2 \frac{h_2\alpha'}{\alpha \beta^2}\frac {\beta'}{\beta}x \phi_x +2\varepsilon \frac{h_2\alpha'}{\alpha \beta^2}\phi_z +\varepsilon^2\frac{h_5}{\beta^2}x\phi_x \nonumber\\[2mm] &+\varepsilon \frac{h_6}{\alpha \beta^2}\frac{x}{\beta}\big[\varepsilon \alpha'\beta \phi_{x}+ \varepsilon\alpha\beta'\phi_{x}+\varepsilon\alpha\beta'x\phi_{xx}+\alpha\beta \phi_{xz}\big] +\varepsilon^2h_7\Big(\frac{x}{\beta}\Big)^2\phi_{xx} \nonumber\\[2mm] &+ \varepsilon h_8\frac{x}{\beta}\phi_{xx} -\left[\, \varepsilon \frac{V_t(0, \varepsilon z)}{\beta^2}\frac{x}{\beta} +\frac{\varepsilon^2}{2} V_{tt}(0, \varepsilon z)\beta^{-2}\Big(\frac{x}{\beta}\Big)^2\right]\phi. \label{B3v} \end{align} Here \begin{equation} \label{B2v} B_2(\phi)\, =\, \frac{1}{\alpha\beta^2}\, \hat{B}_0({\check\phi})+\frac{1}{\alpha\beta^2}\, a_6(\varepsilon s, \varepsilon z)\, \varepsilon^3 \, s^3\, \alpha\, \phi, \end{equation} and $\hat{B}_0({\check\phi})$ is the operator in (\ref{tilteB00}) where derivatives are expressed in terms of $s$ and $z$ through (\ref{vdefine}), $a_6$ is given by (\ref{Vexpan}), and $s$ is replaced by $\beta^{-1}x$.
In the coordinates $(x, z)$, the boundary conditions in (\ref{boundarycondition11})-(\ref{boundarycondition22}) can be recast in the sequel. For $z=0$, \begin{equation} \label{boundary-1-x-z} \eta_{3\delta}^{\varepsilon}\, \big[D_3^0(\phi)+{\mathfrak b}_1\phi_z+D_2^0(\phi)\big]\, =\, -\frac{1}{\alpha}\, \eta^\varepsilon_\delta\, {\mathbf G}_0, \end{equation} where \begin{align} \label{D30} D_3^0(\phi)\, =\, &\varepsilon \Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'}{\beta} \Big]x\phi_{x} +\varepsilon {\mathfrak b}_1\frac{\alpha'}{\alpha}\phi +\varepsilon {\mathfrak b}_4 \frac{x}{\beta}\phi_z \nonumber\\[2mm] &+\varepsilon^2\Big[{\mathfrak b}_3\Big(\frac x \beta \Big)^2\beta+{\mathfrak b}_4\Big(\frac{x}{\beta}\Big)\Big(\frac{\beta'}{\beta}x\Big)\Big]\phi_x +\varepsilon^2{\mathfrak b}_4\frac{\alpha'}{\alpha}\Big(\frac{x}{\beta}\Big)\phi +\varepsilon^2{\mathfrak b}_5\Big(\frac{x}{\beta}\Big)^2\phi_z, \end{align} and \begin{equation}\label{D20} D_2^0(\phi)\, =\, \frac{1}{\alpha}{\hat D}_0^0({\check\phi})+\varepsilon^3\, {\mathfrak b}_5\, \Big(\frac{x}{\beta}\Big)^2\frac{\alpha'}{\alpha}\phi +\varepsilon^3\, {\mathfrak b}_5\, \beta'\, \Big(\frac{x}{\beta}\Big)^3\phi_x. \end{equation} Similarly, for $z=1/\varepsilon$, we have \begin{equation} \label{boundary-2-x-z} \eta_{3\delta}^{\varepsilon}\, \big[D_3^1(\phi)+{\mathfrak b}_6\phi_z+D_2^1(\phi)\big]\, =\, -\frac{1}{\alpha}\, \eta^\varepsilon_\delta\, {\mathbf G}_1, \end{equation} where \begin{align} \label{D31} D_3^1(\phi)\, =\, &\varepsilon \Big[{\mathfrak b}_7+{\mathfrak b}_6\frac{\beta'}{\beta} \Big]x\phi_{x} +\varepsilon {\mathfrak b}_6\frac{\alpha'}{\alpha}\phi +\varepsilon {\mathfrak b}_9 \frac{x}{\beta}\phi_z \nonumber\\[2mm] &+\varepsilon^2\Big[{\mathfrak b}_8\Big(\frac x \beta \Big)^2\beta+{\mathfrak b}_9\Big(\frac{x}{\beta}\Big)\Big(\frac{\beta'}{\beta}x\Big)\Big]\phi_x +\varepsilon^2{\mathfrak b}_9\frac{\alpha'}{\alpha}\Big(\frac{x}{\beta}\Big)\phi +\varepsilon^2{\mathfrak b}_{10}\Big(\frac{x}{\beta}\Big)^2\phi_z, \end{align} and \begin{equation}\label{D21} D_2^1(\phi)\, =\, \frac{1}{\alpha}{\hat D}_0^1({\check\phi})+\varepsilon^3\, {\mathfrak b}_{10}\Big(\frac{x}{\beta}\Big)^2\frac{\alpha'}{\alpha}\phi +\varepsilon^3\, {\mathfrak b}_{10}\, \beta'\, \Big(\frac{x}{\beta}\Big)^3\, \phi_x. \end{equation}
As a conclusion, in local coordinates $(x, z)$, (\ref{equivalent system-1})-(\ref{on the boundary-1}) become \begin{equation}\label{localproblem1}
\eta_{3\delta}^{\varepsilon}\, L(\phi)\, =\, (\alpha \beta^{-2})^{-1}\eta_{\delta}^{\varepsilon}\, \Big[-{\mathbf N}(\eta_{3\delta}^{\varepsilon}\, {\check\phi}+{\check\psi}) -{\mathbf E}-p\, \mathbf{w}^{p-1}{\check\psi}\Big], \end{equation} \begin{equation}\label{localproblem2} \eta_{3\delta}^{\varepsilon}\, \Big[D_3^0(\phi) +{\mathfrak b}_1\phi_z+D_2^0(\phi)\Big]\, =\, -\frac{1}{\alpha}\, \eta^\varepsilon_\delta\, {\mathbf G}_0, \end{equation} \begin{equation}\label{localproblem3} \eta_{3\delta}^{\varepsilon}\, \Big[D_3^1(\phi) +{\mathfrak b}_6\phi_z+D_2^1(\phi)\Big]\, =\, -\frac{1}{\alpha}\, \eta^\varepsilon_\delta\, {\mathbf G}_1. \end{equation}
\subsection{The projected problem}\label{section3.3} For the convenience of presentation, we pause here to give some notation.
\noindent{\textbf{ Notation 3:}} {\it
Observe that all functions involved in (\ref{localproblem1})-(\ref{localproblem3}) are expressed in $(x, z)$-variables, and the natural domain for those variables can be extended to the infinite strip \begin{align} \begin{aligned}\label{domainS} {\mathcal S}\, =\, &\Big\{\, (x, z)\, :\, -\infty<x<\infty, \, 0<z<1/\varepsilon\, \Big\}, \\[2mm] \partial_0{\mathcal S}\, =\, \Big\{\, (x, z)\, :\, -\infty<x<\infty, \, & z=0\, \Big\}, \qquad \partial_1{\mathcal S}\, =\, \Big\{\, (x, z)\, :\, -\infty<x<\infty, \, z=1/\varepsilon\, \Big\}. \end{aligned} \end{align} Accordingly, we define \begin{align} \begin{aligned}\label{domainS1} {\mathcal {\hat{S}}}\, =\, &\Big\{\, (x, \tilde{z})\, :\, -\infty<x<\infty, \, 0<\tilde{z}<{\ell}/\varepsilon\, \Big\}, \\[2mm] \partial_0{\mathcal {\hat{S}}}\, =\, \Big\{\, (x, \tilde{z})\, :\, -\infty<x<\infty, \, & \tilde{z}=0\, \Big\}, \quad \partial_1{\mathcal {\hat{S}}}\, =\, \Big\{\, (x, \tilde{z})\, :\, -\infty<x<\infty, \, \tilde{z}={\ell}/\varepsilon\, \Big\}, \end{aligned} \end{align} where ${\ell}$ is a constant defined as \begin{equation} \label{ell} \ell\equiv \int_0^1 \mathcal{Q}(\theta){\rm d}\theta, \quad\mbox{with }\, \mathcal{Q}(\theta)=\sqrt{\frac{V(0, \theta)}{h_2(\theta)}}
=\sqrt{ \frac{V(0, \theta)\Big({\mathfrak a}_1(0, \theta)|n_1(\theta)|^2+{\mathfrak a}_2(0, \theta)|n_2(\theta)|^2\Big)} {{\mathfrak a}_1(0, \theta){\mathfrak a}_2(0, \theta)}}. \end{equation}
In all what follows, we will introduce some parameters $\{{f}_j\}_{j=1}^N$ and $\{e_j\}_{j=1}^N$ and assume the validity of the following constraints, for $j=1, \cdots, N$,
\begin{equation} \label{constraints of f}
\|f_j\|_{H^2(0, 1)}<C|\ln\varepsilon|^2, \qquad
\, f_{j+1}(\theta)-f_j(\theta)>2|\ln\varepsilon|-4\ln|\ln\varepsilon|, \end{equation} \begin{equation} \label{enorm}
\|e_j\|_{**}\, \equiv\, \|e_j\|_{L^{\infty}(0, 1)}+\varepsilon\|e_j'\|_{L^2(0, 1)}+\varepsilon^2\|e''_j\|_{L^2(0, 1)}\, \leq\, \varepsilon^{\frac{1}{2}}, \end{equation} where we have used the convention $f_0=-\infty$ and $f_{N+1}=\infty$. Set \begin{align} \textbf{f}&=(f_1, \cdots, f_N), \quad f_j \mbox{'s satisfy bounds } (\ref{constraints of f}), \nonumber \\[2mm] \textbf{e}&=(e_1, \cdots, e_N), \quad e_j \mbox{'s satisfy bounds } (\ref{enorm}), \nonumber \\[2mm] \textbf{c}&=(c_1, \cdots, c_N), \quad \textbf{d}=(d_1, \cdots, d_N), \quad c_j \mbox{'s }\, \text{ and } \, d_j \mbox{'s are in } {L^2{(0, 1)}}, \nonumber \\[2mm] \textbf{l}_0&=(l_{0, 1}, \cdots, l_{0, N}), \quad \textbf{l}_1=(l_{1, 1}, \cdots, l_{1, N}), \quad l_{0, j}\mbox{'s }\, \text{ and } \, l_{1, j}\mbox{'s } \mbox { are constants}, \nonumber \\[2mm] \textbf{m}_0&=(m_{0, 1}, \cdots, m_{0, N}), \quad \textbf{m}_1=(m_{1, 1}, \cdots, m_{1, N}), \quad m_{0, j}\mbox{'s }\, \text{ and } \, m_{1, j}\mbox{'s } \mbox { are constants}, \nonumber \end{align} and \begin{equation} \label{mathcalF} \mathcal F\, =\, \big\{\, (\textbf{f}, \textbf{e})\, :\, \{f_j\}_{j=1}^N \mbox{ and } \{e_j\}_{j=1}^N \mbox{ satisfy } (\ref{constraints of f}) \mbox{ and } (\ref{enorm}) \mbox{ respectively}\, \big\}. \end{equation} \qed }
One of the left job is to find the local forms of the approximate solution $\mathbf{w}$ with the constraint (\ref{constraint condition}) and also of the error $\mathbf{E}$. We recall the transformation in (\ref{vdefine})-(\ref{alpha-beta}), and then define the local form of the approximate solution ${\mathbf w}$ by the relation \begin{equation} \label{vdefine1} \eta^{\varepsilon}_{10\delta}(s)\, {\mathbf w}\, =\, \eta^{\varepsilon}_{3\delta}(s)\, \alpha(\varepsilon z)\, v(x, z) \quad\mbox{with}\quad x\, =\, \beta(\varepsilon z)s. \end{equation} The error $\mathbf{E}$ can be locally recast in $(x, z)$ coordinate system by the relation
\begin{equation}\label{locallyE}
\frac{1}{\alpha\beta^2}\, \eta_{\delta}^{\varepsilon}(s)\, \mathbf{E}
=\eta_{\delta}^{\varepsilon}(s)\, \mathcal{E},
\end{equation}
where
\begin{equation}\label{sv}
\mathcal{E}\, =\, S(v) \quad {\text{with}}\quad
S(v)\, =\, \frac{h_2}{\beta^2}v_{zz}+h_1\big[v_{xx}-v+v^{p}\big]+B_2(v)+B_3(v),
\end{equation}
with the operator $B_3$ and $B_2$ defined in (\ref{B3v})-(\ref{B2v}).
In the coordinates $(x, z)$, the boundary errors can be recast as follows. For $z=0$, \begin{equation}\label{g0} \frac{1}{\alpha}\eta_\delta^\varepsilon(s)\, {\mathbf G}_0\, =\, \eta_\delta^\varepsilon(s)\, g_0 \quad\mbox{with}\quad g_0=D^0_3(v)+{\mathfrak b}_1v_z+D_2^0(v), \end{equation} and also for $z=1/\varepsilon$, \begin{align}\label{g1} \frac{1}{\alpha}\eta_\delta^\varepsilon(s)\, {\mathbf G}_1\, =\, \eta_\delta^\varepsilon(s)\, g_1 \quad\mbox{with}\quad g_1=D^1_3(v)+{\mathfrak b}_6v_z+D_2^1(v). \end{align} It is of importance that (\ref{locallyE}), (\ref{g0}) and (\ref{g1}) hold only in a small neighbourhood of $\Gamma_\varepsilon$. Hence we will consider $v$, $S(v)$ as functions of the variables $x$ and $z$ on $\mathcal S$, and also $g_0, ~g_1$ on $\partial_0{\mathcal S}$ and $\partial_1{\mathcal S}$ in the sequel. We will find $v=v_4$ in (\ref{basic approximate}) step by step in Section 4, so that $\mathbf{w}$ will be given in (\ref{globalapproximation}) with the property in (\ref{constraint condition}). In fact, to deal with the resonance, in addition to the parameters $\mathbf{f}$ and $\mathbf{h}$, we shall add one more parameter, say $\mathbf{e}$, in the approximate solution $v_4$. The exact forms of the error terms $S(v_4)$, $g_0$ and $g_1$ will be given in (\ref{E1-d}) and (\ref{E11}).
To make suitable extension of (\ref{localproblem1})-(\ref{localproblem3}), we define an operator on the whole strip $\mathcal S$ in the form \begin{align} \label{nonlocal-1} \begin{split}
\mathcal{L}(\phi)\, \equiv\, &\, \frac{h_2}{\beta^2}\, \phi_{zz}+h_1\big[\phi_{xx}-\phi+pw^{p-1}\phi\big]+\chi(\varepsilon|x|)\, B_3(\phi) \,-\, h_1(1-\eta_{3\delta}^{\varepsilon})pv_4^{p-1}\, \phi\, \quad\mbox{in } \mathcal S, \end{split} \end{align} and also the operators \begin{align}
{\mathcal D}_1(\phi)\, =\, \chi(\varepsilon|x|)\, D_3^1(\phi)+{\mathfrak b}_6\phi_z+\, \chi(\varepsilon|x|)\, D_2^1(\phi) \quad{\rm on}\ \partial_1\mathcal S, \label{D1mathcal} \\[2mm]
{\mathcal D}_0(\phi)\, =\, \chi(\varepsilon|x|)\, D_3^0(\phi)+{\mathfrak b}_1\phi_z+\, \chi(\varepsilon|x|)\, D_2^0(\phi) \quad{\rm on}\ \partial_0\mathcal S, \label{D0mathcal} \end{align} where $\chi(r)$ is a smooth cut-off function which equals $1$ for $0\leq r<10\delta$ that vanishes identically for $r>20\delta$.
Rather than solving problem (\ref{localproblem1})-(\ref{localproblem3}) directly, we deal with the following projected problem: for each pair of parameters $\mathbf{f}$ and $\mathbf{e}$ in $\mathcal F$, finding functions $\phi\in H^2(\mathcal S), ~\mathbf{c}, ~\mathbf{d} \in L^2(0, 1)$ and constants $\mathbf{l}_0, ~\mathbf{l}_1, $~$~\mathbf{m}_0, ~\mathbf{m}_1$ such that \begin{equation} \label{system-1} \mathcal{L}(\phi)\, =\, \eta_{\delta}^{\varepsilon}(s)\, \big[
-\mathcal{E}-{\mathcal N}({\phi})\big]+\sum_{j=1}^Nc_j(\varepsilon z)\chi(\varepsilon|x|) w_{j, x}+\sum_{j=1}^Nd_j(\varepsilon z)\chi(\varepsilon|x|) Z_j\quad\mbox{in } \mathcal S, \end{equation} \begin{equation} \label{system-2}
{\mathcal D}_1(\phi)\, =\, \eta_{\delta}^{\varepsilon}(s)g_1+\sum_{j=1}^Nl_{1, j}\chi(\varepsilon|x|) w_{j, x}+\sum_{j=1}^Nm_{1, j}\chi(\varepsilon|x|) Z_j\quad\mbox{on } \partial_1 \mathcal S, \end{equation} \begin{equation} \label{system-3}
{\mathcal D}_0(\phi)\, =\, \eta_{\delta}^{\varepsilon}(s) g_0+\sum_{j=1}^Nl_{0, j}\chi(\varepsilon|x|) w_{j, x}+\sum_{j=1}^Nm_{0, j}\chi(\varepsilon|x|) Z_j\quad\mbox{on } \partial_0 \mathcal S, \end{equation} \begin{equation} \label{system-4} \int_{{\mathbb R}}\phi(x, z)w_{j, x}\, {\rm d}x \, =\, \int_{{\mathbb R}}\phi(x, z)Z_j\, {\rm d}x \, =\, 0, \quad 0<z<\frac {1}{\varepsilon}, \quad \forall\, j=1, \cdots, N, \end{equation} where we have denoted \begin{equation} {\mathcal N}(\phi)\, =\, \big[v_4+\phi+\psi(\phi)\big]^p-v_4^{p}-pv_4^{p-1}\phi. \end{equation} The functions $w_j, \, Z_j, \, j=1, \cdots, N$ are given in (\ref{wjZj}). This problem has a unique solution $\phi$ such that ${\check\phi}$ satisfies (\ref{decay property}).
\begin{proposition}
\label{prop}There is a number ${\tilde\tau}>0$ such that for all $\varepsilon$ small enough and all parameters $(\mathbf{f}, \mathbf{e})$ in $\mathcal F$, problem (\ref{system-1})-(\ref{system-4}) has a unique solution $\phi=\phi(\mathbf{f}, \mathbf{e})$ which satisfies $$
\|\phi\|_{H^2(\mathcal S)}\, \leq\, {\tilde\tau} \varepsilon^{3/2}|\ln\varepsilon|^q, $$ $$
\big\|\phi\big\|_{L^{\infty}(|x|>\delta/\varepsilon)}+\big\|\nabla \phi\big\|_{L^{\infty}(|x|>\delta/\varepsilon)}\, \leq\, e^{-\rho\delta/\varepsilon}. $$
Moreover, $\phi$ depends Lipschitz-continuously on the parameters $\mathbf{f}$ and $\mathbf{e}$ in the sense of the estimate \begin{align} \label{characteriztion}
\|\phi(\mathbf{f}_1, \mathbf{e}_1)\,-\, \phi(\mathbf{f}_2, \mathbf{e}_2)\|_{H^2(\mathcal S)}
\, \leq\, C\varepsilon^{3/2}|\ln\varepsilon|^q\big [\, \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0, 1)}\, +\, \|\mathbf{e}_1-\mathbf{e}_2\|_{**}\, \big]. \end{align} \end{proposition}
\proof The proof is similar as that for Proposition 5.1 in \cite{delPKowWei2007}, which will be omitted here. \qed
We conclude this section by stating the following announcements: \\[1mm] \noindent $\bullet$ As we have said in the above, we shall construct the approximate solution in Section \ref{section4}. \\[1mm] \noindent $\bullet$ To find a real solution to \eqref{problemafterscaling}, the reduction procedure will be carried out in Sections \ref{section5} and \ref{sectionsolvingreducedequation} to kill the Langrange multipliers in \eqref{system-1}-\eqref{system-4}. This can be done by suitable choice of the parameters $\mathbf{f}=(f_1,\cdots, f_N)$ and $\mathbf{e}=(e_1,\cdots, e_N)$. We will first derive the equations involving the parameters $f_1,\cdots, f_N$ and $e_1,\cdots, e_N$ in Section \ref{section5}, and then solve the coupled system involving the equations in Section \ref{sectionsolvingreducedequation}.
\section{The local approximate solutions} \label{section4} \setcounter{equation}{0}
The main objective of this section is to construct the local form $v$ of the approximation ${\mathbf w}$ (cf. (\ref{globalapproximation})) and then evaluate its error $\mathcal{E}$, $g_0$, $g_1$ in the coordinate system $(x, z)$.
\subsection{The first approximate solution} Recall the notation in Section \ref{section3.3}.
For a fixed integer $N>1$, we assume that the locations of $N$ concentration layers are characterized by sets
$$
\big\{\, (x, z) \, :\, x=\beta(\varepsilon z) f_j(\varepsilon z)+\beta(\varepsilon z)\, h(\varepsilon z)\, \big\}, \qquad j=1, \cdots, N,
$$
in the coordinates $(x, z)$. The function $ h$ satisfies \begin{equation}\label{assumptionofh} {\mathfrak b}_1\, h'(0)\,-\, {\mathfrak b}_2 h(0)\, =\, 0, \qquad {\mathfrak b}_6\, h'(1)\,-\, {\mathfrak b}_7\, h(1)\, =\, 0. \end{equation} In fact, $h$ will be chosen by solving (\ref{equation of h}) and $f_j$'s can be determined in the reduction procedure.
By recalling $w$ given in \eqref{wsolution} and $Z$ in (\ref{lambda0}), we set \begin{equation}\label{wjZj} w_j(x)\, =\, w(x_j), \qquad Z_j(x)\, =\, Z(x_j), \end{equation} with $$ x_j\, =\, x-\beta(\varepsilon z)\, f_j(\varepsilon z)-\beta(\varepsilon z)\, h(\varepsilon z), $$ and then define the first approximate solution by \begin{equation} \label{vvdefine} {v_1}(x, z)\, =\sum_{j=1}^Nw_j(x). \end{equation}
For every fixed $n$ with $1\leq n\leq N$, we consider the following set \begin{equation} \mathfrak{A}_n =\Bigg\{(x, z)\in{\mathcal S} \, :\, \frac {\beta f_{n-1}(\varepsilon z)+\beta f_n(\varepsilon z)}{2}\leq x- \beta h\leq \frac {\beta f_n(\varepsilon z)+\beta f_{n+1}(\varepsilon z)}{2}\Bigg\}. \end{equation} For $(x, z)\in \mathfrak{A}_n$, we expand $S(v_1)$ by gathering terms of $\varepsilon$ and those of order $\varepsilon^2:$ \begin{align}\label{sv1-gather} S(v_1)\, =\, &\sum_{j=1}^N\varepsilon \, \big[h_8(f_j+h)w_{j, xx}- \frac{V_t(0, \varepsilon z)}{\beta^2}\, (f_j+h)w_j\big] \nonumber\\ &+\sum_{j=1}^N\frac{\varepsilon}{\beta}\big[h_3w_{j, x}- \, \frac{V_t(0, \varepsilon z)}{\beta^2} x_jw_j+h_8x_{j}w_{j, xx}\big] \nonumber\\ &-\sum_{j=1}^N\varepsilon^2\Bigg[\Big( -\frac {h_5}{\beta}{f_j} +h_2\frac {f''_j}{\beta} +h_2\frac {2\beta'}{\beta^2}{f_j'} +h_2\frac {2\alpha'}{\alpha \beta}{f_j'} \Big) w_{j, x} \nonumber\\ &\qquad \qquad +h_2\frac {2\beta'}{\beta^2}{f_j'x_jw_{j, xx}} -\frac{2h_7}{\beta}f_jx_jw_{j, xx} +\frac { V_{tt}(0, \varepsilon z)}{\beta^3}{f_jx_jw_j}\Bigg] \nonumber\\ &+\sum_{j=1}^N\varepsilon^2\Big[ \Big( h_2f_j'^2\, +2h_2f_j'h'-h_6\, f_j\, f_j'\, -h_6\, f_j'\, h+h_7f_j^2\Big)w_{j, xx}
-\frac{1}{2}\, \beta^{-2}\, V_{tt}(0, \varepsilon z)\, f_j^2\, w_j\Big] \nonumber\\ &+\sum_{j=1}^N\frac {\varepsilon^2}{\alpha\beta^2}\, \Big[h_6\Big(-\alpha \beta f_j' +\alpha \beta'f_j \Big)x_jw_{j, xx} + h_6\Big( \alpha'\beta f_j+\alpha\beta'f_j \Big)w_{j, x} - h_4\alpha\beta f_j' w_{j, x} \Big] \nonumber\\ &+\sum_{j=1}^N\frac{\varepsilon^2}{\beta^{2}} \Big[\Big(h_5+h_2\frac {\beta{''}}{\beta} +h_2\frac {2\alpha'\beta'}{\alpha\beta} \Big){x_jw_{j, x}}
+h_2\frac {|\beta'|^2}{\beta^2}x_j^2w_{j, xx} +h_2\beta^2\, h'^2\, w_{j, xx} +h_2\frac {\alpha{''}}{\alpha}w_j \nonumber\\ &\quad\quad\quad\quad -\frac {1}{2\beta^2} V_{tt}(0, \varepsilon z)x_j^2w_j -\frac 12 V_{tt}(0, \varepsilon z)(2f_jh+h^2)w_j +h_7\beta^2(2f_jh+h^2)+h_7 x^2w_{j, xx}\Big] \nonumber\\ &-\sum_{j=1}^N\varepsilon^2\Big[\Big( -\frac {h_5}{\beta}{h} +h_2\frac {h''}{\beta} +h_2\frac {2\beta'}{\beta^2}{h'} +h_2\frac {2\alpha'}{\alpha \beta}{h'} \Big) w_{j, x} \nonumber\\ &\qquad \qquad +h_2\frac {2\beta'}{\beta^2}{h'x_jw_{j, xx}} -\frac{2h_7}{\beta}hx_jw_{j, xx} +\frac { V_{tt}(0, \varepsilon z)}{\beta^3}{hx_jw_j}\Big] \nonumber\\ &+\sum_{j=1}^N\frac {\varepsilon^2}{\alpha \beta^2} \Big[ h_6\Big( \alpha'+\frac {\alpha \beta'}{\beta} \Big)x_jw_{j, x} +h_6\frac {\alpha \beta'}{\beta}{x_j^2w_{j, xx}} - h_6\alpha \beta^2(f_jh'+hh')w_{j, xx} +h_4 \alpha'w_j\Big] \nonumber\\ &+\sum_{j=1}^N\, \frac{\varepsilon^2}{\alpha \beta^2}\Big[h_6\Big(-\alpha \beta h' +\alpha \beta'h \Big)x_jw_{j, xx} + h_6\Big( \alpha'\beta h+\alpha\beta'h\Big)w_{j, x} - h_4\alpha\beta h' w_{j, x} \Big] +B_4(v_1) \nonumber\\ \, \equiv\, & \varepsilon\sum_{j=1}^N S_{1, j}\, +\, \varepsilon \sum_{j=1}^NS_{2, j}\, +\, \varepsilon^2\sum_{j=1}^NS_{3, j}\, +\, \varepsilon^2\sum_{j=1}^NS_{4, j}\, +\, \varepsilon^2\sum_{j=1}^NS_{5, j} \, +\, \varepsilon^2\sum_{j=1}^NS_{6, j}\, \nonumber\\ &+\, \varepsilon^2\sum_{j=1}^NS_{7, j}\, +\, \varepsilon^2\sum_{j=1}^NS_{8, j}\, +\, \varepsilon^2\sum_{j=1}^NS_{9, j}\, +\, B_4(v_1), \end{align} where \begin{align}\label{B2v1} B_4(v_1)=&\, \frac{1}{\alpha\beta^2} \big[\, \hat{B}_0(v_1)+a_6(\varepsilon s, \varepsilon z)\, \varepsilon^3 \, s^3\, {v_1}\big] \nonumber\\[2mm] &+h_1\Big[pw_n^{p-1}(v_1-w_n)-\sum_{j\neq n}{w_j}^p+\frac{1}{2}p(p-1)w_n^{p-2}(v_1-w_n)^2\Big]
+\max_{j\neq n}O(e^{-3 |\beta f_j-x|}). \end{align}
Here $B_4(v_1)$ turns out to be of size $(\varepsilon^3+\varepsilon^{\frac {3}{2}}|\ln\varepsilon|^q)$. Let us observe that the quantities $S_{1, j}$, $S_{3, j}$, $S_{5, j}$, $S_{7, j}$ and $S_{9, j}$ are odd functions of $x_j$, while $S_{2, j}$, $S_{4, j}$, $S_{6, j}$ and $S_{8, j}$ are even functions of $x_j$.
Using the assumptions of $h$ in (\ref{assumptionofh}), the boundary errors can be formulated as follows. For $z=0$, the error terms have the expressions \begin{align} \label{boundary-1-v_1-x-z} &\varepsilon \sum_{j=1}^N\beta\Big[{\mathfrak b}_2 f_j-{\mathfrak b}_1f_j'\Big] w_{j, x} +\varepsilon \Big[{\mathfrak b}_2+{\mathfrak b}_1 \frac{\beta'}{\beta} \Big]\sum_{j=1}^Nx_jw_{j, x} +\varepsilon {\mathfrak b}_1\frac{\alpha'}{\alpha}\sum_{j=1}^Nw_{j} \nonumber\\[2mm] &+\varepsilon^2 {\mathfrak b}_4\sum_{j=1}^N\big( -\beta'\, f_j-\beta\, f_j'-\beta'\, h-\beta\, h'\big)\Big(\frac {x_j} \beta+f_j+h\Big)w_{j, x} \nonumber\\[2mm] &+\varepsilon^2\, \sum_{j=1}^N\Big[{\mathfrak b}_3\, \Big(\frac {x_j} \beta+f_j+h\Big)^2\, \beta+{\mathfrak b}_4\, \Big(\frac {x_j} \beta+f_j+h\Big)^2\beta'\Big]\, w_{j, x} \nonumber\\[2mm] &+\varepsilon^2{\mathfrak b}_4\frac{\alpha'}{\alpha}\sum_{j=1}^N\Big(\frac {x_j} \beta+f_j+h\Big)w_{j} \nonumber\\[2mm] &+\varepsilon^3{\mathfrak b}_5\sum_{j=1}^N\big( -\beta'\, f_j-\beta\, f_j'-\beta'\, h-\beta\, h'\big)\Big(\frac {x_j} \beta+f_j+h\Big)^2w_{j, x} +D_2^0(v_1). \end{align} Similarly, for $z\, =\, 1/\varepsilon$, we have the terms \begin{align}\label{boundary-2-v_1-x-z} &\varepsilon \sum_{j=1}^N\, \beta\Big[{\mathfrak b}_7f_j-{\mathfrak b}_6f_j'\Big] w_{j, x} +\varepsilon \Big[{\mathfrak b}_7+{\mathfrak b}_6\frac{\beta'}{\beta} \Big]\sum_{j=1}^Nx_jw_{j, x} +\varepsilon {\mathfrak b}_6\frac{\alpha'}{\alpha}\sum_{j=1}^Nw_{j} \nonumber\\[2mm] &+\varepsilon^2 {\mathfrak b}_9\sum_{j=1}^N\big( -\beta'\, f_j-\beta\, f_j'-\beta'\, h-\beta\, h'\big)\Big(\frac {x_j} \beta+f_j+h\Big)w_{j, x} \nonumber\\[2mm] &+\varepsilon^2\, \sum_{j=1}^N\Big[{\mathfrak b}_8\, \Big(\frac {x_j} \beta+f_j+h\Big)^2\, \beta+{\mathfrak b}_9\, \Big(\frac {x_j} \beta+f_j+h\Big)^2\beta'\Big]\, w_{j, x} \nonumber\\[2mm] &+\varepsilon^2{\mathfrak b}_9\frac{\alpha'}{\alpha}\sum_{j=1}^N\Big(\frac {x_j} \beta+f_j+h\Big)w_{j} \nonumber\\[2mm] &+\varepsilon^3{\mathfrak b}_{10}\sum_{j=1}^N\big( -\beta'\, f_j-\beta\, f_j'-\beta'\, h-\beta\, h'\big)\Big(\frac {x_j} \beta+f_j+h\Big)^2w_{j, x} +D_2^1(v_1). \end{align}
\subsection{ Interior correction layers}
We now want to construct correction terms and establish a further approximation to a real solution that eliminates the terms of order $\varepsilon$ in the errors. Inspired by the method in Section 2 of \cite{delPKowWei2007}, for fixed $z$, we need a solution of \begin{equation}\label{equationofphi1} -\phi_{1, xx}+\phi_{1}-p{w}^{p-1}\phi_{1}\, =\, \sum_{j=1}^N S_{1, j}\, +\, \sum_{j=1}^NS_{2, j}, \qquad \phi_1(\pm \infty)=0. \end{equation} As it is well known, this problem is solvable provided that \begin{equation} \label{orth-condition} \int_{\Bbb R}(S_{1, j}+S_{2, j})w_{j, x}{\rm d}x=0. \end{equation} Furthermore, the solution is unique under the constrain \begin{equation} \label{understrait} \int_{\Bbb R}\phi_1w_{j, x}{\rm d}x=0. \end{equation} Since $S_{1, j}$ is odd in the variable $x_j$, we have \begin{equation*} \begin{split} &\int_{\Bbb R}(S_{1, j}+S_{2, j})w_{j, x}{\rm d}x =\int_{\Bbb R}S_{2, j}w_{j, x}{\rm d}x \\[2mm] &= \frac{1}{\beta}\Big[ h_3\int_{\Bbb R}w_{j, x}^2{\rm d}x- \, V_t(0, \varepsilon z) \beta^{-2} \int_{\Bbb R}x_jw_jw_{j, x}{\rm d}x+h_8\int_{\Bbb R}x_{j}w_{j, xx}w_{j, x}{\rm d}x\Big] \\[2mm] &= \frac{1}{\beta}\Big[ h_3\, +\, \sigma V_t(0, \varepsilon z) \beta^{-2} -\frac{1}{2} h_8\Big]\int_{\Bbb R}w_{j, x}^2{\rm d}x, \end{split} \end{equation*} where we have used the fact \begin{equation}\label{relationofwx} -2\int_{{\mathbb R}}xww_x\, {\rm d}x\, =\int_{{\mathbb R}}w^2\, {\rm d}x \, =\, 2\sigma \int_{{\mathbb R}}w_x^2\, {\rm d}x, \qquad \int_{{\mathbb R}}w_x^2\, {\rm d}x
\, =\, -2\int_{{\mathbb R}}xw_xw_{xx}\, {\rm d}x. \end{equation} Thanks to the stationary condition \eqref{stationary1}, then we obtain \begin{equation} \label{relation-1} h_3\, = \, \frac{1}{2}h_8 -\sigma \frac{V_t(0, \varepsilon z)}{\beta^2}. \end{equation} For more details, the reader can refer the computations in Appendix \ref{appendixA}. Therefore, we have verified the condition \eqref{orth-condition}.
Then the solution $\varepsilon\phi_1$ can be written in the form
\begin{equation}
\label{phi1}
\varepsilon\, \phi_1\, =\, \varepsilon\, \sum_{j=1}^N \phi_{1, j}
=\, \varepsilon\sum_{j=1}^N(\phi_{10, j}+\phi_{11, j}+\phi_{12, j}+\phi_{13, j}),
\end{equation} where \begin{align} \label{phi10} \phi_{10, j}(x, z)\, =\, a_{10}(\varepsilon z)\omega_{0, j}(x)=\, a_{10}(\varepsilon z)\omega_{0}(x_j), \end{align} \begin{align} \label{phi11} \phi_{11, j}(x, z)\, =\, a_{11}(\varepsilon z)\omega_{1, j}(x)=\, a_{11}(\varepsilon z)\omega_{1}(x_j), \end{align} \begin{align}\label{phi12} \phi_{12, j}(x, z)\, =\, \big[f_j(\varepsilon z)+h(\varepsilon z)\big]a_{12}(\varepsilon z)\omega_{2, j}(x)=\, \big[f_j(\varepsilon z)+h(\varepsilon z)\big]a_{12}(\varepsilon z)\omega_{2}(x_j), \end{align} \begin{align}\label{phi13} \phi_{13, j}(x, z)\, =\, \big[f_j(\varepsilon z)+h(\varepsilon z)\big]a_{13}(\varepsilon z)\omega_{3, j}(x)=\, \big[f_j(\varepsilon z)+h(\varepsilon z)\big]a_{13}(\varepsilon z)\omega_{3}(x_j), \end{align} with \begin{align} \label{a10a11} a_{10}(\theta)\, =\, \frac{h_3(\theta)}{\beta(\theta) h_1(\theta)}, \qquad &a_{11}(\theta)\, =\, \frac{h_8(\theta)}{\beta(\theta) h_1(\theta)}, \\[2mm] \label{a12a13} a_{12}(\theta)\, =\, - \frac{V_t(0, \theta)}{\beta^2(\theta) h_1(\theta)}, \qquad &a_{13}(\theta)\, =\, \frac{h_8(\theta)}{h_1(\theta)}. \end{align} The functions $\omega_{0}, \omega_{1}$ are respectively the unique odd solutions to \begin{equation} \label{w{0}} -\omega_{0, xx}+\omega_{0}-p{w}^{p-1}\omega_{0} \, =\, w_{x}+\sigma^{-1} xw, \qquad\int_{{\mathbb R}}\omega_{0}w_{x}\, {\rm d}x \, =\, 0, \end{equation} \begin{equation} \label{w{1}} -\omega_{1, xx}+\omega_{1}-p{w}^{p-1}\omega_{1} \, =\, -\frac{1}{2 \sigma} xw+xw_{xx}, \qquad\int_{{\mathbb R}}\omega_{1}w_{x}\, {\rm d}x \, =\, 0, \end{equation} and $\omega_{2}, \omega_{3}$ are respectively the unique even functions satisfying \begin{align} \label{w{2}} -\omega_{2, xx}+\omega_{2}-p{w}^{p-1}\omega_{2}\, =\, w, \qquad\int_{{\mathbb R}}\omega_{2}w_{x}\, {\rm d}x \, =\, 0, \end{align} \begin{align} \label{w{3}} -\omega_{3, xx}+\omega_{3}-p{w}^{p-1}\omega_{3}\, =\, w_{xx}, \qquad\int_{{\mathbb R}}\omega_{3}w_{x}\, {\rm d}x \, =\, 0. \end{align} Observe that $\varepsilon\, \phi_1$ is of size $O(\varepsilon)$.
\subsection{Boundary corrections}\ In the following, we want to cancel the boundary error terms of first order in $\varepsilon$ given in (\ref{boundary-1-v_1-x-z}) and (\ref{boundary-2-v_1-x-z}), i.e., \begin{equation*} \varepsilon \Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'(0)}{\beta(0)} \Big]x_jw_{j, x} +\varepsilon\, {\mathfrak b}_1\frac{\alpha'(0)}{\alpha(0)}w_j, \end{equation*} and \begin{equation*} \varepsilon \Big[{\mathfrak b}_7+{\mathfrak b}_6 \frac{\beta'(1)}{\beta(1)} \Big]x_jw_{j, x} +\varepsilon\, {\mathfrak b}_6\frac{\alpha'(1)}{\alpha(1)}w_j. \end{equation*} On the other hand, the boundary terms $$ \varepsilon\, \Big[{\mathfrak b}_7f_j(1)\,-\, {\mathfrak b}_6\, f_j'(1)\Big] w_{j, x} \quad\mbox{ and }\quad \varepsilon \, \Big[{\mathfrak b}_2f_j(0)\,-\, {\mathfrak b}_1\, f_j'(0)\Big] w_{j, x} $$ will be dealt with by the standard reduction procedure in Sections \ref{section5}-\ref{sectionsolvingreducedequation}.
This can be done by the methods in Section 2.2 of \cite{JWeiYang2007}. By defining two constants \begin{equation*}
{\bf c}_{1}\, =\, \Big[{\mathfrak b}_7+{\mathfrak b}_6\frac{\beta'(1)}{\beta(1)} \Big]\int_{{\mathbb R}}x\, w_x\, Z\, {\rm d}x \, +\, {\mathfrak b}_6\frac{\alpha'(1)}{\alpha(1)}\int_{{\mathbb R}}\, w\, Z\, {\rm d}x, \end{equation*} \begin{equation*} {\bf c}_{0}\, =\, \Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'(0)}{\beta(0)} \Big]\int_{{\mathbb R}}x\, w_x\, Z\, {\rm d}x \, +\, {\mathfrak b}_1\frac{\alpha'(0)}{\alpha(0)}\int_{{\mathbb R}}\, w\, Z\, {\rm d}x, \end{equation*} and also a function \begin{align}\label{btheta} A({\tilde\theta})\, =\, &\, \frac {{\bf{c}_{0}}\cos\big(\sqrt{\lambda_0}\, {\ell}/\varepsilon\big)-\bf{c}_{1}}{\sqrt{\lambda_0}\sin\big(\sqrt{\lambda_0}\, {\ell}/\varepsilon\big)}\cos\big(\varepsilon^{-1}\, \lambda_0\, {\tilde\theta}\big) \, +\, \frac {\bf{c}_{0}}{\sqrt{\lambda_0}}\sin\big(\varepsilon^{-1}\, \lambda_0\, {\tilde\theta}\big), \end{align} where the constant $\ell$ is given in (\ref{ell}), we choose \begin{align} \label{tildephi}
\phi_{21, j}(x, z)\, =\, A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j=\, A\big({\tilde{\bf d}}(\varepsilon z)\big)Z(x_j), \quad\mbox{with}\ {\tilde{\bf d}}(\theta)\, =\, \int_0^{\theta}\mathcal{Q}(r)\, {\rm d}r, \end{align} where ${\mathcal Q}$ is the function given in \eqref{ell}. On the other hand, by Corollary 2.4 in \cite{JWeiYang2007}, we then find a unique solution $\phi_{*}$ of the following problem: \begin{align*} \Delta \phi_{*}-\phi_{*}+p{w}^{p-1}\phi_{*}\, =\, 0\quad&\mbox{in } \hat{\mathcal S}, \\[2mm] \frac {\partial \phi_{*}}{\partial {\tilde z}}\, =\, \Big[{\mathfrak b}_7+{\mathfrak b}_6\frac{\beta'(1)}{\beta(1) }\Big]xw_{x}+\, {\mathfrak b}_6 \frac{\alpha'(1)}{\alpha(1)}w&-{\bf{c}_{1}}Z \quad\mbox{on } \partial_1\hat{\mathcal S}, \\[2mm] \frac {\partial \phi_{*}}{\partial {\tilde z}}\, =\, \Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'(0)}{\beta(0) }\Big]xw_{x}+\, {\mathfrak b}_1 \frac{\alpha'(0)}{\alpha(0)}w&-{\bf{c}_{0}}Z \quad\mbox{on } \partial_0\hat{\mathcal S}, \end{align*} where $\hat{\mathcal S}$, $\partial_0\hat{\mathcal S}$ and $\partial_1\hat{\mathcal S}$ are defined in (\ref{domainS1}). Moreover, $\phi_{*}$ is even in $x$. By the diffeomophism \begin{align} \label{mathfrak-a-function} {\Upsilon}: [0, {1}/{\varepsilon}]\rightarrow [0, {{\ell}}/{\varepsilon}], \qquad {\Upsilon}(z)\, =\,
\varepsilon^{-1}\int_0^{\varepsilon z}\mathcal{Q}(\theta){\rm d}\theta, \end{align} where ${\mathcal Q}$ is the function given in \eqref{ell}, we define $$ \phi_{22, j}(x, z)\, =\, \phi_{*, j}\, =\, \phi_{*}\big(x_j, {\Upsilon}(z)\big). $$ Hence, $\phi_{22, j}$ satisfies the following problem: \begin{align}\label{boundary-1} \frac{h_2}{\beta^2}\partial_{zz}\phi_{22, j}&+h_1\big[\partial_{xx}\phi_{22, j}-\phi_{22, j}+p{w_j}^{p-1}\phi_{22, j}\big]\, =\, \varepsilon \frac{h_2}{\beta^2}\mathcal{Q}{'}\phi_{{*}, \tilde z}\big(x_j, {\Upsilon}(z)\big) \quad\mbox{in } {\mathcal S}, \nonumber\\[2mm] \frac {\partial \phi_{22, j}}{\partial z}\, =\, &-\frac{\mathcal{Q}(1)}{{\mathfrak b}_6}\Big\{\Big[{\mathfrak b}_7+{\mathfrak b}_6\frac{\beta'(1)}{\beta(1) }\Big]x_jw_{j, x}+\, {\mathfrak b}_6 \frac{\alpha'(1)}{\alpha(1)}w_j-{\bf{c}_{1}}Z_j\, \Big\} \quad\mbox{on } \partial_1{\mathcal S}, \\[2mm] \frac {\partial \phi_{22, j}}{\partial z}\, =\, &- \frac{\mathcal{Q}(0)}{{\mathfrak b}_1}\Big\{\Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'(0)}{\beta(0) }\Big]x_jw_{j, x}+\, {\mathfrak b}_1 \frac{\alpha'(0)}{\alpha(0)}w_j-{\bf{c}_{0}}Z_j\Big\} \quad\mbox{on } \partial_0{\mathcal S}, \nonumber
\end{align} where $\mathcal S$, $\partial_0{\mathcal S}$ and $\partial_1{\mathcal S}$ are defined in (\ref{domainS}). We finally set the boundary correction term \begin{align}
\label{phi2} \varepsilon\ \phi_2(x, z) \, =\varepsilon\, \sum_{j=1}^N\phi_{2, j}(x, z) =\varepsilon\, \sum_{j=1}^N\, \xi(\varepsilon z)\, \Big[\phi_{21, j}(x, z)+\phi_{22, j}(x, z)\Big], \end{align} where \begin{equation*} \xi(\theta)\, \equiv\, \frac{\chi_0(\theta)}{\mathcal{Q}(0)} \, +\, \frac{1-\chi_0(\theta)}{\mathcal{Q}(1)}, \end{equation*} and the smooth cut-off function $\chi_0$ is defined by $$
\chi_0(\theta)\, =\, 1 \quad\mbox{if } |\theta|<\frac 18, \quad {\rm and}\quad
\chi_0(\theta)\, =\, 0 \quad\mbox{if } |\theta|>\frac 38. $$ Note that $\varepsilon\, \phi_2(x, z)$ is of size $O(\varepsilon)$ under the gap condition (\ref{gapconditionofve}).
Let $v_2(x, z)\, =\, v_1+\varepsilon\, \phi_1+\varepsilon\, \phi_2$
be the second approximate solution. A careful computation can indicate that
the new boundary error takes the following form \begin{align} &\varepsilon\sum_{j=1}^N\, \Big[{\mathfrak b}_2\beta f_j\,-\, {\mathfrak b}_1\beta\, f_j'\Big]w_{j, x} \nonumber\\[2mm] &+\varepsilon^2 \Big[{\mathfrak b}_2+{\mathfrak b}_1 \frac{\beta'}{\beta} \Big]\sum_{j=1}^N\Big(\frac{x_j}{\beta}+f_j+h\Big)\Big[a_{10}\omega_{0, j, x}+a_{11}\omega_{1, j, x}+(f_j+h)(a_{12}\omega_{2, j, x}+a_{13}\omega_{3, j, x}) \nonumber\\[2mm] &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\mathcal{Q}^{-1}\big(A(0)Z_{j, x}+\phi_{22, j, x}\big)\Big] \nonumber\\[2mm] &+\varepsilon^2\sum_{j=1}^N\, {\mathfrak b}_1 \Big[a'_{10}\, \omega_{0, j} +a'_{11}\, \omega_{1, j} +(a'_{12}\, \omega_{2, j}+a'_{13}\, \omega_{3, j})(f_j+h) +(a_{12}\omega_{2, j}+a_{13}\omega_{3, j})(f_j'+h')\Big] \nonumber\\[2mm] &-\varepsilon^2\sum_{j=1}^N{\mathfrak b}_1 \big(\beta'f_j+\beta f_j'+\beta'h+\beta h'\big)\Big[a_{10}\omega_{0, j, x}+a_{11}\omega_{1, j, x}+(f_j+h)(a_{12}\omega_{2, j, x}+a_{13}\omega_{3, j, x}) \nonumber\\[2mm] &\qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad+\mathcal{Q}^{-1}\big(A(0)Z_{j, x}+\phi_{22, j, x}\big)\Big] \nonumber\\[2mm] &-\varepsilon^2 \sum_{j=1}^N{\mathfrak b}_4\big(\beta'f_j+\beta f_j'+\beta'h+\beta h'\big) \Big(\frac{x_j}{\beta}+f_j+h\Big)w_{j, x} +\varepsilon^2\sum_{j=1}^N {\mathfrak b}_4\frac{\alpha'}{\alpha}\Big(\frac{x_j}{\beta}+f_j+h\Big)w_j \nonumber\\[2mm] &+\varepsilon^2\sum_{j=1}^N\Big[{\mathfrak b}_3\Big(\frac{x_j}{\beta}+f_j+h\Big)^2\beta+{\mathfrak b}_4\Big(\frac{x_j}{\beta}+f_j+h\Big)\Big(\frac{\beta'}{\beta}x_j+\beta'f_j+\beta'h\Big)\Big]w_{j, x} \nonumber\\[2mm] &+\varepsilon^2 \sum_{j=1}^N {\mathfrak b}_1 \frac{\alpha'}{\alpha} \Big[a_{10}\omega_{0, j}+a_{11}\omega_{1, j}+(f_j+h)(a_{12}\omega_{2, j}+a_{13}\omega_{3, j})+\mathcal{Q}^{-1}(A(0)Z_{j}+\phi_{22, j})\Big] \nonumber\\[2mm] &+\varepsilon^2\sum_{j=1}^N {\mathfrak b}_4 \Big(\frac{x_j}{\beta}+f_j+h\Big)\mathcal{Q}^{-1}(\varepsilon A'(0)\mathcal{Q} Z_{j}+\phi_{22, j, z}) \nonumber\\[2mm]
&+D_2^0(v_1+\phi_1+\phi_2)+O(\varepsilon^3)\quad \quad\mbox{on } \partial_0{\mathcal S}, \label{second-approximated-boundary-new-1}
\end{align} where the functions are evaluated at $\theta=0$. Similar estimate holds on $\partial_1{\mathcal S}$.
\subsection{The second improvement} To deal with the resonance phenomena, which were described in Section 1 of \cite{delPKowWei2007}, and improve the approximation for a solution still keeping the term of $\varepsilon^2$, we need to introduce new parameters $\{e_j\}_{j=1}^N$. In other words, as the methods in \cite{delPKowWei2007} we shall set an improved approximate solution as follows \begin{equation*}
\, v_1+\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\, \sum_{j=1}^N e_j(\varepsilon z)Z_j(x). \end{equation*}
To decompose the coupling of the parameters $\{f_j\}_{j=1}^N$ and $\{e_j\}_{j=1}^N$ on the boundary of $\mathcal S$ (in the sense of projection against $Z$ in $L^2$), by Lemma 2.2 in \cite{JWeiYang2007}, we introduce a new term $\phi^{*}$ (even in $x$) defined by the following problem \begin{align}\label{equationofphi*} \begin{split} \Delta \phi^{*}-\tilde{K}\phi^{*}+p\, {w}^{p-1}\phi^{*}\, =\, &0\quad\mbox{in } \hat{\mathcal S}, \\[2mm] \phi^{*}_{{\tilde z}}\, =\, H_{1}(x)\quad\mbox{on } \partial_0\hat{\mathcal S}, \qquad \phi^{*}_{{\tilde z}}\, =\, &H_{2}(x)\quad\mbox{on } \partial_1\hat{\mathcal S}, \end{split} \end{align} where $\tilde{K}$ is a large positive constant, and the function $H_{1}(x)$ is given by the following \begin{align}\label{H1new} H_1(x)\, =\, &\Big[{\mathfrak b}_2+{\mathfrak b}_1 \frac{\beta'(0)}{\beta(0)} \Big]\big(f(0)+h(0)\big)\Big\{\big[a_{10}(0)\, \omega_{0, x}+a_{11}(0)\, \omega_{1, x}\big] \nonumber\\[2mm] &\, +\, x\big[a_{12}(0)\, \omega_{2, x}+a_{13}(0)\, \omega_{3, x}\big] \, +\, x \frac{1}{\beta(0)} \mathcal{Q} \big[A(0)Z_x+\phi_{22, x}\big]\Big\} \nonumber\\[2mm] &\, +\, 2{\mathfrak b}_3\big[f(0)+h(0)\big]x\, w_x \, +\, {\mathfrak b}_4\left[\frac{\beta'(0)}{\beta(0)}\big(f(0)+h(0)\big)+\Big(f'(0)+\frac{\beta'(0)}{\beta(0)}f(0)\Big)\right]x\, w_x \nonumber\\[2mm] &\, +\, {\mathfrak b}_1\Big[\beta'(0)f(0)+\beta(0)f'(0)+\beta'(0)h(0)+\beta(0)h'(0)\Big]\, \Big[a_{10}(0)\, \omega_{0, x}+a_{11}(0)\, \omega_{1, x}\Big] \nonumber\\[2mm] &\, +\, {\mathfrak b}_1\frac{\alpha'(0)}{\alpha(0)}\Big[(f(0)+h(0))\, (a_{12}(0)\, \omega_2+a_{13}(0)\, \omega_3) +\frac{1}{\beta(0)}\Big(A(0)Z+\phi_{22}(x, 0)\Big)\Big] \nonumber\\[2mm] &+{\mathfrak b}_1\big[f'(0)+h'(0)\big]\Big[a_{12}(0)\omega_2+a_{13}(0)\omega_3\Big] +{\mathfrak b}_1\big[f(0)+h(0)\big]\Big[a'_{12}(0)\omega_2+a'_{13}(0)\omega_3\Big] \nonumber\\[2mm] &+{\mathfrak b}_4\frac{1}{\beta(0)}\big[f(0)+h(0)\big]\big[\varepsilon A'(0)\beta(0)Z+\phi_{22, z}(x, 0)\big]+{\mathfrak b}_3\big[f(0)+h(0)\big]\frac{\alpha'(0)}{\alpha(0)}w. \end{align} The function $H_2(x)$ has a similar expression. We define a boundary correction term again \begin{align} \label{boundary layer} \varepsilon^2\phi_3(x, z)\, =\, \varepsilon^2\sum_{j=1}^N\phi_{3, j}(x, z)\, =\, \varepsilon^2\sum_{j=1}^N\xi(\varepsilon z)\phi^{*}(x_j, {\Upsilon}(z)). \end{align} Note that for $j=1, \cdots, N$, $\varepsilon^2\phi_{3, j}$ is an exponential decaying function which is of order $\varepsilon^2$ and even in the variable $x_j$. Then define the third approximate solution to the problem near $\Gamma_\varepsilon$ as \begin{equation} \label{u3define} v_3(x, z)\, =\, v_1+\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\, \sum_{j=1}^N e_j(\varepsilon z)Z_j(x)+\varepsilon^2\phi_3. \end{equation}
\subsection{The third improvement} By choosing $h$, we will construct a further approximation that eliminates the even terms (in $x_j$'s) in the error $ S(v_3) $. This can be fulfilled by adding a term $\Phi=\sum_{j=1}^N\Phi_j$ and then considering the following term \begin{align} \label{v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi}
S(v_3+\Phi) \, =\, &S(v_1)+L_0(\varepsilon\, \phi_1)+L_0(\varepsilon\, \phi_2)+L_0 \Big( \varepsilon\sum_{j=1}^N{e_j}Z_j \Big)+L_0(\varepsilon^2\, \phi_3)+L_0(\Phi) \nonumber\\ &+B_3(\varepsilon\, \phi_1)+B_3(\varepsilon\, \phi_2)+B_3\Big(\varepsilon\sum_{j=1}^N{e_j}Z_j\Big) +B_3(\varepsilon^2\, \phi_3)+B_3(\Phi) \nonumber\\ &+N_{0}\Big(\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\, \phi_3+\Phi\Big), \end{align} where \begin{equation} L_0(\phi)= \frac{h_2}{\beta^2}\phi_{zz}+h_1\big[\phi_{xx}-\phi+p{v_1}^{p-1}\phi\big], \qquad N_0(\phi)= h_1\big[(v_1+\phi)^p- v_1^p-p v_1^{p-1} \phi\big]. \end{equation} The details will be given in the sequel.
\subsubsection{Rearrangements of the error components} The first objective of this part is to given the details to compute the terms in formula (\ref{v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi}).
\noindent $\bullet$ It is easy to compute that \begin{align}\label{L0phi1} L_0(\varepsilon\, \phi_1)&\, =\, \varepsilon\, \Big[\frac{h_2}{\beta^2}\phi_{1, zz}+h_1\big(\phi_{1, xx}-\phi_1+p{v_1}^{p-1}\phi_1\big)\, \Big] \nonumber\\ &\, =\, -\varepsilon\sum_{j=1}^N(S_{1, j}+S_{2, j}) +\varepsilon\, \frac{h_2}{\beta^2}\phi_{1, zz} +\varepsilon h_1 \Big( \, p\, {v_1}^{p-1}{\phi_1}-\sum_{j=1}^N\, p\, {w_j}^{p-1}{\phi_{1, j}} \Big).
\end{align}
\noindent $\bullet$ Recall the expression of $\varepsilon\, \phi_2$ and $A({\tilde\theta})$ defined in (\ref{phi2}) and (\ref{btheta}). By using the equation of $\phi_{22, j}$ in (\ref{boundary-1}) and the equation of $Z$ in (\ref{eigenvalue}), we get \begin{align} \label{L0phi2} L_0(\varepsilon\, \phi_2)&\, =\, \varepsilon\, \Big[\frac{h_2}{\beta^2}\, \phi_{2, zz}+h_1\big(\phi_{2, xx}-\phi_2+p\, {v_1}^{p-1}\, \phi_2\big)\Big]
\nonumber\\[2mm] &=\sum_{j=1}^N M_{11, j}(x, z)+M_{12}(x, z)+\varepsilon\, h_1\Big(p \, {v_1}^{p-1}{\phi_2}-\, \sum_{j=1}^N\, p\, {w_j}^{p-1}{\phi_{2, j}}\Big),
\end{align} where
\begin{align}
\label{hatL0phi2} M_{11, j}(x, z)\, \equiv\, &\frac{\varepsilon^{2}}{\beta^{2}}h_2\Big\{2\xi'(\varepsilon z)\big[\varepsilon A'\big({\tilde{\bf d}}(\varepsilon z)\big)\mathcal{Q} Z_j+\phi_{22, j, z}\big] +\mathcal{Q}'\xi(\varepsilon z)\big[\varepsilon A'\big({\tilde{\bf d}}(\varepsilon z)\big) Z_j+\phi_{*, j, \tilde{z}}\big]\Big\},
\end{align}
and \begin{equation}\label{tildeL0phi2} M_{12}(x, z)\, \equiv\, \sum_{j=1}^N\frac{\varepsilon^3}{\beta^{2}}\, h_2\, \xi''(\varepsilon z)\big[A'\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\big].
\end{equation}
\noindent $\bullet$ According to the equation of $Z$, we obtain \begin{align*} L_0 \Big( \varepsilon\sum_{j=1}^N{e_j}Z_j \Big) &=\varepsilon\sum_{j=1}^N \Big\{ \frac{h_2}{\beta^2} ({e_j}Z_j)_{zz}+h_1\big[({e_j}Z_j)_{xx}-{e_j}Z_j+p{v_1}^{p-1}{e_j}Z_j \big] \Big\} \\[2mm] &=\varepsilon\sum_{j=1}^N \Big[\, \varepsilon^2 \frac{h_2}{\beta^2} e_j''Z_j+ h_1\lambda_0{e_j}Z_j\, \Big] +\varepsilon\, h_1\, p{v_1}^{p-1}\sum_{j=1}^N {e_j}Z_j \\[2mm] &\quad -\varepsilon\, h_1\, \sum_{j=1}^N p{w_j}^{p-1}{e_j}Z_j+\varepsilon^3 a_7(\varepsilon s, \varepsilon z). \end{align*}
\noindent $\bullet$ Recalling the expression of $\varepsilon^2 \phi_3$ defined in \eqref{boundary layer} and the equation of $\phi^{*}$ in \eqref{equationofphi*}, it follows that \begin{align}\label{L0phi3} L_0(\varepsilon^2 \phi_3) &=\, \varepsilon^2\, \Big[\frac{h_2}{\beta^2}\, \phi_{3, zz}+h_1\big(\phi_{3, xx}-\phi_3+p\, v_1^{p-1}\, \phi_3\big)\Big] \nonumber\\[2mm] &=\sum_{j=1}^N\, \varepsilon^2\, h_1(\tilde{K}-1)\phi_{3, j} +\varepsilon^2\, \Big[\frac{h_2}{\beta^2}\phi_{3, j, zz}+h_1 \big(\phi_{3, j, xx} -\tilde{K}\phi_{3, j}+p\, {v_1}^{p-1}\, \phi_{3, j} \big) \Big] \nonumber\\[2mm] &=\sum_{j=1}^N M_{21, j}(x, z)+M_{22}(x, z) + \varepsilon^2\, h_1 \Big( p\, {v_1}^{p-1}\, \phi_3-\sum_{j=1}^N\, p\, {w_j}^{p-1}\, \phi_{3, j}\Big), \end{align} where \begin{equation} \label{hatL0e^2hatphi} M_{21, j}(x, z)\, \equiv\, \varepsilon^2\, h_1(\tilde{K}-1)\, \phi_{3, j}, \end{equation} and \begin{align} \label{tildeL0e^2hatphi} M_{22}(x, z)\, \equiv\, &\sum_{j=1}^N\frac{h_2}{\beta^2}\Big[\varepsilon^4\xi''(\varepsilon z)\phi^{*}(x_j, {\Upsilon}(z)) +2\varepsilon^3\xi'(\varepsilon z)\phi^{*}_{\tilde{z}}(x_j, {\Upsilon}(z))\mathcal{Q}(\varepsilon z) \nonumber\\ &\qquad \qquad \quad+\varepsilon^3\xi(\varepsilon z)\phi^{*}_{\tilde{z}}(x_j, {\Upsilon}(z)) \mathcal{Q}'(\varepsilon z)\Big].
\end{align}
\noindent $\bullet$ Recall the expression of $B_3$ in (\ref{B3v}), we obtain that \begin{align} \label{Bphi1} B_3(\varepsilon\, \phi_1) \, =\, &\varepsilon^2 \frac{h_3}{\beta} \, \sum_{j=1}^N \phi_{1, j, x} + \varepsilon^2 h_8\, \sum_{j=1}^N \frac{x_j}{\beta}\phi_{1, j, xx} + \varepsilon^2 h_8\, \sum_{j=1}^N (f_j+h)\phi_{1, j, xx} \nonumber\\[2mm] &-\varepsilon^2\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}\, \sum_{j=1}^N \frac{x_j}{\beta}\phi_{1, j} -\varepsilon^2\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}\, \sum_{j=1}^N (f_j+h)\phi_{1, j} +\varepsilon^3a_8(\varepsilon s, \varepsilon z) \nonumber\\[2mm] \, =\, &\sum_{j=1}^N M_{31, j}(x, z)+M_{32}(x, z), \end{align} where \begin{align} \label{M31} M_{31, j}(x, z) \, \equiv\, &\varepsilon^2 \Bigg\{\frac{h_3}{\beta} \big(a_{10}\omega_{0, j, x} +a_{11}\omega_{1, j, x}+a_{12}h\omega_{2, j, x} +a_{13}h\omega_{3, j, x}\big) \nonumber\\[2mm] &\qquad+\frac{h_8}{\beta}\big( a_{10}x_j\omega_{0, j, xx} +a_{11}x_j\omega_{1, j, xx}+a_{12}hx_j\omega_{2, j, xx} +a_{13}hx_j\omega_{3, j, xx}\big) \nonumber\\[2mm] &\qquad+ h_8\big( a_{12}f_jh\omega_{2, j, xx} +a_{13}f_jh\omega_{3, j, xx}\big) \nonumber\\[2mm] &\qquad+ h_8\big[ ha_{10}\omega_{0, j, xx} +ha_{11}\omega_{1, j, xx}+a_{12}(f_jh+h^2)\omega_{2, j, xx} +a_{13}(f_jh+h^2)\omega_{3, j, xx}\big] \nonumber\\[2mm] &\qquad-\frac{ V_t(0, \varepsilon z)}{\beta^{2}}\frac{1}{\beta}\big( a_{10}x_j\omega_{0, j} +a_{11}x_j\omega_{1, j}+a_{12}hx_j\omega_{2, j} +a_{13}hx_j\omega_{3, j}\big) \nonumber\\[2mm] &\qquad-\frac{ V_t(0, \varepsilon z)}{\beta^{2}}\big[h a_{10}\omega_{0, j} +ha_{11}\omega_{1, j}+a_{12}(f_jh+h^2)\omega_{2, j} +a_{13}(f_jh+h^2)\omega_{3, j}\big] \nonumber\\[2mm] &\qquad-\frac{ V_t(0, \varepsilon z)}{\beta^{2}}\big(a_{12}f_jh\omega_{2, j} +a_{13}f_jh\omega_{3, j}\big) \Bigg\}, \end{align} and \begin{align} \label{M32} M_{32}(x, z)\, \equiv\, &\sum_{j=1}^N \, \varepsilon^2\Bigg\{ \frac{h_3}{\beta} \, f_j\big(a_{12}\omega_{2, j, x} +a_{13}\omega_{3, j, x}\big)
+\, \frac{h_8}{\beta}f_j\big( a_{12}x_j\omega_{2, j, xx} +a_{13}x_j\omega_{3, j, xx}\big) \nonumber\\[2mm] &\qquad \quad+\, h_8\, \big( f_ja_{10}\omega_{0, j, xx} +f_ja_{11}\omega_{1, j, xx}+a_{12}f_j^2\omega_{2, j, xx} +a_{13}f_j^2\omega_{3, j, xx}\big) \nonumber\\[2mm] &\qquad \quad-\frac{V_t(0, \varepsilon z)}{\beta^3}f_j\big( a_{12}x_j\omega_{2, j} +a_{13}x_j\omega_{3, j}\big) \nonumber\\[2mm] &\qquad \quad- \frac{V_t(0, \varepsilon z)}{\beta^2}\, \big(f_j a_{10}\omega_{0, j} +f_ja_{11}\omega_{1, j}+a_{12}f_j^2\omega_{2, j} +a_{13}f_j^2\omega_{3, j}\big)\Bigg\} \,+\, \varepsilon^3a_8(\varepsilon s, \varepsilon z).
\end{align}
\noindent $\bullet$ Moreover, we can decompose $B_3(\varepsilon\, \phi_2)$ as follows \begin{align*}
B_3(\varepsilon\, \phi_2) =&\varepsilon^2 \frac{h_3}{\beta} \, \sum_{j=1}^N \phi_{2, j, x} +\varepsilon^2 \frac{ h_4}{\beta^2}\sum_{j=1}^N \phi_{2, j, z} +2\varepsilon^2 h_2 \frac{\beta'}{\beta^{2}}\sum_{j=1}^N \Big(\frac{x_j}{\beta} + f_j +h\Big) \phi_{2, j, xz} \\[2mm] &+2\varepsilon^2 \frac{h_2\alpha'}{\alpha \beta^2}\sum_{j=1}^N \phi_{2, j, z} +\varepsilon^2 \frac{h_6}{\beta} \sum_{j=1}^N \Big(\frac{x_j}{\beta} + f_j +h\Big)\phi_{2, j, xz} \\[2mm] &+ \varepsilon^2 h_8\, \sum_{j=1}^N \Big( \frac{x_j}{\beta}+ f_j+h\Big) \phi_{2, j, xx} -\varepsilon^2\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}\, \sum_{j=1}^N \Big( \frac{x_j}{\beta}+ f_j+h\Big)\phi_{2, j} +\varepsilon^3a_9(\varepsilon s, \varepsilon z)
\\ \, =\, & \, \sum_{j=1}^N M_{41, j}(x, z)+M_{42}(x, z), \end{align*} where \begin{align}\label{M41} M_{41, j}(x, z) \equiv\, & \varepsilon^2 \frac{\xi(\varepsilon z)}{\beta} \Bigg\{h_3\big[A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, x}+\phi_{22, j, x}\big]+\frac{ h_4}{ \beta}\big[A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{j}+\phi_{22, j, z}\big] \nonumber\\[2mm] &\quad+2h_2 \frac{\beta'}{\beta}\Big(\frac{x_j}{\beta}+h\Big)\big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{j, x}+\phi_{22, j, xz}\big] +2\frac{h_2\alpha'}{\alpha \beta}\big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{j}+\phi_{22, j, z}\big] \nonumber\\[2mm] &\quad+ h_6\Big(\frac{x_j}{\beta} +h\Big)\big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q}Z_{j, x}+\phi_{22, j, xz}\big] + \frac{h_8}{\beta}\Big(\frac{x_j}{\beta}+h\Big)\big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, xx}+\phi_{22, j, xx}\big] \nonumber\\[2mm] &\quad-\, \frac{ V_t(0, \varepsilon z)}{\beta}\Big(\frac{x_j}{\beta}+h\Big)\big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j}+\phi_{22, j}\big]\Bigg\}, \end{align} and \begin{align}\label{M42} M_{42}(x, z)\, \equiv\, &\sum_{j=1}^N \varepsilon^2 \xi(\varepsilon z)f_j\frac{1}{\beta} \Bigg\{2h_2 \frac{\beta'}{\beta}\big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{j, x}+\phi_{22, j, xz}\big] +h_6\big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q}Z_{j, x} +\phi_{22, j, xz}\big] \nonumber\\[2mm] &+\frac{h_8}{\beta}\, \big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, xx}+\phi_{22, j, xx}\big] -\, \frac{ V_t(0, \varepsilon z)}{\beta}\big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j}+\phi_{22, j}\big]\Bigg\} +\varepsilon^3a_9(\varepsilon s, \varepsilon z).
\end{align}
\noindent $\bullet$ The computations for next term is the following: \begin{align*} B_3\Big(\varepsilon\sum_{j=1}^N{e_j}Z_j \Big) \, =\, &\varepsilon^2 \frac{h_3}{\beta} \, \sum_{j=1}^N {e_j}Z_{j, x} + \varepsilon^2 h_8\, \sum_{j=1}^N \Big( \frac{x_j}{\beta}+f_j+h \Big) Z_{j, xx} \\[2mm] &-\varepsilon^2\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}\, \sum_{j=1}^N\Big( \frac{x_j}{\beta}+f_j+h \Big){e_j}Z_{j} +\varepsilon^3a_{10}(\varepsilon s, \varepsilon z)
\\[2mm] \, =\, &\sum_{j=1}^N M_{51, j}(x, z)+M_{52}(x, z), \end{align*} where \begin{equation} \label{M51} M_{51, j}(x, z) \, \equiv\, \varepsilon^2 (f_j+h ){e_j} \Big[\, h_8\, Z_{j, xx}-\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}Z_{j} \, \Big], \end{equation} \begin{equation} \label{M52} M_{52}(x, z)\, \equiv\, \sum_{j=1}^N\frac{\varepsilon^2}{\beta} {e_j}\Big[\, h_3 \, Z_{j, x} +h_8\, x_jZ_{j, xx}-\, \frac{ V_t(0, \varepsilon z)}{\beta^{2}}\, x_jZ_{j}\, \Big] +\varepsilon^3a_{10}(\varepsilon s, \varepsilon z).
\end{equation}
\noindent $\bullet$ The main order of the nonlinear term $N_0\Big(\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\, \phi_3+\Phi\Big)$
is in the following $$ \frac {p(p-1)}{2}{v_1}^{p-2}\Big(\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\, \phi_3+\Phi\Big)^2, $$ which can be decomposed in the form \begin{align*} &\quad\frac {p(p-1)}{2}h_1{v_1}^{p-2}\Big(\varepsilon\, \phi_1+\varepsilon\, \phi_2 +\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\, \phi_3+\Phi\Big)^2 = M_{61}(x, z)\, +\, M_{62}(x, z)\, +\, O(\varepsilon^2), \end{align*} where \begin{align}\label{M61} & M_{61}(x, z)=\sum_{j=1}^N M_{61, j}(x, z) \nonumber\\[2mm] &\equiv\sum_{j=1}^N\varepsilon^2h_1\frac {p(p-1)}{2}{w_j}^{p-2} \Big\{ a_{10}^2\omega_{0, j}^2+a_{11}^2\omega_{1, j}^2 +2a_{10}a_{11}\omega_{0, j}\omega_{1, j} \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+2h\big[a_{10}a_{12}\omega_{0, j}\omega_{2, j} +a_{10}a_{13}\omega_{0, j}\omega_{3, j}\big] \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+2h\big[a_{11}a_{12}\omega_{1, j}\omega_{2, j} +a_{11}a_{13}\omega_{1, j}\omega_{3, j}\big] \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+(2f_jh+h^2)\big[a_{12}^2\omega_{2, j}^2 +a_{13}^2\omega_{3, j}^2 +2a_{12}a_{13}\omega_{2, j}\omega_{3, j}\big] \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+e^2Z_j^2 \, +\, \xi^2(\varepsilon z)\left[A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\right]^2 \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+2(a_{10}\omega_{0, j}+a_{11}\omega_{1, j})\xi(\varepsilon z)\left[A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\right] \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad+2 (a_{12}\omega_{2, j} + a_{13}\omega_{3, j})(f_j+h) \left[ e_jZ_j+ \xi(\varepsilon z)\big(A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\big)\right] \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\qquad +2\, \xi(\varepsilon z)\left[A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\right]e_j Z_j \Big\}, \end{align} and \begin{align}\label{M62} &M_{62}(x, z)=\sum_{j=1}^N M_{62, j}(x, z) \nonumber\\[2mm] &\equiv\sum_{j=1}^N\frac {p(p-1)}{2}{w_j}^{p-2} \Big\{ 2\, \varepsilon^2f_j\big(a_{10}a_{12}\omega_{0, j}\omega_{2, j} +a_{10}a_{13}\omega_{0, j}\omega_{3, j} +a_{11}a_{12}\omega_{1, j}\omega_{2, j} +a_{11}a_{13}\omega_{1, j}\omega_{3, j}\big) \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\quad+\varepsilon^2a_{12}^2\omega_{2, j}^2f_j^2 +\varepsilon^2a_{13}^2\omega_{3, j}^2f_j^2 +2\, \varepsilon^2\, a_{11}\omega_{1, j} e_jZ_j \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\quad+\varepsilon^4\, \phi_3^2(x_j, z)+\Phi^2_j +2\, \varepsilon^2\, \phi_{1, j}\, \phi_3+2\, \phi_{1, j}\, \Phi_j+2\, \varepsilon^2\, \phi_2\, \phi_3+2\, \phi_{2, j}\, \Phi_j \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\quad +2\, \varepsilon^3\, e_j\, Z_j\, \xi(\varepsilon z)\phi^{*}(x_j, z)+2\, \varepsilon \, e_j\, Z_j\, \Phi_j+2\, \varepsilon^2 \, \phi_3\, \Phi_j \Big\}. \end{align} For the convenience of notation, we also denote \begin{align} M_{63}(x, z)=N_0\Big(\varepsilon\, \phi_1+\varepsilon\, \phi_2+\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\, \phi_3+\Phi\Big)-M_{61}(x, z)-M_{62}(x, z). \end{align}
Whence, according to the above rearrangements, we rewrite (\ref{v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi}) in terms of \begin{align}\label{S(v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi+varphi)} S(v_3+\Phi) &=\sum_{j=1}^N \Big[ \varepsilon^2S_{6, j}+\varepsilon^2S_{7, j}+\varepsilon^2S_{8, j}+\varepsilon^2S_{9, j}+ h_1\big(\Phi_{j, x_jx_j}-\Phi_j+p{w_j}^{p-1}\Phi_j\big) +M_{11, j}(x, z) \nonumber\\[2mm] &\quad\qquad+M_{21, j}(x, z)+M_{31, j}(x, z)+M_{41, j}(x, z)+M_{51, j}(x, z)+M_{61, j}(x, z) \Big] \nonumber\\[2mm] &\quad+\sum_{j=1}^N\varepsilon^2 \Big( S_{3, j}+S_{4, j}+S_{5, j} \Big) +\sum_{j=1}^N \varepsilon\Big( \varepsilon^2 \frac{h_2}{\beta^2} e_j''Z_j+ h_1\lambda_0{e_j}Z_j\Big) +B_4(v_1)+\varepsilon\frac{h_2}{\beta^2}\phi_{1, zz} \nonumber\\ &\quad+\frac{h_2}{\beta^2}\Phi_{zz}+M_{12}(x, z)+M_{22}(x, z)+M_{32}(x, z)+M_{42}(x, z)+M_{52}(x, z) \nonumber\\[2mm] &\quad+M_{62}(x, z)+M_{63}(x, z)+B_{3}(\varepsilon^2 \phi_3)+B_3(\Phi)+\varepsilon h_1 \Big( p{v_1}^{p-1}{\phi_1}-\sum_{j=1}^N p {w_j}^{p-1}{\phi_{1, j}} \Big) \nonumber\\ &\quad+\varepsilon h_1 \Big( p{v_1}^{p-1}\sum_{j=1}^N {e_j}Z_j -\sum_{j=1}^N p{w_j}^{p-1}{e_j}Z_j \Big) + \varepsilon h_1\Big( p {v_1}^{p-1}{\phi_2}-\sum_{j=1}^N p {w_j}^{p-1}{\phi_{2, j}} \Big) \nonumber\\[2mm] &\quad+\varepsilon^2 h_1 \Big(p{v_1}^{p-1}\phi_3-\sum_{j=1}^Np {w_j}^{p-1} \phi_{3, j} \Big) +h_1\Big(p {v_1}^{p-1}\Phi-\sum_{j=1}^Np{w_j}^{p-1}\Phi \Big). \end{align}
\subsubsection{Finding new correction terms and defining the basic approximation}
In order to eliminate the terms between the first brackets in (\ref{S(v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi+varphi)}), for fixed $z$, we need a solution to the problem \begin{align} \label{equation of Phi} h_1(-\Phi_{j, xx}+\Phi_j-p{w_j}^{p-1}\Phi_j)=&\, \varepsilon^2S_{6, j}+\varepsilon^2S_{7, j}+\varepsilon^2S_{8, j}+\varepsilon^2S_{9, j} \nonumber\\[2mm] &+M_{11, j}(x, z)+M_{21, j}(x, z)+M_{31, j}(x, z) \nonumber\\[2mm] &+M_{41, j}(x, z)+M_{51, j}(x, z)+M_{61, j}(x, z), \quad \forall x\in {\mathbb R}. \end{align} It is well-known that the above problem is solvable provided that \begin{align} \label{solvable of varphi1} \int_{{\mathbb R}}\big[&\varepsilon^2S_{6, j}+\varepsilon^2S_{7, j}+\varepsilon^2S_{8, j}+\varepsilon^2S_{9, j}+M_{11, j}(x, z)+M_{21, j}(x, z) \nonumber\\ &+M_{31, j}(x, z)+M_{41, j}(x, z)+M_{51, j}(x, z)+M_{61, j}(x, z)\big]w_{j, x}\, {\rm d}x \, =\, 0. \end{align} The computations in Appendix \ref{appendixA} give that the validity of (\ref{solvable of varphi1}) holds if the following problem \begin{align} \label{equation of h}
\begin{split} \mathcal{H}_{1}(\varepsilon z)h'' +\mathcal{H}_{1}'(\varepsilon z)h' +\Big[ \mathcal{H}_{2}'(\varepsilon z)-\mathcal{H}_{3}(\varepsilon z) +\frac{\alpha_1(z)}{\zeta(\varepsilon z)}\Big]h =\, \frac{1}{\zeta(\varepsilon z)}\big[G_{1}(z)+G_{3}(z)\big],
\\[2mm] {\mathfrak b}_1\, h'(0)\,-\, {\mathfrak b}_2 h(0)\, =\, 0, \qquad {\mathfrak b}_6\, h'(1)\,-\, {\mathfrak b}_7\, h(1)\, =\, 0,
\end{split} \end{align}
has a solution. Here the functions $\mathcal{H}_{1}$, $\mathcal{H}_{2}$, $\mathcal{H}_{3}$, $\zeta$, $\alpha_1$, $G_1$ and $G_2$ are given in (\ref{mathcalH1})-(\ref{mathcalH3}), \eqref{zeta}, (\ref{alpha1}), (\ref{G1}) and (\ref{G3}). In fact, for the solvability of problem (\ref{equation of h}) the reader can refer to Lemma 6.1 in \cite{weixuyang}. Moreover, $h$ has the following estimate $$
\|h\|_{H^2(0, 1)}\leq C\varepsilon^{\frac{1}{2}}. $$
Now, we can find a function defined by $\Phi=\varepsilon^2\phi_4(x, \varepsilon z)$, such that the terms between the first brackets in (\ref{S(v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi+varphi)}) disappear. Finally, our basic approximate solution to the problem near the curve $\Gamma_\varepsilon$ is \begin{align} \label{basic approximate} v_4\, =v_1+\varepsilon\phi_1+\varepsilon\phi_2+\varepsilon\sum_{j=1}^N{e_j}Z_j+\varepsilon^2\phi_3+\varepsilon^2\phi_4. \end{align}
\subsection{The global approximate solution and errors} Recall the coordinates $(s, z)$ in (\ref{coordinatessz}), $(t, \theta)$ in (\ref{Fermicoordinates-modified}), and also the local approximate solution $v_{4}(s, z)$ in (\ref{basic approximate}), which is constructed near the curve $\Gamma_\varepsilon$ in the coordinates $(s, z)$. By the relations in (\ref{vdefine}), we then make an extension and simply define the approximate solution to (\ref{problemafterscaling}) in the form \begin{align} \label{globalapproximation} \mathbf{w}(\tilde y)\, =\, \eta_{3\delta}^{\varepsilon}(s)\, \alpha(\varepsilon z)\, v_4(x, z). \end{align} Note that, in the coordinates $(\tilde {y}_1, \tilde {y}_2)$ introduced in (\ref{rescaling}), $\mathbf{w}$ is a function defined on $\Omega_{\varepsilon}$ which is extended globally as $0$ beyond the $6\delta/\varepsilon$-neighborhood of $\Gamma_\varepsilon$. The interior error can be arranged as follows \begin{align}\label{new error-2} {\mathcal E}\, \equiv\, S(v_4) \, =&\, \varepsilon^2 \sum_{j=1}^N\big(\, S_{3, j}+\, S_{4, j}+\, S_{5, j}\big) +B_4(v_1) +\sum_{j=1}^N\Big[\, \varepsilon^3 \frac{h_2}{\beta^2} e_j''Z_j+ \varepsilon h_1\lambda_0{e_j}Z_j\, \Big] \nonumber\\[2mm] \, &+\varepsilon\frac{h_2}{\beta^2}\phi_{1, zz} +\varepsilon^2\frac{h_2}{\beta^2}\phi_{4, zz} +M_{12}(x, z)+M_{22}(x, z)+M_{32}(x, z) \nonumber\\[2mm] \, &+M_{42}(x, z)+M_{52}(x, z)+M_{62}(x, z)+M_{63}(x, z) +B_3(\varepsilon^2\phi_3)+B_3(\varepsilon^2\phi_4) \nonumber\\ &+\varepsilon h_1\Big(p\, {v_1}^{p-1}{\phi_1}-\sum_{j=1}^Np{w_j}^{p-1}{\phi_{1, j}}\Big) +\varepsilon h_1 \Big( p\, {v_1}^{p-1}{\phi_2}-\sum_{j=1}^Np{w_j}^{p-1}{\phi_{2, j}} \Big) \nonumber\\ &+\varepsilon^2 h_1 \Big(p\, {v_1}^{p-1}\phi_3-\varepsilon^2\sum_{j=1}^N\, p\, {w_j}^{p-1}\, \phi_{3, j}\Big) +\varepsilon^2 h_1 \Big(p\, {v_1}^{p-1}\phi_4-\varepsilon^2\sum_{j=1}^N\, p\, {w_j}^{p-1}\, \phi_{4, j}\Big), \end{align} where we have used (\ref{S(v_1+phi1+phi2+epsilon sum_{j=1}^N{e_j}Z_j+phi3+Phi+varphi)}) and the equation of $\phi_4$ in (\ref{equation of Phi}). The boundary error term $g_0$ has the form
\begin{align}\label{gonew} g_0(x)\, =&\, \varepsilon\sum_{j=1}^N\Big[{\mathfrak b}_2\beta f_j\,-\, {\mathfrak b}_1\beta\, f_j'\Big]w_{j, x} \nonumber\\ &+\varepsilon^2 \Big[{\mathfrak b}_2+{\mathfrak b}_1 \frac{\beta'}{\beta}\Big]\sum_{j=1}^N(f_j+h) \Big\{ a_{12}\omega_{2, j, x}+a_{13}\omega_{3, j, x}
+\mathcal{Q}^{-1}\big(A(0)Z_{j, x}+\phi_{22, j, x}\big) \Big\} \nonumber\\ &+\varepsilon^2 \Big[{\mathfrak b}_2+{\mathfrak b}_1 \frac{\beta'}{\beta}\Big]\sum_{j=1}^N\Big(\frac{x_j}{\beta}\Big)\big[a_{10}\omega_{0, j, x}+a_{11}\omega_{1, j, x} \big] \nonumber\\ &-\varepsilon^2\sum_{j=1}^N{\mathfrak b}_1\big(\beta'f_j+\beta f_j'+\beta'h+\beta h'\big) \Big\{ (a_{12}\omega_{2, j, x}+a_{13}\omega_{3, j, x})(f_j+h) \nonumber\\ &\qquad\qquad+\, \frac{1}{\beta}\, \big(A(0)\, Z_{j, x}+\phi_{22, j, x}(x_j, 0)\big) + e_j Z_{j, x}\Big\} \nonumber\\ &-\varepsilon^2\sum_{j=1}^N{\mathfrak b}_1\frac{\alpha'}{\alpha} \Big\{ (a_{10}\omega_{0, j}+a_{11}\omega_{1, j})+ e_j Z_j\Big\} +\varepsilon^2{\mathfrak b}_4\frac{\alpha'}{\alpha}\frac{x_j}{\beta}w_j \nonumber\\
&-\varepsilon^2\sum_{j=1}^N\Big\{{\mathfrak b}_3\beta\big(\frac{x_j^2}{\beta^2} +f_j^2+h^2+2f_jh\big) +{\mathfrak b}_4\big[\frac{\beta'}{\beta^2}x_j^2+\beta(f_j+h)^2\big]\Big\}w_{j, x} \nonumber\\
&+\varepsilon^2{\mathfrak b}_1\sum_{j=1}^N\Big[(a'_{10}\omega_{0, j}+a'_{11}\omega_{1, j})+e_j'Z_j+{\mathfrak b}_5\frac{x_j}{\beta^2} \big(\varepsilon A'(0)\beta(0)Z_j+\phi_{22, j, z}(x, 0)\big)\Big] \nonumber\\
&- \varepsilon^2\sum_{j=1}^N (\beta f_j' +\beta 'f_j) \Big[{\mathfrak b}_4(f_j+h)- \varepsilon {\mathfrak b}_5\Big(\frac{x_j}{\beta}+f_j+h\Big)^2 \Big] w_{j, x}
+O(\varepsilon^3).
\end{align} The term $g_1$ has a similar expression.
We decompose \begin{equation} \label{E1-d} {\mathcal E}\, =\, {\mathcal E}_{11}+{\mathcal E}_{12}, \qquad g_0\, =\, g_{01}+g_{02}, \qquad g_1\, =\, g_{11}+g_{12}, \end{equation} with \begin{align} \label{E11} {\mathcal E}_{11}\, =\, \sum_{j=1}^N {\mathcal E}_{11, j} \, =\sum_{j=1}^N\big(\varepsilon^3\, \frac{h_2}{\beta^2} e''_j\, Z_j+\varepsilon\, h_1 \lambda_0\, e_j\, Z_j\big), \quad &{\rm and}\quad {\mathcal E}_{12}\, =\, {\mathcal E}-{\mathcal E}_{11}, \nonumber\\ g_{01}\, =\, \varepsilon\sum_{j=1}^N\beta(0)\big[{\mathfrak b}_2 f_j\,-\, {\mathfrak b}_1f_j'\big]w_{j, x}+\sum_{j=1}^N\varepsilon^2\, {\mathfrak b}_1\, e_j'\, Z_j, \quad &{\rm and}\quad g_{02}\, =\, g_0-g_{01}, \\ g_{11}\, =\, \varepsilon\sum_{j=1}^N\beta(0)\big[{\mathfrak b}_7f_j\,-\, {\mathfrak b}_6\, f_j'\big]w_{j, x}+\sum_{j=1}^N\varepsilon^2\, {\mathfrak b}_6\, e_j'\, Z_j, \quad &{\rm and}\quad g_{12}\, =\, g_1-g_{11}.\nonumber \end{align} For further references, it is useful to estimate the $L^2(\mathcal S)$ norm of ${\mathcal E}$. From the uniform bound of $e_1, \cdots, e_N$ in (\ref{enorm}), it is easy to see that \begin{align} \label{E11norm}
\|{\mathcal E}_{11}\|_{L^2(\mathcal S)}\, \leq\, C\varepsilon^{1/2}|\ln\varepsilon|^q. \end{align} Since $\varepsilon\, \phi_1, \varepsilon\, \phi_2$ and $\varepsilon e_jZ_j$ are of size $O(\varepsilon)$, all terms in ${\mathcal E}_{12}$ carry $\varepsilon^2$ in front. We claim that \begin{align} \label{E12norm}
\|{\mathcal E}_{12}\|_{L^2(\mathcal S)}\, \leq\, C\varepsilon^{ 3/2}|\ln\varepsilon|^q. \end{align}
Similarly, we have the following estimate \begin{align} \label{g02norm}
\|g_{02}\|_{L^2({\mathbb R})}+\|g_{12}\|_{L^2({\mathbb R})}\, \leq\, C\varepsilon^{3/2}|\ln\varepsilon|^q. \end{align} Moreover, for the Lipschitz dependence of the term of error ${\mathcal E}_{12}$ on the parameters $\mathbf{f}$ and $\mathbf{e}$ for the norm defined in (\ref{constraints of f}) and (\ref{enorm}), we have the validity of the estimate \begin{align} \label{E12L}
\|{\mathcal E}_{12}(\mathbf{f}_1, \mathbf{e}_1)-{\mathcal E}_{12}(\mathbf{f}_2, \mathbf{e}_2)\|_{L^2(\mathcal S)}
\, \leq\, C\varepsilon^{ 3/2}|\ln\varepsilon|^q\big[\, \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0, 1)}+\|\mathbf{e}_1-\mathbf{e}_2\|_{**}\, \big]. \end{align} Similarly, we obtain \begin{align} \label{g02L} \begin{split}
&\|g_{02}(\mathbf{f}_1, \mathbf{e}_1)-g_{02}(\mathbf{f}_2, \mathbf{e}_2)\|_{L^2({\mathbb R})}\, +\, \|g_{12}\, (\mathbf{f}_1, \mathbf{e}_1)-g_{12}(\mathbf{f}_2, \mathbf{e}_2)\|_{L^2({\mathbb R})} \\[2mm]
&\, \leq\, C\varepsilon^{3/2}|\ln\varepsilon|^q\big[\, \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0, 1)}+\|\mathbf{e}_1-\mathbf{e}_2\|_{**}\, \big]. \end{split} \end{align}
\section{Derivation of the reduced equations: Toda system}\label{section5} \setcounter{equation}{0}
In this section, we will set up equations for the parameters $\mathbf{f}$ and $\mathbf{e}$ which are equivalent to making $\mathbf{c}(\varepsilon z)$, $\mathbf{d}(\varepsilon z)$, $\mathbf{l}_1$, $\mathbf{l}_0$, $\mathbf{m}_1$ and $\mathbf{m}_0$ are identically zero in the system (\ref{system-1})-(\ref{system-4}).
The equations
$$ \mathbf{c}(\varepsilon z)=0, \quad \mathbf{l}_1=0, \quad \mathbf{l}_0=0,
$$
are then equivalent to the relations, for $ n=1, \cdots, N$,
\begin{align}
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s)\, {\mathcal E}+\eta^\varepsilon_{\delta}(s)\, {\mathcal N}(\phi)+\chi(\varepsilon|x|) B_3(\phi)+h_1p\big(\beta^{-2}\chi(\varepsilon |x|){\mathbf w}^{p-1}&-v_1^{p-1}\big)\phi \, \Big]\, w_{n, x}\, {\rm d}x \, =\, 0,
\label{c=0}
\\[2mm]
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s) g_1-\chi(\varepsilon|x|)D_3^1(\phi)+\chi(\varepsilon|x|) D_2^1(\phi)\, \Big]\, w_{n, x}\, {\rm d}x\, &=\, 0,
\quad z=1/\varepsilon, \label{l1=0}
\\[2mm]
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s) g_0-\chi(\varepsilon|x|) D_3^0(\phi)+\chi(\varepsilon|x|) D_2^0(\phi)\, \Big]\, w_{n, x}\, {\rm d}x\, &=\, 0, \quad z=0. \label{l2=0}
\end{align} Similarly, $$ \mathbf{d}(\varepsilon z)=0, \quad \mathbf{m}_1=0, \quad \mathbf{m}_0=0, $$ if and only if for $n=1, \cdots, N$, \begin{align}
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s)\, {\mathcal E}+\eta^\varepsilon_{\delta}(s)\, {\mathcal N}(\phi)+\chi(\varepsilon|x|) B_3(\phi)+h_1p\big(\beta^{-2}\chi(\varepsilon |x|){\mathbf w}^{p-1}&-v_1^{p-1}\big)\phi \, \Big]\, Z_n{\rm d}x\, =\, 0, \label{d=0}
\\[2mm]
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s) g_1-\chi(\varepsilon|x|)D_3^1(\phi)+\chi(\varepsilon|x|) D_2^1(\phi)\, \Big]\, Z_n\, {\rm d}x\, &=\, 0,
\quad z=1/\varepsilon, \label{m1=0}
\\[2mm]
\int_{{\mathbb R}}\Big[\, \eta^\varepsilon_{\delta}(s) g_0-\chi(\varepsilon|x|) D_3^0(\phi)+\chi(\varepsilon|x|) D_2^0(\phi)\, \Big]\, Z_n\, {\rm d}x\, &=\, 0,
\quad z=0.\label{m0=0} \end{align}
\subsection{Estimates for projections of the error}
For the pair $(\mathbf{f}, \mathbf{e})$ satisfying (\ref{constraints of f}) and (\ref{enorm}),
we denote ${\mathbf b}_{1\varepsilon}$ and ${\mathbf b}_{2\varepsilon}$, generic, uniformly bounded continuous functions of the form $$ {\mathbf b}_{l\varepsilon j}
\, =\, {\mathbf b}_{l \varepsilon j}\big(z, {\mathbf f}(\varepsilon z), {\mathbf e}(\varepsilon z), {\mathbf f}'(\varepsilon z), \varepsilon\, {\mathbf e}'(\varepsilon z)\big), \ l=1, 2, $$ where ${\mathbf b}_{1\varepsilon j}$ is uniformly Lipschitz in its four last arguments, and introduce the notation \begin{align} {\mathfrak S}=\{x\in{\mathbb R}:(x, z)\in \mathcal S \}, \qquad {\mathfrak S}_n=\{x\in{\mathbb R}:(x, z)\in {\mathfrak A}_n\} \quad n=1, \cdots, N. \end{align}
The computations in Appendix \ref{appendixB} lead to the estimate, for $n=1, \cdots, N$, \begin{align} \int_{\mathfrak S}\, {\mathcal E}w_{n, x}{\rm d}x =\, &-\varrho_2h_1\Bigg\{\varepsilon^2\varsigma\Big[\, \mathcal{H}_1f''_n+\mathcal{H}_1'f'_n+ \big(\mathcal{H}_2'-\mathcal{H}_3+\alpha_2(z)\big)f_n\, \Big] \\ &\qquad \qquad+e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)} \Bigg\} \, +\, \mathrm{P}_n(\varepsilon z), \end{align} where \begin{equation}\label{varsigma} \gamma_1(\theta)=\frac{\varrho_1}{\beta(\theta)}, \qquad \varsigma(\theta)= \frac{\gamma_1(\theta)}{\varrho_2\, h_1(\theta)}, \qquad \alpha_2(z)=\frac{\alpha_1(z)}{\zeta(\varepsilon z)}, \end{equation} \begin{align*} \mathrm{P}_n(\varepsilon z)=&\, \varepsilon^2\, \gamma_1(\theta)\, \big[\hbar_3(\varepsilon z)\, e_n+\varepsilon^2\, \hbar_4({\varepsilon}z)\, e''_n\big]
+\varepsilon^{\mu_1}\max_{j\neq n}O(e^{-\beta|f_j-f_n|}) \\ &+\varepsilon^3\sum_{j=1}^N\, \big({\mathbf b}_{1\varepsilon j}\, e_j'+{\mathbf b}_{1\varepsilon j}^2\, f''_j+{\mathbf b}_{2\varepsilon j}\big) +O(\varepsilon^3)\sum_{j=1}^N\, (f_j+f'_j+f''_j+f_j^2), \end{align*} where the functions $\mathcal{H}_1$, $\mathcal{H}_2$, $\mathcal{H}_3$ are defined in \eqref{mathcalH1}-\eqref{mathcalH3}, and $\zeta$ in \eqref{zeta}, $\alpha_1(z)$ in (\ref{alpha1}), $\varrho_1$ and $\varrho_2$ in (\ref{varrho1})-(\ref{varrho2}), $\hbar_3({\varepsilon}z)$ and $\hbar_4({\varepsilon}z)$ in (\ref{hbar3})-(\ref{hbar4}), $h_1$ in \eqref{h1}. For further references we observe that \begin{equation}
\|\mathrm{P}_n\|_{L^2(0, 1)}\leq\, C\varepsilon^{2+\mu_2}, \quad\mbox{for some }\, \mu_2>0, \, n=1, \cdots, N. \end{equation}
On the other hand, the computations in Appendix \ref{appendixC} lead to the estimate, for $n=1, \cdots, N$, \begin{align} \label{E1Z} \int_{{\mathfrak S}_n}{\mathcal E}Z_n\, {\rm d}x \, =\, &\varepsilon^3\frac{h_2}{\beta^2}e''_n+\varepsilon\lambda_0h_1e_n+\varepsilon^3\hbar_5e_n'+\varrho_4h_1\big[e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)}\big] \nonumber\\[2mm] &+\varepsilon^2\varrho_{3}\, (f'_n)^2+\varepsilon^3\rho_1\, (\varepsilon z)+\varepsilon^2\rho_2(\varepsilon z)+\varepsilon^2\rho_3(\varepsilon z)+\mathrm{R}_n(\varepsilon z), \end{align} where \begin{align*} \mathrm{R}_n(\varepsilon z)=\, &\varepsilon^4 f_n\beta^{-2}\, \hbar_6(\varepsilon z)e''_n +2\varepsilon^2\varrho_{3}(f'_nh'+f'_nh) +\varepsilon^3\beta^{-2}\xi''(\varepsilon z)A'\big({\tilde{\bf d}}(\varepsilon z)\big) \\[2mm] &+O(\varepsilon^3)\sum_{j=1}^N\big(f_j^2+{f'_j}^2\big)+O(\varepsilon^5)\sum_{j=1}^N\, e''_j
+\varepsilon^{{\hat\tau}_1}\max_{j\neq n}O(e^{-\beta|f_j-f_n|}) \\ &+\varepsilon^3\sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}f''_j+{\mathbf b}_{2\varepsilon j}e'_j+{\mathbf b}_{2\varepsilon j}\big)+O(\varepsilon^3). \end{align*} Here the constants $\lambda_0$, $\varrho_3$, $\varrho_4$ are given \eqref{lambda0}, (\ref{varrho3}) and (\ref{varrho4}), while the functions $h_1$, $h_2$, $\hbar_5$, $\hbar_6$, $\rho_1$, $\rho_2$ and $\rho_3$ are given in \eqref{h1}, \eqref{h2}, (\ref{hbar5}), (\ref{hbar6}), (\ref{rho1}), (\ref{rho2}) and \eqref{rho3}.
For further references we observe that \begin{align}
\|\mathrm{R}_n\|_{L^2(0, 1)}\leq\, C\varepsilon^{2+\mu_3}, \quad\mbox{for some }\mu_3>0, \, n=1, \cdots, N. \end{align}
\subsection{Projection of errors on the boundary} In this section, we compute the projection of errors on the boundary. Without loss of generality, only the projections of the error components on $\partial_0{\mathcal S}$ will be given. According to the expression of $g_0$ as in \eqref{gonew}, the main errors on the boundary integrated against $w_{n, x}$ and $Z_n$ in the variable $x_n$ can be computed as the following: \begin{align*}
\int_{\mathbb R}& g_0(x)w_{n, x}\, {\rm d}x\, =\, \varepsilon\sum_{j=1}^N \beta(0)\Big[{\mathfrak b}_2f_j\,-\, {\mathfrak b}_1f_j'\Big]\int_{\mathbb R} w_{j, x}w_{n, x}\, {\rm d}x+\varepsilon^2{\mathcal M}_{0, n}^1(\mathbf{f}, \mathbf{e}) +O(\varepsilon^3). \end{align*}
Using the following formulas \begin{equation*} \int_{\mathbb R} Z^2\, {\rm d}x\, =\, -2\int_{\mathbb R} x\, Z_x\, Z\, {\rm d}x\, =\, 1, \end{equation*} we get the following two estimates \begin{equation*}
\begin{split} \int_{\mathbb R} g_0(x)Z_n\, {\rm d}x\, =\, &\varepsilon^2 {\mathfrak b}_1\left[e_n'(0) \, +\, \frac{\alpha'(0)}{\alpha(0)}\, e_n(0) \right] +O(\varepsilon^3). \end{split} \end{equation*}
Higher order errors can be proceeded as follows:
\begin{align*} \int_{\mathbb R}& D_3^0(\phi(x, 0))w_{n, x}\, {\rm d}x\, \nonumber \\ = &\, \sum_{j=1}^N\varepsilon\Big[{\mathfrak b}_2+{\mathfrak b}_1\frac{\beta'}{\beta}\Big]\int_{\mathbb R} x\phi_{x}(x, 0)w_{n, x}\, {\rm d}x -\sum_{j=1}^N\varepsilon\frac{\alpha'}{\alpha}\int_{\mathbb R} \phi(x, 0)w_{n, x}\, {\rm d}x
\\ &\, +\, \sum_{j=1}^N\varepsilon {\mathfrak b}_4\int_{\mathbb R}\Big(\frac{x_j}{\beta}+f_j+h\Big)\phi_{z}(x, 0)w_{n, x}\, {\rm d}x
\\ &+\sum_{j=1}^N\varepsilon^2\int_{\mathbb R} \Big[{\mathfrak b}_1\Big(\frac{x_j}{\beta}+f_j+h\Big)^2\beta+{\mathfrak b}_4\Big(\frac{x_j}{\beta} +f_j+h\Big) \Big(\frac{\beta'}{\beta}x_j+\beta' f_j+\beta' h\Big)\Big]\phi_{x}(x, 0)w_{n, x}\, {\rm d}x \nonumber \\ &+\sum_{j=1}^N\varepsilon^2{\mathfrak b}_4\frac{\alpha'}{\alpha}\int_{\mathbb R}\Big(\frac{x_j}{\beta}+f_j+h\Big)\phi(x, 0)w_{n, x}\, {\rm d}x \, +\, \sum_{j=1}^N\varepsilon^2 {\mathfrak b}_5\int_{\mathbb R} \Big(\frac{x_j}{\beta}+f_j+h\Big)^2\phi_{z}(x, 0)w_{n, x}\, {\rm d}x \nonumber \\ =&\, O(\varepsilon^{2+\mu}), \end{align*}
and also \begin{equation*}
\int_{\mathbb R} D_3^0(\phi(x, 0))Z_n\, {\rm d}x\, = O(\varepsilon^{2+\mu}) \end{equation*} The term $D_2^0(\phi)$ on the boundary integrated against $w_{n, x}$ and $Z_n$ in the variable $x_n$ are of size of order $O(\varepsilon^3)$.
\subsection{The system involving $(\textbf{f}, \textbf{e})$} \label{The system for fe}
As done in \cite{delPKowWei2007} and \cite{delPKowWei2008}, we can estimate the terms that involve $\phi$ in (\ref{c=0}) and (\ref{d=0}) integrated against the functions $w_{n, x}$ and $Z_n$ in the variable $x_n$ in the similar ways. As a conclusion, for $n=1, \cdots, N$, there holds the following estimate \begin{align} -\varepsilon^2\varsigma\Big[\, \mathcal{H}_1\, f''_n+\mathcal{H}_1'\, f'_n+ \big(\mathcal{H}_2'-\mathcal{H}_3+\alpha_2(z)\big)\, f_n\, \Big] +e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)} \, +\, {\mathbb M}_n\, =\, 0. \end{align} Moreover, ${\mathbb M}_n$ can be decomposed in the following way \begin{align} \begin{aligned} {\mathbb M}_n={\mathbb M}_{n1}(\theta, \mathbf{f, f', f'', e, e', e''})+{\mathbb M}_{n2}(\theta, \mathbf{f, f', e, e'}), \end{aligned} \end{align} where ${\mathbb M}_{n1}$ and ${\mathbb M}_{n2}$ are continuous of their arguments.
Functions ${\mathbb M}_{n1}$ and ${\mathbb M}_{n2}$ satisfy the following properties for $n=1, \cdots, N$ $$
\|{\mathbb M}_{n1}\|_{L^2(0, 1)}\, \leq\, C \varepsilon^{2+\mu_0}, \qquad
\|{\mathbb M}_{n2}\|_{L^2(0, 1)}\, \leq\, C \varepsilon^{2+\mu_0}, $$ where $\mu_0$ is a positive constant. For $n=1, \cdots, N$, there also holds the following estimate \begin{align} \begin{aligned} &\varepsilon^3\frac{h_2}{\beta^2}e''_n +\varepsilon h_1\lambda_0e_n +\varepsilon^3\hbar_5e_n' +h_1\varrho_4\big[e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)}\big] \\[2mm] &+\varepsilon^2\varrho_{3}\, (f'_n)^2 +\varepsilon^3\rho_1 +\varepsilon^2\rho_2 +\varepsilon^2\rho_3 +{\mathbf M}_n\, =\, 0. \end{aligned} \end{align} Moreover, ${\mathbf M}_n$ can be decomposed in the following way \begin{align} \begin{aligned} {\mathbf M}_n={\mathbf M}_{n1}(\theta, \mathbf{f, f', f'', e, e', e''})+{\mathbf M}_{n2}(\theta, \mathbf{f, f', e, e'}), \end{aligned} \end{align} where ${\mathbf M}_{n1}$ and ${\mathbf M}_{n2}$ are continuous of their arguments. Functions ${\mathbf M}_{n1}$ and ${\mathbf M}_{n2}$ satisfy the following properties for $n=1, \cdots, N$ $$
\|{\mathbf M}_{n1}\|_{L^2(0, 1)}\, \leq\, C \varepsilon^{2+{\hat\tau}_0}, \qquad
\|{\mathbf M}_{n2}\|_{L^2(0, 1)}\, \leq\, C \varepsilon^{2+{\hat\tau}_0}, $$ where ${\hat\tau}_0$ is a positive constant.
Therefore, using $\theta=\varepsilon z$ and defining the operators \begin{align} {\mathbb L}_{n}(\mathbf{f}) \, \equiv\, &\varepsilon^2\varsigma(\theta)\Big[\, \mathcal{H}_1(\theta)\, f''_n+\mathcal{H}_1'(\theta)\, f'_n+ \big(\mathcal{H}_2'(\theta)-\mathcal{H}_3(\theta)+\alpha_2(\theta/\varepsilon)\big)\, f_n\, \Big]\nonumber \\[2mm] &-e^{-\beta(\theta) (f_n-f_{n-1})} +e^{-\beta(\theta)(f_{n+1}-f_n)}, \quad n=1, \cdots, N, \label{mathbbLn} \end{align} and \begin{equation}
{\mathbb L}(e)\, \equiv\,-\, \varepsilon^2\, h_2(\theta)e''\,-\, \varepsilon^2\tilde{\alpha}(\theta)\, e'\,-\, |\beta(\theta)|^2h_1(\theta)\lambda_0 e, \label{mathbbL} \end{equation} we derive the following nonlinear system of differential equations for the parameters $\mathbf{f}$ and $\mathbf{e}$ \begin{equation}\label{f} {\mathbb L}_{n}(\mathbf{f})={\mathbb M}_n, \quad n=1, \cdots, N, \end{equation} \begin{equation} {\mathbb L}(e_n)=\, \alpha_5+\alpha_{6, n}+\varepsilon^{-1}\, \beta^2\, {\mathbf M}_n, \quad n=1, \cdots, N,
\label{e} \end{equation} with the boundary conditions, $n=1, \cdots, N$, \begin{equation} \label{boundary condition 1}
{\mathfrak b}_6f'_n(1)\,-\, {\mathfrak b}_7\, f_n(1)+{\mathcal M}_{1, n}^1(\mathbf{f}, \mathbf{e})=\, 0,
\qquad
{\mathfrak b}_1f'_n(0)\,-\, {\mathfrak b}_2\, f_n(0)\, +{\mathcal M}_{1, n}^2(\mathbf{f}, \mathbf{e})=\, 0, \end{equation} \begin{equation} \label{boundary condition 3}
e'_n(1)\, +\, \tilde{b}_6\, e_n(1)\, +{\mathcal M}_{2, n}^1(\mathbf{f}, \mathbf{e})=\, 0,
\qquad
e'_n(0)\, +\, \tilde{b}_5\, e_n(0)\, +{\mathcal M}_{2, n}^2(\mathbf{f}, \mathbf{e})=\, 0. \end{equation} The constants ${\tilde b}_6$ and ${\tilde b}_5$ are given by $$\tilde{b}_6\, =\, \frac{\alpha'(1)}{\alpha(1)},
\qquad \tilde{b}_5\, =\, \frac{\alpha'(0)}{\alpha(0)},
$$ where ${\mathcal M}_{j, n}^{i}$'s are some terms of order $O(\varepsilon^{1/2})$. The functions $\varsigma(\theta)\, =\, \frac{\gamma_1(\theta)}{\varrho_2 h_1(\theta) }>0$, and $\alpha_2(\theta/\varepsilon)$ are defined in (\ref{varsigma}) by the relation $\theta=\varepsilon z$. Moreover, we have denoted \begin{equation} \tilde{\alpha}(\theta)\, =\, \beta^2(\theta)\, \hbar_5(\theta), \end{equation}
\begin{equation} \alpha_5(\theta)\, =\, \beta^2(\theta) \big[ \varepsilon^2\, \rho_1(\theta)\, +\, \varepsilon\rho_2\, (\theta)\, +\, \varepsilon\, \rho_3(\theta) \big], \label{alpha5} \end{equation}
\begin{equation} \alpha_{6, n}(\theta)
\, =\, \varepsilon\, \varrho_{3}\, \beta^2(\theta)\, |f'_n(\theta)|^2 \, +\, \varepsilon^{-1}\varrho_{4}\, \beta^{2}(\theta)h_1(\theta) \big[e^{-\beta(\theta)(f_n-f_{n-1})}-e^{-\beta(\theta)(f_{n+1}-f_n)}\big], \end{equation} where $\hbar_5(\theta)$, $\rho_1(\theta)$, $\rho_2(\theta)$, $\rho_3(\theta)$ are defined in (\ref{hbar5}), (\ref{rho1}), (\ref{rho2}), (\ref{rho3}).
\section{Suitable choosing of parameters}\label{sectionsolvingreducedequation} \setcounter{equation}{0}
\subsection{Solving the system of reduced equations} Before solving (\ref{f})-(\ref{boundary condition 3}), some basic facts about the invertibility of corresponding operators will be derived.
Firstly, we consider the following problem \begin{align}\label{equation of e} \begin{split} {\mathbb L}(e)=\, \tilde{g}(\theta), \qquad\forall\, 0<\theta<1, \\ e'(1)\, +\, \tilde{b}_6\, e(1)\, =\, 0, \qquad\quad e'(0)\, +\, \tilde{b}_5\, e(0)\, =\, 0. \end{split} \end{align}
\begin{proposition}\label{proposition7point1} If $\tilde{g}\in L^{2}(0, 1)$, then for all small $\varepsilon$ satisfying (\ref{gapconditionofve}) there is a unique solution $e\in H^2(0, 1)$ to problem (\ref{equation of e}), which satisfies
$$\|e\|_{**}\leq C\, \varepsilon^{-1}\, \|\tilde{g}\|_{L^2(0, 1)}.$$
Moreover, if $\tilde{g}\in H^{2}(0, 1)$, then
\begin{align}
\varepsilon^2\, \|e''\|_{L^2(0, 1)}\, +\, \varepsilon\, \|e'\|_{L^2(0, 1)}\, +\, \|e\|_{L^\infty(0, 1)}\, \leq\, C\, \|\tilde{g}\|_{H^2(0, 1)}.
\end{align}
\end{proposition} \begin{proof} The proof is similar as that for Lemma 8.1 in \cite{delPKowWei2007}. \end{proof}
Secondly, we consider the following problem \begin{equation} \label{ts2a} {\mathbb L}_{n}({\mathbf f})=\varepsilon^2 \tilde{h}_n, \end{equation} \begin{equation} \label{boundary0000} {\mathfrak b}_6f'_n(1)-{\mathfrak b}_7\, f_n(1)=0, \qquad {\mathfrak b}_1f'_n(0)-{\mathfrak b}_2\, f_n(0)=0, \end{equation} for $n=1, \cdots, N$, where $f_0=-\infty, \, f_{N+1}=\infty$.
\begin{proposition}\label{proposition7point2} For given ${\tilde{\mathbf h}}=(\tilde{h}_1, \cdots, \tilde{h}_N)^{T}\in L^2(0, 1)$, there exists a sequence $\{\varepsilon_l:l\in{\mathbb N}\}$ from those $\varepsilon$ satisfying the gap condition (\ref{gapconditionofve}) and approaching $0$ such that problem (\ref{ts2a})-(\ref{boundary0000}) admits a solution ${\bf f}=(f_1, \cdots, f_N)^{T}$ with the form: \begin{align} {\bf f}\, =\, \frac{1}{\beta}\Bigg\{ \rho_{\varepsilon_l}\Big(1-\frac{N+1}{2},\, 2-\frac{N+1}{2},\, \cdots,\, N-\frac{N+1}{2} \Big)^{T} \,+\,\bf{\ddot f} \,+\,\mathbf{P}^T{\hat{\mathfrak u}} \,+\,\mathbf{P}^T{\bf {\tilde w}} \,+\,\mathbf{P}^T{\check{\mathfrak u}} \Bigg\}, \end{align} where the invertible matrix $\bf P$ is defined in \eqref{mathbfP} and the function $\rho_{\varepsilon_l} (\theta) $ satisfies \begin{align}\label{def rhoeps} e^{-\rho_{\varepsilon_l}(\theta)}\, =\, \varepsilon^2 \frac{{\varsigma}(\theta)}{\beta(\theta)}\tau_2 (\theta) \rho_{\varepsilon_l}(\theta), \end{align} with $\tau_2$ given in \eqref{tau2}, and in particular \begin{align*}
\rho_{\varepsilon_l}(\theta)=2|\ln\varepsilon_l|-\ln({2}|\ln\varepsilon_l|)
-\ln\Big(\frac{ \varsigma(\theta) \tau_2(\theta)}{\beta(\theta)}\Big)+O\Big(\frac{\ln\big({2}|\ln\varepsilon_l|\big)}{|\ln\varepsilon_l|}\Big).
\end{align*} The vectors ${\bf{\ddot f}}=(\ddot{f}_{1}, \cdots, \ddot{f}_{N})^T$ defined in Lemma \ref{lemma6point3}, ${\hat{\mathfrak u}}=({\hat{\mathfrak u}}_1, \cdots, {\hat{\mathfrak u}}_N)^T$ defined by \eqref{hbar} and ${\bf {\tilde w}}=({\bf {\tilde w}}_1, \cdots, {\bf {\tilde w}}_N)^{T}$ in (\ref{equationmathbfwn})-(\ref{boundarymathbfw}) do not depend on $\bf{\tilde h}$. There hold the estimates $$ \ddot{f}_{j}=O(1), \qquad
{\hat{\mathfrak u}}_j(\theta) =O\Big(\Big(\frac{1}{\ln|\varepsilon_l|-\ln(\ln|\varepsilon_l|)}\Big)^{\frac{1}{2}}\Big), \qquad j=1, \cdots, N, $$ \begin{align}
\frac{1}{|\ln\varepsilon_l|}\, \|{\bf {\tilde w}}_j''\|_{L^2(0, 1)}
+ \|{\bf {\tilde w}}_j'\|_{L^2(0, 1)}
+\|{\bf {\tilde w}}_j\|_{L^{2}(0, 1)} \, \leq\,
\frac{C}{|\ln\varepsilon_l|}, \quad j=1, \cdots, N-1, \label{mathbfwn} \end{align} \begin{equation}\label{mathbfwN}
\|{\bf {\tilde w}}_N\|_{H^2(0, 1)} \leq C. \end{equation} For the vector ${\check{\mathfrak u}}=({{\check{\mathfrak u}}_1, \cdots, {\check{\mathfrak u}}_N})^T$, we have \begin{align*}
\frac{1}{|\ln\varepsilon_l|}\, \|{\check{\mathfrak u}}''\|_{L^2(0, 1)}
+\frac{1}{\sqrt{|\ln\varepsilon_l|}}\, \|{\check{\mathfrak u}}'\|_{L^2(0, 1)}
+\|{\check{\mathfrak u}}\|_{L^{2}(0, 1)}
\leq C\, \varepsilon^{\mu}\|\tilde{\mathfrak h}\|_{L^2(0, 1)}\, +\, \frac{C}{|\ln\varepsilon_l|}. \end{align*}
\end{proposition} \qed
The proof of Proposition \ref{proposition7point2} will be provided in Section \ref{section6.2}. By accepting this, we here want to finish the proof of Theorem \ref{theorem 1.1}. \\[1mm] {\textbf {Proof of Theorem \ref{theorem 1.1}.}} The profile of the solution given in \eqref{taketheform} can be determined by the approximate solution given in Section \ref{section4}, see \eqref{globalapproximation}. The properties of the parameters $f_j$'s in \eqref{fproperties1}-\eqref{fproperties2} can be derived from Proposition \ref{proposition7point2}.
As we have stated in Section \ref{section3}, we shall complete the last step of suitable choosing the parameters $\mathbf{f}$ and $\mathbf{e}$ by solving (\ref{f})-(\ref{boundary condition 3}). If $\hat{e}$ solves \begin{align}
\begin{split} \mathbb{ L} (\hat{e})=\, \alpha_5(\theta), \qquad\forall\, 0<\theta<1, \\[2mm] \hat{e}'(1)\, +\, {\tilde{b}_6}\, \hat{e}(1)\, =\, 0, \qquad\quad \hat{e}'(0)\, +\, {\tilde{b}_5}\, \hat{e}(0)\, =\, 0, \end{split} \end{align} from the definition of $\alpha_5(\theta)$ in \eqref{alpha5}, we get \begin{equation*}
\|\hat{e}\|_{H^2(0, 1)}\leq C\varepsilon^\frac{1}{2}. \end{equation*} Replacing $e_n$ by $\hat{e}+\tilde{e}_n$, the system (\ref{f})-(\ref{boundary condition 3}) keeps the same form except that the term $\alpha_5(\theta)$ disappear. Moreover, let $\tilde{e}_n$ solves \begin{align}
\begin{split}
\mathbb{L}_2(\tilde{e}_n)=\, \alpha_{6, n}(\theta),
\qquad\forall\, 0<\theta<1,
\\[2mm] \tilde{e}_n'(1)\, +\, {\tilde{b}_6}\, \tilde{e}_n(1)\, =\, 0,
\qquad\quad \tilde{e}_n'(0)\, +\, {\tilde{b}_5}\, \tilde{e}_n(0)\, =\, 0,
\end{split} \end{align} then it derives \begin{equation*}
\|\tilde{e}_n\|_{H^2(0, 1)}\leq C\varepsilon^\mu. \end{equation*}
Define the set $$
\mathcal{D}=\Big\{\, \mathbf{f}, \mathbf{e}\in H^2(\mathcal{S}): \|\mathbf{f}\|_{H^2(0, 1)}\leq D|\ln\varepsilon|^2,
\quad \|\mathbf{e}\|_{**}\leq C\varepsilon^\mu\, \Big\}.
$$ For $(\bar{\mathbf{f}}, \bar{\mathbf{e}})\in \mathcal{D}$, we can set for $n=1, \cdots, N$ \begin{equation*} \tilde{h}_n(\mathbf{f}, \mathbf{e})\equiv \varepsilon^{-2}{\mathbb M}_{n1}(\mathbf{f}, \mathbf{f}', \mathbf{f}'', \mathbf{e}, \mathbf{e}', \mathbf{e}'') \, +\, \varepsilon^{-2} {\mathbb M}_{n2}(\bar{\mathbf{f}}, \bar{\mathbf{f}}', \bar{\mathbf{e}}, \bar{\mathbf{e}}'), \end{equation*} \begin{equation*}
\tilde{g}_n(\mathbf{f}, \mathbf{e})\, \equiv\, \varepsilon^{-1}\, \beta^2 {\mathbf M}_{n1}(\mathbf{f}, \mathbf{f}', \mathbf{f}'', \mathbf{e}, \mathbf{e}', \mathbf{e}'') \, +\, \varepsilon^{-1}\, \beta^2 {\mathbf M}_{n2}(\bar{\mathbf{f}}, \bar{\mathbf{f}}', \bar{\mathbf{e}}, \bar{\mathbf{e}}'). \end{equation*} We now use Contraction Mapping Principle and Schauder Fixed Point Theorem to solve (\ref{f})-(\ref{boundary condition 3}) with the right hand replacing by $\tilde{h}_n$ and $\tilde{g}_n$. Whence, by the fact that ${\mathbb M}_{n1}$, ${\mathbf M}_{n1}$ are contractions on $\mathcal{D}$, making use of the argument developed in Propositions \ref{proposition7point1} and \ref{proposition7point2}, and the Contraction Mapping Principle, we find $\mathbf{f}$ and $\mathbf{e}$ for a fixed $\bar{\mathbf{f}}$ and $\bar{\mathbf{e}}$. In this way, we define a mapping $\mathcal{Z}(\bar{\mathbf{f}}, \bar{\mathbf{e}})=(\mathbf{f}, \mathbf{e})$ and the solution of our problem is simply a fixed point of $\mathcal{Z}$. Continuity of ${\mathbb M}_{n2}$ and ${\mathbf M}_{n2}$, $n=1, \cdots, N$, with respect to its parameters and a standard regularity argument allows us to conclude that $\mathcal{Z}$ is compact as mapping from $H^2(0, 1)$ into itself. The Schauder Fixed Point Theorem applies to yield the existence of a fixed point of $\mathcal{Z}$ as required. This ends the proof of Theorem \ref{theorem 1.1}. \qed
\subsection{Proof of Proposition \ref{proposition7point2}}\label{section6.2}
Note that (\ref{ts2a})-(\ref{boundary0000}) can be concerned as a small perturbation of a simpler problem in the form, for $n=1, \cdots, N$, \begin{align}\label{simpleproblem0} \varepsilon ^2\varsigma\Big[\, \mathcal{H}_1\, f''_n+\mathcal{H}_1'\, f'_n+ \big(\mathcal{H}_2'-\mathcal{H}_3\big)\, f_n\, \Big] -e^{-\beta(f_n-f_{n-1})} +e^{-\beta(f_{n+1}-f_n)} =\varepsilon^{2+\mu} \tilde{h}_n, \end{align} \begin{align} \label{boundary10} {\mathfrak b}_6f'_n(1)-{\mathfrak b}_7\, f_n(1)=0, \qquad {\mathfrak b}_1f'_n(0)-{\mathfrak b}_2\, f_n(0)=0, \end{align} where $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are functions defined in (\ref{mathcalH1})-(\ref{mathcalH3}). By similar arguments as done in Section 6 of \cite{weixuyang}, we can finish the proof of Proposition \ref{proposition7point2} if we can solve (\ref{simpleproblem0})-(\ref{boundary10}).
We now focus on the resolution theory for (\ref{simpleproblem0})-(\ref{boundary10}), whose proof basically follows the methods in \cite{delPKowWeiYang} and \cite{YangYang2013}. However, in this paper, the homogeneous boundary conditions in \eqref{boundary10} make the procedure much more complicated, which will be divided into three steps. In the first step, we will find an approximate solution by solving an algebraic system and then derive the improved equivalent nonlinear system of (\ref{simpleproblem0})-(\ref{boundary10}), see \eqref{equationpftildee}-\eqref{boundaryconditionoftildee}. In step 2, by the decomposition method, the problem can be further transformed into \eqref{equationofui}-\eqref{boundaryconditionofui}. To cancel the boundary error terms $\tilde{\mathbb G}_{1}$ and $\tilde{\mathbb G}_{2}$ (see \eqref{boundaryconditionofui}), we need to find more boundary correction terms ${\hat{\mathfrak u}}_n, \, n=1,\cdots, N$ (see \eqref{hbar}) in the expansions of $\mathfrak{u}_n$'s, which directly leads to the system \eqref{equationofddotu}-\eqref{boundaryconditionofddotu}. Finally, after giving the linear resolution theory in Lemma \ref{lemma6point4}, the proof can be finished by the Contraction Mapping Principle in Step 3.
\noindent{\bf Step 1:} By setting \begin{equation}\label{transformation} {\check f}_n(\theta)=\beta(\theta)f_n(\theta), \end{equation} we get \begin{equation}\label{simpleproblem} \varepsilon ^2\frac{\varsigma}{\beta} \big[\mathcal{H}_1{\check f}''_n+\tau_1{\check f}'_n+\tau_2{\check f}_n\, \big] \,-\, e^{-({\check f}_n-{\check f}_{n-1})} \, +\, e^{-({\check f}_{n+1}-{\check f}_n)} =\varepsilon^{2+\mu} \tilde{h}_n, \end{equation} \begin{align} \label{boundary1} {\check f}'_n(1)+K_2\, {\check f}_n(1)=0, \qquad {\check f}'_n(0)+K_1\, {\check f}_n(0)=0, \end{align} where ${\check f}_0=-\infty, \, {\check f}_{N+1}=\infty$. Here we have denoted \begin{equation}\label{tau1} \tau_1(\theta)=\mathcal{H}_1'(\theta)-2\frac{\beta'(\theta)}{\beta(\theta)}\mathcal{H}_1(\theta), \end{equation} \begin{equation}\label{tau2} \tau_2(\theta)=\mathcal{H}_2'(\theta)-\mathcal{H}_3(\theta)
+2\frac{|\beta'(\theta)|^2}{\beta^2(\theta)}\mathcal{H}_1(\theta) -\frac{\beta''(\theta)}{\beta(\theta)}\mathcal{H}_1(\theta) -\frac{\beta'(\theta)}{\beta(\theta)}\mathcal{H}_1'(\theta), \end{equation} \begin{equation}\label{K1K2} K_1=\frac{\beta'(0)}{\beta(0)}+\frac{{\mathfrak b}_2}{{\mathfrak b}_1}, \qquad K_2=\frac{\beta'(1)}{\beta(1)}+\frac{{\mathfrak b}_7}{{\mathfrak b}_6}, \end{equation} so that $$ \tau_2(\theta)>0, \qquad K_1\, =\, K_2\, =\, 0, $$ due to the assumptions in (\ref{taupositivity})-(\ref{boundaryadmissibility}).
Recall that the assumption \eqref{taupositivity} implies that ${\varsigma}(\theta)\tau_2(\theta)/\beta(\theta)>0$. Let us define two positive functions $\rho_\varepsilon(\theta)$ and $\delta(\theta)$ by \begin{align} e^{-\rho_\varepsilon(\theta)} \, =\, \varepsilon^2\frac{{\varsigma}(\theta)}{\beta(\theta)} \tau_2 (\theta)\rho_\varepsilon(\theta), \qquad \frac{1}{\delta^2(\theta)}\, =\, \tau_2(\theta)\rho_\varepsilon(\theta). \end{align} We can easily obtain that \begin{align}
\rho_\varepsilon(\theta)=2|\ln\varepsilon|-\ln\big(2|\ln\varepsilon|\big)
-\ln\Big(\frac{\varsigma(\theta) \tau_2(\theta)}{\beta(\theta)}\Big)+O\Big(\frac{\ln\big(2|\ln\varepsilon|\big)}{|\ln\varepsilon|}\Big), \end{align} \begin{align}\label{del1} \frac{1}{\delta^2(\theta)}\, =\,
\tau_2(\theta)\, \left[\, 2|\ln\varepsilon|
-\ln\big(2|\ln\varepsilon|\big) -\ln\Big(\frac{\varsigma(\theta)\tau_2(\theta)}{\beta(\theta)}\Big)
+\, O\Big(\frac{\ln\big(2|\ln\varepsilon|\big)}{|\ln\varepsilon|}\Big)\, \right]. \end{align} Then multiplying equation (\ref{simpleproblem}) by $\varepsilon^{-2}\delta^2(\theta)$ and setting \begin{equation*}
{\check f}_n(\theta)\, =\, \Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho_{\varepsilon}(\theta) +{\hat f}_n(\theta), \quad n=1, \cdots, N, \end{equation*} we get an equivalent system, for $n=1, \cdots, N$, \begin{align}\label{ts3a}
&\delta^2\big[\mathcal{H}_1{\hat f}''_n+\tau_1{\hat f}'_n+\tau_2{\hat f}_n\big]
\,-\, e^{-({\hat f}_n-{\hat f}_{n-1})} \, +\, e^{-({\hat f}_{n+1}-{\hat f}_n)} \nonumber\\[2mm] &=\varepsilon^\mu\delta^2\frac{\beta}{\varsigma} \tilde{h}_n- \delta^2\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho_{\varepsilon}'' -\delta^2\tau_1\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho_{\varepsilon}' -\Big(n-\frac{N}{2}-\frac{1}{2}\Big) , \end{align} where ${\hat f}_0=-\infty, \, {\hat f}_{N+1}=\infty$. The boundary conditions become \begin{align}\label{boundarysim} {\hat f}_n'(1)=-\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho_{\varepsilon}'(1), \qquad {\hat f}_n'(0)=-\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho_{\varepsilon}'(0). \end{align}
\begin{remark}\label{remark61}
Note that the terms of order $O(|\ln\varepsilon|)$ in the right hand sides of the equations in (\ref{boundarysim}) disappear so that they are of $O(1)$ due to the assumptions $K_1=K_2=0$ which are exactly given in (\ref{boundaryadmissibility}). \qed \end{remark}
First, we want to cancel the terms of $O(1)$ in right hand side of (\ref{ts3a}). To this end, we will introduce the following lemma
\begin{lemma}\label{lemma6point3} There exists a solution $\ddot{\bf f}=(\ddot{f}_1, \cdots, \ddot{f}_N)^{T}$ to the following nonlinear algebraic system \begin{align}\label{equationofe0}
-e^{- (\ddot{f}_{n}-\ddot{f}_{n-1})}+ e^{-(\ddot{f}_{n+1}-\ddot{f}_{n})}=-\Big(n-\frac{N}{2}-\frac{1}{2}\Big) \end{align} with $n$ running from $1$ to $N$, where $\ddot{f}_0=-\infty, \, \ddot{f}_{N+1}=\infty$. \end{lemma}
\begin{proof} By setting \begin{align}\label{deinfinitionofa_n0} a_0\, =\, a_{N}\, =\, 0, \quad a_n\, =\, e^{-(\ddot{f}_{n+1}-\ddot{f}_{n})}, \quad n=1, \cdots, N-1, \end{align} the proof can be found in the solving method for equation (7.10) in \cite{YangYang2013}. \end{proof}
We set ${\hat f}_n\, =\, \ddot{f}_n + {\tilde f}_n, \, n=1, \cdots, N$, where $\ddot{f}_n=O(1)$ satisfies the system \eqref{equationofe0}. It is obvious that system (\ref{ts3a})-(\ref{boundarysim}) is equivalent to the following nonlinear system of equations, \begin{align}\label{equationpftildee} \begin{split} \delta^2\Bigl[\mathcal{H}_1\, {\tilde f}_n'' +\tau_1{{\tilde f}_n'}+ \tau_2{{\tilde f}_n}\Bigr] &\, +\, a_{n-1}\bigl(\, {\tilde f}_n-{\tilde f}_{n-1} \bigr) \\ &\,-\, a_{n} \bigl(\, {\tilde f}_{n+1}-{\tilde f}_{n} \bigr) \, =\, \delta^2\, \varepsilon^{\mu}\frac{\beta}{\varsigma} \tilde{h}_{n}\, +\, \delta^2\tilde{g}_n+{\mathfrak N}_n({\bf{\tilde f}}), \end{split} \end{align} with boundary conditions \begin{align}\label{boundaryconditionoftildee} {\tilde f}_n'(0)\, =\, G_{1, n}, \qquad {\tilde f}_n'(1)\, =\, G_{2, n}, \end{align} where we have denoted \begin{align} {\tilde g}_n\, = \,-\, \Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho''_{\varepsilon} \,-\, \tau_1\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho'_{\varepsilon} \,-\, \tau_2\ddot{f}_n, \end{align} \begin{align}\label{G1G2} \, G_{1, n}=-\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho'_{\varepsilon}(0), \qquad \, G_{2, n}=-\Big(n-\frac{N}{2}-\frac{1}{2}\Big)\rho'_{\varepsilon}(1). \end{align} Moreover, the nonlinear terms ${\mathfrak N}_n, \, n=1, \cdots, N$, are given by \begin{align} \begin{split} {\mathfrak N}_n({\bf{\tilde f}}) =&\, a_{n-1} \big[e^{-(\, {\tilde f}_n - {\tilde f}_{n-1} )} -1+ \bigl(\, {\tilde f}_n - {\tilde f}_{n-1} \bigr)\big] \,-\, a_n\big[e^{-(\, {\tilde f}_{n+1} - {\tilde f}_{n} )}-1+ \bigl(\, {\tilde f}_{n+1} - {\tilde f}_{n} \bigr)\big], \end{split} \end{align} and ${\tilde f}_0=-\infty, \, {\tilde f}_{N+1}=\infty$.
\noindent{\bf Step 2:} The first try is to decompose the above system. We will denote: \begin{align*} {\bf {\tilde f}}=({\tilde f}_1, \cdots, {\tilde f}_N)^{T}, \qquad {\bf {\tilde h}}=({\tilde h}_1, \cdots, {\tilde h}_N)^{T}, \qquad {\bf {\tilde g}}=({\tilde g}_1, \cdots, {\tilde g}_N)^{T}, \end{align*} \begin{align*} {\mathfrak N}({\bf{\tilde f}})=\big({\mathfrak N}_1({\bf{\tilde f}}), \cdots, {\mathfrak N}_N({\bf{\tilde f}})\big)^{T}, \qquad {\mathbb G}_1=(G_{1, 1}, \cdots, G_{1, N})^{T}, \qquad {\mathbb G}_2=(G_{2, 1}, \cdots, G_{2, N})^{T}. \end{align*} Then system (\ref{equationpftildee}) becomes: \begin{align}\label{linear1} \begin{split} \delta^{2}{\bf I}\Bigl[\, \mathcal{H}_1\, \frac{\mathrm{d}^2}{\mathrm{d}\theta^2} \, +\, \tau_1\frac{\mathrm{d}}{\mathrm{d}\theta} \, +\, \tau_2 \, \Bigr]{\bf {\tilde f}} \, +\, {\mathbb A}{\bf {\tilde f}} \, =\, \delta^2\, \varepsilon^{\mu} \frac{\beta}{\varsigma}{\bf{\tilde h}}\, +\, \delta^2 {\bf{\tilde g}}\, +\, {\mathfrak N}({\bf{\tilde f}}), \end{split} \end{align} where $\bf I$ is a $ N\times N$ unit Matrix and the Matrix ${\mathbb A}$ defined as \begin{align}\label{definitionofbfB} {\mathbb A}\, =\, \left( \begin{array}{ccccccccc} a_1 & -a_1&0& 0&\cdots &0&0&0&0 \\ -a_1 &(a_1+a_2)&-a_2& 0&\cdots &0&0&0&0 \\ \vdots& \vdots&\vdots & \vdots&\ddots& \vdots& \vdots& \vdots&\vdots \\ 0 & 0&0&0&\cdots& 0&-a_{N-2}&(a_{N-2}+a_{N-1})&-a_{N-1} \\ 0 & 0&0&0&\cdots&0& 0&-a_{N-1}&a_{N-1} \end{array} \right). \end{align} For the symmetric matrix ${\mathbb A}$, using elementary matrix operations it is easy to prove that there exists an invertible matrix ${\bf Q}$ such that $$ {\bf Q}{\mathbb A}{\bf Q}^T=\mbox{diag}(a_1, \cdots, a_{N-1}, 0). $$ Since $a_1, \cdots, a_{N-1}$ are positive constants defined in (\ref{deinfinitionofa_n0}), then all eigenvalues of the matrix ${\mathbb A}$ are \begin{align*}
\lambda_1\geq\lambda_2\geq \cdots\geq\lambda_{N-1}>\lambda_N=0. \end{align*} Moreover, since ${\mathbb A}$ is a symmetric matrix, there exists another invertible matrix $\bf P$ independent of $\theta$ with the form \begin{align}\label{mathbfP} \mathbf{P}\mathbf{P}^T={\bf I}, \qquad \mathbf{P}= \left(
\begin{array}{cccc}
p_{11} & \cdots & p_{1N-1} & \frac{1}{\sqrt N} \\
p_{21} & \cdots & p_{2N-1} & \frac{1}{\sqrt N} \\
\vdots & \vdots & \vdots & \vdots \\
p_{N1} & \cdots &p_{NN-1} & \frac{1}{\sqrt N} \\
\end{array} \right), \end{align} in such a way that \begin{align}\label{P} {\bf P}^T{\mathbb A}{\bf P}=\mbox{diag}(\lambda_1, \lambda_2, \cdots, \lambda_{N-1}, \lambda_{N}). \end{align}
We denote \begin{align}\label{alphaanddelta}
\kappa=\frac{1}{ 2|\ln\varepsilon|-\ln(2|\ln\varepsilon|)}, \qquad \frac{\delta^{-2}(\theta)} {\,\kappa\, } \, =\, \tau_2(\theta)\, +\, {\tilde\sigma}(\theta). \end{align} By (\ref{del1}) we have \begin{align}
{\tilde\sigma}(\theta)\, =\, O\Big(\frac{1}{|\ln\varepsilon|}\Big). \label{sigma ep} \end{align} Multiplying (\ref{linear1}) by $\tau_2(\theta)\, +\, {\tilde\sigma}(\theta)$, we get the following system \begin{align}\label{linear} \begin{split} \kappa {\bf I}\Bigl[\, \mathcal{H}_1\, \frac{\mathrm{d}^2}{\mathrm{d}\theta^2} +\tau_1\frac{\mathrm{d}}{\mathrm{d}\theta} \, +\, \tau_2 \Bigr]{\bf {\tilde f}} \, +\, \bigl(\tau_2+{\tilde\sigma}\bigr){\mathbb A}{\bf {\tilde f}} \, =\, \kappa\varepsilon^{\mu}\frac{\beta}{\varsigma} {\bf{\tilde h}} \, +\, \kappa {\bf{\tilde g}} \, +\, \big(\tau_2+{\tilde\sigma}\big){\mathfrak N}({\bf{\tilde f}}). \end{split} \end{align}
Now, define six new vectors \begin{align*} \mathfrak{u}=(\mathfrak{u}_1, \cdots, \mathfrak{u}_{N})^T={\bf P}^T{\bf {\tilde f}}, \qquad \tilde{\mathfrak{h}}=(\tilde{\mathfrak{h}}_1, \cdots, \tilde{\mathfrak{h}}_{N})^T= \frac{\beta}{\varsigma}{\bf P}^T {\bf {\tilde h}}, \qquad \tilde{\mathfrak{g}}=(\tilde{\mathfrak{g}}_1, \cdots, \tilde{\mathfrak{g}}_{N})^T= {\bf P}^T {\bf{\tilde g}}, \end{align*} \begin{align*} {\tilde{\mathfrak N}}(\mathfrak{u})=\big({\tilde{\mathfrak N}}_1(\mathfrak{u}), \cdots, {\tilde{\mathfrak N}}_N(\mathfrak{u})\big)^{T} =\big(\tau_2+{\tilde\sigma}\big){\bf P}^T{\mathfrak N}({\bf{\tilde f}}) =\big(\tau_2+{\tilde\sigma}\big){\bf P}^T{\mathfrak N}({\bf P}\mathfrak{u}), \end{align*} \begin{align*} \tilde{{\mathbb G}}_1\, =\, (\tilde{G}_{1, 1}, \cdots, \tilde{G}_{1, N})^{T}={\bf P}^T {\mathbb G}_1, \qquad \tilde{{\mathbb G}}_2\, =\, (\tilde{G}_{2, 1}, \cdots, \tilde{G}_{2, N})^{T}={\bf P}^T {\mathbb G}_2. \end{align*} Note that the form of ${\mathbf P}$ in (\ref{mathbfP}) and the expressions of ${\mathbb G}_1$ and ${\mathbb G}_2$ in (\ref{G1G2}) imply that $$ \tilde{G}_{1, N}=\tilde{G}_{2, N}=0. $$ Therefore \eqref{equationpftildee}-(\ref{boundaryconditionoftildee}) become \begin{align}\label{equationofui} \kappa\, \Big[\mathcal{H}_1\, \mathfrak{u}''+\tau_1\mathfrak{u}'+ \tau_2\mathfrak{u}\Big] \, +\, \mbox{diag}(\lambda_1, \cdots, \lambda_N)\, (\tau_2+{\tilde\sigma})\mathfrak{u} =\, \kappa\, \varepsilon^{\mu}\tilde{\mathfrak h} \, +\, \kappa \tilde{\mathfrak{g}} \, +\, {\tilde{\mathfrak N}}(\mathfrak{u}), \end{align} with boundary conditions \begin{equation}\label{boundaryconditionofui} \mathfrak{u}'(0)=\tilde{{\mathbb G}}_1, \quad \mathfrak{u}'(1)=\tilde{{\mathbb G}}_2. \end{equation}
For the convenience of notation, we denote \begin{align*}
\ell_{n}\, =\, \Big( \frac{\kappa}{\lambda_n}\Big)^{\frac{1}{2}},
\qquad
\Pi(s)\, =\, \tau_2(s)+{\tilde\sigma}(s), \end{align*} \begin{equation*} \vartheta(\theta)=\int_{0}^{\theta} \Pi(s)^{\frac{1}{2}}\, {\mathrm{d}}s, \qquad l_0=\int_{0}^{1} \Pi(s)^{\frac{1}{2}}\, {\mathrm{d}}s. \end{equation*} In order to cancel the error terms on the boundary in \eqref{boundaryconditionofui}, we introduce the following functions \begin{align}\label{hbar} {\hat{\mathfrak u}}_n(\theta)\, =&\chi(\theta)\frac{\tilde{G}_{1, n}\ell_{n}}{\sqrt{\Pi(0)}}\, \sin\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \,-\, \big(1-\chi(\theta)\big)\frac{\tilde{G}_{2, n}\ell_{n}}{\sqrt{\Pi(1)}} \sin\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big), \quad n=1, \cdots, N-1, \nonumber \end{align} and $$ {\hat{\mathfrak u}}_N(\theta)=0. $$ In the above, $\chi$ is a smooth cut-off function with the properties $$
\chi(\theta)=1\quad \mbox{if } |\theta|<1/8 \qquad\mbox{and}\qquad
\chi(\theta)=0\quad \mbox{if } |\theta|>2/8. $$ It is easy to show \begin{align}
\|{\hat{\mathfrak u}}_n\|_{L^2(0, 1)}\, \leq\, \frac{C}{\sqrt{|\ln\varepsilon|}}. \end{align} For later use, we compute, for $n=1, \cdots, N-1$, \begin{align*} {\hat{\mathfrak u}}_n'(\theta) \, =&\chi(\theta)\frac{\tilde{G}_{1, n}}{\sqrt{\Pi(0)}}\, \sqrt{\Pi(\theta)}\cos\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \, +\, \big(1-\chi(\theta)\big)\frac{\tilde{G}_{2, n}}{\sqrt{\Pi(1)}}\, \sqrt{\Pi(\theta)}\cos\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big) \\[2mm] &\, +\, \frac{\tilde{G}_{1, n}\ell_{n}}{\sqrt{\Pi(0)}}\, \chi'(\theta)\sin\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \, +\, \frac{\tilde{G}_{2, n}\ell_{n}}{\sqrt{\Pi(1)}} \chi'(\theta)\sin\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big). \end{align*} This implies that $\hat{\mathfrak u}_n$ satisfies the following boundary conditions \begin{align} {\hat{\mathfrak u}}'_n(0)\, =\, \tilde{G}_{1, n}, \qquad {\hat{\mathfrak u}}'_n(1)\, =\, \tilde{G}_{2, n}, \quad n=1, \cdots, N. \end{align} For $n=1, \cdots, N-1$, there holds \begin{align*} {\hat{\mathfrak u}}_n''(\theta)\, =& -\chi(\theta)\frac{\tilde{G}_{1, n}}{\sqrt{\Pi(0)}}\, \, \frac{\Pi(\theta)}{\ell_{n}}\sin\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \,+\, \big(1-\chi(\theta)\big)\frac{\tilde{G}_{2, n}}{\sqrt{\Pi(1)}}\, \, \frac{\Pi(\theta)}{\ell_{n}}\sin\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big) \\[2mm] &\,+\,\chi(\theta)\frac{\tilde{G}_{1, n}}{\sqrt{\Pi(0)}}\, \frac{\Pi'(\theta)}{\sqrt{\Pi(\theta)}}\, \cos\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \,+\, \big(1-\chi(\theta)\big)\frac{\tilde{G}_{2, n}}{\sqrt{\Pi(1)}}\, \frac{\Pi'(\theta)}{\sqrt{\Pi(\theta)}}\, \cos\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big) \\[2mm] &\,+\,\chi'(\theta)\frac{\tilde{G}_{1, n}}{\sqrt{\Pi(0)}}\, \Pi(\theta)\cos\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \,-\, \chi'(\theta)\frac{\tilde{G}_{2, n}}{\sqrt{\Pi(1)}}\, \Pi(\theta)\cos\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big) \\[2mm] &\,+\,\chi''(\theta)\frac{\tilde{G}_{1, n}\ell_{n}}{\sqrt{\Pi(0)}}\, \sin\Big(\frac{\vartheta(\theta)}{\ell_{n}}\, \Big) \,+\, \chi''(\theta)\frac{\tilde{G}_{2, n}\ell_{n}}{\sqrt{\Pi(1)}}\, \sin\Big(\frac{l_0-\vartheta(\theta)}{\ell_{n}}\, \Big). \end{align*} Whence, we obtain, for $n=1, \cdots, N$, \begin{align}
\Big\|\kappa\Big[\mathcal{H}_1\, {\hat{\mathfrak u}}_n''+\tau_1 {\hat{\mathfrak u}}_n'+\tau_2{\hat{\mathfrak u}}_n\Big]
+\lambda_n(\tau_2+{\tilde\sigma}){\hat{\mathfrak u}}_n\Big\|_{L^2(0, 1)} \, \leq\,
\frac{C}{|\ln\varepsilon|}. \end{align}
Letting $\mathfrak{u}\, =\, {\tilde{\mathfrak u}}\, +\, {\hat{\mathfrak u}}$ with ${\hat{\mathfrak u}}=({\hat{\mathfrak u}}_1, \cdots, {\hat{\mathfrak u}}_N)^T$, the system \eqref{equationofui}-\eqref{boundaryconditionofui} is equivalent to the following system, for $n=1, \cdots, N$, \begin{align}\label{equationofddotu} \begin{split} \kappa\Big[ \mathcal{H}_1\, {\tilde{\mathfrak u}}_n'' +\tau_1 {\tilde{\mathfrak u}}_n' +\tau_2{\tilde{\mathfrak u}}_n \Big] +\lambda_n(\tau_2+{\tilde\sigma}){\tilde{\mathfrak u}}_n \, =\, \kappa\varepsilon^{\mu}{\mathfrak h}_n \, +\, \kappa \tilde {\mathfrak{g}}_n \, +\, \hat{\mathfrak{g}}_n \, +\, {\tilde{\mathfrak N}}_n({\tilde{\mathfrak u}+{\hat{\mathfrak u}}}), \end{split} \end{align} with boundary conditions \begin{equation}\label{boundaryconditionofddotu} {\tilde{\mathfrak u}}_n'(0)\, =\, 0, \qquad {\tilde{\mathfrak u}}_n'(1)\, =\, 0, \end{equation} where \begin{equation*}
\hat{\mathfrak{g}}_n \, =\, - \kappa\Big[\mathcal{H}_1\, {\hat{\mathfrak u}}_n''+\tau_1{\hat{\mathfrak u}}_n' \, +\, \tau_2{\hat{\mathfrak u}}_n\Big] \,-\, \lambda_n(\tau_2+{\tilde\sigma}){\hat{\mathfrak u}}_n, \quad n=1, \cdots, N-1, \end{equation*} and also \begin{equation*}
\tilde{\mathfrak{g}}_N\, =\, 0. \end{equation*} For later use, we will estimate the terms in the right hand of \eqref{equationofddotu}. For $n=1, \cdots, N-1$, there hold \begin{align*} {\tilde{\mathfrak N}}_n({\tilde{\mathfrak u}+{\hat{\mathfrak u}}}) &=\big(\tau_2+{\tilde\sigma}\big)\big({\bf P}^T{\mathfrak N}({\bf P}(\tilde{\mathfrak{u}}+{\hat{\mathfrak u}}))\big)_n =\big(\tau_2+{\tilde\sigma}\big)\sum_{i=1}^{N}p_{in}{\mathfrak N}_i({\bf P}(\tilde{\mathfrak{u}}+{\hat{\mathfrak u}})), \end{align*} and also \begin{equation*}
\tilde{{\mathfrak h}}_N\, =\, \frac{1}{\sqrt{N}}\, \sum_{i=1}^{N}\, \tilde{h}_i,
\qquad \tilde{{\mathfrak g}}_N\, =\, \frac{1}{\sqrt{N}}\, \sum_{i=1}^{N}\, \tilde{g}_i, \end{equation*} \begin{equation*}
{\tilde{\mathfrak N}}_N({\tilde{\mathfrak u}+{\hat{\mathfrak u}}})
=\big(\tau_2+{\tilde\sigma}\big)\big({\bf P}^T{\mathfrak N}({\bf P}(\tilde{\mathfrak{u}}+{\hat{\mathfrak u}}))\big)_N
=\big(\tau_2+{\tilde\sigma}\big)\frac{1}{\sqrt{N}}\sum_{i=1}^{N}{\mathfrak N}_i({\bf P}(\tilde{\mathfrak{u}}+{\hat{\mathfrak u}}))=0. \end{equation*} According to the definitions of $\tilde{\mathfrak h}_n$'s and $\tilde{\mathfrak g}_n$'s, we can easily get \begin{align}\label{evaluefrakg}
\|\tilde{\mathfrak h}_n\|_{L^2(0, 1)}\, \leq\, C, \quad
\|\tilde{\mathfrak g}_n\|_{L^2(0, 1)}\, \leq\, C, \quad n=1, \cdots, N. \end{align}
\noindent{\bf Step 3:} For the purpose of using a fixed point argument to solve \eqref{equationofddotu}-\eqref{boundaryconditionofddotu}, we concern the following resolution theory for the linear differential equations.
\begin{lemma}\label{lemma6point4}\ {\textbf{(1).}} Assume that the non-degeneracy condition (\ref{nondegeneracy}) holds. For any small $\varepsilon$, there exists a unique solution $v$ to the equation \begin{align}\label{equationofvvv} \Big[\mathcal{H}_1\frac{\mathrm{d}^2}{\mathrm{d}\theta^2}+\tau_1\, \frac{\mathrm{d}}{\mathrm{d}\theta} +\tau_2\Big]v \, =\, h, \qquad v'(0)\, =\, 0, \qquad v'(1)\, =\, 0, \end{align} with the estimate \begin{align}
\|v\|_{H^2(0, 1)} \leq C\, {\|h\|_{L^2(0, 1)}}. \label{l2estimatev} \end{align}
\noindent{\textbf{(2).}} Consider the following system, for $n=1, \cdots, N-1$, \begin{equation}\label{equationofvn} \begin{split} \kappa\, \Big[\mathcal{H}_1\frac{\mathrm{d}^2}{\mathrm{d}\theta^2} +\tau_1\, \frac{\mathrm{d}}{\mathrm{d}\theta} +\tau_2\Big]v_n &\, +\, \lambda_n\Bigl(\tau_2+{\tilde\sigma}\Bigr)v_n \, =\, \mathfrak{p}_n, \\ \qquad v_n'(0)\, =\, & 0, \qquad v_n'(1)\, =\, 0. \end{split} \end{equation} There exists a sequence $\{\varepsilon_l, \, l\in {\mathbb N}\}$ approaching $0$ and satisfying the gap condition (\ref{gapconditionofve}) such that problem (\ref{equationofvn}) has a unique solution $\bf{v}\, =\, \bf{v}(\bf{\mathfrak{p}})$ and \begin{align}\label{h2estimate1}
\frac{1}{|\ln\varepsilon_l|}\, \|{\bf v}''\|_{L^2(0, 1)}
+\frac{1}{\sqrt{|\ln\varepsilon_l|}}\, \|{\bf v}'\|_{L^2(0, 1)}
+\|{{\bf v}}\|_{L^{2}(0, 1)}\leq C\, \sqrt{|\ln\varepsilon_l|}\, \|{\bf \mathfrak{p}}\|_{L^2(0, 1)}, \end{align} where ${\bf v}=(v_1, \cdots, v_{N-1})^{T}$ and ${\bf{\mathfrak{p}}}=(\mathfrak{p}_1, \cdots, \mathfrak{p}_{N-1})^T$. Moreover, if ${\bf{ \mathfrak{p}}}\in H^2(0, 1)$ then \begin{align}
\frac{1}{|\ln\varepsilon_l|}\, \|{\bf v}''\|_{L^2(0, 1)}
+ \|{\bf v}'\|_{L^2(0, 1)}
+\|{{\bf v}}\|_{L^{2}(0, 1)}\leq C\, \|{\bf \mathfrak{p}}\|_{H^2(0, 1)}. \label{h2estimate2} \end{align} \end{lemma}
\begin{proof} We can use the inverse of the transformation in (\ref{transformation}), i.e., $v\, =\, \beta(\theta)\, \tilde{v}$, and then obtain \begin{equation} \frac{\mathrm{d}}{\mathrm{d}\theta}\, v\, =\, \beta'\, \tilde{v}\, +\, \beta\, \frac{\mathrm{d} }{\mathrm{d}\theta}\, \tilde{v}, \qquad \frac{\mathrm{d}^2}{\mathrm{d}\theta^2}\, v\, =\, \beta''\, \tilde{v}\, +\, 2\, \beta'\, \frac{\mathrm{d} }{\mathrm{d}\theta}\, \tilde{v}\, +\, \beta\, \frac{\mathrm{d}^2}{\mathrm{d}\theta^2}\, \tilde{v}. \end{equation} Therefore, we get an equivalent problem of (\ref{equationofvvv}) \begin{equation}\label{equationoftidlev} \begin{split}
\mathcal{H}_1\tilde{v}_{\theta\theta}
\, +\, \mathcal{H}_1'\, \tilde{v}_{\theta}
\, +\, \big( \mathcal{H}_2'-\mathcal{H}_3\big) \, \tilde{v} & \, =\, \frac{1}{\beta} h,
\\[2mm]
{\mathfrak b}_1\tilde{v}'(0) \,-\, {\mathfrak b}_2\, \tilde{v}(0)\, =\, 0, \quad
{\mathfrak b}_6\tilde{v}'(1)& \,-\, {\mathfrak b}_7\, \tilde{v}(1)\, =\, 0. \end{split} \end{equation} Recalling the definition of $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ in \eqref{mathcalH1}-\eqref{mathcalH3} and applying the non-degeneracy condition (\ref{nondegeneracy}), we can solve \eqref{equationoftidlev} directly. The proof of the second part is similar as that for Claim 1 and Claim 2 in \cite{YangYang2013}. The details are omitted here. \end{proof}
In order to solve (\ref{equationofddotu})-(\ref{boundaryconditionofddotu}), we first concern the system, for $n=1, \cdots, N-1$ \begin{align}\label{equationmathbfwn} &\kappa \Big[\mathcal{H}_1{\bf {\tilde w}}_n'' \, +\, \tau_1\, {\bf {\tilde w}}_n' \, +\, \tau_2\, {\bf {\tilde w}}_n\Big] \, +\, \lambda_n\big(\tau_2\, +\, {\tilde\sigma}\big){\bf {\tilde w}}_n \, =\, \kappa\tilde{\mathfrak g}_n \, +\, \hat{\mathfrak{g}}_n, \end{align} \begin{equation}\label{equationmathbfwN} \mathcal{H}_1{\bf {\tilde w}}_N'' \, +\, \tau_1\, {\bf {\tilde w}}_N' \, +\, \tau_2\, {\bf {\tilde w}}_N \, =\, \tilde{\mathfrak g}_N, \end{equation} with boundary conditions \begin{equation}\label{boundarymathbfw} {\bf {\tilde w}}_n'(0)\, =\, 0, \qquad {\bf {\tilde w}}_n'(1)\, =\, 0, \qquad n=1, \cdots, N. \end{equation} Using Lemma \ref{lemma6point4}, we can solve the above system and get the estimates as in (\ref{mathbfwn})-(\ref{mathbfwN}). The substituting $$ {\tilde{\mathfrak u}}_n={\bf {\tilde w}}_n+{\check{\mathfrak u}}_n,\quad n=1, \cdots, N, $$
will then imply that the nonlinear problem (\ref{equationofddotu})-(\ref{boundaryconditionofddotu}) can be transformed into the following system for $n=1, \cdots, N$, \begin{equation}\label{equationofddotu1} \kappa\Big[\mathcal{H}_1\, {\check{\mathfrak u}}_n'' \, +\, \tau_1\, {\check{\mathfrak u}}_n' \, +\, \tau_2\, {\check{\mathfrak u}}_n\Big] \, +\, \lambda_n\, \big(\tau_2\, +\, \tilde{\sigma}\big) {\check{\mathfrak u}}_n \, =\, \kappa\varepsilon^{\mu}{\mathfrak h}_n \, +\, \tilde{\mathfrak N}_n({{\check{\mathfrak u}}+{\bf {\tilde w}}+\hat{\mathfrak u}}), \end{equation} with boundary conditions \begin{equation}\label{boundaryconditionofddotu1} {\check{\mathfrak u}}_n'(0)\, =\, 0, \qquad {\check{\mathfrak u}}_n'(1)\, =\, 0 \qquad n=1, \cdots, N. \end{equation}
Finally, we claim that problem (\ref{equationofddotu1})-(\ref{boundaryconditionofddotu1}) can be solved by using Lemma \ref{lemma6point4} and a Contraction Mapping Principle in the set \begin{align*} {\mathcal{X}}\, =\, \left\{\, {\check{\mathfrak u}}\in H^2(0, 1) \, :\,
\frac{1}{|\ln\varepsilon|}\, \|{\check{\mathfrak u}}''\|_{L^2(0, 1)}
\, +\, \frac{1}{\sqrt{|\ln\varepsilon|}}\, \|{\check{\mathfrak u}}'\|_{L^2(0, 1)}
\, +\, \|{\check{\mathfrak u}}\|_{L^2(0, 1)} \, \leq\,
\frac{C}{\sqrt{|\ln\varepsilon|}}\, \right\}. \end{align*} In fact, this can be done in the following way. According to the definition of ${\tilde{\mathfrak N}} $, we obtain, for any $ \, {\check{\mathfrak u}}^0 \in {\mathcal{X}}$ \begin{align*} {\tilde{\mathfrak N}}({\check{\mathfrak u}}^0+{\bf{\tilde w}}+{\hat{\mathfrak u}})
=\big(\tau_2+{\tilde\sigma}\big){\bf P}^T \left(\begin{array}{c} {\mathfrak N}_1\big({\bf P}(\check{\mathfrak u}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big) \\[2mm] \vdots \\[2mm] {\mathfrak N}_N\big({\bf P}(\check{\mathfrak u}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big) \end{array} \right). \end{align*} This implies that \begin{equation*}
\big\| {\tilde{\mathfrak N}}({\check{\mathfrak u}}^0+{\bf{\tilde w}}+{\hat{\mathfrak u}})\big\|_{L^2(0, 1)}
\leq C\sum_{n=1}^{N} \big\|{\mathfrak N}_{n}\big({\bf P}(\check{\mathfrak u}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big)\big\|_{L^2(0, 1)},
\end{equation*} where the expression of ${\mathfrak N}_{n} $ is \begin{align*} {\mathfrak N}_{n}\big({\bf P}(\check{\mathfrak u}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big) =&\, a_{n-1} \Bigg\{e^{-\left[\, ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_n -\big({\bf P}(\check{\mathfrak u}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big)_{n-1} \right]} -1 \\ & \qquad \quad +\Big[( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_n - ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_{n-1} \Big]\Bigg\} \\[2mm] & \,-\, a_n\Bigg\{e^{-\left[\, ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_{n+1} - ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_{n} \right]} -1 \\ & \qquad \quad+\Big[\, ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_{n+1} - ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_{n} \Big]\Bigg\} \\[2mm]
=&O\Big(\Big|\big( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}}))_n - ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big)_{n-1} \Big|^2\Big) \\[2mm]
&+ O\Big(\Big| \big( {\bf P}({\check{\mathfrak u}}^0+{\mathbf w}+{\hat{\mathfrak u}}))_{n+1} - ( {\bf P}({\check{\mathfrak u}}^0+{\bf {\tilde w}}+{\hat{\mathfrak u}})\big)_{n} \Big|^2\Big). \end{align*} The definitions of ${\hat{\mathfrak u}}$ in \eqref{hbar} and ${\bf P}$ in (\ref{mathbfP}) will imply that \begin{equation*}
\big\| |({\bf P}{\hat{\mathfrak u}})_n|^2\big\|_{L^2(0, 1)}\leq \frac{C}{|\ln\varepsilon|}, \qquad
\big\| |({\bf P}{\check{\mathfrak u}}^0)_n|^2\big\|_{L^2(0, 1)}\leq \frac{C}{|\ln\varepsilon|},
\quad n=1, \cdots, N. \end{equation*} Gathering the above estimates, we get the following estimate \begin{equation*}
\big\|{\tilde{\mathfrak N}}({\check{\mathfrak u}}^0+{\bf{\tilde w}}+{\hat{\mathfrak u}})\big\|_{L^2(0, 1)}\leq \frac{C}{|\ln\varepsilon|}. \end{equation*} Therefore, for any $n=1, \cdots, N-1$, using (\ref{evaluefrakg}) and Lemma \ref{lemma6point4}, we can get a solution ${\check{\mathfrak u}}_n$ to \begin{align*} \kappa\Big[\mathcal{H}_1\, {\check{\mathfrak u}}_n'' \, +\, \tau_1\, {\check{\mathfrak u}}_n' \, +\, \tau_2\, {\check{\mathfrak u}}_n\Big] \, +\, &\lambda_n\, \big(\tau_2\, +\, \tilde{\sigma}\big){\check{\mathfrak u}}_n \, =\, \kappa\varepsilon^{\mu}\tilde{\mathfrak h}_n\, +\, {\tilde{\mathfrak N}}_n({\check{\mathfrak u}}^0+{\bf{\tilde w}}+{\hat{\mathfrak u}}), \\[2mm] {\check{\mathfrak u}}_n'(0)\, =\, &0, \qquad \qquad {\check{\mathfrak u}}_n'(1)\, =\, 0, \end{align*} with the following estimate \begin{align*}
&\frac{1}{|\ln\varepsilon_l|}\, \|{\check{\mathfrak u}}_n''\|_{L^2(0, 1)}
+\frac{1}{\sqrt{|\ln\varepsilon_l|}}\, \|{\check{\mathfrak u}}_n'\|_{L^2(0, 1)}
+\|{\check{\mathfrak u}}_n\|_{L^{2}(0, 1)} \\[2mm]
&\leq C\, \sqrt{|\ln\varepsilon_l|}\,
\bigg\|\kappa\varepsilon^{\mu}\tilde{\mathfrak h}_n \, +\,
{\tilde{\mathfrak N}}_n({\check{\mathfrak u}}^0+{\bf{\tilde w}}+{\hat{\mathfrak u}})\bigg\|_{L^2(0, 1)} \\
&\, \leq\, \varepsilon^{\mu}\|\tilde{\mathfrak h}_n\|_{L^2(0, 1)}
\, +\, \frac{C}{\sqrt{|\ln\varepsilon_l|}} \, \leq \, \frac{C}{\sqrt{|\ln\varepsilon_l|}}. \end{align*} Concerning the $N$-th equation in \eqref{equationofddotu}-(\ref{boundaryconditionofddotu}), i.e., \begin{equation*} \mathcal{H}_1\, {\check{\mathfrak u}}_N'' \, +\, \tau_1\, {\check{\mathfrak u}}_N' \, +\, \tau_2\, {\check{\mathfrak u}}_N \, =\, \varepsilon^{\mu}\tilde{\mathfrak h}_N, \qquad {\check{\mathfrak u}}_N'(0)\, =\, 0, \qquad {\check{\mathfrak u}}_N'(1)\, =\, 0, \end{equation*} using (\ref{evaluefrakg}) and Lemma \ref{lemma6point4}, we can also find a solution satisfying \begin{equation*} \begin{split}
\|{\check{\mathfrak u}}_N\|_{H^2(0, 1)}
\leq \, \varepsilon^{\mu}\|\tilde{\mathfrak h}_N
\|_{L^2(0, 1)}
\leq \frac{C}{\sqrt{|\ln\varepsilon_l|}}. \end{split} \end{equation*} Now, the result follows by a straightforward application of Contraction Mapping Principle and Lemma \ref{lemma6point4}. The proof of Proposition \ref{proposition7point2} is complete. \qed
\noindent {\bf Acknowledgements:} S. Wei was supported by NSFC (No. 12001203) and Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515110622); J. Yang was supported by NSFC (No. 11771167 and No. 11831009). \qed
\begin{appendices}
\section{The derivation of the equation for $h$}\label{appendixA} \setcounter{equation}{0}
The computations of (\ref{solvable of varphi1}) can be showed as follows. Since $S_{6, j}$, $S_{8, j}$, $M_{11, j}(x, z)$, $M_{21, j}(x, z)$, $M_{51, j}(x, z)$ are even functions of $x_j$, then integration against $ w_{j, x}$ therefore just vanish. This gives that \begin{equation*} \begin{split}
\text{LHS of \eqref{solvable of varphi1}}&=\int_{{\mathbb R}}\big[\varepsilon^2S_{7, j}+\varepsilon^2S_{9, j}+M_{31, j}(x, z)+M_{41, j}(x, z)+M_{61, j}(x, z)\big]w_{j, x}{\rm d}x \\[2mm] &\equiv J_1\, +\, J_2\, +\, J_3\, +\, J_4\, +\, J_5. \end{split} \end{equation*} These terms can be computed in the sequel.
\noindent $\bullet$ Recalling the expression of $S_{7, j}$ in (\ref{sv1-gather}), direct computation leads to \begin{align} \label{J1} J_{1}\, =\, &-\varepsilon^2\Big( -\frac {h_5}{\beta}{h} +h_2\frac {h''}{\beta} +h_2\frac {2\beta'}{\beta^2}{h'} +h_2\frac {2\alpha'}{\alpha \beta}{h'} \Big)\int_{{\mathbb R}}w_{j, x}^{2}\, {\rm d}x \nonumber\\[2mm] &-\varepsilon^2h_2\frac {2\beta'}{\beta^2}{h'}\int_{{\mathbb R}}x_j w_{j, x}w_{j, xx}\, {\rm d}x +\varepsilon^2\frac{2h_7}{\beta}h\int_{{\mathbb R}}x_j w_{j, x}w_{j, xx}\, {\rm d}x -\varepsilon^2\frac { V_{tt}(0, \varepsilon z)}{\beta^3}{h}\int_{{\mathbb R}}x_j w_{j}w_{j, x}\, {\rm d}x \nonumber\\[2mm]
\, =\, &\varepsilon^2\, \frac{\varrho_1}{\beta}\Big[\, -h_2\, h'' -h_2(\frac{\beta'}{\beta}+2\frac{\alpha'}{\alpha}){h'}+ (h_5-h_7+\sigma V_{tt}(0, \varepsilon z)\beta^{-2})h\, \Big], \end{align} where we have used the relations \eqref{relationofwx} and \begin{equation} \label{varrho1}
\varrho_{1}\, \equiv\, \int_{{\mathbb R}}w_x^2\, {\rm d}x
\, =\, -2\int_{{\mathbb R}}xw_xw_{xx}\, {\rm d}x. \end{equation}
\noindent $\bullet$ According to the definition of $S_{9, j}$ in (\ref{sv1-gather}) and \eqref{varrho1}, it follows that \begin{align} \label{J2} J_{2}\, =\, & \frac{\varepsilon^2}{\alpha \beta^2}\int_{{\mathbb R}}\big[h_6\Big(-\alpha \beta h' +\alpha \beta'h \Big)x_jw_{j, xx} + h_6\Big( \alpha'\beta h+\alpha\beta'h\Big)w_{j, x} - h_4\alpha\beta h' w_{j, x} \big]w_{j, x}\, {\rm d}x \nonumber\\[2mm]
\, =\, &\varepsilon^2\, \frac{\varrho_1}{\beta}\Big[ h_6\Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big)h +\frac{1}{2} h_6 h'-h_4 h' \Big]. \end{align}
\noindent $\bullet$ By the definition of $M_{31, j}(x, z)$ in \eqref{M31}, the facts $\omega_{0, j}$, $\omega_{1, j}$, $w_{j, x}$ are odd functions of $x_j$ and $\omega_{2, j}, \omega_{3, j}$ are even functions of $x_j$, we obtain \begin{align}\label{J3} J_3 \, =\, &\varepsilon^2 h\int_{{\mathbb R}} \Big\{\frac{h_3}{\beta} \big(a_{12}\omega_{2, j, x} +a_{13}\omega_{3, j, x}\big) + \frac{h_8}{\beta} \big( a_{12}x_j\omega_{2, j, xx} +a_{13}x_j\omega_{3, j, xx}\big) \nonumber\\[2mm] &\qquad \qquad -\, \frac{V_t(0, \varepsilon z)}{\beta^3}\big(a_{12}x_j\omega_{2, j} +a_{13}x_j\omega_{3, j}\big) \nonumber\\[2mm] &\qquad \qquad+h_8 \big( a_{10}\omega_{0, j, xx} +a_{11}\omega_{1, j, xx}\big) -\, \frac{V_t(0, \varepsilon z)}{\beta^2}\big( a_{10}\omega_{0, j} +a_{11}\omega_{1, j}\big)\Big\}w_{j, x}\, {\rm d}x \nonumber\\ \, =\, &\varepsilon^2 h\int_{{\mathbb R}} \Big\{h_1a_{10} \big(a_{12}\omega_{2, j, x} +a_{13}\omega_{3, j, x}\big) + h_1a_{11}\big( a_{12}x_j\omega_{2, j, xx} +a_{13}x_j\omega_{3, j, xx}\big) \nonumber\\ &\qquad \qquad \qquad+h_1( a_{10}-\frac{1}{2}a_{11})\big(a_{12}\sigma^{-1}x_j\omega_{2, j} +a_{13}\sigma^{-1}x_j\omega_{3, j}\big)\Big\}w_{j, x}\, {\rm d}x \nonumber\\ &+\varepsilon^2 h\int_{{\mathbb R}}\Big\{ h_1a_{13}\big( a_{10}\omega_{0, j, xx} +a_{11}\omega_{1, j, xx}\big) +\, h_1a_{12}\big( a_{10}\omega_{0, j} +a_{11}\omega_{1, j}\big)\Big\}w_{j, x}\, {\rm d}x.
\end{align} Here, we have used the \eqref{relation-1} and the definitions of $a_{10}, a_{11}, a_{12}, a_{13}$ as in \eqref{a10a11}-\eqref{a12a13}.
\noindent $\bullet$ By the definition of $M_{41, j}(x, z)$ in \eqref{M41}, we get \begin{align}\label{J4} J_4 \, =\, &\varepsilon^2 \frac{\xi(\varepsilon z)}{\beta} h\int_{{\mathbb R}}
\big( 2h_2 \frac{\beta'}{\beta}+ h_6 \big) \, \Big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{j, x}+\phi_{22, j, xz}\Big]\, w_{j, x}\, {\rm d}x \nonumber\\ &+\varepsilon^2 \frac{\xi(\varepsilon z)}{\beta} \int_{{\mathbb R}} \Big\{ h_3\Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, x}+\phi_{22, j, x}\Big] +\frac{h_8}{\beta}\Big(\frac{x_j}{\beta}\Big)\, \Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, xx}+\phi_{22, j, xx}\Big] \nonumber\\[2mm] &\qquad \qquad \qquad \qquad -\, \frac{ V_t(0, \varepsilon z)}{\beta}\, \Big(\frac{x_j}{\beta}\Big)\, \Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j}+\phi_{22, j}\Big]\Big\}w_{j, x}\, {\rm d}x \nonumber\\ \, =\, &\, \varepsilon^2\, \frac{\varrho_1}{\beta}\, \alpha_1(z)\, h \, +\, \varepsilon^2\, \frac{\varrho_1}{\beta}\, G_{1}(z), \end{align}
where \begin{align} \label{alpha1} \alpha_1(z)\, =\, 2 \frac{\xi(\varepsilon z)}{\varrho_1} \, \int_{{\mathbb R}}\Big[\varepsilon\, A'\big({\tilde{\bf d}}(\varepsilon z)\big)\, \beta (\varepsilon z)\, Z_{j, x}+\phi_{22, j, xz}(x, z)\Big] w_{j, x}\, {\rm d}x, \end{align} \begin{align} \label{G1}
G_{1}(z)\, =\, &\varepsilon^2 \frac{\xi(\varepsilon z)}{\varrho_1}\int_{{\mathbb R}} \Big\{ h_3\Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, x}+\phi_{22, j, x}\Big] +\frac{h_8}{\beta}\Big(\frac{x_j}{\beta}\Big)\, \Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, xx}+\phi_{22, j, xx}\Big] \nonumber\\[2mm] &\qquad \qquad \qquad \qquad -\, \frac{ V_t(0, \varepsilon z)}{\beta}\, \Big(\frac{x_j}{\beta}\Big)\, \Big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j}+\phi_{22, j}\Big]\Big\}w_{j, x}\, {\rm d}x. \end{align}
\noindent $\bullet$ The definition of $M_{61, j}$ is given in \eqref{M61}. Since $\omega_{0, j}$, $\omega_{1, j}$ and $w_{j, x}$ are odd functions of $x_j$, while $\omega_{2, j}$, $\omega_{3, j}$, $\phi_{21, j}$ and $\phi_{22, j}$ are even functions of $x_j$, we then obtain that \begin{align}\label{J5} J_5 \, =\, &\varepsilon^2h_1\int_{{\mathbb R}}\frac {p(p-1)}{2}{w_j}^{p-2}\Big[2h\big(a_{10}a_{12}\omega_{0, j}\omega_{2, j} +a_{10}a_{13}\omega_{0, j}\omega_{3, j} +a_{11}a_{12}\omega_{1, j}\omega_{2, j} +a_{11}a_{13}\omega_{1, j}\omega_{3, j}\big) \nonumber\\[2mm] &\qquad\qquad\qquad\qquad\quad\quad+2\big(a_{10}\omega_{0, j}+a_{11}\omega_{1, j}\big)\xi(\varepsilon z)\big(A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\big)\Big]w_{j, x}\, {\rm d}x \nonumber\\
\, \equiv\, &\varepsilon^2h_1h\int_{{\mathbb R}} \, p(p-1)\, {w_j}^{p-2} \Big[a_{10}a_{12}\omega_{0, j}\omega_{2, j} +a_{10}a_{13}\omega_{0, j}\omega_{3, j} \nonumber\\ &\qquad \qquad \qquad \qquad \qquad \quad+a_{11}a_{12}\omega_{1, j}\omega_{2, j} +a_{11}a_{13}\omega_{1, j}\omega_{3, j}\Big]w_{j, x}\, {\rm d}x +\varepsilon^2 \frac{\varrho_1}{\beta}G_{2}(z), \end{align} where \begin{align} \label{G3}
G_{2}(z)
\, =\, \frac{\beta}{\varrho_1}\, \xi(\varepsilon z) h_1
\, p(p-1)\int_{{\mathbb R}}{w_j}^{p-2}\, \big(a_{10}\omega_{0, j}+a_{11}\omega_{1, j} \big)\Big[A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\, \phi_{22, j}(x, z)\Big]\, w_{j, x}\, {\rm d}x. \end{align}
By differentiating the equation (\ref{w{0}}) and using equations (\ref{w{2}}), \eqref{w{3}}, we obtain \begin{equation}\label{relation1} \int_{{\mathbb R}}p(p-1){w_j}^{p-2}w_{j, x} \omega_{0, j}\omega_{2, j}{\rm d}x =-\int_{{\mathbb R}} w_{j, x}\omega_{0, j}{\rm d}x +\int_{{\mathbb R}}\Big[w_{j, x}+\frac{1}{\sigma}x_j w_{j}\Big]\omega_{2, j, x}{\rm d}x, \end{equation} \begin{equation}\label{relation2}
\int_{{\mathbb R}}p(p-1){w_j}^{p-2}w_{j, x} \omega_{0, j}\omega_{3, j}{\rm d}x =-\int_{{\mathbb R}} w_{j, xxx}\omega_{0, j}{\rm d}x +\int_{{\mathbb R}}\Big[w_{j, x}+\frac{1}{\sigma}x_j w_{j}\Big]\omega_{3, j, x}{\rm d}x. \end{equation} Similarly, by differentiating the equation (\ref{w{1}}) and using equations (\ref{w{2}}), \eqref{w{3}}, we obtain \begin{equation}\label{relation3}
\int_{{\mathbb R}}p(p-1){w_j}^{p-2}w_{j, x} \omega_{1, j}\omega_{2, j}{\rm d}x =-\int_{{\mathbb R}} w_{j, x}\omega_{1, j}{\rm d}x +\int_{{\mathbb R}}\Big[- \frac{1}{2 \sigma }x_j w_{j}+x_j w_{j, xx}\Big]\omega_{2, j, x}{\rm d}x, \end{equation} \begin{equation}\label{relation4}
\int_{{\mathbb R}}p(p-1){w_j}^{p-2}w_{j, x} \omega_{1, j}\omega_{3, j}{\rm d}x =-\int_{{\mathbb R}} w_{j, xxx}\omega_{1, j}{\rm d}x +\int_{{\mathbb R}}\Big[- \frac{1}{2 \sigma }x_j w_{j}+x_j w_{j, xx}\Big]\omega_{3, j, x}{\rm d}x. \end{equation}
Adding (\ref{J3}), (\ref{J5}) and using (\ref{relation1})-\eqref{relation4}, we have \begin{align} \label{J3+J5} J_3+J_5\, =\, & \varepsilon^2 h_1h\Bigg\{a_{10}a_{12}2\int_{{\mathbb R}} w_{j, x}\omega_{2, j, x}{\rm d}x -a_{10}a_{12}\frac{1}{\sigma}\int_{{\mathbb R}} w_{j}\omega_{2, j}{\rm d}x +a_{11}a_{12}\frac{1}{2 \sigma }\int_{{\mathbb R}}w_{j}\omega_{2, j}{\rm d}x \nonumber\\[2mm] &\qquad \quad +a_{10}a_{13}2\int_{{\mathbb R}}w_{j, x}\omega_{3, j, x}{\rm d}x -a_{10}a_{13}\frac{1}{\sigma}\int_{{\mathbb R}}w_{j}\omega_{3, j}{\rm d}x +a_{11}a_{13}\frac{1}{2 \sigma }\int_{{\mathbb R}}w_{j}\omega_{3, j}{\rm d}x \nonumber\\[2mm] &\qquad \quad -a_{11}a_{12}\int_{{\mathbb R}}w_{j, x}\omega_{2, j, x}{\rm d}x -a_{11}a_{13}\int_{{\mathbb R}}w_{j, x}\omega_{3, j, x}{\rm d}x\Bigg\} \nonumber\\[2mm]
=&\, \varepsilon^2 h_1h\Big\{-a_{10}a_{12}\Big(\frac {2}{p-1}+\frac 12\Big) -a_{10}a_{12}\Big(\frac 12-\frac {2}{p-1}\Big) +a_{11}a_{12}\frac{1}{2 }\Big(\frac 12-\frac {2}{p-1}\Big) \nonumber\\[2mm] &\qquad \quad - a_{10}a_{13} +\frac{1}{2}a_{11}a_{13} +a_{11}a_{12}\frac{1}{2}\Big(\frac {2}{p-1}+\frac 12\Big)
\Big\}\int_{{\mathbb R}}w_{x}^2{\rm d}x +\varepsilon^2 \frac{\varrho_1}{\beta}G_{2}(z) \nonumber\\[2mm] \, =&\, \varepsilon^2 \, \rho_1\, h\, h_1 \Big\{-a_{10}a_{12} +\frac{1}{2}a_{11}a_{12} -a_{10}a_{13} +\frac{1}{2}a_{11}a_{13}
\Big\} +\varepsilon^2 \frac{\varrho_1}{\beta}G_{2}(z) \nonumber\\[2mm] \, =\, &\varepsilon^2 \, \rho_1\, h\Big[- \sigma \frac{V_t(0, \theta)}{\beta^3} \Big(\frac{ V_t(0, \theta)}{ V(0, \theta)} -\frac{h_8(\theta)}{h_1(\theta)} \Big)\Big] +\varepsilon^2 \frac{\varrho_1}{\beta}G_{2}(z), \end{align} where we have used (\ref{stationary}) and the following integral identities \begin{align} 2\int_{{\mathbb R}}\omega_{2, j, x}w_{j, x}\, {\rm d}x \, =\, -\sigma\int_{{\mathbb R}}(w_{j, x})^2\, {\rm d}x \, =\, -\sigma\int_{{\mathbb R}}w_{x}^2{\rm d}x, \end{align} \begin{align} \sigma^{-1}\int_{{\mathbb R}}\omega_{2, j}w_j\, {\rm d}x \, =\, \Big(\frac 12-\frac {2}{p-1}\Big)\int_{{\mathbb R}}(w_{j, x})^2\, {\rm d}x \, =\, \Big(\frac 12-\frac {2}{p-1}\Big)\int_{{\mathbb R}}w_{x}^2{\rm d}x, \end{align} \begin{equation} 2 \int_{{\mathbb R}}\omega_{3, j, x} w_{j, x}{\rm d}x\, =\, -\frac{1}{2} \int_{{\mathbb R}} w_{j, x}^2{\rm d}x\, =\, -\frac{1}{2} \int_{{\mathbb R}} w_{x}^2{\rm d}x, \end{equation} \begin{align} \int_{{\mathbb R}}\omega_{3, j} w_{j}{\rm d}x\, =\, \Big( \frac{1}{p-1}+\frac{1}{4} \Big)\int_{{\mathbb R}} w_{j, x}^2{\rm d}x\, =\, \frac{\sigma}{2}\int_{{\mathbb R}} w_{x}^2{\rm d}x. \end{align}
Finally, denote \begin{align}\label{zeta} \zeta(\theta)=\frac{1}{\alpha^2 \beta \sqrt{{\mathfrak h}_1}}, \end{align} \begin{align} \label{hbar1} \hbar_1(\theta)\, =\, h_2\Big[\, \frac {\beta'}{\beta} +\frac {2\alpha'}{\alpha}\, \Big] +h_4 -\frac{1}{2} h_6 \, =\, \zeta(\theta)\mathcal{H}_{1}'(\theta), \end{align} and \begin{align} \label{hbar2} \hbar_2(\theta)\, =\, &-\Big[\, h_5 -h_7+\sigma\frac { V_{tt}(0, \theta)}{\beta^2}\, \Big] -h_6\Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big) +\sigma \frac{V_t(0, \theta)}{\beta^2} \Big(\frac{ V_t(0, \theta)}{ V(0, \theta)} -\frac{h_8}{h_1} \Big) \nonumber\\[2mm] \, =\, &\zeta(\theta) \big[\, \mathcal{H}_{2}'(\theta)-\mathcal{H}_{3} (\theta)\, \big], \end{align} where the last equalities in \eqref{hbar1} and \eqref{hbar2} will be verified in Appendix \ref{appendixD}. We infer that equation (\ref{solvable of varphi1}) becomes \begin{align*} \text{LHS of \eqref{solvable of varphi1}} =&-\varepsilon^2 \frac{\varrho_1}{\beta}\Big[h_2h'' +\hbar_1(\varepsilon z)h'+(\hbar_2(\varepsilon z)+\alpha_1(z))h-G_{1}(z)-G_{2}(z)\Big] \\ =& -\varepsilon^2 \frac{\varrho_1}{\beta}\zeta(\varepsilon z) \Bigg\{\mathcal{H}_{1}(\varepsilon z)h'' +\mathcal{H}_{1}'(\varepsilon z)h' +\Big[\big( \mathcal{H}_{2}'(\varepsilon z)-\mathcal{H}_{3}(\varepsilon z) \big) +\frac{\alpha_1(z)}{\zeta(\varepsilon z)}\Big]h
-\frac{ G_{1}(z)+G_{2}(z)}{\zeta(\varepsilon z)}\Bigg\} \\ =\, &0. \end{align*}
\section{The first projection of error}\label{appendixB} \setcounter{equation}{0}
We do estimates for the term $ \int_{\mathfrak S}{\mathcal E}w_{n, x}\, {\rm d}x$ given in Section \ref{section5}, where ${\mathcal E}$ is defined in (\ref{new error-2}) and $w_{n, x}$ is an odd function of $x_n$. Integration against all even terms of $x_n$, say ${\mathcal E}_{11, n}$ and $S_{4, n}, M_{12, n}, M_{22, n}$ in ${\mathcal E}_{12}$, therefore just vanish. We have \begin{equation} \int_{\mathfrak S}\, {\mathcal E}\, w_{n, x}\, {\rm d}x \, =\Big\{\int_{\mathfrak S_n}+\int_{\mathfrak S\setminus\mathfrak S_n}\Big\}\, {\mathcal E}\, w_{n, x}\, {\rm d}x. \end{equation} We begin with \begin{align} \int_{{\mathfrak S}_n}{\mathcal E}_{12}w_{n, x}{\rm d}x \, =\, &\, \sum_{j=1}^N\int_{{\mathfrak S}_n}\varepsilon^2S_{3, j}w_{n, x}{\rm d}x
+\sum_{j=1}^N\int_{{\mathfrak S}_n}\varepsilon^2S_{5, j}w_{n, x}{\rm d}x
+\int_{{\mathfrak S}_n}B_4(v_1)w_{n, x}{\rm d}x \nonumber \\ & +\int_{{\mathfrak S}_n}\varepsilon\frac{h_2}{\beta^2}\phi_{1, zz}w_{n, x}{\rm d}x +\int_{{\mathfrak S}_n}\varepsilon^2\, \frac{h_2}{\beta^2}\phi_{4, zz}\, w_{n, x}\, {\rm d}x +\int_{{\mathfrak S}_n}M_{32}(x, z)\, w_{n, x}\, {\rm d}x \nonumber \\ &+\int_{{\mathfrak S}_n}M_{42}(x, z)\, w_{n, x}\, {\rm d}x +\int_{{\mathfrak S}_n}M_{52}(x, z)\, w_{n, x}\, {\rm d}x +\int_{{\mathfrak S}_n}M_{62}(x, z)\, w_{n, x}\, {\rm d}x \nonumber \\ &+\int_{{\mathfrak S}_n}M_{63}(x, z)\, w_{n, x}\, {\rm d}x +\int_{{\mathfrak S}_n}\big[B_3(\varepsilon^2\, \phi_3) +B_3(\varepsilon^2\, \phi_4)\big]\, w_{n, x}\, {\rm d}x\, +\, O(\varepsilon^3) \nonumber \\ \, \equiv\, &\, \textrm{I}_1 \, +\, \textrm{I}_2 \, +\, \textrm{I}_3 \, +\, \textrm{I}_4 \, +\, \textrm{I}_5 \, +\, \textrm{I}_6 \, +\, \textrm{I}_7 \, +\, \textrm{I}_8 \, +\, \textrm{I}_9 \, +\, \textrm{I}_{10} \, +\, \textrm{I}_{11}\, +\, O(\varepsilon^3). \label{intE1wx} \end{align} These terms will be estimated as follows.
\noindent $\bullet$ By repeating the same computation used in (\ref{J1}) and (\ref{J2}), we get \begin{equation} \label{I1} \textrm{I}_{1} \, =\, \varepsilon^2\, \frac{\varrho_1}{\beta}\left\{-h_2\, f_{n}'' -h_2\Big[\, \frac {\beta'}{\beta} +\frac {2\alpha'}{\alpha}\, \Big]{f_{n}'} +\Big[\, h_5-h_7+\sigma\frac { V_{tt}(0, \varepsilon z)}{\beta^2}\, \Big]{f_n} \right\}+O(\varepsilon^3)\sum_{j=1}^N\, (f_j+f'_j+f''_j), \end{equation} where $\varrho_{1}$ is a positive constant defined in \eqref{varrho1}.
\noindent $\bullet$ There also holds \begin{equation}\label{I2} \textrm{I}_{2}
\, =\, \varepsilon^2\frac{\varrho_1}{\beta}\, \Big[ h_6\Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big)f_n +\frac{1}{2} h_6f_n'-h_4 f_n' \Big]
+O(\varepsilon^3)\sum_{j=1}^N\, (f_j+f'_j). \end{equation}
\noindent $\bullet$ Recall the expression of $B_4(v_1)$ in (\ref{B2v1}), then \begin{align}\label{I3} \textrm{I}_{3}
\, =\, & \int_{{\mathfrak S}_n}\Big\{\, \frac{1}{\alpha\beta^2}\big[\hat{B}_0(v_1)+\, a_6(\varepsilon s, \varepsilon z)\, \varepsilon^3 \, s^3\, {v_1}\big] \, +\, h_1p(w_n)^{p-1}(v_1-w_n) \nonumber\\[2mm] \, &\qquad
\,-\, h_1\sum_{j\neq n}{w_j}^p\, +\, h_1\frac{1}{2}p(p-1)(w_n)^{p-2}(v_1-w_n)^2\, +\, \max_{j\neq n}O(e^{-3|\beta f_j-x|})\, \Big\}\, w_{n, x}\, {\rm d}x \nonumber\\[2mm]
=\, &\varrho_2h_1\Big[e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)}\Big]+\varepsilon^{\mu_1}\max_{j\neq n}O(e^{-\beta|f_j-f_n|})+\varepsilon^3 \sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}\, f''_j+{\mathbf b}_{2\varepsilon j}\big), \end{align} where $\mu_1$ is a small positive constant, and $\varrho_{2}$ is a positive constant given by \begin{equation}\label{varrho2} \varrho_2\, =\, p\, C_p \int_{0}^{\infty}\, w^{p-1}\, w_x(e^{-x}-e^{x})\, {\rm d}x. \end{equation}
\noindent $\bullet$ Recall the expression of $\varepsilon\, \phi_1$ in \eqref{phi1}, then \begin{equation}\label{I4} \textrm{I}_{4} \, =\, \int_{{\mathfrak S}_n}\varepsilon\, \frac{h_2}{\beta^2}\, \phi_{1, zz}\, w_{n, x}\, {\rm d}x \, =\, \varepsilon^3\sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}f''_j+{\mathbf b}_{2\varepsilon j}\big). \end{equation}
\noindent $\bullet$ It can be derived that \begin{equation} \label{I5} \textrm{I}_5\, =\, \int_{{\mathfrak S}_n}\varepsilon^2\, \frac{h_2}{\beta^2}\, \phi_{4, zz}\, w_{n, x}\, {\rm d}x\, =\, O(\varepsilon^4). \end{equation}
\noindent $\bullet$ From the definition of $M_{42}(x, z)$ in \eqref{M42}, we can estimate the term $\textrm{I}_7$ as the following \begin{align} \label{I7} \textrm{I}_7 \, =\, &\varepsilon^2 \xi(\varepsilon z) \frac{1}{\beta}f_n\, \int_{{\mathfrak S}_n}\, \big(2h_2 \frac{\beta'}{\beta}\, + h_6\big) \Big[ A'\big({\tilde{\bf d}}(\varepsilon z)\big)\varepsilon \mathcal{Q} Z_{n, x}+\phi_{22, n, xz}\Big]\, w_{n, x}\, {\rm d}x \nonumber\\[2mm] &+\varepsilon^3\sum_{j=1}^N\big( {\mathbf b}_{1\varepsilon j}f''_j+{\mathbf b}_{2\varepsilon j}\big) \nonumber\\ \equiv& -\varepsilon^2 \frac{\varrho_1}{\beta}\, \alpha_1(z)f_n +\varepsilon^3\sum_{j=1}^N \big( {\mathbf b}_{1\varepsilon j}f''_j+{\mathbf b}_{2\varepsilon j} \big) +O(\varepsilon^3)\sum_{j=1}^N\big(f_j+f'_j\big), \end{align} where $\alpha_1(z)$ and $\varrho_{1}$ are defined in (\ref{alpha1}) and (\ref{varrho1}).
\noindent $\bullet$ From the definition of $M_{52}(x, z)$ in \eqref{M52}, we need only consider the odd terms and the higher order terms involving $e_j'$ and $e''_j$, so we get \begin{align} \label{I8}
\textrm{I}_8=&\sum_{j=1}^N\frac{\varepsilon^2}{\beta} {e_j}\int_{{\mathfrak S}_n}\Big[ h_3 \, Z_{j, x}+h_8\, x_jZ_{j, xx}-\, \frac{V_t(0, \varepsilon z)}{\beta^2}\, x_jZ_{j}\Big] \, w_{n, x}\, {\rm d}x \nonumber\\
&+\sum_{j=1}^N\, \varepsilon^4\, \frac{1}{\alpha\beta^3} h_9\, e''_j(\varepsilon z)\int_{{\mathfrak S}_n} x_j\, w_{n, x}\, Z_j\, {\rm d}x
+\varepsilon^3\sum_{j=1}^N\, \big({\mathbf b}_{1\varepsilon j}\, e'_j+ {\mathbf b}_{1\varepsilon j}\, f''_j+{\mathbf b}_{2\varepsilon j}\big) \nonumber\\ \, \equiv\, &\varepsilon^2 \frac{\varrho_1}{\beta}\Big[\hbar_3({\varepsilon}z)\, e_n+\varepsilon^2\hbar_4({\varepsilon}z)\, e''_n\Big] +\varepsilon^3\sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}\, e'_j+{\mathbf b}_{1\varepsilon j}\, f''_j+{\mathbf b}_{2\varepsilon j}\big), \end{align} where $\hbar_3(\varepsilon z)$ and $\hbar_4(\varepsilon z)$ are defined like the following \begin{equation}\label{hbar3} \hbar_3(\varepsilon z)\, =\, -\varrho_{1} \int_{{\mathbb R}}\Big[h_3\, Z_x+h_8xZ_{xx}- V_t \beta^{-2}\, x\, Z\Big]\, w_x\, {\rm d}x, \end{equation} \begin{equation}\label{hbar4} \hbar_4(\varepsilon z)\, =\, -\frac{1}{\varrho_1\alpha\beta^2}\, h_9 \int_{{\mathbb R}}\, x\, w_x\, Z\, {\rm d}x. \end{equation}
\noindent $\bullet$ Recalling the definitions of $M_{32}(x, z)$ and $M_{62}(x, z)$ as in \eqref{M32}, \eqref{M62}, we can get \begin{equation*} \textrm{I}_6+ \textrm{I}_9 =\, \varepsilon^2 \, \rho_1\, \Big[- \sigma \frac{V_t(0, \varepsilon z)}{\beta^3} \Big(\frac{ V_t(0, \varepsilon z)}{ V(0, \varepsilon z)} -\frac{h_8}{h_1} \Big)\Big]f_n\, +\, \varepsilon^3\sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}\, f''_j+\, {\mathbf b}_{2\varepsilon j}\big). \end{equation*}
\noindent $\bullet$ According to the fact that the terms in $B_3(\varepsilon^2\, \phi_3)$ and $B_3(\varepsilon^2\, \phi_4)$ are of order $O(\varepsilon^3)$, it follows that \begin{equation} \label{I10} \textrm{I}_{10}+\textrm{I}_{11}
\, =\, \varepsilon^3\sum_{j=1}^N\big({\mathbf b}_{1\varepsilon j}\, f''_j+\, {\mathbf b}_{2\varepsilon j}\big). \end{equation}
The above computations lead to the estimate \begin{align}
\int_{{\mathfrak S}_n}{\mathcal E}w_{n, x}{\rm d}x
\, =\, &-\varepsilon^2\, \frac{\varrho_1}{\beta}\zeta(\varepsilon z)\, \Big\{\mathcal{H}_1(\varepsilon z)\, f''_n+\mathcal{H}_1'(\varepsilon z)\, f'_n+ \Big[\mathcal{H}_2'(\varepsilon z)-\mathcal{H}_3(\varepsilon z)+\frac{\alpha_1(z)}{\zeta(\varepsilon z)}\Big]\, f_n\Big\} \nonumber\\[2mm] &+\varepsilon^2 \frac{\varrho_1}{\beta}\big[\hbar_3(\varepsilon z)e_n+\varepsilon^2\hbar_4({\varepsilon}z)e''_n\big]+h_1\, \varrho_2\big[e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)}\big] \nonumber\\[2mm]
&+\varepsilon^{\mu_1}\max_{j\neq n}O(e^{-\beta|f_j-f_n|})+\varepsilon^3\sum_{j=1}^N\, \big({\mathbf b}_{1\varepsilon j}\, e_j'+{\mathbf b}_{1\varepsilon j}^2\, f''_j+{\mathbf b}_{2\varepsilon j}\big)
\nonumber\\ &+O(\varepsilon^3)\sum_{j=1}^N\big(\, f_j+\, f'_j+\, f''_j+\, f_j^2\big). \end{align}
On the other hand, to compute $\int_{\mathfrak S\setminus\mathfrak S_n}\, {\mathcal E}\, w_{n, x}\, {\rm d}x$ for fixed $n=1, \cdots, N$, we notice that for $(x, z)\in {\mathcal S}_{\delta/\varepsilon}\backslash {\mathfrak A}_n$ with \begin{equation*} {\mathcal S}_{\delta/\varepsilon}\, =\, \big\{\, -\delta/\varepsilon\, <\, x\, <\, \delta/\varepsilon, \ 0\, <\, z\, <\, 1/\varepsilon\, \big\}, \end{equation*} there holds \begin{equation*}
w_{n, x}\, =\, \max_{j\neq n}O(e^{\frac{1}{2}\beta|f_j-f_n|}). \end{equation*} Thus we can estimate \begin{align}
\int_{\mathfrak S\setminus\mathfrak S_n}\, {\mathcal E}\, w_{n, x}\, {\rm d}x=\varepsilon^\frac{1}{2}\max_{j\neq n}O(e^{-\beta|f_j-f_n|})+O(\varepsilon^{\frac{1}{2}})\sum_{i=1}^{11}\textrm{I}_i. \end{align}
\section{The second projection of error}\label{appendixC} We estimate the term $ \int_{\mathfrak S}{\mathcal E}Z_n\, {\rm d}x$ in Section \ref{section5}, where ${\mathcal E}$ and its decomposition are defined in (\ref{new error-2}) and \eqref{E1-d}, and $Z_n$ is an odd function of $x_n$. We have $$\int_{{\mathfrak S}}{\mathcal E}\, Z_n\, {\rm d}x \, =\, \int_{\mathfrak S}{\mathcal E}_{11}\, Z_n\, {\rm d}x +\int_{\mathfrak S}{\mathcal E}_{12}\, Z_n\, {\rm d}x, $$ where \begin{equation*}
\int_{\mathfrak S}{\mathcal E}_{11}\, Z_n\, {\rm d}x
\, =\, \varepsilon^3\, \frac{h_2}{\beta^2}\, e''_n+\varepsilon h_1\lambda_0\, e_n+O(\varepsilon), \end{equation*} and \begin{equation} \int_{\mathfrak S}\, {\mathcal E}_{12}\, Z_n\, {\rm d}x \, =\Big\{\int_{\mathfrak S_n}+\int_{\mathfrak S\setminus\mathfrak S_n}\Big\}\, {\mathcal E}_{12} \, Z_n\, {\rm d}x. \end{equation} Since $S_{6, j}$, $S_{8, j}$, $M_{11, j}(x, z)$, $M_{21, j}(x, z)$, $M_{51, j}(x, z)$ are even functions of $x_j$, then integration against $ w_{j, x}$ just vanish. This gives that \setcounter{equation}{0} \begin{align}
\int_{\mathfrak S_n}{\mathcal E}_{12}\, Z_n\, {\rm d}x \, =\, & \sum_{j=1}^N\int_{\mathfrak S_n}\varepsilon^2\, S_{4, j}\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}B_4(v_1)\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}\varepsilon\, \frac{h_2}{\beta^2}\phi_{1, zz}\, Z_n\, {\rm d}x \nonumber \\[2mm] &+\int_{\mathfrak S_n}\varepsilon^2\, \frac{h_2}{\beta^2}\phi_{4, zz}\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}M_{12}(x, z)\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}M_{22}(x, z)\, Z_n\, {\rm d}x \nonumber \\[2mm] &+\int_{\mathfrak S_n}M_{32}(x, z)\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}M_{42}(x, z)\, Z_n\, {\rm d}x +\int_{\mathfrak S_n}M_{52}(x, z)\, Z_n\, {\rm d}x\nonumber \\[2mm] &+\int_{\mathfrak S_n}\big(M_{62}(x, z)+M_{63}(x, z)\big)Z_n\, {\rm d}x +\int_{\mathfrak S_n}B_3(\varepsilon^2\, \phi_3)Z_n\, {\rm d}x +\int_{\mathfrak S_n}B_3(\varepsilon^2\, \phi_4)Z_n\, {\rm d}x\nonumber \\[2mm] \, \equiv\, & \textrm{II}_1 +\textrm{II}_2 +\textrm{II}_3 +\textrm{II}_4 +\textrm{II}_5 +\textrm{II}_6 +\textrm{II}_7 +\textrm{II}_8 +\textrm{II}_9+\textrm{II}_{10} +\textrm{II}_{11} +\textrm{II}_{12}. \label{E12Z} \end{align} Here are the details of computations.
\noindent $\bullet$ According to the expression of $S_{4, j}$ in (\ref{sv1-gather}) and the assumption of $f_j$, it follows that \begin{align} \textrm{II}_1
=&\sum_{j=1}^N\varepsilon^2\int_{\mathfrak S_n}\Big[ \Big( h_2f_j'^2\, +2h_2f_j'h'-h_6\, f_j\, f_j'\, -h_6\, f_j'\, h+h_7f_j^2\Big)w_{n, xx}
-\frac{1}{2}\, \beta^{-2}\, V_{tt}(0, \varepsilon z)\, f_n^2\, w_n\Big]Z_n\, {\rm d}x \nonumber
\\ \, =\, &\varepsilon^2\, \varrho_{3}\big( h_2f_n'^2\, +2h_2f_n'h'-h_6\, f_n\, f_n'\, -h_6\, f_n'\, h+h_7f_n^2\big) +O(\varepsilon^3)\sum_{j=1}^N\, \big(f_j^2+\, {f'_j}^2+f'_jf_j \big), \label{II1} \end{align} where \begin{equation}\label{varrho3}
\varrho_{3}\, =\, 2\int_{{\mathbb R}}\, w_{xx}\, Z\, {\rm d}x. \end{equation}
\noindent $\bullet$ Recall the expression of $B_4(v_1)$ in (\ref{B2v1}), then \begin{align}\label{II2} \textrm{II}_{2}
\, =\, &\int_{{\mathfrak S}_n}\Big\{\, \frac{1}{\alpha\beta^2}\big[\hat{B}_0(v_1)+\, a_6(\varepsilon s, \varepsilon z)\, \varepsilon^3 \, s^3\, {v_1}\big] \, +\, h_1p(w_n)^{p-1}(v_1-w_n) \,-\, h_1\sum_{j\neq n}{w_j}^p \nonumber\\ \, &\quad\qquad\, +\, h_1\frac{1}{2}p(p-1)(w_n)^{p-2}(v_1-w_n)^2
\, +\, \max_{j\neq n}O(e^{-3|\beta f_j-x|})\, \Big\}\, Z_n\, {\rm d}x \nonumber\\ =\, &
\varrho_4\, h_1\Big[e^{-\beta(f_n-f_{n-1})}-e^{-\beta(f_{n+1}-f_n)}\Big]+\varepsilon^{{\hat\tau}_1}\max_{j\neq n}O(e^{-\beta|f_j-f_n|}) +\varepsilon^3\sum_{j=1}^N \big({\mathbf b}_{1\varepsilon j}\, f''_j+ {\mathbf b}_{2\varepsilon j} \big), \end{align} where ${\hat\tau}_1$ is a small positive constant with $\frac{1}{2}<{\hat\tau}_1<1$ and $\varrho_4$ is positive constant given by \begin{align} \varrho_4\, =\, p\, C_p\int_{0}^{\infty}\, w^{p-1}(x)Z(x)[e^{-x}-e^{x}]\, {\rm d}x. \label{varrho4} \end{align}
\noindent $\bullet$ According to the definition of $\varepsilon\, \phi_1(x, z)$ as in \eqref{phi1},
we obtain \begin{align}\label{II3}
\textrm{II}_{3}
&\, =\, \sum_{j=1}^N\Big[\varepsilon^3 \, \beta^{-2}\, a_{12}(\varepsilon z)\, f''_j\int_{{\mathfrak S}_n} \omega_{2, j}\, Z_n\, {\rm d}x +\varepsilon^3 \, \beta^{-2}\, a_{13}(\varepsilon z)\, f''_j\int_{{\mathfrak S}_n} \omega_{3, j}\, Z_n\, {\rm d}x +\varepsilon^3 \, {\mathbf b}_{2\varepsilon j}\Big] \nonumber\\ &\, =\varepsilon^3 \, \beta^{-2} f''_n\, \int_{{\mathfrak S}_n}\big[\, a_{12}(\varepsilon z) \omega_{2, n}+a_{13}(\varepsilon z)\omega_{3, n}\, \big]\, Z_n\, {\rm d}x +\varepsilon^3\sum_{j=1}^N \big({\mathbf b}_{1\varepsilon j}\, f''_j+ {\mathbf b}_{2\varepsilon j} \big) \nonumber\\ &\, =\varepsilon^3\, \rho_1\, (\varepsilon z)+\varepsilon^3\sum_{j=1}^N \big({\mathbf b}_{1\varepsilon j}\, f''_j+ {\mathbf b}_{2\varepsilon j} \big), \end{align} where \begin{equation}\label{rho1} \rho_1\, (\varepsilon z)= \, \beta^{-2} f''_n\, \int_{{\mathbb R}}\Big[a_{12}(\varepsilon z) \omega_{2, n}+a_{13}(\varepsilon z)\omega_{3, n}\Big]\, Z_n\, {\rm d}x. \end{equation}
\noindent $\bullet$ It is easy to prove that \begin{align} \label{II4} \textrm{II}_{4}\, =\, \varepsilon^2\beta^{-2} h_2\, \int_{{\mathfrak S}_n}\, \phi_{4, zz}\, Z_n\, {\rm d}x\, =\, O(\varepsilon^4). \end{align}
\noindent $\bullet$ According to the expression of $M_{12}(x, z)$ and $M_{22}(x, z)$ as in \eqref{tildeL0phi2} and \eqref{tildeL0e^2hatphi}, hence, it is easy to obtain that \begin{align} \textrm{II}_{5}
&=\sum_{j=1}^N\frac{\varepsilon^3}{\beta^{2}}\, h_2\, \xi''(\varepsilon z)\int_{{\mathfrak S}_n}\big[A'\big({\tilde{\bf d}}(\varepsilon z)\big)Z_j+\phi_{22, j}\big]\, Z_n\, {\rm d}x, \label{II5} \nonumber\\[2mm] &=\varepsilon^3\frac{h_2}{\beta^2}\, \xi''(\varepsilon z)A'\big({\tilde{\bf d}}(\varepsilon z)\big)+O(\varepsilon^3). \end{align}
\noindent $\bullet$ Similarly, there holds \begin{equation}\label{II6} \textrm{II}_{6}=\int_{{\mathfrak S}_n}M_{22}(x, z)\, Z_n\, {\rm d}x=O(\varepsilon^3). \end{equation}
\noindent $\bullet$ The estimate of $\textrm{II}_7$ can be proved by the same way, i.e.,
\begin{align}\label{II7} \textrm{II}_7
\, =\, &\varepsilon^2f_n^2\int_{{\mathfrak S}_n}\Big[\, h_8\big(a_{12}\omega_{2, n, xx} +a_{13}\omega_{3, n, xx}\big) \nonumber\\ &\qquad \qquad -\, \frac{V_t(0, \varepsilon z)}{\beta^2}\, \big(a_{12}\omega_{2, n}+a_{13}\omega_{3, n}\big)\, \Big]Z_n\, {\rm d}x +\varepsilon^3\sum_{j=1}^N \big({\mathbf b}_{1\varepsilon j}\, f''_j+ {\mathbf b}_{2\varepsilon j} \big) \nonumber\\ \, =\, &\, \varepsilon^2\, \rho_2\, (\varepsilon z)+\varepsilon^3\sum_{j=1}^N \big({\mathbf b}_{1\varepsilon j}\, f''_j+ {\mathbf b}_{2\varepsilon j} \big), \end{align} where \begin{equation}\label{rho2} \rho_2\, (\varepsilon z)=f_n^2\int_{{\mathbb R}}\Big[h_8\big(a_{12}\omega_{2, xx} +a_{13}\omega_{3, xx}\big) -\, \frac{V_t(0, \varepsilon z)}{\beta^2}\, \big(a_{12}\omega_{2}+a_{13}\omega_{3}\big)\Big]Z\, {\rm d}x. \end{equation}
\noindent $\bullet$ On the other hand, from the definition of $M_{42}(x, z)$ in \eqref{M42}, we can compute that \begin{align}\label{II8} \textrm{II}_{8}
&=\, \sum_{j=1}^N \varepsilon^2 \frac{1}{\beta}f_j \xi(\varepsilon z)\, \int_{{\mathfrak S}_n} \Big\{\frac{h_8}{\beta}\, \big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j, xx}+\phi_{22, j, xx}\big] \nonumber\\ &\qquad \qquad \qquad \qquad \qquad \qquad-\, \frac{V_t(0, \varepsilon z)}{\beta}\, \big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{j}+\phi_{22, j}\big]\Big\}\, Z_n\, {\rm d}x+O(\varepsilon^3) \nonumber\\[2mm]
&=\, \varepsilon^2\rho_3\, (\varepsilon z)+O(\varepsilon^3), \end{align} where \begin{align}\label{rho3} \rho_3\, (\varepsilon z)\, =\, &\frac{1}{\beta}f_n \xi(\varepsilon z)\, \int_{\mathbb R} \Big\{\frac{h_8}{\beta}\, \big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z_{xx}+\phi_{22, xx}\big] \nonumber\\ &\qquad \qquad \qquad \quad -\, \frac{V_t(0, \varepsilon z)}{\beta}\, \big[ A\big({\tilde{\bf d}}(\varepsilon z)\big)Z+\phi_{22}\big]\Big\}\, Z\, {\rm d}x. \end{align}
\noindent $\bullet$ We need only to compute those parts of $M_{52}(x, z)$ in \eqref{M52}, which are even in $x_n$. It is easy to check that \begin{align} \begin{split} \label{II9}
\textrm{II}_{9}
&=\sum_{j=1}^N\frac{\varepsilon^2}{\beta} {e_j}\int_{{\mathfrak S}_n}\Big[\, h_3 \, Z_{j, x} +h_8\, x_jZ_{j, xx}-\, \frac{V_t(0, \varepsilon z)}{\beta^2}\, x_jZ_{j}\, \Big]\, Z_n\, {\rm d}x+O(\varepsilon^3) =\, O(\varepsilon^3). \end{split} \end{align} Additionally, we also need to consider some higher order terms in $\textrm{II}_{9}$. The ones involving first derivative of $e_j$ are \begin{align} \label{M121important} &\varepsilon^3\sum_{j=1}^N\, e_j'\, \Big(\frac{h_2 \beta'}{\beta^{3}}+\frac{h_6}{\beta^{2}}\, \, \Big)\int_{{\mathfrak S}_n}x_jZ_{j, x}Z_n\, {\rm d}x+\varepsilon^3\sum_{j=1}^N\, e_j'\, \Big(\frac{2\alpha'}{\alpha\beta^{2}}+\frac{h_4}{\beta^{2}}\, \Big)\int_{{\mathfrak S}_n} Z_j Z_n\, {\rm d}x \nonumber\\ &\, =\, \varepsilon^3\, \Big[\, \Big(\frac{2\alpha'}{\alpha\beta^{2}}+\frac{h_4}{\beta^{2}}\, \Big)- \frac{1}{2}\Big(\frac{h_2 \beta'}{\beta^{3}}+\frac{h_6}{\beta^{2}}\, \Big)\, \Big]e_n'+\varepsilon^3\sum_{j=1}^N \, {\mathbf b}_{2\varepsilon j}e'_j(\varepsilon z) \nonumber\\ &\, \equiv\, \varepsilon^3\, \hbar_5({\varepsilon}z)\, e_n'+\varepsilon^3\sum_{j=1}^N \, {\mathbf b}_{2\varepsilon j}e'_j(\varepsilon z), \end{align} \noindent where \begin{equation} \label{hbar5} \hbar_5(\varepsilon z)\, =\, \Big(\frac{2\alpha'}{\alpha\beta^{2}}+\frac{h_4}{\beta^{2}}\, \Big)- \frac{1}{2}\Big(\frac{h_2 \beta'}{\beta^{3}}+\frac{h_6}{\beta^{2}}\, \, \Big). \end{equation} Moreover, the ones involving second derivative of $e_j$ in $\textrm{II}_{9}$ are \begin{equation} \varepsilon^4\, f_n\, \beta^{-2}\, \hbar_6(\varepsilon z)e''_n(\varepsilon z)+O(\varepsilon^5)\sum_{j=1}^Ne''_j(\varepsilon z), \label{hbar6} \end{equation} with $O(\varepsilon^2)$ uniform in $\varepsilon$ and $\hbar_6({\varepsilon}z)$ is a smooth functions of their argument.
\noindent $\bullet$ In the terms of $\textrm{II}_{10}$ and $\textrm{II}_{12}$, we need only to consider those parts which are even in $x$. It is good that the even (in $x$) terms in $\textrm{II}_{10}$ and $\textrm{II}_{12}$ are of order $O(\varepsilon^3)$. Moreover, the terms in $B_3(\varepsilon^2\phi_3)$ are of order $O(\varepsilon^3)$. Consequently, we deduce that \begin{equation} \begin{split}
&\textrm{II}_{10}+\textrm{II}_{11}+\textrm{II}_{12}
=O(\varepsilon^3). \end{split} \end{equation}
To compute $\int_{\mathfrak S\setminus\mathfrak S_n}\, {\mathcal E}_{12}\, Z_n\, {\rm d}x$, we notice that for $(x, z)\in {\mathcal S}_{\delta/\varepsilon}\backslash {\mathfrak A}_n$, \begin{equation*}
Z(x_n)=\max_{j\neq n}O(e^{-\frac{p}{2}\beta |f_j-f_n|}), \end{equation*} and thus we can estimate \begin{equation}
\int_{\mathfrak S\setminus\mathfrak S_n}\, {\mathcal E}_{12}\, Z_n\, {\rm d}x=\varepsilon^\frac{1}{2}\max_{j\neq n}O(e^{-\beta|f_j-f_n|})+O(\varepsilon^{\frac{1}{2}})\sum_{i=1}^{12}\textrm{II}_i. \end{equation}
\section{The computations of \eqref{relation-1}, \eqref{hbar1} and \eqref{hbar2}}\label{appendixD}
We first show the validity of \eqref{relation-1} under the assumption of stationary condition for $\Gamma$ in {\bf (A3)} of Section \ref{section1}. In fact, the stationary assumption means that (c.f. \eqref{stationary1}), i.e., \begin{align}\label{relationoff1f0} \frac{1}{2}{\mathfrak f}_1\, =\, -\sigma\frac{ V_t(0, \theta)}{ V(0, \theta)}{\mathfrak f}_0. \end{align} This gives that \begin{align*}
\frac{1}{{\mathfrak h}_1}{\mathfrak f}_1\beta -\frac{1}{2}\frac{{\mathfrak h}_2}{{\mathfrak h}_1^{2}}{\mathfrak f}_0 \beta \, =\, \frac{1}{2}\frac{1}{{\mathfrak h}_1}{\mathfrak f}_1\beta -\frac{1}{2}\frac{{\mathfrak h}_2}{{\mathfrak h}_1^2}{\mathfrak f}_0\beta -\sigma \frac{V_t(0, \theta)}{\beta}. \end{align*} According to the expressions of $h_3, h_8$ as in \eqref{h3h4} and \eqref{h8}, we can obtain that \begin{align*} h_3\beta\, = \, \frac{1}{2}h_8\beta -\sigma \frac{V_t(0, \theta)}{\beta}, \end{align*} which is exactly the formula \eqref{relation-1}.
Recalling the expression as in \eqref{h1}-\eqref{h2}, we then have \begin{align} {\mathcal H}_1(\theta)= V^{\frac{2}{p-1}}(0, \theta)\frac{\sqrt{ V\big(0, \theta\big)}}{\sqrt{{\mathfrak f}_0}}\, {\mathfrak w}_0\, =\, \alpha^2 \beta \sqrt{{\mathfrak h}_1}\, h_2. \end{align} This gives that \begin{align*} {\mathcal H}_1'(\theta)
=&\alpha^2 \beta\sqrt{{\mathfrak h}_1}\Big[\, h_2 \frac{\beta'}{\beta}
+\frac{1}{\sqrt{{\mathfrak h}_1}} \Big( \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big)'
+2 \frac{\alpha'}{\alpha} h_2\, \Big]. \end{align*} Recalling the definitions of $h_2, h_4, h_5$ as in \eqref{h1}-\eqref{h2}, \eqref{h3h4} and \eqref{h5}, the coefficient of $f'$ is \begin{align*} h_2\Big[\, \frac {\beta'}{\beta} +\frac {2\alpha'}{\alpha}\, \Big]+h_4 -\frac{1}{2} h_6
\, =\, &h_2\Big[\, \frac {\beta'}{\beta} +\frac {2\alpha'}{\alpha}\, \Big] +\frac{1}{\sqrt{{\mathfrak h}_1}} \Big( \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak w}_0\Big)' \\[2mm] \, =\, &\frac{1}{\alpha^2 \beta \sqrt{{\mathfrak h}_1}} {\mathcal H}_1', \end{align*} which is exactly the formula \eqref{hbar1}.
According to the definition of $\beta$ in \eqref{alpha-beta} and $\sigma= \frac{p+1}{p-1}-\frac{1}{2}$, then there holds \begin{align*} {\mathcal H}_2(\theta) =\frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}{\mathfrak l}_1 =\alpha^2 \beta \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1. \end{align*} This implies that \begin{align*} {\mathcal H}_2'(\theta) =&\alpha^2\Big[\beta'\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 \, +\, \beta\frac{1}{\sqrt{{\mathfrak h}_1}}\partial_{\theta} {\mathfrak l}_1 \,-\, \frac{1}{2}\frac{1}{(\sqrt{{\mathfrak h}_1})^3}\partial_{\theta}{\mathfrak h}_1\, \beta{\mathfrak l}_1\Big] +2\alpha \alpha' \beta \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1. \end{align*} Using the fact \eqref{relationoff1f0} and \eqref{h1}-\eqref{h2}, then \begin{align*} {\mathcal H}_2'(\theta)-{\mathcal H}_3(\theta) \, =\, &\alpha^2\Big[\beta'\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 \, +\, \beta\frac{1}{\sqrt{{\mathfrak h}_1}}\partial_{\theta} {\mathfrak l}_1 \,-\, \frac{1}{2}\frac{1}{(\sqrt{{\mathfrak h}_1})^3}\partial_{\theta}{\mathfrak h}_1\, \beta{\mathfrak l}_1\Big] +2\alpha \alpha' \beta \frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1 \\[2mm] &-\alpha^2\frac{ V^{\frac{1}{2}}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_2 -\alpha^2 \sigma \frac{ V_{tt}(0, \theta)\, }{ V^{\frac{1}{2}}\big(0, \theta\big)}\, \sqrt{{\mathfrak f}_0}
-\alpha^2 \sigma(\sigma-1)\frac{\big| V_t(0, \theta)\big|^2\, }{ V^{\frac{3}{2}}(0, \theta)} \, \sqrt{{\mathfrak f}_0} \nonumber \\[2mm] &- \alpha^2 \sigma\frac{ V_t(0, \theta) \, }{ V^{\frac{1}{2}}\big(0, \theta\big)}\, \frac{1}{\sqrt{{\mathfrak f}_0}}{\mathfrak f}_1 \, +\, \alpha^2\frac{1}{4}\frac{ V^{\frac{1}{2}}\big(0, \theta\big)}{\ \big(\sqrt{{\mathfrak f}_0}\, \big)^3\ }{\mathfrak f}_1^2. \\[2mm] \, =\, &\alpha^2 \beta \sqrt{{\mathfrak h}_1} \Bigg\{ \frac{\beta'}{\beta}\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1 \, +\, \frac{1}{{\mathfrak h}_1}\partial_{\theta} {\mathfrak l}_1 \,-\, \frac{1}{2}\frac{1}{{\mathfrak h}_1^{2}}\partial_{\theta}{\mathfrak h}_1{\mathfrak l}_1 \, +\, 2\frac{\alpha'}{\alpha}\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1 \\[2mm] &\qquad \qquad \,-\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 \,-\, \sigma \frac{ V_{tt}(0, \theta)}{\beta^{2}}
\,-\, \sigma \frac{|V_t(0, \theta)|^2}{V \beta} \,-\, \sigma \frac{V_t(0, \theta)}{\beta^2}\frac{{\mathfrak f}_1}{{\mathfrak f}_0} \Bigg\}. \end{align*}
On the other hand, according to the definitions of $h_1, h_5, h_6, h_7, h_8$ as in \eqref{h1}-\eqref{h2}, \eqref{h5}, \eqref{h6h7}, and \eqref{h8}, the coefficient of $f$ is \begin{align}\label{defoff} &-\Big[\, h_5-h_7+\sigma\frac { V_{tt}(0, \theta)}{\beta^2}\, \Big] -h_6\Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big) +\sigma \frac{V_t(0, \theta)}{\beta^2} \Big(\frac{ V_t(0, \theta)}{ V(0, \theta)} -\frac{h_8(\theta)}{h_2(\theta)} \Big) \nonumber\\[2mm] \, =\, & -2\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +\frac{3}{2}\frac{{\mathfrak h}_2}{{\mathfrak h}_1^{2}}{\mathfrak f}_1 -\frac{{\mathfrak h}_2^2}{{\mathfrak h}_1^{3}}{\mathfrak f}_0 +\frac{{\mathfrak h}_3}{{\mathfrak h}_1^{2}}{\mathfrak f}_0 +\frac{1}{\sqrt{{\mathfrak h}_1}} \Big[\frac{1}{\sqrt{{\mathfrak h}_1}}{\mathfrak l}_1\Big]' +\Big[\, \frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 +\frac{{\mathfrak h}_2}{{\mathfrak h}_1^2}{\mathfrak f}_1 -\Big(\frac{{\mathfrak h}_2^2}{{\mathfrak h}_1^3} +\frac{{\mathfrak h}_3}{{\mathfrak h}_1^2}\Big){\mathfrak f}_0\Big] \nonumber\\[2mm] &-\sigma \frac{V_{tt}(0, \theta)}{\beta} \frac {1}{\beta} +2\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1\, \Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big) -\sigma \frac{V_t(0, \theta)}{\beta^2} \Big[\frac{ V_t(0, \theta)}{ V(0, \theta)} -\frac{{\mathfrak f}_1}{{\mathfrak f}_0} +\frac{{\mathfrak h}_2}{{\mathfrak h}_1} \Big] \nonumber\\[2mm]
=&-\frac{1}{{\mathfrak h}_1}{\mathfrak f}_2 -\frac{1}{2}\frac{1}{{\mathfrak h}_1^2}\partial_{\theta}{\mathfrak h}_1{\mathfrak l}_1 +\frac{1}{{\mathfrak h}_1}\partial_{\theta}{\mathfrak l}_1 +\frac{\beta'}{\beta}\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1 +2\frac{\alpha'}{\alpha}\frac{1}{{\mathfrak h}_1}{\mathfrak l}_1 -\sigma \frac{ V_{tt}(0, \theta)}{\beta^2}
+\sigma\frac{| V_t(0, \theta)|^2}{ V\beta^2} -\sigma \frac{ V_t(0, \theta)}{\beta^2}\frac{{\mathfrak f}_1}{{\mathfrak f}_0}. \end{align} Therefore, we obtain that \begin{equation*} \Big[\, h_5-h_7+\sigma\frac { V_{tt}(0, \theta)}{\beta^2}\, \Big] +h_6\Big(\frac{1}{2} \frac{\beta'}{\beta}+\frac{\alpha'}{\alpha}\Big) - \sigma \frac{V_t(0, \theta)}{\beta^2} \Big[\frac{ V_t(0, \theta)}{ V(0, \theta)} -\frac{h_8}{h_2} \Big] \, =\, \frac{{\mathcal H}_2'-{\mathcal H}_3}{\alpha^2 \beta \sqrt{{\mathfrak h}_1}}, \end{equation*} which is exactly the formula \eqref{hbar2}.
\section{The computations of \eqref{boundaryoffunctional1} and \eqref{boundaryoffunctional2}}\label{appendixE} Due to the assumptions in \eqref{a1=a2}, we obtain \begin{equation*} {\mathfrak a}_1(0, 0)={\mathfrak a}_2(0, 0), \qquad {\mathfrak a}_1(0, 1)={\mathfrak a}_2(0, 1), \end{equation*} \begin{equation*} { \tilde{\mathfrak a}}_1(0)={\tilde{\mathfrak a}}_2(0)= {\tilde{\mathfrak a}}_1(1)={\tilde{\mathfrak a}}_2(1)=\frac{1}{\sqrt{2}}, \end{equation*} and \begin{equation*}
-{\mathfrak a}_2(0, 0)|{\tilde{\mathfrak a}}_1'(0)|^2
+ {\mathfrak a}_1(0, 0)|{\tilde{\mathfrak a}}_2'(0)|^2=0, \qquad
-{\mathfrak a}_2(0, 1)|{\tilde{\mathfrak a}}_1'(1)|^2
+ {\mathfrak a}_1(0, 1)|{\tilde{\mathfrak a}}_2'(1)|^2=0. \end{equation*} Using the above facts and expressions of ${\mathfrak w}_0$, ${\mathfrak l}_1$ as in \eqref{m11}, \eqref{m12}, we can derive that \begin{equation} {\mathfrak w}_0(0)
\, =\, {\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_2^2(0) |n_2(0)|^2+{\mathfrak a}_2(0, 0){\tilde{\mathfrak a}}_1^2(0) |n_1(0)|^2
\, =\, \frac{{\mathfrak a}_1(0, 0)}{2}, \label{E24} \end{equation} and \begin{align} {\mathfrak l}_1(0)
\, =\, &\big[{\mathfrak a}_2(0, 0)|n_2(0)|^2+{\mathfrak a}_1(0, 0)|n_1(0)|^2\big]\Theta_{tt}(0, 0)
+\big[{\mathfrak a}_2(0, 0){\tilde{\mathfrak a}}_1(0){\tilde{\mathfrak a}}_1'(0)|n_1(0)|^2
+{\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_2(0){\tilde{\mathfrak a}}_2'(0)|n_2(0)|^2\big]
\nonumber\\[2mm]
&-k\big[-{\mathfrak a}_2(0, 0)|{\tilde{\mathfrak a}}_1'(0)|^2
+ {\mathfrak a}_1(0, 0)|{\tilde{\mathfrak a}}_2'(0)|^2\big]n_1(0)n_2(0)
+\big[-\partial_t{\mathfrak a}_2(0, 0){\tilde{\mathfrak a}}_1(0)
+\partial_t{\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_2(0)\big]n_1(0)n_2(0) \nonumber\\[2mm]
\, =\, &{\mathfrak a}_1\big(0, 0\big)\frac{k_1}{2}
\, +\, \frac{{\mathfrak a}_1(0, 0)}{\sqrt{2}}\bigg[{\tilde{\mathfrak a}}_1'(0)|n_1(0)|^2+{\tilde{\mathfrak a}}_2'(0)|n_2(0)|^2\bigg] \nonumber\\[2mm] &\, +\, \frac{1}{\sqrt{2}}\bigg[-\partial_t{\mathfrak a}_2(0, 0) +\partial_t{\mathfrak a}_1(0, 0)\bigg]n_1(0)n_2(0), \label{E25} \end{align} where we have used \eqref{Thetaderivative1}-\eqref{Thetaderivative2} and \eqref{tildek1}.
The formulas \eqref{mathfrakh1}, \eqref{g12inverse} and \eqref{g22inverse} imply that
\begin{equation*}
{\mathfrak h}_1(0)\, =\, \big[{\tilde{\mathfrak a}}_1(0)|n_1(0)|^2+{\tilde{\mathfrak a}}_2(0)|n_2(0)|^2\big]^2 \, =\, {\tilde{\mathfrak a}}_1^2(0)=\frac{1}{2},
\end{equation*} $$ {\mathfrak g}_3(0)=\frac{1}{{\mathfrak h}_1(0)}\big[{\tilde{\mathfrak a}}_1(0)-{\tilde{\mathfrak a}}_2(0)\big]n_1(0)n_2(0)=0, $$ $$
{\mathfrak g}_6(0)\, =\, \frac{1}{{\mathfrak h}_1(0)}\big[{\tilde{\mathfrak a}}_1^2(0) |n_1(0)|^2 +{\tilde{\mathfrak a}}_2^2(0) |n_2(0)|^2 \big]\, =\, 1, $$ \begin{align*}
{\mathfrak g}_4(0)=&-\frac{1}{{\mathfrak h}_1(0)}\big[{\tilde{\mathfrak a}}_1(0){\tilde{\mathfrak a}'}_1(0)|n_1(0)|^2
+{\tilde{\mathfrak a}}_2(0){\tilde{\mathfrak a}'}_2(0) |n_2(0)|^2\big] \, +\, k\frac{1}{{\mathfrak h}_1(0)}\big[-{\tilde{\mathfrak a}}_1^2(0)+{\tilde{\mathfrak a}}_2^2(0)\big]n_1(0)n_2(0) \nonumber\\[2mm] &\qquad-\frac{1}{{\mathfrak h}_1(0)}\big[-q_1(0)n_2(0)+q_2(0)n_1(0)\big] +{\mathfrak g}_1\big[{\tilde{\mathfrak a}}_1(0)-{\tilde{\mathfrak a}}_2(0)\big]n_1(0)n_2(0) \nonumber\\[2mm]
\, =\, &-\sqrt{2}\Big({\tilde{\mathfrak a}'}_1(0)|n_1(0)|^2+{\tilde{\mathfrak a}'}_2(0) |n_2(0)|^2\Big) -k_1, \end{align*} where we have used \eqref{2.22}, \eqref{Thetaderivative1}-\eqref{Thetaderivative2} and \eqref{tildek1}. Recalling the definition of ${\mathfrak y}_4$, $\mathfrak{p}_8$ as in \eqref{sigma2}, \eqref{p8p9p10}, it is easy to obtain
\begin{equation*} {\mathfrak y}_4(0)\, =\, {\sqrt{{\mathfrak g}_6(0)}}\, =\, 1, \end{equation*} \begin{equation*}
\mathfrak{p}_8(0)\, =\, {\mathfrak a}_2(0, 0){\tilde{\mathfrak a}}_1(0)|n_1(0)|^2+{\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_2(0)|n_2(0)|^2
= {\mathfrak a}_1(0, 0) {\tilde{\mathfrak a}}_1(0)=\frac{{\mathfrak a}_1(0, 0)}{\sqrt{2}}. \end{equation*} Then by using \eqref{E24}, we get \begin{align}
{\mathfrak b}_1=&\frac{1}{\sqrt{{\mathfrak h}_1(0)}}{\mathfrak y}_4(0)\mathfrak{p}_8(0)
={\mathfrak a}_1(0, 0) =2{\mathfrak w}_0(0). \label{b1=mathfrakw0}
\end{align} On the other hand, recalling the definition of $\mathfrak{p}_1$, $\mathfrak{p}_4$, as in \eqref{p1p2}, \eqref{p3p4}, it is easy to obtain \begin{align*}
\mathfrak{p}_1(0)\, =\, {\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_1(0)|n_1|^2\, +\, {\mathfrak a}_2(0, 0){\tilde{\mathfrak a}}_2(0)|n_2|^2
\, =\, {\mathfrak a}_1(0, 0){\tilde{\mathfrak a}}_1(0)
\, =\, \frac{{\mathfrak a}_1(0, 0)}{\sqrt{2}}, \end{align*} \begin{align*} \mathfrak{p}_4(0)\, =\, -\big[\partial_t{\mathfrak a}_1(0, 0)- \partial_t{\mathfrak a}_2(0, 0) \big]n_1(0)n_2(0). \end{align*} The term ${\mathfrak y}_2$ given in \eqref{sigma1} will be evaluated as the following \begin{align*} {\mathfrak y}_2(0)\, =\, \frac{{\mathfrak g}_4}{{\mathfrak g}_6}-\frac{{\mathfrak g}_7{\mathfrak g}_3}{{\mathfrak g}_6^2}
\, =\, -\sqrt{2}\Big({\tilde{\mathfrak a}'}_1(0)|n_1(0)|^2+{\tilde{\mathfrak a}'}_2(0) |n_2(0)|^2\Big) -k_1. \end{align*} Then by using \eqref{E25}, we have
\begin{align} {\mathfrak b}_2=&\frac{1}{\sqrt{{\mathfrak h}_1(0)}}{\mathfrak y}_4(0)\mathfrak{p}_4(0) \, +\, \frac{1}{\sqrt{{\mathfrak h}_1(0)}}{\mathfrak y}_2(0)\mathfrak{p}_1(0) \nonumber\\[2mm] =&-\sqrt{2}\big[\partial_t{\mathfrak a}_1(0, 0)- \partial_t{\mathfrak a}_2(0, 0) \big]n_1(0)n_2(0) \,-\,
\sqrt{2}\Big({\tilde{\mathfrak a}'}_1(0)|n_1(0)|^2+{\tilde{\mathfrak a}'}_2(0) |n_2(0)|^2\Big){\mathfrak a}_1(0, 0) \,-\, k_1 {\mathfrak a}_1(0, 0) \nonumber\\[2mm] \, =\, &-2{\mathfrak l}_1(0). \label{b2=mathfrakw1}
\end{align} The formulas \eqref{b1=mathfrakw0} and \eqref{b2=mathfrakw1} will lead to
\begin{align} \frac{ V^{\sigma}\big(0, \theta\big)}{\sqrt{{\mathfrak f}_0}} \Big[{\mathfrak w}_0(0)f'(0) \, +\, {\mathfrak l}_1(0)f(0) \Big] \, =\, \frac{ V^{\sigma}\big(0, \theta\big)}{2\sqrt{{\mathfrak f}_0}} \big[ {\mathfrak b}_1f'(0) \,-\, {\mathfrak b}_2f(0) \big].
\end{align} This is the formula \eqref{boundaryoffunctional2}. Formula \eqref{boundaryoffunctional1} can be verified in the identically same way.
\end{appendices}
\end{document} | arXiv |
\begin{document}
{{\title{Completions of Implicative Assemblies}
\author[A. Miquel]{Alexandre Miquel} \address{\IMERL} \author[K. Worytkiewicz]{Krzysztof Worytkiewicz} \address{\LAMA} \date{}
\maketitle}}
\begin{abstract}
We continue our work on implicative assemblies by investigating under which
circumstances a subset $M \subseteq \mathscr{S}$ gives rise to a lex full
subcategory $\asm{M}$ of the quasitopos
$\asm{\mathcal{A}}$ of all assemblies such that
$\reglex{(\asm{M})} \simeq
\asm{\mathcal{A}}$. We establish a
characterisation. Furthermore, this latter is relevant to the study of
$\asm{M}$'s ex/lex-completion. \end{abstract}
\tableofcontents
\section{Introduction} \label{sec:intro}
In this note we investigate under which circumstances a subset $M \subseteq \mathscr{S}$ gives rise to a lex full subcategory $\asm{M}$ of the quasitopos $\asm{\mathcal{A}}$ of all assemblies such that $\reglex{(\asm{M})} \simeq \asm{\mathcal{A}}$. It is well-known that we have such a situation in the effective topos $\ensuremath{\mathcal{E}\!f\!f}$ {\cite{hyland1980tripos,hyland1982effective}} where there is the subcategory $\ensuremath{\mathbf{P}}\ensuremath{\mathbf{A}}\ensuremath{\mathbf{s}}\ensuremath{\mathbf{s}} \subseteq \ensuremath{\omega \ens} \subseteq \ensuremath{\mathcal{E}\!f\!f}$ whose objects are known as {\tmem{partitioned assemblies}} (their existence predicates are singletons \cite{robinson1990colimit,rosolini2019elementary}). We establish a characterisation for the implicative case and the case of partitioned assemblies turns out to be quite easy going. We next observe that $\exreg{(\asm{\mathcal{A}})}$ is always a topos, which is a rather immediate consequence of the material in {\cite{menni2000exact}}. This observation implies that $\exlex{(\asm{M})}$ is a topos as well. We then construct a functor \[ \mathcal{K}: \exlex{(\asm{M})} \rightarrow \ensuremath{\tmmathbf{\tmop{Set}}}
[\mathcal{A}] \] the latter being the topos constructed from an implicative tripos {\cite{miquel2020implicative,Castro:2023aa}} by means of the tripos-to-topos construction. It turns out that $\mathcal{K}$ may not always be an equivalence, although it is always the case when $\mathcal{A}$ is compatible with joins. There are undoubtly connections to the material in {\cite{maietti2015unifying}}, we only (very) recently became aware of the latter work. Then again our focus is quite different as we specifically investigate categorical aspects of implicative algebras. The reader is referred to {\cite{miquel2020implicative,Castro:2023aa}} for basic material about implicative algebras and implicative assemblies.
\section{Regular Completions} \label{sec:regular}
\subsection{Memjog on Regular Categories} {$\mbox{}$}\\ \\ The notion of regular category axiomatises image factorisations, as a matter of fact an abelian category is an additive regular category. It is thus not really surprising that regular categories have been studied for some time {\cite{barr1971exact}}.
\begin{definition}
\leavevmode
\begin{enumeratenumeric}
\item A {\tmem{regular}} epi is an epi which is a coequaliser (in any
category).
\item A {\tmem{lex category}} is a category with finite limits.
\end{enumeratenumeric} \end{definition}
\begin{remark}
In a lex category, a regular epi is a coequaliser if its kernel pair. \end{remark}
\begin{definition}
A lex category $\mathbb{D}$ is regular if
\begin{enumerateroman}
\item it has coequalisers of kernel pairs;
\item regular epis are stable under pullback.
\end{enumerateroman} \end{definition}
Despite notorious non-examples like $\ensuremath{\tmmathbf{\tmop{Top}}}$ (always troubles with this one..) or $\ensuremath{\tmmathbf{\tmop{Cat}}}$, regular categories abound in nature: $\ensuremath{\tmmathbf{\tmop{Set}}}$, $\mathbf{G}\mathbf{r}\mathbf{p}$ (all groups), $\ensuremath{\tmmathbf{\tmop{Ab}}}$, any LCCC with coequalisers (thus any quasitopos), any topos etc etc. As mentioned, any abelian category is regular as well. This is the reason why in the context of regular categories we tend to call {\tmem{exact sequence}} a coequaliser diagram of a kernel pair. For, the datum
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=0.793585202676112cm]{regular-1.pdf}} \end{center}
(with $(e_0, e_1)$ the kernel pair of $e$) in an abelian category yields the short exact sequence
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=1.0910730683458cm]{regular-2.pdf}} \end{center}
and vice versa. An equivalent definition of a regular category is then as above but with condition $i \nocomma i.$ replaced by requiring that exact sequences are stable under pullback.
Given any morphism $f : X \rightarrow Y$ in a regular category we can construct the diagram
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=1.98350386986751cm]{regular-3.pdf}} \end{center}
with $\tmop{Im} (f) \cong X / \tmop{Ker} (f)$, $\bar{f}$ the coequaliser of $f$'s kernel pair $(f_0, f_1)$ and $\iota_f$ given by universal property (it is an exercise in diagram chasing to prove that $\iota_f$ is mono). In an abelian category it is just the first isomorphism theorem.
\begin{notation}
Given a morphism $f$ in a regular category, we shall write $f = \iota_f
\circ \bar{f}$ for its factorisation with $\iota_f$ mono and $\bar{f}$
regular epi. \end{notation}
Regular epis and monos form an orthogonal (thus functorial) factorisation system $(\mathcal{E}_{\tmop{reg}}, \mathcal{M})$. Another equivalent definition of a regular category is as a lex category with pullback-stable image factorisations.
\begin{example}
$\mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}$ is a quasitopos, hence
regular. It is nonetheless handy to know how to calculate image
factorisations. Let $f : X \rightarrow Y$ be a morphism in
$\mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}$. Its kernel pair is given by
the pullback square
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.87590187590188cm]{regular-4.pdf}}
\end{center}
Recall that a pullback square in
$\mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}$ is given on carriers by the
forgetful functor $\Gamma : \mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}
\rightarrow \ensuremath{\tmmathbf{\tmop{Set}}}$, with existence and projections inherited from the product
in $\mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}$, so
$\exi{\tmop{Ker} (f)} (x, x') = x \sqcap x'$
while $p_1$ and $p_2$ are tracked by
$\left(\lam{z}{z \left( \lam{x \nocomma y}{x} \right)} \right)^{\mathcal{A}}$
and
$\left( \lam{z}{z \left( \lam{x \nocomma y}{\nocomma y} \right)}
\right)^{\mathcal{A}}$, respectively. Let $[x] := f^{- 1} (f (x))$ be
the inverse image of $f (x)$. The coequaliser diagram
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=1.48770169224715cm]{regular-5.pdf}}
\end{center}
is given by
\begin{enumerateroman}
\item $\car{\tmop{Im} (f)} = X / \sim$ where $\sim$ is the equivalence
relation generated by $f (x) \sim f (x')$, so \ $\car{\tmop{Im} (f)}$ is
the set $\{ [x] |x \in X \}$. The existence predicate is
$\exi{\tmop{Im} (f)} ([x]) = \iex{x' \in [x]} \exi{X} (x')$;
\item $\bar{f} : x \mapsto [x]$ is the canonical projection, tracked by
$\left( \lam{x \nocomma z}{z \nocomma x} \right)^{\mathcal{A}}$.
\end{enumerateroman}
The image factorisation
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=1.8843303161485cm]{regular-6.pdf}}
\end{center}
is in turn given by universal property of $\tmop{Im} (f)$. Specifically, we
have $\iota_f ([x]_f) = f (x)$ independently of the choice of represantant
$x$, tracked by $\ilam{z}{z \tau}$ with $\tau$ some tracker of $f$. It can be
seen without invoking category theory that $\iota_f$ is injective as a map,
thus a mono in $\mathbf{A}\mathbf{s}\mathbf{m}_{\mathcal{A}}$. \end{example}
\subsection{The Universal Property of a Regular Completion} \label{subsec:reg-universal}
\begin{definition} \leavevmode
\begin{enumeratenumeric}
\item A {\tmem{lex functor}} is a functor commuting with finite limits.
\item Let $\mathbb{D}$ and $\mathbb{D}'$ be regular categories. An
{\tmem{exact functor}} $R : \mathbb{D} \rightarrow \mathbb{D}'$ is a lex
functor which preserves regular epis.
\end{enumeratenumeric} \end{definition}
Equivalently, a lex functor preserving exact sequences is exact (whence the name). An exact functor obviously preserves image factorisations.
\begin{notation}
We shall write
\begin{itemizeminus}
\item {\ensuremath{\tmmathbf{\tmop{Lex}}}} for the 2-category of lex categories, lex funtors and their
natural transformations;
\item {\ensuremath{\tmmathbf{\tmop{Reg}}}} for the 2-category of regular categories, exact funtors and
their natural transformations;
\item $| - | : \ensuremath{\tmmathbf{\tmop{Reg}}} \rightarrow \ensuremath{\tmmathbf{\tmop{Lex}}}$ for the forgetful $2$-functor.
\end{itemizeminus} \end{notation}
It is part of the lore that $| - |$ has a left biadjoint $(-)_{\tmop{reg} / \tmop{lex}}$. Given a lex category $\mathbb{C}$ we thus have a lex functor $\ensuremath{\tmmathbf{y}} : \mathbb{C} \rightarrow \mathbb{C}_{\tmop{reg} / \tmop{lex}}$ such that for any lex functor $L : \mathbb{C} \rightarrow \mathbb{D}$ with $\mathbb{D}$ a regular category there is a unique exact functor $\mathcal{U}: \mathbb{C}_{\tmop{reg} / \tmop{lex}} \rightarrow \mathbb{D}$ such that the triangle
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=3.09753377935196cm]{regular-7.pdf}} \end{center}
commutes (up-to equivalence). The objects of $\mathbb{C}_{\tmop{reg} / \tmop{lex}}$ are morphisms of $\mathbb{C}$. A morphism \[ [l] : \arobj{f}{X}{Y} \rightarrow \arobj{f'}{X'}{Y'} \] in $\mathbb{C}_{\tmop{reg}}$ is an equivalence class $[l]$ of morphisms $l : X \rightarrow X'$ such that $f' \circ l$ coequalises $f$'s kernel pair, with $[l_1] = [l_2]$ if $f' \circ l_1 = f' \circ l_2$.
The insertion functor $\ensuremath{\tmmathbf{y}} : \mathbb{C} \rightarrow \mathbb{C}_{\tmop{reg}} \tmop{lex}$ is given by $X \mapsto \arobj{\tmop{id}}{X}{X}$
Given a morphism \[ [f] : \arobj{\tmop{id}}{X}{X} \rightarrow \arobj{\tmop{id}}{Y}{Y} \] in the image of $\ensuremath{\tmmathbf{y}}$, it is obvious that $f$ is the only member of $[f]$ and easy to see that \begin{eqnarray*}
\tmop{Im} ([f]) & = & \arobj{f}{X}{Y} \end{eqnarray*} (where $f$ is seen as an object of $\reglex{\mathbb{C}}$). It would be fair to say that this is the idea behind the construction as the latter freely adds images.
A frequent use-case is when $\mathbb{D} \in \ensuremath{\tmmathbf{\tmop{Reg}}}$ and $\mathbb{C} \subset \mathbb{D}$ is a lex subcategory. The canonical functor $\mathcal{U}$ making the diagram
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.38013249376886cm]{regular-8.pdf}} \end{center}
commute is particularly easy to describe in this situation: \begin{eqnarray*}
\mathcal{U}: \mathbb{C}_{\tmop{reg} / \tmop{lex}} & \longrightarrow &
\mathbb{D}\\
\arobj{f}{X}{Y} & \mapsto & \tmop{Im} (f) \end{eqnarray*} with action on morphisms given by universal property as follows. Assume a morphism \[ [l] : \arobj{f}{X}{Y} \rightarrow \arobj{f'}{X'}{Y'} \] We then have the diagram
\begin{center}
\raisebox{-0.499997244131378\height}{\includegraphics[width=14.8909550045914cm,height=2.97505903187721cm]{regular-9.pdf}} \end{center}
in $\mathbb{D}$. As $f' \circ l$ coequalises the kernel pair $(f_0, f_1)$ of $f$ and $\iota_{f'}$ is mono, $\overline{f'}$ coequalises $(f_0, f_1)$ as well so $\tilde{l}$ is given by universal property of $\bar{f}$. It is easy to see that the assigment $[l] \mapsto \tilde{l}$ is well defined and functorial. Moreover, $\mathcal{U}$ is always faithful in this situation. Finally, we have $\mathbb{C}_{\tmop{reg} / \tmop{lex}} \simeq \mathbb{D}$ if and only if $\mathcal{U}$ is (part of) an equivalence of categories.
\begin{definition}
$\mathbb{D}$ is a {\tmem{regular completion}} of $\mathbb{C}$ if
$\mathcal{U}$ is an equivalence of categories. \end{definition}
In practice we thus need to check if $\mathcal{U}$ is essentially surjective and full. It is well-known that if $\mathbb{D}$ is a {\tmem{regular completion}} of $\mathbb{C}$, closing $\mathbb{C}$ under isomorphisms yields $\mathbb{D}$'s subcategory of projective objects.
\section{Some Implicative Remarks} \label{sec:implicative-remarks}
\subsection{Assemblies as Indexed Families.}
\begin{notation}
Let $\mathcal{A}$ be an implicative algebra.
\begin{enumeratenumeric}
\item $\mathbb{P} [\mathcal{A}]$ stands for the implicative tripos over
$\mathcal{A}$ and $\fibr{X}$ for its fibre over $X$.
\item Let $Q \subseteq \mathscr{S}$ be a subset of $\mathscr{S}$. We shall
write $\mathbb{P} [Q]$ for the full subcategory of $\mathbb{P}
[\mathcal{A}]$ of families of elements of $Q$.
\item $\mathbb{P} [Q]_X := \fibr{X} \cap \mathbb{P} [Q]$.
\end{enumeratenumeric} \end{notation}
\begin{definition}
We call $\mathbb{P} [Q]_X$ {\tmem{the $Q$-fibre}} over $X$. \end{definition}
\begin{proposition}
\label{prop:tracked}The assignment
\begin{eqnarray*}
\Xi : \asm{\mathcal{A}} & \longrightarrow &
\mathbb{P} [\mathscr{S}]\\
X & \mapsto & \exi{X}\\
{\scriptstyle f} \downarrow & & \downarrow {\scriptstyle \left( f, \exi{X} \sqsubseteq_X f^{\ast}
\exi{Y} \right)}\\
Y & \mapsto & \exi{Y}
\end{eqnarray*}
is an isomorphism of categories with inverse
\begin{eqnarray*}
\Xi^{- 1} : \mathbb{P} [\mathscr{S}] & \longrightarrow &
\asm{\mathcal{A}}\\
(u_x)_{x \in X} & \mapsto & (X, u)\\
{\scriptstyle (f, u \sqsubseteq_X f^{\ast} v)} \downarrow & & \text{\quad} \downarrow
{\scriptstyle f}\\
(v_y)_{y \in Y} & \mapsto & (Y, v)
\end{eqnarray*} \end{proposition}
\begin{proof}
$\Xi_0$ and $\Xi^{- 1}_0$ are obviously inverse bijections (on classes).
Furthermore, the maps
\begin{eqnarray*}
\Xi_{X, Y} : \asm{\mathcal{A}} (X, Y) &
\longrightarrow & \mathbb{P} [\mathscr{S}] \left( \exi{X}, \exi{Y}
\right)\\
f & \mapsto & \left( f, \exi{X} \sqsubseteq_X f^{\ast} \exi{Y} \right)
\end{eqnarray*}
and
\begin{eqnarray*}
\Xi^{- 1}_{u, v} : \mathbb{P} [\mathscr{S}] ((u_x)_{x \in X}, (v_y)_{y \in
Y}) & \longrightarrow & \asm{\mathcal{A}} (X,
Y)\\
(f, u \sqsubseteq_X f^{\ast} v) & \mapsto & f
\end{eqnarray*}
are inverse bijections since
\begin{eqnarray*}
\text{$f$ is tracked} & \Longleftrightarrow & \bigM_{x \in X}
\left( \exi{X} (x) \rightarrow \exi{Y} (f (x)) \right) \in \mathscr{S}\\
& \Longleftrightarrow & \bigM \left( \exi{X} \rightarrow_X
f^{\ast} \exi{Y} \right) \in \mathscr{S}\\
& \Longleftrightarrow & \exi{X} \sqsubseteq_X f^{\ast} \exi{Y}
\end{eqnarray*}
\end{proof}
The ismorphism above is obviously rather shallow. Still, it is quite practical to be able to take either point of view according to a given situation.
\subsection{Stability under finite Limits}
\begin{definition}
Let $M \subseteq \mathscr{S}$. The category $\asm{M}$ of
{\tmem{$M$-assemblies}} is the full subcategory of
$\asm{\mathcal{A}}$ with objects assemblies
valued in $M$. \end{definition}
\begin{remark}
$\asm{M} \cong \mathbb{P} [M]$ \end{remark}
Recall that in an implicative algebra, the product $\sqcap$ (with respect to the entailment preorder) is given by the second-order encoding of pairs.
\begin{definition}
A subset $M \subseteq \mathscr{S}$ is {\tmem{algebraic}} if if it is closed under
$\sqcap$. \end{definition}
\begin{remark}
$\asm{M}$ is closed under finite limits if $M$ is algebraic. \end{remark}
\begin{example}
\label{ex:alg}
$\mathcal{A}= \left( \pows{P}, \subseteq, \rightarrow, \mathscr{S} = \ppows{P} \right)$ with $P$
a combinatory algebra. The subset
\begin{eqnarray*}
M & := & \{ \{ p \} |p \in P \}
\end{eqnarray*}
of the separator is manifestly algebraic as in this case we have $\{ p_1 \}
\sqcap \{ p_2 \} = \{ \langle p_1, p_2 \rangle \}$. \end{example}
\subsection{Implicative Existence}
Before we proceed further we need to compile some further useful facts about the operation of existence in implicative algebras.
\begin{remark}
\label{rem:eta}Recall that the implicative existence is given by
second-order encoding as
\begin{eqnarray*}
\bigE U & = & \bigM_{c \in \mathcal{A}} \bigM_{u \in U}
((u \rightarrow c) \rightarrow c)
\end{eqnarray*}
\begin{enumeratenumeric}
\item We have $\vdash \lam{z}{z \nocomma u} : \forall c : \mathcal{A}.
\forall u : U. (u \rightarrow c) \rightarrow c$, hence $\intpr{\lam{z}{z
\nocomma u}} \preccurlyeq \bigE U$ \ for all $u \in U$, so in particular
$\bigE U \in \mathscr{S}$ if $U \cap \mathscr{S} \neq \varnothing$.
\item A consequence of the above item is that if we have $\chi := a
\rightarrow \bigE U$ then for any $a' \preccurlyeq a$ there is an $u \in
U$ such that $\chi a' \preccurlyeq \ilam{z}{z u}$. We systematically
exploit this fact, in particular when building trackers.
\end{enumeratenumeric} \end{remark}
\subsection{Valuations}
\begin{definition}
Let $X$ be a set and $\varnothing \neq M \subseteq \mathscr{S}$. An
{\tmem{$M$-valuation of $X$}} (or just {\tmem{valuation}} if $M$ and $X$ are
understood) is map $\nu : X \rightarrow \ppows{M}$. \end{definition}
\begin{remark}
\label{rem:fib-val}Let $X$ be a set and $M \subseteq \mathscr{S}$. We have the
predicate
\begin{eqnarray*}
\tmop{ex} : \ppows{M} & \longrightarrow & \mathscr{S}\\
U & \mapsto & \bigE U
\end{eqnarray*}
in $\mathbb{P} [\mathscr{S}]_{\ppows{M}}$. Given a valuation $\nu : X
\rightarrow \ppows{M}$, we have the predicate $\nu^{\ast} (\tmop{ex}) \in
\mathbb{P} \left[ \mathscr{S} \right]_X$ where
\begin{eqnarray*}
\nu^{\ast} (\tmop{ex})_x & = & (\tmop{ex} \circ \nu) (x)\\
& = & \tmop{ex} (\nu (x))\\
& = & \bigE (\nu (x))
\end{eqnarray*}
so
\begin{eqnarray*}
\nu^{\ast} (\tmop{ex}) & = & {\left( \bigE (\nu (x)) \right)_{x \in X}}
\end{eqnarray*} \end{remark}
\begin{notation}
Let $\nu : X \rightarrow \ppows{M}$ be a valuation. We shall use the
notation $\bigE \nu : = \nu^{\ast} (\tmop{ex})$. \end{notation}
\begin{lemma}
\label{lem:induced-valuation}Any morphism $f : X \rightarrow Y$ in $\asm{M}$
gives rise to the valuation
\begin{eqnarray*}
\nu_f : \car{\tmop{Im} (f)} & \longrightarrow & \ppows{M}\\
{}[x] & \mapsto & \left\{ \exi{X} (x') |x' \in [x] \right\}
\end{eqnarray*}
and we have $\exi{\tmop{Im} (f)} = \bigE \nu_f$. \end{lemma}
\begin{notation}
$[x] := f^{- 1} (f (x))$. \end{notation}
\begin{proof}
We have $\car{\tmop{Im} (f)} = \{ [x] |x \in X \} $ where $\exi{\tmop{Im}
(f)} ([x]) = \iex{x' \in [x]} \exi{X} (x')$, hence
\begin{eqnarray*}
(\bigE \nu_f)_{[x]} & = & (\nu_f^{\ast} \tmop{ex})_{[x]}\\
& = & \iex{x' \in [x]} \exi{X} (x')
\end{eqnarray*}
\end{proof}
\begin{proposition}
\label{prop:induced-valuation}Let $X$ be a set and $\nu : X \rightarrow
\ppows{M}$ be a valuation. Let $\hat{X} \in \asm{M}$ be the assembly given by
\begin{itemizeminus}
\item $\car{\hat{X}} := \sum_{x \in X} \nu (x)$;
\item $\exi{\hat{X}} (x, m) := m$ for all $x \in X$ and $m \in \nu(x)$.
\end{itemizeminus}
and $\check{X} \in \asm{M}$ be the assembly given by
\begin{itemizeminus}
\item $\car{\check{X}} := X$;
\item $\exi{\check{X}} (x) := m_0$ for all $x \in X$ and some $m_0
\in M$.
\end{itemizeminus}
The obvious surjective map $g_{\nu} : \car{\hat{X}}
\twoheadrightarrow \car{\check{X}}$ is tracked by
$(\lambda x.m_0)^{\mathcal{A}}$
and we have the isomorphism
\begin{eqnarray*}
{}[-]_{\nu} : \left( X, \bigE \nu \right) & \mathop{\longrightarrow}\limits^{\cong} &
\left( \tmop{Im} (g_{\nu}), \bigE g_{\nu} \right)\\
x & \mapsto & g_{\nu}^{- 1} (x)
\end{eqnarray*}
in $\asm{\mathcal{A}}$. \end{proposition}
\begin{proof}
Consider the image factorisation
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.38013249376886cm]{implicative_remarks-1.pdf}}
\end{center}
We have $\tmop{Im} (g_{\nu}) = \{ g_{\nu}^{- 1} (x) |x \in X \}$ as
\begin{eqnarray*}
{}[x_m] & = & g_{\nu}^{- 1} (g_{\nu} (x_m))\\
& = & g_{\nu}^{- 1} (x)
\end{eqnarray*}
for all $m \in \nu (x)$, hence the bijection
\begin{eqnarray*}
{}[-]_{\nu} : X & \mathop{\longrightarrow}\limits^{\cong} & \tmop{Im} (g_{\nu})\\
x & \mapsto & g_{\nu}^{- 1} (x)
\end{eqnarray*}
We have
\begin{eqnarray*}
\nu_{g_{\nu}} (g_{\nu}^{- 1} (x)) & = & \left\{ \exi{\hat{X}} (x_m) |x_m
\in g_{\nu}^{- 1} (x) \right\}\\
& = & \{ m \in M|x_m \in g_{\nu}^{- 1} (x) \}\\
& = & \nu (x)
\end{eqnarray*}
hence
\begin{eqnarray*}
\exi{\tmop{Im} (g_{\nu})} ([x]_{\nu}) & = & {\left( \bigE \nu_{g_{\nu}}
\right)_{g_{\nu}^{- 1} (x)}} \text{\qquad Lemma
\ref{lem:induced-valuation}}\\
& = & \bigE (\nu_{g_{\nu}} (g_{\nu}^{- 1} (x)))\\
& = & \bigE \nu (x)
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\bigM_{x \in X} \left( \left( \bigE \nu \right)_x \rightarrow
\exi{\tmop{Im} (g_{\nu})} ([x]_{\nu}) \right) & = & \bigM_{x \in
X} \left( \left( \bigE \nu \right)_x \rightarrow \left( \bigE \nu
\right)_x \right)\\
& = & \bigM_{x \in X} \left( \exi{\tmop{Im} (g_{\nu})}
([x]_{\nu}) \rightarrow \left( \bigE \nu \right)_x \right)
\end{eqnarray*}
But $\bigM_{x \in X} \left( \left( \bigE \nu \right)_x \rightarrow
\left( \bigE \nu \right)_x \right) \in \mathscr{S}$, hence $[-]_{\nu}$ and \
$[-]_{\nu}^{- 1}$ are tracked.
\end{proof}
\section{Regular Completions of Implicative Assemblies} \label{sec:regular-completion-asm}
\subsection{Density}
\begin{definition}
A non-empty subset $M \subseteq \mathscr{S}$ of the separator is
{\tmem{dense}} in $\mathscr{S}$ (or just {\tmem{dense}} if $\mathscr{S} $ is understood)
if there is a valuation $\nu : \mathscr{S} \rightarrow \ppows{M}$ such that
$\tmop{id}_{\mathscr{S}} \cong_{\mathscr{S}} \bigE \nu$. \end{definition}
\begin{theorem}
\label{th:dense}Let $M \subseteq \mathscr{S}$ be an algebraic subset. The
following are equivalent
\begin{enumerateroman}
\item $M$ is dense;
\item for any $X \in \ensuremath{\tmmathbf{\tmop{Set}}}$ and any $u \in \mathbb{P} \left[ \mathscr{S}
\right]_X$ there is a valuation $\nu : X \rightarrow \ppows{M}$ such that
$u \cong_X \bigE \nu$;
\item the functor $U : \reglex{\left( \asm{M} \right)} \rightarrow
\asm{\mathcal{A}}$ is essentially surjective.
\end{enumerateroman} \end{theorem}
\begin{proof}
We need to insist that $M$ is algebraic only in order to be able to talk
about $\reglex{\left( \asm{M} \right)}$, otherwise it is {\tmem{deus ex
machina}} behind the stage.
$(i) \Longrightarrow (i \nocomma i)$ There is a valuation $\nu : \mathscr{S}
\rightarrow \ppows{M}$ of $\mathscr{S}$ such that $\tmop{id}_{\mathscr{S}} \cong_{\mathscr{S}}
\bigE \nu$. Let $u \in \sfibr{X}$. We have
\begin{eqnarray*}
\tmop{id}_{\mathscr{S}} & = & i_s^{\ast} (\tmop{id}_{\mathcal{A}})
\end{eqnarray*}
with $i_{\mathscr{S}} : \mathscr{S} \rightarrowtail \mathcal{A}$ the insertion map,
hence
\begin{eqnarray*}
u & = & u^{\ast} (\tmop{id}_{\mathcal{A}}) \hspace{4em}
\tmop{id}_\mathcal{A} \in \fibr{\mathcal{A}}
\text{is a split generic object}\\
& = & \left( i_{\mathscr{S}} \circ u \right)^{\ast} (\tmop{id}_{\mathcal{A}})\\
& = & u^{\ast} (i_s^{\ast} (\tmop{id}_{\mathcal{A}})) \hspace{3em}
\mathbb{P} [\mathcal{A}] \rightarrow \ensuremath{\tmmathbf{\tmop{Set}}} \text{ is a split fibration}\\
& = & u^{\ast} (\tmop{id}_{\mathscr{S}})\\
& \cong_X & u^{\ast} \left( \bigE \nu \right) \qquad
\text{functoriality}\\
& = & \bigE (\nu \circ u)
\end{eqnarray*}
so the valuation $\nu \circ u : X \rightarrow \ppows{M}$ witnesses the fact
that $M$ is dense.
$(i \nocomma i) \Longrightarrow (\nocomma i \nocomma i \nocomma i)$ Let $X
\in \asm{M}$. Assume a valuation $\nu : X \rightarrow \ppows{M}$ such that
$\exi{X} \cong_X \bigE \nu$. There is a
morphism $g_{\nu}$ such that $\Xi (\tmop{Im} (g_{\nu})) = \bigE \nu$
(c.f. Lemma \ref{lem:induced-valuation}). We furthermore have $\Xi (X) =
\exi{X}$, hence the isomorphism
\begin{eqnarray*}
\Xi^{- 1} \left( \exi{X} \cong_X \bigE \nu \right) : X &
\mathop{\longrightarrow}\limits^{\cong} & \tmop{Im} (g_{\nu})
\end{eqnarray*}
$(i \nocomma i \nocomma i) \Longrightarrow (i \nocomma)$ We have $\Xi^{- 1}
\left( \tmop{id}_{\mathscr{S}} \right) = \left( \mathscr{S}, \tmop{id}_{\mathscr{S}} \right)$.
There is by hypothesis a morphism $f : X \rightarrow Y$ in $\asm{M}$ such
that $\mathscr{S} \cong \tmop{Im} (f)$. Let $k : \mathscr{S}
\mathop{\longrightarrow}\limits^{\cong} \tmop{Im} (f)$ be a witnessing iso. We have
\begin{eqnarray*}
\Xi (k) & = & \left( k, \tmop{id} \text{ } \nocomma \sqsubseteq_{\mathscr{S}}
\text{ } k^{\ast} \bigE \nu_f \right)\\
& = & \left( k, \tmop{id} \text{ } \nocomma \sqsubseteq_{\mathscr{S}} \text{ }
\bigE (\nu_f \circ k) \right)
\end{eqnarray*}
and
\begin{eqnarray*}
\Xi (k^{- 1}) & = & \left( k^{- 1}, \bigE \nu_f \text{ }
\sqsubseteq_{\tmop{Im} (f)} \text{ } (k^{- 1})^{\ast} \tmop{id}_{\mathscr{S}}
\right)\\
& = & \left( k^{- 1}, \bigE \nu_f \text{ } \sqsubseteq_{\tmop{Im} (f)}
\text{ } k^{- 1} \right)
\end{eqnarray*}
Notice that $k^{- 1} : \tmop{Im} (f) \rightarrow \mathscr{S}$ has the ``right
type'' here in the sense that $k^{- 1} \in \mathbb{P} \left[ \mathscr{S}
\right]_{\tmop{Im} (f)}$. Reindexing the vertical morphism $\bigE \nu_f
\text{ } \sqsubseteq_{\tmop{Im} (f)} \text{ } k^{- 1}$ by $k$ yields
\begin{eqnarray*}
k^{\ast} \left( \bigE \nu_f \text{ } \sqsubseteq_{\tmop{Im} (f)} \text{ }
k^{- 1} \right) & = & k^{\ast} \left( \bigE \nu_f \right) \text{ }
\sqsubseteq_{\mathscr{S}} \text{ } k^{\ast} (k^{- 1})\\
& = & \bigE (\nu_f \circ k) \text{ } \sqsubseteq_{\mathscr{S}} \text{ }
\tmop{id}_{\mathscr{S}}
\end{eqnarray*}
We thus have $\tmop{id}_{\mathscr{S}} \text{ } \cong_{\mathscr{S}} \text{ } \bigE
(\nu_f \circ k)$ by virtue of the valuation $\nu_f \circ k$. \end{proof}
\begin{example}
\label{ex:single-dense} In example \ref{ex:alg} we considered an implicative
algebra $\mathcal{A}= \left( \pows{P}, \subseteq, \rightarrow, \mathscr{S}
\right)$ constructed from an applicative structure $P$ and showed that the
set $M := \{ \{ p \} |p \in P \}$ of all singletons is algebraic. Now
$\mathscr{S} = \ppows{P}$ here so we have the {\tmem{trivial valuation}}
\begin{eqnarray*}
\nu : \mathscr{S} & \longrightarrow & \ppows{M}\\
U & \mapsto & \{ \{ m \} |m \in U \}
\end{eqnarray*}
which yields $\bigE \nu_U = U$, hence
\begin{eqnarray*}
\bigM_{U \in \mathscr{S}} \left( U \rightarrow \left( \bigE \nu
\right)_U \right) & = & \bigcap_{U \in \mathscr{S}} (U \rightarrow U)\\
& = & \bigM_{U \in \mathscr{S}} \left( \left( \bigE \nu \right)_U
\rightarrow U \right)
\end{eqnarray*}
But $\bigcap_{U \in \mathscr{S}} (U \rightarrow U) \neq \varnothing$ as it
contains $\tmop{id}$, hence $\tmop{id} \cong_{\mathscr{S}} \bigE \nu$. \end{example}
\begin{definition}
An implicative structure $\mathcal{A}$ is {\tmem{compatible with joins}}
provided
\begin{eqnarray*}
\bigM_{a \in A} (a \rightarrow b) & = & \left( \bigcurlyvee_{a
\in A} a \right) \rightarrow b
\end{eqnarray*}
for all $A \subseteq \mathcal{A}$ and $b \in \mathcal{A}$. An implicative
algebra is compatible with joins if it is the case for the underlying
implicative structure. \end{definition}
\begin{remark}
In an implicative algebra $\mathcal{A}$ compatible with joins the operations
$\bigE$ and $\bigcurlyvee$are the same. \end{remark}
\begin{proposition}
et $\mathcal{A}$ be an implicative algebra compatible with joins. An
algebraic subset $M \subseteq \mathscr{S}$ is dense provided any element of
$\mathscr{S}$ is a join of elements of $M$. \end{proposition}
\begin{proof}
For each $a \in \mathcal{A}$ we can choose a subset $M_a \subseteq M$ such
that $a = \bigcurlyvee M_a$ by virtue of $(i)$, hence there is the trivial
valuation
\begin{eqnarray*}
\nu : \mathcal{A} & \longrightarrow & \ppows{M}\\
a & \mapsto & M_a
\end{eqnarray*}
We have
\begin{eqnarray*}
\left( \bigE \nu \right)_a & = & \bigE M_a\\
& = & \bigcurlyvee M_a \qquad \text{compatible with joins}\\
& = & a
\end{eqnarray*}
hence
\begin{eqnarray*}
\bigM_{a \in A} \left( a \rightarrow \left( \bigE \nu \right)_a
\right) & = & \bigM_{a \in \mathcal{A}} (a \rightarrow a)\\
& = & \bigM_{a \in A} \left( \left( \bigE \nu \right)_a
\rightarrow a \right)
\end{eqnarray*}
But $\tmop{id} \preccurlyeq \bigM_{a \in \mathcal{A}} (a
\rightarrow a)$ hence $\bigM_{a \in \mathcal{A}} (a \rightarrow a)
\in \mathscr{S}$, which implies that $M$ is dense. \end{proof}
\subsection{Compactness}
\begin{lemma}
\label{lem:pre-compact}Let $M \subseteq \mathscr{S}$ and let $g : A
\rightarrow B$ be a morphism in $\asm{M}$. The following are equivalent
\begin{enumerateroman}
\item any morphism $k : X \rightarrow \tmop{Im} (g)$ in
$\asm{\mathcal{A}}$ with $X{\in}{\asm{M}}$ admits a
lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.87590187590188cm]{regular_completion_asm-1.pdf}}
\end{center}
\item any morphism $l \circ \bar{f} : X \rightarrow \tmop{Im} (g)$ in
$\asm{\mathcal{A}}$ where
\begin{itemizeminus}
\item $f : X \rightarrow Y$ is a morphism in $\asm{M}$;
\item $l : \tmop{Im} (f) \rightarrow \tmop{Im} (g)$ is a morphism in
$\asm{\mathcal{A}}$
\end{itemizeminus}
admits a lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.97507542962088cm]{regular_completion_asm-2.pdf}}
\end{center}
\end{enumerateroman} \end{lemma}
\begin{proof}
The implication $(i) \Rightarrow (i \nocomma i)$ is trivial. Assume $(i
\nocomma i)$ and let $k : X \rightarrow \tmop{Im} (g)$ be a morphism in
$\asm{\mathcal{A}}$. We have the diagram
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=4.85904499540863cm]{regular_completion_asm-3.pdf}}
\end{center}
in $\asm{\mathcal{A}}$ with
\begin{itemizeminus}
\item $X, B \in \asm{M}$;
\item $\exists !$ the unique lift of the lower square (given that regular
epis and monos form an orthogonal factorisation system)
\end{itemizeminus}
hence $k = (\exists !) \circ \overline{\iota_g \circ k}$ lifts. \end{proof}
\begin{definition}
$M \subseteq \mathscr{S}$ is {\tmem{compact}} if for any $u \in \mfibr{X}$
and any valuation $\nu : X \rightarrow \ppows{M}$ such that
\begin{eqnarray*}
u & \sqsubseteq_X & \bigE \nu
\end{eqnarray*}
there is $b \in \mfibr{X}{M}$ with $b_x \in \nu (x)$ for all $x \in X$ such
that
\[ u \text{ } \sqsubseteq_X \text{ } b \text{ } \sqsubseteq_X \text{ } \bigE
\nu \] \end{definition}
\begin{lemma}
\label{lem:compact}Let $M \subseteq \mathscr{S}$. Assuming choice the
following are equivalent
\begin{enumerateroman}
\item $M$ is compact;
\item for any morphisms $g : A \rightarrow B$ in $\asm{M}$ and $k : X
\rightarrow \tmop{Im} (g)$ in
$\asm{\mathcal{A}}$ with $X \in \asm{M}$ the
diagram
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.87590187590188cm]{regular_completion_asm-4.pdf}}
\end{center}
admits a lift.
\end{enumerateroman} \end{lemma}
\begin{proof}
$(i \nocomma) \Rightarrow (i \nocomma i)$ Let $g : A \rightarrow B$ be a
morphism in $\asm{M}$ and $k : X \rightarrow \tmop{Im} (g)$ be a morphism in
$\asm{\mathcal{A}}$ with $X \in \asm{M}$. The
induced valuation $\nu_g : \tmop{Im} (g) \rightarrow \ppows{M}$ verifies
\begin{eqnarray*}
\bigE \nu_g & = & \exi{\tmop{Im} (g)} \qquad (\star)
\end{eqnarray*}
(c.f. Lemma \ref{lem:induced-valuation}). The map $\nu := \nu_{\bar{g}}
\circ k$ is a valuation as well. Furthermore
\begin{eqnarray*}
\bigM_{x \in X} \left( \exi{X} (x) \rightarrow \bigE \nu (x)
\right) & = & \bigM_{x \in X} \left( \exi{X} (x) \rightarrow
\bigE \nu_g (k (x)) \right)\\
& = & \bigM_{x \in X} \left( \exi{X} (x) \rightarrow
\exi{\tmop{Im} (g)} (k (x)) \right)\\
& \in & \mathscr{S}
\end{eqnarray*}
since $k$ is tracked, hence $\exi{X} \sqsubseteq_X \bigE \nu$. $M$ is
discrete by hypothesis, hence there is $b \in \mfibr{X}{M}$ with $b_x \in
\nu (x)$ for all $x \in X$ such that
\[ \exi{X} \text{ } \sqsubseteq_X \text{ } b \text{ } \sqsubseteq_X \text{ }
\bigE \nu \]
We have
\begin{eqnarray*}
\Xi \left( \exi{X} \text{ } \sqsubseteq_X \text{ } b \text{ } \right) & =
& \left( X, \exi{X} \right) \mathop{\longrightarrow}\limits^{\tmop{id}} (X, b)
\end{eqnarray*}
On the other hand
\begin{eqnarray*}
\left( \bigE \nu \right) (x) & = & \bigE (\nu_{\bar{g}} \circ k) (x)\\
& = & \bigE \nu_{\bar{g}} (k (x))
\end{eqnarray*}
for all $x \in X$, hence $\bigE \nu = k^{\ast} \bigE \nu_{\bar{g}}$ which
entails
\begin{eqnarray*}
\Xi \left( b \text{ } \sqsubseteq_X \text{ } \bigE \nu \right) & = & \Xi
\left( b \text{ } \sqsubseteq_X \text{ } k^{\ast} \bigE \nu_{\bar{g}}
\right)\\
& = & (X, b) \mathop{\longrightarrow}\limits^k \left( \tmop{Im} (f), \bigE \nu_g
\right)
\end{eqnarray*}
This in turn entails that $k$ factors as
\[ \left( X, \exi{X} \right) \mathop{\longrightarrow}\limits^{\tmop{id}} (X, b)
\mathop{\longrightarrow}\limits^k \left( \tmop{Im} (f), \bigE \nu_g \right) \]
in $\asm{\mathcal{A}}$. Now $b_x \in \nu (x) =
\left\{ \exi{A} (a) |a \in k (x) \right\}$ for each $x \in X$, hence there
is $\tilde{a} \in k (x)$ such that $b_x = \exi{A} (\tilde{a})$. Notice that
while the {\tmem{value}} $\exi{A} (\tilde{a})$ of $b_x$ is determined,
$\tilde{a}$ itself does not need to be unique. Let
\begin{eqnarray*}
\tilde{A}_x & : = & \left\{ \tilde{a} \in k (x) |b_x = \exi{A} (\tilde{a})
\right\}
\end{eqnarray*}
be the set of those $\tilde{a}$. Assuming choice there is a map $l : X
\rightarrow A$ such that $l (x) \in \tilde{A}_x$ for all $x \in X$. \
Furthermore
\begin{eqnarray*}
\bigM_{x \in X} \left( b_x \rightarrow \exi{A} (l (x)) \right) &
= & \bigM_{x \in X} \left( \exi{A} (\tilde{a}) \rightarrow
\exi{A} (\tilde{a}) \right)\\
& \in & \mathscr{S}
\end{eqnarray*}
so $l$ is tracked. Finally we have $\bar{g} \circ l = k$ as maps since $k
(x) = \bar{g} (a)$ for {\tmem{any}} $a \in k (x)$ hence the lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=3.96664698937426cm]{regular_completion_asm-5.pdf}}
\end{center}
$(i \nocomma i) \Rightarrow (\nocomma i)$ Let $u \in \mathbb{P} [M]_X$ and
$\nu : X \rightarrow \ppows{M}$ be a valuation such that $\bigM_{x
\in X} \left( u_x \rightarrow \left( \bigE \nu \right)_x \right) \in
\mathscr{S}$. We thus have $u \sqsubseteq_X \bigE \nu$ and further $\Xi^{-
1} \left( u \sqsubseteq_X \bigE \nu \right) = \tmop{id} : (X, u) \rightarrow
\left( X, \bigE \nu \right)$. Composing the latter with $[-]_{\nu}$ (c.f
Proposition \ref{prop:induced-valuation}) yields the morphism $[-]_{\nu} :
(X, u) \rightarrow \left( \tmop{Im} (g_{\nu}), \exi{\tmop{Im} (g_{\nu})}
\right)$ in $\asm{\mathcal{A}}$. The latter has a
lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.97507542962088cm]{regular_completion_asm-6.pdf}}
\end{center}
Let $b \in \mathbb{P} [M]_X$ be given by $b_x := \exi{\hat{X}} (l
(x))$. We have by construction $l (x) = x_m$ for some $m \in \nu (x)$, hence
\begin{eqnarray*}
\exi{\hat{X}} (l (x)) & = & \exi{\hat{X}} (x_m)\\
& = & m
\end{eqnarray*}
so $b_x \in \nu (x)$ for all $x \in X$. Now $l$ is tracked so
\begin{eqnarray*}
\bigM_{x \in X} (u_x \rightarrow b_x) & = & \bigM_{x \in
X} \left( \exi{X} \rightarrow \exi{\hat{X}} (l (x)) \right)\\
& \in & \mathscr{S}
\end{eqnarray*}
\end{proof}
\begin{theorem}
\label{th:compact}Let $M \subseteq \mathscr{S}$ be an algebraic subset.
Assuming choice the following are equivalent
\begin{enumerateroman}
\item $M$ is {\tmem{compact}};
\item $U : {\asm{M}}_{\tmop{reg}} \rightarrow \asm{\mathcal{A}}$ is full.
\end{enumerateroman} \end{theorem}
\begin{proof}
$M$ being compact is equivalent to the assertion that any morphism $l \circ
\bar{f} : X \rightarrow \tmop{Im} (g)$ where
\begin{enumerateroman}
\item $f : X \rightarrow Y$ and $g : A \rightarrow B$ are morphisms in
$\asm{M}$;
\item $l : \tmop{Im} (f) \rightarrow \tmop{Im} (g)$ is a morphism in
$\asm{\mathcal{A}}$
\end{enumerateroman}
admits a lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.97507542962088cm]{regular_completion_asm-7.pdf}}
\end{center}
(c.f. Lemma \ref{lem:pre-compact} and \ref{lem:compact}), which is obviously
equivalent to the assertion that $U : {\asm{M}}_{\tmop{reg}} \rightarrow
\asm{\mathcal{A}}$ is full (c.f. Section \ref{sec:regular}). \end{proof}
\begin{example}
We observed that the set of all singletons $M := \{ \{ p \} |p \in P
\}$ is algebraic (c.f. Example \ref{ex:alg}) and dense (c.f. Example
\ref{ex:single-dense}). Taking it from there we now show that $M$ is compact
as well. Let $u \in \mathbb{P} [M]_X$ be a family of singletons $u_x = \{
\check{u}_x \}$ and $\nu : X \rightarrow \ppows{M}$ be a valuation. Suppose
$\bigM_{x \in X} \left( u_x \rightarrow \left( \bigE \nu \right)_x
\right) \in \mathscr{S}$. Unravelling yields
\begin{eqnarray*}
\bigM_{x \in X} (u_x \rightarrow (\bigE \nu)_x) & = &
\bigcap_{x \in X} \left( \{ \check{u}_x \} \rightarrow \bigcup_{s \in \nu
(x)} s \right)\\
& = & \bigcap_{x \in X} \left\{ c \in P|c \cdot \check{u}_x \in \bigcup
\nu (x) \right\}\\
& = & \bigcap_{x \in X} \{ c \in P|c \cdot \check{u}_x \in \nu (x) \}
\qquad \text{elements of $\nu (x)$ are singletons}\\
& \neq & \varnothing
\end{eqnarray*}
Let $c_0 \in \bigcap_{x \in X} \{ c \in P|c \cdot \check{u}_x \in \nu (x)
\}$ and further $b \in \mathbb{P} [M]_X$ given by $b_x := \{ c \cdot
\check{u}_x \}$. We have $b_x \in \nu (x)$ for all $x \in X$ and further
\begin{eqnarray*}
\bigM_{x \in X} (u_x \rightarrow b_x) & = & \bigcap_{x \in X}
(u_x \rightarrow b_x)\\
& = & \bigcap_{x \in X} (\{ \check{u}_x \} \rightarrow \{ c_0 \cdot
\check{u}_x \})\\
& \neq & \varnothing
\end{eqnarray*} \end{example}
\subsection{Generators}
\begin{definition}
An algebraic subset $M \subseteq \mathscr{S}$ is a {\tmem{generator}} of
$\mathscr{S}$ if it is dense and compact. \end{definition}
\begin{theorem}
\label{th:hilbert}Let $M \subseteq \mathscr{S}$ be an algebraic subset. The
following are equivalent
\begin{enumerateroman}
\item $M$ is a generator of $\mathscr{S}$;
\item $\asm{\mathcal{A}}$ is the regular
completion of $\asm{M}$.
\end{enumerateroman} \end{theorem}
\begin{proof}
Theorem \ref{th:dense} and \ref{th:compact}. \end{proof}
\begin{corollary}
If one and thus both of the equivalent assertions of Theorem
\ref{th:hilbert} hold, the closure of $\asm{M}$ under isomorphisms is
$\asm{\mathcal{A}}$'s subcategory of projective
objects. \end{corollary}
\begin{corollary}
If $M$ and $M'$ are generators then $\asm{M} \simeq \asm{M'}$. \end{corollary}
\section{Exact Completions}\label{sec:exact}
\begin{definition} Let $\mathbb{D}$ be a regular category
\begin{enumeratenumeric}
\item An equivalence relation in $\mathbb{D}$ is
{\tmem{effective}} if it is a kernel pair.
\item $\mathbb{D}$ is {\tmem{exact}} if every
equivalence relation is effective.
\end{enumeratenumeric} \end{definition}
\begin{example}
Any topos among others. \end{example}
An exact category in the above sense is sometimes called {\tmem{Barr-exact}} as there is quite a different notion of exactness due to Quillen. We won't need the distinction here as the only exact categories in sight are those in the sense of Barr.
\begin{notation}
We shall write
\begin{itemizeminus}
\item $\ensuremath{\tmmathbf{\tmop{Ex}}}$ for the 2-category of exact categories, exact funtors and
their natural transformations;
\item $| - | : \ensuremath{\tmmathbf{\tmop{Ex}}} \rightarrow \ensuremath{\tmmathbf{\tmop{Reg}}}$ for the forgetful $2$-functor.
\end{itemizeminus} \end{notation}
It is part of the lore that the forgetful 2-functor $| - |$ has a left biadjoint $(-)_{\tmop{ex} / \tmop{reg}} : \ensuremath{\tmmathbf{\tmop{Reg}}} \rightarrow \ensuremath{\tmmathbf{\tmop{Ex}}}$. This entails that the forgetful 2-fiunctor $| - | : \ensuremath{\tmmathbf{\tmop{Ex}}} \rightarrow \ensuremath{\tmmathbf{\tmop{Lex}}}$ has a left biadjoint $(-)_{\tmop{ex} / \tmop{lex}} : \ensuremath{\tmmathbf{\tmop{Lex}}} \rightarrow \ensuremath{\tmmathbf{\tmop{Ex}}}$ as well since biadjoints compose.
Let $\mathbb{C}$ be a lex category.
\begin{definition}
We call $\exlex{\mathbb{C}}$ $\mathbb{C}$'s {\tmem{exact completion}}. \end{definition}
There is an embedding $\ensuremath{\tmmathbf{y}} : \mathbb{C} \rightarrow \exlex{\mathbb{C}}$ enjoying a universal property structurally identical to the one of regular completions (c.f. Section \ref{sec:regular}). There is a well-known direct construction of $\mathbb{C}_{\tmop{ex} / \tmop{lex}}$ {\cite{carboni1982free}} which we now recall.
Let $\mathbb{C}$ be a category.
\begin{definition}
Let $\mathcal{Q}$ be a category $s, t : Q_1 \rightrightarrows Q_0$ with two
objects and two parallel morphisms. A quiver in $\mathbb{C}$ is a functor
$\mathcal{Q} \rightarrow \mathbb{C}$. \end{definition}
As the only quivers of interest here are those internal to a (typically lex) category $\mathbb{C}$, we shall dispense with the qualifier ``in $\mathbb{C}$''.
\begin{notation}
Let $X$ be a quiver.
\begin{enumeratenumeric}
\item $X (x, x') := \pr{s_X}{t_X}^{- 1} (x, x')$.
\item We shall write $e : x \rightsquigarrow x'$ for an edge $e \in X (x,
x')$.
\end{enumeratenumeric} \end{notation}
\begin{definition}
Let $X, Y$ be quivers.
\begin{enumeratenumeric}
\item A quiver morphism $f : X \rightarrow Y$ is given by morphisms $f_0 :
X_0 \rightarrow Y_0$ and $f_1 : X_1 \rightarrow Y_1$ such that
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.97507542962088cm]{exact-1.pdf}}
\end{center}
commutes.
\item Let $f, g : X \rightarrow Y$ be quiver morphisms. A
{\tmem{homotopy}} from $f$ to $g$ is a morphism $h : X_0 \rightarrow Y_1$
such that the morphism $\langle f_0, g_0 \rangle : X_0 \rightarrow Y_0
\times Y_0$ has a lift
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=3.37170405352224cm]{exact-2.pdf}}
\end{center}
with respect to $\pr{s_Y}{t_Y}$.
\end{enumeratenumeric} \end{definition}
\begin{notation}
We shall write $h : f \Rightarrow g$ for a homotopy $h$. \end{notation}
\begin{remark}
\label{rem:edge}Let $f, g : X \rightarrow Y$ be quiver morphisms. A homotopy
$h : f \Rightarrow g$ is just a family
\[ \left( h (x) : f_0 (x) \rightsquigarrow g_0 (x) \right)_{x \in X_0} \]
of edges of $Y_1$. Notice that for any such edge there is an edge $\sigma_Y
(h (x)) : g_0 (x) \rightsquigarrow f_0 (x)$ in the opposite direction. \end{remark}
Suppose now $\mathbb{C}$ is lex.
\begin{definition}
A {\tmem{pseudo-groupoid}} $X$ in $\mathbb{C}$ is a quiver $s_X, t_X : X_1
\rightrightarrows X_0$ equipped with structural morphisms
\begin{itemizeminus}
\item $\rho_X : X_0 \rightarrow X_1$ such that
\begin{enumerateroman}
\item $s_X \circ \rho_X = \tmop{id}$;
\item $t_X \circ \rho_X = \tmop{id}$;
\end{enumerateroman}
\item $\sigma_X : X_1 \rightarrow X_1$ such that
\begin{enumerateroman}
\item $s_X \circ \sigma_X = t_X$;
\item $t_X \circ \sigma_X = s_X$;
\end{enumerateroman}
\item $\tau_X : X_1 \circledast X_1 \rightarrow X_1$ with $X_1
\circledast X_1$ given by pullback of $s_X$ along $t_X$, such that
\begin{enumerateroman}
\item $s_X \circ \tau_X = s_X \circ p_0$;
\item $t_X \circ \tau_X = t_X \circ p_1$.
\end{enumerateroman}
\end{itemizeminus} \end{definition}
Entities we call pseudo-groupoids are also (and perhaps better) known as {\tmem{pseudo-equivalence relations}} {\cite{carboni1995some,carboni1982free}}. As the only pseudo-groupoids of interest here are those internal to a lex category $\mathbb{C}$, we shall once again dispense with the qualifier ``in $\mathbb{C}$''.
Notice that $\rho_X$ is a local section of $\pr{s_X}{t_X} : X_1 \rightarrow X_0 \times X_0$ with respect to the diagonal. The homotopy relation is stable by composition while composition of quiver morphisms among pseudo-groupoids is associative up-to homotopy, so pseudo-groupoids organise themselves in the category $\qgrpd{\mathbb{C}}$ along with homotopy classes of quiver morphisms among them. We have \begin{eqnarray*}
\exlex{\mathbb{C}} & \simeq & \qgrpd{\mathbb{C}} \end{eqnarray*} {\cite{carboni1995some,carboni1982free}}. Quite unsurprisingly, the functor $\tmmathbf{y} : \mathbb{C} \rightarrow \exlex{\mathbb{C}}$ turns then out to be the one sending an object $X \in \mathbb{C}$ on the {\tmem{discrete}} pseudo-groupoid $\tmop{id}, \tmop{id} : X \rightrightarrows X$. As pointed out in {\cite{shulman2021derivator}}, pseudo-groupoids give rise to ``groupoidal'' bicategories internal to $\mathbb{C}$, with homotopies as 2-cells. So if we restrain from quotienting by the homotopy relation, we end up with a tricategory where these data organise themselves (taking into account coherence conditions). The homotopy 1-category of this tricategory is then $\qgrpd{\mathbb{C}}$ (see also {\cite{kinoshita2014category}}).
\section{Exact Completions of Implicative Assemblies} \label{sec:exact-completion-asm}
\subsection{The Exact Completion of $\asm{\mathcal{A}}$}
In this subsection we address the ex/reg completion of $\asm{\mathcal{A}}$ as well as the ex/lex completion of $\asm{M}$, exploiting concepts and results found in {\cite{menni2000exact}}.
\begin{definition}
\label{def:chaos}{$\mbox{}$}
\begin{enumeratenumeric}
\item A {\tmem{chaotic situation}} is given by two lex categories
$\mathbf{S}$ and $\mathbb{C}$ connected by an adjunction $| - | \dashv
\Delta : \mathbf{S} \rightarrow \mathbb{C}$.
\item A morphism $f : X \rightarrow Y$ in $\mathbb{C}$ is a
{\tmem{pre-embedding}} if the square
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=3.4708448117539cm]{exact_completion_asm-1.pdf}}
\end{center}
is a pullback.
\item An object $\Upsilon \in \mathbb{C}$ is {\tmem{generic}} if for every
$X \in \mathbb{C}$ there is a pre-embedding $X \rightarrow \Upsilon$.
\end{enumeratenumeric} \end{definition}
(as a word of caution, we use the symbol $\Delta$ for $\nabla$ in op.cit.). A generic object in the above sense is not the same as (although related to) the homonymous fibred artefact. The notion of chaotic situation is an abstract version of what happens when we have a forgetful functor from a concrete category into $\ensuremath{\tmmathbf{\tmop{Set}}}$.
\begin{proposition}[Menni]
\label{prop:menni}Let $\mathbb{C}$ have a chaotic situation. The following
are equivalent
\begin{enumerateroman}
\item $\mathbb{C}$ has a generic object;
\item $\mathbb{C}$ has a generic mono.
\end{enumerateroman} \end{proposition}
This is Prop 8.1.8 in op.cit.
\begin{theorem}[Menni]
\label{th:menni}Let $\mathbb{D}$ be a regular LCCC with generic mono. Then
$\mathbb{D}_{\tmop{ex} / \tmop{reg}}$ is a topos. \end{theorem}
This is Theorem 11.3.3 in op.cit.
\begin{remark}
\label{rem:chaotic}$(\mathcal{S}, \tmop{id}_{\mathcal{S}})$ is a generic
object in the sense of Definition \ref{def:chaos}. \end{remark}
\begin{theorem}
$(\asm{\mathcal{A}})_{\tmop{ex} / \tmop{reg}}$ is
a topos. \end{theorem}
\begin{proof}
Proposition \ref{prop:menni}, Theorem \ref{th:menni} and Remark
\ref{rem:chaotic}. \end{proof}
\begin{corollary}
\label{cor:ex-lex}Let $M$ be a generator. Then
\[ \left( \asm{M} \right)_{\tmop{ex} / \tmop{lex}} \simeq \left( \left(
\asm{M} \right)_{\tmop{reg} / \tmop{lex}} \right)_{\tmop{ex} /
\tmop{reg}} \simeq
(\asm{\mathcal{A}})_{\tmop{ex} / \tmop{reg}}
\]
is a topos. \end{corollary}
\subsection{Tripos-to-Topos Construction}
\begin{definition}
An {\tmem{implicative set}} $\eset{X}$ is a set equipped with a
{\tmem{non-standard equality}}, that is a predicate $| - \approx - | : X
\times X \rightarrow \mathcal{A}$ which is
\begin{enumerateroman}
\item {\tmem{symmetric:}} $\vld{\bigM_{x, x' \in X} \left(
\eeq{x}{x'} \rightarrow \eeq{x'}{x} \right)}$;
\item {\tmem{transitive:}} $ \vld{\bigM_{x, x', x'' \in X} \left(
\eeq{x}{x'} \sqcap \text{ } \eeq{x'}{x''} \rightarrow
\eeq{x}{x''} \right)}$.
\end{enumerateroman} \end{definition}
Implicative sets are the objects of $\ensuremath{\tmmathbf{\tmop{Set}}} [\mathcal{A}]$. Notice that we do not necessarily have {\tmem{reflexivity}}, that is $\eeq{x}{x} \in \mathcal{S}$ for all $x \in X$. This is in a sense the main point of the construction, as $\eeq{x}{x}$ is interpreted as an {\tmem{existence predicate}}. Accordingly, we call {\tmem{ghosts}} elements $x \in X$ such that $\eeq{x}{x} \not\in \mathcal{S}$.
\begin{notation}
{$\mbox{}$}
\begin{itemizeminus}
\item $\sym{X} := \bigM_{x, x' \in X} \left( \eeq{x}{x'}
\rightarrow \eeq{x'}{x} \right)$;
\item $\trans{X} := \bigM_{x, x', x'' \in X} \left(
\eeq{x}{x'} \sqcap \text{ } \eeq{x'}{x''} \rightarrow
\eeq{x}{x''} \right)$;
\item $\eexi{X} := \eeq{x}{x}$.
\end{itemizeminus} \end{notation}
\begin{definition}
Let $\eset{X}$ and $\eset{Y}$ be implicative sets. A $\tmop{functional}
\tmop{relation}$
\[ F : \eset{X} \rightarrow \eset{Y} \]
is a predicate $F : X \times Y \rightarrow \mathcal{A}$ which is
\begin{enumerateroman}
\item {\tmem{extensional:}} $ \vld{\bigM_{x, x' \in X}
\bigM_{y, y' \in Y} \left( F (x, y) \sqcap \eeq{x}{x'} \sqcap
\eeq{y}{y'} \rightarrow F (x', y') \right)}$;
\item {\tmem{strict:}}
$\vld{
\bigM_{x \in X} \bigM_{y \in Y} \left( F(x, y)
\rightarrow \eexi{X}(x) \sqcap \eexi{Y}(y) \right)
}$;
\item {\tmem{single-valued:}} $\vld{\bigM_{x \in X}
\bigM_{y, y' \in Y} \left( F (x, y) \sqcap F (x, y') \rightarrow
\eeq{y}{y'} \right)}$;
\item {\tmem{total:}} $ \vld{\bigM_{x \in X} \left( \eexi{X} (x)
\rightarrow \iex{y \in Y_0} \left( \eexi{Y} (y) \sqcap F (x, y) \right)
\right)}$.
\end{enumerateroman}
Two functional relations $F, G : \eset{X} \rightarrow \eset{Y}$ are
{\tmem{equivalent}} if they are isomorphic in $\mathbb{P} [\mathcal{A}]_{X
\times Y}$. \end{definition}
\begin{notation}
{$\mbox{}$}
\begin{itemizeminus}
\item $\ext{F} := \bigM_{x, x' \in X} \bigM_{y, y'
\in Y} \left( F (x, y) \sqcap \eeq{x}{x'} \sqcap \eeq{y}{y'}
\rightarrow F (x', y') \right)$;
\item $\str{F} := \bigM_{x \in X} \bigM_{y \in Y}
\left( F (x, y) \rightarrow \eexi{X} (x) \sqcap \eexi{Y} (y) \right)$;
\item $\sv{F} := \bigM_{x \in X} \bigM_{y, y' \in
Y} \left( F (x, y) \sqcap F (x, y') \rightarrow \eeq{y}{y'} \right)$;
\item $\tot{F} := \bigM_{x \in X} \left( \eexi{X} (x)
\rightarrow \iex{y \in Y_0} \left( \eexi{Y} (y) \sqcap F (x, y) \right)
\right)$.
\end{itemizeminus} \end{notation}
\begin{proposition}
\label{prop:topos-morphisms}Two functional relations $f, g : \eset{X}
\rightarrow \eset{Y}$ are equivalent if and only if
\[ \vld{\bigM_{x \in X, y \in Y} (F (x, y) \rightarrow F (x, y))}
\] \end{proposition}
\subsection{The Exact Completion of $\asm{M}$ in Context}
\
Let $M \subseteq \mathscr{S}$ be a generator. We already observed that $\left( \asm{M} \right)_{\tmop{ex} / \tmop{lex}}$ is a topos (c.f. Corollary \ref{cor:ex-lex}), so the question is now if it is the same topos as $\ensuremath{\tmmathbf{\tmop{Set}}} [\mathcal{A}]$.
\begin{notation}
In what follows we make systematic use $\lambda$-terms, some of them with
relatively long arguments lists containing dummy arguments. We shall use the
symbol {\ensuremath{\diamond}} for dummies in order to improve readability. \end{notation}
\subsubsection{A fully faithful functor $\mathcal{K}$.}
\begin{remark}
\label{rem:ex-lex-val}A pseudo-groupoid $X$ induces a valuation
\begin{eqnarray*}
\nu_X : X_0 \times X_0 & \longrightarrow & \ppows{M}\\
(x, x') & \mapsto & \left\{ \exi{X_1} (e) |e \in X (x, x') \right\}
\end{eqnarray*} \end{remark}
\begin{lemma}
\label{lem:impl-set}Let $X$ be a pseudo-groupoid. $X_0$ equipped with the
relation
\begin{eqnarray*}
\ieeq{x}{x'}{X} & := & \exi{X_0} (x) \sqcap \exi{X_0} (x') \sqcap
\bigE \nu_X (x, x')
\end{eqnarray*}
is an implicative set. \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\sym{X} & = & \bigM_{x, x' \in X_0} \left( \exi{X_0}
(x) \sqcap \exi{X_0} (x') \sqcap \bigE \nu_X (x, x') \rightarrow \exi{X_0}
(x') \sqcap \exi{X_0} (x) \sqcap \bigE \nu_X (x', x) \right)
\end{eqnarray*}
Let $e : x \rightsquigarrow x' \in X_1$ be an edge. We then have the
following subquiver of $X$
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=1.78518955791683cm]{exact_completion_asm-2.pdf}}
\end{center}
\
with the ``opposite'' edge $\sigma_X (e) : x' \rightsquigarrow x$. Let
$\xi$ be a tracker of $\sigma_X$. We then have
\begin{eqnarray*}
\lam{u \nocomma v \nocomma w}{} u \sqcap v \sqcap \intpr{\lam{z}{z \left(
\xi \left( w \ensuremath{\tmmathbf{\mathtt{I}}} \right) \right)}} & \preccurlyeq & \sym{X}
\end{eqnarray*}
and therefore $\vld{\sym{X}}$. Next we have
\begin{eqnarray*}
\trans{X} & = & \bigM_{x, x', x'' \in X_0} (\\
& & \text{\qquad} \exi{X_0} (x) \sqcap \exi{X_0} (x') \sqcap \bigE \nu_X
(x, x') \sqcap\\
& & \text{\qquad} \exi{X_0} (x') \sqcap \exi{X_0} (x'') \sqcap \bigE
\nu_X (x', x'')\\
& & \text{\qquad} \rightarrow\\
& & \text{\qquad} \text{$\exi{X_0} (x) \sqcap \exi{X_0} (x'') \sqcap
\bigE \nu_X (x, x'')$}\\
& & \left. \text{ } \right)
\end{eqnarray*}
Let $e_1 : x \rightsquigarrow x'$ and $e_2 : x' \rightsquigarrow x''$ be
edges. We then have the following subquiver of $X$
\begin{center}
\raisebox{-0.50000516673039\height}{\includegraphics[width=14.8909550045914cm,height=1.58685884822248cm]{exact_completion_asm-3.pdf}}
\end{center}
with the ``composed'' edge $\tau_X (e_1, e_2) : x \rightsquigarrow x''$. Let
$\chi$ be a tracker of $\tau_X$. We then have
\begin{eqnarray*}
\lam{u \ensuremath{\diamond} w \ensuremath{\diamond} v w'}{u \sqcap v \sqcap \intpr{\lam{z}{\chi \left(
\left( w \ensuremath{\tmmathbf{\mathtt{I}}} \right) \sqcap \left( w' \ensuremath{\tmmathbf{\mathtt{I}}} \right) \right)}}} &
\preccurlyeq & \trans{X}
\end{eqnarray*}
and therefore $\vld{\trans{X}}$. \end{proof}
Let $f : X \rightarrow Y$ be a morphism of pseudo-groupoids. The relation $\ensuremath{\mathfrak{R}_f}$ from $(X, \approx_X)$ to $(Y, \approx_Y)$ is given by \begin{eqnarray*}
\ensuremath{\mathfrak{R}_f} (x, y) & := & \exi{X_0} (x) \sqcap \exi{Y_0} (y) \sqcap
\bigE \nu_Y (y, f_0 (x)) \end{eqnarray*} \begin{remark}
\label{rem:rf-values}Let $(x, y) \in X \times Y$. We have
$\vld{\ensuremath{\mathfrak{R}_f} (x, y)}$ if and only if there is an edge $e : y'
\rightsquigarrow f_0 (x)$ in $Y$ and $\ensuremath{\mathfrak{R}_f} (x, y) = \bot$
otherwise. \end{remark}
\begin{remark}
\label{rem:rf-implies}Let $\mathsf{Rf} := \bigM_{x \in X_0, y
\in Y_0} \left( \ensuremath{\mathfrak{R}_f} (x, y) \rightarrow \ieeq{y}{f_0 (x)}{Y}
\right)$. We have
\begin{eqnarray*}
\mathsf{Rf} & = & \bigM_{x \in X_0, y \in Y_0} (\\
& & \text{\qquad} \exi{X_0} (x) \sqcap \exi{Y_0} (y) \sqcap \bigE \nu_Y
(y, f_0 (x))\\
& & \text{\qquad} \rightarrow\\
& & \text{\qquad} \exi{Y_0} (y) \sqcap \exi{Y_0} (f_0 (x)) \sqcap \bigE
\nu_Y (y, f_0 (x))\\
& & \left. \text{ } \right)
\end{eqnarray*}
Suppose $\xi$ tracks $f_0$. Then
\begin{eqnarray*}
\intpr{\lam{u \nocomma v \nocomma w}{v \sqcap (\xi u) \sqcap} w} &
\preccurlyeq & \mathsf{Rf}
\end{eqnarray*}
and therefore $\vld{Rf}$. \end{remark}
\begin{lemma}
\label{lem:ext}$\ensuremath{\mathfrak{R}_f}$ is extensional. \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\ext{\ensuremath{\mathfrak{R}_f}} & = & \bigM_{x, x' \in X_0}
\bigM_{y, y' \in Y_0} (\\
& & \text{{\hspace{3em}}$\exi{X_0} (x) \sqcap$} \exi{Y_0} (y) \sqcap
\bigE \nu_Y (y, f_0 (x)) \sqcap\\
& & \text{{\hspace{3em}}} \exi{X_0} (x) \sqcap \exi{X_0} (x') \sqcap
\bigE \nu_X (x, x') \sqcap\\
& & \text{{\hspace{3em}}} \text{} \exi{Y_0} (y) \sqcap \exi{Y_0} (y')
\sqcap \bigE \nu_Y (y, y')\\
& & \text{{\hspace{3em}}} \rightarrow\\
& & \text{{\hspace{3em}}} \text{$\exi{X_0} (x') \sqcap$} \exi{Y_0} (y')
\sqcap \bigE \nu_Y (y', f_0 (x'))\\
& & \left. \text{ } \right)
\end{eqnarray*}
Let $x, x'' \in X_0$ be vertices of $X$ and $y, y' \in Y_0$ be vertices of
$Y$. Let $e : x \rightsquigarrow x'$ be an edge of $X$. Let $e_1 : y
\rightsquigarrow f_0 (x)$ and $e_2 : y \rightsquigarrow y'$ be edges of $Y$
(notice that besides their ``types'' we did not make any hypotheses on the
vertices and edges in question). We then have the following subquiver of $Y$
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=4.26410205955661cm]{exact_completion_asm-4.pdf}}
\end{center}
\
Suppose $\chi$ tracks $f_1$, $\kappa$ tracks $\sigma_Y$ and $\varpi$ tracks
$\tau_Y$. Then
\begin{eqnarray*}
\intpr{\lam{\ensuremath{\diamond} \ensuremath{\diamond} w \ensuremath{\diamond} v w' \ensuremath{\diamond} v' w''}{} v \sqcap v' \sqcap
\lam{z}{z \left( \varpi \left( \left( \varpi \left( \kappa \left( w''
\ensuremath{\tmmathbf{\mathtt{I}}} \right) \right) \sqcap \left( w \ensuremath{\tmmathbf{\mathtt{I}}} \right) \sqcap \left( \chi
\left( w' \ensuremath{\tmmathbf{\mathtt{I}}} \right) \right) \right) \right) \right)}} & \preccurlyeq &
\ext{\ensuremath{\mathfrak{R}_f}}
\end{eqnarray*}
and therefore $\vld{\ext{\ensuremath{\mathfrak{R}_f}}}$. \end{proof}
\begin{lemma}
\label{lem:str}$\ensuremath{\mathfrak{R}_f}$ is strict.\quad \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\str{\ensuremath{\mathfrak{R}_f}} & = & \bigM_{x \in X_0} \bigM_{y
\in Y_0} (\\
& & \text{\qquad$\exi{X_0} (x) \sqcap$} \exi{Y_0} (y) \sqcap \bigE \nu_Y
(y, f_0 (x))\\
& & \text{\qquad} \rightarrow\\
& & \text{\qquad} \exi{X_0} (x) \sqcap \bigE \nu_X (x, x) \sqcap\\
& & \text{\qquad} \exi{Y_0} (y) \sqcap \bigE \nu_Y (y, y)\\
& & \left. \text{ } \right)
\end{eqnarray*}
Suppose $\kappa$ tracks $\rho_X$ while $\kappa'$ tracks $\rho_Y$. Then
\begin{eqnarray*}
\intpr{\lam{u v \ensuremath{\diamond}}{u \sqcap \lam{z}{z (\kappa \nocomma u)} \sqcap v
\sqcap \lam{z}{z (\kappa' \nocomma v)}}} & \preccurlyeq &
\str{\ensuremath{\mathfrak{R}_f}}
\end{eqnarray*}
and therefore $\vld{\str{\ensuremath{\mathfrak{R}_f}}}$. \end{proof}
\begin{lemma}
\label{lem:sv}$\ensuremath{\mathfrak{R}_f}$ is single valued. \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\sv{\ensuremath{\mathfrak{R}_f}} & = & \bigM_{x \in X_0} \bigM_{y, y'
\in Y_0} (\\
& & \text{\qquad} \text{$\exi{X_0} (x) \sqcap$} \exi{Y_0} (y) \sqcap
\bigE \nu_Y (y, f_0 (x)) \sqcap\\
& & \text{\qquad} \text{$\exi{X_0} (x) \sqcap$} \exi{Y_0} (y') \sqcap
\bigE \nu_Y (y', f_0 (x))\\
& & \text{\qquad} \rightarrow\\
& & \text{\qquad} \exi{Y_0} (y) \sqcap \exi{Y_0} (y') \sqcap \bigE \nu_Y
(y, y')\\
& & \left. \text{ } \right)
\end{eqnarray*}
Let $e : y \rightsquigarrow f_0 (x)$ and $e' : y' \rightsquigarrow f_0 (x)$
be edges of $Y$. We then have the subquiver of $Y$
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.28095894004985cm]{exact_completion_asm-5.pdf}}
\end{center}
Suppose $\xi$ tracks $\sigma_Y$ and $\varpi$ tracks $\tau_Y$. Then
\begin{eqnarray*}
\intpr{\lam{\ensuremath{\diamond} v w \ensuremath{\diamond} v' w'}{v \sqcap v' \sqcap \lam{z}{z \left(
\varpi \left( \left( w \ensuremath{\tmmathbf{\mathtt{I}}} \right) \sqcap \left( \xi \left( w' \ensuremath{\tmmathbf{\mathtt{I}}}
\right) \right) \right) \right)}}} & \preccurlyeq & \sv{\ensuremath{\mathfrak{R}_f}}
\end{eqnarray*}
and therefore $\vld{\sv{\ensuremath{\mathfrak{R}_f}}}$. \end{proof}
\begin{lemma}
\label{lem:tot}$\ensuremath{\mathfrak{R}_f}$ is total. \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\tot{\ensuremath{\mathfrak{R}_f}} & = & \bigM_{x \in X_0} (\\
& & \text{\qquad} \exi{X_0} (x) \sqcap \bigE \nu_X (x, x)\\
& & \text{\qquad} \rightarrow\\
& & \text{\qquad} \iex{y \in Y_0} \left( \exi{Y_0} (y) \sqcap \bigE
\nu_Y (y, y) \sqcap \exi{X_0} (x) \sqcap \bigE \nu_Y (y, f_0 (x))
\right)\\
& & \left. \text{ } \right)
\end{eqnarray*}
Suppose $\chi$ tracks $f_0$ while $\kappa$ tracks $\rho_Y$. We know $y \in
Y_0$ that fits, namely $f_0 (x)$. Then
\begin{eqnarray*}
\intpr{\lam{u \nocomma v}{(\chi u) \sqcap} \lam{z}{\kappa (\chi u)} \sqcap
u \sqcap \lam{z}{\kappa (\chi u)}} & \preccurlyeq & \tot{\ensuremath{\mathfrak{R}_f}}
\end{eqnarray*}
and therefore $\vld{\tot{\ensuremath{\mathfrak{R}_f}}}$. \end{proof}
\begin{proposition}
\label{prop:K}The assignment
\begin{eqnarray*}
\mathcal{K}: \left( \asm{M} \right)_{\tmop{ex} / \tmop{lex}} &
\longrightarrow & \ensuremath{\tmmathbf{\tmop{Set}}} [\mathcal{A}]\\
X & \mapsto & (X_0, \approx_X)
\end{eqnarray*}
extends to a faithful functor with action on morphisms $f : X \rightarrow Y$
given by
\begin{eqnarray*}
\mathcal{K} (f) (x, y) & := & \ensuremath{\mathfrak{R}_f} (x, y)
\end{eqnarray*} \end{proposition}
\begin{proof}
$(X_0, \approx_X)$ is an implicative set by Lemma \ref{lem:impl-set} while
$\ensuremath{\mathfrak{R}_f}$ is a functional relation by Lemmas \ref{lem:ext},
\ref{lem:str}, \ref{lem:sv} and \ref{lem:tot}. Let $g : X \rightarrow Y$ be
a further morphism of pseudo-groupoids and $h : f \Rightarrow g$ be a
homotopy. There is thus an edge $h (x) : f_0 (x) \rightsquigarrow g_0 (x)$
for every vertex $x \in X_0$ (c.f. Remark \ref{rem:edge}), which entails
$\vld{\bigE \nu_Y (f_0 (x), g_0 (x))}$ and therefore
\[ \vld{\ieeq{f_0 (x)}{g_0 (x)}{Y}} \]
We further have $\vld{\bigM_{x \in X_0} \bigM_{y \in Y_0}
\left( \ensuremath{\mathfrak{R}_f} (x, y) \rightarrow \ieeq{y}{f_0 (x)}{Y} \right)}$
(c.f. Remark \ref{rem:rf-implies}). Now $\ensuremath{\mathfrak{R}_f} (x, y)$ is either
$\bot$ (so it implies anything) or $\vld{\ensuremath{\mathfrak{R}_f} (x, y)}$ (c.f.
Remark \ref{rem:rf-values}). In the latter case we have
$\vld{\ieeq{y}{f_0(x)}{Y}}$
by (the implicative) modus ponens and further
$\vld{\ieeq{y}{g_0(x)}{Y}}$ by symmetry and transitivity, hence
\[ \vld{\ensuremath{\mathfrak{R}_f} \rightarrow \mathfrak{R}_g} \]
which entails that $\mathcal{K}$ is well-defined on morphisms (c.f.
Proposition \ref{prop:topos-morphisms}). It is easy to see that
$\mathcal{K}$ preserves identities and composition.
\
Let $u, v : X \rightarrow Y$ be morphisms of pseudo-groupoids. Suppose
$\mathcal{K} (u) =\mathcal{K} (v)$, that is
\[ \vld{\mathfrak{R}_u \leftrightarrow \mathfrak{R}_v} \quad (\star) \]
Given $x \in X_0$ $(\star)$ implies $\vld{\ieeq{u_0 (x)}{v_0 (x)}{Y_0}}$, so
there is an edge $e_x : u_0 (x) \rightsquigarrow v_0 (x)$. All these edges
are indexed by $X$ and organise themselves in a homotopy $h : u \Rightarrow
v$, hence $\mathcal{K}$ is faithful. \end{proof}
\begin{notation}
Still seeking the greater good by making things human-readable, we shall
also add the $\letin{x}{e}$ macro to our $\lambda$-calculus. \end{notation}
\begin{proposition}
\label{prop:full}The functor $\mathcal{K}:
\exlex{(\asm{M})} \rightarrow \ensuremath{\tmmathbf{\tmop{Set}}} [\mathcal{A}]$
is full. \end{proposition}
\begin{proof}
Let $X, Y \in \exlex{(\asm{M})}$ and $F :
\mathcal{K} (X) \rightarrow \mathcal{K} (Y)$ be a functional relation.
Unravelling the definition yields in this case
\begin{eqnarray*}
\vld{\sv{F}} & = & \bigM_{x \in X_0} \bigM_{y, y' \in
Y_0} \left( F (x, y) \sqcap F (x, y') \rightarrow \exi{Y_0} (y) \sqcap
\exi{Y_0} (y') \sqcap \bigE \nu_Y (y, y') \right)\\
\vld{\tot{F}} & = & \bigM_{x \in X_0} \left( \left( \exi{X_0} (x)
\sqcap \bigE \nu_X (x, x) \right) \rightarrow \iex{y \in Y_0} \left(
\exi{Y_0} (y) \sqcap \bigE \nu_Y (y, y) \sqcap F (x, y) \right) \right)
\end{eqnarray*}
As implicative sets in the image of $\mathcal{K}$ do not have ghosts, $F$
can be presented as a map
\begin{eqnarray*}
\tilde{F} : X_0 & \longrightarrow & \pows{Y_0}\\
x & \mapsto & \left\{ y \in Y_0 |F (x, y) \in \mathscr{S} \right\}
\end{eqnarray*}
Assuming choice we can extract a map $f_0 : X_0 \rightarrow Y_0$ from
$\tilde{F}$. Suppose $\kappa$ tracks $\rho_X$. For any $x \in X_0$ there is
$y_x$ (in general not unique) such that
\begin{eqnarray*}
\letin{t}{\lam{p}{\tot{F} (p \sqcap (\kappa p))} } & & \\
\intpr{\lam{q}{\lpair{\ensuremath{\tmmathbf{{\pi}}}\tmrsub{0}}{\ensuremath{\tmmathbf{{\pi}}}\tmrsub{2}} \left( (t q) \ensuremath{\tmmathbf{\mathtt{I}}} \right)}} &
\preccurlyeq & \bigM_{x \in X_0} \left( \exi{X_0} (x) \rightarrow
\left( \exi{Y_0} (y_x) \sqcap F (x, y_x) \right) \right)
\end{eqnarray*}
thus
\[ \vld{} \bigM_{x \in X_0} \left( \exi{X_0} (x) \rightarrow \left(
\exi{Y_0} (y_x) \sqcap F (x, y_x) \right) \right) \]
But $\vld{F (x, f_0 (x))}$ for all $x \in X$ by definition of $f_0$, hence
\begin{eqnarray*}
\letin{t'}{\lam{r}{\sv{F} \left( \left( \ensuremath{\tmmathbf{{\pi}}}\tmrsub{1} r \right) \sqcap F (x, f_0
(x)) \right)}} & & \\
\ilam{s}{\ensuremath{\tmmathbf{{\pi}}}\tmrsub{1} (t' s)} & \preccurlyeq & \bigM_{x \in X_0}
\left( \exi{Y_0} (y_x) \sqcap F (x, y_x) \right) \rightarrow \exi{Y_0}
(f_0 (x))
\end{eqnarray*}
thus
\[ \vld{} \bigM_{x \in X_0} \left( \exi{Y_0} (y_x) \sqcap F (x,
y_x) \right) \rightarrow \exi{Y_0} (f_0 (x)) \]
But then
\[ \vld{} \bigM_{x \in X_0} \left( \exi{X_0} (x) \rightarrow
\exi{Y_0} (f_0 (x)) \right) \]
so $f_0$ is tracked.
\
Let $e : x \rightsquigarrow x'$ be an edge of $X$, so we have
$\vld{\ieeq{x}{x'}{X}}$. We have $\vld{F (x, f_0 (x))}$ and $\vld{F (x', f_0
(x'))}$ by definition of $f_0$, hence $\vld{F (x, f_0 (x'))}$ by
extensionality and therefore $\vld{} \ieeq{f_0 (x)}{f_0 (x')}{Y}$ by
single-valuedness. But then $Y (f_0 (x), f_0 (x')) \neq \varnothing$, so
given $e : x \rightsquigarrow x'$ we have
\begin{eqnarray*}
\left( (f_0 \times f_0) \circ \pr{s_X}{t_X} \right) (e) & = & \left(
\pr{s_Y}{t_Y} \circ f_1 \right) (e)
\end{eqnarray*}
for any map such that
\begin{eqnarray*}
f_1 (e) & \in & Y (f_0 (x), f_0 (x'))
\end{eqnarray*}
Assuming choice such a map always exists, what remains to be shown is that
it is tracked. Consider the map
\begin{eqnarray*}
\phi : \mathcal{A} & \longrightarrow & \mathcal{A}\\
a & \mapsto & \left\{\begin{array}{lll}
\exi{Y_1} (f_1 (e)) & & \text{if there are } x, x' \in X_0 \text{and}
e \in X (x, x') \text{such that} a = \exi{X_1} (e)\\
\top & & \text{otherwise}
\end{array}\right.
\end{eqnarray*}
We have
\begin{eqnarray*}
\lamm{\phi} & = & \bigM_{a \in \mathcal{A}} (a \rightarrow \phi
(a))
\end{eqnarray*}
Notice that $\lamm{\phi} \in \mathscr{S}$ as either $a \in \mathscr{S}$ and $\phi (a)
\in \mathscr{S}$ or $\phi (a) = \top$. Furthermore, we have
\begin{eqnarray*}
\ilam{p}{\left( \lamm{\phi} \right) p} & \preccurlyeq & \bigM_{e
\in X_1} \left( \exi{X_1} (e) \rightarrow \exi{Y_1} (f_1 (e)) \right)
\end{eqnarray*}
hence
\[ \vld{} \bigM_{e \in X_1} \left( \exi{X_1} (e) \rightarrow
\exi{Y_1} (f_1 (e)) \right) \]
\end{proof}
\subsubsection{Is $\mathcal{K}$ always essentially surjective?} \mbox{}\\ \\ Let $\eset{X}$ be an implicative set and $\nu : X \times X \rightarrow \ppows{M}$ be a valuation such that $\approx^+ \text{ } \cong_{X \times X} \text{ } \bigE \nu$ (such a valuation always exists since $M$ is dense). Assuming choice there is a map $c : \ngh{X} \rightarrow M$ such that $c (x) \in \nu (x, x)$ for all $x \in \ngh{X}$.
\begin{notation}
\leavevmode
\begin{enumeratenumeric}
\item $\Equ{X} := \left\{ (x, x') \in X \times X| \text{ }
\eeq{x}{x'} \in \mathscr{S} \right\}$ (the set of valid equalitites on $X$);
\item $\ngh{X} := \left\{ x \in X| \eexi{X} (x) \in \mathscr{S} \right\}$
(the set of $X$'s non-ghosts);
\item ${\approx}^{+} := \restr{\approx}{\Equ{X}}$;
\item $\textsf{} \exiplus{X} := \restr{\left( \eexi{X}
\right)}{\ngh{X}}$.
\end{enumeratenumeric} \end{notation}
\begin{lemma}
\label{lem:pseudo}The $M$-assemblies
\begin{itemizeminus}
\item $\hat{X}_1$ given by $\car{\widehat{X_1}} := \sum_{(x, x') \in
\Equ{X}} \nu (x, x')$ and $\exi{\hat{X}_1} ((x, x'), m) := c (x)
\sqcap m \sqcap c (x')$;
\item $\hat{X}_0$ given by $\car{\hat{X}_0} := \ngh{X}$ and
$\exi{\widehat{X_0}} (x) = c (x)$
\end{itemizeminus}
along with the maps $s := \pi_0 \circ \pi_0$ and $t := \pi_1 \circ
\pi_0$ form a pseudo-groupoid $\hat{X}$. \end{lemma}
\begin{proof}
$\widehat{X_1}$ is an $M$-assembly since $M$ is algebraic. The source and
target maps $s$ and $t$ are tracked by $\ensuremath{\tmmathbf{{\pi}}}\tmrsub{0}$ and $\ensuremath{\tmmathbf{{\pi}}}\tmrsub{2}$,
respectively. We have the local section of $\pr{s}{t}$
\begin{eqnarray*}
\rho : & \car{\hat{X}_0} \longrightarrow & \car{\widehat{X_1}}\\
x & \mapsto & ((x, x), c (x))
\end{eqnarray*}
tracked by $\tmop{id}$. Next we have the iso
\[ \approx^+ \text{ } \cong_{X \times X} \text{ } \bigE \nu \qquad (\star)
\]
by density, which entails
\[ \vld{\sym{X}' := \bigM_{x, x' \in \ngh{X}} \left( \bigE \nu
(x, x') \rightarrow \bigE \nu (x', x) \right)} \]
Let $x, x' \in \ngh{X}$ and $m \in \nu (x, x')$. We then have $\left(
\sym{X}' \ilam{z}{z m} \right) \ensuremath{\tmmathbf{\mathtt{I}}} \in \nu (x', x)$, hence the ``twist''
map
\begin{eqnarray*}
\sigma : \car{} & \car{\widehat{X_1}} \longrightarrow &
\car{\widehat{X_1}}\\
((x, x'), m) & \mapsto & \left( (x', x), \left( \sym{X}' \ilam{z}{z m}
\right) \ensuremath{\tmmathbf{\mathtt{I}}} \right)
\end{eqnarray*}
tracked by $\ilam{u \nocomma \nocomma m \nocomma v}{v \sqcap \left( \left(
\sym{X}' \lam{z}{z m} \right) \ensuremath{\tmmathbf{\mathtt{I}}} \right) \sqcap u}$. But $(\star)$ also
entails
\[ \vld{\trans{X}' := \bigM_{x, x', x''} \left( \bigE} \nu (x, x')
\sqcap \bigE \nu (x', x'') \rightarrow \bigE \nu (x, x'') \right) \]
Let $x, x', x'' \in \ngh{X}$, $m \in \nu (x, x')$ and $n \in \nu (x', x'')$.
We then have
\begin{eqnarray*}
\left( \trans{X}' \ilam{z}{z \nocomma m} \ilam{z}{z n} \right) \ensuremath{\tmmathbf{\mathtt{I}}} &
\in & \nu (x, x'')
\end{eqnarray*}
hence the ``connecting'' map
\begin{eqnarray*}
\tau : \car{\widehat{X_1}} \circledast \car{\widehat{X_1}} &
\longrightarrow & \car{\widehat{X_1}}\\
(((x, x'), m), ((x', x''), n)) & \mapsto & \left( (x, x''), \left(
\trans{X}' \ilam{z}{z \nocomma m} \ilam{z}{z n} \right) \ensuremath{\tmmathbf{\mathtt{I}}} \right)
\end{eqnarray*}
tracked by $\ilam{u \ensuremath{\diamond} \ensuremath{\diamond} m n v}{u \sqcap \left( \trans{X}' \ilam{z}{z
\nocomma m} \ilam{z}{z n} \right) \ensuremath{\tmmathbf{\mathtt{I}}} \sqcap v}$. \end{proof}
\label{rem:pseudo}We thus have $\mathcal{K} (\hat{X}) = \left( \ngh{X}, \asymp \right)$ where \begin{eqnarray*}
\eeqq{x}{x'} & = & \left\{\begin{array}{lll}
c (x) \sqcap c (x') \sqcap \bigE \nu (x, x') & & (x, x') \in \Equ{X}\\
\bot & & \text{otherwise}
\end{array}\right. \end{eqnarray*} as $\nu = \nu_{B_X}$ by construction (c.f. Lemma \ref{lem:pseudo}). There are no ghosts here.
\begin{lemma}
\label{lem:pseudo-fun}\label{lem:fun}The relation $K : \left( \ngh{X},
\asymp \right) \rightarrow (X, \approx)$ given by
\begin{eqnarray*}
K : \ngh{X} \times X & \longrightarrow & \mathcal{A}\\
(x, x') & \mapsto & x \ensuremath{\approx} x'
\end{eqnarray*}
is functional. \end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\ext{K} & = & \bigM_{x_1, x_2 \in \ngh{X}} \bigM_{x_1',
x_2' \in X} \left( \eeq{x_1}{x_1'} \sqcap \eeqq{x_1}{x_2} \sqcap
\eeq{x_1'}{x_2'} \rightarrow \eeq{x_2}{x_2'} \right)
\end{eqnarray*}
Suppose $(x_1, x_2) \not\in \Equ{X}$. We then have $\eeqq{x_1}{x_2} = \bot$ \
so
\begin{eqnarray*}
\eeq{x_1}{x_1'} \sqcap \eeqq{x_1}{x_2} \sqcap \eeq{x_1'}{x_2'}
\rightarrow \eeq{x_2}{x_2'} & = & \top
\end{eqnarray*}
in those cases, hence
\begin{eqnarray*}
\ext{K} & = & \bigM_{(x_1, x_2) \in \tmop{Equ}^+_X}
\bigM_{x_1', x_2' \in X} \left( \eeq{x_1}{x_1'} \sqcap
\eeqq{x_1}{x_2} \sqcap \eeq{x_1'}{x_2'} \rightarrow \eeq{x_2}{x_2'}
\right)
\end{eqnarray*}
But $\ext{K}$ is equivalent to
\begin{eqnarray*}
\ext{K}' & := & \bigM_{(x_1, x_2) \in \tmop{Equ}^+_X}
\bigM_{x_1', x_2' \in X} \left( \eeq{x_1}{x_1'} \sqcap c (x_1)
\sqcap c (x_2) \sqcap \eeq{x_1}{x_2} \sqcap \eeq{x_1'}{x_2'}
\rightarrow \eeq{x_2}{x_2'} \right)
\end{eqnarray*}
by density and furthermore
\begin{eqnarray*}
\ilam{u \ensuremath{\diamond} \ensuremath{\diamond} v w}{\trans{X} \left( \sym{X} v \right) \left(
\trans{X} u v \right)} & \preccurlyeq & \ext{K}'
\end{eqnarray*}
Next we have
\begin{eqnarray*}
\str{K} & = & \bigM_{x \in \Equ{X}} \bigM_{x' \in X}
\left( \eeq{x}{x'} \rightarrow | x \asymp x | \sqcap \eeq{x'}{x'}
\right)
\end{eqnarray*}
But $\str{K}$ is equivalent to
\begin{eqnarray*}
\str{K}' & = & \bigM_{x \in \Equ{X}} \bigM_{x' \in X}
\left( \eeq{x}{x'} \rightarrow c (x) \sqcap \exiplus{X} (x) \sqcap
\eexi{X} (x') \right)
\end{eqnarray*}
by density. Consider the map
\begin{eqnarray*}
d : \mathcal{A} & \longrightarrow & \mathcal{A}\\
a & \mapsto & \left\{\begin{array}{lll}
c (x) & & \text{there is } x \in \ngh{X} \text{such that } a =
\exiplus{X} (x)\\
\top & & \text{otherwise}
\end{array}\right.
\end{eqnarray*}
We have $\lamm{d} \in \mathscr{S}$ and furthermore
\begin{eqnarray*}
\letin{\varepsilon}{\lam{p}{} \trans{X} p \left( \sym{X} p \right)}
\text{{\hspace{8em}}} & & \\
\letin{\varepsilon'}{\lam{q}{\left( \lamm{d} \right) (\varepsilon q)}}
\text{{\hspace{7em}}} & & \\
\ilam{r}{(\varepsilon' r) \sqcap (\varepsilon r) \sqcap \left( \trans{X}
\left( \sym{X} p \right) p \right)} & \preccurlyeq & \str{K}'
\end{eqnarray*}
Next we obviously have
\begin{eqnarray*}
\vld{\sv{K}} & = & \bigM_{x_1 \in \ngh{X}} \bigM_{x_1',
x_2' \in X} \left( \eeq{x}{x_1'} \sqcap \eeq{x}{x_2'} \rightarrow
\eeq{x_1'}{x_2'} \right)
\end{eqnarray*}
Finally we have
\begin{eqnarray*}
\tot{K} & = & \bigM_{x \in \ngh{X}} \left( | x \asymp x |
\rightarrow \iex{x' \in X} \left( \eexi{X} (x') \sqcap K (x, x') \right)
\right)\\
& = & \bigM_{x \in \ngh{X}} \left( \left( c (x) \sqcap \bigE \nu
(x, x) \right) \rightarrow \iex{x' \in X} \left( \eexi{X} (x') \sqcap
\eeq{x}{x'} \right) \right)
\end{eqnarray*}
and furthermore $\vld{\xi := \bigM_{x \in X}} \left( \bigE \nu
(x, x) \rightarrow \exiplus{X} (x) \right)$ by density, hence
\begin{eqnarray*}
\ilam{\ensuremath{\diamond} p}{\xi p} & \preccurlyeq & \tot{K}
\end{eqnarray*}
\end{proof}
\begin{proposition}
\label{prop:inj}The morphism $[K] : \left( \ngh{X}, \asymp \right)
\rightarrow (X, \approx)$ in $\ensuremath{\tmmathbf{\tmop{Set}}} [\mathcal{A}]$ is is internally
injective. \end{proposition}
\begin{proof}
Let
\begin{eqnarray*}
\mathsf{Inj} & := & \bigM_{x_1, x_2 \in \ngh{X}}
\bigM_{x \in X} \left( K (x_1, x) \sqcap K (x_2, x) \rightarrow
\eeqq{x_1}{x_2} \right)\\
& = & \bigM_{x_1, x_2 \in \ngh{X}} \bigM_{x \in X}
\left( \eeq{x_1}{x} \sqcap \eeq{x_2}{x} \rightarrow c (x_1) \sqcap c (x_2)
\sqcap \bigE \nu (x_1, x_2) \right)
\end{eqnarray*}
Consider again the map
\begin{eqnarray*}
d : \mathcal{A} & \longrightarrow & \mathcal{A}\\
a & \mapsto & \left\{\begin{array}{lll}
c (x) & & \text{there is } x \in \ngh{X} \text{such that } a =
\exiplus{X} (x)\\
\top & & \text{otherwise}
\end{array}\right.
\end{eqnarray*}
used in the proof of Lemma \ref{lem:pseudo-fun}. We have in particular
$\lamm{d} \in \mathscr{S}$. Furthermore
\[ \vld{\chi :=} \bigM_{x_1, x_2 \in \ngh{X}} \left(
\eeq{x_1}{x_2} \rightarrow \bigE \nu (x_1, x_2) \right) \]
by density, hence
\begin{eqnarray*}
\letin{j}{\lam{p q}{} \trans{X} p \left( \sym{X} q \right)}
\text{{\hspace{16em}}} & & \\
\letin{e_1}{\lam{p}{} \trans{X} \left( \sym{X} p \right) p}
\text{{\hspace{15em}}} & & \\
\letin{e_2}{\lam{p}{} \trans{X} p \left( \sym{X} p \right)}
\text{{\hspace{14em}}} & & \\
\letin{\delta}{\lam{p}{\left( \lamm{d} \right) p}} \text{{\hspace{15em}}} &
& \\
\ilam{p q}{\letin{\varepsilon}{j p q} ((\delta (e_1 \varepsilon)) \sqcap
(\delta (e_2 \varepsilon)) \sqcap (\chi \varepsilon)) } & \preccurlyeq &
\mathsf{Inj}
\end{eqnarray*} \end{proof}
Alas, $[K]$ may fail to be internally surjective. For internal surjectivity means in our case that \[ \bigM_{x' \in X} \left( \eexi{X} (x') \rightarrow \iex{x \in
\ngh{X}} K (x, x') \right) \qquad (\star) \] is in $\mathscr{S}$, yet in general we don't have control over what happens on ghosts. Notice that if $\mathcal{A}$ is compatible with joins, $x'$ being a ghost means $\eexi{X} (x') = \bot$, hence factors indexed by ghosts evaluate to $\top$ so we can ``track'' $(\star)$. $[K]$ is then internally surjective as well, an exemple for this phenomenon being the well-known equivalence $\ensuremath{\mathbf{P}}\ensuremath{\mathbf{A}}\ensuremath{\mathbf{s}}\ensuremath{\mathbf{s}} \simeq \ensuremath{\mathcal{E}\!f\!f}$ {\cite{robinson1990colimit,menni2000exact}}. On the other hand if we co-restrict $\mathcal{K}$ to ``ghostless'' implicative sets, we get an equivalence of categories.
\begin{theorem}
Let $\tmmathbf{\tmop{Set}} [\mathcal{A}]^+$ be the full subcategory of
${\ens}[\ensuremath{\mathcal{A}}]$ with objects implicative sets $\eset{X}$ such that $\eexi{X} (x) \in
\mathscr{S}$ for all $x \in X$. We have
\begin{eqnarray*}
\exlex{(\asm{M})} & \simeq &
\tmmathbf{\tmop{Set}} [\mathcal{A}]^+
\end{eqnarray*} \end{theorem}
\begin{proof}
The image of the functor $\mathcal{K}:
\exlex{(\asm{M})} \rightarrow {\ens}[\ensuremath{\mathcal{A}}]$ is contained
in $\tmmathbf{\tmop{Set}} [\mathcal{A}]^+$. Its co-restriction
$\mathcal{K}^+ : \exlex{(\asm{M})} \rightarrow
\tmmathbf{\tmop{Set}} [\mathcal{A}]^+$ is thus full (c.f. Proposition
\ref{prop:full}), faithful (c.f. Proposition \ref{prop:K} and internally
injective (c.f. Proposition \ref{prop:inj}). Let
\begin{eqnarray*}
\mathsf{Surj} & := & \bigM_{x' \in X} \left( \eexi{X} (x')
\rightarrow \iex{x \in \ngh{X}} K (x, x') \right)
\end{eqnarray*}
Suppose $\eexi{X} (x')$. We then have $x' \in \ngh{X}$ so in particular $K
(x', x') = \eexi{X} (x')$, thus $\tmop{id} \preccurlyeq \mathsf{Surj}$.
Hence $[K]$ is iso, thus $\mathcal{K}^+$ is an equivalence of categories. \end{proof}
Wrapping it up, the following picture begins to emerge
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=2.77676111767021cm]{exact_completion_asm-6.pdf}} \end{center}
All the categories involved but $\asm{M}$ and $\asm{\mathcal{A}}$ are topoi. In particular, we see that $\asm{\mathcal{A}}$ is a subcategory of ${\ens}[\ensuremath{\mathcal{A}}]$ yet the embedding is not the trivial one. As a matter of fact, assemblies embed as kernel pairs of their projective covers. A characterisation of those implicative algebras for which we have $\exlex{(\asm{M})} \simeq {\ens}[\ensuremath{\mathcal{A}}]$ is an open question. We may have more to say about this circle of ideas in subsequent elaborations.
\end{document} | arXiv |
\begin{document}
\title{
Nonconvex Robust High-Order Tensor Completion
Using Randomized Low-Rank Approximation
}
\author{Wenjin~Qin,~Hailin~Wang,~\IEEEmembership{Student Member,~IEEE,}~Feng~Zhang,~Weijun~Ma,~Jianjun~Wang,~\IEEEmembership{Member,~IEEE,}\\
~and~Tingwen~Huang,~\IEEEmembership{Fellow,~IEEE}
\thanks{ This research was supported in part by the National Natural Science Foundation of China under
Grant 12071380, Grant 12101512; in part by the National Key Research and Development Program of China under Grant 2021YFB3101500;
in part by
the China Postdoctoral Science Foundation under Grant 2021M692681;
in part by
the Natural Science Foundation of Chongqing, China, under Grant cstc2021jcyj-bshX0155;
and in part by the Fundamental Research Funds for the Central Universities under Grant SWU120078.
(Corresponding author: Jianjun Wang.) } \thanks{Wenjin Qin and Feng Zhang are with the School of Mathematics and Statistics, Southwest University, Chongqing 400715, China (e-mail: [email protected], [email protected]
).
Hailin Wang is with the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China (e-mail: [email protected]).}
\thanks{Weijun Ma is with the School of Information Engineering, Ningxia University, Yinchuan 750021, China (e-mail: Weijunma\[email protected]). } \thanks{Jianjun Wang is with the School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, and also with the Research Institute of Intelligent Finance and Digital Economics, Southwest University, Chongqing 400715, China (e-mail: [email protected]).}
\thanks{ Tingwen Huang is with the Department of Mathematics, Texas A\&M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). } }
\maketitle \begin{abstract}
Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering.
Nevertheless, these methods involve the T-SVD based low-rank approximation,
which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors.
Against these issues, in this article,
two
efficient low-rank
tensor approximation approaches
fusing
randomized techniques are
first devised
under the order-$d$ ($d\geq3$) T-SVD framework.
On this basis, we then further investigate the robust high-order tensor completion (RHTC) problem, in which a double nonconvex model along with its corresponding
fast optimization algorithms with convergence guarantees are developed.
To the best of our knowledge, this is the first study to incorporate the randomized low-rank approximation
into the RHTC problem.
Empirical
studies on large-scale
synthetic and real tensor data illustrate that
the proposed method outperforms other state-of-the-art approaches in terms of
both computational efficiency and estimated precision.
\end{abstract}
\begin{IEEEkeywords}
High-order T-SVD framework,
Robust high-order tensor completion, Randomized low-rank tensor approximation,
Nonconvex regularizers, ADMM algorithm.
\end{IEEEkeywords} \IEEEpeerreviewmaketitle
\section{\textbf{Introduction}}
Multidimensional data including medical images,
remote sensing images, light field images, color videos, and beyond, are becoming increasingly prevailing in various domains
such as
neuroscience \cite{
beckmann2005tensorial
}, chemometrics \cite{schutt2017quantum},
data mining \cite{
papalexakis2016tensors11 }, machine learning \cite{
sidiropoulos2017tensor11 }, image processing \cite{ marquez2020compressive, lin2022robust, long2021bayesian }, and computer vision \cite{
bibi2017high, zhang2019robust, hou2021robust55
}.
Compared to
vectors and matrices, tensors possess a more powerful capability to
characterize the
inherent
structural information underlying these data from a higher-order perspective.
Nevertheless,
due to various factors such as
occlusions, abnormalities,
software glitches, or sensor failures,
the tensorial data
faced in practical applications
can often suffer from elements loss and noise/outliers corruption.
Hence,
robust low-rank tensor completion (RLRTC) has been widely concerned by a large number of scholars
\cite{ goldfarb2014robust1, huang2015provable22, zhao2015bayesian12, chen2019nonconvex5555, huang2020robust1, chen2021auto, liu2021simulated, li2021robust, he2022coarse,
jiang2019robust,wang2020robust, wang2019robust1,lou2019robust1, song2020robust,ng2020patched,he2020robust, chen2020robust1,zhao2020nonconvex1,qiu2021nonlocal,zhao2022robust1 }.
RLRTC belongs to a
canonical inverse problem, which aims to reconstruct the underlying low-rank tensor from partial observations of target tensor corrupted by noise/outliers. {Mathematically, the RLRTC model can be formulated as follows: }
\begin{align} \label{intro3}
\min_{{\boldsymbol{\mathcal{L}}},{\boldsymbol{\mathcal{E}}} }\;\;
\Psi({\boldsymbol{\mathcal{L}}}) + \lambda\Upsilon({\boldsymbol{\mathcal{E}}}),
\;\; \text{s.t.}\;\;
\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}})= \boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}}), \end{align}
where $\Psi({\boldsymbol{\mathcal{L}}})$
represents the regularizer measuring tensor low-rankness
(also called the
certain relaxation of tensor rank),
$\Upsilon({\boldsymbol{\mathcal{E}}})$ \footnote{
In specific problems, if we assume that the noise/outliers follows the Laplacian distribution or the Gaussian distribution, then $\Upsilon(\boldsymbol{\mathcal{E}})$ can be chosen as $\|\bm{\mathcal{E}}\|_1$ or $\|\bm{\mathcal{E}}\|_F^{2}$, respectively. } denotes the noise/outliers
regularization,
$\lambda > 0 $ is a trade-off parameter that balances these two terms, ${\boldsymbol{\mathcal{M}}}$ is the observed tensor,
and $\boldsymbol{\bm{P}}_{{{\Omega}}}(\cdot )$
is the projection operator
onto the observed index set ${{{\Omega}}}$ such that
\begin{eqnarray*}
\label{intro2} \big(\boldsymbol{\bm{P}}_{{{\Omega}}}(\boldsymbol{\mathcal{M}})\big)_{i_1,\cdots ,i_d}= \begin{cases} {\boldsymbol{\mathcal{M}}}_{i_1,\cdots ,i_d},\;\text{if}\;(i_1,\cdots ,i_d) \in \Omega,\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;
\text{otherwise}. \end{cases}
\end{eqnarray*}
If the index set ${{{\Omega}}}$ is the whole set, i.e., no elements are missing, then the model (\ref{intro3}) reduces to the Tensor Robust Principal Component Analysis (TRPCA) problem \cite{ lu2019tensor,zhang2020low1, gao2020enhanced,li2021nonconvex11,qiu2022fast,wang2022tensor55, zheng2020tensor, shi2021robust1, yang2022nonconvex11 }. If there is no corruption, i.e., ${\boldsymbol{\mathcal{E}}}=0$, then the model (\ref{intro3})
is equivalent to the Low-Rank Tensor Completion (LRTC) problem \cite{ zhang2016exact, kong2018t,
zhang2021low55,
xu2021fast22,
wang2021generalized11,
liu2013tensor,bengua2017efficient,
zhao2021tensor,
xue2022laplacian,
qiu2022noisy }. Therefore,
RLRTC can be viewed as
a generalized form of both LRTC and TRPCA.
Nevertheless,
there exist different definitions of tensor rank and its
corresponding relaxation
within different tensor decomposition frameworks,
which makes the optimization problem (\ref{intro3})
extremely complicated.
The commonly-used
frameworks
contain
CANDECOMP/PARAFAC (CP) \cite{
kolda2009tensor }, Tucker \cite{tucker1966some},
tensor train (TT) \cite{oseledets2011tensor}, tensor ring (TR) \cite{zhao2016tensor},
and tensor singular value decomposition (T-SVD) \cite{kilmer2011factorization,
kernfeld2015tensor
}. Among them, T-SVD presents the first closed multiplication operation
called tensor-tensor product (t-product),
and
derives
a
novel tensor tubal rank \cite{kilmer2013third}
that
well characterizes the intrinsic low-rank structure of a tensor.
In particular,
the recent work \cite{kilmer2021tensor} revolutionarily proved the
best representation and compression theories of T-SVD, making it more notable in capturing the ``spatial-shifting" correlation and the global structure information underlying tensors.
With
these advantages, the
robust low-tubal-rank tensor completion
\cite{jiang2019robust,wang2020robust, wang2019robust1,lou2019robust1, song2020robust,ng2020patched, he2020robust, chen2020robust1,zhao2020nonconvex1,qiu2021nonlocal,zhao2022robust1}
modeled by (\ref{intro3})
and its variants
\cite{ lu2019tensor,zhang2020low1, gao2020enhanced,li2021nonconvex11,qiu2022fast,wang2022tensor55, zheng2020tensor, shi2021robust1, yang2022nonconvex11, zhang2016exact, kong2018t,
zhang2021low55,
xu2021fast22,
wang2021generalized11 }
have recently caught many scholars' attention.
However,
we observe that these approaches are only relevant to third-order tensors. To fix this problem, Qin et al. established an
order-$d$ ($d\geq3$) T-SVD algebraic framework \cite{qin2022low} based on
a family of invertible linear transforms, and then preliminary investigated the model, algorithm, and theory for
the robust high-order tensor completion (RHTC)
\cite{qin2022low,wang2023guaranteed55, qin2022robust , qin2021robust }. The RHTC methods generated by
other tensor factorization frameworks can be found in \cite{ goldfarb2014robust1, huang2015provable22, zhao2015bayesian12, chen2019nonconvex5555, huang2020robust1, chen2021auto, liu2021simulated, li2021robust, he2022coarse}.
Although the
above deterministic
RLRTC investigations
have already made some achievements
in real-world applications,
they encounter enormous challenges in dealing with those tensorial data characterized by large volumes, high dimensions,
complex structures, etc.
This is mainly attributed to the fact that they require
to perform
multiple low-rank approximations based on a specific
tensor factorization.
Calculating such an approximation
generally involves the singular value decomposition (SVD) or T-SVD, which is very time-consuming and inefficient when the data size scales up.
Motivated by the reason that randomized algorithms can accelerate the computational speed of their conventional counterparts at the expense of slight accuracy loss \cite{halko2011finding11,mahoney2011randomized,woodruff2014sketching,martinsson2020randomized55 ,buluc2021randomized55 },
effective
low-rank tensor approximation approaches using randomized sketching techniques (e.g., \cite{ zhang2018randomized,tarzanagh2018fast,che2022fast, wang2015fast,battaglino2018practical, malik2018low,che2019randomized,minster2020randomized,ahmadi2020randomized})
have attracted more and more attention in recent years.
Among these
methods, we obviously find that the ones
based on T-SVD framework are only appropriate
for third-order tensors.
In addition,
RHTC researches
in combination with randomized low-rank approximation are relatively lacking.
With the rapid development of
information technologies, such as Internet of Things, and big data,
large-scale high-order tensors encountered in real scenarios are growing explosively,
like order-four color videos and
multi-temporal remote sensing images, order-five light filed images, order-six bidirectional texture function images. Therefore,
in this work,
we consider developing fast and efficient randomness-based large-scale
high-order tensor
representation and recovery methods under the T-SVD framework.
\subsection{\textbf{Our Contributions}}
Main contributions of this work are summarized as follows:
1) Firstly, within the order-$d$ T-SVD algebraic framework \cite{qin2022low},
two efficient randomized algorithms for low-rank approximation of high-order tensor are devised
considering their adaptability to large-scale tensor data.
The developed
approximation methods
obtain a significant advantage in computational speed against
the optimal $k$-term approximation \cite{qin2022low}
(also called truncated T-SVD)
with a slight loss of precision.
2) Secondly, an effective and scalable model for RHTC is proposed in virtue of nonconvex low-rank and noise/outliers regularizers.
Based on
the
proposed randomized low-rank approximation schemes,
we then design
two
alternating direction method of multipliers (ADMM) framework based fast algorithms with convergence guarantees
to solve the formulated model,
through which
any
low T-SVD rank high-order tensors
with simultaneous elements loss and noise/outliers corruption
can be reconstructed efficiently and accurately.
3) Thirdly, our proposed RHTC algorithm can be applied to a series of large-scale reconstruction
tasks, such as
the restoration of fourth-order color videos and multi-temporal remote sensing images, and fifth-order light-field images.
Experimental results demonstrate that the proposed method
achieves
competitive performance in estimation accuracy
and CPU running time than other state-of-the-art ones.
Strikingly,
in the case of sacrificing a little precision, our versions
combined with
randomization ideas
decrease the CPU time by about $50\%$$\sim$$70\%$ compared with the deterministic version.
\subsection{\textbf{Organization}} \textcolor[rgb]{0.00,0.00,0.00}{ The remainder of the paper is organized as follows.
Section \ref{related} gives a brief summary of related work.
The
main notations and preliminaries are
introduced in Section \ref{nota},
and then
we develop two efficient randomized algorithms for low-rank approximation of high-order tensor
in Section \ref{approximation}.
Section \ref{model} proposes effective nonconvex model
and
optimization
algorithm for RHTC.
In Section \ref{experiments}, extensive experiments
are conducted to evaluate the effectiveness of the proposed method.
Finally, we conclude our work in Section \ref{conclusion}.}
What is noteworthy is that
this article can be regarded as
an expanded version of
our previous
conference paper \cite{qin2022robust}.
Built off
the conference version,
this paper makes the following changes: 1)
the high-order tensor approximation algorithm that fuses the randomized blocked strategy
is added; 2) the original low-rank and noise/outliers regularizers
are further enhanced with more flexible regularizers;
3)
two accelerated algorithms
for solving the newly formulated non-convex model are designed via the proposed low-rank approximation strategies; 4) a large number of experiments concerning with high-order tensor approximation and restoration are added.
\renewcommand{0.0}{0.0} \setlength\tabcolsep{3.0pt} \begin{table*}[!htbp]
\centering
\caption{The main notions and preliminaries for order-$d$ tensor.}
\label{notation_part1}
\small
\scriptsize
\footnotesize
\begin{threeparttable}
\begin{tabular}{l|l|l|l}
\hline
Notations&Descriptions&Notations&Descriptions\cr
\hline
\hline
${{\bm{A}}} \in \mathbb{R} ^{n_{1}\times n_{2}}$ & matrix &
$\bm{I}_{n} \in \mathbb{R} ^{n\times n}$ & $n\times n $ identity matrix
\cr
$\operatorname{trace}(\bm{A})$ & matrix trace &
$ {{\bm{A}}}^{\mit{H}} ({{\bm{A}}}^{\mit{T}})$ & conjugate transpose (transpose)
\cr
$ \langle {{\bm{A}}},{{\bm{B}}} \rangle=\operatorname{trace} ({{\bm{A}}}^{\mit{H}} \cdot {{\bm{B}}})$& matrix inner product&
${\|{{\bm{A}}}\|}_{\bm{w}, {\mathrm{S}}_{p}}= {\big(\sum_{i} {\bm{w}}_{i} \; \big|\sigma_{i}({{\bm{A}}})\big| ^{p}\big)}^\frac{1}{p}$
&matrix weighted Schatten-$p$ norm
\cr
${\boldsymbol{\mathcal{A}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$ & order-$d$ tensor&
${\boldsymbol{\mathcal{A}}}_{i_{1}\cdots i_{d}}$ or ${\boldsymbol{\mathcal{A}}}{(i_{1},\cdots, i_{d})}$ & $(i_1,\cdots,i_d)$-th entry
\cr
${{\bm{A}}_{(k)}} \in \mathbb{R}^{n_k\times
\prod_{j\neq k } {n_j}
} $
& mode-$k$ unfolding of $\boldsymbol{\mathcal{A}}$ &
${\|{\boldsymbol{\mathcal{A}}}\|}_{\infty}= \max_{i_{1} \cdots i_d} |{\boldsymbol{\mathcal{A}}_{i_{1} \cdots i_d}}|$ &tensor infinity norm
\cr
${\|{\boldsymbol{\mathcal{A}}}\|}_{{\boldsymbol{\mathcal{W}}},\ell_q}= ({\sum
}
{\boldsymbol{\mathcal{W}}_{i_{1} \cdots i_d}}
|{\boldsymbol{\mathcal{A}}_{i_{1} \cdots i_d}}|^{q})^{\frac{1}{q}}$ &tensor weighted $\ell_q$-norm &
${\|{\boldsymbol{\mathcal{A}}}\|}_{\mathnormal{F}}= {(\sum_{i_{1} \cdots i_d} |{\boldsymbol{\mathcal{A}}_{i_{1} \cdots i_d}}|^{2})^\frac{1}{2}}$ & tensor Frobenius norm
\cr
$
\boldsymbol{\mathcal{C}}={\boldsymbol{\mathcal{A}}}{*}_{\mathfrak{L}} {\boldsymbol{\mathcal{B}}}
$
& order-$d$
t-product under linear transform $\mathfrak{L}$
&
$ {{\boldsymbol{\mathcal{A}}}}^{\mit{H}} ({{\boldsymbol{\mathcal{A}}}}^{\mit{T}} )$
&transpose (conjugate transpose)
\cr
\hline
$\langle {\boldsymbol{\mathcal{A}}},{\boldsymbol{\mathcal{B}}} \rangle $& \multicolumn{3} {l} {
the inner product between order-$d$ tensors ${\boldsymbol{\mathcal{A}}}$ and ${\boldsymbol{\mathcal{B}}}$, i.e., $\langle {\boldsymbol{\mathcal{A}}},{\boldsymbol{\mathcal{B}}} \rangle = {\sum}_{j=1}^{n_3 n_4 \cdots n_d} \langle {\bm{\mathcal{A}}}^{<j>},{\bm{\mathcal{B}}}^{<j>} \rangle$.
} \cr
${\boldsymbol{\mathcal{A}}}^{<j>} \in \mathbb{R}^{n_1 \times n_{2}}$ &
\multicolumn{3} {l}{
the matrix frontal slice of ${\boldsymbol{\mathcal{A}}}$,
$
{\boldsymbol{\mathcal{A}}}^{<j>}={\boldsymbol{\mathcal{A}}}{(:,:,i_3,\cdots, i_{d})},\;
j={\sum_{a=4}^{d} } {(i_a-1){\Pi}_{b=3}^{a-1}n_b}+i_3$
.}\cr
$\boldsymbol{\mathcal{A}} \;{\times}_{n}\; \bm{M}$ &
\multicolumn{3} {l}{ the mode-$n$ product of tensor $\boldsymbol{\mathcal{A}}$ with matrix ${\bm{M}},\;
\boldsymbol{\mathcal{B}} =\boldsymbol{\mathcal{A}}{\times}_{n} {\bm{M} } $
$\Longleftrightarrow$
${\bm{B}}_{(n)} = {\bm{M}} \cdot {\bm{A}}_{(n)}$.} \cr
${\operatorname{bdiag}} ({{\boldsymbol{\mathcal{A}}}}) \in \mathbb{R}^{n_1n_3\cdots n_d \times n_2n_3\cdots n_d}$& \multicolumn{3}{l} { $ {\operatorname{bdiag}} ({{\boldsymbol{\mathcal{A}}}})$
is a block diagonal matrix whose $i$-th block equals to ${\boldsymbol{\mathcal{A}}}^{<i>}$,
$\forall
i \in \{1,2,\cdots,n_3\cdots n_d\}$.
} \cr
f-diagonal/f-upper triangular tensor ${\boldsymbol{\mathcal{A}}}$& \multicolumn{3} {l} { frontal slice ${\boldsymbol{\mathcal{A}}}^{<j>}$
of ${\boldsymbol{\mathcal{A}}}$ is {a diagonal matrix (an upper triangular matrix)},
$\forall
j \in \{1,2,\cdots,n_3\cdots n_d\}$. } \cr
identity tensor ${\boldsymbol{\mathcal{I}}} \in \mathbb{R}^{n \times n \times n_3\times \cdots \times n_{d}}$ &
\multicolumn{3} {l} { identity tensor ${\boldsymbol{\mathcal{I}}} $ is defined to be a tensor such that $ {{\mathfrak{L}} ({\boldsymbol{\mathcal{I}}}) }^{<j>} =\bm{I}_{n}, \forall j \in \{1,2, \cdots, n_3\cdots n_d\}$.}\cr
Gaussian random tensor ${\boldsymbol{\mathcal{G}}}$&
\multicolumn{3} {l} { the entries of ${{\mathfrak{L}} ({\boldsymbol{\mathcal{G}}})}^{<j>}$
follow the standard normal distribution, $ \forall
j \in \{1,2,\cdots,n_3 \cdots n_d\}$. }
\cr
orthogonal tensor ${\boldsymbol{\mathcal{Q}}} $&
\multicolumn{3} {l} {
orthogonal tensor
satisfies: ${\boldsymbol{\mathcal{Q}}}^{\mit{T}}{*}_{\mathfrak{L}}{\boldsymbol{\mathcal{Q}}} =
{\boldsymbol{\mathcal{Q}}}{*}_{\mathfrak{L}} {\boldsymbol{\mathcal{Q}}}^{\mit{T}}={\boldsymbol{\mathcal{I}}}$,
while
partially orthogonal tensor satisfies: ${\boldsymbol{\mathcal{Q}}}^{\mit{T}}{*}_{\mathfrak{L}}{\boldsymbol{\mathcal{Q}}} = {\boldsymbol{\mathcal{I}}}$.
}
\cr
$\textrm{H-TSVD}({\boldsymbol{\mathcal{A} }}, \mathfrak{L})$&
\multicolumn{3} {l} { order-$d$ T-SVD factorization, i.e.,
${\boldsymbol{\mathcal{A}}}={\boldsymbol{\mathcal{U}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{S}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{V}}}^{\mit{T}}$, where ${\boldsymbol{\mathcal{U}}} $ and ${\boldsymbol{\mathcal{V}}} $ are orthogonal,
${\boldsymbol{\mathcal{S}}} $ is f-diagonal.
}
\cr
$\textrm{H-TQR}({\boldsymbol{\mathcal{A} }}, \mathfrak{L})$& \multicolumn{3} {l} {order-$d$ tensor QR-type factorization, i.e., ${\boldsymbol{\mathcal{A}}}={\boldsymbol{\mathcal{Q}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{R}}}$, where ${\boldsymbol{\mathcal{Q}}} $ is
orthogonal while ${\boldsymbol{\mathcal{R}}} $
is f-upper triangular. } \cr
${\operatorname{rank}}_{tsvd} ({{\boldsymbol{\mathcal{A}}}}) $& \multicolumn{3} {l} {
${\operatorname{rank}}_{tsvd}({\boldsymbol{\mathcal{A}}})
= \sum_{i} {\mathbbm{1} \big[ {\boldsymbol{\mathcal{S}}}(i,i,:,\cdots,:) \neq \boldsymbol{0} \big]}$, where ${\boldsymbol{\mathcal{S}}}$ originates from the middle component of ${\boldsymbol{\mathcal{A}}} = {\boldsymbol{\mathcal{U}}}{{*}_{\mathfrak{L}}}{\boldsymbol{\mathcal{S}}} {*}_{\mathfrak{L}}
{\boldsymbol{\mathcal{V}}}^{\mit{T}}$. } \cr
\hline
\end{tabular}
\end{threeparttable}
\end{table*}
\section{\textbf{Related Work}}\label{related}
Based on
different
factorization schemes,
the representative RLRTC methods can be broadly summarized as follows.
\subsubsection{\textbf{RLRTC Based on \text{T-SVD} Factorization}}
Lu et al. \cite{lu2019tensor}
rigorously deduced
a novel tensor nuclear norm (TNN) corresponding to T-SVD that is proved to be the convex envelope of the tensor average rank.
Sequentially, Jiang et al. \cite{jiang2019robust}
conducted a rigourous study for the RLRTC problem,
which is modeled by the TNN and
$\ell_{1}$-norm penalty terms.
Besides, Wang et al. also adopted this novel TNN or slice-weighted TNN plus
a sparsity measure inducing $\ell_1$-norm to develop the RLRTC methods \cite{wang2020robust,wang2019robust1}.
Theoretically, the deterministic and non-asymptotic upper bounds on the estimation error are established from a statistical standpoint.
However, the previous methods may suffer from disadvantage due to the limitation of Fourier transform \cite{qiu2021nonlocal}.
Aiming at this issue, by utilizing the generalized
transformed TNN (TTNN) and $\ell_1$-norm regularizers,
Song et al. \cite{song2020robust} proposed an unitary transform method for RLRTC
and also analyzed its recovery guarantee. Continuing along this vein, a patched-tubes unitary transform approach for RLRTC was proposed by Ng et al. \cite{ng2020patched}.
Nevertheless, the TTNN
is a loose approximation of the tensor tubal rank, which usually leads to the over-penalization of the optimization problem and hence causes some unavoidable biases in real applications.
In addition,
as indicated by \cite{fan2001variable},
the $\ell_{1}$-norm
might not be statistically optimal in more challenging scenarios.
Recently, to break the shortcomings existing in the TNN
and $\ell_{1}$-norm regularization terms,
some researchers \cite{ chen2020robust1,zhao2020nonconvex1,qiu2021nonlocal,zhao2022robust1
}
designed new nonconvex low-rank and noise/outliers regularization terms to study the RLRTC problem from the model, algorithm, and theory.
But these methods are only limited to the case of third-order tensors
and face the
high computational expense of T-SVD.
\subsubsection{\textbf{RLRTC Based on Other Factorization Schemes}}
Liu et al. \cite{liu2013tensor} primitively developed a new Tucker nuclear norm, i.e., Sum-of-Nuclear-Norms of unfolding matrices of a tensor (SNN),
as convex relaxation of the tensor tucker rank.
Then, the
RLRTC approach
within the Tucker format
was investigated
in \cite{goldfarb2014robust1,huang2015provable22}
via
combining the SNN regularization
with $\ell_{1}$-norm loss function.
Zhao et al. \cite{zhao2015bayesian12} proposed a variational Bayesian inference framework for CP rank determination and applied it to the RLRTC problem.
Within the TT factorization,
Bengua et al. \cite{bengua2017efficient} proposed a novel TT nuclear norm
as the convex surrogate of the TT rank.
Furthermore,
in virtue of an auto-weighted mechanism, Chen et al. \cite{chen2021auto}
studied a new RLRTC method
modeled by the TT nuclear norm and $\ell_{1}$-norm regularizers.
Under the TR decomposition,
by utilizing the TR
nuclear norm and $\ell_{1}$-norm regularizers, the model, algorithm, and theoretical analysis for RLRTC
were developed by Huang et al. \cite{huang2020robust1}.
To be more robust against both missing entries and noise/outliers, an effective iterative $\ell_p$-regression ($0<p\leq2$) TT completion method was developed in \cite{liu2021simulated}.
In parallel,
integrating
TR rank
with $\ell_{p,\epsilon}$-norm ($0<p\leq1$), Li et al. \cite{li2021robust} proposed a new RLRTC formulation.
Besides, He et al. \cite{he2022coarse} put forward a novel two-stage coarse-to-fine framework for RLRTC of visual
data
in the TR factorization. However, these deterministic RLRTC methods mostly involve
multiple SVDs of unfolding matrices,
which
experiences high
computational costs
when dealing with
large-scale tensor data.
\begin{figure*}\label{intro_image22}
\end{figure*}
\section{\textbf{ notations and preliminaries }}\label{nota}
For brevity, the main notations and preliminaries utilized in the whole paper
are summarized in Table \ref{notation_part1}, most of which originate form the literature \cite{qin2022low}.
In this work,
we let
${\mathfrak{L}}(\boldsymbol{\mathcal{A}})$ represent
the result of invertible linear transforms $\mathfrak{L}$ on $\boldsymbol{\mathcal{A}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$, i.e.,
\begin{align}\label{trans}
\mathfrak{L}(\boldsymbol{\mathcal{A}}) =\boldsymbol{\mathcal{A}} \;{\times}_{3} \;\bm{U}_{n_3} \;{\times}_{4} \;\bm{U}_{n_4} \cdots
{\times}_{d} \;\bm{U}_{n_d}, \end{align}
where the
transform matrices $\bm{U}_{n_i} \in \mathbb{C}^{n_i \times n_i}$ of $\mathfrak{L}$ satisfies:
\begin{align}\label{orth}
{{\bm{U}}}_{n_i} \cdot {{\bm{U}}}^{\mit{H}}_{n_i}={{\bm{U}}}^{\mit{H}}_{n_i} \cdot {{\bm{U}}}_{n_i}=\alpha_i {{\bm{I}}}_{n_i}, \forall i\in \{3,\cdots,d\},
\end{align}
in which $\alpha_i>0$ is a constant.
The inverse operator of $\mathfrak{L}(\boldsymbol{\mathcal{A}})$
is defined as
$ \mathfrak{L}^{-1} (\boldsymbol{\mathcal{A}}) =\boldsymbol{\mathcal{A}} \;{\times}_{d} \;\bm{U}^{-1}_{n_d} \;{\times}_{{d-1}} \;{{\bm{U}}_{n}}_{d-1}^{-1} \cdots
{\times}_{3} \;\bm{U}^{-1}_{n_3}$,
and $\mathfrak{L}^{-1} (\mathfrak{L}(\boldsymbol{\mathcal{A}}))=\boldsymbol{\mathcal{A}}$.
\begin{Definition}\label{def10} \textbf{(Order-$d$ WTSN)} Let $\mathfrak{L}$ be any invertible linear transform in (\ref{trans}) and it satisfies (\ref{orth}),
${\boldsymbol{\mathcal{S}}}$ be from the middle component of ${\boldsymbol{\mathcal{A}}} = {\boldsymbol{\mathcal{U}}}{{*}_{\mathfrak{L}}}{\boldsymbol{\mathcal{S}}} {*}_{\mathfrak{L}}
{\boldsymbol{\mathcal{V}}}^{\mit{T}}$. Then,
the weighted tensor Schatten-$p$ norm (WTSN) of ${\boldsymbol{\mathcal{A}}} \in \mathbb{R}^{ n_1\times \cdots \times n_d} $
is defined as
\begin{align*}\label{WTSN}
\| {\boldsymbol{\mathcal{A}}}\|_{ {\boldsymbol{\mathcal{W}}}, {\boldsymbol{\mathcal{S}}}_{p} }
&:= {\Big(\frac{1}{\rho} \big\|\operatorname{bdiag}\big( {\mathfrak{L}}(\boldsymbol{\mathcal{A}} )\big)\big\|_ { {\bm{w}}
,
{\mathrm{S}}_{p} }^{p}\Big)}^{1/p}
\\
&=\Big( \frac{1}{\rho} \sum_{j=1}^{n_3n_4\cdots n_d}
\big\|
{{\mathfrak{L}} (\boldsymbol{\mathcal{A}})}^{< j>}
\big\|_
{{ {\bm{w}}^{(j)},
{\mathrm{S}}_{p}}}^{p}
\Big)^{ {1}/{p} }
\\ &=\Big( \frac{1}{\rho} \sum_{j=1}^{n_3n_4\cdots n_d}
\operatorname{trace}\big( {\boldsymbol{\mathcal{W}}}^{<j>}
\cdot \big| { {\mathfrak{L}}({{\boldsymbol{\mathcal{S}}}})} ^ {<j>}
\big|^{p}\big)
\Big)^{ {1}/{p} }, \end{align*} where $\boldsymbol{\mathcal{W}}$ is the nonnegative weight composed of an order-$d$ f-diagonal tensor, $ \bm{w} = \operatorname{diag} ( \operatorname{bdiag}(\boldsymbol{\mathcal{W}}) )$, $ {\bm{w}^{(j)}} =\operatorname{diag} ( \boldsymbol{\mathcal{W}}^{<j>} )$,
and $\rho=\alpha_3\alpha_4\cdots \alpha_d>0$ is a constant determined by the invertible linear transform $\mathfrak{L}$.
\end{Definition}
\begin{Remark} The high-order WTSN (HWTSN) assigns different weight values to different singular values in the transform domain:
the larger one is multiplied by a smaller weight while the smaller one
is multiplied by a larger weight.
That is,
the weight values should be inversely proportional to the singular values in the transform domain.
In particular, the HWTSN \textbf{1)}
is equivalent to the high-order tensor Schatten-$p$ norm (HTSN) when weighting is not taken into account, \textbf{2)} reduces to the high-order weighted TNN (HWTNN)\cite{qin2021robust} when $p=1$, and \textbf{3)}
simplifies to the high-order TNN (HTNN)\cite{qin2022low} when $p=1$, and $\boldsymbol{\mathcal{W}}$ is not considered.
\end{Remark}
\begin{Theorem}\label{th_app} \textcolor[rgb]{0.00,0.00,0.00}{\textbf{( Optimal
$k$-term approximation \cite{qin2022low})} Let the \text{T-SVD} of ${\boldsymbol{\mathcal{A}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$ be
${\boldsymbol{\mathcal{A}}}={\boldsymbol{\mathcal{U}}}{*}_{\mathfrak{L}}{\boldsymbol{\mathcal{S}}}{*}_{\mathfrak{L}}{\boldsymbol{\mathcal{V}}}^{\mit{T}}$ and define ${\boldsymbol{\mathcal{A}}}_{k}={\sum}_{i=1}^{k}
{\boldsymbol{\mathcal{U}}}(:,i,:,\cdots,:) {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{S}}}(i,i,:,\cdots,:){*}_{\mathfrak{L}}
{\boldsymbol{\mathcal{V}}}(:,i,:,\cdots,:)^{\mit{T}}$ for some $k<\min(n_1,n_2)$. Then, ${\boldsymbol{\mathcal{A}}}_{k}=\arg\min_{
{\tilde{\boldsymbol{{\mathcal{A}}}}} \in \bm{\Theta} }
\|{\boldsymbol{\mathcal{A}}}-{\tilde{\boldsymbol{{\mathcal{A}}}}}\|_{\mathnormal{F}}$, where $ \bm{\Theta}
=\{ {\boldsymbol{\mathcal{X}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{Y}}} | {\boldsymbol{\mathcal{X}}} \in \mathbb{R}^{n_1\times k \times n_3 \times \cdots \times n_d}, {\boldsymbol{\mathcal{Y}}} \in \mathbb{R}^{k\times n_2 \times n_3 \times \cdots \times n_d}\}$.
This
implies that ${\boldsymbol{\mathcal{A}}}_{k}$ is the
approximation of ${\boldsymbol{\mathcal{A}}}$ with the T-SVD rank at most $k$. } \end{Theorem}
\section{\textbf{ Randomized Techniques Based High-Order Tensor Approximation }}\label{approximation}
The optimal $k$-term approximation presented in Theorem \ref{th_app}
is
time-consuming
for large-scale tensors.
To tackle this issue,
an efficient
QB approximation for high-order tensor is developed in virtue of randomized projection techniques.
On this basis, we put forward an effective randomized algorithm for
calculating
the high-order T-SVD (\text{abbreviated as \textbf{R-TSVD}}).
To be slightly more specific,
the calculation
of R-TSVD can be subdivided into the following two steps:
\renewcommand{0.0}{0.1}
\begin{algorithm}[!htbp] \setstretch{0.0}
\caption
{
The Basic \textbf{randQB} Approximation.
}
\label{RTSVD-RANK777}
\KwIn{
$\bm{\mathcal{A}}\in\mathbb{R}^{n_1\times \cdots\times n_d}$, invertible linear transform:
$\mathfrak{L}$, target T-SVD rank: $k$,
oversampling parameter: $s$.
}
{\color{black}\KwOut{
$\bm{\mathcal{Q}},\bm{\mathcal{B}}$.
}}
Set $\hat{l} = k + s$, and generate
a Gaussian random tensor $\bm{\mathcal{G}}\in\mathbb{R}^{n_2\times \hat{l}\times n_3 \times \cdots \times n_d}$\;
Construct
a random projection of
$\bm{\mathcal{A}}$ as
${\boldsymbol{\mathcal{Y}}}={\boldsymbol{\mathcal{A}}}{*}_{\mathfrak{L}} {\boldsymbol{\mathcal{G}}}$\;
Form
the partially orthogonal tensor ${{\boldsymbol{\mathcal{Q}}}}$ by computing the T-QR factorization of $\bm{\mathcal{Y}}$\;
$ {{\boldsymbol{\mathcal{B}}}}= {\boldsymbol{\mathcal{Q}}}^{\mit{T}}{*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{A}}}}$
.
\end{algorithm}
\begin{itemize}
\item \textbf{Step I (Randomized Step):} Compute an approximate basis for the range of the target tensor $\bm{\mathcal{A}}
\in \mathbb{R}^{n_1 \times n_2 \times n_3 \times \cdots \times n_d}
$ via randomized projection
techniques.
That is to say,
we require an orthogonal subspace basis
tensor $\bm{\mathcal{Q}} \in \mathbb{R}^{n_1 \times l \times n_3 \times \cdots \times n_d}
$ which satisfies
\begin{equation}\label{tqb11} {\boldsymbol{\mathcal{A}}}\approx {\boldsymbol{\mathcal{Q}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{B}}}= {\boldsymbol{\mathcal{Q}}}{*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}}^{\mit{T}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{A}}}. \end{equation}
The approximation presented by (\ref{tqb11}) can also be regarded as a kind of low-rank factorization/approximation of ${\boldsymbol{\mathcal{A}}}$, called QB factorization or QB approximation in our work. A basic randomized technique for computing the
QB factorization
is
shown in Algorithm \ref{RTSVD-RANK777},
which is denoted as the basic
randomized QB (randQB)
approximation.
\item \textbf{Step II (Deterministic Step):} Perform the deterministic T-QR factorization on the reduced tensor ${\boldsymbol{\mathcal{B}}}^{\mit{T}}$,
i.e., ${\boldsymbol{\mathcal{B}}}^{\mit{T}}={\boldsymbol{\mathcal{Q}}}_{1} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{R}}}$.
Then, execute the deterministic
T-SVD on the smaller tensor ${\boldsymbol{\mathcal{R}}}$, i.e.,
${\boldsymbol{\mathcal{R}}}=\hat{{\boldsymbol{\mathcal{U}}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{S}}} {*}_{\mathfrak{L}}
{\hat{{\boldsymbol{\mathcal{V}}}}}^{\mit{T}}$. Thus,
\begin{align}\label{tqb} {\boldsymbol{\mathcal{A}}}
&\approx
{\boldsymbol{\mathcal{Q}}} {*}_{\mathfrak{L}} \hat{{\boldsymbol{\mathcal{V}}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{S}}} {*}_{\mathfrak{L}} {\hat{\boldsymbol{\mathcal{U}}}}^{\mit{T}} {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}_{1}}^{\mit{T}}
\notag \\ & = {\boldsymbol{\mathcal{U}}} {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{S}}}} {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{V}}}}^{\mit{T}} (\textit{Let} \; {{\boldsymbol{\mathcal{U}}}}={{\boldsymbol{\mathcal{Q}}}}{*}_{\mathfrak{L}} \hat{{\boldsymbol{\mathcal{V}}}}, {{\boldsymbol{\mathcal{V}}}}={{\boldsymbol{\mathcal{Q}}}_{1}}{*}_{\mathfrak{L}} \hat{{\boldsymbol{\mathcal{U}}}} ) . \end{align}
\end{itemize}
\begin{algorithm}[!htbp] \setstretch{0.01}
\caption{
\text{Transform Domain Version}:
\textbf{R-TSVD}.
}
\label{transform-version-fp11}
\KwIn{$\bm{\mathcal{A}}\in\mathbb{R}^{n_1\times \cdots\times n_d}$,
transform:
$\mathfrak{L}$,
target T-SVD rank: $k$,
oversampling parameter: $s$, power iteration: $t$.
}
{\color{black}\KwOut{ $\bm{\mathcal{U}}$, $\bm{\mathcal{S}}$, $\bm{\mathcal{V}}$.
}}
Set $\hat{l} = k + s$ and initialize a Gaussian random tensor $\bm{\mathcal{G}}\in\mathbb{R}^{n_2\times \hat{l} \times n_3 \times \cdots \times n_d}$\;
Compute the results of $\mathfrak{L}$ on $\boldsymbol{\mathcal{A} }$ and $\boldsymbol{\mathcal{G} }$, i.e.,
$\mathfrak{L}(\boldsymbol{\mathcal{A}}), \mathfrak{L}(\boldsymbol{\mathcal{G}})$\;
\For{$v=1,2,\cdots, n_3\cdots n_d$}
{
$[ \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>},\sim]=\operatorname{qr}(
\mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>}\cdot \mathfrak{L}(\boldsymbol{\mathcal{G}})^{<v>}
)$\;
\For{$ j=1,2,\ldots,t$}
{
$[ \mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>},\sim]=\operatorname{qr}\big(
{( \mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>})}^{\mit{T}}
\cdot \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>}
\big )$\;
$
[ \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>},
\sim]=\operatorname{qr}\big(
\mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>}
\cdot \mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>}
\big)$\;
}
$[\mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>},\mathfrak{L}(\boldsymbol{\mathcal{R}})^{<v>}]= \textrm{qr} \big(
(\mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>})^{\mit{T}}\cdot
\mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>}
\big) $;\\
$[\mathfrak{L}(\hat{\boldsymbol{\mathcal{U}}})^{<v>},\mathfrak{L}(\boldsymbol{\mathcal{S}})^{<v>},\mathfrak{L}(\hat{\boldsymbol{\mathcal{V}}})^{<v>}]= \textrm{svd} \big(\mathfrak{L}(\boldsymbol{\mathcal{R}})^{<v>}\big) $;\\
$
\mathfrak{L}(\boldsymbol{\mathcal{V}})^{<v>} = \mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>}\cdot \mathfrak{L}(\hat{\boldsymbol{\mathcal{U}}})^{<v>} $;\\
$
\mathfrak{L}(\boldsymbol{\mathcal{U}})^{<v>} = \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>}\cdot \mathfrak{L}(\hat{\boldsymbol{\mathcal{V}}})^{<v>} $;\\
} ${\boldsymbol{\mathcal{U}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{U}}})
)$,
${\boldsymbol{\mathcal{S}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{S}}})
)$,
${\boldsymbol{\mathcal{V}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{V}}})
)$.
\end{algorithm}
\begin{Remark} \textbf{(Power Iteration Strategy)} To further improve the accuracy of randQB approximation of ${{\boldsymbol{\mathcal{A}}}}$,
we can additionally apply the power iteration scheme, which multiplies alternately with
${{\boldsymbol{\mathcal{A}}}}$ and ${{{\boldsymbol{\mathcal{A}}}}}^{\mit{T}}$, i.e.,
${({{\boldsymbol{\mathcal{A}}}} {*}_{\mathfrak{L}} {{{\boldsymbol{\mathcal{A}}}}}^{\mit{T}})}^{t}
{*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{A}}}}$,
where $t$ is a nonnegative integer.
Besides, to avoid the rounding error of float point arithmetic obtained from performing the power iteration,
the reorthogonalization step
is required.
Thus,
the randQB approximation algorithm incorporating power iteration strategy
can be obtained by adding the following steps after the third step of Algorithm \ref{RTSVD-RANK777}, i.e.,
\For{$ j=1,2,\ldots,t$}
{
$[ {{\boldsymbol{\mathcal{Q}}}_{1}},\sim]=\operatorname{H-TQR}(
{\boldsymbol{\mathcal{A}}}^{\mit{T}} {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}},
\mathfrak{L})$;\;
$[ {{\boldsymbol{\mathcal{Q}}}},\sim]=\operatorname{H-TQR}(
{\boldsymbol{\mathcal{A}}}{*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}_{1}},
\mathfrak{L})$;\;
} \end{Remark}
\begin{algorithm}[!htbp] \setstretch{0.01}
\caption{
Transform Domain Version:
\textbf{RB-TSVD}.
}
\label{transform-version-fp}
\KwIn{$\bm{\mathcal{A}}\in\mathbb{R}^{n_1\times \cdots\times n_d}$,
transform:
$\mathfrak{L}$,
target T-SVD Rank: $k$,
block size: $b$,
power iteration: $t$.
}
{\color{black}\KwOut{ $\bm{\mathcal{U}}$, $\bm{\mathcal{S}}$, $\bm{\mathcal{V}}$.
}}
Let $\hat{l}$ be a number slightly larger than
$k$, and generate
a Gaussian random tensor $\bm{\mathcal{G}}\in\mathbb{R}^{n_2\times \hat{l}\times n_3 \times \cdots \times n_d}$\;
Compute the results of $\mathfrak{L}$ on $\boldsymbol{\mathcal{A} }$ and $\boldsymbol{\mathcal{G} }$, i.e.,
$\mathfrak{L}(\boldsymbol{\mathcal{A}}), \mathfrak{L}(\boldsymbol{\mathcal{G}})$\;
\For{$v=1,2,\cdots, n_3\cdots n_d$}
{
$ {\mathfrak{L}(\boldsymbol{\mathcal{Q}}) }^{<v>}=[\;\;];\;\;
{\mathfrak{L}(\boldsymbol{\mathcal{B}})}^{<v>}=[\;\;]$\;
\For{$ j=1,2,\ldots,t$}
{
$[ \mathfrak{L}(\boldsymbol{\mathcal{W}})^{<v>},\sim]=\operatorname{qr}(
\mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>}\cdot \mathfrak{L}(\boldsymbol{\mathcal{G}})^{<v>}
)$\;
$
[ \mathfrak{L}(\boldsymbol{\mathcal{G}})^{<v>},
\sim]=\operatorname{qr}\big(
{( \mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>})}^{\mit{T}}\cdot \mathfrak{L}(\boldsymbol{\mathcal{W}})^{<v>}
\big)$\;
}
$ \mathfrak{L}(\boldsymbol{\mathcal{W}})^{<v>}=
\mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>}\cdot \mathfrak{L}(\boldsymbol{\mathcal{G}})^{<v>}$
\;
$\mathfrak{L}(\boldsymbol{\mathcal{H}})^{<v>}=
{( \mathfrak{L}(\boldsymbol{\mathcal{A}})^{<v>})}^{\mit{T}}\cdot \mathfrak{L}(\boldsymbol{\mathcal{W}})^{<v>}$\;
\For{$i=1,2,\cdots,\lfloor\frac{\hat{l}}{ b}\rfloor$}
{
$ \mathfrak{L}\big({\boldsymbol{\mathcal{G}}}^{(i)}\big)^{<v>}=
\mathfrak{L}(\boldsymbol{\mathcal{G}})^{<v>} \big(:,(i-1)b+1:ib \big)$\;
$ \mathfrak{L}\big({\boldsymbol{\mathcal{Y}}}^{(i)}\big)^{<v>} = -
\mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>} \cdot \mathfrak{L}(\boldsymbol{\mathcal{B}})^{<v>} \cdot \mathfrak{L}({\boldsymbol{\mathcal{G}}}^{(i)})^{<v>} +
\mathfrak{L}(\boldsymbol{\mathcal{W}})^{<v>} \big(:,(i-1)b+1:ib \big) $ \;
$[ {\mathfrak{L}({\boldsymbol{\mathcal{Q}}}^{(i)})}^{<v>},\mathfrak{L}({\boldsymbol{\mathcal{R}}}^{(i)})^{<v>}] =\operatorname{qr}\big(\mathfrak{L}({\boldsymbol{\mathcal{Y}}}^{(i)})^{<v>}\big)$\;
$[ {\mathfrak{L}({\boldsymbol{\mathcal{Q}}}^{(i)})}^{<v>}, {( \mathfrak{L}({\hat{\boldsymbol{\mathcal{R}}}}^{(i)} ))}^{<v>} ]=\operatorname{qr}(
{\mathfrak{L}({\boldsymbol{\mathcal{Q}}}^{(i)})}^{<v>} -
{\mathfrak{L}({\boldsymbol{\mathcal{Q}}})}^{<v>}\cdot
{( \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>})}^{\mit{T}}\cdot {\mathfrak{L}({\boldsymbol{\mathcal{Q}}}^{(i)})}^{<v>} )$\;
$ {\mathfrak{L}({\boldsymbol{\mathcal{R}}}^{(i)})}^{<v>}=
{( \mathfrak{L}({\hat{\boldsymbol{\mathcal{R}}}}^{(i)} ))}^{<v>}\cdot {\mathfrak{L}({\boldsymbol{\mathcal{R}}}^{(i)})}^{<v>} $\;
$ {\mathfrak{L}({\boldsymbol{\mathcal{B}}}^{(i)})}^{<v>}=
{\big( \mathfrak{L}( {{\boldsymbol{\mathcal{R}}}^{(i)}})^{<v>}\big)}^{\mit{-T}} \cdot \big[ {\big( {\mathfrak{L}({\boldsymbol{\mathcal{H}}})}^{<v>} (:,(i-1)b+1:ib)\big)}^{\mit{T}} -
{ ({\mathfrak{L}({\boldsymbol{\mathcal{Y}}}^{(i)})}^{<v>}) }^{\mit{T}} \cdot
{\mathfrak{L}({\boldsymbol{\mathcal{Q}}})}^{<v>}\cdot{\mathfrak{L}({\boldsymbol{\mathcal{B}}})}^{<v>} -
{ ({\mathfrak{L}({\boldsymbol{\mathcal{G}}}^{(i)})}^{<v>}) }^{\mit{T}} \cdot
{( \mathfrak{L}(\boldsymbol{\mathcal{B}})^{<v>})}^{\mit{T}} \cdot {\mathfrak{L}({\boldsymbol{\mathcal{B}}})}^{<v>} \big] $\;
$\mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>} =[ \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>}, {\mathfrak{L}({\boldsymbol{\mathcal{Q}}}^{(i)})}^{<v>}]$\; $\mathfrak{L}(\boldsymbol{\mathcal{B}})^{<v>} = [ \mathfrak{L}(\boldsymbol{\mathcal{B}})^{<v>}, {\mathfrak{L}({\boldsymbol{\mathcal{B}}}^{(i)})}^{<v>}]^{\mit{T}}$\;
}
$[\mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>},\mathfrak{L}(\boldsymbol{\mathcal{R}}_{1})^{<v>}]= \textrm{qr} \big(
(\mathfrak{L}(\boldsymbol{\mathcal{B}})^{<v>})^{\mit{T}}
\big) $;\\
$[\mathfrak{L}(\hat{\boldsymbol{\mathcal{U}}})^{<v>},\mathfrak{L}(\boldsymbol{\mathcal{S}})^{<v>},\mathfrak{L}(\hat{\boldsymbol{\mathcal{V}}})^{<v>}]= \textrm{svd} \big(\mathfrak{L}(\boldsymbol{\mathcal{R}}_{1})^{<v>}\big) $;\\
$
\mathfrak{L}(\boldsymbol{\mathcal{V}})^{<v>} = \mathfrak{L}(\boldsymbol{\mathcal{Q}}_{1})^{<v>}\cdot \mathfrak{L}(\hat{\boldsymbol{\mathcal{U}}})^{<v>} $;\\
$
\mathfrak{L}(\boldsymbol{\mathcal{U}})^{<v>} = \mathfrak{L}(\boldsymbol{\mathcal{Q}})^{<v>}\cdot \mathfrak{L}(\hat{\boldsymbol{\mathcal{V}}})^{<v>} $;\\
} ${\boldsymbol{\mathcal{U}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{U}}})
)$,
${\boldsymbol{\mathcal{S}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{S}}})
)$,
${\boldsymbol{\mathcal{V}}} \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{V}}})
)$.
\end{algorithm}
According to the above analysis, the computational procedure of R-TSVD is shown in Algorithm \ref{transform-version-fp11}.
To obtain high performance of linear algebraic computation,
we further
investigate the blocked version of
basic randQB approximation,
and then
derive a randomized blocked algorithm
for computing high-order T-SVD (abbreviated as \textbf{RB-TSVD}, see Algorithm \ref{transform-version-fp}).
\textcolor[rgb]{0.00,0.00,0.00}{\textbf{Owing to the space limitations of this paper,
the detailed derivation of blocked randQB approximation and its induced RB-TSVD are given in the supplementary material.}}
The experimental results shown
in Figure \ref{intro_image22}
indicate that
for various real-world large-scale tensors,
the proposed
RB-TSVD and R-TSVD algorithms achieve up to
$4$$\sim$$8$X
faster running time than
the previous truncated T-SVD
while maintaining
similar accuracy.
Besides, there is no significant difference between RB-TSVD and R-TSVD in
calculation time.
Only for very large-scale tensors (i.e., the spatial dimensions up to about $10^4 \times 10^4$)
does the advantage of RB-TSVD show up,
which is verified in the supplementary materials.
\section{\textbf{Rubust High-Order Tensor Completion}}\label{model}
\subsection{\textbf{Proposed Model}}
In this subsection, we formally introduce
the double nonconvex
model for RHTC,
in which the low-rank component
is constrained by the HWTSN
({see Definition \ref{def10}}),
while the
noise/outlier component
is regularized by its weighted
$\ell_{q}$-norm ({see Table \ref{notation_part1}}).
Specifically,
suppose that we are given a low T-SVD rank
tensor ${\boldsymbol{\mathcal{L}}} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$
corrupted by the noise or outliers. The corrupted part can be represented by the tensor ${\boldsymbol{\mathcal{E}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$.
Here, both ${\boldsymbol{\mathcal{L}}}$ and ${\boldsymbol{\mathcal{E}}}$ are of arbitrary magnitude. We do not know the T-SVD rank of ${\boldsymbol{\mathcal{L}}}$.
Moreover,
we have no idea about the locations of the nonzero entries of ${\boldsymbol{\mathcal{E}}}$, not even how many there are.
Then, the goal of the RHTC problem is to
achieve the reconstruction (either exactly or approximately) of low-rank component ${\boldsymbol{\mathcal{L}}}$ from an observed subset of corrupted tensor ${\boldsymbol{\mathcal{M}}}={\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}}$. Mathematically,
the proposed RHTC model can be formulated
as follows: \begin{equation} \label{orin_nonconvex} \min_{{\boldsymbol{\mathcal{L}}},{\boldsymbol{\mathcal{E}}}}
\| {\boldsymbol{\mathcal{L}}} \|_{{ {\boldsymbol{\mathcal{W}}}_{1}, {\boldsymbol{\mathcal{S}}}_{p} }} ^{p}
+ \lambda\|{\boldsymbol{\mathcal{E}}}\|_{{\boldsymbol{\mathcal{W}}}_{2},\ell_q}^{q},
\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}})= \boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}}), \end{equation} where $0<p,q<1$,
$
\lambda > 0 $ is the penalty parameter,
$\| {\boldsymbol{\mathcal{L}}} \|_{{ {\boldsymbol{\mathcal{W}}}_{1}, {\boldsymbol{\mathcal{S}}}_{p} }} $ denotes the HWTSN of ${\boldsymbol{\mathcal{L}}}$ while
$\|{\boldsymbol{\mathcal{E}}}\|_{{\boldsymbol{\mathcal{W}}}_{2},\ell_q}$
represents the weighted $\ell_q$-norm of ${\boldsymbol{\mathcal{E}}}$,
and ${\boldsymbol{\mathcal{W}}}_{1}$ and ${\boldsymbol{\mathcal{W}}}_{2}$ are the weight tensors, which
will be updated automatically in the subsequent ADMM optimization,
\text{see \ref{optalg} for details.}
\begin{Remark} In the model (\ref{orin_nonconvex}),
the HWTSN not only gives better approximation to the original low-rank assumption, but also considers the importance of different singular components.
Comparing with the previous
regularizers, e.g., HWTNN and
HTNN/HTSN (treat the different
rank components equally),
the proposed one is tighter and more feasible.
Besides, the weighted $\ell_q$-norm
has a superior potential to be sparsity-promoting
in comparison with the $\ell_1$-norm and $\ell_q$-norm.
Therefore, the joint HWTSN and weighted $\ell_q$-norm enables the underlying low-rank structure in the observed tensor ${\boldsymbol{\mathcal{M}}}$ to be well captured, and the robustness against noise/outliers to be well enhanced.
The proposed two nonconvex regularizers
mainly involve several key ingredients: 1) flexible
linear transforms $\mathfrak{L}$;
2)
adjustable parameters $p$ and $q$;
3) automatically updated weight tensors ${\boldsymbol{\mathcal{W}}}_{1}$ and ${\boldsymbol{\mathcal{W}}}_{2}$.
Their various combinations can degenerate to many existing RLRTC models.
\end{Remark}
\subsection{\textbf{HWTSN Minimization Problem}} In this subsection, we mainly present the solution method of HWTSN minimization problem, that is, the method of solving \begin{align}\label{wtsn_prox1}
\arg\min_{ {\boldsymbol{\mathcal{X}}}}
\tau \| {\boldsymbol{\mathcal{X}}}\|_{ { {\boldsymbol{\mathcal{W}}}, {\boldsymbol{\mathcal{S}}}_{p} }}^{p} +
\frac{1}{2}\|{\boldsymbol{\mathcal{X}}}-{\boldsymbol{\mathcal{Z}}}\|^{2}_{\mathnormal{F}}. \end{align}
\noindent Before providing the solution to problem (\ref{wtsn_prox1}), we first introduce the key lemma and definition.
\begin{Lemma} \label{lemm_lp} \cite{zuo2013generalized} For the given $p$ ($0 < p < 1$) and $w > 0$, the optimal solution of the following optimization problem \begin{equation}\label{equ_gst1} \min_{x} w
|x|^{p}
+ \frac{1}{2} {(x-s)}^2 \end{equation} is given by the generalized soft-thresholding (GST)
operator: \begin{align*}\label{lq_shrink}
\hat{x}=\operatorname{ {{GST}} }\big(s,w,p\big) = \begin{cases}
0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{if}\;|s|\leq\delta,\\
\operatorname{sign}(s) {\hat{\alpha}}^{*} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{if}\;|s|>\delta, \end{cases} \end{align*}
where
$\delta={[2w(1-p)]}^{\frac{1}{2-p}}+w p {[2w(1-p)]}^{\frac{p-1}{2-p}}$ is a threshold value, $\operatorname{sign}(s)$ denotes the signum function, and ${\hat{\alpha}}^{*}$ can be obtained by solving
${\alpha}+w p {\alpha}^{p-1}-|s|=0 \; (\alpha>0)$.
\end{Lemma}
\begin{Definition}\label{gtsvt} \textbf{(GTSVT operator)} Let ${\boldsymbol{\mathcal{A}}}={\boldsymbol{\mathcal{U}}}{*}_{\mathfrak{L}}{\boldsymbol{\mathcal{S}}}{*}_{\mathfrak{L}} {\boldsymbol{\mathcal{V}}}^{\mit{T}}$
be the \text{T-SVD} of
${\boldsymbol{\mathcal{A}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$.
For any $\tau>0$, $0<p<1$, then the generalized Tensor Singular Value Thresholding (GTSVT) operator of ${\boldsymbol{\mathcal{A}}}$ is defined as follows \begin{equation}\label{tsvt_operator}
{\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{A}}})
={\boldsymbol{\mathcal{U}}}{*}_{\mathfrak{L}} {\boldsymbol{\mathcal{S}}}_{{\boldsymbol{\mathcal{W}}},p,\tau} {*}_{\mathfrak{L}}{\boldsymbol{\mathcal{V}}}^{\mit{T}}, \end{equation} where
${\boldsymbol{\mathcal{S}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}={\mathfrak{L}^{-1}} \big (GST( {{\mathfrak{L}}({\boldsymbol{\mathcal{S}}})}, \tau {\boldsymbol{\mathcal{W}}},p)\big)$,
${\boldsymbol{\mathcal{W}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$ is
the weight parameter composed of an order-$d$ f-diagonal tensor, and $\operatorname{GST}$
denotes the element-wise shrinkage operator.
\end{Definition}
\begin{Remark} Since the larger singular values usually carry more important information than the smaller ones,
the GTSVT operator
requires to
satisfy:
the larger singular values
in the transform domain
should be shrunk less, while the smaller ones should be shrunk more.
In other words,
the weights are selected
inversely
to the singular values in the transform domain.
Thus,
the original components corresponding to the larger singular values will be less affected.
The GTSVT operator is more flexible than the T-SVT operator proposed in \cite{qin2022low} (shrinks all singular values with the same threshold)
and the WTSVT
operator proposed in \cite{qin2021robust},
and provides more degree of freedom for the approximation to the original problem. \end{Remark}
\begin{Theorem}\label{theorem_svt}
Let $\mathfrak{L}$ be any invertible linear transform in (\ref{trans}) and it satisfies (\ref{orth}), $m=\min{(n_1,n_2)}$. For any $\tau>0$ and ${\boldsymbol{\mathcal{Z}}} \in \mathbb{R}^{n_1\times \cdots \times n_d}$, if the weight parameter satisfies $$ 0\leq {\boldsymbol{\mathcal{W}}}^{<j>}(1,1) \leq \cdots \leq {\boldsymbol{\mathcal{W}}}^{<j>}(m,m),\; \forall j \in [n_3 \cdots n_d], $$
then the GTSVT operator (\ref{tsvt_operator}) obeys \begin{align}\label{eq_svt} {\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{Z}}})= \arg\min_{ {\boldsymbol{\mathcal{X}}}}
\tau \| {\boldsymbol{\mathcal{X}}}\|_{ { {\boldsymbol{\mathcal{W}}}, {\boldsymbol{\mathcal{S}}}_{p} }}^{p} +
\frac{1}{2}\|{\boldsymbol{\mathcal{X}}}-{\boldsymbol{\mathcal{Z}}}\|^{2}_{\mathnormal{F}}. \end{align} \end{Theorem}
\begin{proposition}\label{prop11} Let ${{\boldsymbol{\mathcal{A}}}}= {{\boldsymbol{\mathcal{Q}}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{B}}} \in \mathbb{R}^{n_1 \times n_2 \times \cdots\times n_d}$, where ${{\boldsymbol{\mathcal{Q}}}} \in \mathbb{R}^{n_1 \times k\times \cdots\times n_d}$ is partially orthogonal
and ${{\boldsymbol{\mathcal{B}}}} \in \mathbb{R}^{k \times n_2\times \cdots\times n_d}$. Then, we have
$$ {\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{A}}})={{\boldsymbol{\mathcal{Q}}}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{B}}}). $$
\end{proposition}
\begin{proposition}\label{ppppp6666} Let ${{\boldsymbol{\mathcal{A}}}}= {{\boldsymbol{\mathcal{Q}}}_{1}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{B}}} {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}_{2}^{\mit{T}}} \in \mathbb{R}^{n_1 \times n_2 \times \cdots\times n_d}$, where ${{\boldsymbol{\mathcal{Q}}}_{1}} \in \mathbb{R}^{n_1 \times k\times \cdots\times n_d}$ and ${{\boldsymbol{\mathcal{Q}}}_{2}} \in \mathbb{R}^{n_2 \times k\times \cdots\times n_d}$ are partially orthogonal, ${{\boldsymbol{\mathcal{B}}}} \in \mathbb{R}^{k \times k \times \cdots\times n_d}$. Then, we have
\begin{equation*}
{\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{A}}})={{\boldsymbol{\mathcal{Q}}}_{1}} {*}_{\mathfrak{L}} {\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{B}}}) {*}_{\mathfrak{L}} {{\boldsymbol{\mathcal{Q}}}_{2}^{\mit{T}}}.
\end{equation*}
\end{proposition}
\begin{Remark} From the Definition \ref{gtsvt} and Theorem \ref{theorem_svt},
we can find that the major computational bottleneck of HWTSN minimization problem (\ref{wtsn_prox1})
is to execute
the GTSVT operator involving T-SVD
multiple times.
Based on the previous
Proposition \ref{prop11},\ref{ppppp6666},
we can avoid expensive computation by instead calculating GTSVT on a smaller tensor ${{\boldsymbol{\mathcal{B}}}}$.
In other words, we can efficiently calculate GTSVT operator according to the following two steps: 1) compute two orthogonal subspace basis
tensor $\bm{\mathcal{Q}}_{1}, \bm{\mathcal{Q}}_{2}$ via random
projection techniques;
2) perform the GTSVT operator on a smaller tensor ${{\boldsymbol{\mathcal{B}}}}$.
The computational procedure of GTSVT is shown in Algorithm \ref{RTSVD-RANK12}.
\textbf{What is particularly noteworthy is that the Algorithm \ref{RTSVD-RANK12}
is highly parallelizable because the operations across frontal slices can be readily distributed across different processors. Therefore, additional computational gains can be achieved in virtue of the parallel computing framework. } \end{Remark}
\begin{algorithm}[!htbp]
\setstretch{0.0} \caption{
GST algorithm
\cite{zuo2013generalized}.
}\label{gst}
\KwIn{
$s,w,p,J=3$ or $4$.}
{\color{black}\KwOut{
$\operatorname{ {{GST}} }\big(s,w,p\big)$.}}
${{\delta}}^{GST}_{p}(w)={[2w(1-p)]}^{\frac{1}{2-p}} + wp {{[2w(1-p)]}^{\frac{p-1}{2-p}}}$\;
\eIf{$|s|\leq {{\delta}}^{GST}_{p}(w)$} {
$\operatorname{ {{GST}} }\big(s,w,p\big)=0$ \; }
{$j=0, x^{(j)}=|s|$\;
\For{$ j=0,1,\cdots, J$}
{
$x^{(j+1)}=|s|-wp(x^{(j)})^{p-1}$;\\
$j=j+1$;
}
$
\operatorname{ {{GST}} }\big(s,w,p\big)
=\operatorname{sgn}(s)x^{(j)}$; }
\end{algorithm}
\begin{algorithm}[!htbp] \setstretch{0.0}
\caption{
High-Order GTSVT,
${\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{A}}},\mathfrak{L})$.
}
\label{RTSVD-RANK12}
\KwIn{$\bm{\mathcal{A}}\in\mathbb{R}^{n_1\times \cdots\times n_d}$,
transform: $\mathfrak{L}$, target T-SVD Rank: $k$,
weight tensor:
${\boldsymbol{\mathcal{W}}} \in \mathbb{R}^{n_{1}\times \cdots \times n_d}$,
block size: $b$, $0<p<1$,
$\tau> 0$,
power iteration: $t$.
}
Let $\hat{l}$ be a number slightly larger than $k$
and generate
a Gaussian random tensor $\bm{\mathcal{G}}\in\mathbb{R}^{n_2\times \hat{l}\times n_3 \times \cdots \times n_d}$\;
Compute the results of $\mathfrak{L}$ on $\boldsymbol{\mathcal{A} }$ and $\boldsymbol{\mathcal{G} }$, i.e.,
$\mathfrak{L}(\boldsymbol{\mathcal{A}}), \mathfrak{L}(\boldsymbol{\mathcal{G}})$\;
{
\For{$v=1,2,\cdots, n_3 n_4 \cdots n_d$}
{
\If{ \text{utilize the unblocked randomized technique} } {
Execute
Lines $\textbf{4-12}$ of Algorithm \ref{transform-version-fp11}
to obtain $ {\mathfrak{L}}({\boldsymbol{\mathcal{U}}})^{<v>}$,
$ {\mathfrak{L}}({\boldsymbol{\mathcal{S}}})^{<v>}$, and $ {\mathfrak{L}}({\boldsymbol{\mathcal{V}}})^{<v>}$
\; }
\ElseIf{ \text{utilize the blocked randomized technique} } {
Execute
Lines $\textbf{4-24}$ of Algorithm \ref{transform-version-fp}
to obtain $ {\mathfrak{L}}({\boldsymbol{\mathcal{U}}})^{<v>}$,
$ {\mathfrak{L}}({\boldsymbol{\mathcal{S}}})^{<v>}$, and $ {\mathfrak{L}}({\boldsymbol{\mathcal{V}}})^{<v>}$
\; }
\ElseIf{
\text{not utilize the randomized technique} } {
$[{\mathfrak{L}}({\boldsymbol{\mathcal{U}}})^{<v>},{\mathfrak{L}}({\boldsymbol{\mathcal{S}}})^{<v>},{\mathfrak{L}}({\boldsymbol{\mathcal{V}}})^{<v>}]= \textrm{svd} \big({\boldsymbol{\mathcal{A}}}_{\mathfrak{L}}^{<v>} \big) $
\; }
$ \hat{\bm{S}}
=
\operatorname{ {{GST}} }\big\{ {\operatorname{diag} ( \mathfrak{L}(\boldsymbol{\mathcal{S}})^{<v>} )}, \tau\cdot {\operatorname{diag} ( {{\boldsymbol{\mathcal{W}}}}^{<v>} )}, p \big\} $;\\
${ \mathfrak{L}(\boldsymbol{\mathcal{C}}) }^{<v>}= {\mathfrak{L}(\boldsymbol{\mathcal{U}})}^{<v>}\cdot
\operatorname{diag} ( \hat{\bm{S}})
\cdot { (\mathfrak{L}(\boldsymbol{\mathcal{V}})^{<v>}) } ^{\mit{T}}$ ;\\ } }
{
}
{\color{black}\KwOut{
${\boldsymbol{\mathcal{D}}}_{{\boldsymbol{\mathcal{W}}},p,\tau}({\boldsymbol{\mathcal{A}}},\mathfrak{L}) \leftarrow {\mathfrak{L}}^{-1}(
{\mathfrak{L}}({\boldsymbol{\mathcal{C}}})) $ .}}
\end{algorithm}
\subsection{\textbf{Optimization Algorithm}}\label{optalg} In this subsection, the ADMM framework \cite{boyd2011distributed
} is adopted to solve the proposed model (\ref{orin_nonconvex}).
The nonconvex model (\ref{orin_nonconvex}) can be equivalently reformulated as follows:
\begin{equation}\label{equ_nonconvex}
\min_{{\boldsymbol{\mathcal{L}}},{\boldsymbol{\mathcal{E}}}}
\| {\boldsymbol{\mathcal{L}}} \|_{{ {\boldsymbol{\mathcal{W}}}_{1}, {\boldsymbol{\mathcal{S}}}_{p} }} ^{p}
+ \lambda\|\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})\|_{{\boldsymbol{\mathcal{W}}}_{2},\ell_q}^{q}, \; \; \text{s.t.} \;\; {\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}}= {\boldsymbol{\mathcal{M}}}. \end{equation}
The augmented Lagrangian function of (\ref{equ_nonconvex}) is \begin{align}
\mathcal{F}( {\boldsymbol{\mathcal{L}}},{\boldsymbol{\mathcal{E}}},{\boldsymbol{\mathcal{Y}}},\beta)=
\| {\boldsymbol{\mathcal{L}}} \|_{{ {\boldsymbol{\mathcal{W}}}_{1}, {\boldsymbol{\mathcal{S}}}_{p} }} ^{p}
+ \lambda\|\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})\|_{{\boldsymbol{\mathcal{W}}}_{2},\ell_q}^{q}+ \notag \\ \label{wtsn_admm}
\langle {\boldsymbol{\mathcal{Y}}}, { {\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}}-\boldsymbol{\mathcal{M}}} \rangle + {\beta}/{2}
\| {\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}} -{\boldsymbol{\mathcal{M}}}\|^{2}_{{{\mathnormal{F}}}}, \end{align}
where ${\boldsymbol{\mathcal{Y}}}$ is the dual variable and $\beta$ is the regularization parameter.
The ADMM framework alternately updates each optimization variable until convergence. The iteration template of the ADMM at the ($k+1$)-th iteration is described as follows:
\begin{align} \label{L_hat} {\boldsymbol{\mathcal{L}}}^{k+1} & = \arg \min_{\boldsymbol{\mathcal{L}}} \big\{ \mathcal{F}({\boldsymbol{\mathcal{L}}},{\boldsymbol{\mathcal{E}}}^{k},{\boldsymbol{\mathcal{Y}}}^{k}, \beta^k) \big\}, \\
\label{E_hat} {\boldsymbol{\mathcal{E}}}^{k+1} & = \arg \min_{\boldsymbol{\mathcal{E}} } \;\big\{ \mathcal{F}({\boldsymbol{\mathcal{L}}}^{k+1},{\boldsymbol{\mathcal{E}}},{\boldsymbol{\mathcal{Y}}}^{k}, \beta^k)\big\}, \\
\label{Y_hat} {\boldsymbol{\mathcal{Y}}}^{k+1} &= {\boldsymbol{\mathcal{Y}}}^{k}+\beta^{k}({\boldsymbol{\mathcal{L}}}^{k+1}+{\boldsymbol{\mathcal{E}}}^{k+1}-{\boldsymbol{\mathcal{M}}}),\\
\label{vartheta} \beta^{k+1} &= \min \left( \beta^{\operatorname{max}},\vartheta \beta^k \right), \end{align} \noindent where $\vartheta >1$ is a control constant. Now we solve the subproblem (\ref{L_hat}) and (\ref{E_hat})
explicitly in the ADMM, respectively.
\noindent {\textbf{Update ${\boldsymbol{\mathcal{L}}^{k+1}}$ (low-rank component)}}
The optimization subproblem (\ref{L_hat}) concerning ${\boldsymbol{\mathcal{L}}^{k+1}}$ can be written as \begin{align}\label{L_prox}
\min_{\boldsymbol{\mathcal{L}}}
\big\| {\boldsymbol{\mathcal{L}}} \big\|_{ { { { {\boldsymbol{\mathcal{W}}}_{1}}^{k}}, {\boldsymbol{\mathcal{S}}}_{p}}}^{p} +
{\beta^k}/{2}
\big\|{\boldsymbol{\mathcal{L}}}-{\boldsymbol{\mathcal{M}}}+{\boldsymbol{\mathcal{E}}}^{k} +
{{\boldsymbol{\mathcal{Y}}}^{k}}/{\beta^k}
\big\|^2_{\mathnormal{F}}. \end{align}
Let ${\boldsymbol{\mathcal{G}}}^{k}= {\boldsymbol{\mathcal{M}}}-{\boldsymbol{\mathcal{E}}}^{k} -
{{\boldsymbol{\mathcal{Y}}}^{k}}/{\beta^k}$.
Using the GTSVT algorithm that incorporates the randomized schemes, the subproblem (\ref{L_prox}) can be efficiently solved, i.e., ${\boldsymbol{\mathcal{L}}}^{k+1}= {\boldsymbol{\mathcal{D}}}_{
{{\boldsymbol{\mathcal{W}}}_{1}}^{k},p,\frac{1}{\beta^k}
}({\boldsymbol{\mathcal{G}}}^{k},\mathfrak{L})$.
\begin{Remark}
(\textbf{\textbf{Update \texorpdfstring{${{\boldsymbol{\mathcal{W}}}_{1}}$} \; via reweighting strategy}})
The weight tensor ${\boldsymbol{\mathcal{W}}_{1}}$ can be adaptively tuned at each iteration, and its formula in the $k$-th iteration is given by
\begin{equation*} {({\boldsymbol{\mathcal{W}}_{1}}^{k})} ^{<j>}(i,i)= {\frac{c_{1}} { {
{({ {\boldsymbol{\mathcal{K}}}^{k}})} ^{<j>}(i,i) +\epsilon_{1} }}} \cdot\frac{1}{\beta^{k}} ,\;
\end{equation*}
where $j \in \{1,\cdots,n_3\cdots n_d\}$, $i\in \{1,\cdots,\min( n_1,n_2)\}$, $c_{1} > 0$ is a constant, $\epsilon_{1}$ is a small non-negative constant to avoid division by zero, and the entries on the diagonal of ${({{ {\boldsymbol{\mathcal{K}}} }^{k}})}^{<j>}$
represent the singular values of ${ { {\mathfrak{L}} ( {\boldsymbol{\mathcal{G}}} ^{k} ) }} ^{<j>}$.
In such a reweighted technique, the sparsity performance is enhanced after each iteration and the updated ${{\boldsymbol{\mathcal{W}}}_{1}}^{k}$ satisfy: \begin{equation*} {({\boldsymbol{\mathcal{W}}_{1}}^{k})} ^{<j>}(m,m)
\geq
{({\boldsymbol{\mathcal{W}}_{1}}^{k})} ^{<j>} (n,n)
\geq 0,
\; \forall m\geq n.
\end{equation*}
\end{Remark}
\noindent {\textbf{Update ${\boldsymbol{\mathcal{E}}^{k+1}}$ (noise/outliers component)}} The optimization subproblem (\ref{E_hat}) with respect to $\boldsymbol{\mathcal{E}}^{k+1}$ can be written as \begin{align*}
\min_{
\boldsymbol{\mathcal{E}}}
\lambda\|\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})\|_{{{\boldsymbol{\mathcal{W}}}_{2}}^{k},\ell_q}^{q}
+
{\beta^k}/{2}
\big\| {\boldsymbol{\mathcal{E}}}-{\boldsymbol{\mathcal{M}}}+{\boldsymbol{\mathcal{L}}}^{k+1} +
{{\boldsymbol{\mathcal{Y}}}^{k}} / {\beta^k}
\big\|^2_{\mathnormal{F}}. \end{align*}
Let ${\boldsymbol{\mathcal{H}}}^{k}= {\boldsymbol{\mathcal{M}}}-{\boldsymbol{\mathcal{L}}}^{k+1} -
{{\boldsymbol{\mathcal{Y}}}^{k}}/{\beta^k}$. The above problem can be solved by the following two subproblems
with respect to $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}}^{k+1})$ and $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}}^{k+1})$ , respectively.
Note that the weight tensor ${\boldsymbol{\mathcal{W}}_{2}}$ is updated at each iteration,
and its form at the $k$-th iteration is set as follows:
\begin{equation*} {({\boldsymbol{\mathcal{W}}_{2}}^{k})}(i_1,\cdots,i_d)= {\frac{c_2} { {
\big |{({ {\boldsymbol{\mathcal{H}}}^{k}})} (i_1,\cdots,i_d)\big| +\epsilon_2 }}} \cdot\frac{1}{\beta^{k}} ,\;
\end{equation*} in which $c_{2} > 0$ is a constant, $\epsilon_2>0$ is a small constant to avoid division by zero.
\noindent
\textbf{
Regarding $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}}^{k+1})$
}: the optimization subproblem
with respect to $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}}^{k+1})$
is formulated as following
\begin{align}\label{E_prox_ome}
\min_{
\boldsymbol{\bm{P}}_{{{\Omega}}}(\boldsymbol{\mathcal{E}}) }
\lambda\|\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})\|_{{{\boldsymbol{\mathcal{W}}}_{2}}^{k},\ell_q}^{q}
+
{\beta^k}/{2}
\big\| \boldsymbol{\bm{P}}_{{{\Omega}}}( {\boldsymbol{\mathcal{E}}}-{\boldsymbol{\mathcal{H}}}^{k} )
\big\|^2_{\mathnormal{F}}. \end{align}
\noindent The closed-form solution for subproblem (\ref{E_prox_ome}) can be computed by generalized element-wise shrinkage operator, i.e.,
\begin{align*} \boldsymbol{\bm{P}}_{{{\Omega}}}( {\boldsymbol{\mathcal{E}}}^{k+1})= \operatorname{{GST}} ( \boldsymbol{\bm{P}}_{{{\Omega}}}( {\boldsymbol{\mathcal{H}}}^{k}), {\lambda} \cdot
({\beta^{k}} )^{-1} \cdot \boldsymbol{\bm{P}}_{{{\Omega}}}( {{\boldsymbol{\mathcal{W}}}_{2}}^{k}) ,q
).
\end{align*}
\noindent
\textbf{
Regarding $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}}^{k+1})$}: the optimization subproblem
with respect to $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}}^{k+1})$
is formulated as following
\begin{align}\label{E_prox_ome1} \boldsymbol{\mathrm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}}^{k+1})= \min_{
{\boldsymbol{\bm{P}}}_{{{\Omega}}_{\bot}} (\boldsymbol{\mathcal{E}}) }
{\beta^k/2}
\big\| \boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}( {\boldsymbol{\mathcal{E}}}-{\boldsymbol{\mathcal{H}}}^{k} )
\big\|^2_{\mathnormal{F}}. \end{align}
\noindent The closed-form solution for subproblem (\ref{E_prox_ome1}) can be obtained through the standard least square regression method.
\begin{algorithm}[!htbp] \setstretch{0.0} \caption{
Solve the proposed model (\ref{orin_nonconvex}) by ADMM.
}\label{algorithm1}
\KwIn{
$\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}}) \in \mathbb{R}^{n_1 \times \cdots \times n_d}$, $\mathfrak{L}$,
$\lambda$,
target T-SVD Rank: $k$,
block size: $b$,
power iteration: $t$,
$0<p,q<1$,
$\tau> 0$,
$c_1, c_2, {\epsilon}_1, {\epsilon}_2$.
}
\textbf{Initialize:} ${\boldsymbol{\mathcal{L}}}^{0}={\boldsymbol{\mathcal{E}}}^{0}={\boldsymbol{\mathcal{Y}}}^{0}=\boldsymbol{0}$,
$\vartheta$,
$\beta^0$,
$\beta^{\max}$, $\varpi$, $k=0$\;
\While{\text{not converged}} {
Update ${\boldsymbol{\mathcal{L}}}^{k+1}$ by computing
Algorithm \ref{RTSVD-RANK12};
\\
Update $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}}^{k+1})$ by computing (\ref{E_prox_ome});\\
Update $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}}^{k+1})$ by computing (\ref{E_prox_ome1});\\
Update ${\boldsymbol{\mathcal{Y}}}^{k+1}$ by computing (\ref{Y_hat});\\
Update $\beta^{k+1}$ by computing (\ref{vartheta});\\
Check the convergence conditions \begin{align*}
&\|{\boldsymbol{\mathcal{L}}}^{k+1}-{\boldsymbol{\mathcal{L}}}^{k}\|{_{\infty}}
\leq \varpi,
\;
\|{\boldsymbol{\mathcal{E}}}^{k+1}-{\boldsymbol{\mathcal{E}}}^{k}\|{_{\infty}} \leq \varpi,\\
&\|
\boldsymbol{\boldsymbol{\mathcal{L}}}^{k+1}+
{\boldsymbol{\mathcal{E}}}^{k+1}- {\boldsymbol{\mathcal{M}}}\|{_{\infty}}
\leq \varpi.
\end{align*} } {\color{black}\KwOut{
${\boldsymbol{\mathcal{L}}} \in \mathbb{R}^{n_1 \times \cdots \times n_d}$.
}} \end{algorithm}
\subsection{\textbf{Convergence Analysis}}
In this subsection, we provide a theoretical guarantees for the convergence of the proposed Algorithm \ref{algorithm1},
the detailed proof of which is given in the supplementary material.
\begin{Theorem}\label{conver}
Let $\mathfrak{L}$ be any invertible linear transform in (\ref{trans}) and it satisfies (\ref{orth}), $m=\min{(n_1,n_2)}$.
If the diagonal elements of all matrix frontal slices
on weighted tensor ${{\boldsymbol{\mathcal{W}}}_{1}}^{k}$ are sorted in a non-descending order, i.e.,
$$
{({\boldsymbol{\mathcal{W}}_{1}}^{k})} ^{<j>}(1,1) \cdots \leq {({\boldsymbol{\mathcal{W}}_{1}}^{k})} ^{<j>}(m,m),\; \forall j \in [n_3 \cdots n_d], $$ then the sequences $\{{\boldsymbol{\mathcal{L}}}^{k+1}\}$, $\{{\boldsymbol{\mathcal{E}}}^{k+1}\}$ and $\{{\boldsymbol{\mathcal{Y}}}^{k+1}\}$ generated by Algorithm \ref{algorithm1} satisfy: \begin{align*}
&1)\lim_{k\rightarrow \infty} {\|{\boldsymbol{\mathcal{L}}}^{k+1}-{\boldsymbol{\mathcal{L}}}^{k}\|{_{\mathnormal{F}}}}=0;
\;\; 2)\lim_{k\rightarrow \infty} {\|{\boldsymbol{\mathcal{E}}}^{k+1}-{\boldsymbol{\mathcal{E}}}^{k}\|{_{\mathnormal{F}}}}=0;\\ &3)\lim_{k\rightarrow \infty}{
\|{\boldsymbol{\mathcal{M}}}-{\boldsymbol{\mathcal{L}}}^{k+1}-{\boldsymbol{\mathcal{E}}}^{k+1}\|{_{\mathnormal{F}}}}=0. \end{align*} \end{Theorem}
\subsection{\textbf{Complexity Analysis}} Given an input tensor $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}}) \in \mathbb{R}^{n_1 \times \cdots \times n_d}$, we analyze the per-iteration complexity of Algorithm \ref{algorithm1} with/without randomized techniques.
The per-iteration of Algorithm \ref{algorithm1} needs to update ${\boldsymbol{\mathcal{L}}}$, $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})$, $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}})$, ${\boldsymbol{\mathcal{Y}}}$, respectively. Upadating $\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{E}}})$ requires to perform GST operation with a complexity of
$\boldsymbol{\mathcal{O}}\big( |\Omega| \big)$,
where $|\Omega|$ denotes the cardinality of $\Omega$. $\boldsymbol{\bm{P}}_{{{\Omega}}_{\bot}}({\boldsymbol{\mathcal{E}}})$ and ${\boldsymbol{\mathcal{Y}}}$ can be updated by a low consumed algebraic computation.
The update of ${\boldsymbol{\mathcal{L}}}$ mainly involves
matrix-matrix product, economic QR/SVD decomposition,
linear transforms $\mathfrak{L}(\cdot)$
and its inverse operator $ \mathfrak{L}^{-1}(\cdot)$. Specifically,
for any invertible linear transforms $\mathfrak{L}$,
the per-iteration complexity of ${\boldsymbol{\mathcal{L}}}$
is
\begin{enumerate}
\item $\boldsymbol{\mathcal{O}}\big(\prod_{i=1}^{d}{n_i} \cdot \sum_{j=3}^{d}{n_j} + \hat{l} \cdot \prod_{k=1}^{d}{n_k} \big)$,\; with randomized technique;
\item $\boldsymbol{\mathcal{O}}\big(\prod_{i=1}^{d}{n_i} \cdot \sum_{j=3}^{d}{n_j} + \min \{n_1,n_2\} \cdot \prod_{k=1}^{d}{n_k}\big )$,\; without randomized technique. \end{enumerate}
For some special invertible linear transforms $\mathfrak{L}$, e.g., FFT,
the per-iteration complexity of ${\boldsymbol{\mathcal{L}}}$
is \begin{enumerate}
\item $\boldsymbol{\mathcal{O}}\big(\prod_{i=1}^{d}{n_i} \cdot \sum_{j=3}^{d} \log({n_j})+ \hat{l} \cdot \prod_{k=1}^{d}{n_k}\big)$,\;
with randomized technique;
\item $\boldsymbol{\mathcal{O}}\big(\prod_{i=1}^{d}{n_i} \cdot \sum_{j=3}^{d} \log({n_j})+
\min \{n_1,n_2\} \cdot \prod_{k=1}^{d}{n_k}\big)$,\;
without randomized technique.
\end{enumerate}
It is obvious that
the versions using randomized technique can be advantageous when $\hat{l} \ll \min \{n_1,n_2\}$.
\section{\textbf{EXPERIMENTAL RESULTS}}\label{experiments}
In this section, we perform extensive experiments on both synthetic and real-world tensor data to substantiate the superiority and effectiveness of the proposed approach.
All the experiments are implemented on the platform of Windows 10 and Matlab (R2016a) with an Intel(R) Xeon(R) Gold-5122 3.60GHz CPU and 192GB memory.
\subsection {\textbf{Synthetic Experiments}}
In this subsection, we
mainly perform the efficiency/precision validation and convergence study on the synthetic high-order
tensors, and also compare the obtained results of the proposed method (\textbf{HWTSN+w$\ell_q$})
with the baseline RLRTC method induced by high-order T-SVD framework, i.e., \textbf{HWTNN+$\ell_1$} \cite{qin2021robust}. Two fast versions (i.e., they
incorporate the unblocked and blocked randomized strategies, respectively) of ``\textbf{HWTSN+w$\ell_q$}"
are called \textbf{HWTSN+w$\ell_q$(UR)} and \textbf{HWTSN+w$\ell_q$(BR)}, respectively.
\begin{table*}
\caption{
The CPU time and RelError values
obtained by fourth-order
synthetic tensors restoration.
}
\label{sys_fourth1}
\centering
\renewcommand0.0{0.79}
\setlength\tabcolsep{6pt}
\scriptsize
\begin{tabular}{c cc cc cc cc }
\hline
\multirow{2}{*}{
\text{Algorithm Parameters}
}&
\multicolumn{2}{c }{
HWTNN+$\ell_1$ \cite{qin2021robust}
}&
\multicolumn{2}{c } {HWTSN+w$\ell_q$
}&
\multicolumn{2}{c } {HWTSN+w$\ell_q$(UR)
}&
\multicolumn{2}{c } {HWTSN+w$\ell_q$(BR)
}\\
\cmidrule(rl){2-3} \cmidrule(rl){4-5} \cmidrule(rl){6-7} \cmidrule(rl){8-9}
\qquad&
Time (s) & RelError&Time (s)& RelError&Time (s)& RelError&Time (s)& RelError
\\
\hline
\specialrule{0em}{1pt}{1pt} \hline
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.9$
& \multirow{2}{*}{1119} &\multirow{2}{*}{9.893e$-$8} &1670&4.933e$-$9 &722&4.528e$-$9
&728&4.522e$-$9
\\
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad
&1640&9.179e$-$9 &699&9.325e$-$9
&712& 1.039e$-$8 \\
$\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1230} & \multirow{2}{*}{9.925e$-$8}
&1802&4.949e$-$9 &710&5.189e$-$9
&727& 4.869e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad
&1731&9.218e$-$9 &691&1.042e$-$8
&707& 9.622e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1229} & \multirow{2}{*}{9.085e$-$8} &1784&4.927e$-$9 &706&5.131e$-$9
&720& 5.121e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad
&1749&9.520e$-$9 &700&9.711e$-$9
&716&9.798e$-$9
\\
\hline
\multicolumn{9}{c}{
$N_1=N_2=1000 , N_3=N_4=5$, $R=50, sr=0.5, \tau =0.4$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{844} & \multirow{2}{*}{7.885e$-$8} & 1121&7.208e$-$9 & 462 & 7.395e$-$9 & 456& 7.656e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad
& 1129 & 7.602e$-$9 & 475& 7.843e$-$9
& 454 & 6.756e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{920} & \multirow{2}{*}{7.916e$-$8}
& 1210 &7.259e$-$9 & 452 & 7.785e$-$9
&460 & 7.923e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad
& 1229 &7.419e$-$9 & 474& 6.622e$-$9
& 467 & 6.901e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{921} &\multirow{2}{*}{8.035e$-$8}
&1227 &6.823e$-$9 & 461& 7.978e$-$9
& 452 & 6.965e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.7$ & \qquad&\qquad
& 1126 &8.872e$-$9 & 484 & 6.844e$-$9 &469 & 6.124e$-$9 \\
\hline
\multicolumn{9}{c}{
$N_1=N_2=1000, N_3=N_4=5$, $R=50, sr=0.5, \tau =0.2$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{5141} &\multirow{2}{*}{6.474e$-$8} & 6939 & 3.704e$-$9 & 2234 & 4.051e$-$9 & 2178 & 3.769e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 6696 & 6.926e$-$9
&2191 & 7.619e$-$9
& 2091 & 7.033e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{5871} &\multirow{2}{*}{6.489e$-$8} & 8031 & 3.701e$-$9 & 2853& 4.049e$-$9
& 2775 & 3.766e$-$9\\ $\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 7608 & 6.114e$-$9 & 2673 & 6.909e$-$9
& 2650 & 6.619e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{6137} &\multirow{2}{*}{6.489e$-$8} & 7863 & 3.091e$-$9 & 2731 & 3.672e$-$9 & 2707 & 3.464e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.7$ & \qquad & \qquad
& 7470&6.264e$-$9 & 2610 & 6.933e$-$9 & 2599 &6.672e$-$9
\\
\hline
\multicolumn{9}{c}{
$N_1=N_2=2000 , N_3=N_4=5$, $R=100, sr=0.5, \tau =0.4$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4520} &\multirow{2}{*}{5.588e$-$8} & 4745 &4.872e$-$9 & 1494 & 5.236e$-$9
& 1468 & 5.027e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 4685 & 5.022e$-$9 & 1477 & 5.739e$-$9
& 1461 & 5.488e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4178} &\multirow{2}{*}{5.396e$-$8} & 4509 & 5.358e$-$9 & 1506& 5.959e$-$9
& 1491 & 5.749e$-$9
\\ $\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 4412 &5.121e$-$9 & 1488 & 5.749e$-$9
& 1481 & 5.591e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4143} &\multirow{2}{*}{5.387e$-$8} & 4399 &5.138e$-$9 & 1486 & 6.017e$-$9
& 1457 & 5.932e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 4385 & 4.971e$-$9 & 1483& 5.704e$-$9
& 1470 & 5.369e$-$9 \\
\hline
\multicolumn{9}{c}{
$N_1=N_2=2000, N_3=N_4=5$, $R=100, sr=0.5, \tau =0.2$
}
\\ \end{tabular} \end{table*}
\begin{table*}
\caption{
The CPU time and RelError values
obtained by fifth-order
synthetic tensors restoration.
}
\label{sys_fifth1}
\centering
\renewcommand0.0{0.79}
\setlength\tabcolsep{6.0pt}
\footnotesize \scriptsize
\begin{tabular}{c cc cc cc cc }
\hline
\multirow{2}{*}{
\text{Algorithm Parameters}
}&
\multicolumn{2}{c }{
HWTNN+$\ell_1$ \cite{qin2021robust}
}&
\multicolumn{2}{c } {HWTSN+w$\ell_q$
}&
\multicolumn{2}{c } {HWTSN+w$\ell_q$(UR)
}
&
\multicolumn{2}{c } {HWTSN+w$\ell_q$(BR)
}
\\
\cmidrule(rl){2-3} \cmidrule(rl){4-5} \cmidrule(rl){6-7} \cmidrule(rl){8-9}
\qquad&
Time (s)& RelError&Time (s)& RelError&Time (s)& RelError&Time (s)& RelError
\\
\hline
\specialrule{0em}{1pt}{1pt} \hline
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1286} &\multirow{2}{*}{9.368e$-$8} & 1902 &4.329e$-$9 & 866 & 4.699e$-$9
& 859 & 4.673e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad & 1871 & 8.855e$-$9 & 856 & 1.001e$-$8
& 841 & 9.612e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1424} &\multirow{2}{*}{9.465e$-$8} & 2057 & 5.085e$-$9 & 899 & 4.674e$-$9
& 889 & 4.349e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad & 1980 & 8.539e$-$9 & 886 & 1.032e$-$8
& 866 & 9.428e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1422} &\multirow{2}{*}{9.454e$-$8} & 2069 &4.731e$-$9 & 911 & 4.641e$-$9 & 883 & 4.890e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.7$ & \qquad & \qquad
& 2006&9.273e$-$9 & 982 & 6.675e$-$9 & 947 &4.467e$-$9
\\
\hline
\multicolumn{9}{c}{
$N_1=N_2=1000 , N_3=N_4=N_5=3$, $R=50, sr=0.5, \tau =0.4$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{979} &\multirow{2}{*}{7.865e$-$8} & 1300 & 6.655e$-$9 & 561 & 7.111e$-$9 & 558 & 7.405e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=50, t=1,p=q=0.7$
& \qquad & \qquad & 1301 &6.623e$-$9 & 586 & 7.587e$-$9 & 583 & 7.526e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.9$ & \multirow{2}{*}{1082} &\multirow{2}{*}{7.926e$-$8} & 1411 & 7.174e$-$9 & 582 & 6.688e$-$9 & 577 & 6.746e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad & 1397 & 8.348e$-$9 & 601 & 7.113e$-$9
& 594 & 7.852e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.9$
& \multirow{2}{*}{1088} &\multirow{2}{*}{7.516e$-$8} & 1415 & 7.319e$-$9 & 582 & 6.856e$-$9
& 572 & 6.929e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=50, t=1,p=q=0.7$ & \qquad &\qquad & 1398 & 7.638e$-$9 & 603 & 7.577e$-$9
& 592 & 6.734e$-$9 \\
\hline
\multicolumn{9}{c}{
$N_1=N_2=1000, N_3=N_4=N_5=3$, $R=50, sr=0.5, \tau =0.2$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{5787} &\multirow{2}{*}{6.266e$-$8} & 7789 & 3.062e$-$9 & 2712 & 3.648e$-$9
& 2635 & 3.487e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 7402 & 5.899e$-$9 & 2564 & 6.492e$-$9
&2525 & 6.217e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{5626} &\multirow{2}{*}{6.248e$-$8} & 7603 & 3.041e$-$9 & 2801 & 3.572e$-$9
& 2770 & 3.241e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 7194 & 6.504e$-$9 & 2647 & 7.001e$-$9
& 2621 & 6.898e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{5615} &\multirow{2}{*}{6.395e$-$8}
& 7596 &3.256e$-$9 & 2788 & 3.811e$-$9 & 2764& 3.533e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.7$ & \qquad & \qquad
& 7301 &5.219e$-$9 & 2691 & 5.864e$-$9 & 2669 &5.790e$-$9
\\
\hline
\multicolumn{9}{c}{
$N_1=N_2=2000 , N_3=N_4=N_5=3$, $R=100, sr=0.5, \tau =0.4$
}
\\
\specialrule{0em}{2pt}{2pt} \hline
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4797} &\multirow{2}{*}{5.243e$-$8} & 5210 & 4.995e$-$9 & 1740 & 5.518e$-$9 & 1715 & 5.305e$-$9 \\
$\mathfrak{L}=\text{FFT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 5117 & 5.236e$-$9 & 1726 & 5.769e$-$9
& 1683 & 5.458e$-$9 \\
$\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4636} &\multirow{2}{*}{5.237e$-$8} & 4669 & 5.388e$-$9 & 1646 & 6.104e$-$9
& 1619 & 5.636e$-$9 \\ $\mathfrak{L}=\text{DCT}, k=100, t=1,p=q=0.7$ & \qquad &\qquad & 4596& 5.019e$-$9 & 1642 & 5.887e$-$9
& 1615 & 5.451e$-$9 \\
$\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.9$ & \multirow{2}{*}{4342} &\multirow{2}{*}{4.893e$-$8} & 4658& 5.007e$-$9 & 1669 & 5.705e$-$9
& 1665 & 5.362e$-$9 \\ $\mathfrak{L}=\text{ROT}, k=100, t=1,p=q=0.7$ & \qquad&\qquad & 4637 & 4.899e$-$9 & 1683 & 5.310e$-$9
& 1657 & 5.154e$-$9 \\
\hline
\multicolumn{9}{c}{
$N_1=N_2=2000, N_3=N_4=N_5=3$, $R=100, sr=0.5, \tau =0.2$
}
\\
\end{tabular} \end{table*}
In our synthetic experiments,
the ground-truth low T-SVD rank tensor ${{\boldsymbol{\mathcal{L}}}}$ with ${\operatorname{rank}}_{tsvd} ({{\boldsymbol{\mathcal{L}}}}) = R$ is generated by performing the order-$d$ t-product ${{\boldsymbol{\mathcal{L}}}}={{\boldsymbol{\mathcal{L}}}_{1}}{*}_{\mathfrak{L}}{{\boldsymbol{\mathcal{L}}}}_{2}$, where the entries of ${{\boldsymbol{\mathcal{L}}}}_{1} \in \mathbb{R}^{N_1\times R \times N_3\times \cdots \times N_d}$ and ${{\boldsymbol{\mathcal{L}}}}_{2} \in \mathbb{R}^{R\times N_2 \times N_3 \times \cdots \times N_d}$ are independently sampled from the normal distribution ${\mathcal{N}}(0,1)$.
\textcolor[rgb]{0.00,0.00,0.00}{Three invertible linear transforms $\mathfrak{L}$ are adopted to the t-product: (a) Fast Fourier Transform (FFT); (b) Discrete Cosine Transform (DCT); (c) Random Orthogonal Transform (ROT)}.
Suppose that ${\Omega}$ is the observed index set that is generated uniformly at random while ${{\Omega}_{\bot}}$ is the unobserved index set, in which $|\Omega| =sr \cdot \prod_{a=1}^{d} N_a $, $sr$ denotes the sampling ratio,
$|\Omega|$ represents the cardinality of $\Omega$.
Then,
we construct the noise/outliers
tensor ${{\boldsymbol{\mathcal{E}}}} \in \mathbb{R}^{N_1 \times N_2 \times N_3 \times \cdots \times N_d}$
as follows:
\textbf{1)} all the elements in
${{\Omega}_{\bot}}$ are all equal to $0$;
\textbf{2)}
the $\tau \cdot |\Omega|$ elements randomly selected in ${\Omega}$ are each valued as
$\{\pm1\}$ with equal probability $\frac{1}{2}$, and the remaining elements in ${\Omega}$ are set to $0$.
Finally, we
form the observed tensor as
$\boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}})= \boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{L}}}+{\boldsymbol{\mathcal{E}}})$.
We evaluate the restoration performance by CPU time and Relative Error (RelError) defined as $$ \operatorname{RelError }:=
{\|{\boldsymbol{\mathcal{L}}}- \hat{{\boldsymbol{\mathcal{L}}}}
\|_{{{\mathnormal{F}}}}} / {\|{\boldsymbol{\mathcal{L}}}\|_{{{\mathnormal{F}}}}},
$$ where $\hat{{\boldsymbol{\mathcal{L}}}}$ denote the estimated result of the ground-truth ${{\boldsymbol{\mathcal{L}}}}$.
\subsubsection {\textbf{Efficiency/Precision Validation}} \label{fourth_sys}
Firstly, we verify the accuracy/effectiveness of the proposed algorithm as well as the compared ones on
the following two types of
synthetic tensors:
\textbf{(I)}
$N_1=N_2=N, N \in \{1000,2000\}, N_3=N_4=5$;
\textbf{(II)}
$N_1=N_2=N, N \in \{1000,2000\},
N_3=N_4=N_5=3$.
In our experiments,
we set
$R=0.05 \cdot\min(N_1,N_2), sr=0.5, \tau \in \{0.4,0.2\}$,
$p,q \in \{0.9,0.7\}$, $\mathfrak{L} \in \{\text{FFT}, \text{DCT}, \text{ROT}\}$,
$
b=\lfloor\frac{ k+5}
{ 3}\rfloor,
t=1$, $\vartheta=1.1, \beta^0=10^{-3},
\beta^{\max}=10^{8}, \varpi=10^{-6},\epsilon_{1}=\epsilon_{2}=10^{-16}$, $c_1=\alpha \cdot\min(N_1,N_2)$,
$\alpha \in
\{5,10,15,20,25\}
$, $c_2=1$,
and $\lambda \in \{0.02, 0.03, 0.05,0.08\}$.
The experimental results are presented
in Table \ref{sys_fourth1} and Table \ref{sys_fifth1}, from which we can observed that
the RelError
values obtained
from the proposed method
are relatively small in all case,
which indicates that the proposed algorithm can accurately complete the latent low T-SVD rank tensor ${{\boldsymbol{\mathcal{L}}}}$ while removing the noise/outliers.
Besides,
the versions integrated with
randomized techniques can greatly
shorten the computational time
under different linear transforms $\mathfrak{L}$.
\begin{figure}
\caption{
The convergence behavior of the proposed and competitive RHTC
algorithms.
The x-coordinate is
the number of iterations,
the y-coordinates are the sequence
Chg$1$-Chg$3$.
}
\label{conver_robust11}
\end{figure}
\subsubsection {\textbf{Convergence Study}} Secondly,
we mainly analyze the convergence behavior of the proposed and competitive algorithms
on the following synthetic tensor: $N_1=N_2=1000, N_3=N_4=N_5=3, R=50, sr = 0.5, \tau=0.4$.
The
parameter settings of the proposed algorithm are the same as those utilized in the previous experiments. Then, we record three type values, i.e.,
$\operatorname{Chg1}:={\|{\boldsymbol{\mathcal{L}}}^{k+1}-{\boldsymbol{\mathcal{L}}}^{k}\|{_{\mathnormal{F}}}}$,
$\operatorname{Chg2}:={\|{\boldsymbol{\mathcal{E}}}^{k+1}-{\boldsymbol{\mathcal{E}}}^{k}\|{_{\mathnormal{F}}}}$,
$\operatorname{Chg3}:={\|{\boldsymbol{\mathcal{M}}}-{\boldsymbol{\mathcal{L}}}^{k+1}-{\boldsymbol{\mathcal{E}}}^{k+1}\|{_{\mathnormal{F}}}}$, obtained by various RHTC algorithms at the $k$-th iteration, respectively.
The recorded results are plotted in Figure \ref{conver_robust11}, which is exactly consistent with the Theorem \ref{conver},
i.e.,
the obtained $\operatorname{Chg1}$, $\operatorname{Chg2}$, and $\operatorname{Chg3}$
gradually approach $0$
when the proposed
algorithm
iterates to a certain number of times.
\subsection{\textbf{Real-World Applications}}
\textbf{Experiment Settings:}
In this subsection,
we apply the proposed method (\textbf{HWTSN+w$\ell_q$}) and its two accelerated versions
to several real-world applications,
and also compare it
with other RLRTC approaches:
SNN+$\ell_1$\cite{ goldfarb2014robust1 },
TRNN+$\ell_1$ \cite{huang2020robust1},
TTNN+$\ell_1$ \cite{song2020robust}, TSP-$k$+$\ell_1$ \cite{lou2019robust1},
LNOP \cite{chen2020robust1}, NRTRM \cite{qiu2021nonlocal},
and
HWTNN+$\ell_1$ \cite{qin2021robust}.
In our experiments, we
normalize the gray-scale
value of the
tested tensors to the interval
$[0, 1]$.
For the RLRTC methods based on third-order T-SVD, we reshape the last two or three modes of tested tensors into one mode.
The observed tensor is constructed as follows:
the random-valued impulse noise with ratio $\tau$ is uniformly and randomly
added to each frontal slice of the ground-truth
tensor,
and then we
sample ($sr \cdot \prod_{i=1}^{d} n_i$) pixels from the noisy tensor to form the observed tensor
$ \boldsymbol{\bm{P}}_{{{\Omega}}}({\boldsymbol{\mathcal{M}}})$ at random.
Unless otherwise stated, all parameters involved in the competing methods were optimally assigned or selected as suggested in the reference papers.
The Peak Signal-to-Noise Ratio (PSNR),
the structural similarity (SSIM), and the CPU time are employed to evaluate the recovery performance.
\begin{figure*}
\caption{ Visual comparison of various methods for LFIs recovery. From top to bottom, the parameter pair $(sr, \tau)$ are $(0.05,0.5)$, $(0.1,0.5)$, $(0.05,0.3)$ and $(0.1,0.3)$, respectively. Top row:
the $(6,6)$-th frame of Bench.
The second row: the $(9,9)$-th frame of Bee-1. The third row: the $(12,12)$-th frame of Mini.
Bottom row: the $(15,15)$-th frame of Framed. }
\label{fig_visual_lfi}
\end{figure*}
\begin{table*}[htbp]
\caption{
The PSNR, SSIM values and CPU time
obtained by various RLRTC methods for different fifth-order LFIs.
The best and the second-best results are highlighted in blue and red, respectively.
}
\label{lfi_index}
\centering \scriptsize
\renewcommand{0.0}{0.88} \setlength\tabcolsep{4.5pt}
\begin{tabular}{c c ccccccc cccc cccc |c }
\hline
\multicolumn{1}{c}{LFI-Name}
& \multicolumn{4}{|c|}{Bench} &\multicolumn{4}{c|}{Bee-1} &\multicolumn{4}{c|}{Framed} &\multicolumn{4}{c|}{Mini}
& \multirow{3}{*}{
\tabincell{c}{Average\\Time (s)}
}\\
\cline{1-1}
\cline{2-5}
\cline{6-9}
\cline{10-13}
\cline{14-17}
\multicolumn{1}{c}{$sr$}
&\multicolumn{2}{|c|}{$5\%$} &\multicolumn{2}{c|}{$10\%$} &
\multicolumn{2}{c|}{$5\%$}&\multicolumn{2}{c|}{$10\%$}&
\multicolumn{2}{c|}{$5\%$}&\multicolumn{2}{c|}{$10\%$}&
\multicolumn{2}{c|}{$5\%$}&\multicolumn{2}{c|}{$10\%$}\\
\cline{1-1}
\cline{2-5}
\cline{6-9}
\cline{10-13}
\cline{14-17}
\multicolumn{1}{c}{$\tau$}
&
\multicolumn{1}{|c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$}
&
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$}
\\
\hline
\hline
\multicolumn{1}{c}{SNN+$\ell_1$}
&
\tabincell{c}{ 14.29 \\ 0.176 } & \tabincell{c}{11.94\\ 0.091 }& \tabincell{c}{ 14.81 \\ 0.267 } & \tabincell{c}{ 13.69\\ 0.168}&
\tabincell{c}{12.72 \\ 0.137 } & \tabincell{c}{ 11.07 \\ 0.081 }& \tabincell{c}{ 13.39 \\0.179 } & \tabincell{c}{ 11.74\\ 0.128}&
\tabincell{c}{ 12.76 \\0.203 } & \tabincell{c}{ 11.47 \\0.112 }& \tabincell{c}{ 13.99 \\ 0.269 } & \tabincell{c}{11.67\\ 0.179}&
\tabincell{c}{ 14.29 \\ 0.195 } & \tabincell{c}{ 12.92 \\0.132 }& \tabincell{c}{15.99 \\0.245 } & \tabincell{c}{15.37\\ 0.197 }
&
10871 \\
\hline
\multicolumn{1}{c}{TRNN+$\ell_1$}
&
\tabincell{c}{ 18.16 \\0.446 } & \tabincell{c}{ 14.89 \\ 0.312} & \tabincell{c}{21.27 \\ 0.574 }& \tabincell{c}{ 17.29\\ 0.337}&
\tabincell{c}{ 18.04 \\0.356 } & \tabincell{c}{ 13.71 \\ 0.224} & \tabincell{c}{22.74 \\0.474 }&\tabincell{c}{ 16.66\\ 0.316}&
\tabincell{c}{ 19.07 \\ 0.511 } & \tabincell{c}{ 14.02 \\ 0.355 } & \tabincell{c}{23.07 \\ 0.643 }& \tabincell{c}{17.32\\ 0.469 }&
\tabincell{c}{ 20.83 \\0.483 } & \tabincell{c}{ 16.16 \\0.394 } & \tabincell{c}{ 25.25 \\0.658 }& \tabincell{c}{19.72\\0.408 }
&
9111 \\
\hline
\multicolumn{1}{c}{TTNN+$\ell_1$}
&
\tabincell{c}{ 21.65 \\0.704 } & \tabincell{c}{ 16.48 \\ 0.434 } & \tabincell{c}{23.04 \\ 0.746 }&\tabincell{c}{ 20.72\\0.608 }&
\tabincell{c}{ 22.44 \\ 0.776 } & \tabincell{c}{ 16.24 \\0.353 } & \tabincell{c}{ 24.74 \\ 0.831 }& \tabincell{c}{20.75\\ 0.464 }&
\tabincell{c}{ 23.33 \\0.797 } & \tabincell{c}{ 16.95 \\ 0.516 } & \tabincell{c}{ 25.37 \\ 0.857 }& \tabincell{c}{ 22.13\\0.667 }&
\tabincell{c}{25.67 \\ 0.775 } & \tabincell{c}{ 17.51 \\ 0.486 } & \tabincell{c}{ 27.45 \\ 0.818 }& \tabincell{c}{ 22.92\\ 0.638 } & 4708 \\
\hline
\multicolumn{1}{c}{TSP-$k$+$\ell_1$}
&
\tabincell{c}{22.33 \\0.687 } & \tabincell{c}{17.43 \\0.415 } & \tabincell{c}{24.02 \\ 0.785 }& \tabincell{c}{21.21 \\0.611 }&
\tabincell{c}{ 22.88\\0.734 } & \tabincell{c}{ 17.08\\0.333 } & \tabincell{c}{25.62 \\ 0.843 }&
\tabincell{c}{21.32 \\ 0.456}&
\tabincell{c}{ 23.43\\0.772 } & \tabincell{c}{ 17.81 \\0.491 } &
\tabincell{c}{ 25.67 \\ 0.863 }& \tabincell{c}{22.24\\ 0.661}&
\tabincell{c}{26.36 \\0.757 } & \tabincell{c}{ 18.04 \\0.489 } &
\tabincell{c}{ 29.02 \\ 0.857 }&
\tabincell{c}{ 23.93\\ 0.663} &6798 \\
\hline
\multicolumn{1}{c}{LNOP}
&
\tabincell{c}{ 21.87 \\0.628 } & \tabincell{c}{ 18.35 \\ 0.436 } & \tabincell{c}{ 27.02 \\ 0.813 }& \tabincell{c}{ 21.84\\0.649 }&
\tabincell{c}{ 22.46 \\0.658 } & \tabincell{c}{ 18.08 \\ 0.348 } & \tabincell{c}{28.37 \\ 0.879 }& \tabincell{c}{ 21.87\\ 0.512}&
\tabincell{c}{ 22.88 \\0.674 } & \tabincell{c}{ 19.05 \\ 0.493 } & \tabincell{c}{ 28.83 \\0.839 }& \tabincell{c}{22.79\\ 0.637}&
\tabincell{c}{ 24.58 \\ 0.658 } & \tabincell{c}{ 20.97 \\ 0.529 } & \tabincell{c}{ 30.89 \\ 0.853 }& \tabincell{c}{ 25.73\\ 0.725}
&
6465 \\
\hline
\multicolumn{1}{c}{NRTRM}
&
\tabincell{c}{ 23.67 \\ 0.764 } & \tabincell{c}{ 19.41 \\ 0.458}& \tabincell{c}{ 26.41 \\ 0.838} & \tabincell{c}{22.63 \\ 0.703}&
\tabincell{c}{ 25.46 \\ 0.834 } & \tabincell{c}{19.01\\ 0.362}& \tabincell{c}{ 28.89 \\ 0.891} & \tabincell{c}{ 23.73 \\0.599 }&
\tabincell{c}{ 26.05 \\ 0.864 } & \tabincell{c}{20.53\\ 0.541 }& \tabincell{c}{ 29.11\\ 0.927} & \tabincell{c}{24.84\\ 0.789 }&
\tabincell{c}{ 28.30 \\0.829 } & \tabincell{c}{20.31\\ 0.518}& \tabincell{c}{ 31.92 \\ 0.913 } & \tabincell{c}{ 26.59\\0.773 } &6267 \\
\hline
\multicolumn{1}{c}{HWTNN+$\ell_1$}
&
\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00} {25.63}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.808}} } & \tabincell{c}{ 21.94 \\ 0.543 }
&\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{27.77}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.857}}} & \tabincell{c}{ 23.95 \\ 0.705 } &
\tabincell{c}{ 28.24 \\ 0.839 } & \tabincell{c}{ 22.29 \\ 0.407 } &\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{30.53}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.896}}
} & \tabincell{c}{ 25.25 \\ 0.564 } &
\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{28.45}} \\
{\textcolor[rgb]{1.00,0.00,0.00}{0.893}} } & \tabincell{c}{ 23.03 \\ 0.583 }
&\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{31.08}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.931}}} & \tabincell{c}{ 26.39 \\ 0.767 } &
\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{30.89}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.872}} } & \tabincell{c}{ 25.41 \\ 0.629 }
&\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{33.29}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.919}} } & \tabincell{c}{ 28.44 \\ 0.776 } &
5754
\\
\hline
\multicolumn{1}{c}{\textbf{HWTSN+w$\ell_q$}}
&
\tabincell{c}{{\textcolor[rgb]{0,0,1}{26.85}} \\ \textcolor[rgb]{0,0,1}{0.834} } & \tabincell{c}{ \textcolor[rgb]{0,0,1} {23.44} \\ \textcolor[rgb]{0,0,1} {0.671} }& \tabincell{c}{ \textcolor[rgb]{0,0,1}{30.13} \\ \textcolor[rgb]{0,0,1}{0.892} } & \tabincell{c}{ \textcolor[rgb]{0,0,1}{26.29}\\ \textcolor[rgb]{0,0,1}{0.808} }&
\tabincell{c}{ \textcolor[rgb]{0,0,1}{{29.56}} \\ \textcolor[rgb]{0,0,1}{0.889} } & \tabincell{c}{ \textcolor[rgb]{0,0,1}{25.28} \\ \textcolor[rgb]{0,0,1}{0.651} }& \tabincell{c}{ \textcolor[rgb]{0,0,1}{33.03} \\ \textcolor[rgb]{0,0,1}{0.938} } & \tabincell{c}{ {27.69}\\ {0.723} }&
\tabincell{c}{ \textcolor[rgb]{0,0,1}{29.95} \\ \textcolor[rgb]{0,0,1}{0.916} } & \tabincell{c}{ \textcolor[rgb]{0,0,1}{25.55} \\ \textcolor[rgb]{0,0,1}{0.743} }& \tabincell{c}{ \textcolor[rgb]{0,0,1}{33.78} \\ \textcolor[rgb]{0,0,1}{0.961} } & \tabincell{c}{\textcolor[rgb]{0,0,1}{29.89}\\ \textcolor[rgb]{0,0,1}{0.915} }&
\tabincell{c}{ \textcolor[rgb]{0,0,1}{32.25} \\\textcolor[rgb]{0,0,1}{0.902} } & \tabincell{c}{ \textcolor[rgb]{0,0,1}{28.99} \\ \textcolor[rgb]{0,0,1}{0.766} }& \tabincell{c}{ \textcolor[rgb]{0,0,1}{35.82} \\ \textcolor[rgb]{0,0,1}{0.952} } & \tabincell{c}{ \textcolor[rgb]{0,0,1}{32.78} \\ \textcolor[rgb]{0,0,1}{0.898}}
&
7080 \\
\hline \multicolumn{1}{c}{
\tabincell{c}{\textbf{HWTSN+w$\ell_q$(BR)}}}
&
\tabincell{c}{24.83 \\0.743 } & \tabincell{c}{ 22.86 \\ 0.645 }& \tabincell{c}{ 25.36 \\0.746 } & \tabincell{c}{24.62\\ 0.721}&
\tabincell{c}{ 28.16 \\ 0.846 } & \tabincell{c}{ 24.85 \\ 0.661 }& \tabincell{c}{28.69 \\0.874 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{27.79}}\\ {\textcolor[rgb]{1.00,0.00,0.00}{0.847}} }&
\tabincell{c}{28.15 \\0.863 } & \tabincell{c}{ 25.25 \\ 0.735 }& \tabincell{c}{29.01 \\0.884 } & \tabincell{c}{27.98\\ 0.868}&
\tabincell{c}{ 29.39 \\0.808 } & \tabincell{c}{ 27.29 \\0.723 }& \tabincell{c}{ 29.55 \\0.817 } & \tabincell{c}{29.19\\ 0.811} &
\textcolor[rgb]{0.00,0.00,1.00}{2869}
\\
\hline \multicolumn{1}{c}{
\tabincell{c}{\textbf{HWTSN+w$\ell_q$(UR)}}} &
\tabincell{c}{ 24.92 \\0.741 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{22.96}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.649}} }& \tabincell{c}{25.46 \\0.749 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{24.78}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.728}}}&
\tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{28.74}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.846}} } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{24.94}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.663}}
}& \tabincell{c}{28.76 \\ 0.874 } & \tabincell{c}{ {\textcolor[rgb]{0,0,1}{27.88}}\\ {\textcolor[rgb]{0,0,1}{0.848}} }&
\tabincell{c}{28.23 \\ 0.864 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{25.28}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.736}}}& \tabincell{c}{29.19 \\ 0.886 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{28.29}}\\ {\textcolor[rgb]{1.00,0.00,0.00}{0.869}}}&
\tabincell{c}{29.46 \\ 0.809 } & \tabincell{c}{ {\textcolor[rgb]{1.00,0.00,0.00}{27.38}} \\ {\textcolor[rgb]{1.00,0.00,0.00}{0.725}}}& \tabincell{c}{ 29.67 \\0.816} & \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{29.25}\\\textcolor[rgb]{1.00,0.00,0.00}{0.812}} &
\textcolor[rgb]{1.00,0.00,0.00}{2944}
\\
\hline
\end{tabular}
\begin{tablenotes} \item[**]
In each RLRTC method, the top represents the PSNR values while the bottom denotes the SSIM values.
\end{tablenotes}
\end{table*}
\textbf{Parameters Setting:}
The transform in TTNN+$\ell_1$, NRTRM, HWTNN+$\ell_1$, and {HWTSN+w$\ell_q$}
is set as the FFT for consistency.
The parameter {\bf{$\lambda$}} of \textbf{SNN+$\ell_1$} is set as
$\textbf{$\lambda$}\in \{[10,10,1,1]*\alpha, [10,10,1,1,1]*\alpha\}$,
$\alpha \in \{1, 3,5,8,10\}$.
For \textbf{TRNN+$\ell_1$}, we set $\lambda \in \{0.018, 0.015, 0.012,0.01, 0.009,0.007\}$.
For \textbf{TTNN+$\ell_1$},
we set $\lambda=\varsigma/ (\max{(n_1,n_2)} \cdot \prod_{i=3}^{d} n_i)^{1/2} $,
$\varsigma \in \{1, 1.2, 1.5, 1.8, 2\}$.
For \textbf{TSP-$k$+$\ell_1$}, we set
$k\in \{3,4,5\}, \lambda \in \{100,200,300,400\}$.
For \textbf{LNOP}, the parameter $p$ of $\ell_p$-ball projection is set to be
$0.7$, $\epsilon=500, \lambda=10^{7}$. For \textbf{NRTRM}, the minimax concave penalty (MCP) function is utilized in both regularizers $G_1, G_2$, the parameter $\eta$ of MCP is chosen as $\max(n_1,n_2)/\alpha^{k}$ for $G_1$ and $1/\beta^{k}$ for $G_2$, respectively; $c \in \{0.7,0.9,1.4\}$, and $\lambda={\kappa} / (\max{(n_1,n_2)} \cdot \prod_{i=3}^{d} n_i )^{1/2}
$,
$\kappa \in \{1.2, 1.5, 1.8, 2,2.2, 2.5\}$.
For \textbf{HWTNN+$\ell_1$},
we set $c=\max(n_1,n_2), \epsilon=10^{-16}$ and $\lambda=\theta/ (\max{(n_1,n_2)} \cdot \prod_{i=3}^{d} n_i)^{1/2} $,
$\theta \in \{25,30, 35, 40, 45, 50, 55, 60,65,70\}$.
For \textbf{our algorithms},
we set
$t=1, \vartheta=1.15,
\beta^0=10^{-3}, \beta^{\max}=10^{8},
\varpi=10^{-4},\epsilon_{1}=\epsilon_{2}=10^{-16}$, $c_1= \omega \cdot \max(n_1,n_2)$, $\omega \in \{0.5,1,2,5,10,15,20,25,30\}$, $c_2=1$, $\lambda=\xi/ (\max{(n_1,n_2)} \cdot \prod_{i=3}^{d} n_i)^{1/2} $, $\xi\in \{1, 3,5,6, 8, 10,12,15\}$,
$k=100, b=20$ for
multitemporal remote sensing images
($k=50, b=10$ for color videos and light field images).
The adjustable parameters $p$ and $q$ are set to be inversely proportional to the constant $c_1$, respectively.
In other words, as $p$ and $q$ go up, $\omega$ goes down.
\subsubsection{\textbf{Application in Light Field Images Recovery}}
In this experiment,
we choose four
fifth-order light field images
(LFIs) including Bench, Bee-1, Framed and Mini to showcase the superiority and effectiveness of the proposed algorithms.
These LFIs with the size of $434\times 625\times 3\times 15\times15$ can be downloaded from the lytro illum light field dataset website \footnote{\url{https://www.irisa.fr/temics/demos/IllumDatasetLF/index.html}}.
\begin{figure*}
\caption{ Visual comparison of various methods for CVs restoration. From top to bottom, the parameter pair $(sr, \tau)$ are $(0.1,0.3)$, $(0.1,0.5)$, $(0.2,0.3)$ and $(0.2,0.5)$, respectively. Top row:
the $60$-th frame of Rush-hour.
The second row: the $50$-th frame of Stockholm.
The third row: the $34$-th frame of Intotree.
Bottom row: the $10$-th frame of Johnny. }
\label{fig_visual_cv}
\end{figure*}
\begin{table*}[htbp]
\caption{
The PSNR, SSIM values and CPU time
obtained by various RLRTC methods for different fourth-order CVs.
The best and the second-best results are highlighted in blue and red, respectively.
}
\label{cv_index}
\centering \scriptsize
\renewcommand{0.0}{0.880} \setlength\tabcolsep{4.5pt}
\begin{tabular}{c c ccccccc cccc cccc | c}
\hline
{CV-Name}
& \multicolumn{4}{|c|}{Rush-hour
} &\multicolumn{4}{c|}{Johnny} &\multicolumn{4}{c|}{Stockholm} &\multicolumn{4}{c|}{Intotree}
& \multirow{3}{*}{
\tabincell{c}{Average\\Time (s)}
}\\
\cline{1-1}
\cline{2-5}
\cline{6-9}
\cline{10-13}
\cline{14-17}
$sr$
&\multicolumn{2}{|c|}{$10\%$} &\multicolumn{2}{c|}{$20\%$} &
\multicolumn{2}{c|}{$10\%$}&\multicolumn{2}{c|}{$20\%$}&
\multicolumn{2}{c|}{$10\%$}&\multicolumn{2}{c|}{$20\%$}&
\multicolumn{2}{c|}{$10\%$}&\multicolumn{2}{c|}{$20\%$}\\
\cline{1-1}
\cline{2-5}
\cline{6-9}
\cline{10-13}
\cline{14-17}
$\tau$
&
\multicolumn{1}{|c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$}
&
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$} &
\multicolumn{1}{c|}{$30\%$}&\multicolumn{1}{c|}{$50\%$}
\\
\hline
\hline
SNN+$\ell_1$
&
\tabincell{c}{18.37 \\0.711 } & \tabincell{c}{ 17.26 \\0.559 }& \tabincell{c}{ 21.78 \\ 0.771 } & \tabincell{c}{19.57\\0.689 }&
\tabincell{c}{ 18.03 \\0.758 } & \tabincell{c}{16.61 \\0.734 }& \tabincell{c}{ 21.80 \\0.803 } & \tabincell{c}{ 18.61\\ 0.745}&
\tabincell{c}{ 19.96 \\0.501 } & \tabincell{c}{18.96 \\ 0.484 }& \tabincell{c}{21.42 \\ 0.537 } & \tabincell{c}{ 20.53\\0.502 }&
\tabincell{c}{ 20.81 \\0.591 } & \tabincell{c}{18.88 \\ 0.512 }& \tabincell{c}{23.08 \\0.622 } & \tabincell{c}{ 21.85\\ 0.539 } & 12968 \\
\hline
TRNN+$\ell_1$ &
\tabincell{c}{ 23.79 \\ 0.733 } & \tabincell{c}{20.47 \\ 0.701 }& \tabincell{c}{ 26.28 \\0.811 } & \tabincell{c}{ 23.02\\ 0.718}&
\tabincell{c}{ 24.59 \\ 0.765 } & \tabincell{c}{ 19.84 \\ 0.739 }& \tabincell{c}{27.18 \\0.847 } & \tabincell{c}{23.06\\0.755 }&
\tabincell{c}{ 22.19 \\0.499 } & \tabincell{c}{20.77 \\ 0.488 }& \tabincell{c}{ 23.48 \\0.594 } & \tabincell{c}{ 21.64\\ 0.512}&
\tabincell{c}{ 24.54 \\ 0.599 } & \tabincell{c}{ 22.28\\ 0.567}& \tabincell{c}{ 25.97 \\ 0.631 } & \tabincell{c}{23.64\\ 0.572}&6968 \\
\hline
TTNN+$\ell_1$
&
\tabincell{c}{24.77 \\ 0.758 } & \tabincell{c}{ 22.39 \\ 0.734 }& \tabincell{c}{ 28.13 \\ 0.833 } & \tabincell{c}{ 24.94\\0.753 }&
\tabincell{c}{ 27.29 \\0.868 } & \tabincell{c}{ 24.19 \\ 0.782 }& \tabincell{c}{ 30.57 \\0.911 } & \tabincell{c}{ 26.93\\ 0.791}&
\tabincell{c}{23.02 \\0.609 } & \tabincell{c}{21.32 \\ 0.494 }& \tabincell{c}{ 25.82 \\ 0.649 } & \tabincell{c}{ 22.82\\ 0.567 }&
\tabincell{c}{25.51 \\0.605 } & \tabincell{c}{23.64 \\ 0.577 }& \tabincell{c}{ 27.54 \\0.632 } & \tabincell{c}{24.99\\ 0.602}&5843 \\
\hline
TSP-$k$+$\ell_1$
&
\tabincell{c}{26.59 \\0.819 } & \tabincell{c}{23.46 \\0.741 }& \tabincell{c}{ 28.68 \\0.845 } & \tabincell{c}{25.46\\ 0.808}&
\tabincell{c}{ 29.42 \\0.886 } & \tabincell{c}{24.98 \\ 0.813 }& \tabincell{c}{ 31.55 \\0.912 } & \tabincell{c}{ 27.48\\ 0.835}&
\tabincell{c}{ 24.39 \\0.643 } & \tabincell{c}{21.81 \\ 0.526 }& \tabincell{c}{ 26.08 \\0.686 } & \tabincell{c}{23.24\\0.593 }&
\tabincell{c}{ 26.58 \\0.633 } & \tabincell{c}{23.64 \\ 0.578 }& \tabincell{c}{ 27.85 \\0.637 } & \tabincell{c}{24.85\\ 0.617}&9924 \\
\hline
LNOP
&
\tabincell{c}{ 26.67 \\0.830 } & \tabincell{c}{ 23.31 \\ 0.748 }& \tabincell{c}{29.13 \\0.857 } & \tabincell{c}{ 25.91\\ 0.829}&
\tabincell{c}{ 28.03 \\ 0.768 } & \tabincell{c}{ 23.69 \\ 0.742 }& \tabincell{c}{ 29.55 \\0.861 } & \tabincell{c}{ 28.39\\ 0.832}&
\tabincell{c}{ 24.66 \\ 0.638 } & \tabincell{c}{22.39 \\ 0.533 }& \tabincell{c}{ 26.85 \\0.694 } & \tabincell{c}{ 24.43\\ 0.603 }&
\tabincell{c}{ 26.24 \\ 0.609 } & \tabincell{c}{ 23.78 \\ 0.603 }& \tabincell{c}{27.70 \\0.634 } & \tabincell{c}{ 25.95\\0.623 }&9138 \\
\hline
NRTRM
&
\tabincell{c}{ 26.52 \\ 0.831 } & \tabincell{c}{ 23.94 \\ 0.772 }& \tabincell{c}{ 29.06 \\0.855 } & \tabincell{c}{ 25.16\\ 0.776}&
\tabincell{c}{ 28.91 \\ 0.891 } & \tabincell{c}{ 25.96 \\ 0.838 }& \tabincell{c}{ 31.61 \\0.919 } & \tabincell{c}{ 27.27\\ 0.868 }&
\tabincell{c}{ 24.33 \\ 0.642 } & \tabincell{c}{ 22.19 \\ 0.538 }& \tabincell{c}{ 26.21 \\0.706 } & \tabincell{c}{ 22.89\\ 0.605 }&
\tabincell{c}{ 26.49 \\ 0.639 } & \tabincell{c}{ 24.46 \\ 0.611 }& \tabincell{c}{27.86 \\ 0.661 } & \tabincell{c}{ 25.19\\ 0.632} &7834 \\
\hline HWTNN+$\ell_1$
&
\tabincell{c}{ 29.23 \\0.839 } & \tabincell{c}{26.09 \\ 0.777 }& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{32.22} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.876}} & \tabincell{c}{ 28.22\\ 0.838}&
\tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{31.25} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.893}} & \tabincell{c}{28.16 \\ 0.843 }& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{34.24} \\\textcolor[rgb]{1.00,0.00,0.00}{0.923}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{30.43}\\\textcolor[rgb]{1.00,0.00,0.00}{0.876}}&
\tabincell{c}{26.16 \\ 0.663 } & \tabincell{c}{ 23.34 \\ 0.551 }& \tabincell{c}{ 28.38 \\ 0.718 } & \tabincell{c}{ 24.66\\ 0.627 }&
\tabincell{c}{ 27.49 \\ 0.647 } & \tabincell{c}{ 25.56 \\ 0.619 }& \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{28.72} \\\textcolor[rgb]{1.00,0.00,0.00}{0.687}} & \tabincell{c}{ 26.27\\ 0.643} & 6737 \\
\hline
\textbf{HWTSN+w$\ell_q$}
&
\tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{30.27} \\\textcolor[rgb]{0.00,0.00,1.00}{0.851}} & \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{27.38} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.794}}& \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{33.68} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.885}} & \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{30.02}\\ \textcolor[rgb]{0.00,0.00,1.00}{0.867}}&
\tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{32.36} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.905}} & \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{29.45} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.868}}& \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{35.33} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.931}} & \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{32.07}\\ \textcolor[rgb]{0.00,0.00,1.00}{0.907}}&
\tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{26.86} \\ 0.668 } & \tabincell{c}{ 24.01 \\0.554 }& \tabincell{c}{ 28.62 \\0.741 } & \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{26.89}\\ \textcolor[rgb]{0.00,0.00,1.00}{0.707}}&
\tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{27.92} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.659}} & \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{26.53} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.638} }& \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{28.98} \\\textcolor[rgb]{0.00,0.00,1.00}{0.695}} & \tabincell{c}{\textcolor[rgb]{0.00,0.00,1.00}{27.94}\\\textcolor[rgb]{0.00,0.00,1.00}{0.667}} & 8760 \\
\hline
\tabincell{c}{ \textbf {HWTSN+w$\ell_q$(BR)}
}
&
\tabincell{c}{ 29.15 \\0.841 } & \tabincell{c}{26.66 \\ 0.784 }& \tabincell{c}{ 31.27 \\0.868} & \tabincell{c}{ 28.89\\ 0.850 }&
\tabincell{c}{ 29.85 \\0.868} & \tabincell{c}{ 28.06 \\0.844 }& \tabincell{c}{30.66 \\0.879} & \tabincell{c}{ 29.46 \\ 0.869}&
\tabincell{c}{26.50 \\\textcolor[rgb]{1.00,0.00,0.00}{0.686}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{24.19} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.579}}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{29.21} \\\textcolor[rgb]{1.00,0.00,0.00}{0.755}} & \tabincell{c}{ 26.19\\ 0.688}&
\tabincell{c}{27.60 \\ 0.649 } & \tabincell{c}{ 26.16 \\ 0.623 }& \tabincell{c}{ 28.43 \\0.679} & \tabincell{c}{27.38\\ 0.658}&\textcolor[rgb]{0.00,0.00,1.00}{4243} \\
\hline
\tabincell{c}{\textbf{{HWTSN+w$\ell_q$(UR)}}
}
&
\tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{29.28} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.842}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{26.77} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.789}}& \tabincell{c}{31.41 \\ 0.865 } & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{29.03}\\\textcolor[rgb]{1.00,0.00,0.00}{0.852}}&
\tabincell{c}{ 29.93 \\ 0.869 } & \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{28.19} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.844}}& \tabincell{c}{ 30.76 \\ 0.878 } & \tabincell{c}{29.53\\ 0.870}&
\tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{26.60} \\\textcolor[rgb]{0.00,0.00,1.00}{0.688}} & \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{24.27} \\ \textcolor[rgb]{0.00,0.00,1.00}{0.582}}& \tabincell{c}{ \textcolor[rgb]{0.00,0.00,1.00}{29.27} \\\textcolor[rgb]{0.00,0.00,1.00}{0.758}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{26.30}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.692}}&
\tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{27.68} \\\textcolor[rgb]{1.00,0.00,0.00}{0.651}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{26.21} \\\textcolor[rgb]{1.00,0.00,0.00}{0.625}}& \tabincell{c}{ 28.45 \\0.684 } & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{27.48}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.659}} &\textcolor[rgb]{1.00,0.00,0.00}{4374} \\
\hline
\end{tabular}
\begin{tablenotes} \item[**] In each RLRTC method, the top represents the PSNR values while the bottom denotes the SSIM values. \end{tablenotes}
\end{table*}
Figure \ref{fig_visual_lfi} shows the recovered LFIs and corresponding zoomed regions acquired by
different RLRTC methods
at extremely low sampling rates.
From the enlarged areas, we can observed that the LFIs restored by our method
preserves more details than those achieved by
other competitive algorithms.
The PSNR, SSIM values and average CPU time of various RLRTC methods for four LFIs with different noise levels and
observation ratios are displayed in Table \ref{lfi_index}. The following conclusions can be drawn from above quantitative metrics. \textbf{1)}
Classical methods induced by the Tucker and TR format, i.e., SNN+$\ell_1$ and TRNN+$\ell_1$, perform relative poorer in term of recovery quality. \textbf{2)}
Among those
algorithms
induced by third-order T-SVD, although the ones employing nonconvex schemes (i.e., LNOP and NRTRM) require more running time over the ones utilizing convex methods (i.e., TTNN+$\ell_1$ and TSP-$k$+$\ell_1$), they achieve higher PSNR and SSIM values in most cases. This phenomenon also exists in the
methods based on high-order T-SVD. \textbf{3)} Compared with other methods, the \textbf{HWTSN+w$\ell_q$} achieves an approximately $2$$\sim$$5$ dB
gain in the mean PSNR values, and
its accelerated versions obtains an about $40\%$$\sim$$60\%$ percent drop in the average CPU time. \textbf{4)} In our
algorithms,
the versions fused randomization ideas
shorten the running time by about $55\%$ percent over the deterministic version
with a slight reduction
of psnr and ssim values.
\subsubsection {\textbf{Application in Color Videos Restoration}}
In this experiment, color videos (CVs) are used to evaluate the performance of the proposed algorithm. We download four large-scale CVs from the derf website \footnote{\url{https://media.xiph.org/video/derf/}} for this test. Only the first $100$ frames of each video sequence are selected as the test data owing to the computational limitation, in which each frame has the size $720 \times 1280 \times 3$. For each CV
with $100$ frames, it can be formulated as an $720 \times 1280 \times 3 \times 100$
fourth-order tensor.
Figure \ref{fig_visual_cv} displays the visual comparison of the proposed and competitive RLRTC algorithms for various CVs restoration. From the zoomed regions, we can see that the \textbf{HWTSN+w$\ell_q$} exhibits tangibly better restortation quality
over other
comparative methods according to the color, brightness, and outline. In Table \ref{cv_index}, we report the PSNR, SSIM values and CPU time of ten RLRTC methods for four CVs,
where $sr=0.1,0.2$ and $\tau=0.3,0.5$. These results show that the PSNR and SSIM metrics acquired by \textbf{HWTSN+w$\ell_q$} are higher than those obtained by the baseline method, i.e., {HWTNN+$\ell_1$}.
In contrast to the competitive non-convex methods (i.e., LNOP and NRTRM), the improvements of proposed non-convex algorithm (i.e.,\textbf{HWTSN+w$\ell_q$}) are around $3$ dB
in term of PSNR index while the reductions of its randomized version are about $52\%$ according to the CPU time. Furthermore, under the comprehensive balance of PSNR, SSIM, and CPU time, the proposed randomized RHTC method is always superior to other popular algorithms. Other findings are similar to the case of LFIs recovery.
\subsubsection{\textbf{Application in Multitemporal Remote Sensing Images Inpainting}} This experiment mainly tests three fourth-order multi-temporal remote sensing images (MRSIs), which are named SPOT-5 \footnote{\url{{https://take5.theia.cnes.fr/atdistrib/take5/client/\#/home
}}} ($2000 \times 2000 \times 4 \times 13$),
Landsat-7 ($4500 \times 4500 \times 6 \times 11$),
and T22LGN \footnote{\url{{https://theia.cnes.fr/atdistrib/rocket/\#/home
}}} ($5001\times 5001 \times 4 \times 7$), respectively. To speed up the calculation process,
the spatial size of
these MRSIs
is downsampled (resized) to
$2000\times 2000$.
\begin{figure*}
\caption{ Visual comparison of various methods for MRSIs inpainting.
From top to bottom, the parameter pair $(sr, \tau)$ are $(0.4,0.1)$, $(0.4,0.3)$, and $(0.4,0.5)$, respectively. Top row:
the $(5,1)$-th frame of Landsat-7.
Middle row: the $(2,6)$-th frame of SPOT-5.
Bottom row: the $(3,5)$-th frame of T22LGN. }
\label{fig_visual_rs}
\end{figure*}
\begin{table*}[htbp]
\caption{
The PSNR, SSIM values and CPU time
obtained by various RLRTC methods for different fourth-order MRSIs.
The best and the second-best results are highlighted in blue and red, respectively.
}
\label{mrsi_index}
\centering \scriptsize
\renewcommand{0.0}{0.88} \setlength\tabcolsep{3.0pt}
\begin{tabular}{c cccccc cccccc cccccc |c}
\hline
\multicolumn{1}{c} {MRSI-Name}
& \multicolumn{6}{|c|} {Landsat-7} &\multicolumn{6}{c|}{T22LGN} &\multicolumn{6}{c|} {SPOT-5}
& \multirow{3}{*}{
\tabincell{c}{Average\\Time (s)}
}\\
\cline{1-1}
\cline{2-7}
\cline{8-13}
\cline{14-19}
$sr$
&\multicolumn{3}{|c|}{$20\%$} &\multicolumn{3}{c|}{$40\%$} &
\multicolumn{3}{c|}{$20\%$}&\multicolumn{3}{c|}{$40\%$}&
\multicolumn{3}{c|}{$20\%$}&\multicolumn{3}{c|}{$40\%$}\\
\cline{1-1}
\cline{2-4}
\cline{5-7}
\cline{8-10}
\cline{11-13}
\cline{14-16}
\cline{17-19}
$\tau$
&
\multicolumn{1}{|c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}&
\multicolumn{1}{c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}&
\multicolumn{1}{c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}&
\multicolumn{1}{c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}&
\multicolumn{1}{c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}&
\multicolumn{1}{c|}{$10\%$}&\multicolumn{1}{c|}{$30\%$} &\multicolumn{1}{c|}{$50\%$}
\\
\hline
\hline
SNN+$\ell_1$
&
\tabincell{c}{21.45 \\0.537} & \tabincell{c}{20.86 \\0.453}& \tabincell{c}{20.13 \\ 0.386} & \tabincell{c}{23.12 \\ 0.576}& \tabincell{c}{ 22.49 \\ 0.498} & \tabincell{c}{ 21.44\\ 0.477}&
\tabincell{c}{ 25.94 \\0.637} & \tabincell{c}{25.91 \\ 0.605}& \tabincell{c}{ 25.69\\ 0.592} & \tabincell{c}{27.01\\ 0.718}& \tabincell{c}{ 26.68 \\ 0.666} & \tabincell{c}{ 25.76\\ 0.596}&
\tabincell{c}{22.38 \\0.573} & \tabincell{c}{ 21.58 \\ 0.509}& \tabincell{c}{ 20.91\\ 0.395} & \tabincell{c}{ 25.63 \\ 0.631}& \tabincell{c}{ 24.19 \\ 0.563} & \tabincell{c}{ 22.55\\ 0.492}&23817 \\
\hline
TRNN+$\ell_1$ &
\tabincell{c}{23.96 \\ 0.671} & \tabincell{c}{22.32 \\ 0.589}& \tabincell{c}{ 20.70\\ 0.565} & \tabincell{c}{ 24.61 \\ 0.699}& \tabincell{c}{ 22.67 \\ 0.691} & \tabincell{c}{ 21.30\\ 0.614}&
\tabincell{c}{ 28.57
\\0.716 } & \tabincell{c}{27.22 \\ 0.696}& \tabincell{c}{ 25.02 \\ 0.586} & \tabincell{c}{ 29.15 \\ 0.797}& \tabincell{c}{ 27.52 \\ 0.723} & \tabincell{c}{ 25.89\\ 0.666}&
\tabincell{c}{ 27.06 \\ 0.681} & \tabincell{c}{24.55 \\ 0.568}& \tabincell{c}{ 22.16\\ 0.467} & \tabincell{c}{28.19 \\ 0.716}& \tabincell{c}{ 25.07 \\ 0.584} & \tabincell{c}{ 23.11\\ 0.523}& 22414 \\
\hline
TTNN+$\ell_1$
&
\tabincell{c}{ 23.17 \\ 0.652} & \tabincell{c}{21.89 \\ 0.543}& \tabincell{c}{ 20.19\\ 0.458} & \tabincell{c}{ 24.05 \\ 0.739}& \tabincell{c}{ 22.61 \\ 0.691} & \tabincell{c}{ 21.15\\ 0.575}&
\tabincell{c}{29.88 \\ 0.793} & \tabincell{c}{ 28.34 \\ 0.778}& \tabincell{c}{ 24.58\\ 0.497} & \tabincell{c}{ 31.26 \\ 0.817}& \tabincell{c}{ 29.37 \\ 0.797} & \tabincell{c}{26.29\\ 0.692}&
\tabincell{c}{26.49 \\0.684} & \tabincell{c}{ 24.38 \\0.537}& \tabincell{c}{ 22.22\\ 0.431} & \tabincell{c}{ 28.17 \\ 0.701}& \tabincell{c}{ 25.65 \\ 0.596} & \tabincell{c}{23.49\\ 0.519}&10107 \\
\hline
TSP-$k$+$\ell_1$
&
\tabincell{c}{ 24.07
\\ 0.673} & \tabincell{c}{22.41 \\0.588}& \tabincell{c}{ 20.36\\ 0.461} & \tabincell{c}{25.82 \\ 0.742}& \tabincell{c}{ 23.62 \\ 0.628} & \tabincell{c}{ 21.14\\ 0.472}&
\tabincell{c}{ 29.12 \\0.791} & \tabincell{c}{ 25.22 \\ 0.511}& \tabincell{c}{ 22.54\\ 0.409} & \tabincell{c}{31.14 \\ 0.836}& \tabincell{c}{ 26.03 \\ 0.545} & \tabincell{c}{ 22.59\\ 0.419}&
\tabincell{c}{ 27.07
\\0.657} & \tabincell{c}{24.57 \\ 0.571}& \tabincell{c}{ 22.29\\ 0.413} & \tabincell{c}{29.74 \\ 0.775}& \tabincell{c}{ 26.36 \\ 0.601} & \tabincell{c}{ 23.34\\ 0.506}&27412 \\
\hline
LNOP
&
\tabincell{c}{24.61 \\0.679} & \tabincell{c}{22.72\\ 0.658}& \tabincell{c}{20.83 \\ 0.589} & \tabincell{c}{26.27\\ 0.763}& \tabincell{c}{23.35 \\ 0.688} & \tabincell{c}{21.42\\ 0.592}&
\tabincell{c}{30.98\\0.841} & \tabincell{c}{28.61\\ 0.736}& \tabincell{c}{26.43 \\ 0.703} & \tabincell{c}{33.63\\ 0.919}& \tabincell{c}{29.12 \\ 0.825} & \tabincell{c}{26.89 \\ 0.706}&
\tabincell{c}{27.66 \\0.643} & \tabincell{c}{25.55 \\ 0.593}& \tabincell{c}{23.45\\0.484} & \tabincell{c}{29.88\\ 0.781}& \tabincell{c}{26.63 \\ 0.632} & \tabincell{c}{24.66\\ 0.572}&18418 \\
\hline
NRTRM
&
\tabincell{c}{ 23.76
\\0.669} & \tabincell{c}{22.40\\ 0.626}& \tabincell{c}{20.45 \\ 0.546} & \tabincell{c}{25.72\\ 0.757}& \tabincell{c}{23.84 \\ 0.639} & \tabincell{c}{21.33\\ 0.577}&
\tabincell{c}{
30.34
\\0.833} & \tabincell{c}{ 28.67 \\ 0.782}& \tabincell{c}{ 25.68\\ 0.607} & \tabincell{c}{33.12 \\ {0.893}}& \tabincell{c}{ 30.41 \\ {0.844}} & \tabincell{c}{25.87\\ 0.621}&
\tabincell{c}{ 27.08
\\{0.682}} & \tabincell{c}{24.84 \\ 0.589}& \tabincell{c}{ 22.55\\ 0.432} & \tabincell{c}{29.95 \\ \textcolor[rgb]{0.00,0.00,0.00}{0.799}}& \tabincell{c}{ 26.84 \\ 0.645} & \tabincell{c}{ 23.73\\ 0.569}&18560 \\
\hline HWTNN+$\ell_1$
&
\tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{25.18}
\\\textcolor[rgb]{1.00,0.00,0.00}{0.692}} & \tabincell{c}{23.48 \\ 0.659}& \tabincell{c}{ 21.59\\ 0.630} & \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{ 26.33} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.768}}& \tabincell{c}{ 24.98 \\ 0.693} & \tabincell{c}{22.89\\ 0.665}&
\tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{32.09}
\\ \textcolor[rgb]{1.00,0.00,0.00}{0.854}} & \tabincell{c}{ 29.19 \\ 0.758}& \tabincell{c}{ 25.53\\ 0.639} & \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{34.24} \\\textcolor[rgb]{1.00,0.00,0.00}{0.923}}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{31.46} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.852}} & \tabincell{c}{ 26.46\\ 0.706}&
\tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{28.45}
\\ \textcolor[rgb]{1.00,0.00,0.00}{0.687}} & \tabincell{c}{ {26.24} \\ 0.596}& \tabincell{c}{ 23.61\\ 0.490} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{30.47} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.808}}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{28.36} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.699}} & \tabincell{c}{ 25.49\\ 0.581}&14835 \\
\hline
\tabincell{c}{ \textbf{HWTSN+w$\ell_q$}
}
&
\tabincell{c}{ \textcolor[rgb]{0,0,1}{25.99}
\\\textcolor[rgb]{0,0,1}{0.722}} & \tabincell{c}{ \textcolor[rgb]{0,0,1}{24.71} \\\textcolor[rgb]{0,0,1}{0.683}}& \tabincell{c}{ \textcolor[rgb]{0,0,1}{23.21}\\ \textcolor[rgb]{0,0,1}{0.669}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{28.11}\\ \textcolor[rgb]{0,0,1}{0.777}}& \tabincell{c}{\textcolor[rgb]{0,0,1}{25.99} \\\textcolor[rgb]{0,0,1}{0.724}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{23.61}\\ \textcolor[rgb]{0,0,1}{0.677}}&
\tabincell{c}{\textcolor[rgb]{0,0,1}{32.46}
\\\textcolor[rgb]{0,0,1}{0.888}} & \tabincell{c}{ \textcolor[rgb]{0,0,1}{30.16} \\ \textcolor[rgb]{0,0,1}{0.833}}& \tabincell{c}{\textcolor[rgb]{0,0,1}{27.98} \\ \textcolor[rgb]{0,0,1}{0.746}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{34.98} \\ \textcolor[rgb]{0,0,1}{0.944}}& \tabincell{c}{ \textcolor[rgb]{0,0,1}{31.92}\\ \textcolor[rgb]{0,0,1}{0.896}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{28.64}\\ \textcolor[rgb]{0,0,1}{0.794}}&
\tabincell{c}{\textcolor[rgb]{0,0,1}{29.58} \\ \textcolor[rgb]{0,0,1}{0.713}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{27.79}\\ \textcolor[rgb]{0,0,1}{0.615}}& \tabincell{c}{\textcolor[rgb]{0,0,1}{25.82} \\ \textcolor[rgb]{0,0,1}{0.565}} & \tabincell{c}{ \textcolor[rgb]{0,0,1}{32.47}\\ \textcolor[rgb]{0,0,1}{0.836}}& \tabincell{c}{ \textcolor[rgb]{0,0,1}{30.07} \\ \textcolor[rgb]{0,0,1}{0.756}} & \tabincell{c}{\textcolor[rgb]{0,0,1}{26.77} \\ \textcolor[rgb]{0,0,1}{0.644}}& 18974 \\
\hline
\tabincell{c}{\textbf{HWTSN+w$\ell_q$(BR)}
}
&
\tabincell{c}{ 24.94
\\0.682} & \tabincell{c}{ 24.30 \\ 0.662}& \tabincell{c}{ 22.88\\ 0.641} & \tabincell{c}{25.55 \\ 0.729}& \tabincell{c}{ 24.94 \\ 0.711} & \tabincell{c}{ 23.13\\ 0.674}&
\tabincell{c}{ 30.54 \\0.809} & \tabincell{c}{ 29.32 \\ 0.791}& \tabincell{c}{ 27.79 \\ 0.738} & \tabincell{c}{30.94 \\0.819}& \tabincell{c}{ 30.02 \\ 0.814} & \tabincell{c}{ 28.39\\ 0.777}&
\tabincell{c}{ 27.94 \\ 0.624} & \tabincell{c}{ 27.19\\ {0.599}}& \tabincell{c}{ 25.48\\ 0.551} & \tabincell{c}{28.72 \\ 0.695}& \tabincell{c}{ 28.19 \\ 0.675} & \tabincell{c}{ 26.37\\ 0.614}&5412 \\
\hline
\tabincell{c}{\textbf{HWTSN+w$\ell_q$(UR)}
}
&
\tabincell{c}{25.02
\\ 0.683} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{24.37} \\ \textcolor[rgb]{1.00,0.00,0.00}{{0.676}}}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{22.91}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.645}} & \tabincell{c}{25.69\\ 0.729}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{25.05} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.712}} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{23.15}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.674}}&
\tabincell{c}{ 30.65 \\0.811} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{29.39} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.792}}& \tabincell{c}{\textcolor[rgb]{1.00,0.00,0.00}{27.81} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.739}} & \tabincell{c}{ 31.06\\ 0.822}& \tabincell{c}{ 30.17 \\ 0.816} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{28.45}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.779}}&
\tabincell{c}{27.99
\\0.624} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{27.21} \\ \textcolor[rgb]{1.00,0.00,0.00}{0.605}}& \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{25.51}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.557}} & \tabincell{c}{ 28.82 \\ 0.694}& \tabincell{c}{ 28.26 \\ 0.686} & \tabincell{c}{ \textcolor[rgb]{1.00,0.00,0.00}{26.40}\\ \textcolor[rgb]{1.00,0.00,0.00}{0.617}}&5746 \\
\hline
\end{tabular}
\begin{tablenotes} \item[**] In each RLRTC method, the top represents the PSNR values while the bottom denotes the SSIM values. \end{tablenotes}
\end{table*}
Table \ref{mrsi_index} presents the PSNR, SSIM values and CPU time provided by ten RLRTC methods on three large-scale MRSIs with
$sr \in \{0.4,0.2\}$, $\tau \in \{0.1, 0.3,0.5\}$.
Accordingly,
some visual examples are illustrated in Figure \ref{fig_visual_rs}, which indicates that
the proposed method retains more details and textures over the other state-of-the-art approaches.
Strikingly,
in comparison with the deterministic RLRTC algorithms,
our RHTC algorithms incorporating with randomized technology
can decrease the computational time by about $70\%$
with little or no loss of PSNR and SSIM.
This demonstrates the effectiveness of randomized
approach for processing large-scale tensor data.
Other
conclusions achieved from the quantitative results are similar to those obtained from the tasks of CVs restoration and
LFIs recovery.
Overall, the proposed randomized RHTC method can dramatically shorten the CPU running time
while still achieving reasonable recovery
precision
over other popular approaches,
especially for large-scale inpainting tasks.
\begin{figure}
\caption{
The influence of adjustable parameters $(p,q)$, and
changing linear transforms $\mathfrak{L}$
upon LFIs recovery.
Top row: $sr=0.1, \tau=0.3$, Bottom row: $sr=0.1, \tau=0.5$.
}
\label{disscu1-lfi}
\end{figure}
\begin{figure}
\caption{
The influence of adjustable parameters $(p,q)$, and
changing linear transforms $\mathfrak{L}$
upon CVs restoration.
Top row: $sr=0.2, \tau=0.3$, Bottom row: $sr=0.2, \tau=0.5$.
}
\label{disscu1-cv}
\end{figure}
\begin{figure}
\caption{
The influence of adjustable parameters $(p,q)$, and
changing linear transforms $\mathfrak{L}$
upon MRSIs inpainting.
Top row: $sr=0.4, \tau=0.1$, Bottom row: $sr=0.4, \tau=0.3$.
}
\label{disscu1-rs}
\end{figure}
\subsubsection{\textbf{Discussion}}
In the previous real applications, the proposed methods only employ
the linear transform: FFT, and only set the adjustable parameters $(p,q)$ to be $(0.9,0.9)$.
In this subsection, we additionally
utilize
other linear transforms (e.g., DCT, ROT) and
adjustable parameters $(p,q)$
to perform related experiments on the proposed algorithm and its two accelerated versions.
Our goal is to investigate the influence of adjustable parameters $(p,q)$, and
changing invertible linear transforms $\mathfrak{L}$
upon restoration results of various
tensor data with different noise levels and observed ratios
in both deterministic and randomized approximation patterns.
In our experiments, the values $p, q (p=q)$ are set from $0.2$ to $0.9$ with an interval of $0.1$.
For brevity, the proposed algorithm: ``\textbf{HWTSN+w$\ell_q$}", and its accelerated versions: ``\textbf{HWTSN+w$\ell_q$(UR)}" and ``\textbf{HWTSN+w$\ell_q$(BR)}" are abbreviated as Ours1, Ours2, and Ours3, respectively.
The corresponding
experimental results of the above investigation are shown in Figure \ref{disscu1-lfi},\ref{disscu1-cv},\ref{disscu1-rs}, from which
some instructive conclusions and guidelines can be drawn. \textbf{(I)}
The PSNR or SSIM value obtained by ROT
is always worse than
that
achieved by FFT and DCT
under the same parameters $p$ and
$q$. This implies that ROT may not be a good choice for
the restoration of
real-world high-order
tensors.
\textbf{(II)}
For the recovery of three types of tensors,
with the increase of adjustable parameters $p$ and $q$, the PSNR and SSIM values obtained by various methods gradually increase
whereas
the corresponding running time gradually decreases in most cases.
This suggests that
selecting relatively large $p$ and $q$
may yield better recovery performance
for different
types of high-order
tensors.
\textbf{(III)} Just as we expected, in comparison with
the deterministic version (i.e., Ours1), the randomized methods (i.e., Ours2 and Ours3)
greatly boost the computational efficiency at the premise of compromising a little PSNR and SSIM
for various inpainting tasks.
\textbf{(IV)}
In our randomized versions,
there is remarkably little difference between Ours2 and Ours3 in CPU running time for different recovery tasks. This indicates that it is very likely that only
for very large-scale tensors, the computational cost of Ours3 (i.e., the version fusing blocked randomized scheme) is significantly lower than that of Ours2.
\section{\textbf{Conclusions and Future Work}}\label{conclusion}
In this article, we first develop two efficient
low-rank tensor approximation methods
fusing
random
projection schemes,
based on which
we further study
the effective model and algorithm
for RHTC.
The model construction, algorithm design and theoretical analysis are all based on the algebraic framework of high-order T-SVD.
Extensive experiments on both synthetic and real-world tensor data have verified the effectiveness and superiority of the proposed approximation and completion approaches.
This work will
lay the foundation for many tensor-based data analysis tasks
such as
high-order tensor
clustering,
regression, classification, etc.
In the future,
under the
T-SVD framework,
we first intend to explore the
effective
randomized algorithm
for the fixed-precision low-rank tensor approximation
by devising novel mode-wise projection strategy
that differs from the literature \cite{haselby2023modewise}.
On this basis, we further investigate the
high-order tensor recovery
from the perspective of model, algorithm and theory.
Secondly,
we will exploit the fast high-order tensor clustering, regression and classification approaches
in virtue of some popular randomized sketching techniques (e.g., random projection/sampling and count-sketch).
Finally, we plan to extend the above
batch-based randomized methods to the online versions, which can deal with large-scale streaming tensors incrementally in online mode, and even with dynamically changing tensors.
\ifCLASSOPTIONcaptionsoff
\fi
\ifCLASSOPTIONcaptionsoff
\fi
\end{document} | arXiv |
The use of cognitive enhancers by healthy individuals sparked debate about ethics and safety. Cognitive enhancement by pharmaceutical means was considered a form of illicit drug use in some places, even while other cognitive enhancers, such as caffeine and nicotine, were freely available. The conflict therein raised the possibility for further acceptance of smart drugs in the future. However, the long-term effects of smart drugs on otherwise healthy brains were unknown, delaying safety assessments.
It's been widely reported that Silicon Valley entrepreneurs and college students turn to Adderall (without a prescription) to work late through the night. In fact, a 2012 study published in the Journal of American College Health, showed that roughly two-thirds of undergraduate students were offered prescription stimulants for non-medical purposes by senior year.
"Cavin Balaster knows brain injury as well as any specialist. He survived a horrific accident and came out on the other side stronger than ever. His book, "How To Feed A Brain" details how changing his diet helped him to recover further from the devastating symptoms of brain injury such as fatigue and brain fog. Cavin is able to thoroughly explain complex issues in a simplified manner so the reader does not need a medical degree to understand. The book also includes comprehensive charts to simplify what the body needs and how to provide the necessary foods. "How To Feed A Brain" is a great resource for anyone looking to improve their health through diet, brain injury not required."
Theanine can also be combined with caffeine as both of them work in synergy to increase memory, reaction time, mental endurance, and memory. The best part about Theanine is that it is one of the safest nootropics and is readily available in the form of capsules. A natural option would be to use an excellent green tea brand which constitutes of tea grown in the shade because then Theanine would be abundantly present in it.
Powders are good for experimenting with (easy to vary doses and mix), but not so good for regular taking. I use OO gel capsules with a Capsule Machine: it's hard to beat $20, it works, it's not that messy after practice, and it's not too bad to do 100 pills. However, I once did 3kg of piracetam + my other powders, and doing that nearly burned me out on ever using capsules again. If you're going to do that much, something more automated is a serious question! (What actually wound up infuriating me the most was when capsules would stick in either the bottom or top try - requiring you to very gingerly pull and twist them out, lest the two halves slip and spill powder - or when the two halves wouldn't lock and you had to join them by hand. In contrast: loading the gel caps could be done automatically without looking, after some experience.)
The stop-signal task has been used in a number of laboratories to study the effects of stimulants on cognitive control. In this task, subjects are instructed to respond as quickly as possible by button press to target stimuli except on certain trials, when the target is followed by a stop signal. On those trials, they must try to avoid responding. The stop signal can follow the target stimulus almost immediately, in which case it is fairly easy for subjects to cancel their response, or it can come later, in which case subjects may fail to inhibit their response. The main dependent measure for stop-signal task performance is the stop time, which is the average go reaction time minus the interval between the target and stop signal at which subjects inhibit 50% of their responses. De Wit and colleagues have published two studies of the effects of d-AMP on this task. De Wit, Crean, and Richards (2000) reported no significant effect of the drug on stop time for their subjects overall but a significant effect on the half of the subjects who were slowest in stopping on the baseline trials. De Wit et al. (2002) found an overall improvement in stop time in addition to replicating their earlier finding that this was primarily the result of enhancement for the subjects who were initially the slowest stoppers. In contrast, Filmore, Kelly, and Martin (2005) used a different measure of cognitive control in this task, simply the number of failures to stop, and reported no effects of d-AMP.
Sulbutiamine, mentioned earlier as a cholinergic smart drug, can also be classed a dopaminergic, although its mechanism is counterintuitive: by reducing the release of dopamine in the brain's prefrontal cortex, the density of dopamine receptors actually increase after continued Sulbutiamine exposure, through a compensatory mechanism. (This provides an interesting example of how dividing smart drugs into sensible "classes" is a matter of taste as well as science, especially since many of them create their discernable neural effects through still undefined mechanisms.)
Now, what is the expected value (EV) of simply taking iodine, without the additional work of the experiment? 4 cans of 0.15mg x 200 is $20 for 2.1 years' worth or ~$10 a year or a NPV cost of $205 (\frac{10}{\ln 1.05}) versus a 20% chance of $2000 or $400. So the expected value is greater than the NPV cost of taking it, so I should start taking iodine.
For proper brain function, our CNS (Central Nervous System) requires several amino acids. These derive from protein-rich foods. Consider amino acids to be protein building blocks. Many of them are dietary precursors to vital neurotransmitters in our brain. Epinephrine (adrenaline), serotonin, dopamine, and norepinephrine assist in enhancing mental performance. A few examples of amino acid nootropics are:
…researchers have added a new layer to the smart pill conversation. Adderall, they've found, makes you think you're doing better than you actually are….Those subjects who had been given Adderall were significantly more likely to report that the pill had caused them to do a better job….But the results of the new University of Pennsylvania study, funded by the U.S. Navy and not yet published but presented at the annual Society for Neuroscience conference last month, are consistent with much of the existing research. As a group, no overall statistically-significant improvement or impairment was seen as a result of taking Adderall. The research team tested 47 subjects, all in their 20s, all without a diagnosis of ADHD, on a variety of cognitive functions, from working memory-how much information they could keep in mind and manipulate-to raw intelligence, to memories for specific events and faces….The last question they asked their subjects was: How and how much did the pill influence your performance on today's tests? Those subjects who had been given Adderall were significantly more likely to report that the pill had caused them to do a better job on the tasks they'd been given, even though their performance did not show an improvement over that of those who had taken the placebo. According to Irena Ilieva…it's the first time since the 1960s that a study on the effects of amphetamine, a close cousin of Adderall, has asked how subjects perceive the effect of the drug on their performance.
After trying out 2 6lb packs between 12 September & 25 November 2012, and 20 March & 20 August 2013, I have given up on flaxseed meal. They did not seem to go bad in the refrigerator or freezer, and tasted OK, but I had difficulty working them into my usual recipes: it doesn't combine well with hot or cold oatmeal, and when I tried using flaxseed meal in soups I learned flaxseed is a thickener which can give soup the consistency of snot. It's easier to use fish oil on a daily basis.
"Cavin, you are phemomenal! An incredulous journey of a near death accident scripted by an incredible man who chose to share his knowledge of healing his own broken brain. I requested our public library purchase your book because everyone, those with and without brain injuries, should have access to YOUR brain and this book. Thank you for your legacy to mankind!"
Aniracetam is known as one of the smart pills with the widest array of uses. From benefits for dementia patients and memory boost in adults with healthy brains, to the promotion of brain damage recovery. It also improves the quality of sleep, what affects the overall increase in focus during the day. Because it supports the production of dopamine and serotonin, it elevates our mood and helps fight depression and anxiety.
Specifically, the film is completely unintelligible if you had not read the book. The best I can say for it is that it delivers the action and events one expects in the right order and with basic competence, but its artistic merits are few. It seems generally devoid of the imagination and visual flights of fancy that animated movies 1 and 3 especially (although Mike Darwin disagrees), copping out on standard imagery like a Star Wars-style force field over Hogwarts Castle, or luminescent white fog when Harry was dead and in his head; I was deeply disappointed to not see any sights that struck me as novel and new. (For example, the aforementioned dead scene could have been done in so many interesting ways, like why not show Harry & Dumbledore in a bustling King's Cross shot in bright sharp detail, but with not a single person in sight and all the luggage and equipment animatedly moving purposefully on their own?) The ending in particular boggles me. I actually turned to the person next to me and asked them whether that really was the climax and Voldemort was dead, his death was so little dwelt upon or laden with significance (despite a musical score that beat you over the head about everything else). In the book, I remember it feeling like a climactic scene, with everyone watching and little speeches explaining why Voldemort was about to be defeated, and a suitable victory celebration; I read in the paper the next day a quote from the director or screenwriter who said one scene was cut because Voldemort would not talk but simply try to efficiently kill Harry. (This is presumably the explanation for the incredible anti-climax. Hopefully.) I was dumbfounded by the depths of dishonesty or delusion or disregard: Voldemort not only does that in Deathly Hallows multiple times, he does it every time he deals with Harry, exactly as the classic villains (he is numbered among) always do! How was it possible for this man to read the books many times, as he must have, and still say such a thing?↩
During the 1920s, Amphetamine was being researched as an asthma medication when its cognitive benefits were accidentally discovered. In many years that followed, this enhancer was exploited in a number of medical and nonmedical applications, for instance, to enhance alertness in military personnel, treat depression, improve athletic performance, etc.
Power times prior times benefit minus cost of experimentation: (0.20 \times 0.30 \times 540) - 41 = -9. So the VoI is negative: because my default is that fish oil works and I am taking it, weak information that it doesn't work isn't enough. If the power calculation were giving us 40% reliable information, then the chance of learning I should drop fish oil is improved enough to make the experiment worthwhile (going from 20% to 40% switches the value from -$9 to +$23.8).
Of course, there are drugs out there with more transformative powers. "I think it's very clear that some do work," says Andrew Huberman, a neuroscientist based at Stanford University. In fact, there's one category of smart drugs which has received more attention from scientists and biohackers – those looking to alter their own biology and abilities – than any other. These are the stimulants.
"You know how they say that we can only access 20% of our brain?" says the man who offers stressed-out writer Eddie Morra a fateful pill in the 2011 film Limitless. "Well, what this does, it lets you access all of it." Morra is instantly transformed into a superhuman by the fictitious drug NZT-48. Granted access to all cognitive areas, he learns to play the piano in three days, finishes writing his book in four, and swiftly makes himself a millionaire.
A total of 330 randomly selected Saudi adolescents were included. Anthropometrics were recorded and fasting blood samples were analyzed for routine analysis of fasting glucose, lipid levels, calcium, albumin and phosphorous. Frequency of coffee and tea intake was noted. 25-hydroxyvitamin D levels were measured using enzyme-linked immunosorbent assays…Vitamin D levels were significantly highest among those consuming 9-12 cups of tea/week in all subjects (p-value 0.009) independent of age, gender, BMI, physical activity and sun exposure.
I took 1.5mg of melatonin, and went to bed at ~1:30AM; I woke up around 6:30, took a modafinil pill/200mg, and felt pretty reasonable. By noon my mind started to feel a bit fuzzy, and lunch didn't make much of it go away. I've been looking at studies, and users seem to degrade after 30 hours; I started on mid-Thursday, so call that 10 hours, then 24 (Friday), 24 (Saturday), and 14 (Sunday), totaling 72hrs with <20hrs sleep; this might be equivalent to 52hrs with no sleep, and Wikipedia writes:
Fortunately, there are some performance-enhancing habits that have held up under rigorous scientific scrutiny. They are free, and easy to pronounce. Unfortunately, they are also the habits you were perhaps hoping to forego by using nootropics instead. "Of all the things that are supposed to be 'good for the brain,'" says Stanford neurology professor Sharon Sha, "there is more evidence for exercise than anything else." Next time you're facing a long day, you could take a pill and see what happens.
Medication can be ineffective if the drug payload is not delivered at its intended place and time. Since an oral medication travels through a broad pH spectrum, the pill encapsulation could dissolve at the wrong time. However, a smart pill with environmental sensors, a feedback algorithm and a drug release mechanism can give rise to smart drug delivery systems. This can ensure optimal drug delivery and prevent accidental overdose.
However, normally when you hear the term nootropic kicked around, people really mean a "cognitive enhancer" — something that does benefit thinking in some way (improved memory, faster speed-of-processing, increased concentration, or a combination of these, etc.), but might not meet the more rigorous definition above. "Smart drugs" is another largely-interchangeable term.
The title question, whether prescription stimulants are smart pills, does not find a unanimous answer in the literature. The preponderance of evidence is consistent with enhanced consolidation of long-term declarative memory. For executive function, the overall pattern of evidence is much less clear. Over a third of the findings show no effect on the cognitive processes of healthy nonelderly adults. Of the rest, most show enhancement, although impairment has been reported (e.g., Rogers et al., 1999), and certain subsets of participants may experience impairment (e.g., higher performing participants and/or those homozygous for the met allele of the COMT gene performed worse on drug than placebo; Mattay et al., 2000, 2003). Whereas the overall trend is toward enhancement of executive function, the literature contains many exceptions to this trend. Furthermore, publication bias may lead to underreporting of these exceptions.
Also known as Arcalion or Bisbuthiamine and Enerion, Sulbutiamine is a compound of the Sulphur group and is an analog to vitamin B1, which is known to pass the blood-brain barrier easily. Sulbutiamine is found to circulate faster than Thiamine from blood to brain. It is recommended for patients suffering from mental fatigue caused due to emotional and psychological stress. The best part about this compound is that it does not have most of the common side effects linked with a few nootropics.
Cocoa flavanols (CF) positively influence physiological processes in ways which suggest that their consumption may improve aspects of cognitive function. This study investigated the acute cognitive and subjective effects of CF consumption during sustained mental demand. In this randomized, controlled, double-blinded, balanced, three period crossover trial 30 healthy adults consumed drinks containing 520 mg, 994 mg CF and a matched control, with a 3-day washout between drinks. Assessments included the state anxiety inventory and repeated 10-min cycles of a Cognitive Demand Battery comprising of two serial subtraction tasks (Serial Threes and Serial Sevens), a Rapid Visual Information Processing (RVIP) task and a mental fatigue scale, over the course of 1 h. Consumption of both 520 mg and 994 mg CF significantly improved Serial Threes performance. The 994 mg CF beverage significantly speeded RVIP responses but also resulted in more errors during Serial Sevens. Increases in self-reported mental fatigue were significantly attenuated by the consumption of the 520 mg CF beverage only. This is the first report of acute cognitive improvements following CF consumption in healthy adults. While the mechanisms underlying the effects are unknown they may be related to known effects of CF on endothelial function and blood flow.
Overall, the studies listed in Table 1 vary in ways that make it difficult to draw precise quantitative conclusions from them, including their definitions of nonmedical use, methods of sampling, and demographic characteristics of the samples. For example, some studies defined nonmedical use in a way that excluded anyone for whom a drug was prescribed, regardless of how and why they used it (Carroll et al., 2006; DeSantis et al., 2008, 2009; Kaloyanides et al., 2007; Low & Gendaszek, 2002; McCabe & Boyd, 2005; McCabe et al., 2004; Rabiner et al., 2009; Shillington et al., 2006; Teter et al., 2003, 2006; Weyandt et al., 2009), whereas others focused on the intent of the user and counted any use for nonmedical purposes as nonmedical use, even if the user had a prescription (Arria et al., 2008; Babcock & Byrne, 2000; Boyd et al., 2006; Hall et al., 2005; Herman-Stahl et al., 2007; Poulin, 2001, 2007; White et al., 2006), and one did not specify its definition (Barrett, Darredeau, Bordy, & Pihl, 2005). Some studies sampled multiple institutions (DuPont et al., 2008; McCabe & Boyd, 2005; Poulin, 2001, 2007), some sampled only one (Babcock & Byrne, 2000; Barrett et al., 2005; Boyd et al., 2006; Carroll et al., 2006; Hall et al., 2005; Kaloyanides et al., 2007; McCabe & Boyd, 2005; McCabe et al., 2004; Shillington et al., 2006; Teter et al., 2003, 2006; White et al., 2006), and some drew their subjects primarily from classes in a single department at a single institution (DeSantis et al., 2008, 2009; Low & Gendaszek, 2002). With few exceptions, the samples were all drawn from restricted geographical areas. Some had relatively high rates of response (e.g., 93.8%; Low & Gendaszek 2002) and some had low rates (e.g., 10%; Judson & Langdon, 2009), the latter raising questions about sample representativeness for even the specific population of students from a given region or institution.
Spaced repetition at midnight: 3.68. (Graphing preceding and following days: ▅▄▆▆▁▅▆▃▆▄█ ▄ ▂▄▄▅) DNB starting 12:55 AM: 30/34/41. Transcribed Sawaragi 2005, then took a walk. DNB starting 6:45 AM: 45/44/33. Decided to take a nap and then take half the armodafinil on awakening, before breakfast. I wound up oversleeping until noon (4:28); since it was so late, I took only half the armodafinil sublingually. I spent the afternoon learning how to do value of information calculations, and then carefully working through 8 or 9 examples for my various pages, which I published on Lesswrong. That was a useful little project. DNB starting 12:09 AM: 30/38/48. (To graph the preceding day and this night: ▇▂█▆▅▃▃▇▇▇▁▂▄ ▅▅▁▁▃▆) Nights: 9:13; 7:24; 9:13; 8:20; 8:31.
In most cases, cognitive enhancers have been used to treat people with neurological or mental disorders, but there is a growing number of healthy, "normal" people who use these substances in hopes of getting smarter. Although there are many companies that make "smart" drinks, smart power bars and diet supplements containing certain "smart" chemicals, there is little evidence to suggest that these products really work. Results from different laboratories show mixed results; some labs show positive effects on memory and learning; other labs show no effects. There are very few well-designed studies using normal healthy people.
Besides Adderall, I also purchased on Silk Road 5x250mg pills of armodafinil. The price was extremely reasonable, 1.5btc or roughly $23 at that day's exchange rate; I attribute the low price to the seller being new and needing feedback, and offering a discount to induce buyers to take a risk on him. (Buyers bear a large risk on Silk Road since sellers can easily physically anonymize themselves from their shipment, but a buyer can be found just by following the package.) Because of the longer active-time, I resolved to test the armodafinil not during the day, but with an all-nighter.
In 2011, as part of the Silk Road research, I ordered 10x100mg Modalert (5btc) from a seller. I also asked him about his sourcing, since if it was bad, it'd be valuable to me to know whether it was sourced from one of the vendors listed in my table. He replied, more or less, I get them from a large Far Eastern pharmaceuticals wholesaler. I think they're probably the supplier for a number of the online pharmacies. 100mg seems likely to be too low, so I treated this shipment as 5 doses:
Smart drugs, formally known as nootropics, are medications, supplements, and other substances that improve some aspect of mental function. In the broadest sense, smart drugs can include common stimulants such as caffeine, herbal supplements like ginseng, and prescription medications for conditions such as ADHD, Alzheimer's disease, and narcolepsy. These substances can enhance concentration, memory, and learning.
Between midnight and 1:36 AM, I do four rounds of n-back: 50/39/30/55%. I then take 1/4th of the pill and have some tea. At roughly 1:30 AM, AngryParsley linked a SF anthology/novel, Fine Structure, which sucked me in for the next 3-4 hours until I finally finished the whole thing. At 5:20 AM, circumstances forced me to go to bed, still having only taken 1/4th of the pill and that determines this particular experiment of sleep; I quickly do some n-back: 29/20/20/54/42. I fall asleep in 13 minutes and sleep for 2:48, for a ZQ of 28 (a full night being ~100). I did not notice anything from that possible modafinil+caffeine interaction. Subjectively upon awakening: I don't feel great, but I don't feel like 2-3 hours of sleep either. N-back at 10 AM after breakfast: 25/54/44/38/33. These are not very impressive, but seem normal despite taking the last armodafinil ~9 hours ago; perhaps the 3 hours were enough. Later that day, at 11:30 PM (just before bed): 26/56/47.
Similarly, Mehta et al 2000 noted that the positive effects of methylphenidate (40 mg) on spatial working memory performance were greatest in those volunteers with lower baseline working memory capacity. In a study of the effects of ginkgo biloba in healthy young adults, Stough et al 2001 found improved performance in the Trail-Making Test A only in the half with the lower verbal IQ.
Most research on these nootropics suggest they have some benefits, sure, but as Barbara Sahakian and Sharon Morein-Zamir explain in the journal Nature, nobody knows their long-term effects. And we don't know how extended use might change your brain chemistry in the long run. Researchers are getting closer to what makes these substances do what they do, but very little is certain right now. If you're looking to live out your own Limitless fantasy, do your research first, and proceed with caution.
As for newer nootropic drugs, there are unknown risks. "Piracetam has been studied for decades," says cognitive neuroscientist Andrew Hill, the founder of a neurofeedback company in Los Angeles called Peak Brain Institute. But "some of [the newer] compounds are things that some random editor found in a scientific article, copied the formula down and sent it to China and had a bulk powder developed three months later that they're selling. Please don't take it, people!"
Armodafinil is sort of a purified modafinil which Cephalon sells under the brand-name Nuvigil (and Sun under Waklert20). Armodafinil acts much the same way (see the ADS Drug Profile) but the modafinil variant filtered out are the faster-acting molecules21. Hence, it is supposed to last longer. as studies like Pharmacodynamic effects on alertness of single doses of armodafinil in healthy subjects during a nocturnal period of acute sleep loss seem to bear out; anecdotally, it's also more powerful, with Cephalon offering pills with doses as low as 50mg. (To be technical, modafinil is racemic: it comes in two forms which are rotations, mirror-images of each other. The rotation usually doesn't matter, but sometimes it matters tremendously - for example, one form of thalidomide stops morning sickness, and the other rotation causes hideous birth defects.)
Pharmaceutical, substance used in the diagnosis, treatment, or prevention of disease and for restoring, correcting, or modifying organic functions. (See also pharmaceutical industry.) Records of medicinal plants and minerals date to ancient Chinese, Hindu, and Mediterranean civilizations. Ancient Greek physicians such as Galen used a variety of drugs in their profession.…
No. There are mission essential jobs that require you to live on base sometimes. Or a first term person that is required to live on base. Or if you have proven to not be as responsible with rent off base as you should be so your commander requires you to live on base. Or you're at an installation that requires you to live on base during your stay. Or the only affordable housing off base puts you an hour away from where you work. It isn't simple. The fact that you think it is tells me you are one of the "dumb@$$es" you are referring to above.
The fish oil can be considered a free sunk cost: I would take it in the absence of an experiment. The empty pill capsules could be used for something else, so we'll put the 500 at $5. Filling 500 capsules with fish and olive oil will be messy and take an hour. Taking them regularly can be added to my habitual morning routine for vitamin D and the lithium experiment, so that is close to free but we'll call it an hour over the 250 days. Recording mood/productivity is also free a sunk cost as it's necessary for the other experiments; but recording dual n-back scores is more expensive: each round is ~2 minutes and one wants >=5, so each block will cost >10 minutes, so 18 tests will be >180 minutes or >3 hours. So >5 hours. Total: 5 + (>5 \times 7.25) = >41.
Stayed up with the purpose of finishing my work for a contest. This time, instead of taking the pill as a single large dose (I feel that after 3 times, I understand what it's like), I will take 4 doses over the new day. I took the first quarter at 1 AM, when I was starting to feel a little foggy but not majorly impaired. Second dose, 5:30 AM; feeling a little impaired. 8:20 AM, third dose; as usual, I feel physically a bit off and mentally tired - but still mentally sharp when I actually do something. Early on, my heart rate seemed a bit high and my limbs trembling, but it's pretty clear now that that was the caffeine or piracetam. It may be that the other day, it was the caffeine's fault as I suspected. The final dose was around noon. The afternoon crash wasn't so pronounced this time, although motivation remains a problem. I put everything into finishing up the spaced repetition literature review, and didn't do any n-backing until 11:30 PM: 32/34/31/54/40%.
If you're suffering from blurred or distorted vision or you've noticed a sudden and unexplained decline in the clarity of your vision, do not try to self-medicate. It is one thing to promote better eyesight from an existing and long-held baseline, but if you are noticing problems with your eyes, then you should see an optician and a doctor to rule out underlying medical conditions.
Starting from the studies in my meta-analysis, we can try to estimate an upper bound on how big any effect would be, if it actually existed. One of the most promising null results, Southon et al 1994, turns out to be not very informative: if we punch in the number of kids, we find that they needed a large effect size (d=0.81) before they could see anything:
Kennedy et al. (1990) administered what they termed a grammatical reasoning task to subjects, in which a sentence describing the order of two letters, A and B, is presented along with the letter pair, and subjects must determine whether or not the sentence correctly describes the letter pair. They found no effect of d-AMP on performance of this task.
"The author's story alone is a remarkable account of not just survival, but transcendence of a near-death experience. Cavin went on to become an advocate for survival and survivors of traumatic brain injuries, discovering along the way the key role played by nutrition. But this book is not just for injury survivors. It is for anyone who wants to live (and eat) well."
First half at 6 AM; second half at noon. Wrote a short essay I'd been putting off and napped for 1:40 from 9 AM to 10:40. This approach seems to work a little better as far as the aboulia goes. (I also bother to smell my urine this time around - there's a definite off smell to it.) Nights: 10:02; 8:50; 10:40; 7:38 (2 bad nights of nasal infections); 8:28; 8:20; 8:43 (▆▃█▁▂▂▃).
Nootropics include natural and manmade chemicals that produce cognitive benefits. These substances are used to make smart pills that deliver results for enhancing memory and learning ability, improving brain function, enhancing the firing control mechanisms in neurons, and providing protection for the brain. College students, adult professionals, and elderly people are turning to supplements to get the advantages of nootropic substances for memory, focus, and concentration.
Table 4 lists the results of 27 tasks from 23 articles on the effects of d-AMP or MPH on working memory. The oldest and most commonly used type of working memory task in this literature is the Sternberg short-term memory scanning paradigm (Sternberg, 1966), in which subjects hold a set of items (typically letters or numbers) in working memory and are then presented with probe items, to which they must respond "yes" (in the set) or "no" (not in the set). The size of the set, and hence the working memory demand, is sometimes varied, and the set itself may be varied from trial to trial to maximize working memory demands or may remain fixed over a block of trials. Taken together, the studies that have used a version of this task to test the effects of MPH and d-AMP on working memory have found mixed and somewhat ambiguous results. No pattern is apparent concerning the specific version of the task or the specific drug. Four studies found no effect (Callaway, 1983; Kennedy, Odenheimer, Baltzley, Dunlap, & Wood, 1990; Mintzer & Griffiths, 2007; Tipper et al., 2005), three found faster responses with the drugs (Fitzpatrick, Klorman, Brumaghim, & Keefover, 1988; Ward et al., 1997; D. E. Wilson et al., 1971), and one found higher accuracy in some testing sessions at some dosages, but no main effect of drug (Makris et al., 2007). The meaningfulness of the increased speed of responding is uncertain, given that it could reflect speeding of general response processes rather than working memory–related processes. Aspects of the results of two studies suggest that the effects are likely due to processes other than working memory: D. E. Wilson et al. (1971) reported comparable speeding in a simple task without working memory demands, and Tipper et al. (2005) reported comparable speeding across set sizes.
If the entire workforce were to start doping with prescription stimulants, it seems likely that they would have two major effects. Firstly, people would stop avoiding unpleasant tasks, and weary office workers who had perfected the art of not-working-at-work would start tackling the office filing system, keeping spreadsheets up to date, and enthusiastically attending dull meetings.
Many of the positive effects of cognitive enhancers have been seen in experiments using rats. For example, scientists can train rats on a specific test, such as maze running, and then see if the "smart drug" can improve the rats' performance. It is difficult to see how many of these data can be applied to human learning and memory. For example, what if the "smart drug" made the rat hungry? Wouldn't a hungry rat run faster in the maze to receive a food reward than a non-hungry rat? Maybe the rat did not get any "smarter" and did not have any improved memory. Perhaps the rat ran faster simply because it was hungrier. Therefore, it was the rat's motivation to run the maze, not its increased cognitive ability that affected the performance. Thus, it is important to be very careful when interpreting changes observed in these types of animal learning and memory experiments.
Though their product includes several vitamins including Bacopa, it seems to be missing the remaining four of the essential ingredients: DHA Omega 3, Huperzine A, Phosphatidylserine and N-Acetyl L-Tyrosine. It missed too many of our key criteria and so we could not endorse this product of theirs. Simply, if you don't mind an insufficient amount of essential ingredients for improved brain and memory function and an inclusion of unwanted ingredients – then this could be a good fit for you.
Many people quickly become overwhelmed by the volume of information and number of products on the market. Because each website claims its product is the best and most effective, it is easy to feel confused and unable to decide. Smart Pill Guide is a resource for reliable information and independent reviews of various supplements for brain enhancement.
Fatty acids are well-studied natural smart drugs that support many cognitive abilities. They play an essential role in providing structural support to cell membranes. Fatty acids also contribute to the growth and repair of neurons. Both functions are crucial for maintaining peak mental acuity as you age. Among the most prestigious fatty acids known to support cognitive health are:
(We already saw that too much iodine could poison both adults and children, and of course too little does not help much - iodine would seem to follow a U-curve like most supplements.) The listed doses at iherb.com often are ridiculously large: 10-50mg! These are doses that seems to actually be dangerous for long-term consumption, and I believe these are doses that are designed to completely suffocate the thyroid gland and prevent it from absorbing any more iodine - which is useful as a short-term radioactive fallout prophylactic, but quite useless from a supplementation standpoint. Fortunately, there are available doses at Fitzgerald 2012's exact dose, which is roughly the daily RDA: 0.15mg. Even the contrarian materials seem to focus on a modest doubling or tripling of the existing RDA, so the range seems relatively narrow. I'm fairly confident I won't overshoot if I go with 0.15-1mg, so let's call this 90%.
Prescription smart pills are common psychostimulants that can be purchased and used after receiving a prescription. They are most commonly given to patients diagnosed with ADD or ADHD, as well as narcolepsy. However many healthy people use them as cognitive enhancers due to their proven ability to improve focus, attention, and support the overall process of learning.
Another ingredient used in this formula is GABA or Gamma-Aminobutyric acid; it's the second most common neurotransmitter found in the human brain. Being an inhibitory neurotransmitter it helps calm and reduce neuronal activity; this calming effect makes GABA an excellent ingredient in anti-anxiety medication. Lecithin is another ingredient found in Smart Pill and is a basic compound found in every cell of the body, with cardiovascular benefits it can also help restore the liver. Another effect is that it works with neurological functions such as memory or attention, thus improving brain Effectiveness.
(If I am not deficient, then supplementation ought to have no effect.) The previous material on modern trends suggests a prior >25%, and higher than that if I were female. However, I was raised on a low-salt diet because my father has high blood pressure, and while I like seafood, I doubt I eat it more often than weekly. I suspect I am somewhat iodine-deficient, although I don't believe as confidently as I did that I had a vitamin D deficiency. Let's call this one 75%.
I was contacted by the Longecity user lostfalco, and read through some of his writings on the topic. I had never heard of LLLT before, but the mitochondria mechanism didn't sound impossible (although I wondered whether it made sense at a quantity level14151617), and there was at least some research backing it; more importantly, lostfalco had discovered that devices for LLLT could be obtained as cheap as $15. (Clearly no one will be getting rich off LLLT or affiliate revenue any time soon.) Nor could I think of any way the LLLT could be easily harmful: there were no drugs involved, physical contact was unnecessary, power output was too low to directly damage through heating, and if it had no LLLT-style effect but some sort of circadian effect through hitting photoreceptors, using it in the morning wouldn't seem to interfere with sleep.
Factor analysis. The strategy: read in the data, drop unnecessary data, impute missing variables (data is too heterogeneous and collected starting at varying intervals to be clean), estimate how many factors would fit best, factor analyze, pick the ones which look like they match best my ideas of what productive is, extract per-day estimates, and finally regress LLLT usage on the selected factors to look for increases.
A big part is that we are finally starting to apply complex systems science to psycho-neuro-pharmacology and a nootropic approach. The neural system is awesomely complex and old-fashioned reductionist science has a really hard time with complexity. Big companies spends hundreds of millions of dollars trying to separate the effects of just a single molecule from placebo – and nootropics invariably show up as "stacks" of many different ingredients (ours, Qualia , currently has 42 separate synergistic nootropics ingredients from alpha GPC to bacopa monnieri and L-theanine). That kind of complex, multi pathway input requires a different methodology to understand well that goes beyond simply what's put in capsules. | CommonCrawl |
\begin{document}
\title{Some examples related to the Deligne-Simpson problem
ootnote{Research partially supported
by INTAS grant 97-1644}
\section{Introduction} \subsection{The Deligne-Simpson problem}
In the present paper we consider some examples relative to the {\em Deligne-Simpson problem (DSP)} which is formulated like this:
{\em Give necessary and sufficient conditions upon the choice of the $p+1$ conjugacy classes $c_j\subset gl(n,{\bf C})$, resp. $C_j\subset GL(n,{\bf C})$, so that there exist irreducible $(p+1)$-tuples of matrices $A_j\in c_j$, $A_1+\ldots +A_{p+1}=0$, resp. of matrices $M_j\in C_j$, $M_1\ldots M_{p+1}=I$.}
By definition, the {\em weak DSP} is the DSP in which the requirement of irreducibility is replaced by the weaker requirement the centralizer of the $(p+1)$-tuple of matrices to be trivial.
The matrices $A_j$, resp. $M_j$, are interpreted as {\em matrices-residua} of {\em Fuchsian} systems on Riemann's sphere (i.e. linear systems of ordinary differential equations with logarithmic poles), resp. as {\em monodromy operators} of {\em regular} systems on Riemann's sphere (i.e. linear systems of ordinary differential equations with moderate growth rate of the solutions at the poles). Fuchsian systems are a particular case of regular ones. By definition, the monodromy operators generate the {\em monodromy group} of a regular system.
In the multiplicative version (i.e. for matrices $M_j$) the classes $C_j$ are interpreted as {\em local monodromies} around the poles and the problem admits the interpretation:
{\em For what $(p+1)$-tuples of local monodromies do there exist monodromy groups with such local monodromies.}
\begin{rems} 1) Suppose that $A_j$ denotes a matrix-residuum and that $M_j$ denotes the corresponding monodromy operator of a Fuchsian system. Then in the absence of non-zero integer differences between the eigenvalues of $A_j$ the operator $M_j$ is conjugate to $\exp (2\pi iA_j)$.
2) In what follows the sum of the matrices $A_j$ is always presumed to be 0 and the product of the matrices $M_j$ is always presumed to be $I$. \end{rems}
\subsection{The aim of this paper}
For a conjugacy class $C$ in $GL(n,{\bf C})$ or $gl(n,{\bf C})$ denote by $d(C)$ its dimension (which is always even). Set $d_j:=d(c_j)$ (resp. $d(C_j)$).
For fixed conjugacy classes $C_j$ consider the variety
\[ {\cal V}=\{ (M_1,\ldots ,M_{p+1})~|~M_j\in C_j~,~M_1\ldots M_{p+1}=I\}~.\] This variety might contain $(p+1)$-tuples with non-trivial centralizers as well as with trivial ones. It might contain only the former or only the latter.
\begin{prop}\label{smooth} At a $(p+1)$-tuple with trivial centralizer the variety ${\cal V}$ is smooth and of dimension $d_1+\ldots +d_{p+1}-n^2+1$. \end{prop}
\begin{rem} The proposition is proved at the end of the subsection. A similar statement is true for matrices $A_j$. \end{rem}
For generic eigenvalues (the precise definition is given in the next section) the variety ${\cal V}$ contains only irreducible $(p+1)$-tuples and its dimension remains the same when the eigenvalues of the conjugacy classes are changed but not the Jordan normal forms which they define. We call its dimension for generic eigenvalues the {\em expected} one.
The aim of the present paper is to consider some examples of varieties ${\cal V}$ for non-generic eigenvalues. In the first and in the fifth of them (see Sections~\ref{ex1} and \ref{ex5}) dim${\cal V}$ is higher than the expected one. In the first example we discuss the stratified structure of ${\cal V}$ and we show that ${\cal V}$ consists only of $(p+1)$-tuples with non-trivial centralizers. The latter fact is true for the fifth example as well.
In the second example (see Section~\ref{ex2}) the eigenvalues are not generic and the variety ${\cal V}$ contains at the same time $(p+1)$-tuples with trivial and ones with non-trivial centralizers. The dimension of ${\cal V}$ is the expected one.
In the third example (see Section~\ref{ex3}) the variety ${\cal V}$ contains no $(p+1)$-tuples with trivial centralizers but its dimension equals the expected one.
In the fourth example (see Section~\ref{ex4}) there is coexistence in ${\cal V}$ of $(p+1)$-tuples with trivial centralizers and of $(p+1)$-tuples with non-trivial ones. The dimension of ${\cal V}$ at the former (i.e. the expected dimension) is lower than the dimension at the latter.
In the first and third examples the closure of ${\cal V}$ (topological and algebraic) contains also $(p+1)$-tuples in which some of the matrices $M_j$ belong not to $C_j$ but to their closures, i.e. the eigenvalues are the necessary ones but the Jordan structure is ``less generic''.
Similar examples exist for matrices $A_j$ as well. Before beginning with the examples we recall some known facts in the next section.
{\bf Proof of Proposition~\ref{smooth}:}
It suffices to prove the proposition in the case when $C_j\subset SL(n,{\bf C})$. The variety ${\cal V}$ is the intersection in $C_1\times \ldots \times C_p\times SL(n,{\bf C})$ of the graph of the mapping
\[ C_1\times \ldots \times C_p\rightarrow SL(n,{\bf C})~,~ (M_1,\ldots ,M_p)\mapsto (M_1\ldots M_p)^{-1}\] and of the variety ${\cal C}=C_1\times \ldots \times C_{p+1}$. To prove that ${\cal V}$ is smooth it suffices to prove that the intersection is transversal, i.e. the sum of the tangent spaces to the graph (which is the space $\{ \sum _{j=1}^p[M_j,X_j],X_j\in sl(n,{\bf C})\}$) and the one to ${\cal C}$ (it equals $\{ [M_{p+1},X_{p+1}],X_{p+1}\in sl(n,{\bf C})\}$) is $sl(n,{\bf C})$. This follows from
\begin{prop}\label{commut} The $(p+1)$-tuple of matrices $R_j\in gl(n,{\bf C})$ is with trivial centralizer if and only if the map $(gl(n,{\bf C}))^{p+1}\rightarrow sl(n,{\bf C})$, $(X_1,\ldots ,X_{p+1})\mapsto \sum _{j=1}^{p+1}[R_j,X_j]$ is surjective. \end{prop}
The dimension of ${\cal V}$ is the one of $C_1\times \ldots \times C_p$, i.e. $d_1+\ldots +d_p$, diminished by the codimension of ${\cal C}$ in $C_1\times \ldots \times C_p\times SL(n,{\bf C})$, i.e. by $n^2-1-d_{p+1}$. Hence, dim${\cal V}=d_1+\ldots +d_{p+1}-n^2+1$.$\hspace{1cm}\Box$
{\bf Proof of Proposition~\ref{commut}:}
The map is not surjective exactly if the image of every map $X_j\mapsto [R_j,X_j]$ belongs to one and the same linear subspace of $sl(n,{\bf C})$, i.e. one has tr$(D[R_j,X_j])=0$ for some matrix $0\neq D\in sl(n,{\bf C})$ for $j=1,\ldots ,p+1$ and identically in the entries of $X_j$. One has tr$(D[R_j,X_j])=$tr$([D,R_j]X_j)$ which implies that $[D,R_j]=0$ for all $j$ -- a contradiction with the triviality of the centralizer.$\hspace{1cm}\Box$
\section{Some known facts}
We expose here some facts which are given in some more detail in \cite{Ko}. For a matrix $Y$ from the conjugacy class $C$ in $GL(n,{\bf C})$ or $gl(n,{\bf C})$ set $r(C):=\min _{\lambda \in {\bf C}}{\rm rank}(Y-\lambda I)$. The integer $n-r(C)$ is the maximal number of Jordan blocks of $J(Y)$ with one and the same eigenvalue. Set $r_j:=r(c_j)$ (resp. $r(C_j)$). The quantities $r(C)$ and $d(C)$ depend only on the Jordan normal form of $Y$.
\begin{defi} A {\em Jordan normal form (JNF) of size $n$} is a family $J^n=\{ b_{i,l}\}$ ($i\in I_l$, $I_l=\{ 1,\ldots ,s_l\}$, $l\in L$) of positive integers $b_{i,l}$ whose sum is $n$. The index $l$ is the one of an eigenvalue and the index $i$ is the one of a Jordan block with the $l$-th eigenvalue; all eigenvalues are presumed distinct. An $n\times n$-matrix $Y$ has the JNF $J^n$ (notation: $J(Y)=J^n$) if to its distinct eigenvalues $\lambda _l$, $l\in L$, there belong Jordan blocks of sizes $b_{i,l}$. We usually assume that for each fixed $l$ the numbers $b_{i,l}$ form a non-increasing sequence. \end{defi}
\begin{prop}\label{d_jr_j} (C. Simpson, see \cite{Si}.) The following couple of inequalities is a necessary condition for the existence of irreducible $(p+1)$-tuples of matrices $M_j$:
\[ d_1+\ldots +d_{p+1}\geq 2n^2-2~~~~~(\alpha _n)~~,~~~~~ {\rm for~all~}j,~r_1+\ldots +\hat{r}_j+\ldots +r_{p+1}\geq n~~~~~ (\beta _n)~~~.\] \end{prop}
\begin{rem} The conditions are necessary for the existence of irreducible $(p+1)$-tuples of matrices $A_j$ as well. \end{rem}
We presume that there holds the following evident necessary condition
\[ \sum {\rm Tr}(c_j)=0~,~{\rm resp.~}\prod \det (C_j)=1~.\] In terms of the eigenvalues $\lambda _{k,j}$ (resp. $\sigma _{k,j}$) of the matrices from $c_j$ (resp. $C_j$) repeated with their multiplicities, this condition reads
\[ \sum _{k=1}^n\sum _{j=1}^{p+1}\lambda _{k,j}=0~,~{\rm resp.~} \prod _{k=1}^n\prod _{j=1}^{p+1}\sigma _{k,j}=1~.\] An equality of the kind
\[ \sum _{j=1}^{p+1}\sum _{k\in \Phi _j}\lambda _{k,j}=0~,~{\rm resp.~} \prod _{j=1}^{p+1}\prod _{k\in \Phi _j}\sigma _{k,j}=1\] is called a {\em non-genericity relation}; the sets $\Phi _j$ contain one and the same number $<n$ of indices for all $j$. Eigenvalues satisfying none of these relations are called {\em generic}. Reducible $(p+1)$-tuples exist only for non-generic eigenvalues; indeed, the eigenvalues of each diagonal block of a block upper-triangular $(p+1)$-tuple satisfy some non-genericity relation.
\begin{defi} Denote by $\{ J_j^n\}$ a $(p+1)$-tuple of JNFs, $j=1$,$\ldots$, $p+1$. We say that the DSP is {\em solvable} (resp. that it is {\em weakly solvable} or, equivalently, that the weak DSP is solvable) for a given $\{ J_j^n\}$ and for given eigenvalues if there exists an irreducible $(p+1)$-tuple (resp. a $(p+1)$-tuple with a trivial centralizer) of matrices $M_j$ or of matrices $A_j$, with $J(M_j)=J_j^n$ or $J(A_j)=J_j^n$ and with the given eigenvalues. By definition, the DSP is solvable for $n=1$. Solvability of the DSP imlies its weak solvability, i.e. solvability of the weak DSP. \end{defi}
For a given JNF $J^n=\{ b_{i,l}\}$ define its {\em corresponding} diagonal JNF ${J'}^n$. A diagonal JNF is a partition of $n$ defined by the multiplicities of the eigenvalues. For each $l$ $\{ b_{i,l}\}$ is a partition of $\sum _{i\in I_l}b_{i,l}$ and ${J'}^n$ is the disjoint sum of the dual partitions. We say that two JNFs of one and the same size correspond to one another if they correspond to one and the same diagonal JNF.
\begin{prop}\label{rd} 1) One has $r(J^n)=r({J'}^n)$ and $d(J^n)=d({J'}^n)$.
2) To each diagonal JNF there corresponds a unique JNF with a single eigenvalue. \end{prop}
\begin{ex} To the JNF $\{ \{ 4,3,3\} ,\{ 3,2\} \}$ of size 15 (two eigenvalues, with respectively three Jordan blocks, of sizes 4,3,3 and with two Jordan blocks, of sizes 3,2) there corresponds the diagonal JNF with multiplicities of the eigenvalues equal to 3,3,3,2,2,1,1. Indeed, the partition of $10$ dual to 4,3,3 is 3,3,3,1; the partition of $5$ dual to 3,2 is 2,2,1. After this we arrange the multiplicities in decreasing order.
To the two above JNFs there corresponds the JNF with a single eigenvalue with sizes of the Jordan blocks equal to 7,5,3. Indeed, 7,5,3 is the partition of 15 dual to 3,3,3,2,2,1,1. \end{ex}
For a given $\{ J_j^n\}$ with $n>1$, which satisfies condition $(\beta _n)$ and doesn't satisfy condition
\[ (r_1+\ldots +r_{p+1})\geq 2n~~~~~~~~~~~~~~~~(\omega _n)\] set $n_1=r_1+\ldots +r_{p+1}-n$. Hence, $n_1<n$ and $n-n_1\leq n-r_j$. Define the $(p+1)$-tuple $\{ J_j^{n_1}\}$ as follows: to obtain the JNF $J_j^{n_1}$ from $J_j^n$ one chooses one of the eigenvalues of $J_j^n$ with greatest number $n-r_j$ of Jordan blocks, then decreases by 1 the sizes of the $n-n_1$ {\em smallest} Jordan blocks with this eigenvalue and deletes the Jordan blocks of size 0.
\begin{defi} The quantity $\kappa =2n^2-\sum _{j=1}^{p+1}d_j$ defined for a $(p+1)$-tuple of conjugacy classes is called the {\em index of rigidity}. It is introduced by N. Katz in \cite{Ka}. For irreducible representations it takes the values 2, 0, $-2$, $-4$, $\ldots$. Indeed, every conjugacy class is of even dimension and there holds condition $(\alpha _n)$. If for an irreducible $(p+1)$-tuple one has $\kappa =2$, then the $(p+1)$-tuple is called {\em rigid}. Such irreducible $(p+1)$-tuples are unique up to conjugacy, see \cite{Ka} and \cite{Si}. \end{defi}
\begin{lm} The index of rigidity is invariant for the construction $\{ J_j^n\}\mapsto \{ J_j^{n_1}\}$. \end{lm}
\begin{tm}\label{generic} Let $n>1$. The DSP is solvable for the conjugacy classes $C_j$ or $c_j$ (with generic eigenvalues, defining the JNFs $J_j^n$ and satisfying condition $(\beta _n)$) if and only if either $\{ J_j^n\}$ satisfies condition $(\omega _n)$ or the construction $\{ J_j^n\}\mapsto \{ J_j^{n_1}\}$ iterated as long as it is defined stops at a $(p+1)$-tuple $\{ J_j^{n'}\}$ either with $n'=1$ or satisfying condition $(\omega _{n'})$. \end{tm}
\begin{rems} 1) The conditions of the theorem are necessary for the weak solvability of the DSP for any eigenvalues.
2) A posteriori one knows that the theorem does not depend on the choice(s) of eigenvalue(s) made when defining the construction $\{ J_j^n\}\mapsto \{ J_j^{n_1}\}$. \end{rems}
\section{An example with index of rigidity equal to 2\protect\label{ex1}}
\subsection{Description of the example}
Denote by $J^*$, $J^{**}$ two quadruples of JNFs $J_j$ of size 4, $j=1,\ldots ,4$, in both of which $J_1$, $J_2$ and $J_3$ are diagonal, each with two eigenvalues of multiplicity 2; in $J^*$ the JNF $J_4$ is with a single eigenvalue to which there correspond three Jordan blocks, of sizes 2,1,1; in $J^{**}$ the JNF $J_4$ is diagonal, with two eigenvalues, of multiplicities 3 and 1. The JNFs $J_4$ from the two quadruples correspond to each other.
Hence, both $J^*$ and $J^{**}$ satisfy the conditions of Theorem~\ref{generic} (to be checked by the reader). They are both with index of rigidity 2. In both cases (of matrices $A_j$ or $M_j$) the quadruple $J^{**}$ admits generic eigenvalues and, hence, there exist irreducible quadruples of matrices $A_j$ or $M_j$ with such respective JNFs.
\begin{defi} Suppose that the greatest common divisor of the multiplicities of all eigenvalues of the matrices $M_j$ or $A_j$ equals $q>1$. In the case of matrices $M_j$ denote by $\xi$ the product of all eigenvalues with multiplicities decreased $q$ times. Hence, $\xi$ is a root of unity of order $q$: $\xi =\exp (2\pi il/q)$, $l\in {\bf N}$. Denote by $m$ the greatest common divisor of $l$ and $q$. Hence, for $m>1$ the eigenvalues satisfy the non-genericity relation (called {\em basic}) their product with multiplicities divided $m$ times to equal 1. In the case of matrices $A_j$ the basic non-genericity relation is the sum of all eigenvalues with multiplicities decreased $q$ times to equal 0. Eigenvalues satisfying only the basic non-genericity relation and its corollaries are called {\em relatively generic}. \end{defi}
The quadruple $J^*$ does not admit generic but only relatively generic eigenvalues in the case of matrices $A_j$ because one has $q=2$.
The quadruple $J^*$ admits generic eigenvalues in the case of matrices $M_j$. Indeed, such is the set of eigenvalues of the four matrices $(e,e^{-1})$, $(\sqrt{2},1/\sqrt{2})$, $(3,1/3)$, $i$. In this case $q=2$ and the product of all eigenvalues with multiplicities decreased twice equals $-1$. This is not a non-genericity relation. If the eigenvalue of the fourth matrix is changed from $i$ to $-1$, then the eigenvalues will not be generic -- their product when the multiplicities are decreased twice equals 1. This is the basic non-genericity relation. In this case the eigenvalues are relatively generic but not generic.
In our example we consider conjugacy classes $C_j$ defining the quadruple of JNFs $J^*$, with relatively generic but not generic eigenvalues. Observe that the expected dimension of ${\cal V}$ both in the case of $J^*$ and of $J^{**}$ equals $8+8+8+6-15=15$.
\subsection{The stratified structure of the variety ${\cal V}$ from the example}
The variety ${\cal V}$ from the example
contains at least the following two strata denoted by ${\cal U}$ and ${\cal W}$. The stratum ${\cal U}$ consists of all quadruples defining representations which are direct sums of two irreducible representations, i.e. up to conjugacy one has (for $(M_1,M_2,M_3,M_4)\in {\cal U}$)
\begin{equation}\label{direct} M_j=\left( \begin{array}{cc}N_j&0\\0&P_j\end{array}\right) ~,~N_j,P_j\in GL(2,{\bf C}) \end{equation} where the matrices $N_j$ (resp. $P_j$) are diagonal for $j=1,2,3$. Their quadruples are with generic eigenvalues and for $j=4$ the eigenvalues equal $-1$, $P_4$ is conjugate to a Jordan block of size 2 while $N_4$ is scalar. The existence of irreducible quadruples of matrices $N_j$ and $P_j$ is guaranteed by Theorem~\ref{generic}.
\begin{rem} The matrices $N_j$ (resp. $P_j$) define an irreducible rigid representation (resp. an irreducible representation of zero index of rigidity). \end{rem}
\begin{prop}\label{N_jP_j} 1) The variety of matrices $N_j$ (resp. $P_j$) as above is smooth, irreducible and of dimension 3 (resp. 5).
2) The variety of quadruples of diagonalizable matrices $M_j\in GL(2,{\bf C})$ each with two distinct eigenvalues (the eigenvalues of the quadruple being generic) is smooth, irreducible and of dimension 5. \end{prop}
All propositions from this subsection are proved in Section~\ref{proofofUW}.
The stratum ${\cal W}$ consists of all quadruples defining semi-direct sums of two equivalent rigid representations. Up to conjugacy one has (for $(M_1,M_2,M_3,M_4)\in {\cal W}$)
\begin{equation}\label{semidirect} M_j=\left( \begin{array}{cc}N_j&R_j\\0&N_j\end{array}\right) ~,~N_j,R_j\in GL(2,{\bf C}) \end{equation} with $N_j$ as above. The blocks $R_j$ are such that for $j=1,2,3$ the matrices $M_j$ are diagonalizable while $M_4$ has JNF $J_4$ (i.e. rk$R_4=1$).
The absence of other possible types of representations is guaranteed by the following theorem which follows from Theorem 1.1.2 from \cite{Ka}. The theorem and its proof were suggested by Ofer Gabber.
\begin{tm}\label{Gabber} For fixed conjugacy classes with index of rigidity 2 there cannot coexist irreducible and reducible $(p+1)$-tuples of matrices $M_j$. \end{tm}
The theorem is proved in the Appendix. It follows from the theorem that there can exist only reducible quadruples of matrices $M_j$ in the example under consideration.
\begin{prop}\label{only} One has ${\cal V}={\cal U}\cup {\cal W}$. \end{prop}
\begin{prop}\label{R_4} 1) In a quadruple (\ref{semidirect}) the matrix $R_4$ is nilpotent of rank 1 and for $j=1,2,3$ one has $R_j=[N_j,Z_j]$ with $Z_j\in sl(2,{\bf C})$.
2) If the matrices $N_1$, $N_2$, $N_3$ are fixed, then for every nilpotent rank 1 matrix $R_4$ there exists a quadruple of matrices (\ref{semidirect}). \end{prop}
\begin{prop}\label{centralizer} The centralizers in $SL(4,{\bf C})$ of the quadruples (\ref{direct}) and (\ref{semidirect}) are both of dimension 1. They consist respectively of the matrices \[ \left( \begin{array}{cc}\alpha I&0\\0&\pm \alpha ^{-1}I\end{array}\right) ~ {\rm and~}\left( \begin{array}{cc}\delta I&\beta I\\0&\delta I \end{array}\right) ~,~\alpha \in {\bf C}^*~,~\beta \in {\bf C}~,~\delta ^4=1\] \end{prop}
\begin{prop}\label{UW} The stratum ${\cal W}$ belongs to the closure of the stratum ${\cal U}$. \end{prop}
\begin{prop}\label{dimW} The stratum ${\cal W}$ is an irreducible smooth variety of dimension 15. \end{prop}
\begin{prop}\label{dimU} The stratum ${\cal U}$ is an irreducible smooth variety of dimension 16. \end{prop}
\begin{rem} The closure of the variety ${\cal W}$ (hence, the one of ${\cal U}$ as well) contains the variety ${\cal Y}$ of quadruples which up to conjugacy are of the form (\ref{semidirect}) with $R_j=0$ for all $j$. For such quadruples
1) the matrix $M_4$ is scalar;
2) they define direct sums of two equivalent irreducible rigid representations.
There exist no irreducible such quadruples of matrices $M_j$ or $A_j$ because the conditions of Theorem~\ref{generic} are not fulfilled (neither the necessary condition $(\alpha _n)$). \end{rem}
\begin{prop}\label{dimY} The variety ${\cal Y}$ is smooth and irreducible. One has dim${\cal Y}=12$. \end{prop}
\section{Proofs of the propositions\protect\label{proofofUW}}
{\bf Proof of Proposition~\ref{N_jP_j}:}
$1^0$. The variety of quadruples of matrices $N_j$ is obtained by conjugating one such quadruple by matrices from $SL(2,{\bf C})$ (indeed, rigid $(p+1)$-tuples are unique up to conjugacy, see \cite{Ka} and \cite{Si}). This proves the connectedness. The smoothness and the dimension follow from Proposition~\ref{smooth}.
$2^0$. Denote by $C_j^*$ the conjugacy class of the matrix $P_j$. Prove that the variety $\Pi$ of quadruples of matrices $P_j$ is connected. Denote by $\delta$ the product det$P_1$det$P_2$. By varying the matrices $P_1$ and $P_2$ (resp. $P_3$ and $P_4$) one can obtain as their product $P_1P_2$ (resp. as $P_4^{-1}P_3^{-1}$) any matrix from the set $\Delta (\delta )$ of $2\times 2$-matrices with determinant equal to $\delta$. The set $\Delta (\delta )$ being connected so is the variety $\Pi$ because
$\Pi =\{ (P_1,P_2,P_3,P_4)|P_j\in C_j^*,P_1P_2=P_4^{-1}P_3^{-1}\}$.
$3^0$. The eigenvalues of the matrices $P_j$ being generic the variety $\Pi$ contains no reducible quadruples. Hence, the variety $\Pi$ is smooth, one has dim$\Pi =5$, see Proposition~\ref{smooth}.
$4^0$. Part 2) is proved by analogy with $2^0$ and $3^0$.\hspace{1cm}$\Box$\\
{\bf Proof of Proposition~\ref{only}:}
$1^0$. A quadruple from ${\cal V}$ is block upper-triangular up to conjugacy. The eigenvalues being relatively generic the diagonal blocks can be only of size 2 and the restrictions of the matrices $M_j$ to them can be with conjugacy classes like in the cases of quadruples of matrices $N_j$ or $P_j$.
$2^0$. Show that if one of the diagonal blocks is a quadruple of matrices $N_j$ and the other one of matrices $P_j$, then this is a direct sum conjugate to a quadruple (\ref{direct}). Indeed, for the representations $P$ and $N$ defined by the quadruples of matrices $P_j$ and $N_j$ one has Ext$^1(P,N)=$Ext$^1(N,P)=0$ (to be checked directly). This implies that a block upper-triangular quadruple of matrices $M_j$ with diagonal blocks $N_j$ and $P_j$ is conjugate to its restriction to the two diagonal blocks, i.e. the quadruple is a point from ${\cal U}$. On the other hand, if both diagonal blocks equal $N_j$, then the quadruple is like in (\ref{semidirect}).
Hence, only quadruples like the ones from ${\cal U}$ and ${\cal W}$ can exist in ${\cal V}$.\hspace{1cm}$\Box$\\
{\bf Proof of Proposition~\ref{R_4}:}
$1^0$. The blocks $R_1$, $R_2$, $R_3$ must be of the form $R_j=[N_j,Z_j]$ for some matrices $Z_j\in gl(n,{\bf C})$. Indeed, it suffices to prove this in the assumption that $N_j$ is diagonal: $N_j=\left( \begin{array}{cc}\lambda &0\\0&\mu \end{array}\right)$, $\lambda \neq \mu$. Set $R_j=\left( \begin{array}{cc}g&h\\f&s\end{array}\right)$. One must have $g=s=0$, otherwise $M_j$ will not be diagonalizable. But then $R_j=[N_j,Z_j]$ with $Z_j=\left( \begin{array}{cc}0&h/(\lambda -\mu )\\f/(\mu -\lambda )&0 \end{array}\right)$.
On the other hand, if for $j=1,2,3$ one has $R_j=[N_j,Z_j]$, then the matrices $M_1$, $M_2$, $M_3$ have the necessary JNFs -- one has $M_j=\left( \begin{array}{cc}I&Z_j\\0&I\end{array}\right) ^{-1} \left( \begin{array}{cc}N_j&0\\0&N_j\end{array}\right) \left( \begin{array}{cc}I&Z_j\\0&I\end{array}\right)$ .
$2^0$. If one has rk$R_4=0$, then $R_4=0$ and $M_4$ must be scalar, i.e. $M_4\not\in C_4$. If rk$R_4=2$, then rk$(M_4+I)=2$ and again $M_4\not\in C_4$. Hence, rk$R_4=1$. This leaves two possibilities -- either $R_4$ has two distinct eigenvalues one of which is 0 or it is nilpotent.
$3^0$. The condition $M_1\ldots M_4=I$ restricted to the right upper block and to each of the diagonal blocks reads respectively
\[ R_1N_2N_3N_4+N_1R_2N_3N_4+N_1N_2R_3N_4+N_1N_2N_3R_4=0~,~N_1N_2N_3=-I~.\] Hence, the first of these two equalities takes the form
\[ -R_1-N_1R_2(N_1)^{-1}-(N_1N_2)R_3(N_1N_2)^{-1}-R_4=0~.\] As $R_j=[N_j,Z_j]$, $j=1,2,3$, see $1^0$, one has
\[ {\rm tr}R_1={\rm tr}R_2={\rm tr}(N_1R_2(N_1)^{-1})={\rm tr}R_3= {\rm tr}((N_1N_2)R_3(N_1N_2)^{-1})=0~.\] Hence, tr$R_4=0$. This means that $R_4$ is nilpotent, of rank 1. This proves 1).
$4^0$. To prove 2) one has to recall that $R_j=[N_j,Z_j]$ for $j=1,2,3$, see $1^0$, and that each matrix from $sl(2,{\bf C})$ can be represented as $\sum _{j=1}^3[N_j,Z_j]$, see Proposition~\ref{commut}. Hence, for every nilpotent $R_4$ one can find matrices $Z_j$ such that for $j=1,2,3$ one has $R_j=[N_j,Z_j]$, i.e. $M_j\in C_j$ and $M_1M_2M_3M_4=I$.\hspace{1cm}$\Box$\\
{\bf Proof of Proposition~\ref{centralizer}:}
$1^0$. Denote by $F=\left( \begin{array}{cc}U&V\\W&Y\end{array}\right)$ a matrix from the centralizer of the quadruple. In the case of a quadruple (\ref{direct}) the commutation relations read:
\[ [U,N_j]=[Y,P_j]=0~,~N_jV=VP_j~,~WN_j=P_jW~.\] The representations defined by the matrices $N_j$ and $P_j$ being non-equivalent, these relations imply $V=W=0$. The irreducibility of the quadruples of matrices $N_j$ and $P_j$ and Schur's lemma imply that $U$ and $Y$ are scalar. Hence, $U=\alpha I$, $Y=\xi I$ with $\alpha ^2\xi ^2=1$, i.e. $\xi =\pm \alpha ^{-1}$.
$2^0$. In the case of a quadruple (\ref{semidirect}) the matrix algebra ${\cal A}$ generated by the matrices $M_j$ contains the matrix $M_4+I$ and its left and right products by matrices from the algebra ${\cal B}$ generated by $M_1$, $M_2$ and $M_3$. As ${\cal B}$ contains matrices of the form $\left( \begin{array}{cc}T&\ast \\0&T\end{array}\right)$ for any $T\in gl(2,{\bf C})$ (the Burnside theorem), the algebra ${\cal A}$ contains all matrices of the form $\left( \begin{array}{cc}0&Q\\0&0\end{array}\right)$ with $Q\in gl(2,{\bf C})$.
The commutation relations imply that $WQ=0$, hence, $W=0$, and $UQ=QY$ for any $Q$, i.e. $U=Y=\delta I$. Finally, one has $[N_j,V]=0$ which implies that $V=\beta I$ (use Schur's lemma).
One must have $\delta ^4=1$ because $F\in SL(4,{\bf C})$.\hspace{1cm}$\Box$\\
{\bf Proof of Proposition~\ref{UW}:}
$1^0$. One can deform the matrices $M_j$ from a quadruple from ${\cal W}$ as follows. The deformation parameter is denoted by $\varepsilon \in ({\bf C},0)$ and the deformed matrices by $M_j'$. Assume that $N_4=-I$, $R_4=\left( \begin{array}{cc}0&1\\0&0\end{array}\right)$ (one can achieve this by conjugation of the quadruple with a block-diagonal matrix). Set $M_4'=M_4+\varepsilon (E_{1,2}+w(\varepsilon )E_{1,3})$; the matrix $E_{k,j}$ by definition has a single non-zero entry equal to 1 in position $(k,j)$; $w(\varepsilon )$ is an unknown germ of an analytic function.
$2^0$. For $j=1,2,3$ set $M_j'=(I+\varepsilon X_j(\varepsilon ))^{-1}M_j (I+\varepsilon X_j(\varepsilon ))$ where $X_j=\left( \begin{array}{cc}U_j&V_j\\0&0\end{array}\right)$. Set $X_j(0)=X_j^0$. One must have $M_1'M_2'M_3'M_4'=I$ which in first approximation w.r.t. $\varepsilon$ reads \begin{equation}\label{FA} [M_1,X_1^0]M_2M_3M_4+M_1[M_2,X_2^0]M_3M_4+M_1M_2[M_3,X_3^0]M_4+ M_1M_2M_3(E_{1,2}+w(0)E_{1,3})=0 \end{equation}
$3^0$. Set $U_j^0=U_j(0)$, $U^0=(U_1^0,U_2^0,U_3^0)$, $V_j^0=V_j(0)$, $V^0=(V_1^0,V_2^0,V_3^0)$, $w^0=w(0)$. Equation (\ref{FA}) restricted to the left upper block reads:
\[ {\cal G}(U^0):=[N_1,U_1^0]N_2N_3+N_1[N_2,U_2^0]N_3+ N_1N_2[N_3,U_3^0]=N_1N_2N_3E_{1,2}\] (because $N_4=-I$). Making use of $N_1N_2N_3=-I$ one finds
\begin{equation}\label{N} [N_1,U_1^0N_1^{-1}]+[N_1N_2N_1^{-1},N_1U_2^0N_2^{-1}N_1^{-1}]+ [N_3,N_3^{-1}U_3^0]=E_{1,2} \end{equation} The triple of matrices $N_1$, $N_2$, $N_3$ is irreducible, hence, so is the triple $N_1$, $N_1N_2N_1^{-1}$, $N_3$. By Proposition~\ref{commut}, one can find matrices $U_j^0$ satisfying equation (\ref{N}).
$4^0$. Equation (\ref{FA}) restricted to the right lower block is of the form $0=0$, i.e. it gives no condition at all upon $U_j^0$, $V_j^0$ and $w^0$. Its restriction to the right upper block reads:
\begin{equation}\label{V} {\cal F}(V^0,U^0,w^0):= {\cal G}(V^0)+{\cal H}(U^0)-w^0E_{1,3}=0 \end{equation} where ${\cal H}$ is some linear form in the entries of the matrices $U_j^0$. Hence, if $U_j^0$ are found such that (\ref{N}) holds, then one can find $w^0$ such that tr$({\cal H}(U_1^0,U_2^0,U_3^0))=w^0$. After this one can find matrices $V_j^0$ such that (\ref{V}) hold.
$5^0$. The map $(U^0,V^0,w^0)\mapsto ({\cal G}(U^0),{\cal F}(V^0,U^0,w^0))$ is surjective onto the space of $2\times 4$-matrices. By the implicit function theorem one can find germs of matrices $U_j$, $V_j$ and a germ of a function $w$ holomorphic in $\varepsilon$ at 0 such that $M_1'\ldots M_4'=I$.
Fix $\varepsilon \neq 0$. The quadruple of matrices $M_j'$ is block upper-triangular with diagonal blocks having the properties of $P_j$ and $N_j$ ($P_j$ is above). Moreover, each of the matrices $M_j'$ is conjugate to the block-diagonal matrix whose restriction to the two diagonal blocks is the same as the one of $M_j'$ (to be checked directly). By Proposition~\ref{only}, up to conjugacy the quadruple of matrices is like the one from (\ref{direct}).\hspace{1cm}$\Box$\\
{\bf Proof of Proposition \ref{dimW}:}
$1^0$. Prove the irreducibility. The variety ${\cal W}$ is obtained by conjugating with matrices from $SL(4,{\bf C})$ the quadruples of matrices of the form (\ref{semidirect}) with $R_4$ nilpotent of rank 1. The orbit of $R_4$ is an irreducible variety which implies the irreducibility of ${\cal W}$.
$2^0$. Fix the blocks $N_j$ of a quadruple (\ref{semidirect}). The variety ${\cal S}$ of such quadruples defined modulo conjugacy is of dimension 1. Indeed, the orbit of $R_4$ is of dimension 2. The only conjugations that preserve the form of the quadruple and its restrictions to the two diagonal blocks are with matrices of the form $\left( \begin{array}{cc}aI&V\\0&bI\end{array}\right)$, $ab\neq 0$, $V\in gl(2,{\bf C})$; this is proved in $4^0$. If one requires the matrix to be from $SL(4,{\bf C})$, this means that $b=\pm 1/a$ and factoring out these conjugations decreases the dimension by 1. Indeed, such a conjugation changes $R_4$ to $bR_4/a$, the presence of $V$ does not affect the block $R_4$.
$3^0$. To obtain the variety ${\cal H}$ of all quadruples defining semi-direct sums like (\ref{semidirect}) one has to conjugate the quadruples from ${\cal S}$ by matrices from $SL(4,{\bf C})$. This increases the dimension by 14 (not by 15 because the centralizer of such a quadruple is non-trivial, of dimension 1, see Proposition~\ref{centralizer}). Hence, dim${\cal H}=15$.
$4^0$. Denote by $G$ a matrix the conjugation with which preserves the block upper-triangular form of the quadruple and the blocks $N_j$. If $G=\left( \begin{array}{cc}U&V\\W&Y\end{array}\right)$, then the condition the quadruple to remain block upper-triangular implies that $[W,N_j]=0$, i.e. $W=hI$. The condition the diagonal blocks of $M_4$ to remain the same implies $[N_4,Y]-WR_4=R_4W+[N_4,U]=0$. As $N_4=-I$, one has $[N_4,Y]=[N_4,U]=0$, i.e. $W=0$.
The conditions $[N_j,U]=[N_j,Y]=0$ imply that $U=aI$, $Y=bI$.\hspace{1cm}$\Box$\\
{\bf Proof of Proposition \ref{dimU}:}
$1^0$. The varieties of quadruples of matrices $N_j$ or $P_j$, see Proposition~\ref{N_jP_j}, are smooth, irreducible and of dimensions respectively 3 and 5. Hence, the variety ${\cal P}$ of quadruples of matrices $M_j$ like in (\ref{direct}) is smooth, irreducible and of dimension 8.
$2^0$. The variety ${\cal U}$ is of dimension $8+15-7=16$. Here ``8'' stands for ``dim${\cal P}$'', ``15'' stands for ``dim$SL(4,{\bf C})$'' and 7 is the dimension of the subgroup of $SL(4,{\bf C})$ of block-diagonal matrices with blocks $2\times 2$ conjugation with which preserves the block-diagonal form of quadruple (\ref{direct}) (infinitesimal conjugations {\em only} with such matrices preserve the block-diagonal form of quadruple (\ref{direct})); this subgroup contains the centralizer of quadruple (\ref{direct}), see Proposition~\ref{centralizer}. \hspace{1cm}$\Box$\\
{\bf Proof of Proposition \ref{dimY}:}
The variety ${\cal Y}$ is the orbit of one quadruple of the form (\ref{semidirect}) with $R_j=0$, $j=1,\ldots ,4$, under conjugation by $SL(4,{\bf C})$ (recall that the matrices $N_j$ define a rigid representation, i.e. unique up to conjugacy). Hence, ${\cal Y}$ is irreducible and smooth.
To obtain dim${\cal Y}$ one has to subtract from $15=$dim$SL(4,{\bf C})$ the dimension of the centralizer in $SL(4,{\bf C})$ of the above quadruple. The latter equals 3 -- the centralizer is the set of all matrices of the form $\left( \begin{array}{cc}\alpha I&\beta I\\ \delta I&\eta I \end{array}\right)$ with $\alpha \eta -\delta \beta=\pm 1$.\hspace{1cm}$\Box$
\section{Another example with index of rigidity 2\protect\label{ex2}}
Consider the variety ${\cal V}$ in the case when $p=2$, $n=4$, the three conjugacy classes are diagonalizable and have eigenvalues $(a,a,b,c)$, $(f,f,g,h)$ and $(u,u,v,w)$ (different letters denote different eigenvalues). The index of rigidity equals 2 (to be checked directly).
The eigenvalues are presumed to satisfy the only non-genericity relation $abfguv=1$. Hence, for such conjugacy classes there exist irreducible triples of diagonalizable matrices $L_j\in gl(2,{\bf C})$ (resp. $B_j\in gl(2,{\bf C})$) with eigenvalues $(a,b)$; $(f,g)$; $(u,v)$ (resp. $(a,c)$; $(f,h)$; $(u,w)$) such that $L_1L_2L_3=I$ (resp. $B_1B_2B_3=I$). This follows from Theorem~\ref{generic}. Hence, there exist triples of block-diagonal matrices $M_j$ with diagonal blocks equal to $L_j$ and $B_j$. Denote by ${\cal D}$ the variety of such triples. By Theorem~\ref{Gabber}, irreducible triples of matrices $M_j$ do not exist.
There do exist, however, triples with trivial centralizers which are block upper-triangular: $M_j=\left( \begin{array}{cc}L_j&T_j\\0&B_j \end{array}\right)$ where $T_j=L_jY_j-Y_jB_j$ for some $Y_j\in gl(2,{\bf C})$ because $M_j$ is conjugate to $\left( \begin{array}{cc}L_j&0\\0&B_j \end{array}\right)$. The condition $M_1M_2M_3=I$ restricted to the right upper block reads:
\[ T_1B_2B_3+L_1T_2B_3+L_1L_2T_3=0~~~~~~(*)\] Thus the triple of matrices $T_j$ belongs to the space
\[ {\cal T}=\{ (T_1,T_2,T_3)~|~T_j=L_jY_j-Y_jB_j~,~Y_j\in gl(2,{\bf C})~,~ T_1B_2B_3+L_1T_2B_3+L_1L_2T_3=0\} ~.\]
{\em One has dim${\cal T}=5$.}
Indeed, the conditions $T_j=L_jY_j-Y_jB_j$ imply that each matrix $T_j$ belongs to the image of the map $(.)\mapsto L_j(.)-(.)B_j$ which is a subspace of $gl(2,{\bf C})$ of dimension 3. Condition (*) is equivalent to four linearly independent equalities (we let the reader prove their linear independence using the non-equivalence of the representations defined by the matrices $L_j$ and $B_j$).
Consider the space
\[ {\cal Q}=\{ (T_1,T_2,T_3)~|~T_j=L_jY-YB_j~,~Y\in gl(2,{\bf C})\} ~.\] For such matrices $T_j$ there holds $(*)$, therefore ${\cal Q}\subset {\cal T}$. The space ${\cal Q}$ is the space of right upper blocks of triples of block upper-triangular matrices $M_j$ which are obtained from block-diagonal ones from ${\cal D}$ by conjugation with matrices of the form $\left( \begin{array}{cc}I&Y\\0&I\end{array}\right)$.
{\em One has dim${\cal Q}=4$.}
Indeed, for no matrix from $gl(2,{\bf C})$ does one have $L_jY-YB_j=0$ for $j=1,2,3$ because the triples of matrices $L_j$ and $B_j$ define non-equivalent representations.
Hence, dim$({\cal T}/{\cal Q})=1$. Choose the triple of matrices $Y_j$ to span the factorspace $({\cal T}/{\cal Q})$. Hence, the centralizer ${\cal Z}$ of the triple of matrices $M_j$ will be trivial. Indeed, let $Z=\left( \begin{array}{cc}P&Q\\R&S\end{array}\right) \in {\cal Z}$. Hence, $RL_j=B_jR$ for $j=1,2,3$ (commutation relations restricted to the left lower block), i.e. $R=0$ because the matrices $L_j$ and $B_j$ define non-equivalent representations.
One must have $[P,L_j]=[S,B_j]=0$ (commutation relations restricted to the diagonal blocks), i.e. $P=aI$, $B=bI$. But then one must have (commutation relations restricted to the right upper block) $(a-b)T_j=L_jQ-QB_j$ which means that $a=b$ (otherwise $(T_1,T_2,T_3)\in {\cal Q}$), hence, $L_jQ-QB_j=0$ for $j=1,2,3$, i.e. $Q=0$. Hence, $Z=aI$.
\begin{rems} 1) It is clear that the variety ${\cal D}$ belongs to the closure of ${\cal V}\backslash {\cal D}$ -- the triple of matrices $M_j=\left( \begin{array}{cc}L_j&\varepsilon T_j\\0&B_j\end{array}\right)$ belongs to ${\cal V}\backslash {\cal D}$ for $\varepsilon \neq 0$, for $\varepsilon =0$ it belongs to ${\cal D}$.
2) The variety ${\cal V}$ is connected, hence, irreducible. This follows from $({\cal T}/{\cal Q})$ being a linear space (${\cal V}$ is obtained by conjugating block upper-triangular triples with $(T_1,T_2,T_3)\in ({\cal T}/{\cal Q})$ and with fixed diagonal blocks by matrices from $SL(4,{\bf C})$). \end{rems}
\section{A third example with index of rigidity 2\protect\label{ex3}}
Let $n=4$, $p=2$. Use the notation from the previous section. Define the conjugacy classes $C_j$ as follows: their eigenvalues equal $(a,a,b,b)$, $(f,f,g,g)$, $(u,u,v,v)$, the eigenvalues are relatively generic but not generic (one has $abfguv=1$). To each of the eigenvalues $a$, $b$ and $f$ there corresponds a single Jordan block of size 2, to each of the eigenvalues $g$, $u$, $v$ there correspond two Jordan blocks of size 1. Hence, the index of rigidity equals 2.
The variety ${\cal V}$ contains triples of matrices which up to conjugacy are block upper-triangular with two diagonal blocks equal to $L_j$, see their definition in the previous section. By Theorem~\ref{Gabber}, ${\cal V}$ contains no irreducible triples. Hence, it contains none with trivial centralizer either because the matrices $M_j$ from any such block upper-triangular triple commute with the matrix $E_{1,3}+E_{2,4}$; on the other hand, if a triple of matrices $M_j\in C_j$ is conjugated to a block upper-triangular form, then the diagonal blocks are of size 2 and up to conjugacy they equal $L_j$ -- this follows from the choice of the eigenvalues.
\begin{prop}\label{dimV} One has dim${\cal V}=15$ which is the expected dimension. \end{prop}
\begin{rems} The closure of the variety ${\cal V}$ contains the varieties in which at least one of the two Jordan normal forms $J(M_1)$ and $J(M_2)$ contains instead of some Jordan block(s) of size 2 two Jordan blocks of size 1. We leave the details for the reader. One can prove that ${\cal V}$ is irreducible. \end{rems}
{\bf Proof of Proposition~\ref{dimV}:}
$1^0$. Suppose that one has $M_j=\left( \begin{array}{cc}L_j&T_j\\0&L_j \end{array}\right)$ with $L_1=$diag$(a,b)$, $T_1=$diag$(1,1)$. Fix $L_2$ and $L_3$. Then the couple of blocks $(T_2,T_3)$ belongs to a space of dimension 1.
Indeed, one has $T_3=[L_3,Z_3]$ in order $M_3$ to be diagonalizable and the dimension of the image of the map $Z_3\mapsto [L_3,Z_3]$ in $gl(2,{\bf C})$ equals 2.
The block $T_2$ belongs to an affine space of dimension 2. Indeed, one has $T_2=S+[L_2,Z_2]$, where the dimension of the image of the map $Z_2\mapsto [L_2,Z_2]$ equals 2 and the matrix $S$ is defined as follows. Set $L_2=H^{-1}$diag$(f,g)H$. Then $S=\xi H^{-1}E_{1,3}H$ where $\xi$ satisfies the condition
\[ {\rm tr}(L_2L_3+L_1SL_3)=0~~~~~~(**)\] (If by chance this condition gives $\xi =0$, then one has to choose two diagonal entries of $T_1$ other than $(1,1)$ so that $\xi \neq 0$, otherwise $M_2$ will be diagonalizable.)
$2^0$. The coefficient $\xi$ satisfies condition (**) for the following reason. The condition $M_1M_2M_3=I$ implies that ${\cal H}:=(T_1L_2L_3+L_1T_2L_3+L_1L_2T_3)=0$. In particular, tr${\cal H}=0$. As
\[ L_1L_2L_3=I~,~T_1=I~,~ {\rm tr}(L_1L_2T_3)={\rm tr}(L_1L_2L_3Z_3-L_1L_2Z_3L_3)={\rm tr} (Z_3-L_3^{-1}Z_3L_3)=0\] and tr$(L_1[L_2,Z_2]L_3)=$tr$(L_3^{-1}Z_2L_3-L_1Z_2L_1^{-1})=0$, one has tr$(L_2L_3+L_1SL_3)=0$.
$3^0$. From the dimension $2+2$ of the space to which the couple $(T_2,T_3)$ belongs one has to subtract 3 because the equation ${\cal H}=0$ (after one has chosen $\xi$ so that tr${\cal H}=0$) imposes 3 conditions.
$4^0$. The centralizer ${\cal Z}$ of the triple of matrices $M_j$ in $SL(4,{\bf C})$ is generated by the matrix $E_{1,3}+E_{2,4}$. Moreover, any matrix from $SL(4,{\bf C})$ the conjugation with which preserves the form of the triple belongs to ${\cal Z}$. This can be proved by a direct computation which we leave for the reader.
$5^0$. To find the dimension of ${\cal V}$ one has to conjugate the block upper-triangular triples from $1^0$ whose variety is of dimension 1 by matrices from $SL(n,{\bf C})/{\cal Z}$. The latter variety is of dimension 14. Hence, dim${\cal V}=15$.\hspace{1cm}$\Box$
\section{A fourth example with index of rigidity 2\protect\label{ex4}}
Let $n=p=3$ and let the conjugacy classes $C_j$ define diagonal but non-scalar JNFs the eigenvalues being equal respectively to $(a,1,1)$, $(b,1,1)$, $(c,1,1)$, $(d,1,1)$, with $abcd=1$. Hence, the index of rigidity is 0. There exist reducible such quadruples of matrices $M_j$ with trivial centralizers. Example:
\[ M_1=\left( \begin{array}{ccc}a&0&0\\0&1&0\\0&0&1 \end{array}\right) ~,~M_2=\left( \begin{array}{ccc}b&1&0\\0&1&0\\0&0&1 \end{array}\right) \]
\[ M_3=\left( \begin{array}{ccc}c&0&1\\0&1&0\\0&0&1 \end{array}\right) ~,~M_4=\left( \begin{array}{ccc}d&-1/bc&-1/c\\0&1&0\\0&0&1 \end{array}\right) \] (the reader is invited to check the triviality of the centralizer oneself). Denote by ${\cal T}$ the stratum of ${\cal V}$ of quadruples with trivial centralizers. Hence, dim${\cal T}=8$ (Proposition~\ref{smooth}). By Theorem~\ref{Gabber}, there exist no irreducible quadruples of matrices $M_j\in C_j$.
On the other hand, there exist quadruples defining direct sums of an irreducible representation of rank 2 and of a one-dimensional one. Example:
\[ M_1=\left( \begin{array}{ccc}a&1&0\\0&1&0\\0&0&1 \end{array}\right) ~,~M_2=\left( \begin{array}{ccc}b&-1/a&0\\0&1&0\\0&0&1 \end{array}\right) \]
\[ M_3=\left( \begin{array}{ccc}c&0&0\\-1/d&1&0\\0&0&1 \end{array}\right) ~,~M_4=\left( \begin{array}{ccc}d&0&0\\1&1&0\\0&0&1 \end{array}\right) \] Denote by ${\cal S}$ the stratum of ${\cal V}$ of quadruples defining such direct sums.
{\em One has dim${\cal S}=9$.}
Indeed, the subvariety ${\cal S}'\subset {\cal S}$ of block-diagonal such quadruples is of dimension 5 (Proposition~\ref{smooth}). Hence, ${\cal S}$ is obtained from ${\cal S}'$ by conjugating with matrices from $SL(3,{\bf C})$ (dim$SL(3,{\bf C})=8$) and one has to factor out the conjugation with block-diagonal matrices whose subgroup is of dimension 4. Thus dim${\cal S}=5+8-4=9$.
\begin{rems} 1) Both strata ${\cal S}$ and ${\cal T}$ contain in their closures the variety of quadruples which are diagonal up to conjugacy, also the ones of quadruples defining direct sums of the one-dimensional representation 1, 1, 1, 1 with the semi-direct sums of the representations 1, 1, 1, 1 and $a$, $b$, $c$, $d$.
2) The stratum ${\cal T}$ {\em does not} lie in the closure of the stratum ${\cal S}$ (triviality of the centralizer is an ``open'' property).
3) One can show that at every point of ${\cal V}$ one has dim${\cal V}\leq 9$. \end{rems}
\section{An example with zero index of rigidity\protect\label{ex5}}
By Theorem~\ref{generic}, there exist irreducible quadruples of matrices $A_j$ or $M_j$ of size 2 in which each matrix has two distinct eigenvalues and the eigenvalues are generic. For such quadruples the index of rigidity equals 0 (to be checked directly).
Consider a quadruple of matrices (say, $M_j$; for matrices $A_j$ one can give a similar example) of the form
\[ M_j=\left( \begin{array}{cc}B_j&0\\0&G_j\end{array}\right)\] where each of the quadruples of matrices $B_j$ and $G_j$ is like above, with generic eigenvalues. Moreover, for each $j$ the eigenvalues of $B_j$ and $C_j$ are the same but the quadruples of matrices $B_j$ and $G_j$ define non-equivalent representations. To choose them such is possible because the quadruples are not rigid.
Compute the dimension of the variety ${\cal M}$ of such quadruples of matrices $M_j$. The varieties ${\cal B}$ and ${\cal G}$ of quadruples of $2\times 2$-matrices $B_j$ or $G_j$ are both of dimension 5 (see part 2) of Proposition~\ref{N_jP_j}).
Hence, dim${\cal M}=10$. The variety ${\cal N}$ of quadruples of matrices $M_j$ defining a direct sum of two representations of rank 2 with the properties of ${\cal B}$ and ${\cal G}$ is obtained by conjugating the quadruples from ${\cal M}$ by matrices from $SL(4,{\bf C})$. Infinitesimal conjugation by block-diagonal matrices from $SL(4,{\bf C})$ with two diagonal blocks of size 2 and only by such matrices preserves ${\cal M}$ (their subgroup is of dimension 7 in $SL(4,{\bf C})$). Hence, dim${\cal N}=10+15-7=18$ where $15=$dim$SL(4,{\bf C})$.
The expected dimension of the variety ${\cal N}$ equals 17, see Proposition~\ref{smooth}. In a subsequent paper the author intends to prove that for zero index of rigidity and for relatively generic but not generic eigenvalues the Deligne-Simpson problem is not weakly solvable. Hence, in the above example one has ${\cal V}={\cal N}$ and the dimension of ${\cal V}$ is higher than the expected one.
\begin{conjecture} 1) Is it true that for negative indices of rigidity the dimension of the variety of $(p+1)$-tuples with non-trivial centralizers is always smaller than the expected dimension of the variety of all $(p+1)$-tuples (of matrices $M_j$ or $A_j$) ?
2) Is it true that for negative indices of rigidity if the Jordan normal forms $J^n_1$, $\ldots$, $J^n_{p+1}$ satisfy the conditions of Theorem~\ref{generic}, then the Deligne-Simpson problem is weakly solvable for any eigenvalues ? \end{conjecture}
{\Large\bf Appendix. Proof of Theorem~\ref{Gabber} (by Ofer Gabber)}\\
$1^0$. We use arguments related to the ones from \cite{Ka}. Suppose we are given the conjugacy classes $C_i\subset GL(n,{\bf C})$, $1\leq i\leq p+1$, and we are interested in solutions of
\begin{equation}\label{star} M_1\ldots M_{p+1}={\rm id}~,~M_i\in C_i \end{equation}
We say that a solution $M=(M_1,\ldots ,M_{p+1})$ is {\em rigid} if every solution $M'$ in some neighbourhood of $M$ is $GL(n,{\bf C})$-conjugate to $M$. Here ``neighbourhood'' can be taken in the classical or in the Zariski topology.
$2^0$. Consider distinct points $a_1$, $\ldots$, $a_{p+1}\in {\bf P}_{{\bf C}}^1$ and set $U={\bf P}_{{\bf C}}^1\backslash \{ a_1,\ldots ,a_{p+1}\}$. Choose a base point $x_0\in U$ and a standard set of generators $\gamma _i\in \pi _1(U,x_0)$ where $\gamma _i$ is freely homotopic to a positive loop around $a_i$, $\gamma _1\ldots \gamma _{p+1}=1$ (using $\pi _1$ conventions as in Deligne LNM 163).
Then a solution of (\ref{star}) determines a local system $L$ on $U$, $L_{x_0}\simeq {\bf C}^n$; the local monodromies are given by the matrices $M_i$.
$3^0$. Recall that if $f:X\rightarrow Y$ is an algebraic map of irreducible algebraic varieties, then every irreducible component of a fibre of $f$ has dimension $\geq$dim$(X)-$dim$(Y)$.
Suppose we are given a rigid solution of (\ref{star}). In particular, if $\delta _i$ is the value of the determinant on $C_i$, then $\prod \delta _i=1$, so we have the product morphism
\[ f:C_1\times \ldots \times C_{p+1}\rightarrow SL(n,{\bf C})\] and by assumption the $GL(n,{\bf C})$-orbit of $(M_1,\ldots ,M_{p+1})$ is dense in an irreducible component of $f^{-1}($id$)$. The above orbit is also an $SL(n,{\bf C})$-orbit, so it is of dimension $\leq n^2-1$.
$4^0$. Hence,
\[ \sum _{i=1}^{p+1}d_i\leq 2(n^2-1)~.\] Denote by $j$ the inclusion of $U$ in ${\bf P}_{{\bf C}}^1$ and by ${\cal Z}(M_i)$ the space of matrices commuting with $M_i$. Then $d_i=n^2-$dim${\cal Z}(M_i)$ and by the Euler-Poincar\'e formula (cf. \cite{Ka} p. 16) the above inequality is equivalent to
\[ \chi ({\bf P}_{{\bf C}}^1,{\rm j}_*\underline{{\rm End}}(L))\geq 2~.\]
Now if $F$ is a rank $n$ irreducible local system with local monodromies in the prescribed conjugacy classes, then by the Euler-Poincar\'e formula
\[ \chi ({\bf P}_{{\bf C}}^1,{\rm j}_*\underline{{\rm End}}(L))= \chi ({\bf P}_{{\bf C}}^1,{\rm j}_*\underline{{\rm Hom}}(L,F))\geq 2~,\] so one of the the two cohomology groups $H^0({\bf P}_{{\bf C}}^1,{\rm j}_*\underline{{\rm Hom}}(L,F))\cong {\rm Hom} _U(L,F)$ or
\[ H^2({\bf P}_{{\bf C}}^1,{\rm j}_*\underline{{\rm Hom}}(L,F))\cong H_c^2(U,\underline{{\rm Hom}}(L,F))\cong {\rm Hom} _U(F,L)^{{\sf v}}\] is non-zero, which implies (as $F$ is irreducible) that $F\simeq L$ (cp. \cite{Ka}, Theorem 1.1.2). Hence, if $L$ is reducible, then $F$ does not exist.\hspace{1cm}$\Box$
Author's address: Universit\'e de Nice, Laboratoire de Math\'ematiques, Parc Valrose, 06108 Nice Cedex 2, France,
tel.: (0033) 4 92 07 62 67
fax : (0033) 4 93 51 79 74
e-mail [email protected]
\end{document} | arXiv |
Random walk closeness centrality
Random walk closeness centrality is a measure of centrality in a network, which describes the average speed with which randomly walking processes reach a node from other nodes of the network. It is similar to the closeness centrality except that the farness is measured by the expected length of a random walk rather than by the shortest path.
The concept was first proposed by White and Smyth (2003) under the name Markov centrality.[1]
Intuition
Consider a network with a finite number of nodes and a random walk process that starts in a certain node and proceeds from node to node along the edges. From each node, it chooses randomly the edge to be followed. In an unweighted network, the probability of choosing a certain edge is equal across all available edges, while in a weighted network it is proportional to the edge weights. A node is considered to be close to other nodes, if the random walk process initiated from any node of the network arrives to this particular node in relatively few steps on average.
Definition
Consider a weighted network – either directed or undirected – with n nodes denoted by j=1, …, n; and a random walk process on this network with a transition matrix M. The $m_{ij}$ element of M describes the probability of the random walker that has reached node i, proceeds directly to node j. These probabilities are defined in the following way.
$m_{ij}={\frac {a_{ij}}{\sum _{k=1}^{n}a_{ik}}}$
where $a_{ij}$ is the (i,j)th element of the weighting matrix A of the network. When there is no edge between two nodes, the corresponding element of the A matrix is zero.
The random walk closeness centrality of a node i is the inverse of the average mean first passage time to that node:
$C_{i}^{RWC}={\frac {n}{\sum _{j=1}^{n}H(j,i)}}$
where $H(j,i)$ is the mean first passage time from node j to node i.
Mean first passage time
The mean first passage time from node i to node j is the expected number of steps it takes for the process to reach node j from node i for the first time:
$H(i,j)=\sum _{r=1}^{\infty }rP(i,j,r)$
where P(i,j,r) denotes the probability that it takes exactly r steps to reach j from i for the first time. To calculate these probabilities of reaching a node for the first time in r steps, it is useful to regard the target node as an absorbing one, and introduce a transformation of M by deleting its j-th row and column and denoting it by $M_{-j}$. As the probability of a process starting at i and being in k after r-1 steps is simply given by the (i,k)th element of $M_{-j}^{r-1}$, P(i,j,r) can be expressed as
$P(i,j,r)=\sum _{k\neq j}((M_{-j}^{r-1}))_{ik}m_{kj}$
Substituting this into the expression for mean first passage time yields
$H(i,j)=\sum _{r=1}^{\infty }r\sum _{k\neq j}((M_{-j}^{r-1}))_{ik}m_{kj}$
Using the formula for the summation of geometric series for matrices yields
$H(i,j)=\sum _{k\neq j}((I-M_{-j})^{-2})_{ik}m_{kj}$
where I is the n-1 dimensional identity matrix.
For computational convenience, this expression can be vectorized as
$H(.,j)=(I-M_{-j})^{-1}e$
where $H(.,j)$ is the vector for first passage times for a walk ending at node j, and e is an n-1 dimensional vector of ones.
Mean first passage time is not symmetric, even for undirected graphs.
In model networks
According to simulations performed by Noh and Rieger (2004), the distribution of random walk closeness centrality in a Barabási-Albert model is mainly determined by the degree distribution. In such a network, the random walk closeness centrality of a node is roughly proportional to, but does not increase monotonically with its degree.
Applications for real networks
Random walk closeness centrality is more relevant measure than the simple closeness centrality in case of applications where the concept of shortest paths is not meaningful or is very restrictive for a reasonable assessment of the nature of the system. This is the case for example when the analyzed process evolves in the network without any specific intention to reach a certain point, or without the ability of finding the shortest path to reach its target. One example for a random walk in a network is the way a certain coin circulates in an economy: it is passed from one person to another through transactions, without any intention of reaching a specific individual. Another example where the concept of shortest paths is not very useful is a densely connected network. Furthermore, as shortest paths are not influenced by self-loops, random walk closeness centrality is a more adequate measure than closeness centrality when analyzing networks where self-loops are important.
An important application on the field of economics is the analysis of the input-output model of an economy, which is represented by a densely connected weighted network with important self-loops.[2]
The concept is widely used in natural sciences as well. One biological application is the analysis of protein-protein interactions.[3]
Random walk betweenness centrality
A related concept, proposed by Newman,[4] is random walk betweenness centrality. Just as random walk closeness centrality is a random walk counterpart of closeness centrality, random walk betweenness centrality is, similarly, the random walk counterpart of betweenness centrality. Unlike the usual betweenness centrality measure, it does not only count shortest paths passing through the given node, but all possible paths crossing it.
Formally, the random walk betweenness centrality of a node is
$C_{i}^{RWB}=\sum _{j\neq i\neq k}r_{jk}$
where the $r_{jk}$ element of matrix R contains the probability of a random walk starting at node j with absorbing node k, passing through node i.
Calculating random walk betweenness in large networks is computationally very intensive.[5]
Second order centrality
Another random walk based centrality is the second order centrality.[6] Instead of counting the shortest paths passing through a given node (as for random walk betweenness centrality), it focuses on another characteristic of random walks on graphs. The expectation of the standard deviation of the return times of a random walk to a node constitutes its centrality. The lower that deviation, the more central that node is.
Calculating the second order betweenness on large arbitrary graphs is also intensive, as its complexity is $O(n^{3})$ (worst case achieved on the Lollipop graph).
See also
• Centrality
References
1. White, Scott; Smyth, Padhraic (2003). Algorithms for Estimating Relative Importance in Networks (PDF). ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. doi:10.1145/956750.956782. ISBN 1581137370.
2. Blöchl F, Theis FJ, Vega-Redondo F, and Fisher E: Vertex Centralities in Input-Output Networks Reveal the Structure of Modern Economies, Physical Review E, 83(4):046127, 2011.
3. Aidong Zhang: Protein Interaction Networks: Computational Analysis (Cambridge University Press) 2007
4. Newman, M.E. J.: A measure of betweenness centrality based on random walks. Social Networks, Volume 27, Issue 1, January 2005, Pages 39–54
5. Kang, U., Papadimitriou, S., Sun, J., and Tong, H.: Centralities in Large Networks: Algorithms and Observations. SIAM International Conference on Data Mining 2011, Mesa, Arizona, USA.
6. A.-M. Kermarrec, E. Le Merrer, B. Sericola, G. Trédan: Second order centrality: Distributed assessment of nodes criticity in complex networks. Elsevier Computer Communications 34(5):619-628, 2011.
| Wikipedia |
MIT-Harvard-MSR Combinatorics Seminar
Schedule 1996 Spring
Organizer: Jim Propp
Eighteenth One Day Conference on Combinatorics and Graph Theory
*SUNDAY*, February 4, 1996
10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063
Ethan Coven (Wesleyan Univ.)
Tiling the Integers with One Prototile
Daniel Kleitman (MIT)
Two Problems in Applied Graph Theory: a Vector Matching Problem, and a Shuffling Problem
Emily Petrie (Merrimack College)
The Symmetry Group of an Almost Perfect One-Factorization
Joseph J. Rushanan (MITRE)
Parallel Processing and Cayley Graphs
Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html
We have received an NSF grant to support these conferences. This will allow us to provide a modest transportation allowance to those attendees who are not local.
Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)smith.smith.edu
Karen Collins (Wesleyan Univ.), (203) 685-2169, kcollins(at-sign)wesleyan.edu
Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)smith.smith.edu
Northeastern's Geometry-Algebra-Singularities Seminar, Monday, Feb. 5, 1:30 PM at 509 Lake Hall
Andrei Zelevinsky
Totally positive matrices and pseudo line arrangements
Wednesday, February 7 and Friday, February 9, 4:15 p.m.; MIT, room 2-338
Andrei Okounkov (Institute for Advanced Study)
Edrei's theorem and representations of S(\infty)
Edrei's theorem describes all so-called totally positive (or Polya frequency) sequences. By definition, a sequence (a_i) is called totally positive if
\det [a_{i_p j_q}]_{1 \le p,q \le k}
for all k greater than or equal to 0 and all
i_1 < i_2 < ... < i_k,
j_1 < j_2 < ... < j_k .
Such sequences arise in approximation theory, probability, ..., and representation theory of S(\infty), U(\infty), O(\infty), Sp(\infty). Two proofs of this theorem were known: Edrei's original proof, based on results of Nevanlinna about entire functions, and the "ergodic" proof of Vershik and Kerov, based on the calculation of the asymptotics of the characters of S(n) as n goes to infinity. New methods in the representation theory of infinite-dimensional classical groups provide a new proof of Edrei's theorem as well as a remarkable simplification of the existing proofs.
Wednesday, February 14 1, 4:15 p.m.; MIT, room 2-338
Igor Pak (Harvard)
A new bijective proof of the hook-length formula
We present a new proof of the hook-length formula for the dimension of the irreducible representation of the symmetric group. In order to do that we construct an explicit bijection between two sets of tableaux.
Those who are interested may refer to http://www.labri.u-bordeaux.fr/~betrema/pak/pak.html for definitions and nice examples.
Friday, February 23, 4:15 p.m.; MIT, room 2-338
Morris Dworkin (Brandeis)
Factorization of the cover polynomial
Chung and Graham's cover polynomial generalizes Goldman, Joichi, and White's "factorial" rook polynomial to two variables. We factor the cover polynomial completely for Ferrers boards with either increasing or decreasing column heights. For column permuted Ferrers boards, we find a sufficient condition for its partial factorization. We apply this to column permuted "staircase boards," getting a partial factorization in terms of the column permutation, as well as a sufficient condition for complete factorization.
Wednesday, February 28, 4:30 p.m.; MIT, room 2-338
Special Time, Joint with Lie Groups Seminar
Anatoly Vershik (Steklov Mathematical Institute)
A new version of the representation theory of Coxeter Groups and spectra of Gel'fand-Tsetlin algebras
Classical representation theory of the symmetric groups (Young, Frobenius, Schur, Weyl, von Neumann, et al.) involves from the outset the notion of Young diagrams and some nontrivial combinatorics of the Young lattice.
Since the branching rule for the irreducible representations of S_n (n=1,2,...) is described by the Young lattice, one could wonder: is it possible to find this rule a priori, i.e., before all the representation theory of S_n is constructed? For beginners, the "yes" answer would justify the introduction of the Young diagrams, whereas the experts could say that the representation theory of the symmetric groups at last (a century after its creation) becomes a part of general representation theory.
Now we can say "yes"! Using Coxeter generators, Murphy-Jusys elements, Gel'fand-Tsetlin subalgebra for the symmetric groups, its spectrum, and adding some simple arguments, we obtain a new and very natural version of this remarkable classical theory.
Joint Brandeis-Harvard-MIT-Northeastern Colloquium, Feb. 29
4:30pm, Room 335, New Classroom Building, Northeastern University
Anatoly Vershik
Asymptotic combinatorial and geometric problems from the statistical physics point of view.
Friday, March 1, 4:15 p.m.; MIT, room 2-338
Henry Cohn (Harvard) and Jim Propp (M.I.T.)
A limit law for constrained plane partitions
MacMahon showed that the number of plane partitions with at most n rows, at most n columns, and all parts of size at most n is equal to
n-1 n-1 n-1
------- ------- -------
| | | | | | i+j+k+2
| | | | | | --------
i=0 j=0 k=0
(a generalization of binomial coefficients). The problem can also be viewed as one of counting plane partitions whose solid Young diagram fits inside an n-by-n-by-n box, or as one of counting tilings of a regular hexagon of side-length n by rhombuses of side 1.
Working with Michael Larsen, we have recently shown that for n large, a "typical" tiling of the hexagon (i.e., one chosen uniformly at random from the set of all tilings with n fixed) has one sort of behavior near the boundary of the hexagon and a qualitatively different sort in the interior, where the border between the two regions is asymptotically given by the circle inscribed in the hexagon. The local behavior inside the circle varies from place to place, and we can give a formula for how it varies. Our results can be interpreted as giving an asymptotic law for the typical shape of the solid Young diagram of a constrained plane partition.
Wednesday, March 6, 3:00 p.m.; MIT, room 2-338
(note unusual time)
Sean Carroll (M.I.T.)
Beyond matrix models: a combinatorial approach to discretized two-dimensional quantum gravity
The Feynman path integral for two-dimensional quantum gravity, which is a sum over geometries and matter configurations, can be calculated by taking the continuum limit of a discretized theory of triangulated surfaces with combinatorial data representing matter fields. I will discuss an approach to such a calculation using recursion equations in free variables. The flexibility of this method allows the computation of a number of quantities which would be difficult to compute using traditional "matrix model" approaches to these theories.
(note change of date)
Emily Petrie (Merrimack)
A perfect 1-factorization of the complete graph K2n may be defined as a partition of the edge set into 1-factors, such that the union of any two of the 1-factors is connected. When viewed this way, a natural generalization is to consider 1-factorizations of K2n where the union of any three of the 1-factors is connected. We call these almost perfect 1-factorizations. We examine the automorphism group G of such 1-factorizations. For perfect 1-factorizations on K2n, strong divisibility conditions have been established for the size of the automorphism group, depending only on n. However for other types of 1-factorizations the order of the automorphism group can be relatively large in comparison with the number of vertices 2n. We ask, what restrictions can be placed on the size of the automorphism group G in the case of an almost perfect 1-factorization?
Wednesday, March 13, 4:15 p.m.; MIT, room 2-338
Alex Postnikov (M.I.T.)
Deformed Coxeter hyperplane arrangements
The braid or Coxeter arrangement of type A_{n-1} is the arrangement of hyperplanes in R^n given by the equations x_i - x_j = 0. We study deformations of this arrangement, i.e., hyperplane arrangements of the type
x_i - x_j = a_{ij}^1,a_{ij}^2,...,a_{ij}^k.
We calculate the number of regions and the Poincare polynomial for many arrangements of this form. In particular, we prove a conjecture by Richard Stanley that the number of regions of the arrangement in R^n given by the equations x_i - x_j = 1, i < j, is equal to the number of alternating trees on {1,2,...,n}. The number of regions and the Poincare polynomial have some interesting combinatorial and arithmetical properties. Many of the results presented here are obtained in collaboration with Richard Stanley.
Friday, March 15, 4:15 p.m.; MIT, room 2-338
Christos Athanasiadis (M.I.T.)
The characteristic polynomial of a rational subspace arrangement
Let A be an affine subspace arrangement in R^n, defined over the integers. We give a combinatorial interpretation of the characteristic polynomial chi(A, q) of A that is valid for sufficiently large prime values of q. This result, which generalizes a theorem of Blass and Sagan, reduces the computation of chi(A, q) to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years. The basic idea appeared for the first time in 1970 in a theorem of Crapo and Rota, which unfortunately was overlooked in the later development of the theory of arrangements.
We give applications for various hyperplane arrangements. These include a simple, uniform proof of a result of Blass and Sagan about the characteristic polynomial of a Coxeter arrangement, simple derivations of the characteristic polynomials of the Shi arrangements and various generalizations and a another proof of Stanley's conjecture about the number of regions of the Linial arrangement. We also extend our method to the computation of all face numbers of a rational hyperplane arrangement.
Northeastern Univ. - Geometry-Algebra-Singularities Seminar
Tuesday March 19 at 1 PM, at 509 Lake Hall
Boris Shapiro (U. Stockholm)
Enumeration of connected components of the intersection of two open opposite Schubert cells
\input {amstex}
\advance\voffset by -1.0cm
\NoBlackBoxes
%\nopagenumbers
\magnification=\magstep1
%\hfuzz=3.5pt
\hsize=17truecm
\vsize=24.2truecm
\voffset=0.5truecm
%\hoffset
\document
\define \bZ {\Bbb Z}
\topmatter
\title
On the number of connected components in the intersection of 2 open
opposite Schubert
cells in $SL_n/B$
\endtitle
\author B.~Z.~Shapiro, M.~Z.~Shapiro, A.~D.~Vainshtein
\endauthor
\affil \endaffil
\abstract We consider the space
\endabstract
\abstract
Let $T_n$ denote the group of real unitary uppertriangular matrices and
$\Delta_i,\;i=1,...,n-1$
denote the the hypersurface in $T_n$ given by vanishing of the 'principal'
$i\times i$-minor in the
right upper corner. We study the number of connected components in $\Cal
C_n=T_n
\setminus \bigcup_i\Delta_i$ using ideas from \cite {L} and \cite {BFZ}. At
first the
problem is reduced to a purely combinatorial question about some 'action'
on the group $T_n(\bZ_2)$
of uppertriangular matrices with $0-1$-entries. The final conjecture under
consideration is as follows.
The number $\sharp_n$ of connected components in $\Cal C_n$ equals $3\times
2^n$ for all $n\ge 5$.
(Cases $n=3$ and $n=4$ are exceptional and $\sharp_3=6$, $\sharp_4=52$.)
\endtopmatter
\Refs
\widestnumber \key{ShSh}
\ref \key {BFZ} \by A.~Berenstein, S.~Fomin, A,~Zelevinski
\paper Parametrizations of canonical bases and totally positive matrices
\jour preprint
\yr 1995\pages 1--98\endref
\ref \key {L} \by G.~Lusztig
\paper Total positivity in reductive groups
\inbook Lie theory and geometry: in honor of Bertram Kostant, Progress in Math
\publ Birkh\"auser\vol 123
\yr 1994
\endref
\ref \key {SS} \by B.~Z.~Shapiro, M.~Z.~Shapiro
\paper On the totally positive upper triangular matrices
\finalinfo accepted to Lin. Alg. and Appl
\endRefs
\enddocument$
Volkmar Welker (Essen, Germany)
On divisor posets of affine semigroups
In this talk we give a preliminary report on work on posets that occur as lower intervals in the poset defined on the elements of a sub-semigroup S of N^n by divisibility within S. By work of Laudal to compute the homology of the order complexes over k of these posets is equivalent to compute Tor_i^R(k,k) for R = k[S]. We will show how to reprove some known results about Koszul rings using these techniques and show that the complexes that occur in this context are very closely related to complexes that are associated to quotients of polynomial ring by monomials of degree 2 (e.g., Stanley-Reisner rings of posets).
Come to the nineteenth one day conference on Combinatorics and Graph Theory
Saturday, March 30, 1996; 10 a.m. to 4:30 p.m. at Smith College, Northampton MA 01063
Andrew Kotlov (Yale University)
The rank and chromatic number of graphs
Rodica Simion (George Washington University)
Some relations between polytopes and combinatorial statistics
Sheila Sundaram (University of Miami)
On the homology of partitions with an even number of blocks
Tamas Szonyi (Yale University)
Blocking sets in projective planes
*Our three year NSF grant is ending this spring. Looking at the remaining budget for the two spring conferences, we have to reduce the transportation allowance for non-local participants for the March 30th conference to $\$$40 (from the usual $\$$50). We have applied for a renewal for another 3 years of grant support, and hope to hear soon from NSF.*
A SYMPOSIUM ON EXACTLY SOLUBLE MODELS IN STATISTICAL MECHANICS: HISTORICAL PERSPECTIVES AND CURRENT STATUS
MARCH 30-31, 1996 to be held at Northeastern University, Boston, MA
The purpose of the symposium is to present historical perspectives as well as to assess the current status of the field of soluble models in statistical mechanics. Invited speakers include R. J. Baxter, D. Fisher, V. F. R. Jones, L. H. Kauffman, E. H. Lieb, B. M. McCoy, J. H. H. Perk, S. Sachdev, C. A. Tracy, P. Wiegmann, and others.
There will also be a mini-poster session for contributed papers.
For further inquiries please contact fywu(at-sign)neu.edu, king(at-sign)neu.edu, or circs(at-sign)phyjj4.cas.neu.edu, or write to Ms. M McKeever, Department of Physics, Northeastern University, Boston, MA 02115.
Wednesday, April 3, 4:15 p.m.; MIT, room 2-338
Tony Iarrobino (Northeastern)
The hook algebra
We had shown that given a natural number n, and a sequence T = (1,2,3,...,d,t_d,...,t_i,...,t_j) of integers satisfying t_d \geq t_{d+1} \geq ... \geq t_j and \Sigma t_i = n , then the lattice P(T) of partitions having diagonal lengths T is isomorphic to a product Q(T) = L_d \times ... \times L_j where each L_i is the lattice of partitions having no more than t_i-t_{i+1} rows and 1+t_{i-1}-t_i columns, under inclusion. The map D from P(T) to Q(T): P --> Q(P) arises from arranging the difference-one hooks of P having hands on the i-diagonal into parts according to the number of such hooks having a given hand.
It follows that the knowledge of Q_1(P) = Q(P) --- the difference-one hooks of P --- determines the difference-a hooks of T for all a. In this talk we define difference-a hook partitions and describe a composition Q_a(P) \times Q_b(P) --> Q_{a+b}(P) . Thus we define a "hook difference algebra" such that Q_a(P) = Q_1(P) \times ... Q_1(P) (a times). This algebra is related to the "strand map" S: Q(T) --> P(T) that is the inverse of D.
This is joint work with J. Yam\'eogo.
APPLIED MATHEMATICS COLLOQUIUM
Monday, April 8, 1996, 4:15 p.m. M.I.T., Building 2, Room 105
Professor Rodney J. Baxter (Australian National University)
The hard hexagon model and Rogers-Ramanujanism
Wednesday, April 10, 4:15 p.m.; MIT, room 2-338
Andrei Zelevinsky (Northeastern)
Quasicommuting families of quantum type Plucker coordinates
This is an account of a joint work with Bernard Leclerc. We consider the q-deformation of the coordinate ring of the flag variety of type A_r . This is the algebra with unit over the field of rational functions Q(q) generated by 2^{r+1}-1 generators [J] labeled by nonempty subsets J \subset [1,r+1] := {1,2, ..., r+1} , subject to the quantized Pl\"ucker relations. We refer to the generators [J] as _quantum_flag_minors_ (they can be identified with q-minors of a generic q-matrix whose row set consists of several initial rows). We say that [I] and [J] _quasicommute_ if [J][I] = q^n [I][J] for some integer n. We are concerned with the following problem motivated by the study of canonical bases for quantum groups of type A_r .
Problem A: describe all families of quasicommuting quantum flag minors.
We obtain a combinatorial criterion for quasicommutativity of two quantum flag minors [I] and [J]. As a consequence, we show that the maximal possible size of a quasicommuting family of quantum flag minors is {r+2 \choose 2}. An interesting special class of such families is in a bijection with the set of commutation classes of reduced expressions for the longest permutation w_0 \in S_{r+1}. This result leads to a natural extension of the _second_Bruhat_order_ by Manin-Schechtman.
Friday, April 12, 4:15 p.m.; MIT, room 2-338
Ken Fan (Harvard)
Schubert varieties and short braidedness
The theorem I will prove is this: In a finite type Weyl group, an element w has the property that you can knock out any simple generator from any reduced expression and come up with another reduced expression if and only if w is sts-avoiding. I'll use this fact to exhibit a family of singular Schubert varieties.
One curious thing is that this fact depends on finite type and is not a purely braid relation fact since it isn't true in affine A_2, for instance.
Glenn Tesler (U.C. San Diego)
Plethystic formulas for the Macdonald q,t-Kostka coefficients
Macdonald introduced a two parameter symmetric function basis P_\mu(x;q,t) for which various specializations of q and t yield many of the other well-established bases. The transition matrix expressing a rescaled basis J_\mu(x;q,t) in terms of a modified Schur basis s_\lambda[X(1-t)] has components denoted K_{\lambda,\mu}(q,t), and generalizes the ordinary Kostka matrix. Macdonald conjectured that K_{\lambda,mu}(q,t) are polynomials in q and t with nonnegative integer coefficients. We show that they are polynomials by determining new explicit formulas for them. These formulas separate the dependence on \mu and \lambda, and surprisingly, their structure is entirely determined by a portion of \lambda, and not at all on \mu. These formulas are themselves symmetric functions k_\gamma(x;q,t) indexed by partitions, where if we set \gamma to be \lambda with its largest row deleted, then a certain specialization ``B_\mu'' of x to q,t-monomials depending on \mu essentially expresses K_{\lambda,\mu}(q,t) as k_\gamma(B_\mu;q,t). The coefficients of k_gamma(x;q,t) when expressed in terms of Schur functions are Laurent polynomials in q and t, so that k_\gamma(B_\mu;q,t) is at least a Laurent polynomial, and the simple monomial denominator is easily eliminated to yield a true polynomial.
This is joint work with Adriano Garsia.
Sinai Robins (U.C. San Diego)
The Ehrhart Polynomial of a Lattice Polytope
The problem of counting the number of lattice points inside a lattice polytope in R^n has been studied from a variety of perspectives, including the recent work of Pommersheim and Kohvanskii using toric varieties and Cappell and Shaneson using Grothendieck-Riemann-Roch. Here we show that the Ehrhart polynomial of a lattice n-simplex has a simple analytical interpretation from the perspective of function theory on the n-torus. The methods involve Poisson Summation and Fourier integrals.
We obtain closed forms for the coefficients of the Ehrhart polynomial in terms of the elementary cotangent functions. These expressions are closely related to the formulas of Cappell and Shaneson and Hirzebruch and Zagier.
This is joint work with Ricardo Diaz.
20th meeting of the CoNE conference, Saturday, April 27
Vera Pless (University of Illinois at Chicago)
Constraints on Weight in Binary Codes
Problem Session by Participants
Pizza Lunch!!
Brenda Latka (DIMACS)
Forbidden Subtournaments and Antichains
Linda Lesniak (Drew University)
Tough Graph Theory
*Our three year NSF grant is ending this spring. Looking at the remaining budget, we have to reduce the transportation allowance for non-local participants for the April 27th conference to $40 (from the usual $50). We have applied for a renewal for another 3 years of grant support, and hope to hear soon from NSF.*
Wednesday, May 1, 4:15 p.m.; MIT, room 2-338
Yuval Roichman (M.I.T.)
A recursive rule for Kazhdan-Lusztig characters
The Murnaghan-Nakayama rule is a most useful recursive rule for computing characters of the symmetric groups. We present a generalization of this rule to arbitrary Coxeter groups and their Hecke algebras. The classical version is obtained as a special case, and new combinatorial interpretations follow. The work is done via Kazhdan-Lusztig theory and combinatorics of Coxeter groups.
Sergey Fomin (M.I.T.)
Quantum Schubert polynomials
We compute the Gromov-Witten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring.
This is joint work with S. Gelfand and A. Postnikov. The paper is available from http://www-math.mit.edu/~fomin/papers.html
Discrete Mathematics Dinner, Wednesday, May 8 at 6 p.m. at Helmand's Restaurant
The cost will be $\$$10 for grad students and undergraduates (alcoholic beverages not included), with the rest of us making up the difference.
Wednesday, May 15, 5:00 p.m.; MIT, room 2-338
Sara Billey (M.I.T.)
Vexillary elements in the hyperoctahedral group
The vexillary permutations in the symmetric group have interesting connections with the number of reduced words, the Littlewood-Richardson rule, Stanley symmetric functions, Schubert polynomials and the Schubert calculus. Lascoux and Schutzenberger have shown that vexillary permutations are characterized by the property that they avoid any subsequence of length 4 with the same relative order as 2143. In this talk, we will propose a definition for vexillary elements in the hyperoctahedral group. We show that the vexillary elements can again be determined by pattern avoidance conditions. These vexillary elements share some, but not all, of the "nice" properties of the vexillary permutations in $S_n$.
Friday, May 17, 4:15 p.m.; MIT, room 2-338
Frank Sottile (Toronto)
Symmetries of Littlewood-Richardson coefficients for Schubert polynomials
The Littlewood-Richardson rule is a combinatorial formula for structure constants of the ring of symmetric polynomials in terms of its Schur basis:
s_\mu \cdot s_\nu = \sum_\lambda c^\lambda_{\mu\,\nu} s_\lambda.
Schubert polynomials form a basis for the ring of polynomials in infinitely many variables x_1,x_2,..., so there are similar structure constants for Schubert polynomials, which I also call Littlewood-Richardson coefficients. These generalize the classical coefficients, as every Schur polynomial in x_1,...,x_k is a Schubert polynomial. They are, however, largely unknown.
This talk will discuss recent results (obtained with Nantel Bergeron) on those coefficients which arise when multiplying a Schubert polynomial by a Schur polynomial. We show these coefficients have certain symmetries, similar to symmetries of the classical Littlewood-Richardson coefficients, which facilitates their computation. We apply these results to the enumeration of chains in the strong Bruhat order on the symmetric group.
Rodney Baxter (Australian National University and Northeastern University)
Star-triangle and star-star relations in statistical mechanics
The star-triangle is the simplest form of the "Yang-Baxter" relations and plays a vital role in solvable statistical mechanical models, ensuring that transfer matrices commute.
There are models for which no star-triangle relation is known, but which satisfy a weaker "star-star" relation. These will be discussed, and it will be shown that this weaker relation is still sufficient to ensure the required commutation properties.
Mark Shimozono (M.I.T.)
Monotonicity properties of q-analogues of Littlewood-Richardson coefficients
Certain q-analogues of Littlewood-Richardson (LR) coefficients arise naturally in the resolution of the ideal of a nilpotent conjugacy classes of matrices in a larger nilpotent conjugacy class. These polynomials may be defined using a Kostant-Heckman formula. A conjectural description is given in terms of what we call catabolizable tableaux. In the special case of tensor products of irreducibles corresponding to rectangular partitions, there is another conjectural combinatorial description using classical LR tableaux and a generalization of Lascoux, Leclerc, and Thibon's formula for the charge statistic. Monotonicity properties of these polynomials are studied using families of statistic-preserving injections. Certain compositions of these injections furnish a bijection from the LR tableaux to the catabolizables. This is joint work, part with Jerzy Weyman and part with Anatol N. Kirillov.
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We covered three main techniques for doing this: local gradient-based search (providing a lower bound on the objective), exact combinatorial optimization (exactly solving the objective), and convex relaxations (providing a provable upper bound on the objective).
The order of the min-max operations is important here. Specially, the max is inside the minimization, meaning that the adversary (trying to maximize the loss) gets to "move" second. We assume, essentially, that the adversary has full knowledge of the classifier parameters $\theta$ (this was implicitly assumed throughout the entire previous section), and that they get to specialize their attack to whatever parameters we have chosen in the outer maximization. The goal of the robust optimization formulation, therefore, is to ensure that the model cannot be attacked even if the adversary has full knowledge of the model. Of course, in practice we may want to make assumptions about the power of the adversary: maybe (or maybe not) it is reasonable to assume they could not solve the integer programs for models that are too large. But it can be difficult to pin down a precise definition of what we mean by the "power" of the adversary, so extra care should be taken in evaluating models against possible "realistic" adversaries.
Using lower bounds, and examples constructed via local search methods, to train an (empirically) adversarially robust classifier.
Using convex upper bounds, to train a provably robust classifier.
There are trade-offs between both approaches here: while the first method may seem less desireable, it will turn out that the first approach empircally creates strong models (with empircally better "clean" performance as well as better robust performance for the best attacks that we can produce. Thus, both sets of strategies are important to consider in determining how best to build adversarially robust models.
Perhaps the simplest strategy for training an adversarially robust model is also the one which seems most intuitive. The basic idea (which originally was referred to as "adversarial training" in the machine learning literature, though is also basic technique from robust optimization when viewed through this lense) is to simply create and then incorporate adversarial examples into the training process. In other words, since we know that "standard" training creates networks that are succeptible to adversarial examples, let's just also train on a few adversarial examples.
Note however, that Danskin's theorem only technically applies to the case where we are able to compute the maximum exactly. As we learned from the previous section, finding the maximum exactly is not an easy task. And it is very difficult to say anything formally about the nature of the gradient if we do not solve the problem optimally. Nonetheless, what we find in practice is the following: the "quality" of the robust gradient descent procedure is tied directly to how well we are able to perform the maximization. In other words, the better job we do of solving the inner maximization problem, the closer it seems that Danskin's theorem starts to hold. In other words, the key aspects of adversarial training is incorporate a strong attack into the inner maximization procedure. And projected gradient descent approaches (again, this included the simple variants like projected steepest descent) are the strongest attack that the community has found.
Although this procedure approximately optimizes the robust loss, which is exactly the target we would like to optimize, in practice it is common to also include a bit of the standard loss (i.e., also take gradient steps in the original data points), as this tends to also slightly improve the performance of the "standard" error of the task. It is also common to randomize over the starting positions for PGD, or else there can be issues with the procedure learning loss surface such that the gradients exactly at the same points point in a "shallow" direction, but very nearby there are points that have the more typical steep loss surfaces of deep networks.
Let's see how this all looks in code. To start with, we're going to clone a bunch of the code we used in the previous chapter, including the procedures for building and training the network and for producing adversarial examples.
""" Construct FGSM adversarial examples on the examples X"""
The only real modification we make is that we modify the adversarial function to also allow for training.
"""Standard training/evaluation epoch over the dataset"""
"""Adversarial training/evaluation epoch over the dataset"""
Let's start by training a standard model and evaluating adversarial error.
So as we saw before, the clean error is quite low, but the adversarial error is quite high (and actually goes up as we train the model more). Let's now do the same thing, but with adversarial training.
Ok, so with adversarial training, we are able to get a model that has an error rate of just 2.8%, compared to the 71% that our original model had (and increased test accuracy as well, though this is one are where we want to emphasize that this better clean error is an artifact of the MNIST data set, and not something we expect in general). This seems like a resounding success!
Let's be very very careful, though. Whenever we train a network against a specific kind of attack, it's incredibly easy to perform well against that particular attack in the future: in a sense, this is just the standard statement about deep network performance: they are incredibly good at predicting precisely the class of data they were trained against. What about if we run some other attack, like FGSM? What if we run PGD for longer? Or with randomization? Or what if someone in the future comes up with some amazing new optimization procedure that works even better (for attacks within the prescribed norm bound)?
Let's get a sense of this by evaluating our model against some different attacks. Let's try FGSM first.
Ok, that is good news. FGSM indeed works worse than even the PGD attack we trained against, because FGSM is really just one step of PGD with a step size of $\alpha = \epsilon$. So it's not surprising it does worse. Let's try running PGD for longer.
Also good! Error increases a little bit, but well within the bounds of what we might think in reasonable (you can try running for longer, and see that it doesn't change much … the examples have bit the boundaries of the $\ell_\infty$ ball in most cases, and taking more steps doesn't change things). But what about if we take more steps with a smaller step size, to try to get a more "fine-grained" attack?
Ok, we're getting more confident now. Let's also add randomization.
Alright, so at this point, we've done enough evaluations that maybe we are confident enough to put the model online and see if anyone else can actually break it (note: this is not actually the model that was put online, though it was trained in the roughly the same manner). But we should still probably try some different optimizers, try multiple randomized restarts (like we did in the past section), etc.
Note: one evaluation which is not really relevant (except maybe out of curiosity), however, is to evaluate the performance of this robust model under some other perturbation region, say evaluating this $\ell_\infty$ robust model under an $\ell_2$ bounded attack. The model was trained under one single attack model; of course it will not work well to prevent some completely different attack model. If one does desire a kind of "generalization" across multiple attack models, then we need to formally define the set of attack models we care about, and train the model over multiple different draws from these attack models. This is a topic we won't get in to, except to say that for some classes like multiple different norm bounds, it would be easy to extend the approach to simultaneously defend against e.g. $\ell_1$, $\ell_2$, and $\ell_\infty$ attacks, or something like this. Of course, the real set of attacks we care about (i.e., the set of all images that a human thinks "look reasonably the same") is extremely hard to characterize, and an excellent subject for future work.
What is happening with these robust models?
So why do these models work well against robust attacks, and why have some other proposed methods for training robust models (in)famously come up short in this regard? There are likely many answers to this question, but one potential answer can be seen by looking at the loss surface of the trained classifier. Let's look at a projection of the loss function along two dimensions in the input space (one the direction of the actual gradient, and one a random direction).
Let's look at the loss surface for the standard network.
Very quickly the loss increases substantially. Let's then compare this to the robust model.
The important point to compare here is the relative $z$ axes (the "bumpiness" in the second figure is just to do this much smaller scale; if put on the same scale the second figure would be completely flat). The robust model has a loss that is quite flat both in the gradient direction (that is the steeper direction), and in the random direction, whereas the traditionally trained model varies quite rapidly both in the gradient direction and (after moving some in the gradient direction) in the random direction. Of course, this is no guarantee that there is no direction of steep cost increase, but it at least gives some hint of what may be happening.
In summary, these models trained with PGD-based adversarial training do appear to be genuinely robust, in that the underlying models themselves have smooth loss surfaces, and not by just a "trick" that hides the true direction of cost increase. Whether more can be said formally about the robustness is a quick that remains to be seen, and a topic of current ongoing research.
As a final piece of the puzzle, let's try to use the convex relaxation methods not just to verify networks, but also to train them. To see why we might want to do this, we're going to focus here on the interval-based bounds, though all the same factors apply to the linear programming convex relaxation as well, just to a slightly smaller degree (and the methods are much more computationally intensive).
To start, let's consider using our interval bound to try to verify robustness for the empirically robust classifier we just trained. Remember that a classifier is verified to be robust against an adversarial attack if the optimization objective is positive for all targeted classes. This is done by the following code (almost entirely copied from the previous chapter, but with an additional routine that computes the verified accuracy over batches).
Let's see what happens if we try to use this bound to see whether we can verify that our robustly trained model provably will be insucceptible to adversarial examples in some cases, rather than just empirically so.
Unfortunately, the interval-based bound is entirely vaccous for our (robustly) trained classifier. We'll save you the disappointment of checking ever smaller values of $\epsilon$, and just mentioned that in order to get any real verification with this method, we need values of $\epsilon$ less than 0.001. For example, for $\epsilon = 0.0001$, we finally achieve a "reasonable" bound.
That doesn't seem particularly useful, and indeed, it is a property of virtually all the relaxation-based verification approaches, is that they are vaccuous when evaluated upon a network trained without knowledge of these bounds. Additionally, these errors tend to accumulate with the depth of the network, precisely because the interval bounds as we have presented them also tend to get looser with each layer of the network (this is why the bounds were not so bad in the previous chapter, when we were applying them to a three-layer network).
To do this, we're going to use the interval bounds to upper bound the cross entropy loss of a classifier, and then minimize this upper bound. Specifically, if we form a "logit" vector where we replace each entry with the negative value of the objective for a targeted attack, and then take the cross entropy loss of this vector, it functions as a strict upper bound of the original loss. We can implement this as follows.
Finally, let's train our model using this robust loss bound. Note that training rovably robust models is a bit of a tricky business. If we start out immediately by trying to train our robust bound with the full $\epsilon=0.1$, the model will collapse to just predicting equal probability for all digits, and will never recover. Instead, to reliably train such models we need to schedule $\epsilon$ during the training process, starting with a small $\epsilon$ and gradually raising it to the desired level. The schedule we use below was picked rather randomly, and we can do much better with a bit of tweaking, but it serves our basic purpose.
It's not going to set any records, but what we have here is an MNIST model that where no $\ell_\infty$ attack of norm bounded by $\epsilon=0.1$ will ever be able to cause the classifier to experience more than 9.67% error on the test set of MNIST (acheiving a "clean" error of 5.15%). And just how bad can a real adversarial attack do? It's of course hard to say for sure, but let's see what PGD does.
So somewhere right in the middle. Note also that training these provably robust models is a challenging task, and a bit of tweaking (even still using interval bounds) can perform quite a bit better. For now, though, this is sufficient to make our point that we can obtain non-trivial provable bounds for trained networks.
Even on a dataset like CIFAR10, for example, the best known robust models that can handle a perturbation of $8/255 = 0.031$ color values achieve an (empirical) robust error of of 55%, and the best provably robust models have an error greater than 70%. On the flipside, the choices we have with regards to training procedures, network architecture, regularization, etc, have barely been touched in the robust optimization context. All our architecture choices come from what has been best for standard training, but these likely are no longer optimal architectures for robust training. Finally, as we will highlight in the next chapter, there is substantial benefit to be had from robust models right now, even if true robust performance still remains ellusive. | CommonCrawl |
Can a "free launch" from a space elevator really be free?
The question Benefit of sling shot effect with a space elevator implies that you can get free $\Delta V$ from a space elevator launch. As highlighted in this answer on SciFi quoting Sheffield book "The Web Between the Worlds", and this answer on Space The launch is not really free, it is stealing rotational energy from Earth.
So two questions, given a space elevator launch from 100,000 kilometers, ignoring the energy cost getting mass to and from the launch point, and assuming all mass involved goes up and down the space elevator:
How much mass would need to be launched for the Earth to lose 1 second of rotational speed?
How much mass would need to be lowered to the Earth to gain 1 second of rotational speed?
For prospective, on potential launch mass.
The Space Shuttle Orbiter has a fully loaded weight of about 120 tons
The USS Enterprise (CVN-65) (an American aircraft carrier) has a fully loaded weight of about 94,781 tons
launch planet space-elevator
$\begingroup$ I did not want to assume that raising and lowering would have the same values. For a change of one second, they may be nearly identical but for a change of a minute or an hour, I would expect to there to be a significant difference. $\endgroup$ – James Jenkins Sep 8 '13 at 11:25
$\begingroup$ There is no such thing as a free lunch. Wait, what.. launch! $\endgroup$ – Andrew Thompson Sep 8 '13 at 14:07
$\begingroup$ The problem is your assumption. TANSTAAFL - You cannot ignore the cost to get to the launch point. $\endgroup$ – Rory Alsop Sep 8 '13 at 14:32
$\begingroup$ @RoryAlsop I used "ignoring the energy cost" as in to "not consider" there are many variables to allow for, that would inhibit calculating an answer, and make the question overly broad. $\endgroup$ – James Jenkins Sep 8 '13 at 14:38
$\begingroup$ The funny thing with getting to the launch point is you pay only to get to GEO. "Ascent" beyond GEO produces energy which you can use to lift other cargo or sell - in essence we have a power plant that uses Earth rotation as power source and slowing down Earth is the only inherent cost. So, yes, ignoring technological, staff, maintenance and such costs, the cost of space elevator launch could be actually negative! $\endgroup$ – SF. Sep 8 '13 at 16:21
This is a simple calculation in the conservation of angular momentum.
The angular momentum of uniform sphere is ${2\over 5}MR^2\omega$. We will assume that Earth is a uniform sphere (close enough for this question), so $M=5.972\times 10^{24}\,\mathrm{kg}$, $R=6371\,\mathrm{km}$ (mean), and $\omega=7.292\times 10^{-5}\,\mathrm{s}$. So the angular momentum of the Earth is $L_e=7.071\times 10^{33}\mathrm{kg\,m^2\over s}$.
The angular momentum of a small object rotating about a point is $mr^2\omega$. $\omega$ does not change when climbing the elevator, just $r$. So the change in angular momentum is $\Delta L=m\omega(r_f^2-r_i^2)$. $\omega$ is that of Earth above.
To change the Earth's rotation rate by $1\,\mathrm{s}$, we need to change its angular momentum by $1\over 86164$ of what it was, so $\Delta L=8.206\times 10^{28}\mathrm{kg\,m^2\over s}$. Assuming a space elevator at the equator, and using the equatorial radius of the Earth, $6378\,\mathrm{km}$, and $100000\,\mathrm{km}$ above that, we get $\Delta L=8.206\times 10^{28}\mathrm{kg\,m^2\over s}=m\omega\left(\left(106378\,\mathrm{km}\right)^2-\left(6378\,\mathrm{km}\right)^2\right)$. To the extent that $m$ reduces (or increases) $M$, there is also a small change in the rotation rate for the given angular momentum when it is released. However that is negligible here.
This can be simplified and approximated, in order to better see the sensitivities, as: $m={2\over 5}M{\Delta T\over T}{R^2\over H^2}$, where $T$ is the rotation period of the Earth, $\Delta T$ is the small amount by which you want to change it, and $H$ is the radius of the release point from the elevator. $m$ is the amount of mass to let go of at the release point to effect a change in rotation period of $\Delta T$.
That gives $m\approx 10^{17}\mathrm{kg}$. About a billion (US-style) of your aircraft carriers.
If each aircraft-carrier size spaceship only carried a handful of people, then we could evacuate the entire human population from the Earth, only slowing Earth down by one second. So I would say, for all practical purposes, yes, a space elevator is free. You just have to pay the energy to climb to that altitude, which is quite small compared to the energy from the velocity that you get.
12 revs
Mark Adler
FWIW, I'm with Arthur C. Clarke on this whole "space elevator" business. There is no "free lunch" (or is it "free launch"?) to be had with any "space elevator" that is supposedly capable of taking a payload with non-negligible mass and with non-negligible air resistance in the lower atmosphere to the stratosphere and beyond. It doesn't really matter then where we put this alleged "space elevator", so I'll take a few shortcuts to answer this as easy as possible. Take it as it is, I have no interests in discussing "space elevators".
For your first question, there is no upper limit, short of removing the total of the Earth's mass. The remaining mass would still have radial velocity equal to the total mass from before, only it's kinetic potential will be lowered as the mass spinning decreases, so it would be more prone to changes in its rotation velocity, if we added some of that mass back later on.
The second question is also relatively easy, if we assume the kinetic potential of the added mass in the direction of the Earth's rotation is equal to zero, i.e. it's lowered directly perpendicular to the Earth's surface and dropped at either of the poles (another answer by @MarkAdler already includes calculations for dropping the mass at the equator):
$$\frac{duration\ of\ Earth\ day\ in\ seconds}{Earth's\ total\ mass} = x \left(\frac{duration\ of\ Earth\ day\ in\ seconds + 1\ }{Earth's\ total\ mass}\right)$$
So this comes out as $1/86400$ of the Earth's mass ($5.972 * 10^{23} kg$), which is $6.9123 * 10^{18} kg$, if my data input was correct. But please do check again. ;)
But assuming my pocket calculator doesn't lie, this would come out as:
72,929,329,875 (nearly 73 billion) of USS Enterprise (CVN-65) aircraft carriers
57,602,623,456,791 (a good 57 and a half trillion) of fully loaded Space Shuttle Orbiters
TildalWaveTildalWave
$\begingroup$ You can't have a space elevator at the pole. It would fall down. The question was about a space elevator. $\endgroup$ – Mark Adler Sep 8 '13 at 16:05
$\begingroup$ @MarkAdler - You can have it, but it won't be stationary to the Earth's surface, ie. it will be in a polar orbit. But as far as feasibility goes, you can't have a "free lunch" (or "free launch" as OP put it) space elevator. Full stop. As soon as some mass on it shifts its center of mass towards the Earth and it doesn't reach orbital velocity (the lower to the Earth, the faster it should go) any more, it will drop down just as well. Not to even speak of atmospheric drag. There's even calculations on the net what path its tether would draw on the Earth's surface. ;) $\endgroup$ – TildalWave Sep 8 '13 at 16:12
$\begingroup$ My answer has nothing to do with "dropping" mass at the equator. My answer only works for a space elevator, where by virtue of being attached to the surface, it launches objects by taking small amounts of Earth's angular momentum and angular kinetic energy. The answer for "dropping" mass at the equator or removing mass from the equator would have an entirely different answer. $\endgroup$ – Mark Adler Sep 8 '13 at 16:21
$\begingroup$ @MarkAdler - You were arguing that you can't have a polar space elevator. I say you can't have a space elevator. It only works on paper as a scribble. When you start attaching mass (that so happens has a physical presence, so also air resistance, among other things) to it at various elevations, nothing of it is "free". And if we scratch that "free" part, you can even add mass to the poles, in theory at least. Something that "space elevator" fails anyway. So I'm answering the other part of the question in as simple terms as possible. $\endgroup$ – TildalWave Sep 8 '13 at 16:31
Not the answer you're looking for? Browse other questions tagged launch planet space-elevator or ask your own question.
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Space elevator idea | CommonCrawl |
Integral closure of an ideal
In algebra, the integral closure of an ideal I of a commutative ring R, denoted by ${\overline {I}}$, is the set of all elements r in R that are integral over I: there exist $a_{i}\in I^{i}$ such that
$r^{n}+a_{1}r^{n-1}+\cdots +a_{n-1}r+a_{n}=0.$
It is similar to the integral closure of a subring. For example, if R is a domain, an element r in R belongs to ${\overline {I}}$ if and only if there is a finitely generated R-module M, annihilated only by zero, such that $rM\subset IM$. It follows that ${\overline {I}}$ is an ideal of R (in fact, the integral closure of an ideal is always an ideal; see below.) I is said to be integrally closed if $I={\overline {I}}$.
The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring.
Examples
• In $\mathbb {C} [x,y]$, $x^{i}y^{d-i}$ is integral over $(x^{d},y^{d})$. It satisfies the equation $r^{d}+(-x^{di}y^{d(d-i)})=0$, where $a_{d}=-x^{di}y^{d(d-i)}$is in the ideal.
• Radical ideals (e.g., prime ideals) are integrally closed. The intersection of integrally closed ideals is integrally closed.
• In a normal ring, for any non-zerodivisor x and any ideal I, ${\overline {xI}}=x{\overline {I}}$. In particular, in a normal ring, a principal ideal generated by a non-zerodivisor is integrally closed.
• Let $R=k[X_{1},\ldots ,X_{n}]$ be a polynomial ring over a field k. An ideal I in R is called monomial if it is generated by monomials; i.e., $X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}$. The integral closure of a monomial ideal is monomial.
Structure results
Let R be a ring. The Rees algebra $R[It]=\oplus _{n\geq 0}I^{n}t^{n}$ can be used to compute the integral closure of an ideal. The structure result is the following: the integral closure of $R[It]$ in $R[t]$, which is graded, is $\oplus _{n\geq 0}{\overline {I^{n}}}t^{n}$. In particular, ${\overline {I}}$ is an ideal and ${\overline {I}}={\overline {\overline {I}}}$; i.e., the integral closure of an ideal is integrally closed. It also follows that the integral closure of a homogeneous ideal is homogeneous.
The following type of results is called the Briancon–Skoda theorem: let R be a regular ring and I an ideal generated by l elements. Then ${\overline {I^{n+l}}}\subset I^{n+1}$ for any $n\geq 0$.
A theorem of Rees states: let (R, m) be a noetherian local ring. Assume it is formally equidimensional (i.e., the completion is equidimensional.). Then two m-primary ideals $I\subset J$ have the same integral closure if and only if they have the same multiplicity.[1]
See also
• Dedekind–Kummer theorem
Notes
1. Swanson & Huneke 2006, Theorem 11.3.1
References
• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
• Swanson, Irena; Huneke, Craig (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432, Reference-idHS2006, archived from the original on 2019-11-15, retrieved 2013-07-12
Further reading
• Irena Swanson, Rees valuations.
| Wikipedia |
\begin{document}
\frontmatter
\title{ \textsf{\Huge \hbox{Introduction to Cluster Algebras}}\\[.1in] \textsf{\Huge Chapter 7} \\[.2in] {\rm\textsf{\LARGE (preliminary version)}} }
\author{\Large \textsc{Sergey Fomin}}
\author{\Large \textsc{Lauren Williams}}
\author{\Large \textsc{Andrei Zelevinsky}}
\maketitle
\noindent \textbf{\Huge Preface}
\noindent This is a preliminary draft of Chapter 7 of our forthcoming textbook \textsl{Introduction to cluster algebras}, joint with Andrei Zelevinsky (1953--2013).
Other chapters have been posted as \begin{itemize}[leftmargin=.2in] \item \texttt{arXiv:1608:05735} \hbox{(Chapters~1--3)}, \item \texttt{arXiv:1707.07190} (Chapters~4--5), and \item \texttt{arXiv:2008.09189} (Chapter~6).
\end{itemize} We expect to post additional chapters in the not so distant future.
Anne Larsen and Raluca Vlad made a number of valuable suggestions that helped improve the quality of the manuscript. We are also grateful to Zenan Fu, Amal Mattoo, Hanna Mularczyk, and Ashley Wang for their comments on the earlier versions of this chapter, and for assistance with creating figures.
Our work was partially supported by the NSF grants DMS-1664722, DMS-1854512, and DMS-1854316, and by the Radcliffe Institute for Advanced Study.
Comments and suggestions are welcome.
\rightline{Sergey Fomin} \rightline{Lauren Williams}
\noindent 2020 \emph{Mathematics Subject Classification.} Primary 13F60.
\noindent \copyright \ 2021 by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky
\tableofcontents
\mainmatter
\setcounter{chapter}{6}
\chapter{Plabic graphs} \label{ch:plabic}
\def\mathfrak{X}{\mathfrak{X}}
\def\widetilde{\pi}_{\aff}{\widetilde{\pi}_{\aff}}
In this chapter, we present the combinatorial machinery of \emph{plabic graphs}, introduced and developed by A.~Postnikov~\cite{postnikov}. These are planar (unoriented) graphs with bicolored vertices satisfying some mild technical conditions. Plabic graphs can be transformed using certain \emph{local moves}. A key observation is that each plabic graph gives rise to a quiver, so that local moves on plabic graphs translate into (a subclass of) quiver mutations.
Crucially, the combinatorics underlying several important classes of cluster structures that arise in applications fits into the plabic graphs framework. This in particular applies to the basic examples introduced in Chapter~\ref{ch:tp-examples}. More concretely, we show that the combinatorics of flips in triangulations of a convex polygon (resp., braid moves in wiring diagrams, either ordinary or double) can be entirely recast in the language of plabic graphs. In these and other examples, an important role is played by the subclass of \emph{reduced} plabic graphs that are analogous to---and indeed generalize---reduced decompositions in symmetric groups.
D.~Thurston's \emph{triple diagrams}~\cite{thurston} are closely related to plabic graphs. After making this connection precise and developing the machinery of triple diagrams, we use this machinery to establish the fundamental properties of reduced plabic graphs.
Plabic graphs and related combinatorics have arisen in the study of shallow water waves \cite{kodwil,kodwilpnas} (via the KP equation) and in connection with scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory~\cite{amplitudes}. Constructions closely related to plabic graphs were studied by T.~Kawamura~\cite{kawamura} in the context of the topological theory of graph divides.
A reader interested exclusively in the combinatorics of plabic graphs can read this chapter independently of the previous ones. While we occasionally refer to the combinatorial constructions introduced in Chapters \ref{ch:tp-examples}--\ref{ch:combinatorics-of-mutations}, they are not relied upon in the development of the general theory of plabic graphs.
Cluster algebras as such do not appear in this chapter. On the other hand, reduced plabic graphs introduced herein will prominently feature in the upcoming study of cluster structures in Grassmannians and related varieties, see Chapter~\ref{ch:Grassmannians}.
The structure of this chapter is as follows.
\cref{sec:plabic} introduces plabic graphs and their associated quivers.
In \cref{sec:plabictriangulations}, we recast the combinatorics of triangulations of a polygon and (ordinary or double) wiring diagrams in the language of plabic graphs.
\cref{sec:tri} discusses the version of the theory in which all internal vertices of plabic graphs are \emph{trivalent}. (This version naturally arises in some applications, cf., e.g.,~\cite{fpst}.) As \cref{sec:tri} is not strictly necessary for the sections that follow, it can be skipped if desired.
The important notions of a \emph{trip}, a \emph{trip permutation} and a \emph{reduced plabic graph} are introduced in \cref{sec:reduced-plabic-graphs}. Here we state (but do not prove) some key results, including the ``fundamental theorem of reduced plabic graphs'' which characterizes the move equivalence classes of reduced plabic graphs in terms of associated \emph{decorated permutations}.
\cref{sec:triple-diagrams} introduces the basic notions of \emph{triple diagrams}. We then show that triple diagrams are in bijection with \emph{normal plabic graphs}.
In \cref{sec:mintriple}, we study \emph{minimal} triple diagrams, largely following~\cite{thurston}. These diagrams can be viewed as counterparts of reduced plabic graphs.
In \cref{minred}, we explain how to go between minimal triple diagrams and reduced plabic graphs. We then use this correspondence to prove the fundamental theorem of reduced plabic graphs.
In \cref{sec:bad}, we state and prove the \emph{bad features criterion} that detects whether a plabic graph is reduced or not.
In \cref{sec:affine}, we describe a bijection between decorated permutations and a certain subclass of \emph{affine permutations}.
In \cref{sec:bridge}, a factorization algorithm for affine permutations is used to construct a family of reduced plabic graphs called \emph{bridge decompositions}.
\cref{sec:edge-labels} discusses \emph{edge labelings} of reduced plabic graphs and gives a particularly transparent \emph{resonance criterion} for recognizing whether a plabic graph is reduced.
\cref{sec:face-labels} introduces \emph{face labelings} of reduced plabic graphs.
In \cref{weaksep}, we provide an intrinsic combinatorial characterization of collections of face labels that arise via this construction. Face labels will reappear in Chapter~\ref{ch:Grassmannians} in the study of cluster structures in Grassmannians.
\section{Plabic graphs and their quivers} \label{sec:plabic}
\begin{definition} \label{def:plabic} A \emph{plabic} (planar bicolored) \emph{graph} is a planar graph~$G$ embedded into a closed disk~$\mathbf{D}$, such that: \begin{itemize}[leftmargin=.2in] \item the embedding of $G$ into~$\mathbf{D}$ is proper, i.e., the edges do not cross; \item each internal vertex is colored black or white; \item each internal vertex is connected by a path to some boundary vertex; \item the (uncolored) vertices lying on the boundary of~$\mathbf{D}$ are labeled $1,2,\dots,b$ in clockwise order, for some positive integer~$b$; \item each of these $b$ \emph{boundary vertices} is incident to a single edge.
\end{itemize} Loops and multiple edges are allowed.
We consider plabic graphs up to isotopy of the ambient disk~$\mathbf{D}$ fixing the disk's boundary. The \emph{faces} of~$G$ are the connected components of the complement of
$G$ inside the ambient disk~$\mathbf{D}$. A degree~1 internal vertex which
is connected by an edge to a boundary vertex is called
a \emph{lollipop}. \end{definition}
Two examples of plabic graphs are shown in Figure~\ref{fig:plabic}. Many more examples appear throughout this chapter. In what follows, we will often omit the boundary of the ambient disk when drawing plabic graphs.
\begin{figure}
\caption{(a) A plabic graph $G$. (b) A plabic graph $G'$ with a white lollipop.}
\label{fig:plabic}
\end{figure}
\begin{remark} \label{rem:plabic-history} Plabic graphs were introduced by A.~Postnikov \cite[Section~12]{postnikov}, who used them to describe parametrizations of cells in totally nonnegative Grassmannians. A~closely related class of graphs was defined by T.~Kawamura~\cite{kawamura} in the context of the topological theory of \emph{graph divides}. Our definition is very close to Postnikov's. \end{remark}
The key role in the theory of plabic graphs is played by a particular equivalence relation that is generated by a family of transformations called \emph{(local) moves}.
\pagebreak[3]
\begin{definition} \label{def:moves} We say that two plabic graphs $G$ and $G'$ are \emph{move-equi\-valent}, and write $G \sim G'$, if $G$ and~$G'$ can be related to each other via a sequence of the following \emph{local moves}, denoted (M1), (M2), and~(M3): \begin{itemize}[leftmargin=.4in] \item[(M1)] (The \emph{square move}) Change the colors of all vertices on the boundary of a quadrilateral face, provided these colors alternate and these vertices are trivalent. See Figure~\ref{fig:M1}.
\begin{figure}
\caption{Move (M1) on plabic graphs.}
\label{fig:M1}
\end{figure} \item[(M2)] Remove a bivalent vertex (of any color) and merge the edges adjacent to it; or, conversely, insert a bivalent vertex in the middle of an~edge. See Figure~\ref{fig:M2}. \begin{figure}
\caption{Move (M2) on plabic graphs.}
\label{fig:M2}
\end{figure} \item[(M3)] Contract an edge connecting two internal vertices of the same color; or split an internal vertex into two vertices of the same color joined by an edge. See \cref{fig:M3}.
\begin{figure}
\caption{Move (M3) on plabic graphs. The number of ``hanging'' edges on either side can be any nonnegative integer.}
\label{fig:M3}
\end{figure}
\end{itemize} \end{definition}
\iffalse Besides these moves, there are two kinds of one-way \emph{reductions} that one can apply to plabic graphs, the \emph{loop reduction} (see \cref{fig:R1}) and the \emph{leaf reduction} (see \cref{fig:R2}). The loop reduction applies to any graph which contains a trivalent black vertex and a trivalent white vertex which are joined by two distinct edges. The leaf reduction applies to any graph which has a degree $1$ vertex which is joined to a non-boundary vertex of the opposite color.
\begin{figure}
\caption{The loop reduction on plabic graphs.}
\label{fig:R1}
\end{figure}
\begin{figure}
\caption{The leaf reduction on plabic graphs.}
\label{fig:R2}
\end{figure} \fi
We next explain how to associate a quiver to a plabic graph.
This construction is closely related to the one presented in Definition~\ref{def:graphquiver}.
\begin{definition} \label{def:Q(G)} The \emph{quiver~$Q(G)$ associated to a plabic graph~$G$} is defined as follows. The vertices of $Q(G)$ are in one-to-one correspondence
with the faces of~$G$. A~vertex of $Q(G)$ is declared mutable or frozen depending on whether the corresponding face is \emph{internal} (i.e., disjoint from the boundary of~$\mathbf{D}$) or not.
The arrows of $Q(G)$ are constructed in the following way. Let $e$ be an edge in~$G$ that connects a white vertex to a black vertex and separates two distinct faces, at least one of which is internal. For each such edge~$e$, we introduce an arrow in $Q(G)$ connecting the faces separated by~$e$; this arrow is oriented so that when we move along the arrow in the direction of its orientation, we see the white endpoint of~$e$ on our left and the black endpoint on our right. We then remove oriented $2$-cycles from the resulting quiver, one by one, to get~$Q(G)$. See \cref{fig:plabic2}. \end{definition}
\begin{figure}
\caption{Two plabic graphs and their associated quivers. Shown on the left is the graph~$G$ from \cref{fig:plabic}(a). The quiver on the right has double arrows, corresponding to the instances where a pair of faces
share two boundary segments disconnected from each other. The frozen vertex~$v$ at the top of the picture is isolated: the two arrows between~$v$ and an internal vertex located underneath~$v$ cancel each other.}
\label{fig:plabic2}
\end{figure}
\begin{proposition} \label{pr:plabic-vs-quivers} Let $G$ and $G'$ be two plabic graphs related to each other by one of the local moves {\rm(M1)}, {\rm(M2)}, or~{\rm(M3)}. Additionally, if $G$ and $G'$ are related via the square move~{\rm(M1)}, then we require that \begin{align} \label{eq:square-move-tricky} &\text{among the four faces surrounding the square, the consecutive ones}\\ \notag &\text{must be distinct, see \cref{fig:square-move-tricky}.} \end{align} Then the quivers $Q(G)$ and $Q(G')$ are mutation equivalent. \end{proposition}
\begin{figure}
\caption{ Restriction \eqref{eq:square-move-tricky} allows the square move at~$A$---but not at~$B$, since face~$C$ is adjacent to two consecutive sides~of~$B$.
}
\label{fig:square-move-tricky}
\end{figure}
\begin{proof} It is straightforward to check that a square move in a plabic graph translates into a quiver mutation at the vertex associated to that
square face, provided that condition~\eqref{eq:square-move-tricky} is satisfied.
It is also straightforward to check that the quiver associated with a plabic graph does not change under moves (M2) or (M3), see Figure~\ref{fig:plabic-moves-are-mutations}. \end{proof}
\pagebreak[3]
\begin{figure}
\caption{Fragments of plabic graphs and their associated quivers. The first two plabic graphs are related by a square move (M1); their quivers are related by a single mutation. The second and the third graphs are related by moves of type (M3), and have isomorphic quivers. }
\label{fig:plabic-moves-are-mutations}
\end{figure} \begin{remark} \label{rem:square-move-tricky} Suppose condition~\eqref{eq:square-move-tricky} fails at a square face~$B$, with $B$~incident to another face~$C$ along two consecutive edges, as in \cref{fig:square-move-tricky}. Then the arrows transversal to these edges cancel each other, so they do not appear in the associated quiver. This leads to a discrepancy between the square move and the quiver mutation. \end{remark}
\begin{remark} \label{rem:urban} The key difference between the setting of this section \emph{vs.}\ Section~\ref{urban} is that here we do not require plabic graphs to be bipartite. This distinction is not particularly important, since we can always apply a sequence of moves (M2) to a plabic graph to make it bipartite.
In~the bipartite setting, the square move~(M1) corresponds to urban renewal, see Definition~\ref{def:urban-renewal} as well as \cref{def:urban-normal} below. \end{remark}
\begin{remark} \label{rem:restricted-M1} In light of \cref{pr:plabic-vs-quivers}, one may choose to adjust the definition of the square move~(M1)---hence the notion of move equivalence of plabic graphs---by forbidding square moves violating condition~\eqref{eq:square-move-tricky}. (This convention was adopted in~\cite{fpst}.) We note that for the important subclass of \emph{reduced} plabic graphs (see \cref{def:reduced-plabic} below), condition~\eqref{eq:square-move-tricky} is automatically satisfied, so there is no need to worry about it. \end{remark}
\begin{remark} \label{rem:cluster-algebra-from-plabic} Using \cref{def:Q(G)}, we can associate a seed pattern---hence a cluster algebra---to any plabic graph~$G$. By \cref{pr:plabic-vs-quivers}, this cluster algebra only depends on the move equivalence class of~$G$, assuming that we adopt a restricted notion of move equivalence, cf.\ \cref{rem:restricted-M1}. We will soon see that this family of cluster algebras includes all the main examples of cluster algebras (defined by quivers) introduced in the earlier chapters. This justifies the importance of the combinatorial study of plabic graphs, and in particular their classification up to move equivalence. \end{remark}
The quivers arising from plabic graphs are rather special. In particular, each quiver $Q(G)$ is planar.
On the other hand, the mutation class of any quiver without frozen vertices can be embedded into the mutation class of a quiver of a plabic~graph:
\begin{proposition}[{\cite{FIL}}] \label{pr:embed-into-plabic} Let $Q$ be a quiver whose vertices are all mutable. Then there exists a plabic graph~$G$ such that $Q$ is a full subquiver (see Definition~\ref{def:tildeB_I}) of a quiver mutation-equivalent to~$Q(G)$. \end{proposition}
\section{Triangulations and wiring diagrams via plabic graphs} \label{sec:plabictriangulations}
In this section, we explain how the machinery of local moves on plabic graphs unifies the combinatorial constructions of Chapter~\ref{ch:combinatorics-of-mutations},
including: \begin{itemize}[leftmargin=.2in] \item flips in triangulations of a polygon
(Section~\ref{sec:triangulations}); \item braid moves for wiring diagrams (Section~\ref{sec:mut-wiring}); \item their analogues for double wiring diagrams (Section~\ref{sec:mut-double-wiring}). \end{itemize} As mentioned in Remark~\ref{rem:urban}, the urban renewal transformations (Section~\ref{urban}) can also be interpreted in terms of local moves in plabic graphs.
\begin{example}[{\textbf{Triangulations of a polygon}, see \cite[Algorithm~12.1]{kodwil}}] \label{TriangulationA} Let $T$ be a triangulation of a convex $m$-gon $\mathbf{P}_m\,$. The plabic graph $G(T)$ associated to $T$ is constructed as follows: \begin{enumerate}[leftmargin=.3in] \item Place a white vertex of $G(T)$ at each vertex of~$\mathbf{P}_m$. \item Place a black vertex of $G(T)$ in the interior of each triangle~of~$T$. Connect it by edges to the three white vertices of the triangle. \item Embed $\mathbf{P}_m$ into the interior of a disk~$\mathbf{D}$. \item Place $m$ uncolored vertices of $G(T)$ on the boundary of~$\mathbf{D}$. \item Connect each white vertex of $G(T)$ to a boundary vertex. These edges must not cross. \end{enumerate} We emphasize that the set of edges of $G(T)$ includes neither the sides of $\mathbf{P}_m$ nor the diagonals of~$T$. See \cref{fig:triang-plabic0}. \end{example} \begin{figure}
\caption{A triangulation $T$ of an octagon, and the corresponding
plabic graph $G(T)$, \emph{cf.}\ Figure~\ref{fig:quiver-triangulation}.}
\label{fig:triang-plabic0}
\end{figure}
\begin{exercise}
Show that $Q(G(T)) = Q(T)$, i.e., the quiver associated to the plabic graph of a triangulation $T$
coincides with the quiver $Q(T)$ associated to $T$, as in Definition~\ref{def:Q(T)-polygon}. \end{exercise}
\begin{exercise} Show that if triangulations $T$ and $T'$ are related by a flip, then the plabic graphs $G(T)$ and $G(T')$ are move-equivalent to each~other. More concretely, flipping a diagonal in~$T$ translates into a square move at the corresponding quadrilateral face of~$G(T)$, plus some (M3) moves to make each vertex of that face trivalent. \end{exercise}
\begin{example}[{\textbf{Wiring diagrams}}] \label{def:wiringplabic}
Let $D$ be a wiring diagram, as in Section~\ref{sec:baseaffine}. We associate a plabic graph $G(D)$ to~$D$ by replacing each crossing in $D$ by a pair of trivalent vertices connected vertically, with a black vertex on top and a white vertex on the bottom. We then enclose the resulting graph~in~a~disk.
This construction applies to a more general version of wiring diagrams. Let $s_i$ denote the simple transposition in the symmetric
group $\mathcal{S}_n$ that exchanges $i$ and $i+1$. Given a sequence $\mathbf{w}=s_{i_1} s_{i_2} \dots s_{i_m}$ of simple transpositions, we associate to it a diagram $D(\mathbf{w})$ by concatenating $m$ graphs; here the graph associated to $s_{j}$ consists of $n$ wires, of which $n-2$ are horizontal, while the $j$th and $({j}+1)$st wires cross over each other. See \cref{fig:wiring-plabic0}. \end{example}
\nopagebreak
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\begin{figure}
\caption{Top: the wiring diagrams $D_1$ and $D_2$ associated to reduced expressions $s_2 s_3 s_2 s_1 s_2 s_3$ and $s_3 s_2 s_3 s_1 s_2 s_3$ for $w_0=(4,3,2,1)\in S_4$. These wiring diagrams (resp., reduced expressions) are related via a braid move. Middle: the plabic graphs $G(D_1)$ and $G(D_2)$. Bottom: the quivers $Q(G(D_1))$ and $Q(G(D_2))$, with isolated frozen vertices removed. }
\label{fig:wiring-plabic0}
\end{figure}
\pagebreak[3]
\begin{exercise} \label{exercise:Q(G(D))=Q(D)} Show that after removing isolated frozen vertices at the top and bottom, the quiver $Q(G(D))$ associated to the plabic graph of a wiring diagram $D$ coincides with the quiver $Q(D)$ associated to~$D$, as in Definition~\ref{def:quiverwd}, up to a global reversal of arrows.
\end{exercise}
\begin{remark} If we changed our convention in \cref{def:wiringplabic}, swapping the
colors of the black and white vertices,
we'd recover precisely the quiver $Q(D)$ associated to the wiring diagram.
However, we prefer the convention used in \cref{def:wiringplabic} because it will
lead to a transparent algorithm for recovering the chamber minors,
as shown in \cref{fig:wiring-plabic}.
And as noted in
Remark~\ref{rem:opposite}, the cluster algebra associated to
a given quiver is the same as the cluster algebra associated to the
opposite quiver. \end{remark}
\begin{remark} \label{rem:braid-move=plabic-move} If two wiring diagrams $D$ and $D'$ are related by a braid move, then the corresponding plabic graphs $G(D)$ and $G(D')$ are related by a square move plus some (M3) moves, see \cref{fig:braidsquare0}.
\end{remark}
\begin{figure}
\caption{
A braid move on wiring diagrams, and a
corresponding sequence of moves on plabic graphs. The first two (resp., the last two) plabic graphs are related by two (M3) moves; the two plabic graphs in the middle are related by an (M1) move.}
\label{fig:braidsquare0}
\end{figure}
\begin{example}[{\textbf{Double wiring diagrams}}]
\label{ex:DWD} Let $D$ be a double wiring diagram, as in Section~\ref{sec:matrices}. The plabic graph $G(D)$ associated to~$D$ is defined by adjusting the construction of Example~\ref{def:wiringplabic} in the following way: as before, we replace each crossing in the double wiring diagram by a pair of trivalent vertices connected vertically, and color one of these vertices white and the other black. If the crossing is thin, the top vertex gets colored white and the bottom one black; if the crossing is thick, the colors of the two vertices are reversed.
See \cref{fig:double-wiring-plabic}.
As in \cref{def:wiringplabic}, the above construction works for a more general version of double wiring diagrams. Given two sequences $\mathbf{w} = s_{i_1} s_{i_2} \dots s_{i_m}$ and $\overline{\mathbf{w}} = \overline{s}_{j_1} \overline{s}_{j_2} \dots \overline{s}_{j_{\ell}}$, choose an arbitrary shuffle of $\mathbf{w}$ and $\overline{\mathbf{w}}$. Then we can
associate a (generalized) double wiring diagram to this shuffle, where thick crossings are associated to factors in $\mathbf{w}$, and thin crossings are associated to factors in $\overline{\mathbf{w}}$. So, e.g., the double wiring diagram in \cref{fig:double-wiring-plabic} is associated to the shuffle $\overline{s}_2 s_1 s_2 \overline{s}_1 \overline{s}_2 s_1$. \end{example}
\begin{figure}
\caption{A double wiring diagram $D$, the corresponding plabic graph $G(D)$, and the quiver associated to $G(D)$. If one removes the bottom frozen vertex, one recovers the quiver from Figure~\ref{fig:chamber-quiver2} (up to a global reversal of arrows).}
\label{fig:double-wiring-plabic}
\end{figure}
\begin{exercise} Extend the statements of Exercise~\ref{exercise:Q(G(D))=Q(D)} and Remark~\ref{rem:braid-move=plabic-move} to the case of double wiring diagrams. \end{exercise}
In addition to triangulations and (ordinary or double) wiring diagrams, plabic graphs can also be used to describe Fock-Goncharov cluster structures~\cite{fock-goncharov-ihes}:
\begin{exercise} Construct a plabic graph whose associated quiver is the quiver shown in Figure~\ref{fig:Q_3}. How does this construction generalize to a quiver $Q_3(T)$ associated to an arbitrary triangulation~$T$ of a convex polygon, cf.\ Exercise~\ref{exercise:FG-sl3-d4}? \end{exercise}
\section{Trivalent plabic graphs} \label{sec:tri}
In \cref{sec:plabic}, we introduced plabic graphs and described local moves that generate an equivalence relation on them. In this section, we focus on \emph{trivalent plabic graphs}, i.e., those plabic graphs whose interior vertices are all trivalent. This will require working with an alternative set of moves that preserve the property of being trivalent.
\begin{remark} Trivalent plabic graphs arise naturally in the studies of \begin{itemize}[leftmargin=.2in] \item soliton solutions to the KP equation \cite{kodwil}, \item sections of fine zonotopal tilings of $3$-dimensional cyclic zonotopes~\cite{Galashin}, \item combinatorics of planar divides and associated links~\cite{fpst}, and \item $\pi$-induced subdivisions for a projection $\pi$ from the hypersimplex $\Delta_{k,n}$ to an $n$-gon \cite{PostnikovICM}. \end{itemize} \end{remark}
We begin with the following simple observation.
\begin{lemma} \label{lem:bitri} Any plabic graph with no interior leaves (i.e., no degree $1$ interior vertices) can be transformed by a sequence of moves of type {\rm (M2)} and/or {\rm (M3)} into a plabic graph all of whose interior vertices are trivalent. \end{lemma}
\begin{proof} We can get rid of bivalent vertices using the moves~(M2). If there are any vertices of degree $\ge4$, split those vertices using~(M3) until all internal vertices are trivalent. \end{proof}
The alternative set of moves for trivalent plabic graphs consists of the square move (M1) together with the \emph{flip move} (M4) defined below.
\begin{definition} The \emph{flip move} (sometimes also called the \emph{Whitehead move}) for trivalent plabic graphs is defined as follows: \begin{itemize}[leftmargin=.4in] \item[(M4)] Replace a fragment containing two trivalent vertices of the same color connected by an edge by another such fragment, see \cref{fig:flipmove0}. \end{itemize} \end{definition}
\begin{figure}
\caption{The flip move (or Whitehead move) for trivalent graphs. The four vertices shown should
either all be white, or all be black.}
\label{fig:flipmove0}
\end{figure}
\begin{remark} \label{rem:flip} A flip move (M4) can be expressed as a composition of two moves of type~(M3). \end{remark}
\pagebreak[3]
The main result of this section is the following.
\begin{theorem} \label{thm:newmoves1} Two trivalent plabic graphs $G$ and $G'$ are related via a sequence of local moves of types {\rm (M1)}, {\rm (M2)}, and {\rm (M3)} if and only if they are related by a sequence of moves of types {\rm (M1)} and {\rm (M4)}. \end{theorem}
The rest of this section is devoted to the proof of \cref{thm:newmoves1}.
\pagebreak[3]
\begin{lemma} \label{lem:noM1} If two trivalent plabic graphs $G$ and $G'$ are connected by a sequence of moves of type {\rm(M2)} or {\rm (M3)}, then they are connected by a sequence of moves of type~{\rm(M3)}. \end{lemma}
\begin{proof} We first note that in many cases, move (M2) can be thought of as an instance of move~(M3): instead of using (M2) to add or remove a bivalent vertex that is adjacent via an edge~$e$ to a vertex of the same color, we can use (M3) to (un)contract~$e$, to the same effect.
The only (M2) moves that are genuinely different from (M3) moves are the (M2) moves that add or remove a white (resp., black) vertex in the middle of a black-black or black-boundary (resp., white-white or white-boundary) edge. We call them \emph{creative} or \emph{destructive} (M2) moves, see
the left and middle of \cref{fig:trees}. We need to show that it is never necessary to use a creative or destructive (M2) move to connect $G$ and $G'$ as above.
\begin{figure}
\caption{Using a creative (M2) move and then an (M3) move to grow a white leaf.}
\label{fig:trees}
\end{figure}
Consider a shortest sequence of (M2)/(M3) moves connecting ${G}$ and~${G'}$: \begin{equation} \label{eq:G-to-G'} G = G_0 \sim G_1 \sim \dots \sim G_k = G'. \end{equation} We first claim that each $G_i$ does not contain an internal leaf. Suppose otherwise.
Let $G_{i-1}\sim G_i$ be the last step when we grow a leaf (as in
\cref{fig:trees}). That is, $i$ is maximal such that $G_i$ has a leaf $v'$ attached via edge $e$ to some vertex $v$, where $v'$ and $e$ were not present in~$G_{i-1}$. Since $G'$ has no leaves, there must be some $j>i$ such that $G_j$ is obtained from $G_{j-1}$ by contracting the edge adjacent to~$v'$. By construction, each graph $G_i, G_{i+1}, \dots, G_{j-1}$ contains leaf $v'$ along with a path connecting $v'$ to a vertex
of the same color; this path is obtained by subdividing the edge $e$ by adding bivalent vertices (which must remain bivalent since no new leaves may be added). But then we can construct a shorter sequence of moves connecting $G$ to $G'$ by removing all steps used to create and contract this~path.
Having established that none of the graphs appearing on the shortest path \eqref{eq:G-to-G'} have internal leaves, we will now demonstrate that this path never uses a creative or destructive (M2) move.
Note that we cannot have only destructive (M2) moves because (M3) moves alone cannot create a vertex whose removal requires a destructive (M2) move. Thus, it is enough to show that we cannot have a creative (M2) move along \eqref{eq:G-to-G'}. Suppose there is one, and that the last creative (M2) move adds a bivalent white vertex $w$ along a black-black edge. Along \eqref{eq:G-to-G'}, this bivalent vertex $w$ might split into multiple bivalent white vertices (pairwise adjacent, in a row). However, all these vertices must get removed somewhere along \eqref{eq:G-to-G'}, since $G'$ has no bivalent vertices. We can then shorten \eqref{eq:G-to-G'} by removing all moves involving these bivalent vertices. The lemma is proved. \end{proof}
\pagebreak[3]
\begin{lemma} \label{lem:flipwhite} Let $G$ and $G'$ be two trivalent plabic graphs such that \begin{itemize}[leftmargin=.2in] \item each of the graphs $G$ and $G'$ is connected; \item each of the graphs $G$ and $G'$ has $f$ interior faces, $b$ boundary vertices,
and $b$ boundary faces (the number of boundary vertices
equals the number of boundary faces since the graphs are connected);
\item in each of the graphs $G$ and $G'$, all interior vertices have the same color, and this color is the same in both graphs. \end{itemize} Then $G$ and $G'$ can be connected by a sequence of flip moves~{\rm (M4)}. \end{lemma}
\begin{proof}
The \emph{dual graph} $G_\textup{dual}$ of a trivalent connected plabic graph~$G$ is obtained as follows. Place a vertex of $G_\textup{dual}$ in the interior of each face of~$G$. For each edge~$e$ of~$G$, introduce a (transversal) edge of $G_\textup{dual}$
connecting the vertices of $G_\textup{dual}$ located in the faces of~$G$ on both sides of~$e$, see \cref{fig:dual}. (This new edge may be a loop.) Under the conditions of the lemma, the dual graph $G_\textup{dual}$ is a generalized triangulation~$T$ of a (dual) $b$-gon.
(We note that $T$ may
contain \emph{self-folded} triangles -- loops with
an interior ``pendant'' edge --
coming from the faces of~$G$ enclosed by a loop in~$G$, as in \cref{fig:dual}.) The triangulation~$T$ has $b+f$ vertices: the $b$ vertices of the dual $b$-gon together with the $f$ interior points (``punctures'').
\begin{figure}
\caption{A trivalent plabic graph $G$ and its dual graph $G_\textup{dual}$ (in red); the latter contains a self-folded triangle. Here $b=3$ and $f=1$.}
\label{fig:dual}
\end{figure}
\cref{fig:fliptri} shows that a flip move in a trivalent plabic graph corresponds to a flip in the corresponding triangulation. The claim that $G$ and $G'$ are connected by flip moves can now be obtained from the well-known fact \cite{harer-1986, hatcher} that any two triangulations of a $b$-gon with $f$ interior points are connected by flips. \end{proof}
\begin{figure}
\caption{The flip move for trivalent graphs corresponds to a flip of the corresponding dual triangulations.}
\label{fig:fliptri}
\end{figure}
\begin{definition} \label{def:component} A \emph{white component} $W$ of a plabic graph ${G}$ is obtained by taking a maximal (by inclusion) connected induced subgraph of~${G}$ all of whose internal vertices are white, together with the half-edges extending from the (white) vertices of $W$ towards black vertices outside $W$ or towards boundary vertices of~$G$. \emph{Black components} of ${G}$ are defined in the same way, with the roles of black and white vertices reversed. \end{definition}
\begin{remark} \label{rem:component-as-plabic-graph} Each black or white component $C$ of a plabic graph $G$ can itself be regarded as a (generalized) plabic graph. To this end, enclose~$C$ by a simple closed curve~$\gamma$ passing through the endpoints of the half-edges on the outer boundary of~$C$. If the portion of~$G$ located inside~$\gamma$ is exactly~$C$, then we get a usual plabic graph. It may however happen that $C$ contains ``holes,'' i.e., some of the half-edges on the boundary of~$C$ may be entirely contained in the interior of the disk enclosed by~$\gamma$. In that case, we need to draw simple closed curves through the endpoints of those half-edges, so that $C$ becomes a generalized plabic graph inside a ``swiss-cheese'' shape
(a disk with some smaller disks removed), as in \cref{fig:cheese}. The argument in the proof of \cref{lem:flipwhite} extends to this setting, so \cref{lem:flipwhite} also holds for black/white components
of trivalent plabic graphs. We~will use this generalization in the proof of \cref{prop:flipmoves} below. \end{remark}
\begin{figure}
\caption{A generalized plabic graph inside a ``swiss-cheese'' shape
(in this case, a disk with two smaller disks removed).}
\label{fig:cheese}
\end{figure}
\begin{proposition} \label{prop:flipmoves} Let two trivalent plabic graphs be related to each other by moves {\rm(M2)} or~{\rm (M3)}. Then they are related by a series of flip~moves~{\rm(M4)}. \end{proposition}
\begin{proof} Let $G$ and $G'$ be the plabic graphs in question. Without loss of generality we may assume that $G$ and $G'$ are connected. By \cref{lem:noM1}, $G$ and $G'$ are connected by moves (M3).
Each of the graphs $G$ and $G'$ breaks into disjoint (white or black) components. Each (M3) move only affects a single component. It follows that the white (resp., black) components $W_1,\dots,W_{\ell}$ (resp., $B_1,\dots,B_m$) of~$G$ are in bijection with the components $W'_1,\dots,W'_{\ell}$ (resp., $B'_1,\dots,B'_m$) of~$G'$, so that each $W_i$ (resp., $B_j$) is related to $W_i'$ (resp., $B_j'$) via (M3) moves.
Since an (M3) move preserves both the number of boundary vertices and the number of faces of a graph, both $W_i$ and $W'_i$ (respectively, $B_j$ and~$B'_j$) have the same number of boundary vertices and the same number of faces. It now follows from \cref{lem:flipwhite} (more precisely, from its extension to components of plabic graphs, see \cref{rem:component-as-plabic-graph}) that each pair $W_i$ and $W'_i$ can be connected by flip moves, and similarly for $B_j$ and $B'_j$. The proposition follows. \end{proof}
\begin{proof}[Proof of \cref{thm:newmoves1}.]
The ``if'' direction immediately follows from \cref{rem:flip}.
Suppose that $G$ and $G'$ are related via a sequence of (M1), (M2), and (M3) moves. Let $k$ denote the number of square moves~(M1) in the sequence.
We then have a sequence of move-equivalences \[
G=G_0' \sim G_1 \sim G_1' \sim G_2 \sim G_2'\sim \dots \sim G_k\sim G_k'\sim G_{k+1}=G', \] where for all $i$, \begin{itemize}[leftmargin=.2in] \item $G_i$ is related to $G_i'$ by a single square move; \item $G_i'$ is related to $G_{i+1}$ by a sequence of (M2) and (M3) moves. \end{itemize} Since a square move only involves trivalent vertices, we may assume, applying extra (M2) and (M3) moves as needed, that all plabic graphs $G_i$ and $G_i'$ are trivalent. It then follows by \cref{prop:flipmoves} that for every~$i$, the graphs $G_i'$ and $G_{i+1}$ are related by flip moves alone, and we are done. \end{proof}
\begin{remark} The plabic graphs associated to wiring diagrams and double wiring diagrams as in \cref{def:wiringplabic} and \cref{ex:DWD} are trivalent, and consequently one can express the transformation corresponding to braid moves using square moves and flip moves, as shown in \cref{fig:braidsquare0}.
On the other hand, the plabic graphs associated to triangulations of a polygon (see \cref{TriangulationA} and \cref{fig:triang-plabic0}) are \emph{not} trivalent. \end{remark}
\section{Trips. Reduced plabic graphs} \label{sec:reduced-plabic-graphs}
Reduced plabic graphs are a subclass of plabic graphs that play a critically important role in the study of cluster structures and total positivity in Grassmannians.
In this section, we introduce reduced plabic graphs and various related concepts. The main result of the section, \cref{thm:moves}, provides a criterion for move equivalence of reduced plabic graphs. The proof of this theorem will occupy Sections~\ref{sec:triple-diagrams}--\ref{minred}.
\begin{definition} \label{def:collapsible-tree} We say that $T$ is a \emph{collapsible tree} in a plabic graph~$G$~if \begin{itemize}[leftmargin=.2in] \item
$T$ is a tree with at least one edge not incident
to the boundary of $G$; \item $T$ is an induced subgraph of~$G$; \item $T$ is attached to the rest of~$G$
at a single vertex~$v$, the \emph{root}~of~$T$, so that
\begin{itemize}
\item if $v$ is a boundary vertex, one can use local moves {\rm(M2)--(M3)} to collapse~$T$ to a lollipop based at~$v$.
\item if $v$ is an internal vertex, one can use local moves {\rm(M2)--(M3)} to collapse~$T$ onto the vertex~$v$. \end{itemize} \end{itemize} In other words, local moves can be used to replace the tree~$T$ by either a lollipop based at~$v$, or by~$v$. See \cref{fig:collapsible-tree}. \end{definition}
\begin{figure}
\caption{
A collapsible tree $T$ whose root is the internal vertex~$v$.}
\label{fig:collapsible-tree}
\end{figure}
\begin{definition} \label{def:reduced-plabic} A plabic graph~$G$ is \emph{reduced} if no plabic graph $G'\sim G$ contains one of the following ``forbidden configurations:'' \begin{itemize}[leftmargin=.2in] \item a \emph{hollow digon}, i.e., two edges connecting a pair of distinct vertices,
with no other edges in the region between them; or \item an internal leaf that is not a lollipop and does not belong to a collapsible tree.
\end{itemize}
See \cref{fig:fail} for a slightly modified (but equivalent) version.
\end{definition}
\begin{figure}
\caption{A plabic graph is not reduced if and only if it is move-equivalent to a graph containing a hollow digon, as in (a,b,c), or a graph containing an internal leaf~$u$ adjacent to a trivalent vertex~$v$ of opposite color such that $v$~is not a root of a collapsible tree, see~(d,e).
}
\label{fig:fail}
\end{figure}
\begin{remark} \label{rem:comments-on-reduced-plabic} For plabic graphs, the property of being reduced is, by definition, invariant under local moves. \end{remark}
\begin{remark}
As stated, this property is not readily testable, because the move equiv\-a\-lence class of a plabic graph is usually infinite. We will later obtain criteria (see Theorems~\ref{thm:reduced} and~\ref{thm:resonance}) for testing this property. \end{remark}
\begin{remark} \label{rem:alt-def-reduced} The notion of a reduced plabic graph, as introduced above in \cref{def:reduced-plabic}, may appear artificial, as the choice of forbidden configurations is not well motivated. We will soon present alternative definitions of reducedness (in slightly restricted generality) that are both more conceptual and more elegant; see \cref{pr:reduced-collapse} and \cref{cor:reduced=min-faces}. In~particular, these alternative definitions do not involve
the forbidden configurations in \cref{fig:fail}(d,e). \end{remark}
\pagebreak[3]
\begin{remark} The requirement in \cref{fig:fail}(d,e) concerning the collapsible
tree
cannot be removed: as~shown in \cref{fig:subtle-forbidden}, dropping this requirement would make \emph{any} plabic graph non-reduced. \end{remark}
\begin{figure}
\caption{Any plabic graph can be transformed via moves (M2)--(M3) into a graph containing a configuration as in \cref{fig:fail}(d,e) in which vertex~$v$ is the root of a collapsible tree that contains vertex~$u$.
}
\label{fig:subtle-forbidden}
\end{figure}
\pagebreak[3]
Reduced plabic graphs can be interpreted as generalizations of reduced expressions in symmetric groups:
\begin{exercise} \label{exercise:braid-equivalence} Consider sequences (or ``words'') $\mathbf{w} = s_{i_1} \cdots s_{i_m}$ of simple transpositions in a symmetric group, as in \cref{def:wiringplabic}. We say that two such words are \emph{braid-equivalent} if they can be related to each other using the braid relations $s_{i} s_{i+1} s_i = s_{i+1} s_i s_{i+1}$
and $s_i s_j = s_j s_i$ for $|i-j| \geq 2$. A~word $\mathbf{w}$ is called a \emph{reduced expression} if no word in its braid equivalence class has two consecutive equal entries: $\cdots s_i s_i \cdots$. Show that if $\mathbf{w}$ fails to be a reduced expression, then $G(D(\mathbf{w}))$ fails to be a reduced plabic graph. \end{exercise}
\begin{remark} Conversely, if $\mathbf{w}$ is reduced, then $G(D(\mathbf{w}))$ is reduced; this can be proved using \cref{thm:reduced} or~\cref{thm:resonance}. \end{remark}
\begin{proposition} \label{pr:reduced-collapse} Let $G$ be a plabic graph that has no internal leaves, other than lollipops. Then the following are equivalent: \begin{itemize}[leftmargin=.3in] \item[{\rm (i)}] $G$ is not reduced; \item[{\rm (ii)}] $G$ can be transformed, via local moves that do not create internal leaves, into a plabic graph containing a hollow digon, cf.\ \cref{fig:fail}{\rm(a,b,c)}. \end{itemize} \end{proposition}
\pagebreak[3]
We note that in the absence of internal leaves, the only forbidden configurations are the hollow digons.
The proof of \cref{pr:reduced-collapse} will require some technical preparations, see \cref{def:overline-G} and \cref{lem:nearly-leafless} below.
\begin{definition} \label{def:overline-G} We denote by $\overline G$ the plabic graph obtained from~$G$ by repeatedly collapsing all collapsible trees. It~is not hard to see that $\overline G$ is uniquely defined. \end{definition}
\begin{lemma} \label{lem:nearly-leafless} Let $G$ be a plabic graph such that $\overline G$ has no internal leaves, other than lollipops. Let $G'$ be a plabic graph obtained from $G$ by a local move. Assume that this move is not a square move~{\rm(M1)} where one of the vertices of the square (in~$G$) is the root of a collapsible tree. Then $\overline G'$ has no internal leaves, other than lollipops. Moreover, $\overline G$ and~$\overline G'$ are either equal to each other or related by a single local move. \end{lemma}
\begin{proof} If the changes resulting from the local move $G\to G'$ occur within a collapsible tree, then the tree remains collapsible, so $\overline G'=\overline G$. Otherwise, the same move can be applied in~$\overline G$, yielding~$\overline G'$ (and not creating any leaves). \end{proof}
\begin{proof}[Proof of \cref{pr:reduced-collapse}] The implication (ii)$\Rightarrow$(i) is immediate. Let us establish (i)$\Rightarrow$(ii).
Assume that $G$ is not reduced. Then there exists a sequence of plabic~graphs \begin{equation} \label{eq:G0G1...GN} G=G_0, G_1,\dots, G_N \end{equation} in which each pair $(G_i,G_{i+1})$ is related by a local move and moreover $G_N$ contains a forbidden configuration from \cref{fig:fail}.
\noindent \emph{Case~1:} the sequence \eqref{eq:G0G1...GN} does not include a square move $G_k\stackrel{\text{(M1)}}{\longrightarrow} G_{k+1}$ where one of the vertices of the square in~$G_k$ is the root of a collapsible tree.
Repeatedly applying \cref{lem:nearly-leafless}, we conclude that the graphs $\overline G_i$ do not contain internal leaves, other than lollipops. Moreover, for each~$i$, the plabic graphs $\overline G_i$ and $\overline G_{i+1}$ either coincide or are related via a single local move. It is furthermore easy to see that since $G_N$ contains a forbidden configuration, then the same must be true for~$\overline G_N$. That is, $\overline G_N$ contains a hollow digon. We conclude that $G=G_0=\overline G_0$ is connected by local moves that do not create internal leaves to a plabic graph $\overline G_N$ containing a hollow digon.
\noindent \emph{Case~2:} the sequence \eqref{eq:G0G1...GN} includes a square move $G_k\stackrel{\text{(M1)}}{\longrightarrow} G_{k+1}$ in which one of the vertices of the square in~$G_k$ is the root of a collapsible tree. Let $(G_k,G_{k+1})$ be the first such occurrence (i.e., the one with the smallest~$k$). Repeatedly applying \cref{lem:nearly-leafless}, we conclude that $G=\overline G_0$ is related to $\overline G_k$ via local moves that do not create internal leaves. Moreover, $\overline G_k$ contains a square configuration in which one of the vertices of the square is bivalent. Removing this vertex using (M2) and contracting the resulting edge yields a bicolored hollow digon, and we are done. \end{proof}
The most fundamental result concerning reduced plabic graphs is their classification up to move equivalence (cf.\ \cref{rem:comments-on-reduced-plabic}), to be given in \cref{thm:moves} below. To state this result, we will need some preparation.
\begin{definition} \label{def:trip} A \emph{trip} $\tau$ in a plabic graph $G$ is a directed walk
along the edges of $G$ that either begins and ends at boundary vertices (with all intermediate vertices internal),
or is a closed walk entirely contained in the interior of the disk, which obeys the following
\emph{rules of the road}: \begin{itemize}[leftmargin=.2in] \item at a black vertex, $\tau$ always makes the sharpest possible right turn; \item at a white vertex, $\tau$ always makes the sharpest possible left turn. \end{itemize}
If the trip begins and ends at boundary vertices we call it a \emph{one-way trip}; if it is a closed walk entirely contained in the interior of the disk, we call it a \emph{roundtrip}.
\end{definition} The endpoints of a one-way trip may coincide with each other. For example, the trip corresponding to a lollipop rooted at a vertex~$i$ starts and ends at~$i$.
\begin{remark} Just as different countries have different rules regarding which side of the road one should drive on, different authors make conflicting choices for the rules of the road for plabic graphs. In this book, we consistently use the convention chosen in \cref{def:trip}. \end{remark}
\begin{remark} The notion of a trip and the condition of being reduced have appeared in the study of dimer models in statistical mechanics,
wherein trips have been called \emph{zigzag paths} \cite{Kenyon}. Reduced plabic graphs were called ``marginally geometrically consistent" in \cite[Section 3.4]{Broomhead}, and were said to ``obey condition Z" in \cite[Section 8]{Bocklandt}. \end{remark}
\begin{remark} \label{rem:atmost2} For any edge $e$ in~$G$, there is a unique trip traversing $e$ in each of the two directions.
It may happen that the same trip traverses $e$ twice (once in each direction).
\end{remark}
\begin{exercise} Show that one-way trips starting at different vertices terminate at different vertices. \end{exercise}
\begin{exercise} Let $G(D)$ be a plabic graph associated to some wiring diagram~$D$, see \cref{def:wiringplabic} and \cref{fig:wiring-plabic0}. Show that the trips starting at the left side of~$G(D)$ follow the pattern determined by the strands of~$D$, while the trips starting at the right side of~$G(D)$ proceed horizontally to the left. Describe the trips in a plabic graph associated to a double wiring diagram. \end{exercise}
\begin{definition} Let $G$ be a plabic graph with $b$ boundary vertices. The \emph{trip permutation} $\pi_G: \{1,\dots,b\} \to \{1,\dots,b\}$ is defined by setting \hbox{$\pi_G(i)=j$} whenever the trip originating at $i$ terminates at~$j$.
We will mostly use the one-line notation $\pi_G=(\pi_G(1), \dots, \pi_G(b))$ to represent these permutations. \end{definition}
To illustrate, in \cref{fig:plabic}(a), we have $\pi_G = (3,4,5,1,2)$.
\pagebreak[3]
\begin{exercise} \label{exercise:trip-invariant} Show that move-equivalent plabic graphs have the same
trip permutation. \end{exercise}
The notion of a trip permutation can be further enhanced to construct finer invariants of local moves. For example, we can record, in addition to the trip permutation, the suitably defined \emph{winding number} of each trip. These winding numbers do not change under local moves (with one subtle exception, cf. \cref{fig:M3-white-leaf} below). A~more powerful invariant associates to any plabic graph a particular (transverse) link, see~\cite{fpst}.
\begin{definition} A \emph{decorated permutation} ${\widetilde{\pi}}$ on $b$ letters is
a permutation of the set $\{1,\dots,b\}$ together with a
\emph{decoration} of each fixed point by either an overline or an underline.
In other words, for every~$i$, we have \[ {\widetilde{\pi}}(i)\in\{\overline{i}, \underline{i}\}\cup \{1,\dots,b\}\setminus\{i\}. \] \end{definition}
An example of a decorated permutation on $6$ letters is $(3,4,5,1,2,\overline{6})$.
\begin{exercise} \label{ex:enumeration} Show that the number of decorated permutations on $b$ letters is equal to $b!\sum_{k=0}^b \frac{1}{k!}$.
\end{exercise}
The following statement
will be proved in \cref{minred}.
\begin{proposition} \label{prop:fixedlollipop} Let $G$ be a reduced plabic graph. If $\pi_G(i) = i$, then the connected component of~$G$ containing the boundary vertex~$i$ collapses to a lollipop. \end{proposition}
In \cref{prop:fixedlollipop}, the requirement that $G$ is reduced cannot be dropped, see \cref{fig:fork-not-lollipop}.
\begin{figure}
\caption{A non-reduced plabic graph $G$ with $\pi_G(i)\!=\!i$,
cf.\ \cref{fig:fail}(d). The component containing~$i$ is not collapsible.}
\label{fig:fork-not-lollipop}
\end{figure}
\begin{definition} \label{def:dtp} Let $G$ be a reduced plabic graph with $b$ boundary vertices. The \emph{decorated trip permutation} associated with~$G$ is defined as follows: \[ {\widetilde{\pi}}_G(i) = \begin{cases} \pi_G(i) & \text{if $\pi_G(i) \neq i$;} \\ \ \ \overline{i} & \text{if $G$ contains a tree collapsing to a white lollipop at~$i$;}\\ \ \ \underline{i} & \text{if $G$ contains a tree collapsing to a black lollipop at~$i$.} \end{cases} \] \end{definition}
\cref{fig:plabic-move-equiv} shows two reduced plabic graphs with the same
decorated trip permutation $
(3,4,5,1,2,\overline{6})$.
\begin{figure}
\caption{Two reduced plabic graphs sharing the same decorated trip permutation $(3,4,5,1,2,\overline 6)$. Cf.\ \cref{fig:plabic}(b). }
\label{fig:plabic-move-equiv}
\end{figure}
Exercise~\ref{exercise:trip-invariant} can be strengthened as follows.
\begin{exercise} \label{exercise:forward} The decorated trip permutation of a reduced plabic graph is invariant under local moves. \end{exercise}
We will later show (see \cref{permtoG}) that for each decorated permutation ${\widetilde{\pi}}$ on $b$ letters, there exists a reduced plabic graph whose decorated trip permutation is ${\widetilde{\pi}}$.
Crucially, the move-equivalence class of a reduced plabic graph is completely determined by its decorated trip permutation:
\begin{theorem}[Fundamental theorem of reduced plabic graphs] \label{thm:moves} Let $G$ and $G'$ be reduced plabic graphs. The following statements are equivalent: \begin{enumerate}[leftmargin=.3in] \item[{\rm (1)}] $G$ and $G'$ are move-equivalent; \item[{\rm (2)}] $G$ and $G'$ have the same decorated trip permutation. \end{enumerate} \end{theorem}
To illustrate, the two reduced plabic graphs shown in \cref{fig:plabic-move-equiv} have the same decorated trip permutation and consequently are move-equivalent.
The statement (1)$\Rightarrow$(2) in \cref{thm:moves} is easy, cf.\ \cref{exercise:forward}. The converse implication (2)$\Rightarrow$(1) is much harder. In \cref{minred}, we give a proof of this implication that
utilizes D.~Thurston's machinery of triple diagrams, which is presented in Sections~\ref{sec:triple-diagrams}--\ref{sec:mintriple}.
A very intricate argument justifying the implication (2)$\Rightarrow$(1) was given in A.~Postnikov's original preprint \cite[Section 13]{postnikov}. Another proof of \cref{thm:moves}, involving some difficult results about \emph{plabic tilings} (and relying on \cref{thm:ops} below), was given by S.~Oh and D.~Speyer \cite{oh-speyer}.
\begin{corollary} \label{cor:reduced=min-faces} Let $\pi$ be a permutation on $b$ letters. Consider all plabic graphs $G$ without internal leaves (other than lollipops) whose trip permutation is~$\pi$; in particular, $G$~has $b$ boundary vertices. Among all such~plabic graphs~$G$, the reduced ones are precisely those that have the smallest number of faces. \end{corollary}
\begin{proof} Local moves do not change the number of faces. It follows by \cref{thm:moves} that all reduced plabic graphs with a given decorated trip permutation have the same number of faces.
Changing the color of a lollipop transforms a reduced plabic graph into another reduced graph with the same number of faces and the same trip permutation (but with different decoration). Therefore all reduced plabic graphs~$G$ with $\pi(G)=\pi$ have the same number of faces.
It remains to show that if $G$ is not reduced and has no internal leaves other than lollipops, then there exists a plabic graph $G'$ with $\pi(G')=\pi$ and with fewer faces than~$G$. We note that under our assumptions on~$G$, \cref{pr:reduced-collapse} applies, so $G$ can be transformed by local moves that do not create internal leaves into a plabic graph~$G''$ containing a hollow digon.
We now claim that $G''$ can be replaced by a plabic graph $G'''$ (\emph{not} move-equivalent to~$G''$) such that $G''$ and $G'''$ have the same trip permutation but $G'''$ has fewer faces than~$G''$. The recipe for constructing $G'''$ is as follows. If the vertices of the hollow digon in~$G''$ are of the same color, then remove one of the sides of the digon (keeping its vertices) to get~$G'''$. If, on the other hand, the vertices of the digon have different colors, then remove both sides of the digon; if one of the vertices was bivalent, then remove it as well. It is straightforward to check that in each case, the trip permutation does not change whereas the number of faces decreases by 1 or~2. \end{proof}
In \cref{cor:number-faces}, we will give a formula for the number of faces in a reduced plabic graph in terms of the associated decorated trip permutation.
\begin{remark}
In \cref{cor:reduced=min-faces}, the requirement that $G$ has no internal leaves cannot be dropped. For example, the graph in \cref{fig:fork-not-lollipop} has a single face but is not reduced. \end{remark}
\begin{remark}
Some authors call a plabic graph reduced if it has the smallest number of faces among all graphs with a given decorated trip permutation, cf.\ \cref{cor:reduced=min-faces}. If one adopts this definition, then the graph~$G$ in \cref{fig:fork-not-lollipop} becomes reduced. This leads to a failure of \cref{prop:fixedlollipop}, since the component of~$G$ attached to the boundary vertex~$i$ does not collapse to a lollipop.
Another reason to treat this kind of plabic graph~$G$ as non-reduced will arise in the context of triple diagrams, cf.\ \cref{minred}. While reduced plabic graphs should correspond to minimal triple diagrams (see \cref{red-minimal}), the triple diagram corresponding to~$G$ is not minimal. \end{remark}
\section{Triple diagrams and normal plabic graphs} \label{sec:triple-diagrams}
\emph{Triple diagrams} (or triple crossing diagrams), introduced by D.~Thurston in~\cite{thurston}, are planar topological gadgets closely related to plabic graphs. Our treatment of triple diagrams in Sections~\ref{sec:triple-diagrams}--\ref{sec:mintriple} is largely based on~\cite{thurston}.
\begin{definition} \label{def:triple-diagram} Consider a collection $\mathfrak{X}$ of oriented intervals and/or circles immersed into a disk~$\mathbf{D}$. Their images are collectively called \emph{strands}. The images of immersed intervals (resp., circles) are the \emph{arcs} (resp., \emph{closed strands}) of~$\mathfrak{X}$. A~\emph{face} of~$\mathfrak{X}$ is a connected component of the complement of the union of the strands within~$\mathbf{D}$. We call $\mathfrak{X}$ a \emph{triple diagram} if \begin{itemize}[leftmargin=.2in] \item the preimage of each point in~$\mathbf{D}$ consists of either 0, 1, or 3~points; in the latter case of a \emph{triple point}, the three local branches intersect transversally in the interior of~$\mathbf{D}$; \item the endpoints of arcs are distinct points located on the boundary $\partial \mathbf{D}$; each arc meets the boundary transversally;
\item the union of the strands and the boundary of the disk is connected; this ensures that each face is homeomorphic to an open disk; \item the strand segments lying on the boundary of each face are oriented consistently (i.e., clockwise or counterclockwise); in particular, as we move along the boundary, the \emph{sources} (endpoints where an arc runs away from~$\partial \mathbf{D}$) alternate with the \emph{targets} (where an arc runs into~$\partial \mathbf{D}$).
\end{itemize}
Each triple diagram, say with $b$ arcs, comes with a selection of $b$ points on~$\partial \mathbf{D}$ (called \emph{boundary vertices}) labeled $1,\dots,b$ in clockwise order. There is one such boundary vertex within every other segment of~$\partial \mathbf{D}$ between two consecutive arc endpoints. Specifically, we place boundary vertices so that, moving clockwise along the boundary, each boundary vertex follows (resp., precedes) a source (resp., a target). See \cref{fig:triple}. \end{definition}
\begin{figure}\label{fig:triple}
\end{figure}
Triple diagrams are considered up to smooth isotopy among such diagrams. This makes them essentially combinatorial objects: 6-valent/uni\-valent directed graphs with some additional structure.
\begin{remark} In \cite{thurston}, the definition of a triple diagram does not include the restriction appearing in \cref{def:triple-diagram} that requires the union of the strands and the boundary~$\partial\mathbf{D}$ to be connected. In the terminology of~\cite{thurston}, all our triple diagrams are \emph{connected}. \end{remark}
\begin{remark} In order to ensure consistent orientations along the face boundaries, the orientations of strands must alternate between ``in'' and ``out'' around each triple point. Given a triple diagram with unoriented strands, we can satisfy this condition as follows: start anywhere and propagate out by assigning alternating orientations around vertices. \end{remark}
\begin{definition} \label{def:strandperm} Let $\mathfrak{X}$ be a triple diagram with $b$ boundary vertices (hence $b$ arcs). For each boundary vertex~$i$, let $s_i$ (resp.,~$t_i$) denote the source (resp., target) arc endpoint located next to~$i$ on the boundary of~$\mathbf{D}$.
The \emph{strand permutation} $\pi_{\mathfrak{X}}$ is defined by setting $\pi_{\mathfrak{X}}(i)=j$ whenever the arc originating at~$s_i$ ends up at~$t_j$. Thus, the strand permutation describes the connectivity of the arcs. See \cref{fig:triple}. \end{definition}
We will soon see (cf.\ \cref{def:standard-triple} below) that any permutation can arise as a strand permutation of a triple diagram.
Just as the local moves on plabic graphs preserve the (decorated) trip permutation (see \cref{exercise:forward}), there is a notion of a local move on triple diagrams that keeps the strand permutation invariant.
\begin{definition} \label{def:2-2-move-equivalence} We say that two triple diagrams $\mathfrak{X}$ and $\mathfrak{X}'$ are \emph{move-equivalent} to each other, and write $\mathfrak{X} \sim \mathfrak{X}'$, if one can get between $\mathfrak{X}$ and~$\mathfrak{X}'$ via a sequence of \emph{swivel moves} shown in \cref{fig:2-2-move}. (These moves are called \emph{$2\leftrightarrow2$ moves} in~\cite{thurston}.) \end{definition}
\begin{figure}
\caption{The swivel move replaces one of these fragments of a triple diagram by the other fragment, then smooths out the strands. The orientation of each strand on the left should match the orientation of the strand on the right that has the same endpoints.}
\label{fig:2-2-move}
\end{figure}
\begin{remark} The connectivity of strands on both sides of \cref{fig:2-2-move} is the same, so the strand permutation is invariant under the swivel move. \end{remark}
We will soon see that triple diagrams are essentially cryptomorphic to plabic graphs---or more precisely, to the subclass of normal plabic graphs defined below.
\pagebreak[3]
\begin{definition} \label{def:normal} We say that a plabic graph $G$ is \emph{normal} if the coloring of its
internal vertices
is bipartite, all white vertices in $G$ are trivalent, and each boundary vertex is adjacent to a black vertex. See \cref{fig:normal-plabic}. \end{definition}
\begin{figure}
\caption{A normal plabic graph~$G$. This graph was obtained from the one in \cref{fig:plabic}(a) by inserting several bivalent black vertices.}
\label{fig:normal-plabic}
\end{figure}
\begin{remark} \label{rem:no-white-lollipop} Since a normal plabic graph does not have white leaves, it cannot contain a tree that collapses to a white lollipop. Therefore, in the case of normal graphs, there is no need to decorate the trip permutation. \end{remark}
\begin{definition} \label{def:GX} The triple diagram $\mathfrak{X}(G)$ associated to a normal plabic graph~$G$ is constructed as follows. To each trip in~$G$---either a one-way trip or a roundtrip---we associate a strand in the ambient disk~$\mathbf{D}$ by slightly deforming the trip, as shown in \cref{fig:plabic-to-triple}, so that the strand \begin{itemize}[leftmargin=.2in] \item runs along each edge of the trip, keeping the edge on its left, \item makes a U-turn at each black internal leaf (a vertex of degree~1); \item ignores black vertices of degree~2, \item makes a right turn (as sharp as possible) at each other black vertex, and \item makes a left turn at each white vertex~$v$ along the trip, passing through~$v$. \end{itemize} This collection of strands forms a triple diagram $\mathfrak{X}(G)$,
see \cref{fig:normaltriple} and \cref{lem:normaltriple}. \end{definition}
\begin{figure}
\caption{Constructing a triple diagram from a normal plabic graph.}
\label{fig:plabic-to-triple}
\end{figure}
\begin{figure}
\caption{Left: A normal plabic graph~$G$ (cf.\ \cref{fig:normal-plabic}) together with the associated triple diagram~$\mathfrak{X}=\mathfrak{X}(G)$. Conversely, $G=G(\mathfrak{X})$. Right: The triple diagram~$\mathfrak{X}=\mathfrak{X}(G)$. The trip permutation of $G$ and the strand permutation of $\mathfrak{X}$ are equal: $\pi_G=\pi_\mathfrak{X}=(3,4,5,1,2)$.}
\label{fig:normaltriple}
\end{figure}
\begin{lemma} \label{lem:normaltriple}
The diagram $\mathfrak{X}(G)$
associated to a normal plabic graph~$G$ as in \cref{def:GX} is a triple diagram. \end{lemma} \begin{proof} Since the white vertices in $G$ are trivalent, $\mathfrak{X}(G)$ has a triple point for every white vertex in~$G$, and no other crossings. We need to check that $\mathfrak{X}=\mathfrak{X}(G)$ is connected, or more precisely, that the union of the strands and the boundary of the disk is connected, as required by \cref{def:triple-diagram}.
Let us ignore any component consisting of a single black vertex which is adjacent only to boundary vertices (e.g. a black lollipop), as the corresponding strands are clearly connected to the boundary of the disk. Consider any other strand $S$ in~$\mathfrak{X}$. By construction, $S$~passes through at least one white vertex of the (bipartite) graph~$G$, which is a~triple point on~$S$. It therefore suffices to show that every triple point in~$\mathfrak{X}$ is connected to the boundary within~$\mathfrak{X}$ (i.e., via strand segments of~$\mathfrak{X}$).
Let $u$ be a $k$-valent black vertex in~$G$ and let $v_1,\dots,v_k$ be the white or boundary vertices adjacent to~$u$. The strands of~$\mathfrak{X}$ that run along the $k$ edges of~$G$ incident to~$u$ cyclically connect the triple points $v_1,\dots,v_k$ to each other. (If the list $v_1,\dots,v_k$ includes boundary vertices, then the corresponding strand segments are connected via the boundary.) We conclude that for any two-edge path $v-\!\!\!-u-\!\!\!-v'$ in~$G$ connecting two white or boundary vertices $v$ and~$v'$ via a black vertex~$u$, the triple (or nearby boundary) points $v$ and~$v'$ are connected within~$\mathfrak{X}$. It follows that for any path in the bipartite graph~$G$ connecting a white vertex~$v$ to the boundary, there is a path in~$\mathfrak{X}$ that connects the triple point~$v$ to the boundary.
It remains to note that by \cref{def:plabic}, any white vertex $v$ in~$G$ is connected by a path in~$G$ to some boundary vertex. \end{proof}
We now go in the opposite direction, from a triple diagram to a normal plabic graph.
\begin{definition} \label{def:G(X)} The normal plabic graph $G\!=\!G(\mathfrak{X})$ associated to a triple diagram~$\mathfrak{X}$ is constructed as follows. Place a white vertex of~$G$ at each triple crossing in~$\mathfrak{X}$. Treat each boundary vertex of~$\mathfrak{X}$ as a boundary vertex of~$G$. For each region~$R$ of~$\mathfrak{X}$ whose boundary is oriented counter\-clock\-wise, place a black vertex in the interior of~$R$ and connect it to the white and boundary vertices lying on the boundary of~$R$, so that each white (resp., boundary) vertex is trivalent (resp., univalent). The resulting plabic graph $G\!=\!G(\mathfrak{X})$ is normal by construction. \end{definition}
\begin{proposition} \label{pr:bijplabictriple} The maps $G \mapsto \mathfrak{X}(G)$ and $\mathfrak{X}\mapsto G(\mathfrak{X})$ described in Definitions \ref{def:GX} and~\ref{def:G(X)} are mutually inverse bijections between normal plabic graphs and triple diagrams with the same number of boundary vertices. The trip permutation $\pi_G$ of a normal graph~$G$ is equal to the strand permutation $\pi_{\mathfrak{X}}$ of the corresponding triple diagram $\mathfrak{X}=\mathfrak{X}(G)$. \end{proposition}
\begin{proof} Starting from a normal graph~$G$, let us decompose it into star-shaped subgraphs~$S_v$ each of which includes a black vertex~$v$, all the edges incident~to~$v$, and the endpoints of those edges. Each of these stars will give rise to a fragment of the triple diagram~$\mathfrak{X}(G)$ that ``hugs'' the edges of~$S_v$ and whose boundary is oriented counterclockwise (looking from~$S_v$). Moreover, $\mathfrak{X}(G)$ is obtained by stitching these fragments together. Applying the map $\mathfrak{X}\mapsto G(\mathfrak{X})$ to $\mathfrak{X}(G)$ will recover the original graph~$G$.
One similarly shows that if we start from a triple diagram~$\mathfrak{X}$, construct the normal graph~$G(\mathfrak{X})$, and then apply the map $G\mapsto \mathfrak{X}(G)$ to $G(\mathfrak{X})$, then we recover the original triple diagram~$\mathfrak{X}$. The key property to keep in mind is that each face of~$\mathfrak{X}$ is homeomorphic to an open disk.
The strands of $\mathfrak{X}$ run alongside the trips of~$G$, implying that $\pi_\mathfrak{X}\!=\!\pi_G$. \end{proof}
\begin{figure}
\caption{The six reduced normal plabic graphs with three boundary vertices, shown together with the corresponding triple diagrams, cf.\ \cref{def:GX}. The associated trip (resp., strand) permutations are precisely the six permutations of $\{1,2,3\}$. }
\label{fig:bijreg}
\end{figure}
In \cref{red-minimal}, we will characterize triple diagrams that correspond, under the bijection of \cref{pr:bijplabictriple}, to \emph{reduced} normal plabic graphs.
\pagebreak[3]
The bijective correspondence between triple diagrams and normal plabic graphs can be used to translate the move equivalence of triple diagrams (under the swivel moves, see \cref{def:2-2-move-equivalence}) into a version of the move equivalence of plabic graphs (cf.\ \cref{def:moves}) formulated entirely within the setting of normal plabic graphs:
\begin{definition} \label{def:urban-normal} The \emph{(normal) urban renewal} move is the local transformation of normal plabic graphs described in \cref{fig:urban2}. (This differs slightly from Definition~\ref{def:urban-renewal}. In this chapter, we will consistently use the new definition.) Unlike the square move~(M1), the normal urban renewal move does not require the vertices of the square to be trivalent. They can even be bivalent, see \cref{fig:urban2-bivalent}. \end{definition}
\begin{figure}
\caption{Normal urban renewal replaces one of these configurations by the other. In contrast to the square move of \cref{fig:M1}, where each of the four vertices of the quadrilateral face must have exactly one incident edge leading outside the configuration, we allow each black vertex of the quadrilateral face to have $0$ or more incident edges leading outside the configuration.
Cf\ also \cref{fig:urban2-bivalent}.}
\label{fig:urban2}
\end{figure}
\begin{figure}
\caption{Special case of normal urban renewal: one of the black vertices is not incident to any edges leading outside the configuration.}
\label{fig:urban2-bivalent}
\end{figure}
\begin{definition} \label{def:flip-normal} The \emph{normal flip move} is the local transformation shown in \cref{fig:flipmove}. Ignoring the bivalent black vertices, this local move is the same as the (white) flip move for trivalent plabic graphs. \end{definition}
\begin{figure}
\caption{The normal flip move.}
\label{fig:flipmove}
\end{figure}
\pagebreak[3]
\begin{lemma} \label{lem:normal-moves-via-M1M2M3} Let $G$ and $G'$ be normal plabic graphs related via a sequence of normal urban renewal moves and normal flip moves. Then $G\sim G'$. \end{lemma}
\begin{proof} Each instance of normal urban renewal can be expressed as a square move (M1) together with (M2) and/or~(M3) moves. Each normal flip move can be expressed as a combination of (M2) and (M3) moves. \end{proof}
\begin{theorem} \label{thm:moves-moves} Let $G$ and $G'$ be normal plabic graphs and let $\mathfrak{X}=\mathfrak{X}(G)$ and $\mathfrak{X}'=\mathfrak{X}(G')$ be the corresponding triple diagrams. Then the following are equivalent: \begin{itemize}[leftmargin=.2in] \item $G$ and $G'$ are related via a sequence of urban renewal moves and normal flip moves; \item $\mathfrak{X}$ and $\mathfrak{X}'$ are move-equivalent (i.e., related via swivel moves).
\end{itemize} \end{theorem}
\begin{proof} \cref{fig:flipsplabictriple} shows how one can translate back-and-forth between \begin{itemize}[leftmargin=.2in] \item an arbitrary swivel move in a triple diagram and \item either an urban renewal move or the normal flip move in the corresponding normal plabic graph. \end{itemize} The latter choice depends on the orientations of the strands involved. \end{proof}
\begin{figure}
\caption{Depending on the orientations of the strands involved, a swivel move in a triple diagram may correspond to (a) an urban renewal move or (b) a normal flip move in the associated normal plabic graph.}
\label{fig:flipsplabictriple}
\end{figure}
We next generalize \cref{def:GX} to arbitrary plabic graphs.
\begin{definition} \label{def:triple-non-normal} Let $G$ be a plabic graph (not necessarily normal). The~\emph{generalized triple diagram} $\mathfrak{X}(G)$ associated to~$G$ is defined as follows. (To be precise, we define $\mathfrak{X}(G)$ up to move equivalence.)
The recipe is essentially the same as in \cref{def:GX}, with the following additional rules dealing with non-trivalent white vertices: \begin{itemize}[leftmargin=.2in] \item at a univalent white vertex in~$G$, make a U-turn, see \cref{fig:plabic-low-degree-to-triple}; \item at a bivalent white vertex in~$G$, go straight through, see \cref{fig:plabic-low-degree-to-triple}; \item at a white vertex~$v$ of degree $\ge 4$, replace $v$ by a trivalent tree (with white vertices); then, at each of the vertices of the tree, apply the rule shown in \cref{fig:plabic-to-triple} on the right. See \cref{fig:plabic-high-degree-to-triple}. \end{itemize} Although the trivalent tree replacing~$v$ is not unique, all these trees are related to each other by flip moves, cf.\ \cref{fig:flipmove0}. Hence all triple diagrams constructed from them are move-equivalent to each other, cf.\ \cref{fig:flipsplabictriple}(b) (remove the black vertex in the center). \end{definition}
\begin{figure}
\caption{Constructing a triple diagram around a white vertex of degree~1 or~2 in a general plabic graph.}
\label{fig:plabic-low-degree-to-triple}
\end{figure}
\begin{figure}
\caption{Constructing a triple diagram around a high-degree white vertex in a general plabic graph. We~first replace this vertex by a trivalent tree, then construct the corresponding fragment of the triple diagram following the rule shown in \cref{fig:plabic-to-triple}.}
\label{fig:plabic-high-degree-to-triple}
\end{figure}
The following statement is immediate from the definitions.
\begin{lemma} \label{lem:generalized-triple-diagram-connected} Let $G$ be a plabic graph. If the union of the strands in~$\mathfrak{X}(G)$ and the boundary~$\partial\mathbf{D}$ is connected, then $\mathfrak{X}(G)$~is a triple diagram in the sense of \cref{def:triple-diagram}. \end{lemma}
The connectedness condition in \cref{lem:generalized-triple-diagram-connected} does not hold in general. To be concrete, if~$G$ contains a cycle~$C$ all of whose vertices are black, then the strands located inside~$C$ are disconnected from the rest of~$\mathfrak{X}(G)$.
\begin{remark} \label{rem:problematic-X(G)} Unfortunately, the extension of the definition of the triple diagram~$\mathfrak{X}(G)$ described in \cref{def:triple-non-normal} does not allow a straightforward generalization of \cref{thm:moves-moves}: move-equivalent plabic graphs do not necessarily yield move-equivalent triple diagrams. The only problematic local move is the one shown in \cref{fig:M3-white-leaf}: contracting an edge connecting a white internal leaf to a white vertex of degree~$\ge3$ removes a triple point in the associated triple diagram, thereby altering its move equivalence class. \end{remark}
\begin{figure}
\caption{Contracting (or decontracting) an edge that connects a white internal leaf to a white vertex of degree $\ge 3$ produces a move-inequivalent triple diagram.}
\label{fig:M3-white-leaf}
\end{figure}
This last complication prompts the following definition.
\def\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim}{\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim}} \def$\raisebox{-0.3pt}{$\circ$}$-move{$\raisebox{-0.3pt}{$\circ$}$-move}
\begin{definition} \label{def:omove} Let $G$ and $G'$ be plabic graphs.
We write $G\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G'$ if $G$~and~$G'$ can be related to each other via a sequence of local moves (M1)--(M3) that does not include an instance of the (M3) move shown in \cref{fig:M3-white-leaf}, cf.\ \cref{rem:problematic-X(G)}. \end{definition}
\begin{lemma} \label{lem:M1M2M3-to-triple} Let $G$ and $G'$ be plabic graphs such that $G\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G'$. Then the corresponding (generalized) triple diagrams $\mathfrak{X}(G)$ and~$\mathfrak{X}(G')$ are move equivalent (i.e., related to each other via swivel moves). \end{lemma}
We note that $\mathfrak{X}(G)$ and $\mathfrak{X}(G')$ are defined up to move equivalence, so the statement that they are move-equivalent to each other makes sense.
\noindent\textbf{Proof.} It is straightforward to verify, case by case, that each of the local moves (M1)--(M3), with the exception of the move shown in \cref{fig:M3-white-leaf}, either leaves the associated (generalized) triple diagram invariant or applies a swivel move to it (more precisely, to any of the possible diagrams obtained using the construction in \cref{def:triple-non-normal}). To be specific: \begin{itemize}[leftmargin=.2in] \item a square move (M1) translates into a swivel move, see \cref{fig:flipsplabictriple}(a); \item both the move (M2) and a black (de)contraction move (M3) leave the triple diagram invariant (up to isotopy); \item a white (de)contraction move (M3), other than the instance shown in \cref{fig:M3-white-leaf}, translates into a swivel move, see \cref{fig:flipsplabictriple}(b) (remove the black vertex in the center). \qed \end{itemize}
\begin{corollary} \label{cor:newmoves} Let $G$ and $G'$ be normal plabic graphs.
The following are equivalent: \begin{enumerate}[leftmargin=.3in] \item[{\rm(1)}] $G\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G'$; \item[{\rm(2)}] $G$ and $G'$ are related via a sequence of normal urban renewal moves and normal flip moves; \item[{\rm(3)}] $\mathfrak{X}(G)$ and $\mathfrak{X}(G')$ are move-equivalent (in the sense of \cref{def:2-2-move-equivalence}). \end{enumerate} \end{corollary}
\begin{proof} The implication (2)$\Rightarrow$(1) is an easy enhancement of \cref{lem:normal-moves-via-M1M2M3}. The equivalence (2)$\Leftrightarrow$(3) was established in \cref{thm:moves-moves}. The implication (1)$\Rightarrow$(3) was proved in \cref{lem:M1M2M3-to-triple}. \end{proof}
We note the similarity between \cref{cor:newmoves} and \cref{thm:newmoves1}.
\section{Minimal triple diagrams} \label{sec:mintriple}
\begin{definition} A triple diagram is called \emph{minimal} if it has no more triple points than any other triple diagram with the same strand permutation. \end{definition}
We will show in Section~\ref{minred} that minimal triple diagrams are the natural counterparts of reduced normal plabic graphs.
Much of this section is devoted to the proof of the following key result.
\begin{theorem} \label{thm:domino-flip}
Any two minimal triple diagrams with the same strand permutation are move-equivalent to each other. \end{theorem}
\begin{lemma} \label{lem:min-diag-move-equiv} If a triple diagram $\mathfrak{X}$ is minimal, then so is every triple diagram move-equivalent to~$\mathfrak{X}$. \end{lemma}
\begin{proof} It is easy to see that a swivel move preserves both the number of triple points and the strand permutation. The claim follows. \end{proof}
\iffalse We begin by constructing, for each permutation $\pi$, a particular triple diagram whose strand permutation is $\pi$. We will call these triple diagrams \emph{standard}; we will see in \cref{cor:standardminimal} that they are all minimal. This will be helpful in the proof of \cref{thm:domino-flip}. \fi
We next describe certain ``bad features'' (of a triple diagram) and show that they cannot occur in a minimal triple diagram.
\begin{definition} \label{def:monogon} A strand in a triple diagram that intersects itself forms a \emph{monogon}. A pair of strands that intersect at two points $x$ and $y$ form either a \emph{parallel} or \emph{anti-parallel digon}, depending on whether their segments connecting $x$ and~$y$ run in the same or opposite direction, see \cref{fig:monogondigon}. We use the term \emph{badgon} to refer to either a monogon or a parallel digon. \end{definition}
\begin{figure}
\caption{A monogon, a parallel digon, and an anti-parallel digon. The actual picture will contain (potentially many) additional strands and intersections.}
\label{fig:monogondigon}
\end{figure}
\begin{lemma} \label{lem:no-closed-strands} A triple diagram without badgons has no closed strands. \end{lemma}
\begin{proof} Let $\mathfrak{X}$ be a triple diagram without badgons. Since $\mathfrak{X}$ does not contain monogons, no strand of $\mathfrak{X}$ can intersect itself. Suppose that $\mathfrak{X}$ contains a closed strand $S$.
Let $T$ be another strand of $\mathfrak{X}$ intersecting~$S$ at points~$x$ and~$y$; such~$T$ exists since $\mathfrak{X}$ must be connected to the boundary~$\partial\mathbf{D}$.
Then the segment of~$T$ between $x$ and~$y$ together with one of the segments of~$S$ connecting $x$ and~$y$ form a parallel digon, which is a contradiction. \end{proof}
\pagebreak[3]
\begin{lemma} \label{lem:nomonogondigon} A minimal triple diagram does not contain badgons. Therefore (cf.\ \cref{lem:no-closed-strands}) it does not contain closed strands.
\end{lemma}
\begin{proof} Let $\mathfrak{X}$ be a triple diagram containing a monogon, i.e., a strand~$S$ with a self-intersection at a triple point~$v$. Construct the triple diagram $\mathfrak{X}'$ by deforming $\mathfrak{X}$ around~$v$ so that $S$ ``spins off'' a closed strand while the triple point disappears, see \cref{fig:fewer}. (If the spun-off portion is disconnected from the rest of~$\mathfrak{X}$, then remove it altogether.) The triple diagram $\mathfrak{X}'$ has~the same strand permutation as~$\mathfrak{X}$ but fewer triple points; thus $\mathfrak{X}$ is not minimal.
\begin{figure}
\caption{In the presence of a monogon, we can reduce the number of triple points while keeping the same strand permutation. The triple diagram may contain additional strands intersecting the monogon, as well as additional points of self-intersection.}
\label{fig:fewer}
\end{figure}
\noindent Now suppose that $\mathfrak{X}$ does not contain monogons but does contain two strands $S$ and $T$ that form a parallel digon. Say, $S$ and $T$ contain segments $\overline S$ and~$\overline T$ that run from a triple point~$x$ to a triple point~$y$. Let $U$ (resp.,~$V$) be the third strand passing through~$x$ (resp.,~$y$). We then deform $\mathfrak{X}$ around both $x$ and~$y$ by smoothing each of the two triple points: the strands $U$ and $V$ continue to go straight through, whereas the endpoints of~$\overline S$ (resp.,~$\overline T$) get connected to~$T$ (resp.,~$S$). Thus, the strands $S$ and $T$ swap their segments $\overline S$ and $\overline T$ with each other (with appropriate smoothings), the overall connectivity (i.e., the strand permutation) is preserved, and the triple points at $x$ and~$y$ disappear. (If the diagram becomes disconnected from~$\partial\mathbf{D}$, then remove the disconnected portion.) We then conclude that $\mathfrak{X}$ was not minimal. \end{proof}
\begin{definition} \label{def:boundary-parallel} Let $S$ be an arc in a triple diagram, i.e., a strand whose endpoints $s$ and~$t$ lie on the boundary of the ambient disk~$\mathbf{D}$. We call $S$ \emph{boundary-parallel} if it runs along a segment~$I$ of the boundary~$\partial\mathbf{D}$ between $s$ and~$t$ (in either direction), so that every other strand with an endpoint inside~$I$ runs directly to or from~$S$, without any triple crossings in between. See \cref{fig:standard}. \end{definition}
\begin{figure}
\caption{A boundary-parallel strand $S$ in a triple diagram.}
\label{fig:standard}
\end{figure}
We next describe a particular way to construct, for any given permutation~$\pi$, a triple diagram whose strand permutation is~$\pi$.
\begin{definition} \label{def:standard-triple} Let $\pi$ be a permutation of $b$ letters $1,\dots,b$. A~triple diagram in the disk~$\mathbf{D}$ is called \emph{standard} (for~$\pi$) if it can be constructed using the following recursive process. (The process involves some choices, so a standard diagram for~$\pi$ is not unique.)
We place $b$ boundary vertices on the boundary $\partial\mathbf{D}$ and label them $1,\dots,b$ clockwise. Next to each boundary vertex~$v$, we mark two endpoints of the future strands: a source endpoint that precedes $v$ in the clockwise order and a target endpoint that follows~$v$ in this order. We know which source is to be matched to which target by the strand permutation~$\pi$. Each such pair of endpoints divides the circle~$\partial\mathbf{D}$ into two intervals. Let us partially order these $2b$ intervals by inclusion and select a \emph{minimal interval~$I$} with respect to this partial order.
We start constructing the triple diagram by running a boundary-parallel strand~$S$ along the interval~$I$, introducing a triple crossing for each pair of strands that need to terminate in the interior of~$I$, as shown in \cref{fig:standard}. There will always be an even number (possibly zero) of strands to cross over, so the construction will proceed without a hitch.
Let $\mathbf{D'}$ be the disk obtained from~$\mathbf{D}$ by removing the region between the boundary segment~$I$ and the strand~$S$ together with a small neighborhood of~$S$; so $\mathbf{D'}$ is the shaded region in \cref{fig:standard}. We accordingly remove $S$ and its endpoints from the original pairing of the in- and out-endpoints, and swap each pair that $S$ crossed over. This yields $2(b-1)$ endpoints on the boundary of~$\mathbf{D'}$; note that the in- and out-endpoints alternate, as before. We then determine the new pairing of these endpoints (thus, a new strand permutation, after an appropriate renumbering) and recursively continue the process in~$\mathbf{D'}$ until the desired (standard) triple diagram is constructed. \end{definition}
We shall keep in mind that a standard triple diagram is constructed by choosing a sequence of minimal intervals.
\begin{exercise} \label{lem:useful-moves} For each of the three pairs of triple diagrams shown below, demonstrate that the two diagrams are move-equivalent to each other, i.e., related via a sequence of swivel moves. \begin{align} \label{it:move-1} &\begin{tikzpicture}[baseline={(0,-0.1)}] \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.55 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-4,0.15) to[bend right=30] (-1,0.15); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.55 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-1,-0.15) to[bend right=30] (-4,-0.15); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.45 and 0.9 step 0.5 with {\arrow{Stealth}}}}] (-3.72,-0.3) -- (-3.72,0.3); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.45 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-1.28,0.3) -- (-1.28,-0.3); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-1.95,-0.45) to[bend left=55] (-1.95,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-2.2,0.45) to[bend left=55] (-2.2,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-2.8,-0.45) to[bend left=55] (-2.8,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-3.05,0.45) to[bend left=55] (-3.05,-0.45); \draw[>={Stealth[length=6pt]}, <->] (-0.45,0) -- (0.55,0); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (1.35,-0.45) to[bend right=55] (1.35,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (1.6,0.45) to[bend right=55] (1.6,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.05,-0.45) to[bend right=55] (2.05,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.3,0.45) to[bend right=55] (2.3,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.75,-0.45) to[bend right=55] (2.75,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (3,0.45) to[bend right=55] (3,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.43 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (1,0.3) -- (3.3,0.3); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.7 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (3.3,-0.3) -- (1,-0.3); \end{tikzpicture} \\[.1in] & \label{it:move-2} \begin{tikzpicture}[baseline={(0,-0.1)}] \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.35 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-3.71,-0.35) -- (-3.71,0.35); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-1.3,-0.45) to[bend left=55] (-1.3,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-1.55,0.45) to[bend left=55] (-1.55,-0.45);
\draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-2.15,-0.45) to[bend left=55] (-2.15,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-2.4,0.45) to[bend left=55] (-2.4,-0.45);
\draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-3,-0.45) to[bend left=55] (-3,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (-3.25,0.45) to[bend left=55] (-3.25,-0.45);
\draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.55 and 0.6 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(-1,0.33) (-3,0.33) (-3.7,0) (-3.95,-0.26)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.52 and 0.6 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(-3.95,0.26) (-3.7,0) (-3,-0.33) (-1,-0.33)}; \draw[>={Stealth[length=6pt]}, <->] (-0.45,0) -- (0.55,0); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (1.35,-0.45) to[bend right=55] (1.35,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (1.6,0.45) to[bend right=55] (1.6,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.05,-0.45) to[bend right=55] (2.05,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.3,0.45) to[bend right=55] (2.3,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (2.75,-0.45) to[bend right=55] (2.75,0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.8 step 0.5 with {\arrow{Stealth}}}}] (3,0.45) to[bend right=55] (3,-0.45); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.575 and 0.7 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(1,0.33) (3,0.3) (3.7,-0.26)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.51 and 0.6 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(3.7,0.26) (3,-0.3) (1,-0.33)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.35 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (3.48,-0.35) -- (3.48,0.35); \end{tikzpicture} \\[.1in] &\label{it:move-3} \begin{tikzpicture}[baseline={(0,-0.1)}] \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.5 and 0.7 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(-3.9,0.33) (-1.7,0.3) (-1,-0.26)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.36 and 0.6 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(-3.9,-0.33) (-1.7,-0.3) (-1,0.26)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-3.2,-0.5) -- (-3.55,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-2.5,-0.5) -- (-2.85,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-1.8,-0.5) -- (-2.15,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.6 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-2.7,0.5) -- (-3.33,-0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.6 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (-2,0.5) -- (-2.63,-0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.95 and 0.99 step 0.5 with {\arrow{Stealth}}}}] (-3.4,0.5) to[bend left=30] (-3.9,0); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.62 and 0.88 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(-1,0) (-1.4,-0.01) (-1.6,-0.05) (-1.8,-0.2) (-2,-0.5)}; \draw[>={Stealth[length=6pt]}, <->] (-0.45,0) -- (0.55,0); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.72 and 0.8 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(1,-0.26) (1.3,0) (2,0.33) (4,0.33)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.57 and 0.6 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(1,0.26) (1.3,0) (2,-0.33) (4,-0.33)}; \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (2.2,-0.5) -- (1.85,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (2.9,-0.5) -- (2.55,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.65 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (3.6,-0.5) -- (3.25,0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.6 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (2.68,0.5) -- (2.04,-0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.6 and 0.9 step 0.7 with {\arrow{Stealth}}}}] (3.38,0.5) -- (2.75,-0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.45 and 0.49 step 0.5 with {\arrow{Stealth}}}}] (4,0) to[bend right=30] (3.5,-0.5); \draw[thick, postaction={decorate, decoration = {markings, mark = between positions 0.5 and 0.88 step 0.55 with {\arrow{Stealth}}}}] plot [smooth] coordinates {(2,0.5) (1.8,0.2) (1.55,0.05) (1,0)}; \end{tikzpicture} \end{align} (In each of the three cases, the central section can involve an arbitrary number of repetitions.) \end{exercise}
\begin{exercise} Use \eqref{it:move-3} to prove the move equivalence \eqref{eq:parallel-digon-to-monogon} below:
\begin{equation} \label{eq:parallel-digon-to-monogon} \setlength{\unitlength}{0.35pt} \begin{picture}(435,90)(0,30) \thicklines
\put(70,85){$S$}
\put(95,-20){$U$} \put(70,10){\line(1,0){160}} \put(70,10){\line(4,7){46}} \put(70,10){\line(-4,-7){6}} \put(150,10){\line(4,7){46}} \put(150,10){\line(-4,-7){6}} \put(110,80){\line(1,0){80}} \put(110,80){\line(4,-7){46}} \put(110,80){\line(-4,7){6}} \put(190,80){\line(4,-7){46}} \put(190,80){\line(-4,7){6}} \put(230,10){\line(-2,-3){7}} \put(70,10){\line(2,-3){7}} \put(280,40){\line(1,0){20}} \put(280,40){\line(2,3){20}} \put(280,40){\line(5,-4){20}} \put(20,40){\line(-1,0){20}} \put(20,40){\line(-2,3){20}} \put(20,40){\line(-5,-4){20}} \qbezier(190,80)(230,80)(280,40) \qbezier(110,80)(70,80)(20,40) \qbezier(230,10)(260,10)(280,40) \qbezier(70,10)(40,10)(20,40) \qbezier(230,10)(250,40)(280,40) \qbezier(70,10)(50,40)(20,40) \put(110,10){\vector(1,0){10}} \put(150,80){\vector(1,0){10}} \put(190,10){\vector(1,0){10}} \put(90,45){\vector(-4,-7){5}} \put(170,45){\vector(-4,-7){5}} \put(130,45){\vector(-4,7){5}} \put(210,45){\vector(-4,7){5}} \put(240,66){\vector(2,-1.1){1}} \put(272,30){\vector(1.1,1){1}} \put(250,32){\vector(-2.8,-1){1}} \put(60,66){\vector(2,1.1){1}} \put(13,51){\vector(4,-7){1}} \put(53,30){\vector(-1,1.2){1}} \put(370,45){\makebox(0,0){$\longleftrightarrow$}} \end{picture} \setlength{\unitlength}{0.35pt} \begin{picture}(300,90)(0,30) \thicklines \put(70,10){\line(1,0){230}} \put(70,10){\line(-1,0){70}} \put(70,10){\line(4,7){46}} \put(70,10){\line(-4,-7){6}} \put(150,10){\line(4,7){46}} \put(150,10){\line(-4,-7){6}} \put(110,80){\line(1,0){190}} \put(110,80){\line(4,-7){46}} \put(110,80){\line(-4,7){6}} \put(190,80){\line(4,-7){46}} \put(190,80){\line(-4,7){6}} \put(230,10){\line(-2,-3){7}} \put(70,10){\line(2,-3){7}} \put(280,40){\line(1,0){20}}
\put(20,40){\line(-1,0){20}} \put(20,40){\line(-2,3){20}}
\qbezier(110,80)(70,80)(20,40)
\qbezier(230,10)(250,40)(280,40) \qbezier(70,10)(50,40)(20,40) \qbezier(20,40)(30,25)(20,10) \qbezier(20,40)(0,25)(20,10)
\qbezier(20,10)(15,5)(15,0) \qbezier(20,10)(30,5)(30,0) \qbezier(15,0)(15,-10)(22.5,-10) \qbezier(30,0)(30,-10)(22.5,-10)
\put(25,16){\vector(0,-1){1}}
\put(53,30){\vector(-1,1.2){1}}
\put(45,10){\vector(1,0){10}} \put(110,10){\vector(1,0){10}} \put(150,80){\vector(1,0){10}} \put(190,10){\vector(1,0){10}} \put(90,45){\vector(-4,-7){5}} \put(170,45){\vector(-4,-7){5}} \put(130,45){\vector(-4,7){5}} \put(210,45){\vector(-4,7){5}}
\put(250,32){\vector(-2.8,-1){1}} \put(60,66){\vector(2,1.1){1}}
\put(270,80){\vector(1,0){10}} \put(270,10){\vector(1,0){10}} \put(12.5,51){\vector(4,-7){1}}
\end{picture} \end{equation} \end{exercise}
\iffalse Without loss of generality, we may assume that $\overline S$ and~$\overline T$ do not intersect. (If say both $\overline S$ and~$\overline T$ pass through a triple point~$z$, then we can replace them with shorter segments connecting $x$ and~$z$ that would again form a parallel digon; then iterate.)
We can now apply~\eqref{it:move-3}. Note that we have placed three red dots in the left side of \cref{fig:parallel} to indicate the left boundary of the local region where we are applying~\eqref{it:move-3}. Applying this lemma results in another triple diagram with the same strand permutation which has a monogon, contradicting the fact that $\mathfrak{X}$ is minimal.
\begin{figure}
\caption{If there is a parallel digon, we can apply swivel moves
to find another triple diagram which has a monogon.}
\label{fig:parallel}
\end{figure} \fi
\begin{lemma} \label{lem:straighten2} Let $\mathfrak{X}$ be a triple diagram such that no triple diagram move-equivalent to~$\mathfrak{X}$ contains a monogon. Then the following statements~hold: \begin{itemize}[leftmargin=.3in] \item[{\rm (i)}] No triple diagram move-equivalent to $\mathfrak{X}$ has a badgon or a closed strand. \item[{\rm (ii)}] Let $I$ be a minimal interval for the strand permutation associated with~$\mathfrak{X}$.
Then $\mathfrak{X}$ is move-equivalent to a diagram $\mathfrak{X}'$ in which the strand connecting the endpoints of~$I$ is boundary-parallel along~$I$.
\end{itemize}
\end{lemma}
\begin{proof} We will simultaneously prove statements (i) and~(ii) by induction on the number of triple points in~$\mathfrak{X}$. Thus, we assume that both (i) and~(ii) hold for triple diagrams that have fewer triple points than~$\mathfrak{X}$.
We first prove~(i). Suppose that a triple diagram $\mathfrak{X}'\sim\mathfrak{X}$ contains (non-self-intersecting) strands~$S$ and~$U$ forming a parallel digon. The strand~$S$ cuts the disk~$\mathbf{D}$ into two regions. Let $R$ be the region containing the digon, with a small neighbourhood of~$S$ removed. Since the boundaries of the faces of~$\mathfrak{X}'$ are consistently oriented, the same is true for the portion of~$\mathfrak{X}'$ contained inside~$R$, so this portion can be viewed as a (smaller) triple diagram. Suppose that $U$ bounds a minimal interval within~$S$ (viewed as a portion of the boundary of~$R$). Then by the induction assumption, $U$~can be moved to be boundary-parallel to~$S$. Since $S$ and $U$ are co-oriented, we get the picture on the left-hand side of~\eqref{eq:parallel-digon-to-monogon} (with $U$ running horizontally at the bottom). Applying~\eqref{eq:parallel-digon-to-monogon}, we obtain a monogon, a contradiction.
If the subinterval of~$S$ cut out by~$U$ is not minimal, then there is a strand $T$ that cuts across~$S$ twice, creating a minimal interval within~$S$ and forming a digon inside~$R$. We may assume that this digon is anti-parallel (or else replace $U$ by~$T$ and repeat). By the induction assumption, we can apply swivel moves inside~$R$ to make $T$ boundary-parallel to~$S$. We then apply~\eqref{it:move-1} to remove the digon, as shown in \cref{fig:step-1}. Repeating this operation if necessary, we obtain a triple diagram in which $U$ bounds a minimal interval within~$S$; we then argue as above to arrive at a contradiction.
\begin{figure}
\caption{Removing double intersections with $S$.}
\label{fig:step-1}
\end{figure}
Thus, no triple diagram $\mathfrak{X}'\sim\mathfrak{X}$ contains badgons. By \cref{lem:no-closed-strands}, we conclude that any such $\mathfrak{X}'$ does not contain closed strands either. \iffalse Now suppose that $\mathfrak{X}'\sim\mathfrak{X}$ contains a closed (necessarily non-self\-inter\-secting) strand~$S$.
Let $R$ be the region enclosed by~$S$, with a small neigbourhood of~$S$ removed. The strands intersecting~$R$ form a triple diagram inside~$R$. Since this diagram has fewer triple points than~$\mathfrak{X}'$ (or equivalently~$\mathfrak{X}$), the induction assumption applies. Therefore we can move some strand~$T$ in~$R$ so that it becomes boundary-parallel (to~$\partial R$). If $T$ is parallel to the corresponding segment of~$S$, then we can apply~\eqref{eq:parallel-digon-to-monogon} to create a monogon. Thus, $T$~must be anti-parallel to~$S$. We can then apply \eqref{it:move-1} to move this segment of~$T$ outside~$S$, thereby decreasing the number of triple points inside~$S$. Repeating this step if necessary, we arrive at a situation with no triple points inside~$S$. Applying further swivel moves, we move all strands crossing~$S$ away from~$S$ until there are only two of them left, and then a swivel move creates a self-intersection. \fi This completes the induction step for statement~(i).
We now proceed to proving statement~(ii). In addition to the induction assumption for~(ii),
we may assume that neither~$\mathfrak{X}$ nor any triple diagram move-equivalent to~$\mathfrak{X}$ contains a badgon or a closed strand.
Let $S$ be the strand connecting the endpoints of the minimal interval~$I$.
\noindent \emph{Step 1: Removing double intersections with $S$, see \cref{fig:step-1}.} Let $R$ be the region between the strand~$S$ and the interval~$I$, with a small neigborhood of~$S$ removed.
Suppose there is a strand that intersects $S$ more than once. Among such strands, take one that cuts out a minimal interval along the boundary of~$R$. Let $T$ denote the segment of this strand contained in~$R$. The portion of $\mathfrak{X}$ contained inside~$R$ has fewer triple crossings than~$\mathfrak{X}$, so by the induction assumption, we can make $T$ boundary-parallel to~$S$ by applying swivel moves inside~$R$. Now $T$ and $S$ form a (necessarily anti-parallel) digon, which we then remove using~\eqref{it:move-1}.
We repeat this procedure until there are no strands left that intersect~$S$ more than once. Since the number of triple points along $S$ decreases each time, the process terminates.
\noindent \emph{Step 2: Combing out the triple crossings.} At this stage, no strand crosses $S$ more than once. Since $I$ is minimal, no strand has both ends at~$I$. Since $\mathfrak{X}$ contains no closed strands, every (non-self-intersecting) strand appearing between~$S$ and~$I$ must start or end at a point in~$I$ and cross~$S$. Suppose that $S$ is not boundary-parallel. Then there exists a strand~$T$ with an endpoint at~$I$ that passes through a triple point before hitting~$S$. Among all such~$T$, choose the one with the leftmost endpoint along~$I$, cf.\ Figures~\ref{fig:step-2} and~\ref{fig:step-2p} on the left. Let $R$ be the part of the region between $I$ and $S$ that lies to the right of any strand $T'$ located to the left~of~$T$. (By our choice of~$T$, all such strands $T'$ run directly from $I$ to~$S$, with no crossings in between.) As we have eliminated all double intersections with~$S$, the inter\-val corresponding to~$T$ (looking to the left) is minimal inside~$R$. We can therefore use the induction assumption inside~$R$ to make~$T$ boundary-parallel.
What we do next depends on the orientation of~$T$ relative to~$S$. If $T$ is anti-parallel to~$S$, as in \cref{fig:step-2}, then we apply~\eqref{it:move-2} to make $T$ run directly to~$S$. If $T$ is parallel to~$S$, as in \cref{fig:step-2p}, then we apply~\eqref{it:move-3}.
We repeat this step until $S$ is boundary-parallel.
\end{proof}
\begin{figure}
\caption{Combing out the triple crossings: the anti-parallel case.}
\label{fig:step-2}
\end{figure}
\begin{figure}
\caption{Combing out the triple crossings: the parallel case. }
\label{fig:step-2p}
\end{figure}
\begin{lemma} \label{lem:straighten} For a triple diagram~$\mathfrak{X}$, the following are equivalent: \begin{itemize}[leftmargin=.3in] \item[{\rm(a)}] Any diagram $\mathfrak{X}'$ move-equivalent to~$\mathfrak{X}$ does not contain a monogon. \item[{\rm(b)}] $\mathfrak{X}$ is move-equivalent to any standard triple diagram with the same strand permutation. \item[{\rm(c)}] $\mathfrak{X}$ is minimal. \end{itemize} In particular, any standard triple diagram is minimal. \end{lemma}
\begin{proof} The implication (c)$\Rightarrow$(a) follows from Lemmas \ref{lem:min-diag-move-equiv} and~\ref{lem:nomonogondigon}. To prove the implication (a)$\Rightarrow$(b), choose a sequence of minimal intervals and repeatedly apply \cref{lem:straighten2}. We have now established (c)$\Rightarrow$(b), so any minimal triple diagram is move-equivalent to any standard triple diagram with the same strand permutation. It follows by \cref{lem:min-diag-move-equiv} that any standard triple diagram is minimal, hence so is any diagram move-equivalent to a standard one. Thus (b)$\Rightarrow$(c) is proved. \end{proof}
\begin{proof}[Proof of \cref{thm:domino-flip}] By \cref{lem:straighten}, any two minimal triple diagrams with strand permutation~$\pi$ are move-equivalent to any standard diagram with strand permutation~$\pi$, and therefore to each other. \end{proof}
\begin{lemma} \label{lem:propagate} Let $\mathfrak{X}$ and $\mathfrak{X}'$ be triple diagrams related by a swivel move. If $\mathfrak{X}$ contains a badgon, then so does~$\mathfrak{X}'$. \end{lemma}
\begin{proof} We label the strands and the triple points involved in this swivel move by $a,b, c, d$, and $x, y$, as shown in \cref{fig:22}.
\begin{figure}
\caption{A swivel move relating $\mathfrak{X}$ and $\mathfrak{X}'$.}
\label{fig:22}
\end{figure}
If $\mathfrak{X}$ contains a badgon that involves neither~$x$ nor~$y$, then this badgon persists in~$\mathfrak{X}'$.
Suppose $\mathfrak{X}$ contains a monogon whose self-intersection point is (say)~$x$. Thus, two of the strands $\{a, b, c\}$ coincide. If $a=c$ (resp., $b=c$), then the same monogon persists in~$\mathfrak{X}'$ because in \cref{fig:22}, strands $a$ and~$c$ (resp., $b$ and $c$) intersect in both $\mathfrak{X}$ and~$\mathfrak{X}'$.
If, on the other hand, $a=b$, then $\mathfrak{X}'$ has a parallel digon, see \cref{fig:13}.
\begin{figure}
\caption{A monogon in $\mathfrak{X}$ results in a parallel digon in $\mathfrak{X}'$.}
\label{fig:13}
\end{figure} From now on, we can assume that there is no monogon in~$\mathfrak{X}$. Suppose $\mathfrak{X}$ has a parallel digon whose two intersection points include $x$ but not~$y$. The sides of this parallel digon are either $\{a,b\}$ or $\{a,c\}$ or $\{b,c\}$. The last two cases are easy because such a parallel digon will persist in~$\mathfrak{X}'$, since the strands $a$ and~$c$ (resp., $b$ and~$c$) intersect in both $\mathfrak{X}$ and~$\mathfrak{X}'$.
Now suppose that our parallel digon has sides $a$ and~$b$, see \cref{fig:14} on the left. (If the strands $a$ and $b$ go to the left and meet again there, then we \linebreak[3] get the same picture but with the roles of $x$ and $y$ interchanged.) Note that the end of strand $a$ shown inside the digon must extend outside of it, but it cannot intersect~$a$, as this would create a monogon. So strand $a$ must intersect strand~$b$ again, see \cref{fig:14} in the middle. Then, after the swivel move, we get a parallel digon
as shown in \cref{fig:14} on the right.
Finally, suppose there is a parallel digon in $\mathfrak{X}$ whose two intersection points are $x$ and $y$. We can assume it is oriented from $x$ to $y$. The two arcs of the parallel digon should come from the following list: \begin{itemize}[leftmargin=.4in] \item[{\rm (aa)}] the arc along $a$ from $x$ to $y$; \item[{\rm (bb)}] the arc along $b$ from $x$ to $y$; \item[{\rm (cd)}] an arc leaving $x$ along $c$, and returning to $y$ along $d$ (so $c=d$); \item[{\rm (cb)}] an arc leaving $x$ along $c$, and returning to $y$ along $b$ (so $c=b$); \item[{\rm (bd)}] an arc leaving $x$ along $b$, and returning to $y$ along $d$ (so $b=d$). \end{itemize} In case (bb), we get a closed strand; it will persist in~$\mathfrak{X}'$ and yield a badgon by
\cref{lem:no-closed-strands}. In cases (cb) and~(bd), we get a monogon, contradicting our assumption. The remaining case is when the parallel digon has sides (aa) and~(cd),
in which case we get a monogon in~$\mathfrak{X}'$. (The picture is like
\cref{fig:13}, with the roles of $\mathfrak{X}$ and $\mathfrak{X}'$ swapped and some strands relabeled.) \end{proof}
\begin{figure}
\caption{Persistence of parallel digons under swivel moves.}
\label{fig:14}
\end{figure}
\begin{theorem} \label{thm:minimal} A triple diagram is minimal if and only if it has no badgons.
\end{theorem}
\begin{proof} The ``only if'' direction is \cref{lem:nomonogondigon}.
The ``if'' direction follows from \cref{lem:propagate} and \cref{lem:straighten} (implication (a)$\Rightarrow$(c)).
\end{proof}
\begin{lemma} \label{cor:Dyl} Assume that a triple diagram $\mathfrak{X}$ is not minimal. Then there exists a diagram $\mathfrak{X}'$ move-equivalent to $\mathfrak{X}$ that contains a hollow monogon. \end{lemma}
\begin{proof} We will argue by induction on the number of faces in~$\mathfrak{X}$. If this number is 1 or~2, then the claim is vacuously true.
By \cref{lem:straighten}, there exists $\mathfrak{X}'\sim\mathfrak{X}$ such that $\mathfrak{X}'$ has a monogon. Let $M$ be the segment of a strand in~$\mathfrak{X}'$ that forms a monogon; we may assume $M$ does not intersect itself except at its endpoints (or else replace $M$ by its sub-segment). If the monogon encircled by~$M$ is hollow, we are done. Otherwise, consider the disk~$\mathbf{D}_\circ$ obtained by removing a small neighborhood of~$M$ from the interior of the monogon. Let $\mathfrak{X}'_\circ$ denote the portion of~$\mathfrak{X}'$ contained in~$\mathbf{D}_\circ$; this is a triple diagram with fewer faces than~$\mathfrak{X}'$ (or equivalently~$\mathfrak{X}$).
The rest of the argument proceeds by showing that either we can apply local moves to $\mathfrak{X}'_\circ$ to create a hollow monogon inside $\mathbf{D}_\circ$ or we can apply moves to reduce the number of faces inside the monogon encircled by~$M$ (eventually producing a hollow monogon). If $\mathfrak{X}'_\circ$ is not minimal, then the induction assumption applies, so we can transform~$\mathfrak{X}'_\circ$ (thus~$\mathfrak{X}'$ or~$\mathfrak{X}$) into a move-equivalent triple diagram containing a hollow monogon. Therefore, we may assume that $\mathfrak{X}'_\circ$ is minimal. Let $M_\circ$ denote the interval obtained from the boundary of~$\mathbf{D}_\circ$ by removing a point located near the vertex of our monogon. Let $I\subset M_\circ$ be a minimal interval of the triple diagram~$\mathfrak{X}'_\circ$. Since this triple diagram is minimal, we can, by virtue of~\cref{lem:straighten} (or~\cref{lem:straighten2}), apply local moves inside~$\mathbf{D}_\circ$ to transform $\mathfrak{X}'_\circ$ into a triple diagram in which the strand~$T$ connecting the endpoints of~$I$ is boundary-parallel to~$I$. Let us now look at the digon formed by~$T$ and the portion of~$M$ that runs along~$I$. If this digon is anti-parallel, then we can push~$T$ outside the monogon as in \cref{fig:step-1}, reducing the number of faces enclosed by~$M$.
If, on the other hand, the digon is parallel, then we can use \eqref{eq:parallel-digon-to-monogon} to create a hollow monogon. \end{proof}
\iffalse see \cref{fig:shrinkloop}. If this monogon is hollow, there is nothing to prove. Otherwise, we claim that we can find a sequence of swivel moves on~$\mathfrak{X}'$ that yields a hollow monogon. We will argue by induction on the number of triple points in~$\mathfrak{X}$ (or~$\mathfrak{X}'$).
Let $S$ be the other strand going through the self-intersection point of the monogon. Let $R$ be the part of the ambient disk~$\mathbf{D}$ (shaded in \cref{fig:shrinkloop}) obtained by removing a neighborhood of $S$ together with the region between $S$ and~$\partial\mathbf{D}$ that does not contain the monogon.
Let $\overline\mathfrak{X}$ denote the smaller triple diagram obtained by restricting $\mathfrak{X}'$ to~$R$. Within~$\overline\mathfrak{X}$, the strand~$T$ enclosing (the relevant portion of) the monogon represents a minimal interval~$I$ for the corresponding strand permutation.
If $\overline\mathfrak{X}$ is minimal, then by Lemmas~\ref{lem:straighten2}--\ref{lem:straighten}, we can move $T$ to a strand boundary-parallel along~$I$. This strand will necessarily cut out a hollow monogon, as desired.
If $\overline\mathfrak{X}$ is not minimal, then by \cref{lem:straighten}, we can apply some local moves to produce a monogon whose self-intersection point $x$ lies inside~$R$. We can now invoke the induction assumption (for~$\overline\mathfrak{X}$) to arrive at the desired conclusion.
\begin{figure}
\caption{A triple diagram $\mathfrak{X}'$ that contains a monogon.}
\label{fig:shrinkloop}
\end{figure} \fi
\section{From minimal triple diagrams to reduced plabic graphs} \label{minred}
In this section, we use the machinery of triple diagrams and normal plabic graphs to prove \cref{prop:fixedlollipop} and
\cref{thm:moves}.
\begin{lemma} \label{lem:normal-no-collapse} A normal plabic graph contains no collapsible trees. \end{lemma}
\begin{proof} All internal leaves in a normal plabic graph~$G$ are black. Whatever (M2)--(M3) moves we apply to a tree, it will always have a black leaf (so it can't collapse to a white root or lollipop), and it will always have a white vertex of degree at least $3$ (so it can't collapse to a black root or lollipop).
\end{proof}
\begin{lemma} \label{lem:not-reduced=>osim} Let $G$ be a non-reduced normal plabic graph. Then there exists a plabic graph $G'\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G$ (cf.\ \cref{def:omove}) containing one of the forbidden configurations shown in \cref{fig:fail}. \end{lemma}
\begin{proof} Suppose $G$ has an internal (necessarily black) leaf~$u$ that is not a lollipop. Let $v$ be the unique (white, trivalent) vertex adjacent to~$u$. This gives us a forbidden configuration as in~\cref{fig:fail}(d). (Note that by \cref{lem:normal-no-collapse}, $G$~has no collapsible trees.)
If, on the other hand, $G$ has no such internal leaves, then by \cref{pr:reduced-collapse}, $G$~can be transformed, via local moves that do not create leaves, into a graph containing a hollow digon.
\iffalse The argument is similar to the proof of \cref{pr:reduced-collapse}. By \cref{def:reduced-plabic}, there is a plabic graph $G''\sim G$ containing a forbidden configuration. The sequence of local moves connecting $G$ and~$G''$ may potentially involve growing some collapsible trees. If this does not happen, then the ``problematic move'' shown in \cref{fig:M3-white-leaf} does not occur, and we are done.
Let $G=G_0, G_1,\dots, G_N=G''$ be the sequence of plabic graphs in which each pair $(G_i,G_{i+1})$ is related by a local move. As in \cref{def:overline-G}, we denote by $\overline G_i$ the result of collapsing all collapsible trees in~$G_i$. (Note that $G=G_0=\overline G_0$ by \cref{lem:normal-no-collapse}.) If the sequence $(G_i)$ does not include a square move~(M1) such that one of the vertices in the square is the root of a collapsible tree, then $\overline G_i\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} \overline G_{i+1}$ for all~$i$, and we are done. Otherwise, let $(G_k,G_{k+1})$ be the first occurrence of such a square move (i.e., the one with the smallest~$k$). Then $G=\overline G_0\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} \overline G_k$. Moreover, the sequence of moves relating $G$ and~$\overline G_k$
(which does not involve the problematic (M3) moves) produces a square configuration in which one of the vertices of the square is bivalent. Erasing this vertex and contracting the resulting new edge yields a forbidden configuration (a bicolored digon), and we are done. \fi \end{proof}
Recall that by \cref{pr:bijplabictriple}, the map $G\to \mathfrak{X}(G)$ defined in \cref{def:GX} gives a bijection between normal plabic graphs and triple diagrams with the same number of boundary vertices; moreover, this bijection preserves the associated (resp., trip or strand) permutation.
\begin{theorem} \label{red-minimal} A normal plabic graph $G$ is reduced if and only if the triple diagram $\mathfrak{X}(G)$ is minimal.
Thus the map
$G \mapsto \mathfrak{X}(G)$ restricts to a bijection
between reduced normal plabic graphs and minimal triple diagrams. \end{theorem}
\begin{proof} Suppose $\mathfrak{X}(G)$ is not minimal. By \cref{cor:Dyl}, there is a triple diagram $\mathfrak{X'} \!\sim\! \mathfrak{X}(G)$ such that $\mathfrak{X'}$ has a hollow monogon. By \cref{pr:bijplabictriple}, we have $\mathfrak{X'} \!=\! \mathfrak{X}(G')$ for some normal plabic graph~$G'$. Moreover, by \cref{cor:newmoves}, $G$ and $G'$ are move equivalent. The hollow monogon in~$\mathfrak{X}'$ corresponds in the normal graph~$G'$ to one of the configurations shown in \cref{fig:18}: either a hollow digon or a black leaf adjacent to a white trivalent vertex. Either way, $G'$ contains one of the forbidden configurations from \cref{fig:fail} (cf.\ \cref{lem:normal-no-collapse}), so $G$ is not reduced.
\begin{figure}
\caption{A hollow monogon in a triple diagram yields a forbidden configuration in the corresponding normal plabic graph, cf.\ \cref{fig:fail}.}
\label{fig:18}
\end{figure}
Going in the other direction, suppose that a normal plabic graph $G$ is not reduced. By \cref{lem:not-reduced=>osim}, there exists $G'\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G$ containing a forbidden configuration.
By \cref{lem:M1M2M3-to-triple}, the triple diagram~$\mathfrak{X}(G)$ is move-equivalent to the (generalized) triple diagram~$\mathfrak{X}(G')$. Since $\mathfrak{X}(G)$ is connected, so is~$\mathfrak{X}(G')$. It follows by \cref{lem:generalized-triple-diagram-connected} that $\mathfrak{X}(G')$ is an honest triple diagram.
The remaining argument depends on the type of a forbidden configuration present in~$G'$. Since $G$ is normal and $G'\stackrel{\raisebox{-3pt}{$\scriptstyle\circ$}}{\sim} G$, it follows that $G'$ has no white leaves. If $G'$ contains a digon whose vertices are of the same color, then $\mathfrak{X}(G')$ has a closed strand; hence $\mathfrak{X}(G')$ is not minimal (by \cref{lem:nomonogondigon}) and neither is~$\mathfrak{X}(G)$. Finally, if $G'$ contains one of the configurations shown in \cref{fig:18}, then $\mathfrak{X}(G')$ contains a monogon, hence is not minimal. \end{proof}
\iffalse
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Since $G$ is normal, $\mathfrak{X}(G)$ is connected and hence $G$ has no clockwise roundtrip which is disconnected from the rest of the trips. By \cref{rem:invarianttrip}, the same is true for $G'$. This means that $G'$ cannot contain the local configuration:
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Moreover, following the procedure in the proof of \cref{lem:biptri}, $G'$ is move-equivalent to a normal plabic graph $G''$. Thus $G''$ contains either the configuration \setlength{\unitlength}{1pt} \begin{picture}(45,20)(0,10) \thicklines \put(15,15){\circle{4}} \put(30,15){\circle{4}} \put(22.5,23.5){\circle*{4}} \put(22.5,6.5){\circle*{4}} \put(13,15){\line(-1,0){10}} \put(32,15){\line(1,0){10}} \qbezier(15,17)(22,30)(30,17) \qbezier(15,13)(22,0)(30,13) \end{picture} or one of the configurations shown in \cref{fig:18}.
In the former case, $\mathfrak{X}(G'')$ contains a parallel digon; in the latter case, $\mathfrak{X}(G'')$ contains a monogon. Either way, by \cref{thm:minimal}, $\mathfrak{X}(G'')$ is not minimal. Since the normal plabic graphs $G$ and $G''$ are move-equivalent, by \cref{thm:newmoves}, they are related via a sequence of urban renewal moves and normal flip moves. This implies that $\mathfrak{X}(G)$ and $\mathfrak{X}(G'')$ are related by a sequence of swivel moves. Therefore $\mathfrak{X}(G)$ is not minimal.
\end{proof} \fi
\begin{lemma} \label{lem:biptri2} Let $G$ be a reduced plabic graph that does not contain a white lollipop, nor a tree that collapses to a white lollipop.
Then $G$ is move-equivalent to a normal plabic graph. \end{lemma}
See \cref{fig:make-Bip}.
\begin{figure}
\caption{A reduced plabic graph $G$ and a normal plabic graph move-equivalent to~$G$. }
\label{fig:make-Bip}
\end{figure}
\begin{proof} We use induction on the number of faces of~$G$. Note that each local move keeps this number invariant.
We begin by collapsing all collapsible trees (cf.\ \cref{def:collapsible-tree}) and removing all bivalent vertices. The resulting plabic graph has no white leaves, since a white leaf would either be a lollipop or else be adjacent to a black vertex of degree~$\ge 3$, which is impossible since the plabic graph at hand is reduced and has no collapsible trees.
We then split each white vertex of degree $\ge4$ into a tree made of trivalent white vertices. After that, we insert a bivalent black vertex in the middle of each edge with both endpoints white, as well as near each boundary vertex connected to a white vertex. By an abuse of notation, we keep calling our plabic graph~$G$, even as it undergoes these and subsequent transformations.
It remains to contract the edges with both endpoints black to make the graph bipartite. This step may however be problematic if such an edge~$e$ is a loop connecting some black vertex $v$ to itself. We will demonstrate that this in fact cannot happen.
If $e$ encloses a face (i.e., there are no other vertices/edges of~$G$ inside~$e$), then we can insert a black vertex into~$e$ and obtain a double edge, contradicting the fact that $G$ is reduced. Now suppose that $e$ encloses some nontrivial subgraph of~$G$. Let $\mathbf{D}'$ be the region bounded by~$e$. We split $v$ into two vertices $v_1$ and~$v_2$ using an (M3) move, so that $v_1$ stays incident to~$e$ and to the edges located outside~$\mathbf{D}'$ whereas $v_2$ is incident to the edges inside~$\mathbf{D}'$. See \cref{fig:loop-G'}.
We thus obtain a subgraph~$G'$, viewed as a plabic graph in~$\mathbf{D}'$ with a single boundary vertex~$v_1$. The plabic graph~$G'$ must be reduced, or else $G$ would not~be. Moreover $G'$ has fewer faces than~$G$. Also, $G'$ does not collapse to a white lollipop (even if $v_2$ is bivalent), because this would result (after the collapse) in a forbidden configuration in~$G$. (We note that $G$ has no collapsible trees.) Thus, the induction hypothesis applies, so we can transform $G'$ via local moves into a normal graph~$G''$. Since $G''$ is reduced, its triple diagram $\mathfrak{X}(G'')$ must be minimal by \cref{red-minimal}. Given that $\mathfrak{X}(G'')$ only has one in- and one out-endpoint, this means that $\mathfrak{X}(G'')$ consists of a single strand connecting these endpoints to each other, with no triple points. In other words, the normal graph $G''$ is a black lollipop at~$v_1$. We then contract it into~$v_1$, creating a loop enclosing a face, and arrive at a contradiction with $G$ being reduced. \end{proof}
\begin{figure}
\caption{Excluding a loop $e$ based at a black vertex~$v$.}
\label{fig:loop-G'}
\end{figure}
\begin{proof}[Proof of \cref{prop:fixedlollipop}] Let $G$ be a reduced plabic graph such that $\pi_G(i)\!=\!i$. We need to show that the connected component of~$G$ containing the boundary vertex~$i$ collapses to a lollipop at~$i$, cf.\ \cref{def:collapsible-tree}.
Suppose otherwise. Without loss of generality, we can assume that $G$ has no trees collapsing to other lollipops either. Since $G$ is reduced, \cref{lem:biptri2} applies,
so $G$ is move-equivalent to a normal plabic graph~$G'$. The trip permutations of $G$ and~$G'$ coincide with each other (by \cref{exercise:trip-invariant})~and with the strand permutation of the triple diagram~$\mathfrak{X}(G')$ (by \cref{pr:bijplabictriple}). Since $G$ is reduced, so is~$G'$; hence $\mathfrak{X}(G')$ is minimal by \cref{red-minimal}.
\pagebreak[3]
Let $d$ be the degree of the black vertex adjacent to the boundary vertex~$i$ in~$G'$.
If $d\!=\!1$, then there is a sequence of local moves relating the component~of~$G$ containing~$i$ to the black lollipop at~$i$~in~$G'$. Since local moves preserve the number of internal faces, and a black lollipop has no internal faces, no (M1) move appears in the sequence of moves. Therefore this component must be a (collapsible) tree and we are done.
If $d=2$ (see \cref{fig:baddigon} on the left), then $\pi_G(i)\!=\!i$ implies that $\mathfrak{X}(G')$ has a monogon, so it cannot be minimal, cf.\ \cref{lem:nomonogondigon}.
If $d\ge 3$, then we get a par\-al\-lel digon (see \cref{fig:baddigon} on the right), again contradicting the minimality of~$\mathfrak{X}(G')$. \end{proof}
\begin{figure}
\caption{The vicinity of $i$ in~$G'$.}
\label{fig:baddigon}
\end{figure}
\begin{proof}[Proof of \cref{thm:moves}] Let $G$ and $G'$ be reduced plabic graphs. If $G\sim G'$, then ${\widetilde{\pi}}_G = {\widetilde{\pi}}_{G'}$ by \cref{exercise:forward}. We need to show the converse.
Let $G$ and $G'$ be reduced plabic graphs such that ${\widetilde{\pi}}_G = {\widetilde{\pi}}_{G'}$. If this decorated permutation has a fixed point at some vertex~$i$, then by \cref{prop:fixedlollipop}, after applying local moves if needed, both $G$ and $G'$ have a lollipop (of the same color) in position~$i$. We can delete this lollipop in both graphs; the resulting graphs are still reduced, and their decorated trip permutations still coincide. So without loss of generality, we may assume that ${\widetilde{\pi}}_G = {\widetilde{\pi}}_{G'}$ has no fixed points and correspondingly
$G$ and $G'$ have no trees collapsing to lollipops. Applying local moves as needed, we can furthermore assume, in light of \cref{lem:biptri2}, that both $G$ and $G'$ are normal. Since they are also reduced, \cref{red-minimal} implies that the triple diagrams $\mathfrak{X}(G)$ and $\mathfrak{X}(G')$ are minimal. By \cref{pr:bijplabictriple}, we moreover have $\pi_{\mathfrak{X}(G)} = \pi_G = \pi_{G'} = \pi_{\mathfrak{X}(G')}$. Invoking \cref{thm:domino-flip}, we conclude that $\mathfrak{X}(G)$ and $\mathfrak{X}(G')$ are move-equivalent. Then by \cref{cor:newmoves}, $G$ and $G'$ are move-equivalent as well. \end{proof}
\begin{remark} As we have seen, A.~Postnikov's theory of plabic graphs \cite{postnikov} is closely related to D.~Thurston's theory of triple diagrams \cite{thurston}. In particular, reduced plabic graphs are essentially minimal triple diagrams in disguise. While we have not discussed it here, there are some \emph{reduction \linebreak[3] moves} that can be repeatedly applied to a non-reduced plabic graph (resp., a non-minimal triple diagram) in order to---eventually---make it reduced (resp., minimal).
Here the two theories diverge: reduction moves for triple diagrams preserve the strand permutation, but reduction moves for plabic graphs do not preserve the trip permutation. In spite of that, reduction moves for plabic graphs fit into the theory of the totally nonnegative Grassmannian, as they are compatible with its cell decomposition, cf.\ \cite[Section~12]{postnikov}. We will discuss this in the next chapter. \end{remark}
\section{The bad features criterion}\label{sec:bad}
Here we provide an algorithm for deciding whether a plabic graph is reduced or not. We first explain (see \cref{def:normalize}) how to transform an arbitrary plabic graph~$G$ into a normal plabic graph $N(G)$ move-equivalent to~$G$ (or conclude that $G$ is not reduced). We then use a criterion based on \cref{red-minimal} to determine whether $N(G)$ (hence~$G$) is reduced or not.
\begin{lemma} \label{lem:roundtrips} Let $G$ be a reduced plabic graph. Then $G$ has no roundtrips. Also, $G$ has no loops, i.e., edges whose endpoints coincide. \end{lemma}
\begin{proof} We can assume that $G$ does not contain white lollipops or trees that collapse to white lollipops. (Collapsing such trees and removing lollipops does not affect whether a graph is reduced or whether it has a roundtrip.) Now by~\cref{lem:biptri2}, $G$~is move-equivalent (up to the removal of some white lollipops) to a normal plabic graph~$G'$.
Since $G'$ is reduced, $\mathfrak{X}(G')$ is minimal (see \cref{red-minimal}). Hence $\mathfrak{X}(G')$ has no closed strands (see \cref{lem:nomonogondigon}), so $G'$ has no roundtrips. Since roundtrips persist under local moves, $G$ has no roundtrips either.
Suppose $G$ has a loop~$e$ based at a black (resp., white) vertex. (Some edges and vertices might be enclosed by~$e$.) Then the trip that traverses~$e$ clockwise (resp., counterclockwise) is a roundtrip, a contradiction. \end{proof}
\begin{definition} \label{def:normalize} Let $G$ be a plabic graph. The following algorithm is similar to the procedure employed in the proof of \cref{lem:biptri2}. It either determines that $G$ is not reduced, or outputs a normal plabic graph~$N(G)$ which, up to the addition/removal of lollipops, is move-equivalent to~$G$. In~the latter scenario, the plabic graph~$N(G)$---hence the original graph~$G$---may be either reduced or not.
At each stage of the algorithm, the plabic graph~$G$ undergoes some transformations that do not affect whether it is reduced.
\noindent \emph{Stage 1.} Use moves (M2)--(M3) to collapse all collapsible trees in~$G$.
\noindent \emph{Stage 2.} Use moves~(M2) to remove all bivalent vertices.
\noindent \emph{Stage 3.} Remove all lollipops.
\noindent \emph{Stage 4.} If $G$ has an internal leaf~$u$, then let $v$ be the vertex adjacent to~$u$. By construction, $\deg(v)\ge 3$. Since $G$ has no collapsible trees, we conclude that $u$ and $v$ are of different color and $G$ is not reduced.
If $G$ has no internal leaves, then each internal vertex has degree $\ge3$.
\noindent \emph{Stage 5.} Use moves~(M3) to contract all edges with both endpoints black. If at any point $G$ has a loop, then it is not reduced, by \cref{lem:roundtrips}.
\noindent \emph{Stage 6.} Use moves (M3) to replace each white vertex by a trivalent tree with white vertices.
\noindent \emph{Stage 7.} Use moves (M2) to insert a black vertex into every edge with no black endpoints. The resulting plabic graph $N(G)$ is normal. \end{definition}
\pagebreak[3]
\iffalse See \cref{fig:badfeature+leaf}.
\begin{figure}
\caption{(a) Removing a bivalent vertex and contracting a unicolored edge makes the graph normal. (b) This graph has an internal leaf adjacent to a trivalent vertex of a different color, so it is not reduced.
}
\label{fig:badfeature+leaf}
\end{figure} \fi
\begin{definition} \label{def:bad-features} If a trip passes through an edge~$e$ of a plabic graph twice (in the opposite directions), we call this an \emph{essential self-intersection}.
If the edges $e_1$ and $e_2$ are such that there are two distinct trips each of which
passes first through~$e_1$ and then through~$e_2$, we call this a \emph{bad double crossing}.
We use the term \emph{bad features} to collectively refer to \begin{itemize}[leftmargin=.2in] \item roundtrips (see \cref{def:trip}), \item essential self-intersections, and \item bad double crossings. \end{itemize} These notions are illustrated in \cref{fig:bad-features}. \end{definition}
\begin{figure}
\caption{Plabic graph fragments representing ``bad features:'' (a)~a~roundtrip; (b) essential self-intersection; (c) bad double crossing.}
\label{fig:bad-features}
\end{figure}
\begin{lemma} \label{lem:badfeatures} A normal plabic graph $G$ has a bad feature if and only if the associated triple diagram $\mathfrak{X}(G)$ has a badgon.
\end{lemma}
\iffalse \begin{figure}
\caption{A plabic graph $G$ and the corresponding bipartite graph $\operatorname{Bip}(G)$.
Here we contract a unicolored edge, then remove a bivalent vertex, then contract again, to get $\operatorname{Bip}(G)$.}
\label{fig:Bip2}
\end{figure} \fi
\iffalse
\begin{lemma} \label{lem:bad-features-normal-moves} Let $G$ and $G'$ be normal plabic graphs related to each other via urban renewal and normal flip moves, see Definitions \ref{def:urban-normal}--\ref{def:flip-normal}. Then $G$ contains a bad feature if and only if $G'$ does. \end{lemma}
\begin{proof} Both types of moves preserve the connectivity of trips. A normal flip move only affects the portions of the trips that pass through the two-edge ``bridge'' connecting the two white vertices, cf.\ \cref{fig:flipmove}. Since the edges along the bridge are not essential, the move does not affect the presence of bad features.
We now examine the (normal) urban renewal move. There are a few cases to consider, depending on whether each of the black vertices of the square is bivalent or not. (This affects whether the sides of the square incident to such a vertex are essential or not.) If both black vertices of the square (on one of the sides of the urban renewal move) are bivalent, then both graphs contain a roundtrip. Otherwise, if exactly one of these two black vertices is bivalent (on one of the sides of the urban renewal move), then this graph contains a bad double crossing whereas the graph related to it via urban renewal contains an essential self-intersection. Finally, if none of the squares have bivalent black vertices, then all sides of both squares are essential and moreover each pair of trips that traverse the same side (in opposite directions) in one of the graphs does the same in the other. As the remainder of each trip does not change, the claim follows. \end{proof}
\fi
\begin{proof} Let $G$ be a normal plabic graph. The strands in the triple diagram $\mathfrak{X}=\mathfrak{X}(G)$ closely follow the trips in~$G$. Therefore $\mathfrak{X}$ has a closed strand if and only if $G$ has a roundtrip.
If $G$ has an essential self-intersection (resp., a bad double crossing), then $\mathfrak{X}$ has a monogon (resp., a parallel digon). To see that, take each edge~$e$ involved in a bad feature and consider the white end~$v$ of~$e$. The strands corresponding to the trips involved in the bad feature will intersect at~$v$; thus $v$ will be a vertex of the corresponding badgon. Cf.\ Figures~\ref{fig:18} and~\ref{fig:17}.
\begin{figure}
\caption{
A bad double crossing in $G$ yields a parallel digon in~$\mathfrak{X}(G)$.}
\label{fig:17}
\end{figure}
Conversely, suppose that $\mathfrak{X}$ has a monogon with self-intersection corresponding to the white vertex~$v$ of~$G$. There are three strand segments of $\mathfrak{X}$ that pass through~$v$, each running along two distinct edges incident to~$v$;
because we have a self-intersection, two of these strands segments are part of the same strand $s$.
Since $v$ is trivalent, the pigeonhole principle implies that two of the four edges that $s$ runs along must coincide. This yields an essential self-intersection in~$G$.
A~similar argument shows that if $\mathfrak{X}$ has a parallel digon, then $G$ has a bad double crossing.
\end{proof}
\begin{corollary} \label{cor:bad<=>bad} Let $G$ be a normal plabic graph. Let $\mathfrak{X}=\mathfrak{X}(G)$ be the corresponding triple diagram. Then the following are equivalent: \begin{itemize}[leftmargin=.2in] \item $G$ is reduced; \item $\mathfrak{X}$ is minimal; \item $G$ has no bad features; \item $\mathfrak{X}$ has no badgons. \end{itemize} \end{corollary}
\begin{proof} By \cref{red-minimal}, $G$ is reduced if and only if $\mathfrak{X}$~is minimal. By \cref{thm:minimal}, $\mathfrak{X}$ is minimal if and only if $\mathfrak{X}$ has no badgons. By \cref{lem:badfeatures}, $\mathfrak{X}$ has no badgons if and only if $G$ has no bad features. \end{proof}
\cref{cor:bad<=>bad} provides the following criterion, which is a version of \cite[Theorem 13.2]{postnikov}.
\begin{theorem} \label{thm:reduced} A normal plabic graph
is reduced if and only if it does not contain any bad features. \end{theorem}
For example, any plabic graph containing one of the fragments shown in \cref{fig:bad-features} is necessarily not reduced.
\begin{remark} \label{rem:}
For \emph{any} plabic graph~$G$, \cref{thm:reduced} can be used in conjunction with the procedure described in \cref{def:normalize} to determine whether $G$ is reduced or not. \end{remark}
\begin{remark} Recall from Remark~\ref{rem:reduceddecomp} that a factorization (not necessarily reduced) of an element of a symmetric group into a product of simple reflections can be represented by (a version of) a wiring diagram.
As plabic graphs can be viewed as generalizations of wiring diagrams (see \cref{def:wiringplabic}), reduced plabic graphs may be viewed as a generalization of reduced expressions. In this context, the criterion of \cref{thm:reduced} corresponds to the condition that each pair of lines in the wiring diagram intersect at most once. \end{remark}
\iffalse \begin{exercise} \label{ex:new-bad} Let $G$ and $G'$ be move-equivalent plabic graphs. Show that $G$ has a bad feature if and only if so does~$G'$.
\end{exercise}
\cref{ex:new-bad} can be settled by a careful examination of various cases that can arise while performing a local move in the presence of a bad feature. We note that this is a little subtle since the type of a bad features can change in the process, see \cref{fig:20}.
\begin{figure}
\caption{Persistence of ``bad features'' in plabic graphs under local moves.}
\label{fig:20}
\end{figure}
\begin{proof}[Proof of \cref{thm:reduced}] Let $G$ be a plabic graph without internal leaves. We are going to transform $G$ in various ways that will not affect the presence of bad features, nor whether $G$ is reduced or not. By abuse of notation, we will keep denoting the plabic graph at hand by~$G$, while it undergoes these transformations.
If $G$ has unicolored edges that are not loops, repeatedly contract them. This does not affect whether the graph is reduced and whether it has bad features or not. Moreover it does not create leaves or bivalent vertices.
If we encounter a loop, then $G$ has a roundtrip that includes the loop together with possibly some edges inside the disk it encircles. In this case, $G$ has a bad feature (a roundtrip) and is not reduced (by \cref{lem:roundtrips}).
Now $G$ is bipartite, with all vertices of degree $\ge 3$. We then insert a black vertex into every edge connecting a white vertex (other than a lollipop) to a boundary one. The resulting graph is normal, so by \cref{cor:bad<=>bad}, it is reduced if and only if it does not contain any bad features.
\end{proof} \fi
\section{Affine permutations} \label{sec:affine}
By \cref{thm:moves}, move equivalence classes of reduced plabic graphs are labeled by decorated permutations. An alternative (sometimes more useful) labeling utilizes ($(a,b)$-bounded) \emph{affine permutations}, introduced and studied in this section.
\begin{definition} \label{def:anti} For a decorated permutation ${\widetilde{\pi}}$ on $b$ letters, we say that $i\in \{1,\dots,b\}$ is an \emph{anti-excedance} of ${\widetilde{\pi}}$ if either ${\widetilde{\pi}}^{-1}(i)>i$ or if ${\widetilde{\pi}}(i)=\overline{i}$. The number of anti-excedances of~${\widetilde{\pi}}$ (which we usually denote by~$a$) is equal to the number of values $i\in \{1,\dots,b\}$ such that ${\widetilde{\pi}}(i)<i$ or ${\widetilde{\pi}}(i)=\overline{i}$. \end{definition}
\begin{example} \label{example:523641} The decorated permutation ${\widetilde{\pi}} = (5,\underline{2},\overline{3},6,4,1)$ on $b=6$ letters (see \cref{fig:plabic3}) has $a=3$ anti-excedances, namely, $1$, $4$, and~$\overline{3}$.
Indeed, ${\widetilde{\pi}}^{-1}(1)=6>1$, and ${\widetilde{\pi}}^{-1}(4)=5>4$. \end{example}
\begin{definition} \label{def:affinization} Let ${\widetilde{\pi}}$ be a decorated permutation on $b$ letters with $a$ anti-excedances. The \emph{affinization} of ${\widetilde{\pi}}$ is the map ~$\widetilde{\pi}_{\aff}:\mathbb{Z}\to\mathbb{Z}$ constructed as follows. For $i\in\{1,\dots,b\}$, we set
\begin{equation*}
\widetilde{\pi}_{\aff}(i)= \begin{cases}
{\widetilde{\pi}}(i) &\text{ if }{\widetilde{\pi}}(i)>i,\\
i &\text{ if }{\widetilde{\pi}}(i)=\underline{i}, \\
{\widetilde{\pi}}(i)+b &\text{ if } {\widetilde{\pi}}(i)<i,\\
i+b &\text{ if }{\widetilde{\pi}}(i)=\overline{i}.
\end{cases}
\end{equation*}
We then extend $\widetilde{\pi}_{\aff}$ to~$\mathbb{Z}$ so that it satisfies \begin{equation} \label{eq:affpi+b} \widetilde{\pi}_{\aff}(i+b)=\widetilde{\pi}_{\aff}(i)+b \quad (i\in\mathbb{Z}). \end{equation} We note that \begin{equation} \label{eq:i<f(i)<i+b} i \leq \widetilde{\pi}_{\aff}(i) \leq i+b \quad (i\in\mathbb{Z}) \end{equation} and \begin{equation} \label{eq:sum-aex} \sum_{i=1}^b (\widetilde{\pi}_{\aff}(i)-i) = b\cdot \#\{ i\in\{1,\dots,b\} \mid {\widetilde{\pi}}(i)<i \text{\ or\ } {\widetilde{\pi}}(i)=\overline{i}\} = ab. \end{equation} \end{definition}
\begin{example} \label{ex:affinization} Continuing with ${\widetilde{\pi}} \!=\! (5,\underline{2},\overline{3},6,4,1)$ from \cref{example:523641}, we get
$\widetilde{\pi}_{\aff}(1)\!=\!5$, $\widetilde{\pi}_{\aff}(2)\!=\!2$, $\widetilde{\pi}_{\aff}(3)\!=\!9$, $\widetilde{\pi}_{\aff}(4)\!=\!6$, $\widetilde{\pi}_{\aff}(5)\!=\!10$, $\widetilde{\pi}_{\aff}(6)\!=\!7$, or more succinctly, \begin{equation*} \widetilde{\pi}_{\aff}=(\dots,5,2,9,6,10,7,\dots)=(\cdots\ \boxed{5\ 2\ 9\ 6\ 10\ 7}\ 11\ 8\ 15\ 12\ 16\ 13\ \cdots). \end{equation*} (The boxed terms are the values at $1,\dots,b$. They determine the rest of the sequence by virtue of~\eqref{eq:affpi+b}.) In accordance with \eqref{eq:sum-aex}, we have \begin{equation*} (5+2+9+6+10+7)-(1+\cdots+6)=39-21=18=3\cdot 6=ab. \end{equation*} \end{example}
With the above construction in mind, we introduce the following notion.
\begin{definition} \label{def:affine} Let $a$ and $b$ be positive integers. An~\emph{$(a,b)$-bounded affine permutation} is a bijection $f:\mathbb{Z} \to \mathbb{Z}$ satisfying the following conditions: \begin{itemize}[leftmargin=.2in] \item $f(i+b)=f(i)+b\,$ for all $i\in \mathbb{Z}$; \item $i \leq f(i) \leq i+b\,$ for all $i\in\mathbb{Z}$; \item $\sum_{i=1}^b (f(i)-i) = ab$. \end{itemize} \end{definition}
\begin{lemma}\cite{KLS} \label{lem:affinization-type} The correspondence ${\widetilde{\pi}}\mapsto \widetilde{\pi}_{\aff}$ described in \cref{def:affinization} restricts to
a bijection between decorated permutations on $b$ letters with $a$ anti-excedances and the $(a,b)$-bounded affine permutations. \end{lemma}
\begin{proof} If ${\widetilde{\pi}}$ is a decorated permutation on $b$ letters with $a$ anti-excedances, then \eqref{eq:affpi+b}--\eqref{eq:sum-aex} show that $\widetilde{\pi}_{\aff}$ is an $(a,b)$-bounded affine permutation.
Conversely, given an $(a,b)$-bounded affine permutation $f:\mathbb{Z}\to\mathbb{Z}$, we can define the decorated permutation ${\widetilde{\pi}}$ on $b$ letters by \begin{equation*} {\widetilde{\pi}}(i)=\begin{cases} \underline{i} & \text{if $f(i)=i$;} \\ \overline{i} & \text{if $f(i)=i+b$;} \\
f(i) & \text{if $f(i)\le b$ and $f(i)\neq i$;}\\
f(i)-b & \text{if $f(i)> b$ and $f(i) \neq i+b$.} \end{cases} \end{equation*} We claim that~${\widetilde{\pi}}$ has $a$ anti-excedances. Using the inequality $i\le f(i)\le i+b$, we conclude that the anti-excedances of~${\widetilde{\pi}}$ are in bijection with the values $i\in\{1,\dots,b\}$ such that $f(i)>b$. The claim follows from the observation that
$ab=\sum_1^b (f(i)-i)=
b \cdot \#\{i\in\{1,\dots,b\} \mid f(i)>b\}$.
\end{proof}
\cref{lem:affinization-type} is illustrated in \cref{fig:affinization} (the first two columns).
\begin{figure}\label{fig:affinization}
\end{figure}
Recall from \cref{ex:enumeration} that the number of decorated permutations on $b$ letters is equal to $b!\sum_{k=0}^b \frac{1}{k!}$. The following result, stated here without proof, refines this count by taking into account the number of anti-excedances.
\begin{proposition}[{\rm
\cite[Theorem 4.1]{Williams},
\cite[Proposition~23.1]{postnikov}}] \label{pr:A_{a,b}(1)} Let $D_{a,b}$ be the number of decorated permutations on $b$ letters with $a$ anti-excedances (or the number of $(a,b)$-bounded affine permutations, cf.\ \cref{lem:affinization-type}). Then \begin{align*} &D_{a,b}=\sum_{i=0}^{a-1} (-1)^i \binom{b}{ i} ( (a-i)^i (a-i+1)^{b-i} - (a-i-1)^i (a-i)^{b-i}), \\ & \sum_{0\le a\le b} D_{a,b} \,\, x^a \, \frac{y^b}{b!} = e^{xy}\, \frac{x-1}{x-e^{y(x-1)}} . \end{align*} \end{proposition}
\begin{definition} Let $\widetilde{\pi}_{\aff}$ be an $(a,b)$-bounded affine permutation, an affinization of a decorated permutation~${\widetilde{\pi}}$, cf.\ \cref{lem:affinization-type}.
We refer to a position $i\in\mathbb{Z}$ such that $\widetilde{\pi}_{\aff}(i)\equiv i\bmod b$
(in other words, $\widetilde{\pi}_{\aff}(i)\in\{i,i+b\}$; and if $1\leq i \leq b$ then
${\widetilde{\pi}}(i )\in\{\underline{i},\overline{i}\}$) as a \emph{fixed point} of~$\widetilde{\pi}_{\aff}$. If every $i\in\mathbb{Z}$ is a fixed point of~$\widetilde{\pi}_{\aff}$, then we say that $\widetilde{\pi}_{\aff}$ is \emph{equivalent to the identity modulo~$b$} (or that ${\widetilde{\pi}}$ is a decoration of the identity). \end{definition}
\begin{lemma} \label{lem:affpi-adjacent} If $\widetilde{\pi}_{\aff}$ is not equivalent to the identity modulo~$b$, then there exist $i,j\in\mathbb{Z}$ such that \begin{align} \label{eq:ij-pair-1} &1\le i<j\le b, \\ \label{eq:ij-pair-2} &\widetilde{\pi}_{\aff}(i)<\widetilde{\pi}_{\aff}(j), \\ \label{eq:ij-pair-3} &\text{every position $h$ such that $i<h<j$ is a fixed point of~$\widetilde{\pi}_{\aff}$, and} \\ \label{eq:ij-pair-4} &\text{neither $i$ nor $j$ are fixed points of~$\widetilde{\pi}_{\aff}$.} \end{align} \end{lemma}
\begin{proof} Suppose such a pair $(i,j)$ does not exist. Let $i_1<\dots<i_m$ be the elements of $\{1,\dots,b\}$ that are not fixed points of~$\widetilde{\pi}_{\aff}$. Then \begin{equation*} i_1<\dots<i_m<\widetilde{\pi}_{\aff}(i_m)\le\cdots\le\widetilde{\pi}_{\aff}(i_1). \end{equation*} We conclude that none of the values $\widetilde{\pi}_{\aff}(i_j)$ is of the form~$i_\ell$ and consequently is of the form $i_\ell+b$. In particular, $\widetilde{\pi}_{\aff}(i_j)=i_m+b$ for some~$j\neq m$. This implies $\widetilde{\pi}_{\aff}(i_j) > i_j+b$, a contradiction. \end{proof}
\begin{definition} \label{def:inv-affpi} An \emph{inversion} of $\widetilde{\pi}_{\aff}$ is a pair of integers $(i,j)$ such that $i<j$ and $\widetilde{\pi}_{\aff}(i)>\widetilde{\pi}_{\aff}(j)$.
Two inversions $(i,j)$ and $(i',j')$ are \emph{equivalent} if $i'-i=j'-j\in b\mathbb{Z}$.
The \emph{length} $\ell(\widetilde{\pi}_{\aff})$ of $\widetilde{\pi}_{\aff}$ is the number of equivalence classes of inversions. (These classes correspond to the
\emph{alignments} of~${\widetilde{\pi}}$, as defined in~\cite{postnikov}.) This number is finite since for any inversion $(i,j)$, we have $i<j<i+b$. Indeed, if $j\ge i+b$, then $\widetilde{\pi}_{\aff}(j)\ge j\ge i+b\ge \widetilde{\pi}_{\aff}(i)$. See~\cref{fig:affinization}. \end{definition}
\pagebreak
\begin{lemma} \label{lem:inv-affpi-bound} Let $\widetilde{\pi}_{\aff}$ be an $(a,b)$-bounded affine permutation that is equivalent to the identity modulo~$b$. Then $\ell(\widetilde{\pi}_{\aff})= a(b-a)$. \end{lemma}
\begin{proof} Let $I=\{i\in\{1,\dots,b\}\mid \widetilde{\pi}_{\aff}(i)=i+b\}$ and $\underline{I} =\{i\in\{1,\dots,b\}\mid \widetilde{\pi}_{\aff}(i)=i\}
$.
Then $|I|=a$
and $|\underline{I}|=b-a$. The equivalence classes of inversions of~$\widetilde{\pi}_{\aff}$ are described by the following list of representatives: \begin{equation*} \{(i,j)\in I\times\underline{I} \mid 1\le i<j\le b\} \cup \{(i,j+b)\mid (i,j)\in I\times\underline{I}, 1\le j<i\le b\}. \end{equation*}
The cardinality $\ell(\widetilde{\pi}_{\aff})$ of this set is equal to $|I\times\underline{I}|=a(b-a)$. \end{proof}
We next describe an algorithm for factoring affine permutations.
\begin{definition} \label{def:BCFW0} Let $\widetilde{\pi}_{\aff}$ be an $(a,b)$-bounded affine permutation. If $\widetilde{\pi}_{\aff}$ is not equivalent to the identity modulo~$b$, then by \cref{lem:affpi-adjacent}, there exist positions $i,j\in\mathbb{Z}$ satisfying \eqref{eq:ij-pair-1}--\eqref{eq:ij-pair-4}. We then swap the values of~$\widetilde{\pi}_{\aff}$ in positions $i$ and~$j$ (and more generally, in positions $i+mb$ and $j+mb$, for all~$m\in\mathbb{Z}$).
We repeat this procedure until we obtain an affine permutation that is equivalent to the identity modulo~$b$. The algorithm terminates because each swap increases the length of the affine permutation by~$1$; this number is bounded by \cref{def:inv-affpi}.
See \cref{fig:factorization-bridges}. \end{definition}
\begin{figure}\label{fig:factorization-bridges}
\end{figure}
\begin{remark} \label{rem:l(affpi)-bound} In view of~\eqref{eq:sum-aex}, the affine permutation at hand remains $(a,b)$-bounded after each step of the algorithm in \cref{def:BCFW0}. It then follows from \cref{lem:inv-affpi-bound} that among all $(a,b)$-bounded affine permutations~$\widetilde{\pi}_{\aff}$, the ones that have the maximal possible length $\ell(\widetilde{\pi}_{\aff})=a(b-a)$ are precisely the ones that are equivalent to the identity modulo~$b$. \end{remark}
\section{Bridge decompositions} \label{sec:bridge}
Bridge decompositions \cite[Section 3.2]{amplitudes} provide a useful recursive construction of reduced plabic graphs with a given decorated trip permutation.
\begin{definition} \label{def:BCFW} A \emph{bridge} is a graph fragment shown in \cref{fig:bridges} on the left. Let ${\widetilde{\pi}}$ be a decorated permutation on $b$ letters that has $a$ anti-excedances, and let $\widetilde{\pi}_{\aff}$ be the corresponding affine permutation. To build a plabic graph associated to $\widetilde{\pi}_{\aff}$, we begin by introducing a white (resp., black) lollipop in each position $i$ with ${\widetilde{\pi}}(i)=\overline{i}$ (resp., ${\widetilde{\pi}}(i)=\underline{i}$). If ${\widetilde{\pi}}$ is a decoration of the identity, we are done. Otherwise, we generate a sequence of transpositions $(i,j)$ following the algorithm in \cref{def:BCFW0}, then attach successive bridges in the corresponding positions, as in \cref{fig:bridges}.
The resulting graph is called a \emph{bridge decomposition} of~$\widetilde{\pi}_{\aff}$, or sometimes a BCFW bridge decomposition, due to its relation with the Britto-Cachazo-Feng-Witten recursion in quantum field theory, see~\cite{amplitudes}. \end{definition}
\begin{figure}\label{fig:bridges}
\end{figure}
\begin{proposition} \label{pr:number-of-bridges} A bridge decomposition of an $(a,b)$-bounded affine permutation~$\widetilde{\pi}_{\aff}$ uses $a(b-a)-\ell(\widetilde{\pi}_{\aff})$ bridges. \end{proposition}
\begin{proof} See \cref{lem:inv-affpi-bound} and Definitions \ref{def:BCFW0} and~\ref{def:BCFW}. \end{proof}
\begin{theorem} \label{thm:bridge} Let ${\widetilde{\pi}}$ be a decorated permutation on $b$ letters that has $a$ anti-excedances. Let $\widetilde{\pi}_{\aff}$ be the associated $(a,b)$-bounded affine permutation. Then any bridge decomposition of $\widetilde{\pi}_{\aff}$ is a reduced plabic graph with the decorated trip permutation~${\widetilde{\pi}}$.
\end{theorem}
\begin{proof}
We use induction on the number of bridges $\beta=a(b-a)-\ell(\widetilde{\pi}_{\aff})$. If $\beta=0$, then ${\widetilde{\pi}}$ is a decoration of the identity (see \cref{rem:l(affpi)-bound}), so the bridge decomposition consists entirely of lollipops, and we are done.
Now suppose that ${\widetilde{\pi}}$ is not a decoration of the identity. Proceeding as in \cref{def:BCFW0}, we construct a sequence of transpositions $\sigma_1, \sigma_2,\dots, \sigma_\beta$, where $\sigma_1 = (ij)$ satisfies \eqref{eq:ij-pair-1}--\eqref{eq:ij-pair-4}. Let $G$ be the plabic graph obtained by attaching bridges according to $\sigma_1,\dots, \sigma_{\beta}$ (from top to bottom). By the induction assumption, attaching bridges according to $\sigma_2, \dots, \sigma_\beta$ as in \cref{def:BCFW} produces a reduced plabic graph $G'$ with the trip permutation~${\widetilde{\pi}}'= \sigma_\beta \cdots \sigma_2$. This graph has $\beta-1$ bridges and is obtained by removing the topmost horizontal edge~$e$ from~$G$ and applying local moves (M2) to remove the endpoints of~$e$.
Conversely, $G$ is obtained from $G'$ by attaching a bridge in position $(i,j)$ at the top of~$G'$. (To illustrate, in \cref{fig:bridges} we have $(i,j)=(3,4)$.) When we add this bridge to~$G'$, the trips starting at $i$ and $j$ get their ``tails'' swapped: the trip $T_i$ (resp.,~$T_j$) in~$G$ that begins at~$i$ (resp., at~$j$) traverses~$e$ and continues along the trip that used to begin at~$j$ (resp., at~$i$) in~$G'$; all other trips remain the same. Hence the trip permutation of $G$ is ${\widetilde{\pi}}' \sigma_1 = {\widetilde{\pi}}$.
It remains to show that $G$ is reduced. It is tempting to try to use for this purpose the ``bad features'' criterion of \cref{thm:reduced}. However, this theorem requires the plabic graph to be normal, so we will need to
replace $G$ by a suitable normal graph~$N(G)$. It will also be more convenient to utilize the triple diagram version of the criterion, cf.\ \cref{cor:bad<=>bad}.
We begin by constructing the normal graph $N(G)$ using
\cref{def:normalize}. (Note that stages 1, 4 and 6 of the algorithm are not needed since every vertex in~$G$ has degree 2 or~3, or is a lollipop
that gets removed.)
Up to the addition/removal of lollipops, the
triple diagram $\mathfrak{X}(N(G))$ is isotopic to the triple diagram~$\mathfrak{X}(G)$ constructed directly from~$G$ as in \cref{def:triple-non-normal}. The graph $G$ is reduced if and only if $N(G)$ is reduced, which is
equivalent to the triple diagram $\mathfrak{X}=\mathfrak{X}(N(G))=\mathfrak{X}(G)$ being minimal, or to $\mathfrak{X}$ having no badgons, see \cref{cor:bad<=>bad}. Thus, our goal is to show $\mathfrak{X}$ has no badgons.
By the induction assumption,
the triple diagram $\mathfrak{X}'=\mathfrak{X}(G')$ has no badgons. It follows that any potential badgon in $\mathfrak{X}$ must involve the white endpoint~$w$ of edge~$e$; otherwise this feature would have already been present in~$\mathfrak{X}'$. In particular, this means that $w$ is trivalent, i.e., not an elbow of~$G$.
A monogon in $\mathfrak{X}$ would have to have its vertex at~$w$. Three of the six half-strands at~$w$ run straight to or from the boundary, so we need to use two of the remaining three; moreover, those two half-strands have to be oppositely oriented. There are two such cases to consider. In one case, $i$ would be a fixed point of~${\widetilde{\pi}}$, contradicting our choice of~$(i,j)$. In the other case, $i$ would be a fixed point of~${\widetilde{\pi}}'={\widetilde{\pi}}(G')$, which can also be ruled out since in that case, $i$~would not participate in any bridge in~$G'$, making it impossible to produce the bottom vertex of the vertical edge pointing downwards from~$w$.
Finally, suppose that $\mathfrak{X}$ contains a parallel digon. One of the vertices of the digon has to be~$w$; let $w'$ denote the other vertex. Since the two sides of the digon are oriented in the same way at~$w$, it follows that these sides lie on the strands $S_i$ and~$S_j$ that start near the vertices $i$ and~$j$, respectively.
\pagebreak[3]
By our choice of bridge $(i,j)$, every $h$ with $i<h<j$ is a fixed point of ${\widetilde{\pi}}$, but $i$ and $j$ are not fixed points. It follows that $S_i$ (resp. $S_j$) does not terminate at $i$ (resp. $j$),
and neither terminates between the boundary vertices $i$ and~$j$.
We explained in the monogon case that $S_j$ cannot terminate at~$i$; one can similarly argue that $S_i$ cannot terminate at~$j$. Also, neither $S_i$ nor $S_j$ intersects itself. Moreover the ``tails'' of $S_i$ and~$S_j$ that start at the second vertex~$w'$ of the digon do not intersect each other (since otherwise a parallel digon would have been present in~$\mathfrak{X}'$).
\iffalse The induction assumption implies that $G'$ has no bad double crossings, essential self-intersections, or roundtrips; our goal is to show that the same holds for~$G$. The key observation is that any potential bad feature in $G$ must involve the edge~$e$ or one of the vertical edges incident to the endpoints of~$e$; otherwise this feature would have already been present in~$G'$.
We first observe that $G$ has no roundtrips since every trip in~$G$ that passes through~$e$ or through one of the vertical edges incident to the endpoints of~$e$ either begins or ends at one of the boundary vertices $i$ and~$j$.
An essential self-intersection in $G$ cannot involve~$e$ since the two trips $T_i$ and $T_j$ that pass through $e$ are distinct, as they begin at distinct boundary vertices $i$ and~$j$. Similarly, $G$ cannot have an essential self-intersection involving a vertical edge incident to the white (resp., black) endpoint of~$e$, since this would imply that either $G$ or $G'$ has a fixed point at~$i$ (resp.,~$j$).
Suppose $G$ has a bad double crossing involving $e$ and some other edge~$e'$. The trips $T_i$ and $T_j$ cross over each other both at $e$ and~$e'$; since $G'$ is reduced, $e$ and~$e'$ are the only edges at which they cross over each other.
It follows that either ${\widetilde{\pi}}(i)<i$ and ${\widetilde{\pi}}(j)>j$, or ${\widetilde{\pi}}(i)>{\widetilde{\pi}}(j)>j$, or ${\widetilde{\pi}}(j)<{\widetilde{\pi}}(i)<i$. In each case, we get $\widetilde{\pi}_{\aff}(i)>\widetilde{\pi}_{\aff}(j)$, which contradicts the way we chose $i$ and~$j$.
Finally, $G$ cannot have a bad double crossing involving a vertical edge~$e'$ incident to an endpoint of~$e$. Indeed, $e'$ cannot be the first (resp., second) of the two crossings since one of the trips passing through~$e'$ runs (almost) straight towards the boundary (resp., away from it). \fi
It follows that either ${\widetilde{\pi}}(i)<i$ and ${\widetilde{\pi}}(j)>j$, or ${\widetilde{\pi}}(i)>{\widetilde{\pi}}(j)>j$, or ${\widetilde{\pi}}(j)<{\widetilde{\pi}}(i)<i$. In each case, we get $\widetilde{\pi}_{\aff}(i)>\widetilde{\pi}_{\aff}(j)$, which contradicts the way we chose $i$ and~$j$. \end{proof}
\begin{corollary} \label{permtoG} Let ${\widetilde{\pi}}$ be a decorated permutation on $b$ letters. Then there exists a reduced plabic graph whose decorated trip permutation is ${\widetilde{\pi}}$. \end{corollary}
\begin{proof} Use either \cref{thm:bridge} or the construction in \cref{def:standard-triple} (together with \cref{pr:bijplabictriple} and \cref{red-minimal}). \end{proof}
\begin{corollary} \label{cor:number-faces} Let $G$ be a reduced plabic graph with the decorated trip permutation~${\widetilde{\pi}}$. If ${\widetilde{\pi}}$ has $b$ letters and $a$ anti-excedances, then the number of faces in~$G$ is $a(b-a)-\ell(\widetilde{\pi}_{\aff})+1$. \end{corollary}
\begin{proof} The number of faces is invariant under local moves. Therefore, by \cref{thm:moves}, it suffices to establish this formula for a particular reduced plabic graph with decorated trip permutation~${\widetilde{\pi}}$. By \cref{thm:bridge}, we can use a bridge decomposition of~$\widetilde{\pi}_{\aff}$. Since each bridge adds one face to the graph, the claim follows by \cref{pr:number-of-bridges}. \end{proof}
Let ${\widetilde{\pi}}_{a,b}$ denote the decorated permutation on $b$ letters defined by \begin{equation} \label{eq:dpi-ab} {\widetilde{\pi}}_{a,b}=(a+1,a+2,\dots,b,1,2,\dots,a). \end{equation}
\begin{exercise} \label{exer:length=0} Let ${\widetilde{\pi}}$ be a decorated permutation on $b$ letters that has $a$~anti-excedances. Show that if $\ell(\widetilde{\pi}_{\aff})=0$, then ${\widetilde{\pi}}={\widetilde{\pi}}_{a,b}$. \end{exercise}
\begin{corollary} \label{cor:max-number-faces} Let $G$ be a reduced plabic graph whose decorated trip permutation ${\widetilde{\pi}}_G$ has $b$ letters and $a$ anti-excedances. Then $G$ has at most $a(b-a)+1$ faces. Moreover it has $a(b-a)+1$ faces if and only if ${\widetilde{\pi}}_G={\widetilde{\pi}}_{a,b}$. \end{corollary}
\begin{proof} Immediate from \cref{cor:number-faces} and \cref{exer:length=0}. \end{proof}
\begin{remark} The \emph{permutohedron} $\mathcal{P}_n$ \cite[Exercise~4.64a]{ec1} is a polytope whose $n!$ vertices are labeled by permutations in the symmetric group~$\mathcal{S}_n$. Shortest paths in the $1$-skeleton of~$\mathcal{P}_n$ encode reduced expressions in~$\mathcal{S}_n$, and its $2$-dimensional faces correspond to their local (braid) transformations, cf.\ \cref{exercise:braid-equivalence}. Similarly, paths in the $1$-skeleton of the \emph{bridge polytope}~\cite{WilliamsBridge} encode bridge decompositions of the decorated permutation~$\tilde{\pi}_{a,b}$; its $2$-dimensional faces correspond to local moves in plabic graphs. \end{remark}
\section{Edge labels of reduced plabic graphs} \label{sec:edge-labels}
\begin{definition} \label{def:edgelabeling} Let $G$ be a plabic graph. Let us label the edges of~$G$ by subsets of integers that indicate which one-way trips traverse a given edge; more precisely, for each boundary vertex~$i$, we include~$i$ in the label of every edge contained in the trip that starts at~$i$. By~\cref{rem:atmost2}, each edge will be labeled by at most two integers. See \cref{fig:plabicresonance}.
We say that $G$ has the \emph{resonance property} if after labeling the edges of~$G$ as in \cref{def:edgelabeling}, the following condition is satisfied at each internal vertex~$v$ that is not a lollipop: \begin{itemize}[leftmargin=.2in] \item there exist numbers $i_1 < \dots < i_m$ such that the edges incident to~$v$ are labeled by the two-element sets $\{i_1,i_2\}, \{i_2,i_3\}, \dots, \{i_{m-1},i_m\}, \{i_1,i_m\}$, appearing in clockwise order. \end{itemize} In particular, each edge of $G$ that is not incident to a lollipop is labeled by a two-element subset. See \cref{fig:plabicresonance}. \end{definition}
\begin{figure}
\caption{A reduced plabic graph from Figures~\ref{fig:plabic}(b) and~\ref{fig:plabic-move-equiv}. Its edge labeling
exhibits the resonance property, see \cref{def:edgelabeling}. For example, the edge labels around the vertex~$v$ (resp.,~$v'$), listed in clockwise order, are $\{1,2\}, \{2,4\}, \{1,4\}$ (resp., $\{1,2\}, \{2,5\}, \{1,5\}$). }
\label{fig:plabicresonance}
\end{figure}
\begin{remark} \label{rem:resonance-1-2} If a plabic graph has an internal leaf that is not a lollipop, then the resonance property fails.
At a bivalent vertex~$v$, the resonance condition is satisfied if and only if the two trips passing through~$v$ are distinct and none of them is a roundtrip. \end{remark}
\begin{remark} \label{rem:trivalentresonance} If a plabic graph $G$ is trivalent (apart from lollipops), then the resonance property is equivalent to the following requirement at each interior vertex~$v$ (other than a lollipop): \begin{itemize}[leftmargin=.2in] \item the three edges incident to~$v$ have labels $\{a,b\}$, $\{a,c\}$, and $\{b,c\}$, for some $a<b<c$, and moreover this (lexicographic) ordering of labels corresponds to the counterclockwise direction around~$v$. \end{itemize} For example, in \cref{fig:plabicresonance}, the edge labels around the vertex~$v$ (resp.,~$v'$) are, in lexicographic order, $\{1,2\}, \{1,4\}, \{2,4\}$ (resp., $\{1,2\}, \{1,5\}, \{2,5\}$). The three edges carrying these labels appear in the counterclockwise order around~$v$ (resp.,~$v'$). \end{remark}
\begin{exercise} Verify that none of the plabic graphs shown in \cref{fig:bad-features} (draw a disk around each of the fragments) satisfy the resonance property. \end{exercise}
\begin{theorem}[{\cite[Theorem 10.5]{kodwil}}] \label{thm:resonance} Let $G$ be a plabic graph without internal leaves other than lollipops. Then $G$ is reduced if and only if it has the resonance property. \end{theorem}
\cref{thm:resonance} is proved below in this section, following a few remarks and auxiliary lemmas.
\begin{remark} \cref{thm:resonance} can be used to test whether a given plabic graph (potentially having internal leaves) is reduced or not. To this end, use the moves (M2) and (M3) to get rid of collapsible trees.
If an internal leaf (not a lollipop) remains, then the graph is not reduced. Otherwise, one can apply the criterion in \cref{thm:resonance}. \end{remark}
\begin{remark} We find the resonance criterion of \cref{thm:resonance} easier to check than the ``bad features'' criterion of \cref{thm:reduced}. \end{remark}
\begin{remark} Certain reduced plabic graphs were realized as tropical curves in \cite{kodwil}, where it was shown that the resonance property corresponds to the \emph{balancing condition} for tropical curves. \end{remark}
\begin{lemma} \label{lem:moves-resonance} The resonance property is preserved under the local moves {\rm(M1)--(M3)}, except when the decontraction move {\rm(M3)} creates a new leaf. \end{lemma}
\begin{proof} The square move~(M1) only changes the labels of the sides of the square, see \cref{fig:M1labels}. Moreover, the labels around each vertex match the labels around the opposite vertex after the square move, with the same cyclic order. Hence this move preserves the resonance property.
The case of the local move~(M2) is easy, cf.\ \cref{rem:resonance-1-2}.
For the case of the local move (M3), see \cref{fig:M3labels}. \end{proof}
\begin{figure}
\caption{Transformation of edge labels under a square move~(M1).
}
\label{fig:M1labels}
\end{figure}
\begin{figure}
\caption{Transformation of edge labels under a local move~(M3) at a 4-valent black vertex. (Alternatively, make all the vertices white.)}
\label{fig:M3labels}
\end{figure}
\pagebreak[3]
\begin{lemma} \label{lem:resonance} Any plabic graph obtained via the bridge decomposition construction (see \cref{def:BCFW})
has the resonance property. \end{lemma}
\begin{proof} We will show that, more concretely, the edge labels around trivalent vertices in such a plabic graph~$G$ follow one of the patterns described in \cref{fig:resonance}.
We will establish this result by induction on~$\beta$, the number of bridges, following the strategy used in the proof of \cref{thm:bridge}.
\begin{figure}
\caption{Edge labels near trivalent vertices in a bridge decomposition. At a white vertex, shown on the left, either $r<i<j$ or $i<j<r$. At a black vertex, shown on the right, either $s<i<j$ or $i<j<s$.}
\label{fig:resonance}
\end{figure}
Let $G$ be a bridge decomposition of $\widetilde{\pi}_{\aff}$, associated to the sequence of transpositions $\sigma_1,\dots, \sigma_\beta$. Thus ${\widetilde{\pi}}={\widetilde{\pi}}_G=\sigma_\beta \cdots \sigma_1$. Here $\sigma_1 = (i j)$, where $i$ and $j$ satisfy \eqref{eq:ij-pair-1}--\eqref{eq:ij-pair-4}. Let $G'$ be the bridge decomposition associated to $\sigma_2,\dots, \sigma_\beta$, so that $G$ is obtained from $G'$ by adding a single bridge in position $(i,j)$ at the top of~$G'$.
Suppose the result is true for $G'$. We need to verify it for $G$. Let $r = {\widetilde{\pi}}^{-1}(i)$ and $s={\widetilde{\pi}}^{-1}(j)$. Adding the bridge in position $(i,j)$ at the top of~$G'$ adds at most two trivalent vertices: it adds a white (respectively, black) trivalent vertex provided that $r\neq j$ (respectively, $s\neq i$). The cases when one of the vertices on the bridge is bivalent are easy to verify, so we are going to assume that $r\neq j$ and $s\neq i$. In this case, the local configuration around positions $i$ and $j$ in $G'$ and $G$ is as shown in \cref{fig:resonance2}.
\begin{figure}
\caption{The local configuration around positions $i$ and $j$
in $G'$ and $G$.}
\label{fig:resonance2}
\end{figure}
Recall that $i<j$ and moreover any $h$ such that $i<h<j$ is a fixed point of~${\widetilde{\pi}}$. For the reasons indicated in the proof of \cref{thm:bridge}, adding the bridge $(i,j)$ at the top of $G'$ has the effect of replacing the label $i$ (resp.,~$j$) by~$j$ (resp.,~$i$) in every edge label outside of the bridge.
If $G'$ has an edge with the label $ij$, then $G$ has a bad double crossing involving the trips originating at~$i$ and~$j$. This however is impossible since $G$ is reduced, by \cref{thm:bridge}. Therefore $G'$ has no edge with label~$ij$. Furthermore, $G'$ has no edge with a label $h$ for $i<h<j$. Since all trivalent vertices of $G'$ satisfy the resonance condition of \cref{fig:resonance}, the same remains true after switching the labels $i$ and~$j$. Thus, all trivalent vertices of $G$ that were present in $G'$ satisfy this resonance condition.
Finally, the two new trivalent vertices in $G$ satisfy this condition because $i<j$ and we can exclude $i<r<j$ and $i<s<j$ because of \eqref{eq:ij-pair-3}. \end{proof}
\begin{proof}[Proof of \cref{thm:resonance}] We first establish the ``if'' direction. Let $G$ be a plabic graph without internal leaves that are not lollipops. Suppose that $G$ has the resonance property. We want to show that $G$ is reduced.
Assume the contrary. By \cref{pr:reduced-collapse}, $G$ can be transformed by local moves that don't create internal leaves into a plabic graph~$G'$ containing a hollow digon. Since $G$ has the resonance property, so does~$G'$, by \cref{lem:moves-resonance}. This yields a contradiction because the labels around a hollow digon do not satisfy the resonance property, see \cref{fig:fail2}.
\begin{figure}
\caption{A plabic graph containing a hollow digon fails to satisfy the resonance property.}
\label{fig:fail2}
\end{figure}
We next establish the ``only if'' direction. Suppose that $G$ is reduced, with ${\widetilde{\pi}}_G={\widetilde{\pi}}$. We know from \cref{thm:bridge} that there is a bridge decomposition $G'$---a reduced plabic graph---with trip permutation~${\widetilde{\pi}}$. By \cref{lem:resonance}, $G'$ has the resonance property. By \cref{thm:moves}, $G\sim G'$.
Since $G$ has no internal leaves, there exists a trivalent plabic graph $G_3\sim G$, see \cref{lem:bitri}. Moreover it can be seen from the proof of this lemma that $G$ and $G_3$ are related via moves that do not create internal leaves, so by \cref{lem:moves-resonance} $G$ has the resonance property if and only if $G_3$ does. Similarly, by removing bivalent vertices from~$G'$, we obtain a trivalent graph $G'_3\sim G'$ that has the resonance property. Since $G_3$ and $G'_3$ are trivalent and move-equivalent to each other, they are related via a sequence of (M1) and (M4) moves, see \cref{thm:newmoves1}. By \cref{lem:moves-resonance}, these local moves preserve the resonance property, so $G_3$ has the resonance property and therefore $G$ has it as well. \end{proof}
\section{Face labels of reduced plabic graphs} \label{sec:face-labels}
In this section, we use the notion of a trip introduced in \cref{def:trip} to label each face of a reduced plabic graph by a collection of positive integers. These face labels generalize the labeling of diagonals in a polygon by Pl\"ucker coordinates (cf.\ Section~\ref{sec:Ptolemy}) as well as the labeling of faces in (double or ordinary) wiring diagrams by chamber minors (cf.\ Sections~\hbox{\ref{sec:baseaffine}--\ref{sec:matrices}}). In the following chapter, we will
relate the face labels of reduced plabic graphs to Pl\"ucker coordinates that form an extended cluster for the standard cluster structure on a Grassmannian or, more generally, on a Schubert or positroid subvariety within it.
\begin{remark}
Let $G$ be a reduced plabic graph. Let $T_i$ be the one-way trip in~$G$ that begins at a boundary vertex~$i$ and ends at a boundary vertex~$j$.
If $i\neq j$, then we claim that there are two kinds of faces in~$G$: those on the left side of the trip~$T_i$ and those on the right side of it. If $G$ is normal, then this claim follows from the fact (see \cref{thm:reduced}) that $G$ does not contain essential self-intersections. For a general reduced plabic graph, the claim can be deduced from the case of normal graphs using the procedure described in \cref{def:normalize}.
If $i=j$, then by \cref{prop:fixedlollipop}, the boundary vertex $i$ is the root of a tree that collapses to a lollipop. If this lollipop is white (resp., black), then we declare that all faces of $G$ lie on the left (resp., right) side of the trip~$T_i$. \end{remark}
\begin{definition} \label{def:faces} Let $G$ be a reduced plabic graph with boundary vertices $1,\dots,b$. We define two natural face labelings of $G$, cf.\ \cref{fig:plabic3}: \begin{itemize}[leftmargin=.2in] \item in the \emph{source labeling} $\mathcal{F}_{\operatorname{source}}(G)$, each face $f$ of $G$ is labeled by the set \[ I_{\operatorname{source}}(f)=\{i \mid \text{$f$ lies to the left of the trip starting at vertex $i$} \}; \] \item in the \emph{target labeling} $\mathcal{F}_{\operatorname{target}}(G)$, each face $f$ of $G$ is labeled by the set \[ I_{\operatorname{target}}(f)=\{i \mid
\text{$f$ is to the left of the trip ending at vertex~$i$}\}. \] \end{itemize} \end{definition}
\begin{figure}\label{fig:plabic3}
\end{figure}
\begin{remark} \label{rem:edge-vs-face} The edge labeling and the face labeling of a reduced plabic graph $G$ are related as follows: if two faces $f$ and $f'$ of $G$ are separated by a single edge whose edge label is $\{i,j\}$, then the face label of $f'$ is obtained from that of $f$ by either removing $i$ and adding $j$, or removing $j$ and adding~$i$. \end{remark}
\begin{theorem} \label{thm:faces} Let $G$ be a reduced plabic graph with $b$ boundary vertices. Let $a$ denote the number of anti-excedances in the trip permutation~$\pi_G$. Let us label the faces of $G$ using either the source or the target labeling. Then every face of $G$ will be labeled by an $a$-element subset of $\{1,\dots,b\}$. \end{theorem}
\begin{proof} By \cref{thm:resonance}, every reduced plabic graph $G$ has the resonance property, which in particular means that every edge label of $G$ consists of two distinct numbers. It then follows from \cref{rem:edge-vs-face} that every face label of $G$ has the same cardinality. It remains to show that this cardinality is~$a$, the number of anti-excedances of~${\widetilde{\pi}}={\widetilde{\pi}}_G$.
Furthermore, it is sufficient to establish the latter claim for one particular reduced plabic graph with the trip permutation~${\widetilde{\pi}}_G$, e.g., for a bridge decomposition of~$\widetilde{\pi}_{\aff}$. Indeed, any two reduced plabic graphs with the same trip permutation are related by local moves, and any such move preserves all labels except at most one, see \cref{ex:face-labels-square}.
To prove the theorem for bridge decompositions, we use induction on the number of bridges~$\beta$. In the base case $\beta=0$, the bridge decomposition~$G$ consists of $a$ white lollipops, $b-a$ black lollipops, and no bridges. Thus $G$ has a single face, labeled by the $a$-element subset indicating the positions of the white lollipops.
Consider a bridge decomposition $G$ built from a bridge decomposition $G'$ by adding a bridge in position $(i,j)$ at the top of~$G'$, as in \cref{fig:resonance2}. Both $G$ and $G'$ have trip permutations with $a$ anti-excedances (cf.\ \cref{rem:l(affpi)-bound}), so by the induction assumption, the faces in $G'$ have cardinality~$a$. Since $G$ inherits most of its faces from~$G'$, and all face labels of $G$ have the same cardinality, this cardinality is equal to~$a$. \end{proof}
\begin{exercise} \label{ex:face-labels-square}
Verify that applying a move {\rm (M2)} or {\rm (M3)} does not affect the face labels of a plabic graph, whereas applying the square move {\rm (M1)} changes the face labels as shown in \cref{fig:M1Plucker}.
\end{exercise}
\begin{figure}
\caption{The effect of the square move (M1) on the face labeling. Here $\mathbf{i,j,k,l}$ are the (source or target) labels of the trips that traverse the outer edges towards the central square; $S$~is an arbitrary set of labels disjoint from $\{i,j,k,l\}$; and $abS$ is a shorthand for the set $\{a,b\} \cup S$.}
\label{fig:M1Plucker}
\end{figure}
The face labelings of plabic graphs can be used to recover the labelings of diagonals in a polygon by Pl\"ucker coordinates as well as the labelings of chambers in (ordinary or double) wiring diagrams by minors:
\pagebreak[3]
\begin{exercise} Let $T$ be a triangulation of a convex $m$-gon $\mathbf{P}_m$, and let $G(T)$ be the plabic graph defined in \cref{TriangulationA}. Explain how to label the boundary vertices of $G(T)$ in such a way that the face labeling of $G(T)$ recovers the labeling of diagonals of $\mathbf{P}_m$ by pairs of integers. Cf.\ \cref{fig:triang-plabic}.
\end{exercise}
\begin{figure}
\caption{A triangulation $T$ of an octagon and the corresponding
plabic graph $G(T)$, \emph{cf.}\ Figure~\ref{fig:quiver-triangulation}. }
\label{fig:triang-plabic}
\end{figure}
\begin{exercise} Let $D$ be a wiring diagram with $m$ wires. Let $G(D)$ be the plabic graph defined in \cref{def:wiringplabic}, see also \cref{fig:wiring-plabic}. Label the boundary vertices of $G(D)$ by the numbers $1,\dots,2m$ in the clockwise order, starting with a $1$ at the lower left boundary vertex of $G(D)$. Label the faces of $G(D)$ using the source labeling $\mathcal{F}_{\operatorname{source}}(G)$. Show that intersecting each face label with the set $\{1,2,\dots,m\}$ recovers the labeling of $D$ by chamber minors. See \cref{fig:wiring-plabic}. \end{exercise}
\begin{figure}
\caption{A wiring diagram $D$ and the plabic graph $G(D)$ with the source labeling of its faces.}
\label{fig:wiring-plabic}
\end{figure}
\pagebreak
\begin{exercise} \label{exercise:DWD-source-labeling} Let $D$ be a double wiring diagram with $m$ pairs of wires. Let $G(D)$ be the plabic graph defined in \cref{ex:DWD}. Label the boundary vertices of $G(D)$ by the numbers \begin{equation*} 1,2,\dots,m-1,m,m',\dots,2',1' \end{equation*} in clockwise order, starting with the label $1$ at the lower left boundary vertex of~$G(D)$. Label the faces of $G=G(D)$ using the source labeling $\mathcal{F}_{\operatorname{source}}(G)$, so that each face gets labeled by $I'\cup J$, where $I'\subset \{1',\dots,m'\}$ and $J\subset \{1,\dots,m\}$. Let $I$ denote the set obtained from $I'$ by replacing each $i'$ by $i$. Show that mapping each face label $I'\cup J$ to the pair $([1,m]\setminus I), \mathbf{J})$ recovers the labeling of $D$ by chamber minors. See \cref{fig:better-0119}. \end{exercise}
\begin{figure}
\caption{Double wiring diagram labeling from plabic graphs. The~labeling of a double wiring diagram $D$ is obtained from the source labeling of the associated plabic graph $G(D)$
using the recipe described in \cref{exercise:DWD-source-labeling}.}
\label{fig:better-0119}
\end{figure}
\section{Grassmann necklaces and weakly separated collections}\label{weaksep}
Fix two nonnegative integers $b$ and $a\leq b$. We denote by $\binom{[b]}{a}$ the set of all $a$-element subsets of $\{1,\dots,b\}$.
In this section, we provide an intrinsic combinatorial characterization of the subsets of $\binom{[b]}{a}$ that arise as sets of face labels of reduced plabic graphs. The proofs are omitted.
\begin{definition}[{\rm \cite{leclerc-zelevinsky}}] We say that two $a$-element subsets $I,J\in\binom{[b]}{a}$ are \emph{weakly separated} if there do not exist $i, j, i', j'\in\{1,\dots,b\}$ such that \begin{itemize}[leftmargin=.2in] \item
$i<j<i'<j'$ or $j<i'<j'<i$ or $i'<j'<i<j$ or $j'<i<j<i'$; \item $i,i'\in I \setminus J$ and $j,j'\in J\setminus I$. \end{itemize} Put differently, $I$ and $J$ are weakly separated if and only if, after drawing the numbers $1,2,\dots,b$ clockwise around a circle, there exists a chord separating the sets $I\setminus J$ and $J\setminus I$ from each other. \end{definition}
\begin{theorem}[{\rm \cite{danilov-karzanov-koshevoy,oh-postnikov-speyer}}] \label{th:face-labels-weakly-sep} Let $I$ and $J$ be target face labels of two faces in a reduced plabic graph. Then $I$ and~$J$ are weakly separated. \end{theorem}
\begin{definition} \label{def:wekley-sep-collection} A collection $\mathcal{C} \subset \binom{[b]}{a}$ of $a$-element subsets of $[b]$ is called \emph{weakly separated} if any $I,J\!\in\!\mathcal{C}$ are weakly separated. Thus, \cref{th:face-labels-weakly-sep} asserts that the collection of target face labels of a reduced plabic graph is weakly separated.
A weakly separated collection $\mathcal{C}$ is called \emph{maximal} if it is not contained in any other weakly separated collection. See \cref{fig:plabic4}. \end{definition}
\begin{figure}\label{fig:plabic4}
\end{figure}
\begin{theorem}[{\rm \cite{danilov-karzanov-koshevoy,oh-postnikov-speyer}}] \label{thm:ops} For $\mathcal{C} \subset \binom{[b]}{a}$, the following are equivalent: \begin{itemize}[leftmargin=.2in] \item $\mathcal{C}$ is a maximal weakly separated collection; \item $\mathcal{C}$ is the set of target face labels of a reduced plabic graph~$G$ with ${\widetilde{\pi}}_G\!=\!{\widetilde{\pi}}_{a,b}$ (see \eqref{eq:dpi-ab}). \end{itemize}
In that case, the cardinality of $\mathcal{C}$ is equal to $|\mathcal{C}| = a(b-a)+1$. \end{theorem}
\begin{remark} A general formula for the number of maximal weakly separated collections in $\binom{[b]}{a}$ is unknown. For $a=2$, the maximal weakly separated collections in $\binom{[b]}{2}$ are in bijection with triangulations of a convex $b$-gon, so they are counted by the Catalan numbers~$C_{b-2}$, where $C_n = \frac{1}{n+1} \binom{2n}{n}$. For $a=3$ and $b=6,\dots, 12$, the number of maximal weakly separated collections in $\binom{[b]}{3}$ is equal to $34, 259, 2136, 18600, 168565, 1574298, 15051702$. See~\cite{Early} for more data. \end{remark}
\cref{thm:ops} can be generalized to arbitrary reduced plabic graphs. To state this result, we will need the following notion.
\begin{definition}[{\cite[Definition 16.1]{postnikov}}] A \emph{Grassmann necklace} of type $(a,b)$ is a sequence $\mathcal{I} = (I_1,\dots,I_b)$ of subsets $I_i\in \binom{[b]}{a}$ such that, for $i=1,\dots,b$, we have $I_{i+1}\supset I_i\setminus \{i\}$. (Here the indices are taken modulo $b$, so that $I_1\supset I_b\setminus \{b\}$.) Thus, if $i\notin I_i$, then $I_{i+1}=I_i$. \end{definition}
In other words, either $I_{i+1}=I_i$ or $I_{i+1}$ is obtained from $I_i$ by deleting $i$ and adding another element. Note that if $I_{i+1}=I_i$, then either $i$ belongs to all elements $I_j$ of the necklace, or $i$ belongs to none of them.
\begin{example} \label{ex:grassmann-necklace} The sequence $\mathcal{I}=(126, 236, 346, 456, 156, 126)$ is a Grassmann necklace of type $(3,6)$. \end{example}
\begin{definition} \label{def:ell-order} For $\ell\in\{1,\dots,b\}$, we define the linear order $<_{\ell}$ on $\{1,\dots,b\}$ as follows: \begin{equation*} \ell <_{\ell} \ell+1 <_{\ell} \ell+2 <_{\ell} \dots <_{\ell}b <_{\ell} 1 <_{\ell} \dots <_{\ell} \ell-1. \end{equation*}
For a decorated permutation ${\widetilde{\pi}}$ on $b$ letters, we say that $i\in \{1,\dots,b\}$ is an \emph{$\ell$-anti-excedance} of ${\widetilde{\pi}}$ if either ${\widetilde{\pi}}^{-1}(i) >_\ell i$ or if ${\widetilde{\pi}}(i)=\overline{i}$. Thus, a $1$-anti-excedance is the same as an (ordinary) anti-excedance, as in \cref{def:anti}. \end{definition}
It is not hard to see that the number of $\ell$-anti-excedances does not depend on the choice of $\ell\in \{1,\dots,b\}$, so we simply refer to this quantity as the number of anti-excedances.
\begin{lemma} \label{lem:decpermnecklace} Decorated permutations on $b$ letters with $a$ anti-excedances are in bijection with
Grassmann necklaces $\mathcal{I}$ of type $(a,b)$. \end{lemma} \begin{proof}
To go from $\mathcal{I}$ to the corresponding decorated permutation ${\widetilde{\pi}} = {\widetilde{\pi}}(\mathcal{I})$, we set ${\widetilde{\pi}}(i)=j$ whenever $I_{i+1}=(I_i\setminus \{i\}) \cup \{j\}$ for $i\neq j$. If $i\notin I_i=I_{i+1}$ then ${\widetilde{\pi}}(i) = \underline{i}$, and
if $i\in I_i=I_{i+1}$ then ${\widetilde{\pi}}(i) = \overline{i}$.
Going in the other direction, let ${\widetilde{\pi}}$ be a decorated permutation. For $\ell\in\{1,\dots,b\}$, we denote by $I_\ell$ the set of $\ell$-anti-excedances of~${\widetilde{\pi}}$. Then $\mathcal{I} =\mathcal{I}({\widetilde{\pi}})= (I_1,\dots, I_b)$ is the corresponding Grassmann necklace. \end{proof}
\begin{example} Let $\mathcal{I}\!=\!(126, 236, 346, 456, 156, 126)$, cf.\ \cref{ex:grassmann-necklace}. Then ${\widetilde{\pi}}(\mathcal{I}) = (3,4,5,1,2,\overline{6})$. \end{example}
\begin{example} \label{ex:pi-ab-necklace} Let ${\widetilde{\pi}}_G={\widetilde{\pi}}_{a,b}$, cf.~\eqref{eq:dpi-ab}. The corresponding Grassmann necklace (cf.\ \cref{lem:decpermnecklace}) is given by \begin{equation} \label{eq:I-dpi-ab} \mathcal{I}({\widetilde{\pi}}_{a,b})=(\{1,2,\dots,a\}, \{2,3,\dots,a,a+1\}, \dots, \{b,1,2,\dots, a-1\}). \end{equation} \end{example}
\begin{definition} \label{def:ell-partial-order} We extend the linear order $<_{\ell}$ on $\{1,\dots,b\}$ to a partial order on~$\binom{[b]}{a}$, as follows. Let \begin{align*} &I=\{i_1,\dots,i_a\}, \,\,\quad i_1<_{\ell} i_2 <_{\ell} \dots <_{\ell} i_a; \\ &J=\{j_1,\dots,j_a\}, \quad j_1<_{\ell} j_2 <_{\ell} \dots <_{\ell} j_a. \end{align*} Then, by definition, $I \leq_{\ell} J$ if and only if $i_1 \leq_{\ell} j_1,\dots, i_a \leq_{\ell} j_a$. \end{definition}
\begin{definition} \label{def:necklace-to-positroid}
For a Grassmann necklace $\mathcal{I}=(I_1,\dots,I_b)$ of type $(a,b)$, we define the associated \emph{positroid} $\mathcal{M}_{\mathcal{I}}$ by \begin{equation*} \mathcal{M}_{\mathcal{I}}=\{J\in \textstyle\binom{[b]}{a} \ \vert \ I_\ell \leq_{\ell} J \text{ for all }\ell \in \{1,\dots,b\}\}. \end{equation*} \end{definition}
As we will see in Chapter~\ref{ch:Grassmannians}, positroids are the (realizable) \emph{matroids} that arise from full rank $a\times b$ matrices with all Pl\"ucker coordinate nonnegative. Abstractly, one may also define a \emph{positively oriented matroid} to be an oriented matroid on $\{1,2,\dots,b\}$ whose chirotope takes nonnegative values on any ordered subset $\{i_1< \dots < i_a\}$. By \cite{Ardila-Rincon-Williams}, these two notions are the same, in other words, every positively oriented matroid is realizable.
\begin{example} \label{ex:positroid-ab} Let $\mathcal{I}=\mathcal{I}({\widetilde{\pi}}_{a,b})$, see~\eqref{eq:I-dpi-ab}. Then $\mathcal{M}_{\mathcal{I}}=\binom{[b]}{a}$, i.e., the positroid associated with~$\mathcal{I}$ contains all $a$-element subsets of $\{1,\dots,b\}$. \end{example}
\begin{definition} \label{def:strongly equivalent} Two reduced plabic graphs are called \emph{strongly equivalent} if they have the same sets of face labels.
We note that two plabic graphs which are connected via moves (M2) and (M3) are strongly equivalent. \end{definition}
Recall that $\mathcal{F}_{\operatorname{target}}(G)$ denotes the collection of target-labels of faces of a reduced plabic graph $G$.
\begin{theorem}[{\cite[Theorem~1.5]{oh-postnikov-speyer}}] \label{thm:ops2} Fix a decorated permutation ${\widetilde{\pi}}$ on $b$ letters with $a$ anti-excedances. Let~$\mathcal{I}$ be the corresponding Grassmann necklace of type $(a,b)$, cf.\ \cref{lem:decpermnecklace}. Let $\mathcal{M}_{\mathcal{I}}$ be the associated positroid, cf.\ \cref{def:necklace-to-positroid}. Then the map $G \mapsto \mathcal{F}_{\operatorname{target}}(G)$ gives a bijection~between \begin{itemize}[leftmargin=.2in] \item the strong equivalence classes of reduced plabic graphs $G$ with decorated trip permutation ${\widetilde{\pi}}_G = {\widetilde{\pi}}$ and \item the collections $\mathcal{C} \subset \binom{[b]}{a}$ that are maximal (with respect to inclusion) among the weakly separated collections satisfying $\mathcal{I} \subseteq \mathcal{C} \subseteq \mathcal{M}_{\mathcal{I}}$. \end{itemize} \end{theorem}
\begin{remark} Let $\mathcal{I}=\mathcal{I}({\widetilde{\pi}}_{a,b})$. Then $\mathcal{I}=\mathcal{I}({\widetilde{\pi}}_{a,b})$ is given by~\eqref{eq:I-dpi-ab}. Each of the $b$ cyclically consecutive subsets in \eqref{eq:I-dpi-ab} is weakly separated from every other $a$-element subset of $\{1,\dots,b\}$, so every maximal weakly separated collection $\mathcal{C} \subset \binom{[b]}{a}$ must contain~$\mathcal{I}$. Furthermore, $\mathcal{M}_{\mathcal{I}}=\binom{[b]}{a}$ (see \cref{ex:positroid-ab}), so any such~$\mathcal{C}$ automatically satisfies the inclusions $\mathcal{I} \subseteq \mathcal{C} \subseteq \mathcal{M}_{\mathcal{I}}$. We thus recover \cref{thm:ops} as a special case of \cref{thm:ops2}. \end{remark}
\backmatter
\label{sec:biblio}
\end{document} | arXiv |
Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$
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On existence and numerical approximation in phase-lag thermoelasticity with two temperatures
Marco Campo 1, , Maria I. M. Copetti 2, , José R. Fernández 3,, and Ramón Quintanilla 4,
Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain
Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222 Terrassa, Barcelona, Spain
* Corresponding author: José R. Fernández
Received August 2020 Revised March 2021 Early access April 2021
Fund Project: The work of M. Campo and J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of M.I.M. Copetti has been partially supported by the Brazilian institution CNPq (grant 304709/2017-4). The work of R. Quintanilla has been supported by Ministerio de Economía y Competitividad under the research project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE), and Ministerio de Ciencia, Innovación y Universidades under the research project "Análisis matemático aplicado a la termomecánica" (PID2019-105118GB-I00). The authors want to thank to the anonymous referees their useful comments which have allowed us to improve the paper
In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerical analysis, we restrict ourselves to the case of second-order approximations of the heat flux and first-order approximations for the inductive temperature. First, variational formulation of the corresponding problem is derived and an energy decay property is proved. Then, a fully discrete scheme is introduced by using the finite element method for the approximation of the spatial variable and the implicit Euler scheme for the discretization of the time derivatives. A discrete stability
Keywords: Phase-lag thermoelasticity, existence and uniqueness, finite elements, a priori estimates, numerical simulations.
Mathematics Subject Classification: Primary: 35Q74, 74H10, 80A20, 65M15, 65M60; Secondary: 74F05.
Citation: Marco Campo, Maria I. M. Copetti, José R. Fernández, Ramón Quintanilla. On existence and numerical approximation in phase-lag thermoelasticity with two temperatures. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021130
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Figure 1. Example 1: Asymptotic behavior of the numerical scheme
Figure 2. Example 1: Energy evolution in absolute and semilogarithmic scales
Figure 3. Example 2: Evolution in time of the temperature and inductive temperature at point $ {\boldsymbol{x}} = (4, 0.5) $ for different values of parameter $ m $
Figure 5. Example 2: Evolution in time of the horizontal and vertical displacements at point $ {\boldsymbol{x}} = (1, 0.5) $ for different values of parameter $ m $
Table 1. Example 1: Numerical errors ($ \times 100 $) for some $ nd $ and $ k $
$ n_{el} \downarrow k \to $ 0.02 0.01 0.005 0.001 0.0001
8 0.0986766 0.0985441 0.0993091 0.1004371 0.1018615
16 0.0368700 0.0313313 0.0303459 0.0308803 0.0315329
128 0.0245822 0.0124424 0.0063666 0.0016409 0.0009504
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Marco Campo Maria I. M. Copetti José R. Fernández Ramón Quintanilla | CommonCrawl |
\begin{document}
\maketitle
\section*{Introduction} There are 3 examples in these notes. The first one is the standard example of the cubic resolvent of a quartic. The second example is exactly from Adelmann \cite{Adelmann} and gives a defining polynomial corresponding to the unique $S_4$-quotient of $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$. The splitting field of the Adelmann polynomial over $\mathbb{Q}$ is a subfield of the 4-division field of an elliptic curve, that contains the 2-division field of the elliptic curve. The third example is new and needed in the study of the field theory of quaternion origami. Associated to an elliptic curve defined over $\mathbb{Q}$, with a rational point, is a degree 8 polynomial whose Galois group is a subgroup of $\mathrm{Hol}(Q_8)$. Three defining polynomials corresponding to the three $S_4$-quotients of $\mathrm{Hol}(Q_8)$ are given.
\section{Warm-up example: Resolvent cubic}
\subsection{Roots}
Suppose we are given a quartic polynomial $f(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0$ with roots $\alpha_1, \alpha_2, \alpha_3, \alpha_4$. Suppose also, that the Galois group of the splitting field of $f(x)$ is $S_4$. Group-theoretically, there is exactly 1 $S_3$-quotient of $S_4$. One can associate to $f(x)$, a resolvent cubic, i.e. a degree 3 polynomial $g(x)$ such that the splitting field of $g(x)$ is contained in the splitting field of $f(x)$ and the Galois group of $g(x)$ is $S_3$. There are formulas readily available for this polynomial, but to help systematize this process, we will follow the full procedure used in Adelmann \cite{Adelmann}. Let $G=S_4$, $H=V_4 \leftrightarrow \left\{ (1,4)(2,3), (1,3)(2,4)\right\}$, a normal subgroup, and $N=G/H$ as an abstract group isomorphic to $S_3$. We are looking for a subgroup $F$ of $G$ such that its projection $\overline{F}$ into $N$ has exactly 3 conjugates in $N$. We conclude that the normalizer of $\overline{F}$ in $N$ needs to have index 3. Since $N$ is isomorphic to $S_3$, the normalizer described above is isomorphic to $\mathbb{Z}/2\mathbb{Z}$. There are suitable $F$ having sizes 2, 4, and 8 respectively. In order for the resolvent polynomial to have degree 3, we require $|F|=8$. Since all of the 3 possible subgroups $F$ lead to the same resolvent polynomials, we may select any of these groups as our $F$, for example, take $F=\langle (3,4), (1,4)(2,3), (1,3)(2,4) \rangle $. A system of representatives for $G/F$ is given by $\left\{ (), (2,3), (2,4) \right\}$.
Finally, we need a polynomial $P \in k[x_1,x_2, x_3,x_4]$ having stabilizer subgroup $\mathrm{Stab}_G(P)=F$ which should be homogenous and of lowest possible degree. There is no suitable polynomial in degree 1, but in degree 2, we find $P=\overline{R}_F(x_1x_2)=4(x_1x_2+x_3x_4)$. Then, we can take $\beta_1=\alpha_1\alpha_2+\alpha_3\alpha_4$, $\beta_2=\alpha_1\alpha_3+\alpha_2\alpha_4$, and $\beta_3=\alpha_1\alpha_4+\alpha_2\alpha_3$.
\subsection{Vieta's formulas}
By using Vieta's formulas for relating the roots and the coefficients of a polynomial, we can find the coefficients of the resolvent cubic $g(x)=( x-\beta_1 )( x-\beta_2 )( x-\beta_3 )=x^3 + (-\alpha_1\alpha_2 - \alpha_1\alpha_3 - \alpha_1\alpha_4 - \alpha_2\alpha_3 - \alpha_2\alpha_4 - \alpha_3\alpha_4)x^2 + (\alpha_1^2\alpha_2\alpha_3 +
\alpha_1^2\alpha_2\alpha_4 + \alpha_1^2\alpha_3\alpha_4 + \alpha_1\alpha_2^2\alpha_3 + \alpha_1\alpha_2^2\alpha_4 + \alpha_1\alpha_2\alpha_3^2 + \alpha_1\alpha_2\alpha_4^2
+ \alpha_1\alpha_3^2\alpha_4 + \alpha_1\alpha_3\alpha_4^2 + \alpha_2^2\alpha_3\alpha_4 + \alpha_2\alpha_3^2\alpha_4 + \alpha_2\alpha_3\alpha_4^2)x -
\alpha_1^3\alpha_2\alpha_3\alpha_4 - \alpha_1^2\alpha_2^2\alpha_3^2 - \alpha_1^2\alpha_2^2\alpha_4^2 - \alpha_1^2\alpha_3^2\alpha_4^2 -
\alpha_1\alpha_2^3\alpha_3\alpha_4 - \alpha_1\alpha_2\alpha_3^3\alpha_4 - \alpha_1\alpha_2\alpha_3\alpha_4^3 - \alpha_2^2\alpha_3^2\alpha_4^2$.
For example, the $x^2$-coefficient of $fgx)$ is $-a_2$, the negative of the $x^2$-coefficient of $f$. The $x$-coefficient of $g$ is $a_1a_3 - 4a_0$. The constant coefficient of $g$ is $-a_1^2 + 4a_0a_2 - a_0a_3^2$. Therefore, the resolvent cubic is $g(x)=x^3-a_2x^2+(a_1a_3 - 4a_0)x+(-a_1^2 + 4a_0a_2 - a_0a_3^2)$. This agrees with a standard definition of a resolvent cubic polynomial of $f(x)$.
\section{Adelmann example} \subsection{Roots} This is verbatim from p. 104 of \cite{Adelmann}. Let $E: y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$, the field of rational numbers and consider the degree 6 polynomial $A(Y)=\frac{1}{2}\Lambda_4(Y)=Y^6+5aY^4+20bY^3-5a^2Y^2-4abY-a^3-8b^2$. The Galois group of the splitting field of this polynomial over $\mathbb{Q}$ is a subgroup of $\mathrm{PGL}_2(\mathbb{Z}/4\mathbb{Z})=\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})/\langle \pm I \rangle$. There is a unique $S_4$-quotient of this group. We would like to give the generic polynomial cutting out the splitting field of this quotient. Let $G=\mathrm{PGL}_2(\mathbb{Z}/4\mathbb{Z})=\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})/\langle \pm I \rangle$. Let $H = \left\{ \left(\begin{array}{cc}
1& 0\\
0& 1
\end{array} \right), \left(\begin{array}{cc}
1& 2 \\
2 & -1
\end{array} \right) \right\} \longleftrightarrow \left\{ (), (1,6)(2,4)(3,5) \right\}$. The subgroup $H$ of $G$ is a normal subgroup, the factor group $N=G/H$, as an abstract group, is isomorphic to $S_4$. Hence, the polynomial to be determined should have degree 4. Consequently, we are looking for a subgroup $F$ of $G$, so that its projection $\overline{F}$ into $N$ has exactly 4 conjugates within $N$. We conclude that the normalizer of $\overline{F}$ in $N$ needs to have index 4 in $N$. Since $N$ is isomorphic to $S_4$, the normalizer described is isomorphic to $S_3$, and we discover suitable $F$ having sizes 6 and 12, respectively. In order for the resolvent polynomial to have degree 4 we require $|F|=12$. Since all the 4 possible subgroups $F$ lead to the same resolvent polynomials, we may select any of these groups as our $F$, for example
$F= \left\{ \right.$
\begin{equation*} \begin{split} & \left(\begin{array}{cc}
1& 0\\
0& 1 \end{array} \right),
\left(\begin{array}{cc}
0& -1\\
1& 0 \end{array} \right), \left(\begin{array}{cc}
-1& 0\\
1& 1 \end{array} \right), \\
\noindent & \left(\begin{array}{cc}
-1 & 1 \\
0 & 1 \end{array} \right) \left(\begin{array}{cc}
1& 1\\
1& 0 \end{array} \right), \left(\begin{array}{cc}
0& 1\\
1& -1 \end{array} \right), \\
& \left(\begin{array}{cc}
1& 2\\
2& -1 \end{array} \right), \left(\begin{array}{cc}
2& 1\\
1& 2 \end{array} \right),
\left(\begin{array}{cc}
1& 2\\
1& -1 \end{array} \right), \\
& \left(\begin{array}{cc}
1& 1\\
2& -1 \end{array} \right), \left(\begin{array}{cc}
-1& 1\\
1& 2 \end{array} \right), \left(\begin{array}{cc}
2& 1\\
1& 1 \end{array} \right) \end{split} \end{equation*} $\left. \right\}$
$F$ is generated by
$$ \left\{ \left(\begin{array}{cc}
1& 1\\
1& 0 \end{array} \right), \left(\begin{array}{cc}
0& -1\\
1& 0 \end{array} \right) \right\} \longleftrightarrow \left\{ (1,2,5,6,4,3), (1,4)(2,6) \right\}. $$
A system of representatives of $G/F$ is given by
$$ \left\{ \left(\begin{array}{cc}
1& 0\\
0& 1 \end{array} \right), \left(\begin{array}{cc}
1& 0\\
2& -1 \end{array} \right), \left(\begin{array}{cc}
1& 2\\
0& -1 \end{array} \right), \left(\begin{array}{cc}
1& 0\\
0& -1 \end{array} \right) \right\} \longleftrightarrow \left\{ (), (1,6), (2,4), (3,5) \right\}. $$
Finally, we need a polynomial $P \in k[x_1, \ldots, x_6]$ having stabilizer subgroup $\mathrm{Stab}_G(P)=F$ which should be homogeneous and of lowest possible degree. There is no suitable polynomial in degree 1, but in degree 2 we find $P=\overline{R}_F(x_1x_2)$.
$P=\overline{R}_F(x_1x_2)=x_1x_2+x_2x_5+x_5x_6+x_4x_6+x_3x_4+x_1x_3+x_4x_6+x_5x_6+x_2x_5+x_1x_2+x_1x_3+x_3x_4=2(x_1x_2+x_2x_5+x_5x_6+x_4x_6+x_3x_4+x_1x_3)$.
Take $\beta_1=x_1x_2+x_2x_5+x_5x_6+x_4x_6+x_3x_4+x_1x_3$, $\beta_2=x_2x_6+x_2x_5+x_1x_5+x_1x_4+x_3x_4+x_3x_6$, $\beta_3=x_1x_4+x_4x_5+x_5x_6+x_2x_6+x_2x_3+x_1x_3$, $\beta_4=x_1x_2+x_2x_3+x_3x_6+x_4x_6+x_4x_5+x_1x_5$.
\subsection{Vieta's formulas} By using Vieta's formulas relating the roots and the coefficients of a polynomial, we can find the coefficients of the resolvent quartic $g(x)=( x-\beta_1 )( x-\beta_2 )( x-\beta_3 )(x-\beta_4)$. This time, the formulas involve a little more than the symmetric polynomials, so there is a bit of additional work to do. Let $\omega_f$ be the evaluation homomorphism from $k[x_1, \ldots, x_6] \rightarrow k_f$ given by $x_i \mapsto \alpha_i$ where the $\alpha_i$s are the roots of $f$ (In this case. $f=A(Y)=\frac{1}{2}\Lambda_4(Y)$.
We know that $\omega_f(\overline{R}_G(x_2x_4))$ and $\omega_f(\overline{R}_G(x_1x_2))$ sum to $5a$ since $\overline{R}_G(x_2x_4)$ and $\overline{R}_G(x_1x_2)$ sum to an elementary symmetric polynomial.
Each of these elements is a fundamental invariant and so by Proposition 6.3.3 of Adelmann, \cite{Adelmann}, the coefficients of the resolvent polynomials $\mathcal{RP}_{S_6}(u)(Y)$ are symmetric polynomials in the $x_i$s and can therefore be expressed by elementary symmetric polynomials. By proposition 6.3.8 of Adelmann, the polynomial $\mathcal{R}_{S_6,u}(Y)$ is a polynomial whose coefficients are polynomials in the coefficients of the defining polynomial of $K$ and $\omega_f(u)$ appears as a simple linear factor. Adelmann gives $\mathcal{R}_{u_2}(Y)$, $\mathcal{R}_{u_3}(Y)$, $\mathcal{R}_{u_4}(Y)$ and each has a simple linear factor with root $\omega_f(u)$. The image of $\omega_f$ on all of the fundamental invariants can be determined by combining this information along with the images under $\omega_f$ of the elementary symmetric polynomials. As on p. 102 of Adelmann, \cite{Adelmann}, the result is the following:
\renewcommand\arraystretch{2.0} \begin{center}
\begin{tabular}{| c | c| }
\hline
Fundamental Invariant & Image under $\omega_f$ \\ \hline
$u_1=\overline{R}_G(x_1)=6R_G(x_1)$ & $0$ \\
$u_2=\overline{R}_G(x_2x_4)=3R_G(x_2x_4)$ & $a$ \\
$v_2=\overline{R}_G(x_1x_2)=12R_G(x_1x_2)$ & $4a$ \\
$u_3=\overline{R}_G(x_1x_2x_3)=8R_G(x_1x_2x_3)$ & $-8b$ \\
$u_4=\overline{R}_G(x_2x_2x_3x_4x_5)=3R_G(x_2x_3x_4x_5)$ & $-a^2$ \\
$u_6=\overline{R}_G(x_1x_2x_2x_3x_4x_5x_6)=R_G(x_1x_2x_3x_4x_5x_6)$ & $-(a^3+8b^2)$ \\
$w_3=\overline{R}_G(x_1x_2x_4)=12R_G(x_1x_2x_4)$ & $-12b$ \\
$w_4=\overline{R}_G(x_1x_2x_3x_4)=12R_G(x_1x_2x_3x_4)$ & $-4a^2$ \\
$w_5=\overline{R}_G(x_1x_2x_3x_4x_5)=6R_G(x_1x_2x_3x_4x_5)$ & $4ab$ \\ \hline
\end{tabular} \end{center}
\noindent Using these values gives that $\mathcal{R}_{G,P}(Y)=Y^4-8aY^3+24a^2Y^2+(224a^3+1728b^2)Y+272a^4+1728ab^2$ (See p. 105 of \cite{Adelmann}.). After performing the linear substitution $Y \mapsto Y+2a$ and introducing the quantity $\Delta=-16(4a^3+27b^2)$ (the discriminant of the underlying elliptic curve) we obtain the defining polynomial of the extension:
$$B(Y)=Y^4-4\Delta Y-12a\Delta.$$
\noindent Adelmann also gives the defining polynomials for the division field $\mathbb{Q}(E[n])/\mathbb{Q}$ using resultants. Adelmann \cite{Adelmann} provides these polynomials as follows:
$$T_n(X)=\mathrm{Res}_Y(\Gamma_n(Y), X^2-(Y^3-aY-b))$$
\noindent where $$\Gamma_n(X)=\displaystyle \prod_{d|n} A_d(X,Y)^{\mu(n/d)}$$ {\setlength{\abovedisplayskip}{0pt}
\begin{flalign*} A_1&=1\\
A_2&=2y\\
A_3&=3x^4+6ax^2+12bx-a^2\\
A_4&=4y(x^6+5ax^4+20bx^3-5a^2x^2-4abx-8b^2-a^3)\\
2yA_{2m}&=A_m(A_{m+2}A_{m-1}^2-A_{m-2}A_{m+1}^2)\\
A_{2m+1}&=A_{m+2}A_m^3-A_{m-1}A_{m+1}^3
\end{flalign*}
\noindent Example: $n=4$ $$T_4(X)=\mathrm{Res}_Y(\Gamma_4(Y),X^2-(Y^3-aY-b))$$ \noindent where $\Gamma_4(X) = A_2(X,Y)^{-1}A_4(X,Y)=2(X^6+5aX^4+20bX^3-5a^2X^2-4abX-8b^2-a^3)$.
\noindent Then \begin{gather*} \begin{split} T_4(X)=X^{12} + 54bX^{10} + (132a^3 + 891b^2)X^8 + (432a^3b + 2916b^3)X^6 +\\
(-528a^6 - 7128a^3b^2 - 24057b^4)X^4 + (864a^6b + 11664a^3b^3 +
39366b^5)X^2\\
- 64a^9 - 1296a^6b^2 - 8748a^3b^4 - 19683b^6
\end{split}
\end{gather*}
\section{$\mathrm{Hol}(Q_8)$ has 3 $S_4$-quotients }
\subsection{Roots}
Let $E: y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$ with $P=(z,w) \in E(\mathbb{Q})$ and consider the degree 8 polynomial $f(x)=f_{E,P, Q_8}(x)=x^8-8wx^6+6(2az+3b)x^4-(4a^3+27b^2)$. The Galois group of the splitting field of this polynomial over $\mathbb{Q}$ is a subgroup of $\mathrm{Hol}(Q_8)$ \cite{DavisGoins}. There are 3 distinct $S_4$-quotients of this group. We would like to give the generic polynomials cutting out the splitting fields corresponding to each of these quotients. Let $G=\mathrm{Hol}(Q_8)$.
Then, $G$ is $\mathrm{TransitiveGroup}(8,40)$, as a permutation group. Write $G$ as \\
$\mathrm{sub}\langle S_8 | (1,3,4,8,7,5)(2,6), (2,6)(3,7), (2,4,7)(3,6,5), (3,7)(4,5), (1,7,4,2,8,3,5,6) \rangle$.
Then, consider the following 3 subgroups of $G$:
$H_1=\langle (1,2,8,6)(3,5,7,4), (1,7,8,3)(2,5,6,4), (1,8)(2,6)(3,7)(4,5) \rangle$, and \\
$H_2 = \langle (1,4,8,5)(2,3,6,7), (1,2,8,6)(3,4,7,5), (1,8)(2,6)(3,7)(4,5)\rangle$, and \\
$H_3= \langle (2,6)(4,5), (2,6)(3,7), (1,8)(2,6)(3,7)(4,5) \rangle$. \\
Then, for $1 \leq i \leq 3$, $G/H_i \simeq S_4$. Here, and $H_1 \simeq H_2 \simeq Q_8$ and $H_3 \simeq (\ZZ/2\ZZ)^3$.
Let $N_i=G/H_i$. Then, $N_i=S_4$ for $i=1,2,3$. We are looking for degree 48 subgroups $F_i$ such that the normalizers of $\overline{F}_i$ in $N_i$ have exactly 4 conjugates. We will take
$F_1=\langle (2,4)(3,7)(5,6), (2,3,5)(4,6,7), (1,2,8,6)(3,5,7,4),(1,7,8,3)(2,5,6,4), (1,8)(2,6)(3,7)(4,5)
\rangle$
$F_2 = \langle (2,5)(3,7)(4,6), (2,7,4)(3,5,6), (1,4,8,5)(2,3,6,7), (1,2,8,6)(3,4,7,5), (1,8)(2,6)(3,7)(4,5) \rangle$,
$F_3=\langle(2,4,6,5), (2,7,4)(3,5,6), (2,6)(4,5), (2,6)(3,7), (1,8)(2,6)(3,7)(4,5) \rangle$
One can take representatives for the cosets $G/F_i$ as follows (note that there are many different ways to choose a set of transversals for the $F_i$s):
$$ \langle (), (3, 7)(4, 5), (1, 2, 5, 3, 8, 6, 4, 7),
(1, 3, 5, 8, 7, 4)(2, 6) \rangle, $$
$$ \langle (), (1,3,4,8,7,5)(2,6), (2,6)(3,7), (1,5,7,8,4,3)(2,6) \rangle, $$ and
$$ \langle (), (1, 3, 4, 8, 7, 5)(2, 6), (1, 5, 7, 8, 4, 3)(2, 6),(1, 6, 5, 3, 8, 2, 4, 7) \rangle $$
respectively.
Finally, we need polynomials $P_i \in k[x_1, \ldots, x_8]$ having stabilizer subgroup $\mathrm{Stab}_G(P_i)=F_i$ which should be homogeneous and of lowest possible degree. We find $P_3=x_1x_8$ by studying the primary invariants of $F_3$. Then, we can take $\gamma_1=x_1x_8$, $\gamma_2=x_2x_6$, $\gamma_3=x_3x_7$, $\gamma_4=x_4x_5$.
For $P_1$ and $P_2$, the primary invariants of $F_1$ and $F_2$ suggest using
$P_1=R_{F_1}(x_1x_2x_3)=x_1x_2x_3 + x_1x_2x_4 + x_1x_2x_5 + x_1x_3x_5 + x_1x_3x_6 + x_1x_4x_6 + x_1x_4x_7 + x_1x_5x_7 + x_1x_6x_7 + x_2x_3x_4 + x_2x_3x_8 +
x_2x_4x_7 + x_2x_5x_7 + x_2x_5x_8 + x_2x_7x_8 + x_3x_4x_6 + x_3x_4x_8 + x_3x_5x_6 + x_3x_5x_8 + x_4x_6x_8 + x_4x_7x_8 + x_5x_6x_7 + x_5x_6x_8 + x_6x_7x_8$ and
$P_2=R_{F_2}(x_1x_2x_3)=x_1x_2x_3 + x_1x_2x_4 + x_1x_2x_7 + x_1x_3x_5 + x_1x_3x_6 + x_1x_4x_6 + x_1x_4x_7 + x_1x_5x_6 + x_1x_5x_7 + x_2x_3x_4 + x_2x_3x_5
+ x_2x_4x_8 + x_2x_5x_7 + x_2x_5x_8 + x_2x_7x_8 + x_3x_4x_6 + x_3x_4x_8 + x_3x_5x_8 + x_3x_6x_8 + x_4x_6x_7 + x_4x_7x_8 +
x_5x_6x_7 + x_5x_6x_8 + x_6x_7x_8$.
Let $\alpha_i$ be the $G$-conjugates of $P_1$ with each $x_i$s evaluated with the $r_i$s (the roots of $f(x)$. \\
$\alpha_1=r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_5 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_6 + r_1r_4r_7 + r_1r_5r_7 + r_1r_6r_7 + r_2r_3r_4 + r_2r_3r_8 + r_2r_4r_7 + r_2r_5r_7 + r_2r_5r_8 + r_2r_7r_8 + r_3r_4r_6 + r_3r_4r_8 + r_3r_5r_6 + r_3r_5r_8 + r_4r_6r_8 + r_4r_7r_8+ r_5r_6r_7 + r_5r_6r_8 + r_6r_7r_8$,\\
$\alpha_2= r_1r_2r_4 + r_1r_2r_5 + r_1r_2r_7 + r_1r_3r_4 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_7 + r_1r_5r_6 + r_1r_6r_7 + r_2r_3r_4 + r_2r_3r_5 + r_2r_3r_8 + r_2r_4r_8 + r_2r_5r_7 + r_2r_7r_8 + r_3r_4r_6 + r_3r_5r_8 + r_3r_6r_8 + r_4r_6r_7 + r_4r_6r_8 + r_4r_7r_8 + r_5r_6r_7 + r_5r_6r_8 + r_6r_7r_8$, \\ $
\alpha_3= r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_7 + r_1r_3r_4 + r_1r_3r_5 + r_1r_4r_6 + r_1r_5r_6 + r_1r_5r_7 + r_1r_6r_7 + r_2r_3r_5 + r_2r_3r_8 + r_2r_4r_7 + r_2r_4r_8 + r_2r_5r_7 + r_2r_5r_8 + r_3r_4r_6 + r_3r_4r_8 + r_3r_5r_6 + r_3r_6r_8 + r_4r_6r_7 + r_4r_6r_8 + r_5r_6r_7 + r_5r_7r_8 + r_6r_7r_8$,\\ $\alpha_4= r_1r_2r_3 + r_1r_2r_5 + r_1r_2r_7 + r_1r_3r_4 + r_1r_3r_6 + r_1r_4r_6 + r_1r_4r_7 + r_1r_5r_6 + r_1r_5r_7 + r_2r_3r_4 + r_2r_3r_5 + r_2r_4r_7 + r_2r_4r_8 + r_2r_5r_8 + r_2r_7r_8 +r_3r_4r_8 + r_3r_5r_6 + r_3r_5r_8 + r_3r_6r_8 + r_4r_6r_7 + r_4r_6r_8 +r_5r_6r_7 + r_5r_7r_8 + r_6r_7r_8$.\\
Let $\beta_i$ be the $G$-conjugates of $P_2$ with each $x_i$s evaluated with the $r_i$s (the roots of $f(x)$). \\
$\beta_1=r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_7 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_6 + r_1r_4r_7 + r_1r_5r_6 + r_1r_5r_7 + r_2r_3r_4 + r_2r_3r_5 + r_2r_4r_8 + r_2r_5r_7 + r_2r_5r_8 + r_2r_7r_8 + r_3r_4r_6 + r_3r_4r_8 + r_3r_5r_8 + r_3r_6r_8 + r_4r_6r_7 + r_4r_7r_8 + r_5r_6r_7 + r_5r_6r_8 + r_6r_7r_8$,\\
$\beta_2= r_1r_2r_3 + r_1r_2r_5 + r_1r_2r_7 + r_1r_3r_4 + r_1r_3r_5 + r_1r_4r_6 + r_1r_4r_7 + r_1r_5r_6 + r_1r_6r_7 + r_2r_3r_4 + r_2r_3r_8 + r_2r_4r_7 + r_2r_4r_8 + r_2r_5r_7 + r_2r_5r_8 + r_3r_4r_6 + r_3r_5r_6 + r_3r_5r_8 + r_3r_6r_8 + r_4r_6r_8 + r_4r_7r_8 +
r_5r_6r_7 + r_5r_7r_8 + r_6r_7r_8$, \\
$\beta_3= r_1r_2r_4 + r_1r_2r_5 + r_1r_2r_7 + r_1r_3r_4 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_6 + r_1r_5r_7 + r_1r_6r_7 + r_2r_3r_4 + r_2r_3r_5 + r_2r_3r_8 + r_2r_4r_7 + r_2r_5r_8 + r_2r_7r_8 + r_3r_4r_8 + r_3r_5r_6 + r_3r_6r_8 + r_4r_6r_7 + r_4r_6r_8 + r_4r_7r_8 + r_5r_6r_7 + r_5r_6r_8 + r_5r_7r_8$, \\
$\beta_4= r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_5 + r_1r_3r_4 + r_1r_3r_6 + r_1r_4r_7 + r_1r_5r_6 + r_1r_5r_7 + r_1r_6r_7 + r_2r_3r_5 + r_2r_3r_8 + r_2r_4r_7 + r_2r_4r_8 + r_2r_5r_7 + r_2r_7r_8 + r_3r_4r_6 + r_3r_4r_8 + r_3r_5r_6 + r_3r_5r_8 + r_4r_6r_7 + r_4r_6r_8 + p1
r_5r_6r_8 + r_5r_7r_8 + r_6r_7r_8$.
The problem with these roots is that they are zero. For example, $\alpha_1=(x_1+x_8)(x_2x_3+x_2x_5+x_4x_7)+(x_2+x_6)(x_1x_4+x_5+x_7+x_7x_8)+(x_3+x_7)(x_1x_5+x_1x_6+x_2x_4)+(x_4+x_5)(x_3x_6+x_3x_8+x_6x_8)$ and since $x_1+x_8=0$, $x_2+x_6=0$, $x_3+x_7=0$, and $x_4+x_5=0$, $\alpha_1=0$ and similarly, $\alpha_2=0$, $\alpha_3=0$, $\alpha_4$. Similarly, the $\beta_i$ corresponding to the conjugates of $P_2$ evaluated at the roots of $f(x)$ are also zero.
Instead, we will use the following secondary invariants of $F_2$ and $F_3$, respectively (\cite{magma}) to construct roots:
$P_1=x_1^2x_2x_7 + x_1^2x_3x_4 + x_1^2x_5x_6 + x_1x_2^2x_7 + x_1x_2x_7^2 + x_1x_3^2x_4 + x_1x_3x_4^2 + x_1x_5^2x_6 +
x_1x_5x_6^2 + x_2^2x_3x_5 + x_2^2x_4x_8 + x_2x_3^2x_5 + x_2x_3x_5^2 + x_2x_4^2x_8 + x_2x_4x_8^2 + x_3^2x_6x_8 +
x_3x_6^2x_8 + x_3x_6x_8^2 + x_4^2x_6x_7 + x_4x_6^2x_7 + x_4x_6x_7^2 + x_5^2x_7x_8 + x_5x_7^2x_8 + x_5x_7x_8^2$
and
$P_2=x_1^2x_2x_3 + x_1^2x_4x_6 + x_1^2x_5x_7 + x_1x_2^2x_7 + x_1x_2x_4^2 + x_1x_3^2x_6 + x_1x_3x_5^2 + x_1x_4x_7^2 +
x_1x_5x_6^2 + x_2^2x_3x_5 + x_2^2x_4x_8 + x_2x_3^2x_4 + x_2x_5^2x_7 + x_2x_5x_8^2 + x_2x_7^2x_8 + x_3^2x_5x_8 +
x_3x_4^2x_6 + x_3x_4x_8^2 + x_3x_6^2x_8 + x_4^2x_7x_8 + x_4x_6^2x_7 + x_5^2x_6x_8 + x_5x_6x_7^2 + x_6x_7x_8^2$
Then, the conjugates of these elements evaluated at the roots of $f(x)$ are the following:
$\alpha_1=r_1^2r_2r_7 + r_1^2r_3r_4 + r_1^2r_5r_6 + r_1r_2^2r_7 + r_1r_2r_7^2 + r_1r_3^2r_4
+ r_1r_3r_4^2 + r_1r_5^2r_6 + r_1r_5r_6^2 + r_2^2r_3r_5 + r_2^2r_4r_8 +
r_2r_3^2r_5 + r_2r_3r_5^2 + r_2r_4^2r_8 + r_2r_4r_8^2 + r_3^2r_6r_8 +
r_3r_6^2r_8 + r_3r_6r_8^2 + r_4^2r_6r_7 + r_4r_6^2r_7 + r_4r_6r_7^2 +
r_5^2r_7r_8 + r_5r_7^2r_8 + r_5r_7r_8^2$,
$\alpha_2=r_1^2r_2r_3 + r_1^2r_4r_6 + r_1^2r_5r_7 + r_1r_2^2r_3 + r_1r_2r_3^2 + r_1r_4^2r_6
+ r_1r_4r_6^2 + r_1r_5^2r_7 + r_1r_5r_7^2 + r_2^2r_4r_7 + r_2^2r_5r_8 +
r_2r_4^2r_7 + r_2r_4r_7^2 + r_2r_5^2r_8 + r_2r_5r_8^2 + r_3^2r_4r_8 +
r_3^2r_5r_6 + r_3r_4^2r_8 + r_3r_4r_8^2 + r_3r_5^2r_6 + r_3r_5r_6^2 +
r_6^2r_7r_8 + r_6r_7^2r_8 + r_6r_7r_8^2$,
$\alpha_3=r_1^2r_2r_5 + r_1^2r_3r_6 + r_1^2r_4r_7 + r_1r_2^2r_5 + r_1r_2r_5^2 + r_1r_3^2r_6
+ r_1r_3r_6^2 + r_1r_4^2r_7 + r_1r_4r_7^2 + r_2^2r_3r_4 + r_2^2r_7r_8 +
r_2r_3^2r_4 + r_2r_3r_4^2 + r_2r_7^2r_8 + r_2r_7r_8^2 + r_3^2r_5r_8 +
r_3r_5^2r_8 + r_3r_5r_8^2 + r_4^2r_6r_8 + r_4r_6^2r_8 + r_4r_6r_8^2 +
r_5^2r_6r_7 + r_5r_6^2r_7 + r_5r_6r_7^2$,
$\alpha_4=r_1^2r_2r_4 + r_1^2r_3r_5 + r_1^2r_6r_7 + r_1r_2^2r_4 + r_1r_2r_4^2 + r_1r_3^2r_5
+ r_1r_3r_5^2 + r_1r_6^2r_7 + r_1r_6r_7^2 + r_2^2r_3r_8 + r_2^2r_5r_7 +
r_2r_3^2r_8 + r_2r_3r_8^2 + r_2r_5^2r_7 + r_2r_5r_7^2 + r_3^2r_4r_6 +
r_3r_4^2r_6 + r_3r_4r_6^2 + r_4^2r_7r_8 + r_4r_7^2r_8 + r_4r_7r_8^2 +
r_5^2r_6r_8 + r_5r_6^2r_8 + r_5r_6r_8^2$
and
$\beta_1=r_1^2r_2r_3 + r_1^2r_4r_6 + r_1^2r_5r_7 + r_1r_2^2r_7 + r_1r_2r_4^2 + r_1r_3^2r_6
+ r_1r_3r_5^2 + r_1r_4r_7^2 + r_1r_5r_6^2 + r_2^2r_3r_5 + r_2^2r_4r_8 +
r_2r_3^2r_4 + r_2r_5^2r_7 + r_2r_5r_8^2 + r_2r_7^2r_8 + r_3^2r_5r_8 +
r_3r_4^2r_6 + r_3r_4r_8^2 + r_3r_6^2r_8 + r_4^2r_7r_8 + r_4r_6^2r_7 +
r_5^2r_6r_8 + r_5r_6r_7^2 + r_6r_7r_8^2$,
$\beta_2=r_1^2r_2r_7 + r_1^2r_3r_4 + r_1^2r_5r_6 + r_1r_2^2r_3 + r_1r_2r_5^2 + r_1r_3^2r_5
+ r_1r_4^2r_7 + r_1r_4r_6^2 + r_1r_6r_7^2 + r_2^2r_4r_7 + r_2^2r_5r_8 +
r_2r_3^2r_8 + r_2r_3r_4^2 + r_2r_4r_8^2 + r_2r_5r_7^2 + r_3^2r_4r_6 +
r_3r_5^2r_8 + r_3r_5r_6^2 + r_3r_6r_8^2 + r_4^2r_6r_8 + r_4r_7^2r_8 +
r_5^2r_6r_7 + r_5r_7r_8^2 + r_6^2r_7r_8$,
$\beta_3=r_1^2r_2r_4 + r_1^2r_3r_5 + r_1^2r_6r_7 + r_1r_2^2r_5 + r_1r_2r_7^2 + r_1r_3^2r_4
+ r_1r_3r_6^2 + r_1r_4^2r_6 + r_1r_5^2r_7 + r_2^2r_3r_4 + r_2^2r_7r_8 +
r_2r_3^2r_5 + r_2r_3r_8^2 + r_2r_4^2r_7 + r_2r_5^2r_8 + r_3^2r_6r_8 +
r_3r_4^2r_8 + r_3r_5^2r_6 + r_4r_6^2r_8 + r_4r_6r_7^2 + r_4r_7r_8^2 +
r_5r_6^2r_7 + r_5r_6r_8^2 + r_5r_7^2r_8$,
$\beta_4=r_1^2r_2r_5 + r_1^2r_3r_6 + r_1^2r_4r_7 + r_1r_2^2r_4 + r_1r_2r_3^2 + r_1r_3r_4^2
+ r_1r_5^2r_6 + r_1r_5r_7^2 + r_1r_6^2r_7 + r_2^2r_3r_8 + r_2^2r_5r_7 +
r_2r_3r_5^2 + r_2r_4^2r_8 + r_2r_4r_7^2 + r_2r_7r_8^2 + r_3^2r_4r_8 +
r_3^2r_5r_6 + r_3r_4r_6^2 + r_3r_5r_8^2 + r_4^2r_6r_7 + r_4r_6r_8^2 +
r_5^2r_7r_8 + r_5r_6^2r_8 + r_6r_7^2r_8$
\subsection{Vieta's formulas} Let $d=4a^3+27b^2$.
By using Vieta's formulas relating the roots and the coefficients of a polynomial, we can find the coefficients of the resolvent quartics $h_1(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4)$, $h_2=(x-\beta_1)(x-\beta_2)(x-\beta_3)(x-\beta_4)$, and $h_3=(x-\gamma_1)(x-\gamma_2)(x-\gamma_3)(x-\gamma_4)$. Again, the formulas involve slightly more than the symmetric polynomials, so there is a bit of additional work to do. Let $\omega_f$ be the evaluation homomorphism from $k[x_1, \ldots, x_8] \rightarrow k_f$ given by $x_i \mapsto r_i$ where the $r_i$s are the roots of $f(x)$.
Let $u_{35}= \displaystyle \sum_{\sigma \in G} \sigma(x_1^7x_2^5x_3^3x_5)$. Let $v_{35}=-1/8 u_{35} \displaystyle|_{\langle x_1,x_2,x_3,x_4,-x_4,-x_2,-x_3,-x_1 \rangle}=x_1^7x_2^5x_3^3x_4 - x_1^7x_2^5x_3x_4^3 - x_1^7x_2^3x_3^5x_4 + x_1^7x_2^3x_3x_4^5 + x_1^7x_2x_3^5x_4^3 -
x_1^7x_2x_3^3x_4^5 - x_1^5x_2^7x_3^3x_4 + x_1^5x_2^7x_3x_4^3 + x_1^5x_2^3x_3^7x_4 - x_1^5x_2^3x_3x_4^7 -
x_1^5x_2x_3^7x_4^3 + x_1^5x_2x_3^3x_4^7 + x_1^3x_2^7x_3^5x_4 - x_1^3x_2^7x_3x_4^5 - x_1^3x_2^5x_3^7x_4 +
x_1^3x_2^5x_3x_4^7 + x_1^3x_2x_3^7x_4^5 - x_1^3x_2x_3^5x_4^7 - x_1x_2^7x_3^5x_4^3 + x_1x_2^7x_3^3x_4^5 +
x_1x_2^5x_3^7x_4^3 - x_1x_2^5x_3^3x_4^7 - x_1x_2^3x_3^7x_4^5 + x_1x_2^3x_3^5x_4^7$.
This element is a fundamental invariant and so by Proposition 6.3.3 of Adelmann, \cite{Adelmann}, the coefficients of the resolvent polynomials $\mathcal{RP}_{S_8}(u_{35})(Y)$ are symmetric polynomials in the $x_i$s and can therefore be expressed by elementary symmetric polynomials. By proposition 6.3.8 of Adelmann, the polynomial $\mathcal{R}_{S_8,u_{35}}(Y)$ is a polynomial whose coefficients are polynomials in the coefficients of the defining polynomial and $\omega_f(u_{35})$ appears as a simple linear factor. This polynomial has degree 210 and it too computationally time-consuming. Instead, we can study $\mathcal{RP}_{S_4}(v_{35})(Y)$. This polynomial has degree 2.
Then,
$$v_{35} = \displaystyle \prod_{i=1}^4 x_i \displaystyle \prod_{i \ne j} (x_i-x_j) \displaystyle \prod_{i \ne j} (x_i+x_j)$$
$$\omega_f(v_{35})=-2^6(4a^3+27b^2)(27bz^3-9a^2z^2-a^3).$$
Evaluating the $x_i$ at $r_i$, we get the following:\\
$h_1=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4)=x^4-512dx^2 +2^{15}dw^2x + 2^{16}d(d+w^2(12az-36b)) + 2^{18}d(27bz^3 - 9a^2z^2 - a^3)$ \\
$h_2=(x-\beta_1)(x-\beta_2)(x-\beta_3)(x-\beta_4)=x^4 -512dx^2 +2^{15}dw^2x + 2^{16}d(d+w^2(12az-36b)) $ \\
$h_3=(x-\gamma_1)(x-\gamma_2)(x-\gamma_3)(x-\gamma_4)=x^4-8wx^3+6(2az+3b)x^2-d$.\\
Let $\Delta=-16(4a^3+27b^2)$ and let $g=x^4-4\Delta x-12 a \Delta$, the polynomial for the unique $S_4$-quotient of the $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$-extension of $\mathbb{Q}$ given by $\mathbb{Q}(E[4])/\mathbb{Q}$ when $\overline{\rho}_{E,4}$ is surjective. Let $k_g$ be the (generically degree 4) field extension of $\mathbb{Q}$ given by adjoining a root of $g$ to $\mathbb{Q}$. Let $k_1$ be the (generically degree 4) field extension of $\mathbb{Q}$ given by adjoining a root of $h_1$ to $\mathbb{Q}$. We will show in the quaternion origami paper that there is an isomorphism between $k_g$ and $k_1$.
\begin{comment} > h3; x^4 + (-32x_1^4x2^2x3^2 - 32x1^4x2^2x4^2 - 32x1^4x3^2x4^2 -
32x1^2x2^4x3^2 - 32x1^2x2^4x4^2 - 32x1^2x2^2x3^4 +
384x1^2x2^2x3^2x4^2 - 32x1^2x2^2x4^4 - 32x1^2x3^4x4^2 -
32x1^2x3^2x4^4 - 32x2^4x3^2x4^2 - 32x2^2x3^4x4^2 -
32x2^2x3^2x4^4)x^2 + (-512x1^6x2^2x3^2x4^2 + 512x1^4x2^4x3^4 +
512x1^4x2^4x4^4 + 512x1^4x3^4x4^4 - 512x1^2x2^6x3^2x4^2 -
512x1^2x2^2x3^6x4^2 - 512x1^2x2^2x3^2x4^6 + 512x2^4x3^4x4^4)x +
256x1^8x2^4x3^4 - 512x1^8x2^4x3^2x4^2 + 256x1^8x2^4x4^4 -
512x1^8x2^2x3^4x4^2 - 512x1^8x2^2x3^2x4^4 + 256x1^8x3^4x4^4 +
2048x1^7x2^5x3^3x4 - 2048x1^7x2^5x3x4^3 - 2048x1^7x2^3x3^5x4 +
2048x1^7x2^3x3x4^5 + 2048x1^7x2x3^5x4^3 - 2048x1^7x2x3^3x4^5 -
512x1^6x2^6x3^4 + 5120x1^6x2^6x3^2x4^2 - 512x1^6x2^6x4^4 -
512x1^6x2^4x3^6 - 3072x1^6x2^4x3^4x4^2 - 3072x1^6x2^4x3^2x4^4 -
512x1^6x2^4x4^6 + 5120x1^6x2^2x3^6x4^2 - 3072x1^6x2^2x3^4x4^4 +
5120x1^6x2^2x3^2x4^6 - 512x1^6x3^6x4^4 - 512x1^6x3^4x4^6 -
2048x1^5x2^7x3^3x4 + 2048x1^5x2^7x3x4^3 + 2048x1^5x2^3x3^7x4 -
2048x1^5x2^3x3x4^7 - 2048x1^5x2x3^7x4^3 + 2048x1^5x2x3^3x4^7 +
256x1^4x2^8x3^4 - 512x1^4x2^8x3^2x4^2 + 256x1^4x2^8x4^4 -
512x1^4x2^6x3^6 - 3072x1^4x2^6x3^4x4^2 - 3072x1^4x2^6x3^2x4^4 -
512x1^4x2^6x4^6 + 256x1^4x2^4x3^8 - 3072x1^4x2^4x3^6x4^2 +
15360x1^4x2^4x3^4x4^4 - 3072x1^4x2^4x3^2x4^6 + 256x1^4x2^4x4^8 -
512x1^4x2^2x3^8x4^2 - 3072x1^4x2^2x3^6x4^4 -
3072x1^4x2^2x3^4x4^6 - 512x1^4x2^2x3^2x4^8 + 256x1^4x3^8x4^4 -
512x1^4x3^6x4^6 + 256x1^4x3^4x4^8 + 2048x1^3x2^7x3^5x4 -
2048x1^3x2^7x3x4^5 - 2048x1^3x2^5x3^7x4 + 2048x1^3x2^5x3x4^7 +
2048x1^3x2x3^7x4^5 - 2048x1^3x2x3^5x4^7 - 512x1^2x2^8x3^4x4^2 -
512x1^2x2^8x3^2x4^4 + 5120x1^2x2^6x3^6x4^2 -
3072x1^2x2^6x3^4x4^4 + 5120x1^2x2^6x3^2x4^6 -
512x1^2x2^4x3^8x4^2 - 3072x1^2x2^4x3^6x4^4 -
3072x1^2x2^4x3^4x4^6 - 512x1^2x2^4x3^2x4^8 - 512x1^2x2^2x3^8x4^4
+ 5120x1^2x2^2x3^6x4^6 - 512x1^2x2^2x3^4x4^8 -
2048x1x2^7x3^5x4^3 + 2048x1x2^7x3^3x4^5 + 2048x1x2^5x3^7x4^3 -
2048x1x2^5x3^3x4^7 - 2048x1x2^3x3^7x4^5 + 2048x1x2^3x3^5x4^7 +
256x2^8x3^4x4^4 - 512x2^6x3^6x4^4 - 512x2^6x3^4x4^6 +
256x2^4x3^8x4^4 - 512x2^4x3^6x4^6 + 256x2^4x3^4x4^8
3145728z^4A^4 + 21233664z^4AB^2-9437184z^3A^3B-63700992z^3B^3 + 3145728z^2A^5 + 21233664z^2A^2B^2-6291456zA^4B -42467328zAB^3 +1048576A^6 + 4718592A^3B^2- 15925248B^4; \end{comment}
\end{document} | arXiv |
What is the smallest three-digit multiple of 13?
The smallest three-digit number is $100$. When we divide 100 by 13, we get 7 with a remainder of 9. $$100=13 \cdot 7 + 9$$Because we want the smallest three-digit multiple of $13$, $13\cdot 7$ cannot be our answer because it is less than $100$ and is therefore a two-digit number. Instead we go up one multiple of $13$ and find that $13 \cdot 8= \boxed{104}$ is our smallest three-digit multiple of $13$. | Math Dataset |
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Extending the predictive power of perturbative QCD
The European Physical Journal C, Feb 2019
Bo-Lun Du, Xing-Gang Wu, Jian-Ming Shen, Stanley J. Brodsky
Bo-Lun Du
Xing-Gang Wu
Jian-Ming Shen
Stanley J. Brodsky
The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling \(\alpha _s\) at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this paper, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Padé Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables \(R_{e^+e^-}\), \(R_{\tau }\), and \(\Gamma (H \rightarrow b {\bar{b}})\) which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order \(\alpha _s^p\)) has been determined at order \(\alpha ^n_s\), then the \([N/M]=[0/n-p]\) Padé series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic \(e^+e^-\) annihilation. Thus, by using the combination of the PMC series and the Padé method, the predictive power of pQCD theory can be greatly improved.
https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-019-6704-9.pdf
The European Physical Journal C March 2019, 79:182 | Cite as Extending the predictive power of perturbative QCD AuthorsAuthors and affiliations Bo-Lun DuXing-Gang WuJian-Ming ShenStanley J. Brodsky Open Access Regular Article - Theoretical Physics First Online: 28 February 2019 1 Shares 136 Downloads Abstract The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling \(\alpha _s\) at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this paper, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Padé Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables \(R_{e^+e^-}\), \(R_{\tau }\), and \(\Gamma (H \rightarrow b {\bar{b}})\) which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order \(\alpha _s^p\)) has been determined at order \(\alpha ^n_s\), then the \([N/M]=[0/n-p]\) Padé series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic \(e^+e^-\) annihilation. Thus, by using the combination of the PMC series and the Padé method, the predictive power of pQCD theory can be greatly improved. 1 Introduction Quantum chromodynamics (QCD) is believed to be the fundamental field theory of the hadronic strong interactions. Due to asymptotic freedom [1, 2], the QCD running coupling becomes numerically small at short distances, allowing perturbative calculations of observables for physical processes at large momentum transfer. The fundamental principle of renormalization group invariance requires that the prediction for a physical observable must be independent of both the choice of renormalization scheme and the choice of initial renormalization scale. However, due to the mismatch of the QCD running coupling (\(\alpha _s\)) and the pQCD coefficients at each order, a truncated pQCD series will not automatically satisfy this requirement, leading to well-known ambiguities. The predictive power of pQCD theory thus depends heavily on how to eliminate both the renormalization scheme-and-scale ambiguities and how to predict contributions from unknown higher-order terms. It has become conventional to choose the renormalization scale \(\mu _r\) as the typical momentum flow of the process. The resulting prediction at any fixed order will then inevitably also depend on the choice of the renormalization scheme. The hope is to achieve a nearly scheme-and-scale independent prediction by systematically computing higher-and-higher order QCD corrections; however, this hope is in direct conflict with the presence of the divergent \(n! \alpha _s^n \beta _0^n\) "renormalon" series [3, 4, 5]. It is also often argued that by varying the renormalization scale, one will obtain information on the uncalculated higher-order terms. However, the variation of the renormalization scale can only provide information on the \(\beta \)-dependent terms which control the running of \(\alpha _s\); the variation of \(\mu _r\) gives no information on the contribution to the observable coming from the \(\beta \)-independent terms. We will refer to the \(\beta \)-independent contributions as "conformal" terms – since they match the contributions of a corresponding conformal theory with \(\beta =0\). Obviously, the naive procedure of guessing and varying the renormalization scale can lead to a misleading pQCD prediction, especially if the conformal terms in the higher-order series are more important than the \(\beta \)-dependent terms. For example, the large K-factors for certain processes are caused by large conformal contributions, as observed in the recent analysis of the \(\gamma \gamma ^*\rightarrow \eta _c\) transition form factor [6]. Even if a nearly scale-independent prediction is attained for a global quantity such as a total cross-section or a total decay width, the scale independence could be due to accidental cancellations among different orders, even though the scale dependence at each order could be very large. Worse, even if a prediction with a guessed scale agrees with the data, one cannot explain why it is reliable prediction, thus greatly depressing the predictive power of pQCD. 1.1 The PMC The "Principle of Maximum Conformality" (PMC) rigorously eliminates the conventional renormalization scheme-and-scale ambiguities [7, 8, 9, 10]. It extends the well-known Brodsky–Lepage–Mackenzie (BLM) scale-setting method [11] to all orders in pQCD. The basic PMC procedure is to identify all contributions which originate from the \(\{\beta _i\}\)-terms in a pQCD series; one then shifts the scale of the QCD running coupling at each order to absorb the \(\{\beta _i\}\)-terms and to thus obtain the correct scale for its running behavior as well as to set the number of active quark flavors \(n_f\) arising from quark loops in the gluon propagators. The PMC also agrees with the standard Gell–Mann–Low method (GM-L) [12] for fixing the renormalization scale of \(\alpha (Q^2)\) and the effective number of lepton flavors \(n_\ell \) in Abelian quantum electrodynamics (QED). One can choose any value for the initial renormalization scale \(\mu _r\) when applying the PMC: the resulting scales for the running QCD coupling at each order are in practice independent of its value; thus the PMC eliminates the renormalization scale ambiguity. Moreover, the PMC predictions are scheme-independent due to its conformal nature, and the divergent renormalon behavior of the resulting perturbative series does not appear. The PMC satisfies renormalization group invariance and all of the self-consistency conditions of the renormalization group equation (RGE) [13]. The transition scale between the perturbative and nonperturbative domains can also be determined by using the PMC [14, 15, 16], thus providing a physical procedure for setting the factorization scale for pQCD evolution. The PMC has now been successfully applied to many QCD measurements studies at the LHC as well as other hadronic processes [17, 18, 19]. Within the framework of the PMC, the pQCD approximant can be written in the following form [9, 10], $$\begin{aligned} \rho _{n}(Q)|_{\mathrm{Conv.}}= & {} \sum ^{n}_{i=1} r_i(\mu ^2_r/Q^2) a^{p+i-1}(\mu _r) \end{aligned}$$ (1) $$\begin{aligned}= & {} r_{1,0}{a^p(\mu _r)} + \left[ r_{2,0} + p \beta _0 r_{2,1} \right] {a^{p+1}(\mu _r)} \nonumber \\&+ \big [ r_{3,0} + p \beta _1 r_{2,1} + (p+1){\beta _0}r_{3,1} \nonumber \\&+\frac{p(p+1)}{2} \beta _0^2 r_{3,2} \big ]{a^{p+2}(\mu _r)} {+} \big [ r_{4,0} {+} p{\beta _2}{r_{2,1}} \nonumber \\&+ (p+1){\beta _1}{r_{3,1}} + \frac{p(3+2p)}{2}{\beta _1}{\beta _0}{r_{3,2}} \nonumber \\&+ (p+2){\beta _0}{r_{4,1}}+ \frac{(p+1)(p+2)}{2}\beta _0^2{r_{4,2}} \nonumber \\&+ \frac{p(p+1)(p+2)}{3!}\beta _0^3{r_{4,3}} \big ]{a^{p+3}(\mu _r)} + \ldots ,\nonumber \\ \end{aligned}$$ (2) where \(a={\alpha _s}/\pi \), and Q represents the kinematic scale. The index \(p(\ge 1)\) indicates the \(\alpha _s\)-order of the leading-order (LO) contribution and \(\{r_{i,0}\}\) are conformal coefficients, and the \(\beta \)-pattern at each order is predicted by the non-Abelian gauge theory [20]. Following the standard PMC-s procedure, we obtain $$\begin{aligned} \rho _n(Q)|_{\mathrm{PMC}}=\sum _{i=1}^n r_{i,0}a^{p+i-1}(Q_{*}), \end{aligned}$$ (3) where \(Q_{*}\) is the determined optimal single PMC scale, whose analytical form can be found in Ref. [21]. We emphasize that the factorially divergent renormalon terms, such as \( n!\alpha _s^n \beta _0^n\), do not appear in the resulting conformal series; thus a convergent pQCD series can be achieved. 1 1.2 Padé Resummation The Padé approximation approach (PAA) provides a systematic procedure for promoting a finite Taylor series to an analytic function [22, 23, 24]. In particular, the PAA can be used to estimate the \((n+1)_\mathrm{th}\)-order coefficient by incorporating all known coefficients up to order n. It was shown in Ref. [25] that the Padé method provides an important guide for understanding the sequence of renormalization scales determined by the BLM method and its all-order extension, the PMC. Those scales are the optimal ones for evaluating each term in a skeleton expansion. The leading-order BLM/PMC sequence corresponds to the [0 / 1]-type PAA [26]. After applying the BLM/PMC, the summation over skeleton graphs is then similar to the summation of the perturbative contributions for a corresponding conformal theory. Since the divergent renormalon series does not appear in the conformal \(\beta =0\) perturbative series generated by the PMC, there is an opportunity to use a resummation procedure such as the Padé method to predict higher-order terms and to thus increase the precision and reliability of pQCD predictions. In this paper we will test whether one can use the PAA to achieve reliable predictions for the unknown higher-order terms for a pQCD series by using the renormalon-free conformal series determined by the PMC. For this purpose, we will adopt the PMC single-scale approach (PMC-s) [21], which utilizes a single effective renormalization scale which matches the PMC series via the mean-value theorem. Other applications of resummation methods to pQCD, together with alternatives to the PAA, have been discussed in the literature [25, 26, 27, 28, 29, 30, 31]. However, in our analysis, we will apply the PAA to the scale- and scheme- independent conformal series, whose perturbative coefficients are free of divergent renormalon contributions. 1.3 Applying the PAA to pQCD If we apply the PAA to the PMC prediction, the pQCD series can be rewritten in the following [N / M]-type form $$\begin{aligned} \rho ^{[N/M]}_n(Q)= & {} a^p \times \frac{b_0+b_1 a + \cdots + b_N a^N}{1 + c_1 a + \cdots + c_M a^M} \end{aligned}$$ (4) $$\begin{aligned}= & {} \sum _{i=1}^{n} C_{i} a^{p+i-1} + C_{n+1}\; a^{p+n}+\ldots , \end{aligned}$$ (5) where \(M\ge 1\) and \(N+M+1 = n\). Comparing Eq. (5) with the series (1) or (3), the coefficients \(C_{i}\) can be directly related to \(r_i\) or \(r_{i,0}\), respectively. Furthermore, by using the known \(\hbox {N}^{\mathrm{n-1}}\hbox {LO}\)-order pQCD series, the coefficients \(b_{i\in [0,N]}\) and \(c_{i\in [1,M]}\) can be expressed by using the coefficients \(C_{i\in [1,n]}\). Finally, we can use the coefficients \(b_{i\in [0,N]}\) and \(c_{i\in [1,M]}\) to predict the one-order-higher uncalculated coefficient \(C_{n+1}\) at the \(\hbox {N}^{\mathrm{n}}\hbox {LO}\)-order level. For examples, if \([N/M]=[n-2/1]\), we have $$\begin{aligned} C_{n+1}=\frac{C_n^2}{C_{n-1}}; \end{aligned}$$ (6) if \([N/M]=[n-3/2]\), we have $$\begin{aligned} C_{n+1}=\frac{-C_{n-1}^3+2C_{n-2}C_{n-1}C_{n}-C_{n-3}C_{n}^2}{C_{n-2}^2-C_{n-3}C_{n-1}}; \end{aligned}$$ (7) if \([N/M]=[n-4/3]\), we have $$\begin{aligned} C_{n+1}= & {} \{C_{n-2}^4-(3 C_{n-3} C_{n-1}+2 C_{n-4} C_{n}) C_{n-2}^2 \nonumber \\&\quad +\,2 [C_{n-4} C_{n-1}^2+(C_{n-3}^2+C_{n-5} C_{n-1}) C_{n}] C_{n-2} \nonumber \\&\quad -\,C_{n-5} C_{n-1}^3+C_{n-3}^2 C_{n-1}^2+C_{n-4}^2 C_{n}^2 \nonumber \\&\quad -\,C_{n-3} C_{n} (2 C_{n-4} C_{n-1}+C_{n-5} C_{n})\} \nonumber \\&\quad \{C_{n-3}^3-\left( 2 C_{n-4} C_{n-2}+C_{n-5} C_{n-1}\right) C_{n-3} \nonumber \\&\quad +\,C_{n-5} C_{n-2}^2+C_{n-4}^2 C_{n-1}\}; \mathrm{etc.} \end{aligned}$$ (8) In each case, \(C_{i<1}\equiv 0\). We need to know at least two \(C_i\) in order to predict the unknown higher-order coefficients; thus the PAA is applicable when we have determined at least the NLO terms (\(n=2\)) using the PMC. One can also use the full PAA (4) to estimate the sum of the whole series, e.g. to give the all-oders PAA prediction. As will be found later, the differences for the predictions of the truncated and full PAA series are small for converged perturbative series. In the following, we will apply the PAA for three physical observables \(R_{e^+e^-}\), \(R_{\tau }\) and \(\Gamma (H\rightarrow b{\bar{b}})\) which are known at four loops in pQCD. We will show how the "unknown" terms predicted by the PAA varies when one inputs more-and-more known higher-order terms. The ratio \(R_{e^+e^-}\) is defined as $$\begin{aligned} R_{e^+ e^-}(Q)= & {} \frac{\sigma \left( e^+e^-\rightarrow \mathrm{hadrons} \right) }{\sigma \left( e^+e^-\rightarrow \mu ^+ \mu ^-\right) }\nonumber \\= & {} 3\sum _q e_q^2\left[ 1+R(Q)\right] , \end{aligned}$$ (9) where \(Q=\sqrt{s}\) is the \(e^+e^-\) collision energy. The pQCD approximants for R(Q) are labelled \(R_n(Q)= \sum _{i=1}^{n} r_i(\mu _r/Q)a^{i}(\mu _r)\). The pQCD coefficients at \(\mu _r=Q\) have been calculated in the \(\overline{\mathrm{MS}}\)-scheme in Refs. [32, 33, 34, 35]. For illustration we take \(Q=31.6 \;\mathrm{GeV}\) [36]. The ratio \(R_{\tau }\) is defined as $$\begin{aligned} R_{\tau }(M_{\tau })= & {} \frac{\sigma (\tau \rightarrow \nu _{\tau }+\mathrm{hadrons})}{\sigma (\tau \rightarrow \nu _{\tau }+{\bar{\nu }}_e+e^-)}\nonumber \\= & {} 3\sum \left| V_{ff'}\right| ^2\left( 1+{\tilde{R}}(M_{\tau })\right) , \end{aligned}$$ (10) where \(V_{ff'}\) are Cabbibo–Kobayashi–Maskawa matrix elements, \(\sum \left| V_{ff'}\right| ^2 =\left( \left| V_{ud}\right| ^2+\left| V_{us}\right| ^2\right) \approx 1\) and \(M_{\tau }= 1.78\) GeV. The pQCD approximant, \({\tilde{R}}_{n}(M_{\tau })= \sum _{i=1}^{n}r_i(\mu _r/M_{\tau })a^{i}(\mu _r)\); the coefficients can be obtained by using the known relation of \(R_{\tau }(M_{\tau })\) to \(R(\sqrt{s})\) [37]. The decay width \(\Gamma (H\rightarrow b{\bar{b}})\) is defined as $$\begin{aligned} \Gamma (H\rightarrow b{\bar{b}})=\frac{3G_{F} M_{H} m_{b}^{2}(M_{H})}{4\sqrt{2}\pi } [1+{\hat{R}}(M_{H})], \end{aligned}$$ (11) where the Fermi constant \(G_{F}=1.16638\times 10^{-5}\;\mathrm{GeV}^{-2}\), the Higgs mass \(M_H=126\) GeV, and the b-quark \(\overline{\mathrm{MS}}\)-running mass is \(m_b(M_H)=2.78\) GeV [38]. The pQCD approximant \({\hat{R}}_n(M_H)= \sum _{i=1}^{n}r_i(\mu _r/M_{H}) a^{i}(\mu _r)\), where the predictions for the \(\overline{\mathrm{MS}}\)-coefficients at \(\mu _r=M_H\) can be found in Ref. [39]. In each case the coefficients at any other scale can be obtained via QCD evolution. In doing the numerical evaluation, we have assumed the running of \(\alpha _s\) at the four-loop level. The asymptotic QCD scale is set using \(\alpha _s(M_z)=0.1181\) [40], giving \(\Lambda _{\mathrm{QCD}}^{n_f=5}=0.210\) GeV. After applying the PMC-s approach, the optimal scale for each process can be determined. If the pQCD approximants are known at up to two-loop, three-loop, and four-loop level, the corresponding optimal scales are \(Q_{*}|_{e^+e^-}=[35.36\), 39.68, 40.30] GeV, \(Q_{*}|_{\tau }=[0.90\), 1.01, 1.05] GeV,2 and \(Q_{*}|_{H\rightarrow b{\bar{b}}}=[61.38\), 57.41, 58.84] GeV, accordingly. It is found that those PMC scales \(Q^*\) are completely independent of the choice of the initial renormalization scale \(\mu _r\). Table 1 Comparison of the exact ("EC") \((n+1){\mathrm{th}}\)-order conformal coefficients with the predicted ("[N / M]-type PAA") \((n+1)_\mathrm{th}\)-order ones based on the known \(n{\mathrm{th}}\)-order approximate \(R_{n}(Q=31.6~ \mathrm{GeV})\), where \(n=2,3,4\), respectively \(r_{n+1,0}\) \(n+1=3\) \(n+1=4\) \(n+1=5\) EC \(-\,1.0 \) \(-\,11.0 \) – PAA [0/1]\(+3.4\) [0/2]\(-\,9.9\) [0/3]\(-\,17.8\) – [1/1]\(+0.55\) [1/2]\(-\,18.0\) – – [2/1]\(-\,120\) Table 2 Comparison of the exact ("EC") \((n+1){\mathrm{th}}\)-order conformal coefficients with the predicted ("[N / M]-type PAA") \((n+1)_\mathrm{th}\)-order ones based on the known \(n{\mathrm{th}}\)-order approximate \({\tilde{R}}_{n}(M_{\tau })\), where \(n=2,3,4\), respectively \(r_{n+1,0}\) \(n+1=3\) \(n+1=4\) \(n+1=5\) EC \(+3.4\) \(+6.8\) – PAA [0/1]\(+\,4.6\) [0/2]\(+\,4.9\) [0/3]\(+\,14.7\) – [1/1]\(+\,5.5\) [1/2]\(+\,11.5\) – – [2/1]\(+\,13.5\) Table 3 Comparison of the exact ("EC") \((n+1){\mathrm{th}}\)-order conformal coefficients with the predicted ("[N / M]-type PAA") \((n+1)_\mathrm{th}\)-order ones based on the known \(n{\mathrm{th}}\)-order approximate \({\hat{R}}_{n}(M_H)\), where \(n=2,3,4\), respectively \(r_{n+1,0}\) \(n+1=3\) \(n+1=4\) \(n+1=5\) EC \(-\,1.36\times 10^2\) \(-\,4.32\times 10^2\) – PAA [0/1]\(+3.23\times 10^1\) [0/2]\(-\,7.26\times 10^2\) [0/3]\(\,+\,3.72\times 10^3\) – [1/1]\(\,+\,1.37\times 10^3\) [1/2]\(\,+\,3.20\times 10^3\) – – [2/1]\(-\,1.37\times 10^3\) The remaining task for the PAA is to predict the higher-order conformal coefficients. We present a comparison of the exact \((n+1){\mathrm{th}}\)-order conformal coefficients with the PAA predicted ones based on the known \(n{\mathrm{th}}\)-order approximates \(R_{n}(Q=31.6~ \mathrm{GeV})\), \({\tilde{R}}_{n}(M_{\tau })\) and \({\hat{R}}_{n}(M_H)\) in Tables 1, 2 and 3, respectively. Here the [N / M]-type PAA is for \(N+M=n-1\) with \(N\ge 0\) and \(M\ge 1\). Those Tables show that the \([N/M]=[0/n-1]\)-type PAA provides result closest to the known pQCD result. It is interesting to note that the \([0/n-1]\)-type PAA is consistent with the "Generalized Crewther Relations" (GSICRs) [41]. For example, the GSICR, which provides a remarkable all-orders connection between the pQCD predictions for deep inelastic neutrino-nucleon scattering and hadronic \(e^+e^-\) annihilation shows that the conformal coefficients are all equal to 1; e.g. \({\widehat{\alpha }}_d(Q)=\sum _{i}{\widehat{\alpha }}^{i}_{g_1}(Q_*)\), where \(Q_*\) satisfies $$\begin{aligned} \ln \left. \frac{Q_*^2}{Q^2}\right| _{g_1}= & {} 1.308 + [-\,0.802\, +\, 0.039 n_f] {\widehat{\alpha }}_{g_1}(Q_*) \nonumber \\&+\, [16.100 - 2.584 n_f {+} 0.102 n_f^2] {\widehat{\alpha }}_{g_1}^2(Q_*) {+} \cdots .\nonumber \\ \end{aligned}$$ (12) By using the \([0/n-1]\)-type PAA – the geometric series – all of the predicted conformal coefficients are also equal to 1. The \([0/n-1]\)-type PAA also agrees with the GM-L scale-setting procedure to obtain scale-independent perturbative QED predictions; e.g., the renormalization scale for the electron-muon elastic scattering through one-photon exchange is set as the virtuality of the exchanged photon, \(\mu _r^2 = q^2 = t\). By taking an arbitrary initial renormalization scale \(t_0\), we have $$\begin{aligned} \alpha _{em}(t) = \frac{\alpha _{em}(t_0)}{1 - \Pi (t,t_0)}, \end{aligned}$$ (13) where \(\Pi (t,t_0) = \frac{\Pi (t,0) -\Pi (t_0,0)}{1-\Pi (t_0,0)}\), which sums all vacuum polarization contributions, both proper and improper, to the dressed photon propagator. The PMC reduces in the \(N_C \rightarrow 0\) Abelian limit to the GM-L method [42] and the preferable \([0/n-1]\)-type makes the PAA geometric series self-consistent with the GM-L/PMC prediction. Table 4 Comparison of the exact ("EC") and the predicted ("PAA") pQCD approximants \(R_n(Q=31.6\;\mathrm{GeV})\), \({\tilde{R}}_{n}(M_\tau )\) and \({\hat{R}}_n(M_H)\) under conventional (Conv.) and PMC-s scale-setting approaches up to \(n{\mathrm{th}}\)-order level. The \((n+1)_\mathrm{th}\)-order PAA prediction equals to the \(n{\mathrm{th}}\)-order known prediction plus the predicted \((n+1){\mathrm{th}}\)-order terms using the \([0/n-1]\)-type PAA prediction (The values in the parentheses are results for the corresponding full PAA series). The PMC predictions are scale independent and the errors for conventional scale-setting are estimated by varying the initial renormalization scale \(\mu _r\) within the region of \([1/2\mu _0, 2\mu _0]\), where \(\mu _0=Q\), \(M_\tau \) and \(M_H\), respectively \(\mathrm{EC}\), \(n=2\) \(\mathrm{PAA}\), \(n=3\) \(\mathrm{EC}\), \(n=3\) \(\mathrm{PAA}\), \(n=4\) \(\mathrm{EC}\), \(n=4\) \(\mathrm{PAA}\), \(n=5\) \(R_n(Q)|_{\mathrm{PMC-s}}\) 0.04745 0.04772 (0.04777) 0.04635 0.04631 (0.04631) 0.04619 0.04619 (0.04619) \({\tilde{R}}_{n}(M_{\tau })|_{\mathrm{PMC-s}}\) 0.1879 0.2035 (0.2394) 0.2103 0.2128 (0.2134) 0.2089 0.2100 (0.2104) \({\hat{R}}_{n}(M_H)|_{\mathrm{PMC-s}}\) 0.2482 0.2503 (0.2505) 0.2422 0.2402 (0.2406) 0.2401 0.2405 (0.2405) \(R_n(Q)|_{\mathrm{Conv.}}\) \(0.04763^{+\,0.00045}_{-\,0.00139}\) \(0.04781^{+\,0.00043}_{-\,0.00053}\) \(0.04648_{-\,0.00071}^{+\,0.00012}\) \(0.04632_{-\,0.00025}^{+\,0.00018}\) \(0.04617_{-\,0.00009}^{+\,0.00015}\) \(0.04617_{-\,0.00001}^{+\,0.00007}\) \({\tilde{R}}_{n}(M_{\tau })|_{\mathrm{Conv.}}\) \(0.1527^{+\,0.0610}_{-\,0.0323}\) \(0.1800^{+\,0.0515}_{-\,0.0330}\) \(0.1832_{-\,0.0334}^{+\,0.0385}\) \(0.1975_{-\,0.0296}^{+\,0.0140}\) \(0.1988_{-\,0.0299}^{+\,0.0140}\) \(0.2056_{-\,0.0247}^{+\,0.0029}\) \({\hat{R}}_{n}(M_H)|_{\mathrm{Conv.}}\) \(0.2406^{+\,0.0074}_{-\,0.0104}\) \(0.2475^{+\,0.0027}_{-\,0.0066}\) \(0.2425_{-\,0.0053}^{+\,0.0002}\) \(0.2419_{-\,0.0040}^{+\,0.0002}\) \(0.2411_{-\,0.0040}^{+\,0.0001}\) \(0.2407_{-\,0.0040}^{+\,0.0002}\) Tables 1, 2 and 3 show that as more loop terms are inputted, the predicted conformal coefficients become closer to their exact value. To show this clearly, we define the normalized difference between the exact conformal coefficient and the predicted one as $$\begin{aligned} \Delta _{n} = \left| \frac{r_{n,0}|_{\mathrm{PAA}}-r_{n,0}|_\mathrm{EC}}{r_{n,0}|_{\mathrm{EC}}}\right| , \end{aligned}$$ where "EC" and "PAA" stand for exact and predicted conformal coefficients, respectively. By using the exact terms, known up to two-loop and three-loop levels accordingly, the normalized differences for the \(3{\mathrm{th}}\)-order and the \(4{\mathrm{th}}\)-order conformal coefficients, i.e. those coefficients in the \(n+1=3\) and \(n+1=4\) columns in Tables 1, 2 and 3, become suppressed from \(440.\%\) to \(10\%\) for \(R(Q=31.6~\mathrm{GeV})\), from \(35\%\) to \(28\%\) for \({\tilde{R}}(M_{\tau })\), and from \(124.\%\) to \(68\%\) for \({\hat{R}}(M_H)\). There are large differences for the conformal coefficients if we only know the QCD corrections at the two-loop level; however this decreases rapidly when we know more loop terms. Following this trend, the normalized differences for the \(5{\mathrm{th}}\)-order conformal coefficients should be much smaller than the \(4{\mathrm{th}}\)-order ones. Conservatively, if we set the normalized difference (\(\Delta _5\)) of the \(5{\mathrm{th}}\)-loop as the same one of the \(4_{\mathrm{}}\)-loop (\(\Delta _4\)), we can inversely predict the \(5{\mathrm{th}}\)-loop "\(\mathrm{EC'}\)" conformal coefficients: $$\begin{aligned}&r^{e^+ e^-}_{5,0}|_{\mathrm{EC'}} = -18.0\pm 1.8, \end{aligned}$$ (14) $$\begin{aligned}&r^{\tau }_{5,0}|_{\mathrm{EC'}} = 16.0\pm 4.5, \end{aligned}$$ (15) $$\begin{aligned}&r^{H\rightarrow b{\bar{b}}}_{5,0}|_{\mathrm{EC'}} = (6.92\pm 4.71)\times 10^3, \end{aligned}$$ (16) where the central values are obtained by averaging the two "EC" values derived from \(\frac{r_{5,0}|_\mathrm{PAA}}{(1+\Delta _4)}\) and \(\frac{r_{5,0}|_\mathrm{PAA}}{(1-\Delta _4)}\). The difference between the exact and predicted conformal coefficients is reduced by the \(\alpha _s/\pi \)-power suppression, thus the precision of the predictive power of the PAA should become most useful for total cross-sections and decay widths. We present the comparison of the exact results for \(R_{n}(Q=31.6~\mathrm{GeV})\), \({\tilde{R}}_{n}(M_{\tau })\) and \({\hat{R}}_{n}(M_H)\) with the \([0/n-1]\)-type PAA predicted ones in Table 4. The values in the parentheses are results for the corresponding full PAA series, which are calculated by using Eq. (4). Due to the fast pQCD convergence, the differences between the truncated and full PAA predictions are small, which are less than \(1\%\) for \(n\ge 4\). Similarly, we define the precision of the predictive power as the normalized difference between the exact approximant (\(\rho _n|_{\mathrm{EC}}\)) and the prediction (\(\rho _n|_{\mathrm{PAA}}\)); i.e. $$\begin{aligned} \left| \frac{\rho _{n}|_{\mathrm{PAA}}- \rho _{n}|_{\mathrm{EC}}}{\rho _{n}|_\mathrm{EC}}\right| . \end{aligned}$$ The PMC predictions are renormalization scheme-and-scale independent, and the pQCD convergence is greatly improved due to the elimination of renormalon contributions. Highly precise values at each order can thus be achieved [21]. In contrast, predictions using conventional pQCD series (1) are scale dependent even for higher-order predictions. We also present results using conventional scale-setting in Table 4; it confirms the conclusion that the conformal PMC-s series is much more suitable for applications of the PAA. By using the known (exact) approximants predicted by PMC-s scale-setting up to two-loop and three-loop levels accordingly, the differences between the exact and predicted three-loop and four-loop approximants are observed to decrease from \(3.0\%\) to \(0.3\%\) for \(\rho _n=R_n(Q=31.6~\mathrm{GeV})\), from \(3\%\) to \(2\%\) for \(\rho _n={\tilde{R}}_n(M_{\tau })\), and from \(3.0\%\) to \(\sim 0\%\) for \(\rho _n={\hat{R}}_n(M_H)\), respectively. The normalized differences for \(R_4(Q=31.6~\mathrm{GeV})\), \({\tilde{R}}_4(M_{\tau })\) and \({\hat{R}}_4(M_H)\) are small. If we conservatively set the normalized difference of the \(5{\mathrm{th}}\)-loop to match that of the 4-loop predictions, then the predicted \(5{\mathrm{th}}\)-loop "\(\mathrm{EC'}\)" predictions are $$\begin{aligned}&R_5(Q=31.6~\mathrm{GeV})|_{\mathrm{EC'}} = 0.04619\pm 0.00014, \end{aligned}$$ (17) $$\begin{aligned}&{\tilde{R}}_5(M_{\tau })|_{\mathrm{EC'}} = 0.2100\pm 0.0042, \end{aligned}$$ (18) $$\begin{aligned}&{\hat{R}}_5(M_H)|_{\mathrm{EC'}} = 0.2405\pm 0.0001. \end{aligned}$$ (19) 1.4 Summary The PMC provides first-principle predictions for QCD; it satisfies renormalization group invariance and eliminates the conventional renormalization scheme-and-scale ambiguities. Since the divergent renormalon series does not appear in the conformal (\(\beta =0\)) perturbative series generated by the PMC, there is an opportunity to use resummation procedures such as the Padé method to predict higher-order terms and thus to increase the precision and reliability of pQCD predictions. In this paper, we have shown that by applying PAA to the renormalon-free conformal series derived by using the PMC single-scale procedure, one can achieve quantitatively useful estimates for the unknown higher-order terms based on the known perturbative QCD series. In particular, we have found that if the PMC prediction for the conformal series for an observable (of leading order \(\alpha _s^p\)) has been determined at order \(\alpha ^n_s\), then the \([N/M]=[0/n-p]\) Padé series provides an important estimate for the higher-order terms. The all-orders predictions of the \([0/n-p]\)-type PAA are in fact identical to the predictions obtained from the all-order GSICRs which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic \(e^+e^-\) annihilation. These relations are fundamental, high precision predictions of QCD. Open image in new window Fig. 1 Comparison of the exact ("EC") and the predicted ([0/n-1]-type "PAA") pQCD prediction for \(R_n(Q=31.6\;\mathrm{GeV})\) under the PMC-s scale-setting. It shows how the PAA predictions change when more loop-terms are included, where the five-loop "EC" prediction is from Eq. (17) Tables 1, 2 and 3 show that the difference between the exact and the predicted conformal coefficients at various loops, which decreases rapidly as additional high-order loop terms are included. Table 4 shows that the PAA becomes quantitatively effective even at the NLO level for the pQCD approximant due to the strong \(\alpha _s/\pi \)-suppression of the conformal series. For example, when using the NLO results \(R_2(Q)\), \({\tilde{R}}_2(M_{\tau })\) and \({\hat{R}}_2(M_H)\) to predict the observables \(R_3(Q)\), \({\tilde{R}}_3(M_{\tau })\) and \({\hat{R}}_3(M_H)\) at NNLO, the normalized differences between the Padé estimates and the known results are only about \(3\%\). Taking \(R_{e^+e^-}\) as an explicit example, we show how the PAA predictions change when more loop-terms are included in Fig. 1. In some sense this is an infinite-order prediction for \(R_{e^+e^-}(Q=31.6\;\mathrm{GeV})\), and it is the most precise prediction one can make using our PMC+PAA method, given the present knowledge of pQCD. Thus by combining the PMC with the Padé method, the predictive power of the pQCD theory can be remarkably improved. As a final remark, we show that the way of using PAA basing on the conformal series is consistent with that of the \({{\mathcal {N}}}=4\) supersymmetric Yang-Mills theory. For the purpose, we present a PAA prediction on the NNLO and \(\hbox {N}^{3}\hbox {LO}\) Balitsky–Fadin–Kuraev–Lipatov (BFKL) Pomeron eigenvalues. By using the PAA method together with the known LO and NLO coefficients given in Ref. [43], we find that the NNLO BFKL coefficient is \(0.86\times 10^4\) for \(\Delta =0.45\), where \(\Delta \) is the full conformal dimension of the twisted-two operator. The exact NNLO BFKL coefficient has been discussed in planar \({{\mathcal {N}}}=4\) supersymmetric Yang–Mills theory [44] by using the quantum spectral curve integrability-based method [45, 46], which gives \(1.08\times 10^4\) [44]. Thus the normalized difference between those two NNLO values is only about \(20\%\). As a step forward, we predict the \(\hbox {N}^{3}\hbox {LO}\) coefficient to the Pomeron eigenvalue by using the [0 / 2]-PAA type and the known NNLO coefficient given in Ref. [44], which results in \(-\,3.07\,\times \,10^5\). This value is also consistent with the \({{\mathcal {N}}}=4\) supersymmetric Yang-Mills prediction, since if we adopt the data-fitting prediction suggested in Ref. [47] to predict \(\hbox {N}^{3}\hbox {LO}\) coefficient, we shall obtain \(-3.66\times 10^5\). The normalized difference between those two \(\hbox {N}^{3}\hbox {LO}\) values is also only about \(20\%\). Footnotes 1. Only those \(\{\beta _i\}\)-terms that are pertained to RGE have been absorbed into the PMC scale. There may have cases in which the \(\{\beta _i\}\)-terms are not pertained to RGE and should be treated as conformal coefficients, which may break the pQCD convergence. 2. Because the usually adopted analytic \(\alpha _s\)-running differ significantly at scales below a few GeV from the exact solution of RGE [40], we will use the exact numerical solution of the RGE throughout to evaluate \(R_{\tau }\). Notes Acknowledgements This work is supported in part by the Natural Science Foundation of China under Grant No. 11625520 and No. 11847301, the Fundamental Research Funds for the Central Universities under the Grant No. 2018CDPTCG0001/3, and the Department of Energy Contract No. DE-AC02-76SF00515. SLAC-PUB-17306. References 1. D.J. Gross, F. Wilczek, Phys. Rev. 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B 417, 145 (1998)ADSMathSciNetCrossRefGoogle Scholar 43. M.S. Costa, V. Goncalves, J. Penedones, JHEP 1212, 091 (2012)ADSCrossRefGoogle Scholar 44. N. Gromov, F. Levkovich-Maslyuk, G. Sizov, Phys. Rev. Lett. 115, 251601 (2015)ADSMathSciNetCrossRefGoogle Scholar 45. N. Gromov, V. Kazakov, S. Leurent, D. Volin, Phys. Rev. Lett. 112, 011602 (2014)ADSCrossRefGoogle Scholar 46. N. Gromov, V. Kazakov, S. Leurent, D. Volin, JHEP 1509, 187 (2015)ADSCrossRefGoogle Scholar 47. N. Gromov, F. Levkovich-Maslyuk, G. Sizov, JHEP 1606, 036 (2016)ADSCrossRefGoogle Scholar Copyright information © The Author(s) 2019 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3 Authors and Affiliations Bo-Lun Du1Xing-Gang Wu1Email authorView author's OrcID profileJian-Ming Shen1Stanley J. Brodsky21.Department of PhysicsChongqing UniversityChongqingPeople's Republic of China2.SLAC National Accelerator LaboratoryStanford UniversityStanfordUSA
This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-019-6704-9.pdf
Bo-Lun Du, Xing-Gang Wu, Jian-Ming Shen, Stanley J. Brodsky. Extending the predictive power of perturbative QCD, The European Physical Journal C, 2019, 182, DOI: 10.1140/epjc/s10052-019-6704-9 | CommonCrawl |
Welcome to the official website of the European High Pressure Research Group (EHPRG).
EHPRG is non-profitable academic association established in 1963 which is devoted to science and technology of matter under high pressure. It organizes an annual meeting which brings together hundreds of scientists of various fields, mainly physics, chemistry, Earth and planetary sciences, bio- and food science and technology.
The reasonable size of the group, low conference fees, and a friendly relationship between the attendants have made EHPRG meetings a major event in the European high-pressure community for more than half a century.
Upcoming EHPRG Meeting
The Joint 28th AIRAPT and 60th EHPRG International Conference on High Pressure Science and Technology (AIRAPT-2023), originally scheduled for July 2021, will be held in Edinburgh, UK, from July 23-28, 2023.
The Conference will be held at the Edinburgh International Conference Centre (EICC) and it will be a face-to-face conference only.
Immediately prior to AIRAPT/EHPRG, from the 19th to 23rd of July 2023, a Summer School will take place, also based in Edinburgh, with accommodation provided in University Halls of Residence. The School is aimed at young researchers, and experimentalists wanting a grounding in theory and vice versa.
For more information, please visit the official webpage of AIRAPT-23.
The 59th European High Pressure Research Group Meeting on High Pressure Science and Technology (EHPRG2022) took place in Uppsala, Sweden, from 5th to 8th September 2022.
Meeting sessions were held virtually. The 59th EHPRG intended to be a large scope conference, in line with the rapid increase of high pressure activities in all areas of Physics, Chemistry, Food Science, Geosciences, Material Research, and Biosciences.
For more information, please visit the official webpage of the 59th EHPRG Meeting.
High-pressure in the headlines
Selection of high-pressure related titles from top scientific journals.
Academia's culture of overwork almost broke me, so I'm working to undo it
by Natalia Ingebretsen Kucirkova
Nature, Published online: 31 January 2023; doi:10.1038/d41586-023-00241-8
For young immigrant women like me, the pressures of early career research are even greater than for most. But it doesn't have to be that way.
31 Jan 2023 at 12:00am
A perspective on the microscopic pressure (stress) tensor: History, current understanding, and future challenges
by Kaihang Shi
The Journal of Chemical Physics, Volume 158, Issue 4, January 2023.
The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this Perspective, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and nonequilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties with many-body and long-range potentials; the insufficiency of software and computational tools; and the lack of experimental routes to probe the pressure tensor at the nanoscale. Possible future directions are suggested.
30 Jan 2023 at 12:48pm
Physical mechanisms of the Soret effect in binary Lennard-Jones liquids elucidated with thermal-response calculations
by Patrick K. Schelling
The Soret effect is the tendency of fluid mixtures to exhibit concentration gradients in the presence of a temperature gradient. Using molecular-dynamics simulation of two-component Lennard-Jones liquids, it is demonstrated that spatially sinusoidal heat pulses generate both temperature and pressure gradients. Over short timescales, the dominant effect is the generation of compressional waves, which dissipate over time as the system approaches mechanical equilibrium. The approach to mechanical equilibrium is also characterized by a decrease in particle density in the high-temperature region and an increase in particle density in the low-temperature region. It is demonstrated that concentration gradients develop rapidly during the propagation of compressional waves through the liquid. Over longer timescales, heat conduction occurs to return the system to thermal equilibrium, with the particle current acting to restore a more uniform particle density. It is shown that the Soret effect arises due to the fact that the two components of the fluid exhibit different responses to pressure gradients. First, the so-called isotope effect occurs because light atoms tend to respond more rapidly to evolving conditions. In this case, there appears to be a connection to previous observations of "fast sound" in binary fluids. Second, it is shown that the partial pressures of the two components in equilibrium, and more directly, the relative magnitudes of their derivatives with respect to temperature and density, determine which species accumulate in the high- and low-temperature regions. In the conditions simulated here, the dependence of the partial pressure on density gradients is larger than the dependence on temperature gradients. This is directly connected to the accumulation of the species with the largest partial pressure in the high-temperature region and the accumulation of the species with the smallest partial pressure in the low-temperature region. The results suggest that further development of theoretical descriptions of the Soret effect might begin with hydrodynamical equations in two-component liquids. Finally, it is suggested that the recently proposed concept of "thermophobicity" may be related to the sensitivity of partial pressures in a multicomponent fluid to changes in temperature and density.
Cell-Matrix Elastocapillary Interactions Drive Pressure-Based Wetting of Cell Aggregates
by Muhammad Sulaiman Yousafzai, Vikrant Yadav, Sorosh Amiri, Michael F. Staddon, Youssef Errami, Gwilherm Jaspard, Shiladitya Banerjee, and Michael Murrell
Author(s): Muhammad Sulaiman Yousafzai, Vikrant Yadav, Sorosh Amiri, Michael F. Staddon, Youssef Errami, Gwilherm Jaspard, Shiladitya Banerjee, and Michael Murrell
Observations of pressure-driven motion of cells reveal a novel type of cell migration and cooperation between cellular- and tissue-level forces that may point to unexplored modes of cancer cell movement and early organism development.
[Phys. Rev. X 12, 031027] Published Wed Aug 17, 2022
17 Aug 2022 at 12:00pm
Crystal structures, frustrated magnetism, and chemical pressure in Sr-doped ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ perovskites
by Mélanie Viaud, Catherine Guillot-Deudon, Eric Gautron, Maria Teresa Caldes, Guido Berlanda, Philippe Deniard, Philippe Boullay, Florence Porcher, Carole La, Céline Darie, A. Zorko, A. Ozarowski, Fabrice Bert, Philippe Mendels, and Christophe Payen
Author(s): Mélanie Viaud, Catherine Guillot-Deudon, Eric Gautron, Maria Teresa Caldes, Guido Berlanda, Philippe Deniard, Philippe Boullay, Florence Porcher, Carole La, Céline Darie, A. Zorko, A. Ozarowski, Fabrice Bert, Philippe Mendels, and Christophe Payen
The effects of chemical pressure on the structural and magnetic properties of the triple perovskite ${\mathrm{Ba}}_{3}\mathrm{Ni}{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ are investigated by substituting ${\mathrm{Sr}}^{2+}$ ions for ${\mathrm{Ba}}^{2+}$ ions. Two ${\mathrm{Ba}}_{3−x}{\mathrm{Sr}}_{x}\math…
[Phys. Rev. Materials 6, 124408] Published Tue Dec 20, 2022
20 Dec 2022 at 11:00am
Pressure-induced metallization and superconductivity in the layered van der Waals semiconductor GaTe
by Jin Jiang, Xuliang Chen (陈绪亮), Shuyang Wang, Chao An, Ying Zhou, Min Zhang, Yonghui Zhou, and Zhaorong Yang
Author(s): Jin Jiang, Xuliang Chen (陈绪亮), Shuyang Wang, Chao An, Ying Zhou, Min Zhang, Yonghui Zhou, and Zhaorong Yang
Investigations show that the physical properties of the layered van der Waals semiconductor GaTe are strongly layer dependent. Here, the authors systematically study its properties by using external pressure as a tuning knob. They first find pressure-induced metallization and superconductivity simultaneously occurring at ~3 GPa, presumably owing to a quasi-two-dimensional to three-dimensional structural crossover and to an electronic phase transition due to a change of the bonding nature between layers from weak van der Waals interaction to strong Coulomb coupling. A second distinct superconducting phase is then observed at ~10 GPa due to a structural transition.
[Phys. Rev. B 107, 024512] Published Tue Jan 31, 2023
$GW+\mathrm{EDMFT}$ investigation of ${\mathrm{Pr}}_{1−x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{2}$ under pressure
by Viktor Christiansson, Francesco Petocchi, and Philipp Werner
Author(s): Viktor Christiansson, Francesco Petocchi, and Philipp Werner
Motivated by the recent experimental observation of a large pressure effect on ${T}_{c}$ in ${\mathrm{Pr}}_{1−x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{2}$, we study the electronic properties of this compound as a function of pressure for $x=0$ and 0.2 doping using self-consistent $GW+\mathrm{EDMFT}$. Our …
[Phys. Rev. B 107, 045144] Published Mon Jan 30, 2023
Ramp compression of tantalum to multiterapascal pressures: Constraints of the thermal equation of state to 2.3 TPa and 5000 K
by M. G. Gorman, C. J. Wu, R. F. Smith, L. X. Benedict, C. J. Prisbrey, W. Schill, S. A. Bonev, Z. C. Long, P. Söderlind, D. Braun, D. C. Swift, R. Briggs, T. J. Volz, E. F. O'Bannon, P. M. Celliers, D. E. Fratanduono, J. H. Eggert, S. J. Ali, and J. M. McNaney
Author(s): M. G. Gorman, C. J. Wu, R. F. Smith, L. X. Benedict, C. J. Prisbrey, W. Schill, S. A. Bonev, Z. C. Long, P. Söderlind, D. Braun, D. C. Swift, R. Briggs, T. J. Volz, E. F. O'Bannon, P. M. Celliers, D. E. Fratanduono, J. H. Eggert, S. J. Ali, and J. M. McNaney
The authors use intense laser pulses to compress solid tantalum to pressures in excess of 20 million atmospheres. By combing their experimental measurements with existing high-pressure, high-temperature data on Ta, they can create an experimentally bounded high-temperature equation of state that is valid up to multiterapascal pressures and thousands of degrees kelvin. The equation of state may serve as a useful pressure standard at the extreme compressions and elevated temperatures now achievable in state-of-the-art static compression experiments. This work also provides a clear road map for building an accurate high-temperature equation of state catalogue of materials at extreme conditions.
[Phys. Rev. B 107, 014109] Published Fri Jan 27, 2023
Pressure suppression of the excitonic insulator state in ${\mathrm{Ta}}_{2}{\mathrm{NiSe}}_{5}$ observed by optical conductivity
by H. Okamura, T. Mizokawa, K. Miki, Y. Matsui, N. Noguchi, N. Katayama, H. Sawa, M. Nohara, Y. Lu, H. Takagi, Y. Ikemoto, and T. Moriwaki
Author(s): H. Okamura, T. Mizokawa, K. Miki, Y. Matsui, N. Noguchi, N. Katayama, H. Sawa, M. Nohara, Y. Lu, H. Takagi, Y. Ikemoto, and T. Moriwaki
The layered chalcogenide ${\mathrm{Ta}}_{2}{\mathrm{NiSe}}_{5}$ has recently attracted much interest as a strong candidate for a long-sought excitonic insulator (EI). Since the physical properties of an EI are expected to depend sensitively on the external pressure ($P$), it is important to clarify …
Stability of the Pb divalent state in insulating and metallic $\mathrm{PbCr}{\mathrm{O}}_{3}$
by Jianfa Zhao, Shu-Chih Haw, Xiao Wang, Lipeng Cao, Hong-Ji Lin, Chien-Te Chen, Christoph J. Sahle, Arata Tanaka, Jin-Ming Chen, Changqing Jin, Zhiwei Hu, and Liu Hao Tjeng
Author(s): Jianfa Zhao, Shu-Chih Haw, Xiao Wang, Lipeng Cao, Hong-Ji Lin, Chien-Te Chen, Christoph J. Sahle, Arata Tanaka, Jin-Ming Chen, Changqing Jin, Zhiwei Hu, and Liu Hao Tjeng
PbCrO3 is a perovskite that undergoes a large volume collapse and concurrently an insulator-metal transition with pressure. While one may think that the volume collapse can be associated with conversion of Pb2+ ions to the smaller Pb4+, the spectroscopic measurements in this experiment reveal that the Pb ions remain divalent. Instead, the Cr ions undergo a transition from a charge-disproportionated 2Cr3+/Cr6+ configuration to a homogenous Cr4+ with pressure. Thus, the authors can explain the insulating/metallic nature of PbCrO3 at low/high pressures, respectively.
[Phys. Rev. B 107, 024107] Published Thu Jan 26, 2023
Investigation of null-matrix alloy gaskets for a diamond-anvil-cell on high pressure neutron diffraction experiments
by Shinichi Machida
Volume 42, Issue 4, December 2022, Page 303-317
11 Nov 2022 at 6:06am
Sound velocity anomalies of limestone at high pressure and implications for the mantle wedge
by Fengxia Sun
12 Nov 2022 at 1:16pm
Development of an explicit pressure explicit saturation (EPES) method for modelling dissociation processes of methane hydrate
by Giovanni Luzi
27 Dec 2022 at 4:10am | CommonCrawl |
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Nuclear Talent
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We are Culinary Services.
Who ordered the extra side of 44Ti?
Alex Long (parameters)
MacKenzie Warren (code)
Nathan Parzuchowski (analysis)
Sensitivity studied of 44Ti production in in core-collapse supernova environments.
There are many uncertainties in our understanding of core-collapse supernovae, including the explosion mechanism and nucleosynthesis. One way to gain insight into these phenomena is to study the nucleosynthesis of radioactive isotopes in the shock-heated material. These isotopes, such as 44Ti and 56Ni, determine the features of the supernova light curve. Observations of supernova remnants can be used to put bounds on the production of these isotopes.
Using simulations, we can use these observations to gain insight into the supernova environment. By matching observed abundances, we can gain insight into the environment in which this nucleosynthesis must have taken place and in turn, the details of the explosion mechanism. However, most core-collapse supernova simulations do not include sufficiently large reaction networks to simulate this nucleosynthesis.
If the shock heating is sufficient, the material will be in Nuclear Statistical Equilibrium (NSE). The isotopic abundances will be set by the thermodynamic environment (i.e. temperature and density).
We have chosen to do a parameter space study in peak temperature, density, and electron fraction, tarting with a set parameter space of peak temperatures [T9 = 4 - 7] and densities [$\rho$ = 105 - 107 g/cm3] for three values of the electron fraction [Ye = 0.45, 0.50, 0.55]. This parameter space roughly corresponds with the shock heated region in simulations of Cassiopeia A-like supernovae (Young & Fryer 2007).
We use analytic adiabatic freeze-out trajectories (Hoyle et al. 1964; Fowler & Hoyle 1964) which satisfy the differential equations:
\begin{equation} \frac{dT}{dt} = \frac{-T}{3\tau} \hspace{1cm} \frac{d\rho}{dt} = -\frac{\rho}{\tau} \end{equation}
Where $\tau$ is the free-fall timescale. This leads to temperature and density trajectories:
\begin{equation} T(t) = T_0 exp(-t/3\tau) \hspace{1cm} \rho (t) = \rho_0 exp(-t/\tau) \end{equation} where $T_0$ and $\rho_0$ are the peak temperature and density in the supernova.
We used the XNet reaction network code. Our code included 447 isotopes ranging from hydrogen through germanium. We took the reaction rates from the JINA Reaclib database. We set the threshold temperature for NSE to be 5 GK.
Trends in 44Ti and 56Ni from Core-Collapse Supernovae (Magkotsios et al 2010)
Nuclear Reactions Governing the Nucleosynthesis of 44Ti (The et al 1998)
X-ray and gamma-ray studies of Cas A (Vink 2004) Uncertainties in Supernova Yields I One-dimensional Explosions (Young & Fryer 2007) Asymmetries in core-collapse supernovae from maps of radioactive <sup>44</sup>Ti in Cassiopeia A (Grefenstette et al 2014)
culinary_services.1401995671.txt.gz · Last modified: 2014/06/05 15:14 by warren | CommonCrawl |
A History of Pi
A History of Pi (also titled A History of π) is a 1970 non-fiction book by Petr Beckmann that presents a layman's introduction to the concept of the mathematical constant pi (π).[1]
A History of Pi
Book cover of A History of Pi (3rd ed.)
AuthorPetr Beckmann
CountryUnited States
LanguageEnglish
SubjectMathematics
General Sciences
History of mathematics
PublisherGolem Press (1st, 2nd ed.)
St. Martin's Press (3rd ed.)
Hippocrene Books (Reprint ed.)
Publication date
1970
Pages190 pages
ISBN978-0-911762-07-5
OCLC99082
Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
• Area of a circle
• Circumference
• Use in other formulae
Properties
• Irrationality
• Transcendence
Value
• Less than 22/7
• Approximations
• Madhava's correction term
• Memorization
People
• Archimedes
• Liu Hui
• Zu Chongzhi
• Aryabhata
• Madhava
• Jamshīd al-Kāshī
• Ludolph van Ceulen
• François Viète
• Seki Takakazu
• Takebe Kenko
• William Jones
• John Machin
• William Shanks
• Srinivasa Ramanujan
• John Wrench
• Chudnovsky brothers
• Yasumasa Kanada
History
• Chronology
• A History of Pi
In culture
• Indiana Pi Bill
• Pi Day
Related topics
• Squaring the circle
• Basel problem
• Six nines in π
• Other topics related to π
Author
Beckmann was a Czechoslovakian who fled the Communist regime to go to the United States. His dislike of authority gives A History of Pi a style that belies its dry title. For example, his chapter on the era following the classical age of ancient Greece is titled "The Roman Pest";[2] he calls the Catholic Inquisition the act of "insane religious fanatic"; and he says that people who question public spending on scientific research are "intellectual cripples who drivel about 'too much technology' because technology has wounded them with the ultimate insult: 'They can't understand it any more.'"
Beckmann was a prolific scientific author who wrote several electrical engineering textbooks and non-technical works, founded Golem Press, which published most of his books, and published his own monthly newsletter, Access to Energy. In his self-published book Einstein Plus Two and in Internet flame wars, he claimed that the theory of relativity is incorrect.[3]
Bibliography
A History of Pi was originally published as A History of π in 1970 by Golem Press. This edition did not cover any approximations of π calculated after 1946. A second edition, printed in 1971, added material on the calculation of π by electronic computers, but still contained historical and mathematical errors, such as an incorrect proof that there exist infinitely many prime numbers.[4] A third edition was published as A History of Pi in 1976 by St. Martin's Press. It was published as A History of Pi by Hippocrene Books in 1990.[5] The title is given as A History of Pi by both Amazon[6] and by WorldCat.[7]
1. Beckmann, Petr (1970), A History of π (1st ed.), Golem Press, p. 190, ISBN 0-911762-07-8
2. Beckmann, Petr (1971-01-01), A History of π (2nd ed.), Golem Press, p. 196, ISBN 0-911762-12-4
3. Beckmann, Petr (1976-07-15), A History of Pi (3rd ed.), St. Martin's Press, p. 208, ISBN 0-312-38185-9
4. Beckmann, Petr (1977), A History of π (4th ed.), Golem Press, p. 202, ISBN 0-911762-18-3
5. Beckmann, Petr (1982), A History of π (5th ed.), Golem Press, p. 202, ISBN 0-911762-18-3
6. Beckmann, Petr (1990-06-01), A History of Pi (Reprint ed.), Hippocrene Books, p. 200, ISBN 0-88029-418-3
See also
• History of Pi
References
1. Drum, Kevin (December 2, 1996). "A History of Pi, by Petr Beckman". Archived from the original on July 4, 2007. Retrieved April 13, 2014.
2. Thoreau, Book Recommendation: A History of Pi Archived 2011-07-16 at the Wayback Machine
3. Farrell, John (2000-07-06). "Did Einstein cheat?". Salon. Retrieved 2022-03-25.
4. Gould, Henry W. (1974). "Review of A History of π". Mathematics of Computation. 28 (125): 325–327. doi:10.2307/2005843. ISSN 0025-5718. JSTOR 2005843.
5. "A History of PI by Petr Beckmann ", GoodReads
6. ASIN 0312381859
7. OCLC 472118858
| Wikipedia |
I want to know if this problem can be verified or rejected.
I tried to make a counterexample but I couldn't find anything, I wanted to prove it using the definition too, but I did not get anything. Is there a way to prove, or is there a counterexample?
The left-hand side of your inequality counts the distinct pairs from a size-$k$ set, while the right-hand side takes a partition into size-$x_i$ disjoint subsets, then counts the number of pairs that don't cut across partitions.
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Counting all possible ways to choose $M$ numbers from $1$ to $N$ given some conditions .
given two sets how many ways can we choose 2 subsets of same length?
What is the maximum number of subsets we can choose from a set of size 20 such that no two subsets have more than 2 common elemtns. | CommonCrawl |
Simulation of impacts on elastic–viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes
Thomas Heuzé1
The flux-difference splitting finite volume method (Leveque in J Comput Phys 131:327–353, 1997; Leveque in Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press, 2002) is here employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The formulation is second order accurate in space through flux limiters, embeds the corner transport upwind method, and uses a fractional-step method to compute the relaxation operator. Elastic–viscoplastic constitutive models falling within the framework of generalized standard materials (Halphen and Nguyen in J Mech 14:667–688, 1975) in small strains are considered. Many test cases are proposed and two particular viscoplastic constitutive models are studied, on which comparisons with finite element solutions show a very good accuracy of the finite volume solutions, both on stresses and viscoplastic strains.
The numerical simulation of hyperbolic initial boundary value problems including extreme loading conditions such as impacts requires the ability to accurately capture and track the fronts of shock waves induced in the medium. Indeed, this permits to correctly follow the path of waves and hence understand the mechanical phenomena occuring within that medium. For solid-type media, it also allows for an accurate assessment of the propagation of irreversible strains and hence of residual stresses and distortions within the structure. High speed forming processes like electromagnetic material forming [1,2,3] are some application examples of severe loading conditions in which the track of wave fronts is important both for understanding the development of irreversible strains in the workpiece and optimizing its final shape. Hence, these problems require numerical schemes capable to rewrite the film of history of loading undergone by any material point with sufficient precision to permit the understanding of particular physical phenomena of interest, while freeing oneself from any numerical disturbance that might impair that understanding. In particular, numerical schemes able to represent regular as well as discontinuous solutions are of interest; more precisely, they should meet both high orders of accuracy in regions where the solution is smooth and a high resolution of discontinuities when they occur without any numerical spurious oscillations appearing in their vicinity.
The numerical simulation of impacts on dissipative solids has been and is again mainly performed with the classical finite element method coupled with centered differences or Newmark finite difference schemes in time [4, 5], which is implemented in many industrial codes. Indeed, the finite element method is still popular in the solid mechanics community for, among others, its easy implementation of nonlinearities of partial differential equations, that is for solid-type media it enables to account easily for history-dependent constitutive equations through appropriate integration algorithms [6] and storage of internal variables at integration points in each element. However, on the one hand the amount of artificial viscosity added to numerical time integrators required to reduce the high frequency noise in the vicinity of shocks is hard to assess properly in order to remove the sole spurious oscillations, without destroying the accuracy of the numerical solution. On the other hand, finite elements do not use any feature of the characteristic structure of the set of hyperbolic equations, and is hence not the best suited method to accurately capture discontinuous solutions.
The finite volume method, initially developed for the simulation of gas dynamics [7, 8], has gained recently more and more interest for problems involving impacts on solid media (see e.g. [9,10,11,12,13,14,15,16,17]). This family of methods show some advantages to achieve an accurate tracking of wavefronts; among others (i) the continuity of fields is not enforced on the mesh in its cell-centered version, that allows for capturing discontinuous solutions, (ii) the characteristic structure of hyperbolic equations can be introduced within the numerical solution, either through the explicit solution of a Riemann problem at cell interfaces, or in an implicit way through the construction of the numerical scheme, (iii) the same order of convergence is achieved for both the velocity and stress fields [12], and (iv) the amount of numerical viscosity introduced can be controlled locally as a function of the local regularity of the solution, so that to permit the elimination of spurious numerical oscillations while preserving a high order of accuracy in more regular zones.
Since the early work of Wilkins [18] and Trangenstein et al. [19], several authors have proposed many ways to simulate impacts on dissipative solid media, such as elastic–plastic and elastic–viscoplastic solids, with this class of methods. These can be merely classified into Eulerian approaches, generally based on a fractional-step method to treat the irreversible processes [9, 10, 13, 17, 20] and used for extremely high strain, strain rate and pressure problems, and lagrangian approaches [14, 16, 21] that allow to follow the path of material particles and hence account for refined history-dependent constitutive equations though limited by mesh entanglement, both being coupled with an approximate Riemann or WENO solver. Eulerian approaches are written in a conservative form with a relaxation operator containing inelastic terms [22], and are often based on the so-called Maxwell-type relaxation approach [9,10,11, 13, 17, 23] which actually refers to an adapted version of Perzyna's elastic–viscoplastic solids [24], in fact in its perfect viscoplasticity version since no hardening rules is generally accounted for, obtained by means of a relaxation process of stresses [11, 17]. Lagrangian approaches have been less investigated and have been so far more treated in elastoplasticity [14, 16] using classical integration of constitutive equations [6] coupled with acoustic Riemann solvers, or using simplified elastic–plastic Riemann solvers [21].
We are interested in this work in elastic–viscoplastic systems, whose study is here focused on the isothermal and linearized geometrical framework, leading to a nonhomogeneous system of partial differential equations, generating a system of weakly discontinuous waves beyond the viscoplastic yield, following a discontinuous (elastic) wave due to the transition between elastic and elastic–viscoplastic ranges. This work intends to apply the flux-difference splitting finite volume method, whose formalism has been made popular by Leveque [7, 25], for the simulation of impacts on elastic–viscoplastic solid media on bidimensional non-uniform quadrilateral meshes. Its derivation for these unstructured meshes follows classical ones [26, 27] for first order terms, but the process of limitation of waves required to achieve high resolution methods here accounts for different orientations between the current and upwind edges. Moreover, the approach is here derived using the class of generalized standard materials [28] (GSM) that describes a convenient framework to define thermodynamically consistent viscoplastic constitutive models, which can embed refined viscoplastic models with respect to these already used with such approach [9, 17], some particular creep and hardening rules being considered in this work. The viscoplastic relaxation system is solved by means of a fractional-step method, whose convection part is solved with the flux-difference splitting formalism.
The paper is organized as follows. First, the elastic–viscoplastic constitutive model, the governing balance laws and the characteristic analysis are presented in "Elastic–viscoplastic Initial Boundary Value Problem" section. Next, the flux-difference splitting finite volume method is presented for bidimensional non-uniform quadrilateral meshes in "The flux-difference splitting finite volume method" section. "Computation of the viscoplastic part" section discusses the asymptotic limit of the elastic–viscoplastic system, and the fractional-step method used to compute the viscoplastic part of the behaviour. At last, several test cases are presented in "Applications" section, mainly conducted within the two-dimensional plane strain assumption, on which comparisons with finite element solutions allow to show the good accuracy of finite volume solutions, both on stresses and viscoplastic strains. The viscoplastic flow computed with two viscoplastic constitutive models is compared on the last example. Especially, a very simple Chaboche-type [29] viscoplastic model coupled with Prager's linear kinematic [30] hardening, and a more refined Chaboche–Nouailhas' [31] one coupled with the Armstrong–Frederick's [32, 33] kinematic nonlinear hardening law are considered.
Elastic–viscoplastic Initial Boundary Value Problem
Elastic–viscoplastic constitutive model
Following the local accompanying state approach [29, 34], the thermodynamical state of the material is described by a set of state variables that consists of the linearized strain tensor \(\varvec{\varepsilon }\) and the temperature T, plus some internal state variables \({\mathbf {Z}}\) which describe the evolution of internal microstructure and stored energy due to plastic deformation and other irreversible processes. Under small strain assumption, the total strain \(\varvec{\varepsilon }\) is additively decomposed into an elastic strain (\(\varvec{\varepsilon }^e\)) and a plastic strain (\(\varvec{\varepsilon }^p\)):
$$\begin{aligned} \varvec{\varepsilon } = \varvec{\varepsilon }^e + \varvec{\varepsilon }^p \end{aligned}$$
The set of internal state variables \({\mathbf {Z}}\) consists here of the plastic strain \(\varvec{\varepsilon }^p\) plus some additional variables \(\alpha _I\), \(1\le I \le N\). We assume the existence of a Helmholtz free-energy density potential \(\psi (\varvec{\varepsilon },T,\mathbf {Z})\equiv \psi (\varvec{\varepsilon }-\varvec{\varepsilon }^p,T,\alpha _{I, \, 1\le I \le N})\), concave with respect to temperature and convex with respect to other variables, from which the state laws can be derived:
$$\begin{aligned} \varvec{\sigma } = \rho \frac{\partial \psi }{\partial \varvec{\varepsilon }^e}\, ; \quad s= - \frac{\partial \psi }{\partial T} \, ; \quad A_I = \rho \frac{\partial \psi }{\partial \alpha _I}, \quad 1\le I \le N \end{aligned}$$
where \(\varvec{\sigma }\), s and \(A_I\) refer to the Cauchy stress tensor, the entropy density and the force conjugated to variable \(\alpha _I\) respectively. Thus, the mechanical dissipation reads:
$$\begin{aligned} {\mathscr {D}}= \varvec{\sigma } : \dot{\varvec{\varepsilon }}^p - A_I \cdot \dot{\alpha }_I = \mathbf {Y} \cdot \dot{\mathbf {Z}} \ge 0 \end{aligned}$$
$$\begin{aligned} \mathbf {Y}= \{\varvec{\sigma }, A_I\} ; \quad \mathbf {Z}= \{\varvec{\varepsilon }^p, - \alpha _I \} \end{aligned}$$
where the vector \(\mathbf {Y}\) denotes the thermodynamic forces conjugated to internal variables \(\mathbf {Z}\), and the dot (\({\dot{\square }}\)) applied to quantity \(\square \) stands for a time rate. Following the framework of GSM [28], we assume the existence of a dissipation pseudo-potential \(\Phi (\mathbf {Y})\), convex with respect to flux variables \(\mathbf {Y}\), and containing the origin, such that:
$$\begin{aligned} \dot{\mathbf {Z}}=\frac{\partial \Phi }{\partial \mathbf {Y}} \end{aligned}$$
which ensures the mechanical dissipation
$$\begin{aligned} {\mathscr {D}}=\mathbf {Y} \cdot \frac{\partial \Phi }{\partial \mathbf {Y}} \ge 0 \end{aligned}$$
to be non-negative. The dissipation pseudo-potential \(\Phi (\mathbf {Y})\) defines a family of equipotential surfaces on which any point yields the same dissipation and effective viscoplastic strain rate, and may depend on flux variables \(\mathbf {Y}\) through a yield function \(f(\mathbf {Y})\), so that the surface of zero potential delimits the elastic convex
$$\begin{aligned} {\mathscr {C}} = \{ \mathbf {Y} | f (\mathbf {Y}) \le 0 \} \end{aligned}$$
The flow rule thus reads:
$$\begin{aligned} \dot{\mathbf {Z}}=\frac{\partial \Phi }{\partial f} \frac{\partial f}{\partial \mathbf {Y}} = \dot{p} \frac{\partial f}{\partial \mathbf {Y}} \end{aligned}$$
$$\begin{aligned} \dot{p} = \frac{\partial \Phi }{\partial f} \end{aligned}$$
denotes the viscoplastic flow intensity or effective viscoplastic strain rate, and \(\partial f / \partial \mathbf {Y}\) is the flow direction, normal to the yield function f.
Balance laws and quasi-linear form
Let's consider a continuum body \(\Omega \), of boundary \(\partial \Omega \), and current coordinates \(\mathbf {x}\). The initial boundary value problem driving the motion of the solid in the small strains framework must satisfy the conservation of linear momentum, written here with neglected body forces, the geometric relationships between the strain rate \(\dot{\varvec{\varepsilon }}\) and the velocity field \(\mathbf {v}\), plus the constitutive equations that consist of the strain rate partition (1) combined with the elastic law (2)\(_1\), the viscoplastic flow rule, and evolution equations of other internal parameters:
$$\begin{aligned} \rho \dot{\mathbf {v}}&= \varvec{\nabla } \cdot \varvec{\sigma } \end{aligned}$$
$$\begin{aligned} \dot{\varvec{\varepsilon }}&= \frac{1}{2} ( \varvec{\nabla } \mathbf {v} + (\varvec{\nabla } \mathbf {v})^T) \end{aligned}$$
$$\begin{aligned} \dot{\varvec{\sigma }}&= \mathbf {C}:(\dot{\varvec{\varepsilon }} - \dot{\varvec{\varepsilon }}^p) \end{aligned}$$
$$\begin{aligned} \dot{\varvec{\varepsilon }}^p&= \frac{\partial \Phi }{\partial \varvec{\sigma }} \end{aligned}$$
$$\begin{aligned} - {\dot{\alpha }}_I&= \frac{\partial \Phi }{\partial A_I} \, , \quad 1\le I \le N \end{aligned}$$
which hold \(\forall (\mathbf {x},t) \in \Omega \times ]0,T]\), supplemented with appropriate initial and boundary conditions, and \(\mathbf {C}\) denotes the fourth order elastic stiffness tensor in Eq. (12). In the above system, Eqs. (10) to (13) can be combined in order to write a system of balance equations of the form:
$$\begin{aligned} \frac{\partial {{\mathscr {Q}}}}{\partial t} + \varvec{\nabla } \cdot {\mathscr {F}} = {\mathscr {S({\mathscr {Q}})}} \end{aligned}$$
where \({\mathscr {Q}}\), \({\mathscr {F}}\) and \({\mathscr {S}}\) denote the arrays of balanced quantities, associated fluxes, and the source term respectively, defined as follows assuming a homogeneous medium:
$$\begin{aligned} {{\mathscr {Q}}} = \begin{Bmatrix} \varvec{\sigma }\\ \mathbf {v} \end{Bmatrix} \, ; \quad {\mathscr {F}}= \begin{Bmatrix} - \mathbf {C}: \mathbf {v} \otimes \mathbf {1} \\ - \frac{\varvec{\sigma }}{\rho } \end{Bmatrix} \, ; \quad {{\mathscr {S}} }= \begin{Bmatrix} - \mathbf {C}: \frac{\partial \Phi }{\partial \varvec{\sigma }} \\ \mathbf {0} \end{Bmatrix} \end{aligned}$$
where \(\mathbf {1}\) is the second order identity tensor. More precisely, the system (15) can be derived in full matrix form with cartesian coordinates as:
$$\begin{aligned} \frac{\partial {{\mathcal {Q}}}}{\partial t} + \sum _{k=1}^3 \frac{\partial {\mathcal {F}}_k}{ \partial x_k} = {{\mathcal {S}}({\mathcal {Q}})} \end{aligned}$$
with vector components such that
$$\begin{aligned} \begin{aligned}&{{\mathcal {Q}}} = \begin{Bmatrix} \left\{ \varvec{\sigma } \cdot \mathbf {e}_i \right\} _{1\le i \le 3}\\ \mathbf {v} \end{Bmatrix} \, ; \quad {\mathcal {F}}_k= \begin{Bmatrix} \left\{ -\mathbf {C}: \mathbf {v} \otimes \mathbf {e}_k \cdot \mathbf {e}_i \right\} _{1\le i \le 3}\\ - \frac{\varvec{\sigma }\cdot \mathbf {e}_k}{\rho } \end{Bmatrix} ={\mathcal {F}} \cdot \mathbf {e}_k \\&{{\mathcal {S}}} = \begin{Bmatrix} \left\{ - \mathbf {C}: \frac{\partial \Phi }{\partial \varvec{\sigma }} \cdot \mathbf {e}_i \right\} _{1\le i \le 3}\\ \mathbf {0} \end{Bmatrix} \end{aligned} \end{aligned}$$
where \(\mathbf {e}_i\) (\(1\le i \le 3\)) refer to the cartesian basis vectors, and redundant equations due to symmetry of the stress tensor are of course not considered. The term \(\mathbf {C}: \mathbf {v} \otimes \mathbf {e}_k \cdot \mathbf {e}_i\) in the kth component of the fluxes simplifies into \(C_{pijk}v_j \mathbf {e}_p\), and if elastic isotropy is assumed (leading to the following expression \(C_{ijkl}=\lambda \delta _{ij} \delta _{kl}+\mu (\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk})\), \(\lambda \) and \(\mu \) denoting the Lamé's parameters), it simplifies as follows:
$$\begin{aligned} \mathbf {C}: \mathbf {v} \otimes \mathbf {e}_k \cdot \mathbf {e}_i = \lambda v_k \mathbf {e}_i + \mu (\mathbf {v} \delta _{ik} + v_i \mathbf {e}_k) \end{aligned}$$
where \(\delta _{ik}\) denotes the Kronecker delta symbol. From the system of balance equation (15), a quasi-linear form is conveniently written as follows:
$$\begin{aligned} \frac{\partial {{\mathscr {Q}}}}{\partial t} + \sum _{k=1}^3 {{\mathscr {J}}_k} \frac{\partial {{\mathscr {Q}}}}{\partial x_k} = {{\mathscr {S}} } \end{aligned}$$
$$\begin{aligned} {{\mathscr {J}}_k }= \frac{\partial {\mathscr {F}}_k}{\partial {{\mathscr {Q}}}} = - \begin{bmatrix} {\mathbb {O}}&\mathbf {C}: \mathbf {e}_k \otimes \mathbf {1} \\ \frac{\mathbf {e}_k \cdot {\mathbb {I}}}{\rho }&\mathbf {0} \end{bmatrix} \end{aligned}$$
where \({\mathbb {O}}\) and \({\mathbb {I}}\) denote fourth order zero and identity tensors respectively, the same in full matrix form can be derived from Eq. (17):
$$\begin{aligned} {{\mathcal {J}}_k }= \frac{\partial {\mathcal {F}}_k}{\partial {{\mathcal {Q}}}} = -\, \begin{bmatrix}&&[C_{p1qk}]_{1\le p,q \le 3} \\&\mathbf {0}_{9\times 9}&[C_{p2qk}]_{1\le p,q \le 3} \\&&[C_{p3qk}]_{1\le p,q \le 3} \\ \frac{\delta _{1k}\mathbf {1}_{3\times 3}}{\rho }&\frac{\delta _{2k}\mathbf {1}_{3\times 3}}{\rho }&\frac{\delta _{3k}\mathbf {1}_{3 \times 3}}{\rho }&\mathbf {0}_{3 \times 3} \end{bmatrix} \end{aligned}$$
Characteristic analysis
Since irreversible viscoplastic effects only occur in the source term (16) by means of the viscoplastic flow rule (13), the homogeneous part of this system is governed by the sole elastic part of the elastic–viscoplastic behavior, meaning that any information and in particular irreversible processes propagate along elastic characteristic curves. Let's consider the coordinate \(X = \mathbf {x} \cdot \mathbf {n}\) and \(\mathbf {n}\) refers to an any direction of propagation, along the characteristic curves \(dX = \lambda _p dt\), \(1\le p \le 9\), the eigensystem
$$\begin{aligned} {{\mathscr {J}}_N } \mathbf {K}^{(p)} = \lambda _p \mathbf {K}^{(p)} ; \quad \mathbf {K}^{(p)} = \begin{Bmatrix} \varvec{\sigma }^{(p)} \\ \mathbf {v}^{(p)} \end{Bmatrix} \end{aligned}$$
formed with the Jacobian matrix associated to the quasi-linear form (20)
$$\begin{aligned} {{\mathscr {J}}_N} = \frac{ \partial ({\mathscr {F}} \cdot \mathbf {n}) }{\partial {{\mathscr {Q}}}} = {\mathscr {J}}_k n_k = - \begin{bmatrix} {\mathbb {O}}&\mathbf {C}: \mathbf {n} \otimes \mathbf {1} \\ \frac{\mathbf {n} \cdot {\mathbb {I}} }{\rho }&\mathbf {0} \end{bmatrix} \,, \end{aligned}$$
\(\mathbf {K}^{(p)}\) denoting the right eigenvector associated to the pth eigenvalue (or characteristic speed) \(\lambda _p\) of the Jacobian matrix \({\mathscr {J}}_N\), should be diagonalizable with real eigenvalues for the system (20) to be hyperbolic. The system (23) can be decoupled into the two following systems:
$$\begin{aligned}&- \mathbf {C}:\mathbf {n} \otimes \mathbf {v}^{(p)} =\lambda _p \varvec{\sigma }^{(p)} \end{aligned}$$
$$\begin{aligned}&- \frac{\varvec{\sigma }^{(p)} \cdot \mathbf {n}}{\rho } = \lambda _p \mathbf {v}^{(p)} \end{aligned}$$
Eliminating \(\varvec{\sigma }^{(p)}\) by inserting (26) into (25) yields the symmetric eigenvalue problem
$$\begin{aligned} \mathbf {C}_{NN} \cdot \mathbf {Y}_R^{(K)} = \omega ^{(K)} \mathbf {Y}_R^{(K)} \quad 1 \le K \le 3 \end{aligned}$$
where \([\mathbf {C}_{NN}]_{ij}=C_{ijkl}n_j n_l\) denotes the acoustic elastic tensor, which admits three nonzero eigenvalues and associated right eigenvectors \(\{\omega ^{(K)}; \, \mathbf {Y}_R^{(K)} \}_{1 \le K \le 3}\) thanks to the coercivity property of the elastic stiffness tensor. This leads to six distinct characteristic speeds \(\lambda _p=\pm \sqrt{\omega _K/\rho }\), \(1\le p \le 6\), and right eigenvectors \(\mathbf {v}^{(2K)}=\mathbf {v}^{(2K-1)}=\mathbf {Y}_R^{(K)}\). The characteristic fields can be rewritten for the non zero eigenvalues as:
$$\begin{aligned} \left\{ \pm \sqrt{\frac{\omega ^{(K)}}{\rho }} ; \quad \begin{Bmatrix} - \mathbf {C}: \mathbf {n} \otimes \mathbf {Y}_R^{(K)} \\ \pm \, \sqrt{\frac{\omega ^{(K)}}{\rho }} \mathbf {Y}_R^{(K)} \end{Bmatrix} \right\} _{1\le K \le 3} \end{aligned}$$
These are ordered in three pairs corresponding to pressure waves of speed \(U_{P}=\sqrt{(\lambda + 2 \mu )/\rho }\) and shear waves of speed \(U_{S}=\sqrt{\mu /\rho }\) for an isotropic medium, such that \(\lambda _{1,2}= \pm U_{P}\) and \(\lambda _{3,4}= \lambda _{5,6}= \pm U_{S}\). The last step consists in completing the characteristic basis by finding three independent vectors associated to the kernel of the Jacobian matrix, solving equation (26) for a null eigenvalue.
Combining (23) and (20) yields the characteristic equations satisfied along elastic characteristics lines:
$$\begin{aligned} \mathbf {L}^{(p)}\cdot \left( \frac{d {{\mathscr {Q}}}}{dt} - {{\mathscr {S}}({\mathscr {Q}})} \right) = 0 \end{aligned}$$
where \(\mathbf {L}^{(p)}\) refers to the pth left eigenvector of the Jacobian matrix \({\mathscr {J}}_N\). The characteristic equations are particularized for an elastic–viscoplastic medium as
$$\begin{aligned} \mathbf {Y}_L^{(K)} \otimes \mathbf {n}: \left( d \varvec{\sigma } + \mathbf {C}: \frac{\partial \phi }{\partial \varvec{\sigma }} dt \right) \pm \rho \lambda _p \mathbf {Y}_L^{(K)} \cdot d \mathbf {v} =0\,, \quad 1 \le p \le 9 \end{aligned}$$
where \(\mathbf {Y}_L^{(K)}\) refers to the Kth left eigenvector of the elastic acoustic tensor. Observe that time explicitly occurs in Eq. (30), which is a direct consequence of viscoplasticity, hence in general they do not admit first integrals and thus Riemann invariants.
The flux-difference splitting finite volume method
The finite volume method is based on subdividing the computational domain in elementary cells within which, for the cell-centered version, an approximation \(\mathbf {U}_i\) of the vector of balanced quantities \({\mathscr {Q}}\) is defined in the cell i by integral averaging. Let's consider the quadrangular grid cell i shown in Fig. 1, of area \(|A_i|\), whose edge s (\(1\le s \le 4\)) joining points \(P_s\) and \(P_{s+1}\) is of length \(L_s\), and has an outward unit normal \(\mathbf {n}_s\). Integrating a system of conservation laws, i.e. the homogeneous part of (15), on grid cell i yields the following system of ordinary differential equations:
$$\begin{aligned} \left( \frac{d\mathbf {U}}{dt}\right) _i = -\frac{1}{|A_i|} \sum _{s=1}^N L_s \mathbf {F}_s \end{aligned}$$
where \(\mathbf {F}_s\), \(1\le s \le N\), denote numerical fluxes defined at cell interfaces. The order of accuracy, the physical content and the computation cost of the finite volume method essentially rely on the definition of these numerical fluxes.
Quadrangular finite volume
Commonly, the approach consists in exploiting the solution of Riemann problems defined at cell interfaces to determine these fluxes. In particular the stationary solution (\(x/t=0\)) of the Riemann problem yields the well known Godunov's method [35]. The flux-difference splitting formulation, whose formalism has been made popular by Leveque [7, 25], amounts to rephrase the Godunov's method by splitting numerical fluxes defined at cell interfaces in terms of waves contributions, known as fluctuations, denoted by operators \({\mathcal {A}}_k^{\pm } \Delta \mathbf {U}_k\):
$$\begin{aligned} \sum _{s=1}^N L_s \mathbf {F}_s= \sum _{k=1}^P L_k {\mathcal {A}}_k^+ \Delta \mathbf {U}_k + \sum _{l=1}^Q L_l {\mathcal {A}}_l^- \Delta \mathbf {U}_l \end{aligned}$$
where \(P+Q=N\), N being the number of edges of grid cell i. Assuming edge k has left (L) and right (R) states known (Fig. 1), fluctuations read:
$$\begin{aligned} {\mathcal {A}}_k^{\pm } \Delta \mathbf {U}_k = \sum _{p=1}^{M_w} \lambda _p^{\pm } {\mathcal {W}}_k^{(p)} = \sum _{p=1}^{M_w} \lambda _p^{\pm } \alpha _k^{(p)} \mathbf {K}_k^{(p)} \end{aligned}$$
and quantify the effect of all \(M_w\) waves travelling rightward (\(+\)) or leftward (−) respectively in the local frame of edge k. Theses fluctuations are defined with positive and negative parts of characteristic speeds \(\lambda _p^{\pm }\), and associated characteristic directions \(\mathbf {K}_k^{(p)}\equiv \mathbf {K}^{(p)}(\mathbf {n}_k)\), computed at each edge. Each wave is weighted with a coefficient \(\alpha _k^{(p)}\) determined by the projection of the jump of the state vector \(\Delta \mathbf {U}_k = (\mathbf {U}_R-\mathbf {U}_L)_k\) across the edge k onto the characteristic basis \(\mathbf {K}_k^{(p)}\) (\(1\le p \le M_w\), \(M_w\) being the number of waves, equal to five for bidimensional plane strain case for example):
$$\begin{aligned} \Delta \mathbf {U}_k = \sum _{p=1}^{M_w} {\mathcal {W}}_k^{(p)} = \sum _{p=1}^{M_w} \alpha _k^{(p)} \mathbf {K}_k^{(p)} = \mathbf {K}_k \varvec{\alpha }_k \end{aligned}$$
The wave strength is defined by \({\mathcal {W}}_k^{(p)} = \alpha _k^{(p)} \mathbf {K}_k^{(p)}\). This projection amounts to solve a (linear) Riemann problem associated to edge k. The fluctuations associated to each edge are summed to compute the contribution of first order terms to the update of the state of grid cell i; this summation is performed on negative fluctuations for the Q edges having an outward normal, and on positive fluctuations for the P edges having an inward normal as shown in Fig. 2 for a non-cartesian quadrangle.
Fluctuations
However, the above first order scheme can be improved using the class of total variation non-increasing methods [7, 25], that allow to meet both a high order of accuracy in zones where the solution field is regular and a high resolution of discontinuities without spurious numerical oscillations when they occur. The strength of these methods relies on their ability to introduce a controlled amount of numerical viscosity locally, so that to adapt to the local regularity of the solution. One way among many others to implement them amounts to add additional fluxes to these defined from fluctuations in Eq. (33), which are limited so that a non-increasing total variation of the numerical solution be satisfied at each time step, thus ensuring that no new extrema appear that would not have already exist previously. These additional fluxes are either inward (in) or outward (out) ones, depending on the edge normal, and consist of two types of contributions
$$\begin{aligned} \tilde{\mathbf {F}}_l^{\text {in}} = \tilde{\mathbf {F}}_l^{\text {HO}}+\tilde{\mathbf {F}}_l ^{\text {tran}}. \end{aligned}$$
High order fluxes
The first contribution allows to reach a higher order (HO) of accuracy (order two here), defined with wave strength \({\mathcal {W}}_l^{(p)}\) that has been limited, hence denoted \(\tilde{{\mathcal {W}}}_l^{(p)}= {\tilde{\alpha }}_l^{(p)} \mathbf {K}_l^{(p)}\):
$$\begin{aligned} \tilde{\mathbf {F}}_l^{\text {HO}} = \frac{1}{2} \sum _{p=1}^{M_w} |\lambda _l^{(p)}|\left( 1- \frac{\Delta t}{\Delta s_l} |\lambda _l^{(p)}| \right) \tilde{{\mathcal {W}}}_l^{(p)} \end{aligned}$$
where \(\Delta s_l\) refers to the distance between barycenters of grid cells sharing edge l, as shown in Fig. 3. Waves are limited based on an upwind ratio \(\theta _l^{(p)}\) defined for wave p on edge l as:
$$\begin{aligned} \theta _l^{(p)} = \frac{{\mathcal {W}}_J^{(p)}(\mathbf {n}_l)\cdot {\mathcal {W}}_l^{(p)}}{\Vert {\mathcal {W}}_l^{(p)}\Vert ^2} \end{aligned}$$
where J denotes the upwind edge, that is the opposed edge to l belonging to grid cell L located to the left in the local frame of edge l (see Fig. 3) if \(\lambda _l^{(p)}>0\), or the opposed edge belonging to grid cell R located to the right if \(\lambda _l^{(p)}<0\). The upwind ratio (37) can be understood as a certain measure of the local regularity of the solution. However, for non-cartesian quadrangles, upwind and downwind edges do not necessarily have the same normal. Thus, the computation of the upwind ratio (37) is here performed with wave strengths computed in a same local reference frame, that of edge l. Wave strengths \({\mathcal {W}}_J^{(p)}\) are hence computed from weighting coefficients \(\alpha _J^{(p)}\) recomputed in the local frame of edge l:
$$\begin{aligned} \varvec{\alpha }_J(\mathbf {n}_l) = \mathbf {K}^{-1}(\mathbf {n}_l) \cdot \Delta \mathbf {U}_J \end{aligned}$$
where \(\Delta \mathbf {U}_J\) denotes the jump across edge J of the state vector. Wave strengths associated to edge J expressed in the frame of edge l are then corrected as:
$$\begin{aligned} \mathbf {W}_J (\mathbf {n}_l) = \text {diag }(\varvec{\alpha }_J(\mathbf {n}_l)) \cdot \mathbf {K}(\mathbf {n}_l) = \left[ \text {diag }([\mathbf {K}(\mathbf {n}_l)]^{-1} \cdot \Delta \mathbf {U}_J) \right] \cdot \mathbf {K}(\mathbf {n}_l) \end{aligned}$$
where \(\mathbf {W}_J (\mathbf {n}_l)\) is the matrix made of wave strength vectors \({\mathcal {W}}^{(p)}\), \(1\le p \le M_w\). The upwind ratio (37) can thus be correctly computed from (39). The wave strength of wave p associated to edge l is then limited by means of a limiting function \(\phi (\theta _l^{(p)})\):
$$\begin{aligned} {\tilde{\alpha }}_l^{(p)} = \phi (\theta _l^{(p)}) \alpha _l^{(p)} \end{aligned}$$
Upwind edge for wave comparison
Many limiting functions exist and permit to obtain different known finite volume schemes [36]. Some of them enable the numerical scheme to satisfy a non-increasing total variation, so that the appearance of spurious numerical oscillations can be avoided in the vicinity of discontinuities. The Superbee limiter defined by \(\phi (\theta )= \max (0,\min (1,2\theta ),\min (2,\theta ))\) falls in this family, and is used in the following of this work. More generally, the limitation of wave strength amounts to add locally some numerical viscosity, and to locally lower the order of accuracy to properly capture discontinuities. In zones where the solution field is more regular, the limitation is not active and an accuracy of order two can be reached.
Transverse fluxes
The second contribution to additional fluxes (35) enables to improve the stability of the numerical scheme, so that the Courant number can be set at one. These fluxes allow to account for information travelling in bias with respect to the considered grid cell; this is the contribution to the grid cell to be updated of a cell only sharing a node (but not an edge) with it. This method is known from [37] as the Corner Transport Upwind (CTU) method. These fluxes are of great importance to ensure numerical stability for elastic media; indeed elasticity couples strain components through Poisson's effect, so that a transverse information to the considered grid cell should be introduced in the numerical scheme.
Normal and transverse fluctuations defined from edge i
Let's consider the patch of grid cells shown in Fig. 4. One focuses on the edge denoted (i) whose local frame \((\mathbf {n}_i,\mathbf {t}_i)\) is shown. This edge gives rise to the computation of normal fluctuations \({\mathcal {A}}_i^+ \Delta \mathbf {U}\) and \({\mathcal {A}}_i^-\Delta \mathbf {U}\) contributing to grid cells R and L respectively. These normal fluctuations lead to the computation of transverse fluctuations giving contribution to neighboring cells across edges (j) and (k) for cell L, and across edges (m) and (l) for cell R. These transverse fluctuations are computed by projecting normal fluctuations on the characteristic basis associated to the Riemann problem defined on the adjacent edge; it appears as a transverse Riemann solver. For instance the negative normal fluctuation is decomposed on the characteristic basis associated to edge (j) as
$$\begin{aligned} {\mathcal {A}}_i^- \Delta \mathbf {U} = \sum _{p=1}^{M_w} \beta _p \mathbf {K}_j^{(p)} = \mathbf {K}_j \varvec{\beta } \end{aligned}$$
where \(\mathbf {K}_j\) accounts here for the normal \(\mathbf {n}_j\) of edge (j), but also of different material properties between grid cells L and T. Coefficients \(\beta _p\) are determined analytically for bidimensional plane strain elasticity equations. For outward normals shown in Fig. 4, the transverse fluctuations are computed with the positive operator \({\mathcal {B}}^+\), that is only waves with positive characteristic speeds will contribute to this transverse fluctuation
$$\begin{aligned} {\mathcal {B}}_j^+{\mathcal {A}}_i^- \Delta \mathbf {U} = \sum _{p=1}^{M_w} \lambda _p^{+} \beta _p \mathbf {K}_j^{(p)} \end{aligned}$$
An additional numerical flux defined at edges is hence built from these transverse fluctuations:
$$\begin{aligned} \tilde{\mathbf {F}}_j^{\text {tran}} = \frac{\Delta t}{2 \Delta s_j}{\mathcal {B}}_j^+{\mathcal {A}}_i^- \Delta \mathbf {U}_i \end{aligned}$$
which contributes to grid cell T. The flux \(\tilde{\mathbf {F}}_l^{\text {tran}}\) associated to edge l appearing in Eq. (35) thus denotes transverse contributions from adjacent grid cells to cell i.
Gathering first order fluctuations and additional fluxes, and considering an explicit Euler time integration, the state of grid cell i is updated at time \(t_{n+1}\) through the following formula:
$$\begin{aligned} \mathbf {U}_i^{n+1}= & {} \mathbf {U}_i^{n} -\frac{\Delta t}{|A_i|} \left( \sum _{k=1}^P L_k {\mathcal {A}}_k^+ \Delta \mathbf {U}_k + \sum _{l=1}^Q L_l {\mathcal {A}}_l^- \Delta \mathbf {U}_l \right) \nonumber \\&-\frac{\Delta t}{|A_i|}\left( \sum _{k=1}^P L_k \tilde{\mathbf {F}}_k^{\text {out}} - \sum _{l=1}^Q L_l \tilde{\mathbf {F}}_l^{\text {in}} \right) \end{aligned}$$
More generally, this finite volume scheme is linked, in a cartesian case, to a Taylor expansion of the solution in the vicinity of a grid cell. In this framework, Godunov fluxes are first order terms, high order fluxes are second order terms, and transverse fluxes correspond to cross-derivatives.
Computation of the viscoplastic part
Asymptotic limit of the elastic–viscoplastic relaxation system
The elastic–viscoplastic system of Eqs. (10)–(14) appears to be a system of balance equations due to the viscoplastic flow rule (13) and evolution equation (14), leading to the appearance of the source term \({\mathscr {S}}({\mathscr {Q}})\) in the system (15). This source term can be identified to a relaxation operator of the form
$$\begin{aligned} {\mathscr {S({\mathscr {Q}})} = \frac{{\mathscr {R}}({\mathscr {Q}})}{\tau } } \end{aligned}$$
where \({\mathscr {R}}({\mathscr {Q}})\) is the relaxation term, and \(\tau \) the relaxation time or stiffness parameter. The relaxation term \({\mathscr {R}}({\mathscr {Q}})\equiv {\mathscr {R}}(f({\mathscr {Q}}))\) depends on \({\mathscr {Q}}\) through the yield function \(f(\mathbf {Y}\)), and is such that
$$\begin{aligned} {\frac{\partial {\mathscr {R}} ({\mathscr {Q}})}{\partial f}<0 } \end{aligned}$$
leading to the relaxation of the state of the system towards an equilibrium one associated to the yield condition \(f=0\). The system (10)–(14) can thus be identified to a relaxation system [38, 39], that is a system of hyperbolic conservation laws with relaxation. The latter consists of conservation laws coupled with rate equations, here the set of viscoplastic constitutive equations (12)–(14). The relaxation time \(\tau \) governs the time evolution of the viscoplastic strain and internal parameters, and determines how quickly these nonconserved quantities approach their respective equilibrium values. Moreover, in the context of geometric linearization, this system of conservation laws is linear.
In the asymptotic limit \(\tau \rightarrow 0\), the system (15) tends to the equilibrium system
$$\begin{aligned} {\frac{\partial {\mathscr {Q}}}{\partial t} + \varvec{\nabla } \cdot {\mathscr {G}}({\mathscr {Q}}) = \mathbf {0}} \end{aligned}$$
consistent with the enforced yield ("equilibrium") condition \(f=0\). More precisely, it is well known that viscoplasticity tends to rate-independent plasticity for the limit case of vanishing viscosity, or equivalently here for a vanishing relaxation time. In particular, Haupt [40] gave the solution of the evolution of the overstress for thermomechanical processes, and studied its limit for both slow processes and vanishing viscosity. In these cases, the viscoplastic flow rule (13) reduces to the rate-independent plastic one, within which the expression of the plastic multiplier follows from the consistency condition \(\dot{f}=0\). Hence, in the case of plastic loading, the fluxes \({\mathscr {G}}\) of the equilibrium system (47) are defined from these of \({\mathscr {F}}\) (15) by replacing the elastic stiffness tensor by the tangent moduli \(\mathbf {H}\) of the rate-independent plastic constitutive model obtained for the limit case of vanishing viscosity. Another way to show this result is to perform an expansion analogous to that of Chapman–Enskog for relaxation systems [38, 39, 41,42,43], here considered in the vicinity of the equilibrium condition \(f=0\), as shown in the one-dimensional case in [44] with the elastic–viscoplastic model of [45]. The leading term of this expansion yields (47), while the \({\mathcal {O}}(\tau )\) correction gives an additional dissipative contribution.
Note that the equilibrium system (47) satisfies the subcharacteristic condition [38], meaning that the characteristic speeds of the equilibrium system (47) should be interlaced between these of the relaxation system (15). If isotropic elastic and plastic behaviour is assumed, this means that both plastic shear \(c_s^P\) and pressure \(c_L^P\) wave velocities should be bounded by their elastic counterparts (\(c_s\) and \(c_L\)):
$$\begin{aligned} |c_s^P|\le |c_s|\;\text { and }\;|c_L^P|\le |c_L| \end{aligned}$$
This arises from the definition of tangent moduli \(\mathbf {H}\), whose absolute values of eigenvalues are by construction lower than these of the elastic stiffness tensor. The subcharacteristic condition can be understood as a stability condition, and the consequences of its violation have been studied for the linear case in [41].
Properties of numerical schemes
A system of hyperbolic conservation laws with relaxation is said to be stiff when the relaxation time \(\tau \) is small compared to the time scale determined by the characteristic speeds of the system and some appropriate length scales [42], or put in another way, the wave-propagation behavior of interest occurs on a much slower time scale than the fastest time scales of the ordinary differential equation (ODE) arising from the source term. This means that if the solution is perturbed away from its equilibrium condition, then it rapidly relaxes back towards the equilibrium.
Solving stiff hyperbolic equations with relaxation can be even more challenging than solving stiff ODEs. Indeed in a stiff hyperbolic equation, the fastest reactions are often not in equilibrium everywhere. The stiffness of the relaxation operator can cause many difficulties to numerical schemes, particularly on coarse, underresolved (\(\Delta t \gg \tau \)) grids. Stiff source terms and underresolved numerical methods, though stable, may yield spurious nonphysical or poor numerical solutions [46]. Accordingly, numerical schemes should be designed so that to satisfy some particular properties to ensure asymptotic convergence, accuracy, and stability.
In particular, numerical schemes should (i) use coarse grids [42] that do not resolve the small relaxation time \(\tau \), and still remain bounded by the Courant–Friedrichs–Lewy (CFL) stability constraint, governed by the sole convection part of the system, (ii) be asymptotic preserving [42, 47] meaning that the numerical scheme should be consistent with the asymptotic limit \(\tau \rightarrow 0\) for fixed \(\Delta x\), \(\Delta t\), that is the limiting scheme is a good discretization of the equilibrium system (47) even if the source term is underresolved. The numerical schemes should also (iii) be asymptotic accurate [47], that is preserve the order of accuracy in the stiff limit, (iv) strong stability preserving [47], strong stability is maintained at discrete level, and (v) well-balanced [43], preserving steady state numerically.
Fractional-step or splitting methods
Fractional-step methods are the most commonly used approaches to solve systems of balance equations, and consist in solving alternatively a system of conservation laws with no source term, and a system of ordinary differential equations. The main idea is to take advantage of the numerical methods and mathematical backgrounds already developed both for systems of conservation laws and for stiff ODEs. The simplest (Godunov) fractional-step method takes the form
$$\begin{aligned} \mathbf {U}^{n+1} = S^{(\Delta t)} C^{(\Delta t)} \mathbf {U}^n \end{aligned}$$
where \(C^{(\Delta t)}\) denotes the numerical solution operator over a time step \(\Delta t\) for the homogeneous part of (15), and \(S^{(\Delta t)}\) that for the ODE system
$$\begin{aligned} {\frac{d {\mathscr {Q}}}{dt} = {\mathscr {S}}({\mathscr {Q}}) } \end{aligned}$$
However, the Godunov splitting is only first order accurate, and the well-known Strang splitting [48]:
$$\begin{aligned} \mathbf {U}^{n+1} = S^{(\Delta t/2)} C^{(\Delta t)} S^{(\Delta t/2)} \mathbf {U}^n \end{aligned}$$
provides second order accuracy if each step is at least second order accurate. However, it reduces to first order for very stiff problems [42]. The implicit integration of (50) eliminates any influence of the relaxation time on the CFL condition, which thus depends solely on the convection part of (15). However, any implicit ODE solver may not yield a correct solution. For instance, although the trapezoidal method is second-order accurate and A-stable, it is only marginally stable in the stiff case [7]. Rather, the L-stability property is required [49], for example the simplest L-stable scheme is the Euler implicit method.
In this work, the Godunov splitting (49) coupled with an implicit backward Euler ODE solver will be used for stiff problems (e.g. see [9, 17]), while the Strang splitting and a backward differentiation formula at order two (BDF2) will be used for less stiff problems. However, more complex and higher order time integrators for stiff relaxation terms exist, but are not considered in this work. These are in general not based on splitting methods, and are still the purpose of current researches. Among others, the family of implicit–explicit IMEX Runge–Kutta schemes [47] allows to define high order schemes using an explicit time discretization for numerical flux and an implicit (DIRK [49]) one for the relaxation operator. Other approaches like ADER-WENO schemes [43] are also available.
Plane waves in a one-dimensional finite medium with Riemann-type initial conditions
Chaboche's viscoplastic constitutive model
As a simple, guideline, viscoplastic constitutive model, a Chaboche-type one [29] is considered, derived from Perzyna's work [24], itself generalizing to three dimensions the one-dimensional formulation initially proposed by Sokolowski [50] and Malvern [45], and is based on the following expression of the pseudo-potential of dissipation:
$$\begin{aligned} \Phi = \frac{K}{N+1}\left\langle \frac{f}{K}\right\rangle ^{N+1} \end{aligned}$$
where K and N denote viscosity and sensitivity material parameters, \(\langle x \rangle = (x + |x|)/2\) the positive part of x, and f is the yield criterion (or viscous stress) defined from the elastic convex \({\mathscr {C}}\) and is here associated to Mises' norm:
$$\begin{aligned} f = \sigma _{\text {eq}}(\varvec{\xi }) - \sigma _y\,; \quad \sigma _{\text {eq}} = \sqrt{\frac{3}{2}\varvec{\xi } : \varvec{\xi } } \,; \quad \varvec{\xi } = \mathbf {s}-\mathbf {X} \,; \quad {\mathscr {C}} = \{ (\varvec{\sigma },\mathbf {X}) | f\le 0 \} \end{aligned}$$
where \(\mathbf {s}\) is the deviatoric part of Cauchy stresses \(\varvec{\sigma }\), \(\mathbf {X}\) the variable defining the center of the elastic convex, and \(\sigma _y\) the tensile yield stress. From (9) and the potential (52), the effective viscoplastic strain rate reads:
$$\begin{aligned} \dot{p} = \left\langle \frac{f}{K} \right\rangle ^N \end{aligned}$$
This is coupled with a Prager's [30] linear kinematic hardening:
$$\begin{aligned} \dot{\mathbf {X}} = \frac{2}{3} D \dot{\varvec{\varepsilon }}^p \end{aligned}$$
where D is the hardening parameter. Defining
$$\begin{aligned} \tau = \left( \frac{K}{\sigma _y}\right) ^N \end{aligned}$$
the source term of the viscoplastic relaxation system can be written as a relaxation operator (45) with the following relaxation term:
$$\begin{aligned} {{\mathscr {R}}({\mathscr {Q}})} = \begin{Bmatrix}&-3 \mu \left\langle \frac{\sigma _{\text {eq}}}{\sigma _y} - 1 \right\rangle ^N \frac{\varvec{\xi }}{\sigma _{\text {eq}}}\\&\mathbf {0} \end{Bmatrix} \end{aligned}$$
where elastic isotropy has been considered and \(\mu \) is the elastic shear modulus.
Numerical elastic–plastic asymptotic limit
Let's consider a one-dimensional finite medium of length \(L=6\) m with free boundaries at its two ends, made of the above elastic–viscoplastic material with linear kinematic hardening. Riemann-type initial conditions are prescribed, the velocity is prescribed to \(-\bar{v}\) in the first half of the medium \(x\in [0,L/2[\), and to \(\bar{v}\) in the second half \(x\in ]L/2,L]\), while the stress is considered to be zero everywhere initially. The prescribed velocity is set so that viscoplastic flow occurs:
$$\begin{aligned} \bar{v} = 2 \frac{Y_H}{\rho c_L} \end{aligned}$$
where \(Y_H = (\lambda +2 \mu ) \sigma _y/2 \mu \) denotes the Hugoniot elastic limit and \(c_L\) is the elastic pressure wave velocity. The analytical solution of that problem for an elastic–plastic material with linear (kinematic and/or isotropic) hardening has been introduced in [21]. The solution first consists of two elastic and plastic waves travelling from the middle in opposite directions leading to tensile stress states. These waves are elastically reflected at both free ends, and then interact at the middle of the medium leading to a compressive reloading, first elastically, then plastically.
As \(\tau \) tends to zero, the computed elastic–viscoplastic solution should tend to the elastic–plastic one. On this test case, numerical solutions computed with the Strang and Godunov splitting are compared to that computed with classical P1-finite elements. The ODE system (50) computed for the finite volume numerical solution is solved by means of an implicit backward Euler scheme for the Godunov splitting and a backward differentiation formula at order two (BDF2) for the Strang splitting. Then the viscoplastic strain is updated explicitly (with a forward Euler scheme) at the end of the time step with the viscoplastic flow rule [(combining (13), (52) and (53)]. The finite element solution is coupled with a central difference explicit time integrator, uses a lumped mass matrix [4], and the constitutive equations are integrated with a radial return algorithm [6].
Figure 5 shows a comparison between elastic–viscoplastic finite element (FEM), finite volume (FV) with Strang and Godunov splitting solutions and the elastic–plastic analytical solution. This comparison is performed on axial stress and viscoplastic strain fields at two different times. The first time corresponds to elastic and plastic waves travelling away from the center of the one-dimensional medium, while the second pertains to the compressive reloading with the same wave path. Numerical solutions in Fig. 5 are computed on a mesh that consists of 200 grid cells/elements. The time step is set so that the Courant–Friedrich–Lewy (CFL) number be equal to one. Comparison is performed using the values extracted at integration points for the finite element solution, consistently with centroid values of cells for finite volume solutions. In addition, solutions shown in Fig. 5a–c are computed with a decreasing value of the relaxation parameter \(\tau \), while keeping N fixed. Parameters of Table 1 correspond to results shown in Fig. 5b.
Comparison for different values of the relaxation time \(\tau \), between the elastic–viscoplastic finite element (FEM), finite volume (FV) with Strang and Godunov splitting numerical solutions and the elastic–plastic analytical solution. Comparison is performed on axial stress and viscoplastic strain fields at two different times. a \(\tau = 10^{-4}\) s. b \(\tau = 4.8 \times 10^{-6}\) s. c \(\tau = 10^{-7}\) s
Table 1 Material parameters
In Fig. 5a, a moderately low relaxation time is considered, viscous effects are observable since the numerical elastic–viscoplastic solutions are smoother than the elastic–plastic one. Moreover, an increased apparent tensile yield stress is observable. FEM and Godunov splitting solutions are superposed while the Strang one appears slightly in advance. In Fig. 5b, a lower relaxation time \(\tau \) is set; viscous effects are now less apparent, but the same global behaviour than previously is observed. For a very low relaxation time (see Fig. 5c), the numerical solutions now conform as well as their respective possibility to the elastic–plastic analytical solution. A local overshoot occurs in the viscoplastic strains computed by the Godunov splitting scheme, and small oscillations appear: they do not subsist if the viscoplastic flow rule is solved implicitly together with the source term. The Strang solution shows an error in computing the sound speed of plastic waves, actually it shows a stiff behaviour. Indeed, this very low relaxation time leads to compute a numerical solution on an underresolved grid, since it becomes far smaller than the time step dicted by the convection part of the system. It is well-known that though Strang splitting can yield second-order accuracy for smooth solutions and non-stiff problems, it may fail to correctly compute wave speeds on underresolved grids, even with L-stable ODE solvers (see e.g. [46]).
Energy balance can also be performed in order to compare these schemes. Indeed, the central difference explicit time integrator coupled to FEM is known to conserve the total energy, while some energy is expected to be numerically dissipated for finite volume upwind schemes. The total energy is computed as the sum of kinetic \(W_{\text {kin}}\) and strain \(W_{\text {int}}\) energies, the latter is integrated in time with a trapezoidal rule, as discussed in [4]. These are plotted in Fig. 6 for the three schemes, computed with a CFL number set at unity, and for three values of the relaxation parameter \(\tau \). Figure 6a–c show three stages in their time evolution, respectively associated to the first tensile waves go, the unload return due to reflected waves at the boundaries, and the compressive reloading go from the middle. First, it can be observed that the FEM solution almost conserves the total energy as expected, while this total energy is not constant for finite volumes solutions especially during the first and third stages for which viscoplastic flow occur. It is constant during the second stage because no viscoplastic flow occur, and since the CFL number is set at one, flux limiters are not activated. Therefore, the energy numerically dissipated shown in Fig. 6 arises from the fractional-step algorithms. The total energy of the FEM solution slightly changes during this stage because small oscillations appear with elastic unloading disturbances (see Fig. 5), they do not subsist for a lower CFL number (see Fig. 7).
Time evolution of the total energy (kinetic plus strain energies) shown for different values of the relaxation time \(\tau \), computed with a CFL number of one, with the elastic–viscoplastic finite element (FEM), finite volume (FV) with Strang and Godunov splitting numerical solutions. a \(\tau = 10^{-4}\) s. b \(\tau = 4.8 \times 10^{-6}\) s. c \(\tau = 10^{-7}\) s
As the relaxation parameter \(\tau \) decreases, the energy numerically dissipated by the Godunov splitting slightly increases, while the Strang time evolution of the total energy decreases markedly: it is associated to the appearance of its stiff behaviour (Fig. 5c).
Time evolutions of the total energy are also shown in the stiff case for lower values of the CFL number in Fig. 7a, b. First, a lower CFL number decreases the time step, therefore the Strang splitting do not show a stiff behaviour anymore for that value of relaxation time since this fast time scale can be solved. Second, the energy numerically dissipated by the Godunov splitting is reduced because the numerical error in integrating the ODE system reduces with the time step. Third, since no viscoplastic evolution occur during the second stage, the observed numerical dissipation in that stage only arises from flux limiters which are activated due to the lower values set for the CFL number. However, this numerical dissipation appears negligible with respect to that developed during the propagation of viscoplastic flow, generated by the fractional-step algorithms.
Time evolution of the total energy shown for different values of the CFL number, computed with a relaxation parameter \(\tau =1 \times 10^{-7}\) s, with the elastic–viscoplastic finite element (FEM), finite volume (FV) with Strang and Godunov splitting numerical solutions. a \(CFL=0.4\). b \(CFL=0.04\)
Convergence analysis
To complete the comparison between these schemes, convergence analyses have been performed for both discontinuous and smooth solutions, and for different values of the relaxation parameter \(\tau \). These comparisons are done at the fixed time \(t=4.34 \times 10^{-4}\) s (first go of tensile waves), provided a CFL number set at one. Smooth solutions are obtained by smoothing the velocity profile of the initial condition with a portion of sinus on a length L / 4, centered with respect to the middle of the one-dimensional medium.
Relative \(L^2\) errors of the stress and velocity computed at time \(4.34 \times 10^{-4}\) s with a CFL number set at one for a discontinuous profile of the initial velocity field. Comparison is performed for three values of the relaxation parameter \(\tau \), with the FEM, the Strang and Godunov splitting finite volume solutions. a \(\tau = 10^{-4}\) s. b \(\tau = 4.8 \times 10^{-6}\) s. c \(\tau = 10^{-7}\) s
Relative \(L^2\) errors of the stress and velocity computed at time \(4.34 \times 10^{-4}\) s with a CFL number set at one for a smooth profile of the initial velocity field. Comparison is performed for three values of the relaxation parameter \(\tau \), with the FEM, the Strang and Godunov splitting finite volume solutions. a \(\tau = 10^{-4}\) s. b \(\tau = 4.8 \times 10^{-6}\) s. c \(\tau = 10^{-7}\) s
Figure 8a–c show the convergence curves associated to discontinuous solutions for the three values of the relaxation parameter \(\tau \). A very close convergence rate of about 0.5 for the stress and 0.8 for the velocity is observed for the three schemes in the case of initial discontinuous profile of the velocity. Indeed, because of the discontinuous solution, the source term is expected to be active only over thin regions where there are fast transients that cannot be resolved with a high accuracy [7]. Convergence curves are almost superposed for the highest value of the relaxation parameter. However, the constant in the Strang splitting convergence curves (Fig. 9b, c) becomes larger than these of the FEM and the Godunov splitting for lower values of \(\tau \), yielding a bigger error. Indeed, as \(\tau \) decreases, the Strang splitting shows a too stiff behaviour (Fig. 5c) on an underresolved grid (for example \(\Delta x =3 \times 10^{-2}\) m, 200 grid cells). As expected, this behaviour is improved for a lower grid size associated to a lower time step of the order of the time scale of the source term (for example \(\Delta x = 3 \times 10^{-3}\,\text{ m } \rightarrow \Delta t \approx 5 \times 10^{-7} \text {s} \sim \tau \)). But in general we do not want to use such a fine grid, because larger time steps are preferred to save computational cost.
Figure 9a–c show the convergence curves associated to smooth initial profile of the velocity for the three values of the relaxation parameter \(\tau \). Now, convergence rates of about 1.1 and 1.5 for stress and the velocity are observed, and are higher than these provided for discontinuous solutions. The FEM error appears globally to be the smallest, though that of the Godunov splitting solution is close to it in each case. However, the Strang splitting solution appears less efficient than the two others, although its error has decreased with respect to discontinuous solutions. But, as \(\tau \) decreases, it shows again a too stiff behaviour.
As a first draw of conclusion, for very stiff problems, the Godunov splitting should be preferred to the Strang one. For non-stiff ones, both can be used. However, it should be noticed that the present case of a plane wave is a hard test because of its one-dimensional strain state. Two-dimensional cases below yield less stiff solutions due to their multi-dimensional strain state.
Partial impact on a plane
Let's now consider the bidimensional square domain shown in Fig. 10, submitted to an impact on a part of its top face, by means of a step function of a pressure p. A symmetry condition is considered on the left part, free boundaries are set at the bottom face and on the remaining part of the top face, and a perfect transmission condition is set on its right face. This problem is treated in the bidimensional plane strain framework, for which the expressions of characteristic speeds \(\lambda _p\) and directions \(\mathbf {K}^{(p)}\) (28), \(1\le p \le 5\), are detailed in [7]. For this test case, a comparison between a finite volume numerical solution with Strang splitting and Q1-finite elements is considered. The finite element solution is obtained with the code Cast3M [51], computed with an implicit time stepping, and an absorbing boundary to compute the perfect transmission condition set on the right face. The previously introduced elastic–viscoplastic constitutive model with linear kinematic hardening is also considered with parameters listed in Table 1, to which are added these associated to geometry and loading of the partial impact test case, summarized in Table 2.
Table 2 Parameters of the partial impact test case
Non-uniform quadrilateral mesh
First, the non-uniform quadrilateral mesh shown in Fig. 11 is considered, generated with the free finite element mesh generator Gmsh [52], defined so that to achieve refined elements close to the impact area and coarse ones far from this area. Both numerical solutions are computed with this mesh, and the CFL number is set at one for both solutions. Figures 12 and 13 show the comparison between finite element and finite volume numerical solutions at two different times of computation. These figures consist on the one hand of numerical isovalues of the normal stress \(\sigma _{22}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\equiv p\), for which the left-half of the domain shows the finite element solution while the right one shows the finite volume solution, and on the other hand of superposed plots of the normal stress \(\sigma _{22}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\equiv p\) along the symmetry line. At time \(2.58 \times 10^{-4}\) s (see Fig. 12), pressure and shear waves have been generated by the step pressure prescribed on a part of the top face, and have propagated downward. The curved wave front of the pressure wave is observable on stress isovalues. This front has generated and propagated viscoplastic strains. Finite element and finite volume numerical solutions look close, up to a slight undershoot of the finite element normal stress at the wave front and a slightly delayed progression of viscoplastic strains for the finite volume solution. At time \(9.25 \times 10^{-4}\) s (see Fig. 13), the pressure wave has travelled one round trip from the top face, and still propagates forth. The solution is now smoother, but viscoplastic strains have developed. In particular, close to the connection between loaded and free parts of the top face of the domain, shear has been undergone leading to the spread of viscoplastic strains from that area. Second, the free boundary defined at the bottom face has led to convert the initial pressure wave into a tensile one after reflection, which has generated viscoplastic strains close to that bottom face. Again, finite element and finite volume solutions look close up to few details here and there on stress and cumulated viscoplastic strains fields.
Normal stress \(\sigma _{22}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\equiv p\) at time \(2.58 \times 10^{-4}\) s, computed with finite elements (left-half domain) and finite volumes (right-half domain). Superposed plots are made along the symmetry line (Additional file 1)
Normal stress \(\sigma _{22}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\equiv p\) at time \(9.25 \times 10^{-4}\) s, computed with finite elements (left-half domain) and finite volumes (right-half domain). Superposed plots are made along the symmetry line
Normal stress \(\sigma _{22}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\) at time \(4.89 \times 10^{-4}\) s, computed on a \(100 \times 100\) refined cartesian mesh with finite elements (left-half domain) and finite volumes (right-half domain). Superposed plots are made along the symmetry line
But a clear improvement provided by finite volumes over the finite element solution can be observed by using a refined \(100\times 100\) quadrangle cartesian mesh, which provided the CFL number set at one, enables to properly capture the elastic discontinuity. Figure 14 shows the tensile wave reflected from the bottom side at time \(9.25 \times 10^{-4}\) s, captured in one cell with the finite volume solution, while finite elements exhibit spurious numerical oscillations around this wavefont. Finite volumes benefit here from flux limiters designed to achieve a nonincreasing total variation, applied to the convection part of the system (15).
Double-notched specimen with tensile initial velocity
We consider now a double-notched specimen, whose quarter of geometry is shown in Fig. 15. This double-notched specimen is submitted to a nonzero initial tensile velocity in its extremal parts, within the shaded area (see Fig. 15). This test case is actually extracted from [17], but is here treated with the previously defined elastic–viscoplastic material with linear kinematic hardening. Due to the two planes of symmetry of the problem, the sole quarter of the double-notched specimen is meshed. Its geometrical and loading parameters are listed in Table 3. The finite volume solution is still computed with the Strang splitting, and both finite element and finite volume solutions are computed with a CFL number set at 0.9. Figures 16 and 17 show their comparison at two different times of computation. These figures are organized in the same way than for the partial impact test case, except the normal stress \(\sigma _{11}\) is now considered, and the finite element solution is shown on the half-bottom part of the domain and the finite volume one is shown on the half-top part. Superposed plots are performed along the horizontal symmetry line.
One quarter of a double notched specimen, submitted to an initial tensile velocity on its right part
Table 3 Parameters of the double-notched specimen test case
Normal stress \(\sigma _{11}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\) at time \(3.45 \times 10^{-6}\) s, computed with finite elements (bottom-half domain) and finite volumes (top-half domain). Superposed plots are made along the horizontal symmetry line
The nonzero initial velocity generates a tensile wave, which is first reflected on the right free boundary, as shown in Fig. 16. The finite element solution shows spurious oscillations upstream of the left wavefront, essentially due to the implicit time stepping, while the viscoplastic strains are close for both numerical solutions and show a conical spread pattern due to the plane of symmetry. Then, the left front of the tensile wave is reflected both on the notch and left plane of symmetry, which leads the normal stress \(\sigma _{11}\) to double on the symmetry line and to concentrate at the notch corner. After few waves interactions, Fig. 17 shows viscoplastic strains which have much increased at the notch corners, so does for the normal stress which shows shearing pattern due to multiple wave reflexions. Essentially, the two numerical solutions fit well, though the finite volume one shows less numerical spurious oscillations.
Sudden velocity loading and unloading of a heterogeneous volume
The last test case considered in this work is a square heterogeneous volume, of side length 2a, with an inclusion of circular cross-section of radius R centered within the volume. This volume in an initial natural state is suddenly loaded on its left side at time \(t=0\) with a constant first component of velocity \(\bar{v}\). After time \(t_u\), the applied velocity is set to zero. Symmetry conditions are set at the top and bottom sides of the volume, while transmissive boundary conditions have been set on the right side. Due the symmetry of the problem, only one half of the domain is meshed as shown in Fig. 18. We assume an arbitrary heterogeneous material, that consists of an inclusion made of aluminium embedded in a matrix made of steel. The matrix is here assumed to remain elastic, only the inclusion undergoes viscoplastic strains, driven by the previously introduced Chaboche's viscoplastic constitutive model. The analysis is also carried out in plane strain, with a CFL number set at one. Numerical data used for computations are summarized in Table 4. Figures 19 and 20 show the comparison between finite element and finite volume numerical solutions at two different times of computation. The normal stress \(\sigma _{11}\) and the cumulated viscoplastic strains are shown, and plotted along symmetry line.
Half of a heterogeneous volume that consists of a circular inclusion embedded in a matrix, suddenly loaded then unloaded after a time \(t_u\) on its left side by a prescribed velocity \(\bar{v}\)
Table 4 Parameters values for the heterogeneous volume test case
Normal stress \(\sigma _{11}\) and cumulated viscoplastic strains \(\varepsilon _{eq}\) at time \(1.52 \times 10^{-7}\) s, computed with finite elements (bottom-half domain) and finite volumes (top-half domain). Superposed plots are made along the symmetry line
A slot of compressive normal stress is first formed by the horizontal component of the velocity prescribed on the left side, and travels rightward in the matrix. Then, its first front interacts with the front interface between the matrix and the inclusion, and generates an intermediate state of stress and velocity due to the mismatch of elastic impedances of the matrix and the inclusion. In Fig. 19, the second front of the former stress slot interacts with the generated intermediate state and yields a tensile stress wave, while the first compressive loading keeps on travelling within the inclusion, and propagates viscoplastic strains. One can observe that the finite element stress field shows spurious numerical oscillations in the vicinity of discontinuities, especially close to lateral boundaries where tensile spurious stress states appear. Cumulated viscoplastic strains are almost identical for both numerical solutions, except close to the matrix/inclusion interface. Once the tensile wave has reflected on the left side of the volume, it propagates rightward, following the initial compression slot, as shown in Fig. 20. The former compression slot has interacted with the back side of the inclusion interface, generating an important growth of cumulated viscoplastic strains close to this area. Note also that the front of the tensile slot has been curved after reflexion first on the circular matrix/inclusion interface, second on the boundary of the volume. Generally speaking, the finite volume solution allows to obtain the same viscoplastic strains than these of the finite element solution without the spurious numerical oscillations on stresses obtained with the finite element solution.
Chaboche–Nouailhas' viscoplastic constitutive model
The previous version of Chaboche's viscoplastic constitutive model can be enriched to show the generality and robustness of the method. A viscoplastic model extracted from that of Chaboche and Nouailhas [29, 31] introduces an exponential term in the expression of the pseudo-potential of dissipation:
$$\begin{aligned} \Phi = \frac{K}{\alpha (N+1)} \exp \left( \alpha \left\langle \frac{f}{K}\right\rangle ^{N+1} \right) \end{aligned}$$
which allows, through a parameter \(\alpha \), to saturate the overstress as the effective viscoplastic strain rate increases, which now reads:
$$\begin{aligned} \dot{p} = \left\langle \frac{f}{K} \right\rangle ^N \exp \left( \alpha \left\langle \frac{f}{K}\right\rangle ^{N+1} \right) \end{aligned}$$
The saturation of the overstress driven by expression (60) is shown in Fig. 21, and superposed with the previous one (54). It amounts to consider a power law with a varying apparent exponent:
$$\begin{aligned} N^* = \frac{d \ln \dot{p}}{d \ln f} = N+ \alpha (N+1) \left( \frac{f}{K} \right) ^{N+1} \end{aligned}$$
The variation of this exponent is also shown in Fig. 21. This creep law is coupled with the nonlinear kinematic hardening law of Armstrong–Frederick [32, 33]:
$$\begin{aligned} \dot{\mathbf {X}} = \frac{2}{3} D \dot{\varvec{\varepsilon }}^p - \gamma \mathbf {X}\dot{p} \end{aligned}$$
where \(\gamma \) denotes an additional hardening parameter. A tensile/compression cycle is depicted in Fig. 22 for a strain rate \({\dot{\varepsilon }}=1\)/s, and \(\gamma =200\). The ODE system (50) computed for finite volume numerical solutions is now solved together with the viscoplastic flow rule [combining (13), (59) and (53)] and the hardening rule (62), provided the expression (60) of the effective viscoplastic strain rate, and is still discretized with an implicit backward Euler scheme. In the bidimensional plane strain case, the nonlinear system of equations to be solved consists of eight equations associated to components 11, 22, 12, 33 of \(\varvec{\sigma }\) and \(\mathbf {X}\). Moreover, the finite volume numerical solution is computed with a Godunov splitting, numerical values used in plots of Figs. 21 and 22, and the same CFL number set at one.
Overstress evolutions obtained with the creep laws (54) and (60), and numerical values of Table 1, and \(\alpha =10^{-3}\)
Tensile/compression stress–strain curve at \({\dot{\varepsilon }}=1\) s\(^{-1}\)
Cumulated viscoplastic strain \(\varepsilon _{eq}\equiv p\) in the inclusion at time \(3.35 \times 10^{-7}\) s, computed with Chaboche (right) and Chaboche–Nouailhas' (left) viscoplastic constitutive models. Isovalues have been warpped according to the magnitude of \(\varepsilon _{eq}\) with the same scale factor 0.2
Effective viscoplastic strain rate \(\dot{p}\) in the inclusion at time \(1.89\times 10^{-7}\) s, computed with Chaboche and Chaboche–Nouailhas' viscoplastic constitutive models. Both unwarpped (left) and warpped (right) isovalues are presented, the latter being scaled with the same scale factor \(10^{-8}\)
Figure 23 shows the cumulated viscoplastic strain field in the inclusion after the compression slot is passed, computed by means of finite volume method with Chaboche and Chaboche–Nouailhas' viscoplastic constitutive models. The latter leads to much more important cumulated viscoplastic strain than the former. Indeed, for an overstress close to \(10^2\) MPa, the effective cumulated strain rate has largely increased for that model (see Fig. 21), which yields a more important viscoplastic flow. As the compression slot passes within the inclusion, viscoplastic flow concentrates in the inclusion close to the rear part of the interface with the matrix (see Fig. 23). This is provided by the circular geometry of the inclusion that defines a concentrating profile as the compression wave travels rightward. In Figs. 24 and 25, isovalues of the effective viscoplastic strain rate are shown at two successive instants, as the compression slot passes. The observed profiles appear quite different in terms of the chosen viscoplastic constitutive model. In particular, the saturation of the overstress predicted by expression (60) combined with the particular rear geometry of the interface inclusion/matrix yield an effective strain rate that reaches very high numerical values on a narrow band. This small example illustrates the importance of the chosen viscoplastic constitutive model on the propagated viscoplastic strains in a dynamic process.
In this work, the flux-difference splitting finite volume method [7, 25] has been employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The elastic–viscoplastic system of equations identifies itself with a relaxation system with threshold, whose asymptotic limit yields an elastic–plastic system. The linearized geometrical framework considered here leads to a linear hyperbolic system with a nonlinear source term, driven by the viscoplastic part of the behaviour. This relaxation system is solved by means of a fractional-step method (Strang or Godunov splitting), whose convection part is solved with the flux-difference splitting formalism applied here to bidimensional non-uniform quadrilateral meshes. Several test cases have been proposed, and show the good accuracy of the computed finite volume solutions in terms of both stresses and viscoplastic strains with respect to finite element ones. The flux limiters used to compute the convection part enable to remove spurious numerical oscillations shown in the finite element solution close to the elastic discontinuity. In addition, two viscoplastic constitutive models have been tested to illustrate the genericity of the approach, and their influence on the viscoplastic flow has been shown on the heterogeneous volume test case. Notice that this work is straightforward extendable to three-dimensional meshes following [53] for example.
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TH developed the idea, conducted numerical experiments and wrote the paper. The author read and approved the final manuscript.
The author declares that he has no competing interests.
GeM, UMR 6183 CNRS, École Centrale de Nantes, 1 rue de la Noë, 44321, Nantes, France
Thomas Heuzé
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Additional file
Supplementary material 1 (mp4 2988 KB)
Heuzé, T. Simulation of impacts on elastic–viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes. Adv. Model. and Simul. in Eng. Sci. 5, 9 (2018) doi:10.1186/s40323-018-0101-z
DOI: https://doi.org/10.1186/s40323-018-0101-z
Elastic–viscoplastic solids
Finite volume method
Flux-difference splitting
Non-uniform quadrilateral meshes
Generalized standard materials | CommonCrawl |
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\begin{center} \vskip 1cm{\LARGE\bf M\'enage Numbers and M\'enage Permutations } \vskip 1cm \large Yiting Li\\ Department of Mathematics \\ Brandeis University \\ {\tt [email protected]} \end{center}
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Yiting Li, \textsc{Department of Mathematics, Brandeis University, 415 Soutrh Street, Waltham, MA 02453, USA}\par\nopagebreak
\textit{E-mail address}, Yiting Li: \texttt{[email protected]}
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\begin{abstract} In this paper, we study the combinatorial structures of straight and ordinary m\'enage permutations. Based on these structures, we prove four formulas. The first two formulas define a relationship between the m\'enage numbers and the Catalan numbers. The other two formulas count the m\'enage permutations by number of cycles. \end{abstract}
\section{Introduction} \subsection{Straight m\'enage permutations and straight m\'enage numbers} The straight m\'enage problem asks for the number of ways one can arrange $n$ male-female pairs along a linearly arranged table in such a way that men and women alternate but no woman sits next to her partner.
We call a permutation $\pi\in S_n$ a $straight$ {\it m\'enage} $permutation$ if $\pi(i)\ne i$ and $\pi(i)\ne i+1$ for $1\le i\le n$. Use $V_n$ to denote the number of straight m\'enage permutations in $S_n$. We call $V_n$ the $nth$ $straight$ {\it menage} $number$.
The straight m\'enage problem is equivalent to finding $V_n$. Label the seats along the table as $1,2,\ldots,2n$. Sit the men at positions with even numbers and women at positions with odd numbers. Let $\pi$ be the permutation such that the man at position $2i$ is the partner of the woman at position $2\pi(i)-1$ for $1\le i\le n$. Then, the requirement of the straight m\'enage problem is equivalent to the condition that $\pi(i)$ is neither $i$ nor $i+1$ for $1\le i\le n$.
\subsection{Ordinary m\'enage permutations and ordinary m\'enage numbers} The ordinary m\'enage problem asks for the number of ways one can arrange $n$ male-female pairs around a circular table in such a way that men and women alternate, but no woman sits next to her partner.
We call a permutation $\pi\in S_n$ an $ordinary$ {\it m\'enage} $permutation$ if $\pi(i)\ne i$ and $\pi(i)\not\equiv i+1$ (mod $n$) for $1\le i\le n$. Use $U_n$ to denote the number of ordinary m\'enage permutations in $S_n$. We call $U_n$ the $nth$ $ordinary$ {\it m\'enage} $number$.
The ordinary m\'enage problem is equivalent to finding $U_n$. Label the seats around the table as $1,2,\ldots,2n$. Sit the men at positions with even numbers and women at positions with odd numbers. Let $\pi$ be the permutation such that the man at position $2i$ is the partner of the woman at position $2\pi(i)-1$ for all $1\le i\le n$. Then, the requirement of the ordinary m\'enage problem is equivalent to the condition that $\pi(i)$ is neither $i$ nor $i+1$ (mod $n$) for $1\le i\le n$.
We hold the convention that the empty permutation $\pi_\emptyset\in S_0$ is both a straight m\'enage permutation and an ordinary m\'enage permutation, so $U_0=V_0=1$.
\subsection{Background} Lucas \cite{Lucas} first posed the problem of finding ordinary m\'enage numbers. Touchard \cite{Touchard} first found the following explicit formula \eqref{eq:known_formulas_for_U_n}. Kaplansky and Riordan \cite{Kaplansky} also proved an explicit formula for ordinary m\'enage numbers. For other early work in m\'enage numbers, see \cite{Kaplansky1943,MW} and references therein. Among more recent papers, there are some using bijective methods to study m\'enage numbers. For example, Canfield and Wormald \cite{CW} used graphs to address the question. One can find the following formulas of m\'enage numbers in \cite{Bogart,Touchard} and in Chapter 8 of \cite{Riordan}: \begin{align}\label{eq:known_formulas_for_U_n} U_m=\sum\limits_{k=0}^m(-1)^k\dfrac{2m}{2m-k}{2m-k\choose k}(m-k)!\quad\quad(m\ge2);\\ V_n=\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!\quad\quad(n\ge0).\label{eq:known_formulas_for_V_n} \end{align}
The purpose of the current paper is to study the combinatorial structures of straight and ordinary m\'enage permutations and to use these structures to prove some formulas of straight and ordinary m\'enage numbers. We also give an analytical proof of Theorem \ref{thm:main_theorem_1} in Section \ref{appendix}.
\subsection{Main results}
Let $C_k$ be the $k$th $Catalan$ $number$: \[ C_k=\dfrac{(2k)!}{k!\,(k+1)!} \] and $c(x)=\sum\limits_{k=0}^\infty c_kx^k$. Our first main result is the following theorem. \begin{theorem}\label{thm:main_theorem_1} \begin{align}\label{eq:result_of_straight_menage_numbers} \sum\limits_{n=0}^{\infty}n!\,x^n=\sum\limits_{n=0}^\infty V_nx^nc(x)^{2n+1}, \\ \sum\limits_{n=0}^{\infty}n!\,x^n=c(x)+c'(x)\sum\limits_{n=1}^\infty U_nx^nc(x)^{2n-2}.\label{eq:result_of_ordinary_menage_numbers} \end{align} \end{theorem}
Our second main result counts the straight and ordinary m\'enage permutations by the number of cycles.
For $k\in\mathbb{N}$, use $(\alpha)_k$ to denote $\alpha(\alpha+1)\cdots(\alpha+k-1)$. Define $(\alpha)_0=1$. For $k\le n$, use $C_n^k$ ($D_n^k$) to denote the number of straight (ordinary) m\'enage permutations in $S_n$ with $k$ cycles. \begin{theorem} \begin{align}\label{eq:straight_by_cycles} 1+\sum_{n=1}^\infty\sum_{j=1}^n C_n^j\alpha^jx^n=\sum\limits_{n=0}^\infty(\alpha)_n\frac{x^n}{(1+x)^n(1+\alpha x)^{n+1}}, \end{align} \begin{align}\label{eq:ordinary_by_cycles} 1+\sum_{n=1}^\infty\sum_{j=1}^n D_n^j\alpha^jx^n=\frac{x+\alpha x^2}{1+x}+(1-\alpha x^2)\sum\limits_{n=0}^\infty(\alpha)_n\frac{x^n}{(1+x)^{n+1}(1+\alpha x)^{n+1}}. \end{align} \end{theorem}
\subsection{Outline} We give some preliminary concepts and facts in Section \ref{sec:preliminaries}. In Section \ref{sec:reductions_and_nice_bijections}, we define three types of reductions and the nice bijection. Then, we study the structure of straight m\'enage permutations and prove \eqref{eq:result_of_straight_menage_numbers} in Section \ref{sec:straight_menage_permutation}. In Section \ref{sec:ordinary_menage_permutation}, we study the structure of ordinary m\'enage permutations and prove \eqref{eq:result_of_ordinary_menage_numbers}. Finally, we count the straight and ordinary m\'enage permutations by number of cycles and prove \eqref{eq:straight_by_cycles} and \eqref{eq:ordinary_by_cycles} in Section \ref{count_permutations_by_cycles}.
\section{Preliminaries}\label{sec:preliminaries}
For $n\in\mathbb{N}$, we use $[n]$ to denote $\{1,\ldots,n\}$. Define $[0]$ to be $\emptyset$.
\begin{definition} Suppose $n>0$ and $\pi\in S_n$. If $\pi(i)=i+1$, then we call $\{i,i+1\}$ a $succession$ of $\pi$. If $\pi(i)\equiv i+1$ (mod $n$), then we call $\{i,\pi(i)\}$ a $generalized$ $succession$ of $\pi$. \end{definition} \subsection{Partitions and Catalan numbers}\label{sec:partitions_and_Catalan_numbers}
Suppose $n>0$. A $partition$ $\epsilon$ of $[n]$ is a collection of disjoint subsets of $[n]$ whose union is $[n]$. We call each subset a $block$ of $\epsilon$. We also describe a partition as an equivalence relation: $p\sim_\epsilon q$ if and only if $p$ and $q$ belong to a same block of $\epsilon$.
If a partition $\epsilon$ satisfies that for any $p\sim_\epsilon p'$ and $q\sim_\epsilon q'$, $p<q<p'<q'$ implies $p\sim_\epsilon q$; then, we call $\epsilon$ a $noncrossing$ $partition$.
For $n\in\mathbb{N}$, suppose $\epsilon=\{V_1,\ldots,V_k\}$ is a noncrossing partition of $[n]$ and $V_i=\{a_1^i,\ldots, a_{j_i}^i\}$, where $a_1^i<\cdots<a_{j_i}^i$. Then, $\epsilon$ induces a permutation $\pi\in S_n$: $\pi(a_{r(i)}^i)=a_{r(i)+1}^i$ for $1\le r(i)\le j_i-1$ and $\pi(a_{j_i}^i)=a_1^i$. It is not difficult to see that different noncrossing partitions induce different permutations.
The following lemma is well known. See, for example, \cite{Stanley}. \begin{lemma} For $n\in\mathbb{N}$, there are $C_n$ noncrossing partitions of $[n]$. \end{lemma} It is well known that the generating function of the Catalan numbers is $$c(x)=\sum\limits_{n=0}^\infty C_nx^n=\dfrac{1-\sqrt{1-4x}}{2x}.$$ It is also well known that one can define the Catalan numbers by recurrence relation \begin{align}\label{recurrence_of_Catalan_number} C_{n+1}=\sum\limits_{k=0}^nC_kC_{n-k} \end{align} with initial condition $C_0=1$. \begin{lemma}\label{thm:property_of_Catalan_numbers} The generating function of the Catalan numbers $c(x)$ satisfies $$c(x)=\frac{1}{1-xc(x)}=1+xc^2(x)\quad\text{and}\quad\dfrac{c^3(x)}{1-xc^2(x)}=c'(x).$$ \end{lemma}
\begin{proof}[Proof of Lemma \ref{thm:property_of_Catalan_numbers}] The first formula is well known. By the first formula, \begin{align}\label{www} c'(x)=c^2(x)+2xc(x)c'(x)=\dfrac{c^2(x)}{1-2xc(x)}. \end{align} Thus, to prove the second formula, we only have to show that $\dfrac{c(x)}{1-xc^2(x)}=\dfrac{1}{1-2xc(x)}$ which is equivalent to $c(x)-2xc^2(x)=1-xc^2(x)$. This follows from the first formula. \end{proof}
\subsection{Diagram representation of permutations}\label{sec:diagrams} \subsubsection{Diagram of horizontal type} For $n>0$ and $\pi\in S_n$, we use a diagram of horizontal type to represent $\pi$. To do this, draw $n$ points on a horizontal line. The points represent the numbers $1,\ldots,n$ from left to right. For each $i\in[n]$, we draw a directed arc from $i$ to $\pi(i)$. The permutation uniquely determines the diagram. For example, if $\pi=(1,5,4)(2)(3)(6)$, then its diagram is
\centerline{\includegraphics[width=3.5in]{1.eps}}
\subsubsection{Diagram of circular type} For $n>0$ and $\pi\in S_n$, we also use a diagram of circular type to represent $\pi$. To do this, draw $n$ points uniformly distributed on a circle. Specify a point that represents the number 1. The other points represent $2,\ldots,n$ in counter-clockwise order. For each $i$, draw a directed arc from $i$ to $\pi(i)$. The permutation uniquely determines the diagram (up to rotation). For example, if $\pi=(1,5,4)(2)(3)(6)$, then its diagram is
\centerline{\includegraphics[width=1.5in]{2.eps}}
\subsection{The empty permutation}
The empty permutation $\pi_\emptyset\in S_0$ is a permutation with no fixed points, no (generalized) successions and no cycles.
\section{Reductions and nice bijections}\label{sec:reductions_and_nice_bijections} In this section, we introduce reductions and nice bijections which serve as our main tools to study m\'enage permutations. \subsection{Reduction of type 1} Intuitively speaking, to perform a reduction of type 1 is to remove a fixed point from a permutation. Suppose $n\ge1$, $\pi\in S_n$ and $\pi(i)=i$. Define $\pi'\in S_{n-1}$ such that: \begin{align}\label{eq:expression_of_reduction_of_type_1} \pi'(j)=\begin{cases}\pi(j)&\text{if}\quad j<i\quad\text{and}\quad\pi(j)<i;\\ \pi(j)-1&\text{if}\quad j<i\quad\text{and}\quad\pi(j)>i;\\\pi(j+1)&\text{if}\quad j\ge i\quad\text{and}\quad\pi(j+1)<i;\\\pi(j+1)-1&\text{if}\quad j\ge i\quad\text{and}\quad\pi(j+1)>i\end{cases} \end{align} when $n>1$. When $n=1$, define $\pi'$ to be $\pi_\emptyset$. If we represent $\pi$ by a diagram (of either type), erase the point corresponding to $i$ and the arc connected to the point (and number other points appropriately for the circular case); then we obtain the diagram of $\pi'$. We call this procedure of obtaining a new permutation by removing a fixed point a $reduction$ $of$ $type$ $1$. For example, if $$\pi=(1,5,6,4)(2)(3)(7)\in S_7,$$ then by removing the fixed point 3 we obtain $\pi'=(1,4,5,3)(2)(6)\in S_6$.
\subsection{Reduction of type 2} Intuitively speaking, to do a reduction of type 2 is to glue a succession $\{k,k+1\}$ together. Suppose $n\ge2$, $\pi\in S_n$ and $\pi(i)=i+1$. Define $\pi'\in S_{n-1}$ such that: \begin{align}\label{eq:expression_of_reduction_of_type_2} \pi'(j)=\begin{cases}\pi(j)&\text{if}\quad j<i\quad\text{and}\quad\pi(j)\le i;\\ \pi(j)-1&\text{if}\quad j<i\quad\text{and}\quad\pi(j)>i+1;\\\pi(j+1)&\text{if}\quad j\ge i\quad\text{and}\quad\pi(j+1)\le i;\\\pi(j+1)-1&\text{if}\quad j\ge i\quad\text{and}\quad\pi(j+1)>i+1.\end{cases} \end{align} If we represent $\pi$ by the diagram of the $horizontal$ type, erase the arc from $i$ to $i+1$, and glue the points corresponding to $i$ and $i+1$ together; then, we obtain the diagram of $\pi'$. We call this procedure of obtaining a new permutation by gluing a succession together a $reduction$ $of$ $type$ $2$. For example, if $$\pi=(1,5,6,4)(2)(3)(7)\in S_7,$$ then by gluing 5 and 6 together, we obtain $\pi'=(1,5,4)(2)(3)(6)\in S_6$.
\subsection{Reduction of type 3} Intuitively speaking, to perform a reduction of type 3 is to glue a generalized succession $\{k,k+1\pmod{n}\}$ together. Suppose $n\ge1$, $\pi\in S_n$ and $\pi(i)\equiv i+1\pmod{n}$. Define $\pi'$ to be the same as in \eqref{eq:expression_of_reduction_of_type_2} when $i\ne n$. When $i=n>1$, define $\pi'$ to be \begin{align*} \pi'(j)=\begin{cases}\pi(j)&\text{ if }j\ne\pi^{-1}(n);\\1&\text{ if }j=\pi^{-1}(n).\end{cases} \end{align*} When $i=n=1$, define $\pi'$ to be $\pi_\emptyset$. If we represent $\pi$ by a diagram of the $circular$ type, erase the arc
from $i$ to $i+1$ (mod $n$), glue the points corresponding to $i$ and $i+1$ (mod $n$) together and number the points appropriately; then, we obtain the diagram of $\pi'$. We call this procedure of obtaining a new permutation by gluing a generalized succession together a $reduction$ $of$ $type$ $3$. For example, if $$\pi=(1,5,6,7)(2)(3)(4)\in S_7,$$ then by gluing 1 and 7 together, we obtain $\pi'=(1,5,6)(2)(3)(4)\in S_6$.
\subsection{Nice bijections}\label{sec:nice_bijection}
Suppose $n\ge1$ and $f$ is a bijection from $[n]$ to $\{2,\ldots, n+1\}$. We can also represent $f$ by a diagram of horizontal type as for permutations. The bijection uniquely determines the diagram. If $f$ has a fixed point or there exists $i$ such that $f(i)=i+1$, then we can also perform reductions of type 1 or type 2 on $f$ as above. In the latter case, we also call $\{i,i+1\}$ a $succession$ of $f$. We can reduce $f$ to a bijection with no fixed points and no successions by a series of reductions. It is easy to see that the resulting bijection does not depend on the order of the reductions.
The following diagram shows an example of reduction of type 2 on the bijection by gluing the succession 2 and 3 together.
\centerline{\includegraphics[width=4in]{3.eps}} \begin{definition} Suppose $f$ is a bijection from $[n]$ to $\{2,\ldots, n+1\}$. If there exist a series of reductions of type 1 or type 2 by which one can reduce $f$ to the simplest bijection $1\mapsto2$, then we call $f$ a $nice$ $bijection$. \end{definition} Suppose $\pi\in S_n$ and $p$ is a point of $\pi$. We can replace $p$ by a bijection $f$ from $[k]$ to $\{2,\ldots, k+1\}$ and obtain a new permutation $\pi'\in S_{n+k}$ by the following steps:
(1) represent $\pi$ by the horizontal diagram;
(2) add a point $q$ right before $p$ and add an arc from $q$ to $p$ ;
(3) replace the arc from $\pi^{-1}(p)$ to $p$ by an arc from $\pi^{-1}(p)$ to $q$ ;
(4) replace the arc from $q$ to $p$ by the diagram of $f$.
For example, if $\pi$, $p$ and $f$ are as below,
\centerline{\includegraphics[width=4in]{4.eps}}
then, we can replace $p$ by $f$ and obtain the following permutation:
\centerline{\includegraphics[width=3.5in]{5.eps}}
It is easy to see, by the definition of the nice bijection, that if $f$ is a nice bijection, then we can reduce $\pi'$ back to $\pi$: first, we can reduce $\pi'$ to the permutation obtained in Step (3) because $f$ is nice; then, by gluing the succession $p$ and $q$ together, we obtain $\pi$. Notice that the bijection $f$ in the above example is not nice.
For the circular diagram of a permutation $\pi$, we can also use Steps (2)--(4) shown above to obtain a new diagram. However, to obtain a permutation $\pi'$, we need to specify the point representing number 1 in the new diagram. See the example in the proof of Theorem \ref{gf_of_r}.
\begin{lemma}\label{lemma:count_nice_bijection} For $n\ge2$, let $a_n$ be the number of nice bijections from $[n-1]$ to $\{2,\ldots, n\}$. Define $a_1=1$. The generating function of $a_n$ satisfies the following equation: \[ g(x):=a_1x+a_2x^2+a_3x^3+\cdots=xc(x) \] where $c(x)=\sum\limits_{n=0}^\infty C_kx^k$ is the generating function of the Catalan numbers. \end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:count_nice_bijection}] Suppose $n\ge2$ and $f$ is a nice bijection from $[n-1]$ to $\{2,\ldots, n\}$ such that $f(1)=k$.
Suppose $p<k$. We claim that $f(p)<k$. If not, suppose $q=f(p)>k$. Then, neither $\{1,k\}$ nor $\{p,q\}$ is a succession. Consider the horizontal diagram of $f$. If we perform a reduction of type 1 or type 2 on $f$, then the arc we remove is neither the arc from $1$ to $k$ nor the arc from $p$ to $q$. By induction, no matter how many reductions we perform, there always exists an arc from $1$ to $k'$ and an arc from $p'$ to $q'$ such that $p'<k'<q'$. In other words, we can never reduce the bijection to $1\mapsto2$. Thus, we proved the claim.
Thus, the image of $\{2,\ldots, k-1\}$ under $f$ must be $\{2,\ldots, k-1\}$, and therefore, the image of $\{k,\ldots,n-1\}$ under $f$ must be $\{k+1,\ldots,n\}$.
This implies that $f|_{\{2,\ldots, k-1\}}$ has the same diagram as a permutation $\tau\in S_{k-2}$ and we can reduce $\tau$ to
$\pi_\emptyset$. This also implies that $f|_{\{k,\ldots, n-1\}}$ has the same diagram as a nice bijection
from $[n-k]$ to $\{2,\ldots, n-k+1\}$. By Lemma \ref{lemma:a_permutation_can_be_reduced_to_null_iff...}, the number of nice bijections from $[n-1]$ to $\{2,\ldots, n\}$ such that $f(1)=k$ is $C_{k-2}a_{n-k+1}$. Letting $k$ vary, we obtain $a_n=\sum\limits_{k=2}^nC_{k-2}a_{n-k+1}$.
By \eqref{recurrence_of_Catalan_number}, $(C_n)_{n\ge0}$ and $(a_{n+1})_{n\ge0}$ have the same recurrence relation and the same initial condition $C_0=a_1=1$, so $C_n=a_{n+1}(n\ge0)$. Therefore $g(x)=xc(x)$. \end{proof}
\section{Structure of straight m\'enage permutations}\label{sec:straight_menage_permutation} In Section \ref{sec:straight_menage_permutation}, when we mention a reduction, we mean a reduction of \textbf{type 1 or type 2}.
If a permutation $\pi$ is not a straight m\'enage permutation, then $\pi$ has at least one fixed point or succession. Thus, we can apply a reduction to $\pi$. By induction, we can reduce $\pi$ to a straight m\'enage permutation $\pi'$ by a series of reductions. For example, we can reduce $\pi_1=(1,3)(2)(4,5,6)$ to $\pi_\emptyset$: $(1,3)(2)(4,5,6)\to(1,2)(3,4,5)\to(1)(2,3,4)\to(1,2,3)\to(1,2)\to(1)\to\pi_\emptyset$. We can reduce $\pi_2=(1,5,4)(2)(3)(6)\in S_6$ to $(1,3,2)$: $(1,5,4)(2)(3)(6)\to(1,5,4)(2)(3)\to(1,4,3)(2)\to(1,3,2)$. It is easy to see that the resulting straight m\'enage permutation does not depend on the order of the reductions. Recall that we defined the permutation induced from a noncrossing partition in Section \ref{sec:partitions_and_Catalan_numbers}. \begin{lemma}\label{lemma:a_permutation_can_be_reduced_to_null_iff...} Suppose $\pi\in S_n$. We can reduce $\pi$ to $\pi_\emptyset$ by reductions of type 1 and type 2 if and only if there is a noncrossing partition inducing $\pi$. In particular, there are $C_n$ such permutations in $S_n$. \end{lemma}
\begin{proof} $\Rightarrow$: Suppose we can reduce $\pi\in S_n$ to $\pi_\emptyset$. Then, $\pi$ has at least one fixed point or succession. We use induction on $n$. If $n=1$, the conclusion is trivial. Suppose $n>1$.
If $\pi$ has a fixed point $i$, then by reduction of type 1 on $i$ we obtain $\pi'$ satisfying \eqref{eq:expression_of_reduction_of_type_1}. By induction assumption, there is a noncrossing partition $\Phi=\{V_1,\ldots,V_k\}$ inducing $\pi'$. Now, we define a new noncrossing partition $\Pi_1(\Phi,i)$ as follows. Set
$$\tilde V_r=\{x+1|x\in V_r\text{ and }x\ge i\}\cup\{x|x\in V_r\text{ and }x<i\}$$ for $1\le r\le k$ and $\tilde V_{k+1}=\{i\}$. Define $\Pi_1(\Phi,i)=\{\tilde V_1,\ldots,\tilde V_{k+1}\}$. It is not difficult to check that $\Pi_1(\Phi,i)$ is a noncrossing partition inducing $\pi$.
If $\pi$ has a succession $\{i,i+1\}$, then by reduction of type 2 on $\{i,i+1\}$, we obtain $\pi''$ satisfying \eqref{eq:expression_of_reduction_of_type_2}. By induction assumption, there is a noncrossing partition $\Phi=\{U_1,\ldots,U_s\}$ inducing $\pi''$. Now, we define a new noncrossing partition $\Pi_2(\Phi,i)$ as follows. Set \begin{align*}
\tilde U_t=\begin{cases}\{x+1|x\in U_t\text{ and }x> i\}\cup\{x|x\in U_t\text{ and }x<i\}&\text{ if }t\ne t_0;\\\{x+1|x\in U_t\text{ and
}x> i\}\cup\{x|x\in U_t\text{ and }x<i\}\cup\{i,i+1\}&\text{ if }t=t_0.\end{cases} \end{align*} Define $\Pi_2(\Phi,i)=\{\tilde U_1,\ldots,\tilde U_s\}$. It is not difficult to check that $\Pi_2(\Phi,i)$ is a noncrossing partition inducing $\pi$.
$\Leftarrow$: Suppose there is a noncrossing partition $\{V_1,\ldots,V_k\}$ inducing $\pi$ where $V_r=\{a_1^r,\ldots, a_{j_r}^r\}$ and $a_1^r<\cdots<a_{j_r}^r$. We prove by using induction on $n$. The case $n=1$ is trivial. Suppose $n>1$. For $r_1\ne r_2$, if $[a_1^{r_1},a_{j_{r_1}}^{r_1}]\cap[a_1^{r_2},a_{j_{r_2}}^{r_2}]\ne\emptyset$, then either $[a_1^{r_1},a_{j_{r_1}}^{r_1}]\subset[a_1^{r_2},a_{j_{r_2}}^{r_2}]$ or $[a_1^{r_2},a_{j_{r_2}}^{r_2}]\subset[a_1^{r_1},a_{j_{r_1}}^{r_1}]$; otherwise, the partition cannot be noncrossing. Thus, there exists $p$ such that $[a_1^p,a_{j_p}^p]\cap[a_1^q,a_{j_q}^q]=\emptyset$ for all $q\ne p$. If $j_p=1$, then $a_1^p$ is a fixed point of $\pi$. If $j_p>1$, then $\{a_1^p,a_2^p\}$ is a succession of $\pi$.
For the case $j_p=1$, perform a reduction of type 1 on $a_1^p$ and obtain $\pi'\in S_{n-1}$. Then, $\{\tilde V_r|r\ne p\}$ is a noncrossing partition inducing $\pi'$, where \begin{align}\label{eq:tilde_V_r}
\tilde V_r=\{x-1|x\in V_r\text{ and }x> a_{1}^p\}\cup\{x|x\in V_r\text{ and }x<a_{1}^p\}. \end{align} By induction assumption, we can reduce $\pi'$ to $\pi_\emptyset$; then, we can also reduce $\pi$ to $\pi_\emptyset$.
For the case $j_p>1$, perform a reduction of type 2 on
$\{a_1^p,a_2^p\}$ and get $\pi'\in S_{n-1}$. Then, $\{\tilde V_r|1\le r\le k\}$ is a noncrossing partition inducing $\pi'$, where $\tilde V_r$ is the same as in \eqref{eq:tilde_V_r}. By induction assumption, we can reduce $\pi'$ to $\pi_\emptyset$; then, we can also reduce $\pi$ to $\pi_\emptyset$. \end{proof}
Conversely, for a given straight m\'enage permutation $\pi\in S_m$, what is the cardinality of the set \begin{align}\label{set:permutations_which_can_be_reduced_to_a_given_permutation}
\{\tau\in S_{m+n}\big|\text{we can reduce }\tau\text{ to }\pi\text{ by reductions of type 1 and type 2}\}? \end{align} Interestingly the answer only depends on $m$ and $n$; it does not depend on the choice of $\pi$. In fact, we have \begin{theorem}\label{gf_of_omega} Suppose $m\ge0$ and $\pi\in S_m$ is a straight m\'enage permutation. Suppose $n\ge0$ and $w_m^n$ is the cardinality of the set in \eqref{set:permutations_which_can_be_reduced_to_a_given_permutation}. Set $W_m(x)=w_m^0+w_m^1x+w_m^2x^2+w_m^3x^3+\cdots$; then, \[ W_m(x)=c(x)^{2m+1}. \] \end{theorem}
\begin{proof}[Proof of Theorem \ref{gf_of_omega}] If $m=0$, then $\pi=\pi_\emptyset$, and the conclusion follows from Lemma \ref{lemma:a_permutation_can_be_reduced_to_null_iff...}. Thus, we only consider the case that $m>0$.
Obviously, $w_m^0=1$. Now, suppose $n\ge1$.
Represent $\pi$ by a horizontal diagram. The diagram has $m+1$ gaps: one gap before the first point, one gap after the last point and one gap between each pair of adjacent points.
Let $A$ be the set in \eqref{set:permutations_which_can_be_reduced_to_a_given_permutation}. To obtain a permutation in $A$, we add points to $\pi$ in the following two ways: \begin{enumerate} \item[(a)] add a permutation induced by a noncrossing partition $\Phi_p$ of $[d_p]$ into the $p$th gap of $\pi$, where $1\le p\le m+1$ and $d_p\ge0$ ($d_p=0$ means that we add nothing into the $p$th gap); \item[(b)] replace the $q$th point of $\pi$ by a nice bijection $f_q$ from $[r_q]$ to $\{2,\ldots,r_q+1\}$, where $1\le q\le m$ and $r_q\ge0$ ($r_q=0$ means that we do not change the $q$th point). \end{enumerate} For example, if $\pi$ and $f$ are as below,
\centerline{\includegraphics[width=4in]{6.eps}}
then we can add the permutation $(1,2)$ between $p$ and $q$, add the permutation $(1)(2,3)$ after the last point and replace $p$ by $f$. Then, we obtain a permutation in $A$ that is:
\centerline{\includegraphics[width=4in]{7.eps}}
\textbf{Statement: The set of permutations constructed from (a) and (b) equals $A$.} It is easy to see that we can reduce a permutation constructed through (a) and (b) to $\pi$. Conversely, suppose we can reduce $\pi'\in S_{m+n}$ to $\pi$. Now, we show that one can construct $\pi'$ through (a) and (b). Use induction on $n$. The case that $n=0$ is trivial. Suppose $n>0$. Then, $\pi'$ has at least one fixed point or succession. For a nice bijection $f$ from $[s]$ to $\{2,\ldots,s+1\}$ and $1<w_1<s+1$, $1\le w_2\le s+1$, define nice bijections $B_1(f,w_1)$ and $B_2(f,w_2)$ as \begin{align}\label{eq:B_1(f,w_1)} B_1(f,w_1)=\begin{cases}f(x)&\text{ if }x<w_1\text{ and } f(x)<w_1;\\f(x)+1&\text{ if }x<w_1\text{ and } f(x)\ge w_1;\\f(x-1)&\text{ if }x>w_1\text{ and } f(x)<w_1;\\f(x-1)+1&\text{ if }x>w_1\text{ and } f(x)\ge w_1;\\w_1&\text{ if }x=w_1\end{cases} \end{align} \begin{align} B_2(f,w_2)=\begin{cases}f(x)&\text{ if }x<w_2\text{ and } f(x)\le w_2;\\f(x)+1&\text{ if }x<w_2\text{ and } f(x)>w_2;\\f(x-1)&\text{ if }x>w_2\text{ and } f(x)\le w_2;\\f(x-1)+1&\text{ if }x>w_2\text{ and } f(x)>w_2;\\w_2+1&\text{ if }x=w_2.\end{cases}\label{eq:B_2(f,w_2)} \end{align} We can reduce $B_1(f,w_1)$ to $f$ by a reduction of type 1 on the fixed point $w_1$. We can reduce $B_2(f,w_2)$ to $f$ by a reduction of type 2 on the succession $\{w_2,w_2+1\}$.
\textbf{Case 1: $\pi'$ has a fixed point $i$.} Using a reduction of type 1 on $i$, we obtain $\pi''\in S_{m+n-1}$. By induction assumption, we can construct $\pi''$ from $\pi$ by (a) and (b). According to the value of $i$, there is either a $k$ such that \begin{align}\label{eq:condition_of_k} 1\le i-\sum_{j<k}(d_j+r_j+1)\le d_k+1 \end{align} or a $k'$ such that \begin{align}\label{eq:condition_of_k'} 1<i-(\sum_{j<k'}(d_j+r_j+1)+d_{k'})\le r_{k'}+1. \end{align} If there is a $k$ such that \eqref{eq:condition_of_k} holds, then we can construct $\pi'$ from $\pi$ by (a) and (b), except that we add the permutation induced by $\Pi_1(\Phi_k,i-\sum_{j<k}(d_j+r_j+1))$ instead of $\Phi_k$ into the $k$th gap, where $\Pi_1$ is the same as in the proof of Lemma \ref{lemma:a_permutation_can_be_reduced_to_null_iff...}. Conversely, if there exists a $k'$ such that \eqref{eq:condition_of_k'} holds, then we can construct $\pi'$
from $\pi$ by (a) and (b), except that we replace the $k'$th point of $\pi$ by $B_1(f_{k'},i-\sum_{j<k'}(d_j+r_j+1))$ instead of $f_{k'}$, where $B_1$ is the same as in \eqref{eq:B_1(f,w_1)}.
\textbf{Case 2: $\pi'$ has a succession $\{i,i+1\}$.} By reduction of type 2 on $\{i,i+1\}$ we obtain $\pi''\in S_{m+n-1}$. By induction assumption, we can construct $\pi''$ from $\pi$ by (a) and (b). According to the value of $i$, there is either a $k$ such that \begin{align}\label{eq:condition_of_k_for_succession} 0<i-\sum_{j<k}(d_j+r_j+1)\le d_k \end{align} or a $k'$ such that \begin{align}\label{eq:condition_of_k'_for_succession} 0<i-(\sum_{j<k'}(d_j+r_j+1)+d_{k'})\le r_{k'}+1. \end{align} If there is a $k$ such that \eqref{eq:condition_of_k_for_succession} holds, then we can construct $\pi'$ from $\pi$ by (a) and (b), except that we add the permutation induced by $\Pi_2(\Phi_k,i-\sum_{j<k}(d_j+r_j+1))$ instead of $\Phi_k$ into the $k$th gap, where $\Pi_2$ is the same as in the proof of Lemma \ref{lemma:a_permutation_can_be_reduced_to_null_iff...}. Conversely, if there exists a $k'$ such that \eqref{eq:condition_of_k'_for_succession} holds, then we can construct $\pi'$ from $\pi$ by (a) and (b), except that we replace the $k'$th point of $\pi$ by $B_2(f_{k'},i-\sum_{j<k'}(d_j+r_j+1))$ instead of $f_{k'}$, where $B_2$ is the same as in \eqref{eq:B_2(f,w_2)}.
Thus, we have proved the statement.
Now, add points to $\pi$ by (a) and (b). The total number of points added to $\pi$ is $d_1+\cdots+d_{m+1}+r_1+\cdots+r_m$. To obtain a permutation in $S_{m+n}\cap A$, $d_1+\cdots+d_{m+1}+r_1+\cdots+r_m$ should be $n$. Therefore, the number of permutations in $S_{m+n}\cap A$ is \[ w_m^n=\sum\limits_{d_1,\ldots,d_{m+1}\atop r_1,\ldots, r_m}C_{d_1}\cdots C_{d_{m+1}}a_{1+r_1}\cdots a_{1+r_m} \] where $C_k$ is the $k$th Catalan number and $a_k$ is the same as in Lemma \ref{lemma:count_nice_bijection} and the sum runs over all $(2m+1)$-triples $(d_1,\ldots,d_{m+1},r_1,\ldots,r_m)$ of nonnegative numbers with sum $n$.
By Lemma \ref{lemma:count_nice_bijection}, the generating function of $w_m^n$ is $$c(x)^{m+1}\big(\frac{g(x)}{x}\big)^m=c(x)^{2m+1}.$$ \end{proof}
\begin{proof}[Proof of \eqref{eq:result_of_straight_menage_numbers}] We can reduce each permutation in $S_n$ to a straight m\'enage permutation in $S_i$ ($0\le i\le n$). Thus, we have $$n!\,=\sum\limits_{i=0}^nw_i^{n-i}V_i$$ where $V_i$ is the $i$th straight m\'enage number and $w_i^{n-i}$ is the same as in Theorem \ref{gf_of_omega}. Thus, \begin{align*} \sum\limits_{n=0}^{\infty}n!\,x^n=\sum\limits_{n=0}^\infty \sum\limits_{i=0}^nw_i^{n-i}V_ix^n=\sum\limits_{i=0}^\infty \sum\limits_{n=i}^\infty w_i^{n-i}V_ix^n=\sum\limits_{i=0}^\infty \sum\limits_{n=0}^\infty w_i^nV_ix^nx^i=\sum\limits_{i=0}^\infty c(x)^{2i+1}V_ix^i \end{align*} where the last equality is from Theorem \ref{gf_of_omega}. \end{proof}
\section{Structure of ordinary m\'enage permutations}\label{sec:ordinary_menage_permutation} By definition, a permutation $\tau$ is an ordinary m\'enage permutation if and only if we cannot apply reductions of either type 1 or type 3 on $\tau$. Similarly as in Section \ref{sec:straight_menage_permutation}, we can reduce each permutation $\pi$ to an ordinary m\'enage permutation by reductions of type 1 and type 3. The resulting permutation does not depend on the order of the reductions.
By the circular diagram representation of permutations, it is not difficult to see that we can reduce a permutation $\pi$ to $\pi_\emptyset$ by reductions of type 1 and type 2 if and only if we can reduce $\pi$ to $\pi_\emptyset$ by reductions of type 1 and type 3. Thus, we have the following lemma: \begin{lemma}\label{lemma:equivalence} Suppose $\pi\in S_n$. We can reduce $\pi$ to $\pi_\emptyset$ by reductions of type 1 and type 3 if and only if there is a noncrossing partition inducing $\pi$. In particular, there are $C_n$ permutations of this type in $S_n$. \end{lemma} In the following parts of Section \ref{sec:ordinary_menage_permutation}, when we mention reductions, we mean reductions of \textbf{type 1 or type 3} unless otherwise specified. \begin{theorem}\label{gf_of_r} Suppose $m\ge0$ and $\pi\in S_m$ is an ordinary m\'enage permutation. Let $r_m^n$ denote the cardinality of the set \begin{align}\label{set:permutations_which_can_be_reduced_to_a_given_permutation_by_type_1_and_3}
\{\tau\in S_{m+n}|\text{we can reduce }\tau\text{ to }\pi\text{ by reductions of type 1 and type 3}\}. \end{align} Then, the generating function of $r_m^n$ satisfies \begin{align*} R_m(x):=r_m^0+r_m^1x+r_m^2x^2+r_m^3x^3+\cdots=\begin{cases}c'(x)c(x)^{2m-2}&\text{if}\quad m>0;\\ c(x)&\text{if}\quad m=0.\end{cases} \end{align*} \end{theorem} \begin{proof} When $m=0$, $r_m^n=C_n$ by Lemma \ref{lemma:equivalence}. So $R_0(x)=c(x)$. Now, suppose $m>0$.
Obviously $r_m^0=1$. Suppose $n>0$.
Represent $\pi$ by a circular diagram. The diagram has $m$ gaps: one gap between each pair of adjacent points. Call the point corresponding to number $i$ $point$ $i$.
Let $A$ denote the set in \eqref{set:permutations_which_can_be_reduced_to_a_given_permutation_by_type_1_and_3}. To obtain a permutation in $A$ we can add points into $\pi$ by the following steps: \begin{enumerate} \item[(a)] Add a permutation induced by a noncrossing partition $\Phi_i$ of $[d_i]$ into the gap between point $i$ and point $i+1$ (mod $m$), where $1\le i\le m$ and $d_p\ge0$ ($d_p=0$ means we add nothing into the gap). Use $Q_i$ to denote the set of points added into the gap between point $i$ and point $i+1$ (mod $m$). \item[(b)] Replace point $i$ by a nice bijection $f_i$ from $[t_i]$ to $\{2,\ldots,t_i+1\}$, where $1\le i\le m$ and $t_i\ge0$ ($t_i=0$ means that we do not change the $i$th point). Use $P_i$ to denote the set of points obtained from this replacement. Thus, $P_i$ contains $t_i+1$ points. \item[(c)] Specify a point in $P_1\bigcup Q_m$ to correspond to the number 1 of the new permutation $\pi'$. \end{enumerate} Steps (a) and (b) are the same as in the proof of Theorem \ref{gf_of_omega}, but Step (c) needs some explanation.
In the proof of Theorem \ref{gf_of_omega}, after adding points into the permutation by (a) and (b), we defined $\pi'$ from the resulting $horizontal$ diagram in a natural way, that is, the left most point corresponds to 1, and the following points correspond to 2, 3, 4, \ldots \,respectively. However, now there is no natural way to define $\pi'$ from the resulting $circular$ diagram because we have more than one choice of the point corresponding to number 1 of $\pi'$.
Note that $\pi'$ can become $\pi$ by a series of reductions of type 1 or type 3. Then, for each $1\le u\le m$, the points in $P_u$ will become point $u$ of $\pi$ after the reductions. Thus, we can choose any point of $P_1$ to be the one corresponding to the number 1 of $\pi'$. Moreover, we can also choose any point of $Q_m$ to be the one corresponding to the number 1 of $\pi'$. The reason is that if a permutation in $S_k$ has a cycle $(1,k)$, then a reduction of type 3 will reduce $(1,k)$ to the cycle $(1)$. Thus, we can choose any point in $P_1\bigcup Q_m$ to be the point corresponding to number 1 of $\pi'$. To see this more clearly, let us look at an example. Suppose $\pi$ and $f$ are as below:
\centerline{\includegraphics[width=3.2in]{8.eps}}
Then, we can add the permutation $(1,2)$ between point 2 and point 3, add the permutation $(1)(2,3)$ between point 6 and point 1 and replace point 2 by $f$. Then, we obtain a new diagram:
\centerline{\includegraphics[width=2in]{9.eps}}
Now, we can choose point 1 or any of the three points between point 1 and point 6 to be the point corresponding to number 1 of the new permutation $\pi'$.
For instance, if we set point 1 to be the point corresponding to number 1 of $\pi'$, then $$\pi'=(1,8,2,5)(3,4)(6,7)(9,11,10)(12)(13,14).$$ If we set point w to be the point corresponding to number 1 of $\pi'$, then $$\pi'=(1,14)(2,9,3,6)(4,5)(7,8)(10,12,11)(13).$$
Now, continue the proof. By a similar argument to the one we used to prove the statement in the proof of Theorem \ref{gf_of_omega}, we have that the set of permutations constructed by (a)--(c) equals $A$.
Now, add points to $\pi$ by (a)--(c). Then, $P_1\bigcup Q_m$ contains, in total, $d_m+(t_1+1)$ points. Thus, we have $d_m+(t_1+1)$ ways to specify the point corresponding to number 1 in $\pi'$. The total number added to $\pi$ is $d_1+\cdots+d_m+t_1+\cdots+t_m$. Therefore, the number of permutations in $S_{m+n}\cap A$ is \[ r_m^n=\sum\limits_{d_1,\ldots,d_m\atop t_1,\ldots, t_m}C_{d_1}\cdots C_{d_m}a_{1+t_1}\cdots a_{1+t_m}(d_m+t_1+1) \] where $C_k$ is the $k$th Catalan number, $a_k$ is the same as in Lemma \ref{lemma:count_nice_bijection}, and the sum runs over all $2m$-triples $(d_1,\ldots,d_m,t_1,\ldots, t_m)$ of nonnegative integers such that $\sum_{u=1}^m(t_u+d_u)=n$.
Set $\eta_k=\sum\limits_{r=0}^ka_{r+1}C_{k-r}(k+1)$; from Lemma \ref{lemma:count_nice_bijection}, we have $$1+\frac{\eta_1}{2}x+\frac{\eta_2}{3}x^2+\frac{\eta_3}{4}x^3+\cdots=c^2(x).$$ By Lemma \ref{thm:property_of_Catalan_numbers}, the generating function of $\eta_k$ is $1+\eta_1x+\eta_2x^2+\eta_3x^3+\cdots=(xc^2(x))'=c'(x)$.
Thus, the generating function of $r_m^n$ is $c(x)^{2m-2}c'(x)$. \end{proof}
\begin{proof}[Proof of \eqref{eq:result_of_ordinary_menage_numbers}] We can reduce each permutation in $S_n$ to an ordinary m\'enage permutation. Thus, we have $$n!\,=\sum\limits_{i=0}^nr_i^{n-i}U_i$$ where $U_i$ is the $i$th ordinary m\'enage number and $r_i^{n-i}$ is the same as in Theorem \ref{gf_of_r}. Thus, \begin{multline*} \sum\limits_{n=0}^{\infty}n!\,x^n=\sum\limits_{n=0}^\infty \sum\limits_{i=0}^nr_i^{n-i}U_ix^n=\sum\limits_{i=0}^\infty \sum\limits_{n=i}^\infty r_i^{n-i}U_ix^n\\=\sum\limits_{i=0}^\infty \sum\limits_{n=0}^\infty r_i^nU_ix^nx^i =c(x)+\sum\limits_{i=1}^\infty c'(x)c(x)^{2i-2}U_ix^i \end{multline*} where the last equality follows from Theorem \ref{gf_of_r}. \end{proof}
\section{Counting m\'enage permutations by number of cycles}\label{count_permutations_by_cycles} We prove \eqref{eq:straight_by_cycles} in Section \ref{sec:count_straight_by_cycles} and prove \eqref{eq:ordinary_by_cycles} in Section \ref{sec:count_ordinary_by_cycles}. Our main method is coloring. We remark that one can also prove \eqref{eq:straight_by_cycles} and \eqref{eq:ordinary_by_cycles} by using the inclusion-exclusion principle in a similar way to \cite{Gessel}. \subsection{Coloring and weights} For a permutation $\pi$, we use $f(\pi)$, $g(\pi)$, $h(\pi)$ and $r(\pi)$ to denote the number of its cycles, fixed points, successions and generalized successions, respectively. The following lemma is well known.
\begin{lemma}\label{lemma:well_known_lemma_for_cycles} For $n\ge0$, \begin{align*} \sum\limits_{\pi\in S_n}\alpha^{f(\pi)}=(\alpha)_n. \end{align*} \end{lemma}
For a permutation, color some of its fixed points red and color some of its $generalized$ successions yellow. Then, we obtain a colored permutation. Here and in the following sections, we use the phrase $colored$ $permutation$ as follows: if two permutations are the same as maps but have different colors, then they are different colored permutations.
Define $\mathbb{S}_n$ to be the set of colored permutations on $n$ objects. Set $A_n$ to be the subset of $\mathbb{S}_n$ consisting of colored permutations with a colored generalized succession $\{n,1\}$. Set $B_n=\mathbb{S}_n\backslash A_n$. So we can consider $S_n$ as a subset of $\mathbb{S}_n$ consisting of permutations with no color. In particular, $\mathbb{S}_0=S_0$.
Set $$M_n^\alpha(t,u)=\sum\limits_{\pi\in S_n}\alpha^{f(\pi)}t^{g(\pi)}u^{h(\pi)}\quad\quad\text{and}\quad\quad L_n^\alpha(t,u)=\sum\limits_{\pi\in S_n}\alpha^{f(\pi)}t^{g(\pi)}u^{\tau(\pi)}.$$
For a colored permutation $\epsilon\in\mathbb{S}_n$, define two weights $W_1$ and $W_2$ of $\epsilon$ as: \begin{align*} W_1(\epsilon)&=x^n\cdot\alpha^{f(\epsilon)}\cdot t^{\text{number of colored fixed points}}\cdot u^{\text{number of colored successions}},\\ W_2(\epsilon)&=x^n\cdot\alpha^{f(\epsilon)}\cdot t^{\text{number of colored fixed points}}\cdot u^{\text{number of colored generalized successions}}. \end{align*}
\begin{lemma}\label{lemma:sum_of_the_W1_and_W2_weights} $\sum\limits_{\epsilon\in B_n}W_1(\epsilon)=\sum\limits_{\epsilon\in B_n}W_2(\epsilon)=M_n^\alpha(1+t,1+u)x^n$, $\sum\limits_{\epsilon\in\mathbb{S}_n}W_2(\epsilon)=L_n^\alpha(1+t,1+u)x^n$. \end{lemma}
\begin{proof} The lemma follows directly from the definitions of $M_n^\alpha$, $L_n^\alpha$, $B_n$, $W_1$ and $W_2$. \end{proof}
\subsection{Counting straight m\'enage permutations by number of cycles}\label{sec:count_straight_by_cycles}
In this subsection, we represent permutations by diagrams of \textbf{horizontal type}.
Suppose $n\ge0$ and $\pi\in S_n$. Then, $\pi$ has $n+1$ gaps: one gap before the first point, one gap after the last point and one gap between each pair of adjacent points. We can add points to $\pi$ by the following steps.
(a) Add the identity permutation of $S_{(d_p)}$ into the $p$th gap of $\pi$, where $1\le p\le n+1$ and $d_p\ge0$ ($d_p=0$ means that we add nothing into the $p$th gap).
(b) Replace the $q$th point of $\pi$ by a nice bijection from $[r_q]$ to $\{2,\ldots,r_q+1\}$, which sends each $i$ to $i+1$. Here, $1\le q\le n$ and $r_q\ge0$ ($r_q=0$ means that we do not change the $q$th point).
(c) Color the fixed points added by (a) red, and color the successions added by (b) yellow.
Then (a), (b) and (c) give a colored permutation in $\bigcup_{n=0}^\infty B_n$. \begin{lemma}\label{lemma:sum_of_W1_weight_of_things_constructed_from_pi} Suppose $\pi\in S_n$. The sum of the $W_1$-weights of all colored permutations constructed from $\pi$ by (a)--(c) is \[ x^n\cdot\alpha^{f(\pi)}\cdot\frac{1}{(1-\alpha tx)^{n+1}}\cdot\frac{1}{(1-ux)^{n}}. \] \end{lemma} \begin{proof} In Step (a), we added $d_p$ fixed points into the $p$th gap. They contribute $(xt\alpha)^{d_p}$ to the weight because each of them is a single cycle and a colored fixed point. Because $d_p$ can be any nonnegative integer, the total contribution of the fixed points added into a gap is $\frac{1}{(1-\alpha tx)}$. Thus, the total contribution of the fixed points added into all the gaps is $\frac{1}{(1-\alpha tx)^{n+1}}$.
In Step (b), through the replacement on the $q$th point, we added $r_q$ points and $r_q$ successions to $\pi$ (each of which received a color in Step (c)). Thus, the contribution of this replacement to the weight is $(ux)^{d_q}$. Because $d_q$ can be any nonnegative integer, the total contribution of the nice bijections replacing the $q$th point is $\frac{1}{(1-ux)}$. Thus, the total contribution of the nice bijections corresponding to all points is $\frac{1}{(1-ux)^{n}}$.
Observing that the $W_1$-weight of $\pi$ is $x^n\cdot\alpha^{f(\pi)}$, we complete the proof. \end{proof}
Suppose $\epsilon$ is a colored permutation in $\bigcup_{n=0}^\infty B_n$. If we perform reductions of type 1 on the colored fixed points and perform reductions of type 2 on the colored successions, then we obtain a new permutation $\epsilon'$ with no color. This $\epsilon'$ is the only permutation in $\bigcup_{n=0}^\infty S_n$ from which we can obtain $\epsilon$ by Steps (a)--(c). Therefore, there is a bijection between $\bigcup_{n=0}^\infty B_n$ and $$\bigcup_{n=0}^\infty\bigcup_{\pi\in S_n}\{\text{colored permutation constructed from $\pi$ through Steps (a)--(c)}\}.$$ Because of the bijection, Lemmas \ref{lemma:sum_of_the_W1_and_W2_weights} and \ref{lemma:sum_of_W1_weight_of_things_constructed_from_pi} imply that \begin{align}\label{kkk} \sum\limits_{n=0}^\infty M_n^\alpha(1+t,1+u)x^n=\sum\limits_{n=0}^\infty\sum\limits_{\pi\in S_n}\dfrac{x^n\alpha^{f(\pi)}}{(1-\alpha tx)^{n+1}(1-ux)^{n}}. \end{align} \begin{proof}[Proof of \eqref{eq:straight_by_cycles}] By Lemma \ref{lemma:well_known_lemma_for_cycles} and \eqref{kkk}, the sum of the $W_1$-weights of colored permutations in $\bigcup\limits_{n=0}^\infty B_n$ is \begin{align}\label{lll} \sum\limits_{n=0}^\infty M_n^\alpha(1+t,1+u)x^n=\sum\limits_{n=0}^\infty \frac{x^n(\alpha)_n}{(1-\alpha tx)^{n+1}(1-ux)^{n}}. \end{align} Setting $t=u=-1$, we have \[ \sum\limits_{n=0}^\infty M_n^\alpha(0,0)x^n=\sum\limits_{n=0}^\infty \frac{x^n(\alpha)_n}{(1+\alpha x)^{n+1}(1+x)^{n}}. \]
Recall that straight m\'enage permutations are permutations with no fixed points or successions. Thus, for $n>0$, $M_n^\alpha(0,0)$ is the sum of the $W_1$-weights of straight m\'enage permutations in $S_n$, which is $\sum\limits_{j=1}^nC_n^j\alpha^jx^n$. Furthermore, $M_0^\alpha(0,0)=1$. Thus, we have proved \eqref{eq:straight_by_cycles}. \end{proof}
\subsection{Counting ordinary m\'enage permutations by number of cycles}\label{sec:count_ordinary_by_cycles} In this subsection, we represent permutations by diagrams of the \textbf{horizontal type}.
Suppose $n\ge0$ and $\pi\in S_n$. Define $\mathbb{S}_m(\pi)$ to be the a subset of $\mathbb{S}_m$: $\tau\in\mathbb{S}_m$ is in $\mathbb{S}_m(\pi)$ if and only if when we apply reductions of type 1 on the colored fixed points of $\tau$ and apply reductions of type 3 on the colored generalized successions of $\tau$, we obtain $\pi$. Define $A_m(\pi)=\mathbb{S}_m(\pi)\cap A_m$ and $B_m(\pi)=\mathbb{S}_m(\pi)\backslash A_m(\pi)$.
\begin{lemma}\label{lemma:sum_of_W2_weights_in_all_An} The $W_2$-weights of colored permutations in $\bigcup_{n=0}^\infty A_n$ are \begin{align*} \sum\limits_{n=1}^\infty x^n(\alpha)_n\cdot\frac{1}{(1-\alpha tx)^n}\cdot\frac{1}{(1-ux)^{n+1}}\cdot ux+\alpha utx+\frac{\alpha ux}{1-ux}. \end{align*} \end{lemma} \begin{proof} We first evaluate the sum of $W_2$-weights of colored permutations in $\bigcup_{m=n}^\infty A_m(\pi)$ and then add them up with respect to $\pi\in S_n$ and $n\ge0$.
\textbf{Case 1: $n>0$.} In this case, for $\pi\in S_n$, we can construct a colored permutation in $\bigcup_{m=n}^\infty A_m(\pi)$ by the following steps.
($a^\prime$) Define $\tilde\pi$ to be a permutation in $S_{n+1}$ that sends $\pi^{-1}(1)$ to $n+1$, sends $n+1$ to 1 and sends all other $j$ to $\pi(j)$. Represent $\tilde\pi$ by a horizontal diagram.
($b^\prime$) For $1\le p\le n-1$, add the identity permutation of $S_{(d_p)}$ into the gap of $\tilde\pi$ between number $p$ and $p+1$, where $d_p\ge0$ ($d_p=0$ means we add nothing into the gap).
($c^\prime$) Replace the $q$th point of $\tilde\pi$ by a nice bijection from $[r_q]$ to $\{2,\ldots,r_q+1\}$, which maps each $i$ to $i+1$. Here, $1\le q\le n$ and $r_q\ge0$ ($r_q=0$ means that we do not change the $q$th point).
($d^\prime$) Color the generalized succession consisting of 1 and the largest number yellow. Color the fixed points and generalized successions added by ($b^\prime$)--($c^\prime$) red and yellow, respectively.
Suppose $\epsilon\in\bigcup_{m=n}^\infty A_m(\pi)$. If we perform reductions of type 1 on its colored fixed points and perform reductions of type 3 on its colored generalized successions, we obtain $\pi$. Furthermore, $\pi$ is the only permutation in $\bigcup_{k=0}^\infty S_k$ from which we can obtain $\epsilon$ through ($a^\prime$)--($d^\prime$). Therefore, there is a bijection between $\bigcup_{m=n}^\infty A_m(\pi)$ and $$\{\text{colored permutations constructed from $\pi$ through ($a^\prime$)--($d^\prime$)}\}.$$
Now, we can claim that the sum of $W_2$-weight of the colored permutations in $\bigcup_{m=n}^\infty A_m(\pi)$ is \begin{align}\label{W2_weight_of...} x^n\alpha^{f(\pi)}\cdot\frac{1}{(1-\alpha tx)^n}\cdot\frac{1}{(1-ux)^{n+1}}\cdot ux. \end{align} In \eqref{W2_weight_of...}, $x^n\alpha^{f(\pi)}$ is the $W_2$-weight of $\pi$. The term $\frac{1}{(1-\alpha tx)^n}$ corresponds to the fixed points added to the permutation in ($b^\prime$). The term $\frac{1}{(1-ux)^{n+1}}$ corresponds to the successions added to the permutation in ($c^\prime$). The term $ux$ corresponds to the generalized succession $\{n+1,1\}$ added to the permutation in ($a^\prime$).
\textbf{Case 2: $n=0$ and $m=0$.} In this case, $A_0(\pi_\emptyset)$ is empty.
\textbf{Case 3: $n=0$ and $m\ge2$.} In this case, $A_m(\pi_\emptyset)$ contains one element: the cyclic permutation $\pi_C$, which maps each $i$ to $i+1$ (mod $m$). Each generalized succession of $\pi_C$ has a color, so the sum of the $W_2$-weight of the colored permutations in $A_m(\pi_\emptyset)$ is $\alpha(ux)^m$.
\textbf{Case 4: $n=0$ and $m=1$.} In this case, $A_1(\pi_\emptyset)$ contains one map: $id_1\in S_1$. However, $id_1$ can have two types of color, namely, yellow and red+yellow, because $id_1$ has one fixed point and one generalized succession. Thus, $A_1(\pi_\emptyset)$ contains two colored permutations, and the sum of their $W_2$-weights is $\alpha ux+\alpha tux$.
We remark that the identity permutation $id_1$ actually corresponds to four colored permutations. In addition to the two in $A_1(\pi_\emptyset)$, the other two are $id_1$, with a red color for its fixed point, and $id_1$, with no color. We have considered these colored permutations in $B_1(\pi_\emptyset)$ and $B_1(id_1)$, respectively.
Because $\bigcup_{n=0}^\infty A_n=\bigcup_{n=0}^\infty\bigcup_{\pi\in S_n}\bigcup_{m=n}^\infty A_m(\pi)$, the sum of the $W_2$-weights of the colored permutations in $\bigcup_{n=0}^\infty A_n$ equals the sum of the weights found in Case 1--4. Thus, the sum is \begin{multline*} \Bigg(\sum\limits_{n=1}^\infty\sum\limits_{\pi\in S_n}x^n\alpha^{f(\pi)}\cdot\frac{1}{(1-\alpha tx)^n}\cdot\frac{1}{(1-ux)^{n+1}}\cdot ux\Bigg)+\big(\sum\limits_{m=2}^\infty\alpha(ux)^m\big)+\big(\alpha ux+\alpha tux\big)\\ =\sum\limits_{n=1}^\infty x^n(\alpha)_n\cdot\frac{1}{(1-\alpha tx)^n}\cdot\frac{1}{(1-ux)^{n+1}}\cdot ux+\alpha utx+\frac{\alpha ux}{1-ux} \end{multline*} where we used Lemma \ref{lemma:well_known_lemma_for_cycles}. \end{proof}
\begin{proof}[Proof of \eqref{eq:ordinary_by_cycles}] By Lemma \ref{lemma:sum_of_the_W1_and_W2_weights}, $\sum\limits_{n=0}^\infty L_n^\alpha(1+t,1+u)x^n$ is the sum of the $W_2$-weights of the colored permutations in $\bigcup_{n=0}^\infty\mathbb{S}_n$. By Lemma \ref{lemma:sum_of_the_W1_and_W2_weights} and \eqref{lll}, the sum of the $W_2$-weights of all colored permutations in $\bigcup_{n=0}^\infty B_n$ is $$\sum\limits_{n=0}^\infty \frac{x^n(\alpha)_n}{(1-\alpha tx)^{n+1}(1-ux)^{n}}.$$ Because $\bigcup_{n=0}^\infty\mathbb{S}_n=(\bigcup_{n=0}^\infty B_n)\bigcup(\bigcup_{n=0}^\infty A_n)$, Lemma \ref{lemma:sum_of_W2_weights_in_all_An} implies
\begin{align}\label{eq:equation} &\sum\limits_{n=0}^\infty L_n^\alpha(1+t,1+u)x^n\nonumber\\ =&\Bigg(\sum\limits_{n=0}^\infty \frac{x^n(\alpha)_n}{(1-\alpha tx)^{n+1}(1-ux)^{n}}\Bigg)+\Bigg(\sum\limits_{n=1}^\infty x^n(\alpha)_n\cdot\frac{1}{(1-\alpha tx)^n}\cdot\frac{1}{(1-ux)^{n+1}}\cdot ux+\alpha utx+\frac{\alpha ux}{1-ux}\Bigg)\nonumber\\ =&\sum\limits_{n=0}^\infty \bigg[\frac{x^n(\alpha)_n}{(1-\alpha tx)^{n+1}(1-ux)^{n+1}}(1-\alpha tux^2)\bigg]+\alpha utx+\frac{(\alpha-1) ux}{1-ux}. \end{align} Recall that ordinary m\'enage permutations are permutations with no fixed points and no generalized successions. By definition, $L_0^\alpha(0,0)x^0=1$. When $n\ge1$, $L_n^\alpha(0,0)x^n$ is the sum of the $W_2$-weights of all ordinary m\'enage permutations in $S_n$, which equals $\sum\limits_{j=1}^nD_n^j\alpha^jx^n$. Thus, $\sum\limits_{n=0}^\infty L_n^\alpha(0,0)x^n$ equals the left side of \eqref{eq:ordinary_by_cycles}. When we set $t=u=-1$, the left side of \eqref{eq:equation} equals the left side of \eqref{eq:ordinary_by_cycles}, and the right side of \eqref{eq:equation} equals the right side of \eqref{eq:ordinary_by_cycles}. We have proved \eqref{eq:ordinary_by_cycles}. \end{proof}
\section{An analytical proof of \eqref{eq:result_of_straight_menage_numbers} and \eqref{eq:result_of_ordinary_menage_numbers}}\label{appendix}
Now, we derive \eqref{eq:result_of_straight_menage_numbers} and \eqref{eq:result_of_ordinary_menage_numbers} from \eqref{eq:known_formulas_for_U_n} and \eqref{eq:known_formulas_for_V_n}. By \eqref{eq:known_formulas_for_V_n}, \begin{align}\label{qqq} \sum\limits_{n=0}^\infty V_nx^n&=\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!\,x^n=\sum\limits_{k=0}^\infty\sum\limits_{n=k}^\infty(-1)^k{2n-k\choose k}(n-k)!\,x^n\nonumber\\&=\sum\limits_{k=0}^\infty\sum\limits_{n=0}^\infty(-1)^k{2n+k\choose k}n!\,x^{n+k}=\sum\limits_{n=0}^\infty n!\,x^n\bigg[\sum\limits_{k=0}^\infty(-1)^k{2n+k\choose k}x^k\bigg]\nonumber\\&=\sum\limits_{n=0}^\infty n!\,\dfrac{x^n}{(1+x)^{2n+1}}. \end{align} Letting $x=zc^2(z)$, from Lemma \ref{thm:property_of_Catalan_numbers} we have $1+x=c(z)$, $\dfrac{x}{(1+x)^2}=z$ and \begin{align*} \sum\limits_{n=0}^\infty V_nz^nc^{2n}(z)=\sum\limits_{n=0}^\infty V_nx^n=\sum\limits_{n=0}^\infty n!\,\dfrac{x^n}{(1+x)^{2n+1}}=\sum\limits_{n=0}^\infty n!\,\dfrac{z^n}{c(z)}. \end{align*} which implies \eqref{eq:result_of_straight_menage_numbers}. From \eqref{eq:known_formulas_for_U_n}, \begin{align*} U_n=\begin{cases}1&\text{if }n=0;\\0&\text{if }n=1;\\\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!+\sum\limits_{k=1}^n(-1)^k{2n-k-1\choose k-1}(n-k)!&\text{if }n>1.\end{cases} \end{align*} Thus \begin{align*} \sum\limits_{n=0}^\infty U_nx^n &=x+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!\,x^n+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^n(-1)^k{2n-k-1\choose k-1}(n-k)!\,x^n\\ &=x+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!\,x^n+\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n(-1)^{k+1}{2n-k\choose k}(n-k)!\,x^{n+1}\\ &=x+(1-x)\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n(-1)^k{2n-k\choose k}(n-k)!\,x^n\nonumber. \end{align*} Then, by \eqref{qqq}, \begin{align*} \sum\limits_{n=0}^\infty U_nx^n=x+(1-x)\sum\limits_{n=0}^\infty n!\,\dfrac{x^n}{(1+x)^{2n+1}}. \end{align*} Noticing $U_0=1$ we have $$1-x+\sum\limits_{n=1}^\infty U_nx^n=(1-x)\sum\limits_{n=0}^\infty n!\,\dfrac{x^n}{(1+x)^{2n+1}}$$ and \begin{align}\label{eq:for_analytical_proof_of_ordinary} 1+x+\dfrac{1+x}{1-x}\sum\limits_{n=1}^\infty U_nx^n=\sum\limits_{n=0}^\infty n!\,\dfrac{x^n}{(1+x)^{2n}}. \end{align} Letting $x=zc^2(z)$, from Lemma \ref{thm:property_of_Catalan_numbers} and \eqref{eq:for_analytical_proof_of_ordinary}, we have $1+x=c(z)$, $\dfrac{x}{(1+x)^2}=z$ and \begin{align*} \sum\limits_{n=0}^\infty n!\,z^n=c(z)+\dfrac{c(z)}{1-zc^2(z)}\sum\limits_{n=1}^\infty U_nz^n(c(z))^{2n}=c(z)+\dfrac{(c(z))^3}{1-zc^2(z)}\sum\limits_{n=1}^\infty U_nz^n(c(z))^{2n-2}. \end{align*} Then, \eqref{eq:result_of_ordinary_menage_numbers} follows from the above equation and Lemma \ref{thm:property_of_Catalan_numbers}.
\Addresses
\end{document} | arXiv |
Preparation of a new superhydrophobic/superoleophilic corn straw fiber used as an oil absorbent for selective absorption of oil from water
Yang Xu1 na1,
Haiyue Yang1 na1,
Deli Zang1,
Yan Zhou4,
Feng Liu1,
Xiaochen Huang2,
Jo-Shu Chang2,3,
Chengyu Wang1 &
Shih-Hsin Ho2
Bioresources and Bioprocessing volume 5, Article number: 8 (2018) Cite this article
Oil leakages frequently occur during oil product development and oil transportation. These incidents are a vital factor in water contamination, thus leading to serious environmental destruction. Therefore, superhydrophobic/superoleophilic material is one of the solutions to treat oily wastewater.
This study aimed to develop a simple, fast and low-cost method to treat oily wastewater by synthesizing a new superhydrophobic/superoleophilic corn straw fiber via conventional impregnation. The corresponding results illustrate that abundant homogeneous silica (SiO2) granules evenly accreted on the surface of the prepared fiber were conducive to high surface roughness. Meanwhile, (Heptadecafluoro-1,1,2,2-tetradecyl) trimethoxysilane, a sort of silane coupling agent, could greatly reduce surface free energy by grafting with SiO2 particles on the corn straw fiber surface. The obtained superhydrophobic/superoleophilic corn straw fiber exhibited a water contact angle of 152° and an oil contact angle of 0° for various oils, strongly demonstrating its considerable application as an oil absorbent that can be applied for oil cleanup. In addition, the prepared fiber displayed a great chemical stability and environmental durability.
Due to its high absorption capacity and absorption efficiency, the prepared fiber has great potential as a new oil absorbent for treatment of oily water.
Oil-spill pollution can significantly destroy the human living environment and poses a severe threat to ecological systems (Cojocaru et al. 2011; Deng et al. 2013; Schaum et al. 2010). At present, physical methods (Angelova et al. 2011; Howarter and Youngblood 2007), chemical processing methods and biological treatment (Boopathy et al. 2012) to cleanup oil-bearing wastewater have been reported. Among them, physical adsorption techniques, which employ hydrophilic materials to remove waste oil from water, is considered as the most economic and valid method due to its high absorption capacity and low cost. However, most of the traditional oil absorbents generally encounter problems, such as low separation efficiency, high cost, poor absorption capacity and non-biodegradable characteristics (Karakasi and Moutsatsou 2010). Thus, fabrication of new environmental-friendly oil absorbing material with a higher absorption ability and lower production cost is required and urgent (Yao et al. 2011). Using waste biomass materials as an oil absorbent is of great interest for oil–water separation and is becoming an area of intense research because they are cheap, accessible, and biodegradable (Vlaev et al. 2011; Zang et al. 2016).
Recently, superhydrophobic surfaces have attracted attention because of their unique properties, including self-cleaning, anti-sticking, water proof, chemical stability, and oil recovery (Yuan et al. 2013; Zang et al. 2015). In theory, construction of a hierarchical rough structure with low surface energy are two indispensable parameters for the formation of superhydrophobic surfaces, which are characterized by a water contact angle greater than 150° and a water sliding angle less than 10° (Kim et al. 2011; Li et al. 2010; Wang et al. 2011a). To date, multiple technologies and various materials have been proposed to achieve superhydrophobic and superoleophilic surfaces, such as the solution-immersion process, chemical etching, layer-by-layer assembly, laser treating, and ultrasound irradiation (Liang and Guo 2013; Zhang et al. 2013; Zhou et al. 2013).
Corn straw fiber, a natural and biodegradable biomass from agricultural waste, is usually discarded and burned on the spot, thus resulting in severe air pollution. In fact, efficient utilization of straw resources is vital to solving the serious air pollution caused by straw burning. Zang et al. (2016) removed oil from water using superhydrophobic/superoleophilic corn straw fibers that they prepared with ZnO particles via conventional impregnation. ZnO particles are hollow spheres with an average diameter approximately 5 μm, and a hexadecyltrimethoxysilane (HDTMOS) chemical agent is employed to act as a hydrophobic modifier. In this study, we exploited corn straw fibers as a raw material to prepare a high-efficiency oil absorbent that exhibits great chemical stability and environmental friendliness. The superhydrophobic/superoleophilic properties of the prepared corn straw fiber arose from the combined effects from the deposition of homogeneous SiO2 inorganic particles, with an average particle size about 40–50 nm using the sol–gel method, and the hydrophobic embellishment of (Heptadecafluoro-1,1,2,2-tetradecyl) trimethoxysilane (PTES), thus giving the product the ability to efficiently dislodge oils from oily wastewater. Compared to HDTMOS, the amount of PTES is less; therefore, the production cost is reduced. In view of its intrinsic water-repellence, high absorption capacity, chemical stability and environmental friendliness, the prepared corn straw fiber floats on the surface of the water after absorbing the oil, allowing it to be easily transported and recycled. The information obtained in this study demonstrates that the prepared superhydrophobic/superoleophilic corn straw fiber can be widely applied in the treatment of oily wastewater, thereby providing new insight into producing a sustainable high-efficiency oil absorbent from agricultural waste.
Corn straw from a farm in Harbin Jiangbei District was used in this study. Ethanol (99.7%), sodium hydroxide (96.0%), hydrogen peroxide (30.0%), hydrochloric acid (37%), ammonium hydroxide (25.0%), tetraethoxysilane (98.0%) and glacial acetic acid (99.5%) were purchased from Tianjin Kemiou Chemical Reagent Co., Ltd. (Tianjin, China). (Heptadecafluoro-1,1,2,2-tetradecyl) trimethoxysilane (99.5%), employed to decorate SiO2 granules, was purchased from Aladdin Chemicals Co., Ltd. (Shanghai, China). All chemicals were used as received without any purification. Diesel oil, gasoline, crude oil, bean oil, n-hexane, octane, toluene, and chloroform, supplied by Sigma-Aldrich Co., Ltd. (St. Louis, MO, USA), were used for oil contact angle testing and absorption capacity measurements.
Pre-treatment of the corn straw fiber
Peeled corn straw was placed in a pulveriser to obtain straw fibers, and then sieved through 60 and 80 mesh standard screens to collect fibers (250–425 μm). In addition, corn straw fibers were ultrasonically rinsed three times with deionized water, ethanol and deionized water. The corn straw was then placed in an aqueous solution of NaOH (100 mL, 0.5 wt%) and 30% H2O2 (3.5 mL) with stirring at ambient temperature for 14 h. This helped in removing the resins and additional impurities present in the corn straw and exposed the hydroxy groups. Next, the pH of above solution was adjusted to 6.5–7.0 with HCl (6 mol/L). After washing several times with deionized water to remove chemical residues, the pretreated corn straw was dried at 40 °C for 24 h until its weigh remained constant.
Synthesis of SiO2 particles
The SiO2 particles were fabricated by a sol–gel process. Briefly, 10 mL NH3OH was added dropwise into a beaker containing 180 mL ethanol, 20 mL tetraethoxysilane and 20 mL deionized water. The solution was vigorously stirred for 2 h at room temperature. Next, the mixture was left for 10 h to produce a white suspension. Homogeneous SiO2 particles were dissociated from the white suspension by centrifugation in ethanol and then dried at 60 °C for 6 h.
Fabrication of superhydrophobic/superoleophilic corn straw fiber
In detail, 0.1 g SiO2 was added to a solution of 10 mL anhydrous ethanol, 0.03 mL PTES, 0.025 mL H2O, and 0.005 mL acetic acid. Subsequently, 0.2 g corn straw was added into the mixture. The reaction was maintained under stirring at room temperature for 5 h, and then placed in an oven at 65 °C for 3 h. Finally, the superhydrophobic/superoleophilic corn straw fibers were acquired by drying the corn straw at 50 °C until its weight was consistent.
In this study, the PTES chemical agent acted as a hydrophobic modifier, with its modification mechanism as follows (Fig. 1): silicon hydroxyl groups, which were generated from a hydrolysis reaction with the PTES reagent, reacted with hydroxyl groups on the surfaces of SiO2 particles and pristine corn straw fibers. Therefore, the hydrophobic heptadecafluoro-decylalkyl of PTES was introduced onto the surface of the fibers to induce the low surface energy of superhydrophobic/superoleophilic corn straw.
a Synthesis of superhydrophobic/superoleophilic corn straw and a water droplet on the resulting corn straw fiber surface. b Schematic illustration for the modification of SiO2 particles with PTES. c Chemical structure of (Heptadecafluoro-1,1,2,2-tetradecyl) trimethoxysilane (PTES)
Characterization of pristine corn straw fiber and superhydrophobic/superoleophilic corn straw fiber
Scanning electron microscope (SEM) images were collected on a Hitachi TM3030 tabletop microscope. The chemical composition of the prepared corn straw product was detected by Fourier transform infrared spectroscopy (FTIR, Magna-IR 560, Nicolet, USA), X-ray photoelectron spectroscopy (XPS, PHI Thermo Fisher Scientific, USA) and energy-dispersive X-ray analysis (EDX, Quantax70, USA). Water contact angle (WCA) and oil contact angle (OCA) measurements were carried out on a contact angle instrument (CA-A, Hitachi, Japan) by dropping a 5 μL deionized water or oil onto five or more different positions on the corn straw specimens. The values of the WCA and OCA were determined as averages of those of five measurements.
Evaluation of absorption capability and oil removal efficiency
An absorption capability test was performed in pure oil by suspending a nylon net bag containing 0.5 g corn straw into a beaker with 150 mL oil at room temperature. After 5 h, the nylon bag was taken out from the oil and let to stand for 10 min. Absorption capability calculated using Eq. (1):
$$q = (m_{2} - m_{1} )/m_{1}$$
where q is the absorption capability (g/g); m2 is the weight of corn straw fibers after the absorption; and m1 is the initial weight of corn straw fibers prior to absorption.
Oil removal efficiency experiments were performed in an oil–water system, which was similar to the oil absorption capability test. Nylon net bags containing 0.5 g corn straw were placed in 150 mL oil–water mixtures with different mass ratios, and then stirred (500 r/min) at room temperature. After 5 h, the nylon bags were removed from the oil and let to stand for 10 min. The oil removal efficiency was calculated using Eq. (2):
$$k = (w_{3} - w_{2} - w_{1} )/(w_{3} - w_{2} )$$
where k is the oil removal efficiency (%); w3 is the weight of the corn straw fibers after absorption; w2 is the initial weight of the corn straw fibers prior to absorption; and w1 is the weight of water absorbed in the absorbents.
In this study, one-way analysis of variance (AVONA) was used to analyze the reliability of the experimental data. All experimental data are presented as averages of five measurements. A p value ≥ 0.95 for the Student's t test demonstrates reliability of the experimental data.
Micro-structure analysis of a novel superhydrophobic/superoleophilic corn straw fiber
It is known that surface microtopography is primarily responsible for establishing superhydrophobic surface, similar to the lotus leaf's self-cleaning property that is associated with its micro/nanostructure (Wang et al. 2015; Autumn et al. 2000; Ju et al. 2012; Wei et al. 2010; Gao et al. 2009; Zhang et al. 2008). Therefore, it is crucial to survey the surface micro-profiles of corn straw fiber by SEM. The corresponding surface morphology results of raw corn straw fiber and superhydrophobic/superoleophilic corn straw fiber are shown in Fig. 2. At low SEM magnifications (Fig. 2a, b), there was no marked difference in the fibrous shape between untreated sample and modified product, indicating that the product retained the characteristics of the original corn straw fiber. In contrast with the smooth fiber surface of pristine corn straw, the high magnification SEM image of the prepared superhydrophobic/superoleophilic corn straw fiber showed a surface layer of solid spherical granules, with an average diameter of 40–50 nm (Fig. 2c), which was attributed to the compact deposition of the SiO2 particles.
SEM images of the raw corn straw fiber (a, b) and superhydrophobic/superoleophilic corn straw fiber (c, d) at different magnifications. Compared to raw corn straw fiber, SiO2 particles compactly deposit on the superhydrophobic/superoleophilic corn straw fiber surface
Surface wettability of superhydrophobic/superoleophilic corn straw fiber
The principle of surface wettability can usually be demonstrated by the Young equation, as follows (Wang et al. 2015):
$$\cos \theta = \frac{{\gamma_{\text{sv}} - \gamma_{\text{sl}} }}{{\gamma_{\text{lv}} }}$$
where γsv, γsl and γlv are the interfacial free energy of solid/vapor, solid/liquid, and liquid/vapor, respectively; and θ is the contact angle. In general, the value of the contact angle is an essential to measuring the surface wettability of a superhydrophobic material. Hence, the water/oil contact angles of corn straw fiber were investigated to determine its superhydrophobic/superoleophilic characteristics. For raw corn straw fiber, a water contact angle of 0° was visible on fiber surface, which is ascribed to massive hydroxyl groups on a pristine fiber surface (Fig. 3a). In contrast, a spherical water droplet was observed on the prepared fiber, with a water contact angle of 152° (Fig. 3c), indicating its superior superhydrophobicity. Moreover, when dripped on the surface of the raw corn straw fiber and prepared corn straw fiber, oil droplets instantly spread, indicating an oil contact angle of 0° (Fig. 3b, d). Uniform coverage of sub-micrometer SiO2 microspheres, coupled with micron-sized corn straw fiber, while using the PTES function as a modifier to ornament SiO2 particles would facilitate low surface energy. The combination of a particularly hierarchical rough structure and low surface energy is regarded as an indispensable condition when constructing special superhydrophobic and superoleophilic material from the resulting corn straw fiber surface. Because air was captured and trapped, while falling onto corn straw fiber surface, by the abundant cavities and interspaces among SiO2 particles on the fiber surface, a water droplet could contact the trapped air to manifest a non-wetting phenomenon. As soon as water was dropped on the prepared corn straw fiber surface, it was repelled without leaving a trail, which demonstrates the great waterproof characteristic of the resulting product (Wang et al. 2011b). Accordingly, it could be deduced that the wettability of the corn straw fiber was transformed from superhydrophilicity to superhydrophobicity. Taken together, these results demonstrate the superhydrophobic and superoleophilic properties of the prepared corn straw fiber.
Images of a water droplet (a) and an oil droplet on raw corn straw fiber (b); a water droplet (c) and an oil droplet (d) on prepared superhydrophobic/superoleophilic corn straw fiber. Compared to raw corn straw fiber, the resulting corn straw fiber indicates its superior superhydrophobicity
In this study, the combination of numerous SiO2 particle aggregates and surface modification by PTES could prevent water from wetting the treated fiber surface and result in water droplets on the obtained corn straw fiber surface rolling off without leaving a trace, thereby demonstrating a novel non-wetting material. Hence, the prepared superhydrophobic/superoleophilic corn straw fiber absorbs only oil while completely repelling water.
Surface chemical component analysis
In this study, SiO2 particles were prepared using the Stöber method, where tetraethoxysilane and ammonium hydroxide acted as a precursor and a catalyst, respectively. The synthesis process of SiO2 particles was divided into two stages, which included the hydrolysis of tetraethoxysilane and condensation polymerization of the hydrolyzed intermediate in the presence of the ammonia catalyst. The concrete forming process of the SiO2 particles was as follows (Wang et al. 2011b):
Hydrolysis:
$${\text{Si}} - \left[ {{\text{OC}}_{2} {\text{H}}_{5} } \right]_{4} + 4{\text{H}}_{2} {\text{O}} \to {\text{Si}} - \left( {\text{OH}} \right)_{4} + 4{\text{C}}_{2} {\text{H}}_{5} {\text{OH}}.$$
Alcohol condensation:
$${\text{Si}} - \left( {\text{OH}} \right)_{4} + {\text{Si}} - \left[ {{\text{OC}}_{2} {\text{H}}_{5} } \right]_{4} \to \equiv {\text{Si}} - {\text{O}} - {\text{Si}} \equiv + 4{\text{C}}_{2} {\text{H}}_{5} {\text{OH}}.$$
Water condensation:
$${\text{Si}} - \left( {\text{OH}} \right)_{4} + {\text{Si}} - \left( {\text{OH}} \right)_{4} \to \equiv {\text{Si}} - {\text{O}} - {\text{Si}} \equiv + 4{\text{H}}_{2} {\text{O}}.$$
Large amounts of hydroxyl groups on the surface of the silica particles are critical for the preparation of superhydrophobic/superoleophilic corn straw fibers (Wang et al. 2011b). In addition, due to the great influence of silica size distribution on the generation of a superhydrophobic surface, we strictly abided to the well-known Stöber method for fabrication of silica particles.
FTIR, XPS and EDX were used to analyze the surface chemical composition of superhydrophobic/superoleophilic corn straw fiber. The relevant FTIR spectra of SiO2 particles, pristine corn straw fiber and superhydrophobic/superoleophilic corn straw fiber are listed in Fig. 4. The absorption peak at 955 cm−1 was ascribed to stretching vibration of isolated Si–OH, which was perceptible only in the case of bare silica (Fig. 4a) (Kulkarni et al. 2008). Moreover, corresponding to Si–O–Si asymmetric stretching and symmetric stretching, the bands at 1056 and 795 cm−1 were also observable (Hsieh et al. 2010; Vinogradova et al. 2006). Compared with pristine corn straw fiber, the band at 804 cm−1 was due to Si–O–Si symmetric stretching (Hsieh et al. 2010) and the absorption peak at 1203 cm−1 was a typical characteristic of the C–F stretching vibration of PTES (Zhou et al. 2013), which proves that the SiO2 particle deposition and PTES organic chemistry reagent were observed on the prepared superhydrophobic/superoleophilic corn straw fiber surface (Fig. 4b).
a FTIR spectra of SiO2 particles; b FTIR spectra of raw corn straw fiber (i) and prepared superhydrophobic/superoleophilic corn straw fiber (ii)
The XPS spectra of pristine corn straw fiber and superhydrophobic/superoleophilic corn straw fiber are shown in Fig. 5. With regards to raw corn straw fiber (Fig. 5a), only peaks corresponding to C1s and O1s elements were observed. By contrast, the XPS spectra of superhydrophobic/superoleophilic corn straw fiber demonstrated four new peaks including Si2p, Si2s, F1s and F KLL, which accounted for the generation of SiO2 particles and PTES on the prepared fiber surface. However, the peak intensity of C1s and O1s in curve b was weaker than that in curve a. This can be attributed to the addition of SiO2 particles and PTES, thereby decreasing the relative mass ratio of C1s and O1s.
XPS spectra of pristine corn straw fiber (a) and superhydrophobic/superoleophilic corn straw fiber (b)
Apart from FTIR and XPS characterizations, the elemental composition of superhydrophobic/superoleophilic corn straw fiber was investigated via energy-dispersive X-ray analysis (EDX). The oxygen (O) peak and the carbon (C) peak were observed in corn straw fiber (Fig. 6). In comparison with the raw fiber, there were two new peaks of silica (Si) and fluorine (F) induced by SiO2 and PTES in the prepared superhydrophobic/superoleophilic corn straw fiber, thus providing evidence for the presence of SiO2 particles and PTES on the obtained corn straw fiber surface. Taken together, these results show that SiO2 particles were successfully modified by PTES and truly existed on the surface of the superhydrophobic/superoleophilic corn straw fiber.
The wt% of each element and EDX spectra of pristine corn straw fiber (a) and the prepared superhydrophobic/superoleophilic corn straw fiber (b)
Environmental durability and application in water–oil separation
Considering the importance of environment durability and chemical stability for prepared materials used in practical application, it is necessary to investigate these properties with respect to superhydrophobic/superoleophilic corn straw fiber to confirm its potential as an oil absorbent. The effects of acidic and alkaline conditions on the wettability of superhydrophobic/superoleophilic corn straw fiber were systematically investigated. Contact angle measurements were performed by pipetting 5 μL aqueous solution, from pH 0–14, onto fiber surfaces in order to evaluate chemical stability and durability of the prepared material (Fig. 7a). The measured water contact angle ranged from 152° to 150°, while the oil contact angle remained constant at 0°, implying that the resulting fiber surface still maintained outstanding superhydrophobicity and superoleophilicity properties, even in strong acid and strong alkaline conditions. To evaluate its environmental stability, the contact angles of water and oil of the superhydrophobic/superoleophilic corn straw fiber were monitored over time at ambient temperature and humidity (Fig. 7b). After 150 days, there were no distinct changes in the water and oil contact angles of the prepared corn straw fiber, which clearly demonstrates that the superhydrophobic/superoleophilic corn straw fiber obtained in this study possess excellent environment stability.
a Variation of water contact angle and oil contact angle of superhydrophobic/superoleophilic corn straw surfaces in aqueous solutions with different pH values. b The relationship between contact angles of the resulting superhydrophobic/superoleophilic corn straw fibers and days of storage in air environment
Because the prepared corn straw exhibited favorable superhydrophobic/superoleophilic performance in both acidic solutions and under ambient conditions, these prepared fibers could be utilized as highly selective absorption materials to achieve effective separation of oil–water mixtures. Oil absorption performance is measured by the absorption capacity (g/g), as well as the oil removal efficiency (%). To determine the maximum absorption capacity of superhydrophobic/superoleophilic corn straw fiber, experiments were performed in a variety of pure oils and organic solvents. Figure 8a presents the absorption capacities of raw corn straw fiber, pretreated corn straw fiber and the prepared superhydrophobic/superoleophilic corn straw fiber in various oils and organic solvents. The adsorption capacity of raw corn straw fiber was very low and was always less than 10 g/g. In contrast with the raw fiber, the pretreated corn straw fiber exhibited a higher absorption capacity for all oils and organic solvents, with its value almost 3 times higher than pristine corn straw fiber. Moreover, the oil absorption quantity of the prepared superhydrophobic/superoleophilic corn straw fiber for diesel oil, crude oil, bean oil, and chloroform was relatively large at 17.5, 20.3, 22.6, and 27.8 times their own quality, respectively; however, the oil absorption quantity of the prepared superhydrophobic/superoleophilic corn straw fiber for gasoline, n-hexane, octane, and toluene was relatively small at 15.5, 13.5, 15.1, and 16.3 times their own quality, respectively. The reason is that the oil absorption capacity is relevant to the viscosity and density of the oil products and organic solvents, with a higher viscosity and density resulting in a saturated absorption capacity of prepared corn straw fiber (Guo et al. 2015). For the same oil or organic solvents, the absorption capacity of the prepared superhydrophobic/superoleophilic corn straw fiber was slightly higher than that of the pretreated corn straw fiber, indicating that the superhydrophobic modification can further enhance the oil capacity, which is beneficial for practical application. In addition to the absorption capacity, the absorption efficiency of the prepared corn straw fiber was studied to ascertain the potential of superhydrophobic/superoleophilic corn straw fiber in oil/water separation. In theory, the superhydrophobic/superoleophilic corn straw fibers absorb very little water; however, there are errors in the oil absorption efficiency. When the water content of the oil–water mixture increased, the prepared corn straw fiber absorbed a small amount of water during magnetic stirring. The oil removal efficiency of the resulting fiber for diesel oil and crude oil varied from 100 to 99%, with different mass ratios of water-to-oil (Fig. 8b). The main cause of this phenomenon is that there was a small amount of water absorbed at the same time that the prepared corn straw fibers absorb oil (Wang et al. 2013), which indicates that the prepared fiber can be widely applied for oil removal from water. Taken together, because of the high absorption capacity and oil removal efficiency, we clearly demonstrate that the novel superhydrophobic/superoleophilic corn straw fiber obtained in this study can be regarded as a high-efficient oil absorbent with great chemical stability and environmental durability. Moreover, it has a higher oil adsorption capacity, compared to other biomass-based absorbents (Table 1).
a The absorption capacities of raw corn straw fiber, pretreated corn straw fiber and prepared superhydrophobic/superoleophilic corn straw fiber for various oils and organic solvents; b absorption efficiency of superhydrophobic/superoleophilic corn straw with different mass ratios of water to oil
Table 1 Comparison of the oil absorption capacity of some recently reported oil sorbent and the samples prepared in this study taking diesel oil for example
In this study, we successfully developed a preparation process for a novel superhydrophobic/superoleophilic corn straw fiber by attachment of PTES-modified SiO2 particles onto the fiber surface via the sol–gel and impregnation method. The prepared corn straw fiber exhibited outstanding properties of superhydrophobicity and simultaneous superoleophilicity with a water contact angle of 152° and an oil contact angle of 0° for different oils. In addition, the microtopography, wetting property, chemical composition and oil absorption performance were comprehensively studied. Results revealed that SiO2 granules successfully modified by PTES were robustly attached to the fiber surface, resulting in a hierarchical structure and low surface energy, thus giving rise to the significant phenomenon of both superhydrophobicity and superoleophilicity. Moreover, the prepared superhydrophobic/superoleophilic corn straw fiber displayed great chemical stability and environmental durability. Most importantly, the prepared superhydrophobic/superoleophilic corn straw fiber possessed an excellent absorption capacity and high absorption efficiency. Taken together, these results demonstrate that the prepared fiber obtained in this study exhibits a high application potential to effectively separate oil/water mixtures.
PTES: (Heptadecafluoro-1,1,2,2-tetradecyl) trimethoxysilane; FTIR: Fourier transformation infrared spectroscope; XPS: X-ray photoelectron spectroscopy; EDX: energy-dispersive X-ray analysis; WCA: water contact angle; OCA: oil contact angle; SEM: scanning electron microscope.
Equation parameters
γ sv , γ sl and γ lv : solid–vapor, solid–liquid and liquid–vapor interfacial tensions, respectively; θ: contact angle; q: sorption capability (g/g); m2: the weight of corn straw fibers after absorption; m1: the initial weight of corn straw fibers before absorption; k: oil removal efficiency (%); w3: the weight of corn straw fibers after absorption; w2: the initial weight of corn straw fibers before absorption; w1: the weight of water absorbed in the absorbents.
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YX and HY designed the study, performed experiments, analyzed data, and prepared the manuscript. DZ, FL and XH contributed to the discussion. CY, SHH, JSC and YZ reviewed the results, helped in data analysis, and edited the manuscript. All authors read and approved the final manuscript.
The authors would like to acknowledge English-editing support by Paul Steed at NCKU language center.
This research was supported by the Fundamental Research Funds for the Central Universities (2572015EB01) and the National Natural Science Foundation of China (31470584).
Yang Xu and Haiyue Yang contributed equally to this work and share first authorship
Key Laboratory of Bio-based Material Science and Technology, Ministry of Education, Northeast Forestry University, Harbin, 150040, People's Republic of China
Yang Xu, Haiyue Yang, Deli Zang, Feng Liu & Chengyu Wang
State Key Laboratory of Urban Water Resource and Environment, School of Environment, Harbin Institute of Technology, Harbin, 150090, People's Republic of China
Xiaochen Huang, Jo-Shu Chang & Shih-Hsin Ho
Department of Chemical Engineering, National Cheng Kung University, Tainan, Taiwan
Jo-Shu Chang
President Office, Harbin Medical University, Harbin, 150001, People's Republic of China
Yang Xu
Haiyue Yang
Deli Zang
Xiaochen Huang
Chengyu Wang
Shih-Hsin Ho
Correspondence to Chengyu Wang or Shih-Hsin Ho.
Xu, Y., Yang, H., Zang, D. et al. Preparation of a new superhydrophobic/superoleophilic corn straw fiber used as an oil absorbent for selective absorption of oil from water. Bioresour. Bioprocess. 5, 8 (2018). https://doi.org/10.1186/s40643-018-0194-8
Oil absorption
Corn straw
Superhydrophobic
Superoleophilicity
SiO2 particles | CommonCrawl |
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